Advances in Molecular Vibrations and Collision Dynamics, Volume 3 9781559387903, 1559387904

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ADVANCES IN MOLECULAR VIBRATIONS AND COLLISION DYNAMICS

MOLECULAR CLUSTERS

Volume3

9 1998

This Page Intentionally Left Blank

ADVANCES IN MOLECULAR VIBRATIONS AND COLLISION DYNAMICS MOLECULAR CLUSTERS Series Editon

JOEL M. BOWMAN Department of Chemistry Emory University

Volume Editors: JOEL M. BOWMAN Department of Chemistry Emory University ZLATKO BA(~I(~ Department of Chemistry New York University VOLUME3

9 1998

@ Stamford, Connecticut

JAI PRESS INC. London, England

Copyright 91998 by JAI PRESSINC. 100 Prospect Street Stamford, Connecticut 06901-1640 JAI PRESSLTD. 38 Tavistock Street Covent Garden London WC2E 7PB England All rights reserved. No part of this publication may be reproduced, stored on a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, filming, recording, or otherwise without prior permission in writing from the publisher. ISBN: 1-55938-790-4 ISSN: 1063-5467 Manufactured in the United States of America

CONTENTS

LIST OF CONTRIBUTORS

vii

PREFACE

Zlatko Ba&i~ and Joel M. Bowman

ix

MOLECULAR CLUSTERS: REAL-TIME DYNAMICS AND REACTIVITY

Jack A. Syage and Ahmed H. Zewail ENERGETICS AND DYNAMICS OF ARGON-WATER PHOTODISSOCIATION

Kurt M. Christoffel and Joel M. Bowman

61

INTERACTIONS BETWEEN CN RADICALS AND RARE GAS ATOMS: COLLISIONS, CLUSTERS, AND MATRICES Michael C Heaven, Yaling Chen, and

William G. Lawrence

91

VIBRATIONAL SPECTROSCOPY OF SMALL SIZE-SELECTED CLUSTERS

Udo Buck

127

QUANTUM MONTE CARLO VIBRATIONAL ANALYSIS AND THREE-BODY EFFECTSIN WEAKLY BOUND CLUSTERS

Clifford E. Dykstra

163

VIBRATION-ROTATION-TUNNELING DYNAMICS OF (HF)2 AND (HCI)2 FROM FULL-DIMENSIONAL QUANTUM BOUND-STATE CALCULATIONS

Zlatko Ba~i6 and Yanhui Qiu

183

vi

CONTENTS

SPECTROSCOPY AND QUANTUM DYNAMICS OF HYDROGEN FLUORIDE CLUSTERS

Martin Quack and Martin A. Suhm

205

THE INFRARED SPECTROSCOPYOF HYDROGEN-BONDED CLUSTERS: CHAINS, CYCLES, CUBES, AND THREE-DIMENSIONAL NETWORKS

Timothy $. Zwier

249

AB INITIO CHARACTERIZATION OF WATER AND ANION-WATER CLUSTERS

$otiris S. Xantheas and Thorn H. Dunning, Jr.

281

DIFFUSION MONTE CARLO STUDIES OF WATER CLUSTERS

Jonathon K. Gregory and David C. Clary

311

REARRANGEMENTS AND TUNNELING IN WATER CLUSTERS

David J. Wales

365

SPECTROSCOPY AND MICROSCOPIC THEORY OF DOPED HELIUM CLUSTERS

K. B. Whaley INDEX

397 453

LIST OF CONTRIBUTORS

Zlatko Baff.i#.

Department of Chemistry New York University New York, NY

Joel M. Bowman

Department of Chemistry Emory University Atlanta, GA

Udo Buck

Max-Planck Institut fCir StrOmungsforchung GOttingen, Germany

Yaling Chen

Department of Chemistry Emory University Atlanta, GA

Kurt M. Christoffel

Department of Chemistry Emory University Atlanta, GA

David C Clary

Department of Chemistry University College London London, England

Thom H. Dunning, Jr.

Environmental Molecular Sciences Laboratory Pacific Northwest National Laboratory Richland, WA

Clifford E. Dykstra

Department of Chemistry Indiana UniversitymPurdue University Indianapolis, IN

Jonathon K. Gregory

Department of Chemistry University of Cambridge Cambridge, England vii

viii

LIST OF CONTRIBUTORS

Michael C. Heaven

Department of Chemistry Emory University Atlanta, GA

William G. Lawrence

Department of Chemistry Emory University Atlanta, GA

Yanhui Qiu

Department of Chemistry New York University New York, NY

Martin Quack

Laboratorium fLir Physikalische Chemie Zurich, Switzerland

Martin A. Suhm

Laboratorium ffir Physikalische Chemie Zurich, Switzerland

Jack A. Syage

Syagen Technology, Inc. Tustin, CA

David J. Wales

University Chemical Laboratories Cambridge, England

K.B. Whale;,

Department of Chemistry University of California, Berkeley Berkeley, CA

$otiris S. Xantheas

Environmental Molecular Sciences Laboratory Pacific Northwest National Laboratory Richland, WA

Ahmed H. Zewail

Arthur Amos Noyes Laboratory of Chemical Physics California Institute of Technology Pasadena, CA

Timothy S. Zwier

Department of Chemistry Purdue University West Lafayette, IN

PREFACE

Weakly bound, van der Waals and hydrogen-bonded clusters have received a great deal of attention from experimentalists and theorists alike in the past two decades. As is often the case, the surge of interest in these systems has been driven in part by impressive experimental advances, primarily the development of methods for synthesizing clusters of variable size and a variety of laser spectroscopic techniques for probing cluster properties in time and frequency domains. Another compelling reason to study clusters has been the realization that they provide an exceptional vehicle for exploring the microscopic aspects of a wide range of macroscopic phenomena of fundamental importance in chemistry, physics, and biology. This is due to two key advantages that clusters hold over bulk matter. One is the possibility to vary the cluster size in a controlled, stepwise fashion, and observe experimentally how diverse physical and chemical properties evolve from those characteristic for isolated molecules towards their respective macroscopic, bulk limits. In this sense, the clusters truly constitute a bridge which spans gas-phase molecules and condensed phases. By building condensed matter one particle, atom or molecule, at a time, it is possible to gain quantitative understanding of the forces and dynamical processes operating in the bulk, with clarity and the level of detail that could not be achieved otherwise. The second crucial advantage of molecular clusters stems from extremely low temperatures at which they are formed in supersonic jets. With virtually no internal excitation, such clustersare ideally suited to high-resolution spectroscopy. Moreix

x

PREFACE

over, ultra cold clusters populate appreciably only a narrow range of low-lying isomeric structures, which change little on the time scales of experiments performed in molecular beams. This well-defined environment can provide a particularly clear atomic-scale picture of intermolecular interactions, patterns of energy flow, and chemical reaction dynamics, which is not obscured by the disorder, spatial and temporal inhomogeneities unavoidable in bulk matter. The distinctive properties of clusters mentioned above, which have been so valuable to experimentalists, are of great importance for theorists too. The relatively small number of degrees of freedom, well-characterized geometry, and control over the size, and therefore, the complexity of the system under investigation, make clusters conceptually and computationally simpler than condensed phase, allowing theoretical simulations with degree of rigor and, often, quantum-state-specific information that would not be feasible for bulk liquids and solids. In fact, the hallmark of the field of cluster research, indeed the main reason for its vibrancy, has been the unusual synergy between the most sophisticated experiments and state-of-the-art theory. The rich stream of fascinating experimental findings has spurred the development and implementation of innovative theoretical methods, quantum and classical, for calculating the structural, spectroscopic, and chemical properties of clusters. Theoretical results have proved indispensable for the analysis and interpretation of the experimental data, and theoretical predictions have stimulated and guided further experiments. Research involving clusters has grown in scope so enormously that no single book can hope to cover it completely. This volume focuses on molecular clusters, bound by van der Waals interactions and hydrogen bonds. Twelve chapters review a wide range of recent theoretical and experimental advances in the areas of cluster vibrations, spectroscopy, and reaction dynamics. The authors are leading experts, who have made significant contributions to these topics. The first chapter, by Syage and Zewail, describes the exciting results and new insights in the solvent effects on the short-time photo fragmentation dynamics of small molecules, obtained by combining heteroclusters with femtosecond laser excitation. The contribution by Christoffel and Bowman is on a related theme, and deals with their theoretical work on the effects of a single solvent (argon) atom on the photodissociation dynamics of the solute H20 molecule. Interactions between CN radicals and rare-gas atoms, clusters, and matrices are describeA in the chapter by Heaven, Chen, and Lawrence. The following two chapters cover various experimental and theoretical aspects of the energetics and vibrations of small clusters. The chapter written by Buck gives an overview of the spectroscopy of size-selected neutral clusters, an area in which he has been a pioneer. The theoretical contribution by Dykstra describes diffusion quantum Monte Carlo (DQMC) calculations and non additive three-body potential terms in molecular clusters.

Preface

xi

The next six chapters deal with hydrogen-bonded clusters, reflecting the ubiquity and importance of hydrogen bonds, and the need to understand the structures and intricate dynamics of hydrogen-bonded networks. Ba~i~ and Qiu present a full-dimensional quantum treatment of the vibration-rotation-tunneling dynamics of HF and HCI dimers, while Quack and Suhm review the spectroscopy and DQMC calculations of larger BF clusters. The chapter by Zwier describes his incisive infrared spectroscopy of benzene-water clusters, which has led to experimental determination of the geometries of smaller water clusters, together with the far-infrared (FIR) spectroscopy of water clusters by Saykally and co-workers. Xantheas and Dunning review their high level ab initio characterization of the energetics and vibrations of water and water anion clusters. Gregory and Clary present the DQMC studies of water clusters conducted by them which, among other things, have been essential for establishing the cage structure of the water hexamer observed in the FIR experiments by Saykally. In his contribution, Wales gives an elegant theoretical treatment of the rearrangements and dynamics of water clusters, providing qualitative mechanistic interpretation for the observed tunneling splittings. The final chapter, by Whaley, provides the microscopic theory of the dynamics and spectroscopy of doped helium clusters, highly quantum systems whose unusual properties have been studied extensively in the past couple of years. Joel Bowman Zlatko B a~ir

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MOLECULAR CLUSTERS" REAL-TIME DYNAMICS AND REACTIVITY

Jack A. Syage and Ahmed H. Zewail

1. 2.

3. 4.

5. 6. 7.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bond Dynamics: Dissociation and Caging . . . . . . . . . . . . . . . . . . . . 2.1. I2/Xn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. I 2 / X n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Comparison with Condensed-Phase Studies: Solids and Liquids . . . . . Electron Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Bzn/I2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proton Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. ROH*/Bn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Double Proton Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. (NH3)n and Na(NH3)n . . . . . . . . . . . . . . . . . . . . . . . . . . . Aligned Bimolecular Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . Barrier Crossing: trans-Stilbene Photoisomerization . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Tunneling Model for Proton Transfer in Clusters . . . . . . . . . . . Note Added in Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Advances in Molecular Vibrations and Collision Dynamics, Volume 3, pages 1-60. Copyright 9 1998 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 1-55938-790-4

2 2 5 5 17 19 21 21 31 32 41 44 46 51 53 53 54 56 56

2

JACK A. SYAGE and AHMED H. ZEWAIL

ABSTRACT In this chapter we present a review of the field of real-time chemical dynamics in clusters, with specific examples to illustrate the new level of understanding reached for microscopic solvation and reactivity. The experimental examples presented were chosen to represent a progression of chemical complexity, ranging from elementary bond breaking, to electron transfer, to proton transfer, and to bimolecular chemistry in aligned complexes. The chapter begins with a discussion of the fundamental processes of dissociation and recombination dynamics in solvent cages for the prototypical neutral and ionic cluster systems. We eventually discuss reactions in larger systems, elucidating the elementary steps of proton and double-proton transfer in acid-base and isomerization reactions. In describing recent work, we highlight the new experimental techniques designed to extract new dimensions in the chemical dynamics in clusters, namely time- and state-resolved measurements with product velocity and spatial angular resolutions.

1. I N T R O D U C T I O N Experiments on molecular complexes and clusters formed in supersonic beam expansions are providing new levels of understanding of the effect that individual molecules have on the properties of chemical reactions in particular and on solvation in general. These aggregates can be studied by laser probes and their size-dependence sorted out by mass spectrometry and spectroscopy. A number of books and chapters have appeared on the subject (e.g. refs. 1, 2) and the excellent recent article by Castleman and Bowen gives an overview of the experimental progress made so far.3 Progress in lasers and molecular beams have allowed single-quantum state studies. Ultrafast lasers have reduced the time scale of study to that of elementary bond breaking and bond-making processes. Laser polarization techniques have been used to achieve molecular alignment, making possible measurements of angular distributions of products. And in sophisticated new experiments, all of these laser techniques have been brought to bear on molecular level samples. In fact, techniques to measure chemical properties of clusters have reached the point where direct comparisons with gas-phase and condensed-phase reactions can be made. Solvation under controlled conditions of size and composition offers an opportunity to examine different phenomena of reactivity at the microscopic level. Some of the important new details being learned about solvent influence on reacting molecules include the following: 9 9 9 9

Coherent motion of solvent molecules. Geometry and structure of the solvent about a solute molecule. Energies of interaction by individual molecules. Vibrational mode structure and dynamics.

Molecular Clusters

3

Solvent critical number in phenomena such as electron and proton transfer and caging.

9

Because such processes can be studied at the microscopic level, clusters are an ideal medium for understanding the connection between gas-phase and condensed-phase phenomena and in learning about the breakdown of bulk-phase properties as a system becomes increasingly smaller in size. The structure of the solvent about a reacting molecule is a longstanding issue and the interactions exerted by the solvent often determine the fate of chemical reactions. Learning about solvent interactions in microscopic detail could potentially lead to important advances in the understanding, and possibly control, of bulk-phase chemistry. In this chapter, we present an overview of the rapidly expanding field of real-time dynamics in clusters. Measurements of cluster dynamics in the time domain began in 1983 with experiments probing excited-state lifetime (of isoquinoline) for various hydrogen bonding solvent molecules (Figure 1).4 To date, there are many groups conducting real-time measurements in clusters encompassing a large scope of interests; a number of reviews on time-resolved studies and on general topics in chemical dynamics in molecular clusters have been reported before. 5-9 Metallic clusters represent a branch for different classes of phenomena, interesting in their own right. In keeping with the format of this book, we focus on more detailed accounts of case studies primarily from the authors' laboratories to illustrate the

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Figure 1. Excited state decay times of isoquinoline unsolvated and solvated as measured by time-correlated single-photon countin 8 in a free-jet expansion (ref. 4).

4

JACK A. SYAGE and AHMED H. ZEWAIL

new kinds of information being gained from these studies. We will present a series of studies chosen to illustrate a progression of chemical complexity ranging from elementary bond breaking, to electron transfer, to proton transfer, and to bimolecular chemistry in aligned complexes. This chapter centers largely on bimolecular solute-solvent chemical interactions; those time domain studies that do not fall under this main theme are summarized in Section 7. In Section 2, we consider one of the most classic and fundamental problems in chemistry, namely what happens when a diatomic molecule dissociates in a solvent cage? Key insights have been provided by a history of time-resolved studies of 12 dissociation/recombination in condensed phases. However, only recently have measurements been conducted in real time on solvated cluster systems such as I2/M,, and I2/M,,, where M,, is a solvent of n molecules of M, as a function of specific cluster size. These experiments have provided critical new understanding regarding the precise interaction of each solvent molecule, leading to cage escape vs. recombination probabilities, and measuring the actual coherent motions of bond breakage and caging in the solvent. In Section 3, we extend the discussion of 12 to complexes I2/Bz,,, Bz being benzene, wherein a competition between a charge transfer and a neutral channel to 12 dissociation occurs. In these studies new multi-dimensional probes involving time-resolved angle-velocity measurements of products are introduced. In Section 4, we cover an extensive series of studies on a prototypical acid-base reaction ROH*-B n ~ RO*- H+Bn (excited-state proton transfer), where ROH is an aromatic acid and B~ is a cluster of base molecules. Direct time domain studies have revealed that changing a single solvent molecule can lead to distinct chemical changes, sometimes affecting reaction rates by over 2 orders of magnitude. These results are making possible the validation of a quantum level model of proton tunneling involving solvent dynamics that may be extended to describe condensed-phase behavior. Section 5 covers sophisticated new techniques building on earlier advances in studying bimolecular chemistry using alignments of molecules in van der Waals complexes. Recently, differential reactive cross sections in angle and velocity have been measured in photodissociation of van der Waals complexes with product quantum-state resolution and femtosecond time resolution. We focus on recent studies on (CH3I)2 and I2/Bzn. The quantum-state-resolved work on (CH3I)2 revealed a bimodal angle-velocity distribution for the I atom fragment, indicating an inequivalency of the two individual molecules, and rapid spin-orbit relaxation in I atom. The real-time measurements showed a 150 fs C - I bond scission and a build up of 12 in less than 500 fs. The I2/Bz ~ work has led to direct observation of the time evolution of the transition state and products for the charge transfer reaction Bz-I 2 ~ [Bz§ ~ ~ Bz-I + I, while monitoring the change in anisotropy and fragment translational energy. The latest set of experiments described in Section 5, incorporating time-resolved and state-resolved differential cross-section measurements in aligned complexes, hold great promise of providing information needed to test quantum chemical

Molecular Clusters

5

dynamics in these finite-sized systems. The van der Waals bond in a dimer complex helps to fix the geometry and limit the range of impact parameters under study. The full dimensionality of how molecules collide and break up into products can be followed in real time using femtosecond excitation and angular probing of product velocities. Section 6 describes the effect of microscopic solvation on isomerization reactions, and in Section 7 we conclude the chapter. An appendix has been added describing a state-to-state theory of proton tunneling appropriate to clusters and extendable to the condensed phase, thus helping to unify the connection between gas-phase and condensed-phase chemistry.

2. BOND DYNAMICS: DISSOCIATION AND CAGING 2.1. 12/X. Vibrational Predissociation (n = 1-4)

Dimer systems offer the simplest picture of solute-solvent interactions. In a series of experiments Gutmann et al. directly measured state-to-state rates of vibrational predissociation for 12 complexed to X = He, Ne, Ar, and H2 .1~ Questions of interest included: does the repulsive potential of the van der Waals bond determine the state-to-state rates of predissociation, and do vibrational and electronic predissociation have the same origin? The state-to-state process can be described by the following,

I*2X(vi)' ~~(vl,v))12 . " (Vf) +

X

where excitation is to the B electronic state.

Experimental: Because of the picosecond time scale for these dynamics and the interest in probing state-specific effects, tunable picosecond duration pulses were used. ~~ The 532 nm output of a Nd:YAG laser was split to synchronously pump two cavity-dumped, etalon-tuned dye lasers. The pump laser was operated in the visible in order to excite specific vibration levels in the B-excited electronic states of 12. The probe laser was operated in the ultraviolet in order to pump electronically excited I2 to an ion-pair state, from which fluorescence was detected. Clusters were formed in a supersonic expansion using two arrangements. A continuous nozzle was used for Ar expansions to take advantage of the maximum repetition rate (800 Hz) of the laser. For H2, He, and Ne expansions, higher backing pressures were required, necessitating the use of a pulsed nozzle operated at about 100 Hz. Excitation to a specific vibrational state ~ of the reactant I2 stretching mode is followed by vibrational redistribution to the vdWs mode. In the case of Ne,

(1)

6

JACK A. SYAGE and AHMED H. ZEWAIL a single I2 quantum exceeds the vdW binding energy hence the product I2 is formed with ~ = ~ - 1 quanta. Real-time measurements of the rate of vibrational predissociation (VP) probe the coupling strength of the 12 reactant vibration to the vdW's mode. ~~ Examples of the product formation times for different initial vibrational excitations are shown in Figure 2. The VP rates increase monotonically with increasing vibrational excitation for Ne. The situation for Ar was different in the following ways: (1) because the vdW's bond energy is much greater, the VP process was favored by the ~ = ~ - 3 channel, and (2) an electronic predissociation (EP) competes with VP.

Interesting questions arise when the number of solvent molecules is increased systematically. 12For example, do the rare gas solvent molecules dissociate sequentially, each event being a VP (direct, sequential mechanism), or does intramolecular vibrational redistribution (IVR) to unreactive modes occur first, followed by sequential VP (indirect, evaporative mechanism)? The state-to-state time-resolved dissociation rates indicate that the onset to IVR occurs for just two Ne atoms. These

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Figure 2. Representative pump-probe transients for vibrational levels v~ = 13, 18, and 23. The dashed line is the fit to an exponential rise function, from which the value of the state-to-state rate is given (refs. 10-12).

Molecular Clusters

7

measurements were extended to n = 3 and 4 Ne atoms to test statistical theories for small cluster systems. Campos-Martinez et al. conducted time-dependent Hartreetype calculations on the VP of I2-Ne 2 clusters. 13 The computations indicate a sequential dissociation of Ne, although some behavior bordering on internal vibrational relaxation was observed for high initial vibrational states. The computed trend in the lifetime vs. initial vibrational state was in very good agreement with experiment (Figure 3).

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t 9

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Figure 3. Snapshots at t = 10 ps when the initial state is v' = 17 and for channels (a) v' = 16, (b) v' = 15, and (c) v' = 14. The system is 12Ne2 and R1 (R2) is the distance from 12 to Nel (Ne2). Comparison of computed lifetimes with experimental lifetimes is also shown (ref. 13).

8

JACK A. SYAGE and AHMED H. ZEWAIL

Dissociation~Recombination in 12/XnMacroclusters By increasing the cluster size new solvent dynamics can be probed. In the case of I z, direct measures of dissociation/recombination dynamics can be obtained and compared to the analogous measurements in bulk solvents and condensed phases. 14 These events occur on much faster time scales than the predissociations described above for the smaller cluster systems. Experiments were conducted on clusters and on high-pressure gases.

Experimental: These experiments required femtosecond time resolution. The cluster experiments made use of a mode-locked Nd:YAG synchronously pumped linear, dispersion-compensated dye laser. ~4The 615 nm output was passed through a Nd:YAG pumped, four-stage dye amplifier and then split into pump and probe pulses. The pump was used to generate a white-light continuum and the probe, frequency doubled to 308 nm. The continuum pulse was filtered and amplified by a 355-nm pumped three-stage dye amplifier to give tunable pump pulses (e.g. 480, 510, 520 nm for B state excitation and 614, 620, and 640 nm for A-state excitation of I2). As above, the probe pulse excites an ion-pair state of I2, which is detected by fluorescence. For the high-pressure gas experiments, 60 fs pulses were generated in a collidingpulse mode-locked ring dye laser and amplified in a four-stage dye amplifier system pumped by a Nd:YAG laser. The amplified pulses were recompressed in a double-pass, two-prism arrangement and then separated into pump and probe pulses using a 50:50 beam splitter. The fundamental at 620 nm and the frequency doubled output served as pump and probe, respectively. Other wavelengths were selected by using a continuum generation. Macroclusters consisting of around 40-150 Ar atoms solvating I2 were excited to the A state above its dissociative limit to I + I, and to B-excited electronic states at different energies above and below its dissociative limit to I + I". 14 The A state undergoes direct dissociation whereas the B state predissociates from a bound to a repulsive potential. The direct dissociation of I: in the A state leads to a high-energy impact of the I atoms on the frozen solvent cage. The Ar solvent cage provides an outer potential barrier that causes the I atoms to oscillate back to a nearly molecular state. Initial geometries and snapshots of the structural changes for A-state excitation are calculated in Figure 4 for an I2 solvated by 17 Ar atoms (two isomers presented) and 44 Ar atoms. These dynamics are manifested in the femtosecond transients in Figure 5 for 614 nm excitation. The rise and fall times for the first peak represent the formation of the excited-state wave packet and subsequent decay to I atom separations that no longer absorb the probe light. The signal reaches a minimum in about 250 fs followed by a prompt recovery to an optically absorbing state in about 300 fs. Molecular dynamics calculations show that the recovery represents a coherent bound motion involving

Molecular Clusters

9

the solvent cage. The B state below the direct dissociative limit undergoes predissociation. The measured dynamics for 570 nm excitation in Figure 6 show a decay time of 15 ps, corresponding to the predissociation lifetime, followed by a 30 ps time scale recovery, due to recombination of solvent separated I atoms and vibrational relaxation. These results provided the microscopic picture of the effect of solvation on dissociation and recombination. First, the time scale for direct dissociation, which was measured directly, is essentially unaffected by the solvent. Second, caging as

(a) 17 Ar

t = 0 fs

t = 3 0 0 fs

t = 6 6 0 fs

t = 4 ps

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0

t -- 0 fs

0

t - 6 6 0 fs

t = 3 0 0 fs

(c)44 Ar

Q

t = 0 fs

=

9

t = 6 6 0 fs

Figure 4. Snapshots of the structures of iodine (dark grey) in argon (light grey) solvent cages. (a) For 17 Ar atoms where one I atom is not capped, the subsequent recombination takes more than 4 ps. (b) For 1 7 atoms where 12 is fully enclosed, recombination following dissociation is direct. (c) For larger clusters, the 12 is almost always enclosed leading to caging that is direct and coherent (ref. 14).

10

JACK A. SYAGEand AHMED H. ZEWAIL

a recombination process following direct dissociation, is a coherent process, with the solvent essentially frozen in configuration on this time scale. The wave packet motion is illustrated in Figure 7. The process is in a highly nonequilibrium dynamics with the solvent. Third, vibrational relaxation occurs on a much longer time scale. Finally, unlike the direct dissociation case (A state), the case of predissociation (B state) shows clearly the solvent involvement in the collision-induced predissocia-

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Molecular Clusters

11

tion and in the caging, but now on the picosecond time scale. The solvent reorganization being on the time scale of bond breakage.

I2/X. in the High-Pressure, Supercritical Region To compare with condensed-phase behavior at solvent densities comparable to clusters and liquids, high-pressure, supercritical fluids were studied by Lienau et

B-State D y n a m i c s : 0.4

,.., f

..... , ..........

Experimental

, ....

, ....

, ....

vs MD

, ....

, ......

.T..-

(a) MD

0.3

i'

::3

9~

0.2

0.1

0.0

~ ' "

"

"

I

"

I'

"

"

I

. . . .

I

. . . .

I

. . . .

"!

'"

"

"

"

|

. . . .

I

. . . .

I'

. . . .

I

'"

"

~

"-

(b) Experiment 1.0

0.8 _ r

0.6

0.4

9 ~

oo 9

M 0.2 0.0

-

ao 4o 5o Time (ps)

60

70

8o

90

Figure 6. LIF transients following B-state excitation at 570 nm. (a) Simulations for 12-Ar44 for initial temperature of 30 K. (b) Experimental transient. Probe wavelength is 307 nm for both simulations and measurements (ref. 14).

12

JACK A. SYAGE and AHMED H. ZEWAIL

al. to explore phenomena reflecting details of solute-solvent interactions. 15'16 Femtosecond excitation of 12 at the inner turning point of the B-state potential was achieved using 60 fs pulses from a CPM laser at 620 nm. Pressure was varied from 0 to a few thousand bar for the rare gases He, Ne, Ar, and Kr. The properties that were measured were: coherent nuclear motion of 12, rate of predissociation, and caging time and efficiency. Calculations of the solvent density and structure as a function of pressure for I2/Ne,, are illustrated in Figure 8.

Figure 2". Wave packet motion as a function of time. The wave packet was treated classically for the spatial distributions of I-I distances at given times. The distributions were obtained by averaging over 1000 independent trajectories (ref. 14).

Molecular Clusters

"

0.., 9"

..e

".

13

tb

,..

,

-,

,

..

.

""

.

'

I

""

9

"~

"

,.,

.~

""o

o

"."I" "

"o

P = 100 bar

~o

.

" to

P = 600 bar

6

t0

/~(1

.

9

.,o

,,

Po

P = 900 bar

P = 2000 bar

Figure 8. Snapshots of 12/Nen system at four different pressures. The atom positions are in accordance with M D simulations. Coordinates are in angstroms (ref. 16).

14

JACK A. SYAGEand AHMED H. ZEWAIL o bar

201 bar

201 bar

404 bar l

'"

r

0

1

2

3

4

Time Delay (ps)

0

5

1210 bar

10

15 0

1628 bar

5

10

15

Time Delay (ps)

Figure 9. Femtosecond transients of iodine in compressed supercritical argon at 295 K. Left: experimentally observed transient at Ar pressures of 0, 201, and 594 bar using LIF detection at the "magic angle" (54.7 ~ between the pump and probe pulses. Fluorescence detection wavelength: 340 n m - 0 bar, 351 nm - 201 bar, and 360 nm - 594 bar. Right: the transient behavior at longer times, showing the onset of caging with density changes (ref. 15).

Femtosecond transients as a function of Ar pressure are presented in Figures 9 and 10.15 In Figure 9, 12 vibrational coherence persisted for greater than 1.5 ps up to a pressure of 800 bar. Because the collision frequency between 12 and the bath gas is about 4 ps -1, these results indicate that at least 6 collisions are required to fully quench the vibrational coherence. At similar pressure, the 12 signal decays on the same time scale as the vibrational coherence. The mechanism for this relatively fast decay is collision-induced electronic predissociation. At pressures of 400 bar and above, the transient decays are followed by a slower rise. This trend represents geminate recombination of I atoms by a cage effect. The onset of caging occurs at pressures where the bath gas density begins to deviate from ideal gas behavior. Above 1200 bar, the rise of the transient appears to be biexponential, which is consistent with similar studies in liquid Xe. 17 These observations are assigned as fast vibrational relaxation with the A/A' states and slower curve crossing from the A ~ A' electronic states. In the lighter rare gas solvents He and Ne, the vibrational coherence is only weakly affected by increasing pressure from 100 to 2000 bar. The transients for the

Molecular Clusters

15

I:;. .....

''. .. ';' 'oba~"

.i,., ",,"i

0

',

',

, 4,

."

9

,", ,

',

,8

.',

,.

_,

9

.

! 2 0 ~ i 13-12

1960

8-7

1-0

bar

.

bl

J

0

e

1

|

I

I

.

2

.

!

2 3 4 Time Delay [ps]

~

!

5

!

_

__

100

_

115

130

Frequency (cm "1)

Figure 10. Vibrational dephasing in helium. Left: Oscillations on the femtosecond transients after subtraction of the underlying decay. Note the rapid dephasing at 100 bar compared to 0 bar, but the persistence at higher pressure (ref. 15). I2/Hen system, presented in Figure 10, indicate vibrational persistence of a few picoseconds at the highest pressures studied. The frequency spectrum for the I2/He collisional interaction was examined by evaluating the Fourier transform spectrum of the oscillatory part of the transients obtained by subtracting away the decay component. The Fourier spectrum, plotted alongside the transients in Figure 10, show several interesting trends. At 0 bar, distinct vibrational eigenstates are ob-

16

JACK A. SYAGEand AHMED H. ZEWAIL

served with the maximum corresponding to the frequency co8,7 = 113 cm -]. At a pressure of 100 bar He, the Fourier spectrum broadens significantly due to solventinduced dephasing, and the intensity at higher frequencies increases. At 400 bar, the frequency spectrum is nearly completely broadened and shifted to higher frequency. The broadening and blue shifting increases at 1960 bar He. Much of the underlying dynamics of solute-solvent interactions may be probed by measuring the correlation time for solute-solvent collisions and vibration-rotation couplings relative to the coherence or dephasing time T2 and the energy relaxation time T1.16Two regimes may be defined: slow modulation is when the correlation times are longer than T2, and fast modulation is when they are much shorter than T2. In the limit of very fast modulation, a motional narrowing of the properties of the solute may be observed. A measure of T] and T2 as a function of

A

m

v

M @

0.8

~

m @ @

a

0.6

B

~/T 2

o} Ii,

@

C I11 "O I= m

0.4

lIT 1 0.2

50 mol/l

0

5

10

15

20

25

30

Number Density (atomslnm 3)

Figure 11. The behavior of the rates (1/T2 and 1/11) with solvent density for helium. The lines in this figure are polynomial fits to the experimental data. The densities were obtained from the known pressure-density conversion data reported elsewhere (ref. 16).

Molecular Clusters

17

He pressure revealed interesting behavior as shown in Figure 11. The T 1 behavior can be explained by collision-induced predissociation. The T2 results, however, break into three regions: (1) a low-density region where 1/T2 increases rapidly and linearly, (2) an intermediate-density region where 1/Tz is relatively constant, and (3) a high-density liquid-like region where 1/Te increases again. With the help of molecular dynamics (MD) simulations, the behavior of T1 and T2 with number density can be explained. The two principal forces involved are the real collisional force originating from the solute-solvent intermolecular interaction and a rotation-induced centrifugal force (i.e. vibration-rotation coupling). 18 The motional narrowing regime begins at a number density of about 10 nm -3. At this point the centrifugal force, which rises linearly at lower density, turns over and decreases (slowly) at higher density. This behavior is mirrored in the T2 results up to about 20 nm -3. At this higher density the collisional force dominates and 1/T2 increases again in rate. 2.2.

I~/Xn

The dissociation/recombination dynamics for a charged solute molecule in a neutral solvent is likely to differ markedly from that of a neutral solute/solvent system. Whereas the dynamics in the latter case are mostly dominated by thermally activated collisional interactions, the reaction dynamics in the former case will be driven more by Coulomb interactions between the solute and solvent. The Lineberger group carried out experiments on mass-selected cluster ions of I2/X,, where X is CO 2 or Ar. 19-22 These solvents have similar mass, but differ greatly in their polarity. The principal measurements were the caging fraction, and the recombination rates as a function of solvent cluster size. The Lineberger group employs a tandem mass spectrometer apparatus for time-resolved measurements of mass-selected cluster ions. 19,2~ Cluster ions are formed at the source of a pulsed valve by crossing the free jet expansion by a 1 keV electron beam. The cluster ions grow in size in the expansion and then are allowed to drift before being extracted into a tandem TOF mass spectrometer. In the linear TOF stage, the cluster ion masses separate and a pulsed mass gate allows a single mass through. Laser excitation occurs just beyond this region. The laser system is an Ar ion pumped Ti:Sapphire oscillator that is pulse-stretched and amplified in a regenerative amplifier, then recompressed, producing 150 fs, 1 mJ pulses at 790 rim. 21 Some of the earlier work from this group used a mode-locked Nd:YAG synchronously pumped dye laser and pulsed dye amplifier. 19.20The dynamics of dissociation/recombination are probed by absorption recovery. Following pump excitation of I~, the ion dissociates to I + I- and no longer absorbs the probe pulse, which is of the same wavelength as the pump. IfI + I- recombines to IL and vibrationally relaxes, the cluster again can absorb the probe pulse. Experimental:

18

JACK A. SYAGE and AHMED H. ZEWAIL

This absorption recovery is detected in the following way: I2/Xn

hv~ ; I2 * Xn ~

[I + I-]Xn_ a ---) 12 X n _ a

- hv 2 ; I 2 * Xn_ a -")' [I + I-]X,,~_ b ---) I2Xn__a_ b

(2)

The dissociation initiated by the pump causes some evaporation of solvent (designated as a in Eq. 2). Hence, the resultant mass spectrum leads to fragment ions, which are signatures for 1-photon absorption. If 12 reforms after dissociation, the cluster will absorb the probe pulse, dissociate, and lose additional solvent [designated as b]. Hence a second series of fragment ions are signatures for two-photon absorption, which is the absorption recovery signal. The mass selection of I~/X n before pump-probe excitation enables a measure of the caging fraction as a function of solvent size n. The two product channels I-/X,,_ a and I~/X,,_a in the first line of Eq. 2 are signatures for uncaged and caged dissociation, respectively. For CO 2 solvation, the first evidence of caging appears at n - 4, and the caging fraction increases monotonically reaching unity at about n - 15. 20 For Ar solvation, caging first appears at n - 10 and reaches a maximum caging fraction of about 0.50 at n - 16. ll The completion of the first solvation shell occurs at n - 16 for CO 2, and presumably a similar value for Ar. The results of the caging fraction are consistent with the polarity of the solvent. CO 2, having a larger charge separation (in form of an electric quadrupole moment) than Ar, forms a stronger ion-neutral bond as well as stronger solvent-solvent bonds than Ar. CO 2, therefore, forms a more rigid solvent cage than does Ar, accounting for the greater caging efficiency. Time-resolved measurements of I~(CO2),, dissociation revealed two interesting observations as shown in Figure 12. 20 (1) the absorption recovery times decrease with increasing solvent size over the range n = 12-15; below and above these solvent sizes, the recovery times are asymptotic (---40 ps for n _< 12 and --10 ps for n _ 15), and (2) a recurrence at about 2 ps is observed in the absorption recovery for cluster sizes n = 14-17. These behaviors occur over a size range corresponding closely to the filling of the first solvation shell. The recurrence is attributed to coherent internuclear motion of I~ along the dissociative excited potential, which is now bounded by the solvent. This phenomena is analogous to the recurrence observed and discussed above for I2/Ar,,. 14 As with the I~Ar,, system, I~(CO2),, eventually recombines along the ground-state potential and undergoes vibrational relaxation to a distribution of states that are again optically active, accounting for the longer time dependence in the absorption recovery traces. The combination of experimental time-resolved measurements and theoretical support, particularly molecular dynamics, are providing key understanding of these fundamental processes. The Neumark group recently conducted a femtosecond photoelectron study on size-selected I2(Ar),, clusters. 23 They observed that the photodissociation of the

Molecular Clusters

19

o

eeoo

t "; 0

I..,

9

9

9

9

9

9

9 go.

n=

17

oo,

, ,.,

",~"

p,,,i 9

9

/ /

0

9

n = 16 100~

.. O 0 0

9

9

9

PA 9~

95 9

95~

9

0

o

%,,,,p~mo d,,,,,f "

.
A transfer rate is much faster in Ar, to the extent that B-state emission has not been detected. 1~176Instead, excitation of the B state results in emission from low vibrational levels of CN(A) (1)a < 3). l~ Thus, it appears that relaxation via A - X cascade is very rapid for 1,)a > 3 levels. Only when the energy gaps and Franck-Condon factors become very unfavorable (cf. Figure 1) does relaxation slow down to the point where radiative decay can begin to compete. Lin et al. l~ analyzed the fluorescence decay curves for CN(a)a < 2). As for CN(A) in Ne, kinetic modeling led to the conclusion that relaxation via A ~ X cascade was much faster than vibrational relaxation between successive vibrational levels of the A state. As Figure 5 shows, transfer rate constants derived from the kinetic model were in good agreement with the relationship:

ku ~oux = kelquA,l,x e - f ~

(7)

i012 -

lOll

-

lOlO -

10 9 -

lO s -

10~ 500

!

I

700

900

-

1

I

1

1100

1300

1500

AE/cm"l

Figure 5. Semi-log plot showing the energy gap dependence of scaled A X transfer rates for CN in solid Ar at T = 12 K. The rates are scaled by 1/ClOA,~x as the data are wel I-represented by Eq. 7.

CN + Rg Collisions, Clusters, and Matrices

105

In accordance with simplified theories, and in contrast to the behavior in Ne, transfer in Ar appeared to be sensitive to the Franck-Condon factors. This is a puzzling contrast. Breakdown of the Franck-Condon approximation is expected when the guest is strongly perturbed by the host. The interactions between CN and Ne are generally weaker than those between CN and Ar, so the Franck-Condon approximation should be better, not worse, for Ne. Based on the stronger guest-host interactions in Ar, it would also be expected that the A ~ X transfer rates were generally faster in Ar than Ne. In apparent disagreement with this expectation, Wurfel et al. 19 noted that the intensity of the A --~ X emission, relative to IR emission from ground-state vibrational transitions, was stronger in Ar than Ne. They attributed this result to faster A ~ X transfer in Ne. To explain this behavior they noted that the large blue-shift experienced by CN(A) in solid Ar increased the energy gaps, and suggested that this had a greater influence on the transfer rates than the change in the guest-host interaction strength. However, the time-resolved fluorescence measurements were consistent with faster A --~ X transfer in Ar. 10For example, rates for 1.)A = 2 --~ a9x = 6 of 1.3 x 106 and 2.5 x 107 s-1 were obtained for Ne 18 and Ar l~ matrices, respectively. The implied discrepancy between the relative intensity and time-resolved fluorescence data can be resolved by assuming that the interactions with Ar cause nonradiative vibrational relaxation of CN ~x < 4. The trend of increasing transfer rates with increasing mass of the Rg host continues for Kr and Xe. 2~ In Kr, CN(B) does not emit, and only the lowest vibrational levels of the A-state fluoresce. In Xe even the lowest A-state levels are quenched. The B-X absorption spectrum of CN in Xe shows that there is an unusually strong interaction between CN(B) and Xe 2~ (the transition is red-shifted by roughly 1000 cm-l). This may reflect mixing of the valence CN(B)-Xe state with nearby CN--Xe § charge transfer states.

6. SPECTROSCOPY AND DYNAMICS OF CN-Ne CN-Ne was first detected via bands associated with the B-X transition. 13'15Before describing these results, it will be helpful to define the notation used to label the energy levels. At a typical van der Waals distance, Ne interacts weakly with CN in the X, A, or B states. Consequently, the quantum numbers for the diatom provide useful labels for the levels of the complex. Due to the low anisotropy of the potential energy surfaces, bending of the complex is more realistically described as an hindered internal rotation (HIR) of the CN. For the B and X states of CN-Ne, the spin is so weakly coupled to the molecular frame that it does not produce measurable splittings at the resolution of our measurements (0.06 cm-l). As the spin is a spectator, the levels of the complex are very similar to those of analogous closedshell complexes. In the following, vibrational states of CN(X)-Ne and CN(B)-Ne are labeled by the quantum numbers 09, N K, Vs), where v is the monomer stretch,

106

MICHAEL C. HEAVEN, YALING CHEN, and WILLIAM G. LAWRENCE

v s is the intermolecular stretch, N represents the monomer angular momentum (excluding spin), and K is the unsigned projection of N on the intermolecular axis (note that in ref. 13, N and K were labeled as r and 9). CN-Ne B - X bands were observed in the vicinity of the monomer 0-0 and 1-0 transitions. 13'15 Each monomer transition was accompanied by about eight bands of the complex. The rotational structures were relatively uncongested and easy to assign. Analysis of the bands showed that HIR was the dominant optically active mode in the spectrum; bands terminating on or originating from excited van der Waals stretch states (v s > 0) were not detected. The rotational constants for the various HIR levels were very similar, indicating that there was very little coupling of the radial and angular motions on the B-state IPS. Hence, it was a reasonable approximation to treat the HIR motion independently, using a one-dimensional effective angular potential energy curve. Based on this approximation, Figure 6 shows a schematic energy level diagram that illustrates the prominent bands of the CN-Ne B(0, N K, 0)-X(0, N K, 0) subsystem. The left-hand side of this figure shows the rotational levels of free CN, and the rotational transitions seen at the low temperatures achieved in a free-jet expansion. The associated HIR levels of CN-Ne, and the transitions that dominate the B - X system of the complex are shown on the right-hand side of the figure. Energy spacings between the HIR levels were used to

(0,2~ (0,2t,O) (0,22,0)

s

N--2, +

N~lo

..

o

(0,1~ (0,11,0)

N

(0,0~

N;I,

-

[

i

(o,P,o) (0,I',0)

N----O,+ I

(0,0~ CN

CN-Ne

Figure6. Schematic showing the relationship between the allowed rotational transitions for CN B-X and the prominent hindered internal rotation bands seen in the CN-Ne spectrum.

CN + Rg Collisions, Clusters, and Matrices

107

estimate radially averaged anisotropy parameters. For determination of the qualitative equilibrium geometry of each state, the interval between the (N r) 11-0~ levels proved to be a crucial property. 15 The intervals were consistent with a T-shaped equilibrium geometry for the ground state, and a linear geometry for CN(B)-Ne. Figure 7a shows the effective angular potentials derived from fits to the HIR intervals. Simple electrostatic considerations, based on the occupancy of valence molecular orbitals [B (a'2s) (rt2p) 4 (o'2p) 2 - X (a'2s) 2 (rt2p) 4 (t~2p)], do not provide any obvious reason for the change in geometry on excitation. However, it is reassuring to note that high-level ab initio calculations for CN-He do predict linear and T-shaped geometries for the B and X states, respectively. 33 Despite the change in the equilibrium geometry, it seems that B - X excitation has very little effect on the van der Waals bond strength. The origin band for the complex [B (0,0~176 appeared exactly where the forbidden Q(0) line of CN would occur. Analysis of the CN-NeA-X system yields a ground-state dissociation energy tt of D O = 28 + 8 cm -1 (see below). As there is no complex shift, this range also defines the B-state dissociation energy. The rotational constant for CN-Ne is mostly determined by the average distance of the Ne atom from the CN center of mass (Ray). The zero-point rotational constants for the B and X states were found to be the same, within experimental error, indicating that there was little change in the radial component of the IPS on excitation to the B state. The constants yielded a value for Ray of 7.2 au, which is typical of a system bound by van der Waals interactions. Attempts were made to observe electronic predissociation of CN(B)-Ne. Timeresolved fluorescence measurements for the B (0,0~ level yielded a decay rate that was indistinguishable from the radiative decay rate for CN(~ B = 0). As electronic predissociation is the only nonradiative channel open for B (0,0~ this result showed that processes such as, CN(a9s = 0) - Ne ---) CN(a9A = 9) + Ne

(8)

were much slower than the radiative decay rate of 1.7 x 107 s-1. Vibrational predissociation of CN-Ne B (1,NK,0) was found to be too slow to compete with radiative decay. This implied that the radial-radial coupling in CN(B)-Ne is weak, and was in accord with the observation that the CN(B) AGu2 vibrational interval was unchanged by complex formation (see LeRoy et al. 45 for a discussion of the relationship between changes in AGu2 and vibrational predissociation rates). Bondybey 18 reported emission from CN(a9B = 2) in solid Ne, so it appears that vibrational relaxation of CN(B) in a Ne matrix is also a relatively slow process. CN-Ne A - X bands were observed in association with the monomer 2-0, 3-0, and 4 - 0 transitions. 14'16To facilitate description of the results for CN(A)-Ne, some additional elements of notation are needed. The lower rotational levels of CN(A 2I-1) are near the Hund's case (a) limit. For the monomer, ~, the projection of the total angular momentum (J) on the diatomic axis, is a good quantum number. As Ne

108

MICHAEL C. HEAVEN, YALING CHEN, and WILLIAM G. LAWRENCE

(a)

[

.

i

5...(

0-

-5-

-I0

-

0.0

('b)

40 -I

....

,

l

60.0

120.0

!

.

180.0

i

CN(A)-Ne 3O i

20

Vn

10-

O-

V2-

-10 20 , 0.0

' "j

, 60.0

120.0

. . . . 18q).O

O (degrees)

Figure 7. (a) Effective angular potential energy curves for the X and B states of C N - N e . (b) Effective angular potential energy curves for the average (VII) and difference (V2) potentials of CN(A)-Ne.

109

CN + Rg Collisions, Clusters, and Matrices

approaches CN(A 2I-I), each J level of the monomer splits into (2J + 1) projection states of the complex. These states may be labeled using signed values for P, where the sign is derived from the product f2 x E 46 Alternatively, starting from the perspective of a slightly different basis set, the states may be labeled by an unsigned value for P combined with a symmetry index. 47 The situation for CN(A)-Ne is such that the eigenfunctions do not correspond well to either limiting case. Consequently, the states are labeled using unsigned values for P with the subscript 'T' or "u" appended to denote the lower and upper energy states that have common values for J and P (e.g. a J = 3/2 level will split into P = 1/2 l, 1/2u, 3/2 l, and 3/2 u projection states of the complex). With inclusion of the stretching motions, levels of CN(A 2FI)-Ne are labeled by (1)A, ~, J[P], Vs)" Electronic predissociation processes were readily observed for CN(A)-Ne, to the extent that they complicated the task of obtaining spectroscopic data for the A - X system. Initial attempts to record LIF spectra for the A - X bands yielded a diffuse structure associated with the CN A 21-Iit2 1.)3 -" 3 level. 14'15Features associated with A 2I-I3/2levels were not detected. As the CN A state is inverted (2I-Ilt2 lies above 2I-I3t2), we assumed that the complex bands associated with 2I-Ilt2 were homogeneously broadened by the spin-orbit predissociation process, CN(A 2I'll/2 1)a)

--

Ne --> CN(A 2I-[3t2 1)a)

4"

Ne

(9)

which produces a fluorescing fragment. The lack of detectable fluorescence from the 2FIlr2 levels was attributed to predissociation via IC, which yields nonfluorescing fragments; i.e.: CN(A 2H3t2 1)3)

--

Ne --> CN(X) + Ne

(10)

OODR techniques were used to confirm these speculations and obtain spectra for CN(A 2H3t2 1)A)- Ne. Fortunately, bands associated with 21-I3r2 were found to be sharp, and they provided detailed information about the A-state IPSs. Action spectra for the A (1)A,3/2, J[P], v s) levels were obtained by using a tunable pulsed laser to scan through the A - X bands, while a second pulsed dye laser was used to monitor the appearance of CN(1)X= 1) 3 4- 4 ) fragments via the B - X bands. 16 Figure 8 shows a low-resolution action spectrum of the A (3, 3/2, 3/2[P], v s) - X (0, N K, 0) band system. This congested contour contains overlapping sub-bands that are built on HIR and low-frequency stretch levels. Assignments for the excited state stretch levels are indicated. Figure 9 shows the Vs= 0 bands at the best resolution that could be achieved with the available equipment. This trace shows partial rotational resolution, but it was still too congested to permit unambiguous assignment. Two OODR techniques were used to further simplify this spectrum. 16 Fluorescence depletion (FD) measurements were made by fixing a monitor laser on a previously assigned B - X feature, and sweeping the depletion (hole-burning) laser through the A - X bands. This technique reveals transitions that originate from the specific ground-state level tagged by the monitor laser. The FD spectra were

110

MICHAEL C. HEAVEN, YALING CHEN, and WILLIAM G. LAWRENCE !

1

I

_

1

-0

I

.

1

_

vs=l

.

I. .

I .~

-

-

j

-

I

I

14370

14380

.

1

14390

I

14400 cm- 1

Figure 8. Low-resolution (0.3 cm -1) action spectrum of the CN-Ne A(3, 3/2, 3/2[P], vs) - X(O, N K, 0) band system. Groups of I--IIR levels associated with vs = 0, 1, and 2 are indicated. The features marked with a * originate from the X(0, 11 0) level 1

noisy, but of sufficient quality to permit identification of "hot" bands arising from the ground-state (0, 11, 0) level. Conventional B ~ A ~ X OODR proved to be the most effective technique for simplifying the A - X spectrum. Although lasers with pulse durations of about 10 ns were used for these experiments, the A - X predissociation rates were slow enough that the A-state complexes could be excited before they dissociated. Figure 10 illustrates the spectral simplification achieved by this OODR scheme. For this trace the monitor laser was tuned to the R 1 band head of the B(5, 0 ~ 0) - A(3, 3/2, 3/213/2u],0) transition. Key rotational and HIR level assignments were established through systematic application of this two-dimensional decomposition technique. As for the X and B states, A-state energy level patterns were consistent with weak couplings between the radial and angular motions. Again, HIR was successfully modeled using one-dimensional angular potentials. 16 The quality of this approximation can be gauged from the spectral simulation shown in Figure 9. The HIR transitions that dominated the A - X spectrum, and their relationship to the rotational levels of free CN are shown in Figure 11. Angular potential energy curves for the

CN + Rg Collisions, Clusters, and Matrices ,

111

CN R1(0.5)

hot bands

I

I"

I

i

14378.0

14380.0

14382.0

14384.0

'

'

l

14386.0 cm- 1

Figure 9. High-resolution (0.06 cm -1) action spectrum of the CN-Ne A(3, 3/2, 3/2[P], vs)-X(O, N K, 0) band system. The upper trace is the observed spectrum. Note that the strong line at 14380.7 cm -1 is due to the monomer R1 (0.5)transition. The lower trace is a numerical simulation. The broken lines indicate regions where the rotational structures associated with specific projection states are prominent.

A state, obtained from fits to the spectra, are shown in Figure 7b. Note that the anisotropy of the average potential was much greater than the difference potential, and the former dictates a T-shaped equilibrium geometry. Comparing the potential curves in Figure 7, it is evident that the A state is significantly more anisotropic than the X and B states, at least for the bound regions of the IPSs. This characteristic may be related to the fact that A-X excitation creates a vacancy in the x-bonding orbital [A (~2p) 3 (a2p) 2- X (g2p) 4 (o2p)]. In accordance with theoretical expectations for a radical with a ~3-configuration, 48 the difference potential was found to be small and positive (repulsive). Although the A state of CN-Ne is more anisotropic than the ground state, the bond strength and radial equilibril,m distance do not change much on A-X excitation. The A state Vs= 0 rotational constants gave Ray = 3.8/~ and the origin of the A(3, 3/2, 3/213/21], 0) - X(0, 0 ~ 0) was displaced from the monomer R1(1/2) parent line (cf. Figure 11) by only 0.2 cm -1. Interestingly, the transition to the upper spin-orbit component [A(3, 1/2, 1/2[ 1/21], 0)] was blue-shifted by 3.8 cm -1, showing that Ne binds more tightly to CN(A 21-I3/2)than CN(A 21-11/2).The dissociation energy p of CN(A 2FI1/2) - Ne was bracketed within the range 17 < D O(f2 = 1/2) < 32 cm -1

11 2

MICHAEL C. HEAVEN, YALING CHEN, and WILLIAM G. LAWRENCE I

I

!

I

A(3, 3/2, 3/213/2u], 0)- X (0, 0o, 0)

R(3/'2) R(I/2)

t

R(5/2)

rl

I

14384.0

i

14385.0

R(7/'2)

I

14386.0

I

14387.0

cm" 1

Figure 10. High-resolution (0.06 cm -1 linewidth) OODR spectrum of the CN-Ne A(3, 3/2, 3/213/2u], 0 ) - X(O, 0 ~ O)band. For this trace theaorobe laser (0.3 cm -1 linewidth) was tuned to the R1 band head of the/3 (5, 0 ~, O)- A (3, 3/2, 3/213/2u], O) transition. The spectrum was recorded by monitoring the B---Xemission. by examining the product state distributions resulting from spin-orbit predissociation (see below). Accurate ab initio calculation of potential energy surfaces for open-shell complexes is a computationally demanding task. It is, therefore, impressive to note the level of agreement between CN-Ne properties predicted from the ab initio surfaces of Yang and Alexander 32 and those derived from the spectroscopic data. The ab initio surfaces correctly predicted a T-shaped geometry for the A state, similar well-depths and Re values for both A and X states, and the greater anisotropy of the A state. The A state rotational constant (0.103 cm -l) was close to the measured value (0.109 cm-1). Calculated dissociation limits were within the experimental error bounds. Inevitably, there were details of the ab initio surfaces that were not in such good agreement with experiment. The main problem with the ground-state surface is that it is not sufficiently anisotropic. As can be seen from Figures 2 and 3, the ab initio surface has almost no barrier to CN(X) rotation. The angular characteristics of the

CN + Rg Collisions, Clusters, and Matrices

113 3/2,

J=3/2

~i

3/2 l

b 9

J=l/2

]

_

ti~- - 1/21 + 1 / 2 . !'

! Ji

i

i

"~

,J_,__~':_- 1/2~+1/2 u

i

A 21"[i/2 P J

I

3/2.

.,

_

J=3/2

i i .9 ~ .

-

, ~ , ~

_........

9 I

"

3/2 l i

I/21 +I/2. 21"I3/2

A

.

.

.

.

j

N=0,1=1/2, + CN

~ "X

2E+

N=0, K=0 CN-Ne

Figure 11. Schematic showing the relationship between the allowed rotational transitions for CN A-X and the prominent hindered internal rotation bands seen in the C N - N e spectrum.

A-state surfaces were also slightly in error. In essence, the anisotropy of Vn was underestimated, while that of V2 was overestimated. Some error was present in the radial characteristics of the A-state surfaces as they predicted a - N e stretch interval (AGlt2) of 12.8 cm -1, as compared to the measured value of 8.2 cm -1. As the theoretical binding energy appeared to be correct, this discrepancy indicates that the radial curvature of the average potential was overestimated in the vicinity of the minimum. Despite these shortcomings, Yang and Alexander 32 demonstrated that the ab initio surfaces could be used to predict the structure of the A-X bands with enough accuracy to reliably assign many of the observed features. Spin-orbit predissociation of CN-Ne A(D A, 1/2, J[P], 0) was so rapid that the rotational structures of bands terminating on these levels were obscured by homo-

114

MICHAEL C. HEAVEN, YALING CHEN, and WILLIAM G. LAWRENCE

geneous line-broadening. 14-16Simulations of the unresolved band contours yielded an average linewidth of 0.9 cm -1, corresponding to a predissociation rate of 1.7 x 1011 s -1. The product state distributions resulting from spin-orbit predissociation were examined by probing the B - A bands of the fragments. Figure 12 shows a typical spectrum for the CN(A 21-13/2,1.)a = 3 ) fragment. Product state distributions were somewhat dependent o n 1.)A and the feature within the A(a9 A, 1/2, J[P], v s) X(0, N K, 0) absorption contour that was excited. However, three characteristics of the distributions were always preserved: (1) population was not detected in levels with J > 7/2, (2) diatomic levels with -parity were preferentially populated, and (3) the most populated level was J = 7/2, -parity. The combined population of the J = 7/2 e and f levels accounted for about 60% of the total products. The sharp cutoff in the rotational distribution at J = 7/2 indicated that J = 9/2 levels were not populated because they were energetically inaccessible. Consequently, the monomer energy differences A 21"Ilt2I~A, J = 1/2 - A 21-I3/2DA, J' defined the upper (J' = 9/2) and lower (J' - 7/2) bounds for the dissociation energy noted above. The preference for production of diatomic fragments in -parity states is another manifestation of the rather symmetric CN-Ne IPSs. The reason why -parity states were preferentially populated can be seen by considering the excitation process, and the diatomic components of the HIR states. Model calculations that used the potentials shown in Figure 7 indicate that the X(0, 0 ~ 0) level has 95% N = 0, J = 1/2, +parity diatomic character. Through preservation of the diatomic selection rules, excitation from this level strongly favors transitions to A-state levels of -diatomic parity. Our model showed that the upper levels of the stronger A - X bands had greater than 80% diatomic -parity character. That the diatomic character selected by excitation would be preserved during spin-orbit predissociation is obvious in the limit of a homonuclear diatom. In this case, conservation of the nuclear symmetry (s(-)s, a 0) causes a blue shift, while an elongation of the bond OV/qi < O) leads to a red shift, if the anharmonic force constant is negative. 64 In case of homogeneous clusters, the perturbation theory has to be extended to degenerate states and instead of using Eq. 10 the result is obtained by diagonalization of the corresponding perturbation energy matrix. 64 In this way the manifold of the excited degenerate states in the cluster is coupled. This leads in some cases to a splitting of the frequencies. These modes can be viewed as linear combinations of the respective normal modes of the constituent molecules. In the same way transition dipole moments can also be calculated as a vector sum of the moments of the individual molecules. In second order, the excited modes are coupled to the other vibrational modes of the same molecule. A further improvement in which also the vibrations of the different molecules are coupled has been introduced by Beu. 65

Vibrations of Size-SelectedClusters

139

The disadvantage of this approach lies in the fact that in zeroth order the somewhat unrealistic harmonic approximation of the molecule is used. Therefore we have developed a different procedure that starts from the complete cluster and includes the anharmonic intramolecular force field of the single molecules and the complete intermolecular potential between the monomer constituents already in zeroth order. We call this method the cluster approach. After the normal mode analysis of the complete cluster, the anharmonic corrections are calculated in the usual perturbation theory up to quartic corrections. 66 In cases of very anharmonic potentials a variational calculation using a harmonic oscillator basis has also been used. The procedure has, in addition, the advantage that the complete set of frequencies for all modes is obtained. A method based on similar considerations has been published by Watts. 67

4. METHANOL CLUSTERS 4.1. Resultsfrom IR Spectroscopy Methanol is an important polar solvent with a nearly linear hydrogen bond. The solid phases are known to be made up of parallel hydrogen bonded chains with coordination number z = 2. 68 Similarly, the dominating structures of the liquid are chains and to a lesser extent also rings with two hydrogen bonds per molecule on the average. Structures with one bond per molecule (terminated chains) and three bonds per molecules (branching points) are present but with less probability. 69 On the other hand, it is well known that properties of the gas phase like the thermal conductivity clearly demonstrate the existence of larger clusters in ethanol vapor. 58 Free methanol clusters have been thoroughly investigated by the scattering method both in the range of the CO and the OH stretch mode. We will discuss the results separately.

The CO Stretch Mode In the range of the CO stretch mode near 1033.5 cm -l line-tunable C O 2 lasers were used both for internally excited clusters up to n = 638,70 and also very recently up to n = 871 using cw lasers, and cold clusters up to n = 4 using pulsed lasers. 39 The experimental results agree where they are available very well with each other. The spectra, which are taken in the experimental arrangement of Figure 2 are shown in Figure 3. Here, the laser interacts with slightly excited complexes. In the case of the dimer, the measurements are carried out in the direct beam so that mainly cold ones are probed. 72 This is achieved by making the seeding mixture very dilute to assure that only dimers are in the beam. The results are again in good agreement with the experiment conducted with size-selected, cold species. 39 The dimer spectrum is characterized by a two-peak structure with one peak shifted by -6.8 cm -1 to the red and another one shifted by + 18.5 cm -1 to the blue

140

UDO BUCK -30 -

!

0 -

-

!

30 -

-

1

1.0-

60 -

-

-30

,

9

=

-

0

|

-

9

30

!

-

-

!

60 -

--

-

|

-

n=8

-

1.0

0.5 -

0.0

"

0.5

'

,

0.6 -

c

n=4

. . . . . . . . . .

--

0.0

n=7

1.0

0.3

O

i 0.5 i

O

'U.

i

'

0.0

0.6 -

n=3

0.0

--

n=6

"~ u 0 .~ E3

~

0.8

-10 cm-1

0.3

0.4 ~

0 . 0 "

,

0.50 -

0.25 -

n=2

~

. .

.

.

.

,

, .

-

.

.

.

.

"

n=5

1.0

0.5

~

0.00 . L . ~. , L ~. - ' ~. 1000 1050

i

1100 1000

Frequency / cm

-1

1050

1100

0.0

Figure 3. Measured photodissociation spectra of size-selected methanol clusters near the CO stretch mode at 1033.5 cm -1 (from refs. 38, 71,72). The upper scale marks the frequency shifts. The sticks are calculated spectra with relative intensities using the minimum energy configuration based on the systematic potential and the cluster approach in the anharmonic (n = 4-13) or the variational (n = 2-3) approximation. 59 The values on the right hand side are shifted by the amount indicated in order to match the experimental results.

compared with the frequency of the free CO stretch mode. For the next larger clustersmthe trimer, tetramer, and pentamernthe experimental data have been interpreted as resulting from one peak only, shifted to the blue by +7.5 cm -1, + 10.8 cm -l, and + 14.3 cm -1, respectively. We note that the full width at half maximum F is smaller for the tetramer compared with that for the trimer and the pentamer. This behavior changes with the hexamer. Now a double peak structure appears with one peak shift by only +6.9 cm -l and the other one shifted further to the blue by + 18.5

Vibrations of Size-Selected Clusters

141

cm -I. This double-peak structure is also observed for n = 8, while the spectrum for n = 7 is unstructured similar to the one for n = 5. The explanation for all these spectra is found in the calculated structures of the clusters, as shown in Figure 4. These structures are obtained by using the procedure described in Section 3. The intermolecular interaction potential is a systematic model derived by Wheatley. 59 This potential gives with some exceptions roughly the same minimum structures as the often used empirical site-site OPLS model with parameters fitted mainly to data of the liquid. 52 The calculations of the spectra are, however, in much better agreement with the data than those based on the empirical potential. This is even more so for the reproduction of the OH stretch which will be discussed in the next section.

n=2

n=3

n=4

n=5

-26.8 kJlmol

-67.0 kJlmol

-132.0 kJlmol

-I86.7 kJlmol

n=6

-238.0 kJlmol

n=8

-233.8 kJlmol

n=7

n=9

-280.7 kJlmol

-37 I.I kJlmol

-325.8 kJlmol

-324.0 kJlmol

n=lO

-417.0 kJlmol

-418.5 kJlmol

Figure 4. Calculated minimum energy configurations of methanol clusters using the systematic potential of Wheatley s9 from the dimer to the decamer. For n = 6, 8, and 10 the two lowest energy isomers are shown. The numbers are the energies of the minimum.

142

UDO BUCK

The dimer exhibits a linear hydrogen bond, while the trimer and tetramer are non-planar rings with C l and S4 symmetry, respectively. This is in,contrast to the structures calculated for the OPLS potential 52 which are planar with C3h and C4h symmetry, respectively. For the larger clusters the odd ones n = 5, 7, and 9, exhibit distorted tings, while the even ones continue to show S6 and S8 symmetry. These are very symmetric structures with the methyl groups pointing alternately up and down. The decamer is the first cluster for which, aside from the distorted, elongated, cyclic structure, a doubled five-membered ring also appears which has nearly the same binding energy. It is noted here that for n = 6 and 8 there also exists some isomeric structures close to the just-mentioned minimum-energy configurations. They are also displayed in Figure 4. For the hexamer it is a ring with C2 symmetry with four methyl groups pointing down and two pointing up and for the octamer a folded ring with S4 symmetry. Now let us try to explain the line shifts based on these structure calculations. Here we apply the methods mentioned earlier in Section 3. The procedure used for the calculations is the cluster approach with anharmonic corrections. 59'66 The intramolecular force field is a modified version to that in Ref. 73. The results are presented by the sticks in Figure 3, which also give an indication of the relative intensities. For the dimer, the excited CO stretch mode is in a nonequivalent position with respect to the hydrogen bond. The O atom of the acceptor participates directly in the bond, while this is not the case for the donor. This explains the line splitting. The calculation gives a red shift for the acceptor and a larger blue shift for the donor in nearly complete agreement with the measurements. The red shift originates mainly from the elongation of the C - O distance in the attractive hydrogen bond. The blue shift results from a stronger force constant and the coupling to the OH mode which squeezes the C - O distance. For the larger odd-sized clusters, nonplanar tings are found in the structure calculations. This behavior results in spectra in which the number of lines correspond to the cluster size since each CO oscillator contributes to the spectrum. This gives three lines for the trimer, five for the pentamer, and seven for the heptamer. In contrast, the even-numbered, symmetric structures with Sn symmetry exhibit two lines only. The calculations show that the two peaks originate from the coupled symmetric and antisymmetric motion of all the CO oscillators. The predictions are in good agreement with the measurements for n = 5 to n = 8 as far as the form of the spectra is concerned. Indeed, the hexamer and the octamer exhibit a two-peak structure as predicted, while the pentamer and the heptamer show broad unstructured features caused by several lines. The absolute values of the calculations are systematically shifted to larger frequencies. This is not the case for the results of n = 2 to n = 4. Here the agreement is perfect with respect to the absolute scale. The low resolution of the line-tunable laser, however, prevented us from resolving the three narrow lying lines for the trimer and the two lines for the tetramer, although the lines of largest intensity match very well the measured spectra. The almost perfect agreement of calculated and measured spectra up to n

Vibrations of Size-Selected Clusters

143

= 5 clearly indicates that the new systematic model potential is a very good one. The deviations which occur for the larger clusters in the absolute shift are probably caused by the potential model which is mainly a very accurate dimer potential but neglects, aside from induction effects, repulsive three-body interactions. The explanation given in previous publications saying that the "single" line structure of the trimer and tetramer are an indication of a planar structure of these clusters can definitely be ruled out, since the OPLS potential 52 on which these conclusions for the structures were based, does not predict the correct shifts. All the more realistic potential models predict for the trimer a slightly distorted planar structure which causes the degenerate IR-active band to split into two components and the symmetric IR-inactive vibration to become IR-active. The quantitative comparison of the line-shift calculations based on the different potential models and the different approximations is given in Table 1 for three selected clusters. The best result is obtained as already discussed for the systematic potential model and the cluster approach with anharmonic corrections. The variational calculations do not improve this result. The results based on the empirical OPLS potential gives incorrect shifts or splitting. This is better for the SCF calculation, which predicts the correct splitting but fails for the shifts. This can

Table 1. Measured 38'39'71'72'74and Calculated Line Shifts of the CO and O H Stretch Mode of Selected Methanol Clusters a

OPL5 n, mode

exp

2, CO

+ 18.5 -6.8

3, CO

7.5

6,CO 2,OH 3,OH

6,OH

m o l - 1 mol-2 29 1.3

18 2

4

23

18.5 33 6.9 17 3 21 -107 -239 -172 -211 -248 -361 -451

29 23 -43 -222

Systematic clu-h

clu-a

13 -5 13

14 -5 14

13 10 -35 -261 -262

15 11 -25 -234 -233

-347 -389

-295 -333

clu-h

SCF

clu-a

clu-v

16 -12 8 5 3

20 -7

20 -7

10 7 5

11 9 6

20 3 -40 -162 -187 -194 -225 -387 -426

26 13 -26 -146 -163 -171 -200 -347 -376

29 15 17 -131 -135 --178 -213

scf-h 12 -14 8 1 -2

4 -3 -3 -78 -116 -120 -155 -203 -250

Note: aln cm-1 for different methods: mo1-164; mol-2,6s clu66 with h harmonic, a anharmonic, v variational

calculation of the frequencies using the empirical OPLS potential of Jorgensens2 and the systematic model of Wheatley;s9 scf47 ab initio calculations in the SCF approximation.

144

UDO BUCK

partly be traced back to the harmonic approximation which is used in these calculations. With respect to the line-shift calculations, the cluster approach is, in general, superior to the molecular approach as is the anharmonic correction to the harmonic calculation.

The OH Stretch Mode Recently, results for the excitation of the OH stretch mode at 3681 cm -] have also become available which are based on completely size-selected beams. Huisken and coworkers measured the dimer 41'4~ and the trimer, 74 while in our laboratory spectra from n = 4 to n = 9 were taken. 71 Compared with the CO stretch mode, the OH stretch mode exhibits much larger shifts to the red caused by the lowering of the curvature of the interaction potential of the hydrogen bond. Therefore this system should be a much more critical test both for the potential model and the method for calculating the shifts. In principle, the spectra should display a similar structure as those for the CO stretch mode. The experimental results are shown in Figure 5 for the dimer and trimer and in Figure 6 for the other clusters. The dimer, tetramer, and hexamer show the expected two bands. The octamer, however, has a structured spectrum with three peaks. Regarding the odd cluster sizes, the trimer spectrum consists of three separate lines, while the larger ones, n = 5, n = 7, and n = 9 exhibit spectra with several weak bumps which, in general, are broader than

i

,m,

eD

0.25 ..o < 0.20 c,.

[..

'

n:2

L_

.o

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I'1

'1

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'

n:3

-35

c m "~

.

p

0.15

LL 0.10 "0 -.~ 0.05

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~

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' .. ,,,

v

..=.

i

3550

.

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i

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3650

3700

i

:21400

Frequency / cm

i

:3450

-1

.

i

3500

.

i

3550

.

3600

Figure 5. Measured photodissociation spectra of methanol dimers 39 and trimers 74 near the OH stretch mode at 3681 cm -I . The sticks are calculated spectra using the minimum energy configurations based on the systematic potential and the cluster approach with the variational approximation, s9 The values for the trimer are shifted by the amount indicated in order to match the experimental results.

Vibrations of Size-Selected Clusters

-60O r

u

i

-500 9

145

-400

l

9

-300

I

9

1.0 -

I

-600 9

n-~6

I

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I

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I

-

-

i

~,,o 0~

o

.

~

-

J

0.0

,

.

9

I

.

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,

-

,

m

.

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,

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I

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:

,,I-.I

o 1.0 ci:1 U. "13 (D ,--' 9 0.5 -

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t___

9

-"

"

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-

9 ~176

o O~

i:5 o.o:

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0,5

-

-

n=7

-

go

0,0

-

3100

3200

3300

3400

3100

32~)0

3300

3400

Frequency / cm

Figure 6. Measured pbotodissociation spectra of size selected methanol clusters near the OH stretch mode at 3681 cm -1. The sticks are calculated spectra with relative intensities using the minimum energy configurations based on the systematic potential and the cluster approach in the anharmonic approximation. 59 For the octamer also the lines of the second lowest energy approximation is presented by the sticks with a full point on top. the corresponding even clusters. According to our discussion in the previous section, this is exactly the behavior that we expected. Thus it is not too surprising that the calculations based on the model potential of Wheatley 59 predict essentially the correct trends: large red shifts ranging from 100 to 500 cm -1. For the detailed comparison displayed as stick spectra in the figures, we chose within the cluster approach the anharmonic approximation based on variational calculations for n = 2 and 3 and perturbation theory for n = 4 to 9. Further

146

UDO BUCK

results for the dimer, trimer, and hexamer are again presented in Table 1. For the dimer the large red shift of the donor as well as the nearly unshifted acceptor molecule are reproduced with an error of about 20 cm -1. In case of the trimer, the splitting is predicted correctly with an error in the absolute shift o f - 3 5 cm -1. For the larger clusters the agreement with the pattern is satisfactory, while the absolute shift is usually less than measured. It is noteworthy that the two-band structure of the tetramer, the hexamer with Sn symmetry, and the multi-line spectra of the pentamer, heptamer, and nonamer are correctly predicted. A special case is the octamer. The lowest energy configuration of S8 symmetry should give two lines only. The spectrum, however, is much broader with, at least, one additional peak. This is a clear indication that another isomer contributes to the data. The second lowest isomer of S 4 symmetry, a folded ring, exhibits three additional lines. If they are included in the comparison, the agreement with the data is improved appreciably. In general, the absolute red shifts are too small. The value is largest for the tetramer with about -100 cm -1 and decreases to 80 cm -1 for n = 5, and to about 50 cm -1 for n = 6 and 7. For n = 8 and n = 9 it is nearly correct. We note that the new systematic potential is able to reproduce both the details and the trends in the spectra of the different cluster sizes. It fails to predict the absolute values of the shifts. The largest deviation found for the tetramer might be an indication that the cooperative effect, which is largest here, is not properly taken into account since three body effects in the repulsive part of the potential are left out. There is apparently a contradiction to the results observed for the CO stretch mode where the largest deviations occurred for n = 8, while n = 4 is predicted correctly. We have, however, taken into consideration that the OH stretch mode is a much more sensitive probe of hydrogen bonding than the CO stretch mode for which indirect couplings might lead to compensating effects. The detailed comparison of Table 1 clearly exhibits that the predicted line shift of the OPLS potential are too large for the dimer and too small for the hexamer. As for the different approximations within the cluster approach, the good agreement with the experimental data of dimer and trimer is only obtained in the variational calculation. There is no doubt that at least the anharmonic approximation has to be used in the calculations. Two recent ab initio calculations based on SCF 47 and DFT 75 methods and the harmonic approximation give good agreement with the experimental values for the scaled frequency shift of n = 5 and n = 6. For the smaller ones their predictions are partly better (n = 4) and partly worse (n = 2) than those obtained in the calculations presented here.

4.2. Isomeric Transitions The "melting" of clusters is, in general, described as isomerization among a multitude of isomers. 9'1~The gross features of the size and temperature dependence of this cluster isomerization crucially depend on their interaction and their chemical properties. Weak short-range interactions in rare gas clusters like Ar n lead to a large

Vibrations of Size-SelectedClusters

147

number of isomers and phase transitions occur preferentially between the lowest energy configurations, usually belonging to an icosahedral growth sequence, and the many other isomers representing the solid-liquid transition of the bulk. A special effect of the finite size is the difference between the melting and condensation temperature which disappears for n ~ oo. Strong long -range interactions in small alkali halide clusters involve isomerization among a very small number of well-characterized isomers, a nice example being the cube ---> ring transition of (NaCI)4 .75-77 In both binding types, the melting temperature decreases with decreasing cluster size and is lower than the bulk value, a result which has recently been observed experimentally for metallic clusters 78 and which can be rationalized by the absence of more and more nearest neighbors with an increasing number of surface atoms. Similar calculations for weakly bound molecular clusters are rare. Therefore we will first present calculations on the isomeric transitions which occur for the well-investigated methanol clusters and then demonstrate how they can be measured. A good indicator for such a transition is the relative root-mean-square(rms) fluctuations of distances ~Scmbetween the centers of mass of the molecules as a function of temperature. They are calculated by averaging the results of classical trajectory calculations in molecular dynamics (MD) simulations, _

2

n((~)-(rij>2)l/2

5cm -- n(n - 1----'-~~

(rij)

'

(11)

i 3 different isomers are found than in our experiments with free clusters, namely trimer tings with monomers and dimers attached to it. This is certainly a consequence of the low temperature and the building process of the molecular clusters by diffusion and capture where the stable trimer ring cannot be opened to form further cyclic structures.

ACKNOWLEDGMENTS I acknowledge with gratitude the many contributions of my former students on both the experimental and the theoretical part, Dr. C. Lauenstein, Dr. A. Rudolph, Dr. X. J. Gu, Dr. M. Hobein, Dr. I. Ettischer, Dr. B. Schmidt, and Dr. J. G. Siebers as well as the guest scientists Prof. T. Beu and Dr. R. J. Wheatley. I am especially grateful to my coworkers Dr. I. Ettischer and Dr. J. G. Siebers for their important contribution of the most recent results and their valuable help and advice in preparing the manuscript and the figures. Part of the work was supported by the Deutsche Forschungsgemeinschaft in SFB 357 and in SP "Molekulare Cluster".

REFERENCES 1. Benedek, G.; Martin, T. E; Pacchioni, G. (Eds.). Elemental and Molecular Clusters; Springer: Berlin, 1988. 2. Scoles, G. (Ed.). The Chemical Physics of Atomic and Molecular Clusters, North-Holland: Amsterdam, 1990. 3. The complete issue Faraday Discuss. Chent Soc. 1994, 9Z 4. Nesbitt, D. J. Chent Rev. 1988, 88, 843. 5. Miller, R. E. Science 1988, 240, 447. 6. Saykally,R. J.; Blake, G. A. Science 1993, 259, 1570. 7. Farges, J.; de Feraudy, M. E; Raoult, B; Torchet, C. Adv. Chent Phys. 1988, 70, 45. 8. Bartell, L. S.; Harsanyi, L.; Valente, E. J../. Phys. Chem. 1989, 93, 6201 and references cited therein.

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159

9. Berry, R. S.; Beck, T. L.; Davies, H. L.; Jellinek, J. Adv. Chem. Phys. 1988, 70, 74. 10. Jortner, J.; Scharf, D.; Landmann, U. In Elemental and Molecular Clusters; Benedek, G.; Martin, T. P.; Pacchioni, G., Eds.; Springer: Berlin, 1988, p 148. 11. Janda, K. C. Adv. Chem. Phys. 1985, 60, 201. 12. Zewail, A. H. Science 1988, 242, 1645. 13. Blake, C. A.; Laughlin, K. B.; Cohen, R. C.; Busarow, K. L.; Gwo, D.-H.; Schmuttenmaer, C. A.; Steyert, D. W.; Saykally, R. J. Rev. Sci. lnstrum. 1991, 62, 1693. 14. Cough, T. E.; Miller, R. E.; Scoles, G. J. Chem. Phys. 1978, 69, 1588. 15. Miller, R. E. J. Phys. Chem. 1986, 90, 3301. 16. Coker, D. E; Watts, R. O. J. Phys. Chem. 1987, 91, 2513. 17. Kappes, M.; Leutwyler, S. In Atomic and Molecular Beam Methods; Scoles, G., Ed.; Oxford University Press: New York, 1988, p 380. 18. Habedand, H. Su~ Sci. 1985, 156, 305. 19. Buck, U. J. Phys. Chem. 1988, 92, 1023. 20. Buck, U. In The Chemical Physics of Atomic and Molecular Clusters; Scoles, G., Ed.; North-Holland: Amsterdam, 1990, p 543. 21. Btimsen, K. O.; Lin, L. H.' Selzle, H. L." Schlag, E. W. J. Chem. Phys. 1989, 90, 1299. 22. Leutwyler, S.; Btisinger, J. Chem. Rev. 1990, 90, 489. 23. Gspann, J.; Vollmer, H. In Rarified Gas Dynamics; Karamcheti, K., Ed.; Academic Press: New York, 1974, p 261. 24. Buck, U.; Meyer, H. J. Che~ Phys. 1986, 84, 4854. 25. Buck, U. In Dynamics of Polyatomic van der Waals Complexes; Halberstadt, N.; Janda, K. C. Eds.; NATO ASI Series. Plenum: New York, 1990, p 42. 26. Buck, U.Atomic Physics 1992,13, 557; Walther, H.; H~insch, T. W.; Neizert, B., (Eds.). American Institute of Physics: New York. 27. Buck, U. Ber. Bunsenges. Phys. Chem. 1992, 96, 1275. 28. Buck, U. Dynamical Processes in Molecular Physics; Delgado-Barrio, G., Ed.; IOP; Bristol, 1993, p 275. 29. Huisken, E Adv. Chem. Phys. 1991, 81, 63. 30. Buck, U. J. Phys. Chem. 1994, 98, 5190. 31. Buck, U. Clusters of Atoms and Molecules; Haberland, H., Ed.; Springer: Berlin, 1994b, p 396. 32. Buck, U. Adv. At. Mol. Opt. Phys. 1995, 35, 121. 33. Buck, U.; Ettischer, I. Faraday Disscuss. Cheat Soc. 1994, 97, 215. 34. Gough, T. E." Knight, D. C." Rowntree, P. A." Scoles, C. J. Cheat Phys. 1986, 90, 4026. 35. Vernon, M. E; Krajnovich, D. J." Kwok, H. S.; Lisy, J. M.; Shen, J. R.; Lee, J. T. J. Chem. Phys. 1992, 77, 47. 36. Hoffbauer, M. A.; Liu, K.; Giese, C. E; Gentry, W.R.J. Chem. Phys. 1987, 78, 5567. 37. Buck, U.; Lauenstein, C.; Rudolph, A. Z Phys. D 1991, 18, 181. 38. Buck, U.; Gu, X. J.; Lauenstein, C.; Rudolph, A. J. Chem. Phys. 1990, 92, 6017. 39. Huisken, E; Stemmler, M. Chem. Phys. Lett. 1988, 144, 391. 40. Huisken, E; Stemmler, M. Z Phys. D 1992, 24, 277. 41. Huisken, E; Kulcke, A.; Laush, C.; Lisy, J.M.J. Che~ Phys. 1991, 95, 3924. 42. Bizzari, A.; Stolte, S.; Reuss, J.; van Duijneveldt-van de Rijdt, J. G. C. M.; van Duijneveldt, F. B. Cheat Phys. 1990, 143, 423. 43. de Meijere, A.; Huisken, E J. Chent Phys. 1990, 92, 5826. 44. Xantheas, S. S.; Dunning, T.H., Jr., this edition. 45 Greet, J. C.; Ahlrichs, R.; Hertel, I. V. J. Chent Phys. 1989, 133, 191; Kofranek, M.; Karpfen, A.; Lischka, H. Chem. Phys. 1987, 113, 53. 46. Bone, R. G. A., Amos, R. D.; Handy, N. C. J. Chem. Soc., Faraday Trans. 1990, 11, 193 I. 47. Bleiber, A.; Sauer, J. Pol. J. Cheat 1998. In press. 48. BiShm,H. J.; Ahlrichs, R. J. Chent Phys. 1982, 77, 2028.

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49. Ahlrichs, R.; Brode, S.; Buck, U.; DeKieviet, M.; Lauenstein, C.; Rudolph, A.; Schmidt, B. Z. Phys. D 1990, 15, 341. 50. Wheatley, R. J.; Price, S. L. Mol. Phys. 1990, 71, 1381. 51. Nemenoff, R. A.; Snir, J.; Scheraga, H. A. J. Phys. Chem. 1978, 82, 2504. 52. Jorgensen, W. L. J. Phys. Chem. 1986, 90, 1276. 53. Ahlrichs, R.; Penco, R.; Scoles, G. Chent Phys. 1977, 19, 119. 54. Tang, K. T.; Toennies, J. P. J. Chem. Phys. 1984, 80, 3725. 55. Stone, A. J. Chem. Phys. Lett. 1981, 83, 233; Stone, A. J.; Alderton, M. Mol. Phys. 1985, 56, 1047. 56. Karpfen, A.; Beyer, A.; Schuster, P. Chem. Phys. Lett. 1983, 102, 289. 57. Detrich, J.; Corongiu, G.; Clementi, E. Chem. Phys. Lett. 1984, 112, 426. 58. Curtiss, L. A.; Blander, M. Chem. Rev. 1988, 88, 827. 59. Buck, U.; Siebers, J. G." Wheatley, R. J. J. Chent Phys. 1998, 108, 20. 60. Beu, T. A.; Buck, U.; Siebers, J. G." Wheatley, R. J. J. Chem. Phys. 1997, 106, 6795. 61. Buckingham, A. D. J. Chem. Soc., Faraday Trans. 1960, 56, 753. 62. Westlund, O. O.; Lynden-Bell, R. M. Mol. Phys. 1987, 60, 1189. 63. Buck, U.; Schmidt, B. J. Mol. Liq. 1990, 46, 181. 64. Buck, U.; Schmidt, B. J. Chem. Phys. 1993, 98, 9410. 65. Beu, T. Z Phys. D 1994, 31, 95. 66. Buck, U.; Siebers, J. G. Eur. Phys. J. D 1998, 2, 207. 67. Watts, R. O. In The Chemical Physics of Atomic and Molecular Clusters; Scoles, G., Ed.; North-Holland: Amsterdam, 1990, p 271. 68. Torrie, B. H." Weng, S.-X." Powell, B. M. Mol. Phys. 1989, 67, 575. 69. Palinkas, G.; Hawlicka, E.; Heinzinger, K. J. Phys. Chem. 1987, 91, 4334. 70. Buck, U.; Gu, X. J.; Lauenstein, C.; Rudolph, A. J. Phys. Chem. 1988, 92, 5561. 71. Buck, U.; Ettischer, I. J. Chem. Phys. 1998, 108, 33. 72. Buck, U.; Hobein, M. Z Phys. D 1993, 28, 331. 73. Schlegel, H. B.; Wolfe, S.; Bemardi, E J. Chem. Phys. 1977, 67, 4181. 74. Huisken, E; Kaloudis, M.; Koch, M.; Werhahn, O. J. Chem. Phys. 1996, 105, 8950. 75. Hagemeister, E C.; Gruenloh, C. J.; Zwier, T. S. J. Phys. Chem. A 1998, 102, 82. 76. Jortner, J.; Scharf, D.; Ben-Horin, N.; Even, U.; Landman, U. In The Chemical Physics of Atomic and Molecular Clusters; Scoles, G., Ed.; North-Holland: Amsterdam, 1990, p 43. 77. Martin, T. P. Phys. Rep. 1983, 95, 167; Heidenreich, A,; Jortner, J.; Oref, I. J. Chem. Phys. 1992, 97, 197. 78. Schmidt, M.; Kusche, R.; von Issendorf, B.; Haberland, H. Nature 1998, in press. 79. Buck, U.; Schmidt, B.; Siebers, J. G. J. Chem. Phys. 1993, 99, 9428. 80. Buck, U.; Ettischer, I.; Lohbrandt, P.; Siebers, J. G., unpublished results. 81. Eichenauer, D.; LeRoy, R. J. J. Chem. Phys. 1988, 88, 2898. 82. Kmetic, M. A.; LeRoy, R. J. J. Chem. Phys. 1991, 95, 6271. 83. Gu, X. J." Levandier, D. J.; Zhang, B.; Scoles, G." Zhuang, D. J. Chem. Phys. 1990, 93, 4898. 84. Ben-Horin, N.; Even, U.; Jortner, J. J. Chem. Phys. 1992, 97, 5988. 85. Buck, U.; Ettischer, I. J. Chem. Phys. 1994, 100, 6974. 86. Buck, U. Gu, X. J.; Hobein, M." Lauenstein, C. Chem. Phys. Len. 1989, 163, 455. 87. Buck, U.; Gu, X. J.; Hobein, M." Lauenstein, C.; Rudolph, A. J. Chem. Soc., Faraday Trans. 1990, 86, 1923. 88. Beu, T. A.; Buck, U.; Ettischer, I.; Hobein, M.; Siebers, J. G.; Wheatley, R. J. J. Chem. Phys. 1997, 106, 6806. 89. Huisken, E; Pertsch, T. Chem. Phys. 1988, 126, 213. 90. Huisken, F.; Kaloudis, M.; Kulcke, A.; Laush, C.; Lisy, J. M. J. Chem. Phys. 1995, 103, 5366. 91. Huisken, E; Kaloudis, M.; Kulcke, A. J. Chem. Phys. 1996, 104, 17. 92. Buck, U.; Ettischer, I.; Melzer, M.; Buch, V.; Sadej, J. Phys. Rev. Lett. 1998. 80, 2578.

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93. Amar, E G.; Goyal, S.; Levandier, D. J.; Perera, L.; Scoles, G. Clusters of Atoms and Molecules; H. Haberland, Ed.; Springer: Berlin, 1993. 94. Huisken, E; Stemmler, M. J. Che~ Phys. 1993, 98, 7680. 95. Goyal, S." Schutt, D. L.; Scoles, G. J. Phys. Chem. 1993, 97, 2236. 96. Hartmann, M.; Miller, R. E.; Toennies, J. P.; Vilesov, A. Science 1996, 272, 1631. 97. Whaley, K. B., this edition.

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QUANTUM MONTE CARLO VIBRATIONAL ANALYSIS AND THREE-BODY EFFECTS IN WEAKLY BOUND CLUSTERS

Clifford E. Dykstra

1.

2.

3. 4.

5.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Weak Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Quantum Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . Vibrational Feedback in Generating Potentials . . . . . . . . . . . . . . . . . 2.1. Assessing the Role of Contributing Elements . . . . . . . . . . . . . . 2.2. The Interaction of Argon and Hydrogen Sulfide . . . . . . . . . . . . . Pairwise and M a n y - B o d y Elements in Weak Interaction Potentials Three-Body Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. H C N Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Arn--HF Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Advances in Molecular Vibrations and Collision Dynamics, Volume 3, pages 163-182. Copyright 9 1998 by JAI Press Inc. All fights of reproduction in any form reserved. ISBN: 1-55938-790-4

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164 164 165 167 169 169 170 172 174 174 177 180 180 180

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ABSTRACT Weak intermolecular interaction can be viewed as and can be modeled as arising from a number of competing elements. As the understanding of these elements grows, attention turns to the role of their subtler many-body manifestations, not just the pairwise interactions. The prospect of many-body effects, particularly three-body effects, being important in the behavior of clusters arises because the number of three-body collections increases with the number of monomers faster than the number of pairs. The subtlety of three-body features, especially in trimers and tetramers where they first arise, means a clear resolution of their sizes from experimental data is a challenge. However, rigorous vibrational analysis coupled with adjustable interaction models offers certain capability to delineate the many-body from the pairwise effects. The technique of quantum Monte Carlo is quite useful in this pursuit. Its use in the actual development of an interaction potential surface is discussed along with presentation of calculational information about the sizes and types of certain many-body contributors in weakly bound clusters. The knowledge that results enhances the quality of interaction models and thereby the quality of computational simulation of a host of chemical problems.

1. INTRODUCTION We may think of molecules as able to interact with each other two ways, weakly and chemically. Of course, such distinction does not arise from different terms in a molecular Hamiltonian; the distinction is more phenomenological. Chemical bonding generally means significant changes in electronic structure and energy changes of 50 kJ mo1-1 and more, whereas weak interaction is inclusive of intermolecular interactions not associated with chemical bonding and not associated with the distinct orbital changes that go with bond formation and bond breaking. Where weak interaction yields potential wells, their depth is usually less than 50 kJ mo1-1. Weak interaction has certain features that may be approached classically, and that simpler level of physics offers considerable guidance in representing potential energy surfaces of weakly interacting clusters, guidance we do not have for the potential surfaces of chemically reactive systems. Weak interaction continues to lie at the frontiers of chemistry. It is the link between the detailed nature of chemical bonds in molecules and condensed-phase molecular phenomena ranging from phase transitions to solvent effects on reactions to biological functioning. Forty years ago, Charles Coulson I introduced a review of hydrogen bonding, one form of weak interaction, by saying it "plays a very conspicuous role in human life, for it is responsible for the adherence of dirt to our skin, the structure of proteins, the action of glues and adhesives, the rigidity of many synthetic polymers, such as polyamides, and a good many other biological phenomena." Weak interaction does not explain all such complex phenomena, but it is perhaps the least understood of the elements that together dictate such phenomena. Coulson's review included some very forward-looking notions about the ultimate

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role of simulation for understanding some of these complex phenomena and among the prerequisites for such simulation are explicit weak interaction potentials. Today, computer simulation of biological problems has become widespread, and numerous model potentials exist for chemical and weak interaction. New capability is developing that will lead to much more critical knowledge of weak interaction and ultimately improved interaction potentials. Attention to the cooperative or many-body effects in a cluster of three or more interacting molecules is important. These effects constitute a relatively subtle class but one that is crucial in the link between molecular features and the behavior of aggregations. It is also one whose role has become subject to much more critical elucidation than ever before, and that serves as a focus of this report.

1.1. Weak Interaction Coulson's early review of hydrogen bonding I was one of the several places where the idea being advanced was the partitioning of weak interaction energetics. In the case of each hydrogen bond in water ice, Coulson estimated energetic contributions that he labeled as electrostatic (6 kcal mol-l), delocalization (8 kcal mol-l), repulsive (-8.4 kcal mol-l), and dispersion (3 kcal mol-1). The sum of these contributions was within 2 to 3 kcal mo1-1 of the known water hydrogen bond energy. The important notion we carry today is that weak interaction can be regarded as arising from a number of elements, some attractive and some repulsive. In other words, weak interaction is a juxtaposition of competing effects. That presents a difficulty for theoretical analysis, for it is a sum of positive and negative contributions that must be obtained reliably. Another leap forward in the theoretical picture came within two decades of the Coulson paper. Morokuma 2-4 and then Kollman 5 devised means to partition the electronic energies obtained from ab initio calculations into elements that contribute to hydrogen bonding. This clarified thinking about the pieces that contribute as the elements were more concretely defined. The interaction of permanent charge fields, the energy of polarization (induction), exchange repulsion, charge transfer, and dispersion were recognized as potential players in hydrogen bonding and subsequently in van der Waals clusters. A more contemporary and very thorough discussion of a perturbative approach to partitioning elements in van der Waals complexes using ab initio calculations can be found in the review by Jeziorski, Moszynski, and Szalewicz. 6 Recent claims of a "new type of intermolecular interaction, the H...H or dihydrogen bond" reported by Crabtree and coworkers 7 could very well indicate something other than the usual elements of weak interaction. On the other hand, their observation might prove to I~e a quite normal expression of weak interaction, though perhaps one with a less usual weighting of the same competing elements. There is clearly a delicate balance point among attractive and nonattractive elements that adds complexity to weak interactions. In fact, even the partitioning of ab initio

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energies can be done in different ways with the result that the features of charge transfer and polarization can be rather smoothly blended to suit. A concrete result of efforts at dissecting weak interaction energies is giving direction to the construction of model potentials. For instance, we noticed 10 years ago 8 that by taking the change in a monomer's electronic structure upon cluster formation to be due to polarization, there was usually good accounting for the evolution of cluster properties. This led us to put polarization upfront in modeling weak interaction and include other elements in a simple, direct fashion. The result was a potential energy surface scheme designated molecular mechanics for clusters 9 (MMC) which uses high level ab initio information on molecular electrical properties (permanent moments, multipole polarizabilities, hyperpolarizabilities) in the evaluation of the classical electrical interaction of a cluster. This naturally combines the permanent charge field interaction with polarization energies and it is a rigorous nonempirical evaluation. Though it follows classical laws, the contribution is really semiclassical since it arises from monomer properties associated with, and calculated for, the quantum mechanical electron distribution of the monomers. The other MMC potential elements were treated empirically and not as fully. Mostly, they have been represented collectively via atom-atom "6-12" potential terms with parameters selected so that the overall potential gave the best match with available spectroscopic information. The parameters, like electrical properties, were assigned to monomers, and this inforced transferability meant that once such an MMC representation of a molecule was obtained, an interaction surface could be calculated for it with any other MMC-represented monomer. A strong focus on electrostatics is found in a number of weak interaction models starting with the 1985 model of Buckingham and Fowler. l0 The interaction energy in that model is obtained from point charges, dipoles, and quadrupoles assigned to atoms, with the atoms not allowed to approach closer than their van der Waals radii allow. Intermolecular polarization has been incorporated into molecular mechanics force fields (e.g. refs. 11-14) often with isotropic dipole polarizabilities assigned to non-hydrogen atoms. At a more basic level are models of distributed polarizabilities of molecules (e.g. refs. 15-18) and direct means for extracting distributed polarizability values from ab initio calculations, 19 all of which are available for incorporation into force fields. That so very many of the theoretical and experimental investigations of weak interaction strive to predict and understand structure, energy, and dynamics of clusters implies an underlying "chemistry of weak systems" or a sub-chemistry. Within it, structural understanding means being able to predict and account for cluster structures through physical arguments akin to orbital arguments about molecular structure. A complete "weak chemistry" also requires dynamical insight in view of the floppiness that goes with weakly bound species, and interest in weakly bound complexes has been one of several stimuli for development of dynamical methods.

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1.2. Quantum Monte Carlo Characteristic of weakly bound clusters are shallow potential surfaces with extended troughs or other regions of flatness. These are highly anharmonic surfaces wherein even ground-state vibrational excursions can lead well away from an equilibrium. Quantum Monte Carlo is one technique that is being used increasingly on such problems. Diffusion quantum Monte Carlo (DQMC), one of several QMC approaches, 2~ gained considerable attention with Anderson's development for molecular electronic structure. 21 With DQMC, the computational equivalent of integration of the Hamiltonian is the evaluation of the potential at numerous geometries; the search for the eigenstate is via imaginary-time propagation. DQMC offers great generality in application to different types and sizes of clusters because it does not involve a basis set or integration. The procedure for DQMC comes about 21 from recognizing an equivalence of the differential equation for diffusion 22 and the time-dependent differential Schrtidinger equation on replacing the time variable, t, in the SchriSdinger equation by an imaginary time variable, x = it. Monte Carlo (MC) techniques used to obtain a numerical solution of the diffusion equation can then be used to simulate the modified Schrtidinger equation. In solving a diffusion equation by MC, particles are moved in randomized, discrete time steps and then may be replicated or destroyed depending on the energetic favorability of where a particle has been moved. An exact solution corresponds to the limiting case of an infinite number of diffusing particles, the length of the time steps approaching zero, and the number of time steps approaching infinity. In DQMC, psuedo particles ("psips") 21 propagate in imaginary time x. At a particular instant in imaginary time, each psip represents a particular geometrical arrangement of the atoms/molecules in the system, and at every such arrangement, the potential energy is evaluated. After many time steps, the distribution of the paths of the psips reflects one state (the ground state, generally) because of the exponential decay in imaginary time of higher energy states that may have been mixed initially. Only the potential energy is explicitly evaluated in the DQMC propagation steps. The kinetic energy operator in the Schrticlinger equation plays its role in the randomized selection of the movement of the psips. [For a time step Az, random displacements for a set of pisps are taken from a specific Gaussian distribution of displacements whose mean is zero and whose standard deviation (distribution width) is (AT,/mi) l'r2 where m i is the particle mass.] The kinetic energy must be separable in the geometrical degrees of freedom, as satisfied for a point mass since r - (p x + 2 + p2)/(2m)" Application of DQMC to weakly bound clusters goes back at least to the work of Coker et al. on the water dimer and trimer, 23 followed by a further calculational study of the water dimer by Coker and Watts. 24 Sun and Watts 25 and Quack and Suhm 26 used DQMC to study the HF dimer, including the particularly demanding calculation of the tunneling splitting.

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Analyzing the vibrations of weakly bound clusters does not necessarily call for relaxing the internal structures of the monomers as done in the first DQMC studies of clusters. Often, constituent molecules in a cluster may be treated approximately as rigid species. This amounts to a separation of the relatively fast intramolecular vibrations from the usually slower intermolecular (weak mode) vibrations. Eliminating high-frequency motions allows for the use of longer time steps for a given level of precision and thereby less steps for a simulation to span a fixed interval of time. Also, for other than small molecules, the rigidity assumption reduces the number of degrees of freedom and thereby the calculation cost. Rigid molecule DQMC (or rigid body, RBDQMC) calculations have been reported for several weakly bound dimers. 27-32 Recently, Gregory and Clary have developed a rigid body quantum Monte Carlo technique for directly obtaining tunneling splittings in clusters by simultaneously finding weights of the two tunneling states. 31 Sandier et al. have developed the capability to obtain several excited vibrational states. 32There has also been a recent implementation of rigid body DQMC applied to the problem of a diatomic molecule surrounded by rare gas atoms by Niyaz et al. 33A comparison of DQMC and RBDQMC for the water dimer by Gregory and Clary 30 has shown good agreement for zero-point energies and rotational constants. QMC propagation of rigid bodies is a combination of translation of their mass centers and rotation about their mass centers. For the kinetic energy operator to have the separable form required for equivalence with a diffusion equation, the propagation of rotation about mass centers needs to be done about each of the three principal axes (a,b,c) of the molecule with moments of inertia taking the place of masses. We begin an RBDQMC calculation with the selection of an initial geometry for each molecule, i, in a cluster. This is a selection of the location of the molecule's mass center plus a selection of rotation angles about each of the three principal axes of the molecule. In our procedure 34 a matrix U i is calculated from these orientation angles of the t~hmolecule. Then, at each time step and for each psip, in addition to translating the mass centers, the molecules are rotated about their principal axes by small angular increments selected from a gaussian distribution and designated {8Ra, 8Rb, 8Re} i where the/-subscript refers to the ith molecule. U i is updated via the construction of three matrices which correspond to incremental rotations about the principal axes which may be at any orientation. Note that for some unit vector ~ = (x,y,z), the unitary transformation matrix, W, for a rotation about ~ by an angle 0 is: 35 1 0 0 W= ~ 1 0

0 +sin0 Zy

-z

y

(-y2-z2

0 x

oX + 2 s i n 2 ( 0 / 2 ) /!

xy xz

xy

xz

-x2-z 2

yz -x2-y 2

yz

So, W is found with ~ being the current direction of one of the molecule's principal axes, which means the components of n are simply the elements of the corresponding row of the current U i matrix and 0 is the corresponding increment from the set {SRa, 8R b, 8Re} i. U i is multiplied by W to yield the U i matrix updated for this first

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rotation about a principal axis. The process is repeated for the other two axes. In this way, three successive small-angle rotations about principal axes of a molecule, whatever its current orientation, are performed. This process is derivable from analysis via quaternions, 36 mathematical devices that have sometimes been employed in classical molecular dynamics simulations. 37'38 The expression above for W is valid only if ~ is normalized, and the sequence of rotations about three axes requires that they be strictly orthogonal. After many successive diffusion steps, the limits on a computer's numerical precision can lead to deviations from orthonormality, and ultimately serious errors. This numerical problem is overcome 35 by Schmidt orthogonormalization of the rows of a current U matrix prior to forming the Ws. We have found this need not be done on every step, and the very conservative default choice in our program is to do the orthonorrealization every 10th diffusion step. It then adds little computational cost but insures numerical precision. The explicit construction of U i matrices for RBDQMC facilitates the evaluation of interaction energies involving tensor molecular properties (e.g. electrical response properties). Molecular property tensors are transformed to a laboratory axis system to evaluate fields and field gradients experienced by partner molecules, and the field and field gradient tensors are back-transformed to the principal axis system and the electrical energy is then directly evaluated. Rotation of tensor properties and time-step rotation of molecules are combined. 35 Test calculations on the water dimer to study convergence behavior of both energy and rotational constants with respect to the time step length and number of psips showed it was tractable to carry out calculations to +0.1 cm -1 in the energy and with further simulation to obtain rotational constants with a standard deviation of several MHz. The diffusion simulation of the solution of the Schr6dinger equation yields weights for each of the psips used in a simulation. For a specific time x0 after a steady state has been achieved in the diffusion simulation, one may associate the weights with the specific cluster geometries corresponding to the different psips. The square of the weights are analogous, but not strictly the same as the square of the wavefunction. 3s However, the true probability density may be obtained by descendant weighting, 22'24'39or weighting each psip at time x0 by the number of its descendants in further propagation. Thus, for small clusters, DQMC can yield energetic and certain property information with controllably small errors in the dynamics. This means it can provide for serious tests of interaction potentials.

2. VIBRATIONAL FEEDBACK IN GENERATING POTENTIALS 2.1. Assessingthe Role of Contributing Elements The equilibrium structure on an interaction potential energy surface for a weakly bound cluster tends to differ from the cluster's on-average structure in the ground

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vibrational state as derived from microwave transition frequencies because of the floppiness and anharmonicity of weakly bound clusters. A conclusion about the roles of the various contributing elements of weak interaction in determining an equilibrium structure has limited value. The difference between the on-average and equilibrium structures highlights the possibility that the weighting of elements at any one structure may not be representative, especially for a detailed analysis. On the other hand., the closer connection with experiment achieved through careful vibrational analysis can help assess the quality of the surfaces and perhaps give a more meaningful view of the role of contributing elements. Going one step further, this connection can be used in a feedback cycle to change a model surface and improve it. There are several requisites for using vibrational analysis to develop a model potential. Error introduced in the analysis should not distort the potential, and the adjustability of the potential should be not be so great as to preclude a vibrational "search through parameter space." QMC is one of a number of approaches that are nicely suited. The error in QMC depends on the length of the simulation, and is controllable; no approximations are made in the Hamiltonian, nor is there any basis set truncation. The search among model parameter values can be guided by the partitioning of weak interaction with different types of contributions having different functional forms. In the case of the MMC model, the use of ab initio values for electrical response properties limits the parameter space for searching to those parameters in the nonelectrical terms. We have done parameter searches for the "6-1 2" atom-atom terms in MMC for water 28'34and for nitrogen 28 that end up with values of (B + C)/2 for different isotopic forms within a few percent of experimental values. This suggests that at least that level of reliability is attainable with the relatively simple type of potential used in MMC and that we can distinguish effects of competing elements to that level.

2.2. The Interaction of Argon and Hydrogen Sulfide Whereas ab initio calculations can provide a grid of potential energies for a system, QMC requires evaluation of the potential at numerous non-grid points. Suhm has reported a scheme that accomplishes this by weighting the energies of nearby grid points, 4~ and Brown et al. have employed neural networks to fit an analytic function to the ab initio grid. 41 The idea of the Hartree-Fock plus damped dispersion (HFD) approach, 42 which is quite effective for insuring correct close-in behavior of long-range elements, could also be used. Of course, the explicit or implicit fitting of an ab initio potential surface retains errors in the surface arising from incomplete treatment of electron correlation and basis set deficiencies. For small clusters of small molecules, these error sources in ab initio treatment can be reduced about as far as one likes; however, the practical situation for larger problems means that there are lingering errors in well-depths, location of minima, and curvature of the surface. The Ar-H2S cluster is an interesting example where we

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have used a modest correlation treatment, MP2, to find a 425-point grid of energies, and have refined the surface via a sequence of QMC calculations and comparison with spectroscopically obtained rotational constants. 43 Gutowsky et al. 44 recently reexamined the microwave spectra of the set of dimers, Ar-H2S, Ar-D2S, and Ar-HDS stimulated by the recognition of two internal rotor states of the analog species Ar-H20. 45-49 Ar-H20 has a planar equilibrium structure and a shallow potential surface such that extensive in-plane bending can be accomplished with little change in potential energy. Gutowsky et al. found a corresponding pair of states for both Ar-HES and Ar-D2S, but with an intriguing difference from the argon-water dimers. Deuterium substitution of HES leads to an increase in the rotational constant of the lower state of Ar-HES, not the usual decrease that is seen for Ar-H20. No reason was apparent from the data. Our investigation of the potential surface of Ar-HES with large basis MP2 ab initio calculations 43 showed a wide trough for the argon atom to nearly "orbit" hydrogen sulfide in a "shell" that partially encloses H2S. Selected calculations were performed with larger bases and with higher level correlation treatments to confirm the trough/shell feature. Next, the classical electrical interaction potential which forms half of the MMC model was generated at the 425 grid points using ab initio values for the permanent moments and multipolar polarizabilities 5~ of H2S and for the polarizabilities 51 of At. These energy contributions were found to be small thereby showing the primary source of attraction to be the dispersion interaction between the two heavy atoms. A multisite version of the MMC potential proved to be a workable form of the potential wherein three "6-12" sites for HES, apart from the atomic sites, helped account for the slight anisotropy of the dispersion interaction. The rotational constants for Ar-HES and At-DES at equilibrium are typical, not anomalous. However, since the trough/shell feature of the surface suggests complicated vibrational excursions, the ground vibrational state rotational constants of the two isotopic forms were obtained with RBDQMC. These values were much closer than the equilibrium values. Furthermore, with adjustments to the "6-12" surface parameters made through repeated RBDQMC calculations (parameter space search), adjustments small enough to be within the size of lingering error in the ab initio energies, (B + C)/2 values within 2% of the experimental values for both forms were obtained (Table 1). It was then clear that the intriguing spectroscopic manifestation of deuterium substitution 44 was very much associated with vibrational motion. The surface shows that vibrational excursions along the trough can change the distance between argon and the HES mass center by a considerable amount. With the trough's flatness, the small H-to-D mass change turns out to increase remarkably the vibrational state probability density corresponding to the argon being in the middle of the trough; that is, at a greater c.o.m, separation.

1 72

CLIFFORD E. DYKSTRA

Table 1. Ar-H2S and Ar-D2S (B + C)/2 Rotational Constants (MHz) Ab initio surface equilibrium43 QMC average using fitted ab initio surface43 QMC average using modified surface 43 Experiment44

0

Ar-H2S

Ar-D2S

1917 1644 1706 1681

1825 1598 1699 1707

PAIRWISE AND MANY-BODY ELEMENTS IN WEAK INTERACTION POTENTIALS

From the standpoint of weak interaction, the "body" in many-body effects is usually a molecule; however, sometimes reference is made to many-body terms in systems as simple as the Ar-HF dimer on taking each atom as one body. This latter usage combines intramolecular interaction with intermolecular interaction. Under the approximation of rigid monomers, all intramolecular dependence in the potential can be absorbed into monomer properties leaving only intermolecular elements. We adopt that view in this discussion; the many-body effects in a cluster refer to many-monomer effects. This does not preclude relaxing the assumption of monomer rigidity, as will be seen in one example. The prospect of many-body effects having a significant impact on the behavior of a cluster has much to do with the number of such interactions. Recall that for a collection of N items, the number of groupings of the items into M-membered units is N ( N - 1 ) . . . ( N - M + 1)/M !. For N = 10, there are about three times as many three-membered groups as pairs, and were there some type of three-body interaction that had no distance dependence, it could offset a pairwise interaction three times larger. Of course, all interaction elements diminish with increasing distance, and further analysis is needed. For a uniform distribution of A particles about a single B particle, the number of A-B pairs with a certain separation distance R increases as R2, whereas the number of A - B - B groups increases as R 3. This dependence on numbers of interacting groups is in the opposite direction of the inverse power dependence on R of the contributing elements of weak interaction. Their specific distance dependence completes this analysis. The interaction of permanent charge fields (permanent moments) is strictly pairwise additive; however, polarization energies have many-body contributions. The many-body polarization term with the least fall-off with distance for a system of neutral monomers (no point charges) is that of the field due to two monomers' dipole moments polarizing a third monomer. This has the same R -4 dependence as the pairwise interaction of one dipole with a polarizable center. If we multiply this dependence by the number of pairs and three-membered groups for a given R, then the pairwise polarization in a large assembly falls off more quickly with R than the

Weak Interaction Quantum Monte Carlo

173

three-body polarization. On that basis, polarization must be regarded as a potentially significant source of many-body contributions provided the monomers are polar and polarizable. Continuing the same argument to four-body interactions would require consideration of three monomers' dipoles interacting with the fourth monomer's hyperpolarizability. We have found for a wide range of molecules that hyperpolarizabilities contribute quite small energies for the usual size of the fields arising from neighboring molecules, even fields of highly polar molecules. Though the formal argument might be made that many-body hyperpolarization has a slower fall-off with distance than pairwise hyperpolarization in a large assembly because of the number of four-membered groups as opposed to pairs, both are generally very small parts of the total electrical interaction. Dispersion is the interaction of instantaneous multipoles. Pairwise dipole-dipole dispersion has an R-6 dependence. The slowest fall-off Of many-body dispersion is that of dipole-dipole-dipole (DDD) dispersion which Axilrod and Teller52 showed has an R-9 dependence. The geometrical dependence, foDD, has the following form, taking Ri as the length of a side of the triangle formed by three interacting atoms and 0 i as the angle opposite the side of length R i. 1 + 3 cos 01 cos 0 2 cos 0 3

fDDD =

e~ g 3 g~

Algebraic manipulation yields an expression for fDDD that requires only the distances between centers and can be readily evaluated from position coordinates, 2 e l2R22R32 + 3 g 2R 4 - 6 R 6

fDDD

- -

8

where R" = R7 + R nz + R n3. Meath and Koulis 53 have considered the next most complicated types of multipolar multibody dispersion, such as dipole-dipole-quadrupole (DDQ) dispersion. Each higher order multiple increases k in the overall R -k distance dependence. It is quite clear that many-body effects are manifested in condensed phases. A nice illustration of this was in the work by Morse and Rice 54 on water ice. Essentially, they found that pair potentials are incapable of yielding both the correct separation of two water molecules in a gas-phase dimer and the water-water separation of ice Ih. Many-body effects are crucial to getting the density of ice correct. White and Davidson 55 have used ab initio calculations on (H20)6 with Morokuma partitioning 2-4 to compare two- and three-body interactions. They found that three-body interactions in this cluster could be as sizable as two-body next-nearest neighbor interactions. They applied their results to a unit cell of ice and reached a significant conclusion about ice: "The Morokuma analysis shows that the large three-body terms are almost entirely due to polarization while the

174

CLIFFORD E. DYKSTRA

non-neighbor two-body terms are almost entirely electrostatic." (They use "electrostatic" to refer only to permanent charge field interaction.) There is a fast-growing body of experimental information on multimonomer clusters that are small enough for detailed analysis of microwave and high-resolution infrared spectra (for instance, refs. 56-60). This growing body of information is crucial to the ultimate understanding of the nature of many-body contributions to weak interaction.

4. THREE-BODY EFFECTS 4.1. HCN Chains Clusters of HCN molecules are an interesting testing ground for many-body effects in collections of polar molecules. Crystallographic work shows that HCN forms linear chains. 61 Microwave spectra have been analyzed for (HCN) 2 62 and (HCN)3 ,63 and both clusters are characterized as linear. Infrared spectroscopy has confirmed the linear structure of the trimer, 64 though a cyclic form of (HCN) 3, presumably less stable, has also been observed. 65 Thus, experimental information is available that gives an idea of the contraction in HCN-HCN separation from the dimer to the limit of an infinite chain. That contraction is due to both pairwise and many-body effects, the contraction from pairwise potentials coming from attraction of next-nearest neighbors and so on. MMC calculations done on linear HCN chains show the effect of certain potential elements on the contraction that comes with increasing chain length. 66 The MMC electrical representation of HCN is a point dipole, quadrupole, and octupoleat the molecule's center of mass, a dipole polarizability, dipole-quadrupole polarizability, quadrupole polarizability, and the dipole hyperpolarizability. The atom-atom Lennard-Jones ("6-12") potential parameters were empirically chosen in part to reproduce the experimental structure for (HCN) 2 62 less an estimated adjustment for vibrational averaging effects. Geometry searches for the equilibrium structures of various (HCN),, chains were carried out by successive adjustment of intermolecular structural parameters. The innermost HCN pair was always the one with the closest spacing, and that spacing is used to track the separation distance as chain length increases. Generally, all but the outer eight separations (left and fight) were found to be within 0.0001/~ of the inner pair separation for the longest chains. To compare the contraction due to electrical polarization with that due to many-body dispersion, dipole-dipole-dipole (DDD) dispersion was added to the MMC potential assuming a molecule-centered form (i.e. each fluctuating dipole at the center of mass) and then all polarization energetics were deleted from the potential. The coefficient of the DDD term was chosen to force the contraction in separation distance from (I-ICN)2 to (HCN)3 to be that obtained with the MMC potential. Table 2 lists the MMC equilibrium energies, separations, and dipole moments for a number of HCN chains. The energies are partitioned according to permanent

175

Weak Interaction Quantum Monte Carlo Table 2. Model Potential Results for Linear (HCN)n Clustersa n:

Energy per bond (cm-I) Permanent moment Polarization Total MMC potential Closest pair separation (A) MMC potential without polarization ad hoc DDD Dipole per monomer (D)

2

3

5

10

20

30

50

1633 183 1452

1744 263 1598

1841 340 1728

1916 402 1831

1883

1900

1914

4.421 4.4 78 4.478 3.350

4.395 4.467 4.452 3.501

4.360 4.417 4.424 3.640

4.349 4.448 4.417 3.768

4.346 4.447 4.416 3.834

4.346 4.447 4.416 3.856

4.346 4.446 4.416 3.875

Note: aRef. 66.

4.43 MMC 4.4-

ti ~ "-'-,---'......

10

20

30

Dir. Pol.

. . . .

N o Pol. A-T

4.37-

4.34 0E+0

.....

40

50

N u m b e r of H C N ' s

Figure 1. Contraction in the separation distance (/~) for linear HCN chains as a function of the chain length. 66 The curves show the closest pair separation in a chain with the number of monomers given by the horizontal axis. The solid curve was obtained with the MMC model potential. Also displayed are curves obtained by excluding back polarization (the direct polarization curve), excluding all polarization, and excluding all polarization but adding a dipole-dipole-dipole dispersion term, the Axilrod-Teller term (A-T), whose coefficient was artificially chosen to yield the same dirner to trimer contraction as the MMC curve. Even with this unrealistically huge DDD dispersion, the full extent of contraction is not achieved in the absence of polarization energetics. The three curves obtained with the modified potentials have been shifted along the vertical axis so as to make their values for the dimer coincide with the MMC dimer value and thereby give a comparison of the contraction with each potential.

1 76

CLIFFORD E. DYKSTRA

moment interactions, polarization energetics and the "6-12" term contributions. As one would expect for a linear arrangement, the dipole-dipole interactions are always attractive, whereas the quadrupole interactions are repulsive. The polarization contribution is about one-sixth of the net electrical interaction. The on-average separation distance of the monomers in the ground vibrational state of (HCN) 2 from microwave spectra 62 is 4.447/~, and the average c.o.m, separation in (HCN) 3 is 4.395/~.63 The c.o.m, in the crystal is 4.34/~.61 In comparison, the MMC equilibrium separation for the dimer is 4.421/~, or 0.026/~ shorter than the on-average value, this difference having been built into the MMC parameter selection as an estimate of the vibrational averaging effect. 8 Using this estimate, the contraction in separation distance from the dimer (4.447 - 0.026 = 4.421) to the crystallographic value is 0.08/~. The MMC dimer to 50-mer contraction in separation distance is 0.076/~. With complete neglect of polarization, 42% of this contraction is obtained. With direct polarization (neglect of back polarization) an additional 37% of the contraction results. Mutual or back polarization gives the remaining 21%. Figure 1 displays how the contraction develops with increasing chain length. Campbell and Kukolich measured the dipole moment 67 of the ground rotational state of (HCN)2 and reported a value of 6.552 D. The MMC value is 6.700 D at the MMC equilibrium. The difference between 6.552 D and twice the experimental value 68 for the dipole of HCN is 0.647 D. MMC, which uses an ab initio equilibrium dipole moment for HCN, gives a very similar enhancement of 0.624 D. This agreement is a strong indication of the ability to account for a property change via polarization analysis, and it reinforces the idea that the changes in electronic structures of the monomers due to weak interaction are changes primarily associated with polarization. The average dipole per monomer approaches an infinite chain value of about 3.9 D. This an enhancement of at least 0.8 D over the isolated monomer dipole. Karpfen 69 has carried out ab initio SCF calculations on the HCN dimer and on the single molecule unit cell of an infinite chain using a crystal orbital method. The separation distance in the infinite chain with the largest basis set, a polarized double zeta set, was 4.374 ,~. Though this is very close to the crystallographic value, 61 the dimer separation distance of 4.512 ,~ obtained in these calculations corresponds to a rather large contraction of 0.138 ~. The dipole moment obtained for the dimer using the largest basis was 7.223 D. This is somewhat larger than the experimental value, 67 but with the SCF level treatment, the monomer's dipole is also a bit too large, by 0.26 D. So, the enhancement in the dimer's dipole found in these calculations is 0.72 D. (Kofranek et al. have reported correlated ab initio results showing a dipole enhancement in the dimer of 0.646 D, 7~ almost identical to the experimental results.) For the infinite chain, the enhancement of the dipole per monomer is 0.48 D, 69 which is smaller than our model results. The model results follow from ab initio electrical properties that were obtained with a larger basis than in Karpfen's study, and basis sets certainly play some role in the difference.

Weak Interaction Quantum Monte Carlo

177

An important result of Karpfen's study 69 was the determination that pairwise effects can account for only a fraction of the contraction in separation distance from the dimer to the infinite chain. It was also argued that cooperative effects are important in the energy of interaction. Ab initio results on clusters with up to five HCNs 7~ have confirmed the increasing per monomer stabilization with increasing chain length. MMC results show the role of cooperative or many-body effects quite clearly, too. The results with the ad hoc potential (Table 2) with the dipole-dipole-dipole (DDD) dispersion interaction necessarily show a contribution to contraction of the separation distance in linear chains. The distance dependence of the DDD term is the same as the distance dependence for certain electrostatic terms, and so, the ad hoc calculational experiment that was performed is mostly a test of this as a three-body interaction. The coefficient for this term was chosen so that it alone brought about the contraction in separation distances from the dimer to the trimer; the coefficient was unrealistically huge. One can see that its effect on the separation distance (Figure 1) is very much like the effect of direct polarization, which is also a three-body contribution, but it falls short of the contraction that results from polarization. Realistic treatment of many-body dispersion should have a much smaller effect than this ad hoc term, and at some level of detail, it will not be ignorable. However, polarization is clearly significant in the contraction in HCN chains. If monomers in a cluster have no permanent charge field (i.e. atoms), then a different balance of many-body effects will be found as in the next example.

4.2. Arn--HF Clusters Changes in the vibrational frequencies of molecules are useful markers of weak interaction. A small molecule's vibrational frequencies will tend to shift even for the very weak interaction with a rare gas atom. The size and direction (red or blue) of the shift is related to the nature of the interaction, though in a more complicated manner than being simply proportional to the well-depth (interaction strength). Clusters with several rare gas atoms weakly attached to a molecule are readily formed in molecular beams, and hence, the effects of several weak-bonding partners are observable. To the extent that they can be separated, many-body effects are revealed. Furthermore, the limit of an infinite number of bonding partners and limiting effect of many-body interactions corresponds to vibrational spectra in a rare gas matrix. Spectroscopically determined structures 72-76and HF vibrational frequencies 75-78 of Arl_a-HF clusters are available. Dynamical analysis with different model potentials have quite closely reproduced the HF frequency shifts in one or in several of these clusters very well. 33'78-83 QMC calculations 33 of Niyaz et al. were performed with pairwise additive potentials and gave very good agreement with experimental values, as shown in Table 3. Likewise, calculations of Mcllroy and Nesbitt 80 using pair potentials yielded very good agreement with a collection of

178

CLIFFORD E. DYKSTRA Table 3. ArnHF Clusters: HF Frequency Shifts, 6v Calculations Expt.

ArHF(n = 0) Energy (cm-1) ArHF(n = 1) Energy (cm-1) 8VHFArHF (cm-1) -9.6575 8VHFAr2HF (cm-1) -14.8377 ~VHFAr3HF (cm-1) -19.2677 8VHFAr4HF (cm-1) -19.7077

MMC + DDD

-76.82 -86.29 -9.47 -15.22 -18.63 -19.34

Re(. 33

-101.26 -110.94 -9.67 -15.38 -20.57 -21.08

Ref. 82

-9.65 -15.60 -21.11 -21.72

Ref. 81

-101.26 -110.91 -9.65 -15.35

spectroscopic data for these clusters. However, as Nesbitt has pointed out, 78 the agreement with experiment diverges slightly from that obtained for the dimer, Ar-HF, and it is in a direction that suggests the pairwise potentials alone are too attractive. Hutson and coworkers 81 have included three-body exchange and the three-body effect of the exchange-induced Ar 2 quadrupole interaction with the molecular charge field and in doing so diminished the error in the 14.827 cm -1 HF red-shift in Ar2-HF from 0.527 cm -1 to 0.049 cm -1. We have carried out another set of calculations 84 on the Arn-HF clusters systems to offer a different kind of test of the role of many-body effects on the HF red-shift. The potential function was that of MMC, though with the neglect of the HF hyperpolarizability and of back or mutual polarization, both giving quite small effects in a system wherein only one monomer has a permanent charge field. This means that up to three-body polarization effects are included in the potential, but no higher order effects. A special feature we have exploited with MMC is the use of electrical properties that have been calculated for specific vibrational states. In this case, we have available from our earlier ab initio calculations and diatomic vibrational treatments, sets of dipoles, quadrupoles, dipole polarizabilities, dipolequadrupole polarizabilities, and quadrupole polarizabilities for the v = 0 and v = 1 states, and in fact for higher lying vibrational states. Thus, there is one MMC surface for Arn-HF(v = 0) and a different surface for Ar,,-HF(v = 1). We can find the equilibrium structures and energies on both surfaces. The difference in equilibrium energies is the nondynamical part of the HF frequency shift. Using the original MMC parameters, this nondynamical shift is 17.4 cm -l for Ar2-HF, 18.5 for Ar3-HF, and 18.9 for Ara-HF. Comparison with experimental values (Table 3) shows these are reasonable values for the Ar 3 and Ar 4 clusters, but there is an overshoot for Ar2-HF. At this point, RBDQMC was employed for a more meaningful comparison with experiment and to improve the surfaces. To carry out RBDQMC calculations of Arn-HF clusters using the MMC type of potential requires a specification of the atom positions in the fixed HF(v = 0) and HF(v = 1) monomers. The separations chosen correspond to the on-average

Weak Interaction Quantum Monte Carlo

179

rotational constants of HF in the two vibrational states. Then, using a cycle of RBDQMC calculations, the MMC parameter space was searched for both surfaces so as to reproduce the spectroscopic (B + C)/2 values for Ar-HF (v = 0 and 1) and the red-shift to 2%. The parameters optimized for the two surfaces in this manner were slightly but not significantly different. They were more noticeably different from the original MMC parameters which were obtained without the advantage of DQMC analysis. With these new parameters, an evaluation of the nondynamical contributions to the shifts were 19.9 cm -1 for Ar2-HF, 29.6 cm -1 for Ar3-HF, and 30.8 cm -1 for Ar4-HF, a value that overshoots the observed shift (Table 3). With dynamical effects obtained via DQMC for Ar2-HF, the shift is 18.5 cm -1, which again overshoots the spectroscopic value (Table 3). The next step was a preliminary test of three-body dispersion DDD terms. Pairwise dipole-dipole dispersion is attractive, and one might expect many-body dispersion to be a correction that increases the overall attraction in an Arn-HF cluster. DDD dispersion is attractive for linear arrangements, but this changes as the arrangement of the three centers is bent. As Axilrod and Teller illustrated, simple right and equilateral triangular arrangements of three atoms have repulsive DDD terms. 52 The Ar,-HF clusters are not linear; Ar2-HF is essentially triangular. Thus, DDD terms could diminish the overall stability of the cluster, and the differential amount of this effect for HF(v = 0) versus HF(v = 1) would affect the HF frequency shift. A DDD interaction was added to each of the two potentials with one parameter for A r - A r - A r interaction and one for Ar-Ar-HF. The HF was treated as a single isotropic center. RBDQMC calculations were then carried out as a coarse search through the DDD parameter space so as to improve the agreement with experiment to at least 1 cm -1 for the red-shifts of ArE-HF and Ara-HE This search produces a differential DDD effect between the two states of HF. The DDD parameters are not too much larger than the corresponding parameter we obtained for the DDD term in the He 3 surface, 85 a fewer-electron system that would be expected to have lesser DDD interaction. The small DDD interaction potential terms provide for a several cm -1 effect on the red shifts of Ar2-HF and Ar3-HF. The difference between the Ar3-HF and Ar4-HF shifts, which is the incremental effect of adding the fourth Ar, is small because this Ar forms zhe apex of a tetrahedral Ar 4 cluster opposite the triangular face at which HF is attached. The RBDQMC/MMC + DDD calculated difference in shifts is 0.7 cm -1 compared to the spectroscopic difference value 77 of 0.44 cm -l. The MMC + DDD potential makes it appear reasonable that the size of dipole-dipole-dipole dispersion and the difference between Ar-Ar-Ar and A r - A r - H F repulsive contributions could play a role in how the HF red-shift evolves with addition of Ar atoms, perhaps in concert with exchange multipole three-body effects in these systems. 81 Thus, like the motivation for recent studies with still more argon atoms, 82'83 the objective of calculating the red-shift from that of Ar-HF to the 41 cm -1 shift in an Ar matrix 86

180

CLIFFORD E. DYKSTRA

still holds the promise of valuable insight. With the MMC + DDD potential, this shift value has been very closely approached. 84

5. CONCLUSIONS Partitioning of ab initio electronic energies focuses the problem of weak interaction on different elements such as the interaction of permanent charge fields, dispersion, exchange repulsion, and polarization. As understanding of weak interaction phenomena grows, the attention to contributing elements is at increasingly finer levels of detail. Vibrational analysis has become crucial for assessing potentials, model and ab initio, against experiment. The vibrational analysis can also guide the process of modeling an interaction potential provided the model is carefully chosen. Separating electrical interaction, which can be quite rigorously connected to monomer properties that are obtainable to good accuracy, from the whole interaction allows for the efficient use of vibrational analysis in the refinement of just one part of a potential. This appears to hold considerable promise for one of the most difficult features of weak interaction, many-body effects, particularly those n o t associated with electrical polarization. It is quite likely that considerable modeling progress can be made by incorporation of three-body potential terms associated with polarization, dispersion, and exchange-induced multipoles. Overall, theoretical/computational capabilities are rapidly emerging that can use spectroscopic information on trimers and larger clusters to get solid information on at least three-body and possibly higher order effects.

ACKNOWLEDGMENTS This work was supported, in part, by a grant from the National Science Foundation through the Theoretical and Computational Chemistry Program (CHE-9403545).

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Coulson,C. A. Research 1957, I0, 149. Morokuma,K. J. Chem. Phys. 1971, 55, 1236. Kitaura, K.; Morokuma, K. Int. J. Quant. Chem. 1976, 10, 325. Umeyama,H.; Morokuma, K. J. Am. Chem. Soc. 1977, 99, 1316. Kollman,P. J. Am. Chem. Soc. 1977, 99, 4875. Jeziorski, B.; Moszynski, R.; Szalewicz, K. Chem. Rev. 1994, 94, 1887. Crabtree, R. H.; Siegbahn, P. E. M.; Eisenstein, O.; Rheingold, A. L.; Koetzle, T. E Accts. Chem. Res. 1996, 29, 348; Richardson, T. B.; de Gala, S.; Crabtree, R. H.; Siegbahn, P. E. M. J. Am. Chem. Soc. 1995, 117, 12875. Dykstra,C. E. Accts. Chem. Res. 1987, 21,355. Dykstra, C. E. J. Am. Chem. Soc. 1989, 111, 6168. Buckingham,A. D.; Fowler,P. W. Can. J. Chem. 1985, 63, 2018. Howard,A. E.; Singh, U. C.; Billeter, M.; Kollman,P. A. J. Am. Chem. Soc. 1988, 110, 6984. Caldwell,J.; Dang, L. X.; Kollman, P. A. J. Am. Chem. Soc. 1990, 112, 9144.

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13. Kuwajima, S.; Warshel, A. J. Phys. Chem. 1990, 94, 460. 14. King, G.; Warshel, A. J. Chem. Phys. 1990, 93, 8682. 15. Applequist, J.; Carl, J. R.; Fung, K.-K. J. Am. Chem. Soc. 1972, 94, 2952; Applequist, J. J. Chem. Phys. 1973, 58, 4251. 16. Birge, R. R. J. Chem. Phys. 1980, 72, 5312; Birge, R. R." Schick, G. A.; Bocian, D. E J. Chem. Phys. 1983, 79, 2256. 17. Miller, K. J.; Savchik, J. A. J. Am. Chem. Soc. 1979, 101, 7206; Miller, K. J. J. Am. Chem. Soc. 1990, 112, 8533; 8543. 18. Stout, J. M.; Dykstra, C. E. J. Am. Chem. Soc. 1995, 117, 5127. 19. Stone, A. J. Mol. Phys. 1985, 56, 1065. 20. Ceperley, D. M.; Mitas, L. Adv. Chem. Phys. 1996, 93, 1. 21. Anderson, J. B. J. Chem. Phys. 1975, 63, 1499. 22. Kalos, M. H. Phys. Rev. 1970, A2, 250. 23. Coker, D. E ; Miller, R. E.; Watts, R. O. J. Chem. Phys. 1985, 82, 3554. 24. Coker, D. E; Watts, R. O. J. Phys. Chem. 1987, 91, 2513. 25. Sun, H.; Watts, R. O. J. Chem. Phys. 1990, 92, 603. 26. Quack, M.; Suhm, M. A. J. Chem. Phys. 1991, 95, 28; ibid, Chem. Phys. Lett. 1995, 234, 71. 27. Buch, V. J. Chem. Phys. 1992, 97, 726. 28. Franken, K. A.; Dykstra, C. E. J. Chem. Phys. 1994, 100, 2865; ibid, Chem. Phys. Len. 1994, 220, 161. 29. Sandier, P.; Jung, J. O.; Szczesniak, M. M.; Buch, V. J. Chem. Phys. 1994, 101, 1378. 30. Gregory, J. K.; Clary, D. C. Chem. Phys. Lett. 1994, 228, 547. 31. Gregory, J. K." Clary, D. C. J. Chem. Phys. 1995, 102, 7817. 32. Sandier, P.; Buch, V.; Sadlej, J. J. Chem. Phys. 1996, 105, 10387. 33. Niyaz, P.; Bacic, Z.; Moskowitz, J. W.; Schmidt, K. E. Chem. Phys. Lett. 1996, 252, 23. 34. Dykstra, C. E.; Van Voorhis, T. A. J. Comput. Chem. 1997, 18, 000. 35. Altman, S. L. Rotations, Quaternions and Double Groups; Clarendon Press: Belfast, 1986, p 74. 36. Evans, D. J. Mol. Phys. 1977, 34, 317. 37. Tildesley, D. J.; Madden, P. A. Mol. Phys. 1981, 42, 1137. 38. Reynolds, P. J. J. Chem. Phys. 1990, 92, 2118; East, A. L. U; Rothstein, S. M.; Vrbik, J.J. Chem. Phys. 1990, 92, 2120. 39. Suhm, M. A.; Watts, R. O. Phys. Rep. 1991, 204, 293. 40. Suhm, M. A. Chem. Phys. Lett. 1993, 214, 373; 1994, 223, 474. 41. Brown, D. E R.; Gibbs, M. H.; Clary, D. C. J. Chem. Phys. 1996, 105, 7597. 42. Hepburn, J.; Scoles, G.; Penco, R. Chem. Phys. Lett. 1975, 36, 451; Ahlrichs, R.; Penco, R.; Scoles, G. Chem. Phys. 1977,19, 119; Douketis, C." Hutson, J. M." On', B. J.; Scoles, G. MoL Phys. 1984, 52, 763. 43. de Oliveira, G; Dykstra, C. E.J. Chem. Phys. 1997, 106, 5316. 44. Gutowksy, H. S.; Emilsson, T.; Arunan, E. J. Chem. Phys. 1997, 106, 5309. 45. Fraser, G. T.; Lovas, E J.; Suertrarn, R. D.; Matsumura, K. J. Mol. Spectr. 1990, 144, 97. 46. Cohen, R. C.; Saykally, R. J. J. Chem. Phys. 1991, 95, 7891. 47. Lascola, R.; Nesbitt, D. J. J. Chem. Phys. 1991, 95, 7917; 1992, 97, 8096. 48. Germann, T. C.; Gutowksy, H. S. J. Chem. Phys. 1993, 98, 5235. 49. Arunan, E.; Emilsson, T.; Gutowksy, H. S. J. Am. Chem. Soc. 1994, 116, 8418. 50. de Oliveira, G.; Dykstra, C. E. Chem. Phys. Lett. 1995, 243, 158. 51. Mahan, G. D. Phys. Rev. 1980, A22, 1780. 52. Axilrod, B. M." Teller, E. J. Chem. Phys. 1943, 11,299. 53. Meath, W. J.; Koulis, M. J. Mol. Struct.-Theochem. 1991, 226, 1. 54. Morse, M. D.; Rice, S. A. J. Chem. Phys. 1982, 76, 650. 55. White, J. C.; Davidson, E. R. J. Chem. Phys. 1990, 93, 8029. 56. Peterson, K. I.; Suenram, R. D.; Lovas, E J. J. Chem. Phys. 1989, 90, 5964; ibid, 1995,102, 7807.

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57. Liu, K.; Brown, M. G.; Caner, C.; Saykally, R. J.; Gregory, J. K.; Clary, D. C. Science 1996, 381, 501. 58. Arunan, E.; Emilsson, T.; Gutowksy, H. S. J. Chent Phys. 1994, 101,861. 59. Gutowksy, H. S." Hoey, A. C." Tschopp, S. L.; Keen, J. D.; Dykstra, C. E. J. Chent Phys. 1995, 102, 3032. 60. Gutowksy, H. S." Arunan, E.; Emilsson, T.; Tschopp, S. L.; Dykstra, C. E. J. Chent Phys. 1995, 103,3917. 61. Dulmage, W. L.; Lipscomb, W. N. Acta Crystallogr. 1951, 4, 330. 62. Buxton, L. W.; Campbell, E. J.; Flygare, W. H. Chem. Phys. 1981, 56, 399; Georgiou, K.; Legon, A. C.; Millen, D. J.; Mj6berg, E J. Proc. R. Soc. Lond. 1985, A399, 377. 63. Ruoff, R. S.; Emilsson, T.; Klots, T. D.; Chuang, C; Gutowksy, H. S. J. Chent Phys. 1988, 89, 138. 64. Maroncelli, M.; Hopkins, G. A.; Nibler, J. W.; Dyke, T. R. J. Chem. Phys. 1985, 83, 2129. 65. Jucks, K. W.; Miller, R. E. J. Chent Phys. 1988, 88, 2196. 66. Dykstra, C. E. J. Mol. Struct.-Theochem. 1996, 362, 1. 67. Campbell, E. J.; Kukolich, S. G. Che~ Phys. 1983, 76, 225. 68. Battacharya, B. N.; Gordy, N. Phys. Rev. 1960, 119, 144. 69. Karpfen, A. Chem. Phys. 1983, 79, 211. 70. Kofranek, M.; Lischka, H.; Karpfen, A. Mol. Phys. 1987, 61, 1519. 71. Kofranek, M.; Karpfen, A.; Lischka, H. Chem. Phys. 1987, 113, 53. 72. Gutowksy, H. S.; Klots, T. D.; Chuang, C.; Keen, J. D.; Schmuttenmaer, C. A.; Emilsson, T. J. Chent Phys. 1985, 83, 4817; ibid, 1987, 86, 569. 73. Gutowksy, H. S.; Klots, T. D.; Chuang, C.; Schmuttenmaer, C. A.; Keen, J. D.; Emilsson, T. J. Am. Chem. Soc. 1985, 107, 7174; ibid, 1987, 109, 5633. 74. Gutowksy, H. S.; Klots, T. D.; Chuang, C.; Ernilsson, T.; Ruoff, R. S.; Krause, K. R. J. Chem. Phys. 1988, 88, 2919. 75. Fraser, G. T.; Pine, A. S. J. Chent Phys. 1986, 85, 2502. 76. Lovejoy, C. M.; Schuder, M. D.; Nesbitt, D. J. J. Chem. Phys. 1986, 85, 4890; ibid, Chem. Phys. Lett. 1986, 127, 374. 77. McIlroy, A." Lascola, R.; Lovejoy, C. M.; Nesbitt, D. J. J. Phys. Chem. 1991, 95, 2636. 78. Nesbitt, D. J. Annu. Rev. Phys. Chem. 1994, 45, 367. 79. Nesbitt, D. J.; Child, M. S.; Clary, D. C. J. Chent Phys. 1989, 90, 4855. 80. Mcllroy, A.; Nesbitt, D. J. J. Chertt Phys. 1992, 97, 6044. 81. Hutson, J. M. J. Chent Phys. 1992, 96, 6752; Cooper, A. R.; Hutson, J. M. J. Chent Phys. 1993, 98, 5337; Emesti, A.; Hutson, J. M. Phys. Rev. 1995, A51,239. 82. Liu, S.; Bacic, Z.; Moskowitz, J. W.; Schmidt, K. E. J. Chem. Phys. 1994, 100, 7166; ibid, 1994, 101, 10181; ibid, 1995, 103, 1829. 83. Grigorenko, B. L.; Nemukhin, A. V.; Apkarian, V. A.J. Chem. Phys. 1996, 104, 5510. 84. Dykstra, C. E. J. Chem. Phys. 1998, 108, 6619. 85. Parish, C. A.; Dykstra, C. E. J. Che~ Phys. 1993, 98, 437. 86. Anderson, D. T.; Winn, J. S. Chem. Phys. 1994, 189, 171.

VI BRATIO N-ROTATI O N-TU N N ELI N G DYNAMICS OF (HF)2 AND (HCI)2 FROM FULL-DIMENSIONAL Q U A N T U M BOU N D-STATE CALCU LATIONS

Zlatko Ba~:i?:and Yanhui Qiu

1. 2.

3. 4.

5.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Coordinate System and Hamiltonian . . . . . . . . . . . . . . . . . . . 2.2. Basis Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Basis Set Contraction via Sequential Diagonalization and Truncation 2.4. Vibrationally Adiabatic Approximation . . . . . . . . . . . . . . . . . Potential Energy Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vibration-Rotation-Tunneling Dynamics of (HF)2 and (HC1)2 . . . . . . . . 4.1. Dissociation Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Donor-Acceptor Interchange Tunneling Splittings . . . . . . . . . . . 4.3. Intra- and Intermolecular Vibrations . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Advances in Molecular Vibrations and Collision Dynamics, Volume 3, pages 183-204. Copyright 9 1998 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 1.55938-790-4

183

184 184 185 185 186 . 188 189 190 193 194 195 200 202

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ZLATKO BA~_I(~and YANHUI QIU

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Note Added in Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

203 203 203

ABSTRACT We review significant advances in the full-dimensional (6D) quantum treatment of the rovibrational levels of (HF)2 and (HC1)2 made in the past couple of years. The bound-state methodology employed in these treatments is presented. This is followed by a discussion of various spectroscopic properties obtained from rigorous 6D calculations, such as the dissociation energies, tunneling splittings and their dependence on inter- and intramolecular vibrational excitations of the dimers, frequency shifts of monomer vibrations, and intermolecular vibrations in different intramolecular vibrational manifolds. The theoretical results are compared with the experimental data.

1. INTRODUCTION Since the pioneering molecular-beam study of (HE)2by Dyke et al., l and the first high-resolution infrared (IR) spectroscopy of (HCI)2 by Ohashi and Pine, 2 the two HX (X = F, C1) dimers have become the favorite prototype systems for investigating fundamental aspects of the structure, spectroscopy, and dynamics of hydrogen bonds. 3 Despite their small size and relative simplicity (two intra- and four intermolecular degrees of freedom), HX dimers display extraordinary richness of quantum dynamical behavior, which has constantly pushed experiment and theory to their limits. The two light H atoms in (HX)2give rise to coupled, large amplitude, highly anharmonic intermolecular vibrations, and to vibration-rotation-tunneling (VRT) spectroscopy which differs qualitatively from that of more rigid molecular species. Of particular importance is the large amplitude, multidimensional tunneling between two equivalent equilibrium geometries of (HX) 2. This intricate motion interchanges the roles of the two HX monomers as hydrogen-bond donor and acceptor, and splits the dimer levels by an amount which reflects the tunneling rate. Such tunneling among equivalent configurations is ubiquitous in larger, more complex hydrogen-bonded clusters, such a s (nEO)n .4 Extensive investigations of HF and HCl dimers (and their isotopomers) by means of microwave, near-IR, far-IR, and Raman spectroscopy have produced a wealth of highly accurate experimental data. We refer the reader to some recent experimental papers on (H~2 5-8 and (HCI)2 ,9'10 which also provide excellent reviews of the activity in this area and references to earlier work. Although the data contain detailed information about the potential energy surface (PES) of the dimers, due to the exquisitely complex vibrational dynamics, extracting this information is a formidable task which requires all the resources of modem theoretical chemistry. Recent methodological advances have made possible rigorous full-dimensional

VRT Dynamics of (HF)2 and (HCI)2

185

(6D) calculations of the rovibrational levels of diatom-diatom complexes, including (HF)211-15 and (HCI)2 .16 The progress in dynamical treatments has been matched by the development of ab initio-based 6D PESs of HX dimers. 17-21 The combination of rich experimental data sets and powerful, approximation-free theoretical tools makes (HCI) 2 and (HF) 2 ideal model systems for quantitative characterization of the global PESs of hydrogen bonds, and of the quantum bound state and dissociative dynamics on them. 3 In this chapter we review the quantum 6D bound-state calculations of HF and HCI dimers and compare the theoretical results with the available experimental data. The methodology is outlined in Section 2. Section 3 describes the PESs employed in (HX) 2 calculations. The VRT dynamics of (HF) 2 and (HCI) 2 is discussed in Section 4. A summary is given in Section 5.

2. THEORY 2.1. Coordinate System and Hamiltonian The PESs of hydrogen-bonded dimers such as (HF) 2 and (HC1)2 have multiple minima separated by low barriers. The resulting vibration-rotation-tunneling (VRT) dynamics of the coupled low-frequency modes on these surfaces can not be investigated using the normal-mode coordinates and methods appropriate for the rovibrational spectra of rigid and semi-rigid molecular species. 22'23 Instead, one must employ coordinate systems developed in the context of molecular scattering theory. Such coordinates, whose definition is not based on any particular reference geometry, can describe the full range of nuclear motions, bound and unbound, of the weakly bound complexes, including interconversions among various isomeric structures. 24-26 Multidimensional quantum calculations of the rovibrational levels of weakly bound complexes consisting of two linear molecules are generally carried out in the diatom-diatom Jacobi coordinates {R, 0 l, 0 2, q), r 1, r 2} shown in Figure 1. The first four coordinates are intermolecular; R represents the distance between the centers of mass of the monomers AB and CD, 01 and 0 2 are two in-plane orientation angles, and qo is the out-of-plane torsional angle. The last two coordinates, r I and r 2, describe the intramolecular vibrations of the two monomers. This coordinate set has been routinely used in theoretical treatments of collisions between two diatomic molecules, including the scattering of two HF molecules. 27'28 The full-dimensional (6D) rovibrational Hamiltonian of a diatom-diatom system for a given total angular momentum J, in the body-fixed frame and the Jacobi coordinates of Figure 1 can be written as.ll,14,16

a2

(j-jl9 2

~

H= - ~ ~ +~-+ + ~ +V r2, R) + h I h 2 (r2) 21.t 0R2 21aR2 2~lr 2 2~r22 (rl' (rl) +

(1)

ZLATKO BA~I(~ and YANHUI QIU

186

A

C

r

q~ B

D

Figure 1. Jacobi coordinates for a diatom-diatom system AB-CD. R is the center-of-mass distance between the monomers AB and CD, and g0 is the out-of-plane torsional angle.

In Eq. 1, Ix is the reduced mass associated with the centers of mass of the two monomers, J is the total angular momentum operator, and Jl and j2 are the rotational angular momentum operators of the two monomers, which are coupled to form Jl2" The diatomic reference Hamiltonian h i (ri) (i = l, 2) is defined as,

t~ ~2 hi (ri) = _ 2ixi Or 2 + Vi (ri)

(2)

where Ixi is the reduced mass of the i-th diatom and V i is a reference potential for diatomic vibration. V i is obtained by making a cut along r i (i = 1, 2) through the 6D PES for a large value of the intermolecular coordinate R, when the interaction potential between the two monomers is close to zero.

2.2. Basis Set The matrix representation of the Hamiltonian in Eq. 1 is formed in the following 6D basis. 11'14'16 For the intermolecular coordinate R of the dimer, a pointwise discrete variable representation (DVR) 24'29'3~is used, since it allows a particularly effective implementation of the basis set contraction scheme of Ba~.ie and Light, 24'3~ described below. A sine DVR of Colbert and Miller, 31 consisting of uniformly spaced points {Ri}, has proved to be convenient for this purpose. The angular and intramolecular components of the eigenfunctions of H are expanded in terms of the (primitive) 5D rovibrational basis for diatom-diatom systems, xJMs = vJM~. f~v JM~. vjK " jK = Y),Jri,2 Kc)v, *v~

(3)

VRT Dynamics of (HF)2 and (HCI)2

187

where e = (-1)J,+J2 +l is the parity of the system, v and j denote (k'l, V2) and (J'l, J2, Jl2), respectively, and r (i -- 1,2) are the vibrational eigenfunctions of the monomers AB and CD, i.6. the eigenfunctions of h i (ri) in Eq . 2. Y)Jm~ dv , 2tc in Eq. 3 is the normalized body-fixed (BF) total angular momentum eigenfuniz~ion which can be written as, Y:~ = (1 + ~SK0)-'/2~f-2J + 1 [/~K. yjt,K + E (--1) J+jl+j2+jl2 DJK, M YJ~2 " K] 8~ M -/;2

(4)

where D sK.M ( O ~ ) is the Wigner rotation matrix 32 with three Euler angles (O 9 ~F), and Y~lztc is the angular momentum eigenfunction of jl 2, JIJ2 YJ~zK=:,:2 Z Yjtm, (01' O) yj2g_m, (0 2, qg) m!

(5)

Yjmbeingthesphericalharmonics.NoteinEq.4therestrictionE (-1)/,§ =1 for K = 0. The above rovibrational basis has been discussed in detail by Alexander and DePristo, 33 whose work also provides a useful reference for the BF representation of diatom-diatom systems in general. In the BF rovibrational basis of Eq. 3, the matrix elements of the potential can be written as, < JMF.

yJME

vJME

vJME

Xvjtc IV (R) I ..v,j, tc, > = 8tctc, : ~5~v' < -ix I (J

_

)2 vJMF.

.J~2 I "/x" >

:

,:8>

where the matrix W is given by, WJ;'~ ,K' = {[J (J + 1) + Jr2 (Jl2 + 1) - 2K 2] fitch" - ~ K ~'+ (1 + ~KO)1Ix ~K+I,K" Jl2K

- X~K X~,x (1 + 8K~) 1/2 8K_t,h.} and the quantity ~. is defined as:

(9)

188

ZLATKO BA~I(~ and YANHUI QIU

~'AB = [A (A + 1 ) - B (B + 1)] 1/2.

(10)

When the two monomers AB and CD are identical, as in (HCI) 2 and (HF)2 , the Hamiltonian in Eq. 1 is invariant with respect to the interchange of the two subunits. It is important to take advantage of this additional symmetry, because it can result in major saving of computation time and memory. Following earlier treatments of the exchange symmetry for two identical molecules, 34'35 the basis functions xJM~ vjX in Eq. 3 can be symmetrized as,11,14,16 _ v JM~ ~ } vjK = Avj,v)'{YJKt~ *v + E (-1) J'2 ~']K

(11)

=

(12)

x.~M~

and, A~j,~7

[2 (1 + 5vv ~j~,)]-l/2

=

~ )]-1/2 [2 (1 + 5v, v~ 7,J~

with V and ~" denoting (vz,vl) and (J2, Jl, J12), respectively. The restrictions on quantum numbers of v and j in Eq. 11 are v I > v2 and Jl > J2 for v I = v2.33 Note that for v 1 = v2 and Jl = J2, the allowed J12 quantum numbers must satisfy the condition Pex E (-1)J~2 = 1. Matrix elements of the Hamiltonian in this exchange-symmetry adapted basis are easily expressed in terms of those in the unsymmetrized basis, in Eqs. 6-9.14,36 The symmetrized basis in Eq. 11 permits block diagonalization of the rovibrational Hamiltonian matrix into four symmetry blocks, which correspond to four possible combinations of the total parity (P - + 1) of the system and the monomer exchange symmetry (+).11 The total parity P is defined as the system parity e times the factor (- 1)J. Therefore, the symmetry of (HCI) 2 eigenstates can be labeled with the symbol (P)+. The four symmetry blocks are diagonalized separately, thus reducing greatly the computational effort.

2.3. Basis Set Contraction via Sequential Diagonalization and Truncation The above 6D basis, which is a direct product of the sine-DVR basis in the radial coordinate R and the primitive BF rovibrational basis functions of Eq. 3, would be prohibitively large for direct diagonalization. Our 6D calculations of (H~2 ll and (HC1)216 typically employed about 30 DVR points R i and a 5D rovibrational basis of the size 1000-2000; this would give rise to a final 6D Hamiltonian matrix whose dimension would exceed 30,000. We addressed this problem by implementing the sequential diagonalization and truncation method of BaSiC and Light 24'30'37'38which, as demonstrated by numerous applications to demanding three- and four-atom systems, can drastically reduce the basis set size with no loss of accuracy. For this purpose at each DVR point R i, a matrix representation of the intermediate 5D Hamiltonian H adi (Ri) defined as,

VRT Dynamics of (HF)2 and (HCi)2 (j H adi ( R i ) =

9 2 --,]12) J~ 2l.tR 2 + . . . . .

189

J~2 +

2 t2r

+ V(q, r2, R) + h I (q) + h 2 (r2)

(13)

is constructed in the 5D B F rovibrational basis of Eq. 3, or in its symmetrized version in Eq. 11 for the case of two identical monomers. Its matrix elements (in the unsymmetrized basis) can be obtained as, yJMe Hadi vJME > Hadi (Ri)vjK, vT"K" = < "'vjK [ (Re) I"vTK"

= 8

vZj/sx, c

+ E2)

+ 8jfSrtc {Sv2,v;Jl (Jl + 1) < ~v,

1

21air21 *v;>

1

2bt2r 2 + < yJME "'vjK

I V(R ) {"rtvTK" vJM~ >

vm~l( J -- J12) 9 2 ] v'JMk >

+ < "'vjK

"Xv'fi(

(14)

2 2i

where e i (i = 1, 2) is the eigenvalue of h i (ri) in Eq. 2. The eigenvectors produced by diagonalizing the matrix in Eq. 14 at every DVR point R i constitute a 5D quasiadiabatic basis which depends parametrically on R i, and is well adapted to the features of the PES. As a result, the size of the quasiadiabatic basis can be decreased sharply by retaining only those eigenvectors whose eigenvalues are below a certain energy cutoff (determined by the maximum rovibrational level energies of interest), and discarding the rest. The final matrix of the 6D Hamiltonian, transformed to the (severely) truncated quasiadiabatic basis, is much smaller (about 2000-300011'16) than it would be in the primitive rovibrational function basis. The matrix of this size is readily diagonalized, producing the desired 6D rovibrational energy levels and wave functions.

2.4. Vibrationally Adiabatic Approximation Although the methodology described in the preceding sections practically eliminates the need for approximate, reduced-dimensionality bound state calculations of systems with three and four atoms, such treatments are still useful for interpreting the results of rigorous calculations, and in initial stages of refining PESs. We want to point out that the sequential diagonalization-truncation scheme outlined above, in addition to being an essential component of a full-dimensional method, makes it especially easy to perform some widely used approximate calculations. 16

190

ZLATKO BA~I(~ and YANHUI QIU

One popular method for approximate calculation of bound states of weakly bound systems, introduced by Holmgren et al. for atom-diatom van der Waals (vdW) complexes, 39 assumes an adiabatic or Born-Oppenheimer separation between the radial coordinate (R) and all other internal, angular and intramolecular, degrees of freedom (f2). The angular/intramolecular part of the Hamiltonian which parametrically depends on R, denoted H ang (f2; R), is diagonalized on a grid of R values. The eigenvalues of H ang (if2; Ri) provide a family of effective 1D potentials (adiabats) Ueff (R) for the radial (stretching) motion of the complex. Within this adiabatic approximation, the (ro)vibrational levels of the complex are obtained by solving a 1D radial Schrtidinger equation for the Hamiltonian H rad (R): ff ~)2 H rad (R) = - 2---~---S 0R + Uef~(R)

(15)

This approach has been applied to triatomic vdW complexes 39'4~and to a couple of diatom-diatom systems, including the HCI dimer for the 4D case with fixed HCI bond lengths. 41'42 It is easy to see that the intermediate 5D Hamiltonian H adi (Ri) of Eq. 13 is actually Hang(~; Ri) for diatom-diatom complexes. By connecting the corresponding eigenvalues at different DVR points R i one obtains a set of 1D potentials Ueff(R ) which appear in Eq. 15. The resulting 1D Schrtidinger equation in the coordinate R can be solved numerically, giving the rovibrational levels of the complex in the adiabatic approximation. 16

3. POTENTIAL ENERGY SURFACES The significance of HX (X = E CI) dimers as simple, yet realistic paradigms for understanding the fundamentals of hydrogen bonding has prompted a number of large-scale ab initio calculations aimed at mapping out their global PESs. A 6D PES for HF dimer was first calculated ab initio at 1061 points by Kofranek et al., 43 and subsequently fitted by Bunker et al. 17 to a 6D analytical function (BJKKL). Quack and Suhm 18 used the same ab initio points to develop another analytical representation of the 6D HF dimer PES. The parameters of this 6D PES (SQSBDE) were empirically adjusted to reproduce the experimental dissociation energy and the ground-state rotational constant of ( H ~ 2. Recently, Klopper et al. 19reported a new 6D PES for (HF)2 based on 3284 ab initio points, together with several empirically refined analytical fits to these points. We have been testing these promising new PESs by means of 6D quantum bound-state calculations and comparison with experiment; the results will be presented in a future publication. 44 The 6D PES of the HC1 dimer was calculated ab initio for 1654 geometries by Karpfen et al.; 45 it was then fitted by Bunker et al. 2~ to a 6D analytical function. Elrod and Saykally 21 recently refined the intermolecular (4D) portion of this

VRT Dynamics of (HF)2 and (HCI)2

191

potential function by a direct nonlinear least-squares fit to available microwave, far-IR, and near-IR spectroscopic data, the first such undertaking for a diatom-diatom complex. Tao and Klemperer 46 calculated ab initio the 4D intermolecular PES for (HCI)2 at the MP2 level. We have employed successfully the ES1 PES in the first fully coupled 6D calculations of the rovibrational (J = 0, 1) levels of (HCI)2 ,16 when both HCI subunits are in the ground vibrational state. But, for HCl-excited (HCI) 2, the ES 1 and the ab initio 2~ PES yielded essentially zero tunneling splitting for the lowest energy state of the dimer, 47 in contrast to the experimental value of about 3 cm -~. When one of the subunits is vibrationally excited, the tunneling process involves exchange of the vibrational energy between the two monomers. Closer examination has shown that for the ES1 (and ab initio) PES, the potential matrix elements responsible for such intramolecular energy transfer are 4 to 5 orders of magnitude smaller than for the SQSBDE PES of (HF) 2, which is clearly insufficient to produce appreciable tunneling splittings. In an attempt to correct this deficiency, we decided to add to the ES 1 PES a 6D electrostatic interaction potential which depends linearly on the intramolecular vibrational coordinates r I and r 2 of HCI dimer; the electrostatic terms present in the ES 1 PES are constants, independent of the HC1 stretches. This very simple electrostatic model, described in detail elsewhere 47 and denoted EL2, has greatly improved the description of the tunneling dynamics in HCI dimer as demonstrated by the results discussed below. The new combined semiempirical (ES 1) + electrostatic (EL2) PES will be referred to as ES 1 + EL2. We now discuss those aspects of the PESs of HX dimers essential for understanding their spectroscopy. The equilibrium geometries of the two dimers, for the ES 1 PES of (HC1)2 and the SQSB DE PES of (HF) 2, are displayed in Figure 2. Both dimers are planar, roughly L-shaped, and have nearly linear hydrogen-bonded structures. The two HX subunits are not equivalent in this geometry. The monomer on the left in Figure 2 is referred to as the "bound" subunit, or the hydrogen bond (or proton) donor. The subunit to the right, whose proton does not participate in the hydrogen bond, is referred to as the "free" HX, or hydrogen bond (or proton) acceptor. However, the equilibrium structures in Figure 2 do not do justice to the complexity of the vibrational dynamics of HX dimers. In both dimers, the roles of the two HX subunits as proton donor and acceptor are readily interchanged, largely through the so-called "geared" motion between two equivalent configurations depicted schematically in Figure 3, where the monomers rotate in opposite directions. The motion is in the tunneling regime and is therefore highly quantum; it gives rise to the well-known splitting of the rovibrational levels of HX dimers. For both ( H ~ 2 and (HCI)2, this minimum-energy donor-acceptor interchange tunneling pathway is associated with a well-defined, rather deep potential valley in the (01, 0 2) plane, for planar geometry of the dimers, which connects the two equivalent global minima on the PES. It is shown in Figure 4 for the SQSBDE PES of ( H ~ 2. The tunneling

192

ZLATKO BA(~I(~and YANHUI QIU (21 (21

F

Figure 2. The equilibrium geometries of (HCI)2 (ES1 PES)and (HF)2 (SQSBDE PES).

paths of both HF and HCI dimers exhibit a low-energy saddle at a C2h geometry, but the two tunneling barriers have very different energies, 352 cm -1 in ( H ~ 2 (SQSBDE PES) compared to only 48 cm -1 in (HCI) 2 (ES 1 PES). This has strong implications for the rovibrational level structure of the two dimers, discussed in the following section.

T

x~

x, A W

Xl

A W

tl

Figure 3. Donor-acceptor interchange via geared rotation of the two HX subunits.

VRT Dynamics of (HF)2 and (HCI)2

193

180

160

140

120

t~ Q) "0

100

o~

80

60

40

20

0

20

40

60

80

1 O0

120

140

160

180

01 ( d e g . )

Figure 4. Contour plot of the SQSBDE PES of (HF)2, for n = r2 = 1.744 a0, R = 5.197 a0, and q~ = 180 ~ The first contour is a t - 1 5 5 0 . 0 cm -1, and spacing between the contours is 50 cm -1 .

4. VIBRATION-ROTATION-TUNNELING DYNAMICS OF (HF)2 AND (HCI)2 This section reviews the results of full-dimensional (6D) bound-state calculations of (HF)2 and (HC1)2 and their comparison with the available experimental data. The theoretical results reported here for (HF) 2 have been obtained on the SQSBDE PES of Quack and Suhm, 18 while those for the HCI dimer pertain to the ES1 PES of Elrod and Saykally, 21 with the electrostatic potential (EL2) added for the HC1stretch excited states. 47 As mentioned in Section 2, the eigenstates of HX dimers can be labeled with the symbol (P)• where P = _+1 is the total parity of the system and +_designates the

ZLATKO BA(~I(~ and YANHUI QIU

194

exchange symmetry. In addition, the HX dimer eigenstates are commonly labeled with the set of approximate vibrational quantum numbers (VlV2V3VaV5V6).The first two, v 1 and v 2, stand for the intramolecular high-frequency stretching vibrations of the free and bound HX, respectively. The remaining four quantum numbers denote the intermolecular, low-frequency vibrational modes of the dimers, the in-plane "antigeared" (cis) bend (v3) corresponding to internal rotation of the two HX monomers in the same direction, the van der Waals stretch (v4), the in-plane "geared" (trans) bend (v 5 ), and the out-of-plane torsion (v6). It must be kept in mind that due to coupling between different degrees of freedom, these labels at best provide approximate zero-order descriptions of the dimer eigenstates. The assignments are made by inspecting numerous cuts through the wavefunctions; mode mixing makes the nodal structure often too irregular to allow unambiguous assignment.

4.1. Dissociation Energies The HX dimers have large zero-point energies (ZPEs) as a result of the coupled large amplitude intermolecular vibrations, which mostly involve the motions of the two light H atoms. Consequently, a large difference exists between the equilibrium binding energy (De) obtained simply from the global minimum of the PES, and the dissociation energy (Do) from quantum dynamical calculations, which can be compared directly to experiment. This is evident from Table 1. In the case of HF dimer, the difference between D e and D 0, actually the 6D ZPE, is 506 cm -1 for the SQSBDE PES; the ZPE is 32% of D e. The situation is similar for the HCI dimer; D e and D Odiffer by the 6D ZPE of 276 cm -1 (ES 1 PES), which represents 40% of D e 9

The D Ofrom 6D calculations of HF dimer 11-13 on the SQSBDE PES, 18 1057.33 cm -l, is close to the experimental value 48 of 1062 + 1 cm -1. The new ab initio-based 6D PES by Klopper et al. 19 (SC-2.9) produces even better agreement with experiment; diffusion quantum Monte Carlo (DQMC) 19and variational 6D calculations 44

Table 1. Theoretical and Experimental Dissociation Energies (Do) of (HF)2 and (HCl)2 a

(HF)2 Property

Theory

(HCI)2 Experiment 1062 + 1

Theory

Do

1057.33

De

1563.20

--

416.25

692

ZPE

505.87

m

275.75

Experiment 431 + 22

Note: a Also shown are the calculated equilibrium binding energies (D e) and zero-point energies (ZPE) of

both dimers. The theoretical properties for (HF)2 were calculated on the SQSBDE PES, and for (HCI) 2 on the ES1 PES. All quantities are in cm -1 .

VRT Dynamics of (HF)2 and (HCl)2

195

give Dos of 1062 and 1061.31 cm -1, respectively, which are within the uncertainty of the experimental result. The D O of (HC1)2 has not been determined as accurately as for (HF)2; its experimental value 49 is 431 + 22 crn -1, while the 6D calculation16 on the ES 1 PES gives the D o of 416.25 cm -1. This result, although an improvement over the 6D value obtained 16for the ab initio PES 2~(377.63 cm-l), implies that further improvements of the PES for (HC1)2 are needed.

4.2. Donor-Acceptor Interchange Tunneling Splittings The large amplitude tunneling motion, which reverses the proton donor/acceptor roles of the two HX moieties, splits the levels of (HX) 2 into closely spaced pairs of states with opposite exchange symmetry. Experiments have revealed strong dependence of the tunneling splittings on H/D isotopic substitution and excitation of both intra- and intermolecular vibrational modes of (HE)25-8 and (HCI)2 .9'10 The large body of experimental data can be compared with the predictions from fulldimensional quantum calculations.

Ground-State Tunneling Splittings and Isotope Effect The ground-state tunneling splitting measured for (I--IC1)2,5~ 15.5 cm -l, is more than 20 times that of (HF)2, ~'52 0.66 cm -~. Table 2 shows that these experimental observations, made for the ground-HF- and HCl-stretching states of the respective dimers, are reproduced well by our 6D calculations. The calculated 6D ground-state tunneling splittings for (HCI)216 and (H~211-13 are 0.44 and 14.94 cm -1, respectively. The description of (HF) 2 tunneling splitting is noticeably improved on the SC-2.9 PES, 19for which 6D calculations yield 19'44the value of 0.61 cm -1. The much larger tunneling splitting in HC1 dimer, while undoubtedly a result of complicated intermolecular vibrational dynamics, can be traced to the fact that its C2h tunneling barrier of about 50 cm -1 is much lower than the corresponding barrier in (HF) 2, =340 cm -1.

Table 2. Theoretical and Experimental Ground-State Tunneling Splittings (in cm -1) for (HF) 2, (DF)2, (HCI)2, and (DCI) 2, for the Ground Vibrational State of the Monomers a

Dimer

Theory

(HF)2

0.44

(DF)2

0.03

(HCI)2

14.94

(DCI)2

--

Experiment 0.658690(1 ) 0.0527207(14) 15.476680(4) 5.968(20)

Note: aThetheoretical splittings for (HF)2 and (DF)2 were calculated on the SQSBDE PES,while those of(HCI) 2

and (DCI)2 are for the ES1 PES.

196

ZLATKO BA~_I(~and YANHUI QIU

Elementary quantum mechanics suggests that substituting the hydrogens in (HX) 2 with deuterium atoms should reduce the tunneling splittings. The experimental and theoretical results for (DF)2 and (DC1)2 (experiment only) in Table 2 confirm this expectation. However, deuteration of (HF) 2 decreases the tunneling splitting by 12-fold (experiment, 15-fold from the calculations), while the reduction is less than threefold for (HC1)2, testimony to the complex tunneling dynamics in HX dimers.

Effect ofvl and v2 HX-Stretch Excitations The tunneling coordinate in HX dimers is well-defined, and corresponds approximately to the geared rotation of the HX subunits shown in Figure 3. Nevertheless, the tunneling dynamics is far from being one-dimensional, and in fact involves all internal degrees of freedom of the dimers. This is evident from the experimental observations that the excitation of various intra- and intermolecular vibrational modes changes drastically the tunneling splittings in ( H ~ 2 5-8 and (HC1)2.9'1~ For the discussion which follows, it should be pointed out that for the free-HX (v l, 2vl) excited levels of (HX) 2, the lower energy component of the tunneling doublet has (-) exchange symmetry, 9'53'54 in contrast to the bound-HX (v 2, 21,,2) excited levels, whose lower tunneling sublevel is of (+) exchange symmetry (as in ground-HX-stretching levels). Consequently, the tunneling splittings, defined as the energy difference between the (-) and (+) components of a tunneling pair, are negative for v 1, 2u I excited states. Experimental results in Tables 3 and 4 show that excitation of the free-HF (vl) and bound-HF (u2)-stretching fundamentals of (HF) 2 reduces the ground-state tunneling splitting by a factor of three, while v I and v 2 excitations in (HCI) 2 lead to a fivefold decrease in the tunneling splitting. The calculated ratios of the tunneling splitting for v t and v 2 relative to that for the ground-HX-stretching state, -0.29 (vl) and 0.20 (v2) for (HF)2 ,14'15 and -0.15 (Vl) and 0.16 (v2) for (I-ICI)2,47 agree reasonably well with experiment. Smaller tunneling splittings in v~ and v 2 fundamentals have been generally attributed to the difficulty of transferring the vibrational energy between the two HX subunits in the course of tunneling, s5 The agreement between theory ~4 and experiment 56 deteriorates for the overtone (2vl) excitation of (HF)2. The tunneling splitting calculated for the SQSBDE PES, -0.014 cm -1, is an order of magnitude smaller than the experimental value, -0.21 cm -~. The reasons for this discrepancy are not clear. It is hoped that the calculations 44 employing the SC-2.9 PES will shed light on this problem, also allow comparison with the tunneling splittings measured for the second HF-stretching overtone of (HE)2 .53

Dependence on Intermolecular Vibrational Excitations We discuss separately the effects of intermolecular excitation on tunneling in (HF) 2 and (HCI) 2 since it turns out they are qualitatively different.

VRT Dynamics of (HF)2 and (HCI)2

197

Quantum 6D bound-state calculations (on the SQSBDE PES) for ground -ll and excited 14'15 HF-stretch states of (HF) 2 have predicted strong intermolecular mode specificity of the tunneling splittings. Theoretical results displayed in Table 3 show that for the ground-HF stretch as well as v 1 and v 2 manifolds, the v 5 geared bend excitation causes by far the largest increase (by a factor of 17-23) in tunneling splitting, relative to that in the ground intermolecular state of the same manifold. This is easy to understand, since the geared bend mode correlates strongly with the donor-acceptor interchange tunneling pathway. Not surprisingly in view of this argument, excitation of the v 4 intermolecular stretching mode in different HFstretch manifolds is predicted 14'15to increase the tunneling splitting more modestly, approximately 2-6-fold. The theoretical results in Table 3 are generally in qualitative agreement with the experimental data regarding the mode dependence of tunneling splittings, although the calculated (J = 0) splittings are consistently smaller than the corresponding experimental values. Moreover, the predicted tunneling splittings of the v 4 and v 5 intermolecular modes built on the v 1 fundamental are larger than for the analogous levels in the v 2 manifold, which contradicts the experimental results. Finally, the 6D calculations for the ground-HF-stretch states zl predict a 34-fold increase in the tunneling splitting for the v 3 antigeared bend fundamental, which is in sharp contrast with the observed 1.5-fold decrease between v I and v 1 + v 3. This signals a deficiency of the SQSBDE PES along the antigeared bend coordinate. There are no experimental data on tunneling splittings in excited intermolecular states of (HC1)2, leaving us to discuss the intriguing theoretical predictions. 47 Table 4 shows that the intermolecular mode dependence of tunneling splittings in (HCI) 2 differs profoundly from that for ( H ~ 2 (Table 3). In the ground-HCl-stretch states, the tunneling splitting increases only 3.4-fold in the excited geared bend mode (2v5), compared to the 7.5-fold increase in (HF) 2. Furthermore, the tunneling splitting actually decreases with increasing v 5 excitation, from 51 cm -1 in 2v 5 to 40 cm -l in the 4v 5 state. This can be contrasted with the sixfold increase in tunneling splitting for (HF)2, in going from 2v 5 to 4v 5 (Table 3). Even more surprising are trends in the v 1 and v 2 vibrational manifolds. The tunneling splittings in vl(v2) + v 4 and vl(v2) + 2v 5 states are virtually the same as those of the vl(v2) fundamental. This is very different from the behavior of the analogous levels of (HF) 2, which exhibit pronounced mode dependence. The explanation for this unusual tunneling dynamics of (HCI) 2 is provided in part by Table 5. It is evident from there that all levels of (HC1) 2 having two or more quanta in the v 5 mode are above the C2h tunneling barrier, unlike (HF) 2 where the states with up to five quanta in v 5 lie below the tunneling barrier. The implication is that the (HC1)2 progression in the v 5 mode is not in the tunneling regime at all, while that of (HF)2 is. Consequently, there is no reason to consider the neighboring levels of (HCI)2 with even and odd v 5 quantum numbers, respectively, as forming tunneling pairs, especially since the separation between them is quite uniform (40-50 cm -1) and does not fit into a tunneling pattern.

198

ZLATKO BA~I(~ and YANHUI QIU

This explains the puzzling dependence of the tunneling splitting on w5 excitation (in ground HCl-stretch states) discussed above. It is clear now that the levels involved do not form tunneling doublets; instead, they are rather evenly spaced excitations of the v 5 mode. The nearly constant tunneling splittings in the w1 and w2 manifolds can be understood with the help of Table 6, which shows the vibrational levels of (HC1) 2 associated with v 1 and w2 excitations. It is evident that the intermolecular levels of the (+) and (-) exchange symmetry blocks have very similar energies, measured from the (+) and (-) sublevel, respectively, of the w1 or w2 fundamental. Consequently, the intermolecular levels are split by a nearly constant amount, which is very close to the tunneling splittings in v 1 or v 2 fundamentals.

Table 3. Experimental and Theoretical Tunneling Splittings Aw (in cm -1) of HF dimer for Selected Vibrational (J = 0) States in the Ground (%), wl, and w2 HF-Stretching Manifolds a

State

Aw (Experiment)

Vo

0.658690(I)

Vo + v3

Aw (Theory) 0.44

Aw/Aw I (Theory) I

--

15.02?

Vo + v4

--

0.98

2.23

Wo + 2w5

--

7.48

17.00

~

46.55

Wo + 4Ws W0 + V6

1.626(1) ( K = 1)

1.75

105.8 3.98

Wl

-0.2155(3)

wl + v3

-0.1447(4)

Vl + v4

- I .6639(22)

-0.796

6.22

vl + 2Vs

-2.7391 (I I )

-2.283

17.84

wl+v 6 V2

+

_

0.2334(4)

1#2 + 1'/4 2Ws

v2 + w6

0.089

~

_

--

0.41

3.5868(9) 0.2203(6)

1

~

--

v 2 + w3 u2

-0.128

34.14?

2.028 ( K = I)

--

_ I 4.61 22.79 --

Note: ~he tunneling splittings are defined as Aw = E~,- E~, with the superscript denoting the exchange

symmetry of the state. The tunneling splitting is negative when the lower component of the tunneling pair has (-) exchange symmetry, as in the v 1 manifold. The theoretical splittings were calculated on the SQSBDE PES. Also shown are the ratios of the calculated tunneling splittings Av/Avg where Avs is the tunneling splitting of the lowest energy state in each of the three monomer-stretching manifolds.

VRT Dynamics of (HF)2 and (HCI)2

199

Table 4. Experimental and Theoretical Tunneling Splittings Av (in cm -1) of HCI Dimer for Selected Vibrational (J = 0) States in the Ground (v0), vl, and v 2 HCI-Stretching Manifolds a State

&v (Experiment)

Vo

15.47668(4)

I,'o + I,'4 Vo + 2v5 1,'o + 41,15 vl vl + 1,'4

-

-

-

-

15.26 6.81

Av /Avg (Theory) I 0.45

51.36

3.36

--

40.34

2.64

-3.3237(4)

-2.31

1

-1.89

0.82

-2.28

0.89

-

Vl + 2v5

Av (Theory)

-

--

2.45

1

V2 + V4

v2

--

1.93

0.79

1,'2 + 2Vs

--

2.57

1.05

v 1 + 4v s

~

2.06

0.85

Note:

3.1 760(4)

aThe tunneling splittings are defined as Av = E~,- E~, with the superscript denoting the exchange symmetry of the state. The tunneling splitting is negative when the lower component of the tunneling pair has (-) exchange symmetry, as in the v 1 manifold. The theoretical splittings were calculated on the ES1 + EL2 PES. Also shown are the ratios of the calculated tunneling splittings Av/Avg, where Av8 is the tunneling splitting of the lowest energy state in each of the three monomer-stretching manifolds.

To conclude this section, our calculations 47 predict intermolecular mode dependence of tunneling splitting in HCI dimer very different from that observed in HF dimer, with tunneling splittings of the former being almost insensitive to intermolecular excitation. Spectroscopic data are needed for tunneling splittings in combination states of (HCI) 2 to test these predictions. Table 5. Comparison of the C2hTunneling Barriers and the Calculated Energy Levels of the Geared Bend (Vs) Intermolecular Mode of (HF) 2 and (HCI)2 in the Ground Vibrational State of the Monomers [v 1 v 2 = 00] a (HF)2

C2h Barrier

352

~HCI)2

48

(v3 v4 Vs v6)) (0000)

0.00

0.00

(001 O)

0.44

14.94

(0020)

160.58

53.48

(0030)

168.06

103.81

(0040)

292.65

148.16

(0050)

340.13

188.08

Note: aThedata shown (in cm-1) are for the SQSBDE PESof (HF)2 and the ES1 PESof (HCI)2. The barriers are

relative to the global minima of the respective 6D PESs.

ZLATKO BA(~I(Z and YANHUI QIU

200

Table 6. 6D Vibrational Energy Levels of (HCI) 2 for v 1 and v 2 HCI-Stretching Fundamentals [(v I v~)= (01), (10)] for Even Parity and Total Angular Momentum J= 0 on the ES1 + EL2 PESa (P = +l) + (vlv2v3v4vsv6)

Energy

(P = +l)-

AEol

(010000) (100000)

2856.98 2876.69

0.00

(010020) (010100) (100100)

2912.35 2931.96 2942.48

55.37 74.98

AE10 (vlv2v3v4vsv6)

Energy

gEol

(010010) 0.00 (100010)

2859.43 2874.38

0.00

(010030) (010110) 65.79 (100110)

2914.92 2933.89 2940.59

55.49 74.46

90.53 (100030)

AE10 0.00

(100020)

2967.22

(010120) (010200) (100200)

2 9 7 2 . 1 3 115.15 2 9 9 4 . 9 8 138.00 3002.35

(010130) (010210) 125.66 (100210)

2 9 7 4 . 5 0 115.07 2 9 9 6 . 3 8 136.95 3000.98

2964.94

(010040)

3 0 1 1 . 2 3 154.25

(010050)

3 0 1 3 . 2 9 153.86

66.21 90.56

126.60

Note: aThe energies (in cm -1) are relative to the 6D ground-state energy of-425.259 cm -1 . AE01 (AElo) is the

energy difference between excited and the lowest energy level in the (01) [or (10)] vibrational manifold of that symmetry block.

4.3. Intra- and Intermolecular Vibrations

Intramolecular Vibrational Frequency Shifts The frequencies of the two intramolecular V l and v 2 HX-stretching vibrations in (HX)2 are slightly lower ("red-shifted") than the vibrational frequency of the isolated HX monomer. This indicates softening of the intramolecular HX potential upon complexation and hydrogen bond formation. The theoretical and experimental v 1 and v 2 frequency shifts for (HF)2 and (HCI)2 are displayed in Table 7. The v 2 bound-HX stretch is red-shifted more than the v I free-HX stretch, in both HF and HC1 dimer. This is expected, since the bound-HX subunit directly participates in (and is perturbed by) the hydrogen bond, while the free HX does not. The experimental v 1 and v 2 red shifts of (HF)257 are significantly larger than those for (HCI)2 ,9 consistent with much stronger hydrogen bonding in the former. The v 1 and v 2 red-shifts calculated f o r (HF)214'15 are approximately two-thirds of the experimental values. It may be added that the frequency shifts obtained 44 for the new SC-2.9 PES, 19 -34.1 cm -l (vl) and -96.88 cm -l (rE), are much closer to experiment. Concerning HCl dimer, the ES 1 PES with added model for electrostatic interaction (ES1 + EL2), overestimates the v 1 red-shift and comes close to the measured v 2 red-shift. 44

VRT Dynamics of (HF)2 and (HCI)2

201

Table 7. Theoretica: and Experimental Vibrational Frequency Shifts (in cm -1) of v 1 and v 2 Intramolecular Stretching Fundamentals in (HF)2 and (HCI)2 a

(H~ Vibration

Theory

(HC/)2 Experiment

Theory

Experiment

Vl

-20.91

-30.5195

-11.60

-5.73

V2

-65.03

-93.3432

-29.00

-31.92

Note:

aThe frequency shifts are calculated as the energy difference between the lower tunneling components of the two monomer stretches, the (-) sublevel of v 1, and the (+) sublevel of v 2, respectively, and the monomer origin, which for HF is at 3961.422 490 cm -1 and for HCl at 2885.9777 cm -1. The theoretical frequency shifts for (HF) 2 were calculated on the SQSBDE PES, and for (HCI) 2 on the ES1 + EL2 PES.

Intermolecular Vibrations for Ground and Excited Vibrational States of Monomers Excitation ofv I and V 2 intramolecular vibrational modes changes the frequencies of the intermolecular vibrations in HX dimers. Selected theoretical and experimental intermolecular frequencies of ground-HF-stretching states, and of combination bands built upon v 1 and v 2 HF-stretch fundamentals are shown in Table 8. It is

Table 8. Theoretical and Experimental Intermolecular Vibrational Frequencies (in cm -1) for Selected Vibrational (J = 0) States of HF Dimer in the Ground (Vo), wl, and v 2 HF-Stretching Manifolds a

Intramolecular v0

vl

V2

Intermolecular

Theory

Experiment

w3

425.30

475(3)

v4

126.37

125(5)

2v s

160.58

161(5)

v6

378.72

399.79 (K = 1 )

v3

--

487.0153(4)

1./4

123.56

127.5726(2)

2v s

158.51

166.5232(2)

v6

--

V3

~

V4

138.48

132.6160(19)

2u s

169.57

178.6673(4)

v6

w

Note: aThe calculated frequencies, obtained on the SQSBDE PES, are the differences between the tunneling

components of excited and the lowest energy levels in the HF-stretching manifolds considered.

ZLATKO BA~I(~ and YANHUI QIU

202

Table 9. Theoretical and Experimental Intermolecular Vibrational Frequencies (in cm -I) for Selected Vibrational (J = 0) States of HCl Dimer in the Ground Vibrational State of the Monomers a

Vibration

Theory

v3

Experiment

243.23

v4 2v5

72.53 53.48 (38.54)

v6

164.21

(37.64540) 160.778

Note: aThe theoretical frequencies were calculated on the ES1 PES. The numbers in brackets are for the transition v s = 1 --~ v s = 2.

observed experimentally 6'58 that the frequencies of v 4 and v 5 intermolecular modes increase when v 1 or v 2 is excited, relative to those for ground-HF stretches, with the v 2 mode causing a larger increase. It is evident from Table 8 that the SQSBDE PES is successful in reproducing the observed intermolecular fundamentals for ground-HF-stretching states. 11'12 The only failure is the substantial underestimate of the v 3 fundamental frequency. However, the calculations 14'15 yield lower v 4 and v 5 frequencies in the v 1 than in the ground-HF-stretch manifold, which contradicts the experimental data, and points out the need for improved description of this aspect of mode coupling in (HF)2. The V 4 and v 5 fundamentals calculated for v 2 excitation are in good agreement with experiment. Experimental information about the intermolecular vibrations of HC1 dimer are scant; they are presented in Table 9, together with the theoretical values. 16The only intermolecular fundamental that has been observed directly is the v 6 out-of-plane torsion. Further refinement of the 6D PES for (HCI) 2 is contingent on a more complete experimental characterization of the intermolecular vibrations. 5.

SUMMARY

We have reviewed the recent full-dimensional theoretical treatments of HX (X = F, C1) dimers. The quantum methodology for fully coupled 6D calculations of the rovibrational energy levels of general diatom-diatom complexes was described first. Its key component is a sequential diagonalization and truncation procedure, which generates a compact quasiadiabatic basis sufficiently small to allow construction and diagonalization of the full 6D Hamiltonian matrix. Next, we discussed various aspects of the rich VRT dynamics of (HF) 2 and (HC1)2. The existing 6D PESs yield results which are generally in semiquantitative agreement with the wide range of experimental data. The agreement between theory and experiment is better for ground-HX-stretching vibrational states than for HX-stretch excited states of

VRT Dynamics of (t-1t:)2 and (HCI)2

203

the dimers. Further progress certainly requires higher level ab initio calculations of global PESs of (HX) 2, which can be systematically refined through comparison of spectroscopic observables calculated in 6D and their experimental counterparts. This would produce quantitatively accurate, truly benchmark-quality PESs for these fundamental hydrogen-bonded dimers.

ACKNOWLEDGMENTS We thank Prof. John Z. H. Zhang and Mr. Qian Wu for many helpful and stimulating discussions. This work has been supported by the National Science Foundation, through the Grant CHE-9613641.

NOTE ADDED IN PROOF The list of recent experimental papers on (I-IF)2 cited in the Introduction inadvertently omitted several important earlier contributions by Martin Quack and coworkers (ETH, ZUrich), which were the first to provide high-resolution spectroscopic data about the intermolecular vibrational modes of (HF)2. The relevant references are: Puttkamer, K. v.; Quack, M. Mol. Phys. 1987, 62, 1047; Puttkamer, K. v.; Quack, M. Chem. Phys. 1989, 139, 31; and Quack, M.; Suhm, M. A. Chem. Phys. Lett. 1990, 171, 517.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

Dyke, T. R.; Howard, B. J.; Klemperer, W. J. Chem. Phys. 1972, 56, 2442. Ohashi, N.; Pine, A. S. J. Chent Phys. 1984, 81, 73. Ba~id, Z.; Miller, R. E. J. Phys. Chent 1996, 100, 12945. Liu, K.; Cruzan, J. D.; Saykally, R. J. Science 1996, 271,929. Anderson, D. T.; Davis, S.; Nesbitt, D. J. J. Chent Phys. 1996, 104, 6225. Anderson, D. T.; Davis, S.; Nesbitt, D. J. J. Chent Phys. 1996, 105, 4488. Davis, S." Anderson, D. T.; Farrell, Jr., J. T.; Nesbitt, D. J. J. Chent Phys. 1996, 104, 8197. Davis, S.; Anderson, D. T.; Nesbitt, D. J. J. Chem. Phys. 1996, 105, 6645. Schuder, M. D.; Lovejoy, C. M.; Lascola, R.; Nesbitt, D. J. J. Chem. Phys. 1993, 99, 4346. Schuder, M. D.; Nelson, Jr., D. D.; Nesbitt, D. J. J. Chem. Phys. 1993, 99, 5045. Zhang, D. H.; Wu, Q.; Zhang, J. Z. H.; yon Dirke, M.; Ba~id, Z. J. Che~ Phys. 1995, 102, 2315. Necoechea, W. C." Truhlar, D. G. Chem. Phys. Lett. 1994, 231, 125. Quack, M.; Suhm, M. A. Chem. Phys. Lett. 1995, 234, 71. Wu, Q.; Zhang, D. H.; Zhang, J. Z. H. J. Chem. Phys. 1995, 103, 2548. Volobuev, Y.; Necoechea, W. C.; Truhlar, D. G. J. Phys. Chem. 1997, 101, 3045. Qiu, Y.; BaSiC:,Z. J. Chem~ Phys. 1997, 106, 2158. Bunker, P. R.; Jensen, P.; Karpfen, A.; Kofranek, M.; Lischka, H. J. Che~ Phys. 1990, 92, 7432. Quack, M.; Suhm, M. A. J. Chem. Phys. 1991, 95, 28. Klopper, W.; Quack, M.; Suhm, M. A. Che~ Phys. Lett. 1996, 261, 35. Bunker, P. R.; Epa, V. C.; Jensen, P.; Karpfen, A. J. Mol. Spectrosc. 1991, 146, 200. Elrod, M. J.; Saykally, R. J. J. Chem. Phys. 1995, 103, 933. Califano, S. Vibrational States; Wiley, London, 1976.

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23. 24. 25. 26. 27. 28. 29.

Papou~ek, D.; Aliev, M. W. Molecular Vibrational-Rotational Spectra; Elsevier: Amsterdam, 1982. Ba~ir Z.; Light, J. C. Annu. Rev. Phys. Chem. 1989, 40, 469. Cohen, R. C." Saykally, R. J. Annu. Rev. Phys. Chem. 1991, 42, 369. van der Avoird, A.; Wormer, P. E. S.; Moszynski, R. Chem. Rev. 1994, 94, 1931. DePristo, A. E.; Alexander, M. H. J. Chem. Phys. 1977, 66, 1334. Schwenke, D. W.; Truhlar, D. G. J. Chem. Phys. 1988, 88, 4800. Light, J. C.; Whitnell, R. M.; Park, T. J.; Choi, S. E. In Supercomputer Algorithms for Reactivity, Dynamics and Kinetics of Small Molecules; NATO ASI Ser. C, 277; Lagana, A., Ed.; Kluwer: Dordrecht, 1989, p. 187. Ba~i~, Z. In Domain-Based Parallelism and Problem Decomposition Methods in Computational Science and Engineering; Keyes, D. E.; Saad, Y.; Truhlar, D. G., Eds.; SIAM: Philadelphia, 1995, p. 263. Colbert, D. T.; Miller, W. H. J. Chem. Phys. 1992, 96, 1982. Rose, M. E. Elementary Theory of Angular Momentum; Wiley, New York, 1957. Alexander, M. H.; DePristo, A. E. J. Chem. Phys. 1977, 66, 2166. Davison, W. D. Chem. Soc. Faraday Discuss. 1962, 33, 71. Takayanagi, K. Adv. Mol. Phys. 1965, 1, 149. Zhang, D. H.; Zhang, J. Z. H. J. Chem. Phys. 1993, 99, 6624. BaSiC, Z.; Light, J. C. J. Chem. Phys. 1986, 85, 4594. Ba~i~:,Z.; Light, J. C. J. Chem. Phys. 1987, 86, 3065. Holmgren, S. L.; Waldman, M.' Klemperer, W. J. J. Chem. Phys. 1977, 67, 4414. Hutson, J. M.; Howard, B. J. Mol. Phys. 1980, 41, 1123. Althorpe, S. C." Clary, D. C." Bunker, P. R. Chem Phys. Lett. 1991, 187, 345. Elrod, M. J.; Saykally, R. J. J. Chem. Phys. 1995, 103, 921. Kofranek, M.; Lioschka, H.; Karpfen, A. Chem. Phys. 1988, 121, 137. Qiu, Y., Zhang, J. Z. H.; BaSiC:,Z.; Mtiller, H.; Quack, M.; Suhm, M. Manuscript in preparation. Karpfen, A.; Bunker, P. R.; Jensen, P. Chem. Phys. 1991, 149, 299. Tao, E M.; Klemperer, W. J. Chem. Phys. 1995, 103, 950. Qiu, Y.; Zhang, J. Z. H.; Ba~i~:, Z. J. Chem. Phys. 1998, 108, 4804. Bohac, E. J.; Marshall, M. D.; Miller, R. E. J. Chem. Phys. 1992, 96, 6681. Pine, A. S.; Howard, B. J. J. Chem. Phys. 1986, 84, 590. Schuder, M. D.; Nelson, Jr., D. D.; Nesbitt, D. J. J. Chem. Phys. 1989, 91,4418. Blake, G. A.; Bumgamer, R. E. J. Chem. Phys. 1989, 91, 7300. Below, S. P.; Karyakin, E. N.; Kozin, I. N.; Krupnov, A. E; Polyansky, O. L.; Tretyakov, M. Y.; Zobov, N. E; Suenram, R. D.; Lafferty, W. J. J. Mol. Spectrosc. 1990, 141,204. Chang, H. C.; Klemperer, W. J. Chem. Phys. 1994, 100, 1. Pine, A. S.; Lafferty, W. J. J. Chem. Phys. 1983, 78, 2154. Fraser, G. T. J. Chem. Phys. 1989, 90, 2097. Suhm, M. A.; Farrell, J. T.; Mcllroy, A.; Nesbitt, D. J. J. Chem. Phys. 1992, 97, 5341. Pine, A. S.; Lafferty, W. J.; Howard, B. J. J. Chem. Phys. 1984, 81, 2939. Bohac, E. J.; Miller, R. E. J. Chem. Phys. 1993, 99, 1537.

30.

31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58.

SPECTROSCOPY AND QUANTUM DYNAMICS OF HYDROGEN FLUORIDE CLUSTERS

Martin Quack and Martin A. Suhm

1. 2. 3. 4.

5.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Potential Energy Hypersurfaces (PES) . . . . . . . . . . . . . . . . . . . . . Quantum Dynamical Approaches . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Variational Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Diffusion Quantum Monte Carlo (DQMC) Techniques: The General DQMC Approach for Ground-State Properties . . . . . . . 4.3. Symmetry Restricted DQMC Approach for Excited States . . . . . . . 4.4. Quasiadiabatic Channel Quantum Monte Carlo Method for Rotationally and Vibrationally Excited States . . . . . . . . . . . . . . 4.5. Classical and Harmonic Approximations . . . . . . . . . . . . . . . . . Spectroscopy and Dynamics of the Dimer (HF)2 and Its Isotopomers . . . . . 5.1. Rovibrational States of (HF)2: Spectroscopy and Theory . . . . . . . . 5.2. Hydrogen Bond Interconversion . . . . . . . . . . . . . . . . . . . . . 5.3. Hydrogen Bond Dissociation . . . . . . . . . . . . . . . . . . . . . . . 5.4. Hydrogen Bond Libration . . . . . . . . . . . . . . . . . . . . . . . .

Advances in Molecular Vibrations and Collision Dynamics, Volume 3, pages 205-248. Copyright 9 1998 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 1-55938-790-4

205

206 206 208 209 211 211 213 214 215 217 218 218 218 225 226

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MARTIN QUACK and MARTIN A. SUHM

6. Spectroscopy and Dynamics of the HF "Filmer . . . . . . . . . . . . . . . . . 7. Spectroscopy and Dynamics of Higher HF Oligomers . . . . . . . . . . . . . 7.1. Experimental HF Stretching Spectra as Assigned to Different Cluster Sizes (HF)n . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Stretching Frequency Shift Predictions . . . . . . . . . . . . . . . . . . 7.3. Intracluster Vibrational Redistribution . . . . . . . . . . . . . . . . . . 7.4. Cluster Isomerization . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5. Concerted Hydrogen Exchange . . . . . . . . . . . . . . . . . . . . . . 8. Hydrogen Fluoride Nanocluster Dynamics . . . . . . . . . . . . . . . . . . . 9. Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

227 230 230 234 235 236 237 238 241 242 243

ABSTRACT Hydrogen fluoride clusters (HF)n and their isotopomers are reviewed as prototype systems for hydrogen bond dynamics. Infrared spectroscopy and ab initio calculations of potential hypersurfaces provide deep insights into the quantum dynamics of the hydrogen bonds in these clusters. Infrared spectroscopic developments using cooled cells, supersonic jets, Fourier transform, and laser techniques have contributed to the progress in our understanding of these clusters, as well as new developments in the analytical representation of empirically refined multidimensional potential hypersurfaces based on many body decompositions. The essential link between potential hypersurfaces and spectroscopic data is provided by quantum dynamical techniques allowing for numerically exact (or almost exact) predictions from solutions of the multidimensional rovibrational Schr6dinger equation. The application of quantum dynamical approaches such as quantum Monte Carlo techniques and variational techniques to hydrogen fluoride clusters is summarized. Properties and processes considered include hydrogen bond formation and dissociation, concerted hydrogen bond switching, hydrogen transfer, libration, and intramolecular vibrational-rotational redistribution. Spectral shifts, isotope effects, and the convergence of properties of small clusters to condensed phase properties in large clusters are discussed. We successively review results for the dimer, (HF)2, the trimer, (HF)3, and larger oligomers (HF)n including finally nanocrystalline clusters with n > > 100. Apart from a review of fundamental spectroscopic data we summarize also our current knowledge of the kinetic processes in these clusters, with timescales ranging from femtoseconds to microseconds, as derived from high-resolution spectroscopy.

1. INTRODUCTION Condensed molecular matter is largely shaped by interatomic and intermolecular forces: fluidity, surface tension, volatility, conductivity, diffusivity, phase state, and h a r d n e s s R t h e y all depend on how the constituent atoms and molecules interact with each other. 1 Primary processes of cluster dissociation, rearrangement, and

Spectra and Dynamics of (HF)n

207

further reactions govern these properties. Their detailed investigation profits substantially from small system sizes. This provides the key incentive for studying isolated molecular clusters in the gas phase. By gradually increasing the cluster size, one can hope to approach condensed phase behavior without having to give up the simplicity of finite systems. Towards the same goal, it is advantageous to choose prototype systems which contain as little nonessential complexity as possible. Undoubtedly, the hydrogen bond 2 is among the most important intermolecular interactions. 3 Hydrogen fluoride (I-IF) is the simplest molecule which can undergo such a polar hydrogen bond with itself. In particular the dimer (HF) 2 can be and has been studied by high-resolution rotational-vibrational spectroscopy. 4-7 Furthermore, the series of (I-W)n clusters (n = 2, 3, 4, 5, 6 . . . . ) provides a sequence along which the structure, energetics, and dynamics of the hydrogen bond can be studied particularly well by both experiment and theory. 8 But HF offers additional incentives for a detailed study. Its equilibrium vapor phase exhibits an unmatched clustering tendency at normal temperatures and pressures. 9 In contrast to carboxylic acids, 1~ the clustering does not peak at the dimer. Cooperative effects are very pronounced--the interaction energy in a larger cluster exceeds by far the sum of all molecular pair interactions 8 and the dynamics changes accordingly with cluster size. Finally, HF is a powerful solvent 11 for ionic and biomolecular matter and an important etching agent in semiconductor industry. Experimentally, infrared (IR) spectroscopy provides a sensitive tool for the study of hydrogen bonds. 2'12 Modern supersonic expansion techniques, 13 possibly combined with rovibrational Fourier transform infrared (FTIR) and laser spectroscopy, 14 offer access to hydrogen-bonded clusters in a collision-free environment, while the particular properties of HF also allow a study of these clusters in equilibrium gas cells. 15'16 NMR spectroscopy can provide important complementary information, 17 which has remained largely unexplored for HF in the last decades. 18,19 On the theoretical side, the small number of electrons in HF allows for high-level electronic structure calculations and for the accurate mapping of multidimensional potential energy hypersurfaces (PES), on which the nuclear dynamics can be investigated. 8 The small reduced mass of the HF molecule together with the pronounced anisotropy of its hydrogen bond interaction typically call for a quantum dynamical treatment, 2~with sizeable quantum effects already at the zero-point level. The purpose of this review is to summarize some of the recent IR spectroscopic and dynamical insights which have been obtained for clusters of HF from the dimer to nanometer-size aggregates. Rather than being exhaustive, we will highlight a few key dynamical features. A brief review of the connection to thermodynamic properties of the vapor phase has already been published. 9 Other reviews have concentrated on the PES 8'21 and on early dynamical work 21 on HF clusters, or have embedded HF cluster work in a wider framework of molecular aggregates. 22-33 Collisional energy transfer, 21'34 for which accurate dynamical calculations on the most recent PES remain to be done, is not reviewed here. Reviews of some

208

MARTIN QUACK and MARTIN A. SUHM

time-dependent quantum dynamical techniques and symmetry considerations related generally to our work on molecular and cluster dynamics can be found in refs. 35,36. 2.

SPECTROSCOPY

Hydrogen bonds convert rotational and translational degrees of freedom into intermolecular vibrations and strongly influence the participating intramolecular X - H stretching modes. Hence, rotational-vibrational spectroscopy is a natural choice for their study. As genuine hydrogen bonds are important in electronegative elements of the first Period (X = N, O, F) with relatively small polarizabilities, linear Raman spectroscopy has not played a dominant role, although it can be useful. 37 The lack of low-lying electronically excited states in HF prevents the application of powerful visible and near-UV laser techniques. 38 Radio-frequency and microwave spectroscopy played an important role in characterizing the HF dimer through its large amplitude tunneling motion 4'39'4~at the start of the era of high-resolution spectroscopy of molecular complexes a quarter century ago. In the future, this frequency range may receive revived interest due to the reliable prediction of vibrationally averaged structures of isotopomeric hydrogen fluoride clusters, which carry a small, zero-point motion-induced dipole moment. For the time being, IR spectroscopy remains the single major spectroscopic approach to elucidate the dynamics of HF clusters. One may distinguish two different techniques: 1. In direct absorption methods, the IR attenuation by the clusters is determined as a function of wavenumber ~. This involves FTIR spectroscopy 6'7' 9'16'41'42as well as diode, 43 difference frequency, 44-47 color center 48'49, and Raman-shifted dye lasers. 5~These techniques provide reliable information about the spectra, including band strengths and linewidths. In favorable cases, the latter may be decomposed into instrumental, Doppler, pressure, and lifetime broadening contributions. Sometimes, sensitivity can be a problem, unless pulsed supersonic expansions 43'44'51'52 or long-path cells 7'16'53 are employed. The sensitivity limits inherent in broadband FTIR spectroscopy have been alleviated recently by using a buffered, synchronously pulsed rapid-scan technique. 9'52 In this method, the full low-resolution (0.2-10 cm -1) IR spectrum is measured in a single rapid scan during an intense substance pulse of 50-500 ms duration. The pulse is diluted in a vacuum buffer chamber before entering the mechanical pumping system. By increasing the size of the buffer chamber, the achievable spectral resolution Av (determined by the pulse duration tp via A~ = l/(2tpVm)where vm is the mechanical mirror speed) can be increased up to the instrument limits without the need for a larger pumping system. In this way, FTIR jet spectroscopy 54-56 can be applied routinely to molecular clusters in a wide size range. 52 2. On the other hand, spectroscopies have been employed, which detect the IR absorption indirectly. These include laser-induced fluorescence, 57 bolometric de-

Spectra and Dynamics of (HF)n

209

tection, 48'58'59 as well as size-selective scattering and predissociation experiments. 6~ While bolometric techniques have led to beautifully detailed photofragment distributions for HF dimer, 48 combination with scattering permits discrimination of a given cluster size against all others without requiring rotational analysis (see Section 7). Classical predissociation spectroscopy with mass spectroscopic detection is often misled in its cluster size assignment by extensive cluster ion fragmentation, 62'63 unless it is combined with careful isotope substitution experiments. 64 Double-resonance spectroscopy 65 and saturation spectroscopy 66 have also been applied successfully. Often, these types of spectroscopy have a higher detection sensitivity than direct linear absorption experiments. However, their correct interpretation occasionally requires a detailed knowledge of the available fragmentation channels and typically, they do not resolve the problem of vibrational assignment in any better way than direct absorption techniques.

3. POTENTIAL ENERGY HYPERSURFACES (PES) Our current knowledge about the PES of HF clusters is summarized in a recent review 8. It derives from two complementary approaches. Ab initio supermolecule calculations at various levels of sophistication 67-69 provide important geometrical and energetical trends as a function of cluster size. Although these trends refer to local minimum energy and saddle point structures rather than to experimentally observable (vibrationally averaged) quantities, they yield qualitative information about the convergence of cluster properties towards the condensed phase. 33 At the highest levels available and for small clusters, 7~ supermolecule predictions turn out to be even quantitatively reliable in some cases. However, more global PES scans 72 and representations 72'73 are required to verify this reliability, because comparison to experimental data involves a nonlocal quantum treatment of the nuclear dynamics (see Section 4). Reliable inversion procedures to obtain a fulldimensional PES from spectroscopic data without quantum-chemical guidance do not seem to be in reach for systems of more than three atoms, 74 thus suggesting the use of large-scale ab initio PES scans. For practical reasons, such scans usually have to be carried out at somewhat lower levels. 72'75 In order to be useful, they require empirical adjustments. 72'76 In order to be applicable to larger clusters as well, they should be based on a many-body decomposition scheme. 8'17'68The idea is to separate the total energy of the cluster in a given geometry into three or more parts (illustrated here for a pentamer structure, with circles sketching the cluster structure and filled circles denoting those molecules which contribute to a given energy termS): 1. The energy of the monomers at the geometry in which they are found in the cluster relative to their free equilibrium energy, the so-called one-body potential V1. In HF pentamer there are five such terms, each one corresponding to a process"

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MARTIN QUACK and MARTIN A. SUHM

v,

9

63 9 ~

(I)

Oo d This monomer relaxation contribution is particularly important for strong hydrogen bonds, such as those present in larger HF clusters. It destabilizes the cluster with respect to the monomers, but this is overcompensated by the stabilization effect on the other contributions at the cluster minimum structure. Often, in a rigid-body framework, 77 this term is neglected. 2. The pair potential V2, which describes the interaction of two monomers in a given (monomer and pair) geometry of the cluster relative to separated monomers in the same monomer geometry. In the pentamer there are ten such terms, 8 each one corresponding to a process:

v'

9 Oe

63 9

~

(2)

~O 9

Typically, the sum over all these pair interactions in a cluster is the most important energetical contribution close to the global minimum structure. Often, it is the only contribution considered at all. 78-8~ 3. The three-body potential V3, which describes that part of the interaction of three monomer units in a given cluster geometry which is not captured by monomer relaxation nor by the three pair interactions. In the pentamer there are ten such terms, 8 each one corresponding to a process:

63 De

9 + 9

O 9 Oe

+

O 9

V3~ 9

9

O 9 ~ 9 go 9

+e + Oo + 9

(3)

The three-body potential is often neglected in simplified treatments, 78-81 but it turns out to be essential in hydrogen-bonded systems, reaching up to one-half of the pair interaction in HF clusters at their minimum geometries and much more in some other conformations. 8'17The popular reduction of the three-body potential to simple induction mechanisms is also not applicable here. 68'82-84 In principle, further contributions involving four-, five- and higher body forces would have to be included for clusters beyond the trimer. However, their importance typically decreases quickly after the three-body term. 8'17'68 They can be neglected altogether for some applications (minimum energy structures, hydrogen bond rearrangements, and energetics at moderate accuracies) or may be restricted to four-body terms in others (highly accurate structures and binding energies, collective vibrations, hydrogen transfer reactions). 8'85 The first generation of empirically refined full-dimensional pair potentials for HF clusters is represented by the SQSBDE PES, 76 which was mainly based 86 on

Spectra and Dynamics of (HF)n

211

systematic ab initio scans by the Vienna group. 75 Currently, the best available pair potentials are SC-2.9 and SO-3, two related, empirically refined fits of more than 3000 high-level ab initio points. 72'87 They are usually combined with a monomer potential of generalized Poeschl-Teller type. 72'87'88 The best available analytical three-body potential HF3BG 8'68'89 is a fit to 3000 ab initio points at lower level without empirical refinement, as the three-body term is relatively insensitive to basis size and correlation treatment. For more details, see ref. 8.

4. QUANTUM DYNAMICAL APPROACHES The connection between these potential energy surfaces and spectroscopic results 14'9~requires a careful characterization of the multidimensional dynamics of HF clusters. The cluster sizes discussed here range from the tetratomic dimer to nanometer-scale aggregates. The applicable dynamical methods depend very critically on this size. While complete full-dimensional bound state and quantum scattering calculations are only in reach for a pair of diatomics, the largest of these clusters can at most be treated at a locally harmonic or classical dynamical level, except for maybe the quantum ground state. Figure 1 schematically orders some of the available methods according to their applicability to HF clusters of increasing size. We shall give a very brief outline of the possibilities and limits of some of the more important methods which have been applied to HF clusters, before presenting results for individual cluster sizes.

4.1. Variational Techniques For low-dimensional problems, rigorous variational techniques using finitebasis set or discrete variable representations of the complete configuration space are very powerful. 91'92 Applications to clusters of two flexible diatomic molecules (six vibrational degrees of freedom) represent the state of the art in this field 93 and can provide a very accurate description of the bound and metastable state dynamics of (HE)2 ,94-96 including photofragment formation and distribution. 97-99 Collocation methods have been used as well. 1~176 For larger systems, selective diagonalization techniques are useful, l~176 From the rigorous variational approaches, a series of approximate methods can be derived. For the HF dimer, a powerful approximation consists in adiabatically separating the two high-frequency monomer vibrations from the four low-frequency hydrogen bond modes. 16'72'105-108 The resulting (4 + 2)D adiabatic wavefunctions retain the full dimensionality of the problem but can be obtained much more economically. This also allows for an efficient and accurate treatment of rotationally highly excited states, 72'1~176 which are currently too demanding for rigorous treatmenu. Limitations of the adiabatic approach will be discussed in Section 5. In a "crude adiabatic" (4D) approach, 16the monomer degrees of freedom are kept fixed at their equilibrium value or at some effective, vibrationally averaged geome-

212

MARTIN QUACK and MARTIN A. SUHM full dimensional quantum scattering variational bound states - - - DQMC excited states - - DQMC excited state approximations -

- - subspace grid/variational techniques -- - vibrational SCF + correlation approaches - - ab initio harmonic force fields ....

DQMC ground states and adiabatic channels - - on the fly "ab initio" classical dynamics - - harmonic frequencies - - classical dynamics

I

I

I .....

I

1

2

4

8

'

I-

I

16

32

-I

.....

64

I

I

I

128

256

512

t/

(HF)n

Figure 1. A p p r o x i m a t e ordering of various d y n a m i c a l m e t h o d s a c c o r d i n g to their applicability to HF clusters (HF)n of increasing size n.

try.109,110 While this introduces some arbitrariness, it is a reasonable assumption for weakly interacting monomers if one is mostly interested in the van der Waals modes. HF dimer is a borderline case, whereas larger HF clusters interact too strongly to make this a very useful assumption. 8 For larger molecules such as water and ammonia, the rigid monomer approach is often a prerequisite for the application of variational techniques and for the availability of suitable interaction PES. 9~176 In larger HF clusters, where a full variational treatment is out of reach even for frozen monomers, one can clamp further coordinates and concentrate on reduced subspaces. Such approaches have also been applied to (HF)2 ,112 where one can check their reliability against more rigorous treatments. Investigations of the n-dimensional pure HF stretching subspace for (H~, 0 with pronounced level shifts, while they confirm the rigorous DQMC prediction for K = 0. The coupling is actually so strong that it pushes the K = 1 level below the K = 0 level (see Figure 4), a remarkable and rarely found situation, 192 which had been discussed as a possibility for HF dimer spectra assignments before, 6'193 but without definitive conclusions. Hence, it turns out that inaccuracies in the SQSBDE surface 76 and the underlying ab initio data base 75 are compensated by neglect of Coriolis coupling in the excited K states, a notion which is supported by isotopomeric dissociation energies 48'177 and highlevel ab initio predictions. 7~ The new SC-2.9 and SO-3 PES, which are based on a much larger ab initio basis set than SQSBDE, predict a more anisotropic librational subspace. Now, a much higher v 6 fundamental band c e n t e r I~)6 -.- 420 cm -1 is found to be compatible with the observed K > 0 states 17'72 as well as with combination band data. 47'72'1~176 The K = 0,1 level inversion persists and is thus seen to be a robust feature of the various (HF) 2 PES. In contrast, the K = 1 v 6 tunneling splitting is found to be very sensitive to details of the PES, differing by a factor of 5 between SQSBDE and SC-2.9 and still by about 20% between the very similar SO-3 and SC-2.9 surfaces. 72'87'96We note that a band near 380 cm -l, which several plausible assignments have been proposed, 6'76 may involve the missing K = 0 level of V 6 in a AK = -1 transition from the vibrational ground state. This and the missing direct evidence for the v 3 fundamental near 480 cm -I suggest an experimental reinvestigation of the relevant far-infrared region, which is currently underway in our laboratory by improved FTIR-long path cell absorption techniques.

6. SPECTROSCOPY AND DYNAMICS OF THE HF TRIMER Given a sufficiently complete characterization of the HF dimer spectroscopy and quantum dynamics, the trimer (HF)3 offers the unique opportunity to extract and evaluate three-body contributions to hydrogen bonding. 8'6s Although the number of internal degrees of freedom is doubled relative to the dimer, the C3h symmetry of this cluster should assist a rotationally resolved spectral analysis. Nevertheless,

228

MARTIN QUACK and MARTIN A. SUHM

relatively little is known experimentally about the gas-phase IR spectrum of (HF) 3 and its isotopomers. A comprehensive predissociation study with isotopic substitution 64 indicates that (HF)3 can decay into three monomers as well as into a dimer and a monomer upon excitation of the HF stretching fundamental. From the isotopic substitution pattern, a cyclic structure with C3h symmetry can be inferred, but rapid intramolecular vibrational redistribution (IVR) and predissociation on a 2-20 ps timescale preclude an accurate structure determination. 64 DQMC calculations on a full-dimensional PES including the three-body term 68 confirm that two predissociation channels are open to (HF) 3 after HF stretch excitation, with the three-monomer channel being almost closed for cold clusters, as generated in a supersonic beam. This changes with successive deuteration, 68 and no open predissociation channel is finally predicted for (DF)3 .68 Again, the prediction is borderline, with the dimer + monomer channel being nearly open. However, subsequent improvements in the PES 72'89 confirm the prediction, whereas neglect of anharmonic zero-point contributions or neglect of three-body effects would reverse it. 68 In order to test the cluster stability and structure predictions made by the PES, a high-resolution IR spectrum of the DF stretching fundamental of (DF) 3 was recorded. 31'45 It consists of a dense line pattern including Doppler limited lines, indicative of excitation below (or at best very slightly above) the lowest dissociation channel. From the coarse-grained spectral structure, the cluster symmetry, planarity, and rotational constants can be derived. These are in good agreement with the predictions on the three-body inclusive PES, if (and only if) multidimensional zero-point averaging is taken into account. 31'68'194 The resulting F - F distance is 257-260 pro, where the uncertainty is dominated by possible anharmonic Coriolis contributions to the effective rotational constants. This is significantly shorter than the corresponding estimate for HF dimer 8'72 of 273.5 + 1 pm, in line with the important role of three-body contributions. In addition to providing a crucial test for the quality of the available three-body PES, the experimental spectra contain evidence for a rapid IVR process on a time scale of about 40 ps, 45 apparently involving essentially all available rovibrational states of a given J quantum number and therefore a multitude of states with up to nearly two hydrogen bonds broken. This is an interesting example of highly "statistical, global ''195 rovibrational dynamics near dissociation threshold, for which quantum-dynamical calculations remain to be performed. Little is known experimentally about the low-frequency modes of (HF) 3. An IR double-resonance study provides evidence for two overtone states in the CO 2 laser range, which were tentatively assigned via reduced dimensionality calculations. 65 For the fundamentals themselves, approximate DQMC calculations were carried out on the older PES. 68 They suggest that the anharmonic band centers lie 15-25% below the corresponding harmonic frequencies, although this may change somewhat for the new SO-3 + HF3BG surface. Noteworthy is an inverse isotope effect for the F - F stretching vibration VFF in (HF) 3 and in (DF)3 (I~FF[(H~3]
7 (Figure 3). 153 Green's function Monte Carlo (GFMC) calculations have also been made for very small clusters with C12.163 The number of atoms required for a complete first quantum solvation "shell" is not known for CI 2, but is at least 20 since only a single shell in the radial density profile is seen for all sizes studied to date. For C12Hetr the structural perturbations induced by overall cluster rotation have also been studied. Both the helium density and, to a lesser extent, the dopant density undergo some centrifugal distortion upon rotational excitation, with the effect being more pronounced in the larger cluster where the helium density is less tightly bound to the molecule. 162

Dopant Spectroscopy in Helium Clusters

431

(a)

I

0.080 !

--

0.060 -

:39

-.....

69

. . . . . 111

.=>, r c: 6)

20

. . . .

499 0.040 L-

9

p

o.o o

0.000

~

0

~ 5

~.._':_~.__.h_-~ .... 10 15

~

R

20

(.,&,)

(b) 0.05

[

9

i

9

w -,

9 "',

9

,

r,

.

,

.

w

9

H e nH F

0.04

~

Radial Density -

p..~.

0.02

0.01 1

0.00 0

2

4

6

8

10 12 14 16 18

r / 100 pm

Figure 2. (a) Radial helium density profiles for SF6HeN, with N = 20, 3, 69, 111, and 499 (from ref. 67). The second order DMC extrapolation is used here. The horizontal solid line represents the density of bulk liquid helium. (b) Radial helium density profiles for HFHeN, with N = 1, 3, 5, 7, 9, 11, 13, 15 (solid lines), 20, 25, 30, 35, 40, 45, 50 (dashed lines), and N = 98 and 198 (from ref. 81). The method of descendant weights was used to extract exact expectation values here. The dotted horizontal line represents the density of bulk liquid helium.

432

K.B. WHALEY

0:06 0.05 0.04 0.03 0.02 0.01 0 10 -5 7../100 pm

5

--'-"-~ 0

"- 5 r/lO0 pm

Figure 3. Contour plot of helium density about CI2He8 in cylindrical coordinates (z, r) averaged over the azimuthal angle. 15YThe coordinate origin is located at center of the CI2 bond and the z axis is directed along the bond. The vertical scale is in A -3. Exact DMC results, generated with the method of descendant weights.

The interpretation of such "solvation shells" in doped helium clusters requires some caution. Unlike a classical cluster, the solvating helium atoms are not localized in the T = 0 calculation. The nonzero density between shells is evidence of the quantum delocalization. Furthermore, because of the permutation symmetry, helium atoms can exchange between "shells" even at T = 0 K. The difference between these quantum shell structures and a classical solvation shell is clearly illustrated by the size dependence of the peak densities in Figure 2b. For a classical cluster, the first solvation shell would be completed before the second one begins to accumulate. In contrast, for the quantum cluster HFHe N shown here, the peak density in the first shell continues to increase, while the second shell is growing. 81 Another indicator of the distinction between this quantum shell structure and a classical solvation shell is the participation of the shell atoms in superfluid behavior. Finite temperature PIMC calculations for SF6HeN at T = 2.5 K (the cluster dissociates at about 5K) show nearly identical helium distributions to those at T = 0 K. 111 Microscopic calculation of the superfluid fraction via the contribution of permutation exchange paths shows however that many of the long permutation paths responsible for superfluidity involve atoms in several solvation shells, including atoms very close to the dopant. This is summarized in Figure 4. Thus it is important to realize that this quantum "shell structure" does not correspond to classically localized, solid-like particles. Heavier rare gas atoms (Ar, Xe) and metal atoms (Cu, Ag) embedded in the cluster interior show similar shell structure. 161'214'217Experimental indications confirm an

Dopant Spectroscopy in Helium Clusters

10

"

t

r

r

(a)

433 ,

!

(b)

i

tI

F

-5

F

:

i

-I0 ! -I0

-10

X

-5

0

5

10

X

Figure 4. PIMC "snapshots" of imaginary time paths projected onto the XY-plane for SF6He39 (from ref. 111 ). Thin solid lines---permutation cycles of length 1 He atom; thick dotted line--permutation cycle of length 3 He atoms; thick solid line--permutation cycle of length 1 5 He atoms. The SF6 mass was set to infinity here. Lengths given in ~.

interior location for the rare gases 8'52 and for Ag. 128 From Table 1 we see that the well depth of the Xe-He interaction is approximately three times that for He-He, but the metal atoms have smaller dimeric well depths 218 than He-He. However the heavier masses of these metal atoms results in significant binding in the dimer, so that surface localization is not favored. The presence of the dopant-induced density oscillations around metal atoms, illustrated for Agile N 214 in Figure 5, is a little surprising although not out of line with the magnitude of the (dimer) binding energies. This theoretical prediction is important since it shows that the conventional bubble model for foreign atoms is a severe approximation. The bubble model assumes that impurity atoms form cavities with diameters determined by the repulsive interaction, and that the helium density outside the cavity is unperturbed. Originally proposed for electrons in liquid helium, and later adopted for neutral impurity atoms, 32'219this model has been widely used for the interpretation of metal atom spectroscopy in liquid helium. 41 The intermediate character of the quantum dopant H 2 with regard to its location has been mentioned above. One can regard H2HeN as a mixed quantum cluster, in which both components are extremely delocalized. Correspondingly, there is little noticeable perturbation of the He density by the H2.2~ With the heavier isotope D 2 the onset of a quantum shell structure about the centrally located dopant is evident at larger cluster sizes (N ~ 20). 213 For the dopant series H 2, D 2, HF, CI2, and SF 6 the structural trends thus clearly parallel the energetic trends. All of the above studies dealt with neutral dopant species. Charged dopants have received less theoretical attention. Lewerenz has compared the structure of Ar§

434

K.B. WHALEY Density 0.05 f 0.04 0.03 0.02 I0.01

15

-15

20

10 r/lO0 pm z/lO0 pm

Figure S. Contours of helium density around Ag for ARHe199, in cylindrical coordinates (z, r) averaged over the azimuthal angle.21~rVertical scale in t~-3. Also shown is the Ag density, which forms the sharp central peak. This is suppressed by a factor of 100, to fit on the same scale as the helium density. Exact DMC results generated with the method of descendant weights. with that of the neutral ArHe Nin a VMC/DMC study 212 using an accurate empirical He-Ar + potential. 22~ Quantum solvation shells are found for both species, with approximately 12 atoms in the first shell. The dopant charge was seen to noticeably compact the surrounding helium, resulting in much higher peak density in the first shell for Ar+HeN, as well as larger angular correlations in this shell. This is not surprising in view of the significantly stronger binding of the ionic species (Table 3). Both structural and energetic criteria can be used to characterize shell structure. Changes in the energy derivative with respect to cluster size can be used to correlate shell growth and completion, as well as to isolate magic numbers of high stability. Energy derivatives with respect to cluster size show a minimum for Ar+Hejv at N = 12, indicating a magic number here, and some structure for the ionic species at larger sizes, but relatively smooth behavior for the neutrals. A similar study has been carried out by Wu and Watts for Xe§ using a simple potential model in which the xenon charge is represented by addition of an r -4 charge-induced dipole term to the Xe-He interaction. 217 The analogue species Ne§ has not yet been studied, but would constitute an interesting comparison because of the significantly larger binding energy of the dimer (core) ion which has been cited as the origin of the anomalous magic number N = 13 determined by drift tube experiments. 134 This system would therefore provide a good test of the pairwise additivity assumption made in most potential models. Brown et al. have

Dopant Spectroscopy in Helium Clusters

435

made unbiased DMC calculations for small clusters of N~ HeN,178 (N < 12) using an angle-dependent potential. This size range lies below the completion of a first solvation shell and no evidence for any significant anisotropy in the helium density was found, unlike the situation for C12 above. For all charged dopants, there is a large energetic effect, with the binding energies increased by at least an order of magnitude over that for the neutral analogues. This is what makes an unbiased DMC feasible (see above). No calculation has addressed the role of many-body polarization effects for charged helium clusters yet. While these were found to be significant in charged xenon clusters, 221'222 their magnitude should be considerably reduced for helium. Nevertheless, the relative weight of these with respect to the He-He contribution to the binding would be useful to explore for strongly bound charged dopants such as Ne.

4.3. Related Quantum Clusters-Doped (H2)N Doped clusters of molecular hydrogen display some similarities with doped helium clusters. The structural characteristics of the pure clusters (H2)N are quite difficult to converge, both in zero-temperature calculations 155and in finite-temperature PIMC calculations. 223 The H2-H 2 dimer is more strongly bound than He 2, and as a result the molecular hydrogen clusters possess greater cohesive energy and are also more highly structured. Nevertheless, a comparison of relative accuracies of liquid-like, solid-like, and shadow wavefunctions for VMC studies yielded the conclusion that the liquid-like trial functions are most accurate, despite the considerably higher degree of structure found relative to Heir. 155Zero-temperature, DMC calculations have been made for B(H2)N,179Li(H2)~,, and Li(D2)jv,155 and Hg(H2)12 and Mg(H2)12.177 Finite-temperature PIMC calculations have also been made for Li(HE)N.223'224Boron is an open-shell atom and so for this dopant it is necessary to take into account the potential anisotropy due to the orientation of the p orbital. In the B(H2)N calculations, 179 as in a recent DMC calculation for the analogous open shell system BArE,225 the lowest energy adiabatic interaction potential was derived for each cluster configuration. In all of these calculations the H 2 was treated as a spherical atom, which is assumed justifiable by the low pressure effective in unconfined clusters. The results of these calculations for doped molecular hydrogen clusters show that a lithium atom sits outside the cluster, as in its helium analogues (Section 4.2), while the other foreign atoms listed above are located in the cluster interior. For the latter, the H 2 density appears quite delocalized about the dopant species. However, these studies are currently too limited in size range and not enough microscopic calculations for the pure clusters exist yet for one to determine the extent of structural perturbation caused by the dopant species. Thus the existence of quantum "solvation shells" analogueous to those in doped Hetr has not yet been demonstrated. Finally some work has been done on the mixed isotopic clusters (D2)N.(H2)N. These can be regarded as a special kind of quantum doped quantum cluster, like the HEHeN

436

K.B. WHALEY

cluster mentioned earlier (Section 4.1). Two recent PIMC studies of the mixed hydrogen clusters show segregation effects, 226-22s which could provide a way to control the location of a tertiary dopant.

5.

D O P A N T SPECTROSCOPY CALCULATIONS

Microscopic calculations of dopant spectra is currently a very active and rapidly progressing area of theoretical research on quantum clusters. A number of calculations have been made for dopant vibrational frequency shifts. Rotational spectra of molecular dopants are more difficult to deal with theoretically, yet these appear to hold the key to understanding the role played by superfluidity in the dopant spectral response and thus constitute a key problem for current microscopic theory. Electronic spectra of metal atoms in quantum clusters have been addressed within the semiclassical Franck-Condon approach. Here the main problem is the accuracy of the excited-state potentials, and this is the limiting factor in extending such microscopic calculations of electronic spectra to the molecular dopants which have been recently studied experimentally. In this section we therefore not only summarize the microscopic calculations made to date, but also describe the more approximate treatments which have been employed when quantum Monte Carlo calculations are not possible, whether this is due to technical reasons or to lack of accurate interaction potentials.

5.1. Infrared Spectra of Molecules in HeN Infrared spectra of all molecules studied to date are characterized by small vibrational shifts and by modified rotational structure. Theory has addressed each of these aspects separately. Calculation of vibrational shifts require knowledge of the intramolecular stretching (bending) dependence of the dopant-He interaction potential. This is generally not known~HeHF 229and N~-He 23~are the only systems for which ab initio calculations have explicitly incorporated this. Empirical pair potentials either assume rigid dopant, or impose vibrational averaging, resulting in at best dopant vibrational-state specific interaction potentials. The first vibrational shift calculation was made for a relatively complicated dopant, SF 6, for which no vibrational dependence of the interaction with helium is known. Therefore this shift was calculated approximately, using the instantaneous dipole-induced dipole (DID) model proposed by Eichenauer and LeRoy for the vibrational shifts of molecules in argon clusters. TM In this model the perturbative dopant vibrational energy level shifts AEn = (~0znl~ntlzn~0), which should be given by matrix elements of the quadratic terms in the internal normal mode expansion of the SF6-He interaction potential, are replaced by electrostatic terms deriving from the attractive Component. This results in the following expression for the spectral shift of the triply degenerate v 3 vibration,

Dopant Spectroscopy in Helium Clusters

437 (27)

Alj(i) _-- ~ i ) _ / ~ E o

where AEo is the shift of the ground vibrational state, 2 N

AE~ =

2(o

1

90)

(28)

and AE~i), i = 1-3 are the shifts of the excited vibrational state, which are given by the eigenvalues of the matrix: 2 N

~u

= - 2-'~-

.

II

~j

+48t

~o)

Here Og/i)Q3 ) denotes the intramolecular dipole derivative matrix element, co the gas-phase vibrational energy, and a is the polarizability of helium. When the local solvation structure about the dopant preserves its symmetry, here the octahedral symmetry of SF 6, the excited state remains degenerate. It is important to realize that Eqs. 28 and 29 refer to expectation values over the cluster ground state. Thus from a T = 0 VMC or DMC calculation one can obtain only the three averaged values Av{i), i = 1-3, and no information about heterogeneous line broadening. Despite the formal similarity with the finite temperature averaging made by classical Monte Carlo sampling for SF6ArN,TM the different configurations sampled in Eqs. 28 and 29 all contribute to the spectral shift of a pure state, and have no meaning apart from this average. This becomes clearer when a PIMC average is considered. Here it is straightforward to evaluate the configuration average, 3

1

e -~e-

e(Av) =~ ~ ~] --2--(VmlS(A~0(R)- AV)IVm) i=1

(30)

m

where Av(0 (R) is the difference between the arguments of the expectation values in Eq. 27 (see Eqs. 28 and 29). This can be used to construct the temperaturedependent average shift:

= I AvP(Av)d(Av)

(31)

but not to construct a heterogeneously broadened line profile. The latter is defined by the spectral density, 1

3

e-13E -

~(~v) =~ Y~ ~]-2-~ (10 cm -1) and so correlated sampling is not required. The accuracy of these vibrationally adiabatic DMC calculations based on high-quality multidimensional interaction potentials, for both HFArN and FHe N, is sufficiently high that the comparison with experiment now allows systematic investigation of higher order, nonadditive potential contributions. Clary and coworkers have explored the opposite, inverse adiabatic separation for calculation of the vibrational energies of N~ in HeN.178 A series of unbiased DMC calculations was carried out for just the He atoms, with the N - N vibrational coordinate frozen at a range of values. The DMC energies were used to construct a one-dimensional vibrational potential for N~ which was then diagonalized. This study was restricted to small sizes (N < 12). Because of the relatively strong binding energy of the charged dopant, no importance sampling was necessary, but this could become more significant for larger sizes (the average binding energy per He atom was seen to decrease, which indicates more diffuseness and greater associated sampling problems). The inverse adiabatic shifts were seen to grow linearly with the number of helium atoms for small sizes (N < 8), and to be relatively large, e.g. -5.59(7) cm -1 for N = 8. It would be interesting and indeed feasible to estimate the nonadiabatic terms within this scheme and to estimate the relative error introduced by the approximation. If computational time is not an issue, and the intramolecular stretching dependence of the potential is known, sampling of all degrees of freedom provides a more complete approach to studying vibrational shifts. In a very recent study Beck and Watts used a site-site interaction potential for H20-He constructed to fit ab initio data for the rigid molecule, and used this to study the vibrational shifts of the water monomer in He N within the fixed node approximation. 233 This yielded a red shift of 1.5 cm -~ for the asymmetric stretch mode, in agreement with the experimental measurements, 79 and larger blue shifts (- 4-6 cm -1) for the bend and symmetric stretch modes, indicating a more complex interaction for the latter modes (see discussion in Section 2). Ideally, one would like to go beyond the fixed node approximation. Some exploration in this direction is now beginning. Thus, in a recent study of CI2HeN with model interaction potentials Lewerenz has investigated the explicit calculation of excited vibrational states of a diatomic dopant with a variable node in the diatomic vibration. 153 The cluster wavefunction is based on the usual factorization of Eq. 6, together with the analogue of Eq. 12 for dopant vibration. The dopant vibrational factor possesses a node, whose position is an additional parameter within a constrained VMC calculation for the excited vibrational state. The imposition of constraints, i.e. a penalty function on the minimization functional, is required to ensure that orthogonality with the ground-state wavefunction is maintained. Further exploration of these and otherTM dynamic node approaches (nodal

440

K.B. WHALEY

release) may lead to new insight into the effect of solvation on intramolecular vibrations. The rotational spectra of molecules embedded in He N poses considerable more problems for theoretical analysis than the corresponding vibrational spectra. Molecular vibrations are usually red-shifted by small amounts, consistent with the dominance of the polarization components of the interaction potential in vibrational level shifts (Section 2). Rotational spectra present more of a puzzle, with their empirical interpretation in terms of freely rotating entities which vary from the bare dopant for the lighter molecules (HE al H20, 79) to a "dressed" dopants with small numbers of helium atoms rigidly attached for the heavier molecules (SF 6 and its complexes, 75 OCS76). While the heavier mass dopants would be expected to be hindered to a greater extent, the observation of empirical free rotor behavior is unique to the He matrix environment. As mentioned in Section 2.1, a central question here is to what extent the superfluidity of the He nanomatrix plays a role. In microscopic terms, one can alternatively ask whether the apparent free rotor behavior is merely a result of the weak molecule-He interaction or whether the presence of appreciable quantum mechanical exchange in the environment is crucial. All microscopic calculations for SF 6 which incorporate the anisotropy of the interaction potential show significant density oscillations in the angular distribution of He around the dopant (see Figure 3). In a static picture, such an anisotropic solvation shell would imply a strongly hindered rotation, and splitting of the rotational degeneracy in the upper states. One task for theory therefore is to reconcile the empirical analysis of free rotor-like dopant rotation with the microscopic structural results of extensively delocalized, but still anisotropic quantum solvation shells. This necessarily requires addressing the excited rotational states directly. Ground-state calculations alone do not allow one to draw conclusions about the dynamics. Thus it is not clear whether in a rotational excited state the dopant sees an anisotropic cage or whether the anisotropic helium density adiabatically follows the dopant so that the latter effectively sees an isotropic solvation shell. The role of the helium superfluidity in spectroscopy is related, since the dynamic behavior of the quantum solvation shells will be influenced by this. The first calculation which incorporated dopant rotational kinetic energy was the ground-state calculation of HFHe N using vibrationally adiabatic HF-He potential surfaces. 81 Here the H and F atoms were moved independently and then the HF bond rescaled to impose the vibrational adiabaticity constraint. As mentioned in Section 2, this calculation showed near isotropic helium density distribution about the J = 0 HF, leading to the conclusion that the rotation is unhindered, and that the same rotational constant would be found as in the gas-phase. Fixed-node DMC calculations for rotationally excited states of a general rigid rotor dopant with importance sampling of all degrees of freedom has recently been achieved and applied to several molecules from Table 1 in 4HEN.1s7 Calculation of the first few low-lying excited states yields a free rotor level structure with reduced rotational

Dopant Spectroscopy in Helium Clusters

441

constants, confirming the experimental observations. Fixed-node calculations for the corresponding excitations in 3HEN, which determine the T = 0 spectra in a fermionic cluster are in progress. The recently developed imaginary time spectral evolution method 18s can also be applied to the rotational excitations. As described in Section 3.2, this method currently only yields the rotational excitation energies, and not additional information such as the associated helium density in the excited state. However it is not restricted by the fixed node approximation. For molecules in 4HeN this approach yields similar conclusions for the dopant rotations, i.e. an approximately free rotor level structure with reduced rotational constant, provided that only the first few levels are probed. 246 Other, more phenomenological, approaches to understanding the increased moments of inertia should also be explored. Semiclassical estimates of the kinetic energy of the fluid undergoing a cage pseudo-rotational motion about a rotor have been used to estimate moment of inertia increments for a hindered rotors in conventional matrices. 235 PIMC may allow analysis of kinetic contributions in the surrounding helium, with the advantage that one can switch exchange on and off, and thereby isolate the contributions from superfluidity. Density profiles computed by quantum Monte Carlo can be combined with hydrodynamical formulations of the quantum fluid. 236 It may also be useful to explore generalizations of the fluctuating liquid cage models (Section 3 and refs. 95-97) to a quantum liquid solvent. Two papers have also broached the possibility of additional structure due to cavity vibrations of embedded dopants. 237'238 Dopant delocalization within the effective cavity, to an extent dependent on the dopant-He reduced mass, is expected for all species, and is borne out by the results of VMC and DMC calculations. While the energies of these are relatively easy to estimate, the crucial question of whether these excitations possess any oscillator strength has not really been addressed. To answer this question, it is necessary to know the coupling between the dopant intramolecular vibrations and the dopant-He coordinate. Furthermore, these may couple strongly with the dopant rotation and thereby influence the rotational lineshapes.

5.2. Electronic Spectroscopy of Atoms in HeN and (H2)N Calculations of electronic spectra for atoms attached to or embedded in quantum clusters have been restricted so far to alkali atoms, for which the standard perturbative approach to generate electronically adiabatic excited-state potentials 239 is deemed valid. This can then be combined with the semiclassical generalized Franck-Condon theory to evaluate the electronic absorption spectrum in the presence of the matrix, using either the zero-temperature 24~or finite-temperature 197 densities to derive the Franck-Condon weighting factors. Such microscopic calculations have been made for Li(H2)N 155,197 and Li(D2)N,lss and are in progress for alkali atoms on HeN.241

442

K.B. WHALEY

The basic features of the alkali spectra on helium clusters were explored by Kanorsky and coworkers with an adiabatic model in which the alkali moves in a one-dimensional, static trapping potential provided by the dimple profile of the helium cluster. 58 In the absence of microscopic calculations, the density profile was approximated by that of the infinite surface, lIB In this simplified one-dimensional model both excited- and ground-state wavefunctions can be explicitly calculated and therefore the Franck-Condon overlap matrix elements evaluated directly. However these overlaps will be sensitive to distortions and shifts of the excitedstate potential curves due to fluctuations, both zero-point and finite-temperature induced, and therefore the amplitudes of the calculated spectra are not guaranteed. Comparison of the absorption spectra calculated from this one-dimensional model with experimental LIF spectra showed overall good agreement. The positions of the main spectral features, peaks near the gas-phase absorptions accompanied by long blue tails, are well reproduced for all three alkalis studied~Li, Na, and K. The only significant discrepancy between theory and experiment is the linewidths for the heavier species, Na and K. This is consistent with the structural findings from microscopic calculations which show that all alkalis are located outside the cluster surface, with the Li furthest away as a result of its light mass (despite the strongest binding (Table 3 and refs. 118,242). The Li atom therefore couples less to density fluctuations in the surface which would affect the linewidths. For the related system Li on (H2)N, both zero-temperature VMC/DMC 155 and finite-temperature PIMC 197 spectral calculations have been made. The earlier PIMC calculations employed pseudo potentials for the excited state, while the ground-state calculations used the recently calculated ab initio pair potentials. 243 Li also penetrates bulk hydrogen, and the 2s-2p electronic absorption has been measured both on clusters and in the bulk. This system is a prototypical high energy density material, TM and considerable interest therefore focuses on understanding the relation between the Li dopant spectrum and its location. For Li(H2)N, the zero-temperature semiclassical Franck-Condon analysis (summarized in the corresponding calculations for Li in bulk hydrogen 24~ using the ab initio pair potentials obtains overall good agreement with the experimental absorption spectrum obtained by beam depletion. 126This is shown in Figure 6, where it is however clear that the experimental spectra are systematically red-shifted relative to the theoretical predictions. There are a number of possible reasons for this. Several approximations are made in the description of the interaction potentials~the excited states are dealt with perturbatively and neglect coupling to higher lying states (which can indirectly give rise to many-body effects245), the H 2 is approximated as spherical, and the long-range components of the ab initio potentials are subject to uncertainties for these weakly bound systems. 243 Incorporation of the nuclear coordinate dependence of the dipole moment operator may also have an effect. Theoretical questions also arise in the'use of the semiclassical, mean-value approximations required to make the Franck-Condon analysis computationally tractable for a high-dimensional system. 24~ These are relevant for (H2)N and He N

Dopant Spectroscopy in Helium Clusters 8.0

.

.

.

.

.

.

I i

.

.

443

.

.

.

.

i

i

it 6.0-

=

!

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~ O

4.0-

i

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-Q

!