Solvable One-Dimensional Multi-State Models for Statistical and Quantum Mechanics 9811666539, 9789811666537

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Rajendran Saravanan Aniruddha Chakraborty

Solvable One-Dimensional Multi-State Models for Statistical and Quantum Mechanics

Solvable One-Dimensional Multi-State Models for Statistical and Quantum Mechanics

Rajendran Saravanan · Aniruddha Chakraborty

Solvable One-Dimensional Multi-State Models for Statistical and Quantum Mechanics

Rajendran Saravanan Indian Institute of Technology Mandi Mandi, India

Aniruddha Chakraborty Indian Institute of Technology Mandi Mandi, India

ISBN 978-981-16-6653-7 ISBN 978-981-16-6654-4 (eBook) https://doi.org/10.1007/978-981-16-6654-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

The book is dedicated to my family (R.S) for supporting my interests since childhood

Foreword

Central to chemical physics is the concept of a reaction coordinate, a one-dimensional variable along which we describe the progress of the process. It is often useful to think of it as a geometrical coordinate such as a bond length but it can also be a more general progress variable such as the path towards a folded state of a protein. From chemical kinetics, the idea influenced other key areas where a process unfolds, for example, the paths in a Waddington landscape of epigenetics. It is therefore useful and common to search for a quantitative description of a change along such a onedimensional coordinate. The monograph by Rajendran Saravanan and Aniruddha Chakraborty is a tour de force of the progress made in an analytical quantification of dynamics along a one-dimensional linear reaction coordinate. Nowadays, computer power is such that simulating an all atom description of many simpler chemical reactions is a reality. It is therefore possible to ask why we need one-dimensional descriptions of the dynamics. The answer backed by the history of the progress in theoretical dynamics is clear-cut and unmistakable. One-dimensional models provide understanding of the key features and, in favorable cases, they can lead to essential dimensionless parameters. Such parameters identify and compact the very essence. An early example is the Landau–Zener ‘velocity’ in a non-adiabatic avoided curve crossing problem where the transition is adiabatic at lower physical velocities and diabatic when the physical velocity is much higher. Equally well known and useful is the Marcus free energy parameter for electron transfer in solution. Saravanan and Aniruddha begin their monograph with an introductory chapter that tells what they aim to do and how they will go about it. The two main chapters of the book follow. These chapters deal, respectively, with the physics of dynamics along the reaction coordinate when the motion is heavily damped by the environment and when it is reversible meaning non-dissipative. The damped Brownian-like dynamics is quantified via the Smoluchowski equation. Diverse sinks that bleed population off are explored. Attention is also given to multistate dynamics where the different states share a common reaction coordinate. Arguably, the detailed second chapter that describes the dynamics heavily damped by the environment and the wide and diverse applications is the heart of the book and the part that would be most useful to the larger number of readers. In view of the success of this chapter, I would like vii

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Foreword

to suggest to the authors to follow, in a future volume, with an equally rich treatment of the intermediate and weaker coupling regimes. The third and shorter chapter of the present volume is about wave packet quantum dynamics. The emphasis is on analytical solutions for a linear coordinate which excludes the more computationally oriented but rich in insights works of Heller, Kosloff, and others. What is very well discussed are the essential quantum features of a rectilinear motion such as tunneling, reflection including reflection over the barrier, etc. A useful overall summary is the last chapter. The monograph of Saravanan and Aniruddha provides an extensive set of references several drawing attention to applications away from molecular physics thereby illustrating the broader implications of the work. A special feature of the monograph and a most worthy one are the online Mathematics workbooks that allow the reader to make her/his own computations and examples. A real boon to entering graduate students. Jerusalem September 2021

Raphael D. Levine

Preface

Developments in Statistical mechanics and Quantum mechanics aid in writing down general theories of complex systems that are of interest in natural science, engineering, and even in social sciences. The systems evolve over time according to the governing laws which in turn are subjective to their interactions with numerous subsystem associates. The systems can thus evolve from an initial state to one or more sets of final states which transition can be termed as multi-state transitions. This notion is an interdisciplinary concept and is useful across various disciplines of science. A popular example is a chemical process, the molecules evolve from the initial reactant state to one/more final product states through a change in one or few molecular coordinates. The detailed reasoning behind this idea is, if a molecule is inside the isolated (solvent) environment, the mathematical treatment starting from a complete Hamiltonian description that accounts for every electron, nuclei (solvent molecules) gives rise to a set of Born–Oppenheimer (BO) states that are coupled. Consequently, any molecular process for that matter is (examples such as spectroscopic transitions, chemical reactions, etc.) eventually caused by molecules transiting (called as non-adiabatic transitions) between initial and final BO energy states. Depending on the system size, one can categorize the study in two limits: (1) systems with few numbers (N ) of degrees of freedom, and (2) in the thermodynamic limit of N → ∞. For the first case of isolated molecules with few degrees of freedom, the system consideration can be such that the time evolution is unitary, rendering that the dynamics of the molecular process can be governed by coupledSchrödinger equations. On the other hand, for molecular systems that are immersed in solvents, the energy is continuously dissipated to the thermal solvent environment. The system should achieve thermal equilibrium at longer times, which can be facilitated by the use of a simplified coupled-Smoluchowski equation approach to study molecular kinetics. Taken that the process is effectively one-dimensional (1D) (along the steepest descent change of energy), the relevant configuration that changes during the process is taken to be x, hence simplifying the model equations. Now, 1D coupled-Schrödinger equations used for gas-phase kinetics include energy of the states Ui (x) and the couplings Vi j (x) that induce the i → j state transitions in the molecule. Once the time-dependent wave functions i (x, t) of each state are ix

x

Preface

derived, the population profiles of all states and other spectral profiles can be calculated. Whereas for a condensed-phase process, coupled-1D-Smoluchowski equations govern the probability distribution (Pi (x, t)) over a molecular coordinate (x) at any time t for a given electronic state i. The usefulness of Smoluchowski equation to this case can be appreciated by noting the presence of system parameters inbuilt in the model equation such as the temperature and viscosity of the solvent bath, excitation parameters (if any), and molecular parameters such as potential, initial molecular configurations. The resulting time-domain profiles derived for both the Smoluchowski/Schrödinger case can be useful in studying chemical kinetics. The usefulness of a time-domain profile in understanding and predicting reaction data as a function of parameters is discussed in Chap. 2, which is otherwise a tedious job to obtain the multi-parametric fitting function. The above-mentioned kinetic equations do not always support full analytical solutions even for many simplified models and even in the reduced 1D descriptions. Specialized exceptions are those problems that have special mirror or translational symmetries or when a few problems admit a partial success of obtaining the Laplacedomain solutions, i.e., the wavepacket/probability distribution are obtained as a function of the time-transformed Laplace variable s. This field of non-adiabatic treatment of molecular kinetics in 1D has been evergrowing since the 1920s, so complete surveys of the historical developments are already available as various review articles and books. This monograph is limited in that matter as this details only the recent time-domain mathematical developments that are done for my Ph.D. thesis research with a quick introductory text. It discusses in detail, the mathematical attempts to derive exact results for a few unsolved cases known in the literature. The monograph can be of use to advanced grad students of physics and chemistry who want to get acquainted with the possible methods of treating non-adiabatic transitions in positional coordinates in addition to knowing the detailed introduction and historical research survey. The possible extensions to the presented methods are suggested in respective sections and related open problems are posed that would be of interest to the readers of this monograph. The monograph is organized as follows: Chap. 1 gives a detailed physical background behind the methodology of considering Smoluchowski/Schrödinger equations in molecular physics and poses some useful models that are considered. In Chap. 2, we derive and discuss the time-domain results for molecular models governed by the classical Smoluchowski equation. Appropriate analytical, numerical verifications and limitation of our solution in few cases are discussed in detail in respective sections. In Chap. 3, we apply the methods to solve scattering and multistate problems governed by quantum mechanics (Schrödinger operators). Chapter 4 concludes by enlisting the possibilities arising from the present work and the future scope of the work. I very much hope that the readers would enjoy reading this those who belonging to mathematical physics, chemical physics, and molecular physics backgrounds. The MATHEMATICA notebooks that produces the verifications and graphical results are presented section-wise in the following website of the author: https://sar anotebookarchive.weebly.com/

Preface

xi

Keywords: Mathematical methods • Non-equilibrium statistical physics • Quantum scattering • Time-domain methods • Coupled-partial differential equations (PDEs) • Multi-state problems • Molecular processes • Non-adiabatic transitions • Reactiondiffusion systems • Condensed-phase chemical dynamics • Stochastic processes • Schrödinger equation • Smoluchowski equation • Brownian motion • Spectroscopic and collision processes • Complex systems. Mandi, India May 2021

Rajendran Saravanan Aniruddha Chakraborty

Acknowledgements

The monograph entitled “Solvable One-Dimensional Multi-State Models for Statistical and Quantum Mechanics” is the collection of my Ph.D. research with a brief introduction and detailed literature survey of related works. In the preparation of the book, there had been a direct and indirect influence of several masters, some through their direct teachings, and few through their texts. First and foremost, I would like to thank my Ph.D. supervisor, Dr. Aniruddha Chakraborty who laid the foundation for my interest in chemical physics through his teachings and discussions. I am grateful to the gem of his book collection and for his generosity in letting me use them which resulted in inspiring me to work and write in this area. This monograph is a result of the research conducted under his guidance cum facility support at the Indian Institute of Technology Mandi, India. Next, I would like to thank all my math, physics, course teachers of IIT Mandi whose lectures on the go helped me surpass various steps of my Ph.D. research. I also would like to thank Professor Raphael David Levine for his valuable discussions on scientific topics, specific to molecular physics. The reflection of his interaction with me appears in reshaping the first introductory chapter and future scope chapter of this monograph. Further, there were comments on the manuscript made by Professors Biman Bagchi and Prof. Peter Hanggi, against which I would like to thank them for their motivational comments and for their time spent in educating me on the background studies that the earlier version of this manuscript missed. I am also indebted to them for they suggested the future reach of the work presented here. The influence of several inspiring physics, chemical physics texts on this monograph are not to be overlooked and most of them are also referenced in various sections of the book. I hereby thank my friend Kavita Yadav for being a support while preparing this manuscript, her support by offering a laptop during the COVID lockdown mattered much in the successful preparation of the draft and to carry out my routine remote postdoctoral research. I also thank my friends Prithiv, Alka Kumari, Sonali Kumawat for their support by taking care of the non-academic chores that saved quite my time during the preparation of this manuscript. It was a pleasure to work in a coordination with the springer editor Dr. Loyola, and others Tony Sekar, S Sindhu for their co-operation and support in the publishing process. Thanks reserve to Ms. P. R. Umadevi and Mr. Abishek Pandey xiii

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Acknowledgements

for reading and pointing out english errors. While the limitations of this book and missing information should be solely accredited to my ignorance or carelessness.

Contents

1 Theoretical Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Conservative Versus Dissipative Dynamics in Molecular Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Introduction to Non-equilibrium Systems . . . . . . . . . . . . . . . . 1.1.2 Fokker–Planck Equation Description of Random Walk Approach to Non-equilibrium Systems . . . . . . . . . . . . . 1.2 Arisal of Multi-state Problems for Representing Electron Transfer Process in Condensed Phases . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Born–Oppenheimer Surfaces of the Molecules in Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Derivation of a Multi-state Hamiltonian Involving Reactant and Product States of the System . . . . . . . . . . . . . . . 1.2.3 Libby’s Theory on Electron Transfer: Concepts, Applicability and Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Marcus’ Theory on the Electron Transfer . . . . . . . . . . . . . . . . 1.2.5 Extensions to the Marcus’ Description . . . . . . . . . . . . . . . . . . 1.2.6 Formulation of Multi-state Problems in Electron Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Existing Analytical Methods for Solving Multi-state Reaction-Diffusion Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 The Survey of Already Existing Models Solvable in Time-Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Survey of Laplace-Domain Solvable Models . . . . . . . . . . . . . 1.4 Models Considered in the Monograph . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Effective Single State Models . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Multi-state Reaction Diffusion Systems . . . . . . . . . . . . . . . . . 1.4.3 Few More Applications of Reaction-Diffusion Models . . . . 1.5 Introduction to Quantum Multi-level Systems . . . . . . . . . . . . . . . . . . 1.5.1 Few Examples of Multi-State Processes . . . . . . . . . . . . . . . . . 1.5.2 Equation of Motion Governing the Multi-level Systems . . . .

1 1 4 6 11 12 16 18 20 27 29 31 31 34 35 35 36 36 38 38 39

xv

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Contents

1.5.3 Survey of Existing Analytical Methods to Solve Quantum Multi-state Models . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.4 Models Considered for Quantum Mechanical Case in This Monograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Brief Summary About the Organization of This Monograph . . . . . .

40 42 43

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.1 Prelude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.2 Solution of Single-State Problems in Statistical Physics . . . . . . . . . . 46 2.2.1 Introduction to the Kernel Method . . . . . . . . . . . . . . . . . . . . . . 46 2.2.2 Diffusion Dynamics of a Distribution in Flat Potential with a Dirac Delta-Function Sink . . . . . . . . . . . . . . . . . . . . . . . 47 2.2.3 Exact Diffusion Dynamics of a Distribution in the Presence of Two Competing Sinks: Oster– Nishijima Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.2.4 A General Method to Solve Diffusion in Piece-Wise Linear Potentials in the Time Domain . . . . . . . . . . . . . . . . . . . 65 2.2.5 Understanding Condensed-Phase Dynamics Using Parabolic Potential Models in Presence of Delta-Function Reactive Terms . . . . . . . . . . . . . . . . . . . . . . 75 2.3 Exact Dynamics of Coupled Problems . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.3.1 Exact Diffusion Dynamics of a Distribution in a Coupled System: A Simple Open System . . . . . . . . . . . . 88 2.3.2 Deriving General Characteristics About Open and Closed Systems Using a Simple Model . . . . . . . . . . . . . . 99 2.3.3 Exact Diffusion Dynamics of a Distribution in a Closed System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3 Investigation of Wave Packet Dynamics Using the Presented Time-Domain Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Prelude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Exact Time-Domain Solution of the Schrödinger Equation for a New Scattering Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The Mathematical Methodology to Evaluate the Wave Packet Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Exact Wave Packet Dynamics of Gaussian Wave Packets in Two-Flat States Coupled at a Point . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Methodology: Kernel Method to Calculate Wave Packet Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Asymptotic and Exact Solutions, Results . . . . . . . . . . . . . . . . 3.4 Deriving General Characteristics of a Two-State System Using a Simple Solvable Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Application to Set of Flat Diabatic Curves . . . . . . . . . . . . . . .

119 119 119 120 125 130 130 130 135 140 142

Contents

3.4.2 Application to Linear Diabatic Curves . . . . . . . . . . . . . . . . . . 3.4.3 Application to Harmonic Diabatic Potentials . . . . . . . . . . . . . 3.4.4 Summary of Results, Discussions . . . . . . . . . . . . . . . . . . . . . . 3.5 Exact Solution of a Delta-Function Coupled Two-State System that Exchange Population Through Decay . . . . . . . . . . . . . . . 3.5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xvii

142 143 143 149 153

4 Summary and Future Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Appendix: Miscellaneous Discussions of Sect. 2.2.5 . . . . . . . . . . . . . . . . . . . 157 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Acronyms

1D, 2D, 3D BC BO CDF CM DOF ET FPE NA PDE PDF PES, PEC RC TDSE TISE TPM

One-, two-, three-dimensional Boundary Conditions Born–Oppenheimer Cumulative Density Function/Cumulative Distribution Function Center-of-Mass Degrees of freedom Electron Transfer Fokker–Planck Equations Non-adiabatic Partial Differential Equations Probability Density Function/Probability Distribution Function Potential Energy Surface/Curve Reaction Coordinate Time-Dependent Schrödinger Equation Time-Independent Schrödinger Equation Tri-Phenyl Methane (dye)

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Chapter 1

Theoretical Background and Motivation

Abstract The monograph considers two parts: In Chapter 3, we consider molecular kinetics treated by coupled-Schrodinger equations in relevance to isolated molecules in gas-phase. In Chapter 2, we consider reactive-Smoluchowski equations that can treat molecules inside a dissipative environment. This introductory chapter discusses the background behind the arousal of the above equations in molecular kinetics. We would start by distinguishing the unitary versus dissipative dynamics that motivate the use of the above equations directly from a pure physical discussion. Theoretical motivations are discussed in closer relation to an electron transfer process between metal complexes in solutions, which is relevant to any molecular process for that matter. The detailed origin for the molecular models is presented starting from the Hamiltonian description of a molecular process in gas/solvent phases, which gets further reduced to the study of transitions among adiabatic Born-Oppenheimer states. Now, the coupled-Schrödinger equations become an appropriate tool for studying systems with few DOFs (N) as in a gas-phase process. For molecular systems coupled to the environment, i.e., N → ∞, then classical Smoluchowski equations with reactive terms are of use. The features of the considered models in the monograph are drawn along with a detailed historical survey. Vast applicability of multi-state theory are enlisted at the end of the chapter that spans areas across natural sciences, social sciences, and engineering.

1.1 Conservative Versus Dissipative Dynamics in Molecular Studies The models that are ought to be considered in this monograph are of two distinct type of kinetics: (1) molecules in presence of conservative force fields, (2) molecular motion in presence of dissipative forces. The principal difference can be emphasized using a simplified example: Let us consider a simple pendulum given by, 2 m ddt x2 = −kx, wherein there are no dissipative forces, the dynamics is sinusoidal as we know. On knowing the exact phase-space coordinates at a later time (tl ), one can © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 R. Saravanan and A. Chakraborty, Solvable One-Dimensional Multi-State Models for Statistical and Quantum Mechanics, https://doi.org/10.1007/978-981-16-6654-4_1

1

2

1 Theoretical Background and Motivation

Fig. 1.1 (1) Top panel: Comparison of an electronic wave-packet in presence and absence of dissipation. a In the absence of dissipation, the electronic wave-packet of a diatom oscillates back and forth periodically regaining its shape (This is the situation in which the vibrational coherence is infinitely-lived (Chap. 11 of Ref. [88])). b For molecules in dissipative environment such as the solvent phase, the energy is lost to the environment and the turning point of the distribution diminishes depending on the damping strength due to the solvent. The gray-colored dynamics is underdamped effect of solvents, the dotted distribution dynamics is in the overdamped limit. (2) down panel: Corresponding to the above-shown 3 cases, the respective phase-space dynamics of a single particle is shown that discusses the harmonic motion under 3 damping cases: a an isolated oscillator has a closed phase-space trajectory for a fixed energy E, b, c dissipation-induced underdamped, overdamped oscillations tending to equilibrium position and rest velocity. Adapted with permission from G. A. Voth, R. M. Hochstrasser, The Journal of Physical Chemistry A, 100, no. 31 (1996): 13034-13049. Copyright 1992 American Chemical Society

predict with certainty, the initial coordinates (or for any t < tl ) with which the oscilla2 = −kx, the tor started. On the other hand, when dissipation are present, m ddt x2 − b dx dt oscillator eventually would get damped at a rate depending on the damping strength b. After a finite longer time, the position coordinate would be at equilibrium with velocity coordinate approaching zero in magnitude. Here, no use of this positional and velocity coordinates in retracing to the past. One can see that any possible initial coordinate, would have lead to the same final condition, i.e., the backward time-tracing is not unique. This is what dissipation can do to the dynamics. This is technically referred to as the Hamiltonian not obeying time reversal symmetry. The dissipative forces acts to erase the memory of the system from where it started to evolve. The notice of this simple difference can quickly help to relate the same difference arising in molecular studies. Let us consider the time-evolution of electronic wavepacket of

1.1 Conservative Versus Dissipative Dynamics in Molecular Studies

3

a diatom in response to their internuclear distance (x). The time evolution in presence and absence of dissipation is schematically compared using the Fig. 1.1. The top left panel shows the low-energy dynamics of a diatomic molecule isolated from the environment. The t = 0 Gaussian wave packet periodically oscillates back and forth and would regain its initial original form as far as the isolation is provided. For this conservative evolution, the time-evolution operator is unitary, an indication of conservative systems obeying the time-reversal symmetry. The Schrödinger equation can be used to evaluate the probability of a molecule being with the nuclear coordinate x, schematic dynamics shown in Fig. 1.1a. But if the molecule is coupled to surroundings, the energy can be lost to the surroundings. Although the energy in the overall system plus environment is conserved, so the lost energy should be expected to get back to the system if one waits for long long time, but this backward probability is very very small, rendering that never happens, so the system’s energy can not be expected to be conserved. The top right panel shows that the energy dissipation through which the classical turning point of the oscillator diminishes, and eventually settle to the minimum.1 It is usual to model the above-described energy loss processes to the surroundings through a coupling of bilinear type between the system and the infinite solvent modes, who themselves are assumed to be uncoupled for simplicity. It is sometimes called as the Caldeira–Leggett–Zwanzig model [27, 86]:

Hs B

p2 + U (x), Hs =  22m 2   pj ωj 2 HB = + q , 2 2 j j    γ j2 2 = x , −γ j xq j + 2ω2j j

(1.1) (1.2)

(1.3)

the total Hamiltonian is the sum of the Hamiltonian in parts: corresponding to the system, the bath and the system-bath interactions: H = Hs + H B + Hs B ,

(1.4)

where in the above equation, ω j being the frequency along the solvent modes, γ j is coupling strength between the system and the jth solvent mode. The classical version of the above Hamiltonian is exactly solvable for its position-velocity coordinates. Using the Hamilton’s equation of motion, the coupled set of phase-space coordinates can be derived. Therefrom, the equation of motion for the system momentum coordinate can be picked as given by,

1

In the quantum dissipation, the diminishing is called the vibrational decoherence happens to eventually settle the wavepacket to the minimum of the oscillator. We would also soon discuss that the distribution over the nuclear coordinate becomes what is known as “thermalized” taking up the Boltzmann distribution function in the upcoming Sect. 1.1.2 from the classical perspective.

4

1 Theoretical Background and Motivation

   γj d p(t)  = −U (x(t)) + γj qj − 2 x . dt ωj j

(1.5)

Evaluating the position coordinate of all the solvent modes and substituting in the above equation (for details Refer [181]) rationalizes the phenomenologically introduced Langevin governing equation for a system that undergoes random motion in presence of stochastic forces. The Langevin equation is expressed as, d p(t) p(t) = −U  (x(t)) + F p (t) − ζ , dt m

(1.6)

where F p (t) is the sum of all stochastic forces that act as a noise to the subsystem of interest. In deriving Eq. 1.6, it has been assumed that the force contributions are solely due to the instantaneous position and momentum of the system, which also reflects through the above form. However for a driven-dissipative system, it is known that the system can act from the past memory, because the solvent-system coupling strength can actually be frequency dependent in reality. But in the above equation, the strength is assumed to be frequency independent. In presence of the above-mentioned memory effect, the phenomenological Langevin equation can be generalized:  t p(t − s) d p(t) = −U  (x(t)) + F p (t) − , (1.7) K (s) dt m 0 where the memory function, K (s) can be explicitly related to the microscopic modes in a following manner promoted through the use of Caldeira–Leggett model: K (t) =

 γ j2 ω2j

j

In the continual limit given by,

 j



→



cos ω j t.

(1.8)

dωg(ω), the integral representation of the

cos ωt. In which expression, if the above equation becomes, K (t) = dωg(ω) γ (ω) ω2 coupling strength is constant, i.e., it is independent of the frequency, then K (t) becomes a delta function δ(t) and in this limit, the generalized Langevin Eq. 1.7 reduces to the Markovian Langevin equation given by Eq. 1.6. 2

1.1.1 Introduction to Non-equilibrium Systems A system that is energetically coupled to its surroundings, when subject to a sudden change using an external interaction, it drives the system away from the equilibrium. Then the system relaxes by undergoing a continuous change in order to finally achieve the equilibrium. The branch of physics that addresses this kind of relaxation dynamics is called non-equilibrium statistical mechanics. Examples of such systems are spin relaxation in presence of the magnetic field, movement of chemical particles

1.1 Conservative Versus Dissipative Dynamics in Molecular Studies

5

attached to a heat bath, solvation in liquids, the behavior of stock prices in response to fluctuations, etc. For studies off the equilibrium, the environmental fluctuation forces are time-correlated in contrary to an equilibrium system. An important theorem that forms the backbone of the non-equilibrium mechanics is the fluctuation-dissipation theorem which relates this time-correlation of fluctuations to the dissipative properties of the system such as, resistance in conductors, viscosity in solutions, etc. For example, the fluctuating force term F p (t) in Eq. 1.7 directly relates to the dissipation scaling K (t) through the following manner, F p (t)F p (t  ) = k B T K (t − t  ),

(1.9)

which is a relation between microscopic forces of individual particles relating to the macroscopic observable like temperature T of the solvent, and viscosity ζ , etc. Otherwise for a complete microscopic description, such a non-equilibrium system should be characterized by a Hamiltonian H({qi }, { pi }) in the appropriate phase space. It is extremely cumbersome to solve the Hamiltonian to obtain the exact trajectory of all N-bodies in presence of all degrees of freedom and interactions, and also would be of little relevance in the context of the macroscopic reality. In specific to say, while solving the exact system of equations given by Eqs. 1.1–1.4, the solutions would highly depend on the initial conditions of solvent modes.2 This is where using of the concept of a statistical ensemble would come into the picture. During a measurement, what one measures is actually an time-averaged property of the system,3 and the principle of ergodicity states that when a system evolves for a long long time, the system visits to every possible microstate available for the given macrostate of the system. The use of the above statement is, if we prepare infinite number of mental copies of the system, which are snapshots at different time instant of the system’s evolution. Then the long-time average of an observable should, by construction equal to the average of the observable over those mental copies. Macroscopically, what is relevant is the average of the observable measured from all those copies, that are otherwise known as ensembles. Each ensemble then has its own time-dynamics owing to the difference in their initial starting points of the stochastic environment. Under this ensemble consideration, it makes more sense to speak about the dynamics of a probability distribution, ρ(x, p, t) over the phase-space rather than speaking about the single particle dynamics of individual ensembles. The time-evolution of the distribution of swarm of the ensembles is governed by the classical Liouville equation given by,  ∂H ∂ ∂H ∂ ∂ρ = {H, ρ} = . − . ρ, (1.10) ∂t ∂p ∂x ∂x ∂p

2

It is also that the prior knowledge of every microscopic coordinates at initial time cannot be known precisely due to Heisenberg uncertainty (detailed discussion in Chap. 2 of [14]). 3 The time-average of an observable would not depend on the initial coordinates of the stochastic environment, facilitated by the ergodicity of total system.

6

1 Theoretical Background and Motivation

where in the above equation, the Hamiltonian H should contain all the fluctuation and dissipation drives responsible in the non-equilibrium regime. The probability distribution living in the 6N-dimensional Liouville space is difficult to be solved by tracing the dynamics in all 6N-dimensions. In molecular processes like chemical reactions, solvation dynamics, etc. the number of DOFs (N) typically goes in the order of Avogadro number. Hence, even simulating such processes using a computer also face similar problems: It requires a huge computational budget depending on the degree of the desired accuracy. The inter-particle potential involved in the simulation should be approximated in the cost of the available computational power. One another work in this field is devoted to the Random Matrix Theory [96] using which the systems are characterized by an N*N matrix whose entries are random variables. Diagonalising the N*N Hamiltonian for finding the eigen-modes is another analytically and computationally challenging task. Taking into account that several approaches are difficult to establish the connection from a microscopic theory to reality, one compromises to a minimal description using a great level of intuition. So contextually, one can think of dividing the whole system into a subsystem of interest and the rest of the system as a reservoir (canonical ensemble) as done in Eq. 1.4. The subsystem then undergoes a random motion in response to the stochastic reservoir environment. Now only the probability density that characterizes the subsystem is deterministic provided an equation governing its motion can be derived. The subsystem can be a particle, entity, or a species depending on the context. In such processes, the random motion is resulted due to the competition between environmental fluctuations and dissipation that ultimately leads to a steady state. A special case of such processes where the species of the subsystem undergoes diffusive motion to acquire different states by passing through reactive spots can be classified as ‘Reaction-Diffusion systems’. Examples where such models are useful are, chemical dynamics in condensed phases, biological processes inside solutions, price dynamics of assets and stocks, sports predictions, and in climate analysis. The approach of using Brownian motion to study molecular reaction-diffusion systems started after the work of Kramers’ theory [80].

1.1.2 Fokker–Planck Equation Description of Random Walk Approach to Non-equilibrium Systems In this section, we derive an equation of motion that govern the phase-space probability density related to understanding a reaction-diffusion processes. If it is a multivariate process, it needs to be described by the motion of a set of axes, a = (a1 , a2 . . .) which are drawn from the configuration space spanned by a reaction-diffusion system. The motion of a can be described by the following Langevin equation, which is a multi-dimensional generalization of Eq. 1.6: da = −bv(a) + F(t). dt

(1.11)

1.1 Conservative Versus Dissipative Dynamics in Molecular Studies

7

where b is the viscous drag to the motion and F(t) is the random force from the reservoir. The exact form of the noise F(t) is not known, but for simplicity and tractability, the noise is assumed to take a Gaussian distribution whose first and second moments are given by, F(t) = 0 and F(t)F(t  ) = k B T K (t − t  ) = Bδ(t − t  ),

(1.12)

which is named as the Gaussian white noise. This consideration would physically mean that all type of forces are equally possible and the random walker’s position is influenced only by the force at the instantaneous time, and not from the history i.e., the correlation between impact forces at different instant is zero. Such kind of deltacorrelated driven processes are called Markovian process. In reality, all processes are not Markovian, but nevertheless the assumption, the following derivation does not go through. For non-Markovian processes, where the instantaneous force have a memory, deriving equations are challenging and are still open for exploration in this subject [1, 12, 93, 134, 167, 169]. Because the governing equation given by Eq. 1.11 itself contain random terms, the entity a(t) is a random variable. We cannot obtain the exact trajectory of the a(t), and we are rather interested in the probability distribution function ρ(a, t) over a. We start from the conservation requirement of the probability distribution in absence of any absorbing boundaries as expressed by,  

ρ(a, t)da = 1.

(1.13)

differentiating with respect to time yields the following, 

 ∂ρ(a, t) ∂ ∂a + . ρ(a, t) da = 0. ∂t ∂a ∂t

(1.14)

The validity of the above equation over an arbitrary volume  implies that the integrand should be equal to zero at all points. This leads to the following expression along with the use of the Langevin equation ∂ ∂ρ(a, t) = − .[−bv(a) + F(t)]ρ(a, t). ∂t ∂a

(1.15)

Here one can define an operator, Lψ = −b ∂(v(a)ψ) , ∂a ∂ ∂ρ(a, t) = −Lρ(a, t) − .F(t)ρ(a, t). ∂t ∂a

(1.16)

Multiplying e Lt throughout the equation and integrating over time t leads to the following,

8

1 Theoretical Background and Motivation

t e Lt ρ(a, t) 0 = −



t

e Lτ 0

∂ .F(τ )ρ(a, τ )dτ. ∂a

(1.17)

Substituting for ρ(a, t) from Eq. 1.17 in Eq. 1.16 yields the following,   t ∂ρ ∂ ∂ dτ e−(t−τ )L .F(τ )ρ(a, τ ) = 0. + Lρ(a, t) + .F(t) e−t L ρ(a, 0) − ∂t ∂a ∂a 0 (1.18) Averaging over the noise in all terms: ∂ ∂ρ + Lρ(a, t) + .F(t)e−t L ρ(a, 0) ∂t ∂a  t ∂ ∂ − . e−(t−τ )L F(t)F(τ ). ρ(a, τ )dτ = 0 ∂a 0 ∂a

(1.19)

The term F(t)e−t L ρ(a, 0) = 0 as the term next to F(t) is the solution of the noise free system. The integration term is analytical when the noise is delta-correlated. Using F(t)F(τ ) = Bδ(t − τ ) ∂ρ ∂bv(a)ρ(a, t) ∂ ∂ + − .B. ρ(a, t) = 0 ∂t ∂a ∂a ∂a

(1.20)

The above equation is the general form of a governing equation for probability density of particles in phase-space of any dimension. For a simplest case of one-dimensional phase space, the vectors in the above equation is substituted to be,

a=

x px /m



B=

0 0 0 ζ kB T



b = −1, v(a) =

px /m −U  (x) − ζ p

 x /m

,

(1.21)

to obtain the corresponding one-dimensional Fokker–Planck equation for the probability density in (x, px ) space is given by, ∂ px ∂2 ∂ P(x, px , t) =− P(x, px , t) + ζ k B T 2 P(x, px , t) ∂t ∂x m ∂ px ∂ − (−U  (x) − ζ px /m)P(x, px , t). ∂ px

(1.22)

where P(x, px , t) is the noise-averaged probability distribution equal to ρ(x, vx , t)) , k B is the Boltzmann constant, ζ is the friction, U (x) is the relevant potential. For a system without the heat bath, i.e., H B = 0, the Eq. 1.10 yields the following governing equation, ∂ρ ∂ p ∂ = ρ− (U  (x))ρ ∂t ∂x m ∂p

(1.23)

1.1 Conservative Versus Dissipative Dynamics in Molecular Studies

9

For a system with H B = 0, and with the white noise considerations, the equation of motion was derived in Eq. 1.22. Comparing Eqs. 1.22 and 1.23 shows that the contribution of a solvent bath is just addition of a two operators-diffusive second order, plus a convection term due to viscous drag as given by the last term in the Eq. 1.22 alongside to note that the probability density getting noise-averaged. This was the artifact of the extra term due to F(t) arising in Eq. 1.16 besides the Liouville-like operator. In the limit of high solvent friction, it corresponds to the over-damped limit, momentum relaxation of the solute would be much faster (shown in Fig. 1.12c using single particle phase-space trajectory) than the position relaxation, the non-equilibrium dynamics happen only in space coordinates only as the momentum coordinates have achieved the equilibrium much earlier. In that case, following are the substitutions to Eq. 1.20 that govern the dynamics in 1D position coordinate, a = x, B = D, b = −1, v(a) = −

U  (x) ζ

(1.24)

 ∂ P(x, t) ∂2 1 ∂  = D 2+ U (x) P(x, t). ∂t ∂x ζ ∂x

(1.25)

The above equation is called the Smoluchowski equation which governs a probability distribution. The above equations have wide applications in understanding Brownian motion [60], noise in electric circuits [164], etc. It is interesting to see that in the → 0, the steady state limit t → ∞, the system attains equilibrium given by, ∂ P(x,t) ∂t normalized solution of Eq. 1.25 becomes, peq (x) = P(x, t = ∞) =  ∞

e

− Uk (x) T B

−∞ e

− Uk (x) T B

(1.26) dx

and Eq. 1.22 in the corresponding limit becomes, p2

peq (x, px ) = P(x, px , t = ∞) =

e ∞ ∞ −∞

− 2mkx

−∞ e

BT

− Uk (x) T

p2 − 2mkx T B

B

− Uk (x) T B

(1.27) dxd px

which are the classical versions of the Boltzmann equilibrium.4 In particular to reaction-diffusion systems, the reaction conditions are incorporated through suitable absorbing boundaries. For a reaction-diffusion system, the P(x, t) represents the probability distribution that lies in the diabatic potential state U (x), distributed over a relevant reaction coordinate x. The Eq. 1.25 describes reactant molecules that acquire different configurations through the action of diffusion along the reaction coordinate and D is the diffusivity along x. The absorbing boundaries result in the reduction from the initial population when the reduction conditions are met by the distribution. The 4

The concepts such as normalization of population, fixed temperature T , and energy not being a fixed quantity along with the steady-state form mentioned in Eqs. 1.26, 1.27 relates to the canonical ensemble treatment of our dissipative molecular system.

10

1 Theoretical Background and Motivation

sink function can be introduced by subtracting relevant terms from the Smoluchowski equation as given by, ∂ P(x, t) = ∂t



1 ∂  ∂2 D 2+ U (x) − k0 S(x, t) − kr ∂x ζ ∂x

P(x, t),

(1.28)

The diffusing molecules attain a product state with some probability (characterized by k0 ) by reaching some part of reactive configurations, which is introduced through the exit condition S(x, t). A constant decay (kr ) of population that is irrespective of the spatial configuration is considered in studying the radiative contribution in non-radiative electronic relaxation [10] can be added trivially in the result. The parameter D is the diffusivity of the distribution, U (x) gives the drift along x, and ζ is the viscosity coefficient. The given P(x, 0) is the initial distribution function for which the problem is to be solved. The probability that the distribution survives the decay gives the concentration of the reactants when the U (x) is taken to be a reactant’s diabatic potential curve. The survival probability is given by,  Q(t) =

∞

P(x, t)dx.

(1.29)

−∞

The solution of survival probability for various cases of U (x), S(x) in presence of appropriate boundary conditions such as: • initial distribution: P(x, t = 0), • spatial boundaries, if involved, would be interesting and will have a variety of applications in science. The aspects of the different models and initial conditions will be discussed in the upcoming sections in detail. If there are different states of the system with each state having its own governing equation, the states are coupled and thereby exchange the population. Such models capture a process when the system can switch between states over time for a given environment. For example, all molecular processes are one way of rearranging electron distributions which in turn are transitions between the molecular potential surfaces. The phenomenon where molecular processes are achieved by crossing molecular potential is called curve crossing. In statistical mechanics, a curve crossing system can be represented by a system of coupled equations given by, ⎛

L11 ⎜ S21 ⎜ ⎜ . ⎜ ⎝ . Sn1

S12 L22 . . Sn2

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

... ... . . ...

⎞⎛ ⎞ ⎞ ⎛ P1 (x, t) P1 (x, t) S1n ⎜ ⎟ ⎜ P2 (x, t) ⎟ S1n ⎟ ⎟ ⎜ P2 (x, t) ⎟ ⎟ ⎜ ⎜ ⎟= ∂ ⎜ ⎟ . ⎟ . . ⎟⎜ ⎟ ∂t ⎜ ⎟ ⎠ ⎠ ⎝ . ⎠⎝ . . Lnn Pn (x, t) Pn (x, t)

(1.30)

where Pi (x, t)’s represents the population in different states, Si j is the coupling element from any state i to state j, Lii is the Smoluchowski operator for that particular 2 state taking the form, Lii = Di ∂∂x 2 + ζ1 ∂∂x Ui (x) with each state having own potential form Ui (x) and the diffusivity Di .

1.2 Arisal of Multi-state Problems for Representing Electron …

11

1.2 Arisal of Multi-state Problems for Representing Electron Transfer Process in Condensed Phases Molecular processes inside condensed phases are a continuous change between the continuum of moving molecular potential surfaces. In this section, we derive the microscopic derivation for multi-state problems in the context of electron transfer process in solutions. The theory of electron transfer processes have immense curiosity for both its diverse nature and its adequate complexity. Its ubiquitous nature spans across the fields of physics, chemistry and biology. Some example systems such as photo-induced quantum dot blinking [25, 26], dye-sensitized solar cells [91], chemical systems [20, 59], also life making processes like photosynthesis [166], phosphorylation of ADP to ATP [51], CO binding to heme [2], electrode kinetics [4] etc. are electron transfer systems. There are two kinds of mechanisms involved while electron transfer between reactant molecules. They are, (1) Outer-Sphere-Electron Transfer: In this mechanism, the electron transits between the reactant complexes without affecting the inner sphere (reactant ions+ tightly bound solvents) and no significant ionic motion is involved during the process. This is due to the very small orbital overlap between the complexes which make the electron hop between the molecules leaving the ligands/bonds of the mother species unaffected. This mechanism of transfer is called the outer-sphere-electron-transfer. For example, [Fe(CN)6 ]4− + [IrCl6 ]2− −→ [Fe(CN)6 ] 3− + [IrCl6 ]3− . Here, the coordination sphere remained unaffected. (2) Inner-Sphere-Electron Transfer: On the other hand, if the complexes interact strongly, faster ionic movements can cause a change in the inner sphere and the process is called inner-sphere-electron transfer. It can be said that the charge transfer is mediated through the ligand bridging which may/may not detach after the electron transfer process. Usually, such bridges will be electron donors. For example, [Co Cl(NH3 )5 ]2+ + [Cr(H2 O)6 ]2+ −→ [Co(NH3 )5 (H2 O)]2+ + [CrCl(H2 O)5 ]2+ , where, -Cl- is the bridging ligand. Being a many-body dynamics, the electron transfer processes are intractable analytically. Usually the theories are highly phenomenological and are good qualitatively. For qualitative reasons, one can assume that the electron transfer process in a condensed phase is highly influenced from the solvent relaxation. The reason is obvious as the electric field of the ions are coupled to the polarisation of the solvent sphere. After the development of ultra-fast spectroscopic techniques, the experimental evidences show that there are contributions from other nuclear coordinates as well. Then the role of the vibrational relaxation was included in the theoretical description. In this section, we start from the microscopic description showing the arisal of multistate descriptions and then will show how both the solvent and vibrational modes would contribute significantly to the ET process leading to a many-dimensional multi-state model to completely describe the process.

12

1 Theoretical Background and Motivation

1.2.1 Born–Oppenheimer Surfaces of the Molecules in Solution This section presents the origin of continuum of molecular potential energy surfaces in a poly atomic molecule, surrounded by solvents. The motion of the electron in an electronic process is strongly coupled to the nuclear and solvent modes, due to the interactions between electrons, nuclei, and the solvent. The dynamics is governed by the Schrödinger equation in presence of these interactions. The energy of the electron is a parametric function of nuclear and solvent coordinates. It will be shown that these parametric functions are quantized giving rise to various potential surfaces. Any electronic process inside solvents such as the electron transfer, making and breaking of bonds, or electronic relaxation are mediated through the motion of the nucleus and solvents. Through the motion of the nuclei and solvents, the initial and the final states are either connected by an activation barrier of the process (will be called adiabatic representation of the problem), or through an energy-mediated resonant transfer (diabatic representation). In the end, we see that the process has a multi-state description. Throughout the description of the process in the following, the donor and acceptor molecules will be characterized by a set of nuclear positional coordinates of the molecules (R1 , R2 , R3 ,…Rq ) and positional coordinates of n-electrons as (r1 ’, r2 ’,…rn ’) in respect to a lab-fixed origin. A complete microscopic treatment can be done by considering the total Hamiltonian which have energy contributions from every possible motion. The energy contributions from the solvents are added to the Hamiltonian in terms of solvent coordinates (q1 , q2 ,…qn ) which are the Fourier transform of polarization vectors of the solvent sphere. The total Hamiltonian can be written as, q n   2 2 2 2 ∇ R,i − ∇  + Tˆsol + Vˆ 2Mi 2m e r ,i i i

(1.31)

(q,n) (n,n)  1  V ( Ri − R j )) + V ( Ri − r j ) + V ( ri − r j + Vˆsolvent-species . 2

(1.32)

H =− with, Vˆ =

(q,q)  i, j

(i, j)

i, j

The total Hamiltonian is the sum of energy contributions from kinetic energy of nucleus, kinetic energy of electrons, kinetic energy of solvent coordinates, nucleusnucleus interactions, nucleus-electron, electron-electron, solvent-molecule interaction terms respectively. Once the donor and acceptor molecules approach, the only interest would be the relative motions of the electron and nucleus that mediates the process. Hence, we transform the origin to the centre of mass whose coordinates are, RC M

1 = M

 q  i

Mi Ri +

n  i

 m i ri .

(1.33)

1.2 Arisal of Multi-state Problems for Representing Electron …

13

q with M as the total mass of the system, M = i Mi + nm e . We do transformation of coordinates from lab-fixed to centre of mass fixed frame (3 coordinates) and a set of other 3n + 3 coordinates that give us a physical understanding of motion. The coordinates are, • 3 centre of mass coordinates describing the translational motion of the molecule as a whole (X C M , YC M , Z C M ), • q(q − 1)/2 set of distance vectors pointing between nuclei that describe the rovibrational motions given by, Ri j = R j − Ri , i = j,  • The electron position is transformed as, xi = xi − q1 i X i . This new representation describes the electron motion with respect to centroid of the molecular system. It consists of total 3n coordinates for n-electron system comprising one set (xi , yi , z i ) for each electron. The transformations modify kinetic operators and the interaction terms. After implementing the above mentioned transformations (Eq. 1.33), the kinetic operators become, −

q,q q q n     2 2 2 2 2 2 2 ∇C M − ∇ R,i − ∇r  ,i = − ∇i j .∇ik 2Mi 2m e 2M 2Mi i i i j>i,k>i ⎡ ⎤ q q,n   2 ⎣ 2Mi  − ∇C M .∇i j + (2δiq − 1) ∇i j .∇e ⎦ (1 − 2δiq ) 2Mi M i

j

−

q  i

2 2Mi q 2

 n 

j,e

2 ∇e

(1.34)

−

e

n  2 2 ∇ 2m e r,i i

For a diatomic molecule, i.e., q = 2, the coupling term ∇C M .∇i j between CM and vibrational motions disappear. In such case, the centre of mass motion can be separated unless there is a collision or a potential in translational motion. However for q > 2, i.e., a polyatomic molecule, the problem cannot be reduced into a one-body problem, the centre of mass is coupled to ro-vibrational coordinates. We are dropping the centre of mass motion which is an approximation. In the following way, the total Hamiltonian can be separated into different motions of interest. The total Hamiltonian is,  2 C M + Hˆ 0 + Hˆ  + Tˆsol , H= − 2M

(1.35)

where the first term gives the centre-of-motion. The second term Hˆ 0 is expressed as, Hˆ 0 =

n  −2 i

2m

i + Vˆ .

(1.36)

In the above Hamiltonian operator Hˆ 0 , the influence of centre of mass co-ordinates and the motion of nuclei on the electron’s motion are separated. It consider only

14

1 Theoretical Background and Motivation

the electron’s motion about geometric centre of the molecular system, hence can be impressively called as clamped nuclei Hamiltonian (or electronic Hamiltonian). The Hamiltonian Hˆ  describe the movement of the nuclei that is coupled to the electrons’ motion, neighbouring nuclei motion, and also to the centre-of mass motion which are expressed in the following terms, q q,q  2  ∇i j .∇ik (1.37) 2Mi j>i,k>i i ⎡ ⎤ q q,n   2 ⎣ 2Mi  (1 − 2δiq ) ∇C M .∇i j + (2δiq − 1) ∇i j .∇e ⎦ − 2M M i i j j,e  n 2 q n  2   2 2 − ∇ − ∇r,i + Tˆsol e 2 2M q 2m i e e i i

Hˆ  = −

In the subsequent discussions, we drop the centre-of-mass motion and the interactions between the solvent and chemical species for simplicity. We write the following eigenvalue equation for the total Hamiltonian: Hˆ = ( Hˆ 0 + Hˆ  + Tˆsol ) (r, R, q) = E(R, q) (r, R, q).

(1.38)

The total wave-function (r, R, q) is not analytical. Considering that the electron’s motions are much faster than the nuclear motion, one can consider the electron’s motion parametrically for a given set of nuclear coordinates. The total wave function (r, R, q) can be expanded in the basis of electronic wave functions,5 (r, R, q) =

N 

ψk (r; R, q)χk (R)ζk (q).

(1.39)

k

where χk and ζk are the ro-vibrational and solvent eigenfunctions, which obey the following eigenvalue equations, Hˆ mol χk = E k χk and Tˆsol ζk = E sol ζk respectively. The term Hˆ mol is the Born–Oppenheimer Hamiltonian for the nuclei which is to be defined in this section later. The ψk (r; R, q) are the electronic eigenfunctions of the system that obey the following eigenvalue equation, Hˆ 0 ψk (r; R, q) = E k0 (R, q)ψk (r; R, q), substituting Eq. 1.39 in Eq. 1.38 yields, ( Hˆ 0 + H  + Tˆsol )

N 

ψk (r; R, q)χk (R)ζk (q) = E(R, q)

k 5

For a detailed explanation on this expansion, see [120].

N  k

ψk (r; R, q)χk (R)ζk (q),

1.2 Arisal of Multi-state Problems for Representing Electron …

15

projecting the above equation in the basis of jth electronic state yields, ψ j |( Hˆ 0 + H  + Tˆsol )|

N 

ψk (r; R, q)χk (R)ζk (q) = E(R, q)ψ j |

k

N 

ψk (r; R, q)χk (R)ζk (q).

k

The electronic eigenfunctions are orthonormal, ψ j (r; R, q)|ψk (r; R, q) = δk j and we recall that the ψk ’s are the eigenfunctions of the electronic Hamiltonian. (E 0j (R, q) + E sol )χ j (R)ζ j (q) + ψ j |H 

N 

ψk χk ζk  = E j (R, q)χ j (R)ζ j (q)

(1.40)

k=1

where E j ’s are the sum of the electronic energy plus the corresponding parametric vibrational corrections that give rise to what is called Born–Oppenheimer surfaces. Thus we see that any molecular processes in solution produces a continuum of potential energy surfaces. For an N-atom problem, the potential surfaces form a 3N-5 dimensional hyper surface. Further simplification of the second term in the (1.40) leads to the following equation, 

−

 E 0j (R, q) + E sol − E j (R, q) + ψ j | Hˆ  |ψ j  χ j (R)ζ j (q)

(1.41)

q,q,q q N    2  ∇il .∇im χ j (R)ζ j (q) =  jk 2Mi l>i,m>i i k=1,k= j i,l>i,m>i

where j is the energy level index and  jk is given by,

 jk =

 2 2 ψ j |∇il ψk .∇im χk ζk + ψ j |∇i j .∇im ψk χk ζk − ψ j | Hˆ  |ψk χk ζk . Mi 2Mi

The Eq. 1.41 gives an equation governing the nuclei in the kth electronic state. As the electronic wave function ψk (r; R, q) does not fully diagonalize the total Hamiltonian, there are some corrections which appear in the right hand side of the nuclear Hamiltonian. Those terms are called the non-adiabatic coupling operators. In the adiabatic limit, the potential surfaces are separated with less nuclei-electron coupling. In that case the non-adiabatic coupling operator matrix becomes diagonal. This is called adiabatic approximation. Under this approximation, the nuclear wave function are independent of the electronic coordinates and the total wave function is a separable function, i.e., r,R,q ≈ ψk (r; R, q)χk (R)ζk (q). Further assuming ψ j | Hˆ  |ψ j  0 along with  jk = 0 gives the Born–Oppenheimer approximation. The approximation additionally neglects the correction to the electronic energy from the nuclear Hamiltonian. The main difference between the adiabatic and Born–Oppenheimer approximation is that the adiabatic energy surfaces

16

1 Theoretical Background and Motivation

E 0j (R, q) depends on mass while Born–Oppenheimer potential surface is independent of nuclear mass. Thus in the BO approximation, the total Hamiltonian of the solvated donor and acceptor molecules are written to be, H = Hˆ 0 + Hˆ mol + Tˆsol

(1.42)

with Hˆ mol = −

q q,q,q   2 2  ∇il .∇im − ψ j |∇il ψ j .∇im 2Mi l>i,m>i Mi i i,l>i,m>i

(1.43)

1.2.2 Derivation of a Multi-state Hamiltonian Involving Reactant and Product States of the System The dynamics of the electron’s wave function can be modeled by a multi-state Hamiltonian considering a transition between the initial and final states of the electron transfer process. Let us expand the time-dependent wave function of the total Hamiltonian in the basis of initial and final states of the process as expressed [19], (r, R, q, t) = |ψi (r; R, q)χi (R, t) + |ψ f (r; R, q)χ f (R, t).

(1.44)

the indices i and f represents corresponds to initial and final states respectively. For simplicity and clarity in the derivation, we omitted the solvent part of the wave function. Also we considered that the electronic wave function is time-independent. Substituting the above equation to the Born–Oppenheimer approximated equation gives, i

∂[|ψi χi (R, t) + |ψ f χ f (R, t)] = ( Hˆ 0 + Hˆ mol )[|ψi χi (R, t) + |ψ f χ f (R, t)] ∂t

(1.45)

Multiplying ψi | to the left of the above equation simplifies to, i

∂χi (R, t) = ψi | Hˆ 0 |ψi χi (R, t) + ψi | Hˆ 0 |ψ f χ f (R, t) ∂t +ψi | Hˆ mol |ψi χi (R, t) + ψi | Hˆ mol |ψ f χ f (R, t)

(1.46)

Similarly projecting the Eq. 1.45 onto the final state |ψ f  yields a matrix representation for the coupled initial and final states as given by,

i

∂ ∂t



χi (Rl , q, t) χ f (Rl , q, t)



⎛ ˆ H0,ii + Tˆmol,ii Hˆ 0,i f + Tˆmol,i f ⎜ + Hˆ mol (Rl ) =⎜ ⎝ Hˆ 0, f i + Tˆmol, f i Hˆ 0, f f + Tˆmol, f f + Hˆ mol (Rl )

⎞

 ⎟ χi ⎟ ⎠ χ f . (1.47)

1.2 Arisal of Multi-state Problems for Representing Electron …

(a)

17

(b)

Fig. 1.2 Different representations of the non-adiabatic transitions based on the strength of the interaction between initial and final states: a weakly coupled diabatic potential energy surface (PES), b adiabatic PES are strongly coupled. The two diabatic surfaces Vi (R) and V f (R) combines to produce two equivalent adiabatic surfaces E i (R) and E f (R). The position R ∗ is the avoided crossing where the non-adiabatic transitions takes place at maximum

1.2.2.1

Adiabatic Representation

When choosing the expansion basis |ψx (r; R, q) in Eq. 1.44, one have the choice of choosing a basis set that either diagonalizes the electronic Hamiltonian or a one that does not. In case of choosing a basis set that diagonalizes Hˆ 0 , the off diagonal terms Hˆ 0,i f , Hˆ 0, f i are zero. The transition between the states can just occur by the coupling in the nuclear motion. Upon this consideration, the time-dependent Schrödinger equation simplifies to the following equation as the adiabatic representation of the problem, ⎛

i

∂ ∂t



χadia,i (Rl , q, t) χadia, f (Rl , q, t)



Hˆ 0,ii + Tˆmol,ii Tˆmol,i f ⎜ ⎜ + Hˆ mol (Rl ) =⎜ ⎝ Hˆ 0, f f + Tˆmol, f f Tˆmol, f i + Hˆ mol (Rl )

⎞ ⎟ ⎟ χadia,i ⎟ ⎠ χadia, f

(1.48)

This scenario arises when the individual potential surfaces of the donor and the acceptor molecules are in strong interaction. In the strong interaction limit, the diabatic surfaces form new hybrid surfaces giving rise to adiabatic potential surfaces. In adiabatic problem, coupling between initial and final states are coupled through nuclear motion states that, the process can be achieved by slow movement on the adiabatic potential surface (Fig. 1.2b)

1.2.2.2

Diabatic Representation

Choosing a basis set for |ψi  and |ψ f  that does not diagonalize the electronic Hamiltonian Hˆ 0 means that the exact ψx ’s can be expressed in the following form,

18

1 Theoretical Background and Motivation

|ψx, adiabatic  =

M 

ck |ψx,diabatic ,

(1.49)

k=1

The diabatic wave functions can be chosen to be physically meaningful states with M being the number of such states. The examples of such states can be such as the unperturbed donor and acceptor states or the different possible intermediate states of the process. The transition between the initial and the final states occurs through the non-diagonal terms Hˆ 0,i f , Hˆ 0, f i . Here, the diabatic wave functions (ψi/ f ) need not be parametric functions in R which render the terms such as Tˆmol,i f , Tˆmol, f i to be zero. This assumption mean that the initial and final states of the electron transfer process is induced by a weak coupling. The diabatic representation of the initial and final state of the electron transfer process is expressed as, i

∂ ∂t



χdia,i (Rl , q, t) χdia, f (Rl , q, t)



=

Hˆ 0,ii + Hˆ mol (Rl ) Hˆ 0,i f Hˆ 0, f i Hˆ 0, f f + Hˆ mol (Rl )



χdia,i χdia, f

.

(1.50)

The movement on a diabatic surface does not induce the initial→final state transition. The transition is only induced by the curve crossing (crossing strength is given by diabatic coupling term Hˆ 0,i f ). For M = 2, the adiabatic states is a linear combination of the diabatic states (1.51) ψ adiabatic = |i + | f , The energy of the two new adiabatic surfaces can be calculated using the Ritz method as given by,

E f,i (R) =

Vi (R) + V f (R) ±

 2 (Vi (R) − V f (R))2 + 4 Hˆ 0,i f 2

.

(1.52)

The corresponding adiabatic states when M = 2 is shown schematically in Fig. 1.2. If the chosen diabatic wave functions are complete, usually the results from diabatic and the adiabatic representations match [120]. The transitions between the diabatic or adiabatic potential surfaces is called the non-adiabatic transition.

1.2.3 Libby’s Theory on Electron Transfer: Concepts, Applicability and Limitations As of now it became evident that the electron transfer processes are a curve crossing of electron’s wave function between the molecular potential surfaces. In the earlier days, this curve crossing was believed to be happening similar to a Franck–Condon excitation [89] process in spectroscopy (see Fig. 1.3). It is because during the process, the electron leaves the reactant potential surface transits to the product potential surface. This travel of electron was believed to be faster than the solvation energy transfer, hence the reactants form products in the solvation environment of the reac-

1.2 Arisal of Multi-state Problems for Representing Electron …

19

Fig. 1.3 Schematic picture of Libby’s mechanism of electron transfer between donor and acceptor molecular potentials. During the e− transfer, the solvation atmosphere is unchanged because of the higher mass and thereby the process is nearly Franck–Condon

tants. This sudden jump in the solvation energy difference between the initial and final states would pose a barrier to the electron’s motion. The energy barrier can be calculated as the extra energy needed to add one more charge on the acceptor ionic sphere in the solvent environment (characterized by dielectric constant, r ) as given by, H= =

Z 2 e2 (Z + 1)2 e2 − . 4π 0 (2r R) 4π 0 (2r R)

(1.53)

where H= is the solvation barrier and R is the radius of the ion. The time variation in the electron (e)  transfer and the solvation energy (h) transfer can be roughly te quantified as, th = mmhe . The variation is because of the masses of the two moving entities, hence the crossing process can be schematically depicted as in Fig. 1.3. The above primitive notion of Franck–Condon mechanism qualitatively explained the earlier observations related to e− transfer. It explained the catalytic behavior of substituting ligands, predicted a faster transfer in all symmetric complexes, also predicted a longer time of transfer process in simple ions as was observed. The above predictions were possible only with the consideration of a solvation barrier and by investigating the implications of the molecular nature on the solvation barrier. The quantitative evidence of the e− transfer process was explained parametrically using a simple model in gaseous phase. By calculating the 3dz 2 orbital overlap in a H2+ -molecule, Libby quantitatively characterized the evidence of transfer processes in a transition metal complex. But electron transfer, being a process isolated from external excitations, the solvation barrier has to be overcome by available extra energy (if, it is a exothermic process), or through zero-point motions in case of symmetric reactions, or through minimization of the barrier by solvation in order for the process to be feasible. However, whatever the barrier height is, this theory faced a serious criticism that energy is not conserved for a local time. It was when Marcus [98] identified that for an activated complex should occur, the energy at all stage of the process should be invariably constant. This was essentially true as the electron transfer was driven from the solvation fluctuations occurring from inside

20

1 Theoretical Background and Motivation

Fig. 1.4 Schematic representation of the gradual change in the solvent polarization vectors (Marcus description). The solvation energy should match at the time of transit and will be minimum before the start and after the end of transit

the system and not externally. It was a significant contribution by Marcus by stating that the electron transfer process happens through the gradual change in the solvent polarization. According to Marcus, the transfer occurs when the e− shared the similar solvation sphere between the donor and the acceptor molecules (depicted in Fig. 1.4).

1.2.4 Marcus’ Theory on the Electron Transfer In the work of Marcus [98], the rate of the electron transfer (ET) processes was calculated as a function of molecular parameters. The process can be conceptualized as movement over the free energy surfaces due to the influence of the solvent fluctuations on the collective polarization of the reactants. Once the reactant species attain the activated complex, they form products. Using the knowledge of free energies of the activated complex the rate can be calculated using Fermi Golden rule or Eyring’s equation in the diabatic and adiabatic limits respectively. For quantitative explanation of the ET process and his other contributions to the theory, Marcus won the Nobel prize of Chemistry in 1992. His work [98] comprised by three initial steps: (a) identifying what mediates the reaction (reaction coordinate), (b) calculating the potential energy surfaces of various stages in the process, hence (c) calculating the dependence of the rate as a function of system and molecular parameters. The simplified paraphrasing of his findings [98] are enlisted in the following sections.

1.2 Arisal of Multi-state Problems for Representing Electron …

1.2.4.1

21

Key Propositions of the Theory

Marcus’s initial development of the theory assumes little orbital overlap between the reactant complexes as would be the case of an outer-sphere process. This little overlap limit ignores drastic ionic movements during the process such as in innersphere. Mathematically speaking, in such a limit, the nuclear motion would not couple the initial and final states of the process (we consider the diabatic limit). So, the quantitative applicability of the theory will be as relevant as little the orbital overlap between the species. Marcus conceptualizes the ET reaction to be passing through intermediate states such as Y ∗ , Y in the order driven by solvent fluctuations. The intermediate state Y ∗ corresponds to preparation of the solvation environment needed for the electron process. This state can either lead to the transferred-state (Y ) or could reform the reactants by solvent disorganization. This considerations assume that the solvation happens prior to the ET process. After the electron transferred to form the Y -state, the species move away to form the products. This reaction scheme can be expressed as, k1

Reactants  Y ∗ , k-1

∗ k2

Y  Y, k-2

k3

and, Y −→ Products. If one starts with the Schrödinger representation of molecules and solvents as in Eq. 1.31, it is immediately to find that the reactant states, intermediate states Y ∗ , Y , and product states are all eigenfunctions of the Hamiltonian corresponding to the system. The reactant state can be characterized by a corresponding electronic wave function which can be plotted as a characteristic in the electronic coordinates. Similarly one can characterize the products and other states by their characteristic. The whole ET process is just the transition from one characteristic to another driven by fluctuations within the system. The molecules form an activated complex which is a superposition of the intermediate states. In order that the wave function of the activated complex (can be characterized by |Y  + |Y ∗  for weakly coupled orbitals) to be the eigenfunction of the total Hamiltonian is to assume that all the states constrict to the same total energy of the system E. This was the major correction realized by Marcus over the Libby’s mechanism and hence formalize the process as shown in Fig. 1.4 and not as in Fig. 1.3.

1.2.4.2

Finding and Calculating the Reaction Coordinate Calculation

With all the formalism mentioned in the above section, it is intuitively easy to figure that in order that the state Y ∗ to be formed, the reactant species should approach each other. So, one candidate for mediating the reaction process is of course the distance R

22

1 Theoretical Background and Motivation

between the reactant molecules. However, not just nearing of pairs could ultimately lead to the transfer. The states of the molecule Y and Y ∗ vary in the charge state, and hence the polarization of the surrounding solvent molecules should differ. And in order to form the state Y , at the first stage the solvent molecules should reorganize to form Y ∗ . This leads to state about an important coordinate that determines the reactive motion which is the solvent polarization coordinate. So fixing the approach distance R for time being, the expression for the other reaction coordinate is given by,  X=

dr [D p − Dr ].P(r),

where the P(r) is the orientational polarization of the solvent, whereas Dr and D p are the displacement field due the reactants and products respectively under the dielectric continuum model. An alternative simpler definition for the reaction coordinate is valid [155] for a general type of ET is given by, X = e(VD − V A ),

(1.54)

where D and A are donor and acceptor indices respectively.

1.2.4.3

Calculation of the Free Energy Surfaces

Having identified the energy difference as the reaction coordinate, one can calculate the free energy surfaces that could give the rate of the ET process. The free energy surfaces for the reactant state and final state are assumed to have a parabolic dependence of surfaces on the reaction coordinate X with same spring constant for both the states. Let us suppose that the reacting solutes A and B are embedded in the solvent spheres of radius a and b respectively. The radii a and b are the sum of the crystallographic radius and the extent of the saturated dielectric region around the molecules (radii of the inner-spheres). The potential energy due to solute A placed inside the inner sphere can be written as [155],  VA =

P(r) r a & |r−r B | >b

r − rB |r − r B |

3

r − rA − |r − r A |

 3

.P(r)dr.

(1.55)

The corresponding free energy of a polarisation field can be given by, −1 G 1 = 2π(op − s−1 )−1

 |r−r A | >a & |r−r B | >b

[P(r)]2 dr.

(1.56)

1.2 Arisal of Multi-state Problems for Representing Electron …

23

where op and s are the optical and static dielectric constants respectively. There can be many possible configurations of the solvent i.e., P(r), giving rise to same potential energy of Eq. 1.55. But those states that will be thermodynamically realized are the ones that minimizes the free energy. Hence by minimizing the free energies, along with constraining the energy to be a constant, we calculate the energy of the initial and final states. The corresponding Lagrangian functional for the above goal can be written as,  −1 − s−1 )−1 [P(r)]2 dr G 1 = 2π(op |r−r A | >a & |r−r B | >b

 

 r − rB r − rA .P(r)dr . (1.57) +λ X − − 3 |r − r A | 3 |r−r A | >a & |r−r B | >b |r − r B | The resulting polarization function by solving the above functional is obtained to be, P(r) =

−1 λ(op − s−1 )



4π

r − rA |r − r A |

r − rB − |r − r B |

3

 3

.

(1.58)

Using Eqs. 1.58, 1.55 in Eq. 1.56 yields, G1 =

−1 2π(op − s−1 )

X 2,

I

(1.59)

where the integral I is given by,

 I =

|r−r A | >a & |r−r B | >b

r − rA |r − r A |

3

r − rB − |r − r B |

2 3

dr,

(1.60)

and I can be approximated as [155],  I = 4π

2 1 1 + − a b R

,

where R is the distance between the donor and the acceptor. The free energy surface is,   1 1 2 X2 1 1 1 G1 = + − , (1.61) where λ X = − 4λ X 2 op s a b R λ X is called the solvent reorganization energy. The solvent reorganization energy is defined as the energy spent on the ion pair in order to form the activated complex. Following the same calculations to obtain the final state free energy,  −1 −1 −1 G 2 = 2π(op − s ) [P(r)]2 dr |r−r A | >a & |r−r B | >b

24

1 Theoretical Background and Motivation

Fig. 1.5 Scheme of the ET process between reactant and product’s free energy surfaces. The product’s configuration can be achieved by spending twice the solvent reorganization energy from the reactant state, G is the change in the free energy during the process. The intersection point X c is the amount of energy required to form the activated complex and corresponding free energy of activation is denoted as G ∗N A = G 1 (X c )



 +

|r−r A | >a & |r−r B | >b

r − rB |r − r B |

would yield G 2 as, G2 =

3

r − rA − |r − r A |

 3

.P(r)dr,

(1.62)

(X − 2λ X )2 . 4λ X

In general, it can give rise to free energy surfaces, G2 =

(X − 2λ X )2 + G. 4λ X

turning out that the reorganization energy is same in the initial and final states of the ET as we already assumed the equal spring constants for both harmonic diabatic states. For the above set of free energy surfaces, the scheme of the ET process is shown in Fig. 1.5. If the energy difference X is taken to be at zero reference for the reactant state, then the equilibrium configuration of the products will be displaced twice the solvent reorganization energy. This is because in the ET mechanism, the solvent reorganization happens twice, once before the activated complex called precursor complex and during the successor complex. In order that the reactant molecule has to attain the product’s configuration 2λ X , a free energy equal to solvent reorganization energy should be spent. Similarly if the same amount of energy is taken from the products, the molecule will form the reactant in case of symmetric reaction, and

1.2 Arisal of Multi-state Problems for Representing Electron …

25

would require λ X + G if there were free energy change during the ET process. As already stated, following the conservation of energy, the ET happens only at one localized crossing point (X c ) between the free energy surfaces. The crossing point of the diabatic surfaces is evaluated to be, (X c − 2λ X )2 X c2 = + G, 4λ X 4λ X =⇒ X c = G + λ X . Now depending on the value of −G (the free energy released during the ET), whether if it is lesser than, equal to, or greater than the solvent reorganization energy λ X (the required free energy to complete the ET), the process happens at X c > 0, X c = 0, and X c > 0 respectively. The scheme of such crossings are presented in Fig. 1.6a. Depending on the position of the crossing given by, X c = G + λ X , the process can be categorized (Fig. 1.6a): (a) When X c > 0, the reactive site is in normal region, and requires extra energy to climb the barrier to reach the reactive site, (b) X c = 0, the reactive site is at the minima of the potential curve and no activation energy is needed to attain the reactive site (barrierless reactions), (c) X c < 0, the reactive site is in the inverted region, and similarly requires extra energy to climb the free energy barrier. For a simple reaction like A + B −→ A+ + B− , the free energy gap G is written as, G = I P − E A − h os − h es −

e2 . R

where I P is the ionization potential, E A is the electron affinity, h os is the solvation free energy(orientational mode) of solvents, h es is the electronic mode solvation energy of solvent molecules. Now, with the knowledge of the free energy and reaction coordinate, one may calculate the rate constant for the process. In the diabatic limit, the transition can be given using the Fermi golden rule. Using the definition of rate constant, ∞ k(X )Peq (X ) dX equilibrium flux over barrier ∞ = −∞ (1.63) kN A = initial population −∞ Peq (X ) dX the k(X ) is the intrinsic rate given by the Fermi golden rule of non-adiabatic transitions, 2π 2 H δ(G 2 (X ) − G 1 (x)), (1.64) k(X ) =  12 Using the Boltzmann distribution of the reactant’s free energy surface as Peq (X ) to obtain, 2 (λ +G)2 2π H12 − X e 4λ X k B T . kN A = √  4π λ X k B T

(1.65)

26

1 Theoretical Background and Motivation

Fig. 1.6 a Scheme for the different cases on crossings, b The expected bell-nature as predicted by the Marcus’s theory (Eq. 1.65)

The above equation is the prediction of rate constant behavior by Marcus. The dependence of rate constant on the negative change in free energy is depicted in Fig. 1.6b. When the energy released (−G) exactly matches the solvent reorganization energy i.e., X c = 0, the probability of ET process is maximum. When the deviation increases, a barrier is introduced in the other regions, the rate decreases symmetrically away from the case X c = 0. This symmetric bell-curve dependence persists as long as only one-dimension is considered and the ET reaction is symmetric. Correspondingly in the adiabatic limit, the rate can be calculated using the Eyring equation kA =

∗ A ω0 − G e kB T , 2π

(1.66)

1.2 Arisal of Multi-state Problems for Representing Electron …

27

when the free energy surface in the adiabatic limit is known. In the above equation, ω0 is the frequency of the reactant well and G ∗A is the activation energy.

1.2.5 Extensions to the Marcus’ Description Marcus’ initial treatment was an one-dimensional solvent coordinate mediated ET. It was understood that the reaction was driven by non-equilibrium fluctuations from the solvent coordinate. However, the rate Eq. 1.65 was calculated using the equilibrium statistical mechanics, which simplified the mathematical complexities of coupledequations. In 1980s, Zusman [180], Alexandrov [6] treated the curve-crossing ET problem using a coupled-Fokker–Planck equation description of the reactant and product states. Contrary to the symmetric ET assumption of Marcus’ theory, they considered the different reorganization energies for both the initial and the final states. This problem was further studied in Ref. [29] by deriving long-time solutions and rate constants of a two-way ET reaction that possess two different solvent reorganization energies in the initial and final states. Alexandrov showed that an adiabatic picture as that of Kramers’ [80] was realized to be relaxation from the initially prepared parabolic diabatic states [6]. Further, Zusman [179] described this adiabatic state by using the non-equilibrium Smoluchowski equation in order to account for the dynamic solvent effects, and further, the following rate formula for ET reaction was derived:  (λ +G)2 λX 1 − X e 4λ X k B T , (1.67) kA = τ L 16π k B T where τ L is the longitudinal relaxation time equal to ∞0 τ D , where 0 and ∞ are the static and infinite frequency dielectric constants of the solvent, respectively, and τ D is the Debye relaxation time. Rips et al. [127] treated the two-surface dynamics (with same λ X ) using a real-time path integral formalism to derive the following rate for the process,

 ∗ 2k0 τ L −1 − G k0 1+ kET = √ e kB T , (1.68) λX 4π k B T λ X where k0 is the non-adiabatic coupling strength. The formalism allowed the high barrier and low barrier limit to be treated in a single formalism. The above expression also accommodates the adiabatic rate and non-adiabatic rate expression as special limits. For the case, τλL Xk0 >> 1 we get the adiabatic limit Zusman’s expression for the ET rate as given by Eq. 1.67. On the other hand, the limit τλL Xk0 > S2 , the multi-state equation reduces to an effective single-state description. And so far, the literature considered only single-state models due to the mathematical complexity of solving otherwise in the time domain. Also in singlestate descriptions, the models were solved only if the potential/problem exhibited symmetry. For example, a reactive boundary with an infinite strength introduced a mirror symmetry in some problems. The corresponding solution can be readily written using the method of images when the solution in the absence of boundary is known. Otherwise, for a finite-reactive boundary, the time-domain solution is known only when the potential exhibited a translational symmetry. An example is a free particle diffusion with a δ-function sink. In this chapter, we start solving this simple example and develop the solution over potentials that do not have translational invari2 ance. The potentials like b |x|, kx2 are solved with an arbitrarily placed sink boundary. Then the assumption S1 >> S2 is relaxed, and the solution is obtained for a two-state model, the results of which can be extended to multi-state models. The chapter is organized as follows: Sect. 2.2 considers the single-state description of reaction–diffusion systems. Starting from Sect. 2.2.2, a time-domain method is developed to solve the free particle diffusion with a delta-function sink. Section 2.2.3 considers a diffusion scenario with two finite-reactive sinks and is known as Oster–Nishijima’s model for electronic relaxation. Using the solution developed in Sects. 2.2.2 and 2.2.3, the piece-wise linear potential problems with two boundaries were solved in Sect. 2.2.4. We considered two models: model A as considered by Ghosh [131], and model B as considered by Privman [123] and solved them in time domain. Section 2.2.5 presents the results of reaction–diffusion in presence of a harmonic potential. In Sect. 2.3, we © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 R. Saravanan and A. Chakraborty, Solvable One-Dimensional Multi-State Models for Statistical and Quantum Mechanics, https://doi.org/10.1007/978-981-16-6654-4_2

45

46

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

solved two-state problems of statistical mechanics. In Sect. 2.3.1, a general mathematical methodology is developed to solve a two-state model as considered in Ref. [34]. Then the general properties of open- and closed-type multi-state models are discussed in Sect. 2.3.2 by using the results of a simple solvable multi-state system. Section 2.3.3 solves a closed two-state model and consequently, a general theory for reversible reaction-diffusion systems are derived. Section 2.2.5 provides the experimental use of the analytical concentration profiles of the ultimate model solved in the monograph. The analytical results and verifications are provided in the respective subsections of all the problems. And in the subsections, the results are discussed in detail and mostly in the context of electronic relaxation inside solutions.

2.2 Solution of Single-State Problems in Statistical Physics An effective single-state description for a reaction–diffusion system has been already introduced in the appropriate limit as given in Eq. 1.69, ∂ P(x, t) = (L − S(x) − kr ) P(x, t) ∂t   D ∂ dU (x) ∂2 P(x, t), L= D 2 + ∂x kB T ∂ x d x

(2.1)

where x is the reaction coordinate, diffusivity D = k Bζ T , where T is the temperature of the solvent bath, ζ is the viscosity, U (x) represents the drift along x and S(x) is the sink. In this section, we consider models of various potentials and usually throughout this book, the S(x) would involve some form of Dirac delta functions.

2.2.1 Introduction to the Kernel Method Using the definition of Smoluchowski operator in Eq. 2.1, the time evolution of the probability distribution is expressed as P(x, t) = e(L−S(x)−kr )t P(x, 0).

(2.2)

The initial distribution can be written as the inversion of its Fourier-transformed ˜ 0) as shown version P(k, 1 P(x, 0) = 2π



∞ −∞

˜ 0)e−ikx dx. P(k,

(2.3)

2.2 Solution of Single-State Problems in Statistical Physics

47

In a broader interest of extending the method’s applicability, the above expansion should not be limited to Fourier transformation. It will be useful to choose a general transformation which is the eigenfunction of the L operator. Because the above choice will simplify the expansion of eLt to be easier as would be required in subsequent derivations,  ∞ 1 ˜ 0)eLt F (k)dk. (2.4) P(x, t) = P(k, 2π −∞ The F (k) in the above equation is called the kernel for a considered problem. The kernel in principle contains the effect of boundaries (S(x), kr ) and the effect of potentials incorporated to it. They can be systematically derived by solving the timeindependent form of the L operator.

2.2.2 Diffusion Dynamics of a Distribution in Flat Potential with a Dirac Delta-Function Sink For simplicity, we assume a probability distribution is diffusing in a flat potential, i.e., [U (x) = 0]. The condition for the exit of the population is given by a finite strength delta function, i.e., S(x) = k0 δ(x). This type of models can give a simple theory for chemical reaction in solutions. The model considered here was also previously attempted in the Master’s thesis of the author [135], but the exact answer as of here was not provided, and a complete analysis of different situations was not derived. Hence, it would be convenient to present the method again and to discuss new insights of the model. Equation 2.1 reduces to ∂ 2 P(x, t) ∂ P(x, t) =D − k0 δ(x)P(x, t). ∂t ∂x2

(2.5)

The aim is to find the survival probability of the distribution, which will be the concentration profile for the left out reactants on the flat potential surface. The effect of temperature, viscosity of the solvent, and other molecular parameters would appear in the analytical survival probability. We consider initially that the distribution is fed on the left side of the sink. The probability distribution at t = 0 can be taken to be  P(x, t = 0) = P(x0 ) =

1 4π σ 2

1/2 e

−(x+x0 )2 4σ 2

,

(2.6)

∞ which is a Gaussian function and obeys −∞ P0 (x)dx = 1. Using the method explained in Eq. 2.2.1, the problem can be solved. To derive the time-independent solutions of Eq. 2.5, the separation of variables is expressed as P(x, t) = p(x)T (t),

48

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

to be used. And Eq. 2.5 becomes D

∂ 2 p(x) − k0 δ(x) p(x) = L p(x) = −Sp(x), ∂x2

(2.7)

where S is a constant and L is the corresponding Smoluchowski operator. We solve Eq. 2.7 over whole region other than x = 0 and the solution is  p(x) =

Aeikx + Be−ikx , x < 0

Ceikx ,

x > 0, where k =



S . D

The contribution of x = 0 is added to the solution by using the following requirements at the boundary x = 0: lim [ p(0 − ) − p(0 + )] = 0, and  0+ ∂p = k0 p(0)/D. lim →0 ∂ x 0−

→0

(2.8)

The above equations imply the continuity requirement of probability distribution and flux discontinuity requirement of the delta-function boundary at x = 0. The solution of Eq. 2.7 given by ⎧ 

−ik0 ⎨eikx + e−ikx , x < 0, 2k D+ik 0

p(x) = −ik0 ⎩eikx + eikx , x > 0 2k D+ik0

(2.9)

is the modified Fourier kernel derived for Eq. 2.5. As already mentioned, so far we obtained the stationary solution that can be used as the kernel. Now using Eq. 2.4 to calculate P(x, t) which is expressed as 1 P(x, t) = 2π



∞

Lt −σ 2 k 2 +ikx0



e e −∞

e

ikx

 −ik0 ik|x| dk. + e 2k D + ik0

(2.10)

We see that P(x, t) will have contributions from both the incident part of the stationary solution and the rest (reflected and transmitted) part of the stationary state solution (Eq. 2.9). Time evolution of this two different contributions will be done separately. The time evolution of the distribution corresponding to incident part of the stationary solution is given by P1 (x, t) =

1 2π



∞

eLt e−σ

k +ikx0 ikx

2 2

e

dk,

−∞

where eikx is an eigenfunction of the operator L. After simplifying, we get

2.2 Solution of Single-State Problems in Statistical Physics

P1 (x, t) =



1 2π

∞

e−Dk t e−σ 2

49

k +ikx0 ikx

2 2

e

dk.

−∞

After doing the integration, we get −(x+x0 )2 1 P1 (x, t) =  e 4(σ 2 +Dt) . 2 π(σ 2 + Dt)

(2.11)

It is interesting to note that the survival probability corresponding to incident part ∞ (P1 (x, t)) is Q 1 (t) = −∞ d x P1 (x, t) = 1. So this part (P1 (x, t)) of the solution corresponds to the non-decaying part of the distribution. Using the reflected and transmitted part of the stationary solution as the Fourier kernel, the decaying part of the distribution can be calculated as    ∞ −i0 1 2 2 e−σ k +ikx0 eLt eik|x| dk P2 (x, t) = 2π −∞ k + i0 taking 0 =

k0 2D

and using completion of squares leads to

2 1 4(σ− 2f (x) e +Dt) P2 (x, t) = 2π



∞

e

−(σ 2 +Dt)(k−κ)2

−∞



−i0 k + i0

 dk

(2.12)

(x) with f (x) = |x| + x0 and κ = 2(σi f2 +Dt) . We transform the variables chosen to take care of κ in the integrand, in order to do the Gaussian integration as shown below:

e−(σ +Dt)(k−κ) e−u du dk = k + i0 u + z(x, t) 2

2

2

with u = (σ 2 + Dt)1/2 (k − κ) and z(x, t) = (σ 2 + Dt)1/2 (κ + i0 ). The transformations result in the following expression for P2 (x, t) that also contain residue contributions due to poles in the contour plane, − f (x)2 1 (−i0 )e 4(σ 2 +Dt) P2 (x, t) = 2π



∞

e−u du + −∞ u + z(x, t) 2



 2 e−u du . u + z(x, t)

For finding P2 (x, t) we have to evaluate the above complex integral which has a singularity in the complex plane and thus the integration is evaluated by calculation of residues. The pole z(x, t) is dependent on time and the result of the integration should also depend on time. Initially at time t = 0, the variable u(x, 0) = (σ k − i f2σ(x) ) and its pole is u p (x, 0) = −( i f2σ(x) + iσ 0 ) = −z(x, 0). The contour shown in Fig. 2.1 is expressed as 

e−u du = u + z(x, t) 2







− Re(k)



+ Re(u)

− I m(u)

. I m(u)

50

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

Fig. 2.1 Schematic graph of the integrand in the complex plane at time t = 0. The pole lies inside the contour plane when the pole value 0 satisfies the following (x) , 0]. condition, σ 0 ∈ [− f2σ Reprinted from Reaction-diffusion system: Fate of a Gaussian probability distribution on flat potential with a sink, 536, R. Saravanan, A ,Chakraborty, 120989, Copyright (2019), with permission from Elsevier

The last two terms cancel each other being equal in magnitude but opposite in the sign. The closing point of the contour can be extended till ±∞. In the following, we observe the position of poles with respect to the integration plane for the following cases of 0 : (1) When 0 > 0, the pole lies outside the contour below the real (k) axis in Fig. 2.1. There is no residue to be added in such case. < 0 < 0, the pole lies inside the contour (between Re(k) (2) When −2σf (x) 2  and  Re(u)). The integral value is given by Cauchy’s residue theorem Re(k) − Re(u) =  2πi u p . < 0, the pole lies outside the contour above the Re(u) axis. (3) When 0 < −2σf (x) 2 There are no residual contributions. Therefore, we add the residue only when the 0 is negative and along with another simultaneously necessary condition that the Im(z(x, 0)) > 0 since only then 0 would lie inside the contour. Now, the solution can be expressed as  P2 (x, 0) = A(x, 0)

∞

e−u σ2 du + θ(Im[z(x, 0)])θ (−0 )(2πi)A(x, 0)e −∞ u + z(x, 0) 2

0 +

 f (x) 2 2σ 2

,

(2.13)

− f (x)2 4(σ 2 +Dt)

1 where A(x, t) = 2π (−i0 )e . For t > 0, the poles start moving upwards and so as the contour. Now the residue will be added when Im[z(x, t)] < 0 and 0 > 0. Combining all the results, the final expression for arbitrary time t ≥ 0 is given by − f (x)2 1 (−i0 )e 4(σ 2 +Dt) P2 (x, t) = 2π



+ (0 )(2πi)e

∞

e−u du −∞ u + z(x, t) 2 

|x|+x 2

0 (σ 2 +Dt) 0 + 2(σ 2 +Dt)

(2.14) .

2.2 Solution of Single-State Problems in Statistical Physics

⎧ ⎪ 1 ⎪ ⎪ ⎨

(0 ) = −1 ⎪ ⎪ ⎪ ⎩0

51

  0 ∈ − 2(σf2(x) ,0 . +Dt)   0 ∈ 0, 2(σf2(x) . +Dt)   f (x) 0 ∈ . / − 2(σ 2 +Dt) , 2(σf2(x) +Dt)

The probability distribution P(x, t) is obtained by further simplifying Eq. (2.14) and is given by 2

P(x, t) =

−(x+x0 ) 1 0 2 2 2 2  e 4(σ 2 +Dt) + (0 )0 e0 (σ +Dt)+0 (|x|+x0 ) e0 (σ +Dt) + (2.15) 2 2 π(σ 2 + Dt)      |x| + x0 |x| + x0 + 20 (σ 2 + Dt) + erf . ×e0 (|x|+x0 ) −sgn +  √ 0 2(σ 2 + Dt) 2 σ 2 + Dt

The probability distribution is obtained for the Cauchy problem defined by Eqs. 2.5 and 2.6. The above result P(x, t) for a Gaussian initial data cannot be obtained using other methods such as Green’s function and Laplace transformation. The reason is that in Green’s function method, the integration given by  P(x, t) =

∞ −∞

G(x, x0 , t)P(x0 , 0) dx0

(2.16)

is not analytical. Upon using the Laplace transform method [41, 154], the Laplace inversion has not been analytical so far for non-zero σ . Hence, the presented solution can be reported as the inverse Laplace transform. Also, the Laplace transform of the presented solution P(x, t) matches with the available [41, 154] Green’s function under σ = 0. Equation (2.15) is the general solution of Eq. (2.5). The residue part will be neglected in the subsequent physical considerations, owing to the boundary condition that P(x = ±∞, t) = 0 =⇒ (0 ) = 0. The survival probability Q(t) calculated from the solution (Eq. (2.15)) is given by  Q(t) =

∞

P(x, t)dx. −∞

The analytical form of the above equation can be derived to be  Q(t) = 1 + 2θ(−x0 − 20 (σ 2 + Dt))e−0 (σ 2

2 +Dt)

− erfc



x0



4(σ 2 + Dt)      x0 x0 + 20 (σ 2 + Dt) 2 2 − erf . +e0 x0 +0 (σ +Dt) sgn +  √ 0 2(σ 2 + Dt) 2 σ 2 + Dt

(2.17)

Expression (2.17) has been validated against two possible numerical solutions. One numerical solution (Fig. 2.2b) is obtained using inbuilt numeric function of

52

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

Fig. 2.2 Comparison of analytical result given by Eq. 2.17 with a available result [41] for parameters σ√= 0.053 nm, D = 1 nm2 /ps, k0 = 1 nm/ps, x0 = 1 nm and b numerical result for parameters σ = ( 40)−1 nm, D = 0.4 nm2 /ps, k0 = 3 nm/ps, x0 = 2 nm, xc = 1 nm. Reprinted from Reactiondiffusion system: Fate of a Gaussian probability distribution on flat potential with a sink, 536, R. Saravanan, A ,Chakraborty, 120989, Copyright (2019), with permission from Elsevier

MATHEMATICA, and another (Fig. 2.2a) by numerically integrating the P(x, t)’s expression of Green’s function  P(x, t) = −

∞

G(x, y, t)P(y, 0)dy −∞

(x+x0 )2 4σ 2

with P(x, 0) = 4πσ 2 . Using Green’s function [41], the result of Q(t) is obtained and compared against our analytical solution as shown in Fig. 2.2a. So far, the choice of x0 has to be necessarily positive that models a scenario where the distribution approaches the sink (placed at origin) from negative side of x-axis. For arbitrarily placed sink (xc = 0), and for other possible situations of distribution approaching from the x > 0 side, the calculation can be repeated. The solution of different scenarios was derived which can be combined into single expression by replacing x0 by |x0 + xc | into the Q(t). Only now, the solution will be realized for the translational symmetry and mirror symmetry as expressed by following equivalence relations: e√

• (−x0 , xc ) ↔ (−x0 + c, xc + c) • (x0 , xc ) ↔ (x0 + l, xc ) ↔ (x0 , xc + k), for l and k such that |xc + x0 | is invariant. respectively. The symmetry checks are verified and presented graphically in Fig. 2.3.

2.2.2.1

Arisal of Precision Error in the Survival Probability

In reference to Figs. 2.3 and 2.5a, the Q(t) has a zig-zag behavior for a small interval of time in some parametric regime. This zig-zag behavior is an artifact of precision error in the plotting software. The plot (Fig. 2.4) shows the difference when the second term sign()-erf() of Q(t) is chosen to be two different instances: (a) 1-erf(),

2.2 Solution of Single-State Problems in Statistical Physics

53

Fig. 2.3 Comparison of Q(t) for different situations of initial position at x = −x0 and sink position at x = xc demonstrates the translational symmetry, i.e., dashed↔dotted and mirror symmetry, i.e., thick↔dashed in the model

Fig. 2.4 The illustration of precision error in the survival probability. The instances a and b compare the result when the term sign()-erf() of Eq. 2.17 is taken to be 1-erf() and erfc() respectively

(b) erfc(), and the results (a) and (b) are different. As sign() term in most of the cases is 1, for avoiding precision error in plots, sign()-erf() should be taken to be erfc() unless otherwise needed. It can be realized by calculating the diasspearing flux in order to realize that the slope of the survival probability is always negative and any positive contribution should be unphysical. Consider the diffusion equation with delta-function sink as in Eq. 2.5, ∂ 2 P(x, t) ∂ P(x, t) =D − k0 δ(x)P(x, t). ∂t ∂x2

(2.18)

Integrating the above equation over x from −∞ to ∞ gives the following equation: dQ(t) ∂ P(x, t) ∞ |−∞ − k0 P(0, t), =D ∂t ∂x

(2.19)

54

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

∞ where Q(t) is the survival probability and Q(t) = −∞ P(x, t)dx. The slopes of the probability distribution can be taken to be zero at x = ±∞ which leads to dQ(t) = −k0 P(0, t). ∂t

(2.20)

And remembering the boundary condition of the problem given by Eq. 2.8,   dQ(t) ∂ P 0+h = − lim D = j (0, t), h→0 ∂t ∂ x x=0−h

(2.21)

which conveys that the slope of the survival probability is the flux difference across the boundary and is always negative. Despite the zig-zagness, Q(t) will be infinitely differentiable function following from the property of the probability distribution function P(x, t) ∈ C∞ . Asymptotic solutions: The integral of Eq. (2.12) can be evaluated approximately depending on the values of 0 . Supposing that the 0 is so large (i0 >> k) that it almost approaches infinity. As k is a dynamic variable that takes values between (−∞, ∞), so taking i0 >> k is not a proper case. But the approximation provides some insight in the calculation of the definite integral in Eq. (2.14). In such a limit, κ can be neglected in comparison with 0 . The results become interpretable by the use of the geometric series inside the integral and after using condition (0 ), P(x, t) becomes −(x+x0 )2 1 P(x, t) =  e 4(σ 2 +Dt) 2 π(σ 2 + Dt) +

(|x|+x0 )2 1 − (−i0 )e 4(σ 2 +Dt) 2π

∞ 

√ (n − 1)!! π. n/2 (σ 2 + Dt)(n+1)/2 (κ + i )n+1 2 0 n=2s,s=0

The survival probability is given by  Q(t) = 1 − erfc

x0  4(σ 2 + Dt)



∞  n=2s,s=0

√ (n − 1)!! π. 2n/2 (σ 2 + Dt)n/2 (i0 )n

(2.22)

This solution converges to the result obtained using the method of images on the limits σ → 0 and 0 → ∞. It can be stated that the pinhole sink scenario implicitly assumes i0 > k by assuming 0 → ∞. We say that the solution presented in Eq. 2.15 is a general one and is in accordance with all the available solutions in limiting cases [41, 143]. For the intermediate values of k0 , the plots can be generated using Eq. 2.17 and is shown in Fig. 2.5a. Effective first-order rate constants such as average rate constant (k I ), long time rate constant (k L ), and average survival time (ka−1 ) that are usually calculated from the Laplace-domain solutions can also be calculated by using Q(t) into the following expressions [9]:

2.2 Solution of Single-State Problems in Statistical Physics

55

Fig. 2.5 The survival probability Q(t) versus time is plotted for a various values of sink strength. The parameters are taken to be σ = 0.5 nm, x0 = 3 nm, D = 0.5 nm2 /ps, b for various values of diffusivity with respect to time (ps) for values: σ = 1 nm, x0 = 5 nm, 0 → ∞. Reprinted from Reaction-diffusion system: Fate of a Gaussian probability distribution on flat potential with a sink, 536, R. Saravanan, A ,Chakraborty, 120989, Copyright (2019), with permission from Elsevier

k −1 I =

 ∞ 0

∞ t Q(t) dt d . ln Q(t), and ka−1 = 0∞ t→∞ dt 0 Q(t) dt

Q(t) dt, k L = − lim

(2.23)

Usage of Time Domain Q(t) in the Context of Condensed-Phase Chemical Dynamics For understanding a chemical process, one can start from fitting an available reaction data using the rate equation of the following general form: d[A] = f ([A], k(T, p)). dt

(2.24)

However, the data is for a given set of conditions (solvent, temperature, etc.) and the rate constant k(T, p) is a multi-dimensional parametric function yet to be obtained for different set of conditions. So there will be a need for repeating the experiments for various temperature and pressure, and to find a 2D-function for which k(T, p) fits in and all these procedures have to be repeated for a different solvent. On the other hand, the ease of Smoluchowski approach would be that a rate equation can be derived with inbuilt parameters such as initial molecular configurations, temperature, viscosity of the solvent, activation energy of the molecule, etc. Hence, if exact Q(t) can be obtained, it will be the concentration profile of the reaction model. Whereas asymptotic Laplace-domain solutions give first-order decay fits for concentration data. For this simple model, the theory of the reactions can be understood by plotting the survival probability Q(t) given by Eq. 2.17 for various parameters. The results show that decay is faster for the increase in temperature because it accelerates the distribution to reach Boltzmann equilibrium faster. It is seen graphically in Fig. 2.5b and mathematically seen in the broadening of the Gaussian by the term Dt in Eq. (2.11). The plot (Fig. 2.5a) shows that increase in the strength of the sink

56

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

Fig. 2.6 The survival probability Q(t) versus time is plotted for a various values of x0 . The parameters are taken to be σ = 1 nm, 0 → ∞, D = 0.5 nm2 /ps, b for various values of σ with respect to time (ps) for values: 0 → ∞, x0 = 5 nm, D = 0.5 nm2 /ps. Reprinted from Reaction-diffusion system: Fate of a Gaussian probability distribution on flat potential with a sink, 536, R. Saravanan, A ,Chakraborty, 120989, Copyright (2019), with permission from Elsevier

makes the distribution decay faster and chemical equilibrium is attained in a short time. The reaction in which reacting species are localized near the sink is seen to decay faster is seen evidently in Fig. 2.6a. The parameter σ would characterize the deviation in molecular configuration that helps in achieving the Boltzmann equilibrium. For increase in σ , the reaction starts near the chemical equilibrium as can be deduced from Fig. 2.6b. The exact and approximate results calculated here obey reasonable boundary conditions and the solution has been numerically verified. A full MATHEMATICA notebook statistical_1_state_website_version.nb contains all verifications and results in detail.

2.2.3 Exact Diffusion Dynamics of a Distribution in the Presence of Two Competing Sinks: Oster–Nishijima Model In this subsection, we would consider a simple diffusion equation with two competing absorbing boundaries along with a constant sink term. Random walk problems with multiple boundaries are of interest [44, 124] in many areas of physics. During an electronic relaxation process, the molecule inside a solution can move on a flat potential and may relax to the ground state with or without emitting a photon. The radiative relaxation is modeled by a constant decay term kr that happens irrespective of the molecular configuration. The non-radiative relaxation happens through reactive tunnels that are present in some regions of the excited state. Using such a model with two finite localized tunnels expressed by S(x) = k1 δ(x − a) + k2 δ(x − b) is a generalization to the well-known Oster–Nishijima model [9, 115]. Such model has been useful in understanding fluorescence quenching in tri-phenyl and di-phenyl substi¯ tuted compounds. Introducing the following transformation P(x, t) = P(x, t)e−kr t

2.2 Solution of Single-State Problems in Statistical Physics

57

Fig. 2.7 The schematic picture depicting the diffusion of electronic distribution in the excited state in presence of two reactive channels. The channels are placed at x = a, b that mediates the relaxation of the molecule to the ground state. Reprinted from Diffusion dynamics in the presence of two competing sinks: Analytical solution for Oster-Nishijima’s model, 563, R. Saravanan, A ,Chakraborty, 125317, Copyright (2021), with permission from Elsevier

to the Oster–Nishijima’s model leads to ∂ 2 P(x, t) ∂ P(x, t) =D − k1 δ(x − a) − k2 δ(x − b). ∂t ∂x2

(2.25)

Here, we would like to address the fate of an initially prepared Gaussian distribution given by Eq. 2.6 that would be of interest in studying the electronic relaxation process. The regions are segregated piece-wise due to the presence of boundaries at x = a, b and the corresponding distribution components as shown in Fig. 2.7 are expressed as ⎧ ⎪ ⎨ PA (x, t) + PB (x, t), x < a P(x, t) = PC (x, t) + PE (x, t), a < x < b (2.26) ⎪ ⎩ x > b. PF (x, t), The stationary distribution can be obtained by substituting the separation of variables, P(x, t) = p(x)T (t), into Eq. 2.25 and is given by ⎧ ikx −ikx ⎪ , x a as given by 2

−(x+x0 ) P(x, t) 2 1 2 2 e 4(σ 2 +Dt) + 2 e(σ +Dt)2 +2 f (x) 3 (2 ) + e2 f (x) (2.30) =  N 2 2 π(σ 2 + Dt)      f(x) f(x) + 22 (σ 2 + Dt) 2 2 e(σ +Dt)2 −sgn + erf +  , x >a √ 2 2(σ 2 + Dt) 2 σ 2 + Dt

2.2 Solution of Single-State Problems in Statistical Physics

59

ki with f (x) = xsgn (x − b) + 2bθ (b − x) + x0 and i = 2D . With the suitable introduction of N, the choice of x0 now can be arbitrary. The contributions PA (x, t) and PB (x, t) also can be written similarly using their corresponding stationary distributions and are given by

PA (x, t) 1 = N 2π





∞

ˆ −σ 2 k 2 +ikx 0 Lt

−∞

e e

   ik1 1 − e2i(b−a)k ik1 e2i(b−a)k + 1 eikx dk, + 2Dk 2Dk + ik2

(2.31) and PB (x, t) 1 = N 2π



∞ −∞

 ˆ −σ 2 k 2 +ikx 0 Lt

e e

   ik1 ei2bk − ei2ak i (k2 + k1 ) e2ibk −ikx dk. e − 2Dk 2Dk + ik2

(2.32) In the kernels K i (k) corresponding to PA (x, t) and PB (x, t), 

   ik1 1 − e2i(b−a)k ik1 e2i(b−a)k + K A (k) = + 1 eikx 2Dk 2Dk + ik2     ik1 ei2bk − ei2ak i (k2 + k1 ) e2ibk −ikx − K B (k) = e 2Dk 2Dk + ik2

(2.33)

ik2 there is a pole at k = − 2D , but there is no pole at k = 0. This point is seen by expanding Taylor’s series about the point k = 0,

 ik1 ei2bk 2D

  ik1 ik1 1 − e2i(b−a)k = [1 − 1 − 2i(b − a)k + O(k 2 )] (2.34) 2D  k 2Dk − ei2ak ik1 = [1 + 2ibk + O(k 2 ) − 1 − 2iak + O(k 2 )]. (2.35) k 2Dk

Upon algebraic simplification, the equations for PA (x, t) and PB (x, t) can be written as follows: PA (x, t) 1 = N 2π



 ∞ −α(k−iκ1 )2  ∞ −α(k−iκ2 )2 2 e e dk − i1 e−ακ2 dk k k −∞ −∞ 2  ∞  ∞ 2 2 2 e−α(k−iκ2 ) +e−ακ1 dke−α(k−iκ1 ) + i1 e−ακ2 dk k + i2 −∞ −∞ 2

i1 e−ακ1

with α = (σ 2 + Dt), i = is simplified to PB (x, t) 1 = N 2π

ki ,κ 2D 1

 2

i1 e−ακ3

=

x+x0 , and κ2 2(σ 2 +Dt)

=

x+x0 +2(b−a) . And the 2(σ 2 +Dt)

(2.36)

PB (x, t)

 ∞ −α(k−iκ3 )2  ∞ −α(k−iκ4 )2 2 e e dk − i1 e−ακ4 dk k k −∞ −∞  ∞ −α(k−iκ3 )2  2 e −i(1 + 2 )e−ακ3 dk k + i2 −∞

(2.37)

60

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

Fig. 2.8 Schematic representation of the variables u and k in the Argand plane given by the relation u j = k − κ j . The pole lies inside the contour plane when the value of 2 satisfies the condition 2 ∈ [−κ j , 0]. Reprinted from Diffusion dynamics in the presence of two competing sinks: Analytical solution for Oster-Nishijima’s model, 563, R. Saravanan, A ,Chakraborty, 125317, Copyright (2021), with permission from Elsevier

0 +2b 0 +2a with κ3 = −x+x and κ4 = −x+x . The evaluation of total probability density 2(σ 2 +Dt) 2(σ 2 +Dt) boils down to calculation of all the integrals in Eqs. 2.36, 2.37. A general form of an integral I is introduced using which the solution of all integrals can be calculated similarly. The integral I is expressed by

 I =

∞ −∞

e−α(k−iκ j ) dk, k + i2 2

j ∈ 2, 3.

(2.38)

The above integrand can be variable transformed to u as given by u j = k − iκ j and residues should be added if the contour integrand has poles. The value of the closed contour can be written using the schematic representation given in Fig. 3.2 as   + a

−a

 κ j −α(a+i I m(k)−iκ j )2 2 e−α(k−iκ j ) e dk + idI m(k) I = k + i a + i I m(k) + i2 2 0 −a 2  0 −α(−a+i I m(k)−iκ j )2 e−αu j e du j + d(i I m(k)). u j + iκ j + i2 −a + i I m(k) + i2 κj 

a

(2.39)

In the above equation, the integrals along dI m(k) are equal in magnitude and opposite in direction, hence they cancel up. In the limit a → ∞, they reduce into 

∞

−∞

e−α(k−iκ j ) dk = k + i2 2



∞ −∞

e−αu j du j + u j + κ j + i2 2

 I.

(2.40)

 The closed integral I is equal to sum of the residues. From Fig. 2.8, we can deduce that the pole would lie inside the contour only when 2 ∈ [−κ j , 0], resulting

2.2 Solution of Single-State Problems in Statistical Physics

61

in residue contributions at t = 0. For t > 0, the pole of the u-axis moves upwards in the same amount as the Re(k)-axis moves. Hence, for all time t ≥ 0, the residues will be added only when the value of  lies in the region, [−κ j , 0]. The value of the integral for the simple pole k = −i2 is given by 

I = 2πi j (2 )eα(2 +κ j ) , 2

(2.41)

where j (2 ) is the condition for adding residues given by 

j (2 ) =

1, 2 ∈ (−κ j , 0), / (−κ j , 0). 0, 2 ∈

(2.42)

The probability distributions PA (x, t) and PB (x, t) can be obtained as  1   √ √ 1  PA (x, t) =− −sgn (κ1 ) + erf ακ1 + −sgn (κ2 ) + erf ακ2 N 2 2   √ 1 2 2 − e2 (x+x0 +2(b−a))+2 (σ +Dt) −sgn (κ2 + 2 ) + erf α(κ2 + 2 ) 2 1 2 2 2 + √ e−ακ1 − 1 2 (2 )e2 (x+x0 +2(b−a))+2 (σ +Dt) 2 πα

(2.43)

 1   √ √ PB (x, t) 1  =− −sgn (κ3 ) + erf ακ3 + −sgn (κ4 ) + erf ακ4 N 2 2  √ 1 + 2 2 (−x+x0 +2b)+ 2 (σ 2 +Dt)  2 −sgn (κ3 + 2 ) + erf α(κ3 + 2 ) + e 2 +(1 + 2 ) 3 (2 )e2 (−x+x0 +2b)+2 (σ 2

2 +Dt)

.

(2.44)

The bounded nature of the probability distribution results in the following: P(x = ±∞, t) = 0 =⇒ j (2 ) = 0.

(2.45)

¯ The survival probability can be obtained by integrating the distribution P(x, t) over x,  ∞ ¯ P(x, t)dx. (2.46) Q(t) = −∞

After quite algebraic simplifications, the survival probability can be derived as follows:  

 b + x0 Q(t) − 2 (σ 2 +Dt)  = 1 + 2e 2 θ −b − x0 − 22 (σ 2 + Dt) − erfc N e−kr t 2 σ 2 + Dt      b + x0 + 22 (σ 2 + Dt) b + x0  2 (σ 2 +Dt)+2 (b+x0 )  +e 2 − erf +  sgn 2 2(σ 2 + Dt) 2 σ 2 + Dt

62

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations  + 1 2

 e

−22 (σ 2 +Dt) 2θ(a − x0 − 2b − 22 α) + erf



×e2 (−a+x0 +2b) sgn





−a + 2b + x0  2 (σ 2 + Dt)



 2 (σ 2 +Dt) +e 2

   −a + x0 + 2b x0 + 2b − a + 22 (σ 2 + Dt)  − erf +  2 2(σ 2 + Dt) 2 (σ 2 + Dt)

 a  √  √  √  √   erfc ακ1 + erfc ακ3 − erfc ακ2 − erfc ακ4 dx, + 1 2 −∞

where N =

2.2.3.1

(2.47)

1 . Q(0)

Results and Discussions

In this subsection, we have obtained the time-domain probability distribution and survival probability for the Oster–Nishijima model. So far, the solution has been obtained for this model either in the Laplace domain [133] or in the time domain only when the sinks have infinite strength [7]. This gives the possibility of comparing the obtained solutions with these available solutions. The verification of the derived solution can be done substituting the solution back to original Eq. 2.25 for x = a, b and into the boundary condition given by Eq. 2.28 at x = a, b. These verifications were carried out in MATHEMATICA and are presented in the file two_delta_time-domain.nb. Some other possible checks other than the above tests can be deduced intuitively using the following arguments: (i) Placing both the sinks at the same place or a situation when either of the sink strength is zero should yield the single sink solution. (ii) When the initial distribution is at the midpoint between the sinks, the result should be symmetric about k1 ↔ k2 . (iii) The survival probability is equal to 1 at initial time when σ = 0 and −x0 = a, b. (iv) A condition of equivalence between two situations (|x0 − a|, k1 , k2 , b) ↔ (|x0 − b|, k2 , k1 , a) should hold. Because of the C = 1 consideration, we see that the test (i) is not entirely satisfied in the result given by Eq. 2.47. When k1 is zero, the results reduce to the solution of single sink problem. But the survival probability has a singularity when k2 is zero. This is a parametric limitation, and for correct results one should choose k2 > 0. If one does not have the C = 1 consideration, all the tests will be satisfied. This point is verified by carrying out all the tests on the numerically inverted Laplace-domain solution of this problem (presented in temp1.nb). This suggests that we have obtained a solution that obeys the Smoluchowski equation, boundary conditions but the solution is physical only up to some parametric regimes. Hence, by discussing the regimes for which the tests are satisfied, we discuss the applicability of our solution. From Fig. 2.9, one can see that the symmetry does not hold between the two situations k1 ↔ k2 as mentioned in case (ii). This could convey that the effect of either of the sink is not properly incorporated because of the C = 1 assumption, which point would be clear from the test performed on argument (iv). In Fig. 2.10a, we inspect a case when the distribution is placed near sink 2 at a 2 nm distance and at 8 nm far from sink 1, now

2.2 Solution of Single-State Problems in Statistical Physics

63

Fig. 2.9 Comparison between the two cases: case when a k1 = 1, k2 = 4, case b k1 = 4, k2 = 1 for the parameters: σ = 0.01 nm, D = 1 nm2 /ps, x0 = 5 nm, a = −10 nm, b = 0 nm, kr = 0 ps−1 . Reprinted from Diffusion dynamics in the presence of two competing sinks: Analytical solution for Oster-Nishijima’s model, 563, R. Saravanan, A ,Chakraborty, 125317, Copyright (2021), with permission from Elsevier

the influence of the sink 2 should be remarkable. As expected, increasing the sink strength k2 from 2 → 4 increases the decay. Whereas when the distribution is placed near sink 1 at the same 2 nm distance, the influence of sink 1 is not remarkable as expected (seen from Fig. 2.10b). It is seen that there is no change in the concentration profile between the cases k1 = 2 and k1 = 4. Hence, we get to know that the effect of the k1 sink is not properly incorporated. Combining all the above results, it would be correct to state that the survival probability will be physical when the strength k1 is 0, small number, or can be arbitrary when k2 is a large number. In order to obtain a solution, which would be applicable for all parameters, the problem has to be solved by using A = 1 consideration instead of C = 1 consideration. However that would lead to a transcendental kernel which cannot be decomposed into simple poles and the Fourier inversion is not known within the present knowledge [123]. A new mathematical framework should be developed in order to treat the Fourier/Laplace inversion of such transcendental solutions. Now we compare our analytical result with the numerically inverted Laplacedomain results when k1 is a small number. Figure 2.11 shows there is only a little variation for all values of parameters. It also shows that the radiative decay always dominates the non-radiative decay for sufficiently bigger value kr > 0.01. The curve Expfit exactly fits the Laplace-domain solution, while the presented time-domain solution undermines the expected survival probability. To conclude we state the solution obtained using the approximation C = 1 is a biased solution. The solution also reduces to the exact solution of single sink problem when k1 is taken to be zero. In order to solve the model completely, a new mathematical method that treats a transcendental kernel should be built. The presented time-dependent results with finite k1 , k2 and σ can be useful in electronic relaxation and other reaction–diffusion systems.

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2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

Fig. 2.10 The graph showing the effect of competing sinks on the survival probability for parameters: D = 0.5 nm2 /ps, a = −10 nm, b = 0 nm, kr = 0 ps−1 , a when the distribution is near sink 1 x0 = 2 nm, b when the distribution is near sink 2 x0 = 8 nm. In the second figure, a curve with k1 = 0, k2 = 2 is presented to show that there is an effect of k1 in the overall decay. Reprinted from Diffusion dynamics in the presence of two competing sinks: Analytical solution for OsterNishijima’s model, 563, R. Saravanan, A ,Chakraborty, 125317, Copyright (2021), with permission from Elsevier

Fig. 2.11 Comparison of Q(t) along with numerically inverted Laplace solution. The parameters are σ = 0.01 nm, D = 1 nm2 /ps, x0 = 5 nm, a = −1 nm, b = 3 nm, k1 = 0.2 nm/ps, k2 = 2 nm/ps. The green and blue curves indicate kr = 0 ps−1 and kr = 1 ps−1 respectively. The discrete points are obtained from numerical inversion and continuous curves are the analytical expression obtained in Eq. 2.47. Reprinted from Diffusion dynamics in the presence of two competing sinks: Analytical solution for Oster-Nishijima’s model, 563, R. Saravanan, A ,Chakraborty, 125317, Copyright (2021), with permission from Elsevier

2.2 Solution of Single-State Problems in Statistical Physics

65

2.2.4 A General Method to Solve Diffusion in Piece-Wise Linear Potentials in the Time Domain In the model given by Eq. 2.5, it was shown that the problem had translational and mirror symmetry. In order to remove these symmetries, the presence of potential has to be considered with an arbitrary placement of sink. In particular, this section develops a mathematical method to solve piece-wise linear potentials in presence of reactive/transit boundaries. The derivative of the potential is discontinuous at potential boundaries and they share some similar traits like two boundaries, transcendental Laplace/Fourier-domain solutions, shape shifting in the probability distribution at boundaries, etc. Mathematically, they are challenging due to the presence of transcendental poles in Laplace-domain solutions which are further unsolvable analytically. Regarding contextual importance, these models have a wide variety of applications. As already mentioned, the molecular processes have two different (also equivalent) representations: (1) reactions that proceed by transition between molecular potential surfaces as in diabatic representation [8, 156] and (2) reactions that happen by crossing an activation barrier [80] as in adiabatic representation. For the former case, a simplistic description is a diffusion in the diabatic potential surface in the presence of the sink. So far, there have been no exact time-domain result when the effect of potential surfaces is present. A simple model for such a kind is where the diffusion occurs on the V-shaped potential (V (x) = b |x|) in the presence of finite localized sink, i.e., S(x) = k0 δ(x) (shown in Fig. 2.12a). The potential has been considered in several studies such as for chemical reactions [131], the motion of response signals over T-cells [45], etc., fluorescence yield in molecular crystals [45]. For the latter case, the models involve complicated form of potentials. A simple model that can account for a finite barrier would be a meta-stable potential constructed by three straight lines of opposite slopes placed adjacent to each other (Fig. 2.12b). The model has been useful in the context of surface adsorption process in metals [123], the simplest model to explore the coherence in the Brownian motion [72], etc. Other similarly important approximate potentials are W-shaped potential, bucket potential, and series of piece-wise linear potentials. Here we do not consider the complicated array collection of piece-wise linear potentials, but the method developed here is applicable to a continuous array with the only cost of patience. A time-domain solution has been developed in this study, however the solution has too much parametric constraints for applicability to present an analysis in the light of the applications. It is known that such problems involve roots which are transcendental that complicates the Fourier/Laplace inversion. The further treatment can be proceeded by the use of the Lambert-W function which was tried and left as a future scope. The provided method is of advantage without which an analytical solution cannot be derived in the time domain.

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2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

Fig. 2.12 Schematic representation of the piece-wise linear models A and B

2.2.4.1

Exact Diffusion Dynamics in Model A

The model A (as shown in Fig. 2.12a) can be characterized by Eq. 2.1 with the substi= 2g |x|, D = D/2, and S(x) = k0 δ(x). For convenience, we initially tutions, Uk B(x) T start by considering the absence of sink, i.e., k0 = 0, the distribution will obey the probability continuity requirement and a following flux continuity requirement at the boundary x = 0: 

D ∂ P(x, t) d |x| + g D P(x, t) lim →0 2 ∂x dx

0+ = 0.

(2.48)

0−

Initially, we assume a localized distribution at an arbitrary position given by P(x, 0) = δ(x + x0 ) peaked on the negative side of the well. The solution of the V-shaped potential in absence of absorbing boundaries can be obtained using transformations to the following equation: D

˜ x, ˜ x, ˜ t˜) ∂ P( ˜ t˜) ∂ 2 P( = . 2 ∂ x˜ ∂ t˜

(2.49)

The transformation is given by the following: t 2 2 t v(x, t) = e(gxsgn(x)+Dg 2 )

˜ x, ˜ t (t˜)); P( ˜ t˜) = v(x, t)P0 (x(x),

x˜ = x

t˜ =

(2.50) (2.51)

which upon applying to Eq. 2.49 leads to the following equation: D ∂ 2 P0 ∂ P0 ∂ P0 ± gD = , 2 ∂x2 ∂x ∂t

(2.52)

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67

Fig. 2.13 Comparison of our solution Eq. 2.54 with that of the available solution in Ref. [45]. It can be inspected that both solutions vary around x = 0 because the resulting Eq. 2.54 has not incorporated a necessary boundary condition at x = 0

where ± signs are corresponding to the regions x > 0 and x < 0 respectively. The above equation represents the problem in the piece-wise regions. When the same transformation is applied at the solution level,  P0 (x, t) =

x −∞

˜ x, P( ˜ t˜)d x˜

(2.53)

2

(x+x0 ) 2 t e(−g(xsgn(x)−x0 sgn(x0 ))−Dg 2 ) e− 2Dt = √ 2π Dt

˜ x, with the modified initial condition for P( ˜ 0) = δ(x + x0 ) exp (g(x0 sgn (x0 )). − 2 t e(−g(xsgn(x)−x0 sgn(x0 ))−Dg 2 ) e P0 (x, t) = 2 π D 2t

(x+x0 )2 4D 2t

(2.54)

The random variable transformation form x˜ to x is linear and hence has been obtained by mere substitution [117]. When we compare the above solution (Eq. 2.54) with that of the solution given in Ref. [45] (comparison shown in Fig. 2.13), the solutions are different with an extra term in their solution. The variation between the two would dominate near the minimum of the potential. In piece-wise models, wherever the slope U (x) is discontinuous, the probability distribution undergoes a shape transition [45] owing to Boltzmann thermalization. The shape transition enters through the boundary condition given by Eq. 2.48 and the above solution does not satisfy the condition. We develop ways to enforce the boundary conditions in time domain in the subsequent derivations. From careful inspection, we find that the above

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2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

solution does not satisfy Eq. 2.48 rather a modified form of Eq. 2.48 as given below:  lim

→0

D ∂ P0 + g D P0 sgn (x) 2 ∂x

0+ = Dg P0 (0, t).

(2.55)

0−

The solution can be obtained using the following ways: (i) to find the form of P0 (x, t) from the above equation, (ii) to find the function P1 (x, t) such that P(x, t) = P0 (x, t) + P1 (x, t) and that nullifies the extra term Dg P0 (0, t). The way (ii) can be expressed in the equation as  lim

→0

D ∂ P1 + g D P1 sgn (x) 2 ∂x

0+ = −Dg P0 (0, t).

(2.56)

0−

A corresponding partial differential equation can be obtained for both cases by using the steps given below. Simplification of Eq. 2.55 leads to 

0+ D ∂ lim P0 (x, t) + g D P0 (0 + , t) →0 2 ∂ x 0− +g D P0 (0 − , t) − Dg P(0, t) = 0,

(2.57)

then using the continuity requirement gives us 

D ∂ P0 (x, t) lim →0 2 ∂ x

0+ + g D P0 (0, t) = 0.

(2.58)

0−

Now, the above boundary condition can be alternatively derived from the following PDE: D ∂2 ∂ P0 P0 (x, t) + g Dδ(x)P0 (x, t) = 2 ∂x2 ∂t

(2.59)

for which the solution is known [137], the appropriate function v(x, t) is multiplied to make it an eigenfunction of the linear potential. The solution is given by 2

(x+x0 ) 2 t e(−g(xsgn(x)−x0 sgn(x0 ))−Dg 2 ) e− 2Dt P0 (x, t) = √ 2π Dt      f(x) −g −2gxsgn(x) f(x) − gDt e + + 0 + erf −sgn √ 2 2 D2 t 2Dt

(2.60)

which is exactly the propagator obtained in Ref. [45] when the Sign[] term of above is taken to be 1. The f (x) is given by f (x) = |x| + |x0 |. The PDE resulting from Eq. 2.56 also results in the same solution with appropriate factors. Having established the method, let us add a sink at the minimum of the potential for simplicity. The

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69

corresponding boundary condition in presence of sink can be derived as follows:  lim

→0

D ∂ P + g D Psgn (x) 2 ∂x

0+ = (k0 + Dg)P(0, t).

(2.61)

0−

The corresponding differential equation can be obtained in the similar manner as above, D ∂ 2 P(x, t) ∂P . + (g D − k0 )δ(x)P(x, t) = 2 2 ∂x ∂t

(2.62)

And the solution for such case is given by 2

(x+x0 ) 2 t e(−g(xsgn(x)−x0 sgn(x0 ))−Dg 2 ) e− 2Dt P(x, t) = √ 2π Dt 2 t  −g(xsgn(x)−x sgn(x 0 0 ))−Dg 2 ) 1 1 f (x) e +e( 2      f(x) f(x) + 1 Dt −sgn + 1 + erf √ Dt 2Dt

(2.63)

with 1 = k0 −Dg . As already stated, there are no analytical time-domain solution D available for the finite sink case with piece-wise linear potential. The above solution satisfies Eq. 2.1 and the appropriate continuity and the flux discontinuity equations: P(x, t = 0) = δ(x + x0 ), lim [P(x = 0 + ) − P(x = 0 − )] = 0, →0  0+ D ∂P = k0 P(0, t). lim + g D Psgn (x) →0 2 ∂ x 0−

(2.64)

The solution obtained is compared with the results using other methods. Figure 2.14 compares the survival probability obtained by various methods. The survival probability was obtained numerically using MATHEMATICA. For another benchmark, the Laplace domain results of [131] is also numerically inverted [163]. All these results are then compared. Except for larger and smaller times, the result of Eq. 2.65 has a variation from the other solutions in the order of 10−1 . This variation is explored by comparing the Laplace transform of our solution (Eq. 2.63) with the available solution in Ref. [131]. It is found that the correction term (Sign[]) is responsible for the variation and is more general than 1. The significance of this term is the correction to the Laplace-domain solution and has not been obtained through other methods. This is an advantage of having whole calculation in the time domain only and without going into Fourier/Laplace domain. The results of the survival probability without the Laplace-domain correction are presented as the curve Qtnc[t]. The analytical survival probability is given by

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2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

Fig. 2.14 Comparison of the survival probability along with other results for the following parameters: g = 1 nm−1 , D = 0.2 nm2 /ps, x0 = 5 nm, k0 = 3 nm/ps. The curve NDSolve is obtained by using NDSolve command of MATHEMATICA. The curve NInvLaplace is obtained by numerically inverting the Laplace-domain solution as derived in Ref. [131]. The time-domain solution differs from the other two solutions for intermediate values of time. This variation is the correction to the Laplace-domain inverted solutions. The Qtnc[t] is the when the time-domain correction term Sign[] is removed and replaced by 1

   Dt1 + x0 1 1 1 2 2 e− 2 Dg t+ 2 Dt1 +(g+1 )|x0 | −erf √ √ 1 − g 2 Dt   Dgt + x0 1 −erfc √ √ e 2 (g−1 )(Dt(g+1 )+2x0 ) 2 Dt 

x 0 + 1 +2e(1 −g)(−Dt1 −x0 ) θ (−x0 − Dt1 ) + sgn Dt     g(|x0 |−x0 )  e (−x0 + Dgt) (Dgt + x0 ) + e2 gx0 erfc . erfc √ √ 2 2Dt 2Dt Q(t) =

(2.65) A little more general problem is arbitrarily placed sink problem. The problem is impossible to solve for P(x, t) using integral transformations. When an absorbing boundary is arbitrarily placed, one can see from Ref. [123, 131] that the poles of sor k-domain solution is transcendental. If one wants to treat the poles of the transcendental denominator analytically, Lambert-W-function can be used. But treating the inverse transforms with a complicated function is not of ease. The sink when placed arbitrarily should obey the following boundary conditions simultaneously: 0+ D ∂P + g D Psgn (x) = 0, →0 2 ∂ x 0−  xs + D ∂P lim = k0 P(xs , t), + g D Psgn (x) →0 2 ∂ x xs − 

lim

(2.66)

2.2 Solution of Single-State Problems in Statistical Physics

71

which can be combined into a single equation as done in the previous cases, ∂P D ∂ 2 P(x, t) + g Dδ(x)P(x, t) − k0 δ(x − xs )P(x, t) = . 2 2 ∂x ∂t (2.67) This governing equation of flat potential with two sinks of finite strength is called the Oster–Nishijima model of electronic relaxation considered in Sect. 2.2.3. A special treatment takes care of the transcendental poles and the solution of the above equation has been derived in Ref. [138]. 

2

x+x   − 2Dt0 P(x, t)  x sgn(x − b) + 2bUnit(b − x) + 2 Dt + x0 e + 2 erf − = √ √ v(x, t) 2 2π Dt 2Dt  ! x + x sgn(x − b) + 2bUnit(b − x)  (xsgn(x−b)+2bUnit(b−x)+x0 )+22 Dt/2 sgn 0 e 2 + 2 Dt       2b + Dt2 − x + x0 2b − x + x0   (2b−x+x0 )+ 2 Dt/2 2 − sgn +θ(a − x) 1 e 2 erf + 2 √ 2 Dt 2Dt    2(b − a) + Dt2 + x + x0 1 2 (2(b−a)+x+x0 )+ 2 Dt/2 2 erf + e √ 2 2Dt      2(b − a) + x + x0 1 2b − x + x0 erfc + −sgn + 2 √ Dt 2 2Dt       2(b − a) + x + x0 x0 − x + 2a x + x0 − erfc + erfc √ −erfc √ √ 2Dt 2Dt 2Dt

(2.68)

a = 0, b = xs , and v(x, t) = N e−g(xsgn(x)−|x0 |−Dg t/2) . The   1, x ≥ 0, 1, x > 0, function Unit(x) represents , and the θ (x) = . Figure 2.15 0, x < 0, 0, x ≤ 0. compares the obtained survival probability with the inverted Laplace-domain result [131]. The results vary slightly because the solution was obtained using the solution of Oster–Nishijima model which has parametric limitations. The obtained distribution with 1 = −g, 2 =

k0 , D

Fig. 2.15 Comparison of the time-domain survival probability along with numerical inverted Laplace-domain solution [131] for model A. The parameters are g = 1 nm−1 , D = 0.3 nm2 /ps, x0 = 4 nm, k0 = 3 nm/ps, xs = −1 nm

2

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2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

Fig. 2.16 The comparison of the survival probability Q(t) using a our method, b SK Ghosh [131] for the following parameters: D = 0.3 nm2 /ps, x0 = 4 nm, xs = −1 nm, k0 = 3 nm/ps

obeys all the boundary conditions and the corresponding Smoluchowski equation. However, the presented solution will be numerically accurate only for values k0 > g D. The singularity in the analytical survival probability (Eq. 2.72) suggests the parameters for which the solution is not defined. Whenever k0 is closer to g D the solution will be inaccurate. The discrepancy is the artifact of the solution we use for the Oster–Nishijima model. It arises because of the special treatment involved in solving the model [138]. Hence, it is of use to solve the Oster–Nishijima model without the special consideration C = 1 as the implications of this consideration will pass to the solution for model A. Figure 2.16 gives the effect of the slope of the potential “g” on the probability of the species remaining in the potential well.

2.2.4.2

Exact Dynamics of the Problem B

The section presents the method for solving the model B, however this solution also will share the same limitation. The model B (Fig. 2.12) is characterized by the following boundary conditions: 

D ∂P D ∂U (x) + P lim →0 2 ∂ x kB T ∂x

l+ = 0,

(2.69)

l−

where l = −xm , 0 along with two continuity boundary conditions at −xm , 0. For = θ (−x) |x + xm | + θ (x)(xm − x). model B, the potential function is given by Uk B(x) T All the four boundary conditions can be combined into a single equation as given by ∂P D ∂ 2 P(x, t) . + g Dδ(x + xm )P(x, t) − g Dδ(x)P(x, t) = 2 2 ∂x ∂t (2.70)

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73

Fig. 2.17 The comparison of the survival probability Q(t) using a our method, the solution is numerically inaccurate because of the inaccurate Oster-Nishijima solution we use. b Privman method [123], the results of Privman validate Kramer’s theory in the large barrier limit. The parameters are xm = 2 nm, D = 0.5 nm2 /ps, x0 = 5 nm, b = 0

The solution of the above equation is found to be similar to Eq. 2.68 (the comparison between both models shown in Fig. 2.18) with the following replacements: with 1 = −g, 2 = g, a = −xm , b = 0, and the effect of the potential comes through the transformation function derived as v(x, t) 2 = N e−g(θ(−x)|x+xm |+θ(x)(xm −x)−θ(x0 )|−x0 +xm |+θ(−x0 )(xm +x0 )−Dg t/2) . The aim is to find the survival probability, i.e., the probability that the species in the left side of the well. The entity is calculated by  Q(t) =

0

P(x, t)dx.

(2.71)

−∞

Figure 2.17 shows the difference in the presented solution and the Laplace-inverted results [123]. The result is numerically inaccurate because of the pole in the denominator when 2 = g. Hence, the used Oster–Nishijima solution is not valid. This is the artifact of the solution. Privman’s solution is a correct one and can be explained in two regimes: (1) the reaction mediated through diffusion and (2) the activation-mediated regime. For the case when D > g, the slope of the potential surface increases the rate of the reaction. While for the case, g > D, the reaction is diffusion mediated, the species is yet to climb the barrier through diffusion in order to reach the reactive configuration. The results of the Privman validate Kramer’s asymptotic results. The analytical survival probability for models A and B can be represented in a single equation given by  Q(t) = Q 1 (t) + A(t, x0 ) 2

 2b − a + Dgt + x0 √ g − 2 2Dt sg−2   (2x +(sg+ Dt)) 0 2 b + sDgt + x0 e 2 −2 erfc √ 2 − sg 2Dt

e2bg+

g−2 2



(2x0 +(g+2 )Dt)

erfc

74

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations     (−2x0 +(g−2 )Dt)  b − Dgt + x0 2b − Dgt + x0 − erf + 2 erf √ √ g + 2 2Dt 2Dt       e ga 2b − a + x0 2b − a + x0 + 2 Dt e−ga [ e2 (2b−a) −sgn + ] + 2 + erf √ Dt g − 2 g + 2 2Dt      −gb −gb  e θe b + x b + x +  Dt 0 0 2 + 2 + erf − +2 eb(2 ) −sgn √ Dt −(2 + g) 2 − g 2Dt  g(2b+x0 +2 Dt) 2e θ(a − 2b − x0 − 2 Dt) +2 e2 (−x0 −2 Dt) g − 2 −θ

2 e−2bg+

g+2 2

2e−g(2b+x0 +2 Dt) θ(2b − a + x0 + 2 Dt)θ(−b − x0 − 2 Dt) −(g + 2 )  2e g(x0 +2 Dt) θ(−x0 − 2 Dt − b) +θ 2 − g      1 x0 + 2b − a x0 + 2b − a + 2 Dt − e2 (2b−a)+ag −sgn + 2 + erf √ (2 + g) Dt 2Dt +

1 e22 (b−a) {2e−(g+2 )(2(b−a)+x0 +2 Dt)+22 (b−a) θ(−a + x0 + 2b + 2 Dt) 2 + g !  g+2 a + Dgt − x0 − 2b +e 2 (4a−4b+Dgt−x0 −2 ) erfc √ 2Dt      −a + x0 + 2b −a + x0 + 2b + 2 Dt 1 e2 (2b−a)+ga −sgn + 2 + erf + √ (g − 2 ) Dt 2Dt +

1 e2b2 {−2e(g−2 )(2b+x0 +2 Dt) θ(a − x0 − 2b − 2 Dt) (g − 2 ) !  g−2 −a + x0 + Dgt + 2b +e 2 (4b+2x0 +Dt (g+2 )) erfc √ 2Dt          a x + x0 + 2(b − a) −x + x0 + 2b x + x0 1 gx − erfc + erfc erfc √ + e √ √ 2 −∞ 2Dt 2Dt 2Dt    −x + x0 + 2a g(−θ (x0 )|a−x0 |−θ (−x0 )(a+x0 ))−g 2 Dt/2 dx e −erfc √ 2Dt +

(2.72)

with the following substitutions: Models

s

θ

a

b

1

Model A

1

1

0

xs

−g

2 k0 D

Model B

1

0

−xm

0

−g

g

2.2.4.3

Table of denotations Q 1 (t)  " "   " " e g x0 −x0 [erfc −x√0 +Dgt + 2 2Dt   Dgt+x0 √ ] e2gx0 erfc 2Dt " " "a−x "−θ(−x )(a+x )) g(θ(x ) 0 0  0  0 e g(xm −x0 ) xm +Dgt−x0 √ [e + erfc 2 2Dt   g(x −x ) m x −Dgt−x 0 m√ e 0 ] erfc 2 2Dt

A(t, x0 ) 2 1 2 Dt/2+2 x0 2e " " 2 " " e−g Dt/2+g x0 2 2 1 2 Dt/2+2 x0 −g Dt/2 2e " " " " e g(θ(x0 ) a−x0 e−θ(−x0 )(a+x0 ))

Conclusion

We present exact time-domain results for reaction–diffusion models which were earlier not known for cases when the sink is arbitrarily placed. The solutions obtained through the presented method obey all the boundary conditions and their correspond-

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75

Fig. 2.18 Comparison of the models A and B using the a time-domain solution, and b Laplace-domain solutions [123, 131] for parameters: g = 0.2 nm−1 , D = 0.4 nm2 /ps, |x0 − xs | = 4 nm, when k0 = Dg

ing Smoluchowski equation. The solution is fully exact when the sink is placed at the origin. For the arbitrarily placed sink case, the survival probability will be accurate when the k0 is bigger than g D. However, this discrepancy is not the discrepancy of the method, rather is an artifact of the solution of Eq. 2.67. A fully accurate solutions can be made when the solution for the case of two finite absorbing sinks with flat potential is solved without any special considerations [138]. The detailed illustration of the method and their verifications are presented as a MATHEMATICA notebook modelA.nb and modelB.nb.

2.2.5 Understanding Condensed-Phase Dynamics Using Parabolic Potential Models in Presence of Delta-Function Reactive Terms So far regarding reaction–diffusion systems, exact time-domain solutions have been derived only when the problem either had a translational invariance in potential [41, 137] or a mirror symmetry about the sink trap [143] or both [10, 142, 149]. k 2 = 2D x , it is still When the dimensionless diabatic potential is parabolic, i.e., Uk B(x) T an open problem when the sink is placed arbitrarily (S A (x, t) = k0 δ(x − xc )) and

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2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

only Laplace-domain solutions are available so far for this case [8, 144, 149, 154]. Along with their usefulness in understanding condensed-phase chemical dynamics [10, 97, 144], they are also useful in understanding general properties of nonequilibrium systems such as stochastic resonance [23, 67, 90], self-organized criticality [105], etc. [68, 154]. Here we solve a diffusion problem in the presence of a harmonic potential well (U (x) = 21 kx 2 ) and a finite absorbing sink S(x, t). When the dimensionless diabatic potential of the reactant U (x) is chosen to be parabolic k 2 = 2D x , Eq. 2.1 reduces to as given by Uk B(x) T ∂ 2 P(x, t) ∂ ∂ P(x, t) =D + k [x P(x, t)] − kr P(x, t) − S(x, t)P(x, t), (2.73) ∂t ∂x2 ∂x where k is the spring constant for the potential energy along the reaction coordinate x. Usually, various theoretical treatments [9, 144, 149] that consider electronic relaxation problem in the single-state limit consider that the potential energy curve and the corresponding decay channels to be independent of time. The schematic representation for such a case is as given in Fig. 2.19b. In the context of electronic relaxation, this means that the initially excited molecule moves on the one-dimensional potential energy surface (1D-PES) and gets de-excited from a fixed arbitrary position (xc ) with a finite decay strength (k0 ). This scenario can be mimicked by taking the sink function to be S A (x) = k0 δ(x − xc ) (model A), which is a very interesting model to be considered and it has been an open problem for several years [10, 144, 149, 154] as mentioned above. Here we solve a special exit condition given by S B (x, t) = k0 ekt δ(x − xc e−kt ) (model B) which is akin to the highly sought model

(a)

(b)

Fig. 2.19 Schematic representation of an electronic relaxation process as a curve-crossing process a (Model B): potential curve is moving because of the action of solvation in an initially excited molecule (Refer [146] for examples of this type). b (Model A): A usual representation regarding diffusion-controlled relaxation used in Ref. [9]. The position of the absorbing boundary (xc ) is chosen according to the minimum energy difference between the curves. Reprinted from An analytically solvable reaction-diffusion model for chemical dynamics in solutions, 548, R. Saravanan, 111196, Copyright (2021), with permission from Elsevier

2.2 Solution of Single-State Problems in Statistical Physics

77

A. In a particular context of a photo-induced molecular process, the following physical scenario makes good sense for the choice of model B besides the previously used model A (see Fig. 2.19). Initially, the dye molecules can be assumed to be electronically ground in a solvated environment. And after the illumination by appropriate electromagnetic radiation, molecules get excited. The excitation causes a rearrangement in the electron distribution inside the solution, leading to an extra solvation motion along with the diffusive relaxation in the excited state P(x, t). In some molecules like fluorescein and coumarin 153, it has been observed [146] that due to the solvation effect, the initially excited molecule would undergo a solvation motion, hence lowering its energy curve over time as shown in Fig. 2.19. The explicit time dependence of the excited-state potential curve is not known, but we do choose a solvable form of sink function. As the 1D-PES comes down in energy, the non-adiabatic coupling strength between the surfaces increases over time and can be taken to be k0 ekt for mathematical convenience. If we fix the coordinate to the minima of the moving excited-state potential, the position of the sink would be initially greater far from the minima and would move towards the minima over time (Fig. 2.19a). So, the sink position can be chosen to be xc e−kt for convenience. Hence, the sink function for model B is S B (x) = k0 ekt δ(x − xc e−kt ). The concentration profiles Q(t) of model A and B are similarities except for a small parametric window, so are they akin to each other. In the context of using their profiles for fitting a chemical kinetic data, they have no much difference and can be used interchangeably. In the next section, we attempt the analytical solution for model B and derive the corresponding survival probability. Interesting insights on the dynamics emerge from the interplay between the initial position of the random walker, size of the activation barrier, diffusivity, position of the sink boundary, etc. The derived survival probability profile gives an understanding of chemical dynamics in condensed phases.

2.2.5.1

Mathematical Derivation of the Survival Probability for Harmonic Potential and the Sink SB (x, t)

In the absence of absorbing conditions, the distribution is fixed by the following equation: ∂ 2 P(x, t) ∂ ∂ P(x, t) =D +k (2.74) (x P(x, t)) . ∂t ∂x2 ∂x The above equation can be obtained by applying the following transformations [149, 150]: P0 (x, ˜ t˜) = e−kt P(x(x), ˜ t (t˜)); to the heat equation,

x˜ = xekt ,

t˜ =

∂ P0 (x, ˜ t˜) ˜ t˜) ∂ 2 P0 (x, . =D 2 ∂ x˜ ∂ t˜

1 2kt (e − 1), 2k

(2.75)

(2.76)

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2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

For the initial data, P0 (x, ˜ 0) = δ(x˜ + x0 ), the solution of the above equation is known to be (x+x ˜ 0 )2 1 ˜ t˜) = √ (2.77) e− 4Dt˜ . P0 (x, 2 π D t˜ Transforming back to the original variables of x, t, and P(x, t) gives us the solution of Eq. 2.74 as given below: P(x, t) = 

ekt 4π D(e2kt − 1)/2k

e

−

(xekt +x0 )2 4D(e2kt −1)/2k

.

(2.78)

By rearranging the above equation, the result is derived which is in accordance with the well-known result of Ornstein–Uhlenbeck process (Ref. [162]), P(x, t) = 

1 4π D(1 − e−2kt )/2k

e

−

(x+x0 e−kt )2 4D(1−e−2kt )/2k)

.

(2.79)

Similarly, upon adding a δ-function sink to the free particle problem [3, 168, 170], ˜ t˜) ˜ t˜) ∂ P0 (x, ∂ 2 P0 (x, − k0 δ(x˜ − xc )P0 (x, ˜ t˜), =D 2 ∂ x˜ ∂ t˜

(2.80)

it transforms to the following problem upon using transformations mentioned in Eq. 2.75 (detailed derivation and discussions on the rules of transformation are presented in Appendix A), ∂ 2 P(x, t) ∂ P(x, t) ∂ =D +k (x P(x, t)) − k0 ekt δ(x − xc e−kt )P(x, t). ∂t ∂x ∂x2

(2.81)

The space–time propagator corresponding to the problem given by Eq. 2.80 satisfies the following equation: 

 ∂ ∂2 ˜ t˜|x0 , 0) = δ(x˜ + x0 )δ(t˜) − D 2 − k0 δ(x˜ − xc ) G(x, ∂ x˜ ∂ t˜

(2.82)

whose explicit form is known to be [28, 41, 137, 154] G(x, ˜ t˜|x0 , 0) =

(x+x ˜ 0 )2    |˜x − xc | + abs |xc + x0 | k0 k02 t˜+ k0 (|x−x e− 4Dt˜ k0 − + e 4D 2D ˜ c |+|xc +x0 |) sign √ 4D 2D 2D˜t 2 π D t˜   |˜x − xc | + |xc + x0 | + k0 t˜ − erf . √ 2 D˜t

(2.83)

Transforming to original variables x, t, the solution of Eq. 2.81 for the initial δfunction distribution P(x, 0) = δ(x + x0 ) can be derived. And after rearrangements we obtain the following solution for the problem given by Eq. 2.81:

2.2 Solution of Single-State Problems in Statistical Physics (x+x e−kt )2

k 2 e2kt

79 k ekt

"

"

"

"

0 − 1 k ekt 0 2 f (t) 02D (""x−xc e−kt ""+e−kt "xc +x0 ") 4 f (t) P(x, t) = √ + 0 e e 4D e 4D 2 π f (t) ⎞⎤ ⎡ ⎛ " " " ⎛ "" ⎞ kt k0 e " " " f(t) + "x − xc e−kt " + e−kt |xc + x0 | "x − xc e−kt " + e−kt |xc + x0 | k0 ekt ⎠ ⎟⎥ ⎢ ⎜ D ⎝ + + erf ⎝ √ ⎠⎦ , ⎣−sgn 2f(t) 2D f(t)

(2.84) D (1 − e−2kt ). The corresponding survival probability on the excited with f (t) = 2k ∞ state can be obtained by integrating over all x, i.e., Q(t) = −∞ dx P(x, t) and is evaluated to be Q(t) = 1 + e0 |x0 +xc |+0 e

2 2kt

   −kt e−kt |x0 + xc | e |x0 + xc | + 20 ekt f(t) + 0 ekt − erf √ 2f(t) 2 f(t)  −kt  e |x0 + xc | −kt kt −02 e2kt f (t) |x0 + xc | − 20 e f (t))e +2θ(−e − erfc , √ 2 f(t)

f (t)





sgn

(2.85)

k0 with 0 = 2D . Similarly, the survival probability for an arbitrary initial distribution can be calculated using the following formula:

 Q 1 (t) =

∞

−∞

P(x0 , 0)Q(x0 , t)dx0 .

(2.86)

For the Gaussian initial data given by (x+x0 )2 1 e− 4σ 2 , P(x, 0) = √ 2 πσ2

(2.87)

the result of Q 1 (t) will be obtained to have the same functional form of Q(t) with D (1 − e−2kt )) + σ 2 e−2kt . the replacement of f (t) = 2k 2.2.5.2

Verification for the Analytical Solution Given in Eqs. 2.84 and 2.85

Numerical verification: The numerical solution for Eq. 2.81 is obtained by using NDSolve command of Mathematica which was further numerically integrated to  2 obtain the numerical Q(t). The δ-function sink is mimicked by a Gaussian πα e−αx with a high value of α = 104 . The numerical results are then used to benchmark the obtained analytical results of survival probability Q(t) and the probability distribution P(x, t) and are presented in Fig. 2.20. Verification in the asymptotic limit: Equation 2.74 has known solution in the presence of a perfect absorbing sink, i.e., k0 → ∞, and xc = 0, called as the pinhole sink limit. We see that in the pinhole limit, our solution converges to the available solution [10]. The solution of Eq. 2.80 in the limit xc → 0, k0 → ∞ and σ → 0 can be derived [63, 137] as follows:

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2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

Fig. 2.20 a The numerical verification of the probability distribution given by Eq. 2.84 at different 2 times t = {0, 1, 2, 5, 10, 20, √ 50} for parameters: D = 0.5 nm /ps, x0 = 4 nm, k0 = 3 nm/ps, xc = 1 nm, k = 0.2 ps−1 , σ = ( 40)−1 nm. b Numerical verification of the survival probability Q 1 (t) given by Eqs. k, k0 , xc , x0 , σ ) = a (0.1, 0.2, √ 2.85 and 2.86 for three different √ set of parameters, (D,√ 3, 4, 0, 1/ 400), b (0.5, 0.2, 3, 1, 4, 1/ 40), c (0.5, 1, 3, 2, 4, 1/ 120) in the increasing order of decay. The analytical results agree with the numerical solution. Reprinted from An analytically solvable reaction-diffusion model for chemical dynamics in solutions, 548, R. Saravanan, 111196, Copyright (2021), with permission from Elsevier

2 ˜ −(x+x ˜ 0 )2 −(|x|+x 1 1 0) P f (x, ˜ t˜) = √ e 4Dt˜ − √ e 4Dt˜ . 2 π D t˜ 2 π D t˜

(2.88)

Using the transformation and after rearrangement, we get P(x, t) = 

√ k 2π D(1 − e−2kt )

e

−

k(x+x0 e−kt )2 2D(1−e−2kt )

√ −

k

2π D(1 − e−2kt )

e

−

k(|x|+x0 e−kt )2 2D(1−e−2kt )

,

(2.89) which is exactly the result obtained for the pinhole sink case given in Ref. [10]. The corresponding survival probability is given by / Q(t) = erf (Z(t)) where Z (t) = x0 e

−kt

k . 2D(1 − e−2kt )

(2.90)

For longer times, Z (t) is small, in which case erf (Z(t)) ∼ Z (t). Hence, for longer

k times, the decay is exponential as given by Q(t) ∼ x0 e−kt 2D . The verification of the analytical solution given by Eq. 2.84 satisfying the original equation (Eq. 2.74) and boundary conditions is presented in Section S3 of Appendix A, whereas the numerical verification and a check in the asymptotic limit are shown in this section. The solution when the sink is given by S A = k0 δ(x − xc ) has been interesting contextually [9] as well as mathematically [38, 132, 144, 149]. The difference in our model to that of the conventional model (A) is that our result gives the solution of time-dependent sink strength k0 ekt and the sink moves towards origin over time given by xc e−kt . Still there are some similarities in both models, as they both converge to same solution in

2.2 Solution of Single-State Problems in Statistical Physics

81

Fig. 2.21 The plot of the distribution P(x, t) for different time presented along with the area under 2 the distribution curve. √ a The parameters are D = 0.5 nm /ps, k0 = 3 nm/ps, x0 = 10 nm, xc = 42nm, k = 0.1 ps−1 , σ = ( 40)−1 nm, t = {0, 1, 2, 5, 10, 20, 50}. b√ The parameters are D = 0.5 nm /ps, k0 = 2 nm/ps, x0 = 0 nm, xc = 4 nm, k = 0.3 ps−1 , σ = ( 40)−1 nm, t = {0, 1, 2, 5, 10, 20}. Reprinted from An analytically solvable reaction-diffusion model for chemical dynamics in solutions, 548, R. Saravanan, 111196, Copyright (2021), with permission from Elsevier

the pinhole limit, and they also share differences. Section S4 of Appendix A presents the similarities, and the parametric limitations for applying the presented solution in the place of model A. It is seen that the time-dependent profile Q(t) of model B can be good enough to use in place of model A when k0 and xc are typically 2 nm.ps−1 and 0.2 nm, respectively, when k ≥ 0.2 ps−1 (refer to graphs shown in Appendix A, Sect. S3).

2.2.5.3

Theory of Electronic Relaxation in Solutions

Figure 2.21a, b depicts the dynamics of the distribution on a parabolic potential in the presence of an absorbing boundary. The walker starts at the point x = −x0 at t = 0 and takes a random walk while the mean position of the walker undergoes a translational motion towards the minimum of the potential. The action of diffusion widens the probability distribution. Once the particles reach the potential minimum, there are no more translational motion of the mean position and only diffusion occurs with a scaled diffusion coefficient. At the absorbing boundary, the distribution develops a cusp which is required by the flux discontinuity requirement. The point of the cusp keeps moving towards the origin given by xc = xc e−kt (Fig. 2.21b). Once the distribution encounters the sink in a significant amount, the area of the distribution starts reducing over time. The area in the electronic relaxation sense is the number of molecules surviving on the excited state till time t and is denoted by Q(t). With this picture of distribution dynamics, we study the effect of system parameters on the survival probability of the random walker. In the diffusion–reaction approach for condensed-phase reactions, the solvent parameters are incorporated through the noise that gives rise to a second-order diffusive term. The diffusivity is of the form,

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2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

D = k Bζ T where ζ is the friction coefficient and T is the temperature of the solvent. When the diffusivity is more, the tail of the distribution could reach the sink even for smaller times in significant amounts, and thus have a faster decay (Fig. 2.22a). The flux leaving the state is characterized by the term, −k0 P(xc e−kt , t) meaning that a sharper distribution at x = xc e−kt will have more rate of decay at a particular instant. This results in more rate of decay in intermediate times than over a longer time. The strength of the sink (k0 ) can be related to molecular terms as the non-adiabatic coupling strength between the molecular potential surfaces. Higher the coupling strength, higher the non-radiative decay constant k0 . When the absorbing boundary has high strength k0 , the decay is higher (Fig. 2.22b). The effect of non-radiative absorption strength on the decay gets saturated for bigger values like 3 nm.ps−1 . The effect of excitation wavelength appears through the initial coordinate of the excited-state molecules. The lower the excitation wavelength, far the peak distribution is placed from the potential minima. Figure 2.23a shows that the distribution that is far from the relaxing configuration undergoes a slower decay. The effect of the spring constant along the reaction coordinate is shown in Fig. 2.23b. More the “k”, the molecule gets solvated in a small time, and starts achieving high decay even for small time (Fig. 2.23b). The position of the absorbing boundary determines three cases of reaction propagation, xc < 0 or xc = 0 would mean a barrierless process, with the distribution facing/not facing a translational motion at the boundary respectively. The boundary when placed closer to the distribution leads to faster decay as expected (Fig. 2.24a). For the case xc > 0, the relaxation process faces a potential barrier for smaller times, and hence a lesser decay to ground state. Another interesting excitation parameter is the bandwidth of the source. When the incident light is not of monochromatic character, the dye molecules does not excite to a particular bond length. While most of the molecules excite to a particular bond length, some excite to nearby configurations. So if the initial distribution is assumed to take a Gaussian form as given by (x+x0 )2 1 e− 4σ 2 . P(x, 0) = √ 2 πσ2

(2.91)

Figure 2.24b shows the effect of σ on the electronic relaxation process. It is seen that for smaller times, the distribution with a larger σ decays faster. The effect of σ appears as an extra term σ 2 e−2kt adding to the diffusion process. But in the longer run, the extra term σ 2 e−2kt that adds to the width of the Gaussian vanishes and further there will be no effect of σ on the decay. The equilibrium attaining nature determines that the distribution becomes a Gaussian whose width is determined by D and k for the harmonic potential. So the width σ in the initial Gaussian form disappears (thermalization), which dynamics is fixed by the Boltzmann equilibrium. Regarding classical dissipative systems, the harmonic potential model is the simplest solvable model having realistic effects, and has been used to study several non-equilibrium phenomenon. The celebrated Marcus’ theory for electron transfer utilized the harmonic potential model in presence of a non-adiabatic coupling to describe an outer-sphere

2.2 Solution of Single-State Problems in Statistical Physics

83

Fig. 2.22 The survival probability Q(t) versus time is plotted showing a the effect of diffusivity D while other parameters are k0 = 2 nm/ps, x0 = 0 nm, xc = 4 nm, k = 0.2 ps−1 , σ = 0 nm, b the effect of non-radiative decay strength k0 while other parameters are D = 0.5 nm2 /ps, x0 = 3 nm, xc = 2 nm, k = 0.1 ps−1 , σ = 0 nm. Reprinted from An analytically solvable reaction-diffusion model for chemical dynamics in solutions, 548, R. Saravanan, 111196, Copyright (2021), with permission from Elsevier

Fig. 2.23 The survival probability Q(t) versus time plot showing a the effect of initial position of the walker (x0 ) when the parameters are (D, k, k0 , xc , x0 , σ ) = (0.5, 0.1, 3, 1, x0 , 0), b the effect of slope of the potential when the reactive channel moves from x = 4 towards the initial distribution placed at x = 0 over time. The other parameters are (D, k, k0 , xc , x0 , σ ) = (0.3, k, 2, 4, 0, 0). Reprinted from An analytically solvable reaction-diffusion model for chemical dynamics in solutions, 548, R. Saravanan, 111196, Copyright (2021), with permission from Elsevier

process. Using the equilibrium statistical mechanics, Marcus [98, 155] predicted an equal rate of electron transfer reactions (x0 = 0) in both the inverted region (xc < 0) and in the normal region (xc > 0). The time-dependent and non-equilibrium version of Marcus’ prediction using our result is presented in Fig. 2.25b. The use of Q(t) in electronic relaxation is, it can be used as a profile for fitting the transient experimental data of excited state population (demonstrated in Fig. 2.27). The relaxation data of tri-phenyl methane dyes obtained from Ref. [15] were fitted using the available formula and it fits well with experimental data.

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2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

Fig. 2.24 The survival probability Q(t) versus time plot showing a the effect of the position of the sink. The other parameters are (D, k, k0 , x0 , xc , σ ) = (0.5, 0.2, 3, 5, xc , 0), b the effect of σ . The parameters are (D, k, k0 , xc , x0 , σ ) = (0.5, 0.2, 3, 1, 4, σ ). Reprinted from An analytically solvable reaction-diffusion model for chemical dynamics in solutions, 548, R. Saravanan, 111196, Copyright (2021), with permission from Elsevier

Fig. 2.25 a Marcus’ rate prediction for electron transfer dynamics as a function of xc , b the survival probability Q(t) of the harmonic potential model B demonstrating the time-dependent non-equilibrium version of Marcus’ prediction on electron transfer process. The rate of decay is higher for the barrierless process (xc = 0) and is comparatively lower and is symmetric about the normal (xc > 0) and inverted region (xc < 0). The other parameters are (D, k, k0 , x0 , xc , σ ) = (0.5, 0.2, 3, 0, xc , 0). Reprinted from An analytically solvable reaction-diffusion model for chemical dynamics in solutions, 548, R. Saravanan, 111196, Copyright (2021), with permission from Elsevier

2.2.5.4

Conclusions

We find a time-domain solution for the problem of radiation less decay from a harmonic potential curve modeled by Smoluchowski equation, in presence of a timedependent Dirac delta sink. The derived analytical expression P(x, t) obeys boundary conditions and the solution is verified against numerical and asymptotic analytical results. The time-dependent profile Q(t) is plotted, the features of the model are discussed, and some insights into chemical dynamic processes are derived. We could realize the time-dependent non-equilibrium version of Marcus’ prediction analyt-

2.2 Solution of Single-State Problems in Statistical Physics

85

Fig. 2.26 The survival probability Q(t) of the harmonic potential model B demonstrating the equivalence of results of the survival probability under different situations when |xc − x0 | is a constant = 4, implying the translational and mirror symmetry in the problem. The parameters are (D, k, k0 , xc , x0 , σ ) = (0.5, 0.2, 3, xc , x0 , 0). Reprinted from An analytically solvable reactiondiffusion model for chemical dynamics in solutions, 548, R. Saravanan, 111196, Copyright (2021), with permission from Elsevier

Fig. 2.27 Fitting the electronic relaxation of TPM dyes inside a CH3 CCl3 solvent with the Q(t) expression (Eq. 2.85) using the following values (x0 , xc , σ , k0 , D, k) = (5, 0, 0, 10, 2.975, 0.044), b inside C4 H9 OH solvent with values: (σ , D, k) = (2, 1.05, 0.118)

ically as this is the first time the harmonic potential model is solved in presence of a finite sink placed arbitrarily. However, still the translational symmetry is not removed from the problem, as they have been obtained by transforming from the problem that already has inbuilt translational symmetry (refer Fig. 2.26). Hence, the crossover in the Q(t) based on the interplay between k and xc does not exist in our model which is expected (Fig. 3b of Ref. [149]) for the harmonic potential problem with a finite sink placed arbitrarily (model A). But as a guide, the same transformation can be used to obtain the dynamics of the sink model S(x, t) = k0 ekt δ(x − xc ) which produces knowingly incorrect quantitative results but the solution would satisfy the original equation and boundary conditions (discussed in detail in Sec-

86

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

tions S2, S4 of Appendix A. The comparison between the highly sought model A (S(x) = k0 δ(x − xc )) and the presented model is drawn parametrically. The timedependent profile Q(t) can be useful for fitting the chemical kinetic data as a function of system and molecular parameters. The time-dependent profile Q(t) can be useful for fitting the chemical kinetic data as a function of system, and molecular parameters. All verifications and results are presented as a MATHEMATICA notebook harmonictime.nb. In the following, we will discuss the results that are qualitatively correct for the case, when the sink function is S(x, t) = k0 ekt δ(x − xc ).

2.2.5.5

Qualitative Insights for Model S(x, t) = k0 e kt δ(x − xc ) Drawn by Using xc (t) = xc e−kt While Transforming Eqs. 2.80 to 2.81

If one transforms xc 1 also given by xc (t) = xc ekt while transforming Eqs. 2.80 to 2.81, one can obtain the probability distribution that would correspond to an absorbing boundary that is stationed at x = xc . The analytical expression for the distribution, P(x, t), and the survival probability, Q(t), can be obtained by replacing xc → xc ekt to Eqs. 2.84 and 2.85. All conclusions about the parametric dependence of survival probability and distribution continue to hold with only a slight modification, with only difference in the quantitative results. With one exceptional worth noting result is the combined dependence of xc , x0 , and k in this new case. This can explain the crossover in the survival probability observable in a model (Fig. 3b of Ref. [149]) as the model now does not have both the symmetries (translational and mirror). The cases −x0 ≤ xc < 0 or xc = 0 would mean a barrierless process, with the distribution facing/not facing a translational motion at the boundary respectively. For the case xc > 0, the relaxation process faces a potential barrier, and hence a lesser decay to the ground state. The effect of the spring constant k along reaction coordinate is presented in Fig. 2.28. If the process is barrierless, the slope accelerates the decay (Fig. 2.28a). For a process with an activation barrier, the decay profile exhibits a crossover from activation-controlled regime to diffusion-controlled regime (Fig. 2.28b). For smaller times, the translational motion of the distribution towards the potential minimum is faster for a high k. This time regime is the activation-controlled regime where the potential accelerates the distribution. But soon after the distribution reaches the potential minimum, the rest part of the barrier has to be climbed through the diffusive process to reach the sink at x = xc . Hence, for a higher slope, the decay is lesser in this time regime. This time regime is the diffusion-controlled regime where only diffusion process mediates the reaction (Fig. 2.29).

1

Which one should not, according to the invariance rules of probability while performing random variable transformations [117]. The discussions on the fate of x-related variables such as x0 , xc , σ are given in Sect. S2 of Appendix A.

2.3 Exact Dynamics of Coupled Problems

87

Fig. 2.28 The plots of survival probability Q(t) versus time showing the effect of slope of the potential when the process a is barrierless when other parameters are (D, k, k0 , xc , kr , x0 , σ ) = (0.5, k, 3, 0, 0, 5, 0), and when it b faces a barrier. The parameters are (D, k, k0 , xc , kr , x0 , σ ) = (0.5, k, 3, 1, 0, 4, 0) Fig. 2.29 The symmetry in the rate between normal region (xc > 0) and in the inverted region (xc < 0) in electron transfer process (x0 = 0) was predicted by R. A. Marcus and the time-dependent and non-equilibrium version of the theory. The parameters are (D, k, k0 , kr , x0 , xc , σ ) = (0.5, 0.2, 3, 0, 0, xc , 0)

2.3 Exact Dynamics of Coupled Problems The microscopic arisal of Smoluchowski models has been emphasized in Chap. 1 for an electron transfer process inside condensed phase. The description similarly holds for other chemical dynamics such as chemical reactions [131], electronic relaxation [8, 9], vision transduction process [8, 130], and other photo-reactive processes [21, 43] in condensed phases. In all these processes, the different states of the reactive system are modeled as different potential energy surfaces and the reacting species are transferred through coupling between the surfaces. For an one-dimensional system, with a special case of only two states, i.e., say ground and excited states, the distribution is governed by a coupled Smoluchowski equation as considered in Ref. [34] that has been used in the context of electronic relaxation process as presented by

88

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

 ∂ Pe (x, t) ∂2 De = De 2 + ∂t ∂x kB T  ∂ Pg (x, t) Dg ∂2 = Dg 2 + ∂t ∂x kB T

 ∂ dUe (x) Pe (x, t) − κe (x)Pg (x, t), ∂x dx  ∂ dUg (x) Pg (x, t) − κg (x)Pe (x, t), (2.92) ∂x dx

where Ue (x) and Ug (x) are the diabatic potential curves of the excited state and ground state, De and Dg are the diffusivity of the distribution in the respective surfaces, and x is the reaction coordinate. The functions κe (x) and κg (x) represent population decay from the potential curves. The multi-state problems of various types have been mentioned in the literature so many times [8, 18, 156]. Even Laplacedomain solutions are available only when the non-adiabatic couplings have a special form, i.e., constant [16–18, 61, 94, 156] or a Dirac delta function [8, 34, 53]. Till now, there are no time-domain solutions found for such coupled partial differential equations. In this section, we develop a method that solves flat potentials coupled through Dirac delta functions.

2.3.1 Exact Diffusion Dynamics of a Distribution in a Coupled System: A Simple Open System Here we consider a simplest model of two flat potentials separated by a energy difference of U0 , i.e., [Ue (x) − Ug (x) = U0 ]. The coupling function is a finite strength Dirac delta function [κi (x) = ci δ(x)]. In case of electronic relaxation, more general form of probability distributions is Gaussian functions at initial time. Hence, at t = 0, the states are initially populated with Gaussian probability distributions, 

Pe (x, 0) Pg (x, 0)



⎛ ⎜ = ⎝

1 4πσe2 1 4πσg2

1/2 e 1/2 e

−(x+xe )2 4σe2 −(x+x g )2 4σg2

⎞ ⎟ ⎠

(2.93)

∞ so that the population in each states are normalized, i.e., −∞ Pi (x, 0)dx = 1. Such coupled system of Smoluchowski equations is usually solved numerically [10]. Regarding analytical approaches, the Laplace inversions are not known for the available Laplace-domain solutions [26, 34, 35, 53, 55, 126]. The time-domain solutions would enable the transient analysis on ground-state and excited-state profiles. The coupled Smoluchowski equation for the problem of two flat potential energy states with a δ-function coupling is given by ∂2 ∂ Pe (x, t) = De 2 Pe (x, t) − ce δ(x)Pg (x, t), ∂t ∂x ∂ Pg (x, t) ∂2 = Dg 2 Pg (x, t) − cg δ(x)Pe (x, t). ∂t ∂x

(2.94)

2.3 Exact Dynamics of Coupled Problems

89

We use separation of variables method to get the following stationary state equations: ∂2 pe (x) − ce δ(x) pg (x) = −Spe (x), ∂x2 ∂2 Dg 2 pg (x) − cg δ(x) pe (x) = −Spg (x), ∂x

(2.95)

De

and T (t) = eLi t , with Li = Di ∂∂x 2 . The stationary state solutions are given by 2

 pe (x) =

A3 eikx ,

 pg (x) =

A1 eikx + A2 e−ikx , x < 0 x > 0 with k =



B2 e−ikg x + B1 eikg x , x < 0 B3 eikg x ,

x < 0 with k g =

S . De



S . Dg

We derive the boundary conditions under the continuity requirements of the probability distribution. The following boundary conditions incorporate the effect of the other state into the problem,  lim

h→0

lim [ pe (0 − h) − pe (0 + h)] = 0,   ∂ pe 0+h − ce pg (0 + h)/De = 0, ∂ x 0−h h→0

lim [ pg (0 − h) − pg (0 + h)] = 0 and    ∂ pg 0+h lim − cg pe (0 + h)/Dg = 0. h→0 ∂ x 0−h h→0

(2.96)

All coefficients mentioned above can be obtained only when A1 and B1 are known. The degree of freedom A1 and B1 is the amplitude of whether the states are initially populated or not. As both states were chosen to be populated (Eq. (2.93)) which would mean A1 = 1 and B1 = 1, we would explore the implications of both the possibilities (B1 = 1 and B1 = 0) case by case in the subsequent subsections. Using the boundary conditions (Eq. (2.95)), the following solutions are obtained for the first case B1 = A1 = 1 as given by ⎧ ⎨eikx − pe (x) = ⎩eikx −

ce (cg +2i Dg k g ) −ikx e , ce cg +4De Dg kk g ce (cg +2i Dg k g ) ikx e , ce cg +4De Dg kk g

x < 0, x > 0.

(2.97)

90

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

 cg (ce +2i De k) − ce cg +4De Dg kkg e−ikg x + eikg x , x < 0,

pg (x) =

c (c +2i D k)

e e eikg x + eikg x , − ce cgg +4D e Dg kk g

(2.98)

x > 0.

Using the kernel method, the probability distribution for any state can be written as follows:  ∞ 1 eLi t [ P˜i (k, 0) pi1 (x)dki + P˜ j (k j , 0) pi2 (x)dk j ], i, j ∈ e, g Pi (x, t) = √ 2π −∞ and so the expression for the excited-state distribution becomes Pe (x, t) =



1 Le t e 2π

∞ 

−∞

 ce c g 2 2 eik|x| e−σe k +ikxe dk ce cg + 4De Dg kk g  2 2 2i Dg k g + e−σg kg +ikg x g +ik|x| dk g . ce cg + 4De Dg kk g

eikx −

(2.99)

The stationary distribution pe (x) has three parts as seen from Eq. 2.97. In the subsequent derivations, we calculate different contributions of the distributions using the incident and rest components of the stationary solutions. Use the incident part to evaluate the first part of the distribution, Pe(1) (x, t) =

1 2π



∞

e−σe k

2 2

+ikxe Le t ikx

e

e

dk.

−∞

After doing the integration, we get −(x+xe )2 1 e 4(σe2 +De t) . Pe(1) (x, t) =  2 π(σe2 + De t)

(2.100)

The second part of the distribution is obtained by using the second part of the stationary solution, Pe(2) (x, t) =

1 2π



∞

e−σe k

2 2



+ikxe

−∞

ce cg 2 (2i) kk g De Dg − ce cg



eLe t eik|x| dk.

After completing squares in the exponent and using the relation between k and k g , the equation becomes

Pe(2) (x, t) = − with e2 =

ce cg 4De Dg



1 2  e 2π e

Dg . De

2 − (xe2+|x|) 4(σe +De t)

We take



∞

−∞

e

 −(σe2 +De t) k−i

2

(xe +|x|) ) 2(σe2 +De t)

k 2 + e2

dk,

(2.101)

2.3 Exact Dynamics of Coupled Problems

 I =

∞

e

91

 −(σe2 +De t) k−i

2

(xe +|x|) ) 2(σe2 +De t)

k 2 + e2

−∞

dk.

The integration has poles k = −ie and k = ie in the imaginary k plane. We choose a contour (shown in Fig. 2.30) that encloses the integrand and takes care of the shift in the exponent. Hence, the variable u is defined that will be used for transformation given by u = k − κ. The integration around the contour is given by Cauchy integral formula, 

2 2  a −(σ 2 +De t)(k−κ)2  a+i I m(κ) −(σ 2 +De t)(I m(k)−κ)2 e e e e e−(σe +De t)(k− κ) dk = dk + dI m(k) 2 2 (k + ie )(k − ie ) k + e I m(k)2 + e2 −a a

 −a −(σ 2 +De t)u 2 e e a

(u + κ)2 + e2

du +

 −a

2

e−(σe +De t)(I m(k)−κ) I m(k)2 + e2

−a+i I m(κ)

2

dI m(k)

(2.102)

(xe +|x|) . The integral terms II and IV in the RHS of the above equations with κ = i 2(σ 2 e +De t) get cancelled being equal and opposite values. Extending the limit of the contour to infinity (a → ∞) together reduces Eq. (2.102) as follows:



2 2  ∞ −(σ 2 +De t)(k−κ)2  −∞ −(σ 2 +De t)u 2 e−(σe +De t)u e e e e du = dk + du 2 2 (u + z 1 (x, t))(u + z 2 (x, t)) k + e (u + κ)2 + e2 −∞ ∞

 z 1 (x, t) = i  z 2 (x, t) = i

(xe + |x|) + e 2(σe2 + De t) (xe + |x|) − e 2(σe2 + De t)

 

In the newly defined complex plane, the poles z 1 (x, t) and z 2 (x, t) are time dependent. Evaluating the position of poles at t = 0 one can study their changing position at later time t > 0.   (xe + |x|) u p1 = −z 1 (x, 0) = −i +  e 2σe2  u p2 = −z 2 (x, 0) = −i

 (xe + |x|) − e . 2σe2

The position of the Re(k) axis (Im(k)= 0) defined in the frame of u-axis is I m(u) = −

(xe + |x|) . 2σe2

92

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

The position of the poles on the plane based on the value of e is discussed in Fig. 2.31 at t = 0. For t > 0, the poles start moving upwards. The mid line between the poles moves upwards for t > 0. However, the poles in the cases (i) and (iv) cannot cross the contour even at t → ∞. For t > 0, the pole 2 lies inside the contour in case (ii) and in case (iii) pole 1 lies inside the contour. 

⎡ ⎤ (σe2 +De t)(e − xe2 +|x| )2 (σe2 +De t)(e + xe2 +|x| )2 2 2 2(σe +De t) 2(σe +De t) e e e−(σe +De t)(k−κ) ⎦ dk = 2πi ⎣ (e ) + (e ) (k + ie )(k − ie ) 2ie 2ie



(e ) =

  2 −1 e ∈ − 2(σf2(x) , 0 , +De t) e

0

otherwise,

which gives the following value of I: 

⎡ ⎤ (σe2 +De t)(e − xe2 +|x| )2 (σe2 +De t)(e + xe2 +|x| )2 2 2 2(σe +De t) 2(σe +De t) e e−(σe +De t)(k−κ) ⎣ (e ) e ⎦ dk = 2πi +

( ) e k 2 + e2 2ie 2ie −∞ ∞

 +

+∞ −∞

e−(σe +De t)u du. (u + κ)2 + e2 2

2

(2.103)

The dynamics of the distribution using the third part is given by Pe3 (x, t)

−i2Dg ce = 2π



∞

kg e

−(σg2 +Dg t)k g2 +ik g (xe +|x|



Dg De

ce cg + 4De Dg kk g

−∞

)

dk g .

Upon simplification it becomes −ice  Pe3 (x, t) = 4π De Dg



∞

kg e

−(σg2 +Dg t)k g2 +ik g (x g +|x|



Dg De

k g2 + g2

−∞

)

dk,

(2.104)

c c where g2 = 4Dee Dg g DDge . Use convolution between k and the other part of integral which can be evaluated from Eq. (2.101), ⎡ /  2 (σ 2 +Dg t)  ∞ Dg De e g g −2π (−iδ (|x| − y)) 2 ⎣g (g ) cosh (g ( y + x g )) (2.105) De g −∞ π 4 De Dg2 ⎧ ⎛ ⎛0 ⎞ ⎛0 ⎞⎞ 0 Dg Dg ⎪ 2 ⎪ Dg y + x ⎨ g ⎜ De ⎟ ⎜ De y + xg − g 2(σg + Dg t) ⎟⎟ −g ( D y+x g ) ⎜ g e ⎜−sgn ⎜ ⎟ ⎜ ⎟⎟ e + ⎝ ⎝ 2(σ 2 + D t) − g ⎠erf ⎝ ⎠⎠ 4 ⎪ ⎪ g g 2 σg2 + Dg t ⎩

Pe3 (x, t) =

0

− e

g (

−ice

√

⎛

⎛0

⎝

⎞

⎛0

Dg ⎟ ⎜ De y + xg ⎟ ⎜ ⎝ 2(σ 2 + D t) + g ⎠ + erf ⎝ g g

Dg ⎜ ⎜ De y+x g ) ⎜−sgn ⎜

⎞⎞⎫⎤

Dg ⎪ 2 ⎪ De y + xg + 2g (σg + Dg t) ⎟⎟⎬⎥

2 σg2 + Dg t

⎟⎟ ⎥ dy. ⎠⎠⎪⎦ ⎪ ⎭

2.3 Exact Dynamics of Coupled Problems

93

Results and discussions: The section presents the probability distributions in both states which are calculated using the method given in the previous section. The excited-state probability distribution derived for the case 1 is given by Pe (x, t) = Pe1 (x, t) + Pe2 (x, t) + Pe3 (x, t). 2

−(x+x e ) 1 2 2 2 e 4(σe +De t) − (e )e ee (σe +De t) cosh (e (|x| + xe )) Pe (x, t) =  2 π(σe2 + De t)      |x| + xe − e 2(σe2 + De t) |x| + xe e 2 2  − e + erf − ee (σe +De t) e−e (|x|+xe ) −sgn 2 4 2(σe + De t) 2 σe2 + De t      |x| + xe + 2e (σe2 + De t) |x| + xe e e2 (σe2 +De t) e (|x|+xe )  + e e + e + erf −sgn 2 4 2(σe + De t) 2 σe2 + De t /  Dg ce ∂

+ Abs [x]

( ) cosh ( ( y + x g ))  g g g 2Dg g2 ∂y De ⎧ ⎞⎞ ⎛ ⎞ ⎛ D ⎛ D g g 2 g ⎨ −g ( DDg y+x g ) De y + x g De y + xg − g 2(σg + Dg t) ⎠⎠ ⎝ ⎠ ⎝ ⎝ e e − g + erf −sgn + 4 ⎩ 2(σg2 + Dg t) 2 σ2 + D t

− e

D g ( Dge y+x g )

⎛ D [−sgn ⎝

g

De 2(σg2

y + xg + Dg t)

⎛ D

⎞

+ g ⎠ + erf ⎝

g

g

2 De y + xg + 2g (σg g

2 σg2 + Dg t

(2.106)

+ Dg t)

⎞ ⎫⎤" ⎬ "" ⎠ ] ⎦" ⎭ ""

. y=|x|

b b As we see that a Abs [x] = a sgn (x), ∀a, b ∈ R, we replace Abs [x] by sgn (x). The above replacement is not valid for a discontinuous function but for plotting purposes we take the above weak assumption. It is seen that the part containing Abs [x] does not contribute to the population as the integrand is an odd function with respect to x. Further we impose the following boundary condition: Pe (x = ±∞, t) = 0 =⇒ (i ) = 0, which can be used to neglect all residue contributions based on physical grounds. The survival probability of an i-th state can be calculated using the formula  Q i (t) =

∞ −∞

Pi (x, t)dx,

which leads to    2 2 xe e−e xe +e (σe +De t) sgn −  e 2 2(σe2 + De t) ⎛ ⎞⎤ ⎞ ⎛ 2 + D t) − 2 (σ x x e e e e e ⎠⎦ − erfc ⎝ ⎠ −erf ⎝ 2 σe2 + De t 4(σe2 + De t) ⎛ ⎞⎤ ⎡   2 2 xe + 2e (σe2 + De t) ⎠⎦ ee xe +e (σe +De t) ⎣ xe ⎝ + e − erf (2.107) + sgn 2 2(σe2 + De t) 2 σe2 + De t

  2 2 2 2 +e−e (σe +De t) θ 2e (σe2 + De t) − xe + e−e (σe +De t) θ −2e (σe2 + De t) − xe . Q e (t) = 1 +

94

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

The survival probability of the ground state is qualitatively similar to that of the excited state and can be obtained by following replacements to Eq. 2.107: ce ←→ cg , De ←→ Dg , σe ←→ σg , xe ←→ x g . ⎛

⎞

2

2

−g x g +g (σg +Dg t) ⎠+ e Q g (t) = 1 − erfc ⎝ 2 4(σg2 + Dg t) ⎛ ⎡ ⎞⎤   xg − 2g (σg2 + Dg t) xg ⎝ ⎣sgn ⎠⎦ − g − erf 2(σg2 + Dg t) 2 σg2 + Dg t ⎞⎤ ⎛ ⎡   2 2 xg + 2g (σg2 + Dg t) xg eg x g +g (σg +Dg t) ⎣ ⎠⎦ (2.108) + g − erf ⎝ + sgn 2 2(σg2 + Dg t) 2 σg2 + Dg t

  2 2 2 2 +e−g (σg +Dg t) θ 2g (σg2 + Dg t) − xg + e−g (σg +Dg t) θ −2g (σg2 + Dg t) − xg .

xg

Discussions on the case 2: On the other hand, when the stationary distribution is derived by assuming B1 = 0, the kernels become pe (x) =

 eikx − e

ikx

 pg (x) =

−

ce cg e−ikx , ce cg +4k De Dg k g ce cg eikx , ce cg +4k De Dg k g

x 0

− ce cg +4k gDe eDg kg e−ikg x , x < 0 2ikc D 2ikc D

− ce cg +4k gDe eDg kg eikg x ,

x > 0.

And the corresponding distributions for the excited state are same as known from Eqs. (2.101), (2.100) Pe (x, t) = Pe1 (x, t) + Pe2 (x, t). The dynamics in the ground state can be given by  Pg (x, t) =

∞

−∞

−

2ikcg De −σ 2 k 2 +ik(xe +|x| eLe t e ce cg + 4k De Dg k g



De Dg

)

dk.

(2.109)

Upon rearranging it becomes −icg  Pg (x, t) = 4π Dg De



∞ −∞

ke

−(σ 2 +De t)k 2 +ik(xe +|x|

k 2 + e2

which is similar to Eq. (2.104) and can be re-written as



De Dg

dk,

(2.110)

2.3 Exact Dynamics of Coupled Problems

Pg (x, t) =

2e2

95

cg ∂ (2)  Pe (x, t), De D g ∂ x

(2.111)

which says that the ground-state distribution is the derivative of the even function. The integral over x will be zero. The following Q e (t) and Q g (t) can be obtained, when only the excited state is populated initially 

 2 2 xe e−e xe +e (σe +De t)  + 2 4(σe2 + De t)      xe − 2e (σe2 + De t) xe  − e − erf sgn 2(σe2 + De t) 2 σe2 + De t      2 2 xe xe + 2e (σe2 + De t) ee xe +e (σe +De t)  sgn +  − erf + e 2 2(σe2 + De t) 2 σe2 + De t     2 2 2 2 +e−e (σe +De t) θ 2e (σe2 + De t) − xe + e−e (σe +De t) θ −2e (σe2 + De t) − xe , Q e (t) = 1 − erfc

(2.112)

and Q g (t) = 0.

(2.113)

The ground-state population at all time is zero because B1 = 0 implicitly assumes that there is no initial population in the ground state. Asymptotic solutions: The Integral in Eqs. " (2.109) "can be solved under " and " (2.103) some asymptotic limits. In the limit of "κ 2 + e2 " > "u 2 + 2uκ ", i.e., e → ∞, the condition on (e ) and the use of geometric series expansion to obtain the following solutions: −(x+xe )2 1 e 4(σe2 +De t) Pe (x, t) =  2 π(σe2 + De t)

Fig. 2.30 Schematic graph of the integrand in the complex plane. Reprinted from Exact diffusion dynamics of a Gaussian distribution in one-dimensional two-state system, 731, R. Saravanan, A. Chakraborty, 136567, Copyright (2019), with permission from Elsevier

96

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

−(|x|+x e ) e) , the poles lie outside the contour. In case (ii), −(|x|+x < e < 2σe2 2σe2 (|x|+x e ) 0, pole 1 lies between the Re(k)-axis and Re(u)-axis. In case 0 < e < 2σ 2 , Pole 2 lies inside the e e) contour. In case (iv), where e > 0&&e > (|x|+x both the poles lie outside the contour. Reprinted 2σe2

Fig. 2.31 For the case e
0 only if the ground state is populated initially. There is no population transfer between the states in this model, but still having a decay, this model is an open system consideration. Making either of the sink strength (ce , cg ) to be zero stops the decay. If the decay strength is increased to a very high value, the result exactly matches the solution obtained using the single-state descrip-

2.3 Exact Dynamics of Coupled Problems

99

tion. When both states are equally populated, a symmetry holds in the quantitative profiles when parameters related to e and g states are same.

2.3.1.3

Conclusion

We have obtained the exact analytical solution for the two-state problem which was earlier not known in the time domain for the case of finite coupling k0 and nonzero σ . The time-dependent probabilities of ground state and excited state enable transient analysis of reactive systems having both states explicitly. The solution describes the dynamics of an open system of two states where one state induces decay in the other state [34]. It would be interesting in the context of some processes [35, 53] to consider a closed system where the sum of the population at all time is conserved. The advantage of this model lies in that it accounts for an open system, i.e., population transfer to any other state without having those states explicitly. The possible consideration over this model is for the case of harmonic/absolute (U (x) = b |x|) potential which are realistic and are open problems to solve in the time domain. Any results deducible from the analytical expressions can be checked numerically by running the code presented in stat2state_supplementary.nb. The solution of the above simplified model can ensure the solution of more realistic situations using suitable transformations.

2.3.2 Deriving General Characteristics About Open and Closed Systems Using a Simple Model In this section, we consider a simple multi-state model with constant couplings in order to derive the general properties of multi-state reaction-diffusion systems. The constant coupling models are useful in understanding several processes such as stochastic resonances [30], reversible electron transfer reactions [61, 94, 156], in modeling calcium deficiency [70], in studying effect of gating influence in intraprotein processes [17], etc. Here we show a method to solve constantly coupled surfaces with a limitation that the surfaces and other parameters are all identical. The equation of motion for a general problem can be expressed in the following matrix representation: ˆ n×n |P(x, t) = ∂ |P(x, t). (2.114) [ L] ∂t And when n = 2, i.e., two-state problem, the above equation will look like, 

Lˆ1 − S11 (x) S12 (x) ˆ S21 (x) L 2 − S22 (x)



P1 (x, t) P2 (x, t)



∂ = ∂t



 P1 (x, t) , P2 (x, t)

(2.115)

100

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

where Lˆ i ’s are the Smoluchowski operators and Si j (x) are the coupling functions that represent the population transfer. The coupling terms are taken to be arbitrary constants in general (i.e., (1 − 2δi j )Si j (x) = K i j ), and here we learn that their values decide whether the system is open or closed, and in case of closed systems, it determines whether the reaction are symmetric or asymmetric, etc. Equation 2.115 can be decomposed as follows: ˆ 2×2 = Tˆ + Kˆ = [ L]



Lˆ1 0 0 Lˆ2



 +

K 11 K 12 K 21 K 22

 .

(2.116)

It is to note that for a system with identical states such as when the Ui (x) are in arithmetic difference by constants (i.e., Lˆ 1 = Lˆ 2 ) the matrix operators Tˆ and Kˆ commute. The following methodology holds only for such simple systems.

2.3.2.1

Methodology

The column vector of the time-dependent probability distributions can be expressed using the following time propagation equation given by ˆ

ˆ

|P(x, t) = e(T + K )t |P(x, 0).

(2.117)

It is to remember in every line that the relation [Tˆ , Kˆ ] = 0 holds for the considered case Lˆ 1 = Lˆ 2 . Using Eq. (2.116) in the above equation yields    Lt   ˆ e 0 K 11 K 12 |P(x, 0). |P(x, t) = exp t ˆ K 21 K 22 0 e Lt

(2.118)

ˆ

Let us denote the uncoupled distributions e Lt P(x, 0) to be P0 (x, t), and the above equation reduces to    K 11 K 12 |P(x, t) = exp t |P0 (x, t). K 21 K 22

(2.119)

ˆ

The time propagation matrix e K t can be calculated using the Cayley–Hamilton theorem expressed as n−1  ˆ eK t = ci Kˆ i , (2.120) i=0

where “n” is the number of states and the ci ’s can be obtained by solving the following system of linear equations:

2.3 Exact Dynamics of Coupled Problems

101

eλ j t =

n−1 

ci λij

∀ j.

i=0

Using Eqs. (2.119), (2.120), the time-dependent distribution for an n-state problem can be derived as n−1  |P(x, t) = ci Kˆ i |P0 (x, t). (2.121) i=0

The propagation matrix can be expressed analytically for the model and is given as follows: ⎛

T r ( Kˆ )+ T r ( Kˆ )2 −4det( Kˆ ) ⎜1 ˆ K t ˆ 2 C(t) =e = i + 1| ⎜ ⎝ T r ( Kˆ )− T r ( Kˆ )2 −4det( Kˆ ) i=0 1 2 1 

⎞−1 ⎛ T r ( Kˆ )+ Tr( Kˆ )2 −4det( Kˆ ) t ⎜ ⎟ ⎜e 2 ⎟ ⎜ ⎠ ⎝ 2 ˆ ˆ ˆ T r ( K )− Tr( K ) −4det( K ) t 2 e

⎞ ⎟ ⎟ ⎟ ⎠

|1 Kˆ i .

(2.122)

ˆ For a multi-state problem, the propagation matrix C(t) can be obtained using the MatrixExp command of MATHEMATICA. Using Eq. (2.119), the solution for different cases of potentials can be derived. Application to flat potential surface: When the diffusivity is a constant, Di = D and the potential are constants, Ui (x) = bi , the above methodology applies and using Eq. (2.119), the probability distribution is derived as follows: ˆ |P(x, t) = C(t)



∞

−∞

G 0 (x, xi , t)|P(xi , 0)dxi ,

(2.123)

where G 0 (x, xi , t) is given by ⎛ ⎜ ⎜ ⎜ G 0 (x, xi , t) = ⎜ ⎜ ⎝

√1 e− 2 π Dt

(x−x1 )2 4Dt

0

..

.. .

0 .

0

..

√1 e− 2 π Dt

0 . 0

⎞

..

0 (x−x2 )2 4Dt

√1 e− 2 π Dt

(x−xn )2 4Dt

⎟ ⎟ ⎟ ⎟ (2.124) ⎟ ⎠

which has only diagonal entries. The survival probability is obtained by integrating over all values of x ˆ |Q(t) = C(t)|Q(0), (2.125) where |Q(0) has 1 or 0 depending on whether the state is initially populated or not. Application to linear potential surfaces: When the diffusivity is a constant, Di (x) = D and the potential is Ui (x) =  |x| + bi , the probability distribution becomes  ∞ ˆ |P(x, t) = C(t) |G0 (x, xi , t)P(xi , 0)dxi , (2.126) −∞

102

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

Fig. 2.35 The survival probability Q(t) is plotted for an identical multi-state problem when the  2 initial population is taken to be |Q(0) = 01 for two cases: a For the coupling matrix Kˆ = −2 2 −2 , turns out to be a closed system obeying Q 1 (t) + Q 2 (t) = 1, and b the coupling matrix Kˆ = it −2 1 1 −2 is an open system as expressed by Q 1 (t) + Q 2 (t)  = 1

with G 0 (x, xi , t) = [ √4π1 Dt e−  . D

(x+xi )2 + 2 t 2 4Dt

e−

(|x|−|xi |) 2l

+

e

−|x| l

4l



erfc

|x|+|x √ i |−t 4Dt

 ], and l =

The corresponding survival probability is given by ˆ |Q(t) = C(t)|Q(0).

(2.127)

Application to harmonic potential surface: When the diffusivity is a constant, 2 Di (x) = D, and the potential is Ui (x) = γ 2x + bi , the probability distribution becomes  ∞ ˆ |P(x, t) = C(t) |G0 (x, xi , t)P(xi , 0)dxi , (2.128) −∞

with G 0 (x, xi , t) = probability is



2γ 4π D(1−e−2γ t )



(x+xi e−γ t )2 . The corresponding survival exp − γ4D(1−e −2γ t )

ˆ |Q(t) = C(t)|Q(0).

(2.129)

Hence, using the given method, the time-dependent distribution can be obtained when the solution of the uncoupled system is known. And it is seen from Eqs. (2.125), (2.127), and (2.129) that the survival probability of different states is independent of the parameters appearing in the Smoluchowski operator. Intuitively, it can be understood that the population transfer is nowhere special in space or in time, owing to the constant coupling we employed. Hence, the parameters such as the slope of the potential, diffusivity of the walker, and initial distribution profile do not affect the rate of the reaction. This work proves the intuitive guess. It will be interesting to investigate about the same for a generalized system whose commutator is non-zero. The subsequent discussions are independent of the shape of the potentials considered and should be applicable to Smoluchowski operators of spherical harmonics similar to problems considered in Ref. [156].

2.3 Exact Dynamics of Coupled Problems

103

Fig. 2.36 The survival probability Q(t) of identical five-state problem is plotted when the initial populations |Q(0) = {1, 1, 0, 0, 1} for two cases: a the coupling matrix Kˆ = {{−20, 5, 5, 5, 5}, {5, −20, 5, 5, 5}, {5, 5, −20, 5, 5}, {5, 5, 5, −20, 5}, {5, 5, 5, 5, −20}} is a closed system which 5 ˆ obeys 1, 1, 1}, {1, −5, 1, 1, 1}, {1, 1, −5, 1, 1}, i (Q i (t) − Q i (0)) = 0, and b K = {{−5, 1,  {1, 1, 1, −5, 1}, {1, 1, 1, 1, −5}} is an open system, i.e., i5 (Q i (0) − Q i (t)) = 0

2.3.2.2

Results of General Systems

The results of the constant coupling between identical potential surfaces are presented order to be a closed system  in Figs. 2.35 and 2.36. The figures show that, in i.e., in Q i (t) = 1 the chosen parameters should obey in K i j = 0 ∀ j. In this case, the initially unpopulated states get populated and the populated states would subsequently deplete by equal amounts (50%) in case of symmetric reactions. And the amounts are different for asymmetric reactions. But the sum of all total population n is always conserved for all time. For open systems, when the parameters obey i K i j < 0 ∀ j, the total population is not conserved over time. From this section, we learn that for molecular processes, where  the population should be conserved, the models should be chosen which obeys in K i j = 0 ∀ j.

2.3.3 Exact Diffusion Dynamics of a Distribution in a Closed System To represent a reversible chemical process, the model should be chosen as a closed system. Such a consideration would engage the matter conservation of the chemical species as the process description is complete. This model can explain reversible electron transfer process [53], electronic relaxation in condensed phases [10, 115], and several other chemical processes in condensed phase [64, 76, 141]. The population corresponding to two states are governed by the following coupled equation:

104

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations  De ∂2 ∂ Pe (x, t) = De 2 + ∂t ∂x kB T  ∂ Pg (x, t) Dg ∂2 = Dg 2 + ∂t ∂x kB T

 ∂ dUe (x) Pe (x, t) − κe (x)Pe (x, t) + κg (x)Pg (x, t), ∂x dx  ∂ dUg (x) Pg (x, t) − κg (x)Pg (x, t) + κe (x)Pe (x, t). ∂x dx

(2.130) If we consider flat potential states with a constant energy separation, i.e., [Ue (x) − Ug (x) = U0 ] and the coupling function is taken to be Dirac delta function of finite strength [κi (x) = ci δ(x)]. The solution of Eqs. (2.132) and (2.133) is obtained for two cases: (a) the case when both states are populated initially and (b) the case when only the excited state is populated. The time-domain solution of case (a) applies to reversible reactive processes. On the other hand, the obtained time-dependent probabilities of case (b) give a transient analysis on the electronic relaxation process. Effect of parameters such as temperature, viscosity, and excitation wavelength can be discussed. At t = 0, when both states are assumed to be populated with Gaussian probability distribution (case (a)) is expressed as 

Pe (x, 0) Pg (x, 0)



⎛ ⎜ = ⎝

1 16πσe2 1 16πσg2

1/2 e 1/2 e

−(x+xe )2 4σe2 −(x+x g )2 4σg2

⎞ ⎟ ⎠,

(2.131)

∞ so that total sum of population in both states is normalized, i.e., −∞ (Pe (x, 0) + Pg (x, 0))dx = 1. The coupled Smoluchowski equation for the case of two flat potentials with a δ-function coupling is written as ∂2 ∂ Pe (x, t) = De 2 Pe (x, t) − ce δ(x)Pe (x, t) + cg δ(x)Pg (x, t), ∂t ∂x ∂ Pg (x, t) ∂2 = Dg 2 Pg (x, t) − cg δ(x)Pg (x, t) + ce δ(x)Pe (x, t). ∂t ∂x

(2.132) (2.133)

Using the separation of variables method, the stationary state solutions of Eqs. (2.132) and (2.133) can be obtained as  pe (x) =  pg (x) =

A1 eikx + A2 e−ikx , x < 0 A3 eikx ,

x > 0 with k =



B2 e−ikg x + B1 eikg x , x < 0 B3 eikg x ,

x < 0 with k g =

S , De



S , Dg

by using the substitution Pi (x, t) = pi (x)e−St , and we define T (t) = eLi t , and Li = 2 Di ∂∂x 2 . We derive the following boundary conditions under the continuity and flux discontinuity requirements for a distribution at the localized coupling point. This

2.3 Exact Dynamics of Coupled Problems

105

step incorporates the effect of the other state into the stationary distribution solution, 



lim De

h→0



∂ pe ∂x

0+h

+ cg pg (0) − ce pe (0) = 0, 0−h

lim [ pg (0 − h) − pg (0 + h)] = 0,

h→0

lim Dg

h→0

lim [ pe (0 − h) − pe (0 + h)] = 0,  

h→0



∂ pg ∂x

0+h

and 

+ ce pe (0) − cg pg (0) = 0.

(2.134)

0−h

We explore the implications of two cases where both states are populated, i.e., A1 = 1 and B1 = 1 and another case when only the excited state is populated A1 = 1 and B1 = 0). For the case A1 = 1 and B1 = 1, the distributions are ⎧ Dg k g (ce −cg ) ⎨eikx + e−ikx , x < 0, −kcg De −ce Dg k g +2ik De Dg k g (2.135) pe (x) = Dg k g (ce −cg ) ⎩eikx + eikx , x > 0. −kcg De −ce Dg k g +2ik De Dg k g

⎧ ⎨eikg x + pg (x) = ⎩eikg x +

k De (cg −ce ) e−ikg x , −kcg De +Dg k g (−ce +2ik De ) k De (cg −ce ) eikg x , −kcg De +Dg k g (−ce +2ik De )

x < 0, x > 0.

(2.136)

Using the time-domain methods developed in sections [136, 137], the Fourier-domain probability distributions can be written as Fourier kernels (Eqs. 2.135, 2.136) times corresponding initial distributions in Fourier domain which is expressed as Dg k g (ce P˜0e (k, t) − cg P˜0g (k g , t)) ik|x| e , −kcg De − ce Dg k g + 2ikk g De Dg De k(cg P˜0g (k g , t) − ce P˜0e (ke , t)) ikg |x| P˜g (ke , k g , t) = eikg x P˜0g (k g , t) + e , −k g ce Dg − cg De k + 2ikk g De Dg P˜e (ke , k g , t) = eikx P˜0e (k, t) +

(2.137) (2.138)

where P˜0i (ki , t)’s can be derived using the following definition: P˜0i (ki , t) =



∞

−∞

e−iki x eLi t Pi (x, 0) dx,

to arrive at the following equations:

i ∈ e, g,

(2.139)

106

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

1  ikx −(σe2 +De t)k 2 +ikxe e e P˜e (ke , k g , t) = 2  2 2 2 2 Dg k g (ce e−(σe +De t)k +ikxe − cg e−(σg +Dg t)kg +ikg x g ) ik|x| + e , −kcg De − ce Dg k g + 2ikk g De Dg 1  ikg x −(σg2 +Dg t)kg2 +ikg x g e e P˜g (ke , k g , t) = 2  2 2 2 2 De k(cg e−(σg +Dg t)kg +ikg x g − ce e−(σe +De t)k +ikxe ) ikg |x| + e . −k g ce Dg − cg De k + 2ikk g De Dg Inverting the Fourier-transformed distributions P˜0i (ki , t) of the above equations leads to the solutions as given by −(x+xe )2

2 2 1 2 Pe (x, t) = e 4(σe +De t) + ce ae ee (σe +De t)+e (|x|+xe ) 2 4 π(σe + De t) ⎛ ⎡ ⎞⎤   2 2 |x| + xe + 2e (σe2 + De t) |x| + xe ⎠⎦ − cg bg eg (σg +Dg t) + e + erf ⎝ × ⎣−sgn 2 2(σe + De t) 2 σe2 + De t 0 ⎡ ⎞ ⎞⎤ ⎛ D ⎛ D D |x| Dg + xg |x| Dg + xg + 2g (σg2 + Dg t) g (|x| Dg +x g ) e e e ⎣−sgn ⎝ ⎠⎦ ×e + g ⎠ + erf ⎝ 2(σg2 + Dg t) 2 σg2 + Dg t

0

2

2

g (σg +Dg t)+g (|x| 2 2 +2 1 (e )ce ae ee (σe +De t)+e (|x|+xe ) − 2 2 (g )cg bg e

Dg De +x g )

,

(2.140) and 1

−(x+x g )2 4(σg2 +Dg t) e

0 e2 (σe2 +De t)+e (|x|

De D +x e )

g Pg (x, t) = − ce be e 4 π(σg2 + Dg t) ⎞ ⎞⎤ ⎡ ⎛ D ⎛ D |x| De + xe + 2e (σe2 + De t) |x| De + xe g g ⎠⎦ + cg ag + e ⎠ + erf ⎝ × ⎣−sgn ⎝ 2(σe2 + De t) 2 σe2 + De t ⎞⎤ ⎛ ⎡   |x| + xg + 2g (σg2 + Dg t) |x| + xg g2 (σg2 +Dg t)+g (|x|+x g ) ⎣ ⎠⎦ + g + erf ⎝ e −sgn 2(σg2 + Dg t) 2 (σg2 + Dg t)

−2 3 (e )ce be e

0 De e2 (σe2 +De t)+e (|x| D +xe ) g

2

2

+ 2 4 (g )cg ag eg (σg +Dg t)+g (|x|+x g ) ,

(2.141) with e = (

cg

√

Dg De +ce Dg ), 2De Dg

ai =

1 8Di

, g = (

ce

√

Dg De +cg De ), 2De Dg

and bi = √ 1 8

De D g

. The

i ’s are the conditions for adding residue contributions [136, 137] while evaluating the inverse Fourier transforms. The bounded nature of the probability distribution

2.3 Exact Dynamics of Coupled Problems

107

requires Pi (x = ±∞, t) = 0. This implies that the non-physical contributions can be neglected, i.e., i ( j ) = 0, for plotting purposes [137]. The population probabilities of each state can be obtained by integrating the respective probability distribution function over all values of x,  ∞ Pi (x, t)dx, Q i (t) = −∞

which gives us ⎞ ⎛ ⎡ x 2ai ci ⎣ 1 i 2 −i2 (σi2 +Di t) ⎠ Q i (t) = + − erfc ⎝ 2θ(−xi − 2i (σi + Di t))e 2 i 4(σ 2 + D t) ⎛

+ei xi +i (σi

2 +D t) i

2



⎝sgn



2(σi2

i

i

⎛

⎞⎞⎤

xi + 2i (σi2 + Di t) xi ⎠⎠⎦ + i − erf ⎝ + Di t) 2 σ2 + D t i

(2.142)

i

⎞ ⎛ xj 2c j b j − 2j (σ 2j +D j t) 2 ⎠ − [2θ(−x j − 2 j (σ j + D j t))e − erfc ⎝ j 4(σ 2 + D t) ⎛

+e j x j + j (σ j +D j t) ⎝sgn 2



xj

2

2(σj2

+ Dj t)



⎛

+ j − erf ⎝

j

j

⎞⎞⎤

xj + 2j (σj2 + Dj t) ⎠⎠⎦ . 2 σj2 + Dj t

In the above equation, when i is e, then j is g and is vice versa. The obtained results for Q e (t) and Q g (t) are verified against the numerical solutions and are presented in Fig. 2.37. The numerical solutions were obtained using the NDSolve command of MATHEMATICA. When the excited-state and ground-state species exchange population at the same rate, i.e., cg = ce , and other driving parameters are equal, i.e., xe = x g , σe = σg , De = Dg , the population remains unchanged in both states, i.e., Q i (t) = Q i (0) ∀t. This is an illustration for the dynamic equilibrium seen in reversible chemical reactions. Further discussions on the results will be given in the subsequent discussions. Now, we consider the case (b) in which all the population are in one state leaving none in the other state. This situation can be used for studying electronic relaxation in dye molecules. Hence, this case (b) would mean that all molecules have been excited at the initial time     1/2 −(x+x0 )2 1 Pe (x, 0) 4σ 2 e 2 4πσ = , (2.143) Pg (x, 0) 0 in which case B1 = 0 with A1 = 1. By following similar calculations, the following kernels are obtained:  D k ce eikx − kcg De +ce Dggkgg−2ik e−ikx , x < 0, De D g k g (2.144) pe (x) = ikx Dg k g ce e − kcg De +ce Dg kg −2ik De Dg kg eikx , x > 0.

108

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

Fig. 2.37 Verification of the analytical population probabilities Q e (t) and Q g (t) (Eq. 2.142) against corresponding numerical results. The parameters are (De , Dg ) = (0.2, 1) nm2 /ps, (ce , cg ) = (2, 3) nm/ps, (xe , x g ) = (3, 4) nm, √ √ (σe , σg ) = (1/ 40, 1/ 30) nm. Reprinted from Some exact time-domain results related to reversible reaction-diffusion systems, 539, R. Saravanan, A. Chakraborty, 110955, Copyright (2020), with permission from Elsevier 

pg (x) =

−k De ce e−ikg x , −kcg De +Dg k g (−ce +2ik De ) −ce k De eikg x , −kcg De +Dg k g (−ce +2ik De )

x < 0,

(2.145)

x > 0.

The Fourier-transformed distribution P˜i (k, t)’s can be obtained by multiplying the populating distribution P˜0e (k, t) to the stationary solutions (Eqs. 2.144, 2.145),  D g k g ce 2 2 eik|x| eikxe e−(σ +De t)k , (2.146) kcg De + ce Dg k g − 2ik De Dg k g   k D e ce 2 2 P˜g (k, t) = eikg |x| eikxe e−(σ +De t)k . (2.147) kcg De + Dg k g (ce − 2ik De )

 P˜e (k, t) = eikx −

∞ 1 Using the formula Pi (x, t) = 2π −∞ Pi (k, t) dk, the probability distribution Pi (x, t)’s can be obtained as given by −(x+x )2

0 ce Dg 2 2 1 Pe (x, t) =  e 4(σ 2 +De t) + (e )be ee (σ +De t)+e (|x|+x0 ) + 4De Dg 2 π(σ 2 + De t)      2 2 |x| + x0 + 2e (σ 2 + De t) |x| + x0  + erf ee (σ +De t)+e (|x|+x0 ) −sgn , +  e 2(σ 2 + De t) 2 σ 2 + De t

(2.148) and 0 e2 (σ 2 +De t)+e (|x| −ce Pg (x, t) =  e 4 D g De

De Dg +x 0 )

⎛ + erf ⎝

|x|

⎡

⎛



⎞

De Dg + x0 ⎣−sgn ⎝ + e ⎠ 2(σ 2 + De t)



|x|

De 2 Dg + x0 + 2e (σ + De t)

 2 (σ 2 + De t)

⎞⎤ ⎠⎦ .

(2.149)

2.3 Exact Dynamics of Coupled Problems

109

The population in both the states are derived to get  x0  4(σ 2 + De t)      x0 x0 + 2e (σ 2 + De t) 2 2  , +  +ee x0 +e (σ +De t) sgn − erf e 2(σ 2 + De t) 2 σ 2 + De t

Q e (t) = 1 +

ce 2De e





2θ(−x0 − 2e (σ 2 + De t))e−e (σ 2

2 +D

e t)

− erfc

(2.150)

and Q g (t) =

−ce 2De e

+ee x0 +e (σ 2



 2θ(−x0 − 2e (σ 2 + De t))e−e (σ

2 +D

2

 e t)

 sgn

2 +D



e t)

x0 + e − erf 2(σ 2 + De t)





− erfc

x0



4(σ 2 + De t)  x0 + 2e (σ 2 + De t)  . 2 σ 2 + De t

(2.151)

The system considered is a closed system, hence follows the relation Q g (t) + Q e (t) = 1 ∀t and is shown graphically in Fig. 2.38a. Asymptotic solutions: The population probability can also be obtained as a series solution by expanding the denominator inside the inverse integral transforms of Eqs. (2.146), (2.147). The expansion uses the assumption k < ie and a corresponding solution has been obtained for case (b) to be Q e (t) = 1 +

∞  n=0

−ice 2De (ie )n+1

n 

nC

m=0,m=even

m

(m − 1)!! m+1 2

m

2 2 he



−∂ ∂ fe

 n−m 2

 √  erfc x0 fe , even n-m, √ 2 fe

(2.152) 1 . Deriving Q (t) under same assumptions with h i = (σ 2 + Di t) and f i = 4(σ 2 +D g i t) only gives Q g (t) = 1 − Q e (t) as expected. Since k is an integration variable that takes values from −∞ to ∞, a proper limit for ie > k is e → ∞. This limit c ce ) → ∞ could mean that either cg >> ce , ce >> cg or ce , cg >> 0. ( √ g + 2D e D g De

In the limits, the summation in Eq. 2.152 reduces to the following results: ⎧ 1, ⎪ ⎪   ⎪ ⎪ ⎨ x0 √ Q e (t) ≈ 1 − erfc 2 σ 2 +De t ,  ⎪ ⎪ ⎪ ce Dg ⎪ √ ⎩1 − erfc √ x20 ce Dg +cg

De D g

4

σ +De t

cg >> ce , ce >> cg .  , cg , ce >> 0.

(2.153)

The first case means that the backward excitation rate is much larger than the deexcitation rate, hence the excited-state population is a constant. While the second case shows that in the limit ce >> cg ground-state parameters do not appear in the excited-state population profile Q e (t) of the electronic relaxation model. This is the validity regime for the effective single-state descriptions as used in Refs. [10, 144] which become valid. In this limit, our model exactly reproduces result derived in Ref. [10] for the pinhole sink model of electronic relaxation.

110

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

Fig. 2.38 a The plot of ground-state recovery and excited-state decay (Eqs. 2.150, 2.151) versus time. The curve Q e (t) + Q g (t) = 1 illustrates that the coupled two-state system is a closed system. √ The parameters are (De , Dg ) = (2, 2) nm2 /ps, (ce , cg ) = (2, 3) nm/ps, x0 = 3 nm, σ = ( 40)−1 nm, b The analytical result of Q g (t) (Eq. 2.151) verified against the numerical solution for three √ instances with following parameters: (ce , cg , x0 , σ ) = (2, 2.5, 3, ( 40)−1 ). The three curves represent (De , Dg ) = (0.5, 0.5), (0.1, 1), and (1, 0.1) in the descending order of Q g (t) at t = 100 ps. Reprinted from Some exact time-domain results related to reversible reaction-diffusion systems, 539, R. Saravanan, A. Chakraborty, 110955, Copyright (2020), with permission from Elsevier

2.3.3.1

Results: The Transient Theory for Reversible Chemical Reactions

In reversible chemical reactions, at an arbitrary time, there can be population in both the reactant and product states. So using the result given by Eq. 2.142, a transient analysis for reversible chemical reactions is attempted. It is convenient to use subscripts r and p for denoting the reactants and products instead of the subscripts e and g. Although the population only in the excited-state case also can describe Le Chatelier’s principle on the effect of concentration. However, here we discuss only the results of Eq. 2.142 for studying reversible reactions. Effect of Non-adiabatic Couplings It is straightforward to see analytically that the condition for dynamic equilibrium is cr = c p , Dr = D p , xr = x p , σr = σ p . In order to maintain the dynamic equilibrium, equality should hold for every parameter. An imbalance created in any one parameter would disturb the dynamic equilibrium, then the equilibrium will be shifted to a particular direction until a chemical equilibrium is achieved. The equilibrium concentration of the reactants and products is observed (Fig. 2.39) to be fixed by the ratio of cr and c p , Q p (t) cr = = K eq , (2.154) lim t→∞ Q r (t) cp which is in accordance with the definition of the equilibrium constant in reversible chemical reactions. When other parameters are equal, the ratio K eq determines the direction of the reaction. If forward rate is greater than backward rate, K eq > 1, the

2.3 Exact Dynamics of Coupled Problems

111

Fig. 2.39 Concentration of products when K eq > 1, K eq = ccrp = 1, K eq < 1. In all the curves, the gray line corresponds to the smaller value of constants cr , c p and dashed line corresponds to the greater one. The value of

Q p (t=1010 ) Q r (t=1010 )

is observed to be ≈ 2.49998, 1, 0.66671 for cases K eq = 2.5, 1,

respectively, verifying the relation in Eq. 2.154. The parameters are (D p , Dr , x p , xr , σ p , σr ) = (2, 2, 4, 4, 0, 0). Reprinted from Some exact time-domain results related to reversible reactiondiffusion systems, 539, R. Saravanan, A. Chakraborty, 110955, Copyright (2020), with permission from Elsevier 2 3,

Fig. 2.40 The population probability versus time graph showing the effect of diffusivity in reversible reaction systems. The parameters are (x p , xr , σ p , σr , c p , cr ) = (4, 4, 0, 0, 1, 2), a D p = 1, Dr = 2, b D p = 2, Dr = 1. Reprinted from Some exact time-domain results related to reversible reaction-diffusion systems, 539, R. Saravanan, A. Chakraborty, 110955, Copyright (2020), with permission from Elsevier

concentration of products increases, and otherwise the product decreases to form reactants (Fig. 2.39). Effect of Diffusivity The temperature and the viscosity of the condensed phase enter in the model through the following relation:

112

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

Fig. 2.41 The effect of the distance between initial position-to-reactive point for the species. The parameters are (D p , Dr , σ p , σr , c p , cr )= (2, 2, 0, 0, 1, 2), a x p = 4, xr = 8, b x p = 8, xr = 4. Reprinted from Some exact time-domain results related to reversible reaction-diffusion systems, 539, R. Saravanan, A. Chakraborty, 110955, Copyright (2020), with permission from Elsevier

D=

kB T . ζ

In order to study the effect of the temperature imbalance or the effect of viscosity in the reversible reactions, let us consider an imbalance in the rate, i.e., c p = cr . This c p = cr condition avoids symmetry between the two cases D p > Dr and D p < Dr . The convention that the case cr > c p is the reaction proceeding in forward direction and is vice versa is followed throughout the discussions. So when cr > c p , the reactant reduces and starts to form the products in a closed manner (Fig. 2.40). If the diffusivity in the product molecules is lesser, the newly formed product molecules spend a lot of time near their reactive configurations. This would lead to back-reaction of products decreasing the overall product conversions (Fig. 2.40a). Whereas if the temperature of the product is increased, the diffusive motion is faster and the product immediately moves away from the reactive configuration. This reduces the probability of back-reaction leading to more interconversion as can be seen from Fig. 2.40b. The effect of imbalance in diffusivity is important in studying electron transfer kinetics [53, 156]. The reactants and products are in the same solvent, so the fact that for less viscous solutions, the rate of the reactions are faster can be deduced by plotting Q i (t)’s for different diffusivity. Effect of the Critical-to-Mean Configuration Distance in the Molecule The distance from the mean configuration of the species to the reactive configuration is given by xi ’s. This parameter (xi ) characterize how closer the molecules are to the reactive configuration. If the products are placed close to the reactive configuration, back-reaction happens for small time (Fig. 2.41a). Until this time, the rate of conversion (cr > c p ) does not influence the direction of the reaction, rather the molecular configuration of species does influence the direction. However, in a longer time, the direction of the reaction is decided by the rate of conversions as shown in

2.3 Exact Dynamics of Coupled Problems

113

Fig. 2.42 The effect of wideness of the distribution on the population probabilities. The parameters are (D p , Dr , x p , xr , c p , cr ) = (2, 2, 4, 4, 1, 2), a σ p = 4, σr = 0, b σ p = 0, σr = 4. Reprinted from Some exact time-domain results related to reversible reaction-diffusion systems, 539, R. Saravanan, A. Chakraborty, 110955, Copyright (2020), with permission from Elsevier

Fig. 2.41a. And if the products are placed farther from the reactive configuration, the back-reaction is reduced as can be observed from Fig. 2.41b. Effect of the Deviation from the Mean Configuration The deviation from the mean configuration of the molecules at t = 0 is characterized by the parameter σ . The impact of σ can vary for different potential surfaces. For a flat potential surface, the equilibrium distribution is expected to be a constant given by the Boltzmann distribution. The initial parameter σ adds to the process of achieving the equilibrium distribution. When the distribution is wide, the probability of finding some species near the reactive configuration is possible even for initial smaller times. Due to this advantage, for more σi ’s, the reaction attains chemical equilibrium in short span of time. When the decaying state has more σ , i.e., σr > σ p , the decay of Q r (t) starts near equilibrium (Fig. 2.42b). Hence, in a little time, the system achieves chemical equilibrium. On the other hand, as the dynamic equilibrium was already shifted (by choosing cr > c p ) leading to r → p conversions, but again by shifting the equilibrium tendency to the p-state (σ p > σr ) together drives the system far from equilibrium for smaller times (Fig. 2.42a). However, the system achieves the same equilibrium concentration as in the case of σr > σ p , once the rate factor dominates and influences the direction of the reaction.

2.3.3.2

Results: The Transient Theory of Electronic Relaxation Process

The model given by Eqs. (2.132), (2.133) along with the initial condition Eq. (2.143) can describe electronic relaxation process. This section gives the effect of system and molecular parameters on the electronic relaxation processes using the derived population probabilities (Eqs. 2.150 and 2.151).

114

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

Fig. 2.43 a The graph illustrating the effect of viscosity on the excited-state population. The parameters are (x0 , De , Dg , cg , ce , σ ) = (4, D, D, 2, 2, 0), b the parameters are (x0 , De , cg , ce , σ ) = (4, 1, 2, 2, 0). The curve Q e (t) + Q g (t) = 1 shown when the diffusivity difference is 1, i.e., Dg − De = 1. Reprinted from Some exact time-domain results related to reversible reaction-diffusion systems, 539, R. Saravanan, A. Chakraborty, 110955, Copyright (2020), with permission from Elsevier

Effect of Viscosity of the Solution For a molecule immersed in solution, the viscosity affects the molecule by slowing down its motion along the nuclear coordinates. The solvent parameter appears in this model through the diffusivity “D” as given by D=

kB T , ζ

where ζ is the viscosity and T is the temperature of the system. As the excited molecules and unexcited molecules are in thermal equilibrium with the solution, in this context, De = Dg . The result for this case (Fig. 2.43) shows that in less viscous solutions, the excited-state decay would be faster. And the effect of diffusivity on the decay is observed to be lighter in the two-state model than as in the effective one-state representations [10, 136]. It is experimentally seen that the reactive motion in the excited state is a little bit slower than that of the ground state, and in such cases the Dg can be taken to be slightly greater and the results are shown in Fig. 2.43b. As the difference in Dg − De increases, we see that the excited state has a relatively faster decay. The result predicts more product (g) formation when the difference in the diffusivity, i.e., Dg − De increases. Effect of the Excitation Wavelength When the effect of potential surfaces is considered, a higher excitation wavelength will place the initial distribution near to the potential minimum. Hence, higher the excitation wavelength means the lower is the value of x0 . As the qualitative behavior will not vary over the shape of the potential, the following can be deduced. The result shows that higher the excitation wavelength, faster is the ground-state recovery (Fig. 2.44). The rate calculated using effective single-state representation is greater

2.3 Exact Dynamics of Coupled Problems

115

Fig. 2.44 The graph illustrating the effect of excitation wavelength on the excited-state decay. The parameters are (De , Dg , cg , ce , σ ) = (1, 1, 2, 2, 0). Reprinted from Some exact time-domain results related to reversible reaction-diffusion systems, 539, R. Saravanan, A. Chakraborty, 110955, Copyright (2020), with permission from Elsevier

Fig. 2.45 The graph illustrates the effect of bandwidth of the source incorporated through width (σ ) of the initial distribution. The parameters are (De , Dg , cg , ce , x0 ) = (1, 1, 2, 2, 4). Reprinted from Some exact time-domain results related to reversible reaction-diffusion systems, 539, R. Saravanan, A. Chakraborty, 110955, Copyright (2020), with permission from Elsevier

than when the effect of the ground state is considered [10, 137]. Effect of Bandwidth of the Source If the excitation light has several wavelength components, i.e., the light has a finite bandwidth, then each component would excite the molecule into different configurations. Hence, the width in the initial distribution will be larger if the exciting light is not of a monochromatic character. For a finite bandwidth, if the excited species follow Gaussian distribution, the following results can be deduced. Figure 2.45 shows that when the bandwidth of the source (σ ) is higher, the spectroscopic equilibrium is achieved in a very shorter time scale (Fig. 2.45). This is because the light of high bandwidth prepares an initial distribution as similar to an equilibrium distribution. Effect of Coupling in the Electronic Relaxation The effect of coupling strength between the potential surfaces conveys results contrary to that of single-state representation. Unlike in the result of single-state repre-

116

2 Mathematical Methods for Solving Multi-state Smoluchowski Equations

Fig. 2.46 a The effect of coupling strength on the relaxation process is shown graphically. The parameters are (σ, ce , cg , De , Dg , x0 ) = (0, k0 , k0 , 0.5, 0.5, 4). When both the coupling strengths ce , cg are increased to a larger number, the concentration profiles will remain unchanged as long as the ratio of ccge is same (Eq. 2.154). b The effect of imbalance in the forward and reverse rates on the ground-state recovery. The parameters are (σ, cg , De , Dg , x0 ) = (0, 1, 0.5, 0.5, 4). Reprinted from Some exact time-domain results related to reversible reaction-diffusion systems, 539, R. Saravanan, A. Chakraborty, 110955, Copyright (2020), with permission from Elsevier

sentation, there is a little effect of the coupling strength on the rate of relaxation as seen from Fig. 2.46a. It is because this model takes into account of the back-reaction as well. The reactant (e) has initially more concentration, so even when the rate of both transfers is equal, reactants start to decay favoring the population transfer in one direction (Fig. 2.46b). This effect is what is known as Le Chatelier’s principle on the effect of concentration in a reversible reaction. An initial imbalance in the concentration results in decreasing the reactants to oppose that imbalance. The results can be important in the context of other two-state reactive diffusive systems as well.

2.3.3.3

Conclusion

The time-dependent dynamics of a chemical reaction is usually studied by predicting a rate equation that fits the available reaction data and obtaining the rate constant. Then the effect of temperature, solvent parameters on the rate constant should be observed by repeating several such experiments. The ease of Smoluchowski approach is that a rate equation can be derived that relates concentration profiles to molecular parameters and system parameters. The so far Laplace-domain calculations have been successful in giving different types of first-order rate constants at different time regimes. On the other hand, a time-domain work would give exact concentration profiles without assuming the order of the reaction. The parameters like temperature, solvent are inbuilt in Eqs. (2.132), (2.133) but misses the effect of potential. The possible improvisation to the theory can be attempted by solving a case where the diabatic curves are parabolic/ |x|-potentials which are also open problems in time domain. This is a first attempt to give a time-domain solution

2.3 Exact Dynamics of Coupled Problems

117

when both states are considered explicitly. The results give realizations to concepts like Le Chatelier’s principle, dynamic equilibrium, etc. arising naturally from the statistical description. The results can be useful in several other reaction–diffusion systems as well [35, 53, 156]. All results and verifications are shown in the notebook 2_state_closed_supplementary.nb.

Chapter 3

Investigation of Wave Packet Dynamics Using the Presented Time-Domain Method

3.1 Prelude Investigating quantum scattering models is useful in understanding systems ranging from molecules to nanodevices. This chapter presents time-domain methods useful for solving quantum scattering models. It is known so far that no scattering models other than the delta function barrier have been solved analytically. The Sect. 3.2 introduces a new analytically solvable scattering model in the time-dependent framework. The result for this new ultrashort width model is derived and the statistical properties of the wave packet are calculated and discussed in detail. Section 3.4 discuss general properties of two-state systems using a constant coupling model. Depending on whether the couplings are real and complex, the multi-state systems are categorized as (1) systems that involve real coupling—they exchange population through oscillation (such as vibrational transitions that occurs inside a single potential surface), (2) systems involving complex couplings—they exchange population through decay (such as irreversible transitions that occur between a closed two state system). However, in order to study electronic transitions, a localized coupling model can be useful. A simple model for understanding the effect of a localized coupling is considered regarding both type 1 and type 2, and the results are presented.

3.2 Exact Time-Domain Solution of the Schrödinger Equation for a New Scattering Model Solving quantum scattering models is helpful in understanding systems ranging from nanodevices [175] to quantum chemistry [11, 159, 160]. The available analytical solutions are limited and there are many numerical techniques to solve such systems [11, 102]. For a simple rectangular barrier, the solution is available only in the Laplace/Energy domain [93] and so far no closed form solution in time domain are reported for scattering models other than the Dirac delta function. Here, we consider © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 R. Saravanan and A. Chakraborty, Solvable One-Dimensional Multi-State Models for Statistical and Quantum Mechanics, https://doi.org/10.1007/978-981-16-6654-4_3

119

120

3 Investigation of Wave Packet Dynamics Using the Presented …

a wave packet interacting with a thin rectangular barrier to derive its space–time propagator. The presented model is useful to form a collection scheme to represent an arbitrary potential system and thereby it renders an 1D system be accurately solvable in the time-independent framework [140]. The exact solutions in the Laplace domain was already derived for the same model where the height of the potential is varying over time [104]. But that work [104] involves Green’s function method that employ conventional boundary condition from quantum mechanics. This work employs a new boundary condition which has been found to give physical result [140] for this model. The work is presented in the following organization. The mathematical methodology to evaluate the time-dependent wave function ((x, t)) is presented. Then the explicit wave functions and the graphical results are presented. The future scopes and the advantages of the presented model is discussed in the conclusion part.

3.2.1 The Mathematical Methodology to Evaluate the Wave Packet Dynamics For a scattering scheme shown in Fig. 3.1, the Schrödinger equation that govern the time-dependent wavepacket is expressed as, −2 ∂ 2 ∂ , (x, t) + V (0)θ (|x| < δx)(0, t) = i 2m ∂ x 2 ∂t

(3.1)

where δx is the ultrashort width. The asymptotic energy regions are free, it is possible to derive modified Fourier kernels using the stationary solutions of the Eq. 3.1. The

Fig. 3.1 Schematic representation of the scattering by an ultrashort barrier

3.2 Exact Time-Domain Solution of the Schrödinger Equation …

121

stationary solution ψ(x) in the piece-wise regions (Fig. 3.1) are as follows: ⎧ ik1 x ⎪ + Be−ik1 x , x < −δx, ⎪ ⎨ Ae |x| < δx, ψ(x) = ψ(0),  ⎪ ⎪ E ik x ⎩Ce 1 , x > δx, where k12 = 2m . 2 we use the following boundary conditions to determine the scattering coefficients: ψ(x)|x=−δx = ψc (0),  and

∂ψ ∂x

δx −δx

ψ(x)|x=δx = ψc (0), 2m = − 2 (E − V (0))2δxψc (0). 

(3.2)

to obtain the following kernels:

ψ(x) =

⎧ ik1 x ⎪ + ⎪ ⎨e

im(E−V (0))2δxe−2ik1 δx −ik1 x e 2 k1 −im(E−V (0))2δx im(E−V (0))2δxe−ik1 δx −ik1 δx e + 2 k1 −im(E−V (0))2δx ⎪ ⎪ (0))2δxe−2ik1 δx ⎩eik1 x e−2ik1 δx + im(E−V 2 k1 −im(E−V (0))2δx



x < −δx, |x| < δx,

(3.3)

x > δx.

It is to note that the free particle eigenfunction is not normalizable because the particle being free is delocalized in space, i.e., x = ∞, a curse of being a momentum eigenfunction. In order to localize the particle, one can add further momentum components about the k1 value as its mean position, thus imposing the localization of the wave packet, letting us normalize the wave function. Here, we would choose all momentum components in the range (−∞, ∞) to localize the position to a single point, which would be directly useful in evaluating the space-time propagator for the Cauchy problem. This prompts us to solve the TDSE with a Dirac delta function initial condition (x, 0) = δ(x − a), which facilitates normalization for t > 0. The initial condition in the Fourier integral representation is given by (x, 0) =

1 2π



∞

−∞

eik1 (x−a) dk1 .

(3.4)

The time-dependent wave function can be obtained by propagating the initial wave packet using the modified Fourier kernels. The procedure for obtaining (x, t) can be obtained by the following equation [155, 159]: (x, t) =

1 2π



∞ −∞

k12 t

a(k1 ; x, t)dk1 .

In the above equation, a(k1 ; x, t) = e−i 2m F (k1 )[ is the sum of the following Fourier kernels:

∞

−∞

(3.5)

(x, 0)e−ik1 x dx], and F (k1 )

122

3 Investigation of Wave Packet Dynamics Using the Presented …

F1 (k1 ) = eik1 (x−θ(x−δx)2δx) θ (|x| − δx), F2 (k1 ) = e−ik1 δx θ (− |x| + δx), im −2ik1 δx 2 (E − V (0))2δxe F3 (k1 ) =  eik1 |x| θ(|x| −δx)+ik1 δxθ(δx−|x| ) . im k1 − 2 (E − V (0))2δx

(3.6)

1 x ≥ 0, . The wave function corre0 x < 0. sponding to the first kernel F1 (k1 ) is expressed as where θ (x) is a unit step function, θ (x) = 

1 2π

 (1) (x, t) =

∞

e−i

k12 2m

t+ik1 (x−θ(x−δx)2δx−a)

−∞

θ (|x| − δx)dk,

(3.7)

which upon simplification becomes   (1) (x, t) =

mi(x−θ (x−δx)2δx−a)2 m 2t θ (|x| − δx)e− . 2πit

(3.8)

The second part of the wave packet is expressed by 1  (x, t) = 2π (2)



∞

e−i

k12 2m

t−ik1 (a+δx))

−∞

θ (|x| < δx)dk,

(3.9)

and upon simplification it becomes  (2)

 (x, t) =

mi(a+δx)2 m θ (|x| < δx)e− 2t . 2πit

(3.10)

When we use the reflected and transmitted part of the wave packet as the Fourier kernel, one obtains the rest part of the wave packet  (3) (x, t) =

1 2π



∞ −∞

e−ik1 a−i

k12 2m

im (E t 2

k1 −

− V (0))2δxe−2ik1 δx im (E 2

− V (0))2δx

eik1 |x| θ(|x| −δx)+ik1 δxθ(δx−|x| ) dk.

(3.11)

Upon simplification the above equation becomes 

(3)

1 im2δx (x, t) = 2π 2



∞

−∞

2 k12 2m

k12 − V (0) e−i 2m t+ik(|x| θ(|x| −δx)+δxθ(δx−|x| −a−2δx)) dk. (k1 − k+ )(k1 − k− )

 (0) −i±i 1−8δx 2 mV 2

(3.12)

The poles of the scattering kernel are, k± = . The poles are related 2δx to the bound state energies of the ultrashort potential well [140]. Simplification of the above equation leads to the following:

3.2 Exact Time-Domain Solution of the Schrödinger Equation …

 (3) (x, t) =

  ∂ 2 it i − V (0) e 2m ( f1 (x,t)) I, ∂t  ∞ it 2 e− 2m (k1 − f1 (x,t)) dk. with I = −∞ (k1 − k+ )(k1 − k− )

123

1 im2δx 2π 2

(3.13)

m f 1 (x, t) = t (|x| θ (|x| − δx) + δxθ (δx − |x| ) − a − 2δx). For convenenience, the above equation can be written as

 I = lim

∞

σ →0 −∞

e−(σ + 2m )(k1 − f (x,t)) dk, (k1 − k+ )(k1 − k− ) 2

it

2

where f (x, t) = (σ 2 + it )−1 i (|x| θ (|x| − δx) + δxθ (δx − |x| ) − a − 2δx). The 2m solution of the complex integral gives the time-dependent wave function. The evaluation of the contour can be done using the Cauchy residue theorem. Let us take a transformation in the variables, u = k1 − f (x, t) such that the Re(u) lies above the Re k-axis in the Argand plane representation (Shown in Fig. 3.2). Writing the equation for the contour yields 2  a− f (x,t) −(σ 2 + it )(k1 − f (x,t)2 )  a −(σ 2 + it 2m 2m )(a+I m(u)) e e dk1 + d(I m(u)) k1 − k± −a− f (x,t) a− f (x,t) a + I m(u) + f (x, t) − k±

+

 −a −(σ 2 + it )u 2  −a+i I m( f (x,t)) −(σ 2 + it )(−a+I m(u))2  2m 2m e e du + d(I m(u)) = I u + f (x, t) − k± −a + I m(u) + f (x, t) − k± a −a

Fig. 3.2 Schematic representation of the contour on the Argand plane

(3.14)

124

3 Investigation of Wave Packet Dynamics Using the Presented …



I term is the sum of residues to be added while doing the transformation. In the u-representation, the poles are

u p± (x, t) =

 −i ± i 1 − 8δx 2 mV2(0) 2δx

− f (x, t)

The contour remains a rectangle with decreasing height in increasing time. (1) However, we see that the pole u p+ > − f (x, t)∀t. So, when the I m(u p+ ) < 0, the residue has to be added. (2) The pole u p+ < − f (x, t)∀t, and is always outside the contour. The residue need not be added for this pole. Hence,  I = θ (−I m(u p+ ))

(2πi) 2 it 2 e−(σ + 2m )u p+ (u p+ − u p− )

The terms II and IV cancel being equal in magnitude and opposite signs. Also in the limit of a → ∞, the Eq. 3.14 becomes the following: 

∞

e−(σ

−∞

2 + it 2m

)(k1 − f (x,t)2 )

k1 − k±

 dk1 =

∞ −∞

it

e−(σ + 2m )u du + θ(−I m(u p+ )) u + f (x, t) − k± 2 it 2 (2πi) e−(σ + 2m )u p+ (u p+ − u p− ) 2

2

(3.15)

Using partial fractions one can reduce the integral I to be a following expression:  I =

  ∞ it 2 it 2 1 (2πi) e− 2m u ± + θ(−I m(u p+ )) e−( 2m )u p+ (u p+ − u p− ) ± u + f (x, t) − k (u − u ) ± p+ p− −∞ 1

(3.16) Evaluating the definite integral in the Eq. (3.16) to obtain the exact analytical form for the integral I,   2 (2πi) 1 −(σ 2 + it 2m )u p+ + e (u p+ − u p− ) k+ − k−    it −sgn ((k± )) + erf (−i(f1 (x, t) − k± )) 2m

I = θ (−I m(u p+ )) 

±πie

(i f 1 (x,t)−ik± )2

±



it 2m

(3.17)

The total wave packet is the Green’s function when the initial condition is a Dirac delta function given by the following expression: (x, t) =  (1) (x, t) +  (2) (x, t) +  (3) (x, t)  mi(x−θ(x−δx)2δx−a)2 m m 2t G(x, a, t) = + θ(|x| − δx)e− θ(|x| < δx) 2πit 2πit    2 mi(a+δx) it 2 ∂ 1 im2δx (2πi) i − V (0) e 2m ( f1 (x,t)) θ(−I m(u p+ )) ×e− 2t + 2π 2 ∂t (u p+ − u p− ) 

(3.18)

3.2 Exact Time-Domain Solution of the Schrödinger Equation … e

2 − it 2m u p+

 +



1 k+ − k− 

−sgn ((k± )) + erf

125



±πie(i f1 (x,t)−ik± )

±

it (−i(f1 (x, t) − k± )) 2m

2 it 2m

 

(3.19)

3.2.2 Results and Discussions We calculate the dynamics of two forms of initial wave packets: (1) the frequently used Gaussian form, (2) the time evolution of eigen-state of a δ-function potential, i.e., symmetric exponential function. The section gives the analysis on the fate of wave packets when scattered by the ultrashort potential.

3.2.2.1

Gaussian Initial Wave packetcket

When the initial wave packet is given by a following Gaussian function with a constant momentum and a phase factor iν0 /,  (x, 0) =

1 2π σ 2

1/2

e−

(x+xi )2 4σ 2

+ik0 (x+xi )+i

ν0 

(3.20)

The time evolution of the Gaussian function can be found out to be  g (x, t) = θ (|x| − δx)

e i

e

σ2 σ 2 + it 2m

[k0 (−δx+xi )−

−

(x−θ(x−δx)2δx+xi )2 4(σ 2 + it 2m )

k02 t 2m

]

+

iν0 

1/4

σ2 2π(σ 2 + 

+ θ (δx − |x| )

it 2 2m )

i

e

1/4

σ2

2π(σ 2

+

σ2 σ 2 + it 2m

it 2 2m )

e

[k0 (x+xi )−

−

(−δx+xi )2 4(σ 2 + it 2m )

k02 t 2m

+

]

iν0 

(3.21)

  ∂ im2δx −σ 2 k 2 +iν0 / 2 it 2 0 i − V (0) e(σ + 2m )(κ(x,t)) e 2  ∂t   1 2 2 it ∓πie(−iκ(x,t)−ik± ) (σ + 2m ) k+ − k− ±     it 2 −sgn ((κ(x, t) + k± )) + erf (−i(κ(x, t) + k± )) σ + 2m 

+

σ2 2π 3

1/4

−1 2 with κ(x, t)=(σ 2 + it 2m ) [2k0 σ + i (|x| θ (|x| − δx) + δxθ (δx − |x| ) − a − 2δx)]. The result is compared to that of the Dirac delta potential, i.e., V (x) = V0 δ(x) [155]

126

3 Investigation of Wave Packet Dynamics Using the Presented …

Fig. 3.3 Comparison of probability distribution for a Dirac delta potential, i.e., V (x) = V1 δ(x) and the ultrashort potential for parameters: m = 9.109,  = 1.055, √ k0 = 3 nm−1 , t = 25 fs, σ = 2 nm, ν0 = 0 xi = 8 nm, δx = 0.5 nm, V0 = 10 J, and V1 is V1 = 2V0 δx = 10 J.nm

 D D P = 

1/4

σ2 2 2π(σ 2 + it 2m )

σ2 σ 2 + it 2m

i

e

k 2 t (x+xi )2 iν σ2 [k0 (x+xi )− 2m0 ] − + 0 2 + it ) σ 2 + it 4(σ 2m 2m

e

2 k 2 iν 1/4 − (|x| σ2 +x0 ) +i (k0 (|x| +x0 )− 2m0 t)+ 0 σ 2 + it 4 σ 2 + it 2m 2m e



 −iπ e

 imV0 − 2

σ 2 + it 2m π





mV 2 σ 2 + it −i f (x,t)+ 20 2m

      it mV0 mV0 σ2 + sgn (−if(x, t)) + 2 −if(x, t) + 2 − erf , 2m  



(3.22)

i(|x| +x0 ) with f (x, t) = σ V2 +0 σit + 2(σ 2 + it ) . The solution for the finite barrier (Eq. 3.21) is 2m 2m compared with the solution of the Dirac delta potential. The δ-potential will localize the wave packet in the incident region x < 0, whereas the same wave packet transmits in a significant amount in case for a finite potential barrier (shown in Fig. 3.3). The Fig. 3.4 shows the difference in the wave packet because of the wavelets produced due to different potential strengths. The fate of the Gaussian wave packet is shown in Fig. 3.6. The initially prepared wave packet spreads while moving towards the scatter region. After a time ts = xki m0 , the peak xi encounters the potential which is roughly 25 fs. Some part of the wave packet reflects and some part transmit through the barrier. After the scattering, we find that the scattered wavelets traveling away from the barrier. The probability that a wave packet resides in a region is b given by, P(a, b) = a |(x, t)| 2 dx. The denotations P(>), P(t = −∞ x |(x, t)| 2 dx. In the absence of an barrier, the mean < x >t is expected to increase linearly due to constant velocity of the wave packet given by v = km0 . The trajectory for the mean of the wave packet in presence of barrier 2

3.2 Exact Time-Domain Solution of the Schrödinger Equation …

127

Fig. 3.4 The probability distribution for increasing √ strength of the potential for parameters: m = 9.109,  = 1.055, k0 = 3 nm−1 , t = 25 fs, σ = 2 nm, ν0 = 0 xi = 8 nm, δx = 0.5 nm, V0 = {0, 10, 100}

128

3 Investigation of Wave Packet Dynamics Using the Presented …

Fig. 3.5 The plot of time-dependent probabilities for parameters: m = 9.109,  = 1.055, k0 = 3 nm−1 , √ σ = 2 nm, ν0 = 0 xi = 8 nm, δx = 0.5 nm, V0 = 1 J

Fig. 3.6 The probability distribution of a Gaussian initial wave packet at different times t ={5, 10, 25, 50} for parameters: m = 9.109,  = 1.055, √ k0 = 3 nm−1 , σ = 2 nm, ν0 = 0 xi = 8 nm, δx = 0.5 nm, V0 = 5

is given in Fig. 3.7.1. Before incident on the barrier, the mean moves in a constant velocity and gets deflected with different velocities for different potential strength ‘V0 δx’. Higher the potential strength, higher is the deflection from the trajectory like seen in optical refraction. The other parameters such as σ , k0 have no significant  effect on the peak value trajectory (not shown). The wave packet dispersion, σt = < x 2 >t − < x >2t , is also calculated using numerical integration and shown in Fig. 3.7.2. The dispersion initially remains constant, it grows up, and starts to fall till different time for different strength of the potential. The fall is due to the shrinking of the wave packet while interacting with the potential. Once again, when the wave packet bifurcates into reflected and transmitted portions, the dispersion again starts to increase linearly in time. For high repelling barriers, we see that the dispersion is increasing at a high rate. The time properties of the wave packet can be calculated using the following formula,  τ (a, b) = lim

t→∞ 0



t

dt a

b

|(x, t)| 2 dx.

3.2 Exact Time-Domain Solution of the Schrödinger Equation …

129

Fig. 3.7 (1) The trajectories of the peak of the wave √ packet for the following parameters unless specified: m = 9.109,  = 1.055, k0 = 3 nm−1 , σ = 2 nm, ν0 = 0 xi = 8 nm, δx = 0.5 nm, a V0 = 1 J, b V0 = 2 J, c V0 = 3 J, d V0 = 3 J, δx = 0.2 nm. (2) The dispersion of the wave packet with respect to time for the same parameters

The time of arrival, dwell time can be calculated by making (a, b) as (−∞, −δx), (−δx, δx).

3.2.2.2

Exponential Wave packet

When the initial wave packet was an eigenfunction of the delta-function potential (x, 0) =

√

c0 e−c0 |x| ,

(3.23)

The time evolution was found out to be 



im (x−θ(x−δx)2δx)2 it c −i(x−θ(x−δx)2δx) 2 e (x, t) = θ(|x|π−δx) c3/2 e 2t (−iπ e 2m 0 )

  it c − imδx 2 i ( mδx )2 it (c − i(x − θ(x − δx)2δx)) + θ(δx−|x| ) c3/2 e 2mt t 2m 0 t erfc (−iπ e ) 0 π 2m

 erfc 4  n=1



n 

 it imδx 2m (c0 − t )



j =n, j=1

 1 (cn −c j )

(3.24)

it 2 3/2 ∂ − V (0) e 2m s(x,t) + c π imδx i ∂t 2

−iπ e

(is(x,t)−ici )2 it 2m (−sgn ((cn − is(x, t))

+erf



  it 2m (−i(s(x, t) − cn ))

which shows that the ultrashort potential is not of delta form, from the time-dependent perspective.

130

3 Investigation of Wave Packet Dynamics Using the Presented …

3.2.3 Conclusion Exact time-domain solution for a wave packet interacting with a finite barrier is presented. Using the solution, the dynamics of a Gaussian wave packet and its statistical properties are presented. The method highly simplifies solving finite barrier potentials and in future will find useful application in developing schemes for solving one-dimensional [109, 140] and also multi-state models for studying molecular systems. The results are calculated using MATHEMATICA notebook ultratimedomain.nb.

3.3 Exact Wave Packet Dynamics of Gaussian Wave Packets in Two-Flat States Coupled at a Point We consider a system that exchanges population through real couplings. We attempt a time-domain solution for a simple model when the form of the couplings are a finite strength Dirac delta function [V12 (x) = V21 (x) = k0 δ(x)] and the diabatic states are flat potential curves [V2 (x) − V1 (x) = V0 ]. Owing to the translation symmetry in the potential, we placed the coupling in the origin to reduce the number of parameters involved. Initially, the wave packet is assumed to be in one electronic state which is peaked at x = −x0 . The form of the wave packet is assumed to be a Gaussian  1 (x, 0) =

1 2π σ 2

1/4

e−

(x+x0 )2 4σ 2

+ik1 (x+x0 )

(3.25)

where σ is the width of the initial wave packet and k1 adds momentum to the translation of the wave packet. Using the developed time-domain solution, we analyze the results.

3.3.1 Methodology: Kernel Method to Calculate Wave Packet Dynamics The Schrödinger equation for two flat potential states coupled through a finite strength Dirac delta function becomes ∂1 2 ∂ 2 , 1 + k0 δ(x)2 = i 2 2m ∂ x ∂t

(3.26)

2 ∂ 2 ∂2 . 2 + V0 2 + k0 δ(x)1 = i 2 2m ∂ x ∂t

(3.27)

− −

3.3 Exact Wave Packet Dynamics of Gaussian Wave Packets in Two-Flat …

131

We solve for the stationary state solutions for the Eqs. (3.26) and (3.27) using variable −i Et separable method, i (x, t) = φi (x)e  to yield the following: 2 ∂ 2 φ1 (x) + k0 δ(x)φ2 (x) = Eφ1 (x) 2m ∂ x 2

(3.28)

2 ∂ 2 φ2 (x) + V0 φ2 (x) + k0 δ(x)φ1 (x) = Eφ2 (x) 2m ∂ x 2

(3.29)

− −

The solution for the above equations for x = 0 under physical considerations is given by

ikx Ae + Be−ikx , x < 0,  φ1 (x) = E x > 0, where k = 2m , Ceikx , 2 and

φ2 (x) =



De−ik x , x < 0,

Eeik x ,

where k =

x > 0,



2m(E−V0 ) . 2

Now, we incorporate the effect of coupling at x = 0 through the following boundary conditions: lim [φ1 (0 + h) − φ1 (0 − h)] = 0, h→0

 lim

h→0

∂φ1 ∂x



0+h − 2mk0 φ2 (0)/

2

= 0,

0−h

lim [φ2 (0 + h) − φ2 (0 − h)] = 0, ,

h→0

 and lim

h→0

∂φ2 ∂x



0+h − 2mk0 φ1 (0)/

2

= 0,

0−h

which leads to, ⎧

m 2 k02 ⎨eikx + e−ikx , x < 0 2

4 2 k0

−kk  2−m φ1 (x) = 2 m k0 ⎩eikx + eikx , x > 0, −kk 4 −m 2 k 2 0

and φ2 (x) =

⎧ ⎨

2 (ik)m 2 k02 e−ik x , mk0 (−kk 4 −m 2 k02 ) 2 2

k0 ⎩ 2 (ik)m eik x , mk0 (−kk 4 −m 2 k02 )

x 0, where k =



2m(E−V0 ) . 2

(3.30)

132

3 Investigation of Wave Packet Dynamics Using the Presented …

As known earlier, the different parts of the wave packet can be obtained from different contributions of the Fourier kernel:  1 (x, t) =

σ2 2π 3

1/4 

∞

−∞

  2 2 2 e−i hk t/2m e−σ (k−k1 ) +ikx0 eikx +



m 2 k02

−kk 4 − m 2 k02

 eik|x|

(3.31)

The wave packet corresponding to the eikx can be called 1(1) (x, t) and is evaluated to be  2 k 2 t −(x+x0 )2 + 2iσ it (k1 (x+x0 )− 2m1 ) σ 1 4(σ 2 + it (σ + 2m ) 2m ) e 1(1) (x, t) = (2π )1/4 (σ 2 + it ) 2m The 1(1) (x, t) corresponds  to the initial wave packet and it remains conserved over ∞   time, i.e., −∞ 1(1) (x, t) 2 dx = 1. The effect on the wave packet due to the other state can be evaluated using the reflected and transmitted part of the kernel, 1(2) (x, t) =



σ2 2π 3

1/4 

∞

e−σ

2

(k−k1 )2 +ikx0



−∞

m 2 k02 −kk 4 − m 2 k02



e−i H1 t/ eik|x| dk.

After simplification it becomes 1(2) (x, t)

 =

2 1/4   −iσ 2 2k1 ) m 2 k02 −σ 2 k12 − (|x|4(σ+x20+it/2m) − 4 e e  2  i (|x| +x0 −iσ 2 2k1 )  ∞ −(σ 2 +it/2m) k− 2(σ 2 +it/2m) e dk, m2k2 −∞ kk + 4 0

σ2 2π 3

(3.32)

which simplifies to evaluation of the following integral given by  I =

∞ −∞

e

2  i (|x| +x0 −iσ 2 2k1 ) −(σ 2 +it/2m) k− 2(σ 2 +it/2m)

 k k2 −

2mV0 2

+

m 2 k02 4

dk.

Multiplying by a conjugate in order to move the square root to the numerator leads to,    2 2 m 2 k02 0 e−(σ +it/2m)[k−κ(x,t)] −  ∞ k k 2 − 2mV 2 4   dk I = 4 −∞ i=1 (k − ki ) i |x| +x0 −iσ 2 2k1 ) , where ki ’s are with κ(x, t) = (2(σ 2 +it/2m)

3.3 Exact Wave Packet Dynamics of Gaussian Wave Packets in Two-Flat …

133

Fig. 3.8 Schematic representation of the integrand in the complex plane made by k-space. The poles k3 and k4 are dynamic, and always remain mirror image about the Im(k) axis

 !  2 2 2   ! m k0 mV0 2 " mV0 =± ± + . 2 2 4

k±±

The indices 1,2,3,4 are −+, ++, −−, +− respectively. All parameters in the pole are real, hence the poles 1 and 2 necessarily lie on the real line, while the poles 3 and 4 lies in the complex plane depending on the values of V0 and k0 . The contour depicted in Fig. 3.8 can be expressed as, 

 I =

+

2  j=1

−1 −k1

−T





0

lim

 +

−2 −k2

1 −k1

 +

 ( j eiθ − k j ) ( j eiθ − k j )2 − 4  

−T

+ T

T +κ

+ T

 +

−T

−T +κ

2mV0 2

m2 k2



− 4 0 4

i=1 (k

2 −k2

2mV0 2

i=1 (( j e

 j →0 π



   k k2 −

T

m 2 k02 4

− iθ

2mV0 2

i=1 ((u

2mV0 2

i=1 (k

e−(σ

2 +it/2m)(k−κ(x,t))2

dk

− ki )

2 +it/2m)((

je

iθ −k

2 j )−κ(x,t))

− k j ) − ki )

 (u + κ) (u + κ)2 − 4

  (k + i(k)) k + i(k)2 − 4



e−(σ

−

−

m 2 k02 4



+ κ) − ki )

 j (ieiθ ) dθ e−(σ

2 +it/2m)u 2

du

m 2 k02 −(σ 2 +it/2m)(k+i(k)−κ(x,t))2 e 4

+ i(k) − ki ) d(k), (3.33)

where the second term goes to zero as the contour is around a point of radius that tends to zero and the fourth term goes to zero because the contributions are of equal and opposite sign. In the limit T → ∞ and  j → 0,

134

3 Investigation of Wave Packet Dynamics Using the Presented …    m2 k2 2 2 2mV k k 2 − 2 0 − 40 e−(σ +it/2m)(k−κ(x,t)) 4 I = −∞ i=1 (k−ki )    m2 k2 2 2 2mV0 2 −∞ (u+κ) (u+κ) − 2 − 40 e−(σ +it/2m)u 4 + ∞ i=1 ((u+κ)−ki )



∞

dk

(3.34)

du.

And the closed integral equals to,  I = 2πi

4 

μ(kl )

  kl kl2 −

− 4

2mV0 2

m 2 k02 4

 e−(σ

i =l,i=1 (kl

l=1

2

+it/2m)(kl −κ(x,t))2

(3.35)

− ki )

 # 2 kl ∈ − 2(σf2(x) , 0 . + it ) 2m #  2 kl ∈ 0, 2(σf2(x) . + it 2m ) #  2 f (x)2 kl ∈ . / − 2(σ 2 + it ) , 2(σf2(x) + it )

⎧ ⎪ −1 ⎪ ⎪ ⎨ μ(kl ) = 1 ⎪ ⎪ ⎪ ⎩0

2m

2m

Using Eqs. (3.34) and (3.35) in the Eq. (3.32) to give, ⎡ (2) 1 (x, t)

⎢ = B(x, t) ⎢ ⎣2πi  +



∞

−∞

σ2 2π 3

4 

μ(kl )

  kl kl2 −

2mV0 2

− 4

m 2 k02 4



e−(σ

i =l,i=1 (kl

l=1

2 +it/2m)(k

l −κ(x,t))

− ki )

  ⎤  m 2 k02 −(σ 2 +it/2m)u 2 0 − (u + κ) (u + κ)2 − 2mV e 2 4   ⎥ du ⎥ 4 ⎦ i=1 ((u + κ) − ki )

1/4

with B(x, t) = ground state is given by

2 2 − m2k2 (− 4 0 )e−σ k1 e

(|x| +x0 −iσ 2 2k1 )2 4(σ 2 +it/2m)

2

(3.36)

. The total wave packet in the

1 (x, t) = 11 (x, t) + 12 (x, t) k 2 t 2 −(x+x0 )2 + 2iσ it (k1 (x+x 0 )− 2m1 4(σ 2 + it (σ + ) ) 2m 2m

)

1 (x, t) = Ae +    ⎡ 2 k2 2 2 m 0 kl kl2 − 2mV − 4 0 e−(σ +it/2m)(kl −κ(x,t)) 4  2 ⎢ B(x, t) ⎢ μ(kl ) 4 ⎣2πi i =l,i=1 (kl − ki ) l=1  +

∞

−∞

  ⎤  2 2 m 2 k02 0 − (u + κ) (u + κ)2 − 2mV e−(σ +it/2m)u 2 4 ⎥ du ⎥ 4 ⎦ i=1 ((u + κ) − ki )

(3.37)

3.3 Exact Wave Packet Dynamics of Gaussian Wave Packets in Two-Flat …

with A =

1 (2π)1/4



σ (σ 2 + it 2m )

135

and the wave propagation in the other state can be obtained

by solving the following integral,  2 (x, t) =

σ2 2π 3

1/4 

∞

dk e−σ

2

(k −k1 )2 +ik x0

−∞

2 (ik)mk0

e−i H2 t/ eik |x| , (−kk 4 − m 2 k02 ) (3.38)

3.3.2 Asymptotic and Exact Solutions, Results The integrals in Eqs. (3.37) and (3.38) are not exactly solvable, so in this section, we give solutions for the asymptotic cases and the results are discussed. We also attempt a scheme at the end to evaluate a more general case where the fractional order poles of Eq. 3.38 is constructed using the summation of the simple poles. The asymptotic cases for which (x, t) can be easily solved are the following • When the energy of the wave packet is much greater than the separation between the potential curves i.e., E >> V0 (in a case where high energy states form a continuum), • When the separation between the potential curves are much greater than energy of the system i.e., E > V0 , then k can be safely decomposed 

as k ≈ k −

2mV0 . 2

The above considerations are true for cases when:

• E is comparable to V0 in all magnitudes, √ • for E >> V0 , and for the case V0 )  c2 +

⎧ ⎨ e

icx0 +

(3.40)

m 2 k02 4

c )2 (i(|x| +x0 )+2σ 2 k1 −it m 4(σ 2 + it 2m )

⎩     −sgn  κ − k± + erf i σ 2 +

it

2m (κ

 . − k± ) ,

    i(|x| +x0 ))+σ 2 2k1 −it mc ) m 2 k0 2 ( 2mV0 1 2 −c ± , c = − where κ (x, t)= , k = c + 4 ± 4 it 2  2 2(σ 2 + )

2m

and k0 2 = k02 [θV0 > ]. The functions θV0 >> /θV0 0 with k = 2m E . 1

k1 mVe +k1 mVg −ik1 k1 2

⎧ ⎨eik1 x − φg (x) = ⎩eik1 x −

k1 m (Vg −Ve ) e−ik1 x , k1 mVe +k1 mVg −ik1 k1 2 k1 m (Vg −Ve ) eik1 x , k1 mVe +k1 mVg −ik1 k1 2

2

x