Symmetry of Intramolecular Quantum Dynamics 9783110267648, 9783110267532

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De Gruyter Studies in Mathematical Physics 11 Editors Michael Efroimsky, Bethesda, USA Leonard Gamberg, Reading, USA Dmitry Gitman, São Paulo, Brasil Alexander Lazarian, Madison, USA Boris Smirnov, Moscow, Russia

Alexander V. Burenin

Symmetry of Intramolecular Quantum Dynamics Translated by Alexey V. Krayev

De Gruyter

Physics and Astronomy Classification 2010: 87.15.hp, 31.30.Gs, 33.30.-i, 33.57.+c, 33.15.kr, 33.15.-e, 33.20.-t, 42.50.Lc, 03.65.-w, 33.20.Sn, 33.20.Wr, 33.15.Bh, 11.30.Qc, 33.20.Tp, 33.20.Vq, 31.15.xh, 03.65.Fd,.

ISBN 978-3-11-026753-2 e-ISBN 978-3-11-026764-8 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the internet at http://dnb.dnb.de. © 2012 Walter de Gruyter GmbH, Berlin/Boston Typesetting: PTP-Berlin Protago-TEX-Production GmbH, www.ptp-berlin.eu Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen Printed on acid-free paper Printed in Germany www.degruyter.com

Preface

The molecule is a complex multiparticle system, and in isolated state its internal dynamics can be described, to a good approximation, neglecting the nuclear and electron spin-related contributions to the Hamiltonian. The symmetry properties of a purely spatial Hamiltonian are determined by the symmetry properties of space and time (external symmetry) and by requirements imposed on permutation of identical particles (internal symmetry). However, when we try to solve equations of motion with such a Hamiltonian by the methods of perturbation theory, we unexpectedly face the need to introduce an additional internal geometric symmetry group that characterizes the molecule. This is a matter of principle in this approach, since otherwise it is impossible to write approximate equations of motion. The basic working approximation is the Born–Oppenheimer (BO) approximation [10,14,35], which introduces the concept of the effective potential of nuclear interaction in a given electronic state and, as a consequence, the concept of a set of equilibrium configurations corresponding to the minima of this potential. Qualitatively, molecules can be divided into rigid and nonrigid ones. For rigid molecules in nondegenerate electronic states, the choice of the effective potential with one minimum is quite adequate, whereas for nonrigid molecules several such minima should be taken into account because the internal motion includes the transitions between them. It has long been understood that for rigid molecules an additional geometric group should be selected in the form of a point group of their unique equilibrium configuration, which by definition [60,64] includes all geometric symmetry elements of this structure as a whole. It is commonly assumed that this group and the corresponding inferences are corollaries of the BO approximation, i.e., only in this approximation can we speak of a certain geometric structuring of internal motions. But even in this simplest case there is no clear idea of an applicability domain of the point group. Two essentially different opinions are available in the literature. According to one of them [60, 64], the point group characterizes the total (electronvibration-rotational) internal motion when deviations from the equilibrium position are sufficiently small. However, what a “sufficiently small deviation” means is quite uncertain. The alternative point of view [15, 16, 39] is that the point group describes the symmetry of vibrational and electronic motions only and is inapplicable to rotational motion and, hence, to the total internal motion. As a result, analysis of the total motion is based on the so-called complete nuclear permutation-inversion (CNPI) group [15, 16, 39]. Such contradictions in the status of empirically introduced point groups are connected with the absence of a definite point of view on their nature. Therefore, a very important feature of the book is the statement that these groups are

vi

Preface

implicit or dynamically invariant groups of symmetry of a rigorous problem of internal coordinate motion. Despite the fact that no method is known to date that would enable one to obtain such a group from studies of the equations of rigorous spatial dynamics, this statement can be logically justified on the basis of analysis of the observed properties of a molecular system. It is interesting that such a point of view may change dramatically some general concepts of a molecular system: 1. A characteristic property of a molecular system is the existence of rotational motion of the system as a whole. This means that a molecular system is a certain structure (“microcrystal”), in which internal motions of the particles are basically collective. The symmetry of this structure is characterized by an implicit geometric group. It appears that the concept itself of the structure of a molecular system can be introduced into the description only by using the BO approximation. The correct configuration space of collective motions is constructed separately in each electronic state. In other words, we proceed to the domain of description bounded by one electronic state. For such a bounded domain, implicit symmetry is replaced by its explicit counterpart. 2. The solution of the problem of the discrete spectrum of a molecular system conceptually relies on perturbation theory, both in the analytical and numerical approaches. The point is that the conditions of selection of physically meaningful solutions of a discrete spectrum of collective internal motion against the vast background of formal solutions cannot be formulated without using the BO approximation. This is exactly why we pass from the problem with implicit symmetry to the problem with the same but explicit symmetry in one given electronic state. Since the correct choice of explicit symmetry should be provided, the problem of empirically seeking a geometric group arises. For rigid molecular systems in nondegenerate electronic states, such a group is a point group of their unique equilibrium configuration. However, the symmetry of such a configuration is an elementary consequence of the symmetry of internal dynamics, and not vice versa as is often stated, and these two symmetries coincide only in the aforementioned simplest case. 3. When the Schrödinger equation describing a discrete spectrum of a molecular system is solved by the methods of perturbation theory, a chain of nested (increasingly approximate) models is constructed until the exact solution of the model problem becomes possible. Simultaneously, a chain of symmetry groups characterizing these models arises. In the first place, the difficulties of solving the Schrödinger equation are due to the declarative nature of the obtained series of perturbation theory describing the transitions between the neighboring models. Not only are the properties of the series unknown, but often it is also impossible to correctly calculate even the lower-order corrections. Moreover, the symmetry requirements should also be taken into account. However, the situation changes radically if only the symmetry properties are considered and the transitions between models are de-

Preface

vii

scribed by symmetry matchings. To do this, in the groups of the neighboring models we single out the equivalent elements with respect to which the wave functions and the operators of physical values should be transformed in the same way. In other words, transitions between the neighboring models are accompanied by certain nontrivial constraints on the compliance of symmetry types. The advantages of such an approach are primarily due to the fact that the matchings are rigorous (!). 4. Internal dynamics can be described on the basis of only the symmetry principles with accuracy up to some phenomenological constants which can be determined, for example, from a comparison of theoretical conclusions and experimental data. In this approach, configuration space of a quantum system is not introduced in explicit form at all, and, as a consequence, the wave functions of the coordinates of this space are not explicitly considered. However, due to its wide philosophical and technical difference, this approach is at present the only one that can be used for the solution of many topical problems pertaining to the internal dynamics of molecules. The models obtained rigorously describe all interactions of interesting types of motion that are possible within the framework of a given symmetry and lead to a simple, purely algebraic scheme of calculation for both the position of the levels in the energy spectrum and the transition intensities between them. It is important that the correctness of the models is limited only to the correct choice of the internal dynamics symmetry. Certainly, a change in the general concept concerns not only the molecular system proper, but also a wide variety of other physical systems also requiring the introduction of an additional internal geometric group to describe its basically collective internal motions. Interestingly, an atom does not belong to such systems, and this is exactly the reason why it has no rotational motion of the system as a whole. The main goal of this book is to give a systematic description of quantum intramolecular dynamics on the basis of the symmetry principles only. In this respect, there is no comparable book in the world literature. As compared with the second edition [24], it has been expanded substantially. A number of new problems are considered, among which a discussion of the basic necessity of using the BO approximation for the formulation itself of the problem of finding a discrete molecular spectrum by solving the stationary Schrödinger equation using analytical and/or numerical methods (Chapter 12), the analysis of nonrigid molecular systems with continuous axial symmetry groups (Chapter 15), and a description of the Zeeman and Stark effects (Chapter 19) are worthy of special notice. We have also revised and extended a discussion of the issues studied in the second edition. Importantly, as a result, the range of studied types of nonrigid motions has been completed in a nontrivial way. In particular, we have added analyses of very interesting dynamics of some molecular complexes and the simplest of the carbocations, which are the intermediate molecular ions of many chemical reactions. Finally, the applicability of developed methods in such fields as analysis of

viii

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molecules with more than two identical tops, allowance for hyperfine interactions and parity violation effects, etc. is demonstrated in Chapter 20. The book is basically intended for physicists working in the field of molecular spectroscopy and quantum chemistry. The reader is not expected to know the apparatus of group representation theory needed for application of symmetry methods in quantum intramolecular dynamics since the first part of the book is dedicated to it. For a more detailed study of almost all issues touched upon, one may consult, e.g., the monographs [43, 50, 60, 92]. The problems of using a semidirect product of groups and dynamic groups are the only major exclusions. These are discussed in [8,39]. The second part of the book concerns the state-of-the-art description of quantum intramolecular dynamics employing only symmetry principles. This part now comprises fourteen chapters (instead of nine in the second edition), and the consideration is mainly based on the author’s works [17,18,20,23,28]. The reader is supposed to know at least foundations of the analytical description of intramolecular motions. Various issues in this extensive area can be found in [10, 14–16, 35, 39, 60, 64]. Additional references are given as the statement unfolds. The appendices contain the required reference material. Appendix V about the action of the direction cosines on the rotational unit vectors is added as compared with the second edition. The author is grateful to Professors Yu. S. Makushkin, A. M. Sergeev, B. M. Smirnov, and V. G. Tyuterev for supporting his efforts aimed at developing the methods of symmetry theory.

Nizhny Novgorod, Russia September 17, 2011

Alexander V. Burenin

Contents

Preface

v

I Foundations of the mathematical apparatus 1

2

Basic concepts of group theory

3

1.1 The group postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2 Subgroup, direct product of groups, isomorphism, and homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3 Cosets. Semidirect product of groups . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.4 Conjugacy classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

Basic concepts of group representation theory

10

2.1 Linear vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Operators in configuration and function spaces . . . . . . . . . . . . . . . . . . . 13 2.3 Representations of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4 Characters. Decomposition of reducible representations . . . . . . . . . . . 17 2.5 Direct product of representations. Symmetric power . . . . . . . . . . . . . . 20 2.6 The Clebsch–Gordan coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.7 Basis functions of irreducible representations . . . . . . . . . . . . . . . . . . . . 25 2.8 Irreducible tensor operators. The Wigner–Eckart theorem . . . . . . . . . . 28 3

The permutation group

31

3.1 Operations in the permutation group. Classes . . . . . . . . . . . . . . . . . . . . 31 3.2 Irreducible representations. The Young diagrams and tableaux . . . . . . 33 3.3 Basis functions of irreducible representations . . . . . . . . . . . . . . . . . . . . 35 3.4 The conjugate representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4

Continuous groups

39

4.1 Compact Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.2 Lie group of linear transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.3 Lie algebra. Three-dimensional rotation group . . . . . . . . . . . . . . . . . . . 42 4.4 Irreducible representations of a three-dimensional rotation group . . . . 46

x 5

Contents

Point groups

51

5.1 Operations in point groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.2 Discrete axial groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.3 Cubic groups. Icosahedral groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.4 Continuous axial groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6

Dynamic groups

58

6.1 Invariant dynamic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.2 Noninvariant dynamic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

II Qualitative intramolecular quantum dynamics 7

The philosophy of using the symmetry properties of internal dynamics

67

7.1 Symmetry groups of internal dynamics . . . . . . . . . . . . . . . . . . . . . . . . 67 7.2 Significance of the analysis of symmetry properties . . . . . . . . . . . . . . . 73 7.3 On the domain of the point group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 7.4 The chain of symmetry groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 7.5 The concept of the coordinate spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.6 The influence of numerical methods on the overall description . . . . . . 86 7.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 8

Internal dynamics of rigid molecules

89

8.1 Nonlinear molecules without inversion center . . . . . . . . . . . . . . . . . . . 89 8.2 Nonlinear molecules with inversion center . . . . . . . . . . . . . . . . . . . . . . 101 8.3 Linear molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 8.4 Description of quasidegenerate vibronic states . . . . . . . . . . . . . . . . . . . 115 8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 9

Molecules with torsional transitions of the exchange type

123

9.1 Extended point groups. Intermediate configuration . . . . . . . . . . . . . . . . 123 9.2 Methanol molecule CH3 OH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 9.3 Ethane molecule C2 H6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 9.4 The molecules of complex hydrides LiBH4 and NaBH4 . . . . . . . . . . . 140 9.5 The molecules of dimethyl ether (CH3 )2 O and acetone (CH3 )2 CO . . . 148 9.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Contents

10 Molecules with pseudorotations of the exchange type

xi 157

10.1 Extended point groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 10.2 Cyclobutane molecule C4 H8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 10.3 Molecules of the XPF4 type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 10.4 Phosphorus pentafluoride molecule PF5 . . . . . . . . . . . . . . . . . . . . . . . . 171 10.5 The separation of internal motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 10.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 11 Molecules with transitions of the nonexchange type between equivalent configurations 182 11.1 Extended point groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 11.2 The ammonia molecule NH3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 11.3 The peroxide molecule HOOH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 11.4 The hydrazine molecule N2 H4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 11.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 12 On the meaning of the Born–Oppenheimer Approximation

205

12.1 Nondegenerate electronic states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 12.2 Degenerate electronic states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 12.3 Internal geometric symmetry of the Hamiltonian . . . . . . . . . . . . . . . . . 210 12.4 Definition of the rotational motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 12.5 Selection of physically meaningful states . . . . . . . . . . . . . . . . . . . . . . . 219 12.6 Symmetry methods in the description of intramolecular dynamics . . . 223 12.7 Geometric symmetry and definitions of nonrigid motions . . . . . . . . . . 226 12.8 Nuclear statistical weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 12.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 13 Molecules with transitions of the exchange and nonexchange types between equivalent configurations

235

13.1 Extended point groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 13.2 Methanol molecule CH3 OH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 13.3 Methylamine molecule CH3 NH2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 13.4 Cyclopentane molecule C5 H10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 13.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

xii

Contents

14 On the construction of extended point groups

256

14.1 Hydrogen fluoride dimer (HF)2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 C 14.2 Ionic complexes ArHC 3 and ArD3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

14.3 Carbocation C2 HC 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 14.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 15 Nonrigid molecular systems with continuous axial symmetry groups

278

15.1 Systems of the HCN/HNC type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 15.2 Complexes of the XCO type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 15.3 Nonrigid water molecules H2 O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 15.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 16 Molecules with different isomeric forms in a single electronic state

291

16.1 Distorted molecular systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 16.2 The methanol molecule CH2 DOH . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 16.3 The ethane molecule CH2 D–CH2 D . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 16.4 The ethanol molecule CH3 CH2 OH . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 16.5 The cyclobutane-1,1-d2 molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 16.6 The tetrahydrofuran molecule C4 H8 O . . . . . . . . . . . . . . . . . . . . . . . . . 315 16.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 17 Molecules with different isomeric forms in different electronic states

329

17.1 Formaldehyde molecule H2 CO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 17.2 The ethylene molecule CH2 –CD2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 17.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 18 Algebraic models of the global description of molecular spectrum

349

18.1 The rigid water molecule H2 O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 18.2 The nonrigid methanol molecule CH3 OH . . . . . . . . . . . . . . . . . . . . . . . 354 18.3 The nonrigid water molecule H2 O . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 18.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 19 Description of the Zeeman and Stark effects

363

19.1 External field and symmetry of the stationary states . . . . . . . . . . . . . . . 363 19.2 The Zeeman effect in the case of a rigid molecule . . . . . . . . . . . . . . . . 364 19.3 The Zeeman effect in the case of a nonrigid molecule . . . . . . . . . . . . . 367 19.4 The Stark effect in the case of a rigid molecule . . . . . . . . . . . . . . . . . . 368 19.5 The Stark effect in the case of a nonrigid molecule . . . . . . . . . . . . . . . 373 19.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

Contents

xiii

20 Additional remarks on describing intramolecular dynamics

379

20.1 The nonrigid trimethylborane molecule B(CH3 )3 . . . . . . . . . . . . . . . . . 379 20.2 Hyperfine interactions in the methane molecule CH4 . . . . . . . . . . . . . . 394 20.3 Parity violation effects in molecules with stereoisomers . . . . . . . . . . . . 396 20.4 Parity violation effects in the molecules without stereoisomers . . . . . . 398 20.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 Conclusion

403

Appendices

405

Bibliography

417

Index

423

Part I

Foundations of the mathematical apparatus

Chapter 1

Basic concepts of group theory

1.1 The group postulates Group G is a set of elements A; B; C : : : satisfying the following four conditions: 1. The multiplication law is defined, in which each pair of elements P; Q 2 G (belonging to G) is associated with some element R 2 G, which is written in the form R D PQ:

(1.1)

2. The product of factors is associative: P .QS/ D .PQ/S:

(1.2)

3. Among the elements of the set, there is a unit element E, which has the property EQ D QE D Q

(1.3)

for any Q 2 G. 4. For each element Q 2 G, there exists an inverse element Q1 2 G, such that Q1 Q D QQ1 D E:

(1.4)

It can easily be shown that .PQ/1 D Q1 P 1 . In the general case, the product of elements in the group is noncommutative, i.e., PQ ¤ QP:

(1.5)

If the product is commutative for all elements of the group, then such a group is called abelian. A particular case of abelian groups is cyclic groups, all elements of which can be obtained by raising one element into power, that is, n elements of such a group can be represented as A;

A2 ;

A3 ; : : : ;

An  E:

(1.6)

Theorem 1.1. If Ga runs over all elements of group G, and G0 is a fixed element of group G, then the product G0 Ga (or Ga G0 ) also runs over all elements of this group, and does it one time. Proof. Any element of group Gb is obtained by multiplying G0 on the right by G01 Gb . Among the products G0 Ga not a single one can be repeated since if G0 Ga D G0 Gb , then multiplying this equality on the left by G01 yields Ga D Gb , which is impossible.

4

Chapter 1 Basic concepts of group theory

This theorem implies the invariance of the sum over all elements of the group for an arbitrary function ˆ defined on these elements: X X X ˆ.Ga / D ˆ.G0 Ga / D ˆ.Gb /: (1.7) Ga

Ga

Gb

A simple example of the group is a set of rational numbers (excluding zero) with respect to the common operation of multiplication of numbers. The multiplication of numbers is associative, and a unit element is unity. Naturally, each number of the set has an inverse number belonging to the set. The example of a permutation group of n objects numbered by numbers from 1 to n is much more interesting for the physical applications considered in this book. In total, we can make nŠ different permutations of such numbers. These permutations are often represented in the symbolic form   1 2 3 ::: n P D ; (1.8) i1 i2 i3 : : : in where ik is the number occupying the place of the number k as a result of permutation. The product of two permutations P2 P1 is also a permutation resulting from the sequential action of first the permutation P1 , then P2 . For each permutation (1.8), by interchanging the rows in it, it is possible to obtain the inverse permutation. In the identical permutation, the second row just coincides with the first one. We denote the permutation group of n objects by n . Its nother name is the symmetrical group. The elements of group n can conveniently be divided into a product of cycles. The cycle .i1 i2 i3 ; : : : ; ik / is a permutation of the form   i1 i2 i3 : : : ik ikC1 : : : in : (1.9) .i1 i2 i3 ; : : : ; ik /  i2 i3 i4 : : : i1 ikC1 : : : in The first object i1 in the cycle moves to the place of the last object ik , the second object i2 in the cycle moves to the place of the first object i1 , etc. Objects ikC1 ; : : : ; in that are absent in the cycle designation stay where they are. In particular, six permutations of the group 3 are written as a product of cycles .1/.2/.3/; .1/.23/; .2/.13/; .3/.12/; .123/; .132/:

(1.10)

Another interesting example of the group is a set of spatial transformations of the figure shown in Figure 1.1, such that this figure coincides with itself. The figure is an equilateral triangle, in whose orthogonal plane a straight-line segment is drawn to its geometric center. In total, there are six such spatial transformations: – E; identical transformation; – C31 and C32 , clockwise rotations by 2/3 and 4=3, respectively, about the straightline segment drawn to the geometric center of the triangle;

5

Section 1.1 The group postulates

Figure 1.1. Geometric figure with the symmetry group C3v .

.1/

.2/

.3/

– v , v and v ; reflections in the planes passing through the medians of a triangle and the straight-line segment drawn to its geometric center. These transformations are called symmetry operations of the figure considered. Naturally, there are no transformations, other than the identical transformation, on which the nonsymmetric figures can coincide with themselves. It can easily be found that the sequential use of any two of the six symmetry operations mentioned above is also an operation from this set. The associative law is also fulfilled, and the inverse of each operation exists. This group of geometric transformations is generally denoted as C3v . Table 1.1 shows the results of pair products of its operations. The operation used first is put into the upper row of the table. Table 1.1. The table of multiplication of the group C3v . .1/

v

.2/

v

.1/

v

.2/

v

.2/

v

.3/

v

v

.3/

v

.1/

v

E

C32

C31

.3/

C31

E

C32

.1/

C32

C31

E

C3v

E

C31

C32

v

E

E

C31

C32

v

C31

C31

C32

E

v

C32

C32

E

C31

v.1/

v.1/

v.3/

v.2/

.2/

v

.3/

v

v v

.2/

v

.1/

v

.3/

v

.2/

v

.3/ .3/ .1/ .2/

6

Chapter 1 Basic concepts of group theory

1.2 Subgroup, direct product of groups, isomorphism, and homomorphism If the group G has a subset of elements H forming a group with the same multiplication law as in the group G, then the subset H is called the subgroup of group G. Any group has a trivial subgroup composed of one unit element. In what follows, by a subgroup we mean a nontrivial subgroup. For example, in group 3 such a subgroup is a group of permutations 2 of any two of the three objects available for the group 3 . The number of elements in a group is called its order, which can be either finite or infinite. Consequently, groups are divided into finite and infinite. Let two finite groups G and F do not contain common elements, except for the unit element, and have orders g and f, respectively. If the elements of group G commute with the elements of group F, then the set of products Ga Fb forms a group of order gf. Indeed, by virtue of the mentioned condition of commutation, the product of any two elements from the resulting set belongs to the same set: G1 F1  G2 F2 D G1 G2  F1 F2 D G3 F3 : The fulfillment of other group postulates can also be easily shown. The obtained group is denoted GF and is called the direct product of groups G and F. For a more rigorous definition of the direct product, it is additionally assumed that G and F are subgroups of some abstract general group, in which the products Ga Fb are understood. A simple example of the direct product of groups is the group k  n , where the elements of groups k and n act on different sets of objects. The case of using the direct product to consider the symmetry group of an equilateral triangle, which is generally denoted as D3h and is represented as C3v  CS , is not as trivial. The abelian group CS has two symmetry transformations, namely, the identical transformation E, and the reflection h in the triangle plane. The symmetry operation h is absent in the figure shown in Figure 1.1 because of the presence of a straight-line segment drawn orthogonally to the triangle plane into the geometric center of the triangle. It can easily be shown that this operation commutes with the operations of group C3v , which is necessary for the construction of the mentioned direct product. We now consider the concepts of isomorphism and homomorphism. Two groups of the same order G and F are isomorphic if a one-to-one correspondence, in which the condition that if element Ga corresponds to the element Fa and the element Gb corresponds to the element Fb then the element Ga Gb corresponds to the element Fa Fb is fulfilled, can be established between their elements. In other words, the established one-to-one correspondence should be preserved upon multiplication of the elements within the groups. Isomorphic groups are identical from the viewpoint of their abstract group properties, although the physical meaning of their elements can be various. Examples of such

7

Section 1.3 Cosets. Semidirect product of groups Table 1.2. Isomorphic correspondence between the elements of groups C3v and 3 . .1/

.2/

.3/

C3v

E

C31

C32

v

v

v

3

(1)(2)(3)

(132)

(123)

(1)(23)

(2)(13)

(3)(12)

groups are the geometric group C3v and the permutation group 3 since each geometric operation can be associated with respective permutation of the vertex numbers in an equilateral triangle (Table 1.2). This result is a particular case of the Kelley theorem, according to which any finite group of order n is isomorphic with the subgroup of the permutation group n . Since the number of subgroups in n is finite, it follows immediately from this theorem that the number of different nonisomorphic groups of order n is also finite. Group G is homomorphically mapped onto group F if each element of G is associated with one element of F, while each element of F may correspond to either one or more than one element of G. In this case, the condition that if the element Ga corresponds to the element Fa and the element Gb corresponds to the element Fb then the element Ga Gb corresponds to the element Fa Fb is fulfilled. It can easily be seen that group D3h is homomorphically mapped onto permutation group 3 if each geometric operation is associated with respective permutation of the vertex numbers in an equilateral triangle. In this case, each operation of the permutation group will correspond to two operations of the geometric group. For example, the identical permutation corresponds to the elements E and h .

1.3 Cosets. Semidirect product of groups Let a finite group G of order g have a subgroup H of order h. Choose the element G1 of group G not belonging to H and multiply by it all the elements of H on the left. As a result, we obtain a certain set h of elements, which we denote as G1 H . None of the elements of G1 H belongs to H. Indeed, if the equality G1 Ha D Hb or G1 D Hb Ha1 is fulfilled for some elements Ha , Hb 2 H , then G1 2 H , which is impossible. We now take the element G2 of group G, which does not belong to the sets H and G1 H , and form a set G2 H . It can easily be shown that the three sets do not have common elements. In a similar way, we introduce a set G3 H and do so until all the elements of group G are exhausted. The latter will be divided into m sets: H;

G1 H;

G2 H; : : : ; Gm1 H:

(1.11)

Consequently, the condition g D mh, which determines the Lagrange theorem, is fulfilled, that is, the order of the subgroup of a finite group is the divisor of the order of this group. Hence, the group whose order is a prime number does not have nontrivial subgroups. It can be shown that such a group is necessarily cyclic.

8

Chapter 1 Basic concepts of group theory

The division of the elements of group G into sets (1.11) is unambiguously determined by specifying its subgroup H since any element of the set Gk H .k D 1; 2; : : : ; m  1/ can be used instead of the element Gk . Sets (1.11) are called left cosets of subgroup H. The number of such cosets is called the subgroup index. In a similar way, the same number of right cosets can also be introduced. None of the cosets, except for H, is a subgroup since the coset does not contain a unit element. Consider the division of the elements of group 3 in its subgroup 2 , which includes the elements (1)(2)(3) and (1)(23), into cosets. The index of the subgroup is equal to three, and therefore the cosets are three. Correspondingly, for the left and right cosets we obtain 2 W .1/.2/.3/; .1/.23/ 2 W .1/.2/.3/; .1/.23/ 2 .2/.13/ W .2/.13/; .123/ .2/.13/2 W .2/.13/; .132/ .3/.12/2 W .3/.12/; .123/ 2 .3/.12/ W .3/.12/; .132/:

(1.12)

It is well seen that the left and right cosets do not coincide. If the left and right classes of group G in its subgroup H coincide, that is, Gk H D HGk ;

(1.13)

for any k, then such a subgroup is called invariant. It can easily be shown that the set of h elements Ga HGa1 forms a group for any fixed Ga 2 G and, therefore, a subgroup of group G. Such a subgroup is similar to subgroup H . From equation (1.13) it follows that Ga HGa1 D H; (1.14) i.e., all similar subgroups for the invariant subgroup coincide with the latter. The group not having an invariant subgroup is called simple. Earlier, we introduced the concept of a group G  F , which is the direct product of two groups not having common elements except for the unit one. This requires that the elements of group G be commutative with the elements of group F, but this condition is too rigid for the applications considered in this book. It can be significantly weakened by using the concept of a semidirect product Y D G ^ F . It appears that the products Ga Fb also form a group when the condition of commutation of the elements of group G with the elements of group F is replaced by a considerably weaker condition of the form Fa GFa1 D G; (1.15) where Fa runs over all elements of group F. That is, the group G should be an invariant subgroup of group Y. In this case, the group F is called a factor group for the invariant subgroup G. As an example of the invariant subgroup, we consider a subgroup C3 in group C3v , which comprises three elements, namely, E; C3 , and C32 . It can easily be shown that

9

Section 1.4 Conjugacy classes .k/

although the reflection elements v in group C3v do not commute with the elements of subgroup C3 , the left and right cosets of this subgroup coincide and have the form E; C3 ; C32 I .1/

.2/

.3/

v ; v ; v : As a result, group C3v can be represented as C3 ^ CS , where the factor group CS .k/ consists of an identical element E and any of the three reflection elements v .

1.4 Conjugacy classes Two elements, A and B, of group G are called conjugate if there is such an element Q 2 G that A D QBQ1 . We will show that if the elements A and B are conjugate with some element C, then A and B are conjugate with each other. Indeed, it follows from A D QCQ1 that C D Q1 AQ. Then B D P CP 1 D PQ1 AQP 1 D .PQ1 /A.PQ1 /1 . That is, the elements A and B are conjugate with each other. It follows from this property that the elements of a group are divided into sets of mutually conjugate elements. Such sets are called classes of a group. The class is defined by specifying any of its elements A since the other elements of this class can be obtained by calculating the expressions QAQ1 , where Q runs over all elements of the group. Naturally, some elements of the class are repeated in this calculation. As an example, we consider the group C3v . Using Table 1.1, it can easily be shown that six elements of this group form the following three classes: EI

C3 ; C32 I

v.1/ ; v.2/ ; v.3/ :

Defining the class of conjugate elements yields a simple treatment of condition (1.15) determining the invariant subgroup G in group Y. Indeed, this condition means that the classes of conjugate elements of group Y, which include the elements of subgroup G, should entirely belong to this subgroup. Classes of conjugate elements have the following properties: 1. A unit element forms a class. 2. All classes, except for the class of a unit element, are not subgroups. 3. Each element of the abelian group forms a class. 4. The number of elements in a class is the divisor of the order of the group.

Chapter 2

Basic concepts of group representation theory

2.1 Linear vector spaces Consider a set L, whose elements x; y; : : : can be multiplied by an arbitrary complex number ˛ and summed up with each other, so that 1. if x; y 2 L, then ˛x and x C y D y C x 2 L,

(2.1)

2. .˛ C ˇ/x D ˛x C ˇx,

(2.2)

3. .˛ˇ/x D ˛.ˇx/,

(2.3)

4. 1x D x,

(2.4)

5. ˛.x C y/ D ˛x C ˛y,

(2.5)

6. there is a zero element 0, such that x C 0 D x for any x 2 L. Such a set is called a linear vector space. It is easy to see that L forms an abelian group with respect to the addition operation. The linear vector space of dimension n will be denoted as Ln . In such a space, one can find n linearly independent vectors, and any n C 1 vectors will be linearly dependent. Hence, in Ln it is possible to indicate n linearly independent basis vectors e1 , e2 ; : : : ; en , through which the arbitrary vector x 2 Ln is represented as x D xi ei . The quantities xi are called the components of vector x. If these quantities are complex (real), then Ln is a complex (real) space. Consider the transformation from one system of basis vectors to another: e0k D Si k ei :

(2.6)

Here, Si k is a nonsingular matrix of transformation from the old basis ei to the new basis e0k . We emphasize that the summation on the right-hand side of equation (2.6) is performed over the first index of the transformation matrix. As a result, x D xi ei D xk0 e0k D xk0 Si k ei ;

(2.7)

and for the components of vector x we obtain the following transformation law: xi D Si k xk0

or

x D Sx0 :

(2.8)

11

Section 2.1 Linear vector spaces

The summation on the right-hand side of equation (2.8) is resulted over the second index of the transformation matrix, which corresponds to the regular rules of multiplication of the matrix by a vector. We now consider the relationship y D Ax;

x; y 2 Ln;

(2.9)

where the mapping A can be considered as an operator, which, acting on the vector x of space Ln , transforms it into a vector y of the same space. We will take interest only in linear operators. It is said that A is a linear operator if 1. A.x C y/ D Ax C AyI 2. A.˛x/ D ˛Ax. If the mapping specified by operator A is one-to-one, then there exists an inverse operator A1 , such that AA1 D A1 A D 1; (2.10) where the identical operator 1 leaves all the vectors unchanged. Note that the coordinate system is absent in the definition of operators, i.e., the operators have an intrinsic meaning. In the particular basis, the linear operator is written in the form of a matrix yi D Ai k xk ;

(2.11)

where the matrix Ai k is defined according to the action of the operator on the basis vectors Aei D Aki ek : (2.12) Similarly to equation (2.6), when the action on the basis vectors is defined, the summation on the right-hand side is performed over the first index of the matrix of operator A. The summation on the right-hand side of equation (2.11) is resulted over the second index of the matrix of the operator, which corresponds to the regular rules of multiplication of the matrix by a vector. Rewrite relationship (2.9) in the primed system of basis vectors as y 0 D A0 x0 :

(2.13)

Taking into account that x D Sx0 and y D Sy 0 , we obtain the following relation between the matrices A0 and A: A0 D S 1 AS:

(2.14)

We now introduce metric in space L. Each pair of vectors x;y 2 L will be associated with a complex number .x;y/, which is called a scalar product of these vectors. It is required that this number satisfy the following conditions:

12

Chapter 2 Basic concepts of group representation theory

1. .x;y/ D .y; x/ ,

(2.15)

2. .x;˛y/ D ˛.x;y/,

(2.16)

3. .x1 C x2 ; y/ D .x1 ; y/ C .x2 ; y/,

(2.17)

4. .x;x/  0, where .x;x/ D 0 only if x D 0.

(2.18)

The real number (x, x) is called the square of the norm (or the square of the length) of vector x. The basis was not used when the quantity (x, y) was introduced. Therefore, this quantity is the intrinsic property of the vectors x and y. The space in which the scalar product is defined is called unitary space. Two vectors in unitary space are orthogonal to each other if their scalar product is equal to zero. It is said that an orthonormal basis is introduced into unitary space if all the basis vectors have a unit norm and are orthogonal to each other, i.e., .ei ; ej / D ıij :

(2.19)

Through the scalar product, one can define the distance jx  yj between two vectors: jx  yj2 D .x  y; x  y/:

(2.20)

It is said that a sequence of vectors xn 2 L converges or is a fundamental sequence if lim

m;n!1

jxm  xn j D 0:

(2.21)

We will focus on the linear spaces with a finite or countable basis. In the latter case, an additional requirement of completeness of space L is imposed. This means that for any fundamental sequence there exists x 2 L to which it converges. Complete unitary space is called Hilbert space. Naturally, unitary spaces of finite dimension are necessarily complete. We now consider the concepts of conjugate, self-conjugate, and unitary operators in a unitary space Ln . Operator AC is called conjugate with a linear operator A if .Ax;y/ D .x;AC y/

(2.22)

for any x;y 2 Ln . It follows from equation (2.22) that in the orthonormal basis, .AC /ij D .Aj i / ; that is, the matrix of a conjugate operator is complex-conjugate and transposed with respect to the matrix of operator A. Operator A is called self-conjugate, or Hermitian if AC D A. That is, .Ax;y/ D .x;Ay/

(2.23)

Section 2.2 Operators in configuration and function spaces

13

for any x;y 2 Ln . In the orthonormal basis, the Hermitian operator is represented by a Hermitian matrix which satisfies the expression following from equation (2.22) provided that AC D A. In real space, this matrix is symmetric. Operator U is called unitary if .U x;U y/ D .x;y/

(2.24)

for any x;y 2 Ln. That is, unitary transformation of the vectors preserves the metric. In the orthonormal basis, operator U is represented by a unitary matrix satisfying the condition U C U D U U C D 1. In real space, such a transformation of vectors is called orthogonal. Its matrix O in the orthonormal basis satisfies the orthogonality condition Q D 1, where OQ is the transposed matrix. OO

2.2 Operators in configuration and function spaces Consider an isolated system of n interacting material points. A set of their spatial coordinates forms a configuration space of the system of dimension 3n. Let x be the point of this space. In classical mechanics, the system is characterized by its trajectory x(t) in configuration space. In quantum mechanics, the state of the system is specified by the wave function ‰.x; t /, which is a vector in function Hilbert space. We confine ourselves to function spaces L with a finite or countable basis, which corresponds to considering bound states of a quantum system. Assume that operator R determines a certain nonsingular transformation of the coordinates of configuration space, x 0 D Rx: (2.25) If ‰.x 0 / 2 L, then this transformation can be associated with the linear operator PR in function space L according to the following relationship: ‰.x 0 / D ‰.Rx/ D ˆ.x/  PR ‰.x/:

(2.26)

Note that the time dependence is omitted in equation (2.26) and elsewhere below, which is not of principle for the results considered. Decompose the functions ‰.x/ and ˆ.x/ in orthonormal basis, ‰.x/ D cn 'n .x/;

ˆ.x/ D bm 'm .x/:

(2.27)

Then from equation (2.26) it can easily be obtained that Pkl .R/cl D b k ;

(2.28)

where the matrix Pkl .R/ of operator PR is determined by the action of the latter on the basis unit vectors: (2.29) PR 'l .x/ D Pkl 'k .x/:

14

Chapter 2 Basic concepts of group representation theory

As in equations (2.6) and (2.12), when the action of operator PR on the basis unit vectors is determined, the summation on the right-hand side of equation (2.29) is performed over the first index of matrix Pkl .R/. The summation on the right-hand side of equation (2.28) is resulted over the second index of the operator matrix, which corresponds to the regular rules of multiplication of the matrix by a vector. Theorem 2.1. If transformation R in the configuration space of the system does not change the physical quantity A(x) characterizing this system, then the linear operator, which corresponds to this quantity in the function space of the system, commutes with the operator PR : (2.30) ŒA.x/; PR  D 0: Proof. The linear operator A.x/ can be defined by specifying its action on the arbitrary function .x/ 2 L W A.x/ .x/ D '.x/: (2.31) After the transformation R, instead of equation (2.31) we have A.Rx/ .Rx/ D '.Rx/, or PR1 A.Rx/PR .x/ D '.x/:

(2.32)

With allowance for definition (2.31) of the operator A.x/, it follows from equation (2.32) that (2.33) PR1 A.Rx/PR D A.x/: Since the transformation R does not affect A(x), the relationship A.Rx/ D A.x/ is fulfilled for this quantity and the corresponding linear operator. Hence, the desired result follows from equation (2.33). Note that in the proof of this theorem, the transformations R and PR of the object were introduced for a fixed coordinate system, while in the previous section the matrix change of the linear operator was considered when the coordinate system was transformed. In view of this, equations (2.14) and (2.33) are equivalent since the transformation of an object for a fixed coordinate system can be considered as an inverse transformation of the coordinate system for a fixed object. This theorem makes it very easy to prove the commutativity of some operators. For example, as we will see in Chapter 4, the operator of the components of the angular momentum J˛ on the axis ˛ of the Cartesian laboratory system of coordinates is determined, with accuracy up to a constant factor, by an infinitesimal rotation operator about this axis. Since with such a rotation the square of the angular momentum J2 is preserved, we have (2.34) ŒJ2 ; J˛  D 0: However, the corollaries of the theorem are most interesting when as the operator A.x/ the Hamiltonian H.x/ of the quantum system is chosen. In this case, the operator PR has common eigenfunctions with the Hamiltonian and corresponds to the preserved

Section 2.3 Representations of groups

15

quantity. It is said that this operator defines the symmetry transformation of the quantum system. Its eigenvalues yield good quantum numbers, or symbols of symmetry of the stationary states. A set of such operators PR forms a symmetry group of the quantum system. It should be borne in mind that the various operators in this group generally do not commute with each other and therefore do not have a common system of eigenfunctions. How the symmetry of stationary states should be characterized in this case will be discussed in Section 2.4.

2.3 Representations of groups Homomorphic mapping of group G onto the group of square matrices of the same dimension is called the representation of this group. The dimension of matrices is called the dimension of representation. Thus, each element A of group G is associated with the matrix .A/, so that the product of the elements of group AB D C corresponds to the matrix product obtained using the regular formulas of matrix algebra: i k .A/kl .B/ D i l .C /:

(2.35)

If the mapping of group G onto .A/ is isomorphic, then the representation .A/ is called faithful. In quantum mechanics, representations usually arise in the definition of the action of symmetry transformations on the wave functions from some Hilbert space. Let ‰0 2 L. Then the element A of group G converts this function into ‰A D A‰0 2 L. If A runs over all elements of the group, then the number of built functions is the order of group g. However, some of them may be linearly dependent functions. Assume that the number of linearly independent functions ‰i is f  g. Then under the action of the group elements, they must be transformed through each other, i.e., A‰i D ki .A/‰k ;

(2.36)

where the coefficients ki .A/ form a square matrix of dimension f. As in equation (2.29), summation on the right-hand side of equation (2.36) is performed over the first index of the matrix. In this case, the product AB corresponds to the matrix that is the product of matrices .A/.B/ since AB‰k D A.i k .B/‰i / D li .A/i k .B/‰l D Œ.A/.B/lk ‰l : Thus, the matrices .A/ form an f-dimensional representation of group G. A set f of functions ‰i , on which the matrices .A/ are defined, is called the basis of representation. Consider the nonsingular linear transformation of the basis vectors: ‰i0 D Ski ‰k :

(2.37)

16

Chapter 2 Basic concepts of group representation theory

According to equation (2.15), the representation matrices .A/ then become the matrices  0 .A/ D S 1 .A/S: (2.38) The representations related by equation (2.38) are called equivalent, and the transformation of the representation to an equivalent form is called a similar transformation. It is obvious that there are infinitely many similar transformations and the corresponding equivalent representations. It turns out that among the equivalent representations of any finite group there is always a unitary representation, that is, a representation by a set of unitary matrices. However, it should be borne in mind that in the physical problem, a variety of groups is usually used and this result does not imply that representations of even the finite groups can simultaneously be reduced to a unitary form. Nevertheless, representations in physical applications in most cases are unitary by virtue of the unitarity of the symmetry operators themselves. Exactly such representations will be considered in what follows unless otherwise specified. Representation  of group G is called reducible if there exists a similar transformation reducing all matrices of this representation to the block-diagonal form: 0



0

.2/



0

0

0



0

0

0

    .m/

 .1/  0 .A/ D S 1 .A/S D

0



(2.39)

As a result, the representation splits into a direct sum of representations of smaller dimension, which can be written as : (2.40)  D  .1/ C  .2/ C    C  .m/ : Reducibility of the representation means that by linear transformation of basis vectors the initial space splits into a direct sum of subspaces, each of which transforms according to some representation  .˛/ of group G. That is, the operators of group G transform the vectors only within each subspace without mixing them with the vectors of other subspaces. If there is no transformation reducing all matrices of the representation  .˛/ of group G to the block-diagonal form, then such a representation is called irreducible. Irreducible representations of finite groups have the following properties: 1. The number of nonequivalent irreducible representations is equal to the number of classes of conjugate elements. 2. The sum of the squares of the dimensions of nonequivalent irreducible representations is equal to the order of the group, that is, f12 C f22 C    C fp2 D g; where f˛ is the dimension of the irreducible representation ˛.

(2.41)

Section 2.4 Characters. Decomposition of reducible representations

17

3. The dimension of the irreducible representation is the divisor of the order of the group. 4. For the matrix elements of irreducible representations, the relations of orthogonality have the form X

i k .A/ lm .A/ D .˛/

A

.ˇ /

g ı˛ˇ ıi l ıkm ; f˛

X f˛ .˛/ .˛/  .A/ i k .B/ D ıAB : g ik

(2.42)

(2.43)

˛;i;k

The summation in equation (2.42) is over all g elements of the group, and equation (2.43) summarizes all f˛2 terms of the matrix  .˛/ .A/ for each irreducible representation. In accordance with equation (2.41), the total number of terms in the sum (2.43) is also equal to g. As follows from the first two properties, the irreducible representations of abelian groups are one-dimensional, and their number is equal to the order of the group.

2.4 Characters. Decomposition of reducible representations Let there be a representation  of group G. The sum of diagonal elements of matrix .A/ denoted as .A/ is called the character of operation A in the representation : .A/ D i i .A/:

(2.44)

The characters of the equivalent representation coincide since .S 1 .A/S/i i D .S 1 /i k km .A/Smi D ımk km .A/ D kk .A/: Thus, the assignment of representations by their characters does not distinguish between equivalent representations. This is very convenient for physical applications since it is the nonequivalent representations that usually need to be selected in them. Since the elements of one class are related by an equation similar to equation (2.38), their characters should coincide. That is, the number of different characters of a representation does not exceed the number of classes of the group. For the characters of the irreducible representations, one can easily obtain from equation (2.42) the following orthogonality relation: X .˛/ .A/ .ˇ / .A/ D gı˛ˇ : (2.45) A

18

Chapter 2 Basic concepts of group representation theory

Given that the characters of the elements of one class are identical, equation (2.45) can be rewritten as X gC .˛/ .C / .ˇ / .C / D gı˛ˇ ; (2.46) C

where the sum is over all classes C of the group and g(C) is the number of elements in class C. p The values uC˛ D gC =g.˛/ form a square matrix since the number of classes is equal to the number of irreducible representations. Then instead of equation (2.46) we have uC˛ uCˇ D ı˛ˇ , which coincides with the condition of unitarity uC u D 1 of the matrix u. Using the equivalent condition of unitarity uuC D 1, we obtain another orthogonality relation for the characters of irreducible representations: X g ıi k : .˛/ .Ci / .˛/ .Ck / D (2.47) g.Ci / ˛ The characters of all nonequivalent irreducible representations are written in rows of a table whose columns are denoted by classes of the group. As an example, Table 2.1 shows the characters of the irreducible representations for the groups C3v and CS . Instead of the symbol of a class, one of its elements is presented, and the number before the class symbol indicates the number of elements in it (if the latter is different from unity). One-dimensional representations are given first in the rows, so that the first row is a fully symmetric, or identity representation (representation A1 for the group C3v and representation A0 for the group CS ), then two-dimensional representations follow, etc. Table 2.1. Characters of irreducible representations of the groups C3v and CS . C3v A1 A2 E

E 1 1 2

2C3 1 1 1

3v

CS

E



1 1 0

0

1 1

1 1

A A00

Let some reducible representation  be given. Suppose that using similar transformation (2.39), we decomposed it into irreducible representations  .˛/ : : X .˛/ .˛/ a  ; (2.48) D ˛

where the integer a.˛/ takes into account that the same irreducible representation can be included into the representation  a few times. Given that a similar transformation does not change the character of the representation, we have X a.˛/ .˛/ .A/: (2.49) . / .A/ D ˛

Section 2.4 Characters. Decomposition of reducible representations

19

Multiply equation (2.49) by .ˇ / .A/ and sum up over all elements of the group. Using orthogonality relation (2.45), we obtain the expression a.˛/ D

1 X . /  .A/.˛/ .A/ ; g

(2.50)

A

which is more convenient for use in the form a.˛/ D

1X g.C /. / .C /.˛/ .C / : g

(2.51)

C

It is fairly obvious that the irreducible representation of a group is generally a reducible representation of its subgroup. Indeed, during the transition to a subgroup, part of the transformations that mixed the vectors within the irreducible representation of the group is lost. The process of decomposition of irreducible representations of a group during the transition to its subgroup is called the reduction on subgroup. For example, the reduction of irreducible representations of group C3v in any of its three subgroups CS is as follows: : A1 D A0 ;

: A2 D A00 ;

: E D A0 C A00 :

(2.52)

Consequently, one-dimensional representations of a group convert into one-dimensional representations of its subgroup, and the two-dimensional representation splits into two one-dimensional ones. Let us return to the corollaries of the theorem from Section 2.2 and consider the problem of stationary bound states of an isolated system of n interacting material points: H.x/‰.x/ D E‰.x/: (2.53) Let there be a transformation R in the configuration space of the system, which leaves its Hamiltonian invariant. Then in view of equation (2.33) for A D H , it follows from equation (2.53) that the function PR ‰.x/ also belongs to the eigenvalue E. This means that each eigenvalue can be associated with one of the irreducible representations of the symmetry group of the Hamiltonian. The dimension of the irreducible representations determines the degree of degeneracy of the eigenvalue, and the type of the irreducible representation is exactly the index which characterizes the symmetry of the stationary states corresponding to this eigenvalue. When the symmetry group lowers to its subgroup, the process of reduction on subgroup takes place, which means the splitting of energy levels with the simultaneous lowering of the degree of their degeneracy. It should be noted that the eigenvalues of a Hamiltonian, which correspond to the different irreducible representations of its symmetry group, may coincide in principle (the so-called “accidental” degeneracy, i.e., the degeneracy which is not stipulated by this symmetry). However, the probability of such a coincidence is extremely small unless

20

Chapter 2 Basic concepts of group representation theory

a special reason occurs. In particular, the symmetry of the Hamiltonian may be higher than the considered one. In physical applications, there is also an inverse problem of reduction to the subgroup. That is, it should be determined which irreducible representations of group G can be obtained from a given irreducible representation of its subgroup H . The answer to this question is provided by the Frobenius reciprocity theorem: the reducible representation  of group G, which can be constructed on the basis of the irreducible representation  .h/ of its subgroup H , contains each irreducible representation  .˛/ of group G as many times as the decomposition of representation  .˛/ contains the representation  .h/ during the reduction on subgroup H . For example, using the results (2.52), from irreducible representations of the group CS we obtain the following representations of the wider group C3v A0 ! A1 C E;

A00 ! A2 C E:

(2.54)

2.5 Direct product of representations. Symmetric power Consider two irreducible representations  .˛/ and  .ˇ / of group G defined, re.˛/ .ˇ / spectively, by the sets of basis functions ‰i .i D 1; 2; : : : ; f˛ / and ˆk .k D .˛/

.ˇ /

1; 2; : : : ; fˇ /. Forming all possible products ‰i ˆk , we obtain the basis of an f˛ fˇ dimensional representation of group G, which is denoted as  .˛/   .ˇ / and is called the direct product of representations  .˛/ and  .ˇ / . The matrix elements of the direct product of representations are expressed in the form of products of the matrix elements of representations  .˛/ and  .ˇ / : X .˛/ .˛/ .ˇ / .ˇ / .˛/ .ˇ / mi .Q/nk .Q/‰m ˆn : (2.55) Q‰i ˆk D m;n

The matrix  .˛ˇ / .Q/ from equation (2.55) with four-index matrix elements is called the direct product of matrices  .˛/ .Q/ and  .ˇ / .Q/, and the dimension of the resulting matrix is equal to the product of dimensions of the multiplied matrices. It is obvious that in general  .˛/   .ˇ / ¤  .ˇ /   .˛/ , but these two direct products can be converted into each other by permutation of rows and columns. Therefore, the characters of the direct-product matrices do not depend on the order of multiplication of the matrices of the initial representations, which is also easy to obtain when these characters are calculated: X .˛/ .ˇ / .˛ˇ / .Q/ D i i .Q/kk .Q/ D .˛/ .Q/.ˇ / .Q/; (2.56) i;k

that is, the character of the direct product of two representations is equal to the product of their characters. In general, the direct product forms a reducible representation, and

21

Section 2.5 Direct product of representations. Symmetric power

its decomposition is given by equation (2.50) with . / D .˛ˇ / D .˛/  .ˇ / . If the multiplied representations coincide, then .˛˛/ .Q/ D Œ.˛/ .Q/2 : For example, for the group C3v we have : E  E D A1 C A2 C E:

(2.57)

For the direct product of the coinciding representations, we introduce, instead of the .˛/ .˛/ basis unit vectors ‰i ˆk , their symmetrized combinations: .˛/

.˛/

.˛/

.˛/

‰i ˆk C ‰k ˆi ;

.˛/

.˛/

.˛/

.˛/

‰i ˆk  ‰k ˆi :

Dimensions of the space of symmetric and antisymmetric combinations are .˛/ .˛/ f˛ .f˛ C 1/=2 and f˛ .f˛  1/=2, respectively. If ‰i D ˆi , that is, the bases of multiplied representations coincide, then all antisymmetric combinations vanish. Such a direct product is called a symmetric product, and its dimension is equal to f˛ .f˛ C 1/=2. The characters of such a product, which are denoted as Œ.˛/ 2 .Q/, are not equal to the product of the characters of multiplied representations. The following expression can be obtained for them: Œ.˛/ 2 .Q/ D

1 .˛/ 2 1  .Q / C Œ.˛/ .Q/2 : 2 2

(2.58)

In particular, the dimension of the symmetric product of the representation E of group C3v is three, and the representation A2 is absent in its decomposition as compared with equation (2.57). The symmetric power n of representation  .˛/ arises in the physical applications of interest to us in describing the molecular overtone vibrational states formed in the n-fold excitation of a  .˛/ -type vibration. There are formulas relating the characters of the symmetric power with the characters of the initial representation. In particular, 1 1 1 Œ.˛/ 3 .Q/ D 3 .Q/ C .Q/.Q2 / C .Q3 /; 6 2 3 4

Œ.˛/  .Q/ D

1 4 1 1  .Q/ C 2 .Q/.Q2 / C .Q/.Q3 / 24 4 3

1 1 C 2 .Q2 / C .Q4 / 8 4

(2.59)

For larger n, calculation by such formulas becomes very cumbersome. But it can be greatly simplified for small dimensions of the representation  .˛/ . It is clear that for one-dimensional representations, Œ.n/ .Q/ D n .Q/:

(2.60)

22

Chapter 2 Basic concepts of group representation theory

For the two-dimensional representation  .˛/ , we denote the basis functions as x and y, which we assume to be the components of some vector. It is important that with any transformation Q, the quantities Qx and Qy are converted through x and y by an orthogonal matrix, which formally can always be interpreted as a matrix of rotation or reflection of the xy space. Rotation by an angle 'Q , appropriate to transformation Q, in two-dimensional space is determined by the matrix ˇ ˇ ˇ cos 'Q  sin 'Q ˇ ˇ ˇ (2.61) ˇ sin 'Q cos 'Q ˇ with the sum of diagonal elements 2 cos 'Q . Then from the equality 2 cos 'Q D .Q/ we obtain the value of the rotation angle for the transformation Q. Reflection in twodimensional space is determined by the matrix ˇ ˇ ˇ 1 0 ˇ ˇ ˇ 0 1

(2.62)

with a zero sum of diagonal elements. If .Q/ D 0, then the transformation Q is either a reflection or rotation by an angle 'Q D =2, for which the sum of diagonal elements is also equal to zero. To clarify this, it suffices to consider the value of .Q2 /. Transformation Q is a reflection if the value is 2 and is a rotation if the value is 2. The symmetric power n of a two-dimensional representation is a representation with the basis x n; x n1 y; : : : ; xy n1 ; y n : (2.63) It is easy to see that in such a basis, the character of the operation of reflection with matrix (2.62) is given by 1 C .1/n : (2.64) Œ.n/ .Q/ D 2 To find the character in basis (2.63) corresponding to the rotation, it is convenient to proceed to the spherical components x ˙ iy of a two-dimensional vector. In this case, the rotation matrix (2.61) is diagonal, i.e., ˇ ˇ 0 ˇ exp.i'Q / ˇ ˇ ˇ; 0 exp.i'Q /

(2.65)

and basis (2.63) is rewritten as .x C iy/n ;

.x C iy/n1 .x  iy/; : : : ; .x C iy/.x  iy/n1 ;

.x  iy/n : (2.66)

In such a basis, the character of the operation of rotation with matrix (2.65) is given by (2.67) exp.i n'Q / C expŒi.n  2/'Q  C : : : C exp.i n'Q /;

23

Section 2.6 The Clebsch–Gordan coefficients

whence we finally obtain Œ.n/ .Q/ D

sin.n C 1/'Q : sin 'Q

(2.68)

As an example of using equations (2.64) and (2.68), we make the decomposition into irreducible representations for several symmetric powers of the representation E of group C3v W : ŒE3 D A1 C A2 C E; : (2.69) ŒE4 D A1 C 2E; 5 : ŒE D A1 C A2 C 2E: Formulas obtained in a similar way for the characters of the symmetric power n of the three-dimensional representation  .˛/ are also known. If the transformation Q is equivalent to a rotation in the xyz space, then .n/

Œ

.Q/ D

sin

.nC2/'Q 2

sin

sin 'Q sin

.nC1/'Q 2 'Q 2

;

and if the transformation Q is equivalent to a rotation with reflection orthogonal to the rotation axis in the xyz space, then Œ.n/ .Q/ D .1/n

sin

.nC2/'N Q 2

sin

sin 'NQ sin

.nC1/'N Q 2 'N Q 2

;

'NQ D 'Q C :

Rotation, or rotation with reflection, which formally corresponds to the element Q in these formulas, is determined in the following way. If the coordinates x; y; z really do not belong to the representation  .˛/ in the group considered, then the transformation Q should be understood as the corresponding transformation of such an isomorphic group in which the coordinates x; y; z belong to this representation. For example, for the representation F1 of the point groups O and Td (see Chapter 5) the transformation should be chosen from the group O and for the representation F2 , from the group Td .

2.6 The Clebsch–Gordan coefficients Decomposition of the direct product  .˛/   .ˇ / into irreducible representations X  .˛/   .ˇ / D P a./  ./ (2.70) 

is called the Clebsch–Gordan series. Determine the conditions for the existence of an identity representation A, such that .A/ .Q/ D 1 for any Q 2 G, in this decomposition: 1 X .˛/ a.A/ D  .Q/.ˇ / .Q/ D ıˇ ˛ ; g Q

24

Chapter 2 Basic concepts of group representation theory

where ˛  denotes the representation whose matrix elements are complex conjugate to the matrix elements of the representation ˛. Such representations are called complex conjugate. Thus, an identity representation is contained in decomposition (2.70) only one time and only when the multiplied representations are complex conjugate. For the real representations, an identity representation is only contained in the direct product of an irreducible representation by itself. Decomposition (2.70) is realized by a linear transformation from the basis functions .˛/ .ˇ / ./ ‰i ˆk to sets of basis functions ' t for the irreducible representations : X .a/ .˛/ .ˇ / 't D h˛i; ˇkja t i ‰i ˆk : (2.71) i;k

The coefficients h˛i; ˇkja t i are called the Clebsch–Gordan coefficients. The index a distinguishes the representations  repeated in decomposition (2.70). For the orthonormal bases, the Clebsch–Gordan coefficients form a unitary matrix. In physical applications, this unit matrix usually can be chosen real, that is, the following orthogonality relation is fulfilled for its elements: X h˛i; ˇkja t ih˛i 0 ; ˇk 0 ja t i D ıi i 0 ıkk 0 ; ˛;;t

X

h˛i; ˇkja t ih˛i; ˇkja0  0 t 0 i D ıaa0 ı 0 ı t t 0 :

(2.72)

i;k

In this case, the inverse matrix of the Clebsch–Gordan coefficients is equal to the transposed one, and the transformation inverse of that in equation (2.71) has the form X .˛/ .ˇ / .a/ ‰i ˆk D ha t j˛i; ˇki ' t : (2.73) a;;t

Naturally, h˛i; ˇkja t i  0 for any  not included in decomposition (2.70). Up to now, we have considered the direct product of representations of one group. Now let Y D G  F , and  .˛/ and  .ˇ / be irreducible representations of groups G and F, respectively. Then the direct product of representations  .˛/   .ˇ / is an irreducible representation of group Y, and all irreducible representations of this group can readily be obtained as direct products of irreducible representations of groups G and F. In this case, the element Q D Q1 Q2 of group Y has the character .˛ˇ / .Q/ D .˛/ .Q1 /.ˇ / .Q2 /: We emphasize that the indices of irreducible representations (symbols of symmetry) of groups G and F preserve their meaning in group Y. If Y D G ^ F , then all irreducible representations of group Y cannot be represented in the form of direct products of representations of groups G and F. Consequently, the symmetry symbols of the latter do not preserve their meaning in group Y . The number of irreducible representations of group Y D G ^ F is smaller, and their dimensions can be greater than in the case Y D G  F .

25

Section 2.7 Basis functions of irreducible representations

2.7 Basis functions of irreducible representations Choose such a function ‰0 the action on which by all operations Q 2 G leads to g linearly independent functions ‰Q D Q‰0 (that is, the function ‰0 has no symmetry properties for the operations of group G). Functions ‰Q form the basis of a g-dimensional representation of group G since P ‰Q D PQ‰0 D R‰0 D ‰R :

(2.74)

Such a representation is called regular. According to equation (2.74), the matrices of a regular representation, except for the matrix of identical representation E, have zero diagonal elements. Hence, for the characters of a regular representation we have .Q D E/ D g;

.Q ¤ E/ D 0:

(2.75)

For each irreducible representation, we have .˛/ .E/ D f˛ , where f˛ is the dimension of the representation. Substituting equation (2.75) into equation (2.50), we find that in the decomposition of a regular representation each irreducible representation is contained as many times as its dimension. We will show that for the decomposition of a regular representation, the following linear combinations of the basis functions ‰Q should be introduced: .˛/

‰i k D

f˛ X .˛/ i k .Q/ ‰Q ; g

(2.76)

Q

where the summation is over all elements of the group and i.˛/ is a matrix element k .˛/

of the irreducible representation  .˛/ . To do this, we act on the function ‰i k by an arbitrary operation P 2 G: .˛/

P ‰i k D

f˛ X .˛/ f˛ X .˛/ 1  i k .Q/ PQ‰0 D i k .P R/ R‰0 : g g Q

(2.77)

R

Further we transform the matrix element X .˛/ X .˛/ .˛/ .˛/ .˛/ i m .P 1 / mk .R/ D mi .P /mk .R/ : i k .P 1 R/ D m

(2.78)

m

In the transition to the last expression in equation (2.78) it was taken into account that the matrices of the representation  .˛/ are unitary. Substituting equation (2.78) into equation (2.77), we finally arrive at X .˛/ .˛/ .˛/ mi .P /‰mk : (2.79) P ‰i k D m

26

Chapter 2 Basic concepts of group representation theory .˛/

Consequently, the set f˛ of functions ‰i k with the fixed second index k forms the basis of the irreducible representation  .˛/ . Note that the summation on the right-hand side of equation (2.79) is over the first index of the transformation matrix, as should be the case when considering the transformation of the basis unit vectors. Since the index k runs over f˛ values, a total of f˛ independent bases can be formed. This is exactly what should be expected since each irreducible representation is included in the decomposition of a regular representation as many times as its dimension. It follows from equation (2.76) that to obtain the basis functions of the representation  .˛/ , it suffices to act by the operators D ".˛/ ik

f˛ X .˛/ i k .Q/ Q g

(2.80)

Q

with the fixed second index on an arbitrary function ‰0 not having the symmetry properties. If the function ‰0 has some symmetry properties, then the result of the .˛/ action of the operator "i k on it can lead to zero. Consider the action of this operator on the basis functions of irreducible representations: .˛/

.ˇ / "i k ‰mn D

f˛ X .˛/ .ˇ / i k .Q/ Q‰mn : g Q

Using equality (2.79) and orthogonality relation (2.42) for the matrix elements of irreducible representations, we finally obtain .˛/

.˛/

.ˇ / D ı˛ˇ ıkm ‰i n : "i k ‰mn

(2.81)

.˛/

Consequently, the action of the operator "i k on the basis function of an irreducible representation yields either a basis function of the same representation or zero. In particular, .˛/ .˛/ .˛/ "i i ‰i n D ‰i n : (2.82) .˛/

.˛/

That is, the action of the operator "i i on the basis function ‰i n yields the same function, and its action on other basis functions results in zero. Such operators are called projection operators. Equation (2.82) implies the following equality for them: .˛/ .˛/

.˛/

"i i "i i D "i i :

(2.83)

Using the projection operators, the arbitrary function ‰ can be represented as the decomposition X .˛/ ‰D ‰i i ; (2.84) ˛;i

Section 2.7 Basis functions of irreducible representations

27

.˛/

where the functions ‰i i are given by equation (2.76) for i D k. To prove this, we substitute the expressions for these functions into equation (2.84) and use the definition of the character. Then X f˛ ‰D .˛/ .Q/Q‰: (2.85) g ˛;Q

From orthogonality relation (2.47) for the characters of irreducible representations we have X f˛ 1 X .˛/  .E/.˛/ .Q/ D ıEQ : (2.86) .˛/ .Q/ D g g ˛ ˛ In equation (2.86), it was taken into account that .˛/ .E/ D f˛ is a real value. Substituting equation (2.86) into equation (2.85), we obtain ‰  ‰, which proves equation (2.84). The identity also means that for a set of projection operators the so-called completeness condition X .˛/ "i i D 1 (2.87) ˛;i

is fulfilled. For example, for the group CS the projection operators have the following form: "011 D .1 C /=2; "0011 D .1  /=2: The first operator extracts from an arbitrary function its symmetric part with respect to the reflection operation and the second operator, its antisymmetric part. It is easy to see that the sum of these operators is unity. .˛/

Theorem 2.2. The basis functions ‰i k of irreducible representations of a finite group satisfy the following orthogonality conditions: ² ˇ D E B.˛; k; n/ for k ¤ n; .˛/ ˇ .ˇ / ‰i k ˇ‰mn D ı˛ˇ ıi m (2.88) 1 for k D n; where B.˛; k; n/ is determined by the choice of the bases k and n for the irreducible representation ˛ and does not depend on the numbers of the basis functions in these representations. Proof. We now use the fact that the scalar product is invariant under any unitary transformation of the basis functions. Therefore, ˇ ˇ D E X E D .˛/ ˇ .ˇ / .˛/ .˛/ ˇ .ˇ / .ˇ / ‰i k ˇ‰mn D : (2.89) pi .Q/ sm .Q/ ‰pk ˇ‰sn p;s

The right-hand side in equation (2.89) gives an expression for the scalar product after a unitary transformation of the initial basis functions using the matrix  .˛/ .Q/. We sum up equation (2.89) over all elements of the group and use orthogonality relation

28

Chapter 2 Basic concepts of group representation theory

(2.42) for the matrix elements of irreducible representations on the right-hand side. Then ˇ E D g X D .˛/ ˇˇ .˛/ E .˛/ ˇ .ˇ / D ı˛ˇ ıi m ‰pk ˇ‰pn : (2.90) g ‰i k ˇ‰mn f˛ p The scalar product on the right-hand side of equation (2.90) for k D n is unity, and the sum of these products is f˛ . For k ¤ n, this sum does not depend on i and m, which allows it to be written in the form f˛ B.˛; k; n/. As a result, we obtain equation (2.88).

2.8 Irreducible tensor operators. The Wigner–Eckart theorem .˛/

The irreducible tensor of group G is a set f˛ of values Vi , which is transformed for operations of the group according to its irreducible representation  .˛/ : X .˛/ ki .P /Vk.˛/ : (2.91) P Vi.˛/ D k

Definition (2.91) implies that any set of basis functions of a given irreducible repre.˛/ sentation can be regarded as an irreducible tensor. For the irreducible tensors Vi and .ˇ /

Wk

the operation .a/

Dt

D

X

.˛/

h˛i; ˇkja t iVi

.ˇ /

Wk

i;k

is defined, as a result of which one can form a tensor that transforms according to the irreducible representation  included in the decomposition of the direct product  .˛/   .ˇ / . .˛/ The Irreducible tensor operator is a set f˛ of values Ti with the transformation law X .˛/ .˛/ .˛/ ki .P /Tk : (2.92) P 1 Ti P D k

Transformation laws (2.91) and (2.92) are different because the operators in the old and new bases are related by equation (2.14). However, as was noted in Section 2.2, instead of the transformation of the coordinate system, one can transform the object in an invariable coordinate system. In this case, the transformations P and P 1 on the left-hand side of equation (2.92) should be interchanged. ./ ./ Consider the matrix element h˛i jT t jˇki of the irreducible tensor operator T t , which is specified on the functions belonging to the irreducible representations of the

29

Section 2.8 Irreducible tensor operators. The Wigner–Eckart theorem ./

same group. Using transformation law (2.92), we first show that the function T t jˇki is transformed according to the direct product  ./   .ˇ / . Indeed, X ./ ./ ./ .ˇ / mt .P /nk .P /Tm./ jˇki; P T t jˇki D .P 1 T t P /.P jˇki/ D m;n

as should be the case. Hence, according to equation (2.73), this function is decomposed over basis functions of the irreducible representations in the following way: X ./ hamj t; ˇkijˆ.a/ (2.93) T t jˇki D m .; ˇ/i; a;;m

where the index a distinguishes the irreducible representations repeated in the decomposition of the direct product, and the arguments of the function ˆ indicate that it depends on the multiplied representations. Substituting decomposition (2.93) into the expression for the matrix element, we obtain X ./ hamj t; ˇki h˛i jˆ.a/ (2.94) h˛i jT t jˇki D m .; ˇ/i: a;;m

According to orthogonality condition (2.88) for the basis functions of irreducible representations, we can write for the scalar product on the right-hand side of equation (2.94) h˛i jˆ.a/ (2.95) m .; ˇ/i D ı˛ ıi m Ba .˛; ; ˇ/; where the quantity Ba .˛; ; ˇ/ does not depend on the numbers of the basis functions. Substituting equation (2.95) into equation (2.94), we obtain the analytical content of the Wigner–Eckart theorem: X ./ ha˛i j t; ˇkiBa .˛; ; ˇ/: (2.96) h˛i jT t jˇki D a

This theorem allows the quantities (Clebsch–Gordan coefficients) determined only by the symmetry properties of the considered system to be singled out in the calculation of the matrix elements. In this case, the physical essence of the system is taken into account by the factors Ba . It is clear that matrix element (2.96) immediately vanishes if the decomposition of the direct product  ./  .ˇ / does not contain the representation  .˛/ . We now apply the Wigner–Eckart theorem to the case which is often encountered in physical applications–calculation of the matrix elements hb˛i jT .0/ jb 0 ˇki of the operator T .0/ which is invariant with respect to transformations of the group G (the indices b and b 0 distinguish different realizations of the representations  .˛/ and  .ˇ / , respectively, which are available in the physical problem). Such an invariant operator can, for example, be the Hamiltonian of the system. In this case, in the formulation

30

Chapter 2 Basic concepts of group representation theory

of the theorem the representation  is equal to the identity representation A. Since  .A/   .ˇ / D  .ˇ / , only one term is retained in the sum over the index a. For the corresponding Clebsch–Gordan coefficient we have h˛i jA; ˇki D ı˛ˇ ıi k . As a result, hb˛i jT .0/jb 0 ˇki D ı˛ˇ ıi k hbkT .0/ kb 0 i;

(2.97)

where the double line in the matrix element means that it is independent of the indices of the basis functions. For each irreducible representation ˛, elements (2.97) form a matrix of the operator T .0/ by the indices b and b 0 , whose dimension is equal to the number of different realizations of this irreducible representation in the physical problem.

Chapter 3

The permutation group

3.1 Operations in the permutation group. Classes The group of n permutations of n objects numbered from 1 to n has already been introduced in Section 1.1. Any of the nŠ permutations of this group given by the symbol   1 2 3  n (3.1) P D i1 i2 i3    in can be represented as a product of commuting cycles (1.9). The first cycle starts with the numbers 1 and i1 , then follows the number passing to where the number i1 was, and so on. The number 1 passes to where the last number of the cycle was. Then a similar procedure is done with the rest of the numbers in permutation (3.1). For example,   1 2 3 4 5 6 7 D .1425/.37/.6/: 4 5 7 2 1 6 3 The resulting cycles of this procedure do not contain common elements and therefore commute with each other, that is, their order is arbitrary in the product. By definition, the cycle is invariant with respect to the cyclic permutation of its elements: .i1 i2 i3    ik / D .i2 i3    ik i1 / D .i3    ik i1 i2 /:

(3.2)

The number of elements of the cycle is called its length. Naturally, .i1 i2 i3    ik /k D I;

(3.3)

where I is an identical permutation. From equation (3.3) we easily obtain .i1 i2 i3    ik /1 D .i1 i2 i3    ik /k1 :

(3.4)

The cycle of two elements is called the transposition, and the latter satisfies the relations .i1 i2 / D .i1 i2 /1 D .i2 i1 /: The following properties are also useful for operations in the permutation group: 1. The product QPQ1 is equal to the permutation obtained from P if its elements are rearranged according to the permutation Q, that is, QPQ1  QŒP : For example, .142/.25/.142/1 D .15/, a .25/.142/.25/ D .145/.

(3.5)

32

Chapter 3 The permutation group

2. Two cycles having a common element can be combined into one according to the rule .ab    d m/.mp    q/ D .ab    d mp    q/; where previously the common element m was put at the end of the first and at the beginning of the second cycle. 3. Any cycle can be represented as a product of transpositions. Although such a representation is ambiguous, the number of transpositions that implement it will always have the same parity. Naturally, this rule is preserved when an arbitrary product of cycles is considered. Therefore, all permutations can be divided into even and odd, depending on the parity of the number of transpositions in their representation. 4. Any permutation can be represented as a product of transpositions containing two successive numbers, that is, transpositions such as .i; i C 1/. Obviously, the group n has the subgroups 1 ; 2 ; : : : ; n1 . In addition, all even permutations of group n also form a group called alternating. We show that the alternating group is an invariant subgroup of n . Indeed, if P is an even permutation, then the product QPQ1 is also an even permutation for any Q 2 n . According to equation (1.14), it follows that the even permutations form an invariant subgroup of n . Consider the division of the elements of group n into classes. Recall that any two elements Pi and Pj belong to the same class if they are related by the relationship Pi D QPj Q1 , where Q 2 n . Since relation (3.5) is fulfilled for the permutation group, the cyclic structure of the permutations Pi and Pj should be the same, that is, the number of cycles in these permutations and their lengths should coincide. Permutations Pi and Pj can differ only in numbers in the cycles. Thus, the class of group n is characterized by a certain division of the elements into cycles. The number of different classes is equal to the number of different divisions of the number n into integer nonnegative summands, i.e., is equal to the number of different integer solutions (including zero) of the equation 1 1 C 2 2 C    C n n D n;

(3.6)

where k determines the number of cycles of length k. The sets of integers 1 ; 2 ; : : : ; n satisfying equation (3.6) uniquely identify the classes of n which will be denoted by the symbols ¹11 22    mm º. In these symbols, we will indicate only the cycles whose number for permutations of the designated class is nonzero. Then the class ¹1nº in group n corresponds to the identical permutation, and six permutations of 3 mentioned in equation (1.10) are divided into the following three classes: .1/.2/.3/; ¹13 º W ¹12º W .1/.23/; .2/.13/; .3/.12/; ¹3º W .123/; .132/:

(3.7)

Section 3.2 Irreducible representations. The Young diagrams and tableaux

33

It is clear that when the elements that are conjugate to a given element P of group n are determined using equation (3.5), part of the obtained nŠ elements will be repeated. The number of independent elements determining the number of elements of the class is closely related to its cyclic structure. Indeed, assume there is a class ¹11 22    mm º. We arrange n numbers within the cycles in the order of a natural sequence and act on the permutation obtained in such a way by all elements of group n . It is obvious that if only the relative positions of the cycles of the same length change then the resulting permutation coincides with the initial one. In all, 1 Š 2 Š    m Š such variants exist. Besides, the initial permutation does not change during a cyclic movement of the numbers within any of its cycles. For a cycle of length k there exist k different cyclic movements, and the total number of such variants for all cycles included in the permutation is equal to 22    mm . As a result, for the order of the class we obtain the expression g.¹11 22    mm º/ D

1 Š 2

Š 2 2

nŠ :    m Š m m

(3.8)

Note that in the denominator the factors corresponding to all k D 0 become unity. For class {12} of group 3 , from equation (3.8) we obtain 3Š=2 D 3, which certainly coincides with equation (3.7).

3.2 Irreducible representations. The Young diagrams and tableaux The number of nonequivalent irreducible representations of any finite group is equal to the number of its classes. Therefore, nonequivalent irreducible representations of group n can also be specified by different divisions of the number n into nonnegative integers: (3.9)

.1/ C .2/ C    C .m/ D n; where by tradition the numbers .i / are arranged in decreasing order. Some of .i / in division (3.9) may coincide. Obviously, m  n. The irreducible representation corresponding to such a division will be denoted as Œ   Œ .1/ .2/    .m/ . For convenience, the occurrence of several identical numbers .i / is indicated in the form of a power at one of these numbers. For example, there are irreducible representations Œ  D Œ3; Œ21; Œ13  for the group 3 . The divisions based on the so-called Young diagrams, in which each number .i / is associated with the row of .i / cells, are clearly seen in graphics. The Young diagrams for all irreducible representations of group 4 are presented in Figure 3.1. In what follows it is shown that the Young diagram determines the permutation symmetry of the basis functions of the corresponding irreducible representations. Consider the process of reduction of the group n to its subgroup n1 . As was mentioned in Section 2.4, the irreducible representation of the group in the transition

34

Chapter 3 The permutation group

Figure 3.1. The Young diagrams of irreducible representations of group 4.

to its subgroup becomes, in general, reducible. It appears that the representation [ ] of group n is decomposed into representations Π0  of subgroup n1 which correspond to the Young diagrams obtained from the Young diagrams for the representation [ ] by removal of one cell. For example, it follows from Figure 3.2 that the irreducible representation [22 1] of group 5 is decomposed into irreducible representations [22 ] and [212 ] of its subgroup 4 . In a similar way, one can consider the reduction of the group n1 to its subgroup n2 , etc. up to the reduction of the group 2 to its subgroup 1 .

Figure 3.2. Decomposition of the irreducible representation [22 1] of the group 5 upon reduction on its subgroup 4.

From arbitrary basis functions of the irreducible representation [ ] of group n it is always possible to make up such linear combinations that the matrices of this representation, which are determined by these functions and correspond to the permutations of the subgroups n1 ; n2 ; : : : ; 1 will be of quasidiagonal form. Thus, new basis functions of representation [ ] will simultaneously be the basis functions of all irreducible representations of the subgroups n1 ; n2 ; : : : ; 1 into which the representation [ ] is decomposed upon the sequential reduction n ! n1 !    ! 1 . Such basis functions can conveniently be characterized by the so-called standard Young tableaux. To obtain the standard Young tableau of a given basis function of the representation [ ], the numbers from 1 to n should be so arranged in the cells of the Young diagram that the following condition is fulfilled. The truncation of a cell with the number n should result in a Young diagram according to which this function transforms in the group n1 , and the truncation of a cell with the number n1 should result in a Young diagram according to which this function transforms in the group n2 , and so on, until the Young diagram with a single cell remains. Evidently, the numbers can only be so arranged in the Young diagram that they increase when moving from left to right in the rows and from top to bottom in the columns. Otherwise, unresolved Young diagrams will appear at some stage of the reduction.

35

Section 3.3 Basis functions of irreducible representations

Thus, each basis function of the irreducible representation [ ] is associated with the standard Young tableau, and the dimension f of this representation is determined by the number of such tableaux. A similar irreducible representation is called standard. Its basis functions are automatically orthogonal to each other since they are characterized by different sequences of irreducible representations in the chain n1 , n2 ; : : : ; 1 . Young tableaux are often numbered in the degree of deviation of the placement of the numbers in them from the natural series. For this, the Young tableaux with the number 2 in the first row are placed first, then the Young tableaux with the number 2 in the second row follow. Among a set of tableaux with the number 2 in the same rows, we place the tableaux with the number 3 in the higher row ahead of the other. We do the same with all other numbers. The first standard tableau is called fundamental. As an example, Table 3.1 shows the r .i /-numbered Young tableaux of the irreducible representation [22 1] of group 5 . Table 3.1. Standard Young tableaux of the irreducible representation [22 1] of group 5.

Five basis functions corresponding to these tableaux are uniquely defined by the following sequences of representations of groups 4 ; 3 ; 2 W r .1/ W r

.2/

r .3/

Œ22 

! Œ21 ! Œ2;

W Œ21  ! Œ21 ! Œ2; 2

W

Œ22 

! Œ21 ! Œ12 ;

r

.4/

W Œ21  ! Œ21 ! Œ1 ;

r

.5/

W Œ212  ! Œ13  ! Œ12 :

2

(3.10)

2

All the sequences in (3.10) are different, as should be the case.

3.3 Basis functions of irreducible representations We introduce a symmetrization operator with respect to the operations of the permutation group n X !s D const P; (3.11) where P runs over all operations of the group. Application of the operator !s to an arbitrary function dependent on n objects leads to a symmetric function which does not change under the action of any permutation of group n (of course, if the result

36

Chapter 3 The permutation group

of the symmetrization is not zero). One can also define the antisymmetrization (or alternation) operator with respect to the operations of group n X !a D const .1/p P; (3.12) where p D 0 and 1 for the even and odd permutations P, respectively. From an arbitrary function, the operator !a forms an antisymmetric function that is invariant to the action of any even permutation and changes sign under the action of any odd permutation (of course, if the result of the alternation is not zero). Write out a fundamental Young tableau for the irreducible representation [ ]. We first symmetrize an arbitrary function over the objects included in each row of this tableau and then alternate over the objects contained in each column. Note that after the alternation operation, the function generally ceases to be symmetric for the objects located in one row. Such symmetry is retained only for the objects located in the first-row cells extending beyond the rest of the rows. It appears that when operations of the group n act on this function, there arises a space f of independent functions transformed according to the representation [ ], and the matrices of irreducible representations can always be chosen real. Such a procedure can be performed on the basis of other standard Young tableaux. As a result, f independent bases can be obtained from an arbitrary function not having the symmetry properties with respect to the operations of group n . There are only two one-dimensional irreducible representations, [n] and Œ1n , in the permutation group n . From the construction procedure of basis functions it follows at once that the representation [n], whose Young diagram consists of one row, corresponds to the totally symmetric function, and the representation Œ1n , whose Young diagram consists of one column, corresponds to the totally antisymmetric function. The tables of characters of irreducible nonequivalent representations of the permutation groups 2  8 , which are used in the second part of the book to describe intramolecular dynamics, are given in Appendix 1. When permutation groups are used, it is also important to define the conditions under which the construction of the basis functions of irreducible representations leads to zero. The corresponding problem is formulated as follows. Let there be n identical particles, each of which has a spin s. Construct the product of all one-particle spin functions X D 'i1 .1/'i2 .2/    'in .n/; (3.13) where the number in the parentheses is the particle number and the subscript specifies the different quantum one-particle states, that is, the particle with the number k is in the state ik . It is well known that the particles with spin s have a total of 2s C 1 quantum states. Therefore, if n > 2s C 1, then some subscripts in equation (3.13) will coincide. Let the operations of the group n act on the particle numbers. We will try to construct the basis functions of the irreducible representation [ ] based on function (3.13). It is easy to understand that the operation of alternation will result in zero if the number of

Section 3.4 The conjugate representation

37

rows in the Young diagram exceeds 2sC1. Indeed, in this case, the particles in the same quantum state are subject to antisymmetrization over particle numbers. Consequently, in an ensemble of n identical particles, only the collective spin states which in the permutation group of these particles correspond to the Young diagrams with no more than 2s C 1 rows are allowed.

3.4 The conjugate representation Each standard real irreducible representation [ ] of group n can be associated with Q of the same dimension. The the conjugate (or associated) irreducible representation Œ  Q Œ  matrices  .P / of the conjugate representation differ from the matrices  Œ .P / of the initial representation in the factor .1/p , where p is the parity of the permutation P. From the results of the previous section it follows immediately that for the symmetric representation [n], the antisymmetric representation Œ1n  is conjugate. In this case, the one-column Young diagram of the representation Œ1n  is obtained from the onerow Young diagram of the representation [n] by replacement of a row by a column. It appears that for an arbitrary irreducible representation [ ], the conjugate representaQ is characterized by the so-called dual Young diagram obtained from the Young tion Œ  diagram of the initial representation by replacement of rows by columns. For example, the representations [4], [31], [22 ], [212 ] and [14 ] of group 4 correspond to the conjugate representations [14 ], [212 ], [22 ], [31] and [4], respectively. Rows and columns of the conjugate representation will be numbered by the Young Q Each tableaux rQ .i /, corresponding to the Young diagram of the representation Œ . .i / Young tableau r of the initial representation will correspond to the Young tableau rQ .i / of the conjugate representation, which can be obtained from the tableau r .i / replacing rows by columns. The correspondence between the Young tableaux for the conjugate representations Œ212  and Œ31 of group 4 is shown in Table 3.2. Let there be a direct product Œ 1   Œ 2  of two irreducible representations of group n . Consider under what condition the antisymmetric representation Œ1n  is present in the decomposition of this product. Since the characters of this representation are equal to .1/p , from equation (2.50), with allowance for orthogonality relation (2.45), we obtain 1 X n .1/p Œ1  .P /Œ2 .P / aŒ1  D nŠ P X 1 Q D Œ1  .P /Œ2  .P / D ıQ  : (3.14) 1 2 nŠ P

Consequently, the antisymmetric representation is contained in the decomposition Π1   Π2  only once and only if the multiplied representations are conjugate. That is, only one antisymmetric function can be formed from the basis functions of the

38

Chapter 3 The permutation group

Table 3.2. Correspondence between the Young tableaux of the conjugate representations [212 ] and [31] of group 4 .

Q It is easy to show that this function is representations [ ] and Œ . ‰ Œ1

n



1 X Œ Œ Q Dp ˆr ˆrQ ; f r

(3.15)

p where the summation is over all Young tableaux f , and the factor 1= f is used for the normalization (the indices rQ are rigidly connected to the indices r ). Indeed, acting on function (3.15) by an arbitrary permutation yields P ‰ Œ1

n

Q Q 1 X X Œ Œ Œ Œ Dp kr .P /mQ rQ .P /ˆk ˆmQ f r k mQ   Q 1 X X Œ Œ Œ Œ Dp kr .P /mr .P / .1/p ˆk ˆmQ : f k mQ r

Since the real transformations of group n are orthogonal, the singled-out sum over r is equal to ıkm . The result is P ‰ Œ1

n



n

D .1/p ‰ Œ1  ;

(3.16)

which proves the statement formulated by equation (3.15). Since the symmetric representation [n] is contained in the direct product Œ   Œ , it is easy to show that the expression for the symmetric function ‰ Œn coincides with equation (3.15) with accuQ by Œ . racy up to the replacement of Œ 

Chapter 4

Continuous groups

4.1 Compact Lie groups The finite groups considered in the previous chapters are a particular case of so-called discrete groups. The elements of such groups form discrete sets, i.e., they can be numbered by a natural sequence. However, there is a wide class of groups whose elements form continuous sets. That is, each element of the group is characterized by a set of parameters that vary continuously. Such groups are called continuous groups. Let there be a set of transformations x 0 D a1 x C a2 ;

(4.1)

where two real parameters, a1 and a2 , vary continuously from C1 to 1, so that even an infinitesimal change in their values leads to a new transformation. The sequential use of two transformations x 0 D a1 x C a2 ;

x 00 D b1 x 0 C b2

(4.2)

is equivalent to the third transformation x 00 D c1 x C c2

(4.3)

with the parameters c1 D b1 a1 ;

c2 D b1 a2 C b2 :

(4.4)

Identity transformation has the values of the parameters a1 D 1 and a2 D 0. Since the use of the direct and inverse transformations should yield an identity transformation, the inverse transformation of that in equation (4.1) is a transformation with the parameters b1 D 1=a1 ; b2 D a2 =a1 ; (4.5) which exists if a1 ¤ 0. Thus, the set of transformations (4.1) satisfies all the group postulates and forms a two-parameter continuous group. In general, the elements of the r-parameter continuous group are defined by r real independent parameters. Independence means that all the parameters are essential for specifying the elements of the group. If all elements of the group are determined by a finite number of parameters, then the group is called a continuous finite group. In what follows we will be interested only in finite groups of linear transformations in an n-dimensional vector space.

40

Chapter 4 Continuous groups

Consider the set of transformations xi0 D fi .x1 ; : : : ; xn I a1 ; : : : ; ar /;

i D 1; 2; : : : ; n:

(4.6)

The sequential use of two such transformations xi0 D fi .x1 ; : : : ; xn I a1 ; : : : ; ar /; xi00 D fi .x10 ; : : : ; xn0 I b1 ; : : : ; br / is equivalent to the third transformation xi00 D fi .x1 ; : : : ; xn I c1 ; : : : ; cr /; whose parameters are functions of the parameters of the first two transformations ck D 'k .a1 ; : : : ; ar I b1 ; : : : ; br /:

(4.7)

In fact, equation (4.7) defines the multiplication operation of transformations since it associates each two transformations with the third one. If transformation (4.6) also satisfies the rest of the group postulates and fi are analytical functions of the transformation parameters, then the group is called an r-parameter Lie group of transformations in an n-dimensional vector space. If the parameters of the Lie group vary within finite limits, then the group is called a compact Lie group. For such groups, one can introduce the concept of an invariant integral over the variation range of their parameters. That is, the element of volume in the parameter space can be so introduced that for an arbitrary continuous function ˆ in this space we have Z Z ˆ.P /d P D ˆ.QP /d P ; (4.8) where the transformations P and Q are functions of the parameters and the integral is over the entire range of their variation. Condition (4.8) is a natural generalization of the concept of invariant summation (1.7) for finite discrete groups to continuous compact groups. Each representation of the continuous group contains a continuous set of matrices. However, manipulations with the representations of compact groups are greatly simplified since the number of their nonequivalent irreducible representations, although infinite, is just countable (i.e., these representations form a discrete series), and the dimensions of all irreducible representations are finite. The latter means that the spaces of the basis functions transforming one through another under the action of the compactgroup operations are always finite spaces. Finally, it appears that nearly all the main results presented in Chapters 1 and 2 for finite discrete groups remain valid for compact groups with the formal replacement of the summation over the group elements by invariant integration. For example, orthogonality relations (2.42) and (2.45) for the

Section 4.2 Lie group of linear transformations

41

matrix elements and characters of irreducible representations in the case of compact groups can be written as Z Z 1 .˛/  .ˇ / i k .P / mn .P /d P D ı˛ˇ ıi m ıkn d P ; (4.9) f˛ Z Z (4.10) .˛/ .P / .ˇ / .P /d P D ı˛ˇ d P ; and equation (2.50) for the coefficients in the decomposition of the reducible representation becomes R . /  .P /.˛/ .P / d P .˛/ R : (4.11) a D d P Only the statements that use the finite order of the group (such as the assertion that the number of elements in the class is the divisor of the group order) lose their meaning.

4.2 Lie group of linear transformations Let there be a nonsingular linear transformation xi0 D ai k xk

(4.12)

in the vector space of dimension n. A set of such transformations forms a group. Indeed, the product of linear transformations is also a linear transformation whose matrix is determined by multiplying the matrices of multiplied transformations. Since the transformations are nonsingular, their determinants are nonzero. Therefore, for each transformation a, there exists an inverse transformation a1 . Other group postulates are also fulfilled. Such a group is called a general linear group and is often denoted as GLn . Naturally, the matrices of transformations (4.12) give the n-dimensional representation of this group. In general, these matrices are complex, and the transformation is therefore determined by 2n2 real parameters. Restricting the linear transformations only by unitary transformations, we obtain a group of unitary transformations of n-dimensional space, usually denoted Un . By definition (see Section 2.1), the matrices of unitary transformations ui k in an orthonormal basis satisfy the unitarity condition umi umk D ıi k :

(4.13)

For i D k, equations (4.13) yield n conditions for the modules of matrix elements, and for i ¤ k there are n.n1/=2 conditions of the real parts of equations and the same for the imaginary parts. Thus, a total of n2 conditions binding 2n2 real parameters follow from equations (4.13). Therefore, transformations of a unitary group are specified by n2 real parameters. Transformations of the group Un with determinant equal to unity form its subgroup, called a special unitary group (or a group of unitary unimodular

42

Chapter 4 Continuous groups

transformations) and denoted as SUn . It is clear that transformations of the group SUn are characterized by n2  1 real parameters. If the group Un is limited to real transformations, then a group of orthogonal transformations of n-dimensional space is formed. This group is called an orthogonal group and is denoted as On . The matrices of the orthogonal transformations ai k in the orthonormal basis satisfy the orthogonality condition aa Q D 1;

(4.14)

where aQ is the transposed matrix. Condition (4.14) leads to n.n C 1/=2 links on n2 real parameters of the matrix. Therefore, transformations of the orthogonal group are specified by n.n  1/=2 real parameters. Since the determinant of a matrix is not changed during its transposition, it follows from equation (4.14) that the square of the determinant of the orthogonal transformation is equal to unity. Hence, the determinant itself can take only the values ˙1. The orthogonal transformation with determinant equal to C1 is called the proper transformation and corresponds to the rotation of space around the origin of coordinates. The orthogonal transformation with determinant equal to 1 is called the improper transformation and is the product of transformations of the rotation and inversion of space with respect to the origin of coordinates. Proper transformations of the group On form a subgroup Rn called the group of rotations of n-dimensional space. In physics, the group of rotations of three-dimensional space R3 is very widespread. It is easy to see that its transformations are specified by three real parameters. Any subgroup of the group of orthogonal transformations O3 of a three-dimensional space is called a point group. According to this definition, the group R3 is a continuous point group. Transformations of point groups do not change the position of at least one point of three-dimensional space. In intramolecular quantum dynamics, such a point is the center of mass of the molecular system.

4.3 Lie algebra. Three-dimensional rotation group It can be shown that any finite transformation of the Lie group is uniquely represented as a sequence of infinitesimal transformations, and all matrices of each irreducible representation are expressed through the matrices of this representation which correspond to the infinitesimal transformations. That is, the Lie group is completely characterized by its infinitesimal transformations. To consider them, we write equation (4.6) in a more compact form: xi0 D fi .xI a/; (4.15) and without loss of generality it can be assumed that a set of parameters a D 0 corresponds to the identity transformation xi D fi .xI 0/. Then xi C dxi D fi .xI ıa/:

(4.16)

43

Section 4.3 Lie algebra. Three-dimensional rotation group

We expand the right-hand side into a Taylor series and restrict ourselves to terms of the first order of smallness:  r  r X X @fi .xI a/ dxi D ıa  wi ıa : (4.17) @a aD0 D1

D1

Thus, by specifying r vectors w , we determine the infinitesimal change in position of any point x in n-dimensional space, which is connected with the infinitesimal change in the parameters. For the corresponding change in the arbitrary function ˆ(x) we obtain the following expression: dˆ.x/ D

n r X @ˆ.x/ X i D1

D

r X

@xi ıa

X n

i D1

D1

wi ıa

D1

 r X @ ˆ.x/  wi ıa I ˆ.x/; @xi

(4.18)

D1

where we introduced r operators I D

n X i D1

wi

@ ; @xi

(4.19)

which are called the infinitesimal operators of a Lie group. For these operators, the following important relationship is fulfilled: I I  I I  ŒI ; I  D

r X

 c I ;

(4.20)

D1  are called the structural constants of a Lie group. Consewhere the coefficients c quently, the linear space of infinitesimal operators is closed with respect to the calculation of commutators. Such a space is called the Lie algebra, and the commutator calculation operation is called the multiplication operation in the Lie algebra. We now find infinitesimal operators of the three-dimensional group of rotations R3 . As three real parameters specifying its transformations, one can choose, for example, the Euler angles shown in Figure 4.1, or the polar angles of a rotation axis and the angle of rotation about this axis. Transformation (4.15) for rotation about the axis X of a rectangular Cartesian coordinate system by an angle 'X can be written as

X 0 D X; Y 0 D Y cos 'X  Z sin 'X ; Z 0 D Y sin 'X C Z cos 'X :

(4.21)

44

Chapter 4 Continuous groups

Figure 4.1. Parameterization of rotation transformations in a three-dimensional space by Euler angles. The axis rotation defined by Euler angles is performed in three stages: 1) rotation by an angle ˛ (0  ˛ < 2) about the Z axis, 2) rotation by an angle ˇ (0  ˇ  ) about a new position of the Y axis (of the so-called lines of nodes ON), and 3) rotation by an angle  (0   < 2) about the final position z of the Z axis.

The result is @fX D wXX D 0; @'X

@fY D wYX D Z; @'X

@fZ D wZX D Y; @'X

(4.22)

where the derivatives are taken at the point 'X D 0. Consequently, according to equation (4.19), @ @ Z : (4.23) IX D Y @Z @Y In a similar way, we obtain two other independent infinitesimal operators: IY D Z

@ @ X ; @X @Z

IZ D X

@ @ Y : @Y @X

(4.24)

Calculation shows that these three infinitesimal operators form the algebra with the multiplication operation ŒIX ; IY  D IZ ;

ŒIY ; IZ  D IX ;

ŒIZ ; IX  D IY :

(4.25)

Recall that for a closed physical system, the law of conservation of the angular momentum is a consequence of the isotropy of space. This demonstrates the connection of the angular momentum with the properties of symmetry relative to rotations of the group R3 . This connection is especially important in quantum mechanics since it gives a clear physical meaning to the concept of the angular momentum of a quantum system. The quantum angular momentum is measured in units of ¯, and the operators J˛

Section 4.3 Lie algebra. Three-dimensional rotation group

45

of its three components on the axis of the Cartesian coordinate system are related with infinitesimal operators of the group R3 as follows: J˛ D iI˛ :

(4.26)

Note that unlike the infinitesimal operators, the operators of the angular momentum are self-adjoint operators. From equation (4.25) it is easy to obtain the well-known commutation relations in the algebra of these operators: ŒJ˛ ; Jˇ  D i"˛ˇ J ;

(4.27)

where "˛ˇ is an absolutely antisymmetric tensor of the third rank. Relations (4.27) are exactly the basis for determining the angular momentum operators in quantum mechanics. Operators of a spin momentum not related with the moving of the system particles in space also satisfy these relations. We show how the arbitrary finite rotation operator is expressed in terms of infinitesimal operators. From equations (4.18) and (4.26) it follows that under an infinitesimal rotation ı' about the axis directed along the unit vector n the function ˆ(x) varies as dˆ.x/ D iı'.nJ /ˆ.x/; that is, the function ˆ(x) becomes a function ˆ0 .x/ D Œ1 C iı'.nJ/ˆ.x/:

(4.28)

Rotation by a finite angle ' can be written as the following limit: ik h ' 1 C i .nJ/ ˆ.x/ D expŒi'.nJ /ˆ.x/: k k!1

ˆ0 .x/ D lim

(4.29)

It follows that Rn;' D expŒi'.nJ /

(4.30)

is the operator of finite rotation by an angle ' about the axis n. To determine the result of the action of operator (4.30) on the function, it is needed to represent this operator as a Taylor series and find the result of the action of each term of this series on the function. Rotation by an angle ' about the axis n can be transformed into rotation by the same angle about any other axis k. To do this, one should first perform the rotation P, which converts the axis n into an axis k, then rotate by an angle ' about a new position of the axis n, and finally return the axis n to place using the rotation P 1 , that is, Rn;' D P 1 Rk;' P:

(4.31)

This implies that all rotations by the same angle belong to one class of group R3 . In other words, the classes of group R3 are characterized only by the angle of rotation.

46

Chapter 4 Continuous groups

Finally, we emphasize that the form of the invariant integration element depends on the choice of parameterization of the group R3 transformations. Thus, in the case of the Euler angles we have d R D sin ˇd˛dˇd; and using the polar angles of the rotation axis and the angle of rotations 0  '   about this axis, we obtain d R D 2.1  cos '/d'd ; where d is the element of a solid angle for the rotation axis direction. However, the total volume of the parameter space is the same in both cases and is equal to 8 2.

4.4 Irreducible representations of a three-dimensional rotation group We introduce the operator of the angular momentum squared J 2 D JX2 CJY2 CJZ2 . According to equation (2.34), it commutes with all operators J˛ . The operator quadratic in operators of the algebra and commuting with all these operators is called the Casimir operator. It is easy to show that for compact Lie groups, all the basis functions of an irreducible representation belong to the same eigenvalue of this operator. In other words, the eigenvalues of the Casimir operator can be used to classify irreducible representations. These is only one such operator for the group R3 . The basis functions of the functional space in which the operators of R3 act are specified by a common system of eigenfunctions of commuting operators of the angular momentum squared and one of its components (JZ is usually chosen). In terms of quantum mechanics, these are the operators of a complete set of simultaneously measured angular physical quantities. Consider the action of the operators of group R3 in this function space. Suppose ‰M is the normalized eigenfunction of the operator JZ , which corresponds to some eigenvalue M: (4.32) JZ ‰M D M ‰M : Since both directions of the z axis are equivalent, for each possible positive value of M there is the same negative value. We denote by a symbol J the largest value of jM j for a given eigenvalue of the operator J 2 . The very existence of such an upper limit follows from the fact that the difference J 2  JZ2 D JX2 C JY2 is the operator of a positive physical quantity. We introduce the operators J˙ D JX ˙ iJY , for which from equation (4.27) we have the following commutation rules: ŒJC ; J  D 2JZ ;

ŒJZ ; JC  D JC ;

ŒJZ ; J  D J :

(4.33)

Act by an operator JZ J˙ on the function ‰M . Taking equation (4.33) into consideration, we obtain JZ J˙ ‰M D .M ˙ 1/J˙ ‰M ;

Section 4.4 Irreducible representations of a three-dimensional rotation group

47

which implies that ‰M C1 D constJC ‰M ;

‰M 1 D constJ ‰M :

(4.34)

Therefore, the operators JC and J are called the raising and lowering angular operators, respectively. If in the first equation (4.34) one puts M D J , then there should be JC ‰J D 0 (4.35) as there are no states with M > J . Acting on equation (4.35) by an operator J and taking into account that J 2 D JC J C JZ2  JZ D J JC C JZ2 C JZ ;

(4.36)

.J 2  JZ2  JZ /‰J D 0;

(4.37)

J 2 ‰J D J.J C 1/‰J :

(4.38)

we have which implies that The operators JZ and J have a common system of eigenfunctions, and the eigenvalue of the Casimir operator is J.J C1/. For a given value of J, the eigenvalue of M changes from J to J with step equal to unity. In total, there are 2J C 1 values of M, and this quantity should be an integer, and its minimum is unity. This implies that J takes only integer and half-integer positive values (including zero). According to equations (4.34), for the operators JC and J , only the matrix elements hJ; M kJ; M  1i and hJ; M  1kJ; M i, respectively, are nonzero in the basis of the functions ‰JM (or jJ; M i/. Hence, for the matrix elements on both sides of equation (4.36) we obtain 2

J.J C 1/ D hJ; M jJC jJ; M  1ihJ; M  1jJ jJ; M i C M 2  M:

(4.39)

By virtue of the self-adjoint operators JX and JY , we have hJ; M  1jJ jJ; M i D hJ; M jJC jJ; M  1i : Taking this equality into consideration, it follows from equation (4.39) that p p hJ; M jJC jJ; M  1i D J.J C 1/  M.M  1/ D .J  M C 1/.J C M /; where the positive sign is chosen before the radical. As a result, the action of the angular momentum operators on the basis functions of irreducible representations is determined by the expressions J 2 jJ; M i D J.J C 1/jJ; M i;

(4.40)

JZ jJ; M i D M jJ; M i; p JC jJ; M i D .J  M /.J C M C 1/jJ; M C 1i; p J jJ; M i D .J C M /.J  M C 1/jJ; M  1i:

(4.41) (4.42) (4.43)

48

Chapter 4 Continuous groups

It is very important to note that the calculation of this action does not require the explicit form of the angular functions jJ; M i. A knowledge of commutation relations (4.27), which are determined by the properties of the group R3 , is sufficient. The numbers J and M appear as the symbols of symmetry (the symbol J determines the irreducible representation of group R3 , and the symbol M numbers the basis functions of this representation). The transformation of basis functions for finite rotations can be written as X .J / Rn;' jJ; M i D DM 0 M .Rn;' /jJ; M 0 i; (4.44) 0 M

.J /

where the coefficients DM 0 M .Rn;' / form a matrix of an irreducible representation of dimension 2J C 1, which corresponds to the rotation by an angle ' about the axis n. The explicit form of the matrices of this representation, denoted D .J / by tradition, is tabulated when the transformations of R3 are parameterized by the Euler angles. From equation (4.9) we obtain the following orthogonality relations: Z

0

.J / Di k .P / Dmn .P /d P D .J /

8 2 ıJJ 0 ıi m ıkn : 2J C 1

(4.45)

Since the classes of R3 are characterized only by the rotation angle, to determine the characters of irreducible representations it suffices to consider the rotations about one axis (for simplicity, the axis Z). In this case, from equation (4.30) we have RZ;' D exp.i'JZ /:

(4.46)

Expanding equation (4.46) into a Taylor series and using the fact that JZk jJ; M i D M k jJ; M i; we obtain RZ;' jJ; M i D exp.iM'/jJ; M i:

(4.47)

That is, the matrix of the irreducible representation D .J / for the transformation RZ;' is diagonal, and its character is equal to .J / .'/ D

J X M DJ

exp.iM'/ D

sin.J C 1=2/' : sin.'=2/

(4.48)

From equation (4.47) it follows that for the half-integer values of J, the basis function jJ; M i under rotation by an angle 2 reverses sign and returns to its initial state only after the rotation by an angle 4. Since the rotation by an angle 2 is an identity transformation, such representations are called two-valued. Indeed, for each rotation about any axis by an angle 0  ' < 2 one can specify two matrices of a representation

Section 4.4 Irreducible representations of a three-dimensional rotation group

49

with characters opposite in sign. When the particles moving in ordinary coordinate space are described, only single-valued representations of group R3 are realized, i.e., the value of J is integer. In this case, different irreducible representations of this group are characterized by the values J D 0; 1; 2; : : : and have odd dimensions. The set of rotations about one axis by an arbitrary angle 0  ' < 2 forms an abelian group R2 of two-dimensional rotations (its another designation is C1 ). All irreducible representations of this group are one-dimensional and, according to equation (4.47), are specified by the number M. The basis functions of the irreducible representations D .J / of group R3 belong to different irreducible representation of group R2 . Single-valued representations of R2 correspond to the positive and negative integer values of M (including zero). The form of the basis functions jJ; M i of one particle in the case of integer J results from the solution of the set of equations (4.40) and (4.41) in spherical coordinates. This solution is the well-known spherical functions YJM .; '/, where  and ' are the polar angles of a particle. Note that the choice of the sign before the radical in equations (4.42)–(4.43) and the phase in the normalization constant of spherical functions is ambiguous. Here the choice is made according to [64]. The matrix elements .J / .J / DM 0 .˛; ˇ; 0/ and D0M .0; ˇ; / become, with accuracy up to the constant, the spher.J /

ical functions YJM .ˇ; ˛/ and YJM .ˇ; /, respectively. Hence, the elements DM 0 M .˛; ˇ; / are called generalized spherical functions of order J. Form the direct product D .J1 /  D .J2 / of irreducible representations of group R3 . The basis of the resulting representation with dimension .2J1 C 1/.2J2 C 1/ is formed by the functions jJ1 ; M1 ijJ2 ; M2 i. They are the eigenfunctions of the operator JZ D .J1Z C J2Z /

(4.49)

M D M1 C M2 :

(4.50)

and correspond to its eigenvalue

However, the functions jJ1 ; M1 ijJ2 ; M2 i are not the eigenfunctions of the operator J 2 D .J1 C J2 /2 . This condition is satisfied only by some of their linear combinations. Upon passage to such combinations, the direct product is decomposed into irreducible representations D .J /. It appears that this decomposition has the following simple form: : (4.51) D .J1 /  D .J2 / D D .J1 CJ2 / C D .J1 CJ2 1/ C    C D .jJ1 J2 j/ : Each representation D .J / appears in the decomposition only once, and the number J varies with step equal to unity in the interval jJ1  J2 j  J  J1 C J2 :

(4.52)

If one constructs a triangle with the sides J1 , J2 , and J, then the perimeter J1 CJ2 CJ is an integer (for both integer and half-integer values of J1 and J2 ) and the lengths of

50

Chapter 4 Continuous groups

its sides satisfy condition (4.52). Hence, this condition is called the condition of a triangle with integer perimeter and is denoted as .J1 J2 J /. The transition from the basis functions jJ; M1 iJ; M2 i to the basis functions of irreducible representations appearing in decomposition (4.51) is carried out using the matrix of the Clebsch–Gordan coefficients (see Section 2.6): X hJ1 M1 ; J2 M2 jJ; M ijJ1 ; M1 ijJ2 ; M2 i; (4.53) jJ; M i D M1 ;M2

The nonzero Clebsch–Gordan coefficients hJ1 M1 ; J2 M2 jJ; M i satisfy the conditions of a triangle, and the orthogonality relations following from equation (2.72) are fulfilled in them. Note that the summation over M2 in equation (4.53) is formal since M2 D M  M1 . Interestingly, decomposition (4.51) arises in quantum mechanics when solving the problem of finding the possible values of the angular momentum of a physical system in terms of the angular momenta of its two weakly interacting parts. Such a decomposition is called the rule of summation of angular momenta.

Chapter 5

Point groups

5.1 Operations in point groups By a point group, according to Section 4.2, we mean any subgroup of a three-dimensional group of orthogonal transformations O3 . Such groups can be introduced as partial or complete symmetry groups of rigid bodies. It is believed that a rigid body has the symmetry of a certain point group if under its transformations acting on the rigid body as a whole, the latter coincides with itself. Clearly, the symmetry elements of a rigid body comprise the rotation axes and reflection planes. If the rigid body rotated about some axis by an angle 2 superposes with itself n times, then the axis is called a symmetry axis of the nth order and is denoted as Cn. The smallest rotation angle with such a superposition is equal to 2=n. A sequence of k rotations is denoted as Cnk . Naturally, Cnn  E. The figure considered earlier in Figure 1.1 has a symmetry axis C3 . The reflection plane  is another possible element of symmetry. From a definition of the symmetry plane it follows that  2  E. The plane passing through the axis Cn , is generally denoted as v (in some cases, d ), and the plane orthogonal to the axis Cn , is denoted h . The figure in Figure 1.1 has three reflection planes v with respect to the axis C3 , and if only an equilateral triangle is left in this figure, then a reflection plane h , which coincides with the plane of the triangle, also appears. It may turn out that the rigid body superposes with itself only in the case of the sequential use of two commuting transformations, namely, rotation by an angle 2k=n about some axis Cn and reflection in the plane orthogonal to this axis. Such a transformation of symmetry is called a mirror-rotational transformation (Figure 5.1), and the corresponding axis is denoted Sn and is called a mirror-rotational axis of order n. According to the definition, Snk D Cnk h D h Cnk

(5.1)

the mirror-rotational axis is a new element of symmetry only in the case of even n since for an odd n, from equation (5.1) we have 2mC1 2mC1 2mC1 S2mC1 D C2mC1 h D h :

(5.2)

That is, an independent element of symmetry is reflection in the plane h , and consequently the rotation axis of odd order C2mC1 . It is easy to show that the mirrorrotational axis S2m is simultaneously the rotation axis Cm .

52

Chapter 5 Point groups

Figure 5.1. Mirror-rotational transformation.

Transformation S21 is equivalent to the transformation of inversion I with respect to the point of intersection of the axis S2 with the plane h : S21 D C21 h  I:

(5.3)

It is said that the body with a symmetry operation I has a center of symmetry. In general, the operations of point groups do not commute. In what follows we list all cases where the commutativity still takes place: 1. rotations about one axis; 2. rotations by an angle  about mutually orthogonal axes (the result coincides with the rotation by an angle  about the axis orthogonal to the first two axes); 3. rotation and reflection in the plane orthogonal to the rotation axis; 4. reflections in the mutually orthogonal planes (the result coincides with the rotation by an angle  about the axis passing along the line of intersection of the planes). It follows from items 1 and 2 that the inversion operation commutes with any other operation of a point group. Hence, for example, the point O3 can be represented as R3  CI , where CI D .E; I /. By analogy with the proof of equation (4.31), one can establish a series of rules that are useful for division of the point group operations into classes: 1. Two rotations by one angle about different axes belong to one class if the group has a transformation that superposes these axes. 2. Two rotations about one axis by one angle but in opposite directions belong to one class if the group has a rotation that reverses the axis direction or a v -type reflection relative to this axis (in this case, the axis is called two-sided).

Section 5.2 Discrete axial groups

53

3. Two reflections in different planes belong to one class if the group has a transformation that superposes these planes. In this book, for the point groups we use the Schönflies notation system which is traditionally used to describe the symmetry properties of intramolecular dynamics.

5.2 Discrete axial groups Such point groups may contain only one symmetry axis, whose order is higher than the second, and the lower index n in their notation is determined by the order of the principal axis of symmetry. 1. Group Cn . There is a symmetry axis of order n. The group consists of n rotations by angles 2k=n, and it is abelian and cyclic. Each operation forms a class, and all irreducible representations are one-dimensional. Group. C1 , which contains only an identity transformation, corresponds to the absence of symmetry. 2. Groups S2n . There is only a mirror-rotational axis of order 2n. The group consists k of 2n mirror-rotational operations S2n , and it is abelian and cyclic. It is easy to show that S4mC2 D C2mC1  CI and, in particular, S2 D CI . 3. Groups Cnh . These groups are obtained by adjoining the orthogonal symmetry plane h to the symmetry axis of order n. The group is abelian and contains 2n operations, namely, n rotations Cnk and n mirror-rotational operations Cnk h . Group C1h, which has two elements, E and h , is usually denoted as CS . It is obvious that Cnh D Cn  CS . For an even n, one can also write C2m;h D C2m  CI since the group C2m;h contains an inversion element I. 4. Groups Cnv . These groups are obtained by adjoining the symmetry plane v to the symmetry axis of order n through which the plane v passes. This leads to the appearance of another n  1 such planes intersecting along the axis at angles =n. In general, the group is non-abelian and contains 2n operations, including n rotations Cnk and n reflections v . The axis is two-sided, i.e., the rotations Cnk and Cnk belong to the same class. All of the n reflections belong to the same class only in the case of odd n. For n D 2m, the reflection operations are divided into two classes with m elements in each class since rotations about the C2m axis cannot superpose the neighboring planes with each other. 5. Groups Dn . These groups are obtained by adjoining a second-order axis to the symmetry axis of order n to which it is orthogonal. This leads to the appearance of another n  1 such second-order axes intersecting with each other at angles =n. The group contains 2n operations, including n rotations Cnk about the principal axis, which, for convenience, can be assumed vertical, and rotations by an angle  about n horizontal axes, which are usually denoted U2 . The structure of the group Dn is

54

Chapter 5 Point groups

similar to the structure of the group Cnv which is isomorphic with it. The group D2; which has three mutually orthogonal axes of the second order, is an important specific case. 6. Groups Dnh . These groups are obtained by adjoining the horizontal symmetry plane h to the system of axes of group Dn . Plane h passes through n axes of the second order. This leads to the appearance of n vertical symmetry planes, each of which passes through the vertical axis Cn and one of the horizontal axes U2 . The group contains 4n operations. Namely, 2n operations of group Dn are supplemented by n reflections v and n mirror-rotational operations Cnk h . Since the reflection h commutes with all operations of group Dn, we have Dnh D Dn CS . For an even n, one can write D2m;h D D2m  CI since the group D2m;h contains an inversion element I. 7. Groups Dnd . There is one more method for adjoining the symmetry planes to the system of axes of group Dn. We add a plane passing through the vertical axis Cn midway between any neighboring horizontal axes U2 . This entails the appearance of another n  1 such planes. The group contains 4n operations, namely, 2n operations of group Dn are supplemented by n reflections in the vertically orthogonal planes (denoted d ) and n mirror-rotational operations U21 d . The latter operations 2kC1 about the vertical axis with the subsequent are equivalent to the rotations C2n reflection in the plane h that is orthogonal to it. That is, the axis Cn converts into a mirror-rotational axis S2n , whose order is twice as large. For an odd n, one can write D2mC1;d D D2mC1  CI since the group D2mC1;d contains an inversion element I.

5.3 Cubic groups. Icosahedral groups These groups have several symmetry axes whose order is higher than two. The name of the groups is attributable to the fact that their elements are, correspondingly, the elements of symmetry of a cube and an icosahedron. The cubic groups are T , Td , Th , O and Oh , and the icosaheraon groups are Y and Yh . 1. Group T. This group is purely rotational and can be obtained if to the system of axes of group D2 we add four oblique axes of the third order, the rotations about which convert the three axes of group D2 into each other. The resulting system can conveniently be represented by drawing the third-order axes along four spatial diagonals of a cube and the second-order axes, through the centers of its opposite faces (Figure 5.2). The group has twelve operations divided into four classes: E, three rotations C21 , four rotations C31 , and four rotations C32 . 2. Group Td . This group is obtained if to the system of axes of group T we adjoin six symmetry planes, each of which passes through one axis of the second and two

Section 5.3 Cubic groups. Icosahedral groups

55

Figure 5.2. Transformations of the cubic rotational groups T and O.

axes of the third order. Group Td is a complete group of symmetry of a tetrahedron (Figure 5.3), while the system of axes of group T is a complete system of axes of a tetrahedron. Because of the adjoining of the planes, the second-order axes become mirror-rotational axes of the fourth order, and the third-order axes become twosided. The group contains twenty four operations divided into five classes, namely, E, eight rotations C31 , C32 , three rotations C21  S42 , six reflections d , and six mirror-rotational operations S41 , S43 .

Figure 5.3. Transformations of the cubic groups Td and Th .

3. Group Th . This group is obtained by adjoining the inversion operation I to the system of axes of group T. In this case, three mutually orthogonal planes appear, each of which passes through two axes of the second order, while the axes of the third order become mirror-rotational axes of the sixth order (Figure 5.3). It is obvious that Th D T  CI . 4. Groups O. The system of axes of this purely rotational group is a complete system of axes of a cube. Namely, three axes of the fourth order pass through the centers of

56

Chapter 5 Point groups

the opposite faces, four axes of the third order, along the spatial diagonals, and six axes of the second order, through the middles of the opposite edges (Figure 5.2). All the axes are two-sided. The structure of the group O is similar to the structure of the group Td that is isomorphic with it. Hence, twenty-four elements are also divided into five classes, namely, E, eight rotations C31 , C32 , six rotations C41 , C43 , three rotations C42 , and six rotations C21 . 5. Groups Oh . This group is a complete group of symmetry of a cube and is obtained by adjoining the inversion operation I to the system of axes of group O. In this case, the axes of the third order of group O convert into mirror-rotational axes of the sixth order (spatial diagonals of a cube). Six planes passing through each pair of opposite edges and three planes passing through the center of a cube parallel to its edges also appear. It is obvious that Oh D O  CI . We then proceed with the groups of an icosahedron, which is a regular polyhedron with 20 triangular faces (it has 30 edges and 12 vertices at each of which 5 edges meet). This icosahedron is dual to a dodecahedron, which is a regular polyhedron with 12 pentagonal faces (it has 30 edges and 20 vertices at each of which 4 edges meet). Duality means that polyhedrons are obtained one from another if the geometric centers of the faces of one polyhedron are taken as the vertices of another, and vice versa. This leads to the coincidence of the groups of an icosahedron and a dodecahedron. In this regard, we note that a cube is dual to an octahedron, i.e., a polyhedron with 8 triangular faces, 12 edges, and 6 vertices at each of which 4 edges meet. Therefore, the group Oh is also called a complete symmetry group of an octahedron. As concerns a tetrahedron, it is dual to itself. 6. Groups Y. The system of axes of this purely rotational group is a complete system of axes of an icosahedron. There are six axes of the fifth, ten axes of the third, and fifteen axes of the second order. The group contains sixty operations divided into five classes, namely, E, twenty rotations C31 , C32 , fifteen rotations C21 , twelve rotations C51 , C54 , and twelve rotations C52 , C53 . 7. Groups Yh . This group is a complete group of symmetry of an icosahedron and is obtained by adjoining the inversion operation I to the system of axes of group Y. It is obvious that Yh D Y  CI . The analysis carried out in Sections 5.2 and 5.3 exhausts the finite point groups of transformations of the rigid body as a whole.

5.4 Continuous axial groups We let the order of the axis Cn in discrete axial groups to infinity. This yields the following continuous axial groups: C1 , C1 h , C1 v , D1 , and D1 h . The group C1 , which is the simplest of them, was already discussed in Section 4.4.

Section 5.4 Continuous axial groups

57

1. Group C1 h . This group is obtained if to the rotations relative to the vertical axis of infinite order C1 we adjoin an orthogonal horizontal plane of symmetry h . It is clear that C1 h D C1  CS . Since this group contains an inversion element I, it can also be written as C1 h D C1  CI . 2. Group C1 v . There is a vertical symmetry axis of the infinite order C1 and a continuous set of vertical planes v passing through this axis. Since the axis is twosided, there is a continuous set of classes, each of which contains two operations of rotation by angles ˙' and one class of reflections composed of a continuous set of v operations. 3. Group D1 . There is a vertical symmetry axis of infinite order C1 and a continuous set of horizontal axes of the second order U2 , which orthogonally intersect the vertical axis at one point. The structure of the group D1 is similar to the structure of the group C1 v , which is isomorphic with it. Hence, there is a continuous set of classes each of which contains two operations of rotation by angles ˙' and one class composed of a continuous set of rotations of the second order U21 . 4. Group D1 h . This group is obtained by adjoining the inversion operation I to operations of group C1 v . This leads to the appearance of a continuous set of horizontal axes of the second order, which intersect the axis C1 at one point, and a horizontal plane of symmetry h , which passes through the second-order axes. It is clear that D1 h D C1 v  CI . The group D1 h can also be obtained by adjoining the inversion operation I to operations of group D1 . Hence, it can also be written that D1 h D D1  CI . Appendix II contains the tables of characters of single-valued irreducible representations of the major point groups. Each class is characterized by one element belonging to it, and the numbers before the symbols of these elements indicate the number of elements in the class.

Chapter 6

Dynamic groups

6.1 Invariant dynamic groups There are two definitions of dynamic symmetry. Its first sign is the implicit (or hidden) symmetry of the Hamiltonian and the second is transformations which go beyond the scope of the symmetry of the Hamiltonian. Correspondingly, we call such groups dynamic invariant and noninvariant, thus emphasizing their relation to the Hamiltonian. A dynamic invariant group has properties that make it significantly different from the usual symmetry groups of the Hamiltonian. As a well-known example of its implementation, we mention the motion of a point particle of mass m in a spherically symmetric Coulomb field of attraction V .r / D ˛=r . The problem of particle motion in any spherically symmetric field includes explicit symmetry defined by a group of rotational transformations R3 converting the sphere into itself. As a result, the vector of the angular momentum l of a particle is preserved. However, another vector is also preserved for the Coulomb field: r 1 A D ˛ C Œpl; r m

(6.1)

where p is the momentum of the particle. Vector A is sometimes called the Runge– Lenz vector. In quantum mechanics, it corresponds to the operator r 1 A D ˛ C .Œpl  Œlp/; r 2m

(6.2)

which, as can easily be verified by direct calculation, commutes with the Hamiltonian of a particle p2 ˛ H D  : (6.3) 2m r In 1935, V. A. Fock showed that the preservation of the vector A is related to the fact that the Hamiltonian (6.3) has an implicit symmetry group which is wider compared with the group R3 . An implicit symmetry group has the following important differences. Firstly, this group applies only to the Coulomb field. Secondly, it depends on the particle energy. Thirdly, it leaves the Hamiltonian invariant as a whole rather than treating the kinetic and potential parts separately. In the present case, for bound states, this group is the rotation group R4 in some four-dimensional space, and for the continuous spectrum it coincides with the Lorentz group. However, the conventional groups do not depend on excitation in the system (for example, the group R3 defined by the

Section 6.2 Noninvariant dynamic groups

59

isotropy of the attraction potential is always the same). This change in symmetry is a characteristic feature of the implicit dynamic group. On the one hand, it is specified by the dynamics of a particular system within a certain interval of excitations, and on the other hand, it determines the qualitative properties of the dynamics. Beyond this interval, the dynamics of the system changes qualitatively, which leads to an abrupt change in the dynamic group. Suppose we have a problem with an implicit symmetry group, which admits of a rigorous solution. Then a qualitative analysis with allowance for such a group is unnecessary. This analysis is only desirable as the basis for explaining the qualitative features of the system behavior. In particular, this is the case for the Hamiltonian (6.3). The energy levels of bound states in this problem are rigorously described by a simple formula m˛ 2 En D  2 2 ; (6.4) 2¯ n where n D nr Cl C1 is the principal quantum number, which is expressed through the quantum number l of the angular momentum squared and the quantum radial number nr D 0; 1; 2; : : : . The numbers l and nr enter equation (6.4) only as their sum, and this leads to an additional degeneracy of the levels. Indeed, the same value of the number n, which determines the energy, can be picked up in different ways from the numbers nr and l. This degeneracy is specific only for the Coulomb potential and is attributable to the fact that the Hamiltonian (6.3) in the case of bound states has a symmetry group R4 . However, this item is not worth thinking about since a rigorous solution certainly allows for all consequences of implicit symmetry. However, the situation is radically different if a rigorous solution to the problem with implicit symmetry group is unknown. Then if the methods of perturbation theory are used, it should first of all be ensured that the correct properties of symmetry of internal dynamics are transferred to approximate models. This leaves us with the problem of empirically seeking an implicit group for which the basis functions and operators of the physical quantities allowing for the considered types of motion should be correctly symmetrized. Thus, analysis of the symmetry properties with allowance for the implicit group is fundamentally necessary and should precede the solution of the equations of motion (in fact, without this analysis we have no approximate equations of motion). It will be shown in Part two of this book that such a situation occurs for a multitude of physical systems, and the molecules do belong to such systems, which is most significant for us.

6.2 Noninvariant dynamic groups To this type we assign any group which is an extension of the symmetry group of the Hamiltonian of a system by including transformations that are noninvariant with respect to this Hamiltonian. Since the noninvariant group includes a symmetry group of the Hamiltonian in the form of a subgroup, it contains information on the degener-

60

Chapter 6 Dynamic groups

acy of the energy levels, and the operators of its noninvariant transformations bind the quantum-system states corresponding to the different values of energy. It appears that such groups arise during the formation of a function space of considered intramolecular motions in cases where the symmetry group of the Hamiltonian is too narrow for this. This can be illustrated by an important example of the problem of describing free rotation of a rigid body. The point is that this problem is a physically correct initial approximation for describing all microsystems with a rotational spectrum, to which the molecules also belong. Introduce a laboratory fixed Cartesian coordinate system (FCS) and a moving Cartesian coordinate system (MCS) rigidly connected to the rigid body. We place the origin of both coordinate systems into the center of mass of the rigid body. The axes of the FCS and MCS will be denoted X, Y, Z and x, y, z, respectively (see Figure 4.1). Orientation of the MCS with respect to the FCS and, therefore, the position of the rigid body is determined by three Euler angles, i.e., the dimension of the configuration space is three. The kinetic energy of rotational motion can be written as T D

1 gi k ./Pi Pk ; 2

 D ˛; ˇ; ;

(6.5)

where the form of the tensor gi k ./ depends on the ellipsoid of inertia of the rigid body. The body is called a spherical top if all three principal moments of inertia of the ellipsoid coincide, a symmetric top if two principal moments of inertia coincide, and an asymmetric top if all the principal moments are different. Thus, for the symmetric top, when the axes x, y, z are chosen by the principal moments of inertia, equation (6.5) takes the form T D

I1 2 2 I3 .˛P sin ˇ C ˇP 2 / C .˛P cos ˇ C / P 2; 2 2

(6.6)

where I1 D I2 ¤ I3 are the principal moments of inertia. The analysis shows that the symmetry group of the Hamiltonian (6.5) has a maximum of six independent infinitesimal operators corresponding to rotations with respect to three axes of the FCS and three axes of the MCS. It appears that the components of the angular momentum J˛ on the axis of the FCS commute with its components Ji on the axis of the MCS. To prove this, we use the fact that for any vector quantity A characterizing the physical system, the relations ŒJ˛ ; Aˇ  D i"˛ˇ A

(6.7)

are fulfilled. They are a consequence of the fact that during rotation around the FCS axes, the components A˛ transform to each other and, hence, the commutator of the operators J˛ with the operators Aˇ should be expressed via the components in the FCS of the same operator A. This can easily be obtained from this expression by the commutator calculation in the case where A is simply the radius vector of a particle. In

61

Section 6.2 Noninvariant dynamic groups

particular, relations (6.7) are fulfilled for the unit vectors i directed along the MCS axes: (6.8) ŒJ˛ ; iˇ  D i"˛ˇ i ; where iˇ is the projection of the unit vector i on the axis ˇ of the FCS. Then ŒJ˛ ; Ji  D ŒJ˛ ; iˇ Jˇ  D iˇ ŒJ˛ ; Jˇ  C ŒJ˛ ; iˇ Jˇ D i"˛ˇ iˇ J C i"˛ˇ i Jˇ : Interchanging the dummy indices ˇ and  in the second term and taking into account that "˛ˇ D "˛ ˇ , we finally obtain ŒJ˛ ; Ji  D 0:

(6.9)

We will need some intermediate stages of this proof in what follows, but the result itself in equation (6.9) can be obtained much more easily from the theorem of Section 2.2 since the scalar products .Ji / are invariant with respect to any rotation in the FCS. Due to equation (6.9), the maximum symmetry group can be written as R3  R3 . The group of rotations in the FCS will be called external and a similar group in the MCS, internal. The states of the physical systems implement only those irreducible representations of the group R3  R3 for which the indices of the irreducible representations of the external and internal groups coincide (the J – J representations). Consider the multiplication operation in the algebra of an internal group: ŒJi ; Jk  D Œ i ˛ J˛ ; kˇ Jˇ  D i"˛ˇ i ˛ kˇ J C i"˛ˇ i ˛ k Jˇ  i"ˇ ˛ i kˇ J˛ : Interchanging the dummy indices ˛ and  in the third term, it is easy to show that the first and third terms cancel out. In the second term, we first interchange the dummy indices ˇ and  and then transform the resulting expression before J in the following way: i"˛ˇ i ˛ kˇ D iŒ i  k  D i"i kl l : Here, it was taken into account that the vector product for two unit vectors of the MCS is expressed through the third unit vector of the MCS. As a result, we finally have ŒJi ; Jk  D i"i kl Jl :

(6.10)

Thus, commutation relations (4.27) and (6.10) of the components of the angular momentum in the FCS and the MCS, respectively, differ in sign on the right-hand side. This is because in terms of the action on the wave function of the top, the rotation of the FCS axes is equivalent to the reverse rotation of the MCS. A set of basis functions of the function space, in which the operators of the group R3  R3 act, is specified by a common system of eigenfunctions of the operators J 2 , Jz , and JZ : J 2 jJ; k; M i D J.J C 1/jJ; k; M i;

(6.11)

Jz jJ; k; M i D kjJ; k; M i;

(6.12)

JZ jJ; k; M i D M jJ; k; M i:

(6.13)

62

Chapter 6 Dynamic groups

In quantum mechanics, these are the operators of a complete set of angular physical quantities measured simultaneously. We note that these quantities are three in this problem. As was already shown in Section 4.4 for the external rotation group, calculating the action of infinitesimal operators on the basis functions does not require the explicit form of these functions, and it suffices to know the commutation relations of these operators, which are entirely determined by the properties of the group. Naturally, the action of the operators J˙ of the external group on the functions jJ; k; M i is similar to that described by equations (4.42) and (4.43): p JC jJ; k; M i D .J  M /.J C M C 1/jJ; k; M C 1i; (6.14) p J jJ; k; M i D .J C M /.J  M C 1/jJ; k; M  1i: (6.15) For the operators J˙ of the internal group, one can easily find p JC jJ; k; M i D .J  k/.J C k C 1/jJ; k C 1; M i; p J jJ; k; M i D .J C k/.J  k C 1/jJ; k  1; M i:

(6.16) (6.17)

It appears that the raising and lowering operators of the internal group are defined as J˙ D Jx  iJy , which is opposite of these operators for the external group, and the eigenvalue k, in a similar way to M, is integer and changes from J to J with step equal to unity. Curiously, since the commutation relations of the components of the angular momentum in the FCS and the MCS differ only in sign, the results obtained from these relations in the FCS and the MCS formally convert into each other if all the expressions are replaced by their complex conjugates. In particular, the sign on the right-hand side of equation (4.26) changes, and for the operator of finite rotation by an angle ' about the z axis of the MCS, we have, instead of equation (4.46), Rz;' D exp.i'Jz /:

(6.18)

Rz;' jJ; k; M i D exp.ik'/jJ; k; M i;

(6.19)

As a result, which differs in sign in the exponent compared with the expression for the FCS RZ;' jJ; k; M i D exp.iM'/jJ; k; M i;

(6.20)

which is similar to equation (4.47). The explicit form of the functions jJ; k; M i follows from the solution of the system of equations (6.11)–(6.13) written in the configuration space of Euler angles. With accuracy up to the normalized constant, these .J / functions coincide with the generalized spherical functions DkM .˛; ˇ; /. Note that the choice of the sign before the radical in equations (6.14)–(6.17) and the phase factor in the normalized constant is ambiguous. Here, the choice is made according to [64].

Section 6.2 Noninvariant dynamic groups

63

The presence of the external group is associated with the isotropy of space. Hence, the maximum variant R3 of this group is implemented for any rigid body. However, the internal group characterizes the symmetry of the rigid body itself and includes only the rotations with respect to which the inertia ellipsoid is invariant. Consequently, its maximum variant occurs only for a spherical top. For the symmetric and asymmetric tops, this symmetry is determined by the groups D1 and D2 , respectively. In fact, the presence of the external group allows the rotational states to be characterized using the quantum number J of the total angular momentum. However, the system-related symmetry and, therefore, the specific form of the Hamiltonian are determined by the internal group. Since the rotational dynamics does not depend on the quantum number M determining the orientation of the system in isotropic space, this number is often omitted in the basis functions, and the latter are written as jJ; ki. The space of these functions is specified by the internal group R3 , which naturally is much wider than the symmetry group of the Hamiltonian of the symmetric and, even more so, asymmetric tops. That is, the group R3 for these tops is a noninvariant dynamic group. This group is very important in this role since its irreducible representations define a complete basis set of the rotational function space in the MCS. It is worthy of note that the form of the function space in this fairly simple case is well known, and its basis functions can be written out, as it were, automatically. Generally, attention is not focused on the type of the internal group R3 , but this fact changes nothing in the above-mentioned picture.

Part II

Qualitative intramolecular quantum dynamics

Chapter 7

The philosophy of using the symmetry properties of internal dynamics 7.1 Symmetry groups of internal dynamics An isolated molecule is a complex multi-particle system, whose internal dynamics can be described in a good approximation with neglect of the nuclear and electron spinrelated contributions to the Hamiltonian. The symmetry properties of the resulting purely coordinate Hamiltonian are determined by the symmetry properties of space and time (external symmetry), as well as requirements with respect to permutations of identical particles (internal symmetry). We first consider the consequences of the symmetry of space. Since all the spatial positions of the isolated system of particles as a whole are equivalent, the Hamiltonian H of the system should not change when all of its particles are translated the same distance. This is the so-called space homogeneity property. It implies the law of conservation of the total momentum of the system as a whole. In analyzing the internal dynamics of the system, this momentum can simply be put equal to zero. All directions are also equivalent in space, i.e., the Hamiltonian of an isolated system must not change when it is rotated as a whole at any angle relative to the FCS (fixed Cartesian coordinate system). This is the spatial isotropy property. It implies the law of conservation of angular momentum. In analyzing the internal dynamics, the FSC is placed into the center of mass of the system of particles, and then the angular momentum is determined only by the internal motion. According to quantum mechanics, all stationary states of the system that satisfy the equation H ‰ D E‰

(7.1)

are characterized by quantum numbers of the operators J 2 and JZ according to the law of conservation of angular momentum (see Section 4.4). There is another transformation of space, which leaves the Hamiltonian of an isolated system of particles unchanged in nonrelativistic quantum mechanics. This is transformation of the inversion i, which consists in the simultaneous reversal of sign of the Cartesian coordinates of all particles of the system. Such invariance of the Hamiltonian is due to the symmetry of space with respect to mirror reflections. Interestingly, in classical mechanics, this does not lead to new laws of conservation. In quantum mechanics, the case is significantly different. We introduce the inversion operator i in the function space of the system i‰.r1 ; r2 ; : : :/ D ‰.r1 ; r2 ; : : :/:

(7.2)

68

Chapter 7 The philosophy of using the symmetry properties of internal dynamics

It is obvious that i 2 D 1. Consider the equation for eigenfunctions of the operator i i‰.r1 ; r2 ; : : :/ D p‰.r1 ; r2 ; : : :/:

(7.3)

Acting on equation (7.3) by the operator i on the left, we have p 2 ‰.r1 ; r2 ; : : :/ D ‰.r1 ; r2 ; : : :/ or p D ˙1. Thus, the eigenfunctions are either even or odd. From the theorem of Section 2.2, it follows that ŒH; i D 0;

(7.4)

which expresses the law of conservation of parity. Operators of the angular momentum are also invariant under the inversion (see Section 4.3) since the latter reverses sign of both the Cartesian coordinates and the operators of differentiation with respect to them. The stationary states can simultaneously be chosen eigenfunctions of the operators J 2 , JZ , and i and characterized by the quantum numbers of these operators. Using the relation JC i  iJC D 0, we easily see that the stationary states differing only in the value of the quantum number M have the same parity. We now turn to consequences of the symmetry of time. Since all instants of time relative to the isolated system are equivalent, the Hamiltonian of the system cannot contain time explicitly. This is the so-called time homogeneity property. It implies the law of conservation of energy, which in quantum mechanics is the basis of the concepts of states with a certain value of the energy or stationary states defined by equation (7.1). So far we have considered the transformation of space and time, which are linear and unitary. Consider in more detail why. Let j‰i and jˆi be two states of a quantum system. Then from the fundamental concepts of quantum mechanics it follows that for the system which is in the state j‰i the probability of its detection when measured in the state jˆi is equal to jhˆj‰ij2 . When the symmetry transformation R takes place, the states j‰i and jˆi transform to j‰i D Rj‰i;

jˆ0 i D Rjˆi:

(7.5)

Since the result of measurement should remain unchanged, we have jhRˆjR‰ij2 D jhˆj‰ij2 :

(7.6)

It is clear that equation (7.6) is fulfilled if the operator R is linear and unitary: R.aj‰i C bjˆi/ D aRj‰i C bRjˆi; hRˆjR‰i D hˆj‰i:

(7.7)

However, equation (7.6) includes the square of the modulus of the scalar product, and it is therefore possible to use the so-called antilinear and antiunitary operator R, which satisfies the equations R.aj‰i C bjˆi/ D a Rj‰i C b  Rjˆi; hRˆjR‰i D h‰jˆi D hˆj‰i :

(7.8)

69

Section 7.1 Symmetry groups of internal dynamics

For continuous groups, version (7.7) only is left, since the operator of any of its transformations can be obtained continuously from a linear and unitary unit operator. Therefore, version (7.8) is possible only for transformations of the inversion of coordinates and time reversal. But the spatial inversion operator defined by equation (7.2) is linear and unitary. However, the time reversal operator, which will be denoted as T, has to be introduced as antilinear and antiunitary in nonrelativistic quantum mechanics. This is seen already from the Schrödinger equation i¯

@‰ D H ‰; @t

(7.9)

which is only of the first order in time, while the equations of motion in classical mechanics are of the second order in time. Therefore, the operator T is defined by the relation T ‰.t / D ‰  .t /; (7.10) from which it follows that T 2 D 1. However, in contrast to the spatial inversion operator, it is impossible to introduce a physically meaningful notion of parity of the wave function with respect to the time reversal operation. Indeed, consider the equation for eigenfunctions of T, T j'i D j'i: (7.11) Multiplying the function j'i by an arbitrary phase factor, we obtain T exp.i/j'i D exp.i/T j'i D exp.i/ j'i D Œexp.2i/  exp.i/j'i: It follows that the state exp.i/j'i, which should be physically indiscernible from the state j'i, corresponds to the eigenvalue exp.2i/ . That is, the phase of the eigenvalue , and therefore its sign, do not have a physical meaning. At the same time, the behavior of the operators of physical quantities with respect to the transformation T is physically meaningful. We will show this for the stationary-state problem of an isolated quantum system. In this case, the operators of physical quantities do not depend on time explicitly, and this dependence can be omitted in the wave functions. We define a linear operator A by specifying its action on an arbitrary element of the function space in the form A .x/ D '.x/; where x is the point of configuration space (see Section 2.2). After the transformation T, we have TAT T D T ' or TAT  D '  . It was taken into account that the operator T is given by equation (7.10) and T 2 D 1. Then .TAT / D ', from which it finally follows that TAT D A : (7.12) If A D A, then

TAT D A:

(7.13)

70

Chapter 7 The philosophy of using the symmetry properties of internal dynamics

Such an operator A is called t-even since it is invariant with respect to the time reversal operation. If A D A, then TAT D A (7.14) and the operator A is called t-odd since it changes sign under the action of the time reversal operation. In particular, the operator of the coordinate is t-even, while the operators of the momentum and angular momentum are t-odd, which is fully consistent with the concepts of classical mechanics. Since the Hamiltonian of an isolated quantum system is t-even, it follows from equation (7.1) that H ‰  D E‰  : (7.15) Therefore, if the symmetry group of the Hamiltonian has complex-conjugate irreducible representations, then they belong to the same energy level, and can be regarded as one representation of double dimension in the classification of states. That is why in the tables of characters of point groups in Appendix II such representations are united by a brace and are designated as one representation. We now consider the consequences of internal symmetry. As is well known, the fundamental law of nature is the invariance of the Hamiltonian of a quantum system with respect to simultaneous permutations of the spatial and spin coordinates of its identical particles. Permutation symmetry imposes constraints not only on the Hamiltonian, but also on the wave functions of allowed states of the system, which should obey the socalled forbiddance principle. Namely, only the states of a system of identical particles with spin s, whose total spin-coordinate wave function is multiplied by .1/2s in the permutation of any pair of particles, occur in nature. That is, the system is symmetric for the particles with integer values of s, called bosons, and is antisymmetric for the particles with half-integer values of s, called fermions. From the fundamental permutation symmetry it follows that the purely coordinate Hamiltonian is invariant with respect to permutations of spatial coordinates of identical particles and, therefore, belongs to the fully symmetric coordinate Young diagram. Although this Hamiltonian does not contain spin variables, the forbiddance principle still leads to a strong relationship between the permutation symmetries of the coordinate and spin functions taken separately. Thus, the nuclear spin, although its direct influence on the energy of the molecular stationary states or molecular terms is negligible (the so-called hyperfine interaction), has a strong influence on allowed types of symmetry of the coordinate wave functions of these terms. Indeed, if the identical nuclei are fermions, then, according to Section 3.4, the full spin-coordinate wave function of the stationary state must have the form ‰ Œ1

n



1 X Œ Q Œ Dp ˆr .x; X/XrQ ./; f r

(7.16)

where x and X are the spatial coordinates of electrons and nuclei and  are the spin coordinates of nuclei. Permutation symmetries of the full coordinate wave function

Section 7.1 Symmetry groups of internal dynamics

71

ˆ(x, X) and nuclear spin wave function X(/ are characterized by mutually dual Young diagrams of n cells, where n is the number of identical nuclei. In the case where the identical nuclei are bosons, we have, instead of equation (7.16), 1 X Œ ‰ Œn D p ˆr .x; X/XŒ r ./; f r

(7.17)

where the complete coordinate and nucleus spin functions are characterized by identical Young diagrams. It is important to emphasize that if the hyperfine interaction is neglected, then the type of the coordinate Young diagram becomes a good quantum number. That is, this follows immediately from equation (2.97) since such a Hamiltonian belongs to the coordinate Young diagram [n]. Hence, the right-hand sides of equations (7.16) and (7.17) include only one irreducible representation [ ] (summation over appears if hyperfine interactions are taken into account). As was shown in Section 3.3, the spin Young diagram of identical particles with spin s cannot have more than 2s C 1 rows. This imposes constraints on the form of the coordinate Young diagram. The latter also cannot have more than 2s C 1 rows for bosons and more than 2s C 1 columns for fermions. That is, the solutions of the coordinate Schrödinger equation, which correspond to the forbidden Young diagrams, do not occur in nature. Such forbiddances can only be obtained by the methods of symmetry. Each spin Young diagram is related to definite values of the total spin S of the system of identical nuclei. It is clear that for the nuclei with the spin s D 0, the value of S is also equal to zero. For the nuclei with the spin s D 1=2, each spin Young diagram comprising no more than two rows corresponds only to one value of S, such that S D .n1  n2 /=2; (7.18) where nk is the number of cells in the kth row of the Young diagram. For all other values of the nucleus spin s, each spin Young diagram may correspond to more than one value of the total spin S, and the same value of the latter can be included several times. This is clearly seen in the tables of Appendix III, where this correspondence is shown for identical particles with the number n  4 and the spin s  3. If the hyperfine interactions are neglected, then the energy of a molecular term depends neither on the value of the total spin, nor its orientation in space. The multiplicity of this degeneracy in the state with a given spin Young diagram [ ] is called the nuclear statistical weight ./ nucl of this state. By definition, ./

nucl D

X S

aS .2S C 1/:

(7.19)

Here, aS is the multiplicity of the implementation of the total spin S in the state with the spin Young diagram [ ]. These statistical weights are listed in the tables of Appendix III. As an example, we consider several diatomic molecules with the 2 group of permutations of identical nuclei.

72

Chapter 7 The philosophy of using the symmetry properties of internal dynamics

The molecule H2 . Since s.H/ D 1=2, the hydrogen nuclei are fermions. Then ΠQ spin W

Œ2

Œ12 

1 Œ coord W Œ12 

0 Œ2:

S

(7.20)

Thus, the following coordinate states are allowed: 3Œ12 ; 1Œ2;

(7.21)

where the number before the coordinate Young diagram indicates the degeneracy of the level determined by its nuclear statistical weight. Transitions between the states with different Πspin  are due only to the interactions related to the nuclear spins. In diatomic molecules, these interactions are very small compared with the energy difference of such states. Therefore, the diatomic molecules in the states with different spin symmetry can be considered as different modifications of a substance, which are generally called spin modifications. The modification is called the ortho modification ./ if it corresponds to the larger value of nucl and the para modification if it corresponds ./

to the smaller value of nucl. For the molecule H2 we have ortho=para D 3, so that the ortho modification corresponds to the antisymmetric coordinate wave function. The molecule D2 . Since s.D/ D 1, the deuterium nuclei are bosons. Then Πspin W

Œ2

Œ12 

S

0; 2 1 Œ coord W Œ2 Œ12 :

(7.22)

That is, the allowed coordinate states have the form 6Œ2; 3Œ12 :

(7.23)

As a result, ortho=para D 2, and the ortho modification now corresponds to the symmetric coordinate wave function. The molecule 16 O2. The nucleus of the oxygen isotope 16 O has s D 0. Therefore, only the single-row spin Young diagram [2] is allowed, and the coordinate state 1[2] corresponds to it. Coordinate states of symmetry [12 ] are not implemented (it is often said that they have a zero nuclear statistical weight). As we shall see in Section 8.3, this leads to the forbiddance of half of the solutions of the Schrödinger equation (7.1). This effect is entirely due to the quantum symmetry requirements for permutations of identical particles and is one of the striking manifestations of such symmetry. We emphasize that the transition from a system of identical particles to a system in which this identity is at least partially violated is not smooth, and is accompanied by jumps in the description of the system. In particular, the level forbiddance can appear and be removed only abruptly.

Section 7.2 Significance of the analysis of symmetry properties

73

Group 2 has only nondegenerate irreducible representations. In general, the group of permutations of identical nuclei also contains degenerate irreducible representations, which leads to the permutation degeneracy of solutions of the coordinate Schrödinger equation (7.1). However, such a degeneracy of the energy levels is not physically meaningful and is not in fact observed. This very important fact, which is often ignored, has a simple, although purely quantum, explanation. Indeed, from equations (7.16)–(7.17) it follows that a set of coordinate wave functions corresponding to the degenerate coordinate Young diagram [ ] form only one physically observable spin-coordinate state.

7.2 Significance of the analysis of symmetry properties Obtaining a rigorous solution to the equations of motion of the internal dynamics of molecules is currently impossible if only the nuclear and electron spin-related contributions to the Hamiltonian are neglected. If we try to solve these equations by perturbation methods, we unexpectedly face the need to introduce an additional internal geometric symmetry group to characterize the molecule. This is a matter of principle since otherwise it is impossible to write approximate equations of motion. The basic working approximation is the Born–Oppenheimer (BO) approximation [10, 14, 35], which introduced the concept of the effective potential of nuclei interaction in a given electronic state and, as a consequence, the concept of a set of equilibrium configurations corresponding to the minima of this potential. Qualitatively, molecules can be divided into rigid and nonrigid ones. For rigid molecules in nondegenerate electronic states, the choice of the effective potential with one minimum is quite adequate, whereas for nonrigid molecules several such minima should be taken into account because the internal motion includes the transitions between them. It has long been understood that for rigid molecules an additional geometrical group should be selected in the form of a point group of their unique equilibrium configuration, which by definition [60, 64] includes all geometric elements of symmetry of this structure as a whole. It is commonly assumed that this group and the inferences related with it are corollaries of the BO approximation, i.e., only in this approximation can we speak of a certain geometric structuring of internal motion. However, even in this simplest case there is a question of the applicability domain of a point group. Two essentially different opinions exist in the literature. According to one of them [60, 64], the point group characterizes the total (electron-vibration-rotational) internal motion when deviations from the equilibrium position are sufficiently small. However, what a “sufficiently small deviation” means is quite uncertain. The alternative point of view [15, 16] is that the point group describes the symmetry of vibrational and electronic motions only and is inapplicable to rotational motion and, hence, to the total internal motion. As a result, analysis of the total internal motion is based on the so-called complete nuclear permutation-inversion (CNPI) group [15,16]. Such contradictions in the status of empirically introduced point groups are connected with the absence of a definite point of view on their nature.

74

Chapter 7 The philosophy of using the symmetry properties of internal dynamics

The point group of rigid molecules for a given electronic state appeared as a symmetry group of their unique equilibrium configuration. Then it became clear that this group should also be used in describing the internal excitation spectrum. However, given the reason for its appearance, it was affirmed that this procedure is legitimate only for sufficiently small deviations from the equilibrium position. Such arguments for the operation area of a point group do not look convincing, since even arbitrarily small asymmetric displacements of nuclei from the equilibrium position abruptly lower the geometric symmetry of their configuration. Even more importantly, although this fact is often ignored, the point group may change with the variation in the electron excitation and certainly changes during the transition from a discrete spectrum to a continuous one. In other words, different isomers of the molecule can be realizable in different bound electronic states. As concerns the transition from a discrete to continuous spectrum, the finite multiplicity of degeneracy of the levels changes for the finite one, which is possible only when the type of the geometric symmetry group is changed. However, the conventional groups do not depend on the excitation of the system. Thus, the group of permutations of identical particles of the molecule is always the same. Such a change of symmetry is a characteristic feature of the implicit dynamic group, whose definition was introduced in Section 6.1. On the one hand, this group is specified by the dynamics of a particular system in a certain range of excitations. On the other hand, the group itself determines the qualitative properties of dynamics. Outside the range, the dynamics of the system changes qualitatively, which leads to an abrupt change of the dynamic group. If there is a problem with the implicit symmetry group, which admits of a rigorous solution, then the qualitative analysis with allowance for this group is unnecessary. This analysis is only desirable since it serves as the basis for explaining the qualitative features of the behavior of the system. Otherwise, the situation changes radically because when we use perturbation methods we should first ensure that the correct properties of symmetry of internal dynamics are transferred to approximate models. This gives rise to the problem of empiric search for the implicit group relative to which we should correctly symmetrize the operators of physical quantities and the space of basis wave functions that allow for the considered types of motion. That is, analysis of the symmetry properties with allowance for the implicit group is fundamentally necessary and should precede the solution of the equation of motion (in fact, without this analysis, we simply do not have the approximate equations of motion). Although this conclusion is important for explaining some general qualitative properties of the molecule, it hardly changes the methods and results of the analysis which uses the geometric symmetry groups to characterize the total internal motion. Currently, there is a fairly well developed unified procedure of using these techniques for rigid molecules [17, 60, 64]. Moreover, although this may seem surprising, even the symmetry methods with a physically incorrect (in our opinion) treatment of the applicability range of point groups lead to equivalent results in the qualitative analysis of internal motions of rigid molecules [15, 16]. The point is that in this simple case

Section 7.2 Significance of the analysis of symmetry properties

75

the qualitative properties of the basis wave functions are a priori well known. That is why, using the freedom in the selection of a procedure for the application of symmetry groups, we can fit the result to this answer. The error is excluded here. However, this approach is purely formal and precludes considering rigid molecules in a unified way, although they all have the same types of internal motion (electronic, vibrational, and rotational). But the main problem is that it is very difficult to transfer this approach to nonrigid molecules since very frequently the qualitative properties of the basis wave functions are a priori unknown. Therefore, it is a fundamental requirement that the symmetry methods must be closed (or self-consistent). In this regard, philosophical differences in the treatment of the applicability range of the geometric symmetry group become very significant. It suffices to note that the main reason for the analysis based on the CNPI-group concept in the pioneering work [65] is associated with is that the methods which use geometric symmetry groups to describe the total internal motion cannot be extended to nonrigid molecules. However, later this was invalidated as a result of developing an approach based on the concept of the symmetry group chain [17, 18, 20, 23, 28]. Moreover, at present, this approach is the only one in which the description is based entirely on the symmetry principles. In other words, we can speak of a closed formulation of qualitative intramolecular quantum dynamics, and the geometric symmetry groups play a fundamental role here. In the class of nonrigid molecules, in addition to the electronic, vibrational, and rotational motions, there is a huge variety of observed transitions between different minima of the effective potential of nuclei interaction (the so-called nonrigid transitions). Therefore, it is unclear, at least at this stage, how to construct a general procedure for using the symmetry methods to describe their internal dynamics. As a result, we have to extend the applicability range of the symmetry methods to particular classes of molecules with certain types of nonrigid motions. Although this process seems to be endless, its importance is confirmed by the fact that there is a large number of formulated topical problems for quite simple molecules, whose correct solution encounters the absence of adequate methods for analyzing the symmetry properties. A practical example is the problem of the dynamics in a PF5 molecule with allowance for the Berry pseudorotation, which was formulated four decades ago [34]. In assessing the practical significance of qualitative methods it is necessary to bear in mind one important fact. The stage of analysis at which the symmetry properties of the basis functions of the space of functional states of the molecular system and the symmetry properties of the physical-quantity operators (including the Hamiltonian of the system) specified in this space are determined, is called the construction of a classification of stationary states. Upon completion of this stage, one can try to write the model equation of motion in some internal coordinates of the molecule and find a solution. However, for nonrigid molecules, such a way of describing the internal dynamics spectrum is often too complicated. At the same time, the symmetry methods offer a quite efficient and even a very spectacular alternative means. Its essence is reduced to the following: Any internal motion is specified by a set of symmetry trans-

76

Chapter 7 The philosophy of using the symmetry properties of internal dynamics

formations, which determine the complete set of basis wave functions for this motion. All transformations are included (sometimes in a fairly nontrivial way due to the noncommutativity of the transformations specifying separate motions) into a complete symmetry group of internal dynamics. Thus, using purely algebraic methods, from the basis functions of separate motions we can form the basis functions of the total motion, which transform according to allowed types of symmetry of the complete group. Then the so-called effective operators of physical quantities (including the effective Hamiltonian) defined in the full function space are formed in a similar way from a complete set of self-adjoint operators specified in the subspaces of separate motions. It appears that the calculation of the matrix elements of self-adjoint operators describing separate types of motion requires not the explicit form of the basis wave functions of these types of motion, but only their symmetry properties (this was demonstrated as an example by calculation of the matrix elements of angular operators in Sections 4.4 and 6.2). In addition, the matrix elements are simple elementary functions of the symmetry symbols (quantum numbers of the problem). It can be said that symmetry selects the most appropriate basis. In effective operators, only the spectroscopic constants before combinations of self-adjoint operators admissible by symmetry remain uncertain. The values of these constants are usually found from a comparison of theory with experimental data. The obtained model strictly describes, within a given symmetry, all possible interactions between the different types of internal coordinate motion and leads to a simple, purely algebraic calculation scheme for both the position of levels in the energy spectrum and the intensities of the transitions between them. In this approach to the description of intramolecular dynamics, the configuration space of collective variables is not introduced in explicit form at all, and consequently the wave functions of these variables are not considered in explicit form. But exactly due to its deep philosophical and technical difference, this approach is the only possible solution for any major problem in the internal dynamics of nonrigid molecules. It is important that the correctness of the derived molecular models is limited only to the correct choice of symmetry of the internal dynamics. It is also interesting to note that the approach takes an intermediate position between the description of molecular spectra based on the solution of the nonempirical Schrödinger equation (ab initio methods) and the description based on the empirical form of the energy matrices. According to [46], such intermediate methods are the most optimal.

7.3 On the domain of the point group According to one point of view, the point group of a rigid molecule is a consequence of the BO approximation. Hence, it is concluded that this group is used to describe the internal dynamics only in the neighborhood of the equilibrium configuration. Recall that in the classical treatment, the BO approximation corresponds to the separation of electronic and nuclear motions and is based on the smallness of the ratio of the electron mass to the nucleus mass (see Chapter 12). However, there is a serious

Section 7.3 On the domain of the point group

77

reason to believe that the domain of the point group is much wider. Indeed, the concepts of the electronic state, the effective Hamiltonian in this state, and, accordingly, the effective interaction potential of the nuclei remain when the perturbation theory corrections (corrections to the BO approximation) are taken into account [66]. This is especially important since the formulation of the BO approximation is ambiguous, and sometimes different versions of this approach even lead to different symmetries of equilibrium configurations of the molecule. For example, according to [9], in the formulation of a simple adiabatic approximation for the NH3 molecule the equilibrium configuration is a plane with the point group D3h. Only the nonadiabatic (or vibronic) corrections make the plane configuration unstable and transform it to the actually observable form of a regular triangular pyramid with the point group C3v . At the same time, in the formulation of the adiabatic approximation the equilibrium configuration of the NH3 molecule has the symmetry group C3v at once. Consequently, the perturbation theory corrections can be significant for determining the correct symmetry of equilibrium configuration of the molecule, and therefore relating this symmetry to the BO approximation is not possible in general. Further, there is a consistent procedure for using the point group to describe the total internal dynamics [17,60,64]. It is essential that a comparison of the inferences obtained on its basis with the experimental data does not indicate a violation of symmetry with increasing excitation energy in a nondegenerate electronic state, although the nonadiabatic corrections in this case certainly increase. We discuss the Jahn–Teller effect separately. It is customary to believe that this effect is due to a strong interaction between the electronic and nuclear motions, and therefore its description is beyond the BO approximation. The essence of the effect is associated with the hypothesis, which L. D. Landau put forward in 1934, that the symmetric nuclear configuration of a nonlinear molecule is unstable in an electronic state that is orbitally degenerate due to this symmetry [64]. His idea was the basis for the theorem proven by Jahn and Teller [58]. Corollaries of the theorem have been intensely discussed in the literature. There are two basically different interpretations of this theorem. One of them, coming from the authors of the theorem, states that the instability of the symmetric configuration breaks the symmetry of the system and, as a consequence, the degeneracy of the electronic state is completely removed. However, this is true only in statics, while the interpretation [9,10], according to which all the energetically equivalent minima occurring in the neighborhood of the orbital degeneracy point should be taken into account, is physically correct with allowance for dynamics. These minima convert into each other during transformations of the point group corresponding to the degeneracy point. Therefore, the delocalization of a quantum system over the minima does not destroy its geometric symmetry and the degeneracy is not removed. But the type of degeneracy becomes electron-vibrational, or vibronic (see Section 12.2). Note that in sign of the need to consider several minima, such dynamics corresponds to a nonrigid molecule. As a result, it is logical to assume that for rigid molecules in each electronic state the point group describes a geometric structuring of the internal motion for all dis-

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Chapter 7 The philosophy of using the symmetry properties of internal dynamics

crete levels. From this point of view, the symmetry of the equilibrium configuration is only an elementary consequence of this structuring, but not vice versa. Other physical systems showing a similar behavior also exist. An interesting example is motion of the nucleons in the nucleus. There is strong evidence for structuring of this motion in large nuclei [39, 64]. Structured nuclei feature rotational motion of the nucleus as a whole. For a description of their internal dynamics by perturbation methods we need a point group, for which D1 is usually chosen because of the large number of nucleons ( 150) in such nuclei. This is done when the nucleus does not have a counterpart of a small parameter that is used in molecular dynamics to substantiate the BO approximation and is due to a strong difference in mass of the electrons and nuclei. Crystal is another example of a similar, but macroscopic system. All these systems have a basically collective motion that cannot be destroyed in some finite range of energies. In connection with the use of the concept of basically collective motion, we recall that in unstructured systems (atoms and a part of nuclei), a good zero-order approximation is description of the motion of each separate particle in the average field of all other particles. In the presence of collective motion, such a zero-order approximation is completely unusable. The latter is perfectly understandable since even a zero-order approximation must render correctly the qualitative features of the system behavior (see Section 12.5). Thus, there are strong grounds to believe that the point group of a rigid molecule characterizes the rigorous problem of internal coordinate motion. Since, as was noted above, this group depends on the type of electronic excitation, it can only be a dynamic group of this problem. Unfortunately, it is unclear at present how this group can be derived from the rigorous equations of motion. But since a description of the internal dynamics by perturbation methods cannot be constructed without this group, we must seek it empirically. Naturally, the solution of the problem of the group could give much more information. For example, constraints on the possible form of the effective potential of interaction of nuclei in a molecule could be clarified. It should be emphasized that even the point group C1 has a nontrivial significance. This group indicates that the internal structure in the electronic state under consideration is nonsymmetric. We emphasize that the internal geometric group is defined in the MCS, the very appearance of which in the analysis of the internal dynamics of a microsystem means that all internal motions of the particles of a molecular system are collective. In this regard, it is interesting to note the following. Operators of a complete set of angular physical quantities in a fixed coordinate system are J 2 and JZ . We now introduce the Cartesian coordinate system rotated with respect to the initial system. Naturally, operators of the components of the angular momentum for one coordinate system will not commute with these operators for another coordinate system. However, if the rotated system is introduced as moving (see Section 6.2), then the operators of the components of the angular momentum on the axes of the moving and fixed systems commute with each other. As a result, the third operator Jz appears in the complete set. This means that the internal dynamics changes qualitatively, and the transition from one type of dynamics to an-

Section 7.3 On the domain of the point group

79

other is abrupt. Note in this connection that the transition between the descriptions with different point groups is also accompanied by jumps (see Section 8.5). Hence, generally speaking, the transition to a description with a higher symmetry from the obtained description with a lower symmetry by imposing additional constraints in the latter on the system parameters is erroneous. In other words, the description with a lower symmetry does not include the description with a higher symmetry as a special case. The proposed status of the point group of a molecule can be justified in a somewhat different way. For this, we first present the point of view on such a group described in [39]. Its basis is the proposition that the complete coordinate Hamiltonian has a symmetry group R3 , since it depends only on the relative distances between the nuclei and electrons. The point group corresponding to the equilibrium position of nuclei and which is a subgroup of R3 applies only to describe the electron-vibrational motion. This is interpreted as a spontaneous breaking (lowering) of the symmetry from R3 to the point group due to the exclusion of the system rotation. That is, the symmetry will be restored if in addition to the motion of electrons and the vibrations of nuclei the rotation of a molecule is allowed. As a preliminary comment, we note first of all that if the point group has improper transformations (orthogonal transformations with determinant equal to 1), then it is not a subgroup of R3 . However, a wider group, compared with R3 , can be introduced on the same grounds, as may be desired, additionally allowing for the inversion operation. It is much more significant that the group R3 cannot be used in the above sense as a symmetry group of the complete Hamiltonian. To prove this, we use the solution presented in Section 6.2 to the problem of the symmetry group for the Hamiltonian of the rotational motion of a rigid body. This model is directly related to the internal dynamics of rigid molecules since it is a physically correct initial approximation for the description of their rotational spectra. The desired group has a maximum of six .ex/ .in/ independent infinitesimal operators and can be represented as R3  R3 . The states of the physical systems implement only those of their irreducible representations for which the indices of irreducible representations of the external and internal groups coincide (J – J representations). The existence of the external group is associated with the isotropy of space, and therefore its maximum version of R3 is implemented for any rigid body. The internal group characterizes the symmetry of the rigid body itself and includes only the rotations with respect to which the inertia ellipsoid is invariant. Therefore, the maximum version of the internal group characterizes the symmetry of the rotational motion only if the rigid body is a spherical top [8]. For a rigid body such as a symmetric or asymmetric top, this symmetry is determined by the groups D1 and D2 , respectively, and the group R3 plays the role of a dynamic noninvariant group. Actually, the presence of the external group permits one to characterize the rotational states using the quantum number J of the total angular momentum. However, the system-related symmetry and, therefore, the specific form of the Hamiltonian are determined by the internal group. Even in the limiting case of the rigid top model, the latter does not necessarily coincide with R3 .

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Chapter 7 The philosophy of using the symmetry properties of internal dynamics

We now consider the transition from this limiting case to the model which takes into account all the internal motions and in whose Hamiltonian only the nuclear and electron spin-related contributions are neglected. It is clear that the external group remains the same after such a transition. However, not all of the rotational elements of the internal group remain. Indeed, allowance for the corrections by which this model differs from the BO approximation preserves the concepts of the electronic state and effective Hamiltonian in this state. Hence, it can be argued that a necessary condition for the geometric element of symmetry of the effective Hamiltonian is to transform the potential into itself. Naturally, the equilibrium configuration corresponding to the minimum of this potential also transforms into itself. Thus, the rotational elements belonging to the point group of a molecule remain. Considering also the improper elements of symmetry of the potential, we obtain the point group as a whole. It is logical to assume that the latter is also a symmetry group of the whole effective Hamiltonian. It is essential that such a conclusion is consistent with the procedure used for the point group to describe the total internal motion of a rigid molecule, in which each element of the point group is put into correspondence with the permutation of identical nuclei in a force field that is invariant with respect to these elements. The procedure itself is presented in what follows, and we only mention for now that it leads to a correct description of the internal dynamics of a rigid molecule. According to Section 7.1, the fundamental properties of symmetry imply the invariance of the Hamiltonian of the coordinate motion with respect to permutations of the spatial coordinates of identical nuclei. Since the identical representation for the Hamiltonian in a point group appears already as a consequence of this strict symmetry, the point group is a dynamical invariant group. But we should remember that the very group of permutations of identical nuclei does not contain information on the structuring of the molecule dynamics which takes place for bound states. This fact is manifested most clearly for a linear molecule. Even if the identical nuclei are absent, the point group of such a molecule has a very nontrivial form of the continuous group C1 v . The proof that the action of the point-group transformations on an arbitrary (first of all, nonequilibrium) instantaneous configuration of a rigid molecule is equivalent to permutations of its identical nuclei is given in Section 12.3. The remainder of this section explains such a fundamental property in the behavior of a quantum microsystem as the presence or absence of the rotational spectrum. Generally speaking, the appearance of such a spectrum in the microsystem is quite surprising, since it must rotate as a whole, or in other words must have the properties of a rigid body. To understand such behavior, we note that in describing the internal dynamics of all known microsystems with a rotational spectrum, there is always some internal geometric symmetry group. Furthermore, being a correct initial approximation to describe the rotational motion of the microsystem, the rigid body model is characteristic in that it has an internal geometric group determining the symmetry of the structure of this body. That is, precisely the geometric structuring of the internal dynamics of the microsystem leads to its rotational spectrum stipulated by the rotation of the resulting dynamic structure as

Section 7.4 The chain of symmetry groups

81

a whole. In a given electronic state, the molecule-related moving coordinate system is “frozen” into the effective potential of interaction of the nuclei in this state, which for a rigid nonlinear molecule is equivalent to fixing this coordinate system relative to the equilibrium configuration of the nuclei. Finally, it is interesting to emphasize that in this consideration a molecule and an atom are qualitatively different microsystems and, as a consequence, the rotational motion of the system as a whole is absent in the atom. It is useful to note that the problem of the hydrogen atom strictly reduces itself to the problem of motion of one particle in an external field (see Section 6.1). Therefore, the internal geometric group and the collective effects related with it are absent here in principle.

7.4 The chain of symmetry groups In perturbation theory, the transition to the zero approximation is carried out by constructing a chain of nested (increasingly approximate) models until the exact solution of the model problem becomes possible. At the same time, a chain of symmetry groups describing these models arises. It is a natural problem to determine the evolution of the symmetry properties of the intramolecular motion (i.e., evolution of the symmetry properties of wave functions and operators of physical quantities) in the transition between the neighboring models. Its conventional solution does not exist. In the concept of the symmetry group chain, the problem is solved as follows. For a quantitative calculation, the transition between the neighboring models should be continuous to ensure that the difference between these models can be represented in the form of a small-parameter series. We digress here from the extremely complex problem of substantiating the procedure of using such series. It should only be noted that the presence of a small parameter is not a sufficient or necessary sign of convergence of the series. At the same time, the groups of the neighboring models can be different, that is, the symmetry changes abruptly. This is due to the fact that the approximate model is based on some physical idea and may contain information on less strict types of symmetry of internal motion. However, information on the symmetry of motion in a more rigorous model is at least lost in part in this case. As a consequence, the symmetry groups of all models play a pronounced independent role. Combining them in the connected chain is determined by the matching conditions. Namely, the equivalent elements, with respect to which the wave functions and operators of physical quantities should be transformed in the same way as indicated in the groups of the neighboring models. In other words, transitions between the neighboring models are accompanied by certain nontrivial constraints on the compliance of symmetry types. It is important to emphasize that the main difficulties in solving the stationary Schrödinger equation by the methods of perturbation theory are due to the declarative series describing the transitions between the neighboring models. Not only their properties are unknown, but it is often impossible to correctly calculate even the lower corrections. Symmetry properties should also be taken into account. However, the sit-

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Chapter 7 The philosophy of using the symmetry properties of internal dynamics

uation is absolutely different if only the symmetry properties are considered and the transitions between models are described by symmetry matchings. The point is that the matchings are rigorous(!). This is facilitated by the fact that usually (although not always) there is a finite number of symmetry elements by which the matching is done and also that changing the matching conditions leads immediately to drastic consequences in the description. As a result, we obtain in terms of symmetry a rigorous solution of the discrete-spectrum problem, with a simple algebraic scheme for calculation of the position of the energy levels and intensities of the transitions between them. The solution is rigorous in the sense that its correctness is limited only to the correct choice of the symmetry groups of intramolecular dynamics. Consider the models and the corresponding internal symmetry groups arising in the description of the discrete spectrum of a rigid molecule. Neglecting the molecular particle spin-related contributions to the Hamiltonian, we obtain a purely coordinate problem. The internal symmetry of a rigorously coordinate Hamiltonian is specified by the permutation symmetry of electrons and identical nuclei, as well as the implicit symmetry characterizing the molecular structure. In the BO approximation, internal motion is divided into electronic and nuclear. In a given electronic state, the implicit symmetry is replaced by a similar explicit symmetry coinciding with the point groupdefined symmetry of the equilibrium configuration in this state (see Section 12.3). Allowed irreducible representations of a point group for the complete coordinate function ˆ.x; X/ in equations (7.16)–(7.17) are called the coordinate multiplets of the molecule (or multiplets). The wave functions of electronic and nuclear motions can be separately classified according to irreducible representations of a point group. In the electronic problem, transformations of the point group act on the electron coordinates, and in the nuclear problem on displacements of the nuclei. Nuclear motion is divided into vibrational and rotational, which in a zero approximation are described by the models of a harmonic oscillator and a rigid top. The symmetry of the vibrational model is also determined by a point group, but its transformations act only on the vibrational coordinates. The symmetry of the rotational model is determined by the internal symmetry group of a rigid top (see Section 6.2), whose transformations act on the Euler angles. We note here that the type of rigid top is uniquely related to the form of the point group of the molecule. Formulations of the symmetry group matching the considered models of internal dynamics are general and do not depend on the choice of a rigid molecule. However, the discussion becomes clearer if we consider specific examples, which will be done in the next chapter. For nonrigid molecules, a model allowing for transitions between the minima of the effective interaction potential of the nuclei appears in the chain. These transitions are generally called nonrigid transitions. This model is described by the so-called extended point group, which, compared with the point group, additionally includes the transformations defining the nonrigid transitions. It will be seen later that for a certain type of such transitions the extended point group becomes a dynamic noninvariant group.

83

Section 7.5 The concept of the coordinate spin

7.5 The concept of the coordinate spin In the concept of the symmetry group chain it is very natural to introduce an important notion of the coordinate spin of a molecule, on the basis of which it is easy to form a complete set of self-adjoint coordinate operators in function spaces of finite dimension. These spaces correspond to basically quantum motions and arise, for example, in the description of configuration degeneracy in nonrigid molecules if the latter have several equivalent equilibrium configurations or vibrational and orbital electron degeneracies already available in rigid molecules. Similar spaces also appear in the description of quasidegeneracies. The components of the operator of coordinate spin e are defined in the FCS using the well-known three-dimensional Lie algebra of form (4.27): Œe˛ ; eˇ  D i"˛ˇ e :

(7.24)

Since the spin is a coordinate physical quantity, it follows from equation (6.7) that ŒJ˛ ; eˇ  D i"˛ˇ e :

(7.25)

We assume that the operators of the coordinate spin do not rotate the unit vectors i of the MCS, that is, Œe˛ ; iˇ  D 0: (7.26) From equations (7.24) and (7.26) we can easily obtain the commutation relations for the operators of the coordinate spin components in the MCS. Indeed, Œei ; ek  D Œ i ˛ e˛ ; kˇ eˇ  D i"˛ˇ i ˛ kˇ e : As in Section 6.2, we take into account that "˛ˇ i ˛ kˇ D Œi  k  D "i kl l ; and then we have Œei ; ek  D i"i kl el :

(7.27)

That is, unlike the commutation relations for the components of J, the commutation relations for the components of e are invariant with respect to the transition from the FCS to the MCS. In this respect, the coordinate spin of a molecule behaves like an ordinary spin of the nuclei and electrons. It remains to obtain the commutation relations between the MSC components of the angular momentum and the coordinate spin ŒJi ; ek  D Œ i ˛ J˛ ; kˇ eˇ  D i ˛ kˇ ŒJ˛ ; eˇ  C i ˛ ŒJ˛ ; kˇ eˇ D i"˛ˇ i ˛ . kˇ e C k eˇ /:

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Chapter 7 The philosophy of using the symmetry properties of internal dynamics

By interchanging the dummy indices ˇ and  in the second term and taking into account that "˛ˇ D "˛ ˇ , we finally obtain ŒJi ; ek  D 0:

(7.28)

It is easy to see that equations (7.27) and (7.28) are satisfied in two types of behavior of the spin-component operators with time reversal: either all three components are t-odd or any two components are t-even while the third component is t-odd. In the description of intramolecular motions both variants will be used. As an example, we construct a complete set of self-adjoint operators in some twodimensional function space, in which transformations of the group C2 are specified. Assume that the complete set of basis functions of this space consists of the unit vectors jAi and jBi belonging to the symmetric and antisymmetric representations of the group C2 , respectively. A two-dimensional representation of the Lie algebra (7.27) in the space of these functions is written in terms of the well-known Pauli matrices [64]:       1 1 0 1 0 i 1 0 1 ; e2 D ; e1 D : (7.29) e3 D 2 0 1 2 i 0 2 1 0 Under the standard definition, the unit vectors jAi and jBi are eigenfunctions of the operator e3 . Knowing the symmetry properties of the basis functions and the action of the coordinate-spin operators on these functions, it is easy to obtain the symmetry properties of the operators as well. For example, 1 e1 jAi D jBi; 2

1 e1 jBi D jAi: 2

(7.30)

After the transformation C21 , from equation (7.30) we have 1 C21 e1 C21 jAi D  jBi; 2

1 C21 e1 C21 jBi D  jAi: 2

(7.31)

These expressions determine the action of the transformed operator e1 on the basis unit functions and thus uniquely define it. From a comparison of equations (7.30) and (7.31) it follows that C21 e1 C21 D e1 : (7.32) Thus, the operator e1 belongs to the irreducible representation B of the group C2 . The operator e2 belongs to the same representation, while the operator e3 belongs to the irreducible representation A. In this simple case, these results can be written at once since the action of the operator e3 on the basis vector of a given symmetry leads to the vector of the same symmetry, and the symmetry of the vector changes under the action of the operators e1 and e2 . It is important to know the behavior of the spin operators with respect to the time reversal operation T . Assuming ŒT; C21  D 0 (for all applications considered in this

Section 7.5 The concept of the coordinate spin

85

book, the time reversal operation commutes with the transformations of geometric groups) and acting by the operator T on the left on the equality C21 jAi D jAi, it is easy to see that C21 T jAi D T jAi: (7.33) From equation (7.33) it follows that the vector T jAi belongs to the symmetric representation of the group C2 . Since the problem has only one such independent vector, then T jAi D ajAi: (7.34) In a similar way, we obtain T jBi D bjBi;

(7.35)

and from the condition T 2 D 1 (see Section 7.1) we have jaj2 D jbj2 D 1. Squares of the moduli arise here because of the antilinear operator T . We will show that a D b D 1 can always be chosen. Indeed, if a D e i ı , then multiplying the unit vector jAi by e i ı=2, we will arrive at the variant with a D 1. The same is for the unit vector jBi. Further, from an analysis similar to considering the action of the operation C21 on the spin operators, we find that the components e1 and e3 are t-even, while the component e2 is t-odd. In a two-dimensional space, the complete set of selfadjoint operators consists of three components ei and a totally symmetric unit operator I. If the Hamiltonian of the internal dynamics in this space is t-even and invariant for transformations of the group C2 , then we obtain the following expression for it: H D c0 I C c3 e3 :

(7.36)

Here, c0 and c3 are the real phenomenological constants. Hamiltonian (7.36) describes two levels of symmetry, A and B. When the condition of invariance of the Hamiltonian with respect to transformations of the group C2 is removed, we have, instead of equation (7.36), H D c0 I C c3 e3 C c1 e1 : (7.37) Hamiltonian (7.37) still describes two levels, but the presence of the last term precludes comparing them with the symmetry types A and B. Note that in this case, the group C2 is a noninvariant group. According to the results in Sections 4.4 and 6.2, the standard finite-dimensional function spaces for the coordinate motions correspond to the integer value of the quantum number of the coordinate spin squared and have an odd dimension. In this sense, the two-dimensional space is nonstandard. As will be shown later, this space can be regarded as a subspace in a standard three-dimensional space, which is defined by the eigenvectors of the operator e3 with eigenvalues ˙1.

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Chapter 7 The philosophy of using the symmetry properties of internal dynamics

7.6 The influence of numerical methods on the overall description Modern numerical methods of calculation in intramolecular quantum dynamics do not introduce radical changes in the picture presented since they also rely on perturbation theory, especially the BO approximation [45]. This raises the same problems of transfer of the correct symmetry properties of rigorous dynamics to the approximate model employed. That is, it is necessary to specify in advance the implicit geometric symmetry group, with respect to which the wave functions allowing for the considered types of motion, as well as the operators of physical quantities defined in the space of these functions, should have the correct symmetry. We also emphasize that in quantum mechanics of bound states, it is generally very important to properly define the function space since in this case the physically meaningful solutions should be selected against the vast background of formal solutions. An attempt to find a solution to the Schrödinger equation without the prior selection of the function space of correct solutions results in the discrete spectrum being replaced by a continuous one, that is, a set of additional formal solutions of the power continuum is imposed on a countable (or finite) set of physically meaningful solutions. Therefore, the search for the discrete spectrum by numerical methods without the analytical determination of such a space additionally requires the selection of solutions belonging to the discrete spectrum. Since the numerical solution of the problem of intramolecular dynamics is based on perturbation methods, there are also very nontrivial problems associated with the obligatory account of implicit symmetry. In other words, analysis of the symmetry properties with allowance for the implicit group becomes obligatory and should also precede the writing and solving of approximate equations. Hence, the widespread opinion that the description of this dynamics can be obtained purely numerically (without a substantial analytical support) is erroneous. There issues are considered in detail in Chapter 12.

7.7 Conclusions We have formulated some general conclusions based on the content of this chapter. 1. Almost all the symmetry transformations used in the function space of the problem of bound stationary states of a quantum molecular system are linear and unitary. Only the time reversal transformation T , which in quantum mechanics has to be introduced as antilinear and antiunitary, is the exception. Although for this reason a physically significant behavior of the wave function with respect to T does not exist, such a behavior has physical meaning for the operators of physical quantities and, moreover, plays an important role in this problem.

Section 7.7 Conclusions

87

2. The point group of a rigid molecule in a given electronic state arises as a manifestation of the implicit symmetry of a rigorous problem of internal motion. Thus, to determine the discrete spectrum of a molecule by the methods of perturbation theory, one should first take care to transfer the correct symmetry properties to approximate models. This means that the analysis of the symmetry properties with allowance for the implicit group should be preceded by the solution of approximate equations of motion (actually, we do not have such equations without this analysis). 3. The implicit geometric group is very different in its behavior from the permutation group of identical nuclei, although both of them characterize the internal symmetry. Firstly, the implicit group may change during the electronic excitation since different isomers of the molecule can be realizable in the different electronic states. Clearly, the permutation group is independent of the excitation. Secondly, the implicit symmetry characterizes the rigorous coordinate Hamiltonian as a whole, but not its kinetic and potential parts separately. At the same time, the permutation transformations leave these parts invariant. That is, these groups characterize the different aspects of the internal dynamics, and the geometrical group is responsible for the dynamic structure which arises in each electronic state and leads to a rigid collectivization of internal motions. The sign of such behavior is that the quantum microsystem has a rotational spectrum associated with the rotation of the structure as a whole. An atom does not belong to such microsystems and, therefore, does not have a rotational spectrum. 4. In nondegenerate electronic states of rigid molecules, the effective potential of the nuclear interaction has only one minimum. As a consequence, the symmetry of the equilibrium configuration necessarily coincides with the symmetry of the effective potential. In other words, the symmetry of the equilibrium configuration is an elementary consequence of the symmetry of the internal dynamics, but not vice versa as is quite often stated. 5. For all the applications considered in this book, the operations of external symmetry of a quantum system commute with the operations of its internal symmetry. This is quite obvious, since the external symmetry is determined by the properties of space and time and the internal symmetry, by the properties of the system itself. Relationship (6.9) obtained in Section 6.2 can be regarded as a special case of this statement. 6. When the Schrödinger equation describing the discrete spectrum of a molecular system is solved by the methods of perturbation theory, a chain of nested (increasingly approximate) models is built until the exact solution of the model problem becomes possible. At the same time, a chain of symmetry groups characterizing these models arises. The main difficulties of the solution in terms of perturbation

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Chapter 7 The philosophy of using the symmetry properties of internal dynamics

theory are due to the declarative series describing the transitions between the neighboring models. Not only the properties of these series are unknown, but also it is often impossible to correctly calculate even the lower corrections. Moreover, the symmetry properties should be taken into account. However, the situation is absolutely different if only the symmetry properties are considered and the transitions between the models are described by symmetry matchings. To do this, in the groups of the neighboring models we single out the equivalent elements, with respect to which the wave functions and the operators of physical quantities should be transformed in the same way. In other words, the transitions between the neighboring models obey certain nontrivial constraints on the compliance of symmetry types. The advantages of this approach are primarily due to the fact that the matchings are rigorous(!). As a result, we obtain a rigorous solution of the discrete-spectrum problem in terms of symmetry, with a simple algebraic scheme for calculating the positions of the energy levels and the intensities of the transitions between them. The solution is rigorous in the sense that its validity is limited only to the correct choice of the symmetry groups of intramolecular dynamics. 7. In the concept of the symmetry group chain, we introduce in a natural way an important notion of the coordinate spin of a molecule, on the basis of which a complete set of self-adjoint coordinate operators is easily formed in function spaces of finite dimension. These spaces correspond to the basically quantum motions and arise in the description of configuration degeneracy in nonrigid molecules if they have several energetically equivalent equilibrium configurations or vibrational and orbital electronic degeneracies already available in rigid molecules. Similar spaces also appear in the description of quasidegeneracies.

Chapter 8

Internal dynamics of rigid molecules

8.1 Nonlinear molecules without inversion center The internal dynamics of rigid molecules of different types will be considered with specific examples. In this section, the statement is based on the examples of the molecules of water H2 O, ammonia NH3 , and methane CH4 . The presence of two identical nuclei of H in the H2 O molecule leads to the permutation group 2 . Since s(H) D 1/2, the H nuclei are fermions, and similarly to the H2 molecule (see Section 7.1) we have Œ Q spin W Œ2 Œ12  S 1 0 Œ coord W Œ12  Œ2:

(8.1)

Thus, both spin and, correspondingly, both coordinate Young diagrams are allowed. z

1

2

Figure 8.1. Equilibrium configuration of the H2 O molecule.

The equilibrium configuration of this molecule in the ground electronic state, which is shown in Figure 8.1, belongs to the point group C2v [52] and is an isosceles triangle. Note that all irreducible representations of the group C2v are one-dimensional, and therefore all possible bound states of the molecule are nondegenerate. For rigid molecules in nondegenerate electronic states, the symmetry of equilibrium configuration just coincides with the implicit internal geometric symmetry of the internal dynamics (see Section 7.3). That is, for a given electronic state, by the symmetry of the equilibrium configuration it is possible to determine the geometric symmetry of the effective Hamiltonian. Under transformations of the point group, the effective potential of the nuclear interaction passes into itself. As a result, the spatial position of the equilibrium configuration is not changed, and the identical nuclei either remain in place or exchange places in an invariant force field. Consequently, each transformation of

90

Chapter 8 Internal dynamics of rigid molecules

the point group is related to a certain equivalent permutation of identical nuclei. The resulting matching of the groups C2v and 2 is shown in Table 8.1. The MCS is chosen in such a way that the molecule is located in the xz plane and the z axis coincides with the axis C2 of the group C2v . It is clearly seen in the table that the group C2v is homomorphically mapped onto the group 2 . For the quantities describing the total coordinate motion, only those representations of the point group are realized which have the same behavior for the elements homomorphically mapped onto one element of the permutation group. Such forbiddances will be called geometric. As a consequence, for the H2 O molecule, only two coordinate multiplets (two representations of the group C2v for the complete coordinate wave function) are allowed, namely, 1A1

and 3B1 ;

(8.2)

where the number before the multiplet designation shows its nuclear statistical weight excluding the contribution of the oxygen nucleus spin. The total nuclear statistical weight is determined by the expression ./

nucl D nucl.H/  Œ2s.O/ C 1:

(8.3)

For the main isotope 16 O of the oxygen nucleus, the spin is equal to zero and the second term in equation (8.3) becomes unity. It should be noted that the type of the asymmetric multiplet depends on the binding of the reflection planes v and N v of the group C2v to the reflection planes of the equilibrium configuration. It is seen in Table 8.1 that the plane v coincides with the plane of the molecule. Table 8.1. Homomorphic mapping of the point group C2v of the H2 O molecule onto the permutation group 2 of its identical nuclei. Classes C2v

E

Classes 2

{12 }

Irreducible representations C2v Irreducible representations 2

.z/

.xz/

C2

v

{2}

{12 } A1 [2]

A2 –

.yz/

N v

{2} B1 2

[1 ]

B2 –

Note. For the permutation groups, the coordinate Young diagrams are given.

The interactions of interest to us are invariant with respect to the operation of spatial inversion i of all particles of the molecule. This external operation of symmetry (related to the properties of space) commutes with all operations defined by the internal geometric symmetry of the molecule (see Section 7.7). As a consequence, we can write the group C2v  Ci . The presence of the direct product has a deep physical meaning, because otherwise not all the stationary states of such an isolated system as a molecule will have a definite behavior with respect to the operation i, which violates

Section 8.1 Nonlinear molecules without inversion center

91

one of the basic principles of nonrelativistic quantum mechanics [64]. The behavior of the multiplets with respect to this operation will be denoted by the signs (˙ ) on the top right. In the BO approximation, internal motion is divided into electronic and nuclear, and these motions are separated in degenerate electronic states. That is, the complete coordinate function is written as ˆel.x/ˆnucl .X/, where the electron wave function corresponds to the equilibrium nuclear configuration. Both factors are classified according to irreducible representations of the group C2v . Nuclear motion, in turn, is divided into vibrational and rotational, which in a zero approximation are described by the models of a harmonic oscillator (symmetry C2v / and a rigid asymmetric top (symmetry D2 /. That is, ˆnucl.X/ D ˆvib.Q/ˆrot.˛; ˇ; /, where Q is a set of three vibrational coordinates. To obtain a full qualitative picture of the internal dynamics, it is necessary to match the types of symmetry of the multiplets with the types of symmetry of the zero-approximation wave functions, which we used to construct the multiplets. Naturally, the noncoincidence of the rotational and the point groups should be taken into account in this matching: .mult/C2v Ci D .el /C2v  .vib /C2v  .rot.-in/C2v Ci :

(8.4)

6

Here,  el and  vi b are the irreducible representations of the group C2v for the wave functions of the electronic and vibrational motions and vib.-in are the irreducible rotation-inversion representations of the group C2v  Ci , which for a given rotational representation of the group D2 are determined from the correlation presented in Table 8.2 between the groups D2 and C2v  Ci through their common subgroup D2 . The arrow in matching (8.4) takes into account a quite obvious fact that the behavior relative to the operation i characterizes only the multiplet and has no physical meaning for the wave functions of separate motions. Indeed, otherwise a purely formal agreement about the action of this operation on the position of the MCS is required, and there is an infinite number of options for this agreement. The option in which i does not affect the position of the MCS and, correspondingly, iˆrot D ˆrot:

(8.5)

Table 8.2. Correlation between irreducible representations of the groups D2 and C2v  Ci . D2

C2v  Ci

A B1 B2

A1 ; A2 ./ .C/ A1 ; A2 .C/ B1 ; B2./

B3

B1 ; B2

.C/

./

./

.C/

92

Chapter 8 Internal dynamics of rigid molecules

is the simplest to formulate. The idea of full or partial compensation for i-stipulated variations of the equilibrium positions of nuclei in the MCS by rotation of this coordinate system is often used [15]. Full compensation is possible only for the molecules with linear or planar configuration. For the linear molecules in this case we have iˆrot D .1/J ˆrot;

(8.6)

which coincides with the well-known expression from [64] for the behavior during inversion transformation of the wave function of the stationary state of one particle in a spherically symmetric field. We emphasize that on the one hand, the operation i characterizes the symmetry of space and on the other hand, by i transformation the equilibrium configuration of a molecule converts into a configuration energetically equivalent to it. Equilibrium configurations separated by the potential barrier will be called independent. It is well known that if two independent configurations are mixed in the molecule by i transformation, then the so-called inversion doubling of levels is observed for it. This condition is not satisfied for the molecules with planar and linear equilibrium configuration. For them, the equilibrium configuration obtained by i transformation superposes with the initial free rotation. That is, there is no transition to the independent minimum of the effective nuclear potential by i transformation and the inversion doubling is therefore absent. It is important that in the concept of the symmetry group chain this fact is taken into account automatically due to geometric forbiddances on the possible coordinate multiplets. Expressions (8.2) and (8.4) permit one to obtain a classification of the rotational levels of the H2 O molecule in any electron-vibrational (or vibronic) state. To be more specific, we consider the electronic state of symmetry A1 . For example, this state is the ground electronic state. The water molecule has three normal vibrations, namely, two symmetries A1 and one symmetry B1 (the well-known solution to the problem of division of the normal vibrations of a rigid molecule into irreducible representations of its point group [35, 64] is presented in Appendix IV). The type of nonsymmetric vibration similar to the type of nonsymmetric multiplet depends on the binding of the reflection planes v and N v of the group C2v to the reflection planes of the equilibrium configuration. Figure 8.2 shows a classification of rotational levels for both possible types of symmetry of the vibrational motion. Since the inversion doublets are absent in the energy spectrum, its actual structure is described by the rotational subgroup D2 of the group C2v Ci . Note in this connection that the inversion doublets in the spectrum of nonlinear nonplanar molecules are not split. Indeed, the effective potential of a rigid molecule has one minimum, and the possible independent configurations are therefore separated by an impermeable barrier. Hence, the actual spectrum structure of any rigid molecule is described by the rotational subgroup of the group Gpoint  Ci . Obviously, the effective operator of any physical quantity characterizing the coordinate motion of the H2 O molecule in a given vibronic state belongs to the coordinate

Section 8.1 Nonlinear molecules without inversion center

93

Figure 8.2. Classification of rotational levels in the ground electronic state of a rigid H2 O molecule for both possible types of symmetry of vibrational motion: A1 (left) and B1 (right).

Young diagram [2] of the group 2 . A matching of the groups 2 and C2v shows that such an operator is invariant under transformations of the point group. Then, from a matching of the groups C2v  Ci and D2 within the framework of the latter we obtain representations A and B1 related, correspondingly, to physical quantities that are invariant under the inversion operation i and change sign during this transformation. For example, the effective rotational Hamiltonian refers to the totally symmetric representation A, and the electric dipole moment, to the representation B1 . These effective operators are defined in the function space specified by irreducible representations of the noninvariant internal group R3 (see Section 6.2). Decomposition of the unit vectors jJ; ki of the function space into irreducible representations of the group D2 is well known [64]. Selecting combinations of components of the angular momentum in the MCS of symmetry A in the group D2 and taking into account that the effective rotational Hamiltonian is t-even, we obtain the following general expression for it: HD

1 X nD0

H2n ;

H2n D

X p;s;t

2t c2p;2s;2t J 2p Jz2s .JC C J2t /;

(8.7)

94

Chapter 8 Internal dynamics of rigid molecules

where p C s C t D n for a given n, and c2p;2s;2t are the real phenomenological constants commonly called spectroscopic constants. To simplify the expressions, we omit almost throughout this book the symmetrization over noncommuting rotational operators. The matrix elements of the effective Hamiltonian are easily calculated on the basis of equations (6.11), (6.12), (6.16), and (6.17), which can be derived without the explicit form of the basis functions jJ; ki. The energy matrix for each value of the quantum number J decomposes into a direct sum of matrices corresponding to the different irreducible representations of the group D2 . It is important to emphasize that the operating range of the Hamiltonian is defined by the applicability range of the geometric group C2v . In other words, the Hamiltonian describes the strictly coordinate dynamics. The BO approximation is only necessary for seeking the interpretation of spectroscopic constants, which is beyond the scope of the symmetry methods. That is, the concept of the effective Hamiltonian allows one to separate the construction of a model for the description of excitation spectra and the interpretation of the spectroscopic constants included in this model. In general, each constant like this has both the contributions determined by the BO approximation and the contributions related to the different types of corrections to this approximation. Another thing is that the effective Hamiltonian (8.7) is represented as a Taylor series of components of the angular momentum in the MCS and is thus a series of rotationaldistortion perturbations. In a similar way, it is also easy to construct an operator of the electric dipole moment. Its component ˛ on the ˛ axis of the FCS is written in the form .i / .i / Jk Jl C dklmn Jk Jl Jm Jn C    /; ˛ D ˛i .d .i / C dkl

(8.8)

where ˛i are the direction cosines of the MCS relative to the FCS. In this expression, it is taken into account that the electric dipole moment is a t-even quantity. Besides the components Ji , equation (8.8) also includes the components of the unit vector ˛ in the MCS which are decomposed into the following irreducible representations of the group D2 :

˛z ! B1 ; ˛y ! B2 ; ˛x ! B3 : (8.9) Since the electric dipole moment belongs to the representation B1 of the group D2 , equation (8.8) can be rewritten as ˛ D ˛z  A C ˛y  B3 C ˛x  B2 :

(8.10)

Here, the irreducible representations of the group D2 are used to denote combinations of the components of the angular momentum for the required type of symmetry. The terms not related to the rotational distortion of the molecule give the main contribution to equation (8.10). There is only one such term, namely, ˛ D ˛z d .z/ ;

(8.11)

Section 8.1 Nonlinear molecules without inversion center

95

as it should be. Indeed, from simple geometric considerations it is clear that the H2 O molecule in its equilibrium configuration has the only component of electric dipole moment directed along the z axis. The selection rules for the electric dipole transitions have the form .C/ ./ .C/ ./ (8.12) A1 $ A1 ; B1 $ B1 in terms of the group C2v  Ci and A $ B1 ;

B2 $ B3

(8.13)

in terms of the group D2 . Naturally, the electric dipole transitions between the levels with different statistical weights are impossible. We now consider an ammonia molecule NH3 . The presence of three identical nuclei of H in it leads to the permutation group 3 . For the nucleus of H, the dimension of the spin space is equal to two, and only two spin Young diagrams, [3] and [21], of the three possible Young diagrams of the group 3 are allowed. As a result, we have Œ Q spin  W Œ3 Œ21 S

3=2 1=2 Œ coord W Œ13  Œ21:

(8.14)

The equilibrium configuration of this molecule is a regular triangular pyramid with the point group C3v [52]. We assume that the z axis of the MCS coincides with the thirdorder axis of the group C3v , and the xz plane with one of the three reflection planes of this group. As was mentioned in Section 7.3, the symmetric nuclear configuration of a nonlinear molecule becomes unstable in the electronic state which is orbitally degenerate due to the presence of this symmetry, and this determines the nature of the Jahn–Teller effect. A degenerate electronic state with vibrational modes that destabilize the symmetric nuclear configuration forms the so-called Jahn–Teller system. The vibrational modes leading to the lowering of symmetry of the equilibrium configuration are called active modes. The NH3 molecule has a Jahn–Teller E-E system consisting of a doubly degenerate electronic state of the E type and an active vibration of the same type. In describing such a system, it is necessary to take into account all the energetically equivalent minima of the effective potential of nuclear interaction in the vicinity of the orbital degeneracy point. In the transformations of the group C3v these minima convert into each other, and all transformations are characterized by the equivalent permutation of identical nuclei. Therefore, the delocalization of the NH3 molecule with multiple minima does not break the symmetry group C3v of its internal dynamics. But now the electron-vibrational part of the wave function of the stationary state of the molecule is not represented, even approximately, in the form of a product of the electronic and vibrational parts (see Section 12.2 for more detail). However, as before, the basis functions of these motions can be classified according to irreducible representations of the group C3v . Since it is necessary to consider several minima of the effective potential, the rigid molecule in fact becomes nonrigid in the presence of the Jahn–Teller effect.

96

Chapter 8 Internal dynamics of rigid molecules

Matching the isomorphic groups 3 and C3v (correlations between the irreducible representations of the symmetry groups in different models of the NH3 molecule are presented in Table 8.3), for the allowed coordinate multiplets we obtain 4A2 ; 2E;

(8.15)

where the nuclear statistical weight is given without allowance for the contribution of the nitrogen nucleus spin. The types of symmetry of the basis functions for separate internal motions are matched with the types of symmetry of the allowed multiplets, which are formed on their basis, in the following way: .mult /C3v Ci D .el /C3v  .vib/C3v  .rot.-in/C3v Ci ;

(8.16)

6

where the admissible representations rot.-in for a given rotational representation of the group D1 of a rigid symmetric top are determined from the correlation of the groups D1 and C3v  Ci through their common subgroup D3 . Table 8.3. Correlation tables for the rigid molecule NH3 . 3

C3v

D1

D3

C3v  Ci

[3]

A1

A1

A1

A1 ; A2

[21]

E

A2

A2

A2 ; A1

[13 ]

A2

E1

E

E .˙/

E2

E

E .˙/

E3

A1 C A2

E4

E

E .˙/

E5

E

E .˙/

E6

A1 C A2

...

...

.C/

.C/

./

.C/

./

./

A1 ; A2

.C/

./

A1 ; A2

.C/

./

.C/

./

C A2 ; A1

C A2 ; A1 ...

Figure 8.3 shows a classification of the energy levels for the vibronic state el  vib D A1 . Note that the spectrum has inversion doublets. For the nondegenerate levels, one of the components of such a doublet is absent because of the forbiddance of the multiplet A1 . Since the inversion doublets do not split in a rigid molecule, the actual structure of the energy levels is described by the rotational subgroup D3 of the .k/ group C3v  Ci (one of the axes U2 of the group D3 coincides with the y axis of the MCS). It is seen that the rotational levels with jkj D 3n ¤ 0, which are degenerate in the model of a rigid symmetric top, split (the so-called k-doubling effect).

Section 8.1 Nonlinear molecules without inversion center

97

Figure 8.3. Classification of the rotational levels of a rigid molecule of NH3 in the vibronic state el  vib D A1.

Using a chain of symmetry groups, it is easy to see that for the effective operators of physical quantities characterizing the coordinate motion of the NH3 molecule, representations A1 and A2 belonging, respectively, to the quantities that are invariant under the inversion operation i and change sign during this transformation are realized in the group D3 . That is, the group D3 is a symmetry group of the Hamiltonian. The construction of effective operators in degenerate vibronic states will be discussed in Section 8.3. In nondegenerate vibronic states, all such operators are purely rotational and are defined in the function space specified by irreducible representations of the noninvariant internal group R3 . Decomposition of the unit vectors jJ; ki of this space into irreducible representations of the group D3 is given in Table 8.4. The rotational Hamiltonian contains only t-even combinations of the angular momentum components in the MCS of symmetry A1 in the group D3 and can be written as H D HI C HII : HI D HII D

1 X nD0 1 X nD0

H2n ; H2n D

X

6t c2p;2s;6t J 2p Jz2s .JC C J6t /;

(8.17)

p;s;t

H2nC4 ;

H2nC4 D

X

6t C3 c2p;2sC1;6t C3 J 2p Jz2sC1 .JC C J6t C3 /;

p;s;t

(8.18) where p C s C 3t D n for a given n.

98

Chapter 8 Internal dynamics of rigid molecules

Table 8.4. Decomposition of the rotational basis unit vectors jJ; ki into irreducible representations of the group D3. D3

Basis functions

A1

jJ; 0i, where J is even p .jJ; 3ni C !jJ; 3ni/= 2, n ¤ 0

A2

E

jJ; 0i, where J is p odd .jJ; 3ni  !jJ; 3ni/= 2, n ¤ 0 p E1 D .jJ; 3n C 1i C !jJ; 3n  1i/= 2 p E2 D i.jJ; 3n C 1i  !jJ; 3n  1i/= 2 p E1 D .jJ; 3n C 2i C !jJ; 3n  2i/= 2 p E2 D i.jJ; 3n C 2i  !jJ; 3n  2i/= 2

Note. We have ! D .1/J when one of the second-order axes of the group D3 coincides with the axis x and ! D .1/J Ck when one of these axes coincides with the y axis, and .1/ two components of the representation E are determined by the conditions U2 E1 D E1 and .1/ .1/ U2 E2 D E2 , where the axis U2 is either the axis x or the axis y. ./

The effective operator of the electric dipole moment is of type A1 in the group C3v  Ci and, correspondingly, of type A2 in the group D3 . Hence, the electric dipole selection rules have the form .C/

A2

./

$ A2 ;

E .C/ $ E ./

(8.19)

E$E

(8.20)

in terms of the group C3v  Ci and A1 $ A2 ;

in terms of the group D3 . We emphasize that the so-called forbidden electric dipole transitions [1] (they are forbidden if the interactions of different types of motion are neglected) also obey these rules. Selecting the t-even rotational combinations of type A2 in the group D3 in equation (8.8), it is also easy to construct the effective rotational operator itself of the electric dipole moment. We now consider a methane molecule CH4 . The existence of four identical nuclei of H in this molecule leads to the permutation group 4 . Only three schemes, namely [4], [31], and [22 ], of the five possible spin Young diagrams are allowed. Hence, we obtain Œ Q spin W Œ4 Œ31 Œ22  S

2

1

0

Œ coord W Œ14  Œ212  Œ22 :

(8.21)

99

Section 8.1 Nonlinear molecules without inversion center z

H2 H1 C y H4

x

H3

Figure 8.4. Equilibrium configuration of the CH4 molecule.

The equilibrium configuration of the molecule in the ground electronic state is shown in Figure 8.4 and corresponds to the point group Td [28]. Matching the groups 4 and Td (correlations between the irreducible representations of symmetry groups in different models of the CH4 molecule are given in Table 8.5), for the allowed coordinate multiplets we have (8.22) 5A2 ; 1E; 3F1 : The types of symmetry of the basis functions for separate internal motions are matched with the types of symmetry of the allowed multiplets, which are formed on their basis, in the following way: .mult/Td Ci D .el /T  .vib /Td  .rot.-in/Td Ci ; d

(8.23)

6

where the admissible representations rot.-in for a given rotational representation of the group R3 of a rigid spherical top are determined from the correlation of the groups R3 and Td  Ci through their common subgroup O. Figure 8.5 shows a classification of the energy levels for el  vib D A1 . Inversion doublets do not split in a rigid molecule, and the actual structure of the levels is described by the rotational subgroup O of the group Td  Ci . For the effective operators of the coordinate physical quantities, representations A1 and A2 belonging, respectively, to the quantities that are invariant under the operation of spatial inversion i and change sign during this transformation are realized in the group O. That is, the effective Hamiltonian relates to the representation A1 and the effective operator of the electric dipole moment, to the representation A2 . As a consequence, the electric dipole selection rules in terms of the group O have the form A1 $ A2 ;

E $ E;

F1 $ F2 :

(8.24)

The construction of effective rotational operators of physical quantities in the ground vibrational state is discussed in Section 20.4.

100

Chapter 8 Internal dynamics of rigid molecules

R3

R3

Td × Ci

Td × Ci

3F1( − )

5A2( −)

J=2

J=0

1E ( ± )

O

O

F2

A1 E

3F1( + )

3F1( − )

J=1 O

J=3

3F1( + )

F1

O

5A2( + )

F2 F1 A2

Figure 8.5. Energy-level classification for the CH4 molecule in the type A1 vibronic state.

Table 8.5. Correlation tables for the rigid molecule CH4 . 4

Td

R3

Œ4 [14 ] [22 ] [212 ] [31]

A1 A2 E F1 F2

JD0

A1

1

F1

2

E C F2

3

A 2 C F1 C F2

4

A 1 C E C F1 C F2

5

E C 2F1 C F2

6

A1 C A2 C E C F1 C 2F2

7

A2 C E C 2F1 C 2F2

8

A1 C 2E C 2F1 C 2F2

9

A1 C A2 C E C 3F1 C 2F2

10

A1 C A2 C 2E C 2F1 C 3F2

11

A2 C 2E C 3F1 C 3F2

12

2A1 C A2 C 2E C 3F1 C 3F2 D reg C A1

Td  Ci

O

./ A.C/ 1 ; A2 .C/ ./ A2 ; A1 .˙/ E F1.C/ ; F2./ .C/ ./ F2 ; F1

A1 A2 E F1 F2

......

O

..........................................

101

Section 8.2 Nonlinear molecules with inversion center

8.2 Nonlinear molecules with inversion center For molecules with inversion center, there are some differences in the qualitative description of the internal dynamics, which will be shown by examples of the ethylene 12 C2 H4 and ethane 12 C2 H6 molecules. These molecules are also useful for considering some features of analysis not related to the presence of the inversion center but not encountered in the examples of the previous section. The permutation group of identical nuclei of the 12 C2 H4 molecule is a direct product of the permutation group 2 of carbon nuclei and the permutation group 4 of hydrogen nuclei. Since the spin of the 12 C nucleus is zero, the group 4 can, without loss of generality, be used as a permutation group of identical nuclei. Only three schemes, namely [4], [31], and [22 ], of the five possible spin Young diagrams are allowed. As a result, Œ Q spin W Œ4 Œ31 Œ22  S

2

1

(8.25)

0

Œ coord W Œ14  Œ212  Œ22 :

H3

H2 1

H1

2

H4

Figure 8.6. Equilibrium configuration of the C2 H4 molecule (the molecule lies in the yz plane).

The equilibrium configuration of the molecule in the ground electronic state is planar (Figure 8.6) and corresponds to the point group D2h [84], which can conveniently be written in the form C2v  CI . We assume that this molecule lies in the N v plane of the group C2v . Only the representations A1 and B2 of the group C2v are allowed in the coordinate multiplets of the group D2h because of the planarity of the molecule (it is useful to compare it with the H2 O molecule which was chosen lying in the plane v of the group C2v /. We emphasize that the most common recipe for matching the point group with the permutation group of identical nuclei is implemented for the ethylene molecule. Namely, the point group is homomorphically mapped onto the subgroup of the permutation group (correlations between the irreducible representations of symmetry groups in different models of the 12 C2 H4 molecule are given in Table 8.6). As a result of this mapping, the degenerate coordinate irreducible representations of the

102

Chapter 8 Internal dynamics of rigid molecules

group 4 are formally written as a decomposition into one-dimensional irreducible representations of the group D2h. For example, : Œ212  D A1u C B2g C B2u:

(8.26)

Table 8.6. Correlation tables for the rigid molecule 12 C2 H4. 4

{14 } E;  .yz/

{22 }  .xz/

C2.z/ ;

{22 } I; C2.x/

{22 }  .xy/ ; C2.y/

D2h D C2v  CI

Œ4

1

1

1

1

A1g

[31] [22 ]

3 2

–1 2

–1 2

–1 2

A1u C B2g C B2u 2A1g

[212 ] [14 ]

3 1

–1 1

–1 1

–1 1

A1u C B2g C B2u A1g

D2

D2h

H2  D2

D2h  Ci

A

A1g , A2u

As

A1g ; A2u

B1

A1u , A2g

Aa

A2u ; A1g

B2

B1g , B2u

B1s

.C/ ./ A2g ; A1u

B3

B1u , B2g

B1a

A1u ; A2g

B2s

B1g ; B2u

B2a

B2u ; B1g

B3s

.C/ ./ B2g ; B1u

B3a

B1u ; B2g

.C/

./

.C/

./

.C/

./

.C/

./

.C/

./

.C/

./

Such a decomposition cannot be interpreted as the splitting of a level of the [212 ] type into three sublevels due to the symmetry breaking of the group 4 . Naturally, the fundamental permutation symmetry cannot be broken in the case of transition to an approximate model. Actually, decomposition (8.26) should be used in the sense of the Frobenius theorem (see Section 2.4). That is, in the case of transition from the group D2h to the group 4 , the stationary state of symmetry A1u may pass into the state of symmetry [212 ] at the expense of freedom of choice ofbehavior with respect to permutations not having a counterpart in the group D2h. The same can be said about the symmetry states B2g and B2u. It is important that any of these three representations of the group D2h passes into a complete coordinate representation [212 ] of the group 4 , which together with the complete spin representation [31] gives one physically

Section 8.2 Nonlinear molecules with inversion center

103

observable spin-coordinate state. This interpretation of the matching of irreducible representations of the groups 4 and D2h leads to the following allowed coordinate multiplets: .5 C 1 C 1/A1g ; 3A1u; 3B2g ; 3B2u: (8.27) Note that the notation of multiplets in equation (8.27) does not include the irreducible representations A2 and B1 of the group C2v which are forbidden because of the planarity of the C2 H4 molecule. The types of symmetry of the basis functions for separate internal motions are matched with the types of symmetry of the allowed multiplets, which are formed on their basis, in the following way: .mult/D2hCi D .el /D2h  .vib /D2h  .rot.-in/D2h Ci ;

(8.28)

6

where the admissible representations rot.-in for a given rotational representation of the group D2 of a rigid asymmetric top are determined from the correlation of the groups D2 and D2h Ci through their common subgroup D2 . However, it is important here that the group D2h  Ci D C2v  CI  Ci contains two different inversion operations. Inversion i is the external symmetry operation, determined by the space properties, for the total Hamiltonian of the internal dynamics of any molecule and, as a consequence, is the external symmetry operation of the purely coordinate part of this Hamiltonian. At the same time, inversion I is the internal symmetry operation of the effective Hamiltonian for a given electronic state, and only for the molecules with inversion center in this state. The result is that each element of the group D2 has two .yz/ counterparts in the group D2h  Ci . For example, there are the counterparts N v i .yz/ .x/ and N v I for the element C2 . A natural requirement of their coinciding leads to the need to fulfill the following correspondence: g $ C;

u $ :

(8.29)

Here, the indices g and u belong to the rotational part, and the indices ˙ to the multiplet. It follows from equation (8.29) that for the purely rotational motion the operation Ii is equivalent to the identical operation E. Thus, to obtain the admissible representations rot.-in for a given rotational representation of the group D2 the latter should be matched with the group D2h through their common subgroup D2 , and the signs ˙ should be added in accordance with equation (8.29). Figure 8.7 shows a classification of the energy levels for the vibronic state el  vib D A1g . Due to the planarity of the molecule, the inversion doublets are absent in the spectrum, and the actual structure of the levels is described by the rotational subgroup H2 D2 of the group D2h Ci . For the group H2 D .E; I i/, the symmetric and antisymmetric irreducible representations are denoted as s and a. Note that only one of these representations is implemented for each vibronic state (s for the type g vibronic state and a for the type u vibronic state).

104

Chapter 8 Internal dynamics of rigid molecules

Figure 8.7. Classification of the rotational levels of a rigid 12 C2 H4 molecule in the vibronic state el  vib D A1g .

All irreducible representations of the group D2h are one-dimensional. Therefore, the effective operators of physical quantities in any vibronic state are purely rotational. From a chain of symmetry groups it follows that representations As and Aa belonging, respectively, to the quantities that are invariant under the inversion operation i and change sign during this transformation are realized for these operators in the group H2  D2 . The rotational Hamiltonian relates to the representation As . Since the operation Ii is equivalent to the identical operation E for any rotational quantity, the rotational Hamiltonian is determined by equation (8.7) obtained for the water molecule. The decomposition of the unit vectors of rotational space into irreducible representations of the symmetry group of the rotational Hamiltonian also coincides with the case of the water molecule since all the rotational functions belong to the s type. The situation is absolutely different for the effective rotational operators related to the representation Aa . All of them are identically equal to zero. In particular, the effective rotational operator of the electric dipole moment is also zero. This result indicates that not only the constant component of the electric dipole moment is zero, as is clear even from simple geometric considerations, but also that all the contributions to it due to the rotational distortion of the molecule vanish. That is, the forbidden rotational transitions are also absent in the spectrum of the ethylene molecule C2 H4 . This is confirmed

105

Section 8.2 Nonlinear molecules with inversion center

by the electric dipole selection rules having the form .C/

./

A1g $ A1g ;

.C/

./

A1u $ A1u ;

.C/

./

B2g $ B2g ;

.C/

./

B2u $ B2u

(8.30)

B3s $ B3a

(8.31)

in terms of the group D2h  Ci and As $ Aa ;

B1s $ B1a ;

B2s $ B2a ;

in terms of the group H2  D2 . It is easy to see that the electric dipole transitions within a single vibronic state are impossible. This is true for all rigid molecules with inversion center. We now consider an ethane molecule 12 C2 H6 . As in the ethylene molecule, the group 2 for the carbon nuclei can be omitted in the permutation group of identical nuclei. In the group 6 of permutations of H nuclei, only four of the eleven possible spin Young diagrams are allowed: ΠQ spin  W

Œ6

Œ51

Œ42

Œ32 

S

3

2

1

0

(8.32)

Œ coord W Œ16  Œ214  Œ22 12  Œ23 : The equilibrium configuration of the C2 H6 molecule in the ground electronic state, which is shown in Figure 8.8, corresponds to the point group D3d [84], which can conveniently be written in the form C3v CI . In a matching of the point group with the permutation group of identical nuclei, the group D3d is isomorphically mapped onto the subgroup of the group 6 (correlations between the irreducible representations of symmetry groups in different models of the 12 C2 H6 molecule are given in Table 8.7). As a result, for the allowed coordinate multiplets we obtain .5C1C1/A1g ; .7C3C3/A1u ; 3A2g ; 1A2u; .5C3C1/Eg ; .5C3C3/Eu : (8.33) The types of symmetry of the basis functions for separate internal motions are matched with the types of symmetry of the allowed multiplets, which are formed on their basis, in the following way: .mult /D3d Ci D .el /D3d  .vib/D3d  .rot-in/D3d Ci ; 6

Figure 8.8. Equilibrium configuration of the ethane molecule C2 H6 .

(8.34)

106

Chapter 8 Internal dynamics of rigid molecules

where the admissible representations rot.-in for a given rotational representation of the group D1 of a rigid symmetric top are determined from the correlation of the groups D1 and D3d through their common subgroup D3 , with additional account for the correspondence given in equation (8.29). Figure 8.9 shows an energy-level classification for the vibronic state el  vib D A1g . In this case, all the levels are unsplit inversion doublets. The actual structure of the levels is described by the rotational subgroup H2  D3 of the group D3d  Ci . As in the ethylene molecule, only one representation of the group H2 is implemented for each vibronic state (s for the type g vibronic state and a for the type u vibronic state). Table 8.7. Correlation tables for the rigid molecule 12 C2 H6. 6

D3d

H2  D3

D3d  Ci

Œ16 

A1u

A1s

A1g ; A2u

[214 ] [22 12 ] [23 ]

A1g C Eg C Eu A2g C Eg C 2A1u C 2Eu 2A1g C Eg C A2u

A1a

./ .C/ A1g ; A2u

A2s

A2g ; A1u

A2a

A2g ; A1u

.C/

./

.C/

./

./

.C/

.C/

./

./

.C/

D1

D3

D3d

Es

Eg ; Eu

A1 A2

A1 A2

A1g ; A2u A2g ; A1u

Ea

Eg ; Eu

E1 E2

E E

Eg ; Eu Eg ; Eu

E3 ...

A1 C A2 ..........

A1g ; A2u C A2g ; A1u ............

From a chain of symmetry groups it follows that for the effective operators of physical quantities, representations A1s and A1a belonging, respectively, to the quantities that are invariant under the inversion operation i and change sign during this transformation are realized in the group H2  D3 . That is, the effective Hamiltonian relates to the representation A1s and the effective operator of the electric dipole moment, to the representation A1a . In nondegenerate vibronic states, these operators are purely rotational. The decomposition of the rotational unit vectors into irreducible representations of the group H2 D3 coincides with the decomposition into irreducible representations of the group D3 , which is presented in Table 8.4, since all the rotational unit vectors belong to the s type. The form of the effective rotational Hamiltonian is partially dif.k/ ferent from that obtained for the ammonia molecule, since in ethane one of U2 axes of the group D3 coincides with the x axis, but not with the y axis. As a result, equation

107

Section 8.2 Nonlinear molecules with inversion center

Figure 8.9. Classification of the rotational levels of a rigid molecule of 12 C2 H6 in the vibronic state el  vib D A1g .

(8.18) is replaced by HII D H2nC4 D

X

1 X

H2nC4 ;

 6t C3  ic2p;2sC1;6t C3 J 2p Jz2sC1 JC  J6t C3 : nD0

(8.35)

p;s;t

Since it is impossible to construct a rotational quantity of the a type, the effective rotational operators related to the representation A1a are identically equal to zero. In particular, this is an effective rotational operator of the electric dipole moment, which is very easy to confirm by the electric dipole selection rules A1s $ A1a ;

A2s $ A2a ;

Es $ Ea :

(8.36)

108

Chapter 8 Internal dynamics of rigid molecules

8.3 Linear molecules In molecules of this type in equilibrium configuration, all the nuclei lie on a straight line, which leads to the continuous point groups C1 v and D1 h . In molecules with the point group C1 v , the identical nuclei are either absent or nonsymmetric about the center of mass of the molecule. In both cases, in a given electronic state, all transformations of the point group are matched with the identical permutation of nuclei, that is, the group C1 v is homomorphically mapped onto 1 , as is shown in Table 8.8. As a consequence, only the identity representation A1 is allowed for the coordinate multiplets. Table 8.8. Homomorphic map of the point group C1 v onto the permutation group 1 of identical nuclei. Classes C1 v

E

2C'

v

Classes 1

{1}

{1}

{1}

Irreducible representations C1 v

A1

A2

E1



En



Irreducible representations 1

[1]

–

–



–



As in the nonlinear molecules, the basis electronic and vibrational functions of a linear molecule are classified by irreducible representations of a point group. The situation with the symmetry group of the rotational problem is more difficult for a linear molecule. The fact is that here the zero approximation corresponds to the motion of a point over a spherical surface, which is parameterized by only two polar angles. Hence, the symmetry group of the rotational Hamiltonian consists only of the external group R3 , while the internal structure of the molecule is completely ignored in this model. However, it is easy to add a noninvariant internal group R3 taking into account that only the J–J representations are implemented for the physical system (see Section 6.2). This noninvariant group just defines the complete basis set jJ; ki of the rotational function space in the MCS. As a result, the types of symmetry of the basis functions for separate internal motions are matched with the types of symmetry of the allowed multiplets, which are formed on their basis, in the following way: .mult/C1 v Ci D .el /C1 v  .vib /C1 v  .rot.-in/C1 v Ci ;

(8.37)

6

where the admissible representations rot.-in for a given representation of the internal group R3 follow from the correlation of the groups R3 and C1 v  Ci through their

109

Section 8.3 Linear molecules

common subgroup D1 , which is shown in Table 8.9. Matching (8.37) on account of the fact that only one coordinate multiplet A1 is admissible permits one to construct an energy-level classification for any vibronic state. Figure 8.10 shows this classification for el D A1 ; vib D E and el D vib D E1 . The inversion doublets are absent, and the actual structure of the rotation-vibronic (or rovibronic) levels are described by the group D1 . Part of the levels with low J are forbidden in the classification. This is explained by the well-known fact [64] that the projection of the angular momentum onto the C1 axis of a linear molecule is connected only with the electron-vibrational motion. Naturally, the angular momentum cannot be less than this projection which is different from zero in degenerate vibronic states. The latter fact is taken into account by the types of symmetry in el vib . We also note that the nuclear statistical weights of all levels of a linear molecule with the point group C1 v are the same. Table 8.9. Correlation table of the groups R3 and C1 v  Ci . R3

D1

C1 v  Ci

JD0

A1

A1 ; A2

1

A2 C E1

2

A1 C E1 C E2

3

A2 C E1 C E2 C E3

A1 ; A2

......

.....................

.......................................

.C/

./

A1./ ; A.C/ C E1.˙/ 2 .C/

./

.˙/

A1 ; A2 C E1 ./

.C/

.˙/

C E1

.˙/

C E2 .˙/

C E2

.˙/

C E3

Consider the construction of effective operators of physical quantities for el D vib D E1 , that is, in the presence of both the electronic and vibrational degeneracies. We assume that the x, y, and z axes of the MCS are fixed with respect to the effective nuclear potential (see Section 12.4), so that the z axis coincides with the axis of symmetry of infinite order. As the basis unit vectors of the electronic representation E1 we choose the unit vectors j ˙ 1e i belonging to the following pair of complex-conjugate representations of the group C1 : C' j ˙ 1e i D exp.˙i'/j ˙ 1e i:

(8.38)

Here, we used the behavior of the basis unit vector in the case of rotation by an angle ' in the MCS, which corresponds to equation (6.20). The point is that the electronic degeneracy is described by the operator of coordinate spin whose components in the MCS satisfy the commutation relations (7.27) having the plus sign on the right-hand side. In the group C1 v , the rotational operations C' and the reflection operations v do not commute with each other. It is easy to show that v C' v D C' 2 C1 :

(8.39)

110

Chapter 8 Internal dynamics of rigid molecules

Consequently, the group C1 is an invariant subgroup of the group C1 v . Thus, the latter can be represented by the following semidirect product (see Section 1.3): C1 v D C1 ^ CS ; .xz/

where CS D .E; v

(8.40)

/. Using equation (8.38), we obtain

C' v.xz/ j ˙ 1e i D v.xz/ C' j ˙ 1e i D exp i'v.xz/ j ˙ 1e i; which implies  .yz/ j1e i D aj  1e i

(8.41)

where a D 1. One can always choose a D 1 since otherwise it suffices to change sign in one of the vectors. Thus, the unit vectors j ˙ 1e i realize the two-dimensional representation E1 of the group C1 v . Since these unit vectors are transformed by a pair of complex-conjugate representations of the group C1 , the time reversal operator T mixes them (see Section 7.5): (8.42) T j1e i D bj  1e i; 2

where jbj2 D 1. The modulus squared comes from the antilinearity of the operator T , which permits one to put b D 1 without changing the value of a in equation (8.41). Indeed, if b D e i ı , then multiplying both unit vectors by e i ı=2, we arrive at a variant with b D 1 and the value of a preserved. The basis vectors j ˙ 1v i of the vibrational representation E1 have similar properties. .y/ The group D1 can be represented as C1 ^ U2 , where U2 D .E; U2 D  .xz/i/. Therefore, the symmetry properties of the vibronic quantities in the group D1 coincide with those in the group C1 v if  .xz/ is replaced by  .xz/i. Allowing for the known behavior of the rotational functions in the internal group R3 [64], we obtain the following complete set of basis rovibronic unit vector of type A1 in the group D1 : p js0 i D jJ; 0i.j  1e ij1v i C !j1e ij  1v i/= 2; (8.43) p js2 i D .jJ; 2ij1e ij1v i C !jJ; 2ij  1e ij  1v i/= 2; where ! D .1/J . It should be emphasized that during rotation by an angle ' about the z axis in the MCS, the rotational functions transform according to equation (6.19), which differs from equation (8.38) in the sign of the exponent. The complete basis set of unit vectors ja0 i and ja2 i of type A2 follows from equation (8.43) with ! replaced by !. It is important that the rovibronic unit vectors contain only the products jJ; kijƒe ijlv i for which k D ƒe C lv . That is, the construction scheme allows for the fact that the projection of the angular momentum on the z axis is due to the vibronic motion (all the four basis unit vectors for a given value of J are available only at J  2). Electronic parts of the operators of physical quantities are constructed on the basis of the operator of coordinate spin ƒ with an equal-to-unity quantum number of

Section 8.3 Linear molecules

111

Figure 8.10. Energy-level classification for a linear molecule with the point group C1 v in the vibronic states el D A1; vib D E (left) and el D vib D E1 (right). The chain .rot /R3 ! .rot.-in/C1 v Ci ! .mult/C1 v Ci ! ./D1 is shown for each multiplet.

the operator of the spin squared. The unit vectors j ˙ 1e i are considered as eigenvectors of the operator ƒ3 having the eigenvalues ˙1. Since the eigenvector with a zero eigenvalue is excluded from consideration, of all the admissible spin operators in

112

Chapter 8 Internal dynamics of rigid molecules

Table 8.10. Symmetry properties of the complete set of independent spin operators for describing the electronic degeneracy in a linear molecule. C1; v

t -even

t -odd

A1

Ie

–

A2 E2

E2;1

– D ƒ2C C ƒ2

E2;2 D i.ƒ2C  ƒ2 /

ƒ3 –

Note. Two components of the representation E2 are determined by the conditions  .xz/ E2;1 D E2;1 and  .xz/ E2;2 D E2;2 .

three-dimensional space, only the operators Ie ; ƒ3 ; ƒ2C ; ƒ2

(8.44)

are independent. Here, Ie is a unit operator and ƒ˙ D ƒ1 ˙ iƒ2 are the raising and lowering operators. Following the analysis in Section 7.5 and using the symmetry properties of the electronic basis functions for transformations of the group C1 v and for the operation T, it is easy to obtain the symmetry properties of spin operators (8.44) presented in Table 8.10. In a similar way, we also introduce the operator of coordinate spin l to construct the vibrational parts of the operators of physical quantities. The effective electron-vibration-rotational (or rovibronic) Hamiltonian belongs to the .C/ representation A1 of the group C1 v  Ci and, consequently, to the representation A1 of the group D1 . Taking into account that the Hamiltonian is t-even, its general expression for the state el D vib D E1 can be written in the form     2 2 2 2 ƒ h3 C lC JC C J2 l2 h4 H D h1 C .ƒ3 C l3 /2 h2 C ƒ2C l2 C lC     2 2 4 C ƒ2CJC C J2 ƒ2 h5 C ƒ2ClC JC C J4 l2 ƒ2 h6 ; (8.45) 1 X .i / 2p ap J : hi D pD0

It was additionally taken into account that in the space of allowed rovibronic functions, the operator Jz is equivalent to the operator ƒ3 C l3 . It is easily seen that the action of the Hamiltonian on the basis unit vectors does not violate the condition k D ƒe C lv . For the levels with J  1, the energy matrices are one-dimensional and E.J / D h1 .J / ˙ 4h3 .J /;

(8.46)

where the upper and lower signs correspond to the states js0 i and ja0 i. For J  2, the energy matrices become two-dimensional. Because of the mixing of the basis unit vectors with jkj D 0; 2, a nonpolynomial dependence of the energy on J(J C 1) appears and the quantity k2 is no longer a good quantum number. The domain of Hamiltonian (8.45) is determined by the applicability range of the geometrical group C1 v . In

113

Section 8.3 Linear molecules

this description, the electronic and vibrational motions are not separated even approximately, as it should be (see Section 12.2). Such manifestations in the linear molecules are called the Renner effect [14]. For the effective operator of the electric dipole moment in the group D1 we obtain the representation A2 , which leads to the following selection rules in terms of this group: A1 $ A2 : (8.47) The operator itself of the electric dipole moment is also easy to construct. For linear molecules with the point group D1 h D C1 v CI it is useful to consider a very interesting example of the oxygen molecule 16 O2 . Its permutation group of identical nuclei is 2 . It was already shown in Section 7.1 that only the coordinate Young diagram [2] of this group is allowed since s(16 O) D 0. In the matching of the point group with the permutation group of identical nuclei, the group D1 h is homomorphically mapped onto the group 2 (correlations between the representations of symmetry groups in different models of the 16 O2 molecule are given in Table 8.11). As a result, we obtain the only allowed coordinate multiplet 1A1g . Table 8.11. Correlation tables for a linear molecule with the point group D1 h . Irreducible representations of 2

[2]

[12 ]

–

–

–

–

–

–

...

Irreducible representations of D1 h

A1g

A1u

A2g

A2u

E1g

E1u

E2g

E2u

...

D1

D1 h D C1 v  CI

H2  D1

D1 h  Ci

A1

A1g ; A2u

A1s

A1g ; A2u

A2

A1u ; A2g

A1a

./ .C/ A1g ; A2u

E1

E1g ; E1u

A2s

A2g ; A1u

E2

E2g ; E2u

A2a

A2g ; A1u

...

............

...

..........

.C/

./

.C/

./

./

.C/

The types of symmetry of the basis functions of separate internal motions are matched with the types of symmetry of the only allowed multiplet formed on their basis: .mult /D1 h Ci D .el /D1 h  .vib /D1 h  .rot.-in/D1 h Ci ; 6

(8.48)

114

Chapter 8 Internal dynamics of rigid molecules

where the admissible representations rot.-in for a given representation of the internal group R3 follow from the correlation between the groups R3 and D1 h through their common subgroup D1 with the further account for the correspondence given in equation (8.29) which takes place for all molecules with inversion center. As a result, we have a classification of the energy levels in an arbitrary vibronic state. Consider the ground electronic state, which in the case of diatomic molecules usually belongs to the identity representation of the point group. However, the O2 molecule is one of the few exceptions. The ground electronic state of this molecule has the type of symmetry † g [64] (or A2g in the notation used here). It is clear that the only vibrational mode of this molecule belongs to the type A1g . Therefore, the vibronic state is of the type A2g . The energy-level classification obtained for such a state is presented in Fig 8.11. Inversion doublets are absent in the spectrum, and the actual structure of the levels is described by the group H2  D1 . It is clearly seen that all the levels with even values of the quantum number J are forbidden. This is fully consistent with the result obtained in [64] on the basis of a method that is suitable only for the diatomic molecules. The forbiddance of one-half of the levels is entirely due to the quantum symmetry for permutations of identical particles and is one of its striking manifestations. As was noted in Section 7.1, the transition from the system of identical particles to a system in which this identity is at least partially violated is not smooth and is accompanied by jumps in the system description. In particular, the forbiddance of levels appears and disappears only abruptly.

Figure 8.11. Energy-level classification for the 16 O2 molecule in the type A2g vibronic state.

For the effective Hamiltonian in the group H2  D1 we have a representation A1s , and for the effective operator of the electric dipole moment, a representation A1a . Therefore, the electric dipole selection rules have the form A1s $ A1a :

(8.49)

Section 8.4 Description of quasidegenerate vibronic states

115

It can be easily verified that in the ground electronic state of the 16 O2 molecule, both the rotational and the vibration-rotational electric dipole transitions are forbidden.

8.4 Description of quasidegenerate vibronic states For rigid molecules, describing the whole set of “closely spaced” vibronic states in their energy spectrum is a fairly regular problem. The point is that a separate (or, as is commonly said, isolated) consideration of each vibronic state of the set leads to an effective Hamiltonian in the form of a perturbation series, in which, as a rule, the smallness parameter is absent. This means that the wave functions of isolated states are far from reality. It is sometimes said that “strong interactions,” which are commonly called accidental resonances, exist between these states. As a consequence, summing the perturbation series is a major problem. To avoid these difficulties, such a set of states is considered jointly. Naturally, the joint description is also easily carried out by the symmetry methods. This will be shown on the example of the first triad of vibrational states in the ground electronic state of the H2 O molecule. The H2 O molecule has already been discussed in Section 8.1. For the chosen binding of the elements of the point group C2v to the equilibrium configuration, all vibrational excitations refer to the symmetry types A1 and B1 . The first triad includes the vibrational levels (100) and (020) of symmetry A1 and the vibrational level (001) of symmetry B1 . The energy difference between these levels is small compared with the average energy of the triad. Therefore, it can be assumed, although this is not necessary for the further description, that the observed quasidegeneracy is due to the fact that a wider symmetry group compared with the group C2v is present for a certain main contribution to the internal dynamics. That is, when only the main contribution is taken into account, the triad collapses to the only degenerate level. To describe the quasidegeneracy, we introduce the operator of coordinate spin l. Its components implement a three-dimensional representation of Lie algebra (7.27) in the basis of unit vectors j0i and j ˙ 1i. We assume that l3 in this basis is a diagonal operator with the eigenvalues 0 and ˙1, and l˙ D l1 ˙ il2 are the raising and lowering operators. First of all, the levels of the first triad should be related with the basis unit vectors. By definition of the operator lC , we have p p (8.50) lC j0i D 2j1i; lC j  1i D 2j0i: Acting by an arbitrary transformation R of the group C2v on equations (8.50) from the left, we obtain p p RlC RRj0i D 2Rj1i; RlC RRj  1i D 2Rj0i; (8.51) where it is taken into account that R2 D 1. Equations (8.51) are compatible for the triad if the unit vectors j ˙ 1i belong to the A1 type and the unit vector j0i, to the B1 type. In this case, lC is an operator of the B1 type. Then we easily see that l is also an

116

Chapter 8 Internal dynamics of rigid molecules

operator of the B1 type and l3 is an operator of the A1 type. Since the representations of the group C2v are real, we can write T j0i D j0i;

T j1i D j1i;

T j  1i D j  1i:

(8.52)

Clearly, this space is nonstandard, although it has an odd dimension. Following the analysis in Section 7.5 and using equation (8.52) as the basis, we find that l3 and l1 are t-even operators while l2 is a t-odd operator. The effective operators of physical quantities for the triad may include self-adjoint vibrational operators whose total power over the components of l is not higher than two. Symmetry properties of the complete set of these operators are given in Table 8.12. Table 8.12. Symmetry properties of the complete set of self-adjoint vibrational operators for the first triad of the H2 O molecule. C2v A1 B1

t -even Ie ; l32; l3 2 lC C l2

lC C l Œl3 ; lC C l C

t -odd 2 i.lC  l2 /

i.lC  l / iŒl3 ; lC  l C

Note. Π; C is the anticommutator.

A classification of rotational levels for both types of vibrational excitation in the ground electronic state of the H2 O molecule is shown in Figure 8.2. The actual structure of the energy levels is described by the group D2 . Symmetry properties of the vibrational variables in this group follow from those in the group C2v after the replacement .y/ .x/ v.xz/ ! v.xz/ i D C2 ; N v.yz/ ! N v.yz/ i D C2 : (8.53) For the vibrational quantities, this leads to the following correspondence between the irreducible representations of the group C2v and the group D2 that is isomorphic with it: (8.54) A1 $ A; B1 $ B2 : In particular, the unit vectors j ˙ 1i belong to the A type and the unit vector j0i, to the B2 type of the group D2 . Decomposition of the basis vibration-rotational functions into irreducible representations of this group is given in Table 8.13. The effective vibration-rotational Hamiltonian of the triad transforms according to the representation A of the group D2 . Hence, the following two vibration-rotational structures are possible for this Hamiltonian: .A/vib  .A/rot;

.B2 /vib  .B2 /rot :

(8.55)

117

Section 8.4 Description of quasidegenerate vibronic states

Table 8.13. Decomposition of the basis vibration-rotational functions for the first triad of the H2 O molecule into irreducible representations of the D2 group. D2 A

B1

Functions j ˙ 1iA j0iB2 j ˙ 1iB1 j0iB3

D2 B2

B3

Functions j ˙ 1iB2 j0iA j ˙ 1iB3 j0iB1

Note. The rotational parts of the basis functions are denoted by irreducible representations of the group D2 .

Selecting combinations of the components of the angular momentum in the MCS of symmetry types A and B2 in the group D2 and taking into account that the effective Hamiltonian is t-even, we obtain the following general expression for it: H D

1 X  .A/ .A/ .B2 / .B2 /  H2n C H2nC3 : C H2nC2 C H2nC1

(8.56)

nD0

On the right-hand side, the superscript of the contribution to the Hamiltonian determines the type of irreducible representation of the group D2 , by which the combinations of components of the angular momentum in the MCS are transformed, and the subscript specifies the total degree of the combinations of these components. These contributions have the form X .A/ 2t H2n D cO2p;2s;2t J 2p Jz2s .JC C J2t /; (8.57) p;s;t .A/ H2nC3

D

X

2t C2 i dO2p;2sC1;2t C2 J 2p Jz2sC1 .JC  J2t C2 /;

(8.58)

2t C1 fO2p;2sC1;2t C1 J 2p Jz2sC1 .JC C J2t C1 /;

(8.59)

2t C1 i gO 2p;2s;2t C1 J 2p Jz2s .JC  J2t C1 /;

(8.60)

p;s;t .B /

2 D H2nC2

X

p;s;t .B /

2 D H2nC1

X

p;s;t

where p Cs Ct D n for a given of n. Due to allowance for the vibrational quasidegeneracy, the parameters c; O fO and dO ; gO are t-even and t-odd spin operators, respectively: 2 cO D c .1/ I C c .2/ l32 C c .3/ l3 C c .4/ .lC C l2 /; 2  l2 /; dO D id .1/ .lC

fO D f .1/.lC C l / C f .2/ Œl3 ; lC C l C ; gO D ig .1/ .lC  l / C ig .2/ Œl3 ; lC  l C :

(8.61)

118

Chapter 8 Internal dynamics of rigid molecules

Here ; c .k/ ; d .k/ ; f .k/ , and g .k/ are the real spectroscopic constants (the subscripts are omitted). We emphasize that the Hamiltonian (8.56) automatically includes all types of resonances in this triad in an arbitrary order of perturbation theory. Formulation in operator form for both the rotational and vibrational types of motion is another important advantage of this approach. Recall that in the standard approach to the description of accidental resonances, the Hamiltonian is written in matrix form for the vibrational type of motion [66], which substantially complicates manipulations with it. In particular, we are talking about the solution of the Hamiltonian reduction problem. The effective operator of the electric dipole moment, which belongs to the representation B1 of the group D2 , is also easy to construct. Its component ˛ on the ˛ axis of the FCS is written in the form .i / ˛ D ˛i .dO .i / C dOk.i / Jk C dOkl Jk Jl C    /;

(8.62)

where the parameters dO .i / are vibrational operators. Therefore, two vibration-rotational structures are possible for the electric dipole moment: .A/vib  .B1 /rot ;

.B2 /vib  .B3 /rot :

(8.63)

As a consequence, the full expression for ˛ has the form ˛ D ˛z Œ.A/vib A C .B2 /vibB2  C ˛y Œ.A/vib B3 C .B2 /vib B1  C ˛x Œ.A/vib B2 C .B2 /vibA:

(8.64)

This notation means that the ˛i -proportional terms of the effective electric dipole moment are combinations of the components of the angular momentum in the MCS of symmetries A, B1 , B2 , or B3 in the group D2 , and the parameters before these combinations are linearly dependent on the vibrational operators correlating with them through symmetry. Equation (8.64) is not difficult to unwind in full. It should only be taken into account that the effective dipole moment is t-even. The main contribution comes from terms that do not depend on the components of the angular momentum. Keeping only these terms, we obtain 2 ˛ D ˛z Œd1 I C d2 l32 C d3 l3 C d4 .lC C l2 /

C ˛x Œd5 .lC C l / C d6 Œl3 ; lC C l C :

(8.65)

Equation (8.65) contains six independent constants, as it should be. These constants determine the main contribution for purely rotational transitions in the three vibrational states and for the three branches of vibration-rotational transitions within the triad. We also note that the contributions of the odd total power over the components of the angular momentum in equation (8.64) are due to the existence of resonant interactions in the triad. Naturally, the condition of closely spaced vibrational states has nowhere been used in the qualitative construction of effective operators for the triad. When accidental

Section 8.5 Conclusions

119

resonances are small, the Hamiltonian (8.56) can be reduced to diagonal form for the vibrational operators by using a unitary transformation. As a result, each vibrational state of the triad is described as isolated based on the Hamiltonian of form (8.7). Formally, such a description can be done for any value of the accidental resonances. However, as was mentioned in the beginning of this section, great difficulties arise with the summation of the perturbation series for the rotational Hamiltonian. We also note that a joint description is sometimes necessary for the weakly coupled vibronic states. For example, this is needed to construct an effective operator of electric dipole moment responsible for the transitions between these states. In Chapter 18, we consider a general approach to the construction of algebraic models (with an algebraic calculation scheme for both the position of the levels in the energy spectrum and the intensities of the transitions between them) for the description of an arbitrary set of vibrational states. Finally, we emphasize that the analysis is much more complicated if we need to describe jointly the states related to different isomers of the molecule. This variant is discussed in Chapters 16 and 17.

8.5 Conclusions We have formulated some general conclusions based on the content of this chapter. 1. In a qualitative description of the internal dynamics of rigid molecules, two matchings are used to obtain a chain of bound symmetry groups of various models. Physically, these matchings are completely uniform for the entire class of rigid molecules, although their technical implementation of course depends on the type of a concrete molecule. Importantly, the qualitative description is internally closed since it is based solely on the symmetry properties. This is largely possible because the approach based on the symmetry group chain does not require the explicit form of the basis wave functions (or wave functions of the zero approximation). We also note that the situation is diametrically opposite in the routine approach based on the CNPI group [15, 16]. This group is defined as the direct product of the permutation group of identical nuclei and the inversion group of spatial coordinates of all particles of a molecule (all nuclei and electrons) with respect to its center of mass. To simplify the analysis, in view of the limited accuracy of experimental data, only the so-called feasible elements (corresponding to the molecule transformations in which the energy barrier is zero or sufficiently small compared with the dissociation energy), which form a molecular symmetry group, or an MS group, are usually selected from the CNPI group. For a rigid molecule, the MS group includes only transformations equivalent to the rotation of the molecule as a whole. It is believed that the CNPI group (in most cases, the MS group) contains all the symmetry elements necessary to describe the problem of a discrete spectrum of the molecule. To construct a classification of energy levels, we should first write out explicitly at least the approximate wave functions of stationary states, taking

120

Chapter 8 Internal dynamics of rigid molecules

into account all required types of motion, and then specify the action of the elements of the CNPI or MS group on the coordinates of the configuration space of these motions and, finally, calculate the action of the elements of the group on the approximate wave functions. Thus, the symmetry properties in such an approach are knowingly “secondary” since their use is based on knowledge of approximate solutions for the wave functions. The consideration of planar or linear molecules faces major philosophical difficulties, and they must be separated from the general class of rigid molecules. For planar molecules, the problem is that the geometric symmetry element corresponding to the reflection in the plane of a molecule cannot be specified as a permutation of its identical nuclei. Hence, such a reflection is specified by using the spatial inversion operation since this is the only possible option. However, the spatial inversion is an external symmetry operation related to the properties of space and is not applicable to the internal symmetry of the molecule. In the case of the linear molecules it is still more serious. For example, for a linear molecule without identical nuclei the CNPI group is 1  Ci , but it cannot, even formally, describe a very wide geometric symmetry defined by the continuous point group C1 v . Hence, an extremely artificial group, the so-called extended CNPI group, 1"  Ci is introduced, where 1" is the permutation group of a single element, which is continuous with respect to some parameter " (?!). Such physically incorrect methods of analysis are effective only if we know in advance which qualitative properties of the approximate wave functions should be obtained (see also Section 12.7). 2. Any electron-vibration-rotational level of a rigid molecule corresponding to a degenerate irreducible representation of its point group corresponds to a single physically observable state. This, at first glance, very strange and thus often ignored statement has a simple, although purely quantum, rationale. The fact is that each transformation of a point group corresponds to the permutation of identical nuclei. As a consequence, the spin-coordinate wave function should satisfy the fundamental requirements of symmetry for permutations of these nuclei. Therefore, as was explained in Section 8.2, different wave functions belonging to a degenerate coordinate multiplet are included in one regular spin-coordinate function. Rather special in this sense are the linear molecules for which the degenerate coordinate multiplets are impossible because of the homomorphic mapping of the point group onto the permutation group of identical nuclei. From the fact that the degenerate multiplet corresponds to one physically observable state a very important consequence follows. First we consider a purely model problem of changes in the spectrum of a rigid molecule occurring due to the distortion of its internal dynamics connected with the symmetry breaking (e.g., the point group decreases from Td to C3v /. We assume that the distortion is so weak that the position of the levels in the spectrum can be considered virtually unchanged. However, even a weak distortion leads to a partial destruction of the coordinate degeneracy. In place of, say, a triply degenerate level of the group Td , there appear one doubly degenerate and one nondegenerate

Section 8.5 Conclusions

121

level of the group C3v , which almost coincide in energy. Therefore, the number of different physically observable states changes abruptly. We can say that with the lowering of symmetry the efficiency of using the coordinate wave functions for the formation of physically observable states increases. As a result, a variety of characteristics of the internal dynamics of a molecule (including such an integral characteristic as statistical sum [60]) also changes abruptly. Consequence: the transition between descriptions with different point groups is not continuous. Hence, in general, it is erroneous to derive a description with higher symmetry from the existing description with lower symmetry by imposing restraint conditions on the system parameters. In other words, the description with lower symmetry does not include a description with higher symmetry as a particular case. A continuous transition takes place only for the smoothly varying characteristics. These include the position of energy levels (except for the cases of the occurrence of singularities, for example upon passage from the nonlinear to a linear molecule), but the intensity of the transitions between the levels abruptly changes. 3. Most of the nonlinear molecules considered in this chapter are typical nonrigid molecules. This means that in a given electronic state, even with excitations much less than the dissociation energy, one should take into account the so-called nonrigid transitions, i.e., transitions between different minima of the effective potential of nuclear interaction. Therefore, their qualitative analysis clearly has a rather narrow range of applicability. To extend it, the chain of groups should be supplemented by another type of internal geometric symmetry group, which includes not only the elements that characterize the different equilibrium configurations of the molecule, but also the elements that define the nonrigid transitions. Such groups will be called the extended point groups. Generally speaking, the latter contain not only the transformations of the molecule as a whole and, hence, are not subgroups of the group O3 of orthogonal transformations of three-dimensional space (that is, they do not satisfy the mathematical definition of the point group given in Section 4.2). At the same time, transformations of extended point groups also preserve the position of at least one point of three-dimensional space, namely, the position of the center of mass of the molecule. It is important to understand that the extended point groups are no longer defined by symmetry transformations as a whole of the equilibrium configurations of nonrigid molecules, and their finding may be a very nontrivial problem. In addition, they are often not among the tabulated groups. 4. Isotropy of space is described by the external symmetry group R3 . As a consequence, good quantum numbers are the quantum number J of the square of the total angular momentum of internal motion of the quantum system and the quantum number M of the projection of this momentum on one of the FCS axes. The states with a given value of J are transformed according to the representation D .J / of the external group R3 . Since the electric dipole moment refers to the representa-

122

Chapter 8 Internal dynamics of rigid molecules

tion D .1/ of this group, we obtain the following additional electric dipole selection rules:

J D 0; ˙1: Naturally, these rules are the same for all the isolated quantum systems. 5. Effective operators of physical quantities are described in this book in the form of an infinite Taylor series of rotational distortion. However, it should be borne in mind that a vast number of molecules have a finite number of levels in their bound electronic states. Therefore, only a finite number of the first terms are independent in these series. We also note that other types of the rotational distortion expansion can be used as well.

Chapter 9

Molecules with torsional transitions of the exchange type 9.1 Extended point groups. Intermediate configuration In this and the next two chapters we consider the molecules in the chosen electronic state of which it is necessary to allow for nonrigid transitions between several energetically equivalent minima of the effective interaction potential between nuclei. It is clear that the equilibrium configurations in such minima belong to the same point group G0 . It was mentioned in Section 8.5 that when passing from the analysis of a rigid molecule localized in one of such minima to the analysis of a nonrigid molecule it is necessary to supplement the chain with an internal extended point group G, which characterizes both the local motions in the minima and the transitions between them. This immediately implies that the group G includes G0 as its subgroup. From a qualitative point of view, the elements extending the group G0 to the group G can conveniently be divided into exchange and nonexchange types. Under the action of the elements of the exchange type, the effective potential of nuclear interaction passes into itself, and the spatial position of the equilibrium configuration is preserved. Hence, each of such elements can be associated with the permutation of identical nuclei in a force field that is invariant with respect to it. In this sense, the case is completely similar to that for the point group of a rigid molecule. If all the elements of extension are of the exchange type, then the group G is a dynamic invariant symmetry group of the rigorous model of the coordinate motion in a nonrigid molecule. The elements of the nonexchange type are characterized exactly by the fact that they are not symmetry transformations of the force field of the nuclei and thus change the spatial position of the equilibrium configuration. Naturally, none of these elements can be associated with the permutation of identical nuclei. This means that the belonging of the Hamiltonian of the total coordinate motion to the totally symmetric coordinate Young diagram of the permutation group of identical nuclei implies that this Hamiltonian is invariant only with respect to the elements of the exchange type which form a subgroup GH of the G group. It is clear that GH includes G0 as a subgroup, and the group G itself becomes a dynamic noninvariant group. This chapter discusses the analysis of nonrigid molecules with torsional transitions of the exchange type. The role of the extended point group for the internal dynamics of a nonrigid molecule is similar to that of the point group for a rigid molecule since they both characterize the geometric structuring of the total coordinate motion for bound states. However, their practical significance is incomparable since the symmetry properties of the basis unit vectors of the function space of internal motions, as well as of the ef-

124

Chapter 9 Molecules with torsional transitions of the exchange type

fective operators of physical quantities (including the effective Hamiltonian) specified in this space, are most often a priori unknown for the nonrigid molecules. However, these properties are of principle for a description by the methods of perturbation theory. Since the internal geometric group of a nonrigid molecule is no longer determined by transformations of the equilibrium configurations as a whole, the construction of this group can be quite a nontrivial problem. Information on the geometry of nonrigid transitions, which can conveniently be specified in terms of symmetry of the so-called intermediate configuration, is usually used. Consider this concept with specific examples of constructing an extended point group. It is well known that the molecule of ethane C2 H6 , whose equilibrium configuration is shown in Figure 8.8, has an internal rotation of two identical tops CH3 about the chemical bonding of C atoms, which penetrates a low potential barrier [72]. Each internal top rotated by angles 2/3 and 4/3 from the equilibrium position comes again to an equivalent equilibrium position. All these equilibrium positions are taken into account by the c3v  c3 group, where two groups c3v are determined by the symmetry of two CH3 tops. However, the c3v  c3v group has no elements that allow for the tops identity. It is exactly why this group does not include the point group D3d D C3v CI of the ethane molecule as a subgroup and cannot be regarded as an extended point group allowing for the internal rotation. It is necessary to add to it an operation I, which does not commute with an arbitrary element p  q of the c3v  c3 group. More precisely, .p  q/I D I.q  p/: (9.1) Of course, this relation satisfies the commutation requirements of operation I with all transformations of the molecule as a whole in the c3v  c3v group, that is, with transformations of the form p  p. From equation (9.1) it follows immediately that c3v  c3v is an invariant subgroup of the G72 group obtained by extension of the c3v  c3v group using the element I. Therefore, G72 D .c3v  c3v / ^ CI :

(9.2)

Extended point group (9.2) is not a tabulated group. Division of the elements into classes and the table of characters for this group are shown in Table 9.1. All elements of this group are given in terms of the equilibrium configuration. That is, in this case, the intermediate configuration coincides with the equilibrium one, and the extended point group is different from the point group in that it includes not only the transformations of the molecule as a whole. A characteristic feature of the coincidence of these two configurations is the fact that the identical nuclei participating in the nonrigid motion are located in the equilibrium configuration in a geometrically equivalent way. Since the intermediate configuration is introduced to define the geometry of nonrigid motion, it should not necessarily correspond to the top of the barrier for this motion. However, as will be seen later, such a correspondence is sometimes the case. It is clear that the elements of the G72 group belong to the exchange type and do not affect the spatial position of the equilibrium configuration.

125

Section 9.1 Extended point groups. Intermediate configuration

Table 9.1. Division of the elements into classes and the table of characters of the extended point group G72 of the ethane molecule. .E  E/I

E E

{16 }

.c3  c32 /I; .c32  c3 /I

{23 } {13 3}

E  c3k ; c3k  E

{32 }

c3k  c3l

.p/

.v

.p/

 v /I

.E  c3k /I; .c3k  E/I .c3k  c3k /I

{6} {14 2}

E  v.p/ ; v.p/  E

.v.p/  v.t/ /I; p ¤ t

{12 22 }

v.p/  v.t/

{123}

c3k  v.p/ ; v.p/  c3k

.E  v.p/ /I; .v.p/  E/I

{24}

.c3k  v.p/ /I; .v.p/  c3k /I

G72 D .c3v  c3v / ^ CI {16 } {13 3} {32 } {14 2} {12 22 } {123} {23 } {6}

{24}

Class order

1

4

4

6

9

12

6

12

18

A1 A2 A3 A4 E T1 T2 T3 T4

1 1 1 1 2 4 4 4 4

1 1 1 1 2 1 1 –2 –2

1 1 1 1 2 –2 –2 1 1

1 1 –1 –1 0 –2 2 0 0

1 1 1 1 –2 0 0 0 0

1 1 –1 –1 0 1 –1 0 0

1 –1 1 –1 0 0 0 –2 2

1 –1 1 –1 0 0 0 1 –1

1 –1 –1 1 0 0 0 0 0

Note. The indices k, l D 1; 2 and p, t D 1; 2; 3. Designation of classes is similar to that for the 6 group.

As another example, we consider the ethanol molecule CH3 OH, which has a pronounced torsional motion of the methyl top CH3 . The existence of this motion is entirely due to the distortion of the equilibrium configuration of the molecule from the most symmetric with the point group C3v to the actually observed configuration (shown in Figure 9.1) with the point group CS [52]. As a consequence, there appear three energetically equivalent configurations related by rotations of the CH3 top through a low potential barrier [72]. These transitions belong to the exchange type, and the extended point group has the form G12 D c3v  cS ;

(9.3)

126

Chapter 9 Molecules with torsional transitions of the exchange type

Figure 9.1. Equilibrium configuration of the methanol molecule CH3 OH (the nuclei of H3 , C, O, and H4 and the C3 axis of the methyl group lie in the symmetry plane of the point group CS /.

where the groups c3v and cS characterize the structures of CH3 and COH. Torsional motion is determined by rotations through angles 2/3 and 4/3 about the c3 axis. However, the three nuclei of H that exchange places do not occupy equivalent positions in the equilibrium configuration. More precisely, the position of the H nucleus in the symmetry plane of the molecule is not equivalent to the position of the H nuclei outside this plane, which indicates a slight distortion of the regular pyramidal structure of the CH3 top. Hence, the c3v group refers to some intermediate unstable configuration, through which the symmetry of the problem is defined with allowance for the three minima. This intermediate configuration allows for their equivalence. Thus, it can have a higher geometric symmetry compared with the equilibrium configuration which allows for only one minimum. In other words, the intermediate configuration is characterized by the fact that the identical nuclei participating in the nonrigid motion are located in this configuration equivalently with respect to the geometric operations that interchange these nuclei. If the equilibrium configuration does not satisfy this condition, then the intermediate configuration does not coincide with it and has a higher geometric symmetry. The molecules of complex hydrides of light alkali metals LBH4 .L D Li; Na/, in which the LC cation readily changes its position relative to the BH 4 anion, are an interesting example. In the equilibrium configuration of the LBH4 molecule [32], which is shown in Figure 9.2, the LC cation is above the middle of the face of the C tetrahedral anion BH 4 deformed by it. Since the position of the L cation is coordinated by three atoms of H, such a configuration belonging to the point group C3v is called the tridentate configuration (or t-configuration). There are four independent energetically equivalent equilibrium t-configurations corresponding to the position of the cation above each of the four tetrahedral faces. Analysis of the experimental data shows that the cation transits between these positions, migrating around the entire

Section 9.1 Extended point groups. Intermediate configuration

127

Figure 9.2. Equilibrium configuration of the LBH4 molecule.

space of the anion. From calculations by the methods of quantum chemistry it follows [32] that the transitions occur in two channels. In one channel, the top of the barrier corresponds to the bidentate configuration (or b-configuration) of the symmetry of C2v (the LC cation is above the middle of the edge of the tetrahedral anion BH 4, where its position is coordinated by two atoms of H/. In another channel, the top of the barrier corresponds to the monodentate, or m-configuration of symmetry of C3v (the LC cation is above the vertex of the tetrahedral anion BH  4 , where its position is coordinated by one atom of H). The dominant channel is t ! b ! t , and the barrier height is small (about 4 kcal/mol for LiBH4 and 2 kcal/mol for NaBH4 ). Of course, the transitions should not be represented as the migration of the cation on a fixed anion since the position of the center of mass of the molecule changes in this case. In reality, such motions correspond to rotations of the anion about its center of mass for a fixed cation, which leads to an extended point group of the form c3v  Td , where the groups c3v and Td characterize the cation and the anion, respectively. Here, the group c3v is required only to ensure that the extended point group includes the point group C3v as a subgroup. In addition, the anion in the equilibrium configuration is deformed by the cation, and the group Td refers to the intermediate unstable configuration of the molecule with a geometrically equivalent position of the H nuclei in the anion. The high symmetry of this configuration means that the equilibrium configurations bound by rearrangements (and operations in the Td group) are equivalent. As a final example, we consider the molecules of dimethyl ether (CH3 )2 O and acetone (CH3 )2 CO. In the ground electronic state, their equilibrium configurations are of .z/ .xz/ .yz/ the same type (Figure 9.3) and belong to the point group C2v D .E; C2 ; v ; N v / [52, 84]. In both molecules, there occurs the torsional motion of two identical methyl tops, whose axes do not coincide. For the acetone molecule, the height of the barrier of this motion is significantly lower ( 250 cm1 [49]). Torsional transitions

128

Chapter 9 Molecules with torsional transitions of the exchange type

between nine minima of the effective potential are taken into account of the group Qr D c3v  cS  c3v , where two groups c3v belong to two CH3 structures, and the group cS to the O structure for the dimethyl ether molecule and the CO structure for the acetone molecule.

Figure 9.3. Equilibrium configuration of the acetone molecule (CH3 )2 CO (the nuclei of H1 , C1 , O, C3 , C2 , and H4 lie in the yz plane).

The group cS was introduced only to ensure that the group Qr includes the trans.yz/ of the molecule as a whole. Since transformations of the cS group formation N v correspond to the identical permutation of nuclei (O or C and O), this group can, without loss of generality, be omitted, assuming that the Qr includes the transformation .yz/ .yz/ .yz/ in the form v  v . It is easy to show that N v .z/

.z/

C2 .p  q/ D .q  p/C2 ;

(9.4)

where p  q is an arbitrary element of the group c3v  c3v . From equation (9.4) it follows immediately that Qr is an invariant subgroup of the Q72 group obtained by .z/ addition of the element C2 to the elements of the Qr group. As a result, Q72 D .c3v  c3v / ^ C2 ; .z/

(9.5)

where C2 D .E; C2 /. Since the group Q72 includes the C2v group as a subgroup, Q72 is an extended point group in this problem. It is easy to show that the group Q72 is isomorphic with the group G72 of the ethane molecule, which is defined by equation (9.2) (division of the group Q72 elements into classes follows from Table 9.1 with .z/ I replaced by C2 /. Note that unlike the ethane molecule, now the H nuclei in each methyl top in the equilibrium configuration occupy the nonequivalent positions. More .yz/ precisely, the position of the nucleus in the N v plane is not equivalent to the position of the nuclei outside this plane, which is indicative of a slight distortion of the regular pyramidal structure of the CH3 top. Hence, the groups c3v belong to some unstable configuration, by which the symmetry of the problem is specified with allowance for

129

Section 9.2 Methanol molecule CH3 OH

all the minima associated with the torsional motion. This intermediate configuration takes into account that these minima are equivalent, and thus have a higher geometric symmetry than the geometric symmetry of the equilibrium configuration which allows for only one minimum. We emphasize that although the intermediate configurations of the molecules CH3 OH, LBH4 , (CH3 )2 O and (CH3 )2 CO differ from the equilibrium configuration, they do not correspond to the top of the barrier for the nonrigid motion. To conclude this section, we note that torsional motions of the exchange type are perhaps the most widespread in molecules and their complexes. In addition, the capabilities of the proposed methods in such a broad area as analysis of the molecular systems with more than two identical torsional tops are demonstrated in Section 20.1 using the trimethylborane molecule B(CH3 )3 as a classic example.

9.2 Methanol molecule CH3 OH As a permutation group of identical H nuclei of the CH3 OH molecule, it suffices to use 3  1 , where 3 specifies the permutation symmetry of a methyl top. A matching of the 3  1 group with the point group CS D .E;  .yz/ / of the equilibrium configuration of this molecule shown in Figure 9.1 is determined by an isomorphic mapping of the point group onto a subgroup of the permutation group (correlations between the irreducible representations of symmetry groups in different models are given in Table 9.2). Table 9.2. Correlation table for the CH3 OH molecule. 3  1

CS

D2

CS  Ci

CS

c3v  cS

Œ3  Œ1 Œ21  Œ1

A0 A0 C A00

A, B3 B1 , B2

A0.C/ ; A00./ A0./ ; A00.C/

A0 A00

A1 A0 ; EA0 A2 A0 ; EA0

Œ13   Œ1

A00

3

c3v

H6

G12  Ci

C2

H6

Œ3 [21]

A1 E

A1 A2

.A1 A0 /.C/ ; .A2 A0 /./ .A1 A0 /./ ; .A2 A0 /.C/

A B

A1 , E A2 , E

[13 ]

A2

E

.E  A0/.˙/

Note. Notation for the irreducible representation of the H6 group is similar to that for the D3 group that is isomorphic with it.

For the allowed coordinate multiplets we obtain 4A0 ;

.8 C 4/A00 ;

(9.6)

130

Chapter 9 Molecules with torsional transitions of the exchange type

where the nuclear statistical weight allows only for the spins of the H nuclei. Types of symmetry of the zero-approximation wave functions for the rigid molecule are matched with the types of symmetry of the allowed multiplets formed on their basis: .mult /CS Ci D .el /CS  .vib/CS  .rot.-in/CS Ci ;

(9.7)

6

where the admissible representations rot.-in for a given rotational representation of the D2 group of a rigid asymmetric top are determined from the relation of the groups .x/ D2 and CS  Ci through their common subgroup C2 D .E; C2 D  .yz/ i/. As a result, we have a classification of the energy levels of a rigid molecule of methanol in any vibronic state. This classification for el  vib D A0 is shown in the first two columns of Figure 9.4. Since the inversion doublets are not split, the actual structure of the levels is described by the rotational subgroup C2 of the CS  Ci group.

Figure 9.4. Classification of the torsion-rotational levels of the CH3 OH molecule in the type A0 vibronic state.

For the nonrigid molecule CH3 OH, an extended point group, which takes into account the torsional motion of the methyl top CH3 , needs to be introduced into a chain between the groups 3  1 and CS . According to equation (9.3), this group has the

131

Section 9.2 Methanol molecule CH3 OH

form G12 D c3v  cS . Matching it with the 3  1 group, we obtain allowed irreducible representations of the geometric group for a complete coordinate wave function with the torsional motion taken into account or, in other words, we obtain allowed coordinate multiplets of a nonrigid molecule. Matching is divided into an isomorphic mapping of c3v onto 3 and a homomorphic mapping of cS onto 1 . As a result, the desired allowed coordinate multiplets can be written as 8.A2  A0 /;

4.E  A0 /:

(9.8)

Note that the multiplet A1 A0 has zero nuclear statistical weight since the spin Young diagram [13 ] of the 3 group is forbidden, and the representations   A00 are not realized as multiplets because of the planarity of the COH structures. Operation i commutes, as previously, with all internal geometric elements of symmetry, and each multiplet in equation (9.8) is characterized by the signs (˙). Matching the geometrical groups G12 and CS , we obtain a picture of split levels of a rigid molecule with allowance for the torsional motion, which is shown for el  vib D A0 in Figure 9.4. Since the torsional motion corresponds to the rotational transformations c31  E and c32 E in the G12 group, the inversion doublets remain unsplit, and the actual structure of the levels is described by the complete rotational subgroup H6 of the G12 Ci group. Note that for the one-dimensional states of a nonrigid molecule, one of the levels of the inversion doublets is absent since the multiplet A1  A0 is forbidden. The rotational group H6 that is isomorphic with the group D3 can be represented as H6 D H3 ^ C 2 ;

(9.9)

where the group H3 D .EE; c31 E; c32 E/ includes the torsional motion operations and the group C2 , rotations of the molecule as a whole. Indeed, the group H3 is an invariant subgroup of H6 since .x/

.x/

C2 .c31  E/C2

D  .yz/ i.c31  E/ .yz/ i D .c32  E/:

(9.10)

The complete basis set of torsional unit vectors j0i and j ˙ 1i is determined by three irreducible representations of the H3 group that is isomorphic with the C3 group, such that .c31  E/j0i D j0i;

.c31  E/j1i D "j1i;

.c31  E/j  1i D "2 j  1i;

(9.11)

where " D exp.2 i=3/. Using equation (9.10), it can easily be shown that .c31  E/C2.x/ j0i D C2.x/j0i; .x/

.c31  E/C2.x/j1i D "2 C2.x/j1i;

.x/

.c31  E/C2 j  1i D "C2 j  1i:

(9.12)

From a comparison of equations (9.12) and (9.11) we obtain .x/

C2 j0i D aj0i;

.x/

C2 j1i D bj  1i;

(9.13)

132

Chapter 9 Molecules with torsional transitions of the exchange type

where a2 D b2 D 1. That is, by virtue of the properties of the semidirect product, the action of the elements of a factor group is also specified in the space of basis unit vectors of the invariant subgroup. It follows from the classification that a D 1 should be chosen for el  vib D A0 , which will be clear after the construction of the basis torsion-rotational functions. The value of b can always be chosen equal to 1 since otherwise it suffices to change the sign of one of the unit vectors j ˙ 1i. Since the unit vectors j0i and j ˙ 1i belong, respectively, to the real and a pair of complex-conjugate representations of the H3 group, after the time reversal (see Section 7.5) we have T j0i D cj0i;

T j1i D d j1i;

(9.14)

where jcj2 D jd j2 D 1. The squares of the moduli appear due to the antilinear operator T , which makes it possible to take c D d D 1 with the preservation of the chosen values of a and b in equation (9.13). From transformations (9.11) and (9.13) we find that the unit vector j0i implements the representation A1 of the H6 group and the unit vectors j˙1i, the representation E (notation for the group H6 representations is similar to that for the D3 group). As concerns the rotational functions, only operations of the rotation of the molecule as a whole from the C2 subgroup act on them in the H6 group. As a result, the rotational functions belonging to the representations A and B of the C2 group are transformed, respectively, according to the representations A1 and A2 of the H6 group. It is now easy to decompose the torsion-rotational basis functions into irreducible representations of the H6 group (see Table 9.3). In this decomposition, the function j0i0; 0i belongs to the representation A1 , which is consistent with the classification for el vib D A0 (for a D –1, this function belongs to the representation A2 , which corresponds to the classification for el  vib D A00 /. The torsional parts of the operators of physical quantities are constructed on the basis of the operator of coordinate spin e with quantum number of the operator of the spin squared equal to unity. The unit vectors j0i and j ˙ 1i correspond to the eigenvectors of the operator e3 with the eigenvalues 0 and ˙1. Using the symmetry properties of the torsional unit vectors in transformations of the H6 group and in operation T, we find that the component e3 is a t-odd operator which transforms according to the representation A2 of the H6 group, while the components e1 and e2 are t-even operators which transform according to the representation E. A complete set of independent self-adjoint operators in three-dimensional torsional space includes the products of the components e with a total power not higher than two. The symmetry properties of such a set of operators are given in Table 9.4. Obviously, the effective operator of any physical quantity that characterizes the internal coordinate motion in the methanol molecule belongs to the coordinate Young diagram [3]  [1] of the 3  1 group. A matching of the groups 3  1 and G12 shows, that such an operator is invariant under transformations of the G12 group. Then, matching the groups G12  Ci and H6 , we obtain, within the framework of the latter, the representations A1 and A2 related, respectively, to the physical quantities that are invariant under the operation of spatial inversion i and change sign during this trans-

133

Section 9.2 Methanol molecule CH3 OH

Table 9.3. Decomposition of the torsion-rotational basis functions for the vibronic state el  vib D A0 of the CH3 OH molecule into irreducible representations of the H6 group. H6

Basis functions

A1

j0ijJ; 0i, J is even p j0i.jJ; ki C !jJ; ki/= 2

A2

E

j0ijJ; 0i, J is odd p j0i.jJ; ki  !jJ; ki/= 2 p 2 E1 D jJ; 0i.j1i C !j  1i/= p E2 D ijJ; 0i.j1i  !j  1i/= 2 p E1 D .j1ijJ; ki C !j  1ijJ; ki/= p 2 E2 D i.j1ijJ; ki  !j  1ijJ; ki/= 2 p 2 E1 D .j1ijJ; ki C !j  1ijJ; ki/= p E2 D i.j1ijJ; ki  !j  1ijJ; ki/= 2

Note. ! D .1/J , and the components of the representation E are specified by the conditions .x/ .x/ C2 E1 D E1 and C2 E2 D E2 . Table 9.4. Complete set of spin operators for describing the torsional motion in the CH3 OH molecule. H6

t-even

t-odd

A1 A2

I; e32 –

– e3

E1 D eC C e E2 D i.eC  e / 2 2 C e E1 D eC 2 2 E2 D i.eC  e /

E1 D i Œe3 ; eC  e C E2 D Œe3 ; eC C e C .x/

Note. The components of the representation E are specified by the conditions C2 E1 D E1 .x/ and C2 E2 D E2 .

formation. In particular, we have a representation A1 for the torsion-rotational Hamiltonian and a representation A2 for the torsion-rotational operator of the electric dipole moment. These effective operators are defined in the space of torsion-rotational basis functions given in Table 9.3. Since the rotational operators implement only the representations A1 and A2 of the H6 group, only two torsion-rotational structures, .A1 /tors  .A1 /rot and .A2 /tors  .A2 /rot ;

(9.15)

134

Chapter 9 Molecules with torsional transitions of the exchange type

are possible for the effective Hamiltonian. Thus, a full expression for this Hamiltonian can be written as H D .I; e32 /  .A; B3 / C .e3 /  .B1 ; B2 /:

(9.16)

This expression means that each term of the Hamiltonian is a certain combination of the angular momentum components in the MCS of the symmetries A and B3 or B1 and B2 in the D2 group (symmetries A and B, respectively, in the C2 group), and the parameters before the combination are linearly dependent on the torsional operators correlating with this combination through symmetry. Since the Hamiltonian is t-even, the rotational combinations of the A and B3 types should be t-even and of the B1 and B2 types, t-odd. Unwinding equation (9.16), we obtain HD

1 X  .A/ .B3 / .B1 / .B2 /  H2n C H2nC2 : C H2nC1 C H2nC1

(9.17)

nD0

On the right-hand side, the superscript of the contribution to the Hamiltonian determines the type of the irreducible representation of the D2 group, by which combinations of the angular momentum components in the MCS are transformed. The subscript specifies the total power of combinations of these components. These contributions have the form X .A/ 2t H2n D cO2p;2s;2t J 2p Jz2s .JC C J2t /; (9.18) p;s;t .B /

3 H2nC2 D

X

2t C1 i cO2p;2sC1;2t C1 J 2p Jz2sC1 .JC  J2t C1 /;

(9.19)

2t dO2p;2sC1;2t J 2p Jz2sC1 .JC C J2t /;

(9.20)

2t C1 i dO2p;2s;2t C1 J 2p Jz2s .JC  J2t C1 /;

(9.21)

p;s;t .B /

1 H2nC1 D

X p;s;t

.B /

2 H2nC1 D

X p;s;t

where p C s C t D n for a given n. Due on account of the torsional motion, the parameters cO and dO are t-even and t-odd spin operators, respectively: cO D c .1/ I C c .2/ e32 ;

dO D de3 ;

(9.22)

where c .k/ and d are the real spectroscopic constants (the subscripts are omitted). Hamiltonian (9.17)–(9.22), represented as a series of perturbations due to the rotational distortion, automatically includes all the torsion-rotational interactions. The operator form of this Hamiltonian for both the rotational and torsional types of motion is one of its important advantages. Moreover, the Hamiltonian does not contain

135

Section 9.3 Ethane molecule C2 H6

in explicit form operators in the space of the internal rotation angle of a methyl top, which radically simplifies the structure of the Hamiltonian. This effect is based on the fact that unlike the usual elementary torsional operators, the elementary operators of the coordinate spin are introduced with allowance for the required number of independent equilibrium configurations and correct symmetry properties for the transitions between them. Consequently, the elementary spin operators are integral in the sense that at the same time they allow for the large-amplitude motion throughout their entire range. The energy matrix for each quantum number J decomposes into a direct sum of matrices corresponding to the different irreducible representations of the H6 group. The application domain of the obtained description is determined by the domain of applicability of the geometrical group G12 , which, in principle, relates to the rigorous coordinate dynamics. However, two comments are warranted here. First, it may be difficult to sum up a series of perturbations for the effective Hamiltonian. If this is due to the presence of accidental resonances (see Section 8.4), then the description can be extended to the entire set of resonance-related vibronic states. A common approach to solving this problem is considered in Chapter 18. In particular, we obtained an algebraic model to describe the total spectrum of a nonrigid molecule of methanol CH3 OH in the ground electronic state. Second, with increasing excitation energy, it may be necessary to allow for other nonrigid transitions as well. For example, there exists one more intermediate configuration for the nonrigid motion corresponding to the maximum point group C3v possible, which in this case plays the role of an extended point group. The barrier for such a motion is entirely due to the distortion of the equilibrium configuration compared with the intermediate one and is about 30 kcal/mol. This value is much larger than the barrier in the already considered variant, in which it is only 1.07 kcal/mol. Selection rules for the electric dipole transitions in a nonrigid molecule of methanol have, in terms of the H6 group, the following simple form: A1 $ A2 ;

E $ E:

(9.23)

Taking into account that only two torsion-rotational structures, namely, .A1 /tors  .A2 /rot and .A2 /tors  .A1 /rot ;

(9.24)

are possible for the effective operator of the electric dipole moment, this operator is also easy to construct (see Section 18.2).

9.3 Ethane molecule C2 H6 The ethane molecule 12 C2 H6 in a rigid approximation was considered in Section 8.2. To allow for the torsional motion of two identical methyl tops relative to the chemical bond of carbon atoms, an extended point group G72 D .c3v  c3v / ^ CI , which was defined in Section 9.1, should be introduced into a chain between the groups 6 and

136

Chapter 9 Molecules with torsional transitions of the exchange type

D3d . A matching of the 6 and G72 groups (the correlations additionally required to describe the torsional motion are given in Table 9.5) yields the following allowed coordinate multiplets of a nonrigid molecule: .5 C 1/A3 ; .7 C 3/A4 ; .5 C 3/T1 ; 3T3 ; 1T4 :

(9.25)

Table 9.5. Additional correlation tables for the nonrigid molecule 12 C2 H6 . 6

G72

H2

B6

B6  D3

G72  Ci

Œ16 

A4

s

A1 , E

A1  A1

A1.C/ ; A3.C/ ; E ./

[23 ]

A 3 C T4

a

A2 , E

4

A 3 C T1

A1  A2

A2./ ; A4./ ; E .C/

[21 ] 2 2

[2 1 ]

A1  E

A4 C T1 C T3

D3d

G72

A1g

A1, A3 , T4

A2g

E, T3

A1u

A 2 , A 4 , T3

A2u

E, T4

Eg

T1 , T2 , T4

Eu

T1 , T2 , T3

./

.C/

T3 ; T4 ./

./

.C/

.C/

A2  A1

A1 ; A3 ; E .C/

A2  A2

A2 ; A4 ; E ./

A2  E

.C/

T3

./

; T4

.˙/

E  A1

T4

E  A2

T3

E E

.˙/

.˙/

T1

.˙/

; T2

The operation of spatial inversion i commutes with all internal geometric symmetry elements, and each multiplet in equation (9.25) is characterized by the signs (˙). In this case, it is clearly seen that the elements i and I enter the group G72  Ci nonsymmetrically. This is not surprising, since these elements belong, respectively, to the external and internal symmetries of the problem. Now matching the geometrical groups G72 and D3d , we obtain a complete picture of the multiplet splittings of a rigid molecule taking into account the internal rotation, which is given in Figure 9.5 for some rotational levels in the vibronic state el  vib D A1g . Since the torsional motion is determined by rotational transformations of the tops in the G72 group, the actual structure of the levels is characterized by the total rotational subgroup of the G72  Ci group, and this subgroup can be written as the following direct product: H36 D B6  D3 :

(9.26)

Section 9.3 Ethane molecule C2 H6

137

Figure 9.5. Classification of the torsion-rotational levels of the 12 C2 H6 molecule in the type A1g vibronic state.

Group B6 can be represented as B3 ^ H2 , where B3 D .E  E; c31  c32 ; c32  c31 /. Such a representation is based on equation (9.1). Irreducible representations of the B6 group are denoted similarly to the group D3 that is isomorphic with it. The group G72 multiplets corresponding to one torsion-rotational level, form unsplit blocks, which is due to the fact the torsional motions mix only part of the independent configurations. The nuclear statistical weight of the torsion-rotational level is equal to the sum weights of the levels of this block. All the splittings of rotational levels described by the group B6 D3 are due to the k-doubling effect, which exists in the model of a rigid molecule, and the internal rotation effect. In the ethane molecule, as well as in any molecule with a linear backbone and two identical tops, there is the problem of separating the torsional and rotational motions. The point is that the molecule as a whole can be rotated about the linear backbone even by means of only the torsional motions. Equation (9.26) suggests a very elegant way of motions separation for the ethane molecule, in which the groups B6 and D3 describe independently the torsional motion and the rotation of the molecule as a whole, respectively. The total torsion-rotational motion is uniquely decomposed into the torsional motion determined by the elements of the B3 group and the rotation of the molecule as a whole. The complete basic set of torsional unit vectors j0i and j ˙ 1i is determined by three irreducible representations of the B3 group that is isomorphic

138

Chapter 9 Molecules with torsional transitions of the exchange type

with the C3 group, so that .c31  c32 /j0i D j0i;

.c31  c32 /j1i D "j1i;

.c31  c32 /j  1i D "2 j  1i;

(9.27)

where " D exp.2 i=3/. By analogy with equations (9.13) and (9.14), it is easy to find that I ij0i D j0i; I ij1i D j  1i; (9.28) T j0i D j0i;

T j1i D j  1i:

(9.29)

The choice of sign in the transformation of the unit vector j0i in equation (9.28) corresponds to the vibronic state el  vib D A1g . As a result, the unit vector j0i belongs to the representation A1 of the B6 group and the unit vectors j ˙ 1i, to the representation E: p p E1 D jsi D .j1i C j  1i/= 2; E2 D jai D .j1i  j  1i/= 2i; (9.30) where the components E1 and E2 are transformed according to the representations s and a of the H2 group. Rotational functions are classified by the group D3 (see Table 8.4), which for el vib D A1g leads to the decomposition, shown in Table 9.6, of the torsion-rotational basis functions into irreducible representations of the B6 D3 group. The function j0ij0; 0i belongs to the representation A1 A1 , which is consistent with the classification for el  vib D A1g (sign reversal in the transformation of the unit vector j0i in equation (9.28) corresponds to the type u vibronic states). The torsional parts of the operators of physical quantities are constructed on the basis of the operator of coordinate spin e with the quantum number of the spin operator square equal to unity. The unit vectors j0i and j˙i correspond to the eigenvectors of the operator e3 with the eigenvalues 0 and ˙1. Using the symmetry properties of the torsional unit vectors in transformations of the B6 group and in the operation T, we find that the component e3 is a t-odd operator, which is transformed by representation A2 of the B6 group, and the components e1 and e2 are t-even operators transformed

Table 9.6. Decomposition of the basis torsion-rotational functions for the vibronic state el  vib D A1g of the C2 H6 molecule into irreducible representations of the B6  D3 group. B6  D3

Basis functions

A1  A1

A1  A2

A1  E

E  A1

E  A2

E E

j0iA1

j0iA2

j0iE1 j0iE2

jsiA1 jaiA1

jsiA2 jaiA2

jsiE1 jsiE2 jaiE1 jaiE2

Note. The rotational parts are specified by irreducible representations of the D3 group.

139

Section 9.3 Ethane molecule C2 H6

by representation E. A complete set of independent self-adjoint operators in threedimensional torsional space includes the products of the components e with a total power not higher than two. Symmetry properties of such a set of operators are similar, with accuracy up to the replacement of the H6 group by the B6 group, to those indicated in Table 9.4. From the chain of symmetry groups it follows that for the effective operators of physical quantities, the representations A1 A1 and A2 A1 belonging, respectively, to the quantities that are invariant under the inversion operation i and change sign during this transformation are realized in the B6  D3 group. As a result, a full expression for the torsion-rotational Hamiltonian can be written as H D .I; e32 /  A1 :

(9.31)

Since the Hamiltonian is t-even, the type A1 rotational combinations of the angular momentum components in the D3 group should also be t-even. Hence, the torsionrotational Hamiltonian fully corresponds, in its rotational structure, to the rotational Hamiltonian of a rigid molecule obtained in Section 8.2. On account of the torsional motion, however, the parameters c become spin operators cO of the form cO D c .1/ I C c .2/ e32 ;

(9.32)

where c .k/ are the real spectroscopic constants (the subscripts are omitted). The energy matrix for a given value of the quantum number J decomposes into a direct sum of the matrices determined by six possible representations of the B6  D3 group for the torsion-rotational basis functions shown in Table 9.6. Since the effective operator of the electric dipole moment transforms according to the irreducible representation A2  A1 of the B6  D3 group, the electric dipole rules of selection in terms of this group have the form A1   $ A2  ;

E   $ E  :

(9.33)

where the irreducible representation  of the D3 group does not change during the transition. The first rule of selection in equation (9.33) does not give transitions in the vibronic state el  vib D A1g , and the second rule of selection leads to the appearance of the forbidden electric dipole transitions. The latter are so named because such transitions are forbidden in the approximation of a rigid molecule of ethane. To construct the torsion-rotational operator of the electric dipole moment, we note that its component ˛ on the ˛ axis of the FCS is written as .i / .i / ˛ D ˛i .dOk Jk C dOklm Jk Jl Jm C    /:

(9.34)

Here, we take into account that the parameters dO are expressed through the spin operators of the A2 type in the B6 group, and the only operator e3 of this type is t-odd. However, since the electric dipole moment is t-even, only t-odd combinations of the

140

Chapter 9 Molecules with torsional transitions of the exchange type

angular momentum components can appear in equation (9.34). The components on the MCS axis of the unit vector ˛ are decomposed into irreducible representations of the D3 group as follows:

˛z ! A2 ;

˛y ! E2 ;

˛x ! E1 :

(9.35)

Taking into account that the electric dipole moment belongs to the representation A1 of the D3 group, equation (9.34) can be rewritten as ˛ D ˛z  A2 C . ˛y  E2 C ˛x  E1 /;

(9.36)

where combinations of the angular momentum components are denoted by the symmetry types of the D3 group, according to which they should be transformed. The parameters before each admissible combination in equation (9.36) are linearly dependent on the spin operator e3 .

9.4 The molecules of complex hydrides LiBH4 and NaBH4 For the molecules of complex hybrids of light alkali metals LBH4 (L D Li, Na), the permutation group of identical nuclei is 4 . Matching this permutation group with the point group C3v of the equilibrium configuration shown in Figure 9.2 (the necessary correlations are given in Table 9.7), we obtain for a rigid molecule the following allowed coordinate multiplets: .5 C 3/A2 ;

.3 C 1/E;

(9.37)

where the nuclear statistical weight allows only for the spins of the H nuclei. Multiplet A1 has a zero statistical weight since the spin Young diagrams Œ212  and Œ14  are forbidden. Symmetry types of the zero-approximation wave functions are matched with the symmetry types of the allowed multiplets formed on their basis: .mult /C3v Ci D .el /C3v  .vib/C3v  .rot.-in/C3v Ci ;

(9.38)

6

where the admissible representations rot.-in for a given rotational representation of the D1 group of a symmetric top are determined from the relation between the groups D1 and C3v  Ci through their common subgroup D3 . As a result, we have a classification of energy levels in an arbitrary vibronic state. For el  vib D A1 , this classification is shown in the first two columns in Figure 9.6. The inversion doublets are not split, and the actual structure of the levels is described by the rotational subgroup D3 . To describe the nonrigid molecule LBH4 , it is necessary to introduce an extended point group allowing for the torsional motion of the BH 4 cation. According to Section 9.1, this group has the form c3v  Td . Of course, a matching of the c3v group

141

Section 9.4 The molecules of complex hydrides LiBH4 and NaBH4 Table 9.7. Correlation tables for the LBH4 molecule. 4

C3v

4

Td

D1

D3

O

Td  Ci

Œ4

A1

Œ4

A1

A1

A1

A1

A1 ; A2

[31]

A1 C E

[31]

F2

A2

A2

A2

A1 ; A2

[22 ]

E

[22 ]

E

E1

E

[212 ]

A2 C E

[212 ]

E2

E

F1

E3

A1 C A2

[14 ]

A2

[14 ]

A2

:::

:::::::::

D3

C3v  Ci .C/

.C/

./

./

.C/

E

E.˙/

F1

F1.C/ ; F2./

F2

F1./ ; F2.C/

C3v

Td

D3

O

A1

A1 ; A2

./

A1

A 1 , F2

A1

A 1 , F2

A2

.C/ ./ A2 ; A1

A2

A 2 , F1

A2

A 2 , F1

E

E.˙/

E

E, F1 , F2

E

E, F1 , F2

with the permutation group 1 of the cation leads to a unique trivial contribution of the form A1 into the multiplet of a nonrigid molecule. This part of the multiplet can be omitted, leaving only the Td group for the further work (see Sections14.2 and 14.4). However, it should be remembered that transformations of the molecule as a whole in this group are only transformations of the C3v group. From a matching of the groups 4 and Td , for the allowed multiplets of a nonrigid molecule we find 5A2 ;

1E;

3F1 ;

(9.39)

and the multiplets A1 and F2 have a zero nuclear statistical weight since the spin Young diagrams Œ14  and Œ212  are forbidden. Matching the groups Td and C3v , we obtain a picture of the multiplet splitting of a point group due to the torsional motion, which is given for a totally symmetric vibronic state in the third column in Figure 9.6. It is clear that the regrouping channels indicated in Section 9.1 correspond to the rotational transformations of the Td group. Therefore, the nonrigid motions belong to the torsional type, and consequently the inversion doublets remain unsplit. As a result, the actual structure of the levels is determined by the rotational subgroup O of the group Td  Ci . It is clearly seen in Figure 9.6 that in the transition from the D3 group to the O group (which corresponds to the transition from a rigid to a nonrigid molecule), the dimension of the function space increases fourfold. That is the way it should be in the presence of four independent equilibrium configurations.

142

Chapter 9 Molecules with torsional transitions of the exchange type

Figure 9.6. Energy-level classification for the LBH4 molecule. in the type A1 vibronic state.

Operations of the isomorphic groups Td and O can conveniently be determined on the basis of Figure 9.7, in which the tetrahedron is inscribed into a cube. For the further analysis it is important that the group O can be represented as O D D2 ^ D3 ; .3/

.2/

.1/

(9.40) .k/

where D2 D .E; C2 ; C2 ; C2 /, and the elements C2 group. By definition of the class, we have QC2 Q1 2 3C2 .k/

form a 3C2 class of the O

(9.41)

for any Q 2 O. It follows immediately from equation (9.41) that D2 is an invariant subgroup of the O group, which is necessary and sufficient for the validity of representation (9.40). Thus, all elements of the O group are written as a product of transformations of the rotation of the molecule as a whole (group D3 / and the torsional motion of the anion (group D2 /. This means that any torsion-rotational motion is uniquely decomposed into independent torsional and rotational motions defined in such a way. The complete

143

Section 9.4 The molecules of complex hydrides LiBH4 and NaBH4

Figure 9.7. Operations of the symmetry groups Td and O for the LBH4 molecule.

basis set of unit vectors jAi; jB1 i; jB2 i and jB3 i of the torsional space is specified by four irreducible representations of the D2 group. Of course, the number of unit vectors coincides with the number of independent equilibrium configurations of the molecule. Determining the action of the elements of the D3 group in torsional space requires permutation relations between the elements of the groups D3 and D2 , which, according to equation (9.41), should have the form .k/

QC2

.m/

D C2

Q;

(9.42)

where the superscript m depends on both the superscript k and the element Q belonging to the D3 group. In particular, it can easily be shown that .1/

C3 C2 .1/

.1/

U2 C2

.2/

D C2 C3 ; .1/

.1/

.1/

D v

D U2 C2 ;

where the axis U2

C3 C2

.2/

D C2 C3 ;

.1/

.2/

D C2 U2 ;

U2 C2

.xz/

.3/ .3/

.1/

.3/

C3 C2

.1/

.1/

D C2 C3 ; .3/

U2 C2

.2/

(9.43) .1/

D C2 U2 ; (9.44)

i coincides with the axis y of the MCS. .1/

Using equations (9.43) and (9.44), we obtain the action of the elements C3 and U2 (and thus the action of the elements of the whole group D3 / on the unit vector jAi: C3 jAi D jAi;

.1/

U2 jAi D ajAi;

(9.45)

where a2 D 1. In a similar way, for the unit vectors jB1 i; jB2 i, and jB3 i we have C3 jB1 i D b1 jB2 i; .1/

U2 jB1 i D c1 jB2 i;

C3 jB2 i D b2 jB3 i; .1/

U2 jB2 i D c2 jB1 i;

C3 jB3 i D b3 jB1 i; .1/

U2 jB3 i D c3 jB3 i;

(9.46) (9.47)

144

Chapter 9 Molecules with torsional transitions of the exchange type

where b1 b2 b3 D 1, c1 c2 D 1, and c32 D 1. For a D 1, the unit vector jAi in equation (9.45) belongs to the representation A1 in the O group, and for a D 1, to the representation A2 . The unit vectors jB1 i; jB2 i and jB3 i for c3 D 1 in equation (9.47) belong to the representation F2 and for c3 D 1, to the representation F1 . It follows from the classification that for the totally symmetric vibronic state it is necessary to choose a D c3 D 1, as will be clear after the construction of the basis torsionrotational functions. As concerns the rotational functions, only the operations of rotation of the molecule as a whole from the D3 subgroup act on them in the O group. For these functions we have (9.48) .A1 /D3 ! .A1 /O ; .A2 /D3 ! .A2 /O ; .E/D3 ! .E/O : As a result, we decompose the basis torsion-rotational unit vectors into irreducible representations of the O group (see Table 9.8). In this decomposition, we have four independent torsion-rotational unit vectors for each rotational state, as should be the case. In equations (9.45) and (9.47), we chose a D c3 D 1 to ensure that the unit vectors jAij0; 0i and jF2 ij0; 0i belong to the representations A1 and F2 , respectively, which is consistent with the classification in Figure 9.6. To construct a complete set of operators in the torsional space of the D2 group, we .1/ .2/ write this group in the form C2  C20 , where C2 D .E; C2 / and C20 D .E; C2 /. Symmetric and antisymmetric irreducible representations of the C2 group will be de-

Table 9.8. Decomposition of the torsion-rotational basis unit vectors into irreducible representations of the O group. O

Basis functions

A1

jAiA1

A2

jAiA2

E

jAiE p jB1 i. p 3E1  E2 /=2 jB2 i. 3E1  E2 /=2 jB3 iE2 p jB1 i.E1 C p 3E2 /=2 jB2 i.E1  3E2 =2 jB3 iE1

F1

jB1 iA2 jB2 iA2 jB3 iA2

F2

jB1 iA1 jB2 iA2 jB3 iA3

Note. The rotational parts of the unit vectors are specified by irreducible representations of the D3 group, in which the components of the representation E are determined by the conditions .1/ .1/ U2 E1 D E1 and U2 E2 D E2 .

Section 9.4 The molecules of complex hydrides LiBH4 and NaBH4

145

noted a1 and b1 and those of the C20 group, a2 and b2 . Consequently, the torsional space is represented as a direct product of two two-dimensional spaces: jAi D ja1 ija2 i;

jB1 i D jb1 ijb2 i;

jB2 i D jb1 ija2 i;

jB3 i D ja1 ijb2 i; (9.49)

We assume that the operator of coordinate spin e.1/ is specified in the space of unit vectors ja1 i and jb2 i and the operator e.2/ is specified in the space of unit vectors ja2 i and jb2 i. The two-dimensional representation of Lie algebra (7.27) can be written as (7.29). Using the symmetry properties of the unit vectors ja1 i and jb1 i with respect to transformations of the C2 group and the time reversal operation T , namely, that T ja1 i D ja1 i;

T jb1 i D jb1 i;

(9.50)

.1/

it is easy to show that the component e3 belongs to the representation a1 of the C2 .1/

.1/

group and is t-even, while the components e1 and e2 belong to the representation b1 and are t-even and t-odd, respectively. Symmetry properties of the components of the e.2/ operator are similar if the group C2 is replaced by the group C20 . It is clear that the effective torsion-rotational operators of physical quantities belong to the coordinate Young diagram Œ4 of the 4 group and, therefore, to the representation A1 of the Td group. Passing to the group O, we obtain the representations A1 and A2 for the quantities that are invariant under the inversion operation i and change sign during this transformation, respectively. According to equation (9.48), there are constraints on types of representations in the O group for the rotational operators. Hence, for the operators of physical quantities in the O group, only the following torsionrotational structures are possible: .Ak /tors  .Am /rot .k; m D 1; 2/;

.E/tors  .E/rot:

(9.51)

That is, it is necessary to single out the torsional operators of types A1 , A2 , and E in the O group. It is easy to show that in the D2 group such operators can belong only to the type A. Consequently, there are four independent torsional operators: I1 I2 ;

.2/

I1 e3 ;

.1/

e3 I2 ;

.1/ .2/

e3 e3 ;

(9.52)

and all of them are t-even. Clearly, the operator I1 I2 belongs to the type A1 in the O group. In addition, it can easily be concluded from the analysis of Table 9.8 that combinations of three other operators can belong only to representations A1 and E of the O. group. The action of the type A1 operator on the components of the state jF2 i should be reduced to multiplying each component by the same constant. This requirement makes it possible to directly specify this expression: .1/ .2/ .2/ .1/ RO D 2e3 e3 C I1 e3 C e3 I2 :

(9.53)

Two other operators of the form .1/ .2/

.2/

.1/

d1 e3 e3 C d2 I1 e3 C d3 e3 I2

(9.54)

146

Chapter 9 Molecules with torsional transitions of the exchange type

belonging to the representation E when they act on the unit vector jAi must give zero, since otherwise we obtain a unit vector of the jEi type, which is absent in the torsional space. Hence, the condition d1 C 2d2 C 2d3 D 0 is imposed on the constants .1/ in equation (9.54). Further, we require that in the transformations C3 and U2 the components E1 and E2 behave as follows: p .1/ .1/ C3 E1 C32 D 1=2E1  3=2E2 ; U2 E1 U2 D E1 ; (9.55) p .1/ .1/ C3 E2 C32 D 3=2E1  1=2E2; U2 E2 U2 D E2 : It is easy to verify that all of these conditions on the constants d1 ; d2 ; d3 are consistent and lead to the following final result: E1 D 2e3 e3 C I1 e3  2e3 I2 D PO ; p .1/ .2/ .2/ O E2 D 3.2e3 e3  I1 e3 / D Q: .1/ .2/

.2/

.1/

(9.56)

.1/

For the operations C3 and U2 we used the transformation matrices corresponding to the choice b1 D b2 D b3 D 1 in equation (9.46) and c1 D c2 D 1 in equation (9.47). The effective operators of coordinate physical quantities in a nondegenerate vibronic state are torsion-rotational. The effective Hamiltonian belongs to the representation A1 of the O group, and only two torsion-rotational structures are possible for it, namely, (9.57) .A1 /tors  .A1 /rot; .E/tors  .E/rot : Hence, a full expression for the Hamiltonian is written as follows: O  A1 C .PO  E1 C QO  E2 /; H D .I1 I2 ; R/

(9.58)

where E1 and E2 are two components of the representation E, which are defined by .1/ .1/ the conditions U2 E1 D E1 and U2 E2 D E2 . In unwinding equation (9.58) we take into account that the Hamiltonian is a t-even quantity. As a result, H D H .A1/ C H .E/ :

(9.59)

The first term includes type A1 combinations of the angular momentum components in the MCS in the D3 group: H .A1/ D

1 X

.1/

.2/

.H2n C H2nC4 /;

(9.60)

nD0

where the subscript specifies the total power of combinations of these components: X .1/ 6t D cO2p;2s;6t J 2p Jz2s .JC C J6t /; (9.61) H2n p;s;t .2/

H2nC4 D

X p;s;t

6t C3 cO2p;2sC1;6t C3 J 2p Jz2sC1 .JC C J6t C3 /:

(9.62)

147

Section 9.4 The molecules of complex hydrides LiBH4 and NaBH4

Here, p C s C 3t D n for a given n. The parameters cO are torsional operators of the form O (9.63) cO D c .1/ I1 I2 C c .2/ R; where c .k/ are the real spectroscopic constants (the subscripts are omitted). The second term in equation (9.59) includes type E combinations of the angular momentum components in the D3 group and has a more complex structure: H

.E/

D

1 X

.1/

.2/

.3/

.4/

.H2nC2 C H2nC4 C H2nC2 C H2nC6 /:

(9.64)

nD0

Here, .1/

H2nC2 D

X

.1/

b2p;2sC1;6t C1J 2p Jz2sC1

p;s;t

.2/

H2nC4

6t C1 6t C1 O Œ.JC C J6t C1 /PO C i.JC  J6t C1 /Q; X .2/ D b2p;2s;6t C4J 2p Jz2s

(9.65)

p;s;t

.3/

H2nC2

6t C4 6t C4 O Œ.JC C J6t C4 /PO C i.JC  J6t C4 /Q; X .3/ D b2p;2s;6t C2J 2p Jz2s

(9.66)

p;s;t

.4/ H2nC6

6t C2 6t C2 O Œ.JC C J6t C2 /PO  i.JC  J6t C2 /Q; X .4/ D b2p;2sC1;6t C5J 2p Jz2sC1

(9.67)

p;s;t 6t C5 6t C5 O Œ.JC C J6t C5 /PO  i.JC  J6t C5 /Q;

(9.68)

where b.k/ are the real spectroscopic constants. The Hamiltonian (9.57) certainly includes all the torsion-rotational interactions. The operator form of this Hamiltonian for both the rotational and torsional types of motion is its important merit. Energy matrix decomposes into a direct sum of the matrices corresponding to different irreducible representations of the O group. For a purely torsional effective Hamiltonian, from equation (9.59) we have .1/ .2/ O Htors D c000 I1 I2 C c000 R:

(9.69)

The relative position of the torsional levels is specified by one spectroscopic con.2/ stant c000 . Exactly one constant can be determined from information on the relative position of two torsional levels, one of which is nondegenerate and another is triply degenerate. Note that the purely torsional unitary transformations of the Hamiltonian are absent since all the torsional operators are t-even.

148

Chapter 9 Molecules with torsional transitions of the exchange type

The operator of the electric dipole moment in the O group belongs to the irreducible representation A2 , which gives the selection rules A1 $ A2 ;

E $ E;

F1 $ F2 :

(9.70)

It is also easy to construct the electric dipole moment operator itself.

9.5 The molecules of dimethyl ether (CH3 )2 O and acetone (CH3 )2 CO For the main isotopes of the carbon 12 C and oxygen 16 O nuclei, the spin is equal to zero. In this case, it suffices to write the permutation group of identical nuclei in the form of a permutation group 6 of the H nuclei. Matching this group with the point group C2v (the required correlations are given in Table 9.9), we have the following allowed coordinate multiplets of a rigid molecule: 16A1 ;

12A2 ;

12B1 ;

24B2 :

(9.71)

Then, by analogy with equation (8.4), we relate the symmetry types of the multiplets and of the zero-approximation wave functions used for their construction. As a result, we have a classification of the energy levels in an arbitrary vibronic state for el  vib D A1 , which is shown in the left two columns in Figure 9.8. Since the inversion doublets are not split in a rigid molecule, the actual structure of the levels is specified by the rotational subgroup D2 of the C2v  Ci group. The effective operator of any coordinate physical quantity characterizing the internal motion of a molecule transforms according to the totally symmetric coordinate Young diagram of the 6 group. Passing to the group C2v , for such an operator, we have a representation A1 . Finally, from a matching of the groups C2v  Ci and D2 we have the representations A and B1 of group D2 belonging to the quantities that are invariant under the inversion operation i and change sign during this transformation, respectively. That is, the effective Hamiltonian is of the A type, and the operator of the electric dipole moment is of the B1 type. Therefore, the electric dipole selection rules have the form (8.13). The component ˛ of the electric dipole moment operator in an arbitrary vibronic state on the ˛ axis of the FCS can be represented as (8.8). The term (8.11) gives the main contribution to this expression, as should be the case. To take into account the torsional motion of two identical methyl tops, we introduce an extended point group Q72 into a chain between the groups 6 and C2v (see Section 9.1). Matching the group Q72 with the group 6 , we obtain the following allowed multiplets for a nonrigid molecule: 6A3 ; 10A4 ; 8T1 ; 3T3 ; 1T4 :

(9.72)

Then, matching the geometrical groups Q72 and C2v , we have a picture of the level splittings of a rigid molecule in an arbitrary vibronic state due to nonrigid motions,

149

Section 9.5 The molecules of dimethyl ether (CH3 )2 O and acetone (CH3 )2 CO Table 9.9. Correlation tables of the (CH3 )2 O and (CH3 )2 CO molecules. 6

C2v

C2v  Ci

D2

6

Q72

Œ16 

B2

A1.C/ ; A2./

A

Œ16 

A4

2A1 C A2 C B1 C B2 A1 C 2A2 C 2B1 C 4B2 3A1 C A2 C B1

./ .C/ A1 ; A2 .C/ ./ B1 ; B2 B1./ ; B2.C/

B1 B2 B3

4

4

Œ21  Œ22 12  Œ23 

Œ21  Œ22 12  Œ23 

A 3 C T1 A 4 C T1 C T3 A 3 C T4

C2v

Q72

D2

B6  A6

A1 A2

A1 , A3 , T1 , T2 , 2T4 E, T1 , T2 , T3 , T4

A

A1  A1 , A1  E, E  A1, E  E

B1 B2

E, T1 , T2 , T3 , T4 A2 , A4 , T1 , T2 , 2T3

B1 B2 B3

A1  A2 , A1  E, E  A2, E  E A2  A1 , A2  E, E  A1, E  E A2  A2 , A2  E, E  A2, E  E

B6  A6

Q72  Ci

A1  A1 A1  A2

A1 ; A3 ; E ./ A1./ ; A3./ ; E .C/

A2  A1 A2  A2

A2 ; A4 ; E .C/ .C/ .C/ A2 ; A4 ; E ./

B6  A6

Q72  Ci

.C/

.C/

A1  E A2  E

T4 ; T4 T3.C/ ; T3./

.C/

./

./

./

E  A1 E  A2

T3 ; T4 .C/ ./ T3 ; T4

./

.C/

E E

T1

.˙/

.˙/

; T2

which is shown for el  vib D A1 in Figure 9.8. Since these motions are specified by rotational transformations in the Q72 group, the inversion doublets remain unsplit, and the actual structure of the levels is described by the total rotational subgroup of the Q72  Ci group. For the further analysis it is important that this subgroup, as the group H36 of the ethane molecule in equation (9.26), can be written as a direct product of two groups, (9.73) B6  A6 ; where the groups A6 and B6 , which are isomorphic with the group D3 , have the form A6 D A3 ^ U2 ; B6 D B3 ^ C2 :

(9.74)

Here, A3 D .E  E; c31  c31 ; c32  c32 / and B3 D .E  E; c31  c32 ; c32  c31 / include the torsional operations, while U2 D .E; U2.y/ D v.xz/ i/ and C2 are the rotations of the molecule as a whole. That is, the geometry of the motions has changed as compared with the ethane molecule. It is clearly seen in Figure 9.8 that the transition from the group D2 (rigid molecule) to the group B6  A6 (nonrigid molecule) increases the

150

Chapter 9 Molecules with torsional transitions of the exchange type

Figure 9.8. Energy-level classification for the (CH3 )2 O and (CH3 )2 CO molecules in a totally symmetric vibronic state.

dimension of the function space by a factor of nine. It should be so, since the torsional motion binds nine independent equilibrium positions. The effective operator of any coordinate physical quantity belongs to the representation A1 of the Q72 group. Passing to the group B6  A6 , we obtain representations A1  A1 and A1  A2 belonging to the quantities that are invariant under the inver-

Section 9.5 The molecules of dimethyl ether (CH3 )2 O and acetone (CH3 )2 CO

151

sion operation i and change sign during this transformation, respectively. That is, the Hamiltonian of a nonrigid molecule is of the A1  A1 type, and the operator of the electric dipole moment is of the A1 A2 type. As a consequence, for the electric dipole selection rules we obtain   A1 $   A2 ;

  E $   E;

(9.75)

where the irreducible representation  of the B6 group is preserved in the electric dipole transition. The basis unit vectors of the function space in any vibronic state are classified by irreducible representations of the B6  A6 group and are constructed from the unit vectors of the torsional and rotational spaces. A complete set of unit vectors in the torsional space is formed by the products of the unit vectors specified by irreducible representations of the groups A3 and B3 that are isomorphic with the C3 group. We then consider the ground vibronic state for which el  vib D A1 . A set of torsional unit vectors specified by three irreducible representations of the A3 group will be denoted as j0a i, j ˙ 1a i. By virtue of the properties of the semidirect product, the action of the elements of the U2 factor group is also specified in the unitvector space of the invariant subgroup: .y/

U2 j0a i D cj0i;

.y/

U2 j1a i D d j  1a i;

(9.76)

where c 2 D d 2 D 1. It will be clear in the further analysis that for the ground vibronic state c D 1. The value d can always be chosen equal to 1, since otherwise it suffices to change the sign of one of the unit vectors j ˙ 1a i. We then find that the unit vector j0a i belongs to the representation A1 of the A6 group and the unit vectors j ˙ 1a i, to the representation E: p E1 D .j1a i C j  1a i/= 2 D jsa i; (9.77) p E2 D i.j1a i  j  1a i/= 2 D jaa i; where the components E1 and E2 are determined by the relations .y/

U2 E1 D E1 ;

.y/

U2 E2 D E2 :

(9.78)

Passing to the group B6 , we introduce the torsional unit vectors j0b i; jsb i, jab i, which also belong to the irreducible representations A1 and E. As concerns the rotational functions, only operations of the rotation of the molecule as a whole from the D2 subgroup act on them in the B6  A6 group. Hence, for these functions, A ! A1  A1 ;

B1 ! A1  A2 ;

B2 ! A2  A1 ;

B3 ! A2  A2 ;

(9.79)

where the representations of the D2 group are on the left and their counterparts in the B6  A6 group are on the right. Multiplying the unit vectors of separate motions, we

152

Chapter 9 Molecules with torsional transitions of the exchange type

decompose the complete set of torsion-rotational unit vectors into irreducible representations of the B6  A6 group (see Table 9.10). Since we choose c D 1 in equation (9.76), the unit vector j0b ij0a iA belongs to the representation A1  A1 , which is consistent with the classification. We note that the geometric group coinciding with equation (9.73) was constructed in [70] as a symmetry group of the model Hamiltonian that allows only for the torsional motions and the rotation of the molecule as a whole. The potential energy was chosen as a finite segment of the two-dimensional Fourier expansion over two angles of internal rotation. In the construction, the authors had to require that the action of the group D2 operations on the Euler angles be accompanied by a certain variation of the internal rotation angles. This was explained by the interaction between the torsional motion and the motion of the molecule as a whole. However, it follows from equation (9.76) that the action of the group D2 operations in the torsional space is determined by the noncommutativity of the torsional operations with the rotational operations of the molecule as a whole and is therefore related only with the geometry of the problem.

Table 9.10. Decomposition of the torsion-rotational unit vectors into irreducible representations of the B6  A6 group.

B6  A6 A1  E E  A1

B6  A6

Unit vectors

A1  A1 A1  A2 A2  A1 A2  A2

j0b ij0a iA j0b ij0a iB1 j0b ij0a iB2 j0b ij0a iB3

B6  A6

Unit vectors

E E

jsb ijsa iA jsb ijaa iB1 jab ijsa iB2 jab ijaa iB3 jsb ijaa iA jsb ijsa iB1 jab ijaa iB2 jab ijsa iB3 jab ijsa iA jab ijaa iB1 jsb ijsa iB2 jsb ijaa iB3 jab ijaa iA jab ijsa iB1 jsb ijaa iB2 jsb ijsa iB3 Unit vectors

j0b ijsa iA j0b ijaa iA jsb ij0a iA jab ij0a iA

j0b ijaa iB1 j0b ijsa iB1 jab ij0a iB2 jsb ij0a iB2

B6  A6

Unit vectors

A2  E

j0b ijsa iB2 j0b ijaa iB3 j0b ijaa iB2 j0b ijsa iB3 jsb ij0a iB1 jab ij0a iB3 jab ij0a iB1 jsb ij0a iB3

E  A2

Note. The rotational parts of the unit vectors are specified by representations of the D2 group.

Section 9.5 The molecules of dimethyl ether (CH3 )2 O and acetone (CH3 )2 CO

153

Assume that in the space of the unit vectors j0a i and j ˙ 1a i the operator of coordinate spin a is given and these unit vectors correspond to the eigenvectors of the operator a3 with the eigenvalues 0 and ˙1, respectively. Using the symmetry properties of the unit vectors relative to transformations of the group A6 and the time reversal operation T , namely, that T j0a i D j0a i;

T j1a i D j  1a i;

(9.80)

it is easy to show that the component a3 belongs to the representation A2 of the group A6 and is t-odd, while the components a1 and a2 belong to the representation E and are t-even. A complete set of independent self-adjoint operators in three-dimensional space, which includes the products of the a components with a total power not higher than two is given in Table 9.11. A similar set is introduced in the space of the vectors j0b i and j ˙ 1b i by using the operator of coordinate spin b. The effective operators of coordinate physical quantities in the B6 A6 group belong only to the representations A1 A1 and A1 A2 , and the rotational operators are limited by representations (9.79). That is, the torsion-rotational quantities comprise only the torsional operators belonging to one-dimensional representations of the groups A6 and B6 . The effective Hamiltonian is contributed by the torsion-rotational structures .A1  A1 /tors  .A1  A1 /rot;

.A1  A2 /tors  .A1  A2 /rot ;

.A2  A1 /tors  .A2  A1 /rot;

.A2  A2 /tors  .A2  A2 /rot :

(9.81)

Hence, a full expression for these contributions is written as H D .Ib Ia ; Ib a32 ; b32 Ia ; b32 a32 /  A C .Ib a3 ; b32 a3 /  B1 C .b3 Ia ; b3 a32 /  B2 C .b3 a3 /  B3 :

(9.82)

Each term of the Hamiltonian is a combination of the products of the angular momentum components in the MCS, which belongs to one of the representations of the

Table 9.11. Symmetry properties of the complete set of torsional operators in the a space. A6

t-even

t-odd

A1

Ia ; a32

–

A2

–

a3

E

E1 D aC C a E2 D i .aC  a / 2 2 C a E1 D aC  2 2 E2 D i aC  a

E1 D i Œa3 ; aC  a C E2 D Œa3 ; aC C a C

Note. Ia is a unit operator.

154

Chapter 9 Molecules with torsional transitions of the exchange type

D2 group, and the parameters before its combination are linearly dependent on the torsional operators correlating with them through symmetry. Since the Hamiltonian is t-even, combinations of the A and B3 types are t-even and combinations of the B1 and B2 types are t-odd. The Hamiltonian (9.82) includes all interactions of the torsional and rotational motions in explicit form and implicitly (via contributions into phenomenological constants) allows for the influence of other types of internal motion. The correctness of this Hamiltonian is limited only to the correctness of the choice of the internal dynamics symmetry. The operator form of the Hamiltonian for both types of motion is also an advantage of this approach. The energy matrix decomposes into a direct sum of matrices corresponding to nine types of irreducible representations of the B6  A6 group. The calculation of matrix elements is elementary. For the molecules considered here, the construction of an effective torsion-rotational Hamiltonian by traditional methods is a complex problem. The Hamiltonian was constructed in [48] in matrix form. The model Hamiltonian allowing for only the torsional motions and the rotation of the molecule as a whole was used as the initial one. The potential energy was chosen in the form of a two-dimensional Fourier expansion by two angles of internal rotation. The resulting matrix elements have a very cumbersome form, and most importantly, it is very difficult to estimate the applicability domain of the approximations made in the construction. For example, the effective Hamiltonian does not have terms with combinations of the angular momentum components with an odd total power. The effective operator of the electric dipole moment is contributed by the torsionrotational structures .A1  A1 /tors  .A1  A2 /rot;

.A1  A2 /tors  .A1  A1 /rot;

.A2  A1 /tors  .A2  A2 /rot;

.A2  A2 /tors  .A2  A1 /rot:

(9.83)

Its component ˛ is written in the form (8.62), where now the operators dO .i / are torsional. The irreducible representation in the B6  A6 group for the quantities ˛i can easily be obtained from equations (8.9) and (9.79). Of most interest are the main contributions into equation (8.62) obtained by neglecting the weak rotational distortion of the molecule: .z/

.z/

.z/

.z/

˛ D ˛z .d1 Ib Ia C d2 Ib a32 C d3 b32 Ia C d4 b32 a32 / C ˛y d .y/ b3 a3 ; (9.84) .z/

where dk and d .y/ are the real phenomenological constants. Thus, the z and y components are active. The first component leads to the usual rotational transitions, which .z/ obey selection rules (8.13), in each of the four torsional states. The constant d1 corresponds to the constant component of the electric dipole moment along the z axis for the unexcited torsional state, and the other three z constants are corrections in this component for three excited torsional states. The second component leads to transitions which obey the rotational rules of selection A $ B2 ;

B1 $ B3 ;

(9.85)

Section 9.6 Conclusions

155

These transitions occur only between the states of the E  E type and are entirely due to the torsional distortion of the molecule. Note that the selection rules for the electric dipole transitions were obtained in [70] on the basis of the assumption that the electric dipole moment is directed along the z axis (the x axis in the notation of [70]). However, the latter statement is true only for the undistorted configuration. Although these selection rules formally coincide with equation (9.75), they do not include the transitions stipulated by the y component.

9.6 Conclusions We have formulated some general conclusions based on the content of this chapter. 1. An extended point group G, which characterizes both the local motions in the minima and the transitions between them, appears additionally in a chain of groups when a nonrigid molecule is considered. Hence, the group G comprises the point group G0 as a subgroup. If the nonrigid transitions belong to the exchange type, then each element of G is associated with the permutation of identical nuclei in the force field that is invariant with respect to this element. In this respect, the situation is similar to that of the point group of a rigid molecule. However, a simple recipe for determining the geometric symmetry of internal dynamics by the equilibrium configuration symmetry no longer works, and there is generally a nontrivial problem of finding the group G. Information on the geometry of nonrigid transitions is usually used in this case. Often, such a group does not belong to tabulated groups, for example, such as those for the molecules of ethane, dimethyl ether, and acetone. 2. To describe the internal dynamics, we have obtained simple algebraic models (with an algebraic scheme for calculation of both the position of the levels in the energy spectrum and the intensities of the transitions between them), and their correctness is limited only to the correctness of the choice of symmetry. In fact, these models automatically take into account all symmetry-allowed intramolecular interactions, including weak (e.g., relativistic) ones. In Chapter 19, we consider the constructions of such models to describe the complete energy spectrum of nonrigid molecules in a nondegenerate electronic state. Perhaps such an approach is currently the only possible one for the correct solution of any major problem in the internal dynamics of nonrigid molecules. 3. The operator form of the effective operators of physical quantities for all considered types of internal motion is an important advantage of this approach. The fact that the set of internal coordinates is absent in explicit form in the construction of a description leads to a radical simplification of the structure of these operators. This effect is based on the fact that unlike the usual elementary operators

156

Chapter 9 Molecules with torsional transitions of the exchange type

in internal coordinates of nonrigid motions, the elementary operators of the coordinate spin are introduced with allowance for the necessary number of independent equilibrium configurations and correct symmetry properties for the transitions between them. That is, the elementary spin operators are integral in the sense that they allow at once for the large-amplitude motion throughout their entire range.

Chapter 10

Molecules with pseudorotations of the exchange type 10.1 Extended point groups The nonrigid motions specified by the operations of rotation of the molecule as a whole in an extended point group will be called pseudorotations. We first consider such a motion using the cyclobutane molecule C4 H8 as an example. This molecule has the shape of a four-membered cycle formed by carbon nuclei. In its equilibrium configuration corresponding to the point group D2d [38] and shown in Figure 10.1, the cycle is nonplanar and the H nuclei are arranged geometrically nonequivalently. Hence, we obtain a pronounced nonrigid motion, whose geometry is qualitatively represented as follows. First, using the motions of the nuclei along the z axis, we arrive at an intermediate configuration with planar cycle (the nuclei C1 , C3 and C2 , C4 together with the H nuclei move up and down, respectively) and geometrically equivalent arrangement of the identical nuclei, which lies about 500 cm1 above the equilibrium configuration [38]. In the intermediate configuration, the molecule rotates through a zero barrier about the z axis by an angle =2. Then, the nuclei keep moving along the z axis until they attain their new positions in the equilibrium configuration. Due to a rotation, the spatial arrangement of the equilibrium configuration is retained. That is, nonrigid motion belongs to the exchange type. The extended point group is determined by the geometric symmetry of the intermediate configuration and has the form

Figure 10.1. Equilibrium configuration of the cyclobutane molecule C4 H8 (the nuclei H1 , H3, H5 , and H7 are equatorial and the nuclei H2 , H4, H6 , and H8 are axial).

158

Chapter 10 Molecules with pseudorotations of the exchange type

D4h, with the nonrigid transitions specified by the operations C41 and C43 of rotation of the molecule as a whole about the z axis. In this case, the nuclei C exchange places and the equatorial nuclei H exchange places with the axial ones. The barrier for such a nonrigid motion is due to the distortion of the equilibrium configuration compared with the intermediate configuration. The case of the motion which takes place in the phosphorus pentafluoride molecule PF5 and is called the Berry pseudorotation is much more complicated. Its equilibrium configuration belongs to the point group D3h and is given in Figure 10.2. In this configuration, the F nuclei have two nonequivalent positions, namely, in the plane of the triangle (equatorial position) and on the straight line perpendicular to this plane (axial position). It is well established [34] that the nonrigid motion, called the Berry pseudorotation, takes the nuclei from the equatorial position to the axial position, and vice versa. In the motion shown in Figure 10.2, the equatorial nucleus F1 retains its position, while the other four F nuclei pass through an intermediate configuration with the point symmetry group C4v with the simultaneous rotation of the molecule as a whole by an angle /2 about the pseudorotational axis C4 (the F1 P axis). In the intermediate configuration, the nuclei participating in the nonrigid motion occupy geometrically equivalent positions. Due to a rotation by an angle /2, the equilibrium configuration retains its position in space and the pseudorotation belongs to the exchange type. The barrier for such a motion is due to the difference in energy between the equilibrium and intermediate configurations.

Figure 10.2. Initial (left) and final (right) equilibrium configurations of the PF5 molecule bound by the Berry pseudorotational motion about the F1 P axis (arrows on the left indicate the motion of the nuclei through an intermediate configuration of the C4v symmetry without allowance for the rotation of the molecule by an angle /2 about the F1 P axis).

An important feature of the nonrigid molecule PF5 is the presence of as many as three equivalent pseudorotational axes passing through three vertices of an equatorial triangle. As a result, there are three equivalent intermediate configurations occupying

159

Section 10.2 Cyclobutane molecule C4 H8

different spatial positions, resulting in a very complicated motion with transitions between twenty independent equilibrium configurations. Naturally, the symmetry group C4v describing only the transition through one intermediate configuration does not include the point group D3h as its subgroup and cannot be regarded as an extended point group. It is therefore necessary to solve a very nontrivial problem of constructing a geometric group that would allow for the three pseudorotational axes. This is done in Section 10.4. Simpler dynamics with one Berry pseudorotational axis is considered in Section 10.3 using as an example the XPF4 molecules, in which the X group (in particular, CH3 or (CH3 /2 N) replaces one of the equatorial F atoms.

10.2 Cyclobutane molecule C4 H8 For the primary isotope 12 C of a carbon nucleus, the spin is zero. In this case, as a permutation group of identical nuclei of the molecule, a permutation group 8 of hydrogen nuclei can be used in the analysis without loss of generality. Matching this group with the point group D2d of equilibrium configuration (the necessary correlations are given in Table 10.1), we obtain the following allowed coordinate multiplets of a rigid molecule: 55A1 ;

15A2 ;

21B1 ;

45B2 ;

60E:

(10.1)

Now we match the symmetry types of the zero-approximation wave functions with the symmetry types of the allowed multiplets formed on their basis: .mult/D2d Ci D .el /D2d  .vib /D2d  .rot.-in/D2d Ci ;

(10.2)

6

where the admissible representations rot.-in for a given rotational representation of the D1 group of a rigid symmetric top follow from the correlation of the groups D1 and D2d  Ci through their common subgroup D4 D .E; C2 ; 2C4 D 2S4 i; 2U2 ; 2UN 2 D 2d i/: As a result, we have a classification of the energy levels in an arbitrary vibronic state for el vib D A1 , which is shown in Figure 10.3. A splitting of the inversion doublets is absent, and the actual structure of the levels is specified by the rotational group D4 . The effective operator of any physical quantity describing the intramolecular motion belongs to the coordinate Young diagram Œ8 of the 8 group and, consequently, to the representation A1 of the D2d group. Passing to the group D4 , we obtain representations A1 and B2 for the quantities that are invariant under the operation i and change sign in this transformation, respectively. That is, the effective Hamiltonian is of type A1 and the effective operator of the electric dipole moment is of type B2 . Hence, the electric dipole selection rules have the form A1 $ B2 ;

A2 $ B1 ;

E $ E:

(10.3)

160

Chapter 10 Molecules with pseudorotations of the exchange type

Table 10.1. Correlation table for the cyclobutane molecule C4 H8 . D2d

D2d  Ci

D4

D1

D4

Œ18  Œ216 

A1 A1 C 2B2 C 2E

A1 B2

A1 A2

A1 A2

Œ22 14  Œ23 12 

5A1 C A2 C 3B1 C 3B2 C 4E 3A1 C 3A2 C B1 C 5B2 C 8E

A2 B1

E2nC1 E4nC2

E B1 C B2

Œ24 

5A1 C A2 C 3B1 C B2 C 2E

.C/ ./ A1 , B1 A1./ , B1.C/ .C/ ./ A2 , B2 ./ .C/ A2 , B2 .˙/

E

E4nC4

A1 C A2

8

E

8

D40  CI

Œ18 

A1g

Œ21  Œ22 14  Œ23 12 

A2u C B1g C B2u C Eg C Eu 4A1g C A1u C A2u C 2B1g C B1u C 2B2g C B2u C 2Eg C 2Eu A1u C 2A2g C 3A2u C 2B1g C B1u C 3B2u C 4Eg C 4Eu

Œ24 

5A1g C A1u C B1g C B1u C 2B2g C Eg C Eu

6

D40  CI

D2d

H2  D40

D4

H2  D40

D40  CI  Ci

A1g ; B2u

A1

A1s ; B2a

A1

s

g.C/ ; u./

A1u ; B2g A2g ; B1u

B1 A2

A1a ; B2s A2s ; B1a

B2 A2

a

g ; u

A2u ; B1g Eg ; Eu

B2 E

A2a ; B1s Es ; Ea

B1 E

./

.C/

Component ˛ of the electric dipole moment in a nondegenerate vibronic state on the ˛ axis of the FCS can be represented in the form (8.8). Allowing for the decomposition of ˛i into irreducible representations of the D4 group

˛z ! A2 ;

. ˛y ; ˛x / ! E;

(10.4)

we find that the main contribution not associated with the weak rotational motion of the molecule to equation (8.8) is zero, as it should be. Indeed, even from simple geometric considerations it is clear that a constant electric dipole moment is absent in the equilibrium configuration. To account for the pseudorotation, it is necessary to introduce in the chain an extended point group D4h (see Section 10.1), which can be written as D4h D D40  CI ;

(10.5)

where D40 D .E; C2 ; 2C40 ; 2U2 ; 2UN 20 /. All the elements of the D4h group belong to the exchange type, and the group D40 is therefore different from the D4 group since

Section 10.2 Cyclobutane molecule C4 H8

161

Figure 10.3. Energy-level classification for a rigid molecule of 12 C4 H8 in the type A1 vibronic state.

the elements of classes 2C4 and 2UN 2 of the latter belong to the nonexchange type. Nonrigid motion is specified by the exchange elements C41 and C43 , which form a class 2C40 of the D40 group. Note that unlike the equilibrium configuration, the intermediate configuration has an inversion center. From a matching of the groups 8 and D4h, for the allowed coordinate multiplets of a nonrigid molecule we have 34A1g ; 9A1u; 6A2g ; 21A2u; 24B1g ; 9B1u; 12B2g ; 21B2u; 30Eg ; 30Eu: (10.6) Then, matching the geometrical groups D4h and D2d , we obtain a picture of the level splittings of a rigid molecule in an arbitrary vibronic state with allowance for the nonrigid motion for el  vib D A1 , which is shown in Figure 10.4. Since the nonrigid motion is specified by rotational transformations in the D4h group, the inversion doublets remain unsplit, and the actual structure of the levels is determined by the rotational subgroup H2  D40 of the D4h  Ci group. Symmetric and antisymmetric representations of the H2 D .E; I i/ group are denoted as s and a. The effective operator of any physical quantity describing the internal motion of a nonrigid molecule belongs to the representation A1g of the D4h group. As a consequence, in the group H2  D40 we have representations A1s and A1a belonging to the

162

Chapter 10 Molecules with pseudorotations of the exchange type

Figure 10.4. Energy-level classification for a nonrigid molecule of vibronic state.

12

C4 H8 in the type A1

quantities that are invariant under the inversion operation i and change sign during this transformation, respectively. That is, the effective Hamiltonian is of type A1s and the effective operator of the electric dipole moment is of type A1a . This implies the electric dipole selection rules s $ a ; (10.7) where  is an irreducible representation of group D40 , which preserves in the electric dipole transition.

163

Section 10.2 Cyclobutane molecule C4 H8

We now construct the function space to describe the entire spectrum of the nonrigid motion excitations of a cyclobutane molecule in the type A1g electronic state. To do this, one should write a complete set of basis unit vectors and determine their behavior with respect to transformations of the H2  D40 group. We use the fact that the group D40 can be represented as (10.8) D40 D C4 ^ U2 ; .x/

where U2 D .E; U2 /. It is important that in the cyclic group C4 the nonrigid motion is specified only by the elements C41 and C43 , while the element C42 determines the usual rotation of the molecule. The point is that the elements C41 and C43 take the molecule into independent local minima of the effective nuclear potential, while the element C42 keeps the molecule in the initial minimum. In this case, the subgroup of an extended point group, which includes only the pseudorotational elements, cannot be indicated. To separate the rotational and pseudorotational motions, the basis pseudorotational unit vectors should be constructed only from the vectors which are invariant under the action of the element C42 . Such vectors belong to the irreducible representations A and B of the C4 group, and they can be written as j4ni and j:4nC2i, respectively, where n is any integer (including zero), such that C41 jmi D exp.im=2/jmi:

(10.9)

The action of the elements of the factor group U2 in the basis specified by irreducible representations of the C4 group is chosen in the form .x/

U2 jmi D j  mi:

(10.10)

We now form the basis pseudorotational unit vectors p jsmD0 i D j0i; jsm¤0 i D .jmi p C j  mi/= 2; jsm6D0 i D i.jmi  j  mi/= 2:

(10.11)

From equations (10.9) and (10.10) it follows that these unit vectors for an even m belong to the following irreducible representations of the D40 group: js4n i ! A1 ;

ja4nC4 i ! A2 ;

js4nC2 i ! B1 ;

ja4nC2 i ! B2 :

(10.12)

It remains to determine the effect on unit vectors (10.11) of the operation I i of the H2 group. For this we note that .x/ .yz/ (10.13) I i D U2 d i: .yz/

The operation d i also corresponds to a rotation by an angle  about the x axis, but it pertains to the nonexchange type and is valid only in the rotational space. Hence, we obtain I ijsm i D jsm i; I ijam i D jam i: (10.14)

164

Chapter 10 Molecules with pseudorotations of the exchange type

That is, the s and a unit vectors belong to the representations s and a, respectively, of the H2 group. According to equation (10.13), for the rotational basis unit vectors the operation I i is equivalent to the identical operation, as it should be. Therefore, the rotational unit vectors belong to the representation s of the H2 group, and their behavior relative to transformations of the D40 group is well known (the effect of the elements of the groups D4 and D40 in the rotational space coincides due to the condition I i D E/. In the rigid-molecule limit, the pseudorotation converts into a type B2 vibration in the D2d group [38]. The ground vibrational state is determined by the pseudorotational unit vectors js0 i and ja2 i. Indeed, the functions j0; 0ijs0 i and j0; 0ija2 i then belong to the representations A1s and B2a , in accordance with the classification in Figure 10.4. The singly excited state B2 is determined by the pseudorotational unit vectors js2 i and ja4 i, in accordance with the classification for vib D B2 . The doubly excited state B2  B2 D A1 is determined by the unit vectors js4 i and ja6 i. This series of excitations is easy to continue. As a result, we decompose a complete set of basis unit vectors of the function space for the nonrigid motion excitations of interest to us. In Table 10.2, this decomposition is given for four lower pseudorotational states. Note that all the results concerning the structure of the function space are easy to extend to an arbitrary nondegenerate electronic state. Table 10.2. Decomposition of the basis unit vectors of the function space of four lower pseudorotational states into irreducible representations of the H2  D40 group. H2  D40

Basis unit vectors

H2  D40

Basis unit vectors

A1s A2s B1s B2s Es

js0 iA1 ; js2 iB1 js0 iA2 ; js2 iB2 js0 iB1 ; js2 iA1 js0 iB2 ; js2 iA2 js0 iE; js2 iE

A1a A2a B1a B2a Ea

ja2 iB2 ; ja4 iA2 ja2 iB1 ; ja4 iA1 ja2 iA2 ; ja4 iB2 ja2 iA1 ; ja4 iB1 ja2 iE; ja4 iE

Note. The rotational parts of the unit vectors are specified by irreducible representations of the D40 group.

The complete set of self-adjoint pseudorotational operators is constructed on the basis of the operator of coordinate spin e. For this, we need to find the representation of Lie algebra (7.27) in the space of vectors jmi (an upper limit on the number jmj in explicit form will not be set). Let e3 be a diagonal operator with the eigenvalues m and e˙ D e1 ˙ ie2 be the raising and lowering operators. From equation (10.9) it is easy to obtain (10.15) C41 T jmi D exp.im=2/T jmi: Hence, we can choose T jmi D j  mi:

(10.16)

165

Section 10.2 Cyclobutane molecule C4 H8

Based on the symmetry properties of the vectors jmi we find that e3 belongs to the representation A of the C4 group and is t-odd, while eC and e belong to the representations "1 and "2 , respectively (as the vectors j1i and j  1i/, and T eC T D e . The description includes only the pseudorotational operators of types A and B. Operator e3 2 and e 2 belonging to the representation B serve as the generators and the operators eC  for their construction. When the group C4 is extended to H2  D40 , we obtain e3 ! A2a ;

2 2 eC C e ! B1s ;

2 2 i.eC  e / ! B2a :

(10.17)

A complete set of self-adjoint operators with a total power not higher than four in components e is presented in Table 10.3. Table 10.3. Complete set of self-adjoint pseudorotational operators. H2  D40

t-even

t-odd

A1s A2a B1s B2a

4 4 I; e32 ; e34 ; eC C e 4 4 i.eC  e / 2 2 2 2 eC C e ; Œe32 ; eC C e C 2 2 2 2 i.eC  e /; iŒe32 ; eC  e C

 e3 ; e33 2 2 iŒe3 ; eC  e C 2 2 Œe3 ; eC C e C

The effective Hamiltonian is of type A1s , and the contributions to it are formed by the structure .A1s /sp  .A1s /rot; .B1s /sp  .B1s /rot: (10.18) A full expression for these contributions can be written as 4 4 C e /  A1 H D .I; e32 ; e34 ; eC 2 2 2 2 2 2 C .eC C e ; Œe32 ; eC C e C ; iŒe3 ; eC  e C /  B1 :

(10.19)

That is, each term of the Hamiltonian is a combination of the products of the angular momentum components in the MCS belonging to the representations A1 or B1 of the D40 group, and the parameters before this combination are linearly dependent on the spin operators correlating with this combination. Naturally, the Hamiltonian should be t-even. Equation (10.19) is easy to unwind as a series of rotational distortions. This Hamiltonian includes all allowed interactions of the pseudorotational and rotational motions. The operator form in both types of motion is also an important merit of this Hamiltonian. The energy matrix for a given value of the quantum number J of the angular momentum squared decomposes into a direct sum of matrices corresponding to the different types of irreducible representations of the H2  D40 group. The effective operator of the electric dipole moment is of type A1a , and its component ˛ on the ˛ axis of the FCS is written as (8.62), where the parameters dO .i / are

166

Chapter 10 Molecules with pseudorotations of the exchange type

now pseudorotational operators. The components on the MCS axis of the unit vector ˛ belong to the following irreducible representations of the H2  D40 group:

˛z ! A2s ;

. ˛y ; ˛x / ! Es :

(10.20)

The type A1a contributions to equation (8.62) are formed by the structures .A2a /sp  .A2s /rot;

.B2a /sp  .B2s /rot :

(10.21)

For the strongest electric dipole transitions not stipulated by a weak rotational distortion of the molecule, we find, with accuracy up to operators in Table 10.3, that 4 4  e /; ˛ D ˛z d .z/ i.eC

(10.22)

where d .z/ is a real phenomenological constant. That is, only the z component of the electric dipole moment is active, and the purely rotational transitions are absent. Thus, using the symmetry methods, we have constructed a rigorous model describing the most important excitation band for a study of pseudorotations in the cyclobutane molecule C4 H8 . Unfortunately, the vibrational electric dipole transitions in this band lie in the far IR range and are fairly weak, while the rotational transitions due to the constant component of the electric dipole moment are absent. Hence, the direct spectroscopic observations are very difficult, and for this reason the microwave studies were performed for the purely rotational spectra of cyclobutane-d1 [87] and cyclobutane-1,1-d2 [30], whose small constant component of the electric dipole moment is stipulated by the substitution of one or two H nuclei, respectively, for deuterium nuclei. The symmetry methods discussed in this book make it possible to describe the isotopically nonsymmetric combinations of a molecule as a distortion of its most symmetric combination. In Chapter 16, this is done for cyclobutane-1,1-d2 . It is important that the approach in which molecular systems of different symmetry are considered from a unified point of view explains the qualitative features of the cyclobutane-1,1-d2 spectrum which follow from the analysis of its microwave data. Here we are referring primarily to the observed selection rules. Finally, it should be emphasized that we have obtained a unified model describing the entire spectrum of pseudorotational excitations in a nondegenerate electronic state. As a result, the influence of the frequently encountered accidental resonances between different pseudorotational states is automatically taken into account in the experimental data fitting. We are speaking here of the influence of both the position of the energy levels and the intensity of the transitions between them. The latter is often no less important, which is due to the fact that accidental resonances can change these intensities by orders of magnitude.

10.3 Molecules of the XPF4 type In this section, we consider the internal dynamics of the XPF4 molecules, where the X group (in particular, CH3 or (CH3 /2 N) substitutes one of the equatorial atoms. It is

167

Section 10.3 Molecules of the XPF4 type

known that the Berry pseudorotation plays a dominant role in the transition of the F nuclei between the equatorial and axial positions [81]. The description of this motion is greatly simplified by the fact that only one pseudorotational axis is present here. Let the X group substitute the F1 nucleus in Figure 10.2. Choose the MCS in such a way that the z axis coincides with the XP axis, and the reflection plane yz coincides with the plane of the equatorial triangle. Matching the permutation group 4 of identical .z/ nuclei F with the point group C2v D .E; C2 ;  .xz/ ; N .yz/ / of the equilibrium configuration of the XPF4 molecule (the necessary correlations are given in Table 10.4), for the allowed coordinate multiplets of a rigid molecule we obtain .5 C 3 C 1/A2 ;

1A1 ;

3B1 ;

3B2 :

(10.23)

Further, by analogy with equation (8.4), we match the symmetry types of the zeroapproximation wave functions with the symmetry types of the allowed multiplets formed on their basis. As a result, we have a classification of the energy levels in an arbitrary vibronic state, which is shown for el  vib D A1 in the first two columns in Figure 10.5. The inversion doublets are not split, and the actual structure of the levels is described by the rotational subgroup D2 of the C2v  Ci group. Table 10.4. Correlation tables for the molecules of the XPF4 type.

4 4

C2v

4 4

C4v

Œ1  Œ212  Œ22  Œ31 Œ4

A2 A2 C B1 C B2 A1 C A2 A1 C B1 C B2 A1

Œ1  Œ212  Œ22  Œ31 Œ4

B2 A2 C E A1 C B2 B1 C E A1

D4

C4v  Ci

C2v

C4v

A1 A2 B2 B1 E

A1.C/ , A2./ ./ .C/ A1 , A2 .C/ ./ B1 , B2 ./ .C/ B1 , B2 .˙/ E

A1 A2 B1 B2

A1 , B1 A2 , B2 E E

On account of the Berry pseudorotation, the molecule is delocalized in two independent equilibrium configurations, and an extended point group is the point group C4v of a single intermediate configuration. Matching the groups 4 and C4v , we obtain the following allowed coordinate multiplets for a nonrigid molecule: 1A1 ;

3A2 ;

.5 C 1/B2 ;

3E:

(10.24)

168

Chapter 10 Molecules with pseudorotations of the exchange type

Finally, matching the geometrical groups C2v and C4v , we obtain a picture of splittings of the rigid-molecule multiplets due to the pseudorotation (see Figure 10.5). Since the nonrigid motion is specified in an extended point group C4v by the rotational operations C41 and C43 , the inversion doublets remain unsplit, and the actual structure of the levels is described by the rotational subgroup D4 of the C4v  Ci group. Note that for the one-dimensional states of a nonrigid molecule, one of the levels of the inversion doublet is absent since the multiplet B1 of the C4v group is forbidden.

Figure 10.5. Energy-level classification for a nonrigid molecule of the XPF4 type in the type A1 vibronic state.

We write the group D4 as C4 ^ U2 , where the factor group has the form U2 D .y/ .E; U2 /. For a unique determination of the action of the group D4 elements, we note that rotations about the axes x and y belong to the 2U2 class. Elements C41 and C43 of the C4 group specify the pseudorotation, but they are simultaneously the rotations of the molecule as a whole, which act on the rotational unit vectors. At the same time, the element C42 describes the rotation of the molecule as a whole through a zero barrier, and its action should be determined only in the rotational subspace. It is important here that the elements C41 and C43 take the molecule into independent local minima of the effective nuclear potential, while the element C42 keeps the molecule at the initial minimum. The complete basis set in the pseudorotational subspace consists of only two unit vectors, jai and jbi, belonging in the C4 group to the totally symmetric

169

Section 10.3 Molecules of the XPF4 type

representation A and the representation B that is antisymmetric with respect to the elements C41 and C43 . Only these unit vectors are invariant under the action of the purely rotational element C42 . The dimension of this space determines the number of mixed independent equilibrium configurations (we obtain two configurations, as it should be). Since the representations A and B are real, one can choose T jai D jai;

T jbi D jbi:

(10.25)

.y/

The action of the element U2 in the basis of the invariant subgroup C4 for these unit vectors is trivial: .y/ .y/ (10.26) U2 jai D jai; U2 jbi D jbi; where the choice of signs corresponds to the vibronic state el vib D A1 . As a result, in the D4 group the unit vectors jai and jbi belong to the irreducible representation A1 and B2 , respectively. Allowing for the known behavior of the rotational functions for transformations of the D4 group, we decompose the complete set of basis unit vectors into irreducible representations of this group (see Table 10.5). Table 10.5. Decomposition of the complete set of basis unit vectors with allowance for the Berry pseudorotation of the molecule of the XPF4 type. D4

Unit vectors

A1 A2

A1 jai; B2 jbi A2 jai; B1 jbi

B1 B2

B1 jai; A2 jbi B2 jai; A1 jbi

E

E1 jai

E1 jbi

E2 jai

E2 jbi

Note. The rotational parts are specified by irreducible representations of the D4 group, and .y/ the components of the representation E are defined by the conditions U2 E1 D E1 and .y/ U2 E2 D E2 .

For Lie algebra (7.27), the two-dimensional representation in the space of unit vectors jai and jbi can be written as equation (7.29). Assume that they are eigenfunctions of the operator e3 . Then it is easy to see that the operator e3 belongs to the representation A1 of the D4 group and is t-even, while the operators e1 and e2 belong to the representation B2 and are t-even and t-odd, respectively. The complete set of selfadjoint operators also includes a totally symmetric unit operator I. For the effective operators of physical quantities, representations A1 and A2 are allowed in the D4 group for the quantities that are invariant under the operation of spatial inversion i and change sign, respectively, during this transformation. The effective

170

Chapter 10 Molecules with pseudorotations of the exchange type

Hamiltonian belongs to the type A1 , which is implemented using two spin-rotational structures: (10.27) .A1 /sp  .A1 /rot; .B2 /sp  .B2 /rot: As a result, we obtain a full expression for this Hamiltonian: H D

1  X .A1 / .B2 / .B2 / C H2nC2 C H2nC3 H2n ;

(10.28)

nD0

where the superscript of the contributions to the Hamiltonian specifies the type of the irreducible representation of the D4 group, to which combinations of the angular momentum components in the MCS belong. These contributions have the form X .A / 4t H2n 1 D cO2p;2s;4t J 2p Jz2s .JC C J4t /; (10.29) p;s;t .B2 / H2nC2

D

X

4t C2 dO2p;2s;4t C2J 2p Jz2s .JC C J4t C2 /;

(10.30)

4t C2 i gO 2p;2sC1;4t C2 J 2p Jz2sC1 .JC  J4t C2 /:

(10.31)

p;s;t .B /

2 D H2nC3

X

p;s;t

Here, c, O dO , and gO are the t-even and t-odd spin operators, respectively, cO D c .1/ I C c .2/ e3 ;

dO D de1 ;

gO D ge2 ;

(10.32)

where c .k/ ; d , and g are the real spectroscopic constants (the subscripts are omitted). It is interesting to note that the conditions of limiting the process to the case of an impermeable barrier for the pseudorotation can be formulated in terms of spin operators. Indeed, in the case of such a barrier, the description should not depend on the symmetry of a nonrigid transition. Hence, only the spin operators invariant with respect to the index permutation in the spin subspace are preserved. These are only I and e1 . It is important that their space is closed relative to the multiplication operation in the Lie algebra. Otherwise, the condition of preservation of the description with admissible unitary transformations will be violated. As a result, only the rotational contributions with cO D c .1/ I; dO D de1 (10.33) will be retained in the effective operators. In the basis set from Table 10.5 we turn to the spin vectors of the form p p j1i D .jai C jbi/= 2; j2i D .jai  jbi/= 2: When choosing the parameters cO and dO in the form (10.33), the terms of the effective operator of any physical quantity do not mix the basis vectors j1i and j2i. That is, the basis set decomposes into a direct sum of two independent sets. Since the spin vector

171

Section 10.4 Phosphorus pentafluoride molecule PF5

in each set is single, it can be omitted, and therefore the spin parts of the effective operators can also be omitted. Then we obtain the effective operators (in particular, the effective Hamiltonian) corresponding to a rigid molecule with the point group C2v . In other words, the vectors j1i and j2i describe the localization of a molecule in two equivalent minima of the potential with equilibrium configurations of symmetry C2v .

10.4 Phosphorus pentafluoride molecule PF5 We now consider the internal dynamics of the PF5 molecule. From a matching of the permutation group 5 of identical nuclei with the point group D3h D D3  CS of equilibrium configuration (the necessary correlations are given in Table 10.6), for the allowed coordinate multiplets we obtain .6 C 4 C 2/A001 ;

4A02 ;

2E 0 ;

.4 C 2/E 00 :

(10.34)

Then the symmetry types of the zero-approximation wave functions are matched with the symmetry types of the allowed multiplets formed on their basis: .mult/D3h Ci D .el /D3h  .vib/D3h  .rot.-in/D3h Ci ;

(10.35)

6

where the admissible representations rot.-in for a given rotational representation of the D1 group of a rigid symmetric top follow from the correlation of the groups D1 and D3h  Ci through their common subgroup D6 , which can conveniently be written as .z/ D3  C2 . For the group C2 D .E; C2 D h i/, the symmetric and antisymmetric representations are denoted by the indices a and b. A classification of the energy levels for the vibronic state el  vib D A01 is presented in Figure 10.6. The inversion doublets are not split, and the actual structure of the levels is described by the rotational group D3  C2 . As mentioned in Section 10.1, the symmetry group C4v describing the Berry pseudorotation through one intermediate configuration does not include the point group D3h as a subgroup and cannot be considered an extended point group of the PF5 molecule. Since each element of pseudorotation can be associated with the permutation of identical nuclei, we use the well-developed apparatus of a permutation group to construct a geometric group allowing for all three independent intermediate configurations. We proceed from the elements of the D3 group and the pseudorotation shown in Figure 10.2. The elements conjugate to them are easy to construct on the basis of equation (3.5). We obtain 66 elements completely filling the classes {15 }, {12 3}, {122 }, and {14} of the 5 group. Now it is easy to form one element in each of the remaining classes {13 2}, {23}, and {5} as a product of two elements from the already filled classes. Defining the elements which are conjugate to this single element, we obtain a complete filling of the last three classes. Thus, the extended point group GH , which is a symmetry group of the coordinate Hamiltonian, is isomorphic with the

172

Chapter 10 Molecules with pseudorotations of the exchange type

Table 10.6. Correlation tables for the PF5 molecule. 5

A001

Œ15  Œ21 

A02

2

3

Œ2 1 2

Œ31 

D3  C2

D3h

A02

A001

A1a CE

00

A001

CE CE

00

C

A002

C

0

0

CE CE

[32]

A01

0

CE CE

[41]

A01

C A002 C E 0

00

00

A01

[5]

R3 J D0 1 2 3 4 5 6 ...

D3h  Ci 0 .C/

A1

0 .C/

A2a

A2

A1b

A1

A2b

A2

Eb

E

Ea

E

Y A F1 T F2 C D DCT F1 C F2 C T A C F1 C D C T ...

0 ./ 0 ./

0 ./

Aa Da Ta Fa

; A1

00./

; A2

00.C/

; A1

00.C/

; A2 ;E

0 .C/

Y^C2

00./

00 .C/

;E

00 ./

D1

D3  C2

A1 A2 E1 E2 E3 E4 E5 E6 ...

A1a A2a Eb Ea A1b C A2b Ea Eb A1a C A2a ...

5  Ci .C/

[5] , [15 ]./ [41].C/ , [213 ]./ [32].C/ , [22 1]./ [312 ].˙/

Note. Correlation of the representation  b of the Y ^ C2 group with representations of the 5  Ci group follows from the correlation of the representation a by replacing the indices C $  in representations of the 5  Ci group.

5 group. Since the rotational operations of the D3 group do not mix the independent equilibrium configurations, the total number of the latter is 5!/6 D 20. In the 5 group, we single out an alternating subgroup which includes only the even permutations (see Section 3.1) to which 60 elements of the classes {15 }, {12 3}, {122}, and {5} belong. The first three classes are also the classes of the alternating subgroup, and the class {5} in the alternating subgroup is divided into classes {5}1 and {5}2 with 12 elements in each class. This is an invariant subgroup, and it can be shown [69] that it is isomorphic with the icosahedron group Y comprising 60 rotations about its symmetry axes. The following correspondence exists between the classes of the alternating subgroup and of the Y group: ¹12 3º $ C31 ; C32 I

¹122 º $ C2 I

¹5º1 $ C51 ; C54 I

¹5º2 $ C52 ; C53 :

Section 10.4 Phosphorus pentafluoride molecule PF5

173

Figure 10.6. Classification of the rotational levels of a rigid molecule of PF5 in the type A01 vibronic state.

As a result, the extended point group can be represented as GH D Y ^ CS :

(10.36)

The nontrivial element of the factor group CS D .E; h / is a reflection in the equatorial triangle plane, and the symmetric and antisymmetric representations of this group will be denoted p and q. In fact, equation (10.36) yields a geometric interpretation of the elements of the GH group. It is important to emphasize the following: In the construction of an extended point group we proceeded from purely rotational elements of the D3 group and the pseudorotation also specified by a rotational element in the symmetry group of the intermediate configuration. Nevertheless, only the elements of the Y subgroup are the rotational elements in the GH group. Indeed, when this subgroup is extended to the GH group, the undivided class {5} comprises the elements corre-

174

Chapter 10 Molecules with pseudorotations of the exchange type

sponding to rotations by angles different in absolute value (for example, C51 and C52 /, which is impossible for the rotational group (see Section 4.3). However, such a conclusion does not contradict the construction procedure for the GH group. The point is that the pseudorotation occurs via three intermediate configurations, not existing simultaneously and differently arranged in space, with different sets of four equivalently located nuclei F. Hence, it cannot be stated that all the elements of the GH group are necessarily rotational. A good confirmation of this is the fact that the GH group includes as its subgroup a point group D3h , which has improper transformations as well. Using the results of matching the groups D3h and 5 , it is easy to obtain a full picture of the level splittings of the rigid molecule PF5 with allowance for the Berry pseudorotation. For el  vib D A01 , this picture is shown in Figure 10.7. Unlike the Berry pseudorotation in the XPF4 molecule, all independent equilibrium configurations are mixed in this case. However, the spectrum of the nonrigid molecule PF5 is strongly depleted since the spin Young diagrams with more than two rows (and therefore the coordinate Young diagrams with more than two columns) are forbidden. The levels of a nonrigid molecule are denoted by representations of the 5  Ci group. However, the corresponding space of the wave functions and the operators of physical quantities specified in it should be written in terms of the geometric interpretation of its elements: 5  Ci ! .Y ^ CS /  Ci D Y ^ .CS  C2 /:

(10.37)

By virtue of the properties of the semidirect product, the elements of the factor group CS  C2 , which do not commute with the elements of the invariant subgroup Y, act in the basis of the latter as well. It suffices to indicate such an action for the element h : h jAi D jAi;

h jDi D jDi;

h jT i D jT i;

h jF1 i D jF2 i:

(10.38)

The sign in the first three expressions in equation (10.38) is chosen for the vibronic state el  vib D A01 . Writing the functions of a certain type of symmetry in the Y ^ .CS  C2 / group in the form of products of the basis unit vectors of the groups Y, CS and C2 , we obtain, with a glance to equation (10.38), the correspondence between these functions and the representations of the 5  Ci group (Table 10.7). Consider the procedure for constructing the function space in the type A01 vibronic state of the PF5 molecule to describe its rotational motion with allowance for transitions of the Berry pseudorotation type between twenty independent equilibrium configurations. We start with the configuration subspace (describing a set of 20 independent configurations), whose basis unit vectors are specified with respect to transformations of the GH D Y ^ CS group. Using the results of the classification for the rotational state with a zero angular momentum, it is easy to obtain : .conf /Y ^CS D Ap C Aq C Dp C Dq C Tp C Tq :

(10.39)

Section 10.4 Phosphorus pentafluoride molecule PF5

175

Figure 10.7. Classification of the rotational levels of the PF5 molecule with allowance for the Berry pseudorotation in the type A01 vibronic state.

We did not take into account the forbiddance of the levels due to their zero nuclear statistical weights. Naturally, the dimension of the resulting subspace is twenty. Passing in equation (10.39) to the subgroup Y, we also obtain : .conf /Y D A C D C T:

(10.40)

The situation with the rotational subspace is nontrivial, since the group GH  Ci D Y ^ .CS  C2 / has two rotational subgroups incompatible with each other. One of them is the subgroup Y and another is the subgroup C2 . The incompatibility arises from the fact that these subgroups cannot be combined into one rotational group. Indeed, as in the group Y ^ CS , one class of the Y ^ C2 group comprises the elements corresponding to rotations by angles different in absolute value. In other words, the

176

Chapter 10 Molecules with pseudorotations of the exchange type

Table 10.7. Correspondence between the representations of the 5 Ci group and the functions of a certain type of symmetry in the Y ^ .CS  C2 / group. 5  Ci

Y ^ .CS  C2 /

Functions

Œ5.C/ Œ5./ Œ41.C/ Œ41./ Œ32.C/ Œ32./

Apa Apb Dpa Dpb Tpa Tpb

2 .C/ 31

Fpa

Ajpijai Ajpijbi Djpijai Djpijbi T jpijai T jpijbi F1 jpijai F2 jpijai F1 jqijbi F2 jqijbi

Fqb

5  Ci

Y ^ .CS  C2 /

Functions

5 .C/ 1

5 ./ 1

3 .C/ 21

3./ 21

2 .C/ 2 1

2 ./ 2 1

Aqb Aqa Dqb Dqa Tqb Tqa

Ajqijbi Ajqijai Djqijbi Djqijai T jqijbi T jqijai F1 jpijbi F2 jpijbi F1 jqijai F2 jqijai



31

 2 ./

Fpb Fqa

purely rotational function cannot be transformed according to representations of the Y ^ C2 group. Therefore, the configuration-rotational functions are constructed as follows. First, the irreducible rotational representation of the R3 group for a given value of the quantum number J of the total angular momentum is decomposed into a direct sum of irreducible representations  rot of the Y group. Then for each rotational component  rot we construct configuration-rotational components for the Y group according to the decomposition : X .˛/ c˛ conf.-rot; (10.41) conf  rot D ˛

where the representation  conf is specified by equation (10.40). It is exactly the components  conf.-rot on the right-hand side of equation (10.41) that determine the basis unit vectors of the Y group in the functions from Table 10.7 symmetrized according to transformations of the Y ^ .CS  C2 / group. Here, the following circumstance plays an important role. According to the stated procedure, on passage from the rotational component  rot of the Y group to the functions symmetrized according to transformations of the Y ^ .CS  C2 / group, the dimension of the function space increases by a factor of 40 (the factor 10 is specified by the dimension  conf in the Y group, and the factor 4 D 2  2, by the number of basis unit vectors in the CS  C2 group). However, the number of independent equilibrium configurations is 20, and therefore only onehalf of the resulting basis unit vectors of the Y ^ .CS  C2 / group are implemented. In other words, there is a correlation between the representations of the groups Y, CS and C2 in these unit vectors, which is clearly seen in Figure 10.7 (an auxiliary group Y ^ C2 was introduced to clarify the behavior of the levels with respect to transformations of the groups Y and C2 /. We also emphasize that a given representation  rot of the Y group generally corresponds to the multiplets of both a and b types.

Section 10.4 Phosphorus pentafluoride molecule PF5

177

The consequences of this correlation are the most simple for the wave functions belonging to the sixfold degenerate representations Fpa and Fqb (which correspond to the representation [312 ].C/ / and Fpb and Fqa (which correspond to the representation [312 ]./ /. We single out these representations because only during their formation do we simultaneously use two representations of the Y group (F1 and F2 /. The Hamiltonian of the internal coordinate motion belongs to the representation [5].C/ or the representation Apa . It is clear that the Hamiltonian must bind the functions corresponding to the representation [312 ].C/ . However, this requires that only one of the representations Fpa and Fqb should be preserved. The situation is similar with the functions corresponding to the representation [312 ]./ . The choice of functions for the representations [312 ].C/ and [312 ]./ is not independent, since the coordinate operator belonging to the representation [5]./ or Apb (for example, the operator of the electric dipole moment) must bind them. We assume for definiteness that the wave functions of the representations Fpa and Fpb are preserved. For the rest of the wave functions of the Y ^.CS C2 / group, we give a few specific examples of their construction to consider the correlation consequences. We start with the states with J D 0. In this case, .rot /Y D A and equation (10.41) takes the form : conf  rot D A C D C T:

(10.42)

In the classification, the levels with J D 0 correspond to the representation Aa of the rotational group D3 C2 . Allowance for the classification results leads to the following extension of the function space given by the right-hand side of equation (10.42) when we pass to the Y ^ .CS  C2 / group: Apa C Aqa C Dpa C Dqa C Tpa C Tqa :

(10.43)

Since only the type a functions are present here, the dimension of space during its extension increased only twofold and became twenty, as it should be. However, unlike the wave functions of types Fpa and Fpb , the second half of the functions may occur, in principle, in the extension of the right-hand side of equation (10.42), but it is implemented for other rotational levels. Note that the nuclear statistical weights of the states Apa , Dpa , and Tpa are equal to zero, and thus they are absent in Figure 10.7. Beginning with J D 1, some additional difficulties appear in this consideration. Indeed, for J D 1 we have .rot/Y D F1 , and equation (10.41) can be written as : conf  rot D 2.F1 C F2 C D C T /:

(10.44)

Taking into account the results of classification of the states with J D 1 (which correspond to the representation A2a C Eb of the rotational group D3  C2 /, we obtain the following extension of the function space on passage to the group Y ^ .CS  C2 /: 2.Fpa C Fpb C Dpb C Dqb / C Tpa C Tqa C Tpb C Tqb :

(10.45)

178

Chapter 10 Molecules with pseudorotations of the exchange type

The dimension of space (10.45) is sixty, that is, the dimension again increases twofold during the extension. However, most of the states in this space have zero nuclear statistical weights, and only its subspace of dimension eighteen is actually implemented: It is essential here that decomposition (10.44) contains two representations of one type for configuration-rotational functions in the Y group, and it is necessary to specify exactly which representation of the Y ^ .CS  C2 / group is the result of the passage of each representation of such a pair. This gives rise to difficulties if representations of one type of the Y group convert into different representations of the Y ^ .CS  C2 / group. This takes place for the type T representations. Then it is necessary to know already the algebraic properties of the configuration-rotational functions in the Y group and, in particular, their expression through rotational functions jJ; ki of the R3 group in the MCS and the configuration basis functions jli in subspace (10.40). In a similar way, we can continue by considering the higher values of the quantum number of the total angular momentum. In the energy-level classification of the nonrigid molecule PF5 in Figure 10.7, representations of the 5 Ci group can be replaced by representations of the geometric group Y ^ .CS  C2 /. The latter, in particular, are easy to form from representations of the auxiliary group Y ^ C2 by adding the indices p and q with allowance for the following correlation conditions: .pa; qb/ ! C;

.pb; qa/ ! :

Concluding this section, we emphasize that there is a very important distinction between the descriptions of the Berry pseudorotation in the molecules XPF4 and PF5 . Indeed, the XPF4 molecule is a typical system with distorted equilibrium configuration, and the distortion is due to the geometrically nonequivalent location of the identical nuclei. The internal dynamics symmetry with allowance for the transition of the Berry pseudorotation type between two equilibrium configurations is characterized by the symmetry group C4v of the intermediate configuration with equivalent location of the nuclei. This group includes as a subgroup the point group C2v of both equilibrium configurations and hence characterizes both the equilibrium configurations and the transitions between them. In addition, this group characterizes one geometric figure, making it possible to easily single out the rotation of the molecule as a whole. The case is absolutely different with the PF5 molecule. It is not that the number of independent equilibrium configurations mixed with the Berry pseudorotation increases dramatically. It is important that this mixing occurs through three intermediate configurations not existing simultaneously and differently arranged in space, that is, the transformations belong to the different geometric figures, and their rotational interpretation can no longer be extended to the system as a whole. Hence, the identification of rotational transformations is a very complicated problem whose solution requires the detailed analysis of the structure of the total geometric group.

179

Section 10.5 The separation of internal motions

10.5 The separation of internal motions Rotational subgroup Grot of the direct product of the point group of a molecule and the spatial inversion group plays an important role in describing the internal dynamics of rigid molecules. Group Grot, all transformations of which correspond to rotations without the barrier of the molecule as a whole, determines the symmetry of the rotational motion on account of all internal interactions. Hence, it is exactly this group, and not the rotational subgroup of the point group, that characterizes the type of the top to which the rigid molecule belongs. For example, the point group D2d contains only the second-order rotational axes. At the same time, the group D2d  Ci has a fourth-order rotational axis; and therefore the molecules with the point group D2d belong to the symmetric top type. Besides, since the inversion doublets in rigid molecules are not split (or are absent), the group Grot also describes the actual structure of the energy levels of the internal motion. In the nonrigid molecules with one intermediate configuration, we construct two .1/ rotational groups. One of them is a complete rotational subgroup Grot of the direct product of an extended point group and a spatial inversion group. Since the group .1/ Grot generally includes not only the rotations of the molecule as a whole, singling out .2/ . When all considered nonrigid motions such rotations in it leads to its subgroup Grot .1/ are specified by rotational operations, the group Grot describes the actual structure of the energy spectrum of the internal motion. The problem of separation of internal motions arises quite often in the nonrigid molecules. A well-known example is the molecules with two identical tops on the linear backbone (see Section 9.3), where the torsional motion should be separated from the rotation of the molecule as a whole about the linear backbone. Nontriviality of this problem even leads to philosophical difficulties with the application of the CNPI group [15,16], because of which some purely coordinate motions are described by twovalued wave functions that change their sign during rotation by 2. As a consequence, the so-called double groups, introduced to describe spin motions of the half-integer spin systems [64], are employed. Certainly, this is physically incorrect (see Sections 12.5 and 12.7). In the concept of the symmetry group chain, no difficulties arise, and a general recipe for separating the rotational motion is very simply formulated. It suffices .2/ to require that the group Grot operations corresponding to rotations of the molecule as a whole without the barrier be defined only in the rotational subspace. For example, for the ethane molecule C2 H6 we have .1/

.2/

Grot D B6  Grot ; .2/

.2/

Grot D D3 ;

(10.46)

where the group Grot includes the transformations corresponding to the rotations with.1/ out the barrier. Representing Grot as (10.46) makes it possible to describe independently the torsional and rotational motions by the groups B6 and D3 , respectively. The case of independent description is possible only if rigid requirements are imposed on

180

Chapter 10 Molecules with pseudorotations of the exchange type

the symmetry of the problem. A less symmetric case is implemented in the molecules with planar torsion tops, such as ethylene C2 H4 (point group D2h/ and allene C3 H4 (point group D2d /. Here, the axis of the top is only a second-order axis, and the group .1/ Grot can no longer be represented as a direct product of the groups describing the different types of motion. The point is that the identity of the tops requires the symmetric use of the torsional operation c21  E and E  c21 . However, their product c21  c21 describes the rotation of the molecule as a whole without the barrier since this element does not bind independent local minima of the effective nuclear potential and therefore does not take the molecule from one minimum to another. To separate the rotational and torsional motions, it suffices to require that the operation c21  c21 be defined only in the rotational subspace. The case where the pseudorotation in the cyclobutane molecule C4 H8 and in molecules of the XPF4 type is described (see Sections 10.2 and 10.3) is interesting. For example, for the molecule of the XPF4 type we have .1/

Grot D D4 D C4 ^ U2 ;

.2/

.1/

Grot D Grot :

(10.47)

Pseudorotation is specified by operations C41 and C43 of the C4 group, which simultaneously are rotations of the molecule as a whole through a barrier corresponding to the pseudorotation. Therefore, these rotations are defined in the pseudorotational subspace and the rotational subspace. At the same time, the element C42 describes the rotation of the molecule as a whole through a zero barrier and is defined only in the rotational subspace. It is important to note here that the elements C41 and C43 bind the independent minima of the effective potentials of the nuclei and therefore take the molecule from one minimum to another, while the element C42 keeps it at the initial minimum. As a consequence, the complete basis set in the pseudorotational subspace comprises only two unit vectors belonging to the totally symmetric representation A in the C4 group and the representation B which is antisymmetric with respect to the elements C41 and C43 . Only such unit vectors are invariant with respect to the purely rotational element C42 . The dimension of this subspace determines the number of mixed independent equilibrium configurations (we obtain two configurations, as it should be). Thus, the prior recipe for separating motions is employed. The procedure is similar for the cyclobutane molecule. Generally speaking, the situation is much more difficult when the nonrigid motions characterized by a few intermediate configurations are described. The problem is that these configurations can be differently arranged in space. In this case, the interpretation of the transformations specified relative to the different intermediate configurations is very difficult to extend to the system as a whole. Here we are referring primarily to the identification of rotational transformations of the molecule as a whole. This is the situation for the phosphorus pentafluoride molecule PF5 (see Section 10.4).

Section 10.6 Conclusions

181

10.6 Conclusions We have formulated some general conclusions based on the content of this chapter. 1. Using a description of the pseudorotation in the cyclobutane molecule C4 H8 (Section 10.2) as an example, we show how to construct a rigorous algebraic model for the entire excitation band of nonrigid motion in a nondegenerate electronic state (the model is rigorous in the sense that its correctness is limited only to the correctness of choosing the symmetry of the internal dynamics). Such a model in the fitting of experimental data allows for the influence of the frequently encountered accidental resonances between the pseudorotational states. We are speaking here of the influence of not only these resonances on the position of the energy levels, but also on the transition intensities between them, which is sometimes no less important. The latter is due to the fact that accidental resonances can change these intensities by orders of magnitude. In Chapter 18, we consider the construction of similar models for the total energetic spectrum of a nonrigid molecule in a nondegenerate electronic state. Perhaps such an approach is currently the only one possible for a correct description of major problems of the internal dynamics of nonrigid molecules. 2. To a large extent, the efficiency of the qualitative methods set forth is due to the fact that the trajectory of the nonrigid motion should not be considered in some internal coordinates. One needs only to specify the intermediate configuration, by transformations of which the geometry of this motion is defined. This advantage becomes still more important with increasing complexity of the nonrigid motion. 3. It is easy to solve the problem of separation of the molecule rotation as a whole without the barrier with torsional and pseudorotational nonrigid motions. It suffices to require that the rotation operations of the molecule as a whole without the barrier be defined only in the rotational subspace. It is important here that there operations do not bind the independent minima of the effective nuclear potential and therefore do not take the molecule from one minimum to another. In particular, there are no problems with the separation of the rotational and torsional motions in the molecules with two identical tops on the linear backbone. On application of the CNPI group, the solution of this problem leads to philosophical difficulties, because of which some coordinate motions are described by two-valued wave functions, which is physically incorrect (see Sections 12.5 and 12.7).

Chapter 11

Molecules with transitions of the nonexchange type between equivalent configurations 11.1 Extended point groups Motions of the nonexchange type between equivalent configurations are quite common in nonrigid molecules. A simple example is the so-called inversion motion in the ammonia molecule NH3 [64], in which the N nucleus tunnels through the plane of the H nuclei and emerges in the energetically equivalent position on another side. Naturally, this plane also moves to preserve the position of the center of mass of the molecule, that is, the equilibrium configuration in the form of a regular triangular pyramid “rotates outwards” relative to the point of the center of mass and changes its position in space in such a way that all the nuclei occupy a new position. The extended point group has the form F12 D c3v  CS ; (11.1) where C3v is a point group of the molecule, and the nontrivial element of the CS group, which determines the inversion motion, is a reflection in the plane passing through the center of mass of the molecule orthogonally to the third-order axis of the C3v group. Thus, the group F12 , which allows for the presence of two independent configurations bound by the inversion motion, is a dynamic noninvariant group with the symmetry subgroup FH of the Hamiltonian, which coincides with the point group C3v . As another example, we consider a molecule of hydrogen peroxide HOOH. Its equilibrium configuration, shown in Figure 11.1, corresponds to the point group C2 D .x/ .E; C2 / [52]. Hence, the molecule has stereoisomers. Recall that the existence of stereoisomers in a molecule means [64] that its inverted equilibrium configuration by rotating the configuration as a whole cannot be superposed spatially with the initial configuration. This is possible only if the point group has no improper transformations (see Section 4.2). In rigid molecules, the stereoisomers are not mixed, so one can speak of their “right” and “left” modifications, whose energy levels coincide. It is easy to understand that only the nonrigid motions of the nonexchange type can mix the stereoisomers since this requires change in the spatial position of the initial configuration. In the hydrogen peroxide molecule, the OH structures tunnel between the energetically equivalent positions. Since the center of mass of the molecule in equilibrium configuration does not lie on the chemical bond OO, the nonrigid motion can be represented as a synchronous rotation of these structures about the z axis towards each other (cis transition) or from each other (trans transition), with the trans transition

Section 11.1 Extended point groups

183

Figure 11.1. Equilibrium configuration of the HOOH molecule (0 is the point of the center of mass and 2˛ is a dihedral angle between the planes H1 O1 O2 and H2 O2O1 ).

dominating [72]. It is clear that the extended point group has the form F8 D C2  CS  CS0 ;

(11.2)

where the group CS D .E;  .yz/ / specifies the trans transition and the group CS0 D .E;  .xz/ /, the cis transition. The symmetry subgroup of the Hamiltonian in the noninvariant group F8 is a point group of the C2 molecule. In this respect, the situation is similar to the ammonia molecule. Nonrigid motions of the nonexchange type in the hydrazine molecule N2 H4 are a much more interesting example. An important feature of this molecule is that its .y/ equilibrium configuration corresponding to the point group C2 D .E; C2 / [52] is close to the intermediate configuration shown in Figure 11.2. In the latter configuration, two equivalent structures of NH2 are isosceles triangles, and the dihedral angle between them is /2. As a consequence, each of such structures can be transferred through a fairly low potential barrier to an energetically equivalent position using either the reflection 2 in the plane passing through the z axis in parallel to the segments H1 H2 or H3 H4 (inversion motion) or rotation by an angle  about the z axis (internal rotation) [62,85]. These motions of the nonexchange type delocalize the molecule in eight independent equilibrium positions, which is taken into account by the group .z/ Fr D c2v  c2v . Here, two groups c2v D .E; c2 ; 1 ; 2 / characterize the geometry of the nonrigid motions of two structures of NH2 , such that the following relation is .y/ satisfied for an arbitrary element p  q of the group F and the element C2 : .p  q/C2.y/ D C2.y/ .q  p/:

(11.3)

We emphasize that if the element p  q changes the spatial position of one structure of NH2 , then the position of the symmetry axis of the point group also changes from

184

Chapter 11 Nonexchange transitions between equivalent configurations

Figure 11.2. Intermediate configuration of the N2 H4 molecule (the z axis coincides with the straight line between the centers of mass of the equivalent structures of NH2).

y to x. However, equation (11.3) holds since its right-hand side is still written in terms of the symmetry problem: .y/

.x/

.z/

.x/

.z/

.z/

C2 .q  p/ D C2 C2 .q  p/ D C2 .c2 q  c2 p/:

(11.4)

Here, it was taken into account that the element c2.z/  c2.z/ changes the position of both structures of NH2 and therefore leaves the position of the symmetry axis of the point group the same. It follows from equation (11.3) that the extended point group of the N2 H4 molecule can be represented as (11.5) F32 D .c2v  c2v / ^ C2 ; since the group Fr is an invariant subgroup of the F32 group. Division into classes and the table of characters of the F32 group are presented in Table 11.1. It is clear that F32 is a noninvariant dynamic group with the symmetry subgroup of the Hamiltonian FH D .cS  cS / ^ C2 ;

(11.6)

where cS D .E; 1 /. The element 1 belonging to the exchange type results from a sequential implementation of two motions of the nonexchange type of the NH2 structure. As a consequence, the group FH is much wider than the point group of the molecule. The molecules with transitions of the nonexchange type have very important specific features in the construction of a qualitative description, since the required internal geometric group no longer coincides with the symmetry group of the coordinate Hamiltonian in a given electronic state, but only includes it as a subgroup.

185

Section 11.2 The ammonia molecule NH3

Table 11.1. Division of the elements into classes and the table of characters of an extended point group F32 of the hydrazine molecule. E

E E

c2

c2  c2

1

1  1

2

2  2

L1

E  c2 c2  E

L4

c2  1 1  c2

L2

E  1 1  E

L5

c2  2 2  c2

L3

E  2 2  E

L6

1  2 2  1

M1

C2  C2.y/ .c2  c2 /C2 .1  1 /C2 .2  2 /C2

M2

.E  c2/C2 .c2  E/C2 .1  2 /C2 .2  1 /C2

Note. c2  c2.z/ .

M3

.E  1 /C2 .1  E/C2 .c2  2 /C2 .2  c2 /C2

M4

.E  2 /C2 .2  E/C2 .c2  1 /C2 .1  c2 /C2

F32

E

c2

1

2

L1

L2

L3

L4

L5

L6

M1

M2

M3

M4

A1 A2 A3 A4 A5 A6 A7 A8 E1 E2 E3 E4 E5 E6

1 1 1 1 1 1 1 1 2 2 2 2 2 2

1 1 1 1 1 1 1 1 2 2 –2 –2 –2 –2

1 1 1 1 1 1 1 1 –2 –2 2 2 –2 –2

1 1 1 1 1 1 1 1 –2 –2 –2 –2 2 2

1 1 1 1 –1 –1 –1 –1 2 –2 0 0 0 0

1 1 –1 –1 1 1 –1 –1 0 0 2 –2 0 0

1 1 –1 –1 –1 –1 1 1 0 0 0 0 2 –2

1 1 –1 –1 –1 –1 1 1 0 0 0 0 –2 2

1 1 –1 –1 1 1 –1 –1 0 0 –2 2 0 0

1 1 1 1 –1 –1 –1 –1 –2 2 0 0 0 0

1 –1 1 –1 1 –1 1 –1 0 0 0 0 0 0

1 –1 1 –1 –1 1 –1 1 0 0 0 0 0 0

1 –1 –1 1 1 –1 –1 1 0 0 0 0 0 0

1 –1 –1 1 –1 1 1 –1 0 0 0 0 0 0

11.2 The ammonia molecule NH3 Inversion motion in the NH3 molecule is perhaps the best known example of nonrigid motion of the nonexchange type. In a rigid approximation, this molecule was considered in Section 8.1. To allow for the inversion motion, it is necessary to introduce an extended point group F12 D C3v  CS into a chain between the groups 3 and C3v (see Section 11.1). This is a dynamic noninvariant group, whose symmetry subgroup of the Hamiltonian coincides with the point group C3v . Since the matching of the groups F12 and 3 reduces to a matching of the groups C3v and 3 , the allowed coordinate multiplets of a nonrigid molecule have the following form: 4A2s ;

4A2a ;

2Es ;

2Ea :

(11.7)

Compared with multiplets (8.15) for a rigid molecule, there appear the indices s and a, pointing to the symmetry and antisymmetry of the multiplet with respect to the reflection in the h plane of the CS group. The symmetry types of these multiplets are

186

Chapter 11 Nonexchange transitions between equivalent configurations

matched with the symmetry types of the basis functions of separate internal motions as follows: .mult/C3v CS Ci D .el /C3v  .vib/C3v  .rot.-conf/C3v CS Ci :

(11.8)

6

The rotation-configuration representation rot.-conf allows for all configurations arising in this problem. Operation h is not included in the point group, and therefore, similarly to the operation i, it characterizes only the multiplet. The admissible type of rot.-conf for a given rotational representation of the D1 group of a rigid symmetric top follows from the relation between the groups D1 and F12 Ci through their common subgroup D6 (see Table 11.2). The latter can be written as D3  C2 . Symmetric and .z/ antisymmetric representations of the C2 D .E; C 2 / group are denoted as e and o. Table 11.2. Correlation between irreducible representations of the groups D1 and F12  Ci . D1

D3  C2

C3v  CS  Ci

A1

A1e

A1s , A2a

A2

A2e

.C/ ./ A2s , A1a

E1

Eo

Es , Ea

E2

Ee

Es

E3

A1o C A2o

E4

Ee

Es

E5

Eo

Es , Ea

E6

A1e C A2e

A1s , A2a C A2s , A1a

...

...............

..............................

.C/

./

.C/

./

./

.C/ ./

, Ea

.C/

./

.C/

A2s , A1a C A1s , A2a .C/ ./

.C/

./

./

, Ea

.C/ .C/

./

Figure 11.3 shows a classification of the inversion-rotational levels for el  vib D A1 . The introduction of rot.-conf instead of rot.-in makes it possible to allow for the dependence of the configurations obtained by using the operations i and h acting simultaneously on the electronic and nuclear coordinates. These configurations are related by the operation of rotation of the molecule as a whole through a zero barrier. In total, the group F12  Ci “sees” four equivalent configurations with the point group C3v , of which only two are independent. The matching of type (11.8) allows for this fact, and as a consequence, no extraneous levels are present in the classification. Indeed, the indices s, a, and ˙ in Figure 11.3 are not independent, and there is a fairly nontrivial correlation between them. The group F12  Ci can also be represented as D6  CS . Consequently, all of its elements are expressed through operations defining the considered types of motion. In other words, all operations correspond to zero

Section 11.2 The ammonia molecule NH3

187

Figure 11.3. Classification of the inversion-rotational levels of the NH3 molecule in the type A1 vibronic state.

or sufficiently low barriers. As a result, the actual structure of the energy levels is described by the total group F12  Ci . In particular, this leads to a splitting of the inversion doublets. In this case, the rotational subgroup D6 plays only a subsidiary role (it permits one, e.g., to single out the splitting stipulated by the k doubling effect). The complete basis set of inversion unit vectors jsi and jai is determined by two irreducible representations of the CS group. Since these representations are real, one can choose T jsi D jsi; T jai D jai: (11.9) Rotational unit vectors are specified by irreducible representations of the D6 group, which includes rotations of the molecule as a whole from the groupF12  Ci . As a

188

Chapter 11 Nonexchange transitions between equivalent configurations

result, we decompose the basis inversion-rotational unit vectors for the vibronic state el vib D A1 into irreducible representations of the F12 Ci group (see Table 11.3). A two-dimensional representation (7.27) of Lie algebra in the space of the basis inversion unit vectors jsi and jai is written as (7.29), assuming that these unit vectors are eigenfunctions of the operator e3 . Knowing the symmetry properties of the basis unit vectors for the transformations h and T, it is easy to find that the operator e3 belongs to the representation s of the CS group and is t-even, while the operators e1 and e2 belong to the representation a and are t-even and t-odd, respectively. In the two-dimensional space, a complete set of self-adjoint operators also includes a totally symmetric unit operator I. Table 11.3. Decomposition of the inversion-rotational unit vectors for the vibronic state el  vib D A1 of the NH3 molecule into irreducible representations of the C3v  CS  Ci group. F12  Ci .C/ A1s .C/ A1a .C/ A2s .C/ A2a .C/ Es

Ea.C/

Functions

F12  Ci

Functions

A1e jsi A1o jai

A2o jsi A2e jai

Ee jsi

./ A1s ./ A1a ./ A2s ./ A2a ./ Es

Eo jai

Ea./

Ee jai

A2e jsi A2o jai

A1o jsi A1e jai Eo jsi

Note. The rotational parts are specified by irreducible representations of the D3  C2 group.

The effective inversion-rotational operator of any physical quantity (including the Hamiltonian) belongs to the coordinate Young diagram [3] of the 3 group, and the following representations are possible for it in the F12 group: A1s ;

A1a :

(11.10)

Since noninvariant terms are present in the Hamiltonian for transformations of the CS group, the indices s and a are not the symmetry indices for stationary wave functions. This is exactly the reason why the wave functions, for which the irreducible representations differ in only these indices, are written together in Table 11.3. Correspondingly, in the group D6  CS we obtain the representation A1e s; A1o a

(11.11)

for the physical quantities that are invariant under the inversion i and the representations (11.12) A2o s; A2e a

189

Section 11.3 The peroxide molecule HOOH

for the quantities that change their sign in this transformation. A full expression for the inversion-rotational Hamiltonian can be written as H D .I; e3 /  A1e C .e1 ; e2 /  A1o :

(11.13)

Unwinding this expression with allowance for the t-parity of the Hamiltonian, we obtain 1  X .A1e / .A1o/ .A1o/ H2n : (11.14) C H2nC4 C H2nC3 H D nD0

On the right-hand side, the superscript of the contribution to the Hamiltonian determines the type of the irreducible representation of the D3  C2 group, by which combinations of the angular-momentum components in the MCS are transformed and the subscript specifies the total power of combinations for these components. These contributions have the form X .A / 6t cO2p;2s;6t J 2p Jz2s .JC C J6t /; (11.15) H2n 1e D p;s;t .A

/

1o H2nC4 D

X

6t C3 C J6t C3 /; dO2p;2sC1;6t C3 J 2p Jz2sC1 .JC

(11.16)

6t C3 i gO2p;2s;6t C3 J 2p Jz2s .JC  J6t C3 /;

(11.17)

p;s;t .A

/

1o H2nC3 D

X p;s;t

where pCsC3t D n for a given n. On account of the inversion motion, the parameters c, O dO , and gO are spin operators: cO D c .1/ I C c .2/ e3 ;

dO D de1 ;

gO D ge2 ;

(11.18)

where c .k/ , d, and g are the real spectroscopic constants (the subscripts are omitted). Contributions to the Hamiltonian (11.16) and (11.17) are noninvariant for the transformation h and mix the basis functions, of which irreducible representations differ only in the indices s and a. Therefore, the matrix of the Hamiltonian (11.14) for a given value of the quantum number J is divided into a direct sum of only six different types of the energy matrices. The effective operator of the electric dipole moment is transformed according to representations (11.12) of the D6  CS group. This makes it easy to write the electric dipole selection rules, as well as to construct the electric dipole moment operator itself.

11.3 The peroxide molecule HOOH For the primary isotope 16 O of the oxygen nucleus, the spin is zero. In this case, as the permutation group of identical nuclei of the molecule, one can use, without loss of

190

Chapter 11 Nonexchange transitions between equivalent configurations

generality, the permutation group 2 of the hydrogen nuclei. From its matching with the point group C2 of the equilibrium configuration (the necessary correlations are given in Table 11.4) we have the following allowed coordinate multiplets for a rigid molecule: 1A; 3B: (11.19) Then the symmetry types of the zero-approximation wave functions are matched with the symmetry types of the allowed multiplets formed on their basis: .mult/C2 Ci D .el /C2  .vib/C2  .rot.-in/C2 Ci ;

(11.20)

6

where the admissible representations rot.-in for a given rotational representation of the group D2 D .E; C2.3/; C2.2/, and C2.x/) of a rigid asymmetric top are determined from the correlation of the groups D2 and C2  Ci through their common subgroup C2 . As a result, we have a classification of the energy levels in an arbitrary vibronic state, which is shown for el  vib D A in Figure 11.4. The inversion doublets are not split, and the actual structure of the levels is described by the rotational subgroup C2 . Note that for a molecule with stereoisomers, such a subgroup just coincides with the point group of the molecule, and the inversion doublet includes the states differing only in behavior relative to the operation of spatial inversion i. Table 11.4. Correlation table for the HOOH molecule. H2  D20

2

C2

F8  Ci

Œ2

A

Acp ; Adq

./

As

2

B

./ Acp ; A.C/ dq

Aa

./ A.C/ cq ; Adp

B3s

[1 ]

D2

.C/

C2  Ci .˙/

A, B3

A

B1 , B2

B.˙/

Acq ; Adp

./

.C/

B3a

.C/ Bcp ;

./ Bdq

B1a

./ Bcp ;

.C/ Bdq

B1s

.C/ Bcq ;

./ Bdp

B2a

.C/

B2s

./

Bcq ; Bdp

For a nonrigid molecule, the extended point group F8 has the form (11.2), and the nontrivial elements of the groups CS and CS0 , which specify the trans and cis transitions, belong to the nonexchange type. Therefore, the matching of the groups 2 and

191

Section 11.3 The peroxide molecule HOOH

Figure 11.4. Energy-level classification for a rigid molecule of HOOH in the type A vibronic state.

F8 reduces to a matching of the groups 2 and C2 , which for a nonrigid molecule leads to the allowed multiplets 1Ai k ;

3Bi k ;

(11.21)

where the values of the indices i D c; d and k D p; q determine the symmetric and antisymmetric representations of the groups CS and CS0 , respectively. The rotational motion of a nonrigid molecule is specified by the group D20 D .E; C2

.z/

.y/

D  .yz/  .xz/ ; C2

.x/

.x/

D C2  .yz/  .xz/; C2 /;

which includes the rotation of the molecule as a whole from the group F8  Ci . Note that the group D2 is not preserved since the corresponding ellipsoid of inertia changes their orientation in the nonrigid transitions (among the elements of the D20 group, only .x/ the element C2 , which is now in the group D2 , remains). For the construction of multiplets (11.21) we have, instead of equation (11.20), .mult/C2 CS CS0 Ci D .el /C2  .vib/C2  .rot.-conf/C2 CS CS0 Ci ;

(11.22)

6

where the possible types of rot.-conf for a given rotational representation of the D20 group of a rigid asymmetric top are determined from the correlation of the groups D20 and F8  Ci through their common subgroup D20 . In addition, the fact that each element of the D20 group has two counterparts in the F8  Ci group plays an important .x/

.x/

role. Thus, for the element C2 there are counterparts C2 and  .yz/ i. From the requirement of coincidence of the counterparts, we obtain the correlation .z/

C2  .yz/  .xz/ $ E; .z/

(11.23)

where the action of the operation C2 is determined by the symmetry of the rotational part, and the action of the operations  .yz/ and  .xz/ by the symmetry of the multiplet

192

Chapter 11 Nonexchange transitions between equivalent configurations

(this situation is similar to that considered in Section 8.2 for rigid molecules with inversion center). Interestingly, the constraints selecting physically correct solutions of the Schrödinger equation for a nonrigid molecule of HOOH, which were obtained in [42], follow immediately from equation (11.23). The energy-level classification of a nonrigid molecule, which was obtained according to equations (11.21)–(11.23), is shown in Figure 11.5 for el  vib D A. The actual structure of the levels is described by the total group F8  Ci , since all of its elements are expressed through operations defining the considered types of motion. The rotational subgroup H2  D20 of the F8  Ci group is used as an auxiliary. Symmetric .z/ and antisymmetric representations of the group H2 D .E; C2  .yz/  .xz/ / are de.z/

noted as s and a. It is clear that in the rotational space the operation C2  .yz/  .xz/ is equivalent to E. From the picture of splittings of the rotational levels it is clearly seen that nonrigid transitions mix only two independent configurations. This means

Figure 11.5. Energy-level classification for a nonrigid molecule of HOOH in the type A vibronic state.

193

Section 11.3 The peroxide molecule HOOH

that the configurations obtained from the initial one using trans and cis transitions are dependent. We emphasize that this conclusion is a consequence of the construction procedure, and it agrees with the experimental data. The existence of only two independent configurations leads to a fairly nontrivial correlation between the symmetry indices characterizing the energy levels in this problem. Consider a description of the energy levels of a nonrigid molecule in the type A vibronic state. Complete sets of the basis trans unit vectors jci and jd i and cis unit vectors jpi and jqi are specified by irreducible representations of the groups CS and CS0 , respectively. Basis rotational unit vectors are specified by irreducible representations of the group D20 . Now it is ncessary to decompose the products of the unit vectors of different motions into irreducible representations of the group F8 Ci . All of its operations are easy to express as products of the elements whose action is defined in the space of the basis trans-cis-rotational unit vectors. Indeed, operations of the groups CS , CS0 and D20 act only on the trans, cis, and rotational unit vectors, respectively. Operation i is written in the form .x/

i D  .yz/ C2

.y/

D  .xz/ C2 :

(11.24)

It follows that both the trans and cis transitions mix the stereoisomers. Naturally, the admissible trans-cis-rotational unit vectors (given in Table 11.5) should satisfy the correlation condition (11.23). To describe the trans transition, we introduce the operator of coordinate spin e. Two-dimensional representations (7.27) of Lie algebra in the space of the trans unit vectors jci and jd i are written in the form (7.29), assuming that these unit vectors are eigenfunctions of the operator e3 . Knowing the symmetry properties of the basis unit vectors for transformations of the group CS and operation T, T jci D jci;

T jd i D jd i;

(11.25)

it is easy to find that the operator e3 belongs to the representation c of the group CS and is t-even, while the operators e1 and e2 belong to the representation d and are t-even and t-odd, respectively. For the cis transition, we introduce the operator of coordinate spin  . Symmetry properties of its components are similar to those for the components of the operator e with accuracy up to the replacement of the group CS by the group CS0 . The unit operator in the space of the trans and cis transitions will be denoted I1 and I2 . The effective operator of any physical quantity of a nonrigid molecule relates to the coordinate Young diagram [2] of the 2 group. Hence, in the group F8 for these operators we have the following set of representations: Ai k

.i D c; d I

k D p; q/:

However, the correlation condition (11.23) and the limitedness of the admissible types of symmetry for rotational operators in the H2  D20 group lead to the fact that only

194

Chapter 11 Nonexchange transitions between equivalent configurations

Table 11.5. Decomposition of the complete set of trans-cis-rotational unit vectors into irreducible representations of the F8  Ci group. F8  Ci .C/

Acp

.C/ Acq ./ Adp ./

Adq

Basis unit vectors

F8  Ci

jcijpiA

Bcp

./

jcijpiB1

jcijqiB3

./ Bcq

jcijqiB2

jd ijpiB3

.C/ Bdp

jd ijpiB2

jd ijqiA

Bdq

.C/

jd ijqiB1

Basis unit vectors

Note. The rotational parts are specified by irreducible representations of the D20 group.

the representations A.C/ cp ;

A.C/ cq ;

./

Adp ;

./

Adq

(11.26)

are possible for the operators of physical quantities in the F8  Ci group. In particular, the effective Hamiltonian belongs to the first pair of representations in equation (11.26) and the effective operator of the electric dipole moment, to the second pair. Taking into account that the Hamiltonian is t-even, its full expression can be written as 1 X .A/ .B3 / .B3 / HD .H2n C H2nC2 C H2nC1 /; (11.27) nD0

where the superscript specifies the type of the irreducible representation of the D20 group, according to which combinations of the angular momentum components in the MCS are transformed, and the subscript determines the total power of the combinations of these components: X .A/ 2t H2n D cO2p;2s;2t J 2p Jz2s .JC C J2t /; (11.28) p;s;t .B /

3 H2nC2 D

X

2t C1 i dO2p;2sC1;2t C1 J 2p Jz2sC1 .JC  J2t C1 /;

(11.29)

2t C1 gO 2p;2s;2t C1 J 2p Jz2s .JC C J2t C1 /;

(11.30)

p;s;t .B /

3 H2nC1 D

X p;s;t

195

Section 11.4 The hydrazine molecule N2 H4

where p C s C t D n. On account of the nonrigid motions, the parameters c, O dO , and gO are spin operators of the form cO D.c .1/ I1 C c .2/ e3 /I2 C .c .3/ I1 C c .4/ e3 /3 ; dO D.d .1/ I1 C d .2/ e3 /1 ; gO D.g

.1/

I1 C g

.2/

(11.31)

e3 /2

with the real spectroscopic constants c .k/ , d .k/ , and g .k/ . Note that the trans transition contributes only to the diagonal terms of Hamiltonian (11.27) in the space of spin states, while the cis transition contributes to the nondiagonal (or resonant) terms as well. Due to the noninvariant contributions, the matrix of the Hamiltonian decomposes into a direct sum of only four different types of the energy matrices. These matrices belong to four irreducible representations of the C2  Ci group, where the group C2 is present as a symmetry subgroup of the Hamiltonian in the noninvariant group F8 . Hence, the unit vectors in Table 11.5 are decomposed into four units. Correspondingly, the electric dipole selection rules in the F8  Ci group ./

./

.C/ .A.C/ cp ; Acq / $ .Adp ; Adq /;

.C/

.C/

./ ./ .Bdp ; Bdq / $ .Bcp ; Bcq /

(11.32)

are also written by using four families of states belonging to the different irreducible representations of the C2  Ci group.

11.4 The hydrazine molecule N2 H4 Nonrigid motions of the nonexchange type in the ammonia and hydrogen peroxide molecules have one common property simplifying their description. The point is that although the independent equilibrium configurations related to these motions occupy different positions in space, the elements of their point groups spatially coincide (in the ammonia molecule, the axis C3 and three planes v are the same for both configurations, and in the hydrogen peroxide molecule, the spatial position of the C2 axis is retained). In this respect, the situation is similar to that for the nonrigid motions of the exchange type. Naturally, this is more the exception than the rule. Using the hydrazine molecule 14 N2 H4 as an example, we consider the case where the nonrigid motions do not preserve the spatial position of the point-group elements. The permutation group of identical nuclei of the 14 N2 H4 molecule has the form 2  4 . Taking into account that s.14 N/ D 1 and s.H/ D 1=2, it is easy to find the admissible spin Young diagrams and respective coordinate Young diagrams of this group. The equilibrium configuration shown in Figure 11.2 belongs to the point group C2 D .E; C2.y/ /. Therefore, the hydrazine molecule has stereoisomers. The matching of the groups 2  4 and C2 (the necessary correlations are given in Table 11.6) yields the following coordinate multiplets of a rigid molecule: 78A;

66B:

(11.33)

196

Chapter 11 Nonexchange transitions between equivalent configurations

Table 11.6. Correlation tables for the 14 N2 H4 molecule. 2  4

C2

2  4

FH

FH

F32

Œ2  Œ14 

A

Œ2  Œ14 

a3

a1

A1 ; A5; E3

Œ2  Œ21 

A C 2B

Œ2  Œ21 

a4 C e

a2

A2 ; A6; E3

Œ2  Œ22 

2A

Œ2  Œ22 

a1 C a3

a3

A3 ; A7; E4

Œ1   Œ1 

B

Œ1   Œ1 

a4

a4

A4 ; A8; E4

Œ1   Œ21 

2A C B

Œ1   Œ21 

a3 C e

e

E1 ; E2 ; E5 ; E6

Œ12   Œ22 

2B

Œ12   Œ22 

a2 C a4

2

2

2

4

2

2

2

2

D2

C2  Ci

B2  H8

A; B2

A.˙/

a1a

B1 ; B3

4

B .˙/

a2a a3a a4a ea

2

F32  Ci .C/

.C/

D4 ./

A1 ; A3 ; E1

.C/ .C/ ./ A2 ; A4 ; E1 .C/ .C/ ./ A5 ; A7 ; E2 ./ A6.C/ ; A.C/ 8 ; E2 .C/ .C/ ./ ./ E3 ; E4 ; E5 ; E6

B2  H8

A1

a1a ; a3b

A2

a2b ; a4a

B1

a1b ; a3a

B2

a2a ; a4b

E

ea ; eb

Note. The correlation between the representation  b of the B2  H8 group and representations of the F32  Ci group follows from the correlation of the representation  a by replacement of the indices C $  in the representations of the F32  Ci group.

Then the symmetry types of zero-approximation wave functions are matched with the symmetry types of the allowed multiplets formed on their basis: .mult/C2 Ci D .el /C2  .vib/C2  .rot.-in/C2 Ci ;

(11.34)

6

where the admissible types of rot.-in for a given rotational representation of the D2 group of a rigid asymmetric top is determined from the correlation of the groups D2 and C2  Ci through their common subgroup C2 . A classification of the rotational levels of a rigid molecule of hydrazine for both possible types of the vibronic state is shown in Figure 11.6. For an account of nonrigid motions of the NH2 structures, a noninvariant extended point group F32 in the form (11.5) with a symmetry subgroup of the Hamiltonian FH in the form (11.6) is introduced into a chain between the groups 2  4 and C2 . Figure 11.7 shows an intermediate configuration of the N2 H4 molecule and its stereoisomer, which are related by the operation of spatial inversion i. From a comparison of the configurations one can obtain .1/

i D .2  1 /U2

.2/

D .1  2 /U2 ;

(11.35)

197

Section 11.4 The hydrazine molecule N2 H4

Figure 11.6. Classification of the rotational levels of a rigid molecule of 14 N2 H4 in vibronic states of type A (left) and type B (right).

.k/

where U2 are rotations of the molecule by an angle  about the axes k D 1, 2 passing along the bisectors of the angles between the axes x and y. That is, the group F32 .z/ allows for the possibility of stereoisomer mixing. The element 1 D c2 2 belonging to the exchange type describes the sequential implementation of two motions of the nonexchange type for the NH2 structure. Therefore, the stereoisomers are mingled through a fairly low potential barrier. .k/ For a complete determination of the U2 operations it is necessary to identify for them the permutation relations with the operations of the F32 group. It is easy to do this by using a commutation of the operation i with all elements of the F32 group. .k/ As a result, it follows from equation (11.35) that the operations U2 commute with .y/

operations of the group Fr D c2v  c2v , and their permutation relations with C2 the usual rotational relations: .y/

.1/

C2 U2

.2/

.y/

D U2 C2 :

are

(11.36)

The symmetry group of the Hamiltonian FH (division of the elements into classes and the table of characters are given in Table 11.7) determines the coordinate degeneracy of the levels in the energy spectrum and their nuclear statistical weights. From a matching of the groups 2  4 and FH it follows that a complete coordinate wave

198

Chapter 11 Nonexchange transitions between equivalent configurations

Figure 11.7. Intermediate configurations of the stereoisomers of the N2 H4 molecule (the z axis passes through the centers of mass of the equivalent structures of NH2/.

Table 11.7. Division of the elements into classes and the table of characters for the symmetry groupof the Hamiltonian of a nonrigid molecule of N2H4 . E 1 r1

E E 1  1 E  1 1  E

r2

r3

C2  C2.y/ (1  1 /C2 .E  1 /C2 .1  E/C2

FH a1 a2 a3 a4 e

E 1 1 1 1 2

1 1 1 1 1 2

r1 1 1 1 1 0

r2 1 1 1 1 0

r3 1 1 1 1 0

function relates to the following irreducible representations of the latter: 6a1 ; 3a2 ; 45a3 ; 36a4 ; 27e:

(11.37)

Matching the groups FH and F32 , we obtain the following allowed coordinate multiplets of a nonrigid molecule: 6.A1 ; A5 ; E3;2 /; 36.A4 ; A8 ; E4;1 /;

3.A2 ; A6 ; E3;1 /;

45.A3 ; A7 ; E4;2 /;

27.E1 ; E2 ; E5 ; E6 /:

(11.38)

It was taken into account in equation (11.38) that the two-dimensional representations E3 and E4 of the F32 group split into one-dimensional ones when the F32 group is reduced to the FH subgroup. Therefore, the components of these representations, given by the relations E3;1 ! a2 ;

E3;2 ! a1 ;

E4;1 ! a4 ;

correspond to individual nondegenerate levels.

E4;2 ! a3 ;

(11.39)

199

Section 11.4 The hydrazine molecule N2 H4

The symmetry of the rotational motion with allowance for nonrigid transitions is determined by rotations of the molecule as a whole from the group F32  Ci , which form the group D4 . For a unique determination of the action of the elements of the D4 group we note that the classes C2 and 2C4 contain rotations about the axis z and the classes 2U2 and 2UN 2 , rotations about the axes 1 and 2 and the axes x and y, respectively. The symmetry types of the coordinate multiplets for a given rotational representation of the D4 group are matched with the symmetry types of the basis functions of separate internal motions as follows: .mult/.Fr ^C2 /Ci D .el /C2  .vib /C2  .rot.-conf/.Fr ^C2 /Ci : 6

(11.40)

6

The rotation-configuration representation rot.-conf allows for all configurations arising in this problem, and the behavior relative to the Fr group operations is characterized by a multiplet, since these operations do not belong to a point group. The possible types of symmetry of rot.-conf for a given rotational representation of the D4 group follow from the correlation of the groups D4 and F32  Ci . Figure 11.8 shows a classification of the energy levels in the vibronic state el  vib D A. The introduction of rot.-conf in the matching made it possible to allow for the existence of dependent configurations. In total, the group F32  Ci “sees” 32 equivalent configurations corresponding to the point group C2 , of which only eight are independent according to the classification. Allowing for equation (11.35), it is easy to show that all the elements of the group F32  Ci are expressed through operations defining the considered types of motion. Hence, the actual structure of the energy spectrum is described by the total group F32  Ci . Its rotational subgroup, which plays only a subsidiary role, is the group B2  H8 , where .z/

.2/

B2 D .E; .E  c2 /U2 /;

H8 D .c2  c2 / ^ C2 :

(11.41)

The rotational subgroup is represented as a direct product of groups B2 and H8 since the nontrivial element of the B2 group is the product of operations 1  1 and i that commute with all operations of the H8 group. Symmetric and antisymmetric representations of the B2 group are denoted by the indices a and b, and for irreducible representations of the H8 group we use the notation of the group FH which is isomorphic with it. As already mentioned, the levels of types E3 and E4 split into doublets. A picture of the levels for the vibronic state el  vib D B follows from that presented in Figure 11.8 by formal replacement of the D4 group representations: A1 $ B2 ;

A2 $ B1 :

(11.42)

The elements specifying the torsional motion of the NH2 structures form the group c2  c2 . As a result, the complete basis set describing this motion consists of the unit vectors jc1 i; jc2 i and jc2 i; jd2 i related to the symmetric and antisymmetric representations of the groups c2  E and E  c2 , respectively. In the F32 group, by virtue of

200

Chapter 11 Nonexchange transitions between equivalent configurations

Figure 11.8. Energy-level classification for a nonrigid molecule of 14 N2 H4 in the type A vibronic state.

the properties of the semidirect product, the elements of the factor group C2 also act on the basis unit vectors of the torsional motion: .y/

C2 jc1 ijc2 i D jc1 ijc2 i;

.y/

C2 jd1 ijd2 i D jd1 ijd2 i;

.y/

C2 jc1 ijd2 i D jd1 ijc2 i: (11.43) The sign in the first two of equations (11.43) is chosen for the vibronic state el vib D A. The group specifying the inversion motion contains 2 as the generator instead of .z/ c2 . Therefore, in a similar way, we introduce a set of unit vectors js1 i; ja1 i and js2 i; ja2 i to describe the torsional motion. Rotational functions are classified by the group D4 . As a result, all operations of the F32  Ci group can easily be expressed as products of the elements whose action on the basis inversion-torsion-rotational func.y/ tions is defined. For example, the operation C2 acts on the inversion, torsional, and rotational parts of these functions. Operation i is expressed in the form (11.35). Spec.z/ .z/ .z/ ifying the action of the operation C2 D c2  c2 is important here. Although this

201

Section 11.4 The hydrazine molecule N2 H4

operation arises as a description of the sequential torsional motion of both NH2 structures by an angle , it corresponds to the rotational motion by the same angle of the molecule as a whole through a zero barrier, and its action should therefore be determined only in the rotational subspace. In fact, this condition is a solution to the problem of separation of the torsional and rotational motions in the hydrazine molecule (see Section 10.5). Decomposition of the inversion-torsion-rotational basis functions into irreducible representations of the F32  Ci group for the vibronic state el  vib D A is presented in Table 11.8. Table 11.8. Decomposition of the basis functions for the vibronic state el  vib D A of a nonrigid molecule of N2 H4 into irreducible representations of the F32  Ci group. F32  Ci .C/

Basis functions

A1

A 1 c 1 c 2 s1 s2

A.C/ 3 ./ A7 ./ A5 A2./ ./ A4 .C/ A8 .C/ A6 A1./ ./ A3 .C/ A7 A.C/ 5 .C/ A2 .C/ A4 ./ A8 A6./

A1 c1c2 a1 a2 A 1 d 1 d 2 s1 s2

F32  Ci .C/

E1

E1./

A1 d1 d2 a1 a2 A 2 c 1 c 2 s1 s2

.C/

E2

A2 c1c2 a1 a2 A 2 d 1 d 2 s1 s2

./

E2

A2 d1 d2 a1 a2 B1 c1 c2 s1 s2 B1 c1 c2 a1 a2 B1 d1 d2 s1 s2 B1 d1 d2 a1 a2 B2 c1 c2 s1 s2 B2 c1 c2 a1 a2 B2 d1 d2 s1 s2 B2 d1 d2 a1 a2

E3.C/ ./ E3 .C/ E4 E4./ .C/ E5 ./ E5 .C/ E6 E6./

Basis functions A2 c1 c2 .s1a2 C a1 s2 / B1 c1 c2 .s1 a2  a1 s2 / A1 c1 c2 .s1a2  a1 s2 / B2 c1 c2 .s1 a2 C a1 s2 / A1 d1 d2 .s1 a2  a1 s2 / B2 d1 d2 .s1 a2 C a1 s2 / A2 d1 d2 .s1 a2 C a1 s2 / B1 d1 d2 .s1 a2  a1 s2 / B3 c1 d2 s1 a2 C B2 d1 c2 a1 s2 B3 d1 c2 a1 s2 C B2 c1 d2 s1 a2 B3 c1 d2 a1 s2 C B2 d1 c2 s1 a2 B3 d1 c2 s1 a2 C B2 c1 d2 a1 s2 .B3 d1 c2 C B2 c1 d2 /s1 s2 .B3 c1 d2 C B2 d1 c2 /s1 s2 .B3 d1 c2 C B2 c1 d2 /a1 a2 .B3 c1 d2 C B2 d1 c2 /a1 a2

Of the two components of two-dimensional representations En defined by the con.y/ .y/ ditions C2 En;1 D En;1 and C2 En;2 D En;2 , only the component En;1 is given p (the normalization factor 1= 2 is omitted), and the component En;2 is obtained by changing the sign in symmetrization of the expression for the component En;1 . The

202

Chapter 11 Nonexchange transitions between equivalent configurations

rotational parts of all one-dimensional representations and two-dimensional representations E1 and E2 are specified by irreducible representations of the D4 group, while for other two-dimensional representations, by the representations B2 and B3 of the D2 ˙ group in the axes (z, 2, 1), where .B2 ; B3 / D ‰J;2nC1 [64] and  D i 2nC1 . The torsional parts of the effective operators of physical quantities will be constructed on the basis of the coordinate spin operators e.1/ and e.2/ (specified in the unit-vector spaces jc1 i; jd1 i and jc2 i; jd2 i respectively), and the inversion parts, on the basis of the coordinate spin operators  .1/ and  .2/ (specified in the unit-vector spaces js1 i; ja1 i and js2 i; ja2 i respectively). The algebraic and symmetry properties of these operators are similar to the properties of the operator e, which was used in Section 11.2 to describe the inversion motion in the ammonia molecule. A complete set of self-adjoint operators in two-dimensional space also comprises a totally symmetric unit operator (E1 , E2 for the internal motions and I1 , I2 for the inversion motion). The effective operators of coordinate physical quantities are invariant under transformations of the FH group. Hence, the following types of symmetry are admissible for them in the F32 group: (11.44) A1 ; A5 ; E3;2 : Here, the appearance of one component of the irreducible representation E3 does not lead to contradictions, since only transformations of the noninvariant (or nonexchange) type mix the components of this representation, and these transformations also change the position of some geometric elements of the FH group (the position of the symmetry axis of the point group 2 changes from y to x, or vice versa). We write the effective Hamiltonian in the form H D HI C HII C HIII , where three types of contributions correspond to three allowed symmetry types from equation (11.44). In their construction it is also necessary to take into account that the Hamiltonian is invariant with respect to the spatial inversion and time reversal operations. Contributions of the A.C/ type are formed on the basis of spin operators of the 1 c1 c2 s1 s2 type in the c2v  c2v group (Table 11.9). All fourteen of such operators are .y/ t-even, ten of them are invariant with respect to the operation C2 (type A), and four operators change their sign (type B). A general expression for such contributions to the Hamiltonian is represented as HI D

1  X

.A / .B2 / H2n 1 C H2nC2 :

(11.45)

nD0

On the right-hand side of equation (11.45) the superscript determines the type of the irreducible representation of the D4 group, according to which combinations of the angular-momentum components in the MCS are transformed and the subscript speci-

203

Section 11.4 The hydrazine molecule N2 H4

Table 11.9. Complete set of independent spin operators of the c1 c2 s1 s2 type in the Fr group. Type A

Type B .e/

E1 E2 I1 I2

r.e/ I1 I2

rC I1 I2

.1/ .2/

.e/ .1/ .2/

e3 e3 I 1 I 2

.1/ .2/

r.e/ 3 3

rC 3 3

E1 E2 3.1/ 3.2/

. / E1 E2 rC

.1/ .2/ .1/ .2/

E1 E2 r. /

.1/ .2/ . /

e3 e3 3 3

.1/ .2/

e3 e3 r. /

e3 e3 rC .e/ . /

rC rC

r.e/ r. / .e/

.2/

.1/

. /

.2/

.1/

Note. r˙ D E1 e3 ˙ e3 E2 and r˙ D I1 3 ˙ 3 I2.

fies the total power of the combinations of these components, X .A / 4t H2n 1 D cO2p;2s;4t J 2p Jz2s .JC C J4t /;

(11.46)

p;s;t .B /

2 D H2nC2

X

4t C2 C J4t C2 /; dO2p;2s;4t C2J 2p Jz2s .JC

(11.47)

p;s;t

where p C s C 2t D n for a given n. On account of nonrigid motions, the parameters cO and dO linearly depend on ten spin operators of type A and four spin operators of type B, respectively. In a similar way, we construct contributions to the Hamiltonian for the other two types of symmetry. Their specific expressions will not be presented here (see [19]). We only note that it is exactly these contributions that are responsible for the splitting of the levels of the E3 and E4 types into doublets. Because of the noninvariant contributions, the matrix of the Hamiltonian decomposes into a direct sum of only ten different types of the energy matrices (belonging to the different irreducible representations of the FH  Ci group). Five irreducible representations of the FH group correspond to five families (11.38) of representations of the F32 group. In the group F32  Ci , they convert into ten families since each family in equation (11.38) may have the behavior ˙ with respect to the operation of spatial inversion i. An example of a hydrazine molecule clearly shows that analysis of the symmetry properties of the internal dynamics can be highly nontrivial even for the molecules with a relatively simple geometry of the nonrigid motions.

204

Chapter 11 Nonexchange transitions between equivalent configurations

11.5 Conclusions We have formulated some general conclusions based on the content of this chapter. 1. In a description of the molecules with transitions of the nonexchange type between the energetically equivalent configurations, an extended point group F, which characterizes both the local motions in the minima and the transitions between them, additionally appears in a chain of groups. Hence, the group F includes the point group F0 as a subgroup. The elements specifying the nonrigid motions are not the symmetry transformations of the effective Hamiltonian. That is, the group F is a noninvariant group. A characteristic feature of such elements is that they change the spatial position of the equilibrium configuration. If there are several motions of the nonexchange type, then the symmetry subgroup of the Hamiltonian FH in the group F becomes wider than F0 provided that elements of the exchange type not belonging to F0 appear as a result of the sequential implementation of these motions (see Section 11.4). We emphasize that the group FH is not enough to describe the internal dynamics since there are no elements specifying the nonrigid transitions. It should also be mentioned that the analysis based on the CNPI group is also an attempt to limit the analysis to the symmetry group of the Hamiltonian. Naturally, this can lead to major philosophical difficulties. 2. For all the molecules considered in this chapter, the actual structure of the energy levels is determined by the F  Ci group since all of its elements are expressed through operations defining the considered types of motion. Coordinate degeneracy of the energy levels and their nuclear statistical weights are determined by the group FH . Therefore, if the components of a degenerate representation of the F group belong to different irreducible representations of its subgroup FH , then they correspond to individual levels. Coordinate degeneracy of the energy levels in the group FH , is not physically meaningful, since each transformation of this group is associated with the permutation of identical nuclei. Therefore, the spin-coordinate wave function should satisfy the fundamental requirements of symmetry with respect to such permutations. As a consequence, all the wave functions belonging to the degenerate coordinate representation of the FH group form only one physically observable spin-coordinate state. 3. Since the effective Hamiltonian of the internal motion of the molecule contains noninvariant contributions with respect to the F group transformations, not all of the symmetry symbols of this group are good quantum numbers. Therefore, the energy matrix decomposes only into a direct sum of matrices corresponding to the different irreducible representations of the FH  Ci group. This is also the reason why in the general case only the type of irreducible representations of the FH subgroup is retained in the electric dipole transitions.

Chapter 12

On the meaning of the Born–Oppenheimer Approximation Isolated atoms and molecules, neutral or ionized, are described to a good approximation by neglecting spin-related electronic and nuclear contributions (called fine and hyperfine interactions [64]). Using an FCS, the coordinate Hamiltonian is usually written out in the form H D Tel C Tnucl C Vel-el C Vnucl-nucl C Vel-nucl;

(12.1)

where Tel and Tnucl are the kinetic energies of the electrons and nuclei, and Vel-el, Vnucl-nucl, and Vel-nucl are the Coulomb interaction energies between the electrons, between the nuclei, and between the electrons and the nuclei, respectively. As already mentioned in Chapter 7, the internal dynamics of the atomic and molecular systems differ radically. In particular, a characteristic feature of the atomic system is the absence of the rotational motion of the system as a whole, while a characteristic feature of the molecular system is the presence of this rotational motion. The model of free rotation of a rigid body, which is considered in Section 6.2, is a physically correct initial approximation for describing any microsystem having this motion. This means that the molecular system is a certain structure (“microcrystal”), whose particles should be considered as performing collective motions. It is to be emphasized in this connection that for atoms a good initial approximation is to describe each electron separately as being subject to the averaged field of the nucleus and the remaining electrons. This means that the configuration space of an atom can be formed as a direct sum of the configuration spaces of individual particles. In the presence of collective motions, configuration space cannot, in principle, be formed in this way. In fact, the highly nontrivial problem of constructing this space arises, requiring definitions for separate types of motion including rotation. The point to note here is that the structure under consideration should feature some implicit geometric symmetry group. It turns out that the concept itself of molecular structure cannot be introduced without using the BO approximations. The correct configuration space of collective motion should be constructed separately for each electronic state. In other words, we pass to a description area restricted to one electronic state. For such a limited area, implicit symmetry is replaced by its explicit counterpart. In fact, the BO approximation allows certain conditions to be formulated for the stationary Schrödinger equation which select physically meaningful discrete-spectrum solutions of collective motion against the vast background of formal solutions. It is only then that the problem of describing the spectrum of intramolecular dynamics is formulated and can be solved analytically and/or numerically. Consequently, it is due to the BO approximation that the very def-

206

Chapter 12 On the meaning of the Born–Oppenheimer Approximation

inition of this problem can be formalized. Importantly, constraints on the definition of some types of internal motion arise which are related to the dynamic symmetry properties and whose violation leads to physically incorrect consequences.

12.1 Nondegenerate electronic states Consider the problem of bound stationary molecular states. The BO approximation [10,14,35] is based on the large mass difference between electrons and nuclei, namely, mel Mnucl:

(12.2)

Due to condition (12.2), nuclei and electrons can be considered a slow subsystem and a fast subsystem, respectively, that is, the state of the electron subsystem adiabatically follows the state of the nuclear subsystem. Hence, the first step is to solve the equation for the motion of electrons in a field of fixed nuclei in a given configuration: .Tel C Vel-el C Vnucl-nucl C Vel-nucl/‰el.n/ D Eel.n/ ‰el.n/ ;

(12.3)

where the superscript n specifies the set of electron quantum numbers. This is followed by solving, in each electronic state, the nuclear motion problem with the effective in.n/ teraction potential Eel .Xnucl/, which, by definition, is dependent on the instantaneous nucleus configuration: .n/

.n;v/

.n;v/

.n;v/

ŒTnucl C Eel .Xnucl /ˆnucl D Enucl ˆnucl ;

(12.4)

where the index v specifies the set of nuclear quantum numbers. This approach appears to be, but is not, conceptually simple to implement; the point is that the dynamic structure of the molecule has not yet been specified. To do this, it is additionally necessary to make the following important comments. 1. In fact, the equation for nuclear motion has not yet been specified. The point is that the existence of the molecular structure leads to the appearance of the rotational motion of the molecule as a whole, so that a molecule-related MCS should also be introduced “manually” into the effective nuclear Hamiltonian in model (12.4) to define the very concept of such a motion, and then the remainder of the nuclear motion should be divided into individual types. In this way, a correct, complete, nuclear configuration space is formed. This procedure is nontrivial [14] and requires the transition to collective variables to be realized. A physically correct definition of the rotational motion requires that the MCS should be “frozen” into the effective nuclear potential in a given electronic state, which for a rigid nonlinear molecule is equivalent to fixing the MCS with respect to its equilibrium configuration (see Section 12.4). We emphasize that a criterion for a molecule to undergo no rotational motion as a whole is not equivalent to the requirement that its angular momentum be zero [64].

207

Section 12.2 Degenerate electronic states

2. The kinetic and potential parts of the nuclear Hamiltonian written in collective variables should be invariant with respect to the geometric group transformations defined in the MCS. Thus, we pass from the description with implicit geometric symmetry (see Section 7.2) to the model with a similar explicit symmetry. We emphasize that the kinetic term Tel in equation (12.1) is used to obtain the effective nuclear potential in equation (12.4). The price for such a transition is the narrower range of applicability, since the structure and the geometric group characterizing it depend, generally speaking, on the electronic excitation. It is useful to note that the Hamiltonian (12.1) describes the continuous spectrum as well. On passage to the continuous spectrum, the geometric group certainly changes, since the multiplicity of the energy-level degeneracy becomes infinite. In bound electronic states of rigid molecules, the effective potential has one minimum. As a consequence, the symmetry of the equilibrium configuration necessarily coincides with the symmetry of the effective potential. In other words, the symmetry of the equilibrium configuration is an elementary consequence of the symmetry of internal dynamics, but not vice versa, as is often stated. Only for a rigid molecule in a nondegenerate electronic state do these two symmetries coincide. In all other cases, the geometric symmetry of the internal dynamics is wider than the symmetry of the equilibrium configuration. 3. The regular coordinate wave function of a stationary state in the BO approximation should belong to one of the irreducible representations of the geometric group. Only for the nondegenerate electronic state is this condition fulfilled if this function is chosen in the form of a simple product, .n/

.n;v/

‰el ˆnucl ;

(12.5)

where the electronic function corresponds to the equilibrium configuration. The form of equation (12.5) means separation between electronic and nuclear motions. In other words, independent electronic and nuclear structures are formed. This result is violated only by small nonadiabatic corrections to the effective nuclear Hamiltonian Hnucl in model (12.4). However, irrespective of the approximation degree of the expression for Hnucl , the same rigorous internal geometric symmetry group, which is actually implemented in a given electronic state, is employed.

12.2 Degenerate electronic states Let us start by noting that the stationary Schrödinger equation for a degenerate electronic state is also solved in two stages, using equations (12.3) and (12.4) written under the assumption that the electronic subsystem adiabatically follows the nuclear subsystem. However, in the solution obtained in this way, the electronic and nuclear motions are no longer separated (!). The reason is that in writing out the coordinate wave function in the form (12.5) one cannot satisfy the requirement that this function belong

208

Chapter 12 On the meaning of the Born–Oppenheimer Approximation

to one of the irreducible representations of the geometric group. The correct solution constitutes a certain linear combination of products of the electronic and nuclear functions, with the coefficients completely determined by the requirements of geometric symmetry and having nothing to do with condition (12.2). This means the formation of a unified electronic-nuclear structure, which cannot be divided, even approximately, into separate electronic and nuclear structures. The distinguishing features of these stationary states (known as vibronic) are related to the Jahn–Teller [9, 10] and Renner effects [14] in the nonlinear and linear molecules, respectively. It is traditionally assumed that in degenerate electronic states, the electronic and nuclear motions strongly interact, and the BO approximation is therefore violated. Actually, the symmetry requirements lead here to a rigid nonforce relation between these two types of motion, while the force interaction is small, as before, due to condition (12.2). Such a rigid relation is similar to the relation between the coordinate and spin types of motion due to the symmetry requirements for permutations of identical particles. In other words, it is due to the symmetry alone that the condition of the electronic subsystem adiabatically following the nuclear subsystem cannot be reduced to the separation condition between electronic and nuclear motions. Consider a very nontrivial question about the geometric group of a rigid molecule in a degenerate electronic state. According to the Jahn–Teller theorem [58], the symmetric equilibrium nuclear configuration of a nonlinear molecule is always unstable in an electronic state that is orbitally degenerate due to this symmetry. Or more precisely, there will exist nuclear displacements for which the effective potential in this configuration has its first-order derivative different from zero. The degenerate electronic state, together with the vibrational modes that destabilize the symmetric nuclear configuration, form the Jahn–Teller system. Such modes are called active in the sense of the Jahn–Teller effect. For example, the ammonia molecule NH3 in a nondegenerate ground electronic state corresponds to the geometric group C3v . In a doubly degenerate electronic E state, an equilibrium configuration of symmetry C3v is already unstable with respect to the same type of vibrational displacements (Jahn–Teller E-E system). For several decades it was believed that the lowering of symmetry of the equilibrium configuration leads to the removal of degeneracy. However, the experimental verification of this has been attempted many times, but with no success (for a brief historical review see [10]). The point is that a symmetric molecule always has several equivalent directions of distortion of the equilibrium configuration. For example, the NH3 molecule has three such directions (corresponding to three NH bonds in a symmetric equilibrium configuration). Therefore, the effective potential already has three equivalent minima with an equilibrium configuration of symmetry CS in each [9]. That is, the molecule becomes nonrigid, and the geometric symmetry of the Hamiltonian is now determined by the extended point group of the form C3v D C3 ^ CS ;

(12.6)

Section 12.2 Degenerate electronic states

209

where the invariant group C3 allows for the symmetry of the motion between three minima and the factor group CS , the symmetry of the motion in one minimum. As a consequence, degeneracy is determined, as before, by the group C3v , which now corresponds to the unstable configuration. The latter plays the role of the intermediate configuration, through which the geometric symmetry is specified with allowance for all three minima (see Section 9.1). In this case, the type of degeneracy becomes electronnuclear. This means that again we come to the unified electron-nuclear structure. Thus, if transitions between the Jahn–Teller minima of the effective potential need to be taken into account, the molecule should simply be considered nonrigid. If the molecule also exhibits other nonrigid transitions, the overall picture can be very complicated. However, for a correct analysis it suffices to construct a correct extended point group. For the linear molecules, degeneracy of the electronic states can be only twofold, and the effective potential in degenerate states has a zero first-order derivative over all nuclear displacements in a linear configuration. However, in the case of a strong Renner effect, the effective potential can be at a maximum in the linear configuration, with the minimum corresponding to the curved configuration. Nevertheless, the axial symmetry of internal dynamics is reconstructed due to transitions through a barrier that corresponds to the linear configuration. Naturally, in this case, these transitions bind the dependent configurations, since the final configuration can also be obtained by conventional rotation of the initial configuration with respect to the linearization axis. Geometric symmetry is a characteristic of a rigorous coordinate Hamiltonian and is therefore preserved on account of the corrections to the vibronic state description based on the approximation given by equations (12.3) and (12.4). Clearly, the corrections are calculated in a different way compared to the case of a nondegenerate electronic state. However, the initial approximation itself, as before, plays a decisive role in obtaining the solutions with correct symmetry properties. Finally, we note that describing the whole set of quasidegenerate electronic states is a fairly regular problem. The fact is that a separate or isolated consideration of these states on the basis of equations (12.3) and (12.4) leads to solutions the corrections to which, as a rule, do not contain the smallness parameter. That is, such solutions are far from reality (it is said sometimes that “strong interactions” remain between the states described by equations (12.3) and (12.4)). Therefore, there is a major problem of the summation of corrections. To avoid this problem, the quasidegenerate states are considered on a unified basis. This means that for nonlinear molecules the Jahn– Teller pseudoeffect should necessarily be taken into account. In terms of symmetry, such effects can be treated as the existence of a wider geometric group for some main contribution to the rigorous coordinate Hamiltonian.

210

Chapter 12 On the meaning of the Born–Oppenheimer Approximation

12.3 Internal geometric symmetry of the Hamiltonian As a simple example, we first consider a rigid water molecule H2 O. We choose the coordinate system in which the point group transformations are most convenient to consider. In the conventional, easiest method, such transformations are introduced as transformations of an object with a constant position on the coordinate system. Since the MCS does not satisfy this requirement (its axes change orientation on transformations corresponding to the rotation of the molecule as a whole), we will use the Cartesian coordinate system, whose axes x; y, and z are fixed with respect to the elements of the point group. The equilibrium configuration of the H2 O molecule in the ground electronic state, which is shown by light circles in Figure 12.1, belongs to the point group .z/

C2v D .E; C2 ; v.xz/; N v.yz/ /

(12.7)

and is an isosceles triangle [52]. All irreducible representations of this group are onedimensional, and the possible bound electronic states are therefore nondegenerate. In this case, the symmetry of the equilibrium configuration, which is a consequence of the symmetry of internal dynamics, just coincides with the latter. That is, the effective

Figure 12.1. Action of the transformation C2.z/ on the nuclear configuration of the H2O molecule corresponding to the vibrational mode B1 .

Section 12.3 Internal geometric symmetry of the Hamiltonian

211

nuclear Hamiltonian is represented as the sum of kinetic and potential parts that are invariant under transformations of the C2v group and describe the vibration-rotational motion of the nuclei. We emphasize that although the concept of effective nuclear Hamiltonian arises due to the BO approximation given by equations (12.3) and (12.4), this concept remains valid with the corrections taken into account [66]. The molecule H2 O has three vibrational modes, namely, two symmetries A1 and one symmetry B1 . The case of asymmetric nuclear displacements is the most interesting. Therefore, we consider the action of the C2v group transformations on the nuclear configuration corresponding to the mode B1 , which is shown by dark circles in the upper part of Fig.z/ ure 12.1. We start with the transformation of C2 . Since the effective nuclear potential is invariant under transformations of the point group, it remains to determine .z/ the action C2 in the vibration-rotational configuration space. Transformation of the vibrational coordinates (the result is shown in the middle part of Figure 12.1) leads .z/ to a simple change in the displacement of the nuclei according to the operation C2 , where the equilibrium configuration preserves its position since the MCS does not rotate. The question of rotational transformations is more difficult. The point is that the rotational motion is set by the MCS rotation, and in their action on the rotational coordinates and, therefore, the rotational wave function, infinitesimal rotations of the MCS are equivalent to the inverse infinitesimal rotations of the fixed coordinate system [64]. Therefore, in the x; y; z coordinate system, the rotations corresponding to the rotational transformations should be inverse of the rotations corresponding to the vibrational transformations. This is quite an important point, although specifically for the rigid molecule H2 O it does not matter (here, all geometric symmetry elements co.z/ incide with their inverses). The overall result of the action of the operation C2 on the vibrational and rotational coordinates is shown in the lower part of Figure 12.1. It is clearly seen that this result is equivalent to the permutation of the identical nuclei of hydrogen. We emphasize the following. .z/

1. In the transformation C2 , the vibrational and rotational changes are compensated in such a way that the spatial position of an arbitrary nuclear configuration is preserved, and only the identical nuclei change places. That is, this transformation is of the exchange type. Invariance of the effective vibration-rotational Hamiltonian in a given electronic state with respect to such transformations follows from the fundamental symmetry properties for permutations of the identical nuclei. .z/

2. The action of the transformation C2 on the instantaneous configuration in the case of asymmetric nuclear displacements does not coincide with its usual geometric action. Naturally, the latter is not equivalent to the permutation of identical nuclei. However, it is not correct to consider this as an argument in favor of the inapplicability of the point group to describe the rigorous symmetry of the intramolecular motion problem [16].

212

Chapter 12 On the meaning of the Born–Oppenheimer Approximation .z/

3. The identical-nuclei permutation equivalent to the transformation C2 can easily be derived from the equilibrium configuration of the molecule since in this case the .z/ action of the transformation C2 coincides with its usual geometric action. .yz/

We now turn to the transformation N v . Here, it should additionally be taken into account that the motions corresponding to the improper transformations of the point .yz/ group are not implemented in a rigid molecule. Hence, the operation N v is not defined in the complete configuration space of a rigid molecule, and the actual structure of its energy spectrum is described by the rotational subgroup of the Gpoint  Ci group (see Chapter 8). The action of the external operation i of symmetry of the nonexchange type, which is related to the properties of space, on the spatial Cartesian coordinates of all particles of the molecule is a common geometric action. For a given electronic state in the coordinate system x; y; z, this operation inverts the effective interaction potential of the nuclei and their coordinates. The rotational subgroup in this case has the form .z/ .y/ .x/ D2 D .E; C2 ; C2 D iv.xz/ ; C2 D i N v.yz/ /: (12.8) .yz/

It suffices to determine the action of the transformation i N v on the position of the nuclei since the spatial position of the effective potential follows directly from the spatial position of the equilibrium configuration. This transformation admits only of an identity representation in the vibration-rotational configuration space: .x/

.i N v.yz/ /nucl D .C2 /rot  .N v.yz/ /vib ;

(12.9)

and the vibrational transformation preserves the position of the equilibrium configuration, as it should be. From equation (12.9) we obtain .x/

.N v.yz/ /nucl D .C2 /rot  .N v.yz/ /vib  inucl :

(12.10)

Figure 12.2 shows from top to bottom all three stages of action of the transformations on the right-hand side of equation (12.10) on the initial nuclear configuration presented in the upper part of Figure 12.1. It is easy to see that the action of the trans.yz/ is equivalent to the permutation of the identical nuclei of hydrogen formation N v in the effective nuclear potential that is invariant to this transformation. Clearly, this .yz/ does not coincide with its usual geometric action. action of N v .xz/ .z/ .yz/ From the expression v D C2 N v it is easy to see that the action of the trans.xz/

formation v is equivalent to the identical permutation of nuclei in the effective nuclear potential that is invariant to this transformation. For a water molecule, this result is obvious since all the three of its nuclei always lie in the xz plane. We emphasize that this consideration of the action of the point-group operations on the instantaneous nuclear configuration of the molecule is substantially based on the use of a physically correct recipe for introducing the very concepts of vibrational and rotational motions.

Section 12.3 Internal geometric symmetry of the Hamiltonian

213

.yz/

Figure 12.2. The action of the transformation N v on the nuclear configuration of the H2 O molecule corresponding to the vibrational mode B1 .

Namely, when the center of mass is fixed, the rotational motion is the motion responsible for changes in the spatial position of the equilibrium configuration, and the vibrational motion is the motion responsible for changes in the displacements of the nuclei from their equilibrium positions (see Section 12.4). Importantly, this recipe is free of constraints on the nuclear displacements from equilibrium positions. From the foregoing it is clear that the case of symmetric nuclear displacements does not require separate analysis. Thus, the action of the transformations of the point group C2v is equivalent to permutations of the identical nuclei in the effective nuclear potential that is invariant to these transformations. As a consequence, we have a matching of the groups C2v and 2 , which is shown in Table 8.1. The operators of the physical quantities characterizing the total internal coordinate motion relate to the totally symmetric coordinate Young diagram of the 2 group (see Section 7.1). It follows from Table 8.1 that all such operators (including the Hamiltonian) belong to the identity representation of the point group. Thus, the point group in the electronic state which it characterizes is a geometric group of rigorous symmetry of the electron-vibrationrotational motion. This conclusion is free of constraints on the nuclear displacements from equilibrium positions. We extend these results to an arbitrary rigid molecule. Generally, the point group may have transformations of the rotations, reflections, and mirror rotations. The ef-

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fective nuclear potential is invariant under all of these transformations. It remains to consider their action in the vibration-rotational configuration space. We start with the transformation of the rotation. Its action in the vibrational and rotational spaces are opposite in direction, and its action in the vibrational space does not change the position of the equilibrium configuration. As a result, the vibrational and rotational changes compensate for each other in such a way that the spatial position of an arbitrary nonequilibrium nuclear configuration is preserved with accuracy up to the interchange of identical nuclei. The appropriate permutation of identical nuclei can easily be derived from the action of the rotation on the equilibrium configuration of the molecule. Note that the point group of a molecule with linear equilibrium configuration includes a continuous set of rotations about an infinite-order axis, and for such rotations the vibrational and rotational changes fully compensate for each other. Therefore, all these rotations are equivalent to the identical permutation of nuclei. The motions corresponding to the improper transformations of a point group are not implemented in a rigid molecule and, correspondingly, the operations of reflection and mirror rotation are not defined in the vibration-rotational configuration space. The rotational subgroup of the Gpoint Ci group, which describes the actual structure of the energy spectrum, includes, instead of the improper elements, their products with the element i of spatial inversion. By analogy with the results of the previous section, for the transformation i we have the following representation in the vibration-rotational configuration space: .i/nucl D .C21 /rot  ./vib ; (12.11) where C21 is the rotation by an angle  about the axis of the orthogonal plane . With respect to the reflection operation, the nuclei of a molecule can be either single or paired, depending on whether they lie in the equilibrium configuration in the plane  or are symmetric to it. Obviously, the paired nuclei are necessarily identical. In both cases, we used the H2 O molecule as an example to consider the transformation . By definition (see Section 5.1), the mirror rotation Sn1 by an angle 2=n is written as Sn1 D Cn1 h ; (12.12) where the reflection plane h is orthogonal to the rotation axis Cn. That is, the mirror rotation is a combination of the above-considered types of transformations. Correspondingly, for the transformation iSn1 in the vibration-rotational configuration space we have (12.13) .iSn1 /nucl D .Cn1 C21 /rot  .Sn1 /vib; where Cn1 C21 is the rotation by an angle .2=n/ C . Thus, all transformations of the point group of an arbitrary molecule are equivalent to the permutations of its identical nuclei, which can easily be derived from the action of these transformations on the equilibrium configuration. Generally, in a matching by equivalent elements of the permutation group and the point group, the latter is homomorphically mapped onto the subgroup of the permutation group, in such a way that

Section 12.3 Internal geometric symmetry of the Hamiltonian

215

the totally symmetric coordinate Young diagram of the permutation group corresponds to the identity representation of the point group. Therefore, the point group characterizing the molecule in the chosen electronic state should be used as a group of rigorous symmetry of the total (electron-vibration-rotational) motion in this state. Importantly, this conclusion is free of the constraints on displacements of the rigid-molecule nuclei from their equilibrium positions. Homomorphism in the matching leads to geometric forbiddances for the coordinate multiplets and takes place only for the molecules with planar or linear equilibrium configuration. In the latter case, this fact plays the strongest role. Thus, for the multiplets of a molecule with the point group C1 v , among an infinite (although countable) number of irreducible representations, only one identity representation is possible (see Table 8.8). As a consequence, only the stationary states in which the projection of the total coordinate angular momentum on the C1 axis is due to the vibronic motion are admissible. For many molecules in the chosen electronic state, the transitions between the different minima of the effective nuclear potential should be taken into account, even if the excitation energy is much less than the dissociation energy. Thus, it is of principle to clarify the question on the compliance between the symmetry transformations when the rigid model of the molecule is replaced by a nonrigid one. For a nonrigid molecule, we construct an extended point group that characterizes the local motions in each minimum and the transitions between them. Clearly, the extended point group should include the point groups corresponding to the minima of the potential as its subgroups. The elements that define the transitions may pertain both to the exchange and nonexchange types. The latter are not the symmetry elements of the Hamiltonian of the total coordinate motion, and will not be considered here. Transitions of the exchange type can bind only the energetically equivalent minima of the effective potential with the same point group. In the action of the elements that define such transitions, the effective nuclear potential transforms into itself, and the spatial position of the equilibrium configuration is preserved. Consider the methane molecule C2 H6 . The equilibrium configuration of this molecule in the ground electronic state corresponds to the point group D3d D C3v  CI (see Figure 8.8). This molecule features the internal rotation of two identical tops CH3 about the chemical bond of carbon atoms hindered by a low potential barrier. The rotation of each internal top from the point of the minimum of the potential by angles 2=3 and 4=3 leads to an equivalent minimum, and the spatial position of the equilibrium configuration is preserved. According to equation (9.2), the extended point group has the form G72 D .c3v  c3v / ^ CI . The group G72 operations not included in its subgroup D3d act on the equilibrium positions of the nuclei and mix the rigid molecules belonging to the different minima of the effective nuclear potential. It is clear that in this case the position of an arbitrary nuclear configuration in space is preserved with accuracy up to the change of numbers of the identical nuclei. Thus, the group G72 is a symmetry group of the total internal coordinate motion of a nonrigid molecule. Operations of the

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point group D3d can be formed from the group G72 operations not included in the point group. For example, C31 D .c31  E/  .E  c31 /. It is important that the actions of the left- and right-hand sides of the equalities on an arbitrary nuclear configuration should coincide. It can easily be verified that this is really so. From a qualitative point of view, the configuration space of nuclear displacements of nonrigid molecules in the chosen electronic state is divided into regions corresponding to individual rigid molecules. Naturally, the displacements of the nuclei of each rigid molecule are bounded by its region. It should also be emphasized that the improper operations of the point group, which are not implemented in the vibration-rotational space of a rigid molecule, are the operations of mixing with equivalent rigid molecules. It is easily seen that the aforementioned action of such operations on an arbitrary nuclear configuration is similar to the permutation of equilibrium positions of the identical nuclei.

12.4 Definition of the rotational motion In the case of a rigid molecule, internal motion of the nuclei is divided into rotational and vibrational. For their determination it is required to physically correctly relate the MCS to the molecule. For the molecule with nonlinear equilibrium configuration, the traditional choice of the MCS is defined by the Eckart conditions X Ma Œr0.a/ ır .a/  D 0; (12.14) a

.a/

where r0 is the radius vector of the equilibrium position of the nucleus à and ır .a/ is the displacement of the nucleus from this position. Eckart derived these conditions from the requirement of the minimum kinetic interaction between the vibrational and rotational motions introduced [37]. However, their geometric interpretation presented in [14] is important to us. Let some motion of the nuclear system bind the initial and final configurations in a fixed position of its center of mass in space. To divide this motion into rotational and vibrational parts, we convert this system from the initial to the final configuration in two stages. We begin with the rotation of the initial configuration that minimizes the quadratic form X .a/ .a/ Ma Œrcon  rin .˛; ˇ; /2 D min; (12.15) a

.a/ rin .˛;

where ˇ; / is the radius vector of the nucleus a after the rotation of the initial configuration, which is determined by three Euler angles. In this case, we fix the MCS with respect to the equilibrium configuration of the nuclei. We emphasize that such a configuration within the limits of one electronic state is a rigid body. This part is called the rotational motion. Principle (12.15), which also leads to the Eckart conditions, relates real displacements to the rotational type in the maximum degree possible. At the second stage, the remaining motions are related to the vibrational type. In fact, the

Section 12.4 Definition of the rotational motion

217

rotational motion is responsible for the rotation of the equilibrium configuration with preserved displacements of the nuclei relative to their equilibrium positions, while the vibrational motion only changes the displacements of the nuclei relative to their equilibrium positions. The important point is that such rotational motion is free (there are no fields specified in the space of Euler angles) – as it must be due to the isotropy of space. As was mentioned in Section 12.1, a criterion for a molecule to undergo no rotational motion as a whole is not equivalent to the requirement that its angular momentum be zero. The reason is that the expression for the angular momentum is not the total time derivative of some function of the nuclear coordinates alone and that the equality to zero of the angular momentum cannot be integrated with respect to time in such a way that to be reduced to the equality to zero of a certain function of these coordinates. In [64], where it was assumed that this reduction is necessary for a reasonable formulation of the “pure rotation and pure vibration” concepts, a requirement identical to equation (12.14) was obtained for choosing an MCS for a nonlinear molecule. Finally, it can be shown [14] that the formulation of the Eckart conditions does not involve restrictions on the displacements ır .a/ from the equilibrium positions. If the equilibrium configuration of a molecule is linear, it is impossible to fix an MCS relative to this configuration (the Euler angle specifying rotation about the molecule axis remains arbitrary), and special analysis is needed [14]. We will therefore use that definition of rotational motion in which this motion is responsible for the rotation of the effective potential with the nuclear positions unchanged in this potential. This means that the MCS is fixed relative to the effective potential. Clearly, such a motion is free. It is also seen that this definition poses no restrictions on displacements of the nuclei from their equilibrium positions. For a nonlinear molecule, both definitions are equivalent since in this case the positions of the effective potential and equilibrium configuration are rigidly related. However, for a linear molecule, this relation is violated because the equilibrium configuration is now linear, whereas the effective potential remains nonlinear. When fixing the MCS relative to the nonlinear potential, an additional condition for determining the earlier arbitrary Euler angle arises, namely, k D ƒ C l;

(12.16)

where k; ƒ, and l are the projections of the total angular, electronic, and vibrational momenta, respectively, on the axis of a linear molecule. Expression (12.16) is consistent with the well-known fact that the projection of the total angular momentum on the axis of a linear molecule is determined only by the electron-vibrational motion (see Section 8.3). Note that ƒ  0 for the nondegenerate electronic state and l  0 for the nondegenerate vibrational state. This definition of the rotational motion easily extends to nonrigid molecules. Finally, it is important that in a given electronic state the concept of an effective nuclear Hamiltonian including the effective potential remains valid upon introducing corrections to the BO approximation [66]. Therefore, all of the above conclusions regarding the choice of the MCS remain valid, too. With the correct configuration space

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formed, the requirement should be made that the kinetic and potential parts of the nuclear Hamiltonian in equation (12.4), expressed in terms of vibrational and rotational variables, be invariant under transformations of the point group of the molecule. It is only now that the problem of the spectrum of intramolecular dynamics is formulated and can be solved by analytical and/or numerical methods. The key problem with applying Hamiltonian (12.1) for this purpose is that its expression does not involve the concept of the molecular structure. It is of principle that this concept can be introduced only by using perturbation theory. The opinion is therefore wrong (in spite of its great popularity) that, given sufficient computational resources, the description of the spectrum of intramolecular dynamics can be obtained exclusively by numerical methods using Hamiltonian (12.1). Unfortunately, from the formulation (12.14) it is difficult to understand all the consequences of choosing an MCS not satisfying the Eckart conditions. Only this can explain why so many papers have appeared over the past three decades on simplifying the description of the internal dynamics of a rigid nonlinear molecule by abandoning the Eckart conditions (see, e.g., [16, 45] and references therein). Paper [16] describes the so-called embedding procedure which rigidly relates the MCS with the instantaneous rather than equilibrium nuclear configuration. As a result, the expression for Tnucl in equation (12.4) becomes simpler, but the kinetic vibration-rotational interaction increases in this case, which is considered a drawback. It is believed that such an approach is easier for small multiatomic molecules. In particular, it is widely used to describe the spectrum of triatomic molecules in nondegenerate electronic state (see, e.g., [77, 78]). At the same time, it is argued in [45] that by abandoning the Eckart conditions, the MCS can be chosen in such a way that the kinetic vibration-rotational interaction is totally absent. Clearly, this leads to a significant simplification of the description. Refraining from comments on the results presented in [16, 45], we emphasize that the most important consequences of abandoning the Eckart conditions were ignored when those results were obtained. Namely, in this case, the rotational motion will change the position of the nuclei in the effective potential. In other words, the rotational motion is no longer free. As a specific example, consider the choice of the MCS using the embedding procedure for the water molecule, H2 O. In the upper part of Figure 12.1, the light circles show the equilibrium configuration, which corresponds to the point group C2v , for the H2 O molecule in the ground electronic state and the dark circles, its instantaneous configuration for the vibrational mode B1 . We assume that the axes shown in this figure are the MCS axes fixed with respect to the equilibrium configuration (the z axis coincides with the second-order axis of the C2v group). The spatial arrangement of these axes remains unchanged in the vibrational motion. In the embedding procedure, the z axis is usually considered passing through the O nucleus and the center of mass of the system of two H nuclei. Both MCS axes spatially coincide when the nuclei are in the equilibrium position. However, during a vibrational displacement of the B1 type, the axes z and x in the embedding procedure rotate about the y axis that is orthogonal to the plane of the molecule, and the max-

Section 12.5 Selection of physically meaningful states

219

imum rotation angle depends on the vibration amplitude. The same rotations of the MCS can also be obtained by free rotation of the molecule with fixed (e.g., equilibrium) positions of the nuclei. That is, identical rotations of an MCS may correspond to different displacements of nuclei in the effective potential. This means that a field in the space of Euler angles at a given rotation depends on the specific trajectory of the nuclear motion (the trajectory determines which part of a nuclear displacement in the effective potential can be ascribed to rotation). Nevertheless, the rotation of an MCS in the embedding procedure is considered to be free, and nuclear displacements in the effective potential, to be exclusively vibrational. It should be noted that in applying this procedure, vibrational motions are specified by internal coordinates (for example, by the lengths of the two OH bonds and the angle between bonds). This leaves our conclusions totally unchanged, however.

12.5 Selection of physically meaningful states In the quantum description of bound stationary states, it is very important to correctly determine the basis wave functions since in this case not all the solutions of the Schrödinger equation are considered, but only the so-called physically meaningful ones. With this done, the resulting description produces the discrete energy spectrum observed in such states. For example, for a one-dimensional harmonic oscillator with the mass m and frequency ! the stationary Schrödinger equation has the form [64] 

¯2 d 2 ‰ m! 2 x 2 C ‰ D E‰: 2m dx 2 2

(12.17)

The solution to equation (12.17) for a discrete energy spectrum is well known to be En D ¯! .n C 1=2/;

(12.18)

where n D 0; 1; 2; : : : is the quantum number of the oscillator. Note, however, that in obtaining solution (12.18) an additional condition was taken into account, namely, that the wave functions of the stationary states belong to the function space L2 (the space of square-integrable functions) since only they satisfy the normalization condition. Otherwise, a continuum power set of formal solutions is superimposed on a countable set of physically meaningful solutions. Therefore, the search for bound states by numerical methods without prior analytical writing out of the space of correct basis wave functions requires, in addition, selecting the solutions belonging to a discrete spectrum. Importantly, the procedure used to select the discrete spectrum depends substantially on the geometry of motions, which is specified by the coordinates of the configuration space of the system under consideration. For example, for the oscillator one has 1 < x < 1;

(12.19)

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Chapter 12 On the meaning of the Born–Oppenheimer Approximation

and the particle moves along a straight line. However, if the particle is taken to be moving along a circumference of radius r , x D r ';

(12.20)

then for ! D 0 equation (12.17) describes the free rotation of a one-dimensional rotator with a discrete spectrum of the form El D

¯2 2 l ; 2I

(12.21)

where l D 0; 1; 2; : : : is the quantum number of the rotator and I D mr 2 is its moment of inertia. In selecting this spectrum, the condition was additionally used that the wave function be single-valued ‰.' C 2/ D ‰.'/: (12.22) The term “single-valued” is understood here in the physical sense: the wave function should be the same on the intervals 0  2, 2  4, and so on, because these intervals are all physically equivalent. In fact, this condition should be regarded as a law of nature for coordinate motions (spin motions in half-integer spin systems require using two-valued functions for their description [64]). Importantly, this condition does not reduce here to the requirement that the wave functions belong to space L2 . As a result, the periodic wave functions of the stationary states of the rotator, which depends on the variable ' D x=r , and the wave functions of the stationary states of the harmonic oscillator are not expanded in terms of each other even though each family of stationary wave functions forms a complete basis in its respective function space. Actually, the Hilbert space, to which the solutions of a discrete spectrum belong, is defined differently. Therefore, discrete spectra in these examples belong to different segments of the solutions of equation (12.17). In other words, changing the geometry of the motion leads to a totally different type of a discrete spectrum. Selecting the discrete spectrum of an atom is a rather simple procedure. When the motion of the nucleus is neglected (i.e., the mass is assumed to be infinite), the configuration space of the atom forms as a direct sum of the configuration spaces of individual electrons: .xi ; yi ; zi / ! .ri ; i ; 'i /; (12.23) where ri is the radius vector of the ith electron, and i and 'i are its two polar angles. Therefore, the basis functions of the discrete spectrum can be written as linear combinations of the products of basis functions of all the electrons, the precise form of this combination being determined by the symmetry of the problem. The basis functions of an electron have the well-known form [64] R.r /Ylm .; '/:

(12.24)

Here, the radial wave function R.r / belongs to the space L2 and Ylm .; '/ are spherical functions. Note that the problem of the motion of one electron involves no geometric constraints on its displacements, and the angular coordinates were introduced just

Section 12.5 Selection of physically meaningful states

221

to provide the simplest way of taking into account the spherical symmetry of the problem. However, the coordinates x; y, and z can also be used. Therefore, the procedure of the discrete spectrum selection reduces here to the condition that the complete coordinate wave function belongs to the space L2 . Because the basis functions are defined by the quantum numbers of individual electrons, it follows that the state of an atom is a superposition of the states of individual particles. This last statement means that for all the realizable internal motions of the atom, no particle is geometrically limited to its displacements by other particles. The finite mass of the nucleus is taken into account by placing the origin of the coordinate system at the center of mass of the atom. As a result, the system of electrons is replaced by a system of quasiparticles (with a mass somewhat different from the electron mass). Then, the configuration space is, as before, a direct sum of configuration spaces (12.23), but now of those related to individual quasiparticles. It is important to stress here the large role in quantum mechanics of the determination of the assemblage of motions into which the total motion of the microsystem is decomposed. The reason is that it is precisely for separate motions in this set that conditions for selecting the discrete spectrum are formulated. That is, the problem of selecting the discrete spectrum is closely related to that of defining separate internal motions. In the case of an atom, the total motion represents the set of independent single-particle motions. For a molecule, selecting the discrete spectrum is much more complicated. The characteristics of the observed molecular spectrum cannot be adequately accounted for by a description which places the origin of the coordinate system at the molecular center of mass and in which the configuration space is formed as a direct sum of configuration spaces of individual quasiparticles. The reason is that in collective particle motions the displacements of one particle are limited by the positions of other particles and that these limitations depend on the geometry of the motions. As a result, the basis functions of separate molecular motions cannot be specified by a superposition of the basis functions of independent single-particle motions. A configuration space of collective motions which allows the formulation of physically correct conditions for selecting the discrete spectrum of the molecule can only be constructed by using the BO approximation. Only then is the problem of describing such a spectrum formulated and can be solved analytically and/or numerically. The underlying concept of this approach is the molecular structure, which arises in the BO approximation and whose characteristics depend on its electronic state. Rotational motion is defined using the molecule-related MCS whose introduction has already been considered in Section 12.4. The complete set jJ; k; M i of rotational wave functions, where k and M are the quantum numbers of the angular momentum projection on the MCS and FCS axes, respectively, is selected by the conditions of their being single-valued. It is important that they do not reduce here to the condition of the basis functions belonging to the space L2 . That is, the rotational wave functions can by no means be expanded in the basis functions of single-particle motions.

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For a rigid molecule, the condition that the basis functions belong to the space L2 is applied only to select the basis functions of vibrational motion. When writing out the correct wave functions of internal motions, it is also necessary that the symmetry of the problem be taken into account. The above suggests that, in using Hamiltonian (12.1) in the Cartesian coordinates to describe the discrete molecular spectrum, it is not known how to introduce information which would allow the selection of physically meaningful solutions in the presence of collective motions. It is for this reason that such a Hamiltonian is impossible to use. Note that the molecular structure and, therefore, the selection procedure depend on the electronic state. However, the Hamiltonian expression (12.1) involves electron coordinates, but not electronic states. It is to be emphasized in connection with the above that there have been recent attempts to avoid the BO approximation when describing bound stationary states of some simple molecular systems (see [41, 61] and references therein). Specifically, three-particle, namely two nuclei and one electron, systems were considered, by analogy with the HC 2 system shown in Figure 12.3. The initial Hamiltonian is taken in the Cartesian coordinates in the form (12.1), but then it is followed by the transition to collective variables by introducing an MCS directly into the three-particle problem. Recall that when the BO approximation is used, for each bound electronic state we have a two-particle problem of the motion of nuclei in the effective potential, relative to which the MCS is fixed with the origin at the center of mass of the system of two nuclei. That is, the axes of the MCS are fixed based on the solution for the motion of the electronic subsystem according to equation (12.3). Because the geometric symmetry of internal dynamics is specified by the group D1 h , it is convenient to choose as the axis z the axis C1 passing through the equilibrium positions of the nuclei (the specific choice of the axes x or y affects nothing due to the axial symmetry of the problem). In the three-particle system, the origin of the MCS is also placed in the center of mass of the system of two nuclei. The axis passing through the instantaneous nuclear positions is taken to be the z axis. But in this simple case, the z axis necessarily coincides with the axis passing through the now unknown equilibrium positions of the nuclei. As for the x or y axes, one of them is chosen to lie in the plane given by three particles, and

Figure 12.3. Three-particle molecular system HC 2 (the electron lies in the xz plane).

Section 12.6 Symmetry methods in the description of intramolecular dynamics

223

the other is orthogonal to this plane. Clearly, rotations about such x, y, and z axes do not change the potential energy of the three-particle system. It is easily seen that the choice of an MCS is performed similarly to the two-particle problem (using, as far as possible, the simplicity of the system). As a result, however, a contradiction arises because the origin of the MCS does not coincide with the center of mass of the three particles, i.e., rotations about the MCS axes change the position of the center of mass of an isolated system, which is impossible for the actual rotational motions. That is, rotations about the MCS axes change the position of the center of an isolated system, which is impossible in the actual rotational motions. In other words, choosing an MCS in this way necessitates that a certain compensating motion should accompany the rotation to preserve the position of the center of mass. This clearly shows the difference between the independent and collective motions of particles. The point is that in the description under consideration, the motion of the center of mass was separated, so that the three-particle problem reduced to the problem of motion of two quasiparticles. If the quasiparticles are considered independent, no constraints on their displacements arise (the “expelled” particle moves in such a way as to maintain the spatial position of the center of mass). The situation, however, is completely different when considering collective (in this case, rotational) motion. Then the motions of all particles are rigidly connected, and the MCS axes must necessarily pass through the center of mass of the system. It is useful to note in this connection that the molecular structure is formed only under conditions of a certain spatial “smearing” of the electron, that is, in certain electronic states, and in these states the position of the center of mass of the molecule coincides with the center of mass of the nuclear subsystem. This is confirmed by the absence of violations of the geometric symmetry D1 h. The transition to a “smeared” electron (or the effective nuclear potential) is performed by using the BO approximation. Note that in this approximation, it is more correct to consider the nuclei as quasiparticles since their masses not only do not coincide with the masses of isolated nuclei due to nonadiabatic corrections, but also are functions of the coordinates of the configuration space.

12.6 Symmetry methods in the description of intramolecular dynamics Currently there are two, philosophically very different, approaches to using the symmetry properties of intramolecular dynamics. In the first approach, explained for a number of examples for rigid nonlinear molecules even in an early edition of “Quantum mechanics. Nonrelativistic theory” [63], a Course of Theoretical Physics by L. D. Landau and E. M. Lifshitz, the symmetry of the electron-vibration-rotational motion in a given electronic state is determined by the point group. Such application is justified and generalized within the framework of the symmetry group chain concept, which is presented in this book, and the generalization to a large extent concerns the extension

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to nonrigid molecules. The concept is essentially as follows. As was already shown, the methods for solving the stationary Schrödinger equation are fundamentally based on perturbation theory. The transition to the zero approximation is carried out by construction of a chain of nested models (increasingly approximate ones) until the exact solution of the model problem becomes possible. Simultaneously, a chain of symmetry groups characterizing these models arises. In the first place, the difficulties of solving the Schrödinger equation are due to the declarative nature of the obtained series of perturbation theory describing the transitions between the neighboring models. Not only their properties are unknown, but often it is impossible to correctly calculate even the lower-order corrections. The situation changes radically, however, if we limit ourselves to the symmetry properties of the molecule. Consider the models and their corresponding symmetry groups for a rigid molecule. The rigorous internal symmetry of the Hamiltonian is defined by the permutation group of the electrons and identical nuclei, as well as by the implicit geometric group which characterizes the molecule structure. With the BO approximation, internal motion is divided into electronic and nuclear. In a given electronic state, the implicit symmetry is replaced by its explicit counterpart coinciding with the symmetry of the equilibrium configuration, which is determined by the point group, in this state. The wave functions of electronic and nuclear motions can be separately classified by irreducible representations of the point group. In the electronic problem, transformations of the point group act on the electronic coordinates and in the nuclear problem, on displacements of the nuclei. Nuclear motion is divided into vibrational and rotational, which in the zero approximation are described by the models of a harmonic oscillator and a rigid top. Symmetry of the vibrational model is also determined by the point group, but here its transformations affect only the vibrational coordinates. Symmetry of the rotational model is determined by the internal symmetry group of a rigid top, whose transformations act on the Euler angles. All groups are combined into a connected chain by the matching conditions. To do this, in the groups of the neighboring models, we single out the equivalent elements, with respect to which the wave functions and operators of physical quantities should be transformed in the same way. In other words, transitions between the neighboring models are accompanied by certain nontrivial constraints on the compliance of symmetry types. An extended point group characterizing both the local motions at the minima of the effective nuclear potential and the motions between these minima additionally appears for a nonrigid molecule in the BO approximation. This group can be noninvariant (see Section 6.2). Advantages of the approach based on a chain of groups, as was mentioned in Section 7.4, are due primarily to the fact that the matchings are rigorous (!). A chain of groups has very wide capabilities. First, the symmetry symbols of its groups specify the basis unit vectors of the function spaces of separate motions, of which symmetrized basis unit vectors of the total function space are constructed. Second, the complete set of self-adjoint operators is specified in the function space of each

Section 12.6 Symmetry methods in the description of intramolecular dynamics

225

motion, whose matrix elements are functions of only the symmetry symbols of the basis unit vectors. From these, we construct operators of physical quantities specified already in the complete function space. In this case, the symmetry properties of the physical quantities are determined by their known behavior with respect to the rigorous fundamental laws. As a result, we obtain, in terms of symmetry, an exact solution to the problem of bound molecular states, the positions of the energy levels and transition intensities between them being calculated with a simple geometric scheme. The correctness of this approach is limited only to the correct choice of symmetry groups, which is a very important point. Actually, in the usual approach the effective Hamiltonian Hnucl is written out only approximately, and the manipulations with it (even numerical) are performed by approximate methods. But the group chain approach is free of such problems. It can be said that the chain automatically takes into account all symmetry-allowed interactions, including the weak ones (for example, relativistic). The second approach, proposed in the pioneering work [65] (a state-of-the-art analysis is given in [16]), assumes that the geometric group cannot characterize the symmetry of the total intramolecular motion. For this purpose, the complete nuclear permutation-inversion (CNPI) group is introduced, defined as a direct product of the permutation group of identical nuclei and the inversion group of nuclear and electron coordinates with respect to the center of mass of a molecule. To simplify the consideration, in view of the limited accuracy of experimental data, only the so-called feasible elements (corresponding to molecule transformations in which the barrier is zero or fairly small compared with the dissociation energy), which form a molecular symmetry group or an MS group, are selected from the CNPI group. It is assumed that the MS group contains all the symmetry elements necessary to describe the problem of the molecular energy spectrum. For a rigid nonlinear molecule in a nondegenerate electronic state, the MS group is isomorphic with the point group, and the efficiency of the latter in describing the internal dynamics is due to this fact alone and is considered to be accidental. However, the use of the MS group leads to very serious problems. We dwell on the major one. It is easy to verify that molecules may have qualitatively different internal dynamics but identical MS groups. Examples are ethane C2 H6 and acetone (CH3 )2 CO molecules for which, taking into account the torsional motion of both methyl tops, one obtains the MS group G36 [16]. The experimentally confirmed difference in their internal dynamics is due to the fact that in the ethane molecule the tops rotate about one axis, while in the acetone molecule, about different axes (see Sections 9.3 and 9.5). That is, the difference is determined by the different geometry of the internal motions. However, such information is entirely absent in the MS group. Therefore, to use this group, it is necessary first to construct the configuration space of the molecule with allowance for all motions needed for describing the experimental data and to write out in explicit form the complete set of basis wave functions dependent on the variables of this space (some set of approximate wave functions of the stationary states is a usual choice).

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Then the action of the elements of the MS group is specified “manually” in the configuration space, and their action on the basis functions is calculated. As a result, a classification of stationary states can be constructed. In fact, this very cumbersome procedure allows one to take into account the geometry of internal motions. The symmetry here is clearly of a “secondary” nature since its use is based on knowledge of at least an approximate solution for the wave functions of the energy spectrum. However, such a solution is often unknown for the nonrigid molecules. Moreover, the MS group is, by definition, a symmetry group of the Hamiltonian and cannot allow for the nonrigid motions corresponding to the noninvariant transformations. Note also that, as will be shown in Section 12.7, even when realized, this approach can lead to major errors in defining separate motions. Naturally, constructing the description of the energy spectrum also requires writing an expression for Hnucl to the desired accuracy, which is a separate and complex problem. Classification of the wave functions in the MS group only simplifies the obtaining of eigenvalues of this Hamiltonian.

12.7 Geometric symmetry and definitions of nonrigid motions Since a solution to the problem of stationary states of the molecule is fundamentally based on perturbation theory, it should of course be ensured that the correct properties of symmetry of internal dynamics are transferred to approximate models. There appear certain requirements for defining separate types of motion, and their violation also leads to physically incorrect consequences. The point regarding the rotational and vibrational motions was discussed in Section 12.4. Here, this point will be considered for nonrigid motions. The internal geometric symmetry of a nonrigid molecule is characterized by an extended point group, which, besides the local motions at the minima of the effective potential (applicability range of the point groups), also allows for transitions between the minima. There occurs a nontrivial problem of defining the action of the transformations of an extended point group in the function spaces of separate types of internal motion. Only after this problem is solved can the definitions of the motions be considered complete. Consider a methanol molecule CH3 OH in the ground electronic state (see Section 9.2). The equilibrium configuration of the molecule corresponds to the point group CS . The actual structure of the energy levels is described by the rotational subgroup C2 of the group CS  Ci . Basis unit vectors of the electronic, rotational, and vibrational function spaces are defined with respect to the elements of the C2 group. The molecule becomes nonrigid because of the torsional motion of the methyl top CH3 , which is taken into account by the extended point group G12 . In this case, the actual structure of the energy levels is described by the complete rotational subgroup H6 D H3 ^ C2 of the G12  Ci group, and the torsional motion is determined by the

Section 12.7 Geometric symmetry and definitions of nonrigid motions

227

elements of the H3 group. That is, the basis unit vectors of the electronic, vibrational, and rotational motions are defined with respect to transformations of the C2 group, and the basis unit vectors of the torsional motion to transformations of the H3 group. It is important that these groups do not intersect. However, due to the geometry of the problem, the group C2 transformations, according to equation (9.13), also act in a nontrivial way on the torsional unit vectors, which is crucial for the construction of a correct description. Thus, only due to the requirement (9.13), the necessary doubly degenerate torsional states appear in the description. In applying group chain methods, all operators of the physical quantities of the molecule are constructed based only on the symmetry properties of the problem. However, even if the description uses a certain model Hamiltonian Hnucl, it is necessary to introduce a requirement like that in equation (9.13) when forming the function space. In this sense, symmetry is always “primary.” The following example clearly demonstrates the importance of this conclusion. Consider an ethane molecule C2 H6 in the ground electronic state (see Section 9.3). The equilibrium configuration of this molecule belongs to the point group D3d D D3  CI . The actual structure of the spectrum is described by the rotational subgroup H2  D3 of the D3d  Ci group. Electronic, vibrational, and rotational basis unit vectors are defined with respect to transformations of the H2  D3 group. This molecule also features internal rotations of two identical methyl tops CH3 about the same axis passing through the carbon nuclei. These internal rotations are taken into account by the extended point group G72 . The inversion doublets remain unsplit, and the actual structure of the energy levels is described by the complete rotational subgroup H36 D B6  D3 of the G72  Ci group, where the group B6 has the form B3 ^ H2 . Writing the H36 in this way makes it much easier to answer the nonsimple question “What transformations of the molecule define the torsional motion?” These are the transformations of the B3 group. Then the total internal motion uniquely decomposes into motions defined by the nonintersecting groups H2  D3 and B3 . The answer is very nontrivial since according to it, the rotation of one top is not a purely torsional motion. For example, the rotation c31  E of the left top is the superposition of the torsional motion c32  c31 and the rotation C32 of the molecule as a whole. Due to the geometry of the problem, the elements of the H2 group do not commute with the elements of the torsional group B3 . Therefore, the elements of the H2 group also act in the torsional space according to equation (9.28). However, the operations C3k of free rotation of the molecule as a whole commute with the elements of the B3 group and do not act in the torsional space, although C3k D c3k  c3k . Importantly, the operations C3k describe namely the free rotation since they do not relate independent local minima of the effective nuclear potential and therefore do not take the molecule from one minimum to another. For a nonrigid ethane molecule, the G36 group is used as an MS group [16]. The groups G36 and H36 are isomorphic, but the latter as a direct product of the two groups was not written. Although the groups H36 and G36 are differently employed, the prob-

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Chapter 12 On the meaning of the Born–Oppenheimer Approximation

lem of determining the internal motions remains. In the MS approach, the approximate wave functions of the vibration-rotation-torsional energy levels are specified in explicit form and the action of the elements of the G36 group on the arguments of these functions is indicated. As a result, the vibrational, rotational, and torsional wave functions are two-valued in the G36 group. Hence, it is replaced by its double group G36 (EM). Of course, such a behavior of the wave functions of separate coordinate motions is impossible (see Section 12.5). In fact, this two-valuedness is a consequence of the incorrect account of symmetry. We then consider a nonrigid water molecule H2 O in the ground electronic state. The equilibrium configuration of the molecule, shown in Figure 8.1, is nonlinear and corresponds to the point group C2v . Such a configuration may be considered to result from the distortion of a linear configuration with axial symmetry group D1 h. There are numerous high-precision experimental data, for which it is necessary to take into account excitations with respect to the angle between the OH valence bonds above the potential barrier corresponding to the linear configuration [29], i.e., the description should include nonrigid transitions through this configuration. The presence of these transitions, which is due entirely to the distortion of the equilibrium configuration, leads to the restoration of the axial symmetry of internal dynamics. This conclusion can easily be drawn based on the geometry of the problem. It is important that here the nonrigid transitions do not lead to the energy-level splitting since the final configuration can also be obtained from the initial one by rotation of the molecule as a whole by an angle  about the linearization axis. That is, a nonrigid transition does not take the molecule into an independent local minimum. Chain group methods automatically take this into account. The fact that the description of a nonrigid molecule involves the continuous group D1 h D C1 v CI instead of the finite group C2v for a rigid molecule implies that the internal dynamics changes radically if nonrigid transitions are taken into account. Indeed, the group C2v leads to a 3C3 and the group D1 h , a 2C4 division into rotational and vibrational degrees of freedom. In fact, the role of nonrigid transitions reduces here exactly to the rearrangement of the vibration-rotational configuration space, and this point should be the basis for constructing a function space and the effective operators of physical quantities acting in this space. This problem is discussed in Sections 15.3 and 18.3. We only note that a similar consideration based on the CNPI group leads to philosophical difficulties. The point is that the CNPI group D1 h .M / used for a symmetric linear triatomic molecule has the form D1 h .M / D 2  Ci

(12.25)

However, this group is too poor, and even formally it cannot allow for the axial geometric symmetry of the problem. Therefore, one has to introduce a highly artificial extended CNPI group D1 h .EM / [16], namely, D1 h .EM / D 2"  Ci ;

(12.26)

Section 12.8 Nuclear statistical weights

229

where 2" is the permutation group of two identical nuclei, both elements of which are continuous with respect to some parameter " (?!). Then the action of the elements of the D1 h .EM / group is defined “manually” in the configuration space, and their action on the basis functions written in explicit form is calculated. This analysis gives the necessary result only if both the basis functions themselves and the required symmetry properties are known beforehand. Nonrigid motions in the molecules considered above have a fairly simple geometry. Nevertheless, even these motions are subject to nontrivial constraints in terms of defining the action of symmetry transformations in the function space of bound states. The violation of these constraints leads to physically incorrect consequences in the description. Because of the huge variety of nonrigid molecular motions, it is hardly possible to obtain a general formulation for such constraints. Most likely, one will have to extend the understanding of this issue to individual classes of molecules with certain types of nonrigid motions.

12.8 Nuclear statistical weights Significant differences in the philosophy of two existing approaches to the use of the symmetry properties of intramolecular dynamics can lead to trouble when the results of one approach are used in another. A misunderstanding with the calculation of nuclear statistical weights is quite typical. As mentioned in Section 8.5, the coordinate multiplet degeneracies related with the point group do not lead to the actually observed degeneracy of the energy levels. However, to a coordinate multiplet of a given symmetry, several coordinate Young diagrams may correspond, and each of these diagrams is convolved with a spin Young diagram that is similar or dual to coordinate diagrams, depending on whether the identical nuclei are bosons or fermions. Each spin Young diagram, in turn, corresponds to one or several values of the total spin of the identical nuclei. With the nuclear spin-related hyperfine interactions neglected, the energy of the molecular term depends neither on the value of the total spin nor its orientation in space. The multiplicity of this degeneracy is called the nuclear statistical weight (see Section 7.1). For example, for a rigid molecule of ammonia NH3 , the coordinate and dual spin Young diagrams of the 3 group have the form (8.14), which leads to the coordinate multiplets (8.15). To account for the contribution of the primary isotope 14 N of the nitrogen nucleus, the nuclear statistical weights in equation (8.15) should be increased threefold. A method greatly simplifying the procedure of determining the nuclear statistical weights of the coordinate multiplets of any rigid molecule was obtained in [64]. In this method, the characters Y .Q/ D .2sa C 1/.1/2sa .na 1/ (12.27) are calculated first for the operations Q of the point group. Here, na is the number of nuclei that exchange places in the Q operation in each group of identical nuclei and sa

230

Chapter 12 On the meaning of the Born–Oppenheimer Approximation

is the spin of the nuclei in this group. The product is taken over all groups of the nuclei. Next, decomposing the representation (12.27) into irreducible ones, we have all admissible types of coordinate multiplets. The nuclear statistical weight of the multiplet is equal to the number of its occurrence in this representation. Certainly, for the NH3 molecule, this method gives the same nuclear statistical weights. It should be emphasized that no constraints were imposed on the intramolecular motions when equation (12.27) was derived. Hence, this formula is valid not only for the point groups of rigid molecules, but also for the symmetry subgroups of the Hamiltonian of extended point groups of nonrigid molecules. This method is especially important if permutation groups of more than eight identical nuclei need to be taken into account in the description of the internal dynamics. The fact is that the tables of characters of these groups are difficult to find, whereas equation (12.27) does not require the use of such tables. Meanwhile, it is asserted in [59] that a “basic mistake” in deriving equation (12.27) was made in [64], since the behavior of the multiplet with respect to the spatial inversion operation i was neglected during the action of the operations Q. Therefore, this formula is assumed valid only for the positive states, and it is corrected for the negative states. The point in [59] is erroneous since in that paper this formula is used for the MS group. Of course, for a rigid nonlinear molecule, the MS group is isomorphic with the point group, but these groups are different in their physical nature. From this point of view, the MS group is much closer to the rotational subgroup of the Gpoint  Ci group and is obtained from this subgroup by replacement of the elements of Gpoint by equivalent permutations of identical nuclei. In particular, the MS group of a rigid molecule of ammonia NH3 has the form C3v .M / D .¹13 º; ¹3º; ¹12ºi/;

(12.28)

where the notations in braces specify the classes of the 3 group. It is essential that, unlike the point group C3v of the ammonia molecule, the MS group given by equation (12.28) includes the elements containing the spatial inversion operation i, which exactly leads to errors for the negative levels. Therefore, the results in [59] can be treated only as the adaptation of the method obtained in [64] to allow working with the MS group. Moreover, such an adaptation was carried out previously [93]. Unfortunately, the interpretation proposed in [59] is fairly widespread (see, e.g., [16]).

12.9 Conclusions We have formulated some general conclusions based on the content of this chapter. It appears that the very concept of structure of the molecular system is introduced into the description only by the BO approximation. 1. The fact that the molecular system has a rotational motion of the system as a whole means that the molecule is a certain structure (“microcrystal”), in which the motions of the particles should be considered as basically collective. The symmetry

Section 12.9 Conclusions

231

of this structure is characterized by an implicit internal geometric group. It turns out that the very concept of structure of the molecular system can be introduced into the description only by using the BO approximation. With this approximation, the correct configuration space of the internal collective motions is constructed separately in each electronic state. In other words, a transition to the excitation region limited by a single electronic state takes place. For such a limited region, the implicit symmetry is replaced by a similar explicit symmetry. Since the correct choice of the explicit symmetry should be provided, the geometric group must be sought empirically. For rigid molecules in nondegenerate electronic states, such a group is a point group of their unique equilibrium configuration. However, the symmetry of the latter is an elementary consequence of the symmetry of internal dynamics, but not vice versa as is often stated, and only in this simplest case do these two symmetries coincide. In fact, the BO approximation makes it possible to formulate for the stationary Schrödinger equation the conditions for selecting the physically meaningful solutions of the discrete spectrum of internal collective motions against the vast background of formal solutions. It is only then that the problem of describing the spectrum of intramolecular dynamics is formulated and can be solved by analytical and/or numerical methods. In other words, the BO approximation is fundamentally necessary for formalizing the very definition of such a problem. Therefore, the widespread opinion that, given sufficient computational resources, the discrete spectrum of a molecule can be obtained based on Hamiltonian (12.1) in the Cartesian coordinates by purely numerical methods is erroneous. We also emphasize that the corrections to the BO approximation do not violate the problem formulation based on the BO approximation. 2. Currently, there are two philosophically very different approaches to using the symmetry properties of the intramolecular dynamics. The approach based on the concept of the CNPI group is better known. The principled disadvantage of this concept is that the CNPI group does not contain information on the geometry of internal motions. Therefore, it is first necessary to construct the configuration space of the molecule taking into account all motions needed for describing the experimental data, and write out explicitly the complete set of basis wave functions dependent on the variables of this space (a set of approximate wave functions of the stationary states is a usual choice). Then, the action of the elements of the MS group is specified “manually” in the configuration space, and their action on the basis functions is calculated. As a result, a classification of stationary states can be constructed. In fact, this very cumbersome procedure allows one to take into account the geometry of internal motions. The symmetry here is clearly of a “secondary” nature since its use is based on knowledge of at least an approximate solution for the wave functions of the energy spectrum. However, such a solution is often unknown for nonrigid molecules. Moreover, the CNPI group is, by definition, a symmetry group of the Hamiltonian and cannot allow for nonrigid motions corresponding to the noninvariant transformations.

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Chapter 12 On the meaning of the Born–Oppenheimer Approximation

In an alternative approach based on the concept of a chain of symmetry groups, the geometry of the internal motions is taken into account by the internal geometric group characterizing the molecular structure in a given electronic state. The main advantage of this approach is that its symmetry requirements are “primary.” That is, both the classification of bound stationary states and the description of their discrete spectrum are constructed on the basis of the symmetry principles alone. The resulting models lead to a simple algebraic scheme for calculating the position of the energy levels and transition intensities between them. It is important that the correctness of such models is limited only to the correct choice of the symmetry groups of internal dynamics. That is, a chain of groups automatically takes into account all the symmetry-allowed interactions, including the weak ones. 3. It is traditionally believed that in degenerate electronic states the electronic and nuclear motions strongly interact, and the BO approximation is therefore violated. Actually, in this case the symmetry requirements lead to a rigid nonforce relation between these motions. The force interaction is small and is taken into account as a perturbation in what follows. Such a rigid relation is similar to the relation between the coordinate and spin types of motion due to the symmetry requirements for permutations of identical particles. In other words, it is due to the symmetry alone that the condition of the electronic subsystem adiabatically following the nuclear subsystem cannot be reduced to the separation condition between electronic and nuclear motions. 4. According to the Jahn–Teller theorem, the symmetric equilibrium nuclear configuration of a rigid nonlinear molecule is always unstable in an electronic state that is orbitally degenerate because of this symmetry. As a result, the geometric symmetry of the equilibrium configuration is lowered, but the geometric symmetry of the internal dynamics does not change and coincides with the geometric symmetry of the unstable configuration. However, the molecule becomes nonrigid. 5. The action of the point-group transformations of a rigid molecule in a given electronic state on its arbitrary (especially nonequilibrium) instantaneous nuclear configuration is equivalent to permutations of identical nuclei in the effective potential of nuclear interaction that is invariant under these transformations. Generally, in a matching by equivalent elements of the permutation group and the point group, the latter is homomorphically mapped onto the subgroup of the permutation group, in such a way that the totally symmetric coordinate Young diagram of the permutation group corresponds to the identity representation of the point group. Therefore, it follows immediately from the fundamental symmetry properties with respect to permutations of identical particles that the point group characterizing the molecule in the chosen electronic state should be used as a rigorous symmetry group of the total (electron-vibration-rotational) motion in this state. Importantly, this conclu-

Section 12.9 Conclusions

233

sion is free of the constraints on displacements of the rigid-molecule nuclei from their equilibrium positions. 6. From a qualitative point of view, the configuration space of nuclear displacements of nonrigid molecules in the chosen electronic state is divided into regions corresponding to individual rigid molecules. Certainly, the displacements of the nuclei of each rigid molecule are bounded by its region. The improper operations of a point group, which are not implemented in the vibration-rotational configuration space of a rigid molecule, are the operations of mixing with equivalent rigid molecules. 7. The concept of the rotational motion of the molecule as a whole in a given electronic state introduces a molecule-related MCS. That is, by definition, the rotational motion is a motion that changes the orientation of the MCS axes with respect to the FCS axes. Of course, this motion should be free for an isolated molecule. Hence, it follows immediately that rotational displacements should not change the positions of the nuclei in their effective interaction potential (all other internal motions change the nuclei positions in the effective potential, that is, they are “inhibited” by this potential). Therefore, the definition of rotational motion, in which the effective nuclear potential is rotated as a whole with the nuclear positions unchanged in this potential, is the only physically correct definition. In other words, the MCS should be “frozen” into the effective nuclear potential, which for a rigid nonlinear molecule is similar to fixing the MCS with respect to the equilibrium configuration of the nuclei. The latter formulation is equivalent to the Eckart conditions. It is important that any other choice of the MCS violates the requirement of free rotational motion in isotropic space. 8. Quantum mechanics is based on the linear Schrödinger equation. Nevertheless, quantum mechanics copes with the qualitative jumps that occur in the description of the discrete spectrum of a quantum system when an implicit internal geometric symmetry group of this system appears or changes. The point is that in the case of the presence of a qualitative jump we change the conditions of selection of physically meaningful solutions. That is, we pass to another segment of solutions for the discrete spectrum. 9. The accuracy of the BO approximation, which is mainly determined by nonadiabatic corrections, is too small to allow working with the experimental data from high-resolution molecular spectroscopy. Indeed, the analytical estimates and direct numerical calculations of the nonadiabatic interaction of electronic states show that the relative contribution of these corrections is of the order of 102 for the energy levels of small polyatomic molecules in the ground nondegenerate electronic state (see, e.g., [45]). However, to describe the present-day experimental data on the spectral line frequencies, the relative accuracy of calculation of the energy levels should be of the order of 107  108 or even better. At present, it is unclear how to

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Chapter 12 On the meaning of the Born–Oppenheimer Approximation

calculate the nonadiabatic corrections by ab initio methods to the necessary accuracy. Naturally, under such requirements, corrections much smaller compared with nonadiabatic ones (for example, relativistic corrections) should also be taken into account. Therefore, even for triatomic molecules, a large number of phenomenological constants are introduced to improve the description of experimental data when the vibration-rotational spectrum of the molecule in a nondegenerate electronic state is calculated by ab initio methods. Moreover, as was rightly emphasized in [45], already the choice of a finite segment of the infinite-dimensional function space in solving electronic equation (12.3) is essentially a phenomenological procedure. This procedure is also used in precision calculations for the purpose of improving predictions. As a result, spectral line frequencies are described with accuracy several orders of magnitude better than the value of the contribution of nonadiabatic corrections without their being correctly taken into account. This means that the physical meaning of the calculation is lost. In this situation, the group chain methods, in which internal interactions are allowed for in full and at once (in particular, without separating between adiabatic and nonadiabatic) seem to be very promising.

Chapter 13

Molecules with transitions of the exchange and nonexchange types between equivalent configurations 13.1 Extended point groups The methanol molecule CH3 OH is a methodologically interesting example of a molecule with transitions of the exchange and nonexchange types between equivalent equilibrium configurations. This molecule was considered in Section 9.2 as a nonrigid system with the CH3 methyl top rotating about the planar COH structure. The existence of this motion is due entirely to the distortion of the equilibrium configuration of the molecule from the maximum symmetric configuration with the point group C3v to the actually observed one with the point group CS . As a result, there are three equivalent equilibrium configurations connected by a rotation of the exchange type of the methyl top. The corresponding extended point group G12 has the form (9.3). However, the planar COH structure can also rotate relative to the methyl top, a motion that group (9.3) disregards. This raises an important question on whether such a motion can lead to qualitatively new contributions to the effective operators of physical quantities. The geometry of nonrigid motions is now determined by an intermediate configuration, in which the CH3 top is a regular pyramid having the symmetry group c3v , whose axis c3 passes through the center of mass of the planar COH structure. As a result, the extended point group has the form F36 D c3v  c3v ;

(13.1)

where the left and right groups c3v characterize the structures CH3 and COH, respectively. Since the torsional motion of the COH structure, which is specified by the subgroup c3 of the right group c3v , is of the nonexchange type, F36 is already a noninvariant group with the Hamiltonian symmetry subgroup G12 . Group G12 determines only the coordinate degeneracies and nuclear statistical weights of the energy levels. The methylamine molecule CH3 NH2 is another example. The equilibrium configuration of this molecule, shown in Figure 13.1 for the ground electronic state, corresponds to the point group CS [52]. This molecule features a torsional motion of the CH3 methyl top and a much more complex inversion motion of the CH3 and NH2 structures [72]. To specify the geometry of these motions, we introduce an intermediate configuration, in which the axis z passing through the centers of mass of CH3 and NH2 , coincides with the axis c3 of the undistorted methyl top (the equilibrium configuration mentioned in [52] is very close to intermediate). In terms of symmetry of such an intermediate configuration, the torsional motion is taken into account by an

236 Chapter 13 Exchange and nonexchange transitions between equivalent configurations

Figure 13.1. Equilibrium configuration of the methylamine molecule CH3 NH2 (the nuclei H1 , C, and N lie in the symmetry plane xz, and the axis z passes through the centers of mass of the CH3 and NH2 structures). Dashed lines show a configuration connected to the initial configuration by inversion motion.

extended point group similar to that given by equation (9.3) G12 D c3v  cS ;

(13.2)

where the groups c3v and cS now belong to the CH3 and NH2 structures, respectively. As concerns the inversion motion, it looks like a mirror reflection of the NH2 structure relative to the xz plane with the methyl top rotated simultaneously by an angle  about the z axis in order to reconstruct the correct position of the CH3 and NH2 structures with respect to each other. This motion of the nonexchange type is specified by .z/ the noninvariant transformation F D c2   .xz/ . Therefore, both nonrigid motions are taken into account by the extended point group F24 D c3v  cS  F2 ;

(13.3)

where F2 D .E; F /. Obviously, the Hamiltonian symmetry subgroup in the group F24 is G12 . The internal dynamics of the cyclopentane molecule C5 H10 is very interesting. It is generally accepted (see [51] and references therein) that the equilibrium configuration of this molecule in the ground vibronic state is the so-called twist structure, which is shown in Figure 13.2. The most important feature of this structure is that the nuclei C1 and C2 move the same distance up and down, respectively, from the plane of the nuclei C3 , C4 , and C5 . That is, the five-membered cycle of the carbon nuclei becomes nonplanar. Therefore, the point group of the equilibrium configuration reduces from .x/ D5h to the purely rotational group U2 D .E; U2 /. As a consequence, there occurs

Section 13.1 Extended point groups

237

Figure 13.2. Equilibrium configuration of the cyclopentane molecule C5 H10 (the nuclei C3 , C4 , and C5 lie in the xy plane).

a nonrigid motion of the pseudorotation type, which can be represented locally as a sequence of correlated displacements of the neighboring nuclei of the cycle along the z axis. Namely, let the nuclei C1 and C2 move down until the nucleus C1 is in the plane of the nuclei C3 , C4 , and C5 (of course, this plane has a negative motion to preserve the position of the center of mass of the molecule). In this configuration, which corresponds to the top of the barrier for the pseudorotation and is called the envelope structure, the nucleus C2 is moved downwards from the plane of the remaining nuclei. The envelope structure has a symmetry plane passing through the z axis and the nucleus C2 . Further, the nuclei C2 and C3 move up and form a twist structure, in which the left nucleus C2 lies below the plane of the nuclei C4 , C5 , and C1 , whereas the right nucleus C3 lies above this plane. Importantly, this twist structure now belongs to another stereoisomer of the cyclopentane molecule. Therefore, one pseudorotation step is also a transition between two stereoisomers. In the next step, a twist structure of the first stereoisomer again results, but already with the C3 and C4 nuclei displaced from the plane. In each step, besides moving locally, the molecule rotates as a whole by an angle 2=5 in direction opposite to the transition of the twist structure about the ring during the local motion. It is easy to see that the pseudorotation binds a total of ten energetically equivalent equilibrium configurations (five for each of the two stereoisomers). During this motion, the molecule always remains nonplanar, and the barrier for such a motion is anomalously low ( 1 cal/mol). Note that the concept of pseudorotation in the cyclopentane molecule was introduced as early as in 1947 to explain the anomalous thermodynamic properties of cyclopentane. The intermediate configuration, in terms of which the geometry of nonrigid motions is specified, is characteristic in that the identical nuclei participating in the nonrigid motion are arranged geometrically equivalently in this configuration. This requirement

238 Chapter 13 Exchange and nonexchange transitions between equivalent configurations leads immediately to a configuration with the point group, D5 D C5 ^ U2 ;

(13.4)

where C5 is the group of fifth-order rotations about the z axis. Indeed, the even number of pseudorotation steps, which binds all five configurations of one stereoisomer, corresponds to the elements C52 ;

C54 ;

C56 D C51 ;

C58 D C53 ;

C510 D C55

(13.5)

Consider the interpretation of their action using the element C52 as an example. The initial configuration in Figure 13.2 after the local displacements of two pseudorotation steps transforms to a configuration with the nonplanar position of the C3 and C4 nuclei. The latter configuration comes back to its initial spatial position through a zero barrier by rotation of C52 . As a result, the spatial position of the configuration remains unchanged, which corresponds to a transformation of the exchange type. The extension of the U2 group to the D5 group with elements of the exchange type reflects the simple fact that the effective Hamiltonian of internal dynamics is invariant with respect to permutations of five independent twist structures of one stereoisomer. The element that specifies one pseudorotation step will be written as C51 P; where P is an element of the nonexchange type which allows for the transition between two stereoisomers. Since P 2 D E and C51 P C51 P D C52 ; the elements P and C51 commute with each other. Therefore, the odd number of steps of the cyclic pseudorotation group F10 corresponds to the elements C51 P;

C53 P;

C55 P D P;

C57 P D C52 P;

C59 P D C54 P

(13.6)

According to equations (13.5) and (13.6), one pseudorotation cycle includes two rotations about the z axis. The first rotation leads to a twist structure with the nonplanar position of the same carbon nuclei (the transformation C55 P D P /. But now the nucleus C1 lies below and the nucleus C2 above the plane of the nuclei C3 , C4 , and C5 . That is, this structure belongs to a stereoisomer. It is easily seen that F10 D C5  P2 ;

(13.7)

where P2 D .E; P /. Symmetric and antisymmetric irreducible representations of the P2 group will be denoted p1 and p2 . It is also necessary to take into account that the operation B D P  .xy/ is of the exchange type (it corresponds to the permutation of the hydrogen nuclei from the class ¹25 º of the 10 group). To allow for such an operation, we introduce an additional group CS D .E;  .xy/ / with the element  .xy/ of the nonexchange type. Then the extended point group can be written as F40 D D5  P2  CS :

(13.8)

The transition between stereoisomers also occurs as an inversion motion of the C nucleus cycle through its planar configuration. Beginning with the classical work [53],

239

Section 13.2 Methanol molecule CH3 OH

such a motion is taken into account in the construction of models describing the internal dynamics of cyclopentane. The result of the motion looks like a reflection of the C nuclei in the xy plane, but the accompanying motion of the H nuclei is not consistent with this operation. In the inversion motion, the displacements of all nuclei are small enough to ensure a small height of the barrier (about 5 kcal/mol). Conversely, this reflection operation leads to large displacements of the H nuclei and corresponds to a very high barrier. We now introduce an operation Q to describe the inversion motion. This operation is of the nonexchange type, and Q2 D E. Symmetric and antisymmetric representations of the group Q2 D .E; Q/ will be denoted q1 and q2 . For the further analysis, it is important that taking into account the inversion motion does not change the number of independent equilibrium configurations bound by nonrigid transitions. Their number is ten, as before. This is a consequence of the simple fact that the action of the operations P and Q on some initial configuration leads to the same final configuration, but in different ways. Therefore, PQ D E:

(13.9)

Condition (13.9) is sufficient for writing an extended point group of the cyclopentane molecule, on account of pseudorotation and inversion motion, in the form F80 D D5  P2  CS  Q2 ;

(13.10)

so that the following correlation rules between representations of the groups P2 and Q2 are fulfilled for allowed multiplets of this molecule: p1 $ q1 ;

p2 $ q2 :

(13.11)

Thus, there are no additional energy levels compared to the case where only pseudorotation is taken into account.

13.2 Methanol molecule CH3 OH We first note that nonrigid motions in a methanol molecule, specified by an extended point group F36 in the form (13.1), lead to the delocalization of the plane of the COH structure. However, the space of wave functions and the operators of physical quantities specified in this space can be written for any position of this structure. We choose the position of the COH structure in the  .yz/ plane as its reference position. Corresponding to this position is the equilibrium configuration with the point group CS D .E;  .yz/ /, which is shown in Figure 9.1. The complete coordinate wave function should relate only to irreducible representations (9.8) of the G12 group. Matching the latter with the group F36 , we obtain the following allowed types of transformations for the multiplets in the F36 group: 8.A2  A1 /;

8.A2  E1 /;

4.E  A1 /;

4.E  E1 /:

(13.12)

240 Chapter 13 Exchange and nonexchange transitions between equivalent configurations Here, the components of the representation E of the noninvariant group c3v describing the torsional motion of the COH structure are given by the relations describing the reduction of the group c3v on its symmetry subgroup cS of the Hamiltonian: E1 ! A0 ;

E2 ! A00 :

(13.13)

Equation (13.12) has only the component E1 . As concerns the component E2 ; it participates in the maintaining of the description in cases where the position of the COH structure is different from the reference one (see Section 17.2). The symmetry of the rotational motion is specified by the group D3 , which includes all rotations of the molecule as a whole from the group F36  Ci . We emphasize that all elements of the D3 group describe the rotations through a zero barrier (such operations are characteristic in that they do not bind the independent local minima of the effective nuclear potential and therefore do not take the molecule from one minimum to another). It is clearly seen in Chapters 9, 10, and 11 that the schemes for taking into account nonrigid transitions of the exchange and nonexchange types are different. In order to superpose these schemes in a unified description it is convenient to introduce an intermediate stage in the construction of multiplets: .coord/.C3 ^CS /Ci D .el /CS  .vib/CS  .rot.-conf/.C3 ^CS /Ci ; 6

(13.14)

6

where C3 ^ CS D C3v is a subgroup of transformations of the intermediate configuration as a whole in the F36 group. It is important that the group C3v satisfies two conditions. First, this group is wider than the point group CS only due to the elements of the nonexchange type. Second, the group D3 is a subgroup of the C3v  Ci group. The rotation-configuration representations rot.-conf, which are admissible for a given rotational state of the D3 group, result from a correlation of the groups D3 and C3v Ci . In this case, the behavior relative to operations C31 and C32 characterizes only the multiplet since these operations are not included in the point group (the necessary correlations are given in Table 13.1). The introduction of rot.-conf made it possible to allow for the presence of dependent configurations bound by the operation of rotation of the molecule as a whole through a zero barrier (see Section 11.2). Therefore, the classification obtained after stage (13.14) by matching the groups C3v and F36 has no extraneous levels. In fact, with the intermediate stage, the differences in describing nonrigid transitions of the exchange and nonexchange types are easy to superpose. Figure 13.3 shows a picture of the torsion-rotational levels for el  vib D A0 . Since the rotational transformations correspond to the torsional motion, the inversion doublets are not split. Therefore, the actual energy-level structure is specified by the complete rotational subgroup H18 of the F36  Ci group. Division of the elements into classes and the table of characters of the H18 group are presented in Table 13.2. This is also a noninvariant group with the Hamiltonian symmetry subgroup H6 in the form (9.9). When the group H18 reduces to subgroup H6 ; the two-dimensional representation E1

241

Section 13.2 Methanol molecule CH3 OH

is decomposed into one-dimensional ones: E1;1 ! A1 ; E1;2 ! A2 :

(13.15)

Therefore, the components of the representation E1 correspond to the individual nondegenerate levels. Note that with stage (13.14) it is easy to obtain changes in the energy-level picture shown in Figure 13.3 for the type A00 vibronic state. Passing to the Hamiltonian symmetry subgroups of the groups used in the classification, namely, D3 ! C 2 ;

C3v ! CS ;

F36 ! G12 ;

H18 ! H6 ;

it is easily seen that this classification is fully consistent with that obtained in Section 9.2 for the case of torsional motion of the methyl top alone. Table 13.1. Correlation table for the CH3 OH molecule. C3v  Ci

D3

.C/

A1

./

A1 ; A2 ./

.C/

A1 ; A2 E .˙/

.A1  A1 /

; .A2  A2 /

c3v  c3v

A1

A1  A1 ; A2  A2 ; E  E

A2

A1  A2 ; A2  A1 ; E  E A1  E; A2  E; E  A1

A2 E

E

c3v  c3v  Ci .C/

C3v

H18

H18

H6

A1

A1

A1

A2

A2

E1

A1 C A2

.C/

.A1  A2 / ; .A2  A1 /./ .A1  A1 /./ ; .A2  A2 /./ ./

E  A2 ; E  E

A2

.A1  A2 /.C/ ; .A2  A1 /.C/ .A1  E/.˙/ ; .A2  E/.˙/ .E  A1 /.˙/ ; .E  A2/.˙/

E1 E2

E2

E

E3

E

.E  E/.˙/

E3 C E4

E4

E

Next, it is important that the rotational group H18 can be represented as H18 D H3 ^ D3 ;

(13.16)

where the group H3 is similar to that introduced in Section 9.2 and includes the operations of torsional motion of a methyl top, and the group D3 includes the operations of rotation of the molecule as a whole. That is, unlike the ethane molecule with identical tops of symmetry c3v , the total rotational group can no longer be written as a direct product of the groups responsible for the symmetry of the torsional motion and

242 Chapter 13 Exchange and nonexchange transitions between equivalent configurations Table 13.2. Division of the elements into classes and the table of characters for the H18 group. E

E E

L1

2c3  E

L2

E  2c3

L3

L4 M

H18

E

L1

L2

L3

L4

M

A1

1

1

1

1

1

1

A2

1

1

1

1

1

–1

c32  c32

E1

2

2

–1

–1

–1

0

c31  c32

E2

2

–1

2

–1

–1

0

c32  c31

E3

2

–1

–1

2

–1

0

E4

2

–1

–1

–1

2

0

c31  c31

.3v  3v /i

Figure 13.3. Energy-level classification for the methanol molecule CH3 OH in the type A0 vibronic state.

243

Section 13.2 Methanol molecule CH3 OH

rotation of the molecule as a whole. The complete basis set of torsional unit vectors j0i and j ˙ 1i is determined by three irreducible representations of the group H3 as in equation (9.11). The elements of the factor group D3 , which do not commute with the elements of the invariant subgroup H3 ; act not only in the rotational, but also in the torsional space. It is important to emphasize that although the elements C31 and C32 of the D3 group, which commute with the elements of the H3 group, appear as the products c31  c31 and c32  c32 ; they correspond to rotations of the molecule as a whole through a zero barrier and should be defined only in the rotational space. This is the condition of separation of the rotational and torsional motions (see Section 10.5). As a result, it suffices to indicate the action on the torsional unit vectors for the element .x/ U2 of the D3 group: .x/

U2 j0i D aj0i;

.x/

U2 j1i D bj  1i;

(13.17)

where a2 D b 2 D 1. It is seen in the classification that for the type A0 vibronic state one should choose a D 1; which will be clear after the construction of the basis torsion-rotational functions. The value b can always be set equal to 1 since otherwise, it suffices to change the sign of one of the unit vectors in the pair j ˙ 1i. It follows from transformations (9.11) and (13.17) that the unit vector j0i transforms according to the representation A1 of the H18 group and the pair of unit vectors j ˙ 1i, according to the representation E3 , and E3;1 D jai D

j1i C j  1i p ; 2

E3;2 D jbi D

j1i  j  1i p : 2i

(13.18)

Only the transformations of rotation of the molecule as a whole from the subgroup D3 act on the rotational functions in the group H18. Therefore, for the rotational functions we have .A1 /D3 ! .A1 /H18 ;

.A2 /D3 ! .A2 /H18 ;

.E/D3 ! .E1 /H18 :

(13.19)

As a result, we have the basis torsion-rotational unit vectors decomposed into irreducible representations of the group H18 , as shown in Table 13.3. This decomposition involves three independent torsion-rotational functions for each rotational state, as it should be. The value a D 1 was chosen in equation (13.17) to ensure that the function j0ij0; 0i is transformed according to the representation A1 ; which is consistent with the classification in Figure 13.3. Note that the basis unit vectors in Table 13.3 for two components of the representation E1 of the group H18 correspond to two individual nondegenerate levels. In the effective operators of physical quantities, the torsional part will be constructed on the basis of the operator of coordinate spin e as in Section 9.2. From the symmetry properties of the torsional unit vectors j0i and j ˙ 1i for transformations of the group H18 and the operation T we find that the operator e3 applies to the representation A2

244 Chapter 13 Exchange and nonexchange transitions between equivalent configurations Table 13.3. Decomposition of the torsion-rotational basis unit vectors for the type A0 vibronic state into irreducible representations of the H18 group. H18

Basis unit vectors

A1

j0iA1

A2

j0iA2 E1;1 D j0iE1

E1

E1;2 D j0iE2

p E2;1 D .jaiE1  jbiE2 /= 2 p E2;2 D .jbiE1 C jaiE2 /= 2

E2

E3

E3;1 D jaiA1

E3;1 D jbiA2

E3;2 D jbiA1

E4

E4;1 E4;2

E3;2 D jaiA2 p D .jaiE1 C jbiE2 /= 2 p D .jbiE1  jaiE2 /= 2

Note. The rotational parts of the unit vectors are specified by irreducible representations of the D3 group.

of the group H18 and is t-odd, while the operators e1 and e2 apply to the representation E3 and are t-even. All independent torsional operators, with their symmetry properties indicated, are presented in Table 13.4. As for the rotational operators, they belong to the representations A1 ; A2 ; and E1 of the group H18 according to equation (13.19).

Table 13.4. Complete set of independent torsional operators. H18

t-even

t-odd

A1

I; e32



A2



e3

E3;1 D eC C e E3

E3;2 D i.eC  e /

E3;1 D iŒe3 ; eC  e C

2 2 C e E3;1 D eC

E3;2 D Œe3 ; eC C e C

E3;2 D

2 i.eC



2 e /

245

Section 13.3 Methylamine molecule CH3 NH2

Effective operators of the physical quantities that are invariant under the inversion operation i belong to the representation A1 of the group H6 and, as a consequence, to the transformation types A1 ; E1;1 (13.20) of the group H18 . Correspondingly, for the quantities that change their sign under the operation i, we have a representation A2 of the group H6 and the transformation types A2 ;

E1;2

(13.21)

of the group H18 . The effective Hamiltonian transforms according to equation (13.20). : Since the relationship E3 E1 D E2 CE4 is fulfilled in the group H18, spin operators of type E3 are not included in the effective Hamiltonian, and its full expression can be written in the following form: H D .I; e32 /  .A1 ; E1;1 / C .e3 /  .A2 ; E1;2 /;

(13.22)

where combinations of components of the angular momentum are given by the types of transformation of the group D3 (see Table 8.4 for the case ! D .1/J /. That the Hamiltonian includes nonsymmetric contributions is due to the fact that H18 is the noninvariant group. It is exactly these contributions that are responsible for the splitting of the type E1 levels into doublets. It can easily be verified that Hamiltonian (13.22) is equivalent to Hamiltonian (9.16) which takes into account the torsional motion of the methyl top alone. Consequently, the additional consideration of the internal rotation of the COH structure does not lead to qualitative effects, but only gives corrections to the spectroscopic constants.

13.3 Methylamine molecule CH3 NH2 It suffices to write the permutation group of identical nuclei in the methylamine molecule as 3  2 . The nonrigid motions in this molecule are taken into account by a noninvariant group F24 in the form (13.3) with the Hamiltonian symmetry subgroup G12 in the form (13.2). Matching the groups 3  2 and G12 (the necessary correlations are given in Table 13.5), we find that the complete coordinate wave function should be related to the following irreducible representations of the G12 group: 4.A2  A0 /;

12.A2  A00 /;

2.E  A0 /;

6.E  A00 /;

(13.23)

where the nuclear statistical weights are presented for the H nuclei. Matching the groups G12 and F24 , we find allowed representations for the multiplets of a nonrigid molecule in the group F24 . They are easily obtained from equation (13.23) if each representation is doubled by multiplying it by p and q; the symbols of symmetric and antisymmetric representations of the F2 group.

246 Chapter 13 Exchange and nonexchange transitions between equivalent configurations Table 13.5. Correlation tables for the CH3 NH2 molecule. 3

C3v

2

CS

C20  CS  Ci

Œ3 [21] [13 ]

A1 E A2

Œ2 [12 ]

A0 A00

A  A .C/ , A  A

D2

0

00 ./

AA

0 ./

00 .C/

B A

0 .C/

B A

0 ./

,AA

,B A

,B A

A B1

00 ./

00 .C/

B3 B2

CS C20

G12  F2

D2

C20  H6

A0  A A00  A A0  B A00  B

.A2  A00 /q, .E  A0 /p, .E  A00 /q .A2  A0 /p, .E  A0 /p, .E  A00 /q .A2  A00 /p, .E  A00 /p, .E  A0 /q .A2  A0 /q, .E  A00 /p, .E  A0/q

A B1 B2 B3

A  A1 , A  E A  A2 , A  E B  A2 , B  E B  A1 , B  E

G12  F2  Ci

H6  C20

.A1  A0 /.C/ p, .A1  A00 /./ q .A2  A0 /./ p, .A2  A00 /.C/ q

A1  A

.A1  A0 /.C/ q, .A1  A00 /./ p .A2  A0 /./ q, .A2  A00 /.C/ p

A1  B

.A1  A0 /./ p, .A1  A00 /.C/ q .A2  A0 /.C/ p, .A2  A00 /./ q

A2  A

.A1  A0 /./ q, .A1  A00 /.C/ p .A2  A0 /.C/ q, .A2  A00 /./ p

A2  B

.E  A0 /.˙/ p, .E  A00 /.˙/ q

E A

.E  A0 /.˙/ q, .E  A00 /.˙/ p

E B

The rotational motion of a nonrigid molecule in the group F24  Ci is specified by the subgroup of rotational transformations of the molecule as a whole D2 D ŒE; C2.z/ D F  .E   .yz/ /; C2.y/ D C2.z/ C2.x/; C2.x/ D  .yz/ i; (13.24) .z/

where the element C2 is formed as a product of elements of the groups F2 and G12 . We emphasize that all elements of the group D2 correspond to the rotations through a zero barrier (these operations are characteristic in that they do not bind the independent local minima of the effective nuclear potential and, therefore, do not take the molecule

247

Section 13.3 Methylamine molecule CH3 NH2

from one minimum to another). In order to superpose in a unified description the differences of taking into account nonrigid transitions of the exchange and nonexchange types, we introduce an intermediate stage in the construction of the multiplets (see the preceding section): .coord/CS C 0 Ci D .el /CS  .vib/CS  .rot.-conf/CS C 0 Ci ; 2

2

6

(13.25) .z/

where CS D .E;  .yz/ / is the point group of the molecule, and C20 D .E; C2 /. The rotation-configuration representations rot.-conf, which are admissible for a given rotational state of the group D2 , result from a correlation between the groups D2 and .z/ CS C20 Ci . In this case, the behavior relative to the operation C2 characterizes only the multiplet since this operation is not included in the point group. Finally, the groups CS  C20 and F24 are matched at the closing stage. Figure 13.4 gives an energy-level picture for el  vib D A0 . Here, it should be taken into account that all considered types of motion are specified in the group F24 Ci by the elements of the noninvariant subgroup H24 D C20  H6  F2 ; (13.26) where the group H6 has the form (9.9). Therefore, it is exactly the group H24 that describes the actual energy-level structure, and its Hamiltonian symmetry subgroup .z/ is the group B2  H6 , where B2 D .E; C2 F /. Note that each irreducible representation of the group H24 corresponds to only one energy level. The point is that the degenerate irreducible representations of the group H24 do not decompose into representations of a smaller dimension when this group reduces to its subgroup B2  H6 . It is clearly seen in Figure 13.4 that allowance for the nonrigid motion, called the inversion motion in a methylamine molecule [72], does not lead to the inversion doublet splitting. Let us construct a description for the vibronic state el  vib D A0 . We introduce a complete set of torsional unit vectors j0i and j ˙ 1i as in Section 9.2. That is, the unit vector j0i belongs to the representation A1 of the group H6 , and the pair of unit vectors j ˙ 1i, to the representation E: E1 D jsi D

j1i C j  1i ; p 2

E2 D jai D

j1i  j  1i : p 2i

(13.27)

.x/

The components E1 and E2 are specified by the conditions C2 E1 D E1 and .x/

C2 E2 D E2 . It follows from the structure of the group H24 that the operations of the group C20  F2 do not act on these unit vectors, and they are of the Ap type. The complete basis set of inversion unit vectors jpi and jqi is determined by two irreducible representations of the group F2 . Accordingly, in the group H24 , jpi ! .A  A1 /p;

jqi ! .A  A1 /q:

(13.28)

248 Chapter 13 Exchange and nonexchange transitions between equivalent configurations

Figure 13.4. Energy-level classification for the CH3 NH2 molecule in the type A0 vibronic state.

Only the transformations of rotation of the molecule as a whole from the subgroup D2 act on the rotational functions in the group H24. Passing from the group D2 to the group H24 we have B3 ! .B  A1 /p: (13.29) As a result, we have the inversion-torsion-rotational basis unit vectors decomposed into irreducible representations of the group H24 , as is shown in Table 13.6 for one A ! .A  A1 /p;

B1 ! .A  A2 /p;

B2 ! .B  A2 /p;

249

Section 13.3 Methylamine molecule CH3 NH2

Table 13.6. Decomposition of the inversion-torsion-rotational basis unit vectors into irreducible representations of the H24 group. H24

Unit vectors

.A  A1 /p .A  A2 /p

jpij0iA jpij0iB1

.B  A1 /p .B  A2 /p

jpij0iB3 jpij0iB2

H24

.A  E/p

.A  E/p

Unit vectors E1 D jpijsiA E2 D jpijaiA E1 D jpijaiB1 E2 D jpijsiB1 E1 D jpijaiB2 E2 D jpijsiB2 E1 D jpijsiB3 E2 D jpijaiB3

Note. The rotational parts of the unit vectors are denoted by irreducible representations of the D2 group.

half of the unit vectors. The second half is obtained by replacing p with q in terms of representations, and replacing jpi with jqi in terms of unit vectors. It is easily seen that the symmetry of the basis unit vectors is consistent with the classification in Figure 13.4. In order to describe the torsional part of the effective operators of physical quantities, we introduce the operator of coordinate spin e as in Section 9.2. The complete set of independent self-adjoint torsional operators with indicated symmetry properties in the group H6 is presented in Table 9.4. In the group C20  F2 , these operators are of the Ap type. In order to describe the inversion part, we introduce the operator of coordinate spin p. Two-dimensional representation of this operator in the space of unit vectors jpi and jqi is obtained from equation (7.29) by replacement of ek by pk . The complete set of self-adjoint inversion operators with indicated symmetry properties in the group F2 has the form I; p3 ! p;

p1 ; p2 ! q;

(13.30)

where I is the identical operator. In the group C20  H6 , operators (13.30) are of type A  A1 . Besides, the operators I , p3 , and p1 are t- even, and the operator p2 is t-odd. As for the rotational operators, they are transformed according to equation (13.29). Effective operators of any coordinate physical quantities belong to the representation A1  A0 of the group G12 . Passing to the group F24 , we obtain the representations .A1  A0 /p and .A1  A0 /q for these operators. Finally, matching the groups F24  Ci and H24 , we have, within the framework of the latter, the representations .A  A1 /p;

.B  A1 /q

(13.31)

250 Chapter 13 Exchange and nonexchange transitions between equivalent configurations for the quantities that are invariant under the inversion operation i and the representations (13.32) .A  A2 /p; .B  A2 /q for the quantities that change their sign under the action of i. The effective inversion-torsion-rotational Hamiltonian transforms according to equation (13.31). Therefore, its full expression can be written in the form H D .I; e32 /  .I; p3 /  A C .e3 /  .I; p3 /  B1 C .e3 /  .p1 ; p2 /  B2 C .I; e32 /  .p1 ; p2 /  B3 :

(13.33)

Each term of the Hamiltonian is a combination of products of the components of the angular momentum in the MCS belonging to one of the representations of the group D2 , and the parameters before this combination are linearly dependent on the products of inversion and torsion operators that correlate with this combination through symmetry. Hamiltonian (13.33) explicitly includes all interactions of the inversion, torsion, and rotational motions and implicitly takes into account (through contributions to the phenomenological constants) the remaining types of internal motion. The correctness of this Hamiltonian is limited only to the correct choice of symmetry of the internal dynamics. The operator form for all considered types of motion is another advantage of this Hamiltonian. In unwinding equation (13.33) it should be taken into account that the Hamiltonian is t-even. The energy matrix decomposes into a direct sum of matrices corresponding to six types of irreducible representations of the group B2  H6 or six families from two irreducible representations of the group H24 : Œ.A  /p; .B  /q;

Œ.B  /p; .A  /q;

(13.34)

where  D A1 ; A2 ; E is an irreducible representation of the group H6 . The effective operator of the electric dipole moment is transformed according to equation (13.32), which makes it possible to write the electric dipole rules through families (13.34): Œ.A  A1 /p; .B  A1 /q $ Œ.A  A2 /p; .B  A2 /q; Œ.A  E/p; .B  E/q $ Œ.A  E/p; .B  E/q:

(13.35)

It is also easy to construct the electric dipole operator itself.

13.4 Cyclopentane molecule C5 H10 The permutation group of identical nuclei of the cyclopentane molecule C5 H10 is 5  10 . For the primary isotope 12 C of the carbon nucleus, the spin is equal to zero, and the group 10 is sufficient in the analysis. Moreover, transformations of the point group U2 have counterparts in the group 10 only for its subgroup 2  2  2  2  2 .

251

Section 13.4 Cyclopentane molecule C5 H10

Matching the group U2 with this subgroup, for the allowed coordinate multiplets of a rigid molecule we obtain 496e; 528o; (13.36) where e and o are, respectively, the symmetric and antisymmetric irreducible representations of the group U2 . Symmetry types of the multiplets are formed from symmetry types of the zero-approximation wave functions on the basis of matching: .mult/U2 Ci D .el /U2  .vib /U2  .rot.-in/U2 Ci ;

(13.37)

6

where the admissible representation rot.-in for a given representation of the group D2 of a rigid asymmetric top follow from a correlation of the groups D2 and U2  Ci through their common rotational subgroup U2 . As a result, we have a classification of the energy levels in an arbitrary vibronic state, which is shown in Figure 13.5 for el vib D e. The actual energy-level structure is described by the rotational subgroup of the group U2  Ci . This subgroup coincides with the point group, which is a typical feature of the molecules with stereoisomers.

Figure 13.5. Energy-level classification for a rigid molecule of cyclopentane C5 H10 in the type e vibronic state.

The effective operator of any coordinate physical quantity characterizing the rigid molecule belongs to the representation e of the point group U2 . Since it is exactly the point group that determines the actual energy-level structure in the case of a rigid molecule with stereoisomers, the behavior of the physical quantities with respect to the inversion operation i is not essential in constructing effective operators of these quantities. By virtue of the fact that the irreducible representations of the group U2 are onedimensional, the effective operators are purely rotational in any vibronic state. Since the effective electric dipole moment belongs to the representation e, for the electric dipole selection rules we have e $ e;

o $ o:

(13.38)

252 Chapter 13 Exchange and nonexchange transitions between equivalent configurations The component ˛ of the electric dipole moment operator in the FCS has the form (8.8). The terms not related to the rotational distortion of the molecule give the main contribution to this operator. There is one such term, namely, ˛ D ˛x d .x/;

(13.39)

as it should be. Indeed, it follows from simple geometric considerations that only one component of the electric dipole moment in the cyclopentane molecule is x axisdirected. The extended point group F40 has the form (13.8). Since the operation B D P  .xy/ is of the exchange type, the expression for the extended point group is better written as F40 D D5  P2  B2 :

(13.40)

Symmetric and antisymmetric irreducible representations of the group B2 D .E; B/ will be denoted c and d. The symmetry subgroup of the Hamiltonian in the noninvariant group F40 has the form (13.41) FH D D5  B2 : Matching the groups 10 and FH , for the complete coordinate wave function we have the following irreducible representations of the FH group: 88 .A1 c C A1 d /;

120 .A2 c C A2 d /;

204 .E1 c C E1 d /;

204 .E2 c C E2 d /: (13.42) Here, we indicated the total nuclear statistical weight for the representations given in the parentheses. As will be seen later, these weights are quite sufficient to characterize the energy levels. To determine all allowed coordinate multiplets of the molecule in the F40 group, each representation in equation (13.42) should be doubled by multiplying it by p1 and p2 . Now it is necessary to construct the obtained multiplets from the wave functions of separate types of motion. Since the operations of the C5 group describe the transitions through a pseudorotation barrier, the rotations of the molecule as a whole through a zero barrier, which take place in the group F40 Ci ; form its subgroup D2 D U2 C2 ; .z/ where C2 D .E; C2 D  .xy/ i/. In order to superpose in a unified description the differences between taking into account nonrigid transitions of the exchange and nonexchange types, we introduce an intermediate stage in the construction of the multiplets (see Section 13.2): .coord /U2 CS Ci D .el /U2  .vib/U2  .rot.-conf/U2 CS Ci ;

(13.43)

6

where the rotation-configuration representations rot.-conf, which are admissible for a given rotational state of the D2 group, result from a correlation of the groups D2 and U2  CS  Ci through their common subgroup D2 . The behavior relative to the operation  .xy/ characterizes only the multiplet since this operation is not included in

Section 13.4 Cyclopentane molecule C5 H10

253

the point group. Finally, the groups U2  CS and F40 are matched at the closing stage. Figure 13.6 shows an energy-level classification for the type e vibronic state. Here, it should be taken into account that all considered types of motion are specified in the group F40  Ci by the elements of the noninvariant subgroup H40 D D5  P2  C2 :

(13.44)

Figure 13.6. Energy-level classification for a nonrigid molecule of cyclopentane C5 H10 in the type e vibronic state.

Therefore, it is exactly the group H40 that describes the actual energy-level structure. The Hamiltonian symmetry subgroup of this group is the group D5  B20 , where B20 D .E; Bi/. Irreducible representations of the group H40 correspond to one energy level since the degenerate irreducible representations of this group do not decompose into representations of a smaller dimension when the group H40 reduces to its subgroup D5  B20 . In this case, each energy level in the group H40 corresponds to the unsplit doublet in the group F40  Ci , whose levels behave identically in the groups

254 Chapter 13 Exchange and nonexchange transitions between equivalent configurations D5 and P2 , but differently in the groups B2 and Ci . Consequently, the nuclear statistical weights given in equation (13.42) are sufficient for the spectrum analysis. We also note that when passing from the group D2 to the group D5  P2  C2 the dimension of the function space increases tenfold, as should be the case in the presence of ten independent equilibrium configurations. A classification for the rotational states B1 and B2 of the group D2 results from a classification for the rotational states A and B3 , accordingly, by replacement of the representation A1 by the representation A2 in the group D5 . The effective operator of an arbitrary coordinate physical quantity describing a nonrigid molecule belongs to the representation A1 c of the FH group. Accordingly, in the group D5 P2 C2 , for the physical quantities that are invariant under transformation of the spatial inversion i we have the representations A1 p1 A;

A1 p2 B;

(13.45)

and for the physical quantities that change sign under the transformation i, the representations A1 p1 B; A1 p2 A: (13.46) The effective Hamiltonian transforms according to equation (13.45), and the effective operator of the electric dipole moment according to equation (13.46). Unlike the case of a rigid molecule, transformations (13.45) and (13.46) are different, since pseudorotation mixes the stereoisomers. Due to the noninvariant contributions in the Hamiltonian with respect to the groups P2 and C2 , the symbols of their irreducible representations are not good quantum numbers. The electric dipole selection rules are a simple consequence of equation (13.46). The basis unit vectors of the function space in a given vibronic state are constructed from unit vectors of the pseudorotation and rotation subspaces. The effective operators of physical quantities are defined in this space. Thus, for the component ˛ we have equation (8.62), where the parameters dO .i / are now operators in the pseudorotation space. The components of the unit vector ˛ in an MCS transform according to irreducible representations of the group D5  C2 as follows:

˛z ! A2 A;

. ˛x ; ˛y / ! E1 B:

(13.47)

The terms not related to the rotational distortion of the molecule give the main contribution to equation (8.62): ˛ D ˛i dO .i / : (13.48) If one chooses dO .i / D d .i / I , where I is a totally symmetric unit operator in the pseudorotation space, then equation (13.48) has no terms of admissible symmetry types for the electric dipole moment. That is, although the rotational transitions are present in a rigid molecule of cyclopentane, they are nevertheless absent in the nonrigid molecule. Of course, we refer here only to the strongest transitions stipulated by the constant component of the electric dipole moment.

Section 13.5 Conclusions

255

13.5 Conclusions We have formulated some general conclusions based on the content of this chapter. 1. When molecules with the exchange and nonexchange type transitions between energetically equivalent equilibrium configurations are described, the chain of groups has additionally a noninvariant extended point group F, which characterizes both the equilibrium configurations and the transitions between them and, therefore, includes the point group F0 of these configurations as its subgroup. Since part of the elements extending the group F0 to the group F are of the exchange type, the Hamiltonian symmetry subgroup FH in the group F is always wider than F0 . Naturally, internal dynamics cannot be described by the group FH alone since this group has no elements defining the transitions of the nonexchange type. In the general case, the actual energy-level structure is described by the noninvariant subgroup FL of the group F  Ci . The group FL includes only the elements that specify the considered types of motion. 2. The degeneracy and nuclear statistical weights of the energy levels are described by the subgroup FH . As a consequence, the components of a degenerate representation of the group FL , which reduce to the different irreducible representations of its Hamiltonian symmetry subgroup FLH , correspond to the individual levels. Such splittings are due to contributions in the effective Hamiltonian that are noninvariant under transformations of the group FL . That is, not all of the symmetry symbols of this group are good quantum numbers. Therefore, the energy matrix decomposes only into a direct sum of the matrices corresponding to the different irreducible representations of the FLH group. For this reason, only the type of irreducible representation of the subgroup FLH is preserved for the electric dipole transitions in the general case. 3. The rigid molecule of cyclopentane has rotational transitions stipulated by the constant component of the electric dipole moment, but taking into account the pseudorotation leads to their forbiddance. The efficiency of the analysis of pseudorotation in molecules with a five-membered cycle is demonstrated in Chapter 16, where a unified algebraic model is constructed to describe the entire excitation band of pseudorotation in the tetrahydrofuran molecule C4 H8 O. For this molecule, unlike the cyclopentane molecule, there are numerous high-precision experimental data on this excitation band, which is most important from the point of view of pseudorotation research. The model explained for the first time a number of features observed in the spectrum of the tetrahydrofuran molecule.

Chapter 14

On the construction of extended point groups

In this chapter, the internal dynamics of the molecular complexes (HF)2 , ArHC 3 , and C C ArD3 , and the simplest compound C2 H3 of the family of carbocations, which are intermediate molecular systems in many chemical reactions, is described by group chain methods. These objects are convenient for considering a number of features to construct extended point groups which were absent or not explained in the examples of the preceding chapters. Thus, for the dimer (HF)2 it is necessary to take into account the presence of two nonequivalent intermediate configurations. The extended point group C of the ionic complexes ArHC 3 and ArD3 has the form h1 h2 , where the groups h1 and h2 characterize the structures consisting of one nucleus Ar and three nuclei of hydrogen, respectively. A single nucleus gives only a trivial totally symmetric contribution to the multiplet of a nonrigid complex. The conditions where this contribution can be omitted and a much simpler extended point group can be applied have been formulated. As for the carbocation C2 HC 3 , it features a very complex internal dynamics even with only five nuclei. Nevertheless, the corresponding extended point group is fairly easy to construct.

14.1 Hydrogen fluoride dimer (HF)2 The hydrogen fluoride dimer (HF)2 is one of the best studied molecular complexes. Its planar equilibrium configuration presented in the upper part of Figure 14.1, which corresponds to the point group CS [52], is characterized by a nonequivalent position of the HF monomers. That is, information on their identity is absent in the point group. However, due to such a low symmetry, the dimer has two energetically equivalent configurations differing in the permutation of monomers. It is well known [55] that these configurations are mixed by transitions of the exchange type. The so-called trans and cis transitions, which are specified by intermediate configurations with the symmetry groups C2h and C2v , respectively, with geometrically equivalent positions of the monomers, are the most probable transitions (Figure 14.1). Since the monomers exchange places due to nonrigid motions, the trans transition is given by the operation .y/ C2 in the extended point group C2h and the cis transition is given by the operation .x/ C2 in the extended point group C2v . That is, these operations constitute the rotation of the dimer as a whole. Therefore, according to the definition given in Chapter 10, these nonrigid motions are pseudorotations. It is important to note that during the nonrigid motion the equilibrium configuration does not change its position in space. This

Section 14.1 Hydrogen fluoride dimer (HF)2

257

Figure 14.1. Equilibrium and intermediate configurations of the trans and cis transitions (from top to bottom) for the (HF)2 dimer (the nuclei in all configu-rations lie in the xz plane, and the z axis coincides with the straight line between the centers of mass of the monomers).

can be represented as follows. First, the monomers move to their positions in the intermediate configuration under conservation of the position of the center of mass of the dimer. In the intermediate configuration, the equivalently located monomers exchange places by means of rotation of the system as a whole. Then the monomers continue to move to their new equilibrium positions. As a result, the monomers exchange places under conservation of the position of the equilibrium configuration in space. For example, the left monomer in the upper part of Figure 14.1 is, as before, at a large angle to the z axis, but now this monomer has the index 2 for the nuclei H and F. The methods of quantum chemistry show that the trans transition occurs through a lower potential barrier and is therefore more probable. Clearly, the simultaneous allowance for both transitions requires that two extended point groups be superposed in a unified description. It is quite obvious that these groups correspond to the nonequivalent intermediate configurations. The permutation group of identical nuclei is 2  2 , and the spins of the nuclei F and H are equal to 1/2. From a matching of this group with the point group CS D .E;  .xz/ / (the necessary correlations are given in Table 14.1) we obtain one allowed coordinate multiplet 16A0 since the representation A00 is forbidden because of the planarity of the equilibrium configuration. Symmetry types of the wave functions of separate motions are matched with the symmetry type of the allowed multiplet

258

Chapter 14 On the construction of extended point groups

Table 14.1. Correlation tables for the (HF)2 dimer. 2  2

CS

D2

U2

Œ2  [2] [2]  [12 ] [12 ]  [2] [12 ]  [12 ]

A0 A0 A0 A0

A, B2

e

B1 , B3

o

CS  Ci 0

00 ./

0 ./

00 .C/

A .C/ , A A

,A

2  2

D2

Œ2  [2] [2]  [12 ] [12 ]  [2] [12 ]  [12 ]

A B1 B1 A

formed on their basis: .mult /CS Ci D .el /CS  .vib /CS  .rot.-in/CS Ci :

(14.1)

6

The representations rot.-in, which are admissible for a given rotational state in the group D2 of a rigid asymmetric top, are determined from a correlation of the groups D2 and CS  Ci through their subgroup U2 D .E; U2.y/ D  .xz/i/. A classification of the energy levels of a rigid dimer (HF)2 for el  vib D A0 is shown in the first two columns in Figure 14.2. The actual structure of the levels is described by the group U2 . Symmetric and antisymmetric representations of this group are denoted by the indices e and o.

Figure 14.2. Energy-level classification for the (HF)2 dimer in the type A0 vibronic state.

When passing to a nonrigid dimer (HF)2 it is necessary to take into account that its two nonequivalent intermediate configurations do not exist simultaneously. In this respect, the situation is similar to the Berry pseudorotation in the PF5 molecule, but with the important difference that the intermediate configurations of the PF5 molecule are equivalent (see Section 10.1). Nevertheless, for the dimer, one can also construct

259

Section 14.1 Hydrogen fluoride dimer (HF)2

an extended point group that would take into account both nonrigid transitions. This group has the form (14.2) G8 D CS  D2 ; where D2 D .E; C2.z/ ; C2.y/ ; C2.x//. The operation C2.z/ specifies two sequential nonrigid transitions through different intermediate configurations. Therefore, this operation does not change the initial configuration of the dimer and corresponds to the identity element in the group 2  2 . As a result, we have two allowed multiplets: 10.A0  A/;

6.A0  B1 /:

(14.3)

The representation A00 of the CS group and the representations B2 and B3 of the D2 group are forbidden in equation (14.3) because of the homomorphic mapping of these groups on the group 2  2 . Matching now the geometrical groups G8 and CS , we obtain a full picture of the level splitting of a rigid dimer in the presence of the trans and cis transitions, which is shown in Figure 14.2 for the type A0 vibronic state. Since the nonrigid motions are specified by rotational operations, the actual structure of the levels is described by the rotational subgroup U2  D2 of the G8  Ci group. Complete basis sets of the trans unit vectors jAi and jBi and cis unit vectors jai and jbi are specified by irreducible representations of the groups C2 D .E; C2.y/ / .x/

and C20 D .E; C2 /, respectively. However, their operations are also rotations of the dimer as a whole, which act on the rotational unit vectors. Operations of the group U2 specify rotations of the dimer as a whole through a zero barrier and are therefore defined only in the rotational space (see Section 10.5). The actions of the elements .y/ C2 and  .xz/ i on the rotational unit vectors should coincide, since they define the same rotation of the dimer as a whole. Thus, we obtain a complete set of basis transcis-rotational unit vectors decomposed into irreducible representations of the group U2  D2 , which is shown in Table 14.2 for the type A0 vibronic state. The two-dimensional representation of Lie algebra (7.27) in the space of unit vectors jAi and jBi can be written in the form (7.29). We assume that the unit vectors are

Table 14.2. Decomposition of a complete set of basis unit vectors into irreducible representations of the group U2  D2 . U2  D2

Unit vectors

U2  D2

Unit vectors

Ae

AjAijai B2 jAijbi

B1e

AjBijbi B2 jBijai

Ao

B1 jBijbi B3 jBijai

B1o

B1 jAijai B3 jAijbi

Note. The rotational parts are specified by irreducible representations of the D2 group.

260

Chapter 14 On the construction of extended point groups

eigenfunctions of the operator e3 . Knowing the symmetry properties of the basis unit .y/ vectors for transformations C2 and T, T jAi D jAi;

T jBi D jBi;

(14.4)

it is easily seen that the operator e3 belongs to the representation A of the C2 group and is t-even, while the operators e1 and e2 belong to the representation B and are t-even and t- odd, respectively. In two-dimensional space, the complete set of selfadjoint operators also includes a totally symmetric unit operator I. Passing to the group U2  D2 , we obtain .I; e3 / ! Ae ; .e1 ; e2 / ! B3e : (14.5) In the space of basis unit vectors jai and jbi, we determine the coordinate spin  with components of the form (7.29). Symmetry properties of the components k are similar to those for ek with accuracy up to the replacement of the group C2 by the group C20 . Therefore, in the group U2  D2 we have .I ; 3 / ! Ae ;

.1 ; 2 / ! B2e :

(14.6)

.y/

The behavior relative to the operations C2 and  .xz/ i is the same for the rotational operators. Therefore, for these operators in the group U2  D2 we have Ae ; B2e ; B1o; B3o :

(14.7)

For the effective operators of physical quantities, admissible in the group U2  D2 are the representations Ae and Ao , related, respectively, to the quantities that are invariant under the spatial inversion operation i and change sign during this transformation. The effective Hamiltonian belongs to the type Ae , which leads to the following expression for it: H D .I; e3 /  .I ; 3 /  A C .I; e3 /  .1 ; 2 /  B2 :

(14.8)

Unwinding this expression with allowance for the fact that the Hamiltonian is t-even, we obtain 1  X .A/ .B2 / .B2 / HD H2n C H2nC2 : (14.9) C H2nC1 nD0

Here, the superscript of the contribution to the Hamiltonian specifies the type of irreducible representation of the D2 group, according to which the combinations of components of the angular momentum in the MCS are transformed: X .A/ 2t cO2p;2s;2t J 2p Jz2s .JC C J2t /; (14.10) H2n D p;s;t .B2 / H2nC2

D

X

2t C1 C J2t C1 /; dO2p;2sC1;2t C1J 2p Jz2sC1 .JC

(14.11)

2t C1 i gO 2p;2s;2t C1 J 2p Jz2s .JC  J2t C1 /:

(14.12)

p;s;t .B /

2 D H2nC1

X

p;s;t

C Section 14.2 Ionic complexes ArHC 3 and ArD3

261

Here, p C s C t D n for a given n. Due to allowance for the nonrigid transitions, c, O dO , and gO are spin operators, namely, cO D .c .1/ I C c .2/ e3 /I C .c .3/ I C c .4/ e3 /3 ; dO D .d .1/I C d .2/ e3 /1 ;

(14.13)

gO D .g .1/ I C g .2/e3 /2 : It is characteristic that equation (14.13) has nontrivial cross terms for two types of nonrigid motion. Note that the Hamiltonian (14.9) coincides with that obtained in [24], where a more cumbersome scheme was used to superpose the nonequivalent intermediate configurations in a unified description (in that scheme, separate descriptions with allowance for only trans and only cis transitions are additionally required) The effective operator of the electric dipole moment belongs to the type Ao in the group U2  D2 . This implies the electric dipole selection rules: e $ o ;

(14.14)

where  is the irreducible representation of the group D2 , which is retained during transition. The operator of electric dipole moment itself is also easy to construct.

14.2 Ionic complexes ArHC and ArDC 3 3 C The complexes ArHC 3 and ArD3 were the first weakly bound ionic systems whose rotational absorption spectra were observed by the methods of high-resolution spectroscopy. An analysis [11, 12] leads to the planar equilibrium configuration with the .z/ point group C2v D .E; C2 ;  .xz/ ; N .yz/ /, which is shown on the left in Figure 14.3. The observed splitting of the rotational lines is due to the internal rotation of a structure of three identical hydrogen nuclei about the axis passing through the center of mass of this structure orthogonally to the xz plane. The top of the internal rotation barrier corresponds to the planar configuration shown on the right in Figure 14.3. In this case, the extended point group has the form

c2v  D3h;

(14.15)

where the groups c2v and D3h characterize the structures consisting of the Ar nucleus and three hydrogen nuclei, respectively. Here, the group c2v is needed only to ensure that group (14.15) includes the point group C2v as a subgroup. Since the structure of the hydrogen nuclei is deformed in the equilibrium configuration, the group D3h refers to some intermediate unstable configuration with a geometrically equivalent position of the hydrogen nuclei at the vertices of an equilateral triangle. The intermediate configuration takes into account that the effective potential minima connected by torsional transitions (and operations of the cyclic subgroup C3 in the group D3h/ are

262

Chapter 14 On the construction of extended point groups

Figure 14.3. Equilibrium configuration (left) and the configuration of the top of the planar internal rotation barrier (right) for the ArHC 3 complex.

equivalent. Therefore, this configuration has a higher geometric symmetry than the equilibrium configuration which allows for only one minimum. From a matching of the point group with the permutation group 1  3 of identical nuclei, for the allowed multiplets of a rigid complex (the necessary correlations are given in Table 14.3) we obtain 2A1 ; 6B1 : (14.16) Here nuclear statistical weights are given for the H nuclei (for the D nuclei, we have 18A1 and 9B1 ). Symmetry types of the wave functions of separate motions are matched with the symmetry types of the allowed multiplets formed on their basis: .mult/C2v Ci D .el /C2v  .vib /C2v  .rot.-in/C2v Ci ;

(14.17)

6

where the admissible representations rot.-in for a given rotational state of the D2 group of a rigid asymmetric top are determined from a correlation of the groups D2 and C2v  Ci through their common subgroup D2 . A classification of the energy levels of a rigid complex for el  vib D A1 is shown in the first two columns in Figure 14.4. The actual energy-level structure is described by the group D2 . A matching of the groups c2v and 1 , leads to one possible contribution A1 of the Ar nucleus to the multiplet of a nonrigid complex. This contribution can be omitted, and instead of the group given by equation (14.15), the group D3h can be used. But it should be remembered that the transformations of the complex as a whole in the latter group are only transformations of the group C2v . Matching the groups 3 and D3h D D3  CS , for the allowed multiplets of a nonrigid complex ArHC 3 we obtain 4.A2  A0 /;

2.E  A0 /:

(14.18)

For the nonrigid complex ArDC 3 , the number of multiplets increases: 10.A1  A0 /;

1.A2  A0 /;

8.E  A0 /:

(14.19)

C Section 14.2 Ionic complexes ArHC 3 and ArD3

263

Table 14.3. Correlation tables for the ionic complex ArHC 3. 1  3

C2v

Œ1  Œ3

A1

C2v  Ci .C/

Œ1  Œ21

A1 C B1

Œ1  Œ1 

B1

3

./

A1 ; A2

./ .C/ A1 ; A2 B1.C/ ; B2./ ./ .C/ B1 ; B2

CS  Ci

D2

0

00 ./

0 ./

00 .C/

A .C/ , A

A

A

B1

,A

C2 A B

B2 B3

3

D3  CS

D3  CS

C2v

D2

D3  C2

Œ3

A1  A0

A1  A0

A1

A

A1  A, E  A

[21]

E A

0

A2  A

0

B1

B1

A1  B, E  B

[13 ]

A2  A0

E  A0

A1 CB1

B2

A2  A, E  A

B3

A2  B, E  B

Now from a matching of the geometrical groups D3h and C2v we obtain a picture of splitting of the energy levels of a rigid complex due to the internal rotation, which is shown in Figure 14.4 for the ArHC 3 complex in the state el  vib D A1 . The nonrigid motion is specified by the rotational operations, and the actual energylevel structure is described by the rotational subgroup of the D3h  Ci group. This subgroup has the form (14.20) D6 D D3  C 2 ; where C2 D .E; C2.y/ D  .xz/ i/. Note that although the elements C2.y/ and C3k describe the rotations about the axes orthogonal to the xz plane, these axes do not coincide. The axis C2 passes through the center of mass of the complex, and the axis C3 through the center of mass of a top comprising three hydrogen nuclei. The dotted line in Figure 14.4 shows the levels with zero statistical weights (they become allowed for the ArDC 3 complex). In the splittings, the levels are given in increasing order jmj D 0; 1, where m is the quantum number of the problem for one-dimensional cyclic motion. The following correspondence between jmj and the torsional types of symmetry in the group D3  C2 was taken into account: A1  A ! 0;

E  A ! 1:

(14.21)

We also emphasize that although the geometrical group D6 is isomorphic with the MS group D3h.M /, which was introduced in [12] to analyze the nonrigid complexes C ArHC 3 and ArD3 , these groups are used differently.

264

Chapter 14 On the construction of extended point groups

Figure 14.4. Energy-level classification for the ArHC 3 complex in the type A1 vibronic state.

It should be mentioned that the trivial behavior of the multiplet with respect to one group in a direct product of the form (14.15) is generally insufficient to omit this group. The point is that the actual structure of the levels is described by the rotational subgroup of the group .c2v  D3h/  Ci D .c2  D3 /  CS  Ci :

(14.22)

Part of the elements of this subgroup may include the products of improper operations of the group c2v and the operation i. The behavior relative to such products is no longer trivial. That is, the improper operations of the group c2v should not be omitted. However, in this case, the rotational subgroup can be written as .c2  D3 /  C2 :

(14.23)

Omitting the group c2 in equation (14.23), the behavior relative to which is trivial, we obtain a rotational subgroup in the form (14.20). This is due to the fact that the inversion operation i in equation (14.23) is included only in the elements of rotation of the molecule as a whole. It can be shown that the situation is similar for the molecules

C Section 14.2 Ionic complexes ArHC 3 and ArD3

265

of complex hydrides considered in Section 9.4. If this condition is not fulfilled, then a complete group of the form (14.15) has to be used. Such examples are considered in the next chapter. The effective operator of any physical quantity transforms according to the representation A01 of the group D3h . In the group D3  C2 , we have the representations A1  A and A1  B, related, respectively, to the quantities that are invariant under the inversion i and change sign during this transformation. That is, the Hamiltonian belongs to the type A1  A and the effective operator of electric dipole moment, to the type A1  B. Therefore, the electric dipole selection rules in the group D3  C2 have the form   A $   B; where  is the irreducible representation of the group D3 , which is retained during transition. For the further analysis, it is important that the group D3 can be represented as D3 D C3 ^ U2 ;

(14.24)

.z/

where U2 D .E; U2 /. The complete basis set of torsional unit vectors is specified by three irreducible representations of the cyclic group C3 : C31 j0i D j0i;

C31 j ˙ 1i D "˙1 j ˙ 1i;

(14.25)

with " D exp.2 i=3/. It follows from equations (14.24) and (14.25) that .z/

.z/

U2 j0i D aj0i;

U2 j1i D bj  1i;

(14.26)

where a2 D b 2 D 1. For the type A1 vibronic state, one should choose a D 1 (b can always be set equal to 1/. Then the unit vector j0i belongs to the representation A1 of the D3 group and the unit vectors j ˙ 1i, to the representation E: E1 D js1 i D

j1i C j  1i ; p 2

E2 D ja1 i D

j1i  j  1i p 2i

(14.27)

where the components E1 and E2 are determined by the relations .z/

U2 E1 D E1 ;

.z/

U2 E2 D E2 :

As for transformations of the group C2 , they do not act in the torsional space. The rotational unit vectors are specified by irreducible representations of the group D2 . Passing to the group D3  C2 we obtain A ! A1  A;

B1 ! A1  B;

B2 ! A2  A;

B3 ! A2  B:

(14.28)

The found decomposition of a complete set of torsion-rotational unit vectors into irreducible representations of the group D3  C2 is given in Table 14.4. With the value

266

Chapter 14 On the construction of extended point groups

Table 14.4. Decomposition of the torsion-rotational unit vectors into irreducible representations of the D3  C2 group. D3  C2

Unit vectors

D3  C2

Unit vectors

A1  A

j0iA

A2  A

j0iB2

E A

js1 iA ja1 iA

E A

ja1 iB2 js1 iB2

A1  B

j0iB1

A2  B

j0iB3

E B

js1 iB1 ja1 iB1

E B

ja1 iB3 js1 iB3

Note. The rotational parts of the unit vectors are denoted by irreducible representations of the D2 group. Table 14.5. Complete set of independent torsional operators corresponding to one-dimensional representations of the D3 group.

D3

t-even

t-odd

A1 A2

I; e32

– e3

–

a D 1 chosen in equation (14.26), the symmetry properties of the functions given in this table are consistent with the classification given in Figure 14.4. In order to describe the torsional part of the effective operators of physical quantities, we introduce the coordinate spin e. It is necessary to write a three-dimensional representation of Lie algebra (7.27) in the space of unit vectors j0i and j ˙ 1i. Assume that e3 j0i D 0; e3 j ˙ 1i D ˙j ˙ 1i: (14.29) From the properties of the unit vectors for the group D3 transformations and time reversal operation T T j0i D j0i; T j1i D j  1i (14.30) it follows that the operator e3 is of the type A2 and is t-odd, while the operators e1 and e2 are of the type E and are t-even. In three-dimensional space, the complete set of independent operators includes the products of the components ek of a total power not higher than two. In the further analysis, we will need only the torsional operators corresponding to one-dimensional representations of the group D3 . All such operators are given in Table 14.5.

C Section 14.2 Ionic complexes ArHC 3 and ArD3

267

The effective torsion-rotational Hamiltonian is formed by two spin-rotational structures: (14.31) .A1  A/sp  .A1  A/rot ; .A2  A/sp  .A2  A/rot : Therefore, a full expression for this Hamiltonian can be written in the following form: H D .I; e32 /  A C .e3 /  B2 :

(14.32)

Each term of the Hamiltonian comprises a certain combination of products of the angular momentum components of symmetries A or B2 in the group D2 , and the parameters before this combination are linearly dependent on the spin operators that correlate with it. Naturally, the Hamiltonian is t-even. Therefore, combinations of the A type should be t-even and combinations of the B2 type, t-odd. Equation (14.32) is easy to unwind as a series of rotational distortion perturbations. Such a Hamiltonian includes all symmetry-allowed interactions of the torsional and rotational motions. The energy matrix for a given value of the quantum number J of the operator of the angular momentum squared decomposes into a direct sum of matrices corresponding to the different types of irreducible representations of the group D6 . The component ˛ of the electric dipole moment on the ˛ axis of the FCS has the form ˛ D ˛i dO .i / ; (14.33) where the parameters dO .i / depend on the torsional operators and operators of the angular momentum components on the MCS axis. Admissible contributions to equation (14.33) are formed by the structures .A1  A/sp  .A1  B/rot ;

.A2  A/sp  .A2  B/rot :

(14.34)

For the strongest transitions, one can neglect the dependence of the parameters dO .i / on the angular momentum operators. Then  .z/ .z/  (14.35) ˛ D ˛z d1 I C d2 e32 ; where dk.z/ are the real phenomenological constants. Here, it was taken into account that the electric dipole moment is a t-even quantity and

˛z ! A1  B;

˛y ! A2  A;

˛x ! A2  B:

(14.36)

Equation (14.35) determines the purely rotational transitions corresponding to the z component of the electric dipole moment. In the group D2 , these transitions correspond to the selection rules A $ B1 ;

B2 $ B3 :

(14.37)

It is clearly seen from the classification in Figure 14.4 that the torsional splitting of rotational transitions (14.37) in the ArHC 3 complex occurs only for the second type,

268

Chapter 14 On the construction of extended point groups

and in the form of doublets. At the same time, in the ArDC 3 complex, such doublets take place for both types of transitions. In [25], it is demonstrated that the effective Hamiltonian (14.32) makes it easy to describe the available experimental data on the torsional splitting within the measurement errors. Note that such a description made earlier in [6] is based on a model that violates the requirement of self-adjointness of the operators of physical quantities in quantum mechanics (see Section 14.3). The obtained results also apply for a wide class of planar complex molecules of the L[MX3 ] type, where L is an alkali metal, M D Be; Mg; P, and X D H; F; O. It should be borne in mind that here, the equilibrium configuration and the configuration corresponding to the top of the barrier of planar internal rotation of the anion [MX3 ] relative to the cation LC exchange places [71]. But in this case the results of the analysis are independent of such a replacement.

14.3 Carbocation C2 HC 3 Carbocations are intermediate objects in many chemical reactions. Here, we consider a very interesting internal dynamic of the simplest carbocation C2 HC 3 . Calculations by the methods of quantum chemistry (see [83] and references therein) give two planar equilibrium structures shown in Figure 14.5. The bridge (B) structure corresponds to the global minimum of the effective nuclear interaction potential, and the classical (C) structure corresponds to the local minimum lying about 1300 cm1 higher. Both structures correspond to the point group C2v . Since the barrier between the B and C structures is small, the cation C2 HC 3 is nonrigid to the motion of the H nuclei about the carbon backbone, which is usually depicted as a ring in Figure 14.6 (see, e.g., [13]). For the initial structure, we will choose the C structure shown in the upper part of the ring on the left. This structure transits through the upper structure B to the structure C located in the upper part of the ring on the right, mainly due to the motion of the H1 nucleus. Then mainly the H3 nucleus begins to move in the same direction about the backbone, and so on. As a result, we have a closed cycle connecting 12 equilibrium structures (six C type and six B type ones), the motion along which, called the internal rotation, lies in one plane. We will focus on the torsion-rotational spectrum of a B isomer. Therefore, it is necessary that the extended point group explicitly takes into account the B structures alone. Consider two sequential steps between the B structures clockwise in the cycle. The spatially final structure coincides with the initial one, differing from it only by a permutation of the H nuclei. Such a transition is of the exchange type and corresponds to the geometric operation c31 of clockwise rotation of the H nuclei about the y axis (the MCS is chosen such that the cation lies in the xz plane, while its axis C2 is the x axis). This operation refers to some intermediate configuration, in which the H nuclei form a regular triangle with the center of mass coinciding with the center of mass of the

Section 14.3 Carbocation C2 HC 3

269

Figure 14.5. The bridge (left) and the classical (right) planar equilibrium structures of the C2 HC 3 cation.

Figure 14.6. The cycle of planar nonrigid motion of the C2 HC 3 cation.

molecule. The intermediate configuration takes into account that the transition-bound minima of the effective potential are equivalent and, therefore, this configuration has a higher geometric symmetry than the equilibrium configuration which allows for only one minimum.

270

Chapter 14 On the construction of extended point groups

It is seen in Figure 14.6 that with one step between the B structures, the final structure does not coincide spatially with the initial structure. However, such a coincidence is necessary for a transition of the exchange type since in this case the identical nuclei exchange places in space. Therefore, the transition also includes the motion in which the cation is rotated as a whole by an angle  about the ó axis (this axis is chosen since the nonrigid motion lies in the xz plane). As a result, we assign the actually geometrically complex motion of one step to such a combination of the rotation of the .y/ three H nuclei structure and the motion C2 of the cation as a whole that the square of this combination yields a correct expression for two steps of the transition. This combination has the form C2.y/ c31  c21  c61 ; (14.38) where the elements c21 and c61 act on the three H nuclei and two C nuclei structures, respectively. Operation (14.38) allows for all significant features of the geometry of one step and is a generating element of the cyclic group: P6 D C2  H3 ;

(14.39)

.y/

where C2 D .E; C2 / and H3 D .E  E; E  c31 ; E  c32 /. Analysis of three steps of the cyclic motion .C2.y/ c31 /3 D C2.y/ with the use of Figure 14.6 shows that the .y/

operation C2 is of the exchange type and corresponds to the permutation ¹2º  ¹13 º of the 2  3 group. .x/ .xz/ The extended point group should include the point group C2v D .E; C2 , v , .yz/ N v / as a subgroup. Therefore, we obtain for the extended point group the following expression: (14.40) G24 D C2  H6  CS : Here, the group H6; that is isomorphic with the group D3; can be represented as H6 D H3 ^ C20 ;

(14.41)

where C20 D .E; C2.x//. From a matching of the point group with the permutation group 2  3 of identical nuclei (the necessary correlations are given in Table 14.6), for the allowed multiplets of a rigid cation we obtain 2A1 ; 6B1 : (14.42) The nuclear statistical weights are given for the primary isotope of the carbon nucleus 12 C (for the isotope 13 C; we have 20A1 and 12B1 , respectively). Symmetry types of the wave functions of separate motions are matched with the symmetry types of the allowed multiplets, which are formed on their basis, as in equation (14.17). A classification of the energy levels of a rigid cation in the type A1 vibronic state is shown in the first two columns in Fig 14.7. The actual structure of the levels is described by the group D2 , which includes only the rotational operations of the C2v  Ci group.

Section 14.3 Carbocation C2 HC 3

271

Table 14.6. Correlation table for the C2 HC 3 cation. 2C  2H  1H

C2v

Œ2  [2]  [1]

A1

[12 ]  [2]  [1]

B1

[2]  [12 ]  [1]

B1

[1 ]  [1 ]  [1]

A1

2

2

C2v  Ci .C/

./

A

./ .C/ A1 ; A2 B1.C/ ; B2./ ./ .C/ B1 ; B2

B3

A1 ; A2

2  3

G24

CS  Ci

Œ2  Œ3

A  A1  A0

A .C/ , A ./ 0 00 A ./ , A .C/

0

0

Œ1   Œ3

B  A2  A

Œ2  Œ21

A  E  A0

Œ12   Œ21

B  E  A0

Œ2  Œ13 

A  A2  A0

Œ12   Œ13 

B  A1  A0

2

D2

00

B2 B1

U2

D2

H6  U2

a b

A B1 B2 B3

A1  a, E  a A2  b, E  b A2  a, E  a A1  b, E  b

From a matching of the groups 2  3 and G24 , for the allowed coordinate multiplets of a nonrigid cation we have 4.A  A2  A0 /;

2.A  E  A0 /:

(14.43)

The number of admissible multiplets is small, and this leads to a strong depletion of the spectrum. For the isotope 13 C, the number of allowed multiplets doubles: 4.A  A2  A0 /;

2.A  E  A0 /;

12.B  A1  A0 /;

6.B  E  A0 /: (14.44)

Matching now the geometrical groups G24 and C2v , we obtain a picture of the level splitting of a rigid cation with allowance for the nonrigid motion, which is shown in Figure 14.7 for the type A1 vibronic state. The energy spectrum for the 12 C isotope is strongly depleted, and the levels with zero nuclear statistical weights are shown by a dotted line for completeness (part of these levels become allowed for the 13 C isotope). The actual structure of the levels is described by the group H24 , which includes only the torsion-rotational operations of the G24  Ci group: H24 D C2  H6  U2 ;

(14.45)

272

Chapter 14 On the construction of extended point groups .y/

where U2 D .E; U2 D  .xz/ i/. Symmetric and antisymmetric representations of U2 group are denoted as a and b. In the splittings, the levels are given in increasing order jmj D 0; 1; 2; 3, where m is the quantum number of the problem for onedimensional cyclic motion. The following correspondence between jmj and the torsional types of symmetry in the C2  H6 group was used: A  A1 ! 0;

B  E ! 1;

A  E ! 2;

B  A1 ! 3:

It is easily seen in Figure 14.7 that the allowed torsional levels in different rotational states differ in the values of jmj. We also note that, although the geometric group H24 is isomorphic with the CNPI group which was introduced to analyze the nonrigid cation C2 HC 3 [33, 56], these group are differently used.

Figure 14.7. Classification of stationary states of the B isomer of the C2 HC 3 cation in the type A1 vibronic state.

The effective operator of any physical quantity is transformed according to the representation A  A1  A0 of the G24 group. Passing to the group H24 , we obtain the

Section 14.3 Carbocation C2 HC 3

273

representations A  A1  a and A  A1  b related, respectively, to the quantities that are invariant under the inversion operation i and change sign during this transformation. That is, the Hamiltonian belongs to the type AA1 a and the effective operator of the electric dipole moment, to the type A  A1  b. Therefore, the electric dipole selection rules in the H24 group have the form   a $   b;

(14.46)

where  is the irreducible representation of the group C2  H6 , which is retained during transition. We now construct the torsion-rotational function space of a B isomer for el vib D A1 . The basis set of torsional unit vectors is specified by six irreducible representations of the group described by equation (14.39): .y/

C2 jAi D jAi; .E 

c31 /j0i

D j0i;

.y/

C2 jBi D jBi;

.E 

c31 /j

˙ 1i D "

˙1

(14.47) j ˙ 1i;

(14.48)

where " D exp.2 i=3/. It follows from equation (14.41) that C2.x/j0i D cj0i;

C2.x/j1i D d j  1i;

(14.49)

where c 2 D d 2 D 1. For the ground vibrational state, c D 1 (the value of d can always be chosen equal to 1/. Then the unit vector j0i belongs to the representation A1 of the group H6 and the unit vectors j ˙ 1i, to the representation E: E1 D js1 i D

j1i C j  1i p ; 2

E2 D ja1 i D

j1i  j  1i p 2i

where the components E1 and E2 are determined by the relations .x/

C2 E1 D E1 ;

.x/

C2 E2 D E2 :

As for transformations of the group U2 , they do not act in the torsional space. The rotational unit vectors will be specified by irreducible representations of the group D2 . When passing to the group H24 , it should be taken into account that the .y/ .y/ behavior relative to the operations C2 and U2 must coincide for any rotational quantities. Therefore, A ! A  A1  a;

B1 ! B  A2  b;

B2 ! A  A2  a;

B3 ! B  A1  b; (14.50) where the representations of the D2 group are on the left and their counterparts in the H24 group are on the right. As a result, we have the complete set of torsion-rotational unit vectors decomposed into irreducible representations of the H24 group, which is shown in Table 14.7. In equation (14.49), the value c D 1 was chosen to ensure that the unit vectors given in this table are consistent with the classification in Figure 14.7.

274

Chapter 14 On the construction of extended point groups

Table 14.7. Decomposition of the torsion-rotational unit vectors for the rotational states A and B1 into irreducible representation of the H24 group. C2  H6  U2

Unit vectors

C2  H6  U2

Unit vectors

A  A1  a

jAij0iA

B  A2  b

jAij0iB1

B  A1  a

jBij0iA

A  A2  b

jBij0iB1

AE a

E1 D jAijs1 iA E2 D jAija1 iA

B E b

E1 D jAijs1 iB1 E2 D jAija1 iB1

B E a

E1 D jBijs1 iA E2 D jBija1 iA

AE b

E1 D jBijs1 iB1 E2 D jBija1 iB1

Note. For the rotational states B2 and B3 , one should make the replacements A $ B and a $ b in the representations of the H24 group and the replacements A ! B3 and B1 ! B2 in the unit vectors.

In order to describe the torsional part of the effective operators of physical quantities, we introduce a coordinate spin p in the space of the unit vectors jAi and jBi and a coordinate spin q in the space of the unit vectors j 0i and j ˙ 1i. The operators pk are written as in equation (7.29). From the properties of the unit vectors for the group C2 transformations and time reversal operation T T jAi D jAi;

T jBi D jBi

(14.51)

it follows that the operator p3 is of the type A and is t-even, while the operators p1 and p2 are of the type B and are t-even and t-odd, respectively. In two-dimensional space, the complete set of operators also includes a totally symmetric unit operator I . We assume that the unit vectors j0i and j ˙ 1i are the eigenvectors of the operator q3 having the eigenvalues 0 and ˙1, respectively. From the properties of the unit vectors for transformations of the group H6 and the time reversal operation T T j0i D j0i;

T j1i D j  1i

(14.52)

it follows that the operator q3 is of the type A2 and is t-odd, while the operators q1 ; q2 are of the type E and are t-even. In three-dimensional space, the complete set of independent operators includes the products of the components q of a total power not higher than two. In the further analysis, we will need the torsional operators corresponding to one-dimensional representations of the H24 group. All such operators are presented in Table 14.8. Contributions to the effective Hamiltonian are formed by the following two spinrotational structures: .A  A1  a/sp  .A  A1  a/rot;

.A  A2  a/sp  .A  A2  a/rot :

(14.53)

Section 14.3 Carbocation C2 HC 3

275

Table 14.8. Complete set of independent torsional operators corresponding to one-dimensional representations of the H24 group. C2  H6  U2

t-even

t-odd

A  A1  a B  A1  a A  A2  a B  A2  a

I; p3 ; q32 ; p3 q32 p1 ; p1 q32 – p2 q3

– p2 ; p2 q32 q3 ; p3 q3 p1 q3

A full expression for these contributions can be written as H D .I; p3 ; q32 ; p3 q32 /  A C .q3 ; p3 q3 /  B2 :

(14.54)

Each term of the Hamiltonian is a combination of products of the angular momentum components in the MCS related to the representations A1 or B2 of the D2 group, and the parameters before this combination are linearly dependent on the spin operators that correlate with this combination. Naturally, the Hamiltonian should be t-even. Equation (14.54) is not difficult to unwind as a series of rotational distortion perturbations. This Hamiltonian includes all symmetry-allowed interactions of the torsional and rotational motions. The energy matrix for a given value of the quantum number J of the operator of the angular momentum squared decomposes into a direct sum of matrices corresponding to the different types of irreducible representations of the H24 group. The component ˛ of the effective operator of the electric dipole moment on the ˛ axis of a FCS is written as (14.55) ˛ D ˛i dO .i / ; where the parameters dO .i / depend on the spin operators and operators of the angular momentum components on the MCS axis. The admissible contributions in equation (14.55) are formed by the structures .B  A2  a/sp  .B  A2  b/rot ;

.B  A1  a/sp  .B  A1  b/rot: (14.56)

For the strongest transitions, one can neglect the dependence of the parameters dO .i / on the angular momentum operators. Then ˛ D ˛z d .z/ p2 q3 C ˛x .d1.x/ p1 C d2.x/p1 q32 /;

(14.57)

.x/

where d .z/ and dk are the real phenomenological constants. It was taken into account in equation (14.57) that the electric dipole moment is t-even and the direction cosines belong to the following representations of the H24 group:

˛z ! B  A2  b;

˛y ! A  A2  a;

˛x ! B  A1  b:

(14.58)

276

Chapter 14 On the construction of extended point groups

According to equation (14.57), the purely rotational transitions are absent. The transitions corresponding to the z component of the electric dipole moment are of the types A $ B1 ;

B2 $ B3

(14.59)

in the D2 group. These transitions are possible only between the degenerate torsional sublevels. Therefore, for the 12 C isotope, the torsional splittings are absent in the rotational spectrum of the z component. For the x component of the electric dipole moment, the transitions in the D2 group are of the types A $ B3 ;

B1 $ B2 :

(14.60)

From a classification in Figure 14.7 it is easily seen that for the 12 C isotope, the torsional splittings occur only for the second type of transitions in the rotational spectrum of the x component, and these splittings have the form of doublets. In [26], it is shown that the effective Hamiltonian (14.54) makes it easy to describe the available experimental data on the torsional splittings within the measurement errors. Such a description made earlier in [33] uses the model taken from [56]. We will show that this model violates the requirement of self-adjointness of the operators of physical quantities in quantum mechanics. The approximate construction of the effective Hamiltonian in [56] is based on the assumption that the matrix elements of the Hamiltonian of the nonrigid motion (called the tunneling motion in [56]) are proportional to the tunneling “overlap integral.” For the C2 HC 3 cation, the following expression is obtained for the matrix elements: hn; J; kjH jn C 1; J; k 0 i hnjn C 1iDk 0 ;k .ˇ/;

(14.61)

where D.ˇ/ is the operator of finite rotation by an angle ˇ about the y axis. The product of the vibrational and rotational overlap integrals is on the right-hand side. The vibrational integral specifies the overlap of the vibrational wave functions localized in the neighboring minima along the nonrigid motion coordinate . In [33], these functions were chosen in the form n ./

D exp¹aŒ  .2n  1/=62º;

(14.62)

where n D 1; : : : ; 6. The matrix element Dk 0k .ˇ/ specifies the overlap of the rotational basis wave functions jJ; k; M i and jJ; k 0 ; M i for the coordinate systems in the minima n and nC1, which are rotated relative to each other by an angle ˇ that describes one step of nonrigid motion. It is exactly Dk 0 k .ˇ/ that determines the dependence on the rotational quantum numbers in equation (14.61). The principal disadvantage of this method is that the tunneling Hamiltonian is not self-adjoint. The point is that D.ˇ/ is a unitary operator, which can be written in the form D.ˇ/ D exp .iˇJy / D cos .ˇJy /  i sin .ˇJy /;

(14.63)

Section 14.4 Conclusions

277

where Jy is the component of the angular momentum along the y axis. The first operator on the right-hand side is self-adjoint and is of the A type in the group D2 , while the second operator is of the B2 type and is antiself-adjoint (it reverses sign in the adjoining operation). The symmetry properties admit the appearance of a type B2 operator only for the degenerate torsional states. Equation (6) in [33] gives a matrix of the Hamiltonian for this operator, and this one is a typical matrix of the antiselfadjoint operator. We emphasize that the levels bound by an antiself-adjoint interaction operator are always “attractive,” and they are always “repulsive” for the self-adjoint interaction operator. The model used in [6] to describe the torsional splittings in the C rotational spectrum of the ionic complexes ArHC 3 and ArD3 has a similar drawback.

14.4 Conclusions We have formulated some general conclusions based on the content of this chapter. 1. The power of the discussed qualitative methods is to a considerable degree due to the fact that to consider a nonrigid motion, one does not need to specify its trajectory in a set of internal coordinates, but should only indicate the intermediate configuration, by transforming which the geometry of this motion can be determined. It appears that the molecular system may have several different geometries of motion of the exchange type which lead to the same permutation of identical nuclei. The trans and cis transitions in the (HF)2 dimer are the typical examples. Note that philosophical problems occur in such cases when the CNPI group is used since the latter does not differentiate between geometrically different transitions leading to the same permutation of identical nuclei. C 2. The extended point group of the ionic complexes ArHC 3 and ArD3 has the form h1 h2 , where the groups h1 and h2 characterize the structures consisting of one Ar nucleus and three hydrogen nuclei, respectively. One nucleus gives only a trivial totally symmetric contribution to the multiplet of a nonrigid complex. Conditions where this contribution can be omitted, and a much simpler extended point group in the form h2 can be used, have been formulated. In particular, these conditions C are fulfilled for the ArHC 3 and ArD3 complexes. The situation is similar for the molecules of complex hydrides of the LBH4 type, which were considered in Section 9.4. If the conditions are not fulfilled, then a total group of the form h1  h2 has to be used. Such examples are presented in the next chapter. C C 3. For the ionic complexes ArHC 3 and ArD3 and for the carbocation C2 H3 , we obtained rigorous algebraic models which make it easy to describe the experimental data on torsional splittings of the rotational transitions within the measurement errors. Prior descriptions used the approximate models violating the requirement of self-adjointness of the operators of physical quantities in quantum mechanics.

Chapter 15

Nonrigid molecular systems with continuous axial symmetry groups Nonrigid molecular systems with continuous axial groups appearing in a description of the geometric symmetry of their internal dynamics are numerous enough. The HCN / HNC system in the ground electronic state is a well-known example. The isomeric forms HCN and HNC corresponding to the local minima of the effective nuclear interaction potential have a linear equilibrium configuration [52]. The symmetry of motions in the local minima is characterized by groups coinciding with the point groups of equilibrium configurations in these minima. This group is C1 v for both isomers. A nonrigid system is characterized by an extended point group that allows for both the motions in the local minima and the transitions between them. Since the transformations specifying a transition between nonequivalent isomers are of course not the transformations of symmetry of the Hamiltonian, an extended point group in this case is a noninvariant group. It is to be emphasized in this respect that considering nonrigid systems in terms of the CNPI group [16] is not applicable here since this group by definition includes only the symmetry elements of the Hamiltonian (see Sections 11.5 and 12.6). Such dynamics is also inherent in other molecular systems, e.g., hydroxide HOX/ HXO, where X D Li, Be, B, Na, Mg, Al [32]. These molecular systems also include a variety of weakly bound complexes, in particular, ArHX/ArXH, where X D F, Cl, Br, I [31]. These systems are characteristic in that their equilibrium configurations are linear. Therefore, the continuous axial groups appear for such systems even if the nonrigid motions are neglected, although taking the latter into account, of course, leads to the enlargement of such groups. However, there are nonrigid molecular systems for which the point groups are only finite, whereas the extended point groups that allow for nonrigid motions become continuous axial groups. Clearly, a finite point group always corresponds to a nonlinear equilibrium configuration. A continuous axial group appears when considering such a system if the analysis of experimental data requires that the nonrigid transitions through a barrier corresponding to the linear configuration of this system be taken into account. The water molecule H2 O, whose equilibrium configuration in the ground electronic state is nonlinear and corresponds to the point group C2v (see Section 8.1), is a classic example. However, the transitions through a linear configuration due to a change in the angle between the two valence bonds of OH are significant for a correct description of the existing experimental data obtained by the methods of high-resolution spectroscopy [29]. Weakly bound complexes of the form XCO, where X D He, Ne, Ar, Kr, Xe, are another example. Their equilibrium configuration is nonlinear and corresponds to the point group CS [88]. However, the

279

Section 15.1 Systems of the HCN/HNC type

weakly hindered nonrigid motion due to the rotation of the CO structure about its center of mass moves the system through the linear configuration. Thus, a list of nonrigid molecular systems with the continuous axial groups determining the geometric symmetry of their internal dynamics is not only extensive, but also includes the topical objects of research.

15.1 Systems of the HCN/HNC type Consider first a classification of the HCN and HNC isomers, assuming them to be independent rigid molecules. The construction of a classification for the linear molecules with the point group C1 v can be found in Section 8.3. For el D A1 and vib D A1 ; E1 the classification is shown in the first two columns in Figure 15.1. The actual structure of the levels is described by the rotational subgroup D1 of the C1 v  Ci group. The forbiddance of some of the levels with small values of J is explained by the fact that the projection of the angular momentum on the axis C1 is due only to the electron-vibrational motions [64]. Naturally, nonrigid motions in the HCN / HNC system must not be represented as the migration of H with the CN structure fixed, since this would change the position of the center of mass of the system. Actually, corresponding to such motions are rotations of the CN structure by an angle  about any axis passing through its center of mass orthogonally to the linearization axis. Hence, the extended point group can be written as F D c1 v  d1 h ; (15.1) where the groups c1 v and d1 h D c1 v  cI characterize the H and CN structures, respectively. The group c1 v of the structure with one nucleus is needed in the extended point group to include the group C1 v as its subgroup. Clearly, the rotations that specify the nonrigid transitions between nonequivalent isomers are certainly not the Hamiltonian symmetry transformations. As a consequence, F is a noninvariant group with the Hamiltonian symmetry subgroup G D c1 v  c1 v :

(15.2)

All elements of the G group correspond to the identical nuclear permutation. As a result, only the representation A1  A1 is admissible for a complete coordinate wave function in the G group. By matching the groups G è F (the necessary correlations are given in Table 15.1), for the allowed multiplets of a nonrigid system we have A1  A1g ;

A1  A1u ;

(15.3)

where the subscripts g and u denote the symmetric and antisymmetric representations of the cI group, respectively. From a matching of the point group C1 v of any isomer and the group F, we obtain a picture of the level splitting with allowance for nonrigid motions, which is shown for el D A1 and vib D A1 ; E1 in Figure 15.1.

280

Chapter 15 Nonrigid molecular systems with continuous axial symmetry groups

Table 15.1. Correlation table for the HCN/HNC system. c1 v  c1 v

C1 v

DQ 1  d1

D1

A1  A1 , A2  A2

A1

A1  A1 , A2  A2

A1

A1  A2 , A2  A1

A2

A1  A2 , A2  A1

A2

c1 v  d1 v  Ci

DQ 1  d1

(A1  A1g /.C/ , (A1  A2u /./ (A2  A2u /.C/ , (A2  A1g /./

A1  A1

(A2  A2g /.C/ , (A2  A1u /./ (A1  A1u /.C/ , (A1  A2g /./

A2  A2

(A1  A2g /.C/ , (A1  A1u /./ (A2  A1u /.C/ , (A2  A2g /./

A1  A2

(A2  A1g /.C/ , (A2  A2u /./ (A1  A2u /.C/ , (A1  A1g /./

A2  A1

Since nonrigid motions are specified by rotational operations, the actual energylevel structure is described by the rotational subgroup H of the F  Ci group. This subgroup can be written in a fairly simple form H D DQ 1  d1 ;

(15.4)

where DQ 1 D .E  E; 2c'  E; 1 u2  I i/ is isomorphic with the group D1 (the right and left elements act on the H and CN structures, respectively), and d1 is the rotational subgroup of the d1 h group. Here, it is important that the spatial inversion operation i is given relative to the center of mass of the system. Therefore, the axes 1 u2 D 1 v i of the DQ 1 group pass through the center of mass of the system, while the axes 1 u2 D 1 v I of the d1 group pass through the center of mass of the CN structure. It is clearly seen in Figure 15.1 that each level of a rigid isomer splits into two levels since the extended point group allows for the existence of two isomers. We note that in group (15.4) the spatial inversion operation i is included in the elements which are not rotations of the system as a whole (in this case, the operation I inverts only the CN structure relative to the center of its mass). Therefore, unlike the C ionic complexes ArHC 3 and ArD3 considered in Section 14.2, extended point group (15.1) cannot be simplified by omitting in it the group c1 v of the structure with one nucleus. That is, one has to manipulate the complete rotational group (15.4). The case is similar for the LiReO4 molecule [22].

281

Section 15.1 Systems of the HCN/HNC type

Figure 15.1. Energy-level classification of the HCN/HNC system for el D A1 , vib D A1 (left) and vib D E1 (right).

The effective operator of any physical quantity characterizing the molecular system belongs to the representation Œ1  Œ1  Œ1 of the 1  1  1 group and, as a consequence, representations (15.3) of the F group. Passing to the group H, we have the representations A1  A1 ; A2  A2 (15.5) for the quantities that are invariant under the operation i and the representations A2  A1 ;

A1  A2

(15.6)

for the quantities that change sign in this transformation. That is, the effective Hamiltonian transforms according to equation (15.5), and the effective operator of the electric dipole moment according to equation (15.6). That the Hamiltonian has a nonsymmetric representation is due to the fact that nonrigid motions are of the nonexchange type. Therefore, the matrix of the Hamiltonian for a given value of the quantum J is divided into a direct sum of only two blocks with the symmetry types .A1  A1 ; A2  A2 /;

.A2  A1 ; A1  A2 /;

(15.7)

through which the electric dipole selection rules can easily be written: .A1  A1 ; A2  A2 / $ .A2  A1 ; A1  A2 /:

(15.8)

282

Chapter 15 Nonrigid molecular systems with continuous axial symmetry groups

15.2 Complexes of the XCO type The equilibrium configuration of the XCO complex, where X D He, Ne, Ar, Kr, and Xe, which is shown in Figure 15.2, is close to the T-shape and corresponds to the point group CS D .E;  .yz/ /. It can be assumed that the nonlinear equilibrium configuration results from a distortion of the highly symmetric linear configuration. Weakly hindered nonrigid motion, which consists in the rotation of the CO structure about its center of mass in the yz plane, moves the system through the linear configuration, thereby recovering the axial symmetry of the internal dynamics. This conclusion follows immediately from the geometry of the problem. Therefore, the symmetry group of the Hamiltonian of a nonrigid complex XCO can be written in the form (15.2), where now the left and right groups c1 v characterize the X and CO structures, respectively. The noninvariant group F given by equation (15.1) is different from group (15.2) in that the group F takes into account explicitly the existence of two nonequivalent isomers. Thus, the energy levels are divided into two parts. Although these parts are interaction-bound, such a division is logical for the spectrum of the HCN / HNC system. However, the internal rotation potential of the CO structure in the XCO complex has no minima corresponding to the independent isomers. In cases where it is not necessary to divide the energy levels into parts, group G in the form (15.2) is sufficient to describe the internal motions of the complex. It should be mentioned that the equilibrium configuration corresponding to the minimum of the effective potential with the CO structure arranged symmetrically with respect to the xz plane can also be obtained by no-barrier rotation of the initial equilibrium configuration by an angle  about the z axis. The existence of such a minimum is taken into account by the axial symmetry of the problem on the spectrum of one isomer.

Figure 15.2. Equilibrium configuration of the XCO complex.

283

Section 15.3 Nonrigid water molecules H2 O

Since nonrigid motions are specified by rotational operations, the actual structure of the levels is described by the rotational subgroup H of the G  Ci group. This subgroup can be represented in the form H D .c1  c1 / ^ U2 ;

(15.9)

.x/

where the elements of the group U2 D .E; U2 D  .yz/ i/ act on the complex as a whole. As aleady mentioned, the group c1 v of the one-nucleus structure is necessary in equation (15.2) for an extended point group to include the group C1 v as its subgroup. A matching of the group c1 v with the permutation group 1 of this nucleus gives a trivial contribution of the form A1 to the multiplet of a nonrigid complex. In this case, such a part of the behavior of the multiplet can be omitted, which leads to the replacement of group (15.2) by C1 v and the replacement of group (15.9) by D1 . Such replacements are possible since in equation (15.9) the operation i is included only in the elements of rotation of the complex as a whole (see Section 14.2). As a result, the classification is constructed as in the case of rigid molecules HCN and HNC. Since the actual energy-level structure is described by the group D1 , the effective Hamiltonian relates to the representation A1 , and the effective operator of the electric dipole operator to the representation A2 of this group. Accordingly, the electric dipole selection rules have the form (8.47). The fact that instead of the finite group CS for a rigid complex, a continuous group C1 v appears in the description of a nonrigid complex indicates radical changes in the internal dynamics when nonrigid transitions are taken into account. This issue will be considered in detail in the next section.

15.3 Nonrigid water molecules H2 O A rigid molecule of water H2 O was considered in Section 8.1. Its equilibrium configuration in the ground electronic state corresponds to the point group C2v . However, numerous experimental data obtained by the methods of high-resolution spectroscopy require that the excitations be taken into account by the angle between valence bonds of OH above the barrier corresponding to a linear configuration of the molecule [29]. This means that the description should allow for nonrigid transitions through the linear configuration. The existence of these transitions is due entirely to the distortion of symmetry of the equilibrium configuration (from the maximum possible with a continuous axial group D1 h to the actually observed with the group C2v /. Therefore, taking these transitions into account leads to the restoration of the axial symmetry of internal dynamics. This conclusion easily follows from the geometry of the problem. Importantly, in this case nonrigid transitions do not lead to level splitting since the final configuration of a transition can also be obtained from the initial configuration by no-barrier rotation of the molecule as a whole by an angle  about the linearization axis. That is, a nonrigid transition does not move the H2 O molecule to an independent local minimum. This is automatically taken into account in group chain methods.

284

Chapter 15 Nonrigid molecular systems with continuous axial symmetry groups

The fact that instead of the finite group C2v for a rigid molecule, a continuous group D1 h appears in the description of a nonrigid molecule indicates radical change in the internal dynamics when nonrigid transitions are taken into account. Indeed, the group C2v leads to a 3 C 3 and the group D1 h , a 2 C 4 division into rotational and vibrational degrees of freedom. Here, the role of nonrigid transitions reduces exactly to the rearrangement of the vibration-rotational configuration space, which should underlie the construction of a correct description. Matching the permutation group 2 of hydrogen nuclei with the group D1 h D C1 v  CI (the necessary correlations are given in Table 15.2), for the nonrigid molecule H2 O we obtain the following allowed multiplets: 1A1g ;

3A1u :

(15.10)

Then, the symmetry types of the basis functions of separate internal motions are matched with the symmetry types of the allowed multiplets formed on their basis, as in equation (8.48). As a result, we obtain a classification of the energy levels of a nonrigid molecule H2 O, which is shown for el D A1g and vib D A1g ; E1u in Figure 15.3. The inversion doublets are absent in the spectrum. The actual structure of the levels is described by the subgroup H2  D1 of the D1 v  Ci group. Symmetric and antisymmetric representations of the group H2 D .E; I i/ are denoted s and a. According to equation (15.10), only the representations A1s ;

A1a ;

A2s ;

A2a

(15.11)

are admissible for the complete coordinate function in the H2  D1 group. Note that only one representation of the H2 group is possible for a given vibronic state (s for the type g state and a for the type u state). The forbiddance of some of the levels with small values of J is explained by the fact that as in the case of a linear Table 15.2. Correlation tables for the nonrigid molecule H2O. 2

D1 h

D1

D1 h

H2  D1

D1 h  Ci

Œ2

A1g

A1

A1g , A2u

A1s

A1g ; A2u

[12 ]

A1u

A2

A1u , A2g

A1a

A1g ; A2u

–

A2g

E1

E1g , E1u

A2s

A2g ; A1u

–

A2u

E2

E2g , E2u

A2a

A2g ; A1u

–

E1g

E3

E3g , E3u

E1s

E1g ; E1u

–

E1u

E4

E4g , E4u

E1a

E1g ; E1u

...

...

...

...

...

Note. The coordinate Young diagrams are given for the group 2.

.C/

./

./

.C/

.C/

./

./

.C/

.C/

./

./

.C/

...

285

Section 15.3 Nonrigid water molecules H2 O

Figure 15.3. Energy-level classification for a nonrigid water molecule H2 O for el D A1g ; vib D A1g (left) and E1u (right).

molecule, the projection of the angular momentum on the axis C1 is due only to the electron-vibrational motions. Clearly, the angular momentum cannot be smaller than its projection of this which is nonzero in degenerate electron-vibrational states and is taken into account by the symmetry types in el  vib . It is easily seen that the effective Hamiltonian relates to the representation A1s of the group H2  D1 and the effective operator of the electric dipole moment, to the representation A1a . Therefore, the electric dipole selection rules have the form A1s $ A1a ;

A2s $ A2a :

(15.12)

It follows, in particular, that the electric dipole transitions are not possible within the limits of one vibronic state. Note that such a forbiddance is absent when the consideration is based on the C2v group and nonrigid transitions are not taken into account. However, there is no contradiction here since the division into vibrational and rotational degrees of freedom is different in these two models. In the group D1 h D C1 v  CI , the ground electronic state of the H2 O molecule belongs to the totally symmetric representation A1g and four vibrational degrees of freedom, to the representation [64] : vib D A1g C A1u C E1u ;

(15.13)

286

Chapter 15 Nonrigid molecular systems with continuous axial symmetry groups

where one-dimensional representations relate to the valence vibrations and the twodimensional representation, to the bending vibration. Passing to the group H2  D1 , for the electronic state we have the type A1s and for the vibrational modes, : vib D A1s C A2a C E1a :

(15.14)

Consider the spectrum corresponding to the ground vibrational state in the rigidmolecule model. This spectrum is entirely determined by the mode E1a , whose v-fold excitation is specified by the symmetric power ŒE1a v (see Section 2.5). Decomposition of this power into irreducible representations of the group H2  D1 is shown in Table 15.3. The levels of interest to us are given by a sequence .0; A1s /;

.1; E1a /;

.2; E2s /;

.3; E3a /;

.4; E4s /;

:::;

(15.15)

where the numbers before the designations of representations are the values of the excitation index v. Table 15.3. Decomposition of the symmetric power ŒE1a v into irreducible representations of the H2  D1 group. ŒE1a v

H2  D1

vD0 1 2 3 4 ...

A1s E1a E2s C A1s E3a C E1a E4s C E2s C A1s ...

The basis vibrational set of unit vectors jli for the levels in sequence (15.15) is specified by irreducible representations of the group C1 : C'.z/ jli D exp.il'/jli:

(15.16)

The axes x; y, and z of the molecule-related MCS are fixed with respect to the effective nuclear potential (see Section 12.4), and the axis z coincides with the linearization axis of the molecule. Group D1 can be represented as D1 D C1 ^ U2 ;

(15.17)

where U2 D .E; U2.x/ /. By virtue of the properties of the semidirect product, we have .x/

U2 j0i D aj0i;

.x/

U2 jli D bj  li;

(15.18)

287

Section 15.3 Nonrigid water molecules H2 O

where a2 D b 2 D 1. It will be seen later that for sequence (15.15) it is necessary to choose a D 1 (the value of b can always be set equal to 1/ and require that I ijli D .1/l jli:

(15.19)

Then the unit vector j0i relates to the representation A1s of the H2 D1 group and the unit vectors j˙li, to the representations Els and Ela (for even and odd l, respectively). Note that other sequences in Table 15.3 specify the levels corresponding to the excited vibrational states of the bending mode in the rigid-molecule model. For example, for the first bending state, we have a sequence .2; A1s /;

.3; E1a /;

.4; E2s /; : : : :

The behavior of the rotational unit vectors relative to the group D1 transformations is well known [64]: C'.z/ jJ; ki D exp.ik'/jJ; ki;

U2.x/jJ; ki D .1/J jJ; ki:

(15.20)

As for the operation I i, its action on any rotational quantities is equivalent to an identical operation. Using equations (15.16) and (15.18)–(15.20), it is easy to obtain the vibrationrotational unit vectors corresponding to representation (15.11) of the H2  D1 group. These unit vectors are decomposed into irreducible representations of the D1 group in Table 15.4. The transition to the group H2  D1 is easily accomplished using equation (15.19). It is important that all admissible unit vectors contain only the products jJ; kijli, for which k D l. That is, this construction takes into account that the projection of the angular momentum on the z axis is due entirely to the vibrational motions. Accordingly, l can be interpreted as a quantum number of the projection of the vibrational angular momentum onto this axis. The value a D 1 in equation (15.18) and relation (15.19) were chosen on the basis of compliance between the symmetry properties of the vibration-rotational unit vectors and the energy-level classification.

Table 15.4. Decomposition of the vibration-rotational unit vectors into irreducible representations of the D1 group. D1

Note. ! D .1/J .

Basis unit vectors

A1

jJ; 0ij0i; J is even p .jJ; lijli C !jJ; lij  li/= 2;

l¤0

A2

jJ; 0ij0i; J is odd p .jJ; lijli  !jJ; lij  li/ = 2;

l¤0

288

Chapter 15 Nonrigid molecular systems with continuous axial symmetry groups

We will construct a complete set of self-adjoint vibrational operators on the basis of the operator of coordinate spin l. It is required to find the representation of Lie algebra of the form (7.27) in the space of unit vectors jli (the upper bound on jlj is equal to the number v of considered excitations). Let l3 be a diagonal operator in the space of unit vectors jli with the eigenvalues l, and l˙ D l1 ˙ il2 be the raising and lowering operators, respectively, such that p lC jli D .v  l/.v C l C 1/jl C 1i; (15.21) p l jli D .v C l/.v  l C 1/jl  1i: From the symmetry properties of the unit vectors jli for the group H2  D1 transformation and time reversal operation T , T jli D jli;

(15.22)

it follows that the operator l3 belongs to the representation A2s of this group and is t-odd and the operators l˙ belong to the representation E1a (they transform as the vectors j ˙ 1i/ and T lC T D l . On the basis of these generators, one can write out a complete set of independent self-adjoint vibrational operators with the symmetry properties indicated, and the operators of such a set may include the products of the components l of a total power not higher than 2v. Admissible contributions to the effective Hamiltonian are formed by the following vibration-rotational structures: .A1s /vib  .A1s /rot;

.E2ls /vib  .E2ls /rot ;

(15.23)

where l D 1; 2; : : : ; v. Since the Hamiltonian is t-even, we obtain the following expression for it: HD

1 X

H2n ;

H2n D

X

2t 2t c2p;2s;2t J 2p .l32s lC JC C J2t l2t l32s /;

(15.24)

p;s;t

nD0

where p C s C t D n, and c2p;2s;2t are the real phenomenological constants. Here, it was additionally taken into account that in the space of admissible basis unit vectors the operator Jz is equivalent to the operator l3 . It is easily seen that the action of Hamiltonian (15.24) on the basis unit vectors does not violate the condition k D l. In the model of a rigid molecule, the analog of such a Hamiltonian includes only the operators of the angular momentum in the MCS, H D

1 X nD0

H2n;

H2n D

X

2t c2p;2s;2t J 2p .Jz2s JC C J2t Jz2s /;

(15.25)

p;s;t

where the axis z lies in the plane of the molecule orthogonal to the axis C2 of the C2v group. Equations (15.24) and (15.25) are different in that the Hamiltonian of a

Section 15.3 Nonrigid water molecules H2 O

289

nonrigid molecule additionally contains the operators l˙ . Since l D k, this changes the dependences on the quantum number k in the description. It is also easy to write out the expressions for the generators of unitary transformations transforming the Hamiltonian (15.24) to a reduced form. Symmetry of the generators in the group H2  D1 is similar to the Hamiltonian, but these are t-odd. Therefore, for a family of generators we have 2t C2 2t C2 is2p;2sC1;2t C2 J 2p .l32sC1 lC JC  J2t C2 l2t C2 l32sC1 /;

(15.26)

where s2p;2sC1;2t C2 are the real phenomenological constants. The component ˛ of the electric dipole moment operator on the axis ˛ of the FCS can be written in the form (8.62). The direction cosines relate to the following representations of the H2  D1 group:

˛z ! A2s ;

. ˛x ; ˛y / ! E1s :

(15.27)

The dependence on the angular momentum operators in equation (8.62) can be neglected for the strongest electric dipole transitions. Then the contributions to the electric dipole moment are formed by a single structure: .E1a /vib  .E1s /rot :

(15.28)

Taking into account that the electric dipole moment is t-even, we find that ˛ D d .xy/ . ˛C lC C l ˛ /;

(15.29)

where d .xy/ is the real constant and ˛˙ D ˛x  i ˛y are the raising and lowering operators in number k. That is, the electric dipole transitions stipulated by the z component of the electric dipole moment are absent in this approximation. This is consistent with the rigid-molecule model in which the strongest rotational transitions are due to a constant component of the electric dipole moment along the C2 axis orthogonal to the z axis. Finally, we note that the action of operator (15.29) on the basis unit vectors does not violate the condition k D l. Clearly, the obtained description is also fully applicable to the spectrum corresponding to an arbitrary vibrational state (including the excited valence vibrations) in the rigid-molecule model. The only restriction is that each of such states is considered as isolated. However, the approach used here makes it possible to remove this restriction and extends the description to the whole set of vibrational states of interest to us (see Section 18.3).

290

Chapter 15 Nonrigid molecular systems with continuous axial symmetry groups

15.4 Conclusions We have formulated some general conclusions based on the content of this chapter. 1. Nonrigid molecular systems with the continuous axial groups appearing in a description of the geometric symmetry of internal dynamics can be divided into two types. The first type consists of the systems with linear equilibrium configurations. For these, the continuous axial groups occur even if the nonrigid motions are neglected, although taking the latter into account, of course, leads to the enlargement of this group. The second type consists of nonrigid systems with nonlinear equilibrium configurations and, correspondingly, with finite point groups. But the extended point groups that allow for nonrigid motions already become continuous axial groups. This occurs if the analysis of experimental data requires that the nonrigid transitions through the barrier corresponding to a linear configuration of the system be taken into account. 2. The finding that instead of the finite point group for a rigid system, a continuous axial group appears in the description of a nonrigid molecular system indicates radical changes in the internal dynamics when nonrigid transitions are taken into account. In particular, the finite point group leads to a 3 C 3 and the continuous axial group, a 2 C 4 division into rotational and vibrational degrees of freedom. It is exactly the rearrangement of the vibration-rotational configuration space that is here the most important consequence of taking the nonrigid transitions into account.

Chapter 16

Molecules with different isomeric forms in a single electronic state 16.1 Distorted molecular systems There is a wide class of nonrigid molecular systems, for which it is necessary to take into account the transitions between their different isomeric forms. Partially deuterated species of the methanol molecule CH2 DOH is a simple example of such a system. In the ground electronic state of the CH2 DOH species, the torsional motion of an isotopically distorted methyl top also mixes three equilibrium configurations. But now two of them are stereoisomers with the point group C1 and correspond to the position of the D nucleus on the left and right of the plane of the COH structure, and the third equilibrium configuration with the point group CS has the nucleus D in the plane of COH. That is, we are dealing with different isomeric forms of this species, which are bound by nonrigid transitions. The description of internal dynamics of the CH2 DOH species will be constructed as the distortion of the corresponding description of the isotopically symmetric CH3 OH species with energetically equivalent equilibrium configurations. This means that the invariant extended point group given by equation (9.3) and characterizing the internal dynamics with allowance for the torsional motion in the CH3 OH species is used in the same capacity for the CH2 DOH species, but already as a noninvariant group. In this connection, we will call the CH2 DOH species the distorted molecular system. That a noninvariant group appears in the CH2 DOH species description is understandable, since the operations of the Hamiltonian symmetry group cannot bind different isomeric forms of the molecule. Interestingly, this approach made it possible for the first time to solve the problem of nuclear statistical weights of the energy levels of distorted systems. Although this problem is classic for the symmetry methods, its nontriviality here emphasizes the obvious fact that the nuclear statistical weights of stationary states of the isomers are generally different. The proposed approach will be demonstrated using a simple example of the rotational spectrum of the isotopic species NH2 D of a rigid ammonia molecule. Note that the correct answer is well known for this case. The isotopically symmetric species NH3 of the ammonia molecule was considered in Section 8.1. Passing to the distorted species NH2 D, we assume that the changes occurring in the qualitative analysis are independent of the mass ratio of the nuclei H and D and are determined by the fact that there are only two identical nuclei H instead of previously three. Then the point group C3v of the NH3 molecule is conserved after transition to the NH2 D molecule as a noninvariant group, in which the symmetry subgroup of the Hamiltonian is a point group CS D (E, /. From a matching of the groups 1D 2H and C3v , for the allowed

292

Chapter 16 Molecules with different isomeric forms in a single electronic state

coordinate multiplets we obtain 3A1 ;

9A2 ;

3E1 ;

9E2 ;

(16.1)

where the components E1 and E2 of the representation E go over to the representations A0 and A00 , respectively, when the group C3v reduces on a subgroup CS . That these components have different nuclear statistical weights is a special case of the statement that the coordinate degeneracies and nuclear statistical weights of the energy levels are determined by the symmetry group of the Hamiltonian. That is, each level of the E type splits into two sublevels. The rotational group D3 describing the actual energylevel structure also becomes a noninvariant group, and representation E of this group determines two energetically different levels. They correspond to the components E1 and E2 , which go over to the representations A and B when the group D3 reduces on its Hamiltonian symmetry subgroup C2 D (E, i). Figure 16.1 shows how a classification of the rotational levels of a rigid molecule NH3 , which is presented in Figure 8.3, is modified after the transition to its isotopic species NH2 D. The effective operators of physical quantities belong to the coordinate Young diagram Œ1  Œ2 of the 1D  2H group. Matching the groups 1D  2H and C3v , we obtain for such operators the possible symmetry types A1 and E1 in the noninvariant group C3v . Finally, a matching of the groups C3v  Ci and D3 gives, within the framework of the latter, the symmetry types A1 ;

E1

(16.2)

for the quantities that are invariant under the spatial inversion operation i and A2 ;

E2

(16.3)

for the quantities that change sign during this transformation. The occurrence of individual components of the irreducible representation E should not be regarded as a contradiction since these components are mixed only in transformations of the noninvariant (or nonexchange) type, which, simultaneously, change the position of the reflection plane of the point group CS . Thus, the symmetry types given by equation (16.2) are admissible for the effective rotational Hamiltonian in the group D3 . It is exactly the transformation of noninvariant type E1 in equation (16.2) that leads to a distortion of the picture corresponding to a symmetric top with the point group C3v . Simple construction shows that the distorted picture is completely equivalent to the description of a rigid molecule of an asymmetric top type with the point group CS . Taking also the degenerate vibronic states of the C3v group into consideration means taking into account the resonant interactions for a quasidegenerate set of states that occur when the group C3v reduces on the group CS . However, the advantages of the description scheme with the use of a distortion are small for rigid molecules. The reason for this is that in any scheme of description a

Section 16.2 The methanol molecule CH2 DOH

293

Figure 16.1. Classification of the rotational levels of the isotopic species NH2 D of a rigid molecule of ammonia.

three-dimensional group of rotations R3 in the MCS, which specifies a complete set of unit vectors jJ; ki in the rotational space, is implicitly used as a noninvariant group. Actually, the proposed approach is based on the specific role of the highly symmetric molecular system in the qualitative analysis. It is interesting that the role of such a system is probably also important in the analytical solution of the equations of motion. To confirm this, we can point to the calculations of the binding energy of various forms of four-particle mesomolecules in [80].

16.2 The methanol molecule CH2 DOH The torsional transitions between three energetically equivalent configurations of the isotopically symmetric species CH3 OH of the methanol molecule are described in Section 9.2. On passage to the CH2 DOH molecule, the extended point group G12 D c3v  cS , which allows for the torsional motion in the CH3 OH molecule, becomes a noninvariant group. Indeed, in the group c3v only the in-plane reflection that interchanges the H nuclei has a counterpart in the permutation group 1D  2H of identical nuclei of the distorted methyl top. Therefore, in the group G12 , the Hamiltonian sym-

294

Chapter 16 Molecules with different isomeric forms in a single electronic state

metry subgroup is GH D cS  cS . From a matching of the groups 1D  2H  1 and GH (the necessary correlations are given in Table 16.1) it follows that the complete coordinate wave function may relate to the following irreducible representations of the latter: 6.A0  A00 /; 18.A00  A0 /: (16.4) Matching now the groups GH and G12 , we obtain the following allowed coordinate multiplets of the CH2 DOH molecule with allowance for the torsional motion: 6.A1  A0 /;

18.A2  A0 /;

6.E1  A0 /;

18.E2  A0 /;

(16.5)

where the components E1 and E2 of the representation E go over to the representations A0 and A00 , respectively, when the group c3v reduces on its Hamiltonian symmetry subgroup cS . The actual energy-level structure is described by the rotational subgroup H6 of the G12  Ci group. This subgroup is also a noninvariant group, and its representation E determines two energetically different levels. Figure 16.2 shows how the energy-level classification of a nonrigid molecule CH3 OH, which is presented in Figure 9.4, is modified after the transition to its isotopic species CH2 DOH. Table 16.1. Correlation tables for the asymmetric species CH2 DOH of the methanol molecule. 1D  2H  1

cS  cS

cS  cS

Œ1  Œ2  Œ1

A0  A0

A0  A0

A1  A0 ,

E  A0

Œ1  Œ12   Œ1

A00  A0

A00  A0

A2  A0 ,

E  A0

c3v  cS  Ci

H6

´ E

c3v  cS

A1

.A1  A0 /.C/ ;

.A2  A0 /./

A2

.A2  A0 /.C/ ;

.A1  A0 /./

E1

.E1  A0 /.C/ ;

.E2  A0 /./

E2

.E2  A0 /.C/ ;

.E1  A0 /./

μ .E  A0 /.˙/

We note that the nuclear statistical weights for the group CS representations are absent in Figure 16.2 since there are more than one set of the weights in this approximation, which is related with the realization of different point groups for isolated equilibrium configurations of the CH2 DOH species. In all, there are three independent configurations, namely, a reference one and two configurations obtained from it by using the transformations c31  E and c32  E of the H3 group, which specify the torsional motion. We choose the configuration with the point group CS as the reference configuration. The other two configurations are stereoisomers with the point

Section 16.2 The methanol molecule CH2 DOH

295

Figure 16.2. Classification of the torsion-rotational levels of the isotopic species CH2 DOH of the methanol molecule.

group C1 . Naturally, the qualitative description should not depend on the magnitude of the torsional-motion barrier. Therefore, the nuclear statistical weights in the limit of an impermeable barrier should go over into well-known expressions determined by three rigid molecules. Since the energy levels of the stereoisomers coincide, it is easy to see that this limiting process is correct. The basis torsion-rotational functions of the CH2 DOH molecule are constructed in a similar way as the CH3 OH molecule, for which the decomposition of these functions into irreducible representations of the H6 group is shown in Table 9.3. We only note that in this case, different energy levels belong to the components E1 and E2 of the representation E, and the definition of these components depends on the chosen symmetry plane of the Hamiltonian in the c3v group. This plane is delocalized during

296

Chapter 16 Molecules with different isomeric forms in a single electronic state

the torsional motion, but the wave function space, and the operators of physical quantities specified in this space, can be written out for any position of the plane. It is most convenient to choose the position  .yz/ corresponding to the reference configuration. Then in the H6 group the Hamiltonian symmetry subgroup is C2 D .E; C2.x/ /. In this .x/

.x/

case, C2 E1 D E1 and C2 E2 D E2 , which is consistent with the definition of the components in Table 9.3. The effective operators of physical quantities belong to the coordinate Young diagram [1]  [2]  [1] of the group 1D  2H  1 . From a matching of the groups 1D  2H  1 and G12 , we obtain the possible types of symmetry A1 and E1 for such operators in the noninvariant group G12 . Then, a matching of the groups G12  Ci and H6 gives, within the framework of the latter, the transformation types A1 , E1 and A2 , E2 for the quantities that are invariant under the spatial inversion operation i and that change sign during this transformation, respectively. The torsional parts of the operators of physical quantities are constructed as for the CH3 OH molecule on the basis of the operator of coordinate spin e. The symmetry properties of the complete set of independent torsional operators are given in Table 9.4. For the torsion-rotational Hamiltonian in the H6 group, the admissible types of transformations are A1 and E1 . It is precisely the presence here of the noninvariant type E1 that leads to the distortion of the CH3 OH molecule description. Since the rotational operators belong only to the representations A1 and A2 in the H6 group, the following four torsion-rotational structures are possible for the effective Hamiltonian: .A1 /tors  .A1 /rot ;

.A2 /tors  .A2 /rot ;

.E1 /tors  .A1 /rot ;

.E2 /tors  .A2 /rot :

(16.6)

Thus, in comparison with the structures given in equation (9.15) for the torsion-rotational Hamiltonian of the CH3 OH molecule, there are two additional structures stipulated by the fact that the group H6 is noninvariant now. As a result, we obtain the following general expression for the torsion-rotational Hamiltonian of the CH2 DOH molecule: HD

1 X 

.A/ .B3 / .B1 / H2n C H2nC2 C H2nC1

nD0

.B2 / .B1 / .B2 / .A/ .B3 /  ; C H2nC1 C H2nC2 C H2nC2 C H2nC3 C H2nC1

(16.7)

where the superscript determines the type of the irreducible representation of the D2 group, according to which the used combinations of the angular momentum components in the MCS are transformed, and the subscript specifies the total power of the combinations of these components. In writing equation (16.7) it was taken into account that the Hamiltonian is t-even. The first four types of contributions are also present in equation (9.17) for the isotopically symmetric species of methanol. However, the sets

Section 16.3 The ethane molecule CH2 D–CH2D

297

of spin operators become broader for these contributions: 2 2 /; C e cO D c .1/ I C c .2/ e32 C c .3/ .eC C e / C c .4/ .eC

dO D d .1/e3 C d .2/ Œe3 ; eC C e C : For the remaining four types of contributions in equation (16.7) we obtain X .B1 / 2t C2 D i fO2p;2s;2t C2 J 2p Jz2s .JC  J2t C2 /; H2nC2

(16.8)

(16.9)

p;s;t .B /

2 H2nC2 D

X

2t C1 fO2p;2sC1;2t C1J 2p Jz2sC1 .JC C J2t C1 /;

(16.10)

2t C2 i gO 2p;2sC1;2t C2 J 2p Jz2sC1 .JC  J2t C2 /;

(16.11)

2t C1 gO 2p;2s;2t C1J 2p Jz2s .JC C J2t C1 /;

(16.12)

p;s;t .A/ D H2nC3

X

p;s;t .B3 / H2nC1

D

X

p;s;t

where p C s C t D n for a given n. Due to the torsional motion, the parameters fO and gO are spin operators of the form 2 2 fO D if .1/.eC  e/ C if .2/.eC  e /;

gO D ig .1/ Œe3 ; eC  e C :

(16.13)

To determine the action of the Hamiltonian on the basis unit vectors in Table 9.3, it is useful to take into account the following rules of reduction of the H6 group on its Hamiltonian symmetry subgroup C2 : A1 ; E1 ! A;

A2 ; E2 ! B:

(16.14)

It follows immediately from these rules that the energy matrix of the Hamiltonian (16.7) for a given value of the quantum number J decomposes into a direct sum of the matrices corresponding to the symmetry types (A1 , E1 / and (A2 , E2 / in the H6 group.

16.3 The ethane molecule CH2 D–CH2 D The case of an isotopic species CH2 D–CH2 D of the ethane molecule, where the identity of the distorted methyl tops should be taken into account in the description of its torsional transitions, is much more interesting. An isotopically symmetric molecule of ethane C2 H6 was considered in Section 9.3. On passage to the CH2 D–CH2 D species, the extended point group G72 D .c3v  c3v / ^ CI becomes noninvariant. To single out the Hamiltonian symmetry subgroup GH in the G72 group, we take into account that in the noninvariant group c3v of the methyl top, only the in-plane reflection that

298

Chapter 16 Molecules with different isomeric forms in a single electronic state

interchanges the H nuclei has a counterpart in the permutation group of the identical nuclei 2H  1D . This plane is delocalized during the torsional motion. However, the wave function space, and the operators of physical quantities specified in this space, can be written for any positions of the plane. The form, in which the nucleus D of both methyl tops lies in the yz plane in Figure 8.8, is the most convenient choice. Then GH can be written in the form GH D .cS  cS / ^ CI ;

(16.15)

where cS D .E;  .yz/ /. Division of the elements into classes and the table of characters of the GH group are presented in Table 16.2. From a matching of the groups 2D  4H and GH (the necessary correlations are given in Table 16.3) it follows that the complete coordinate wave function may relate to the following irreducible representations of the latter: 6a1 ;

3a2 ;

45a3 ;

36a4 ;

27e

(16.16)

Matching now the groups GH and G72 , we obtain the following allowed coordinate multiplets of the CH2 D–CH2 D species with allowance for the torsional motion: 6.A1 ; T2;1 ; T4;1 /; 36.A4 ; T1;2 ; T3;2 /;

3.A2 ; T2;2 ; T3;1 /;

45.A3 ; T1;1 ; T4;2 /;

(16.17)

27.E; T1;3 ; T2;3 ; T3;3 ; T4;3 /:

The components Ti;k of the four-dimensional irreducible representation Ti (two onedimensional with the indices k D 1, 2 and one two-dimensional with the index k D 3) are defined in Table 16.3 by the condition of the transition to irreducible representations of the GH subgroup when the group G72 reduces to it. Thus, in principle, each level of the Ti type splits into three sublevels. However, since the nonrigid motion is of the torsional type, the actual structure of the levels is described by the noninvariant rotational group B6  D3 , in which the Hamiltonian symmetry subgroup is H2  C2 , .x/ where C2 D .E; C2 /. Figure 16.3 shows how the energy-level classification of a nonrigid molecule C2 H6 , which is shown in Figure 9.5, is modified after the transition

Table 16.2. Division of the elements into classes and the table of characters for the Hamiltonian symmetry group of a nonrigid molecule CH2 D–CH2 D. E

E E



 

r1

E   E

r2

r3

I .  /I D

.x/ C2

(E  /I (  E)I

GH

E



r1

r2

r3

a1 a2 a3 a4 e

1 1 1 1 2

1 1 1 1 –2

1 1 –1 –1 0

1 –1 1 –1 0

1 –1 –1 1 0

299

Section 16.3 The ethane molecule CH2 D–CH2D

Table 16.3. Correlation tables for the asymmetric species CH2 D–CH2D of the ethane molecule. 2  4

GH

GH

G72

Œ2  Œ1  Œ2  Œ212 

a3 a4 C e

a1

A1 , T2;1 , T4;1

Œ2  Œ22  Œ12   Œ14  [12 ]  [212 ]

a1 C a3 a4 a3 C e

a2

A2 , T2;2 , T3;1

a3

A3 , T1;1 , T4;2

a4

A4 , T1;2 , T3;2

Œ1   Œ2 

a2 C a4

e

E, T1;3 , T2;3 , T3;3 , T4;3

4

2

2

H2  C2

GH  Ci

As

a1.C/ ; a3.C/ ; e ./

Aa Bs

a1 ; a3 ; e .C/ ./ ./ a2 ; a4 ; e .C/

Ba

a2 ; a4 ; e ./

./

./

.C/

.C/

to its isotopic species CH2 D–CH2 D. The nuclear statistical weights for representations of the D3d group are absent here since, as for the CH2 DOH molecule, there are more than one set of the weights in this approximation. Indeed, in total, there are three independent configurations, namely, one reference and two configurations obtained from it by using the transformations c31  c32 and c32  c31 of the B3 group, which specify the torsional motion. The reference configuration has a point group C2h, and .p/ two other configurations are stereoisomers with a single element of symmetry C2 , where p is the symmetry axis of the DCCD structure. Degeneracy of the levels in the H6  D3 group is completely removed, and the resulting splittings are denoted using the components E1 and E2 of the representation E of the groups B6 and D3 , and these components go over to symmetric and antisymmetric representations of the groups H2 and C2 , that is, .x/

I i.or C2 /E1 D E1 ;

.x/

I i.or C2 /E2 D E2 :

The symmetry of the Hamiltonian lowers significantly in the case of a transition to the distorted species CH2 D–CH2 D of the ethane molecule. However, the nuclear statistical weights of the torsion-rotational levels have not one, as in the case of the CH2 DOH molecule, but two values, namely, 78 and 66 (they are formed by summation of these weights for the group G72 multiplets corresponding to one torsion-rotational representation of the group B6  D3 /. It is easy to show that in the limit of an impermeable torsion barrier, the nuclear statistical weights of the energy levels go over to the well-

300

Chapter 16 Molecules with different isomeric forms in a single electronic state

Figure 16.3. Classification of the torsion-rotational levels of the isotopic species CH2 D–CH2D of the ethane molecule.

known expressions determined by a set of three rigid molecules, two of which are stereoisomers. The basis torsion-rotational functions of the CH2 D–CH2 D molecule and a complete set of independent spin operators defined in the torsional space are similar to the C2 H6 molecule. It should only be taken into account that each component of the degenerate representations for the basis functions now determines an individual energy level. Effective operators of the coordinate physical quantities belong to an identity representation of the group GH . A matching of the groups GH  Ci and B6  D3 gives,

301

Section 16.3 The ethane molecule CH2 D–CH2D

within the framework of the latter, the symmetry types A1  A1 ;

A1  E1 ;

E1  A1 ;

E1  E1

(16.18)

for the physical quantities that are invariant under the operation i and A2  A1 ;

A2  E1 ;

E2  A1 ;

E2  E1

(16.19)

for the quantities that change sign under the action of i. According to equation (16.18), a full expression for the effective torsion-rotational Hamiltonian is written in the following form: 2 2 C e ; iŒe3 ; eC  e C /  .A1 ; E1 /: H D .I; e32 ; eC C e ; eC

(16.20)

It is not difficult to unwind this expression in full. It should only be taken into account that the Hamiltonian is t-even. It is exactly the presence of noninvariant contributions in equation (16.20) that leads to a complete removal of the degeneracy of the levels in the B6  D3 group. In determining the action of the Hamiltonian on the basis unit vectors in Table 9.6, it is useful to take into account, as in equation (16.14), the rules of reduction of the B6  D3 group to its Hamiltonian symmetry subgroup H2  C2 . It follows immediately from these rules that there are only four different types of energy matrices (two for the group B6 and two for the group D3 /, into a direct sum of which the matrix of the Hamiltonian decomposes for a given value of the quantum number J. Transformation types (16.19) are admissible for the effective operator of the electric dipole moment. Therefore, for component ˛ of the electric dipole moment on the ˛ axis of the FCS, one can write a general expression in the following form: 2 2  e /; Œe3 ; eC C e C ; e3 / ˛ D .i.eC  e /; i.eC

. ˛z .A2 ; E2 / C ˛y .A2 ; E2 / C ˛x .A1 ; E1 //:

(16.21)

Here, the symmetry types of the D3 group specify the admissible combinations of the angular momentum components in the MCS. As an example, we write out the terms not related with a fairly weak rotational distortion of the molecule. Since such terms are of the type A1 for the D3 group, we have 2 2  e /; ˛ D ˛x  iŒd .1/ .eC  e/ C d .2/ .eC

(16.22)

where d .k/ are the real phenomenological constants. Since the electric dipole moment is t-even, the t-odd operators Œe3 ; eC C eC and e3 are absent in equation (16.22). Purely torsional motion contributes only to the x component of the electric dipole moment. The reason is that the chosen reference configuration does not have the electric dipole moment, and the configurations of two stereoisomers obtained from the reference configuration by using the torsional transformations c31  c32 and c32  c31 of the B3 group have a nonzero component of the electric dipole moment exactly along the x axis.

302

Chapter 16 Molecules with different isomeric forms in a single electronic state

16.4 The ethanol molecule CH3 CH2 OH Up to now, the nonrigid motions between the nonequivalent equilibrium configurations in a single electronic state were analyzed for the molecules whose different isomers appeared due to isotopic distortion. However, there is a wide class of molecules in which the existence of such a motion is not associated with isotopic distortion. The ethanol molecule CH3 CH2 OH is a classic representative. It is well known that there are three isomers in the ground electronic state of this molecule. One isomer is generally called a trans conformer and two other, gauche conformers [52]. The equilibrium trans configuration shown in Figure 16.4 belongs to the point group CS D .E;  .yz/ /. In the transition to the energetically equivalent equilibrium gauche configurations belonging to the point group C1 , OH hydroxyl goes out of the symmetry plane to the left and to the right by rotation of hydroxyl by an angle of approximately 120°. The torsional barrier having an average height of about 400 cm1 for the trans $ gauche transition is 10% smaller than for the gauche $ gauche transition because of the energy nonequivalence of the trans and gauche configurations ( E  40 cm1 / [74]. Analysis of very numerous experimental data for the torsion-rotational transitions in the ground electronic state shows [75] that resonant interactions between the three conformers play a large role in the description.

Figure 16.4. Equilibrium configuration of the trans conformer of the ethanol molecule CH3 CH2 OH (the nuclei H1 , C1 , C2 , O, and H6 lie in the symmetry plane yz).

Consider first the ethanol molecule as a set of three rigid systems corresponding to individual conformers. For the primary isotopes of the carbon 12 C and oxygen 16 O nuclei, the spin is equal to zero. In this case, the permutation group of identical nuclei of the molecule can, without loss of generality, be written in the form 3 2 1 , where 3 ; 2 , and 1 are the permutation groups of the hydrogen nuclei in the structures CH3 , CH2 , and OH, respectively. Matching the group 3  2  1 with the point group CS of the trans conformer (the necessary correlations are given in Table 16.4), we have

303

Section 16.4 The ethanol molecule CH3 CH2 OH

Table 16.4. Correlation tables for the trans conformer of the ethanol molecule CH3 CH2 OH. 3  2  1

CS

D2

C2

CS  Ci

Œ21  Œ2  Œ1

A0 C A00

A; B3

A

A0.C/ ; A00./

Œ21  Œ12   Œ1 Œ13   Œ2  Œ1 Œ13   Œ12   Œ1

A0 C A00 A00 A0

B1 ; B2

B

A0./ ; A00.C/

for it the following allowed coordinate multiplets: 40A0 ;

24A00 :

(16.23)

Symmetry types of the multiplets are formed from the symmetry types of the zeroapproximation wave functions on the basis of matching: .mult/CS Ci D .el /CS  .vib /CS  .rot.-in/CS Ci :

(16.24)

6

The admissible representations rot.-in for a given rotational representation of the D2 group of a rigid asymmetric top are determined from a correlation of the groups D2 .x/ and CS  Ci through their common rotational subgroup C2 D .E; C2 D  .yz/ i/. As a result, we obtain a classification of the energy levels in an arbitrary vibronic state, which is shown for el vib D A0 in Figure 16.5. The inversion doublets are not split, and the actual energy-level structure is therefore described by the rotational subgroup C2 of the CS  Ci group. The effective operator of any physical quantity describing the internal motion of a trans conformer belongs to the representation A0 of the CS group. As a consequence, in the group C2 we obtain the representations A and B related to the quantities that are invariant under the operation i and that change sign during this transformation, respectively. The effective Hamiltonian is of the type A and the effective operator of the electric dipole moment is of the type B. For the electric dipole selection rules, A $ B. For a rigid gauche conformer, the only coordinate multiple 64A of the C1 group is allowed. The energy-level classification is trivial. In any vibronic state, each rotational level is an unsplit inversion doublet of the form 64A.˙/. It is clear that the ethanol molecule has a stereoisomer in the gauche configuration. It is obvious that for a unified description of the conformers, the extended point group should be noninvariant since the symmetry elements of the Hamiltonian cannot bind the nonequivalent configurations. Here, this group has a fairly simple form: F12 D c3v  cS ;

(16.25)

where the group c3v characterizes hydroxyl OH, and one of planes of this group coincides with the yz plane, and the group cS D .E;  .yz/ / is a point group of the

304

Chapter 16 Molecules with different isomeric forms in a single electronic state

Figure 16.5. Classification of the rotational energy levels of a trans conformer in the type A0 vibronic state.

CH3 CH2 structure. In the c3v group, noninvariant elements are included in the subgroup c3 , which specifies the torsional motion of hydroxyl. It is clear that the angle of rotation between the positions of hydroxyl in the neighboring configurations is different from 2=3. Also, the nonequivalent configurations have different lengths of the bonds. However, all this can be considered as a consequence of the distortion of the molecular system with three independent nonequivalent configurations bound by the torsional motion. The Hamiltonian symmetry subgroup in the c3v group is cS0 , in the reflection plane of which hydroxyl is located. Therefore, the Hamiltonian symmetry subgroup in the F12 group has the form FH D cS0  cS :

(16.26)

Naturally, the hydroxyl plane is delocalized during the torsional motion, but the wave function space, and the operators of physical quantities specified in this space, can be written for any reference position of this plane. It is convenient to choose the reference position in the plane  .yz/ . Matching the groups 3 2 1 and FH , for a complete coordinate wave function we obtain the following irreducible representations of the FH group: 40.A0  A0 /;

24.A0  A00 /:

(16.27)

From a matching of the groups FH and F12 , for the allowed coordinate multiplets of a nonrigid molecule we have 40.A1  A0 /;

40.E  A0 /;

24.A1  A00 /;

24.E  A00 /:

(16.28)

305

Section 16.4 The ethanol molecule CH3 CH2 OH

Here, it should be taken into account that the two-dimensional representation E of the c3v group in the case of reduction on its Hamiltonian symmetry subgroup cS0 decomposes into one-dimensional representation. Therefore, the components of such a representation, which are given by the reduction rules E1 ! A0 ;

E2 ! A00 ;

(16.29)

correspond to the individual nondegenerate levels with the same nuclear statistical weights. The minimum group that includes the point groups of equilibrium configurations of all conformers is CS . The construction of a classification starts with the stage shown in equation (16.24). However, definite statistical weights can no longer be ascribed to the irreducible representations of the CS group. A single set of statistical weights is absent in this approximation because different point groups are realized for isolated equilibrium configurations. Then, matching the groups CS and F12 (the necessary additional correlations for a nonrigid molecule are given in Table 16.5), we obtain an energy-level classification, which is shown in Figure 16.6 for the type A0 vibronic state. Table 16.5. Correlation tables for a nonrigid molecule of ethanol CH3 CH2 OH.

CS

c3v  cS

C2

H6

A0 A00

A1  A0 ; A2  A00 ; E  A0 ; E  A00 A1  A00 ; A2  A0 ; E  A0 ; E  A00

A B

A1 ; E A2 ; E

.c3v  cS /  Ci

H6 A1 A2 E1 E2

.A1  A0 /.C/ ; .A1  A0 /./ ; .E1  A0 /.C/ ; .E1  A0 /./ ;

.A2  A0 /./ ; .A2  A0 /.C/ ; .E2  A0 /./ ; .E2  A0 /.C/ ;

.A1  A00 /./ ; .A1  A00 /.C/ ; .E1  A00 /./ ; .E1  A00 /.C/ ;

.A2  A00 /.C/ .A2  A00 /./ .E2  A00 /.C/ .E2  A00 /./

Since the nonrigid motion is of the torsional type, the actual structure of the levels is described by the rotational subgroup H6 of the F12  Ci group. Group H6 has the form (9.9) and is a noninvariant group with the Hamiltonian symmetry subgroup C2 . Therefore, the components of the irreducible representation E of the H6 group, which are given by the reduction rules on a subgroup C2 , E1 ! A;

E2 ! B

correspond to different nondegenerate levels.

(16.30)

306

Chapter 16 Molecules with different isomeric forms in a single electronic state

Figure 16.6. Energy-level classification for a nonrigid molecule of ethanol CH3 CH2 OH in the type A0 vibronic state.

It is easy to show that in the limit of an impermeable torsional barrier, the nuclear statistical weights of the energy levels go over to the well-known expressions determined by a set of three rigid molecules. It should only be taken into account that in this limit, the energy levels of two gauche conformers and their two independent stereoisomers coincide. The algebraic model for describing the spectrum of the ethanol molecule CH3 CH2 OH with allowance for the torsional motion of hydroxyl OH is fully similar to the model of an isotopic species CH2 DOH of the methanol molecule with allowance for the torsional motion of an isotopically distorted methyl top CH2 D, which is considered in Section 16.2. However, besides the torsional motion of hydroxyl, the ethanol molecule also has a torsional motion of the methyl top CH3 , with the barrier in the ground vibronic state equal to 1170 cm1 for the trans conformer and 1330 cm1 for the gauche conformer. This leads to the actually observed additional splitting of the

Section 16.5 The cyclobutane-1,1-d2 molecule

307

energy levels about 1 MHz [74]. Clearly, taking account of the second torsional motion considerably complicates the analysis. However, the methods discussed in this book make it easy to cope with this problem [21].

16.5 The cyclobutane-1,1-d2 molecule In Section 10.2, we analyze an isotopically symmetric species of cyclobutane C4 H8 , and this analysis was the basis for the construction of a unified algebraic description of the entire band of pseudorotation excitations in the ground electronic state. It is clear that this band is most interesting. However, vibrational electric dipole transitions of this band lie in the far IR range and are weak, while the rotational transitions associated with the constant component of the electric dipole moment are absent. Therefore, microwave studies were performed for the rotational spectra of the cyclobutane-d1 [87] and cyclobutane-1,1-d2 molecules [30], in which the small constant component of the electric dipole moment is due to the replacement of one or two H nuclei, respectively, by deuterium nuclei. The second variant is more interesting since here, as in the case of a symmetric combination, the splitting of the levels into doublets is entirely due to the pseudorotation effect. The point is that in this case pseudorotation mixes the equivalent configurations. Nevertheless, the cyclobutane-1,1-d2 molecule is included in this chapter since its description is constructed as a distorted description of the most symmetric isotopic species. Of course, the considered approach also applies to the deuterium-substituted species of a cyclobutane molecule for which the configurations mixed by pseudorotation are nonequivalent.

Figure 16.7. Equilibrium configuration of the cyclobutane-1,1-d2 molecule.

The equilibrium configuration of a cyclobutane-1,1-d2 molecule is shown in Figure 16.7. The replacement of two H nuclei by D nuclei leads to the destruction of the point group D2d to the group CS D .E;  .xz/ /. A description of the internal dynamics of a rigid combination 1,1-d2 will be constructed as a distorted description of an

308

Chapter 16 Molecules with different isomeric forms in a single electronic state

Table 16.6. Correlation table for the cyclobutane molecule -1,1-d2 . 2.D/  6.H /

CS

D2d

CS

D4

C2

Œ2  Œ16 ; Œ12   Œ16 

A0

A1 ; B2

A0

A1 ; B1

A

A2 ; B1

00

A2 ; B2

B

E

ACB

Œ2  Œ21 ; Œ1   Œ21  4

2

Œ2  Œ2 1 ; Œ1   Œ2 1  2 2

2

2 2

Œ2  Œ2 ; Œ1   Œ2  3

2

3

0

00

0

00

0

00

3A C 2A

4

5A C 4A 3A C 4A

E

A 0

A CA

00

2.D/  6.H /

C2v

D40  CI

C2v

H2  D40

D2

Œ2  Œ16 

B2

A1g ; B1g

A1

A1s ; B1s

A

Œ2  Œ21 

2A1 C A2 C B1 C B2

A1u; B1u

A2

A1a ; B1a

B3

Œ2  Œ22 12 

A1 C 2A2 C 2B1 C 4B2

A2g ; B2g

B1

A2s ; B2s

B1

Œ2  Œ23 

3A1 C A2 C B1

A2u; B2u

B2

A2a ; B2a

B2

Œ1   Œ1 

A1

Eg

B2 C A2

Es

B2 C B3

Œ1   Œ21 

A1 C A2 C B1 C 2B2

Eu

A1 C B1

Ea

A C B1

Œ1   Œ2 1 

4A1 C 2A2 C 2B1 C B2

Œ1   Œ2 

A2 C B1 C 3B2

4

2

2

2

6

4

2 2

2

3

isotopically symmetric rigid combination. That is, the point group D2d will be preserved as a noninvariant group with the Hamiltonian symmetry subgroup CS . For the primary isotope of the carbon 12 C nucleus, whose spin is equal to zero, it suffices to .D/ .H / use the permutation group of identical nuclei in the form 2  6 . Matching this group through the Hamiltonian symmetry subgroup CS with the group D2d (the necessary correlations are given in Table 16.6), for the allowed coordinate multiplets we obtain 360.A1 ; B2 ; E1 /; 216.A2 ; B1 ; E2 /: (16.31) Since the representation E of the group D2d decomposes into one-dimensional ones in the case of reduction of the group D2d on a subgroup CS , the components E1 and E2 of this representation, which are given by the reduction rules E1 ! A0 ;

E2 ! A00 ;

(16.32)

correspond to the individual nondegenerate levels. The further construction of the energy-level classification is similar to that for an isotopically symmetric combination of the cyclobutane molecule. It should only be taken into account that the group D4 , which determines the actual structure of the energy levels, is already a noninvari-

309

Section 16.5 The cyclobutane-1,1-d2 molecule .y/

ant group with the Hamiltonian symmetry subgroup C2 D .E; C2 D  .xz/i/. When the group D4 is reduced to this subgroup, the representation E decomposes into onedimensional representations, and the components E1 and E2 given by the reduction rules (16.33) E1 ! A; E2 ! B correspond to the individual nondegenerate levels. Figure 16.8 shows how the energylevel classification for el vib D A1 of a rigid molecule of cyclobutane C4 H8 , which is presented in Figure 10.3, is modified upon transition to isotopic combination 1,1-d2.

Figure 16.8. Energy-level classification for a rigid molecule of cyclobutane -1,1-d2 in the type A1 vibronic state.

The effective operator of any physical quantity belongs to the coordinate Young .D/ .H / scheme Œ2  Œ6 of the 2  6 group. From a matching of this group with the group D2d it follows that such operator belongs to the symmetry types A1 ; B2 and E1 of the D2d group. Then, from a matching of the groups D2d  Ci and D4 we have, within the framework of the latter, the transformation types A1 ;

B1 ;

E1

(16.34)

310

Chapter 16 Molecules with different isomeric forms in a single electronic state

for the quantities that are invariant under the operation i and the transformation types A2 ;

B2 ;

E2

(16.35)

for the quantities that change sign under the action of i. The effective Hamiltonian transforms according to equation (16.34) and the effective operator of the electric dipole moment, according to equation (16.35). The presence of the noninvariant types in equation (16.34) distorts the level pattern corresponding to the isotopic species of cyclobutane C4 H8 . The energy matrix decomposes into a direct sum of the matrices corresponding to only two sets of symmetry types: .A1 ; B1 ; E1 /;

.A2 ; B2 ; E2 /;

(16.36)

and the electric dipole selection rules have the form .A1 ; B1 ; E1 / $ .A2 ; B2 ; E2 /:

(16.37)

Component ˛ of the operator of the electric dipole moment in a nondegenerate vibronic state has the form (8.8). The terms not stipulated by the rotational distortion of the molecule give the main contribution. Taking into account the decomposition of the direction cosines into transformation types of the D4 groups,

˛z ! A2 ;

˛y ! E1 ;

˛x ! E2 ;

(16.38)

we obtain two such terms, ˛ D ˛x d .x/ C ˛z d .z/ ;

(16.39)

as it should be. Indeed, it is apparent from simple geometric considerations that the isotopic combination 1,1-d2 in the equilibrium configuration has the electric dipole moment components directed along the x and z axes. It is easy to show that the picture with the distortion introduced is equivalent to describing the 1,1-d2 species as a rigid molecule of the asymmetric top type with the point group CS . The scheme with distortion offers only a few technical advantages. For example, by considering degenerate vibronic states in the D2d group, the resonant interactions can explicitly be taken into account for a set of quasidegenerate states occurring upon transition to the group CS . The situation changes radically in the analysis of a nonrigid species 1,1-d2 . Here, using the scheme with distortion is of principle since it permits one to determine the very concept of nonrigid motion as of a counterpart of this motion in the symmetric species. The extended point group D4h D D40  CI , which allows for pseudorotation, upon transition to the 1,1-d2 species becomes a noninvariant group with the Hamiltonian symmetry subgroup in the form .x/

C2v D .E; C2 ;  .xy/ ; N .xz/ /

(16.40)

311

Section 16.5 The cyclobutane-1,1-d2 molecule

Part of the transformations of the exchange type change during the pseudorotation, and the configuration shown in Figure 16.7 will be chosen as reference. From a matching .D/ .H / of the groups 2 6 and C2v it follows that a complete coordinate wave function relates to the following irreducible representations of the latter: 168A1 ;

108A2 ;

108B1 ;

192B2 :

(16.41)

Matching the groups C2v and D4h, we obtain the allowed coordinate multiplets of a nonrigid molecule in the form 168.A1g ; B1g ; Eu;1 /;

108.A1u; B1u; Eg;2 /;

108.A2g ; B2g ; Eu;2 /;

192.A2u; B2u; Eg;1 /:

(16.42)

When the group D4h is reduced to a subgroup C2v the two-dimensional representations decompose into one-dimensional ones. Therefore, their components given by the relation  .xz/ E1 D E1 ;  .xz/E2 D E2 (16.43) correspond to the individual nondegenerate levels. The further construction of the classification is similar to an isotopically symmetric species of the cyclobutane molecule. It should only be taken into account that the group H2  D40 , which determines the actual structure of the levels of a nonrigid molecule, becomes a noninvariant group with the Hamiltonian symmetry subgroup in the form .z/

D2 D .E; C2

.y/

D  .xy/ i; C2

.x/

D  .xz/i; C2 /

(16.44)

When the group H2  D40 is reduced to its subgroup D2 , the two-dimensional representations decompose into one-dimensional ones. Therefore, their components given by the relations C2.y/ E1 D E1 ; C2.y/E2 D E2 (16.45) correspond to the individual nondegenerate levels. Figure 16.9 shows how the energylevel classification for el  vib D A1 of a nonrigid molecule of cyclobutane C4 H8 , which is presented in Figure 10.4, is modified upon transition to its isotopic species 1,1-d2 . It is easily seen that in the group H2  D40 , the transformation types A1s ;

B1s ;

Ea;1

(16.46)

are admissible for the physical quantities that are invariant under the operation i and the transformation types (16.47) A1a ; B1a ; Es;2 are admissible for the physical quantities that change sign under the action of i. The effective Hamiltonian transforms according to equation (16.46) and the effective operator of the electric dipole moment, according to equation (16.47). The presence of the

312

Chapter 16 Molecules with different isomeric forms in a single electronic state

noninvariant types in equation (16.46) distorts the level pattern corresponding to the isotopic species of the cyclobutane molecule C4 H8 . The energy matrix decomposes into a direct sum of the matrices corresponding to four sets of symmetry types: .A1s ; B1s ; Ea;1 /;

.A2s ; B2s ; Ea;2 /;

.A2a ; B2a ; Es;1 /;

.A1a ; B1a ; Es;2 /; (16.48)

and the electric dipole selection rules have the form .A1s ; B1s ; Ea;1 / $ .A1a ; B1a ; Es;2 /;

.A2s ; B2s ; Ea;2 / $ .A2a ; B2a ; Es;1 /: (16.49) Pseudorotation in the species 1,1-d2 of a cyclobutane molecule, as well as in its symmetric isotopic species, is specified by the elements C41 and C43 . The only difference is that now these elements are of the nonexchange type. This fact almost does not matter in the construction of the function space for describing the excitation spectrum of pseudorotation. It should only be taken into account that the basis unit vectors of this space, which belong to different components, given by equations (16.45), of the two-dimensional representations Es and Ea of the H2  D40 group, belong to different sets of symmetry types (16.48). The complete set of self-adjoint pseudorotational operators, which is required for the construction of effective operators of physical quantities, is similar to the case of a symmetric isotopic species. The type A1s contributions to the effective Hamiltonian have already been constructed for the symmetric isotopic species and have the form (10.18). The type B1s contributions are formed by the structures .A1s /sp  .B1s /rot;

.B1s /sp  .A1s /rot;

(16.50)

and a full expression for these contribution can be written in the following form: 2 2 2 2 2 2 4 4 C e ; Œe32 ; eC C e C ; iŒe3 ; eC  e C /  A1 C .I; e32 ; e34 ; eC C e /  B1 ; H2 D .eC (16.51) where the rotational operators are specified by the representations of the D40 group. Finally, the type Ea;1 contributions are formed by the structures

.A2a /sp  .Es;1 /rot ;

.B2a /sp  .Es;1 /rot ;

(16.52)

which leads to an expression of the form 4 4 2 2 2 2 2 2 e /; i.eC e /; iŒe32 ; eC e C ; Œe3 ; eC Ce C ºE1 : (16.53) H3 D ¹e3 ; e33 ; i.eC

In the case of the formation of the symmetry types, specified in equations (16.46) and (16.47) by the components of the representations Es and Ea , it is required to correctly choose the index of the component of the rotational part in structures of equation (16.52) type. Since the components of the representations Es and Ea are specified by the transformations of the D2 group presented in equation (16.44), the easiest way to

Section 16.5 The cyclobutane-1,1-d2 molecule

313

Figure 16.9. Energy-level classification for a nonrigid molecule of cyclobitane-1,1-d2 in the type A1 vibronic state.

check the index is to use the table of correlation between the representations of the groups H2  D40 and D2 . For example, .B2a /sp  .Es;1 /rot ! B2  B2 D A ! Ea;1 ; where the first arrow means the transition to the group D2 and the second, coming back to the group H2  D40 . It is not difficult to unwind the effective Hamiltonian as a series of perturbations due to the rotational distortion. The resulting Hamiltonian automatically includes all interactions of the pseudorotational and rotational motions.

314

Chapter 16 Molecules with different isomeric forms in a single electronic state

The component ˛ of the electric dipole moment operator can be written in the form (8.62), where the parameters dO .i / are now the pseudorotational operators. The direction cosines ˛i correspond to the following types of transformation of the H2  D40 group: (16.54)

˛z ! A2s ; ˛y ! Es;1 ; ˛x ! Es;2 : In the further analysis, we focus on the electric dipole transitions not stipulated by the rotational distortion of the molecule. Then the type A1a contributions are formed by the only structure (16.55) .A2a /sp  .A2s /rot ; which, with accuracy up to the operators in Table 10.3, leads to the expression 4 4 ˛ D ˛z d .z/ i.eC  e /;

(16.56)

where d .z/ is the real phenomenological constant. That is, the z component of the electric dipole moment is active and the purely rotational transitions are absent. Clearly, equation (16.56) exhausts the contributions to the electric dipole operator for a symmetric isotopic species of the cyclobutane molecule. The type B1a contributions are also formed by the only structure .B2a /sp  .A2s /rot;

(16.57)

which makes it possible to write .z/

.z/

2 2 2 2  e / C d2 iŒe32 ; eC  e C º: ˛ D ˛z ¹d1 i.eC

(16.58)

As before, only the z component of the electric dipole moment is active and the purely rotational transitions are absent. Finally, the type Es;2 contributions are formed by two structures, (16.59) .A1s /sp  .Es;2 /rot ; .B1s /sp  .Es;2 /rot ; whence .x/

.x/

.x/

.x/

4 4 ˛ D ˛x ¹d1 I C d2 e32 C d3 e34 C d4 .eC C e / .x/

.x/

2 2 2 2 Cd5 .eC C e / C d6 Œe32 ; eC C e C º:

(16.60)

Here, the x component of the electric dipole moment is active and the purely rotational transitions are present. In particular, this analysis leads to the following important conclusion: Although in the equilibrium configuration of the 1,1-d2 species the electric dipole moment components directed along the x and z axes are nonzero, the symmetry of internal dynamics forbids the rotational transitions stipulated by the z component. This is exactly the reason why despite a thorough search, these transitions have not been observed in [30].

Section 16.6 The tetrahydrofuran molecule C4 H8 O

315

Thus, an algebraic model has been constructed to describe the whole excitation spectrum of pseudorotation for the isotopic species 1,1-d2 of a cyclobutane molecule as the distortion of a similar model for its isotopically symmetric species. This explains the observed selection rules for the electric dipole microwave transitions in the isotopic species 1,1-d2 . Moreover, in the experimental data fitting the influence of the accidental resonances between the pseudorotational states on the position of the energy levels and transition intensities between them is easily taken into account in this model. The operator form of the formulation for all considered types of motion is also an important advantage of this model.

16.6 The tetrahydrofuran molecule C4 H8 O Pseudorotation in the tetrahydrofuran molecule C4 H8 O is very interesting. As the cyclopentane molecule (see Section 13.4), it has a five-membered cycle, which is nonplanar in the equilibrium configuration. Here, this cycle is formed by four carbon nuclei and one oxygen nucleus. The studies show (see [67] and references therein) that the effective interaction potential of the nuclei with four minima in one pseudorotation cycle is sufficient for experimental data analysis. Two minima correspond to the twist structures, one of which is shown in Figure 16.10. The most important feature of this structure is that the nuclei C1 and C2 are moved the same distance up and down, respectively, out of the plane of the C3 , O and C4 nuclei. The second twist structure is different in that the nucleus C1 is moved downward and the nucleus C2 , upward. Both .x/ twist structures correspond to the point group U2 D .E; U2 / and are stereoisomers. The other two energy minima correspond to the envelope structures, in which the nucleus O is moved the same distance up or down from the plane of the nuclei C1 , C2 , C3 and C4 . These structures correspond to the point group CS D .E;  .xz/ /. Thus, pseudorotation is very nontrivial. It mixes not only the stereoisomers of one of the structures, but also the energetically nonequivalent structures. Consider the tetrahydrofuran molecule as a set of rigid molecules. Begin with the twist molecule. In the case of the main isotope 12 C, it suffices to use as the permutation group of identical nuclei the group 8 . The elements of the point group U2 have counterparts only in the subgroup 2  2  2  2 . From a matching of the group U2 with this subgroup, for the allowed coordinate multiplets of a rigid twist molecule we have 136A; 120B: (16.61) These multiplets are formed from the zero-approximation wave functions on the basis of matching: .mult/U2 Ci D .el /U2  .vib /U2  .rot.-in/U2 Ci ;

(16.62)

6

where the admissible representations rot.-in for a given representation of the group D2 of a rigid asymmetric top follow from a correlation of the groups D2 and U2  Ci

316

Chapter 16 Molecules with different isomeric forms in a single electronic state

Figure 16.10. Equilibrium twist structure of the tetrahydrofuran molecule (the dashed line shows the xy plane in which the nuclei C3 , C4 and O lie).

through their common subgroup U2 . As a result, we have a classification of the energy levels in an arbitrary vibronic state, which is shown for el vib D A in Figure 16.11. The actual structure of the levels is specified by the rotational subgroup of the U2  Ci group. This subgroup just coincides with the point group, which is characteristic of the molecules with stereoisomers.

Figure 16.11. Energy-level classification for a twist molecule in the type A vibronic state.

The effective operator of any physical quantity describing the internal motion of a twist molecule belongs to the representation A of the U2 group. Since the actual structure of energy levels for the molecules with stereoisomers is specified by the point group, the behavior of the physical quantities relative to the inversion operation i does not affect the construction of their effective operators. Both the effective Hamiltonian and the electric dipole moment operator belong to the identity representation A.

Section 16.6 The tetrahydrofuran molecule C4 H8 O

317

Therefore, the electric dipole selection rules have the form A $ A;

B $ B:

(16.63)

The component ˛ of the electric dipole moment operator in an arbitrary vibronic state can be represented in the form (8.8). The terms not related with the rotational distortion of the molecule give the main contribution. There is only one such term, namely, ˛ D ˛x d .x/;

(16.64)

as it should be. Indeed, it is apparent from simple geometric considerations that the twist molecule in the equilibrium configuration has only an electric dipole moment component directed along the x axis. For the allowed coordinate multiplets of an envelope molecule, we have 136A0 ;

120A00 :

(16.65)

These multiplets are formed from the zero-approximation wave functions on the basis of the matching obtained from equation (16.62) with the group U2 replaced by CS . The admissible types rot.-in for a given representation of the D2 group follow from a correlation of the groups D2 and CS  Ci through their common subgroup U20 D .y/

.E; U2 D  .xz/ i/. The obtained energy-level classification is given for el  vib D A0 in Figure 16.12. The actual structure of the levels is specified by the group U20 , whose symmetric and antisymmetric representations are denoted a and b. It is easily seen that the physical quantities that are invariant under the operation i relate to the representation a, and the physical quantities that change sign under the action of i to the representation b. The effective Hamiltonian is of the type a, and the effective operator of the electric dipole moment is of the type b. As a consequence, the electric dipole selection rules have the form a $ b: (16.66) The terms ˛ D ˛x d .x/ C ˛z d .z/

(16.67)

give the main contribution to the expression (8.8) for the electric dipole moment. Transitions stipulated by the z component of the electric dipole moment appear additionally compared with the twist molecule. As in the cyclopentane molecule, we specify pseudorotation by a cyclic group, which in this case binds four equilibrium points: P4 D .E; C2 ; P; C2 P /:

(16.68)

In determining the action of the elements of this group, we assume that the twist structure is the initial one. The element C2 converts the initial structure into an envelope structure with the simultaneous rotation of the molecule by an angle  about the z axis.

318

Chapter 16 Molecules with different isomeric forms in a single electronic state

Figure 16.12. Energy-level classification for an envelope molecule in the type A0 vibronic state.

The element C22 D P converts the initial structure into a stereoisomer and rotates the molecule by an angle 2. The element C23 D C2 P converts the initial structure into the second envelope structure and rotates the molecule by an angle 3. Finally, the element C24 D P 2 D E returns to the initial twist structure and rotates the molecule by an angle 4. Thus, the pseudorotation cycle includes two rotations by an angle 2 about the z axis. All nontrivial elements of the P4 group (isomorphic group C4 / are of the nonexchange type. We now extend the group P4 by adding symmetry elements of the equilibrium configurations. It is easy to show that .x/

.x/

U2 C2 U2

D C2 P;

 .xz/ C2  .xz/ D C2 P:

(16.69)

.x/

These relations reflect the simple fact that the operations U2 and  .xz/ change the direction of pseudorotation. By virtue of equation (16.69), the extended point group can be written as F16 D P4 ^ C2v ; (16.70) .x/

where the group C2v D .E; U2 ;  .xz/ ; N .xy/ / characterizes the symmetry of equi.x/

librium configurations. It should be emphasized that U2 is the symmetry element of only twist structures,  .xz/ is the symmetry element of only envelope structures, and N .xy/ is not at all a symmetry element of a stable structure. As in the case of a cyclopentane molecule, the operation of the nonexchange type N .xy/ was introduced to take into account the operation of the exchange type B D P N .xy/ , which corresponds to the permutation of the hydrogen nuclei from class ¹24 º of the 8 group. The group F16 can also be represented in the form H8  CS0 , where the group H8 D P4 ^ U2 is isomorphic with the group D4 , and CS0 D .E; N .xy/ /. The designations of irreducible representations of the F16 group are given in Table 16.7. The form of the Hamiltonian symmetry subgroup in the noninvariant group F16 changes upon passing from the twist structure to an envelope structure. In what follows,

319

Section 16.6 The tetrahydrofuran molecule C4 H8 O

Table 16.7. Division of the elements into classes and the table of characters of the F16 D H8  CS0 group. E

E

P

P

.x/

U2 L2

.x/

U2 P .x/

C2 L1

C2 U2 L3

C2 P

.x/

C2 U2 P

H8

E

P

L1

L2

L3

A1

1

1

1

1

1

A2 B2

1 1

1 1

1 1

1 1

1 1

B1 E

1 2

1 2

1 0

1 0

1 0

F16

H8  CS0

A1 A2 A3 A4 A5 A6 A7 A8 E1 E2

A1  A0 A1  A00 A2  A0 A2  A00 B2  A0 B2  A00 B1  A0 B1  A00 E  A0 E  A00

the first structure is chosen as reference. Then the Hamiltonian symmetry subgroup has the form FH D U2  B2 : (16.71) Symmetric and antisymmetric representations of the group B2 D .E; B/ will be denoted as c and d. From a matching of the groups 8 and GH (the necessary correlations are given in Table 16.8), for a complete coordinate wave function we have the following irreducible representations of the GH group: 76Ac;

60Ad;

60Bc;

60Bd:

(16.72)

Matching now the group FH and F16 , for the allowed multiplets of a nonrigid molecule we have 76.A1 ; A5 ; E2;1 /;

60.A2 ; A3 ; A4 ; A6 ; A7 ; A8 ; E1;1 ; E1;2 ; E2;2 /;

(16.73)

where it was taken into account that the representations E1 and E2 of the F16 group are decomposed into one-dimensional ones when this group is reduced on its subgroup FH . Therefore, the components of these representations, which are given by the reduction rules E1;1 ! Ad;

E1;2 ! Bd;

E2;1 ! Ac;

correspond to individual nondegenerate levels.

E2;2 ! Bc;

(16.74)

320

Chapter 16 Molecules with different isomeric forms in a single electronic state

Table 16.8. Correlation table for the tetrahydrofuran molecule C4 H8 O. U2  B2

8 8

Œ1  Œ216  Œ22 14  Œ23 12  Œ24  F16 A1 ; A5 A2 ; A6 A3 ; A7 A4 ; A8 E1 E2

U2  CS  Ci

Ac Ac C 2.Ad C Bc C Bd / 8Ac C 4.Ad C Bc C Bd / 4Ac C 8.Ad C Bc C Bd / 8Ac C 2.Ad C Bc C Bd / U2  B2 Ac Ad Bc Bd Ad C Bd Ac C Bc

F16 A1 ; A5 A2 ; A6 A3 ; A7 A4 ; A8 E1 E2

.A  A0/.C/ ;

D2

.A  A00 /./

A

00 .C/

.A  A /

B3

.B  A0/.C/ ;

.B  A00 /./

B2

.B  A0/./ ;

.B  A00 /.C/

B1

F16  Ci

H16

A1.C/ ; A2./

A1

./ A1 ; .C/ A3 ; A3./ ; .C/ A5 ; ./ A5 ; .C/ A7 ; A7./ ; .C/ E1 ; ./ E1 ;

A2

0 ./

.A  A /

U2  CS 0

AA A  A00 B  A00 B  A0 A  A0 C B  A00 A  A00 C B  A0

;

.C/ A2 ./ A4 A4.C/ ./ A6 .C/ A6 ./ A8 A8.C/ ./ E2 .C/ E2

A3 A4 A5 A6 A7 A8 E1 E2

It should be pointed out how to construct multiplets (16.73) from the wave functions of separate types of motion. In the group F16 Ci , rotations of the molecule as a whole through a zero barrier form a subgroup given by .z/

D2 D .E; U2

.y/

D N .xy/ i; U2

.x/

D  .xz/ i; U2 /:

(16.75)

First we perform the construction .coord /U2 CS Ci D .el /U2  .vib/U2  .rot.-conf /U2 CS Ci :

(16.76)

6

The rotation-configuration representations rot.-conf, which are possible for a given rotational representation of the D2 group, are obtained from a correlation of the groups D2 and U2  CS  Ci through their common subgroup D2 . The groups U2  CS and F16 are matched at the final stage. Figure 16.13 shows an energy-level classification for the type A vibronic state. The use of rotation-configuration representations

Section 16.6 The tetrahydrofuran molecule C4 H8 O

321

in matching (16.76) made it possible to take into account that the configurations obtained using the nonexchange type operations  .xz/ and i are dependent. As a result, the classification has no extraneous levels. Then it should be taken into account that all considered types of motion in the F16  Ci group are specified by the elements of the noninvariant subgroup H16 that is isomorphic with the group F16 . Therefore, it is exactly the group H16 that describes the actual energy-level structure. It is easily seen that due to pseudorotation the rotational levels split into four sublevels (which correspond to four equilibrium points in one pseudorotation cycle), as it should be.

Figure 16.13. Energy-level classification for a nonrigid molecule of tetrahydrofuran in the type A vibronic state.

322

Chapter 16 Molecules with different isomeric forms in a single electronic state

The effective operator of any physical quantity describing the internal coordinate motion of a nonrigid molecule transforms according to the totally symmetric Young diagram of the 8 group and, as a consequence, according to the identity representation Ac of the FH group. Accordingly, in the noninvariant group F16 we obtain the following transformation types: A1 ;

A5 ;

E2;1 :

(16.77)

Finally, passing to the group H16, we have the transformation types A1 ;

A5 ;

E2;1

(16.78)

for the physical quantities that are invariant under the inversion operation i and the transformation types (16.79) A2 ; A6 ; E1;1 for the physical quantities that change sign under the operation i. The effective Hamiltonian transforms according to equation (16.78) and the effective operator of the electric dipole moment, according to equation (16.79). The presence of the noninvariant transformation types in equation (16.78) is due to the fact that the group H16 is noninvariant. Therefore, the energy matrix decomposes into a direct sum of only four matrices corresponding to the following sets of symmetry types of this group: .A1 ; A5 ; E2;1 /;

.A2 ; A6 ; E1;1 /;

.A3 ; A7 ; E2;2 /;

.A4 ; A8 ; E1;2 /;

(16.80)

.A3 ; A7 ; E2;2 / $ .A4 ; A8 ; E1;2 /:

(16.81)

and the electric dipole selection rules have the form .A1 ; A5 ; E2;1 / $ .A2 ; A6 ; E1;1 /;

Naturally, any electric dipole transitions, including very weak ones stipulated by the rotational distortion of the molecule, obey the selection rules described by equation (16.81). The strongest transitions can be singled out by analysis of the effective operator of the electric dipole moment. The basis unit vectors of the function space in an arbitrary vibronic state are classified according to irreducible representations of the group H16 D P4 ^ D2 and are constructed from the unit vectors of pseudorotational and rotational spaces. The complete set of unit vectors in the pseudorotational space is specified by four irreducible representations of the pseudorotational group P4 . In the ground vibronic state, these unit vectors are formed from the vectors j0i; j ˙ 1i and j ˙ 2i, which are defined by the condition  m jmi: (16.82) C2 jmi D exp i 2 Using the properties of the semidirect product, one can write .x/

U2 j0i D aj0i;

.x/

U2 j1i D bj  1i;

.x/

U2 j2i D cj  2i;

(16.83)

323

Section 16.6 The tetrahydrofuran molecule C4 H8 O

where a2 D b 2 D c 2 D 1. It will be seen in what follows that the value a D 1 needs to be chosen for the ground vibrational state. The values of b and c can always be taken equal to 1 since otherwise it is possible to change the sign of one of the vectors in the pairs j ˙ 1i and j ˙ 2i. Passing to symmetrized combinations of the form jsm i D

jmi C j  mi ; p 2

jam i D i

jmi  j  mi p 2

(16.84)

at m ¤ 0, we write four unit vectors of pseudorotation as j0i;

js1 i;

a1 i;

js2 i:

(16.85)

It follows from equations (16.82) and (16.83) that in the group H16 these unit vectors belong to the transformation types A1 ;

E1;1 ;

E1;2 ;

A5 :

(16.86)

The rotation of the molecule as a whole through a zero barrier is specified by the group D2 given of equation (16.75), and the rotational unit vectors are defined by four irreducible representations of this group. However, in the H16 group, besides the elements (16.75), the elements .z/

C2 U2 ;

C2 ;

.x/

C2 U2 ;

.y/

C2 U2

(16.87)

also act in the rotational space. Unlike the elements (16.75), these are associated with the transitions through a pseudorotation barrier. Clearly, the elements (16.75) and (16.87) corresponding to a rotation by the same angle, should have the same action on the rotational unit vectors. Therefore, for the rotational unit vectors we have .A/D2 ! .A1 /H16 ;

.B1 /D2 ! .A3 /H16 ;

.B2 /D2 ! .A8 /H16 ;

.B3 /D2 ! .A6 /H16 :

(16.88)

Using equations (16.85), (16.86), and (16.88), we obtain the basis unit vectors of the function space in the type A vibronic state decomposed into irreducible representations of the H16 group (see Table 16.9). The symmetry of the unit vectors is consistent with the classification due to that the value a D 1 was chosen in equation (16.83). The dimension of the pseudorotational space is easy to extend. To this end, it is necessary to take into account the vibrational states B; B 2 D A; B 3 D B, and so on, which correspond to the sequential excitation of motion along the cycle. Allowance for each excitation yields additionally four pseudorotational unit vectors. The first additional unit vector ja2 i relates to the representation A7 of the H16 group. The self-adjoint pseudorotational operators will be constructed on the basis of the operator of coordinate spin e. For this, one should find the representation of algebra (7.27) in the space of unit vectors jmi (an upper bound on the number jmj will not

324

Chapter 16 Molecules with different isomeric forms in a single electronic state

Table 16.9. Decomposition of the basis unit vectors of the function space in the type A vibronic state into irreducible representations of the H16 group. H16

Unit vectors

H16

Unit vectors

H16

Unit vectors

A1

j0iA

A5

js2 iA

E1;1

js1 iA; ja1 iB1

A2

js2 iB3

A6

j0iB3

E1;2

js1 iB1 ; ja1 iA

A3

j0iB1

A7

js2 iB1

E2;1

js1 iB3 ; ja1 iB2

A4

js2 iB2

A8

j0iB2

E2;2

js1 iB2 ; ja1 iB3

Note. The rotational parts of the unit vectors are specified by the irreducible representations of the D2 group.

be set in explicit form). Let e3 be a diagonal operator with the eigenvalues m and e˙ D e1 ˙ ie2 be the raising and lowering operators. Using the symmetry properties of the vectors jmi for transformations of the group P4 and operation T , T jmi D j  mi;

(16.89)

we find that e3 belongs to the representation A of the P4 group and is t-odd, while eC and e belong to the representations "1 and "2 , respectively (they are transformed as the vectors j1i; j  1i/, and T eC T D e . Passing from the group P4 to the group H16 , we have e3 ! A3 ; eC C e ! E1;1 ; i.eC  e / ! E1;2 : (16.90) A complete set of self-adjoint pseudorotational operators of a total power not higher than three over the components e with the symmetry properties indicated is presented in Table 16.10. The type A1 contributions to the effective Hamiltonian are formed by the structures .A1 /sp  .A1 /rot;

.A3 /sp  .A3 /rot;

(16.91)

which allows a full expression for these contributions to be written as H1 D .I; e32 /  A C .e3 ; e33 /  B1 :

(16.92)

Each term of the Hamiltonian is a combination of the products of the angular momentum components in the MCS, which relates to the representations A or B1 of the D2 group, and the parameters before this combination are linearly dependent on the spin operators correlating with the combination. Since the Hamiltonian is t-even, combinations of the A type should be t-even and combinations of the B1 type, t-odd. The type A5 contributions to the Hamiltonian are formed by the structures .A5 /sp  .A1 /rot;

.A7 /sp  .A3 /rot;

(16.93)

325

Section 16.6 The tetrahydrofuran molecule C4 H8 O Table 16.10. Complete set of self-adjoint pseudorotational operators. H16

t-even

t-odd

A1 A3

I; e32 –

– e3 ; e33

A5 A7 E1;1

E1;2

2 2 eC C e 2 2 i.eC  e / eC C e

2 2 iŒe3 ; eC  e C 2 2 Œe3 ; eC C e C

Œe32 ; eC C e C 3 3 eC C e

iŒe3 ; eC  e C

i.eC  e / iŒe32 ; eC  e C 3 3 i.eC  e /

Œe3 ; eC C e C

and a full expression for these contributions can be written as 2 2 2 2 2 2 2 2 H2 D ¹eC C e ; iŒe3 ; eC  e C º  A C ¹i.eC  e /; Œe3 ; eC C e C º  B1 ; (16.94)

where, in comparison with equation (16.92), t-odd rotational combinations of the A type and t-even combinations of the B1 type additionally appear. Finally, the type E2;1 contributions to the Hamiltonian are formed by the structures .E1;1 /sp  .A6 /rot ;

.E1;2 /sp  .A8 /rot;

(16.95)

which leads to an expression of the form 3 3 ; iŒe ; e  e  º  B H3 D ¹eC C e ; Œe32 ; eC C e C ; eC C e 3 C  C 3 3 3 C¹i.eC  e /; iŒe32 ; eC  eC ; i.eC  e /; Œe3 ; eC C e C º  B2 : (16.96) Unwinding equations (16.92), (16.94), and (16.96) in the form of a series of rotationaldistortion perturbations encounters no difficulties. The Hamiltonian automatically includes all interactions of the pseudorotational and rotational motions. The operator form of the formulation of the Hamiltonian for both motions is its important advantage. In particular, this makes the solution of the reduction problem much easier, which plays a significant role here due to the complex structure of the Hamiltonian. For the effective operator of the electric dipole moment, transformation types (16.79) are admissible. The component ˛ of this operator can be written in the form (8.62), where the parameters dO .i / are now the pseudorotational operators. The direction cosines relate to the following irreducible representations:

˛z ! .B1 /D2 ! .A3 /H16 ;

˛x ! .B3 /D2 ! .A6 /H16 :

˛y ! .B2 /D2 ! .A8 /H16 ;

326

Chapter 16 Molecules with different isomeric forms in a single electronic state

In the further analysis, we focus only on the main contributions not associated with a fairly weak rotational distortion of the molecule. The type A2 contributions to equation (8.62) are formed by the structures .A5 /sp  .A6 /rot;

.A7 /sp  .A8 /rot;

which leads to the expression 2 2 2 2 C e / C ˛y d .y/ i.eC  e /; ˛ D ˛x d .x/ .eC

(16.97)

where d .x/ and d .y/ are the real phenomenological constants. The type A6 contributions are formed by the only structure .A1 /sp  .A6 /rot; which yields an expression given by .x/

.x/

˛ D ˛x .d1 I C d2 e32 /:

(16.98)

The contributions given by equation (16.98) are responsible for the purely rotational transitions in an arbitrary pseudorotational state, which are stipulated by the x component of the electric dipole moment. The second term describes the dependence on the pseudorotational number m2 . Finally, the type E1;1 contributions are also formed by the only structure .E1;2 /sp  .A3 /rot; whence .z/

.z/

.z/

3 3 ˛ D ˛z ¹d1 i.eC  e / C d2 iŒe32 ; eC  e C C d3 i.eC  e /º:

(16.99)

Clearly, the selection rules specified by operators (16.97)–(16.99) are much narrower compared with those described by equation (16.81). Assuming that the pseudorotational states are not bound by accidental resonances, we formulate two conclusions. Firstly, the rotational transitions are stipulated only by the x component of the electric dipole moment. That is, the selection rules for such transitions in the case of a nonrigid molecule are not the superposition of the corresponding selection rules in the case of rigid twist and envelope molecules, but coincide with the selection rules for a rigid twist molecule. Experimentally, this fact has been known for a long time [40], but its correct explanation requires the correct analysis of the symmetry properties and can be obtained only on the basis of the methods considered in this book. Secondly, the transitions from the ground pseudorotational state j0i to the excited states with jmj D 1 are stipulated by the z component of the electric dipole moment, and only the transition j0i $ ja1 i is allowed. Of course, the actual analysis should take into account that there are pairs of closely spaced pseudorotational levels bound by accidental resonances [67]. In particular, such a pair is two lowest pseudorotational states j0i and

Section 16.7 Conclusions

327

js1 i which are resonantly bound by the term .eC C e /B3 in equation (16.96). It is easy to show that this leads to the appearance of transitions between them, which are due to the x component of the electric dipole moment. Such transitions were observed in [67], and this fact was explained by the resonant interaction. With the resonance taken into account, the rotational transitions are still related only with the x component of the electric dipole moment. Finally, this resonance maintains the forbiddance of the transitions j0i $ js1 i associated with the z component of the electric dipole moment. This explains why such transitions were not found in [67].

16.7 Conclusions We have formulated some general conclusions based on the content of this chapter. 1. The extended point group underlying the description of the internal dynamics of a molecule with allowance for the transitions between its different isomeric forms in a single electronic state is necessarily a noninvariant group since the operation of the geometrical symmetry group of the Hamiltonian cannot bind such forms. In other words, the geometrical symmetry group of the Hamiltonian “does not see” different isomers of the molecule simultaneously. 2. The noninvariant group is constructed on the basis of the requirement of correct transfer of the topology of internal dynamics. That is, the number of different isomeric forms, their symmetry properties, as well as the number and geometry of the independent nonrigid motions binding these forms, should correctly be taken into account. In fact, it is exactly this requirement that underlies the concept of distorted molecular system. 3. Degeneracy of the energy levels and their nuclear statistical weights are determined by the Hamiltonian symmetry subgroup of the extended point group. The problem of nuclear statistical weights of the energy levels of a molecule with transitions between its different isomeric forms has been solved for the first time. Although this problem is classical for the symmetry methods, its nontriviality is emphasized by the obvious fact that the nuclear statistical weights of stationary states of the isomers are generally different. It is clear that any level corresponding to the degenerate irreducible representation of the Hamiltonian symmetry subgroup corresponds to only one physically observable state. 4. The symmetry methods discussed above make it possible to construct a description of the nonrigid isotopically nonsymmetric combinations of a molecule as the distortion of a description of its most symmetrical combination. Use of the scheme with distortion permits one to determine the very concept of nonrigid motion as of a counterpart of such motion in the symmetric combination. It is important here that the approach which considers the molecular systems with different symmetries on

328

Chapter 16 Molecules with different isomeric forms in a single electronic state

a unified basis explains the qualitative features of the spectrum of a less symmetric system. 5. A rigorous algebraic model has been constructed to describe the entire excitation spectrum of pseudorotation for the isotopic combination 1,1-d2 of a cyclobutane molecule as the distortion of a similar model for its isotopically symmetric combination. In particular, this explained for the first time the observed selection rules for the electric dipole microwave transitions in the isotopic combination 1,1-d2 . Moreover, in the experimental data fitting the influence of the accidental resonances between pseudorotational states on the position of the energy levels and transition intensities between them is easily taken into account in this model. We also note that a rigorous algebraic model describing the torsional motion in the ionic complexes ArH2 DC and ArD2 HC as the distortion of the model for an isotopically symmetric complex, which is considered in Section 14.2, was developed in [27]. 6. Pseudorotational motion in a tetrahydrofuran molecule mixes the twist and envelope isomers. It is shown that the electric dipole selection rules are not the superposition of the electric dipole selection rules for rigid twist and envelope molecules, but coincides with the selection rules for a rigid twist molecule. Experimentally, this fact has been known for a long time, but its correct explanation was obtained only on the basis of the methods considered in this book. 7. Since the CNPI group was introduced as a symmetry group of a rigorous Hamiltonian of internal motion, it is obvious that transformations of such a group cannot mix the energetically nonequivalent isomeric forms of the molecule. Therefore, the correct analysis of the internal dynamics on the basis of the CNPI group is in principle impossible if nonrigid transitions between such forms need to be described.

Chapter 17

Molecules with different isomeric forms in different electronic states A very common situation is when a change in the electronic state of a molecule is accompanied by changes in its isomeric form. Constructing a unified model to describe such states is very useful. The fact is that even if the electronic states with different isomeric forms are fairly well isolated, their analysis within a unified model is needed, for example, to describe the electric dipole transitions between them. In this chapter, the statement is carried out using the formaldehyde molecule H2 CO and isotopic species of the ethylene molecule CH2 –CD2 as examples.

17.1 Formaldehyde molecule H2 CO The formaldehyde molecule is one of the classic objects of high-resolution molecular spectroscopy. It is well known [52] that the equilibrium configuration of this molecule in the ground electronic state is planar and belongs to the point group C2v (the left part of Figure 17.1). But in some excited electronic states, the CO bond in the equilibrium configuration does not lie in the plane of the CH2 structure (the right part of Figure 17.1), and the point group CS is realized [54].

Figure 17.1. Equilibrium configurations of the isomers of the H2CO molecule (for the nonplanar isomer, the z axis passes through the center of mass of the CH2 structure).

In a nonplanar isomer, by virtue of the relatively small deviation from the planarity, inversion motion is observed between two energetically equivalent configurations which convert into each other when the molecule is reflected in the yz plane. Obviously, a unified description of such isomeric forms can be constructed only by using noninvariant geometrical groups. We first present the results of the analysis of isolated isomers of the H2 CO molecule, which are necessary for what follows. We begin with a nonplanar isomer. Matching

330

Chapter 17 Molecules with different isomeric forms in different electronic states

the permutation group 2 of identical nuclei with the point group CS D .E;  .xz/ /, for the allowed coordinate multiplets we obtain 1A0 ;

3A00 ;

(17.1)

where the nuclear statistical weight allows for the spins of only the H nuclei. Then, as in equation (16.24), the zero-approximation wave functions are matched with the allowed multiplets formed on their basis. The admissible types of symmetry rot.-in for a given rotational representation of the D2 group of a rigid asymmetric top follow from a correlation of the groups D2 and CS  Ci through their common subgroup .y/ C2 D .E; C2 D  .xz/ i/. As a result, we have a classification of the energy levels of a rigid isomer. It is clear that the inversion motion belongs to the nonexchange type. To take this motion into account, we introduce a noninvariant extended point group F4 D CS  CS0 , where CS0 D .E;  .yz/ /. Matching the group F4 through its symmetry subgroup of the Hamiltonian CS with the group 2 , we obtain for the allowed coordinate multiplets of a nonrigid isomer 1A0s ;

1A0a ;

3A00s ;

3A00a :

(17.2)

Here, the subscripts s and a specify the symmetric and antisymmetric representations of the CS0 group. These multiplets are formed from the basis functions of separate internal motions in the following way: .mult/CS C 0 Ci D .el /CS  .vib/CS  .rot.-conf/CS C 0 Ci : S

S

(17.3)

6

The possible types of symmetry rot.-conf for a given rotational representation of the .z/ .y/ .x/ group D2 D .E; C2 ; C2 ; C2 / follow from a correlation of the groups D2 and F4  Ci through their common subgroup D2 . The behavior relative to the  .yz/ operation characterizes a multiplet, since this operation is not included in the point group. Figure 17.2 shows a classification of inversion-rotational levels in the vibronic state el  vib D A0 . Note that the group F4  Ci “sees” four equivalent configurations with the point group CS , but only two of them are independent. As was mentioned in Section 11.2, this fact can be taken into account by using rot.-conf in the matching, so that there are no extraneous levels in the classification. The actual structure of the spectrum is described by the total group F4  Ci since all of its elements can be expressed through operations specifying the motions considered. The rotational subgroup D2 is only auxiliary. In the case of a planar isomer, from a matching of the group 2 with the point group C2v D CS  CS0 , for allowed coordinate multiplets we have 1A0s ;

3A00s :

(17.4)

The type a representations are absent in equation (17.4) because of the planarity of the isomer. Then, as in equation (8.4), the zero-approximation wave functions are matched

Section 17.1 Formaldehyde molecule H2 CO

331

Figure 17.2. Classification of the inversion-rotational levels of a nonplanar isomer of the H2 CO molecule in the type A0 vibronic state.

with the allowed multiplets formed on their basis. The classification of rotational levels for el  vib D A0s formally coincides with that shown in Figure 17.2 on condition that the type a multiplets are forbidden. For a unified description of two electronic states with different isomeric forms, we introduce a noninvariant group F12  CS , where F12 D c3v  cS . The group c3v specifies the transformations of the CH2 structure, the c3 axis passes through its center of mass in parallel to the y axis, and one of the symmetry planes coincides with the yz plane. Here, the fact that the angle between the CH2 planes in the planar and nonplanar isomeric forms is different from 2/3 is not significant, since this fact can be interpreted, for example, as the distortion of the nonplanar form. Naturally, such a distortion is also due to the different lengths of the bonds in the planar and nonplanar isomers. Here, it is only important that the group c3v gives a qualitatively correct picture of the transitions between three considered configurations, two of which are energetically equivalent. As a consequence, this group determines a complete set of unit vectors of the function space for the description of these transitions. The group cS D .E;  .yz/ /, which specifies the O structure transformations, permits one to describe in the group F12 the reflection of the molecule as a whole in the yz plane using the element  .yz/   .yz/ . Only the identity representation of the cS , group is admissible, and when writing the symmetry types of the F12 group this trivial behavior will

332

Chapter 17 Molecules with different isomeric forms in different electronic states

be omitted. The group c3v can be represented as c3 ^ cS0 , where cS0 D .E;  .yz/ /. This is convenient for taking into account the planarity of the CH2 structure. Indeed let the molecule be localized in the yz plane. Then the planarity of the CH2 structure results in that of the two representations of the cS0 group, only the identity representation is realizable. Delocalization of the initial configuration is described by representations of the cyclic group c3 , and the forbiddance of the antisymmetric representation of the cS0 group leads to the forbiddance of the representation A2 of the c3v group in the multiplets of the F12 group. As a result, the allowed multiplets of the noninvariant group F12  CS have the form 1.A1  A0 /;

3.A1  A00 /;

1.E  A0 /;

3.E  A00 /:

(17.5)

We emphasize that the coordinate degeneracies and the nuclear statistical weights of the levels in the energy spectrum are determined by the symmetry group of the Hamiltonian. In this problem, the components of the degenerate multiplets in equation (17.5) correspond to different energy levels, but they have the same nuclear statistical weights. On account of nonrigid motions between three equilibrium configurations, the rotational symmetry is specified by the subgroup D2 of rotations of the molecule as a whole in the F12  CS  Ci group. Figure 17.3 shows a picture of configuration splittings of the rotational levels of the D2 group (the necessary correlations are given in Table 17.1). Table 17.1. Correlation tables in a unified description of the isomers of the H2CO molecule. F12  CS  Ci

D2

F12

CS0

.A1  A0 /.C/ , .A2  A00 /./ .A1  A0 /./ , .A2  A00 /.C/

A B1

A1 A2

A0 A00

.A1  A00 /.C/ , .A2  A0 /./ .A1  A00 /./ , .A2  A0/.C/

B3 B2

E1 E2

A0 A00

.E1  A0 /.C/ , .E2  A00 /./ .E1  A00 /./ , .E2  A0 /.C/ .E1  A0 /./ , .E2  A00 /.C/

A B2 B1

.E1  A00 /.C/ , .E2  A0 /./

B3

In addition, this figure shows the behavior of the levels on passage to the description of isolated isomers (the passage from the group F12 CS Ci to the group CS CS0 Ci /. Note that the correspondence between the representations of the CS  CS0  Ci group and the rotational representations of the D2 group depends on the type of vibronic states of isomers in the point group (CS or C2v /.

333

Section 17.1 Formaldehyde molecule H2 CO

Figure 17.3. Energy-level classification in a unified description of two isomers of the H2 CO molecule.

The function space is constructed by using the basis functions of the D2 group to describe the rotation of a nonrigid molecule as a whole and the basis configuration functions of the c3v group to describe the nonrigid transitions. The latter are constructed only from the unit vectors j0i and j ˙ 1i, which are determined by three irreducible representations of the c3 group. The point is that due to the planarity of the CH2 structure, of the unit vectors determined by two irreducible representations of the cS0 group, only the symmetric unit vector is retained, and it can be omitted in the construction. By virtue of the properties of the semidirect product, the action of the elements of the factor group cS0 is specified in the basis of the invariant subgroup c3 as well:  .yz/ j0i D j0i;

 .yz/ j1i D j  1i:

(17.6)

That is, the unit vector j0i belongs to the representation A1 of the c3v group and the unit vectors j ˙ 1i, to the representation E: E1 D jsi D

j1i C j  1i p ; 2

E2 D jai D

j1i  j  1i p : 2i

(17.7)

334

Chapter 17 Molecules with different isomeric forms in different electronic states

The choice of the sign for transformation of the unit vector j0i in equation (17.6) corresponds to the absence of the A2 unit vector, which is consistent with the classification. Then, the correlation of representations of the groups D2 and F12  CS  Ci leads to the decomposition of the configuration-rotational unit vectors into irreducible representations of the F12  CS  Ci group indicated in Table 17.2. Table 17.2. Decomposition of the configuration-rotational functions into irreducible representations of the F12  CS  Ci group. F12  CS  Ci

Function

F12  CS  Ci

Function

(A1  A0 /.C/

A  j0i

(A1  A0 /./

B1  j0i

A  jsi

0 ./

B1  jsi

00 .C/

0 .C/

.E1  A /

.E1  A /

00 ./

.E2  A /

A  jai

.E2  A /

B1  jai

(A1  A00 )./

B2  j0i

(A1  A00 ).C/

B3  j0i

B2  jsi

00 .C/

B3  jsi

0 ./

B3  jai

00 ./

(E1  A )

0 .C/

(E2  A )

B2  jai

(E1  A ) (E2  A )

Note. The rotational parts are specified by the group D2 representations.

On account of the two isomers of the H2 CO molecule, the effective operators of physical quantities belong to the coordinate Young diagram [2] of the 2 group. Therefore, the following types of transformations of the F12  CS group are allowed for them: .  A0 /;  D A1 ; A2 ; E1 ; E2 : (17.8) It is important that the transformation types in equation (17.8), which are antisymmetric relative to the operation  .yz/ of the c3v group, are realized due to antisymmetric combinations of the operators determined in the basis of the unit vectors j0i and j ˙ 1i of the invariant subgroup c3 of the c3v group. These operators are formed on the basis of the operator of coordinate spin e. Using the behavior of the unit vectors in transformations of the c3v group and relative to the time reversal operation T j0i D j0i;

T j1i D j  1i;

(17.9)

we obtain the symmetry properties of the complete set of independent spin operators. This set is similar to that shown in Table 9.4 with accuracy up to the replacement of the H6 group by the isomorphic group c3v . Interestingly, there is a spin operator of type A2 , although such a spin function cannot be constructed. However, this fact does not cause contradiction since e3 jA1 i D 0. In constructing the effective configuration-rotational Hamiltonian, we also take into account that the interactions considered are invariant under the operation of spatial

Section 17.1 Formaldehyde molecule H2 CO

335

inversion i. As a result, its full expression can be written as follows: H D .A1 ; E1 /  A C .A2 ; E2 /  B2 :

(17.10)

Here, each term of the Hamiltonian is a certain combination of components of the angular momentum in the MCS of the A (or B2 / symmetry in the D2 group, and the parameters before this combination are linearly dependent on the spin operators of the A1 , E1 (or A2 , E2 / symmetry in the c3v group, which correlate with this combination. The noninvariant contributions in equation (17.10) describe, in particular, the splitting of two-dimensional multiplets from equation (17.5). As an example, we present the type A contributions with t-even rotational combinations HD

1 X nD0

.A/

H2n ;

.A/

H2n D

X

2t cO2p;2s;2t J 2p Jz2s .JC C J2t /;

(17.11)

p;s;t

where p C s C t D n for a given n. On account of nonrigid motions, the parameters cO are spin operators of the form 2 2 cO D c .1/ I C c .2/ e32 C c .3/ .eC C e / C c .4/ .eC C e /:

(17.12)

Here, c .k/ are the real spectroscopic constants (the subscripts are omitted). Since the Hamiltonian is t-even, the t-odd spin operator iŒe3 ; eC  e C of the E1 type does not contribute to equation (17.12). The effective operator of the electric dipole moment is perhaps the most interesting in this case. Taking into account that in addition to equation (17.8) this operator reverses sign under the operation i, we obtain ˛ D ˛z Œ.A1 ; E1 /  A C .A2 ; E2 /  B2  C ˛y Œ.A1 ; E1 /  B3 C .A2 ; E2 /  B1 

(17.13)

C ˛x Œ.A1 ; E1 /  B2 C .A2 ; E2 /  A; where the rotational parts in the form of irreducible representations of the D2 group include combinations of the angular momentum in the MCS. Write out a specific expression for the terms not related with a fairly weak rotational distortion of the molecule: 2 2 / ˛ D ˛z Œd .1/I C d .2/ e32 C d .3/ .eC C e / C d .4/.eC C e 2 2 C ˛x iŒd .5/ .eC  e / C d .6/.eC  e /;

(17.14)

where d .k/ are real spectroscopic constants. Since the electric dipole moment is t-even, the t-odd spin operators are absent in equation (17.14). Only one vibronic state was considered in the presented analysis for each isomeric form. However, the analysis can also be extended to a set of such states, which is

336

Chapter 17 Molecules with different isomeric forms in different electronic states

formed, for example, by a certain set of vibrational excitations in each electronic state. For this, in addition to the group c3v we introduce a group c1 v , the number of accountable irreducible representations (the number of excitations) of which is related with the number of considered vibronic states. High flexibility in the application of the apparatus of dynamic noninvariant groups within the concept of the symmetry group chain should be emphasized. Indeed, in the formaldehyde molecule, the transitions between two isomeric forms cannot be described at all by symmetry transformations in the traditional sense. The difficulties are also due to the fact that for a nonplanar form it is necessary to take into account the inversion motion, whereas for a planar form this motion is impossible in principle. Also, the considered case is technically one of the easiest. This is illustrated by the example of the ethylene molecule.

17.2 The ethylene molecule CH2 –CD2 The rigid molecule of ethylene C2 H4 was considered in Section 8.2 assuming that its equilibrium configuration is planar and belongs to the point group D2h. In particular, this situation occurs for the ground electronic state. However, in some excited electronic states in the equilibrium configuration, the planes of the CH2 tops are orthogonal to each other [15] and the point group D2d is realized. To focus on the physical aspect of the unified description of two isomers, we simplify the problem by going to the isotopic species CH2 –CD2 (Figure 17.4). Then there is no need to take into account the identity of the tops. It is sufficient that the permutation group of identical nuclei is written in the form 2H  2D .

Figure 17.4. Equilibrium configurations of the isomers of the ethylene molecule.

The analysis of isolated isomers is rather trivial, and we present here only the main results that are necessary for what follows. Matching the group 2H 2D with the point .z/

group C2v D .E; C2 ;  .xz/ ; N .yz/ / of a planar isomer, for its allowed coordinate multiplets we obtain .9 C 6/A1 ; .18 C 3/B2 : (17.15) The representations A2 and B1 are absent in equation (17.15) because of the planarity of the isomer. Then, as in equation (8.4), the zero-approximation wave functions are matched by the allowed multiplets formed on their basis. In the two left columns,

337

Section 17.2 The ethylene molecule CH2 –CD2

Figure 17.5 shows a classification of the rotational levels for el  vib D A1 . The torsional motion of the tops CH2 and CD2 is strongly hindered [72] and belongs to the exchange type. However, as will be seen in what follows, this motion should be taken into account in a unified description of the isomers. The extended point group H D  c2v . From its matching with the allowing for the torsional motion is G16 D c2v H D group 2  2 , we obtain the following allowed coordinate multiplets of a nonrigid planar isomer: 6.A1  A1 /;

3.A1  B2 /;

18.B2  A1 /;

9.B2  B2 /:

(17.16)

H D and c2v groups do not appear in equation The A2 and B1 representations of the c2v (17.16) because of the planarity of the torsional tops. Matching now the groups G16 and C2v , we have a full picture of splittings with allowance for the torsional motion, which is shown in Figure 17.5 for el  vib D A1 . The actual structure of the levels is described by the rotational subgroup H8 D B2  D2 of the G16  Ci group, where the group D2 specifies the symmetry of the rotation isomer as a whole and the group B2 D .E  E; c2  E/, the symmetry of its torsional motion.

Figure 17.5. Classification of the torsion-rotational levels of a planar isomer of the CH2 – CD2 molecule in the type A1 vibronic state (the subscripts a and b denote the symmetric and antisymmetric representations of the B2 group).

The point group of a nonplanar isomer is again C2v . However, due to changes in the geometry of the equilibrium configuration, we have, instead of equation (17.15), 6A1 ;

9A2 ;

18B1 ;

3B2 :

(17.17)

338

Chapter 17 Molecules with different isomeric forms in different electronic states

The extended point group also has the previous form, but for its allowed multiplets we now obtain 6.A1  A1 /;

3.A1  B2 /;

18.B1  A1 /;

9.B1  B2 /:

(17.18)

Due to the planarity of the tops, equation (17.18) now does not include the representaH group and, as before, the representations A and B of the tions A2 and B2 of the c2v 2 1 D group. With allowance for these changes, we obtain a classification of the torsionc2v rotational levels of a nonplanar nonrigid isomer, which is shown in Figure 17.6 for el  vib D A1 . We pay attention to the fact that despite the differences in geometry, both isomers have the same picture of the energy levels in the H8 group. For a unified description of two electronic states with different isomeric forms, we H D  c4v . Rotations of one of the two tops introduced a noninvariant group F64 D c4v by an angle /2 (or 3/2) correspond to the noninvariant transformations between two isomers. Since two such sequential rotations define the torsional motion within the isomer, the motions between the isomers cannot be considered without allowance for their internal torsional motions. The principal point is that the group F64 correctly allows for the topology of the nonrigid motions. For example, from a given configuration of a planar isomer, we can go over to two independent energetically equivalent configurations of a nonplanar isomer, differing in the torsional rotation of one of the tops. Then, from any configuration of a nonplanar isomer, it is possible to come back either to the initial configuration of a planar isomer or the energetically equivalent configuration differing from the initial one of the torsional rotation.

Figure 17.6. Classification of the torsion-rotational levels of a nonplanar isomer of the CH2 – CD2 molecule in the type A1 vibronic state.

339

Section 17.2 The ethylene molecule CH2 –CD2

Matching the group G16 of the planar isomer shown in Figure 17.4 with the noninvariant group F64 , for the allowed coordinate multiplets of the latter we obtain 6.A1  A1 ; A1  B1 ; B1  A1 ; B1  B1 /; 18.E2  A1 ; E2  B1 /;

3.A1  E2 ; B1  E2 /;

9.E2  E2 /;

(17.19)

where E2 is one of the two components of the representation E of the c4v group, which are defined by the conditions  .xz/ E1 D E1 ;

 .xz/ E2 D E2 :

(17.20)

That is, the components E1 and E2 of the representation E of the c4v group, go over to the representations B1 and B2 when this group is reduced on the group c2v . From a similar matching for the nonplanar isomer presented in Figure 17.4 we obtain the allowed multiplets of the noninvariant group F64 in a form that follows from equation (17.19) upon replacement of the last two multiplets by 18.E1  A1 ; E1  B1 /;

9.E1  E2 /:

(17.21)

This expression gives new types of multiplets compared to those in equation (17.19). It should also be taken into account that by using the noninvariant motions, both isomers can be rotated as a whole by an angle /2 (or 3/2) about the z axis. From similar matchings for the rotated isomers, it is easy to obtain all the rest types of multiplets of the noninvariant group F64 . Then the complete set of multiplets can be written as 6.A1  A1 ; A1  B1 ; B1  A1 ; B1  B1 /; 18.E  A1 ; E  B1 /;

9.E  E/:

3.A1  E; B1  E/;

(17.22)

H and Due to the planarity of the tops, the representations A2 and B2 of the groups c4v D c4v are absent in this expression. Moreover, in this problem the components of the degenerate irreducible representations of a noninvariant group are described to different energy levels, but have the same nuclear statistical weights. On account of nonrigid motions within and between the isomers, the rotational symmetry is specified by the subgroup D4 of rotations of the molecule as whole in the F64  Ci group. The picture of configuration splittings of the rotational levels specified by irreducible representations of the D4 group is shown in Figure 17.7 (the necessary correlations are given in Table 17.3). The picture of the levels for the rotational states A2 and B2 of the D4 group is easy to construct by analogy with the rotational states A1 and B1; since with such a change of the rotational state, the multiplets of the F64  Ci group simply change sign with respect to the operation i. Nonrigid motions belong to the torsional type, and the actual structure of the levels is described by the total rotational subgroup H32 of the F64  Ci group. Division of the elements into classes and the table of characters of the H32 group

340

Chapter 17 Molecules with different isomeric forms in different electronic states

Figure 17.7. Energy-level classification in a unified description of two isomers of the CH2 – .˙/ .˙/ CD2 molecule (the expressions of T .˙/ , Sk and Rk through representations of the F64 Ci group follow from the corresponding rows of the correlation table of the groups H32 and F64  Ci ; the replacement of «˙» by «» in these expressions means the sign reversal in the representations of the F64  Ci group).

are given in Table 17.4. It should be emphasized that the states of the F64  Ci group, which go over to one state of the H32 group, do not exist simultaneously and maintain the description of the same physical state for different orientations of the torsional tops. Indeed, such multiplets in equation (17.22) that occur independently when the group F64 is matched with the groups G16 of equilibrium configurations of the ethylene molecule with different spatial position go over jointly to one state of the H32 group. Naturally, the group H32 is also a noninvariant group with the symmetry subgroup of the Hamiltonian H8 D B2  D2 . Since the two-dimensional representations decompose into one-dimensional when the group H32 is reduced on subgroup H8 , the components of two-dimensional representations specify the different energy levels. In addition, Figure 17.7 displays the behavior of the levels on passage to the description

341

Section 17.2 The ethylene molecule CH2 –CD2

Table 17.3. Correlation tables in a unified description of the isomers of the CH2 –CD2 molecule. H32

F64  Ci

H32

F64  Ci

A1

(A1  A1 /.C/

E1;1

A2

(A1  A1 /

./

A3

D4

H32

(A1  E1 /.C/ , (A1  E2 /./

A1

A1 , A7 , E5

E2;1

(B1  E1 /

./

A2

A2 , A8 , E5

(A1  B1 /.C/

E3;1

(E1  A1 /.C/ , (E2  A1/./

B1

A4 , A6 , E6

A4

(A1  B1 /./

E4;1

(E1  B1 /.C/ , (E2  B1 /./

B2

A3 , A5 , E6

A5

(B1  A1 /

(E1  E1 /

E

E1 , E2 , E3 , E4

A6

(B1  A1 /

A7

(B1  B1 /.C/

A8

(B1  B1 /./

.C/ ./

E5;1

E6;1

.C/

.C/

(E1  E2 /

./

, (B1  E2 /

, (E2  E2 /

.C/

, (E2  E1 /

./

(E1  E1 /.C/ , (E2  E2 /.C/ (E1  E2 /./ , (E2  E1 /./

Note. For the group F64  Ci , only the representations realized as allowed multiplets are pointed out, and two components of the representation Ek of the H32 group are specified by the conditions C2.y/ Ek;1 D Ek;1 and C2.y/ Ek;2 D Ek;2 . The correlation row for Ek;2 is obtained from the correlation row for Ek;1 by the replacement C $ .

of isolated isomers (the passage from the group H32 to the group H8 /. In this case, the correlation between the irreducible representations of the H8 group and the rotational states of the D2 group of separate isomers depends on the type of their vibronic states in the point group C2v . It is important for what follows that the group H32 can be represented as H32 D H4 ^ D4 :

(17.23)

Indeed, the group H4 D .E  E; c4  E; c2  E; c43  E/ is an invariant subgroup of the H32 group, which is necessary and sufficient for using the semidirect product (see Section 1.3). It is exactly the H4 group that specifies all nonrigid motions, and the transitions between isomers correspond to the noninvariant elements c4  E and c43  E. The element c2  E, which specifies the torsional motion within one isomer, arises here as a consequence of taking into account the mixing of isomers. The complete basis set of torsional unit vectors jAi; jBi; j"1 i, and j"2 i is defined by four irreducible representations of the H4 group that is isomorphic with the C4 group. Designations of the latter are exactly those used for the irreducible representations of the H4 group. By virtue of the properties of the semidirect product, the elements of the factor group D4 , which do not commute with the elements of the invariant subgroup H4 , act not only in the rotational, but also in the torsional space. It is enough to indicate this action in

342

Chapter 17 Molecules with different isomeric forms in different electronic states

Table 17.4. Division of the elements into classes and the table of characters of the H32 group. E

EE

K1

c2  c2

K2 K3

E  c2 c2  E

E  c4

L1

E

L4

c43

c4  c2 c34  c2

.x/

.y/

.x/

.y/

M1

.c2 ; c2 /  .c2 ; c2 /

L2

c2  c4 c2  c43

L5

c4  c4 c43  c43

M2

.c2.1/ ; c2.2/ /  .c2.x/ ; c2.y/ /

L3

c4  E c43  E

L6

c4  c43 c43  c4

M3

.c2 ; c2 /  .c2 ; c2 /

M4

.c2 ; c2 /  .c2 ; c2 /

.x/

.y/

.1/

.2/

.1/

.2/

.1/

.2/

Note. In the elements of rotation about the z axis, the axial index is omitted and the axes 1 and 2 are obtained from the axes x, y by rotation by an angle /4 about the z axis. H32

E

K1

K2

K3

L1

L2

L3

L4

L5

L6

M1

M2

M3

M4

A1 A2 A3 A4 A5 A6 A7 A8 E1 E2 E3 E4 E5 E6

1 1 1 1 1 1 1 1 2 2 2 2 2 2

1 1 1 1 1 1 1 1 2 2 –2 –2 –2 –2

1 1 1 1 1 1 1 1 –2 –2 2 2 –2 –2

1 1 1 1 1 1 1 1 –2 –2 –2 –2 2 2

1 1 1 1 –1 –1 –1 –1 2 –2 0 0 0 0

1 1 –1 –1 1 1 –1 –1 0 0 2 –2 0 0

1 1 –1 –1 –1 –1 1 1 0 0 0 0 2 –2

1 1 –1 –1 –1 –1 1 1 0 0 0 0 –2 2

1 1 –1 –1 1 1 –1 –1 0 0 –2 2 0 0

1 1 1 1 –1 –1 –1 –1 –2 2 0 0 0 0

1 –1 1 –1 1 –1 1 –1 0 0 0 0 0 0

1 –1 1 –1 –1 1 –1 1 0 0 0 0 0 0

1 –1 –1 1 1 –1 –1 1 0 0 0 0 0 0

1 –1 –1 1 –1 1 1 –1 0 0 0 0 0 0

the torsional space for the element C2.y/: .y/

C2 jAi D jAi;

.y/

C2 jBi D jBi;

.y/

C2 j"1 i D j"2 i:

(17.24)

which mixes the unit vectors belonging to a pair of complex-conjugate representations of the H4 group. Now, allowing for the known behavior of the rotational functions for the group D4 transformations, we obtain a decomposition of the torsion-rotational basis unit vectors into irreducible representations of the H32 group shown in Table 17.5. The rotational parts of the unit vectors are specified by the irreducible representations of the D4 group. The components of the representation E are defined by the conditions .y/ .y/ C2 E1 D E1 , C2 E2 D E2 . The effective operators of coordinate physical quantities belong to the coordinate Young diagram [2]  [2] of the 2H  2D group. From a matching of this group with

343

Section 17.2 The ethylene molecule CH2 –CD2

Table 17.5. Decomposition of the torsion-rotational unit vectors into irreducible representations of the H32 group. H32

Functions

H32

Functions

A1

A1  jAi

E3;1

Œ.E1 C iE2 /j"1 i C .E1  iE2 /j"2 i=2

A2

A2  jAi

E3;2

Œ.E1 C iE2 /j"1 i  .E1  iE2 /j"2 i=2

A3

B2  jAi

E4;1

Œ.E1  iE2 /j"1 i C .E1 C iE2 /j"2 i=2

A4

B1  jAi

E4;2

A5

B2  jBi

A6

B1  jBi

A7

A1  jBi

A8

A2  jBi

E1;1

E1 jAi

E1;2

E2 jAi

E2;1

E1 jBi

E2;2

E2 jBi

Œ.E1  iE2 /j"1 i  .E1 C iE2 /j"2 i=2 p A1 .j"1 i C j"2 i/= 2 p A2 .j"1 i  j"2 i/= 2 p A1 .j"1 i  j"2 i/= 2 p A2 .j"1 i C j"2 i/= 2 p B1 .j"1 i  j"2 i/= 2 p B2 .j"1 i C j"2 i/= 2 p B1 .j"1 i C j"2 i/= 2 p B2 .j"1 i  j"2 i/= 2

E5;1

E5;2

E6;1

E6;2

H D the extended point group G16 D c2v c2v of either from the two isolated isomers with internal torsional motion, we obtain for such operators one admissible representation A1  A1 of the G16 group. Then, as in constructing the classification, all the groups G16 are matched with the noninvariant group F64 , which, within the framework of the latter, already yields four admissible representations

A1  A1 ;

A1  B1 ;

B1  A1 ;

B1  B1 :

(17.25)

Now, passing from the group F64  Ci to the group H32 , we obtain the allowed representations (17.26) A1 ; A3 ; A5 ; A7 for the physical quantities that are invariant under the operation of spatial inversion i and the allowed representations A2 ;

A4 ;

A6 ;

A8

(17.27)

for the physical quantities that change sign under the operation i. Note that the sets of allowed representations include all the eight one-dimensional representations of the H32 group. In the operators of physical quantities, the torsional part will be constructed on the basis of the operator of coordinate spin e. To do this, the torsional unit vectors

344

Chapter 17 Molecules with different isomeric forms in different electronic states

jAi; jBi; j"1 i, and j"2 i should be related with the eigenvectors of the operator e3 . In a standard definition, the required properties in the H4 group are provided by a set of five eigenvectors with the eigenvalues 0, ˙1, and ˙2: j0i ! jAi;

j1i ! j"1 i;

j  1i ! j"2 i;

j ˙ 2i ! jBi:

(17.28)

That is, there is one extraneous vector of the B type. An attempt to immediately exclude this vector leads to major difficulties with the coordination of the algebraic properties and the symmetry properties of the components of the coordinate spin. Therefore, we first construct a complete set of independent spin operators in a five-dimensional space and then reduce this set to the actual four-dimensional torsional space. Using the behavior of the eigenvectors for transformations of the H4 group and relative to the time reversal operation T j0i D j0i;

T j1i D j  1i;

T j2i D j  2i;

(17.29)

we find that the operator e3 belongs to the representation A of the H4 group and is t-odd, while the raising eC and lowing e operators belong, respectively, to the representations "1 and "2 of the H4 group, and T eC T D e :

(17.30)

It follows from equation (17.30) that the operators e1 and e2 are t-even. We now consider the extension of the H4 group to the H32 group. For the eigenvectors, we obtain, instead of equation (17.24), .y/

C2 j0i D j0i;

.y/

C2 j1i D j  1i;

.y/

C2 j2i D j  2i:

(17.31)

Then, the operator e3 belongs to the representation A2 of the H32 group and the operators e1 and e2 , to the representation E5 . For the irreducible representations of the H32 group, the following relations are fulfilled: : E5  E5 D A1 C A2 C A7 C A8 ; : Ak  E5 D E5 .k D 1; 2; 7; 8/:

: A2  A1 D A2 ;

: A2  A7 D A8 ;

Therefore, the products of the spin components can belong only to the one-dimensional representations A1 , A2 , A7 , and A8 and the two-dimensional representation E5 . Note that in a five-dimensional space, the products of the spin components of a total power not higher than four are independent. Further, it is important that the rotational parts belonging to the representations A1 , A2 , B1 , B2 , and E of the D4 group when the group D4 is extended to the group H32 are transformed according to the representations A1 , A2 , A4 , A3 , and E1 , respectively, Taking into account that : .E5 /sp  .E1 /rot D E3 C E4 ;

345

Section 17.2 The ethylene molecule CH2 –CD2

we obtain that the effective operators of physical quantities can contain only the spin and rotational operators belonging to the one-dimensional representation of the H32 group. In total, there are thirteen such independent spin operators. They all are presented in Table 17.6. Importantly, the space of these spin operators is closed relative to the operation of commutators calculation (this space is a Lie algebra) since otherwise the condition that the picture of description is invariant for the unitary transformations not breaking the symmetry of the problem is violated. The fact that the space is closed follows from the simple fact that the operation of direct multiplication does not lead out from the set of representations A1 , A2 , A7 , and A8 . We now consider the reduction to the actual four-dimensional space, choosing j0iI

js1 i D

j1i C j  1i p I 2

ja1 i D i

j1i  j  1i p I 2

js2 i D

j2i C j  2i p (17.32) 2

as its unit vectors. Naturally, the space given by equation (17.32) is closed relative to transformations of the H32 group and the time reversal operation T. In fact, pthe parts mixing the unit vectors (17.32) with the unit vector ja2 i D i.j2i  j  2i/= 2 should be omitted in the operators shown in Table 17.6. This problem is easy to solve by writing the operators through bra and ket unit vectors [36] and omitting the terms with the unit vector ja2 i. Table 17.6. Complete set of independent spin operators transformed according to onedimensional representations of the H32 group. H32 A1

t-even I; e32;

e34;

t-odd

4 4 eC C e

–

A2

4 4 i.eC  e /

A7

2 2 2 2 eC C e ; Œe32; eC C e C

2 2 iŒe3 ; eC  e C

A8

2 2 2 2 i.eC  e /; iŒe32 ; eC  e C

2 2 Œe3 ; eC C e C

e3 ;

e33

Importantly, the terms remaining in each operator for the operations of the H32 group and the operation T convert only into each other, that is, the symmetry of the operators is preserved. However, the lowering of the dimension of the spin space certainly decreases the number of linearly independent operators. Using the operator written through bra and ket unit vectors, it can be shown that the following linear relations take place for the operators from Table 17.6 in space (17.32): 4 4 eC C e D 2.e34  e32 /; 4 4  e / D 0; i.eC

e33 D e3 ;

2 2 2 2 iŒe32 ; eC  e C D 2i.eC  e /;

2 2 Œe3 ; eC C e C D 0:

(17.33)

346

Chapter 17 Molecules with different isomeric forms in different electronic states

Each of the five relations in equation (17.33) contains equal-symmetry operators for transformations of the H32 group and operation T, which ensures the invariance of these bonds. Consequently, there remain eight linearly independent spin operators contributing to the torsional parts of the effective operators of physical quantities. A possible set of these operators is listed in Table 17.7, where the following notation is adopted: e3 ! K1 D js1 iha1 j C ja1 ihs1 j; 2 C e2  ; K2 D Œe32 ; eC  C

(17.34)

2  e2  ; K3 D iŒe3 ; eC  C 2  e 2 / ! K D 6i.ja ihs j  js iha j/: i.eC 4 i 1 1 1 

The arrows in equations (17.34) indicate a significant modification of some operators from Table 17.6 when we go over to the space given by equations (17.32), since many contributions for these operators were omitted. The space of eight independent operators in Table 17.7 is closed relative to the operation of commutators calculation. This follows even from the symmetry considerations, but can easily be shown by straightforward calculation of the commutators. Note that the operators K1 and K4 are nonzero for the unit vectors js1 i; ja1 i and the operator K3 is nonzero for the unit vectors j0i; js2 i. Table 17.7. Complete set of independent spin operators included in the effective torsionrotational operators of physical quantities. H32

t-even

t-odd

A1

I; e32; e34

–

–

K1

A2 A7 A8

2 eC

C

2 e ;

K4

K2

K3 –

The torsion-rotational Hamiltonian belongs to representations (17.26) of the H32 group, which are realized by the following eight spin-rotational structures: A1 D .A1 /sp  .A1 /rot ;

.A2 /sp  .A2 /rot;

A3 D .A1 /sp  .A3 /rot ;

.A2 /sp  .A4 /rot;

A5 D .A7 /sp  .A3 /rot ;

.A8 /sp  .A4 /rot;

A7 D .A7 /sp  .A1 /rot ;

.A8 /sp  .A2 /rot:

(17.35)

Therefore, a full expression for this Hamiltonian can be written as follows: 2 2 H D .I; e32 ; e34 ; eC C e ; K2 ; K3 /  .A1 ; B2 / C .K1 ; K4 /  .A2 ; B1 /:

(17.36)

347

Section 17.2 The ethylene molecule CH2 –CD2

Each contribution to equation (17.36) is a combination of the angular momentum components in the MCS, which is transformed according to one of the one-dimensional representations A1 , A2 , B1 , and B2 of the D4 group, and the parameters before this combination are linearly dependent on the spin operators correlating with it. As an example, we give the terms with t-even rotational operators of type A1 : H D

1 X nD0

H2n;

H2n D

X

4t cO2p;2s;4t J 2p Jz2s .JC C J4t /:

(17.37)

p;s;t

Here, p C s C 2t D n for a given n, and the parameters cO are spin operators of the form 2 2 cO D c .1/ I C c .2/ e32 C c .3/ e34 C c .4/ .eC C e / C c .5/ K2 ;

(17.38)

where c.k/ are the real spectroscopic constants (the subscripts are omitted). Since the Hamiltonian is t-even, the t-odd spin operator K3 is absent in equation (17.38). The effective torsion-rotational operator of the electric dipole moment belongs to representations (17.27) of the H32 group, which are also realized by eight spin-rotational structures: A2 D .A1 /sp  .A2 /rot ;

.A2 /sp  .A1 /rot;

A4 D .A1 /sp  .A4 /rot ;

.A2 /sp  .A3 /rot;

A6 D .A7 /sp  .A4 /rot ;

.A8 /sp  .A3 /rot;

A8 D .A7 /sp  .A2 /rot ;

.A8 /sp  .A1 /rot:

(17.39)

Therefore, a full expression for this operator can be written as follows: 2 2 ef D .I; e32 ; e34 ; eC C e ; K2 ; K3 /  .A2 ; B1 / C .K1 ; K4 /  .A1 ; B2 /: (17.40)

This expression can easily be rewritten for the component ˛ of the electric dipole along the axis ˛ of the FCS with the direction cosines ˛k explicitly singled out in the rotational operators. In particular, for the contributions that are proportional to ˛z we obtain 2 2 ˛ D ˛z Œ.I; e32 ; e34 ; eC Ce ; K2 ; K3 /.A1 ; B2 /C.K1 ; K4 /.A2 ; B1 /: (17.41)

Here, the rotational parts written in the form of irreducible representations of the D4 group include only combinations of the angular momentum components in the MCS. Note that only the contributions proportional to ˛z contain the terms not related to a fairly weak rotational distortion of the molecule: 2 2 C e / C d .5/K2 ; ˛ D ˛z  Œd .1/ I C d .2/ e32 C d .3/ e34 C d .4/ .eC

(17.42)

where d.k/ are the real phenomenological constants. Since the electric dipole moment is t-even, the t-odd spin operator K3 is absent in equation (17.42).

348

Chapter 17 Molecules with different isomeric forms in different electronic states

17.3 Conclusions We have formulated some general conclusions based on the content of this chapter. 1. The approach to describing the internal dynamics of a molecule with allowance for nonrigid transitions between its different isomeric forms in different electronic states is philosophically similar to the case where different isomeric forms mixed in one electronic state are taken into account. Naturally, the geometrical symmetry group of the Hamiltonian “does not see” the different isomers of the molecule simultaneously and, as a consequence, the extended point F is necessarily a noninvariant group. 2. Constructing the noninvariant group F is based on the requirement of correct consideration of the topology of internal dynamics. It is necessary to correctly take into account the number of different isomeric forms and their symmetry properties, as well as the number and geometry of the independent nonrigid transitions connecting these forms. In general, the actual energy-level structure is specified by the noninvariant group FL , which is a subgroup of the F  Ci group. The components of the degenerate representation of the FL group, which reduce into different irreducible representations of the Hamiltonian symmetry subgroup FLH , correspond to individual energy levels. 3. The use of the apparatus of noninvariant groups within the concept of the symmetry group chain is very flexible. Indeed, even in one of the technically simplest cases of a formaldehyde molecule, the transitions between two isomeric forms cannot be described at all by symmetry transformations in the traditional sense. The difficulties are also due to the fact that for a nonplanar form it is necessary to take into account the inversion motion, whereas for a planar form this motion is impossible in principle. 4. Since the CNPI group is introduced as a symmetry group of a rigorous Hamiltonian of the internal motion, it is obvious that its transformations cannot mix the energetically nonequivalent isomeric forms of the molecule. Thus, if the nonrigid transitions between such forms need to be described, the correct analysis of internal dynamics in terms of the CNPI group is impossible in principle.

Chapter 18

Algebraic models of the global description of molecular spectrum So far, marked progress has been made in developing approaches, at least for small polyatomic molecules, capable of describing within the framework of a unified model all the available precision experimental data on the high-resolution spectra in a nondegenerate ground electronic state. Such a description is commonly called global, and it plays an important role in creating a spectroscopic high-temperature data bank. The existing approaches can be divided into three groups. The first group comprises approaches in which the potential energy surfaces of the nuclear interaction and the electric dipole moment are first determined in the adiabatic approximation by ab initio methods. Then, the problem of the energy spectrum of the nuclear motion in this potential and of the intensities of the electric dipole transitions in this spectrum is solved by numerical methods. However, the available accuracy of description (first of all, for the transition frequencies) is absolutely unsatisfactory in comparison with the accuracy of experimental data. As a consequence, the obtained surfaces are additionally varied to fit the experimental data, and such variations should be “small.” As a result, the calculation accuracy increases substantially, but the description itself becomes empirical. This approach requires very large computational resources, and even with supercomputers, it has been realized only for triatomic molecules (see, e.g., [73, 86]). The second group of approaches includes those that use effective Hamiltonians and effective operators of the electric dipole moment obtained by the methods of perturbation theory [66]. The effective Hamiltonian is written as a matrix whose elements depend on the vibrational quantum numbers and rotational operators. The constants in such dependences, called the spectroscopic constants, are determined by comparing the model predictions and the experimental data. Naturally, the approaches of the second group require much less computational resources since a considerable part of the calculations is performed analytically, which is the main difficulty for these approaches. The possibility of their successful application to global description was demonstrated by examples of linear triatomic molecules (see, e.g., [76]). However, the approaches of these two groups are difficult to use for more complex molecules even if they are rigid (for the approaches from the first group, the amount of computational resources needed increases rapidly, while for the second group, the analytical calculations become more and more cumbersome). As for nonrigid molecules, it is as yet unknown how to realize this description correctly within the framework of these approaches. In this respect, it is much more promising to use the symmetry methods set forth in this book. We relate such approaches to the third group. In the first

350

Chapter 18 Algebraic models of the global description of molecular spectrum

section of this chapter, we construct an algebraic model (with an algebraic scheme for calculation of the position of the levels and transition intensities between them) for the global description of the energy spectrum of a rigid molecule using the water molecule H2 O in the ground electronic state as an example.

18.1 The rigid water molecule H2 O Classification of the spectrum of the rigid molecule H2 O and the effective rotational operators of physical quantities for the description of one isolated vibronic state were considered in Section 8.1. In addition, a unified description of the first triad of the vibrational states was obtained in Section 8.4. It is necessary to generalize the results to the case of describing the entire discrete spectrum in the ground electronic state. Three vibrational degrees of freedom of the H2 O molecule refer to the following irreducible representations of its point group C2v : : vib D 2A1 C B1 :

(18.1)

According to the results of the classification, in the group D2 describing the actual structure of the spectrum we have, instead of equation (18.1), : vib D 2A C B2 :

(18.2)

Consider the function space of the type A vibrational mode. We will construct this space from the jpi vectors defined by the relations C'.z/ jpi D exp.ip'/jpi;

.y/

C2 jpi D .1/p j  pi;

T jpi D j  pi;

(18.3) (18.4)

where p is an arbitrary integer. To do this, we will form the basis unit vectors as js0 i D j0i;

jsp6D0 i D

jpi C j  pi ; p 2

jap6D0 i D i

jpi  j  pi : p 2

(18.5)

All the unit vectors are t-even and decompose into irreducible representations of the D2 group in the following way: js2p i ! A;

ja2p i ! B1 ;

ja2pC1 i ! B2 ;

js2pC1 i ! B3 :

(18.6)

Since all excitations of the considered mode belong to the A type, we have v D 0 ! js0 i;

v D 1 ! js2 i;

v D 2 ! js4 i    :

(18.7)

Such a chain of excitations is consistent with the classification, and this chain contains no missing unit vectors of A type.

351

Section 18.1 The rigid water molecule H2 O

For the type B2 mode, we will form the basis unit vectors, as in equation (18.5), from the jqi vectors defined by equations (18.3) and (18.4). It is clear that the chain of excitations can be written as v D 0 ! js0 i;

v D 1 ! ja1 i;

v D 2 ! js2 i;

v D 3 ! ja3 i    :

(18.8)

The rotational basis functions are determined by the relations C'.z/ jJ; ki D exp.ik'/jJ; ki;

C2.y/ jJ; ki D .1/J Ck jJ; ki;

(18.9)

and their decomposition into irreducible representations of the D2 group is well known [64]. The basis unit vectors of the total function space are obtained by multiplying the basis unit vectors of the function spaces of the vibrational modes and the rotational motion. The complete set of self-adjoint operators in the function space of the type A vibrational mode will be constructed on the basis of the operator of coordinate spin p. Since this space includes only the j2pi vectors, as the generators it is necessary to 2 2 2 2 choose, besides the diagonal operator p3 , the operators pC C p and i.pC  p /. It is easy to show that in the D2 group, these operators refer to the following irreducible representations: p3 ! B1 ;

2 2 pC C p ! A;

2 2 i.pC  p / ! B1 ;

(18.10)

where the first operator is t-odd and the other two are t-even. Since the function space is formed entirely by the type A states, only the type A vibrational operators are feasible. Their complete set is listed in Table 18.1 (in the tables of this section, m; n D 0; 1; 2; : : :/. As an example, we give the expression for a vibrational Hamiltonian that comprises terms of power not higher than four of the components of p: 2 2 4 4 2 2 H D c1 I Cc2 p32 Cc3 .pC Cp /Cc4 p34 Cc5 .pC Cp /Cc6 Œp32 ; pC Cp C ; (18.11)

where ck are the real spectroscopic constants. Here, it was taken into account that the Hamiltonian belongs to the A type in the D2 group and is t-even. Table 18.1. Complete set of vibrational operators of the type A mode of the H2 O molecule. D2 A

t-even I;

2n 2n Œp32m ; pC C p C

t-odd 2nC2 2nC2 iŒp32mC1 ; pC  p C

It is also easy to write the expressions for the generators of unitary transformations transforming the Hamiltonian (18.11) to a reduced form. The symmetry of these operators in the D2 group is similar to the Hamiltonian, but they are t-odd: 2 2  p C : S D isŒp3 ; pC

(18.12)

352

Chapter 18 Algebraic models of the global description of molecular spectrum

Using the freedom in the choice of the constant s, the spectroscopic constant before the fourth-power term of the components of coordinate spin can be taken equal to zero in the reduced Hamiltonian. However, this conclusion may change in the model for the description of an isolated fragment of the vibrational states. Indeed, let only the states js0 i and js2 i be interesting. Then jpj  2, and the vibrational operators of power not higher than four over the components of coordinate spin are admissible. Since the states js1 i; ja1 i and ja2 i are absent, not all of these operators are independent. This problem, as in Section 17.2, is easy to solve by writing the operators in space of unit vectors js0 i and js2 i through the bra and ket vectors: I D js0 ihs0 j C js2 ihs2 j; p32 D 4js2 ihs2 j; p 2 2 C p D 4 3.js0 ihs2 j C js2 ihs0 j/; p34 D 16js2 ihs2 j; pC p 4 4 2 2 pC C p D 24js2 ihs2 j; Œp32 ; pC C p C D 16 3.js0 ihs2 j C js2 ihs0 j/: (18.13) It is clearly seen that only three operators are independent, and the Hamiltonian of a two-level model can be written as 2 2 H D c1 I C c2 p32 C c3 .pC C p /:

Accordingly, for the generator given by equation (18.12) we have p S D 8 3is.js2 ihs0 j  js0 ihs2 j/:

(18.14)

(18.15)

Using the constant s, the spectroscopic constant before the term of the second power of the components of coordinate spin can be taken equal to zero in the reduced Hamiltonian. Therefore, the reduced Hamiltonian has two independent spectroscopic constants, as should be the case in the presence of two levels. We will use the operator of coordinate spin q as the basis to construct a complete set of self-adjoint operators in the function space of the type B2 vibrational mode. Now, as the generators we have, together with the diagonal operator q3 , the operators qC C q and i.qC  q /. These operators belong to the following irreducible representations of the D2 group: q 3 ! B1 ;

q C C q  ! B3 ;

i.qC  q / ! B2 ;

(18.16)

where the first operator is t-odd and the other two are t-even. Since the function space of the mode is formed by the type A and type B2 states, vibrational operators of only the same types are feasible. A complete set of these operators is listed in Table 18.2. Finally, the rotational operators written in the form of combinations of the components of the total angular momentum J in the MCS decompose into irreducible representations of the D2 group on the basis of the symmetry properties in this group of rotational functions jJ; ki. It is clear that all the three components of the angular momentum are t-odd.

353

Section 18.1 The rigid water molecule H2 O

Table 18.2. Complete set of vibrational operators of the type B2 mode of the H2 O molecule. D2 A B2

t-even

t-odd

2n 2n Œq32m ; qC C q C

2nC2 2nC2 iŒq32mC1 ; qC  q C

2nC1 2nC1 iŒq32m ; qC  q C

2nC1 2nC1 Œq32mC1 ; qC C q C

I;

On the basis of the vibrational and rotational operators, we form the effective operators of any physical quantities of interest. In a full version, the model includes the operators formed on the basis of two vectors p.i / (two vibrational modes of the A type), the vector q (vibrational mode of the B2 type), and the vector of the angular momentum J (rotational motion). It is important that the vibrational and rotational operators introduced in such a way commute with each other. We now show how in this approach a model can be constructed for the description of an arbitrary isolated fragment of the vibrational states. As an example, we consider the first triad of the vibrational states which includes two symmetry A states and one symmetry B2 state (see Section 8.4). As the vibrational basis, we use the unit vectors from equation (18.5). Assume that two A states are specified by the unit vectors js0 i and js2 i. Note that the latter no longer relate to the ground and excited states of one mode. The state B2 for jpj  2 can be specified by the unit vector ja1 i or the unit vector ja2 i The resulting descriptions are equivalent. Choose the unit vector ja2 i. The admissible vibrational operators (of power not higher than four of the components of coordinate spin) belong to the irreducible representations A and B2 of the D2 group. As in equation (18.13), but in the space of unit vectors js0 i js2 i, and ja2 i we write all the vibrational operators through the bra and ket vectors. As a result, there remain four t-even operators of the A type I;

p32 ;

2 2 pC C p ;

4 4 pC C p ;

(18.17)

one t-odd operator of the A type 2 2  p C ; iŒp3 ; pC

(18.18)

two t-even operators of the B2 type 2 2  p /; i.pC

4 4 i.pC  p /;

(18.19)

and two t-odd operators of the B2 type p3 ;

2 2 Œp3 ; pC C p C :

(18.20)

In all, we have nine independent operators. The description of the first triad, defined by these operators, is completely equivalent to that in Section 8.4. We emphasize that a

354

Chapter 18 Algebraic models of the global description of molecular spectrum

description of any isolated fragment implicitly takes into account all the intramolecular interactions admissible for a given symmetry. Naturally, a similar global description can be constructed for any rigid molecule. It should be kept in mind that the n-fold excited degenerate vibration is described by the symmetric nth power of the point-group irreducible representation to which this vibration belongs (see Section 2.5). However, it is most important that this approach is easy to generalize to the case of nonrigid molecules. This will be shown in the next two sections.

18.2 The nonrigid methanol molecule CH3 OH The classification of the spectrum of the nonrigid molecule CH3 OH and the construction of an algebraic model describing the torsion-rotational spectrum in the type A0 vibronic state were considered in Section 9.2. It is necessary to generalize this model to the entire discrete spectrum in the ground electronic state. Twelve vibrational degrees of freedom of the methanol molecule refer [84] to the following irreducible representations of its point group CS : : (18.21) vib D 8A0 C 4A00 : Torsional motion in the limit of a rigid molecule goes over to a type A00 vibration. We will call this vibration nonrigid, and all the remaining vibrations rigid. Consider first the function space of the torsional excitation spectrum. It is necessary to write a complete set of basis unit vectors and determine their properties for transformations of the H6 D H3 ^ C2 group describing the actual structure of the nonrigid molecule spectrum. The basis unit vectors will be constructed from the vectors jmi specified in the torsional group H3 and with respect to the T operation by the relations .c31  E/jmi D exp.im2=3/jmi;

(18.22)

T jmi D jmi;

(18.23)

where m is an integer. By virtue of the properties of the semidirect product, the elements of the C2 group act in the basis determined by the transformations of the H3 group. We choose this action in the form .x/

C2 jmi D j  mi

(18.24)

and form the following basis torsional unit vectors: js0 i D j0i;

jsm6D0 i D

jmi C j  mi ; p 2

jam6D0 i D i

jmi  j  mi : p 2

(18.25)

Their decomposition into irreducible representations of the H6 group is shown in Table 18.3, where the components of the representation E are defined by the conditions .x/

C2 E1 D E1

.x/

and C2 E2 D E2 :

(18.26)

355

Section 18.2 The nonrigid methanol molecule CH3 OH

Table 18.3. Decomposition of the torsional unit vectors into irreducible representations of the H6 group. H6

Torsional unit vectors

A1

js3n i

A2

ja3nC3 i

E

E1 D js3nC1 i

E1 D js3nC2 i

E2 D ja3nC1 i

E2 D ja3nC2 i

The rotational functions belong to the irreducible representations A1 and A2 of the H6 . group. Therefore, we find that the ground electronic state is specified by the unit vectors js0 i and js1 i; ja1 i. Indeed, then the functions j0; 0ijs0 i and j0; 0i.js1 i; ja1 i/ refer to the representations A1 and E, which is consistent with the classification for vib D A0 . The singly excited state A00 of a nonrigid mode is specified by the unit vectors js2 i; ja2 i, and ja3 i, which is consistent with the classification for vib D A00 . The doubly excited state A00  A00 D A0 is specified by the unit vectors js3 i; js4 i, and ja4 i. This chain of excitations can readily be continued. Thus, any excitation of the vibrational mode corresponding to the torsional motion is specified by three torsional unit vectors. Naturally, their number coincides with the number of independent equilibrium configurations of the methanol molecule mixed by the torsional motion. One unit vector describes the nondegenerate level (the A1 or A2 symmetry types in the H6 group), while the remaining two vectors describe the doubly degenerate level (the E symmetry type). Note that all the chosen torsional unit vectors are t-even. We now consider the function space of a rigid mode of the A0 type. It follows from the classification that this mode in the C2 group describing the actual structure of the rigid molecule spectrum is of the A type, to which all of its excitations also belong. The complete set of basis unit vectors will be formed from the vectors jpi, as in equation .x/ (18.25). The action of the operations T and C2 on these vectors will be specified by the relations of forms (18.23) and (18.24). Then the s and a unit vectors belong, respectively, to the irreducible representations A and B of the C2 group, and all the unit vectors are t-even. The chain of unit vectors describing the sequential excitations of the type A mode has the form v D 0 ! js0 i;

v D 1 ! js1 i;

v D 2 ! js2 i;

v D 3 ! js3 i    :

(18.27)

Since the operations of the H3 group do not act on the basis unit vectors of a rigid mode, the s unit vectors refer to the irreducible representation A1 of the H6 group.

356

Chapter 18 Algebraic models of the global description of molecular spectrum

It remains to consider the function space of a rigid mode of the A00 type. According to the classification, this mode in the C2 group is of the B type. The complete set of basis unit vectors will be formed from the vectors jqi by analogy with the type A mode. Then the chain of unit vectors describing the sequential excitations of the type B mode has the form v D 0 ! js0 i;

v D 1 ! ja1 i;

v D 2 ! js2 i;

v D 3 ! ja3 i    :

(18.28)

It is clear that the a unit vectors in the H6 group refer to the irreducible representation A2 . The basis unit vectors of the total function space are obtained by multiplication of the basis unit vectors of the function spaces of separate motions. It is then necessary to write out the complete sets of self-adjoint operators in the function spaces of considered motions and decompose them according to symmetry properties in the H6 group and relative to the T operation. We begin with the operators in the torsional space, the complete set of which will be constructed on the basis of the operator of coordinate spin e. It is required to find the representation of Lie algebra (7.27) in the space of the vectors jmi (an upper bound on the number jmj will not be set in explicit form). Let e3 be a diagonal operator with the eigenvalues m, and e˙ D e1 ˙ ie2 be the raising and lowering operators. From the symmetry properties of the vectors jmi it follows that the operator e3 refers to the representation A of the H3 group and is t-odd, while the operators eC and e refer to the representations "1 and "2 , respectively (as the vectors j1i and j  1i/, and T eC T D e . On passage to the group H6 , we obtain e3 ! A2 ;

eC C e ! E1 ;

i.eC  e/ ! E2 ;

(18.29)

where the components of the representation E are defined by conditions (18.26). Only the type A1 and type A2 torsional operators will be needed in what follows. A complete set of such operators with power not higher than four of the components of coordinate spin is listed in Table 18.4. Table 18.4. Complete set of torsional operators of the CH3 OH molecule with power not higher than four of the components of coordinate spin. H6

t-even

t-odd

A1

3 3 I; e32 ; e34 ; eC C e

3 3 iŒe3 ; eC  e C

A2

3 3 i.eC  e /

3 3 e3 ; e33; Œe3 ; eC C e C

The complete set of self-adjoint operators in the function space of the rigid vibrational mode of the A type in the C2 group will be constructed on the basis of the operator of coordinate spin p. The generators, besides the diagonal operator p3 are the

357

Section 18.2 The nonrigid methanol molecule CH3 OH

operators pC C p and i.pC  p /. It is easy to show that in the C2 group, these operators refer to the following irreducible representations: p3 ! B;

pC C p ! A;

i.pC  p / ! B;

(18.30)

where the first operator is t-odd and the other two are t-even. The function space of the mode is formed entirely by the type A states. Therefore, only the type A vibrational operators are feasible in the C2 group and, as a consequence, only the type A1 vibrational operators are feasible in the H6 group. A complete set of these operators is listed in Table 18.5. Table 18.5. Complete set of vibrational operators of the type A mode of the CH3 OH molecule. H6

t-even

t-odd

A1

n n I; Œp32m; pC C p C

nC1 nC1 iŒp32mC1 ; pC  p C

The complete set of self-adjoint operators in the function space of a rigid vibrational mode of the B type will be constructed on the basis of the operator of coordinate spin q. For the generators, we have the following symmetry properties in the C2 group: q3 ! B;

qC C q ! A;

i.qC  q / ! B;

(18.31)

where the first operator is t-odd and the other two are t-even. The function space of the mode includes the states of both types in the C2 group. Therefore, vibrational operators of both types are feasible as well. However, it should be taken into account that even excitations always belong to the type A and odd excitations, to the type B. Therefore, the type A vibrational operators are admissible only with even powers of q˙ and the type B vibrational operators, only with odd powers of q˙ . A complete set of these operators belonging to the types A1 and A2 in the H6 group is listed in Table 18.6. Table 18.6. Complete set of vibrational operators of the type B mode of the CH3 OH molecule. H6

t-even

t-odd

A1

2n 2n I; Œq32m; qC C q C

2nC2 2nC2 iŒq32mC1 ; qC  q C

A2

2nC1 2nC1 iŒq32m ; qC  q C

2nC1 2nC1 Œq32mC1 ; qC C q C

For the effective operators of physical quantities which are invariant under the operation i and change sign in this transformation, the representations A1 and A2 , respectively, are allowed in the H6 group. In particular, the effective Hamiltonian refers to

358

Chapter 18 Algebraic models of the global description of molecular spectrum

the representation A1 and the effective operator of the electric dipole moment, to the representation A2 . First we construct these effective operators when only the torsional and rotational motions are explicitly taken into account. Since the rotational operators in the H6 group belong only to the representations A1 and A2 , only the structures given by equation (9.15) contribute to the torsion-rotational Hamiltonian. That is, the torsional operators of the E type are not involved in the construction, and such a situation occurs for the effective operators of any physical quantities. According to equation (9.15), the expression for the Hamiltonian comprising terms of power not higher than four of the coordinate spin components can be written as 3 3 4 3 3 C e ; e3 ; iŒe3 ; eC  e C º  A H D ¹I; e32 ; eC 3 3 3 3  e /; e33 ; Œe3 ; eC C e C º  B: C¹e3 ; i.eC

(18.32)

Each term in equation (18.32) is a combination of products of the angular momentum components in the MCS belonging to the representations A or B of the C2 group, and the parameters before the combination are linearly dependent on the torsional operators correlating with it. The resulting Hamiltonian explicitly includes all interactions of the torsional and rotational motion and implicitly takes into account all the intramolecular interactions admissible for a given symmetry. The effective electric dipole operator is formed by the structures given by equation (9.24). Component ˛ can be written as (8.62), where the parameters dO .i / are now the torsional operators. The quantities ˛i refer to the following irreducible representations of the H6 group:

˛x ! A1 ;

. ˛y ; ˛z / ! A2 :

(18.33)

Consider the contributions to equation (8.62) not related with a weak rotational distortion of the molecule. Then with allowance for the terms of power not higher than four of the components of coordinate spin we have 3 3 3 3 ˛ D ˛x d .x/i.eC  e / C ˛y Œd1.y/ I C d2.y/ e32 C d3.y/ .eC C e / C d4.y/ e34  3 3 C e / C d4.z/ e34 ; C ˛z Œd1.z/ I C d2.z/ e32 C d3.z/ .eC .y/

.z/

(18.34)

where d .x/ , dk , and dk are the real phenomenological constants. Thus, we have constructed a rigorous algebraic model describing the most important excitation band to study the torsional motion in the methanol molecule CH3 OH in the ground electronic state. Naturally, as a particular case this model includes a description of the torsion-rotational spectrum in the ground vibronic state, which was obtained in Section 9.2. For passage to the latter description, it is needed to retain only the unit vectors with jmj  1 in the torsional space and only the torsional operators I; e3 , and e32 in the effective operators of physical quantities. Then the description of the torsional excitation band should be extended to take into account the excitations of all rigid vibrational modes of interest, which is a fairly

359

Section 18.3 The nonrigid water molecule H2 O

simple procedure. The full version includes the operators formed on the basis of eight coordinate spins p.i / (eight rigid modes of the A0 type), three coordinate spins q.i / (three rigid modes of the A00 type), the coordinate spin e (torsional motion) and the angular momentum J (rotational motion). It is important that the vibrational, torsional, and rotational operators introduced in such a way commute with each other. Naturally, such a model in the form of cross terms for different types of motion rigorously takes into account the interactions between them in explicit form. We emphasize that the correctness of the model is limited only to the correct choice of the internal dynamics symmetry.

18.3 The nonrigid water molecule H2 O Classification of the spectrum of a nonrigid molecule H2 O and the construction of an algebraic model describing the spectrum in the type A0 vibronic state were considered in Section 15.3. It is required to generalize this model to the entire discrete spectrum in the ground electronic state. The fact that instead of the finite group C2v for a rigid molecule, a continuous group D1 h appears in the description of a nonrigid molecule indicates that the internal dynamics radically changes when nonrigid transitions are taken into account. Indeed, group C2v leads to a 3C3 division into rotational and vibrational degrees of freedom, while the group D1 h leads to a 2C4 division. Here, the role of the nonrigid transitions reduces exactly to the rearrangement of the vibration-rotational configuration space, which should underlie the construction of a correct description. The actual energy-level structure of a nonrigid molecule of water H2 O is described by the subgroup H2  D1 of the D1 v  Ci group. As was mentioned in Section 15.3, the ground electronic state in the H2  D1 group refers to the representation A1s and three vibrational modes, to the representations A1s ; A2a , and E1a . The first two modes will be called rigid modes and the third one, a nonrigid mode. The group D1 can be represented in the form (15.17), and in this group only the group U2 transformations act on the wave functions of rigid modes. Consider first the function spaces of rigid modes. The complete set of basis unit vectors of the type A1s mode will be formed from the vectors jpi determined by the relations U2.x/ jpi D exp.ip/jpi D .1/p jpi;

(18.35)

I ijpi D .1/ jpi;

(18.36)

p

where p is an integer. Since the irreducible representations of the groups U2 and H2 are real, transformation with respect to the T operation can be chosen in the form T jpi D jpi:

(18.37)

Then it is easy to see that in the H2  D1 group, the vectors j2pi refer to the representation A1s and the vectors j2p C 1i, to the representation A2a . Therefore, the chain

360

Chapter 18 Algebraic models of the global description of molecular spectrum

of unit vectors describing the sequential excitations of the type A1s mode has the form v D 0 ! j0i;

v D 1 ! j2i;

v D 2 ! j4i;

v D 3 ! j6i    :

(18.38)

In this case, the set of unit vectors is chosen in such a way that the vectors with the negative values of p are not used in the formation of this set. Importantly, the set is closed relative to the group H2 D1 transformations and the T operation. We will not specify explicitly the upper boundary of the integer p (the boundary can be connected, for example, with the number of considered excitations of this mode). The coordinate multiplets in the H2  D1 group may have only representations (15.11). Therefore, the vibration-rotational unit vectors have the form A1s ! jJ; 0ij2pi; A2s ! jJ; 0ij2pi;

J is even, J is odd.

(18.39)

Note that they are consistent with the energy-level classification obtained in Section 15.3. The complete set of self-adjoint operators in the function space of the type A1s rigid mode will be constructed on the basis of the operator of coordinate spin p. The generators are the diagonal operator p3 , as well as the operators pC C p and i.pC  p /. It is easy to show that in the H2 D1 group, these operators refer to the following irreducible representations: p3 ! A1s ;

pC C p ! A2a ;

i.pC  p / ! A2a ;

(18.40)

where the first two operators are t-even and the third one is t-odd. The effective Hamiltonian in the H2 D1 group belongs to the type A1s . Therefore, a general expression for the Hamiltonian can be written as H D

1 X

2t 2t c2p;s;2t J 2p Œp3s ; pC C p C ;

(18.41)

p;s;t D0

where c2p;s;2t are the real phenomenological constants. Clearly, the parts that mix unit vectors (18.38) with the missing unit vectors with p < 0 are removed in the offdiagonal operators in equation (18.41) (see Section 17.2). Note that with this choice of the vibrational unit vectors, the effective Hamiltonian includes both even and odd powers in operator p3 . It should be emphasized that there is freedom in the choice of the relations that specify the symmetry of the basis unit vectors. It is natural that a good choice leads to a higher efficiency of using the obtained algebraic models in the experimental data fitting. As for the operator of the electric dipole moment, it is of the A1a type and is identically equal to zero for this mode. The complete set of basis unit vectors of the type A2a rigid mode will be formed from the vectors jqi defined as in equations (18.35)–(18.37). Therefore, the chain of unit vectors describing sequential excitations of the type A2a mode has the form v D 0 ! j0i;

v D 1 ! j1i;

v D 2 ! j2i;

v D 3 ! j3i    :

(18.42)

361

Section 18.4 Conclusions

Accordingly, for the vibration-rotational unit vectors of this mode we have A1s ! jJ; 0ij2qi;

J is even,

A2s ! jJ; 0ij2qi;

J is odd,

A1a ! jJ; 0ij2q C 1i;

J is odd,

A2a ! jJ; 0ij2q C 1i;

J is even.

(18.43)

Naturally, they are also consistent with the classification. The complete set of selfadjoint operators in the space of unit vectors jqi will be constructed on the basis of the operator of coordinate spin q. In this case, the generators are introduced by analogy with equation (18.40). Further, it is easy to write out the expressions for the effective operators of physical quantities. Since the vibrational unit vectors in equation (18.42) are of the types A1s and A2a , vibrational operators of these two types are also included into the effective operators. We now proceed to a nonrigid mode. It is shown in Section 15.3 that the excitations of this mode decompose into sequences corresponding to the excited vibrational states of the bending mode in the rigid molecule model. Such an arbitrary sequence is also described there under the assumption that it can be considered as isolated. The coordinate spin l is used for this purpose. In order to construct a unified description of these sequences, it is necessary to introduce an additional chain of unit vectors by analogy with equation (18.38) for the type A1s rigid mode. The first unit vector in this chain is the first unit vector of a sequence corresponding to the unexcited bending mode in the rigid molecule model. The second unit vector in the chain is the first unit vector of a sequence corresponding to the first excited state of the bending mode, and so on. The complete set of vibrational operators specified on the additional chain of unit vectors will be constructed on the basis of the operator of coordinate spin p0 . Thus, the full version of the description of excitations in the ground electronic state of a nonrigid molecule of water H2 O includes the operators formed on the basis of the coordinate spin p (rigid mode of the A1s type), the coordinate spin q (rigid mode of the A2a type), coordinate spins l and p0 (nonrigid mode of the E1a type), and the angular momentum J (rotational motion). Naturally, by cross terms over considered motions, this model rigorously takes into account in explicit form the interactions between these motions.

18.4 Conclusions We have formulated some general conclusions based on the content of this chapter. 1. Using equations (12.3)–(12.4) of the BO approximation to describe the internal dynamics of polyatomic molecules is a very complicated problem, especially in the presence of nonrigid motions. Firstly, calculation of the effective nuclear potential by the methods of quantum chemistry with an adequate accuracy for the

362

Chapter 18 Algebraic models of the global description of molecular spectrum

high-resolution spectroscopy requires huge computational resources even in the simplest case of a nondegenerate electronic state. Even with supercomputers, this can be done only for triatomic molecules. Therefore, the effective potential of a polyatomic molecule is mainly represented in the form of different empirical expansions. Secondly, it is required to write the adiabatic Hamiltonian of the nuclear motion in collective variables (taking into account the comments in Section 12.1) and then be able to solve the obtained equation analytically or numerically. The implementation of such an approach is extremely difficult for nondegenerate electronic states even of the simplest nonrigid molecules. Moreover, the corrections to BO approximation equations, which are even more difficult to calculate, often far exceed the accuracy of the experimental data. The symmetry methods of analysis of nonrigid molecules, which are based on the MS group concept, change the picture presented only slightly. The reason is that to use these methods one has to write the nuclear Hamiltonian in collective variables, and at least approximate solutions for its stationary states. Then one specifies the action of the symmetry elements on the collective variables to calculate their action on approximate solutions. The symmetry properties of the approximate solutions, which are obtained as a result of this cumbersome procedure, can only be used for some simplification of the solution refinement scheme. 2. At the same time, the methods presented in this book, which describe the intramolecular dynamics only on the basis of the symmetry principles, make it possible to realize a fairly effective, and one might even say elegant, alternative. The method is essentially as follows. Each internal motion is given by a set of symmetry transformations, which define a complete set of basis wave functions for this motion using the symmetry symbols. All transformations enter the complete symmetry group of internal dynamics, which makes it possible to construct the basis functions of the total motion from the basis functions of separate motions. Then complete sets of self-adjoint operators are formed in the subspaces of separate motions, and these sets are used to construct the effective operators of physical quantities (including the effective Hamiltonian). It appears that the matrix elements of the self-adjoint operators characterizing separate motions need for their calculation only the symmetry properties of the basis wave functions of these motions, but the explicit form of their functions is not needed. The point is that the matrix elements are simple elementary functions of only the symmetry symbols (quantum numbers of the problem). It can be said that symmetry selects the most appropriate basis. Only the spectroscopic constants before the symmetry-admissible combinations of self-adjoint operators remain undetermined in the effective operators. The values of these constants are as a rule found to best fit the experimental data. The obtained model rigorously describes all interactions of different types of internal coordinate motion which are possible within a given symmetry and leads to a simple, purely algebraic scheme for calculation of both the position of the levels in the energy spectrum and transition intensities between them.

Chapter 19

Description of the Zeeman and Stark effects

19.1 External field and symmetry of the stationary states Suppose that the molecule is in a constant magnetic or electric field. It is clear that this molecule undergoes a distortion. A question arises: “What occurs with the symmetry of the internal dynamics in this case?” In fact, it is assumed by default, without discussion, that the external field changes only the symmetry properties of space and time, or in other words, the external symmetry of the problem (see, e.g., [16, 44]). Therefore, in the case of a rigid molecule in a homogeneous magnetic field we have Gext D C1  Ci ;

Gint D Gpoint

(19.1)

and in a homogeneous electric field, Gext D C1 v ;

Gint D Gpoint ;

(19.2)

where the infinite-fold axis coincides with the field direction. Of course, the magnetic field also breaks the symmetry of internal dynamics with respect to the operation of the time sign reversal. Apparently, the assumption that only the external symmetry is broken relies on a fairly large set of purely empirical facts supporting this assumption. It should be noted that using the argument that the molecular distortion is small in moderate fields to explain the applicability of the internal symmetry group of an isolated molecule in this case is not correct. As was mentioned in Section 8.5, the transition from one internal symmetry group to another is not smooth from the viewpoint of changes in the description under small distortions, but leads to a jump. For example, the lowering of the point group C2v of the water molecule H2 O to the minimum possible group CS is accompanied by a jump in the redouble statistical sum because of the change in the nuclear statistical weights. Such a jump should take place even if an arbitrarily small external field is applied. However, the jump has not been observed, and the statistical sum corresponds to the group C2v . Actually, the rigid molecule becomes nonrigid upon application of the external field, and the number of equivalent minima of the effective nuclear potential is determined by the number of equivalent directions of distortion. For example, there are three such directions for the NH3 molecule in the ground electronic state (they correspond to three NH bonds in the equilibrium configuration of symmetry C3v ). The distortiondetermined barriers between the minima can be so small that they will not have a noticeable effect on the internal dynamics. However, the internal geometric symmetry

364

Chapter 19 Description of the Zeeman and Stark effects

is conserved exactly because of the existence of such minima bound by the point-group transformations of a rigid molecule. That is, the mechanism is the same as in the case of conservation of the point group of a rigid molecule if the Jahn–Teller effect occurs (see Section 12.2). In nondegenerate electronic states, the distortions are connected only with the field, whereas in degenerate electronic states, the Jahn–Teller distortions are added. The general picture can be very complex, but the point group of the molecule is conserved. The situation is similar for the molecules which are nonrigid even in the absence of the field, but in this case the extended point group is conserved. As a result, a description of the Zeeman and Stark effects in the rigid and nonrigid molecules can be obtained only on the basis of the symmetry principles. The same mechanism also works in the case of other external actions (e.g., in the case of molecular collisions). This is exactly why the geometric internal symmetry is used with great success in describing precision experimental data of high-resolution spectroscopy.

19.2 The Zeeman effect in the case of a rigid molecule As an example, we will choose a rigid ammonia molecule NH3 . Consider first the excited electronic state of type E. The energy of interaction with the magnetic field in the dipole approximation can be written as [64] mH D mZ H ;

(19.3)

where m is the operator of the coordinate magnetic dipole moment of the molecule and H is the magnetic field vector, whose direction coincides with the direction of the Z axis of the FCS. It is obvious that the operator m belongs to the totally symmetric coordinate Young diagram of the permutation group 3 of identical nuclei. Matching this group with the point group C3v (see Section 8.1), we obtain for the operator m the representation A1 of the C3v group. Such a conclusion is a direct consequence of the result of Section 19.1, according to which each transformation of the C3v group corresponds to the permutation of identical nuclei. The magnetic moment is also in.C/ variant with respect to the operation i, which yields the representation A1 . Thus, the invariance of the coordinate Hamiltonian of an isolated molecule in the C3v Ci group is not violated by the interaction with the magnetic field, which makes it possible to use the results of the classification of stationary states of an isolated molecule from Section 8.1. In particular, the energy-level degeneracies stipulated by the point group are preserved, and their actual structure can be described by the rotational subgroup D3 of the C3v  Ci group. Of course, it should additionally be taken into account that all the levels with a given value of the quantum number J split into a 2J C 1 component over the quantum number M of projection of the angular momentum on the Z-axis in the FCS (in accordance with the one-dimensional irreducible representations of the external group C1 /. In addition, there is no invariance of the Hamiltonian of the molecule with respect to the time reversal operation T .

Section 19.2 The Zeeman effect in the case of a rigid molecule

365

In constructing the function space of bound levels we use the fact that the group D3 can be represented as D3 D C3 ^ U2 ; .y/

where U2 D .E; U2 D  .xz/ i/. As the basis unit vectors of the electronic state E we will choose the unit vectors j ˙ 1e i belonging to a pair of complex-conjugate representations of the C3 group: C3 j ˙ 1e i D exp.˙2 i=3/j ˙ 1e i:

(19.4)

The action of the elements of the factor group U2 is also specified in the space of the basis unit vectors of the invariant subgroup .y/

U2 j1e i D aj  1e i;

(19.5)

where a2 D 1. Thus, the unit vectors j ˙ 1e i realize the representation E of the D3 group, and we can choose a D 1 (otherwise, it suffices to change the sign of one of the unit vectors). Since the unit vectors are specified by a pair of complex-conjugate representations, T j1e i D bj  1e i; (19.6) where jbj2 D 1. One can always take b D 1 (see Section 7.5). We will restrict ourselves to the rotational band in the ground electronic state. Taking into account the behavior of the rotational functions for the group D3 transformations in the MCS (Table 8.4), we find that the electron-rotational basis unit vectors p .jJ; 3n C 1; M ij1e i ˙ jJ; 3n  1; M ij  1e i/= 2 (19.7) p .jJ; 3n  2; M ij1e i ˙ jJ; 3n C 2; M ij  1e i/= 2 refer to the irreducible representations A1 or A2 and the pairs of basis unit vectors (over the signs ˙/ p jJ; 0; M i.j1e i ˙ j  1e i/= 2; p .j1e ijJ; 3n C 3; M i ˙ j  1e ijJ; 3n  3; M i/= 2; p (19.8) .j1e ijJ; 3n  3; M i ˙ j  1e ijJ; 3n C 3; M i/= 2; p .j1e ijJ; 3n  1; M i ˙ j  1e ijJ; 3n C 1; M i/= 2; p .j1e ijJ; 3n C 2; M i ˙ j  1e ijJ; 3n  2; M i/= 2 refer to the irreducible representation E. It is also possible to add the vibrational excitations (see Chapter 18) and obtain the total function space in the required electronic state. The complete set of independent self-adjoint electronic operators is constructed using the operator of coordinate spin ƒ.

366

Chapter 19 Description of the Zeeman and Stark effects

Assume that the unit vectors j ˙ 1e i are the eigenvectors of the ƒ3 operator: ƒ3 j ˙ 1e i D ˙j ˙ 1e i:

(19.9)

Since the eigenvector with the zero eigenvalue is excluded from consideration, of all possible self-adjoint operators only the operators I;

ƒ3 ;

ƒ2C C ƒ2 ;

i.ƒ2C  ƒ2 /

(19.10)

are independent. Here, I is a totally symmetric unit operator and ƒ˙ D ƒ1 ˙ iƒ2 are the raising and lowering operators. Using the symmetry properties of the electronic unit vectors with respect to the group D3 transformations and the operation T, we obtain that the operator ƒ3 belongs to the representation A2 of the D3 group and is t-odd, while the operators ƒ2C C ƒ2 and i.ƒ2C  ƒ2 / belong to the representation E and are t-even. From a matching of the groups C3v  Ci and D3 , we find that the magnetic moment in the D3 group refers to the representation A1 . The component m˛ of this moment on the axis ˛ of the FCS can be written as m˛ D ˛i m O i;

(19.11)

where m O i are the components of the magnetic dipole moment in the MCS, which depend on the electronic operators (19.10) and the angular momentum operators. The direction cosines relate to the following irreducible representations of the D3 group:

˛z ! A2 ;

. ˛y ; ˛x / ! E :

(19.12)

Since the magnetic moment is t-odd, we obtain the following expression for the main contribution to the Z component: mZ D g Zz ƒ3 :

(19.13)

Consequently, this contribution is due to the electronic motion. The value of the real phenomenological constant g is therefore determined by the Bohr magneton 0 . The field-linear contribution to the splitting of stationary states is specified by the diagonal matrix element of interaction operator (19.3). The matrix element Zz of the direction cosine, which is needed for the calculation, has the form (see Appendix V) hJ; k; M j Zz jJ; k; M i D kM=J.J C 1/ :

(19.14)

Of course, the stationary states are usually combinations of the unit vectors of equal symmetry, but this effect, requiring quantitative calculation, does not change the general qualitative picture. The basis unit vectors (19.7) are divided into two parts with the electronic unit vector coinciding or not coinciding with k in sign. For one part, the positive values of M correspond to the energy downshift and the negative values of

Section 19.3 The Zeeman effect in the case of a nonrigid molecule

367

M , to the energy upshift, while for other part the situation is exactly the opposite. The basis unit vectors (19.8) are divided already into three parts. The first is the unit vectors of the first row. For them, there is no field-linear contribution to the splitting. The remaining unit vectors are divided into two parts like the unit vectors (19.7). Thus, in a set of unit vectors with one value of M , the linear Zeeman effect abruptly changes even in sign. Note that the choice of unit vectors is ambiguous and, consequently, the Zeeman effect is ambiguous for them. We now consider the nondegenerate ground electronic state of a rigid molecule NH3 , limiting ourselves to the rotational band in the ground vibrational state. In this case, for the main contribution to the Z component of the magnetic dipole moment we obtain the following expression: mZ D g3 Zz Jz C g? . Zy Jy C Zx Jx /:

(19.15)

Consequently, this contribution is due to the rotation of the molecule as a whole. The values of the real phenomenological constants g3 and g? are therefore determined by the nuclear magneton n , which is about three orders of magnitude smaller than the Bohr magneton. In equation (19.15), it was taken into account that ŒJi ; ˛k  D i"i kl ˛l :

(19.16)

That is, the rotational operators Ji and Zi with the same values of the index i commute with each other. Elementary calculations yield the following result: h hJ; k; M jmZ jJ; k; M i D .g3  g? /

i k2 C g? M: J.J C 1/

(19.17)

The off-diagonal elements of the direction cosines, which are necessary for the calculations, are given in Appendix V. This result is equivalent to the previously known one [44] obtained on the basis of the semiclassical consideration. Note that such a consideration leads to more cumbersome calculations and, most importantly, is difficult to adapt both for the refinement of the results and for more complex cases.

19.3 The Zeeman effect in the case of a nonrigid molecule We now consider briefly a description of the linear Zeeman effect in the presence of nonrigid motions using the methanol molecule CH3 OH as an example. In the ground electronic state, this molecule corresponds to the point group CS , in which the degenerate representations are absent. Such a low symmetry leads to the appearance of a pronounced torsional motion of the methyl top CH3 through a low potential barrier between three energetically equivalent configurations. The analysis of the internal dynamics of an isolated nonrigid molecule is based on an extended point group G12 (see Section 9.1). With the external field applied, all transformations of this group as

368

Chapter 19 Description of the Zeeman and Stark effects

before correspond to the permutations of identical nuclei in the effective nuclear interaction potential that is invariant with respect to these transformations. Therefore, the magnetic moment refers to the totally symmetric representation of the G12 group. The main contribution to the Z component of this moment has the form mZ D g Zz e3 ;

(19.18)

where e3 is the component of the coordinate spin describing the torsional motion (see Section 9.2). From equation (19.18) we readily obtain that this contribution leads to the linear Zeeman effect for degenerate torsional states.

19.4 The Stark effect in the case of a rigid molecule The problem of describing the Stark effect is more complicated since in the electric field, the internal dynamics of the molecule changes much more significantly than in the magnetic field, because of the violation of the invariance of the Hamiltonian with respect to the spatial inversion operation. As an example, we will choose rigid molecules of ammonia NH3 and methanol CH3 OH in their ground electron-vibrational state. For the isolated molecule NH3 in ground electronic state, the total geometric symmetry group has the form (see Section 8.1) R3  C3v  Ci ;

(19.19)

and the actual energy-level structure is described by its rotational subgroup R3  D3 :

(19.20)

The external group R3 can be omitted in the case of an isolated molecule. It should only be taken into account that the isotropy of the space leads to an additional degeneracy of the levels over the quantum number M of the projection of the total angular momentum on Z-axis in the FCS. With the external field applied, the total geometrical group becomes noninvariant. In the case of a homogeneous constant electric field, the Hamiltonian symmetry subgroup of group (19.19) is C1 v  C3v ;

(19.21)

where the rotation axis of the C1 v group coincides with the direction of the electric field (we assume that the field is directed along the Z axis). That is, the external symmetry group of the Hamiltonian reduces to the subgroup C1 v , while the internal group C3v is conserved. It is obvious that the group (19.20), which determines the actual energy-level structure, also becomes a noninvariant group. The Hamiltonian symmetry subgroup of this group is the group H1 . Division of the elements into classes and the table of characters of the latter are given in Table 19.1. The elements of the H1

369

Section 19.4 The Stark effect in the case of a rigid molecule

Table 19.1. Division of the elements into classes and the table of characters of the H1 group. E

E E

2C'

2C'  E

2C3

E  2C3

R1

C'  C3 C'  C31

R2

C'  C3 C'  C31

H1 E A1 A2 E Em 0 Em 00 Em

2C'

1 1 1 1 2 2 2 2 cos m' 2 2 cos m' 2 2 cos m'

2C3 1 1 –1 2 –1 –1

R1

R2

M

1 1 1 1 1 –1 –1 –1 0 2 cos m' 2 cos m' 0 2 cos.m'  2=3/ 2 cos.m' C 2=3/ 0 2 cos.m' C 2=3/ 2 cos.m'  2=3/ 0

M 1 U2 3U2 Note. m D 1; 2; : : :

group are written through products of the elements of the groups D1 and D3 , where the group D1 is a rotational subgroup of the external group C1 v  Ci . Note that the table of characters of the H1 group is easier to construct using the fact that all of its irreducible representations are realized by the rotational unit vectors jJ; k; M i since H1 is a subgroup of the D1  D3 group. It is exactly the group H1 that specifies the energy-level degeneracies in the presence of the electric field. We emphasize that this group is isomorphic with the group obtained in [90] within the framework of the CNPI concept for the analysis of the energy-level degeneracies of rigid molecules with the point group C3v in a homogeneous constant electric field. The reduction of group (19.20) on subgroup H1 , which is necessary to construct a description, is convenient to carry out down the chain: R3  D3 ! D1  D3 ! H1 :

(19.22)

The reduction of the group R3 to a subgroup D1 is given in Table 8.9 and the reduction of the group D1  D3 to a subgroup H1 , in Table 19.2. Decomposition of the rotational basis functions jJ; k; M i into irreducible representations of the H1 group for the ground vibrational state is given in Table 19.3.

Table 19.2. Reduction of the D1  D3 group on a subgroup H1 . D1  D3

H1

A1  A1 , A2  A2

A1

A1  A2 , A2  A1 A1  E, A2  E Em  A1 , Em  A2

A2 E Em

Em  E

0 00 Em C Em

370

Chapter 19 Description of the Zeeman and Stark effects

Table 19.3. Decomposition of the rotational unit vectors into irreducible representation to the H1 group. H1 A1 A2 E EM 0 EM

00 EM

Rotational unit vectors p .jJ; 3n C 3; 0i C .1/k jJ; 3n; 3; 0i/= 2 p .jJ; 3n C 3; 0i  .1/k jJ; 3n  3; 0i/= 2 p .jJ; 3n C 1; 0i ˙ jJ; 3n  1; 0i/=p 2 .jJ; 3n C 2; 0i ˙ jJ; 3n  2; 0i/= 2 p .jJ; 0; M i ˙ jJ; 0; M i/= 2 p .jJ; 3n C 1; M i ˙ jJ; 3n  1; M i/=p 2 .jJ; 3n C 2; M i ˙ jJ; 3n  2; M i/= 2 p .jJ; 3n C 2; M i ˙ jJ; 3n  2; M i/=p2 .jJ; 3n C 1; M i ˙ jJ; 3n  1; M i/= 2

jJ; 0; 0i;

Note. The pair of basis unit vectors over the signs ˙ refers to the two-dimensional representation.

Effective operators of the coordinate physical quantities characterizing the molecule transform according to the totally symmetric representation A1 of the H1 group. On passage to the noninvariant group D1  D3, for such operators we have the representations (19.23) A1  A1 ; A2  A2 : In the group R3 D3 , we additionally obtain that the operators of the A1 A1 and A2  A2 types respectively transform according to the representations D .2n/ and D .2nC1/ in the R3 group, where n D 0; 1; 2; : : :. The main term of interaction with the field for an electrically neutral molecule can be written as [64] E D Z E; (19.24) where  is the operator of the coordinate electric dipole moment and E is the electric field vector. The component ˛ on the ˛ axis of the FCS can be written in the form (8.8). The direction cosines in the external group D1 relate to the irreducible representations

Zi ! A2 ; . Y i ; Xi / ! E1 ; (19.25) and in the internal group D3 to the representations

˛z ! A2 ;

. ˛y ; ˛x / ! E :

(19.26)

371

Section 19.4 The Stark effect in the case of a rigid molecule

As a result, for the main contribution to the Z component of the electric dipole moment we have (19.27) Z D Zz d .z/ ; in compliance with simple geometric considerations. We note that contribution (19.27) refers to the representation A2  A2 of the D1  D3 group and to the representation D .1/ of the R3 group. For the linear Stark effect, a nonzero diagonal matrix element of equation (19.27) is required in the basis of the symmetrized functions from Table 19.3. The diagonal matrix element of the direction cosine has the form (19.14). As a result, 0 and E 00 levels of the H we find that the linear effect takes place only for the EM 1 M group (for the isolated molecule, these levels in the group D3 belong to the type E/, which coincides with the result in [90]. We emphasize that this approach permits one to easily obtain in algebraic form any further corrections in the description of the Stark effect. These results are rigorous since their validity is limited only to the correct choice of the symmetry of internal dynamics of the molecule. We note here two additional points. Firstly, as in the case of an isolated molecule, the inversion doublets are not split in the presence of the field since the independent molecular configurations bound by the v reflections of the C3v group are assumed, as before, to be separated by an impermeable barrier. It is another matter that the levels in the doublets no longer have a definite behavior relative to the operation of spatial inversion i. Indeed, interaction with the field breaks the symmetry of the Hamiltonian with respect to this operation and mixes the (C/ and (–) states. Secondly, the nuclear statistical weights are also similar to the case of an isolated molecule since the group C3v determining them is conserved. For a rigid molecule of methanol CH3 OH in the ground electronic state, the rotational subgroup of the total geometrical group has the form (see Section 9.2) R3  C2 ;

(19.28)

.x/

where C2 D .E; C2 D  .yz/ i/. On application of the electric field, the group (19.28) becomes noninvariant. The Hamiltonian symmetry subgroup of this group is .x/ DQ 1 D .E  E; 2C'  E; 1 U2  C2 / :

(19.29)

The elements of the DQ 1 group are written through products of the elements of the external group D1 and the internal group C2 , where D1 is the rotational subgroup of the C1 v  Ci group. Obviously, the group DQ 1 is isomorphic with the D1 group. The reduction of group (19.28) to a subgroup DQ 1 , which is necessary to construct a description, is convenient to carry out down the chain: R3  C2 ! D1  C2 ! DQ 1 :

(19.30)

The reduction of the group D1  C2 to a subgroup DQ 1 is given in Table 19.4.

372

Chapter 19 Description of the Zeeman and Stark effects

Table 19.4. Reduction of the group D1  C2 group to a subgroup DQ 1 . DQ 1

D1  C2 A1  A, A2  B

A1

A1  B, A2  A Em  A, Em  B

A2 Em

Decomposition of the rotational basis functions into irreducible representations of the DQ 1 group is given in Table 19.5. It is important that, simultaneously, these functions are also symmetrized with respect to the irreducible representations of the C2 group. The effective rotational operators of any coordinate physical quantities belong to the representation A1 of the DQ 1 group. On passage to the noninvariant group D1  C2 , for such operators we have the representations A1  A;

A2  B:

(19.31)

In the group R3 C2 , we additionally obtain that the operators of the A1 A and A2 B types transform, respectively, according to the representations D .2n/ and D .2nC1/ of the R3 group. Table 19.5. Decomposition of the rotational unit vectors into irreducible representations of the DQ 1 group. DQ 1

Rotational unit vectors p .jJ; k; 0i C jJ; k; 0i/= 2 p .jJ; k; 0i  jJ; k; 0i/= 2 p .jJ; k; M i C jJ; k; M i/= 2 p .jJ; k; M i C jJ; k; M i/= 2 p .jJ; k; M i  jJ; k; M i/= 2 p .jJ; k; M i  jJ; k; M i/= 2

A1

jJ; 0; 0i;

A2

8