Transitions in Molecular Systems

Hans J. Kupka

Transitions in Molecular Systems

Hans J. Kupka Transitions in Molecular Systems

Hans J. Kupka

Transitions in Molecular Systems

The Author Prof. Hans J. Kupka Düsseldorf, Germany [email protected] Cover Picture A representation of the multi-dimensional FC factor at different rotations between the normal coordinates.

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V

Contents Preface

1 1.1 1.1.1 1.1.2 1.1.3 1.1.4 1.2 1.3 1.4 1.4.1 1.4.2 1.4.3

IX

Introduction 1 The Adiabatic Description of Molecules 1 Preliminaries 1 The Born–Oppenheimer Approximation 3 The Crude Born–Oppenheimer Basis Set 6 Correction of the Crude Adiabatic Approximation 7 Normal Coordinates and Duschinsky Effect 9 The Vibrational Wavefunctions 13 The Diabatic Electronic Basis for Molecular Systems 14 Preliminaries 14 Conical Intersection Between the States B˜ 2B2/2A0 and A˜ 2A1/2A0 of H2Oþ 16 The Linear Model for Conical Intersection 18

2.2

Formal Decay Theory of Coupled Unstable States 21 The Time Evolution of an Excited State 21 Some Remarks About the Decay of a Discrete Molecular Metastable State 26 The Choice of the Zero-Order Basis Set 27

3 3.1 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.2.6

Description of Radiationless Processes in Statistical Large Molecules Evaluation of the Radiationless Transition Probability 31 The Generating Function for Intramolecular Distributions I1 and I2 The Generating Function G2(w1,w2,z1,z2) 36 Properties of dm1m2, n1n2, am1m2, n1n2, and bm1m2, n1n2 41 Case w1 ¼ w2 ¼ 0 42 Case w1 6¼ w2 6¼ 0 42 Symmetry Properties of I2 45 Case w ¼ 0 47

2 2.1 2.1.1

31 36

VI

Contents

3.3 3.4

Derivation of the Promoting Mode Factors Kg(t) and Ig(t) 48 Radiationless Decay Rates of Initially Selected Vibronic States in Polyatomic Molecules 52

4 4.1 4.1.1 4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.2.5 4.2.6 4.3 4.3.1 4.3.2 4.3.3 4.3.3.1 4.4 4.4.1 4.4.2 4.4.3 4.4.3.1 4.4.4 4.4.5 4.4.6 4.4.7 4.4.8 4.5 4.5.1 4.5.2 4.6

Calculational Methods for Intramolecular Distributions I1, I2, and IN 57 The One-Dimensional Distribution I1(0, n; a, b) 57 The Addition Theorem 60 The Distributions I1(m, n; a, b) 61 Derivation of I1(m, n; a, b) 61 The Addition Theorem for I1(m, n; a, b) 65 The Recurrence Formula 65 Case b ¼ 0 67 Case b 6¼ 0 68 Numerical Results 69 Calculation of the Multidimensional Distribution 71 Preliminary Consideration 71 Derivation of Recurrence Equations 75 The Calculation Procedure 78 Some Numerical Results 79 General Case of N-Coupled Modes 82 The Generating Function GN 82 Properties of dm,n, am,n, and bm,n 87 The Distribution and its Properties 89 Symmetry Property of IN 91 A Special Case 92 Concluding Remarks and Examples 93 Recurrence Relations 94 The Three-Dimensional Case 96 Some Numerical Results 97 Displaced Potential Surfaces 102 The Strong Coupling Limit 102 The Weak Coupling Limit 106 The Contribution of Medium Modes 107

5 5.1 5.2

The Nuclear Coordinate Dependence of Matrix Elements The q-Centroid Approximation 111 Determination of the q-Centroid 123

6 6.1 6.2 6.3 6.3.1 6.3.2

Time-Resolved Spectroscopy 129 Formal Consideration 129 Evaluation of the Radiative Decay Probability of a Prepared State The Sparse Intermediate Case 137 Preliminary Consideration 137 The Molecular Eigenstates 139

111

131

Contents

6.4 6.5 6.5.1 6.5.2 6.5.3 6.5.4 7 7.1 7.1.1 7.1.2 7.1.2.1 7.2 7.2.1 7.2.2 7.2.3 7.2.3.1 7.2.4 7.3 7.4 7.4.1 7.5 7.5.1 7.5.2 7.5.3 7.5.4 7.5.5 7.5.6 7.6 7.6.1 7.6.2 7.6.2.1 7.6.2.2

8 8.1 8.1.1 8.1.2 8.2

Radiative Decay in Internal Conversion by Introduction of Decay Rates for {y1} 142 Dephasing and Relaxation in Molecular Systems 145 Introduction 145 Interaction of a Large Molecule with a Light Pulse 146 Free Induction Decay of a Large Molecule 149 Photon Echoes from Large Molecules 151 Miscellaneous Applications 155 The Line Shape Function for Radiative Transitions 155 Derivation 155 Implementation of Theory and Results 160 Excited-State Geometry 169 On the Mechanism of Singlet–Triplet Interaction 171 Phosphorescence in Aromatic Molecules with Nonbonding Electrons 171 Radiative T1 (pp
)!S0 Transition 172 Nonradiative Triplet-to-Ground State Transition 178 Theory and Application 179 Remarks on the Intersystem Crossing in Aromatic Hydrocarbons Comment on the Temperature Dependence of Radiationless Transition 184 Effect of Deuteration on the Lifetimes of Electronic Excited States Partial Deuteration Experiment 186 Theory and Experiment of Singlet Excitation Energy Transfer in Mixed Molecular Crystals 191 Transport Phenomena in Doped Molecular Crystals 191 The System Pentacene in p-Terphenyl 191 Techniques 194 Nature of the Energy Transfer: Theory 198 Time Evolution of the Guest Excitations 201 The Decay of the Transient Grating Signal 208 Electronic Predissociation of the 2B2 State of H2Oþ 211 Evaluation of the Nonadiabatic Coupling Factor 211 The Basis State Functions 216 The Initial-State Wavefunction xi 216 The Final Vibrational Wavefunction xf : The Closed Coupled Equations 217 Multidimensional Franck–Condon Factor 225 Multidimensional Franck–Condon Factors and Duschinsky Mixing Effects 225 General Aspects 225 Derivation 228 Recursion Relations 238

183

186

VII

VIII

Contents

8.3 8.4 8.4.1 8.4.2 8.4.3 8.5

Some Numerical Results and Discussion 241 Implementation of Theory and Results 244 The Resonance Raman Process and Duschinsky Mixing Effect 244 Time-Delayed Two-Photon Processes: Duschinsky Mixing Effects 247 Results 249 The One-Dimensional Franck–Condon Factor (N ¼ 1) 255 Appendices

A.1 A.2

259

Appendix A: Some Identities Related to Green’s Function The Green’s Function Technique 261 Evaluation of the Diagonal Matrix Element of Gss 264

261

Appendix B: The Coefficients of the Recurrence Equation

267

Appendix C: The Coefficients of the Recurrence Equations Appendix D: Solution of a Class of Integrals

271

273

Appendix E: Quantization of the Radiation Field

277

Appendix F: The Molecular Eigenstates 281 Appendix G: The Effective Hamiltonian and Its Properties

285

H.1 H.2

Appendix H: The Mechanism of Nonradiative Energy Transfer Single-Step Resonance Energy Transfer 287 Phonon-Assisted Energy Transfer 289

I.1

Appendix I: Evaluation of the Coefficients bmn, cmn, and bm in the Recurrence Equations 8.28 and 8.29 293 Application 294

287

Appendix J: Evaluation of the Position Expectation Values of xsm(qs) 299 Appendix K: Vibronic Coupling Between Two Electronic States: The Pseudo-Jahn–Teller Effect 301 References Index

327

313

IX

Preface The analysis of electronic relaxation processes, especially of radiationless transitions in molecular systems, has rapidly evolved in the last few decades and today plays a central role in almost all investigations of molecular physics and spectroscopy. The development of lasers has significantly contributed to this evolution. The purpose of this book is to give a self-contained and unified presentation of this development, with applications to molecular and solid-state physics. It is primarily intended for graduate students in theoretical physics and chemistry, who are beginning their research careers, although it is hoped that any physicist and chemist working with lasers, molecular spectroscopy, and solid-state physics will also find it useful. The greatest possible emphasis has been placed on clarity, and to this end, presentation is often made in strict mathematical detail. I hope that the reader will thus be able to rederive many of the formulas presented without much difficulty. Some basic understanding of symmetry principles in solid state and molecular physics may be helpful for the reader. The book consists of eight chapters and several appendices. In Chapter 1, the different basis sets used to classify molecular eigenstates and to study molecular dynamics, including molecular vibrations, are discussed within the context of the Duschinsky mixing effect. This mixing caused by the normal coordinate rotation has been elucidated further in following chapters. In Chapter 2, the treatment of radiationless transition probability is presented on the basis of Green’s function formulation for the transition amplitude, in which the states of interest are selected by suitable projection operators. A discussion of the proper basis set for describing electronic relaxation processes in large molecules is given for each of the cases treated. Chapter 3 provides a detailed description of radiationless processes in a statistical large molecule embedded in an inert medium. In this chapter, we are for the first time able to express the vibrational overlap between the electronic states under consideration in terms of intramolecular distributions in the full harmonic approximation taking into account the effects of vibrational frequency distortion, potential surface displacement, and the Duschinsky rotation. Chapter 4 deals in greater detail with the symmetry properties, the evaluation and presentation of the intramolecular distributions for arbitrary vibrational degrees of freedom.

X

Preface

An important example of the utility and power of the aforementioned intramolecular distributions is presented in Chapter 5. This chapter, which is of a more advanced nature, is entirely devoted to the investigation of the nuclear dependence of the electronic matrix element for radiationless transitions. It leads the reader, employing a class of integrals found in Appendix D, to a fix-point theorem for determining the q-centroid at which the electronic matrix element is to be evaluated. It is not recommended that the reader uninterruptedly attempt to master all of these derivations that lead to the proofs of the fundamental theorems. Instead, this chapter or a part of it, may be bypassed on the first reading, proceeding to the less complex following chapters and referring back, as necessary. Chapter 6 deals with the time evolution of radiative decaying states of polyatomic molecules with special emphasis on radiative decay in internal conversion. The decay of a manifold of closely spaced coupled states is handled by the Green’s function formalism, where the matrix elements are displayed in an energy representation that involves either the Born–Oppenheimer or the molecular eigenstate basis set. The features of radiationless transitions in large, medium-sized, and small molecules are elucidated, deriving general expressions for the radiative decay times and for the fluorescence quantum yields. Chapter 7 introduces the reader to solutions of many selected problems in molecular physics. In particular, the following important problems are studied in detail: the fluorescence spectrum of p-terphenyl crystal, the vibrational fine structure of the spin-allowed absorption band of trans-[Co(CN)2(tn)2]Cl3H2O, and transport phenomena of electronic excitation in pentacene-doped molecular crystals. It is followed by an analysis of phosphorescence and radiationless transition in aromatic molecules with nonbonding electrons as well as predissociation of the 2B2 state of H2Oþ by nonadiabatic interaction via conical intersection. Finally, Chapter 8 deals with the evaluation of multidimensional Franck–Condon integrals. As an illustration of the complexity of the latter upon the normal mode rotation, a study of sequential two photon processes is presented. At the beginning of each chapter, there is a brief summary of what the reader will find in the particular chapter. These summaries provide a detailed survey of the subject matter covered in this book. No attempt was made to provide all-inclusive references. References are not prioritized and are presented as supplementary reading for students. Some people have made important contributions to this book at various stages of its development. In particular, I would like to mention here my scientific colleagues G. Olbrich, C. Kryschi, D. Gherban, A. Urushiyama, J. Degen, Th. Ledwig, and P.H. Cribb. In addition, I wish to express my deep appreciation to G. Moss for suggested improvements to text readability and to G. Pauli for preparing most of the graphics, which form an essential part of the presentation. December 2008 Düsseldorf, Germany

Hans J. Kupka

j1

1 Introduction

In this chapter we shall provide a brief overview of a number of different basis sets to classify molecular eigenstates and study molecular dynamics. The basic procedure is described in Section 1.1, where the solution of the Schr€ odinger equation for the molecular system is given by separating the electronic motion from the nuclear motion in the molecule. This procedure, called the adiabatic description, represents the basis set that most often describes the initially excited states in large molecules. Alternatively, Section 1.1.3 introduces the crude Born–Oppenheimer (BO) basis, and Section 1.1.4 gives a description of the Herzberg–Teller adiabatic approximation. Sections 1.2 and 1.3 are devoted to the vibrational wavefunctions and their normal coordinates as well as to the Duschinsky effect. Section 1.4 concludes the chapter with a mathematical analysis of two strongly coupled adiabatic states, one of the fundamental and difficult problems of physics. The analysis is performed by using a diabatic basis set, and as an application a formal and compact solution is derived for the predissociation of a triatomic molecule via a conical intersection. We assume that the reader is familiar with the basic notions of quantum theory. However, to make our study reasonably self-contained, we have included some of the derivations in the appendices.

1.1 The Adiabatic Description of Molecules 1.1.1 Preliminaries

In the treatment of electronic states in large molecules, one usually neglects the details concerning the rotation and translation motions and rather concentrates on the dynamics of the electronic and vibrational motions. The starting point for the description of these motions in a molecule consisting of electrons and K nuclei is the complete Hamiltonian H of the molecule. To write down the Hamiltonian, the origin of the molecular coordinate system is placed at the center of mass. It is assumed that the positions of the K nuclei will deviate only by small amounts from some reference

j 1 Introduction

2

configuration. The molecules with large amplitude motions, such as internal rotations, are therefore explicitly excluded. The nuclear inertia tensor is then approximated by the inertia tensor of the reference configuration and the axes of the internal coordinate system are directed along the principal axes of this reference inertia tensor. If now the center of mass motion is removed, the nuclear motion can be described by a vector of 3K
6 dimensions for a (nonlinear) system with K atoms. The latter are normally taken as linear combinations of mass-weighted vectors describing the displacements from the reference configuration [1–3]. With this approximation and, for the sake of simplicity, taking only the electrostatic Coulomb interaction, the vibronic Hamiltonian can be written as H ¼ Te ðrÞ þ TN ðqÞ þ Uðr; qÞ:

ð1:1Þ

Here the vector r ¼ (r1 ; r2 ; . . . ; rn ) where ri ¼ ðxi ; yi ; zi Þ denotes collectively all electronic coordinates and the coordinates of the nuclei are specified by q ¼ ðq1 ; q2 ; . . . ; qN Þ, where N ¼ 3K
6. In the following, we shall adopt the convention that the components of the vector q are labeled by Greek indices if they range from 1 to N, and the Latin ones denote the components of the electronic coordinates. The electronic kinetic energy operator Te ðrÞ and the nuclear kinetic operator TN ðqÞ are presented in a diagonal form: X

h2 
q2  Te ðrÞ ¼
ð1:2Þ 2m qri2 i and TN ðqÞ ¼


X

h2  m

2

! q2 ; qq2m

ð1:3Þ

where m is the mass of the electron and qm are mass-weighted (dimensioned) nuclear coordinates given by R ¼ R0 þ M
1=2 Aq;

where R and R0 are ð3K
6Þ-dimensional column vectors of the instantaneous and equilibrium Cartesian coordinates, respectively, associated with the nonzero frequency normal modes. M is the ð3K
6Þ  ð3K
6Þ mass-weighted matrix, A is the orthogonal transformation that diagonalizes the mass-weighted Cartesian force constant matrix, and q is the dimensioned normal coordinate vector. Uðr; qÞ in Equation 1.1 is the total (internal) potential energy and includes all the electron–electron, nucleus–nucleus, and electron–nucleus interactions. In spite of the approximation already made, the exact molecular vibronic eigenstates Yðr; qÞ in a stationary state satisfy the time-independent Schr€ odinger equation ½Te ðrÞ þ TN ðqÞ þ Uðr; qÞ Yðr; qÞ ¼ E Yðr; qÞ:

ð1:4Þ

Serious approximations become necessary when one tries to solve Equation 1.4. One of these solutions is the adiabatic separation, which will be outlined below. This

1.1 The Adiabatic Description of Molecules

outline will serve as a guide to the possible classifications of molecular states and as an aid to the solution of specific quantum mechanical problems. 1.1.2 The Born–Oppenheimer Approximation

The first step of the adiabatic description is the Born–Oppenheimer approximation, according to which the nuclear kinetic energy is neglected, and the nuclear configuration is fixed at the position R. The adiabatic approximation is based on the fact that typical electronic velocities are much greater than typical nuclear (ionic) velocities. (The significant electronic velocity is v ¼ 108 cm=s, whereas typical nuclear velocities are at most of order 105 cm=s.) One therefore assumes that, because the nuclei have much lower velocities than the electrons, at any moment the electrons will be in their ground state for that particular instantaneous nuclear configuration. Under circumstances where TN ðqÞ ¼ 0, and at particular arrangement of the ion cores, we can separate electronic and nuclear motions. This can be accomplished by selecting some basis set of electronic wavefunctions ja ðr; qÞ, which satisfy the partial Schr€odinger equation ½Te ðrÞ þ Uðr; qÞ ja ðr; qÞ ¼ Ea ðqÞ ja ðr; qÞ;

ð1:5Þ

where Ea ðqÞ corresponds to the electronic energy at this fixed nuclear configuration. The configuration q is chosen arbitrarily, but for the solution of Equation 1.5 it must be fixed. In other words, the electronic wavefunction ja ðr; qÞ depends on the electronic coordinate r and parametrically on the nuclear coordinates. For any value of q, the ja are assumed to be orthonormal and complete (i.e., span the subspace defined by the electronic coordinates r). They are also assumed to vary in a continuous manner with q. The total (molecular) wavefunction Yðr; qÞ can be expanded in terms of the electronic basis function [4, 5] X Yv ðr; qÞ ¼ jb ðr; qÞxbv ðqÞ; ð1:6Þ b

where the nuclear wavefunctions xbv ðqÞ are initially treated as coefficients in the series (1.6). These coefficients are selected such that Equation 1.4 is satisfied. We have to substitute Equation 1.6 for Yðr; qÞ in Equation 1.4. Remarking that ! ! q2 ðjb xbv Þ q2 jb qjb qxbv q2 xbv xbv þ 2 ð1:7Þ ¼ þ jb qq2m qqm qqm qq2m qq2m and
2  q2 ðjb xbv Þ q jb ; ¼ xbv qri2 qri2

j3

j 1 Introduction

4

we find according to Equation 1.6 that ( X X Eb ðqÞjb ðr; qÞxbv ðqÞ þ TN ðqÞjb ðr; qÞxbv ðqÞ H jb xbv ¼ b

b

) X

h2  qj ðr; qÞ qx ðqÞ bv b
2 þ jb ðr; qÞTN ðqÞxbv ðqÞ qqm qqm 2 m X ¼E jb ðr; qÞxbv ðqÞ: b

In deducing this result, we have used Equation 1.5 and the fact that the wavefunction jb is an eigenfunction of Equation 1.5. Multiplying from the left by ja and integrating over the electronic coordinates, we obtain the usual set of coupled equations for the xav [4, 5] (see also Ref. [6] with modifications given by McLachlan [7] and Kolos [8]): ½TN ðqÞ þ Ea ðqÞ þ hja jTN jja i
E xav ðqÞ  X X 2   þ ð
h =2Þhja q=qqm jb ir q=qqm xbv ðqÞ ¼ 0: hja jTN ðqÞjjb ir
2 b6¼a

m

ð1:8aÞ

The restriction b 6¼ a in Equation 1.8a is a consequence of the orthonormality of the jb ; hjb jja ir ¼ dab . Here and in Equation 1.8a, angle brackets indicate integration over the electronic coordinates only. To avoid confusion resulting from numerous subscripts, it is often convenient to adopt a matrix notation, writing Equation 1.8a as ½TN ðqÞ þ Ea ðqÞ þ hja jTN ðqÞjja i
E xav ¼

X b6¼a

Xab xbv ;

ð1:8bÞ

 where Xab ¼
hja ½TN ; jb ir and ½A; B ¼ AB
BA. The adiabatic approximation (or BO adiabatic approximation in the nomenclature of Ballhausen and Hansen) is obtained by neglecting the coupling term in Equation 1.8a (the expression in the curly brackets). The molecular wavefunction now reduces to the simple product Yav ðr; qÞ ¼ ja ðr; qÞxav ðqÞ

ð1:9Þ

and the corresponding equation for the nuclear function xav ðqÞ in this approximation has the form ½TN ðqÞ þ Ea ðqÞxav ðqÞ ¼ Eav xav ðqÞ;

ð1:10Þ

where Eav is the eigenvalue for the nth vibrational level in the ath electronic state. Thus, from Equations 1.5 and 1.10, we see that, in the BO approximation, the nuclei move in an effective potential Ea ðqÞ generated by the electron distribution, while the electron distribution is a function of the nuclear configuration q. Ea ðqÞ is designated as the adiabatic potential surface of ja . The additional diagonal term hja jTN ðqÞjja ir in Equation 1.8 is omitted in the BO approximation, as we have done in Equation 1.10. Alternatively, if this term (designated as the adiabatic correction to the potential energy surface) is taken into account, we speak of the Born–Huang approximation [5].

1.1 The Adiabatic Description of Molecules

From numerical calculations of the low-lying electronic states of H2 þ and H2 , it is known that this correction is invariably small [9, 10] and can usually be neglected. The approximate wavefunctions of the adiabatic approximation are characterized by the following off-diagonal matrix elements between different electronic states [11]: ðhYav jH jYav0 ir Þq ¼ Eav dvv0

ð1:11Þ

(i.e., the adiabatic basis set is diagonal within the same electronic configuration) and     hYav jHjYbv0 ir q ¼ xav hja jTN jjb ir xbv0 q  X   

h2 xav ja q=qqm jb r qxbv0 =qqm : ð1:12Þ m

q

In Equation 1.12, we have indicated convenient abbreviations for the two integrals: hj ji for the integral over electronic coordinates and ðj jÞ for the integral over nuclear coordinates. Equation 1.12 represents the so-called Born–Oppenheimer coupling, which promotes transitions between potential energy surfaces via the nuclear kinetic energy operator. If these terms in the basis defined by Equation 1.9 are small relative to the separation of vibronic states Eav
Ebv0 , the BO approximation will give a very good approximation and will lead to tremendous simplification. In the case of close lying vibronic states belonging to different electronic configurations Eav  Ebv0 , the adiabatic approximation can fail. The interaction of nuclear vibrations with the electronic motion in molecules gives rise to interesting effects that have been attributed to linear and quadratic terms in the nuclear displacements from the equilibrium configuration. Linear vibronic coupling terms lead to vibrational borrowing, an effect that appears most clearly with forbidden electronic transitions made allowed through the simultaneous excitation of certain asymmetric vibrations. The other physical situations associated with linear displacements along certain asymmetric normal coordinates lead to the Jahn–Teller [12–25] and the pseudo-Jahn–Teller effects (see Appendix K). The effect of quadratic nuclear displacement terms is manifested in the Renner effect [26]. Although the study of these effects is of considerable interest, their observation is limited to systems of high symmetry that have degenerate or nearly degenerate electronic states. Going back to expression (1.12) for the coupling term, we shall now elucidate the situation that occurs when the potential energy surfaces belonging to different electronic states cross. This is easily obtained on introducing the following expressions [27]:       ½Eb ðqÞ
Ea ðqÞ ja q=qqm jb ¼ ja qU=qqm jb ð1:13Þ and  E D   E   D       ½Eb ðqÞ
Eb ðqÞ ja q2 =qq2m jb ¼ ja q2 U=qq2m jb þ 2 ja ðqU=qqm Þq=qqm jb : ð1:14Þ

In the region where the two potential energy surfaces do not cross, Ea ðqÞ 6¼ Eb ðqÞ; Equation 1.13 may obviously be rewritten as       ja q=qqm jb ¼ ja qU=qqm jb =½Eb ðqÞ
Ea ðqÞ ð1:15Þ

j5

j 1 Introduction

6

and relation (1.15) is well behaved. At the surface intersections Ea ðqÞ ¼ Eb ðqÞ, relation (1.15) is not as such without further ado valid. To see this, we differentiate the general expression (1.13) with respect to qm and then evaluate the result at the surface intersection to yield      



ð1:16Þ ja q=qqm jb ¼ q ja qU=qqm jb =qqm = qEb =qqm
qEa =qqm ; where we have assumed, for simplicity, that the intersection surface results from the variation of a single coordinate qm and that ðqEb =qqm Þc 6¼ ðqEa =qqm Þc at the intersection point c. This means that hja jq=qqm jjb i is well behaved over the whole range of values of qm . Indeed, Equation 1.16 can be rederived directly from Equation 1.15 by applying l’Hospital’s rule. Expression (1.15) should likewise be well behaved (nonsingular) in the more general case of multidimensional surface intersections, where qm in Equation 1.16 denotes the coordinate normal to the intersection surface defined by Ea ðqÞ ¼ Eb ðqÞ. The property of hja jq2 =qq2m jjb i and its nonsingularity clearly follow in a completely analogous way. The behavior of hja jq=qqm jjb i and hja jq2 =qq2m jjb i has been examined in Ref. [28] for H2 þ as a function of the internuclear distance R. Both these quantities were shown to vary smoothly with R. Subsequently, Nitzan and Jortner [29] have used Equation 1.15 in the whole range of values of qm , including the region of the intersection of the adiabatic surfaces by assuming the principal value for ½Ea ðqÞ
Eb ðqÞ
1 at the intersection point. This leads to a finite but peaked value of (1.15) at the surface intersection. A representative example of a similar situation will be shown in Sections 1.6 and 7.6, where the nonadiabatic coupling (1.15) near the conical intersection between states 2 B2 and 2 A1 of H2 O þ is shown. 1.1.3 The Crude Born–Oppenheimer Basis Set

In this and the following sections, we will discuss ways of selecting the basis function ja by separating the nuclear and electronic motions in a manner different from that in the previous section. In the present approach, the electronic Hamiltonian is assumed to be Helec ¼ Te ðrÞ þ Uðr; q0 Þ þ DUðr; qÞ;

ð1:17Þ

where q0 is a reference configuration and DU ¼ Uðr; qÞ
Uðr; q0 Þ is taken as a perturbation. In what follows, we will first briefly discuss the crude approximation and then the improvement of the crude BO basis set by using the Herzberg–Teller approximation. In addition to its practical utility, the Herzberg–Teller approximation provides an instructive way of viewing the (improved) crude BO basis complementary to that of the adiabatic basis derived in Section 1.1, permitting a reconciliation between the apparently contradictory features of both the crude BO basis set and the BO adiabatic basis set. The situation we have in mind occurs in the case of widely separated electronic states, which when mixed with each other give rise to vibronically induced allowed electronic transitions [30, 31] (see, for example, the mixing of odd parity states with the even parity states of transition metal complexes).

1.1 The Adiabatic Description of Molecules

In the crude adiabatic (CA) approximation [1, 32–40], the electronic wavefunctions odinjCA a ðr; qÞ defined at a specific nuclear configuration q0 satisfy the following Schr€ ger equation: CA CA ½Te ðrÞ þ Uðr; q0 Þ jCA a ðr; q0 Þ ¼ Ea ja ðr; q0 Þ;

ð1:18Þ

where EaCA is the ath eigenvalue and q0 implies all the nuclear coordinate positions of the reference configuration. Since these wavefunctions form a complete set (which span the Longuet–Higgins space), the eigenstate of the total Hamiltonian Yv ðr; qÞ may be expanded (analogous to Equation 1.6) in terms of jCA a ðr; q0 Þ: X Yv ðr; qÞ ¼ jCA ð1:19Þ b ðr; q0 Þ xbv ðqÞ: b

As before, xbv ðqÞ are initially treated as expansion coefficients, which must be determined. Inserting Equation 1.19 in Equation 1.4 results in the usual infinite set of coupled equations for the xbv ðqÞ: 

CA TN ðqÞ þ EaCA þ jCA a ðr; q0 Þ jDU j ja ðr; q0 Þ
Et xav ðqÞ X ð1:20Þ CA jCA þ a ðr; q0 Þ jDU j jb ðr; q0 Þ xbv0 ðqÞ ¼ 0: b6¼a

The functions xav ðqÞ are therefore determined by the set of coupled equations (1.20).  CA are usually represented as power series DU The potential functions jCA j jj a b expansions in the normal coordinates qm around q0 , where q0 is usually chosen at the minimum of the ground state. Provided that  CA ¼0 ð1:21Þ ja jDU jjCA b for a 6¼ b, Equation 1.19 is simply written as a product CA CA YCA av ðr; qÞ ¼ ja ðr; q0 Þxav ðqÞ;

ð1:22Þ

where the coefficient xCA av is the eigenstate of the following equation:  CA CA CA CA TN ðqÞ þ EaCA þ jCA xav ðqÞ: ð1:23Þ xav ðqÞ ¼ Eav a jDU jjb  CA The diagonal matrix elements ja jDU jjCA are the effective potential energy a surface that governs nuclear motion. From Equations 1.10 and 1.23, it is evident BO that the vibrational wavefunction xCA av differs from the adiabatic wavefunction xav : As ðr; q Þ is complete in the electronic space, the CA basis is long as the basis set jCA 0 a perfectly adequate (independent of the choice of q0 ). The two matrix representations 1.8 and (1.20) are merely two different representations of the same operator. 1.1.4 Correction of the Crude Adiabatic Approximation

The electronic wavefunction in the crude adiabatic approximation is defined according to Equation 1.18 at a specific nuclear configuration q0 and therefore it does not

j7

j 1 Introduction

8

depend on the nuclear coordinates fqm g. To calculate corrections to this extreme case, we apply the Rayleigh–Schr€odinger (RS) perturbation calculation, taking DU as perturbation operator. This leads to X ja ðr; qÞ ¼ jCA jCA ð1:24aÞ a ðrÞ þ b ðrÞcba ðqÞ; b6¼a

where  CA   CA CA X jCA ðrÞjDU jjCA jb ðrÞjDU jjCA a ðrÞ c ðrÞ jc ðrÞjDU jja ðrÞ b     þ EaCA
EbCA EaCA
EbCA EaCA
EcCA c6¼a;b

cba ðqÞ ¼

ð1:25Þ

and CA Ea;b ¼ Ea;b ðq0 Þ:

The same procedure gives for the eigenvalues (in second order) Ea ðqÞ ¼ EaCA þ ðDUÞaa þ

X ðDUÞ ðDUÞ ab ba ; CA
E CA E a b b6¼a

ð1:26Þ

where  CA ðDUÞab ¼ jCA a ðrÞjDU jjb ðrÞ :

ð1:27Þ

Expansion of DU in the vicinity of q0 in terms of nuclear coordinates fqm g gives DU ¼

X
qUðqÞ m

qqm

qm þ

q0


 1 X q2 UðqÞ q m qn þ    : 2 m;n qqm qqn q

ð1:28Þ

0

After inserting (1.28) into (1.26), we have Ea ðqÞ ¼

EaCA

þ

X m

Umaa qm

( ) X Umab Unba 1X aa þ qm qn Um;n þ 2 2 m;n E CA
EbCA b6¼a a

ð1:29Þ

with * Umab

¼

 
+  qUðr; qÞ   CA jb ðrÞ   qqm q

 jCA a ðrÞ

ð1:30Þ

0

and quadratic terms in qm : ab Umn ¼

  2   q Uðr; qÞ CA j ðrÞ :  jCA ðrÞ a  qq qq  b m n

ð1:31Þ

In writing Equation 1.29, we have taken into account the linear terms from Equation 1.28 in second order and quadratic terms in q in first order. Correspondingly, the coefficients cba ðqÞ in (1.25) are expressed as (in second order)

1.2 Normal Coordinates and Duschinsky Effect

j9

" # ba X Umba X 1 Umn X Umbc Unca     qm qn : cba ðqÞ ¼ q þ þ CA m CA 2 EaCA
EbCA c6¼a EaCA
EbCA EaCA
EcCA m Ea
Eb m;n ð1:32Þ

The correction of the CA approximation performed above is known as “vibronic coupling” and the wavefunction (1.24a) is sometimes designated as the Herzberg– Teller approximation. In this approximation, the corrected molecular eigenfunction can be written as " # X CA CA Yav ðr;qÞ ¼ ja ðrÞþ jb ðrÞcba ðqÞ xBO ð1:33Þ av ðqÞ b6¼a

and is still (of product form) adiabatic. Ballhausen and Hansen [1] have introduced the term Herzberg–Teller adiabatic approximation to emphasize the adiabatic nature of Equation 1.33 [40]. An obvious generalization of Equation 1.24a results if we choose X ja ðr; qÞ ¼ jCA cba ðqÞ ð1:24bÞ b ðrÞ
b

for an adiabatic electronic wavefunction ja ðr; qÞ. Upon substituting Equation 1.24b into Equation 1.6, we obtain X X
ccb ðqÞxBO Yv ðr; qÞ ¼ jCA ð1:34Þ c ðrÞ bv ðqÞ; c

b

which can be compared with Equation 1.19 to yield the relation X
c cb ðqÞxBO xCA cv ðqÞ ¼ bv ðqÞ

ð1:35Þ

b

between the vibrational wavefunction in the CA approximation and the vibrational wavefunction in the BO approximation. The classic cases of the Herzberg–Teller mechanism relate to coupling between two electronic states of different symmetry. An important example of this case occurs when electric dipole transitions of one of the two states are forbidden (e.g., the Laporte-forbidden d–d and f–f transitions). In this case, the forbidden transition may acquire absorption intensity by Herzberg–Teller mixing with an allowed transition via a nontotally symmetric mode of appropriate symmetry (the irreducible representation of the active mode must be contained in the direct product of the irreducible representations for the two states coupled by the Herzberg–Teller mechanism). We shall illustrate our results in Chapter 7 by evaluating the vibronic induced d–d transitions in transition metal complexes.

1.2 Normal Coordinates and Duschinsky Effect

Let us now return to Equation 1.29 for the potential energy surface of the ath electronic state and reformulate it in a more suitable (canonical) form:

j 1 Introduction

10

Ea ðqÞ ¼ Ea ðq0 Þ þ

X m

lam qm þ

1X a 1 fmn qm qn ¼ Ea0 þ ðIa Þt q þ qt Fa q; 2 m;n 2

ð1:36Þ

with lam ¼ Umaa

and a aa fmn ¼ Umn þ2

X Umab Unba b6¼a

EaCA
EbCA

:

ð1:37Þ

In Equation 1.36, the boldface letters q and F are column vector and square matrix, respectively. The superscript t indicates matrix transposition. Apart from the linear terms in qm , the potential of the ath electronic state contains in the harmonic approximation pure and mixed quadratic terms. The linear terms lm 6¼ 0, especially for total symmetric vibrational modes m (see Equations 1.30 and 1.37) is closely related to the geometrical displacement associated with the electronic transition between the electronic ground state and the ath electronic state. The pure quadratic a force constant coefficients fmm describe the curvature of the potential energy surface of the ath electronic state along the axes of the nuclear coordinate system, whereas the mixed quadratic terms fmna ðm 6¼ nÞ are responsible for the mixing of vibrational coordinates (modes) upon electronic excitation (see later). Therefore, our first goal is to transform them from the expression of Ea ðqÞ. Since Fa is a real and symmetric square matrix, it can be diagonalized by the following transformation: qa ¼ Aa q þ ka ;

ð1:38Þ

where Aa is an orthogonal matrix that diagonalizes the mass-weighted force constant matrix F. Applying Equation 1.38 on the electronic ground state a ¼ 0 and noting that ðABÞt ¼ Bt At for any two matrices, we have E0 ðqÞ ¼E0 ðq0 Þ þ ðl0 Þt ðA0 Þ
1 q0
ðl0 Þt ðA0 Þ
1 k0 1 0
1 0 0 t 0 0
1 0 0 ½ðA Þ ðq
k Þ F ½ðA Þ ðq
k Þ 2 1 1 ¼E0 ðq0 Þ þ ðl0 Þt ðA0 Þ
1 q0 þ ðq0 Þt A0 F0 ðA0 Þ
1 q0
ðq0 Þt A0 F0 ðA0 Þ
1 k0 2 2 1 0 t 0 0 0
1 0 0 t 0
1 0 1 0 t 0 0 0
1 0
ðk Þ A F ðA Þ q
ðl Þ ðA Þ k þ ðk Þ A F ðA Þ k 2 2 1 ¼ E0 ðq0 Þ þ ððl0 Þt ðA0 Þ
1
ðk0 Þt A0 F0 ðA0 Þ
1 Þq0 þ ðq0 Þt A0 F0 ðA0 Þ
1 q0 2 1
ðl0 Þt ðA0 Þ
1 k0 þ ðk0 Þt A0 F0 ðA0 Þ
1 k0 : ð1:39Þ 2 þ

In deriving (1.39), we have made use of the relation A
1 ¼ At for A being orthogonal. (The inverse of the matrix is its transpose At A ¼ E.) The linear term in q in

1.2 Normal Coordinates and Duschinsky Effect

Equation 1.39 vanishes, if ðl0 Þt ðA0 Þ
1
ðk0 Þt A0 F0 ðA0 Þ
1 ¼ 0

and hence ðl0 Þt ðA0 Þ
1 k0 ¼ ðk0 Þt A0 F0 ðA0 Þ
1 k0 :

ð1:40Þ

Inserting (1.40) into (1.39) yields E0 ðqÞ ¼ E0 ðq0 Þ þ

1 0 t 0 0 1 0 t 0 0 ðq Þ L q
ðk Þ L k ; 2 2

ð1:41Þ

where L0 ¼ A0 F0 ðA0 Þ
1 ¼ diag ðl01 ; l02 ; . . . ; l0N Þ is composed of diagonal elements lm , which arises from the nonzero frequency normal modes. Thus, Equation 1.41 represents the potential surface of the ground electronic state in the diagonal (canonical) form in mass-weighted ground-state normal coordinates. The last term in Equation 1.41 is a constant and can be included in E0 ðq0 Þ. Thus, the transformation (1.38) that diagonalizes the potential energy for the nuclear motion is determined uniquely by the coefficients l0m and fmn0 of the respective electronic state. Equation 1.41 pertains to the normal coordinates in the ground electronic state; an analogous expression holds for any electronic state a, where again La ¼ Aa Fa ðAa Þ
1 ¼ diag ðla1 ; la2 ; . . . ; laN Þ and Aa is the transformation matrix to mass-weighted coordinates, defined by qa ¼ Aa q þ ka :

ð1:42Þ

It can be proved that Ea ¼ Ea0 þ

1 a t a a ðq Þ L q : 2

ð1:43Þ

Combining (1.42) for a 6¼ 0 and (1.38) leads to qa ¼ Aa ðA0 Þ
1 q0 þ ka
Aa ðA0 Þ
1 k0 ;

which we abbreviate to qa ¼ Wq0 þ k0a ;

ð1:44Þ

where W ¼ Aa ðA0 Þ
1 thus formed is known as the Duschinsky rotation matrix associated with the 0 ! a electronic transition and k0a is related to the geometrical displacement vector between these states. (To simplify notation, we shall henceforth drop the 0a superscript on k.) According to Equation 1.44, the normal coordinates of an excited electronic state qa relative to those of the ground electronic state q0 are rotated (rotation matrix W) and displaced by the vector k. This rotation is called the Duschinsky rotation or Duschinsky mixing effect [41–44] (of the vibrational modes among each other). This mixing effect is subject to symmetry rules of the molecular symmetry group. Since in the most common instances vibrational modes of the same symmetry are mixed with each other (Equations 1.29–1.31 and 1.37), the matrix W assumes the

j11

j 1 Introduction

12

quasi-diagonal form indicated below:     =====     ==== 0 ; W ¼   0 ===    ==  where the elements outside the shaded area are zero, since they correspond to modes aa 6¼ 0 if qm qn of different symmetry. Simple symmetry arguments show that Umn ab ba transform as the totally symmetric transformation and Um Un 6¼ 0 if qm qn transform as the direct product of ja and jb . Rotations of normal coordinates in an excited electronic state relative to the ground-state normal coordinate space can therefore be expected for such a molecule if it possesses at least two different modes transforming as the same irreducible representation [41]. Cross-terms, and hence rotations in aa term in Equation 1.29, whereas totally symmetric modes, are generated by the Umn rotations in the nontotally symmetric modes are generated by the terms in the summation over b 6¼ a. As we shall see, this feature plays a crucial role in the derivation of transition probabilities. Vibrational modes of the same symmetry species assigned to the same shading fields in the matrix W cannot be represented as single separable modes and are said to be mixed or nonseparable modes. A very thorough survey of the Duschinsky effect is given in Refs [45–57]. A 8  8 Duschinsky matrix W has been determined by quantum mechanical calculation of the potential energy surfaces to interpret the vibronic structure of the 1 Bu 1 Ag transition of trans,trans-1,3,5,7-octatetraene in alkane matrices at 4.2 K [45]. The mixed modes are of a1g symmetry. Supersonic jet excitation and single vibronic level dispersed fluorescence spectra of a- and b-methyl naphthalene (S1 state) presented in Ref. [46] reveal that mode mixing of the ground-state normal coordinates and energy redistribution appear to be active in the S1 state. The vibronic spectra and related phenomena such as fluorescence–absorption mirror symmetry breakdown are found in azulene and certain azaazulenes [47–49]. On the basis of an analysis of vibronic spectra and calculation of normal vibrations, a complete assignment of the vibrational frequencies of s-tetrazine-d0 and s-tetrazine-d8 in the 1B3u excited state is given in Ref. [50]. In this connection, the rotation matrix W calculated from the data on the intensities of the vibronic band is used to estimate the force field in the excited electronic state. In Ref. [51], a Duschinsky effect that results from two nontotally symmetric vibrations involved in the vibronic coupling in the S1 --S0 systems of benzonitrile and phenyl acetylene is reported. An ab initio calculation of multidimensional FC (MFC) factors used to analyze the vibronic spectrum of ethylene corresponding to the p–p excitation was presented in Ref. [52] taking into account 12 normal coordinates of ethylene among which 4 totally symmetric modes are mixed. ~ 2 B2 Þ and D2 O þ , Jia-Lin Recently, in a study of the photoelectron spectra of H2 O þ ðB Chang has calculated MFC integrals including the Duschinsky effect. He found that the photoelectron spectra were mainly composed of v2 progressions and combination bands of v1 and v2 vibrations [53]. The idea that the Duschinsky effect plays a crucial role in the identification of band structures is also confirmed by the vibrational ~ 1 B2 --X ~ 1 A1 transition in tropolone. This molecule possesses an assignment for the A

1.3 The Vibrational Wavefunctions

intramolecular hydrogen bond and the hydroxyl proton tunnels from one oxygen atom to the other. Spectroscopic studies indicate that tropolone exhibits a double minimum energy potential along the tunneling coordinate and tunneling doublings have been detected in many vibronic bands in the absorption and laser excitation spectra [54–56]. However, the band structure is exceedingly complex and some of the complexity in the vibrational bands around the 0–0 band is due to strong Duschinsky mixing involving the two lowest b1 modes v25 and v26 . The importance of determining the potential energy surfaces of molecular states is clear. The shape of these BO surfaces of the molecular states is intimately involved in the electronic transitions between vibronic states belonging to crossing BO surfaces. This was applied by Paluso et al. to the dynamics of electronic transfer in neutral mixed valence monoradicals, using the diabatic representation (as will be illustrated in Section 1.4) and by considering a significant Duschinsky effect [57]. Generally, a significant Duschinsky effect might be expected when the change in equilibrium geometry upon excitation has components of large and comparable magnitude along two or more totally symmetric (ground-state) coordinates. This point will be clarified in detail in Chapter 3 and also in our subsequent consideration, where the Duschinsky effect will play an important role.

1.3 The Vibrational Wavefunctions

The introduction of mass-weighted normal coordinates fqm g obtained by diagonalizing the mass-weighted force constant matrix F in the respective electronic states allows us to solve Equation 1.10 for the vibrational wavefunctions in these states in a very convenient manner. Indeed, writing the kinetic operator (1.3) in mass-weighted coordinates qm and taking the diagonalized form (1.43) of the potential energy surface Ea ðqa Þ, Equation 1.10 is written as "
 #
h2 X q2 1 X a a2
þ l q x ðqa Þ ¼ Eav xav ðqa Þ; a2 2 m m m av 2 m qqm

ð1:45Þ

where the superscript a denotes the respective electronic state to which xav is assigned. In our formulation of Equation 1.45, we have made use of the fact that the kinetic energy operator remains invariant under the transformation (1.44). The Hamiltonian of the nuclear motion in Equation 1.45 separates now into parts, each of which is represented by an individual harmonic oscillator Hm with
2 2
h q 1 2 Hm ¼
þ lam qam : 2 qqam2 2

ð1:46Þ

For local stable molecules, lam > 0 (m ¼ 1; 2; . . . ; N), and we can set lam ¼ va2 m , where vam is the vibrational frequency of the mth oscillator. The eigenfunctions and eigenvalues of (1.46) are known and given by

j13

j 1 Introduction

14

 1=2   1=2  Hm xanm bam qam ¼ Eanm xanm bam qam ;

ð1:47Þ

Eanm ¼
hvam ðnm þ 1=2Þ;

ð1:48Þ

where nm ¼ 0; 1; 2; . . .

and xanm



1=2 bam qam



bam pffiffiffi nm p2 nm ! 1=2

¼

!1=2




1=2  1 2 exp
bam qam Hnm bam qam : 2

ð1:49Þ

Here Hnm is the Hermite polynomial of the degree nm : Hn ðxÞ ¼

X ð
1Þk n! ð2xÞn
2k k!ðn
2kÞ! k¼0

and bam ¼ ðvam =
hÞ. The qam ’s are dimensioned mass-weighted normal coordinates and the transformation to dimensionless normal coordinates is accomplished by the corresponding frequency factors of the vibrational modes m in the electronic state 1=2 a, ba ¼ ðvam =
hÞ1=2 . The eigenfunction and eigenvalue of the total Hamiltonian P m m Hm are  1=2   1=2   1=2  xan1 n2 ...nN ðqa1 ; qa2 ; . . . ; qaN Þ ¼ xan1 bam qa1 xan2 bam qa2    xanN bam qaN

and Ean1 n2 ...nN ¼ Ean1 þ Ean2 þ    þ EanN ;

ð1:50Þ

where the vibrational quantum numbers n1 ; n2 ; . . . ; nN assume in mutual independence of one another the values nm ¼ 0; 1; 2; . . ..

1.4 The Diabatic Electronic Basis for Molecular Systems 1.4.1 Preliminaries

The Born–Oppenheimer adiabatic approximation derived in Section 1.1 is very useful in classifying molecular eigenstates and calculating molecular dynamics. As long as adiabatic potential energy surfaces remain well separated, it is generally a good approximation to consider the nuclear motion to be confined to one such surface. When two or more surfaces intersect or pass close to one another, it becomes necessary to consider more than one surface in the calculation of nonradiative transition probabilities. When more than one surface must be considered, the adiabatic function is not necessarily more advantageous than various possible linear combinations of these functions. In particular, if the combination is taken over a small number of electronic states presumed to be of interest for a particular problem,

1.4 The Diabatic Electronic Basis for Molecular Systems

they can be chosen by means of a proper transformation such that the nuclear derivative coupling terms (e.g., in Equation 1.12) vanish with respect to the new basis functions. The coupling for the new so-called diabatic basis occurs then as a potential operator. Consider an example in which only two adiabatic states (say j1 and j2 ) are strongly coupled and assume that the coupling involving the other states can be safely neglected. In this case, the nuclear derivative coupling term in Equation 1.12 is given by the components    gm ¼ j2 q=qqm j1 ð1:51Þ of a nuclear momentum vector gðqÞ ¼ ðrq Þ21. Here rq stands for the vectorial operator rq ¼ ðq=qq1 ; q=qq2 ; . . . ; q=qqN Þ. In this two-state approximation, it is convenient to consider as an alternative a diabatic basis set ðw1 ; w2 Þ defined by the condition [58–65]    ð1:52Þ w2 rq w1 ¼ 0: In the diabatic representation, the nuclear coupling is eliminated or drastically reduced. The new diabatic states ðw1 ; w2 Þ are thereby allowed to move along with the nuclei [7, 65]. They are not the fixed functions considered in Section 1.2 and referred to as crude adiabatic. Hence, they will be denoted by wi ðr; qÞ to emphasize the fact that rwi 6¼ 0. In this two-state approximation, it is always possible to transform the pair of adiabatic functions ðj1 ; j2 Þ by a q-dependent orthogonal transformation [7, 63, 65]: ! ! ! w1 j1 cos qðqÞ sin qðqÞ ¼ : ð1:53Þ j2 w2
sin qðqÞ cos qðqÞ Substituting Equation 1.53 into Equation 1.52 now leads to    gðqÞ ¼ rq qðqÞ þ w2 rq w2 :

ð1:54Þ

Note that Equation 1.54 is a vector equation. Thus, condition (1.52) implies that there should exist an angle q such that gðqÞ ¼ rq qðqÞ:

ð1:55Þ

As discussed in Ref. [63], this can only be the case if curl gðqÞ ¼ 0;

ð1:56Þ

which means that if qm and qn are two of the nuclear coordinates, we can have a solution only if q q gn
gm ¼ 0: qqm qqn

ð1:57Þ

As proved in Ref. [63], this follows trivially for a polyatomic molecule if one adopts as diabatic functions a set of q-independent functions for which rwi ¼ 0. This can be seen if we use Equation 1.53 to prove another very useful result [66, 67]:

j15

j 1 Introduction

16

X   rw i ¼ ðgðqÞ
rqÞw i þ ½cos q hja jrj1 i
sin q hja jrj2 ijja i; 1 2

ð1:58Þ

a

which follows from the fact that ðrq Þ21 ¼
ðrq Þ12 . Here ja are adiabatic eigenfunctions of Hel other than j1 and j2 . This is a most remarkable fact. It asserts that there can be no solution if rw1 6¼ 0. Then, one must be either ðg
rqÞ 6¼ 0 or hja jrjji i 6¼ 0 (i ¼ 1; 2), or both of them. In the first case, this is in striking contrast to the concept of a strictly diabatic basis (see Equation 1.55). In the second case, there is an interaction with the other (higher) electronic states. The possibility remains, however, that it might be possible to eliminate the largest part of the coupling through transformation, so that the remainder can be neglected and Equation 1.55 can be replaced by the less stringent condition: 

  w2 rq w1  0:

ð1:59Þ

A particularly nice discussion to this subject can be found in Refs [7, 60, 63, 68, 69]. Let us now return to Equation 1.55. The angle qðqÞ; which depends on N internal nuclear coordinates, can be obtained by a multidimensional integration of the coupling matrix element gm [62]. For example, in a two-dimensional configuration space ðqx ; qy Þ; one has ð qx ð qy qðqx ; qy Þ ¼ qðqx0 ; qy0 Þ þ gx ðx; qy0 Þdx þ gy ðqx0 ; yÞdy: ð1:60Þ qx0

qy0

In the case where g is an irrotational vector and q a single-valued function, the value of q should be independent of the integration path. We shall return in Chapter 7 to a more extensive discussion of applications of the approximation just described. Now we consider the coupling between two electronic states, the potential energy surfaces of which cross in one point. 1.4.2 Conical Intersection Between the States B~ 2B2/2A0 and A~ 2A1/2A0 of H2O þ

The method we have described for studying unimolecular decay of electronically excited molecules may be applied to a variety of problems. These include, for example, the study of properties of conical and Jahn–Teller intersections. Illustrative calculations are presented for the H2 O þ ion, whose dissociation mechanisms are controlled by a conical intersection between the states 2B2 and 2A1 ðC2v Þ, the potential energy surfaces of which have been calculated in Refs [70, 71]. A schematic view of these surfaces is given in Figure 1.1. The coordinates ðr; aÞ are defined as follows: coordinate r is the asymmetric stretch r ¼ R1
R2 , where R1 and R2 are the two OH bond lengths and a is the valence angle between R1 and R2 . When the two OH bond lengths are equal (r ¼ 0; C2v point group) and q ¼ 1=2ðR1 þ R2 Þ, the symmetric  ~ and B ~ belong to the 2A and 2B stretching coordinate is equal to 1.15 A, the states A 1 2 representations. Their potential energy surfaces cross at ac ¼ 71:6 [70] and the energy at the conical intersection is equal to 74.4385 hartree. The minimum of the

1.4 The Diabatic Electronic Basis for Molecular Systems

~ 2 A1 potential surfaces of H2 O þ in the adiabatic ~ 2 B2 and A Figure 1.1 The actual form of the B representation. The upper adiabatic potential energy surface E2 consists of parts of the surfaces of the states 2 B2 and 2 A1 . (After Ref. [35].)

potential function of the state 2 B2 lies at an absolute energy of
75.4435 hartree and the 2 B2 state equilibrium conformation has an H
O
H bond angle of 55.7 . When the antisymmetric stretching coordinate r differs from zero, that is, when the two OH bond lengths are unequal, both electronic states then belong to the 2 A0 representation of the Cs point group and therefore the corresponding potential energy surfaces repel each other. As a result, a region of strong nonadiabatic interaction is centered around ~ the apex of the double cone. It is important to note that the two interacting states A ~ and B are well separated in energy from the remaining states. If it did, one is dealing here with a two-state conical intersection problem (with the second term in Equation 1.58 having been omitted). The ab initio calculations at the SCF level of the nonadiabatic coupling matrix element ga ¼ qq=qa for different cross sections r ¼ constant have shown [70] that ga has a Lorentzian shape with a unique maximum centered at the crossing between the two surfaces (e.g., at ac ¼ 71:6 ). Along the direction r, the function gr exhibits again a unique maximum and a Lorentzian shape, although the two contributions grMO and grCI sometimes add and sometimes subtract. The resulting gr ¼ grMO þ grCI is therefore found to be positive at values of a smaller than ac and negative for a larger than ac , where ac denotes the value of the valence angle at the apex of the cone. The closer the cross section lies to the apex of the cone, the narrower the g-function becomes. In the case, where the cross section passes through the apex of the cone, the linear model (which will be encountered later) predicts that the g-function should become a Dirac delta function with an area close to the theoretical value of p=2 [70, 73–76]. Adiabatic surfaces are defined as the eigenvalues of the electronic Hamiltonian (see Equation 1.5). In the diabatic representation defined above, the potential energy

j17

j 1 Introduction

18

surfaces are defined by the diagonal elements H11 and H22 . They can cross freely and are coupled by an off-diagonal matrix element H12 . Adiabatic surfaces are related to diabatic matrix elements Hii by the equation

1 1 2 1=2 Ei ðqÞ ¼ ðH11 þ H22 Þ  ðH11
H22 Þ2 þ 4H12 ; 2 2

i ¼ 1; 2:

ð1:61Þ

This expression is the equation of a double cone with expansion of the Hij functions around the apex and retention of first-order terms only. The linear approximation will be discussed later in this section. The unitary transformation (1.53) (of the Hamiltonian matrix in the diabatic basis ðw1 ; w2 ÞÞ leading to Ei ðqÞ is determined by the angle 1 q ¼ arctanf2H12 =ðH11
H22 Þg; 2

ð1:62Þ

which depends on three internal nuclear coordinates q ¼ qðq; r; aÞ. Furthermore, one has from (1.55) gm ¼ qq=qqm ;    with gm ¼ j2 q=qqm j1 or equivalently (see Equation 1.60) ðr ða qðr; aÞ ¼ drgr ðr; aÞ þ da ga ðr0 ; aÞ r0

a0

ð1:63Þ

ð1:64Þ

if we restrict ourselves to an integration at fixed value of q (the symmetric stretching  coordinate). In the case of H2 O þ ; q ¼ 1:15 A , that is, the value of q at the apex of the cone. The integration in Equation 1.64 of the ab initio calculated gm matrix elements can be performed numerically. Before giving a more quantitative discussion, we first note that Equation 1.64 defines generally a multivalued function (i.e., qðqÞ is of modulus p). This behavior of q is a result of the singularities of the functions ga and gr at a ¼ 71:6 and r ¼ 0 (the apex of the cone). This problem is better understood within the framework of the linear model of the conical intersection, which will be dealt with later. Once the angle q is known as a function of the internal coordinates, it is not difficult to obtain the diabatic energies H11 and H22 and the coupling matrix element H12 by inverting the orthogonal transformation (1.53). A schematic view of the diabatic surfaces H11 and H22 is given in Figure 1.2. 1.4.3 The Linear Model for Conical Intersection

The linear model of a conical intersection [58, 66, 77–79] is obtained by neglecting terms of order higher than one in the expansion of the matrix elements Hij around the apex of the cone ðr ¼ 0; ac ¼ 71:6 Þ: H11
H22 ¼ Fa ða
ac Þ;

ð1:65Þ

1.4 The Diabatic Electronic Basis for Molecular Systems

~ states. The thin line marks the seam. ~ and A Figure 1.2 The diabatic representation of the B

H12 ¼ ð1=2Þ Fr r:

ð1:66Þ

Note that ac depends on the value of q ¼ 1=2ðR1 þ R2 Þ. The same applies to the quantities Fa and Fr . This leads to a particular simple model of conical intersections, the features of which are as follows. Equation 1.61 can be cast in the convenient form 1 qða; rÞ ¼ arctan ½Fr r=Fa ða
ac Þ: 2

ð1:67Þ

The coupling matrix elements ga and gr along cross sections parallel to the symmetry lowering a and symmetry conserving r axes derived from Equation 1.63 have the form
 qq
Fa =ð2Fr rÞ ga ðaÞ ¼ ¼ ð1:68Þ qa q;r 1 þ ðFa =Fr rÞ2 ða
ac Þ2 and
gr ðrÞ ¼

 qq Fr =½2Fa ða
ac Þ ¼ : qr q;a 1 þ ½Fr =Fa ða
ac Þ2 r 2

The parameters Fa and Fr can be estimated directly from the shape of the adiabatic potential energy surfaces. Fa can be read directly on a cross section along the axis a at r ¼ 0. The result is Fa ¼ 4:5  10
3 hartree/deg, compared to the value of 4:2  10
3 hartree/deg obtained from Refs [71, 72]. Fr can be determined from the shape of the adiabatic curves along the axis r, via Equation 1.66 and DE ¼
2H12 =sin q, where DE ¼ E2
E1 is the energy difference between the adiabatic curves. This leads to Fr ¼ 0:045 hartree/bohr [70], compared to a value of 0.054 hartree/bohr obtained from Refs [71, 72]. Hence, for any cross section obtained by varying only one nuclear degree of freedom a or r, the nonradiative coupling elements are Lorentzians, which is in accord with the ab initio prediction [70].

j19

j 1 Introduction

20

Equation 1.67 represents graphically in a coordinate system with the axes ða; rÞ a series of straight lines around the apex of the cone for the locus of constant q in conformity with the result of numerical integration of (1.64) [70]. In particular, the line of intersection between diabatic states corresponds to the locus H11 ¼ H22 or q ¼ p=4 þ kp=2. It thus follows that as q ! p=4, the intersection coincides precisely with axis r at a ¼ 71:6 . At a complete rotation around the apex of the cone, the angle q increases from 0 to p only. According to Equation 1.68, the closer the cross section lies to the apex of the cone, the sharper the Lorentzian. Therefore, Equation 1.68 correctly describes the nonradiative coupling matrix elements gm well in accordance with the numerical calculations of the g-function cited above.

j21

2 Formal Decay Theory of Coupled Unstable States

The special methods for solving the Schr€odinger equation for molecular systems, which were discussed in Chapter 1, are used in the study of the decay of metastable states. This includes such familiar examples as the decay by nonradiative transitions of excited molecular states. Prior to the principal discussion, a description of the time evolution of a single excited state of an isolated molecule based on the Green’s function technique is given in Section 2.1. This leads to an expression for the rate constant of nonradiative transitions with a matrix element of the transition operator T. Following the basis of this treatment, Section 2.2 addresses the question of the proper choice of the basis set for calculating transition probabilities in greater detail than that presented in Chapter 1.

2.1 The Time Evolution of an Excited State

We begin by considering an isolated molecule that is initially in a given discrete electronic state ys .1) Let Es be the energy of this state. The state ys interacts with a dense manifold of vibronic states belonging to a lower lying electronic state ylv0 by the presence of a perturbation that induces a transition from the original state. The state ylv0 is assumed to be of the same spin multiplicity as the original state ys (internal conversion) or of different spin multiplicity (intersystem crossing). No other electronic states ybv or different vibronic components of the same electronic state are close to ys in the energy region under study. If, for example, the prepared single state ys is excited by an electromagnetic field, we assume that only ys carries oscillator strength from the ground state j0i, whereas the states fyln0 g are optically inactive (Figure 2.1). When the manifold fylv0 g is of triplet species and ys is a singlet, spin selection rules are sufficient to fulfill this condition. On the other hand, if the manifold fylv0 g is singlet and the energy gap between ys and the lowest vibronic level of fylv0 g is sufficiently large, then the Franck–Condon factor for excitation of 1) Strictly speaking, truly isolated systems are not known in physics. They are extremely useful idealization.

j 2 Formal Decay Theory of Coupled Unstable States

22

|b> Vbs Vsl

|s>

µ0s

{|l>}

µ0l = 0

0 Figure 2.1 A schematic representation of the molecular states and relevant couplings. The zero-order molecular levels h0j, jsi, jbi, and fjlig are BO states for intrastate dynamics. They correspond, respectively, to the ground state h0j, the optically accessible doorway state jsi,

and the background manifold fjlig. jbi is a higher lying excited state separated from the state jsi. Arrows indicate dipole coupling via the interaction with radiation field. Wave lines represent intramolecular coupling.

the states that are quasi-degenerate with ys are small, so that hykv jmjys i
hykv jmjylv0 i, where m is the transition moment operator and ykv is the wavefunction of the ground state. In Section 6.4, we return to the general case for which the last inequality is relaxed. To make the discussion simple, we use Dirac notation and accordingly denote the state vector ys by jsi and the basis set ylv0 by fjlig, where jli stands for jlv0 i. We suppose that the state ys ¼ jsi of the unstable system described above to be represented by an approximate Hamiltonian H0 at time t ¼ 0: H0 jsi ¼ Es jsi:

ð2:1Þ

Our central problem is now the study of the dynamical behavior of the system, as determined by the exact Hamiltonian H ¼ H0 þ V;

ð2:2Þ

with V being the perturbation. The development of jsi in time has the explicit form2) yðtÞ ¼ expðiHtÞjsi;

ð2:3Þ

which may be formally integrated to yield ð 1 yðtÞ ¼ dEeiEt GðEÞys ð0Þ: 2pi c

ð2:4Þ

The function GðEÞ that appears inside the integrand is the Green’s function GðEÞ ¼ ðEH þ ieÞ1 ;

e ! 0;

ð2:5Þ

2) In this chapter, we use a system of units in which
h ¼ 1, except when the value of
h is important for our discussion.

2.1 The Time Evolution of an Excited State

and the contour c runs from þ 1 to 1 above the real E-axis, since the eigenvalues of H are real or they lie below the real axis (see Appendix A). To find the probability that the system remains in the initial state jsi (or in other words, the evolution at time t of the state jsi), we write the amplitude ð 1 Iss ðtÞ ¼ dEeiEt hsjGðEÞjsi: ð2:6Þ 2pi c The probability that the system is in the state jsi at time t is then Ps ðtÞ ¼ jIss ðtÞj2 :

ð2:7Þ

To simplify the calculation of Iss ðtÞ, we make use of the Feshbach projection formalism [80, 81] by introducing the operators P and Q satisfying the relations P þ Q ¼ 1;

QP ¼ PQ ¼ 0;

P2 ¼ P:

ð2:8Þ

Introducing the operator P ¼ jsihsj;

ð2:9Þ

which projects any vector on the isolated state function jsi, then the states jli are selected out by Q. To specify the matrix element Gss ¼ hsjGðEÞjsi corresponding to the state jsi, we must know the projection of the Green’s function PGP. The latter can be expressed (see Appendix A.1) in the form PGP ¼ ½EPH0 PPRP1 :

ð2:10Þ

The quantity R in Equation 2.10 is called the level shift operator or “self-energy” and is given by R ¼ V þ VQðEQHQÞ1 QV:

ð2:11Þ

To evaluate this operator, we rewrite Equation 2.10 in a modified form. To do this, we use an operator identity. Let A and B be two operators for which the respective reciprocals ðA þ BÞ1 and A1 are defined. Then,
 1 1 1 ¼ 1þB : AþB A AþB On taking A ¼ EPH0 P and B ¼ PRP, we find PGP ¼ ðEPH0 PÞ1 þ ðEPH0 PÞ1 PRPðEPH0 PPRPÞ1 :

Thus, we may write Equation 2.10 as hsjPGPjsi ¼

1 ; EEs hsjRjsi

where RðEÞ ¼ V þ VQ½ðEQH0 QÞ1 þ ðEQH0 QÞ1 QVðEQHQÞ1 QV:

ð2:12Þ

j23

j 2 Formal Decay Theory of Coupled Unstable States

24

This equation permits us to expand it in powers of V. Then, RðEÞ ¼ V þ VQðEQH0 QÞ1 QV þ higher order terms in V

and the matrix element of PRP becomes X

hsjRðEÞjsi ¼

hsjV jli

l

1 hljV jsi þ    : EEl

ð2:13Þ

Here use has been made of the relationships QH0 Q jli ¼ El jli and ðEQH0 QÞ1 jli ¼ ðEEl Þ1 jli. From Equation 2.13 it is clear that as E approaches some El , then one term in the sum on l will become very large. Using the identity 0 1 1 1 1 1 1 1 A ¼ lim @ þ þ  EEl 2 e ! 0 EEl þ ie EEl ie EEl þ ie EEl ie ¼ PP

1 1 e 1 ip limþ ¼ PP ipdðEEl Þ; 2 EEl EEl e ! 0 p ðEEl Þ þ e2

ð2:14Þ

Equation 2.13 can be written as [81] hsjRðEÞjsi ¼ DðEÞiCðEÞ=2;

ð2:15Þ

where DðEÞ, the level shift, is given by Equation 2.13, replacing there the energy denominator 1=ðEEl Þ by its principal part PPð1=ðEEl ÞÞ and X CðEs Þ ¼ 2p jhsjV jlij2 dðEs El Þ þ    l

¼ 2p

X

ð2:16Þ

jhsjTðEÞjlij2 dðEs El Þ;

l

where hsjTðEÞjli ¼ hsjV jli þ

X hsjV jcihc jV jli c

EEc

þ  ;

ð2:17Þ

with jc i being the intermediate states belonging to the space spanned by Q. The last relation follows by observing that the inclusion of higher order terms in the probability amplitude is achieved by replacing V by the transition operator TðEÞ ¼ V þ VGV ¼ V þ TG0 V ¼ V þ VG0 T. The first term in Equation 2.16 is the well-known Fermi Golden width. Inserting Equation 2.15 into Equation 2.12 yields Gss ðEÞ ¼

1 1 ¼
s þ iCðEÞ=2 ; EEs DðEÞ þ iCðEÞ=2 EE

ð2:18Þ


s ¼ Es þ DðEÞ. where we have absorbed the level shift DðEÞ in the term E It follows from Equation 2.18 that the integrand in Equation 2.6 has poles, so we need a well-defined prescription to avoid these singularities and give a meaning to the

2.1 The Time Evolution of an Excited State

integral. Some aspects of these questions will be explored in somewhat greater detail in later chapters of this book, where in addition to manifold fjlig, the state jsi interacts with a radiation field. For this it is important to know the analytical properties of the level shift operator RðEÞ in the complex E-plane. For the case of a single isolated state with a projection operator defined by Equation 2.9, the analytical properties of RðEÞ have been presented by Goldberger and Watson [81]. Returning finally to Equation 2.16, we use d functions to ensure energy conservation. Evaluation of the trace (sum) in Equation 2.16 evidently requires some care since the individual terms diverge. This is purely a mathematical problem and the first indication of the subtlety of our limiting procedure; we get a contribution to the width CðEs Þ as e ! 0, when the energy El of the set ðlÞ overlaps Es . Before contemplating the passage to the limit e ! 0, we must discuss the nature of the singularity of (2.16) near the energy Es ¼ El to give a precise instruction as to how this singularity is to be dealt with. First of all, the reader will note that the d function used in Equation 2.16 was adopted as a bookkeeping expedient and has meaning only in terms of distributions. However, it is really illogical to interpret Equation 2.16 in terms of distributions, whereas the width CðEs Þ is finite and continuous. The idea of a transition rate becomes somewhat absurd. On the other hand, the introduction of d function is convenient but not essential. To gain some insight into radiationless processes, Chock et al. [82] show that Lorentz distributions with a width directly related to the spectroscopic linewidth (resulting from the radiative and nonradiative decay width) are much more suitable to use. We here adopt this latter procedure, interpreting the density of state function rl as a row of Lorentz functions instead of a row ofXsharp equal d function peaks at the allowed final energies E ¼ El , dðEEl Þ. This is most easily accomplished by the definition of generrl ðEÞ ¼ l alized functions, according to which the d function is defined as a regular sequence of “good” functions [83], such as was derived in Equation 2.14 in terms of Lorentzian’s. With this tacit understanding, which will play an important role later, we shall interpret the limit in Equation 2.14 and the sum in Equation 2.16. The d functions are used further only in abbreviation. The introduction of a width associated with each state of fyl g may be seen in somewhat greater detail as follows. The states yl in Figure 2.1 are isoenergetic with ys and vibrationally hot, so that they would spontaneously emit infrared radiation leading to a molecule with energy too low to “cross back” to ys . Resulting from the time limitation on the duration of our experiment, we can question whether irreversibility behavior prevails for the time limit tmax . Our physical intuition tells us that if a relaxation for time tmax can be found, then the phenomena may be considered irreversible. In this case, the energy levels of states fyl g become continuous during portion of time that our experiment is being performed. Since the system is considered for a finite time tmax , all energy levels, except the ground state of the system (molecule and radiation field), have a vibrational relaxation width in energy DEl, as required by the uncertainty principle [84]: tl DEl ¼ ð
h=Cl ÞDEl 
h;

where Cl is directly related to the level width DEl .

j25

j 2 Formal Decay Theory of Coupled Unstable States

26

This is the case of an isolated molecule. Our discussion can readily be extended to include other dissipation channels due to collisions of the molecule with one another after some time (tcollis  104 s at 1 mm pressure) in the system (molecule and radiation) and interaction of the molecule with the wall of the vessel containing them, Hwall . Then the appropriate Hamiltonian is described by an “effective Hamiltonian” with a small non-Hermitian component. Thus, the energies are complex, E ¼ El iCl =2, with a negative imaginary part, Cl > 0, directly related to the spectroscopic linewidth. The lifetimes are then just tl ¼
h=Cl and the above equation has a clear physical sense. It is instructive to give each state an average lifetime due to the existence of these other dissipative channels. In practice this can be quite difficult to do and a more practical approach is to consider the quantity Cl as impirical parameter, to be measured directly by experiment (Chapter 7). 2.1.1 Some Remarks About the Decay of a Discrete Molecular Metastable State

The Green’s function method described in the previous section is based on the observation that the time evolution of the initially prepared state jsi is given by the Fourier transformation of the diagonal matrix element hsjGjsi of the Green’s operator for the system G ¼ ðEH þ ieÞ1 , and that the decay characteristic of this state is determined by the complex pole of this diagonal element: hsjGjsi ¼ ðEEs iCs =2Þ1 ;

ð2:19Þ

where Cs is the half width of energy distribution given by the Fermi Golden Rule: Cs ðEs Þ ¼ 2

X

jhsjTðEs Þjlij2

l

C ðEs El Þ2 þ C2

:

ð2:20Þ

According to our remarks about the d functions, we have here used Equation 2.16 to set 1 C ¼ dðEs El Þ p ðEs El Þ2 þ C2

consistent with the finite lifetime of the state or the finite duration of any real experiment. We are now in position to calculate the probability for the transition between states jsi and jli per unit time as a result of V acting during the infinite period 1 to 1: 2X C wnonrad ¼ ; ð2:21Þ jhsjTðEs Þjlij2
h l ðEs El Þ2 þ C2 where the matrix element of the transition operator TðEÞ contains terms to all order in perturbation theory [84], that is, hsjTðEs Þjli ¼ hsjV jli þ

X hsjV jcihc jV jli c6¼s

Es Ec0

þ

X hsjV jcihcjV jc0 ihc0 jV jli þ  : ðEs Ec0 ÞðEs Ec00 Þ c;c 0 6¼s ð2:22Þ

2.2 The Choice of the Zero-Order Basis Set

Here the intermediate states fjc ig arising from the projector 1P ¼ 1jsihsj may include the final states fjlig. The energies Ec0 are the corresponding eigenvalues of H0 . Equation 2.21 together with Equation 2.22 has a very attractive look, but an exact evaluation of the expression (2.22) is not feasible. To provide a rather remarkable amount of physical insight into this matter, let as return to the results of Chapter 1, where we have solved the Schr€odinger equation for the molecular system by expanding the molecular wavefunction in terms of either the adiabatic or the crude adiabatic basis set.

2.2 The Choice of the Zero-Order Basis Set

In this section we will discuss the problem of the nature of intramolecular interstate coupling and criteria for the choice of a basis set and consider the aspect whether these basis sets mentioned above is appropriate for describing the electronic relaxation processes. We begin by considering a complete set of zero-order functions, where electronic and nuclear motions have been separated arbitrarily. The index a refers to electronic state, while the second index v labels the vibrational state. X Utilizing the completeness assumption janihanj ¼ 1 and the trivial relation a;v X X 0 0 0 0 H¼ j av i h va j H ja v i h v a j , we decompose the total Hamiltonian in the form a;v a0 ;v0 H ¼ H0 þ V;

where both H0 ¼

ð2:23Þ

X javihvajHjavihnaj

ð2:24Þ

a;v

and the perturbation XX V¼ javihvajHja0 v0 ihv0 a0 j

ð2:25Þ

a;v ¼ 6 a0 ;v0

are specified by the zero-order basis set. In the BO representation, javi ¼ jAa ðr; qÞxAav ðqÞ, where the electronic jAa and the nuclear xAav wavefunctions satisfy (Equations 1.5 and 1.10) the equations ½Te þ Uðr; qÞjAa ðr; qÞ ¼ Ea ðqÞjAa ðr; qÞ

and

   TN þ Ea ðqÞ þ jAa jTN jjAa xAav ðqÞ ¼ Eav xAav ðqÞ;

ð2:26Þ

while (Equation 1.12) VA ¼

 
 A  A  X X   jA xA
h2 xA  ja jqU=qqjja0  q xA0 0 a av av  E ðqÞE 0 ðqÞ  qq a v a;v a0 ;v0

a

 A  A 

 A A

A  A  þ xav ja jTN jja0 xa0 v0 ja0 xa0 v0

a

ð2:27Þ

j27

j 2 Formal Decay Theory of Coupled Unstable States

28

is the so-called nonadiabatic operator. In the crude adiabatic representation, CA CA CA javi ¼ jCA a ðr; q0 Þxav ðqÞ, where the electronic ja and nuclear xav wavefunctions satisfy (Equations 1.18 and 1.23) CA ½Te þ Uðr; q0 ÞjCA a ðr; q0 Þ ¼ Ea ðq0 Þja ðr; q0 Þ

and    CA CA CA CA TN þ Ea ðq0 Þ þ jCA xav ðqÞ; xav ðqÞ ¼ Eav a jDUðr; qÞjja

ð2:28Þ

where DU ¼ Uðr; qÞUðr; q0 Þ and q0 is a fixed nuclear configuration (presumably the equilibrium configuration of the electronic ground state). In this representation, the perturbation is V CA ¼

X X   CA CA    CA jCA xCA jCA xCA jTN þ DUðr; qÞjjCA j 0 x 0 0 : 0 x 0 0 a;v6¼ a0 ;v0

a

av

a

av

a

av

a

av

ð2:29Þ

Before we return to our problem, we must note that the nonradiative probability wnonrad (Equation 2.21) as a spectroscopic observable is independent of the zero-order basis set. Thus, we have to take into account all terms of the transition operator (2.22). In this case, both A and CA sets (untruncated and complete) are adequate. Since this condition is very difficult to fulfill, we decide to choose the basis set in such a way that the second term and the higher order terms in Equation 2.22 are small and negligible. More specifically, which choice of basis gives rise to the smallest higher order corrections and/or leads to the fastest convergence of (2.22)? This can be achieved if the off-diagonal coupling matrix elements in Equation 2.22 are small relative to the first near-resonance coupling term. Thus, since only the leading term in the series (2.22) is ordinarily used, Equation 2.21 reduces to the physically interesting quantity, namely, the (first order) transition probability per unit time. It is given by w¼

2X C jðhjs xsv jV jjl xlv0 iÞj2 :
h v0 ðEsv Elv0 Þ2 þ C2

ð2:30Þ

This formula, first obtained by Dirac, has played such an important role in timedependent perturbation theory that Fermi called it the Golden Rule. The interstate coupling matrix element (Equation 2.27) was conventionally treated by the application of the Condon approximation, (a) replacing the denominator in Equation2.27 by the constant electronic energy gap and (b) assuming that the matrix  element jAa jqU=qqjjAa0 is independent of the nuclear configuration. This is almost certainly incorrect and unnecessary. A much more thorough discussion of such matrix elements is given in Chapter 5. Another formulation of this subject was given by Nitzan and Jortner [85] in an approximate scheme, relaxing both assumption (a) and (b). The point of view of that formulation will be followed here. For

2.2 The Choice of the Zero-Order Basis Set

near-resonance coupling between states jsi and jli having energy in the neighborhood of Es , they obtained for the nonadiabatic element (2.27)  D s  s E  jCA r; q0 ðqU=qqk Þq0 jCA r; q0 

 A A  A  A A 

s l k

xAsv ðqÞxAlv0 ðqÞ ; js xsv V jl xlv0 ¼ g DEsl =
hv ð2:31Þ

where DEslk is the effective electronic energy gap between states s and l modified by a promoting mode k (to be explained later) and v is an average vibrational frequency. The correction factor g in Equation 2.31 is a linear function of DEslk =
hv and exhibits a weak dependence on the coupling strength. Thus, for near-resonance coupling, Equation 2.31 is practically independent of the energy gap. For off-resonance coupling (e.g., with state jbi having an energy far from Es ) in the weak coupling limit, g  1, so that   D E   CA jCA A  A

 A A  A  A A 

s ðqU=qqk Þq0 jb xsv ðqÞxbv0 ðqÞ : js xsv V jb xbv0 ¼ ð2:32Þ ðDEsbk =
hvÞ Thus, the off-resonance coupling matrix element in the A basis is small. In the CA basis, the coupling terms are of the same form for both near-resonance and off-resonance couplings:  CA CA  CA  CA CA   CA  CA js xsv V jc xcv0  js jDU jjCA Fv;v0 ; c

c ¼ l; b;

ð2:33Þ

CA where Fv;v 0 is the Franck–Condon factor between the CA states. Comparing (2.31) with (2.33), we see that off-resonance matrix elements in the CA basis are relatively large. The ratio of the off-resonance coupling terms in the CA and A basis is of the order of magnitude DE=
hv ffi 10. Before we conclude this section, let us summarize what we have learned from this discussion. The choice of the basis set, which describes the bound-level structure of molecules is merely a matter of convenience. Assuming that the zero-order basis set is formulated such that it minimizes off-resonance coupling terms and thus allows to describe electronic relaxation processes in a two electronic level system, (a) the adiabatic basis set is superior to the crude adiabatic basis. At the same time, if the effect of off-resonance interactions can be disregarded, then (b) only a single or a few “doorway” states (which carry oscillator strength from the ground state) are accessible by transition on light absorption (accessible criterion). On the other hand, the appreciable contamination of the zero-order states jsi and jli by other states jbi in the crude adiabatic basis set implies that it is meaningless to consider the decay of an initial crude adiabatic state. (c) Generally, the appropriate basis set has to be chosen to satisfy the so-called similarity criteria according to which the basis set of H0 should be close to that of H (the entire Hamiltonian). (d) Another important point concerns the vibrational wavefunctions. The adiabatic potential surface Ea ðqÞ required for the calculation of the vibrational wavefunction xAan ðqÞ is known from molecular calculations, whereas little information is available about the crude adiabatic potential

j29

j 2 Formal Decay Theory of Coupled Unstable States

30

  CA and the vibrational wavefunctions xCA surface jCA a jDUðr; qÞjja an ðqÞ. This last point is very crucial in view of the fact that the vibrational overlap enters into the calculation of the interstate matrix elements and thus can influence the latter to an appreciable extent. This depends on the manner in which the potential energy surfaces of the electronic states under study differ in their linear and (all) quadratic interaction terms (as we shall see later). Finally, we note that only in the case in which all off-resonance states are included in the calculations of relaxation processes, the crude adiabatic basis set is perfectly adequate. These conclusions remove the considerable confusion and intense discussions in the literature at the turn of the seventies, concerning the nature of the interaction responsible for the intramolecular radiationless transitions and the choice of the proper zero-order vibronic wavefunctions [86–91].

j31

3 Description of Radiationless Processes in Statistical Large Molecules In this chapter, a general scheme for calculating nonradiative rates of polyatomic molecules embedded in an inert medium is developed. Here we are concerned with the case of fast vibrational relaxation limit within an electronically excited state. These descriptions explain the experimental phenomena observed in widely varying environments, such as high-pressure gases, liquid solution, and solid hosts. The transition probability is calculated by taking proper account of the effects of geometry and frequency distortions as well as Duschinsky mixing of modes involved in the transition. Finally, a discussion of the decay of initially selected vibronic levels in the large molecule statistic limit is presented.

3.1 Evaluation of the Radiationless Transition Probability

We now apply the formalism of Chapter 2 to calculate the nonradiative decay probability of statistical large molecules embedded in an inert medium. The basic ideas used to describe the transition probability follow logically from the following assumptions: 1)

two-manifold electronic system is considered consisting of a limited number of zero-order vibronic states ys1 ; ys2 ; . . . ; and so on belonging to one electronic state s and a dissipative manifold of levels yln belonging to a lower electronic state l of the same multiplicity (see Figure 2.1). It is assumed that the density of states in fylnm g is so large in the region of fysmm g that we can safely assume that the (large molecule) statistical limit ensures. If this condition is not satisfied, then the situation corresponds to the “small molecule” case or to the “intermediate” case situations, and more consideration is required (see Section 6.3). The

j 3 Description of Radiationless Processes in Statistical Large Molecules

32

zero-order Born–Oppenheimer (BO) wavefunctions are ysm ðr; qÞ ¼ js ðr; qs Þxsm ðqs Þ; yln ðr; ql Þ ¼ jl ðr; ql Þxln ðql Þ;

xsm ðqs Þ ¼ xln ðql Þ ¼

N Y m

N Y m


1=2  xsmm bsm qsm ;


1=2  xlnm blm qlm ;

ð3:1Þ

where the first index refers to the electronic state, while the second corresponds to the vibronic levels within a given electronic manifold a ¼ s; l. The variable r stands for the totality of electronic coordinates and qa ¼ fqam g labels the set of normal modes. The molecular vibrational wavefunctions are here represented in terms of products of 1=2 1=2 harmonic–oscillator wavefunctions xsmm ðbsm qsm Þ and xlnm ðblm qlm Þ for the individual vibrational modes m. The qam ’s are dimensioned mass-weighted normal coordinates and the transformation to dimensionless normal coordinates is accomplished by the frequency factor of the corresponding vibrational mode m in the electronic state a, bam ¼ ðvam =
hÞ. 2) Interference effects between resonances are neglected. It is thus assumed that the spacing between consecutive resonances centered about the energies Esm of the zero-order states ysm considerably exceeds the widths of these resonances. 3) The “inert” medium is characterized (with some exception) by the following features [92]:

4)

a. It does not modify the energy levels (e.g., it produces negligible level shift). b. It does not provide promoting modes influencing the radiationless transitions. The promoting modes correspond only to intramolecular vibrations. c. The inert medium does not enhance the coupling between electronic states. d. The inert medium may contribute low-frequency vibrational modes that can only accept energy, that is, acts as aheat  bath. The inert medium contributes to the widths of the levels lnm in the dense molecular manifold by vibrational relaxation (see chapter 4). Acting as a heat bath  the  inert medium enhances the population of higher vibronic states in the smm manifold. Only in special cases, which are referred to as Shpolskii matrices, the medium does not contribute to accepting modes for electronic relaxations. Direct evidence supporting this vibrational relaxation is obtained from the phonon broadening of single lines in optical spectra. However, in the statistical limit, these effects are of no importance as the intramolecular density of states is sufficiently large. Therefore, electronic relaxation processes of large molecules in inert host matrices hardly reveal any medium effects. In the case of small molecules, which are characterized by a small number of vibrational degrees of freedom, the situation is quite different and the low-frequency medium vibrations may act as accepting modes in the electronic relaxations. As a further assumption, we adopt the so-called “fast vibrational relaxation limit,” according to which the vibrational relaxations in the excited electronic states are more efficient than electronic relaxations. Provided that vibrational relaxation rates considerably exceed the nonradiative decay probability, thermal equilibri-

3.1 Evaluation of the Radiationless Transition Probability

um prevails and the nonradiative decay probability is expressed by a thermal   weighted sum of transitions originating from individual vibrational states smm . The coupling operator in Equation 1.12 involves first and second derivatives of js and jl , qj=qqg and q2 j=qqg qqm . The former are generally larger than the latter, so only they are retained. In the event that the second derivative is also of importance, it can easily be treated analogously to the first one. Hence the form taken by Vsm;ln of (1.12) is Vsm;ln ¼ ðhysm jH0 BO jyln iÞ ¼

h2

X g

   D E   xsm js ðr; qs Þq=qqlg jl ðr; ql Þ qxln =qqlg : ð3:2Þ

As we have discussed in Section 2.2 and shall treat more precisely in Chapter 5, the coupling matrix element (3.2) for nonradiative transition between the Born–Oppenheimer adiabatic states can be calculated at a configuration q0 , the so-called qcentroid. This permits Equation 3.2 to be rewritten in the form g

Vsm;ln ¼
h2

 E     X D     js iq=qqlg jl xsm iq=qqlg xln ; g

q0

ð3:3Þ

where the electronic matrix element is evaluated at q0 . These matrix elements will be nonvanishing only for vibrations qg that corresponds to the same representation of the molecular point group belonging to the direct product js xjl . Such modes are designated as promoting modes. The remaining modes of a totality of N modes, which belong to the same molecular point group, are accepting modes. The specification of what constitutes an accepting mode will be given later. The thermal average nonradiative decay probability from the manifold fysm g to the manifold fyln g in the two-level approximation is given by the “Golden Rule” expression hwiT ¼

 2 2X C pðm; TÞ Vsm;ln  ;
m;n h ðDE þ Esm
Eln Þ2 þ C2

ð3:4Þ

where DE ¼ Es0
El0 is the energy gap between the lowest vibronic levels of the two s and l electronic states and Esm ¼

X m

mm
hvsm

ð3:5Þ

nm
hvlm

ð3:6Þ

and Eln ¼

X m

are the energies of the vibronic levels in each electronic manifold measured from the zeroth level of that manifold. When the temperature is not zero, it is necessary to

j33

j 3 Description of Radiationless Processes in Statistical Large Molecules

34

properly weight the sum over the states m ¼ fmm g. According to the basic principles of statistical mechanics, if a system (molecule) is in thermal equilibrium at temperature T, then its properties should be calculated by averaging over all states ysm , assigning to each state a weight pðm; TÞ proportional to expð
Esm =kB TÞ: expð
Esm =kB TÞ pðm; TÞ ¼ P : m expð
Esm =kB TÞ

ð3:7Þ

The sum indexed by m in the denominator of Equation 3.7 shall always be understood to run over all vibrational levels of the s electronic state. pðm; TÞ is nothing more than the probability that the molecule will initially be in the zero-order state ysm . The geometric series in the denominator is easily summed to give Y

1 ðsÞ Z
1 ¼ Zm
1 ; Zm ¼ 1
exp

hvm =kB T ; ð3:8Þ m

Q where kB is the Boltzmann constant. The product Z
1 ¼ m Zm
1 is known as the partition function. The double sum in Equation 3.4 weighted with the Lorentz distribution represents a density of states function over the manifold of initial and final states. The energies El0 þ Eln do not differ from the initial energies Es0 þ Esm by more than C. Hence, the transitions approximately conserve the energy, the spread in energy being given by C. This is the difference to dðDE þ Esm
Eln Þ. The latter strictly vanishes when Es0 þ Esm 6¼ El0 þ Eln , so that it disappears for transitions between different states. For calculation of this weighted sum of density of states, we utilize the generating function (GF) approach derived for different potential parameters in a fully harmonic approximation. The latter involves the effects of frequency distortion and surface displacement, as well as the normal coordinate rotation on the two electronic states under consideration. The latter will be induced (as we have shown in Section 1.4) by a matrix (to be called W) that leads to mixing of normal coordinates belonging to these (different) electronic states. We begin by making use of the following Fourier transformation: 2C 2

ðDE þ Esm
Eln Þ þ C

2

¼
h


1

1 ð



dt exp ith
1 ð
DE þ Eln
Esm Þ
C
h
1 jtj

1 1 ð



Y 


¼ h

1 dt exp
iVt
C
h
1 jtj exp it nm vlm
mm vsm ; 1

m

ð3:9Þ

where
hV ¼ DE ¼ Es0
El0 is the energy gap between both s and l electronic states. Upon substituting Equations 3.7 and 3.9 into Equation 3.4, we obtain 1 ð 
 X
1
1 hwiT ¼ 2 Z dt exp
it V
C
h
1 jtj Vsmm ;lnm exp itnm vlm Vlnm ;smm
h mm ;nm
1
s 

s exp
mm itvm þ
hvm =kB T :

3.1 Evaluation of the Radiationless Transition Probability

j35

If the explicit form for the matrix element (3.3) with the states (3.1) is used, then we are left with ( 1 ð

1
1 X  g 2 hwiT ¼ 2 Z Rsl dt exp
itV
C
h
1 jtj Kg ðtÞGN
1 ðtÞ h
g
1 ð3:10Þ ) 1 X g g0  ð


1  þ Rsl Rsl dt exp
itV
C
h jtj Ig ðtÞIg0 ðtÞGN
2 ðtÞ ; g;g0

where


1

    Rgsl ¼ i
h js q=qqg jl q0 ; Y m Y m G1 ðtÞ; GN
2 ðtÞ ¼ G1 ðtÞ; GN
1 ðtÞ ¼ m6¼g;g0

m6¼g

with m

G1 ðtÞ ¼

X

ð3:11Þ ð3:12Þ





 exp
mm itvsm þ
hvsm =kB T exp itnm vlm

mm ;nm 1 ð ð 1




1=2 
1=2
1=2 
1=2  dqlm d
qlm xsmm bsm
qsm xlnm blm
qlm Þxsmm bsm qsm xlnm blm qlm :


1
1

ð3:13Þ

Here the square of the vibrational matrix element is written as a double integral in respect to qm and
qm , the latter of which represents the same nuclear coordinate. The Kg ðtÞ corresponds to the single-mode generating function that involves the nuclear momentum operator for the promoting mode qg : X
 


Kg ðtÞ ¼ exp
mg itvsg þ
hvsg =kB T exp itng vlg mg;ng

1 ð 1 ð

 h
2
1
1


1=2  q
1=2 
1=2  q
1=2  dqlg d
qlg xsmg bsg
qsg i l xlng blg
qlg xlng blg qlg i s xsmg bsg qsg : qqg q
qg ð3:14Þ

Finally, analogous to (3.14), the mixed-type single-mode generating function Ig ðtÞ appearing in Equation 3.10 is given by X 



 Ig ðtÞ ¼ exp
mg itvsg þ
hvsg =kB T exp itng vlg mg ;ng

1 ð ð 1


h
1
1


1=2  q
1=2 
1=2 s  
l1=2 l  dqlg d
qlg xsmg bsg
qsg i l xlng blg
qlg xsmg bs1 g qg xlng bg qg : q
qg ð3:15Þ

Ig0 ðtÞ is defined in the same manner as Ig ðtÞ except with qxlng =qqlg replaced by xlng and xsmg replaced by qxsmg =qqsg. We note at this point that Equation 3.10 contains a direct contribution from each promoting mode g and an interference term between pairs of promoting modes g and g0 . Analysis of optical spectra of aromatic hydrocarbons [87, 93] and of transition

j 3 Description of Radiationless Processes in Statistical Large Molecules

36

metal ions [94–105] have shown that molecular normal modes include a small number p (p  N) of promoting modes g and a large number of accepting modes m. According to the definition given above, the classification is based on the magnitude of the g electronic factor Rsl, Equation 3.14, the distortion of the excited electronic state relative to the lower (ground) electronic state Dm , and in smaller measure on the frequency factor bm ¼ bsm =blm ¼ vsm =vlm. For the promoting modes g, g

Rsl 6¼ 0;

Dg ¼ 0;

bg  1 ðor strictly bg ¼ 1Þ

(writing Dg as the dimensionless nuclear displacement Dqg associated with the s ! l transition, see below). Modes m characterized by m

Rsl ¼ 0;

Dm 6¼ 0;

bm 6¼ 1;

are called accepting modes. If bg 6¼ 1, the promoting mode can simultaneously act as an accepting mode. In aromatic hydrocarbons, the displacement Dm 6¼ 0 for totally symmetric carbon–hydrogen stretching modes and skeletal C–C symmetric stretching modes [93]. In transition metal complexes having an octahedral or nearly octahedral skeleton, accepting modeswith Dm 6¼ 0 arerepresented by the even parity modes, presumably the totalsymmetricstretchingmodes [94, 99, 103]. It is shown below that Ig ðtÞ andtherefore the second term in Equation 3.10 vanishes if the promoting modes are nontotally symmetric or if there is only one promoting mode as in this case Dg ¼ 0. Hence, Ig ðtÞ ¼ 0

and the transition probability of Equation 3.10 becomes 9 8 1 ð =

1
1 <X  g 2 hwiT ¼ 2 Z Rsl dt exp
itV
C
h
1 jtj Kg ðtÞGN
1 ðtÞ ; ; : g
h

ð3:16Þ


1

m

where N is the totality of vibrational modes. For some of them, Rsl ¼ 0, Dm ¼ 0, and bm  1 and their contribution to Equation 3.16 will be unity. Such modes are completely inactive and will not affect the nonradiative decay probability.
1 Aswasobservedearlier,theoccurrenceofexpð
C
jtjÞ in(3.10)impliesthedamping
h
1  of the integrand in the Fourier integral. (The exp
C
h jtj factor makes the integrand “good” atinfinity.)Ontheotherhand, byassigningthevibrationalrelaxationwidths dueto     smm and lnm , the damping the coupling with the mediumto the zero-order states
 factor exp
C
h
1 jtj ensures irreversible decay.

3.2 The Generating Function for Intramolecular Distributions I1 and I2 3.2.1 The Generating Function G2(w1,w2,z1,z2)

Before we analyze in more detail the expression for the transition probability (3.16), let us consider the generating function of Equation 3.12 for the case N
1 ¼ 2,

3.2 The Generating Function for Intramolecular Distributions I1 and I2

j37

that is, for two accepting modes. To put Equation 3.12 into a more useful form, we write 
wm ¼ expð
ivsm t

hvsm =kB TÞ; zm ¼ exp itvlm ; ð3:17Þ and X

1=2 
1=2  m rs qsm ;
qsm ; wm ¼ xsmm bsm qsm xsmm bsm
qsm wm m mm

0 11=2

bsm


1=2 2 2
1 i 1 h

m þ qsm

qsm w
m ; ¼@ A exp
bsm qsm þ
qsm w 1
wm2 4 p ð3:18aÞ

where
m ¼ w

1
wm ; 1 þ wm



1 w m ¼

1 þ wm ; 1
wm

ð3:19aÞ

have been set. Analogously, X

1=2 
1=2  n rl qlm ;
qlm ; zm ¼ xlnm blm qlm xlnm blm
qlm zmm nm

0 11=2

blm

1=2 2 2 i
1 h
1
z2m ¼ @ A exp
blm qlm þ
qlm
zm þ qlm

qlm
z
1 ; m 4 p ð3:18bÞ

with
zm given by
zm ¼

1
zm : 1 þ zm

ð3:19bÞ

The calculation of the sum in (3.18) is performed in close analogy to the Slater sum (Mehler formula) [106]. Using Equations 3.17 and 3.18, Equation 3.12 can be recast for N
1 ¼ 2 as !  1  D12 D212 b1 b2  G2 w1 ; w2 ; z1 ; z2  12 ; ¼ b12 b21 D1 D12 2
s s l l 1=2  ðð 1 ð ð 2 2
1 b1 b2 b1 b2 1
1

1
bs1 qs1

qs1 w
1 exp
bs1 qs1 þ
qs1 w





1=2 4 4 p2 1
z2 1
z2 1
w 2 1
w 2 1

2

1

2


1

2 2
1 1 l
l 2 2 1
1
1

2
bs2 qs2

qs2 w
2
b1 q1 þ
ql1
z1
bl1 ql1

ql1
z
1
bs2 qs2 þ
qs2 w 1 4 4 4 4  2 2 1
1

bl2 ql2 þ
ql2
z2
bl2 ql2

ql2
z
1 dql1 d
ql1 dql2 d
ql2 : 2 4 4 ð3:20Þ

j 3 Description of Radiationless Processes in Statistical Large Molecules

38

In our mathematically lax manner, we ignore in Equation 3.20 possible difficulties in interchanging orders of integration and summation. To calculate this fourfold integral, we need the transformation (1.44) of Chapter 1 between the coordinates qs and ql , which in the two-dimensional case is written as     l  s    kð1Þ   q1   w11 w12   q     12 1  þ  ¼ : ð3:21Þ   ql   ð2Þ   qs   w  21 w22 k 2 2 12

In dealing with several modes, which we shall study in Chapter 4, it is customary to ð1Þ ð2Þ use this notation for the displacement vector k12 ¼ colðk12 ; k12 Þ. To facilitate integration, we introduce new coordinates
 qm ¼ 2
1=2 qlm þ
qlm
 ð3:22aÞ
qm ¼ 2
1=2 qlm

qlm ; m ¼ 1; 2; and correspondingly
 qm0 ¼ 2
1=2 qsm þ
qsm

qm0 ¼ 2
1=2 qsm

qsm ; m ¼ 1; 2: If we combine the transformations (3.22) with (3.21), we find that     1=2 ð1Þ   0     q 1   w11 0 w12 0   q1   2 k12            
q0   0 w  0 w12  
q1   0  11  1  :  þ  0 ¼ 1=2 ð2Þ   q 2   w21 0 w22 0   q2   2 k  12          
q   
q0   0 w  0 w 21 22 0 2 2

ð3:22bÞ

ð3:23Þ

With this transformation, the integral appearing in (3.20) is easily performed, giving [107]    l 2 s
   w 2 bs w 1
1 þ w21 w22 bs2 w
2  z1 w11 w12 bs1 w 2 þ b1
11 1
1 þ w21 b2 w   q1  dq1 d
q1 dq2 d
q2 exp
kq1 ; q2 k s l   q2   w w bs w 2 s
2 s
2


þ w w b w b þ w b þ b w w w z 11 12 1 1 21 22 2 2 12 1 1 22 2 2 2 2
1     l
1 s
1 s
1  
1 2 s

1    w 2 bs w 1
q


þ w b þ b w w b w þ w w b w z w 11 12 21 22 1 1 1 1 2 2 11 1 1 21 2 2  1 

q1 ;
q2    s
1  w w bs w  
q 
1 2 s

1 2 s

1 2
2 w12 b1 w 1 þ w22 b2 w 2 þ bl2
z
1 2 11 12 1
1 þ w21 w22 b2 w 2  

    2 2
 q s ð1Þ 1
1 þ bs2 kð2Þ
2 :
21=2  w11 bs1 kð1Þ
1 þ w21 bs2 kð2Þ
2 w12 bs1 kð1Þ
1 þ w22 bs2 kð2Þ
2   12 w  q2 
b1 k12 w 12 w 12 w 12 w 12 w 1 ð

ð3:24Þ

We see that the fourfold integral (3.24) is written as a product of two Gaussian integrals of the form 1 ð


1

    1 1 dq1 dq2 exp
qt xq þ yt q ¼ 2pðdet xÞ
1=2 exp yt x
1 y ; 2 2

where q ¼ colðq1 ; q2 Þ

ð3:25Þ

3.2 The Generating Function for Intramolecular Distributions I1 and I2

j39

and



ð1Þ
1 þ w21 bs2 kð2Þ
2 ;
2
1=2 w12 bs1 kð1Þ
1 þ w22 bs2 kð2Þ
2 y ¼ col
2
1=2 w11 bs1 k12 w 12 w 12 w 12 w

are the column vectors in the first integral of Equation 3.24. Correspondingly,   q ¼ colð
q1 ;
q2 Þ and y ¼ colð0; 0Þ are in the second integral. The quantity x ¼ xij  stands for the square matrices in Equation 3.24. Note that the matrix x in the second

1
m and
zm by w integral is obtained from that in the first integral by replacing w m and
z
1 ðm ¼ 1; 2Þ. Carrying out the integration in (3.24) by twice applying formum la (3.25), the two-dimensional generating function G2 now becomes (by a lot of algebra) [107]  1   D D212 b1 b2 1 1=2 1=2
1 Þð1 þ w
2Þ G2 w1 ; w2 ; z1 ; z2  12 ; ¼ b1 b2 ð1 þ w 12 12 4 D1 D2 b12 b21  

2 ;
z1 ;
z2 Þ w1 ; w A1 ð
exp
~
2 ;
z1 ;
z2 Þ w1 ; w B 1 ð
ð1 þ
z1 Þð1 þ
z2 Þ 
;
2 ð

2 ;
z1
z2 Þ
2 ;
z1 ;
z2 ÞB ½B1 ð
w1 ; w w1 w ð3:26Þ where the following abbreviations have been introduced: 2

2

2


1 ð

2 ;
z1 ;
z2 Þ ¼ b1 Dð1Þ
1
z1
z2 þ b2 Dð2Þ
2
z1
z2 þ b2 Dð12Þ
1w
2
z1 w1 ; w w A 12 w 12 w 1 ð12Þ2

þ b1 D2


1w
2
z2 ; w

2 2 2
1 ð

1w
2 þ w11
1
z2 þ w12
1
z1 þ w22
2
z1
2 ;
z1 ;
z2 Þ ¼ b1 b2 w B w1 ; w b1 w b12 w b2 w 2
2~ þ w21 b21 w z 2 þ
z1
z2 ; 2 2 2
2 ð

2 ;
z1 ;
z2 Þ ¼ w
1w
2 þ w11
2
z1 þ w12
2
z2 þ w22
1
z2 B w1 ; w b1 w b12 w b2 w 2
1
z1 þ b1 b2
z1
z2 : þ w21 b21 w

ð3:27Þ

Here bm ¼ bsm =blm ;

bmn ¼ bsm =bln ;

ð3:28Þ

m; n ¼ 1; 2;

are the changes of the vibrational frequencies in the s state relative to those in the ðmÞ are the dimensionless displacement parameters scaled as state l and D12 ; Dð12Þ m ðmÞ2

ðmÞ2

ð12Þ2

D12 ¼ blm k12 ; Dð12Þ ¼ bsm km m

where

   kð12Þ   w11  1    ð12Þ  ¼   k   w21 2

2

;

ð3:29Þ

m ¼ 1; 2;

 
1   kð1Þ  w12  12     ð2Þ : k  w22 

ð3:30Þ

12

ðmÞ

ðmÞ

The dimensionless displacement parameters D12 describe the shift k12 of the origin of the electronic state s relative to that of the electronic state l along the displacements coordinate qlm in the unit of ðblm Þ1=2 . The dimensionless Dð12Þ m ð12Þ are related according to Equation 3.29 to the dimensioned displacements km .

j 3 Description of Radiationless Processes in Statistical Large Molecules

40

The latter in turn are obtained from the inverse transformation (3.30) of those ðmÞ ð12Þ given by k12 ðm ¼ 1; 2Þ. Thus, km (the so-called reciprocal displacement compoð12Þ nents), or strictly
k , describes the difference of the equilibrium positions of l and s electronic states in the coordinate system with base vectors qlm . We may restate ðmÞ this by saying that an electron sees the displacements D12 during the transition ð12Þ fsmm g ! flnm g in emission and the parameters Dm ðm ¼ 1; 2Þ in absorption flnm g ! fsmm g. We shall return to this question in Section 3.2.5, where the symmetry properties of G2 are discussed. We have purposely used here a rather ðmÞ detailed (and therefore sometimes cumbersome) notation for the parameters D12 ð12Þ and Dm . This has been done to make the discussion in the following sections all the more clear. In Chapter 8, it will often be convenient to extend this notation to higher vibrational degrees of freedom. Returning to our original variables wm and zm ðm ¼ 1; 2Þ, we may also write   A1 ðw1 ; w2 ; z1 ; z2 Þ exp
B1 ðw1 ; w2 ; z1 ; z2 Þ 1=2 1=2 G2 ðw1 ; w2 ; z1 ; z2 Þ ¼ 4b1 b2 ; ð3:31Þ ½B1 ðw1 ; w2 ; z1 ; z2 ÞB2 ðw1 ; w2 ; z1 ; z2 Þ1=2 where w1 ; w2 ; z1 ; z2 are generally complex variables in the polydisc [108] D4 ð0; 1Þ [|wm | < 1, |zm | < 1 ðm ¼ 1; 2Þ] (Equation 3.17), and A1 ðw1 ; w2 ; z1 ; z2 Þ ¼

1 X

1 X

m1 ;m2 ¼0 n1 ;n2 ¼0

B1 ðw1 ; w2 ; z1 ; z2 Þ ¼

1 X

1 X

m1 ;m2 ¼0 n1 ;n2 ¼0

m

m

ð3:32Þ

am1 m2 ;n1 n2 w1 1 w2 2 zn11 zn22 ;

m

m

ð3:33Þ

m

ð3:34Þ

dm1 m2 ;n1 n2 w1 1 w2 2 zn11 zn22 ;

and B2 ðw1 ; w2 ; z1 ; z2 Þ ¼

1 X

1 X

m1 ;m2 ¼0 n1 ;n2 ¼0

m

bm11 m2 ;n1 n2 w1 1 w2 2 zn11 zn22 :

The quantities dm1 m2 ;n1 n2 ; am1 m2 ;n1 n2 ; and bm11 m2 ;n1 n2 in Equations 3.32, 3.33 and 3.34, respectively, which constitute tensors of the fourth rank, are real and given by ð1Þ2

ð2Þ2

dm1 m2 ;n1 n2 ¼ ð
Þm1 þ n1 þ n2 b1 D12 þ ð
1Þm2 þ n1 þ n2 b2 D12 ð12Þ2

þ ð
1Þm1 þ m2 þ n1 b2 D1

ð12Þ2

þ ð
1Þm1 þ m2 þ n2 b1 D2

;

2 2 b1 þ ð
1Þm1 þ n1 w12 b12 am1 m2 ;n1 ;n2 ¼ ð
1Þm1 þ m2 b1 b2 þ ð
1Þm1 þ n2 w11 2 2 þ ð
1Þm2 þ n1 w22 b2 þ ð
1Þm2 þ n2 w21 b21 þ ð
1Þn1 þ n2

ð3:35Þ

ð3:36Þ

and bm1 m2 ;n1 n2 ¼ ð
1Þm1 þ m2 þ n1 þ n2 am1 m2 ;n1 n2 :

ð3:37Þ

3.2 The Generating Function for Intramolecular Distributions I1 and I2

Equation 3.35 defines in the representation of nonseparable modes a new set of displacement parameters, whereas Equation 3.36 defines the new set of terms, which depend on all frequency factors. Equations 3.35–3.37 are functions of eight dimensionless parameters, namely, four positive b-parameters b1 ; b2 ; b12 ; and b21 and four D-parameters, which are real numbers. In our special case, where the matrix in Equation 3.21 is orthogonal, the latter are reduced to three independent parameters D1 ; D2 , and w, where w is a rotation angle, which (see later) parameterizes the matrix (3.21). 3.2.2 Properties of dm1m2,

n1n2,

am1m2, n1n2, and bm1m2,

n1n2

First, note that the 16 components of the tensor dm1 m2 ;n1 n2 are not independent and we can reduce them to 8 by noting that dm1 m2 ;n1 n2 ¼
d1
m1 ;1
m2 ;1
n1 ;1
n2 :

ð3:38Þ

A further reduction is accomplished with the help of the relations d00;00 þ d00;10 þ d00;01 þ d00;11 d01;00 þ d01;10 þ d01;01 þ d01;11 d10;00 þ d10;10 þ d10;01 þ d10;11 d11;00 þ d11;10 þ d11;01 þ d11;11

¼0 ¼0 ¼0 ¼0

ð3:39Þ

in which the last two relations are obtained from the first two by taking into account Equation 3.38. This indicates that six independent components are required to specify the generating function. It can be reduced still further to five in the special case if we assume that the matrix (3.21) is orthogonal. According to the definition (3.35), the parameters dm1 m2 ;n1 n2 can be treated as new displacement parameters. Similarly, it follows directly from (3.36) that am1 m2 ;n1 n2 is unaltered by the interchange m1 ! 1
m1 ; m2 ! 1
m2 ; n1 ! 1
n1 ; and n2 ! 1
n2 . Thus, it is sufficient to specify the values of am1 m2 ;n1 n2 for the two values of the pair ðm1 m2 Þ and the same two values of the pair (n1 n2 ). The reader may also convince himself that a00;00 þ a00;10 þ a00;01 þ a00;11 a01;00 þ a01;10 þ a01;01 þ a01;11 a10;00 þ a10;10 þ a10;01 þ a10;11 a11;00 þ a11;10 þ a11;01 þ a11;11

¼ 4b1 b2 ¼
4b1 b2 ¼
4b1 b2 ¼ 4b1 b2 ;

ð3:40Þ

which reduces the number of independent components to two. Finally, according to Equation 3.37, we obtain in the same manner b00;00 þ b00;10 þ b00;01 þ b00;11 b01;00 þ b01;10 þ b01;01 þ b01;11 b10;00 þ b10;10 þ b10;01 þ b10;11 b11;00 þ b11;10 þ b11;01 þ b11;11

¼4 ¼
4 ¼
4 ¼ 4:

ð3:41Þ

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42

3.2.3 Case w1 ¼ w2 ¼ 0

Before we analyze the generating function G2 and the corresponding distributions in greater detail, let us consider the special case of the so-called zero-temperature limit, which is adequate for many practical purposes. In that case, w1 ¼ w2 ¼ 0 and we obtain from (3.31) !  ð1Þ ð2Þ  D12 D12 b1 b2  G2 0; 0; z1 ; z2  ð12Þ ; ð12Þ b b21 D1 D2 12   ð3:42Þ d00;00 þ d00;10 z1 þ d00;01 z2 þ d00;11 z1 z2 exp
a00;00 þ a00;10 z1 þ a00;01 z2 þ a00;11 z1 z2 1=2 1=2 ; ¼ 4b1 b2 ½ða00;00 þ a00;11 z1 z2 Þ2
ða00;10 z1 þ a00;01 z2 Þ2 1=2 which represents the kernel of G2 ðw1 ; w2 ; z1 ; z2 Þ. The mathematical subtleties in two dimensions though are more difficult than they are in one. We are restating here conclusions obtained in Equation 4.4. The functions G1 ð0; zÞ and G2 ð0; 0; z1 ; z2 Þ exhibit a remarkable regularity. This is manifested in the similarity of the functions with respect to their exponents, which have many of the same structural features. (They have the topology of a torus of the dimension one for N ¼ 1 and of two for N ¼ 2, respectively). Both of them vanish when z ¼ 1 or z1 ¼ z2 ¼ 1, respectively. Analogous to the one-dimensional case, the mapping of the bidisc [108] D2 ð0; 1Þ [|z1 | 1, |z2 | 1] by G2 ð0; 0; z1 ; z2 Þ constitutes a domain confined by exp½
b1 D21
b2 D22   jG2 ð0; 0; z1 ; z2 Þj  1 and in particular G2 ð0; 0; z1 ; z2 Þ is 1, when both z1 and z2 reach 1. The point G2 ¼ 0 is a bifurcation point of infinity order. This behavior of the generating function is valid, as will be further substantiated, for higher dimensions N. 3.2.4 Case w1 6¼ w2 6¼ 0

The G2 ðw1 ; w2 ; z1 ; z2 Þ of Equation 3.31 is holomorphic in the polydisc D4 ð0; 1Þ and has the expansion in a series of w1m1 w2m2 zn11 zn22 : !  ð1Þ ð2Þ  D12 D12 b1 b2 G2 w1 ; w2 ; z1 ; z2  ð12Þ ; ð12Þ b D1 D2 12 b21 ! ð3:43Þ  ð1Þ 1 1 X X b1 b2 m1 ; m2  D12 Dð2Þ 12 m1 m2 n1 n2 w 1 w2 z 1 z 2 ; I2 ; ¼ n1 ; n2  Dð12Þ Dð12Þ b12 b21 m1 ;m2 ¼0 n1 ;n2 ¼0 2 1   m1 m2 is, as will be shown below, for each pair of nonnegative integers where I2 n1 n2 (m1 ; m2 ), a two-dimensional probability distribution of n1 ; n2 and vice versa. On the other hand, from Equations 3.17–3.20, it follows that  I2

m1 n1

m2 n2



ð ð1 ¼
1

dq1 d
q1 dq2 d
q2 xsm ðqs Þxsm ð
qs Þxln ðql Þxln ð
ql Þ:

ð3:44Þ

3.2 The Generating Function for Intramolecular Distributions I1 and I2

In Equation 3.43, the dependence of the generating function and their distributions on the parameters are
exhibited explicitly. For
the physical  situation described above, ðlÞ  we can set zm ¼ exp ivm t and wm ¼ exp
a=T
ivsm t ðm ¼ 1; 2Þ, with a being a positive
number, especially a ¼
h
vsm =kB T,  or in another modification, ðlÞ  zm ¼ exp
a=T þ ivm t and wm ¼ exp
ivsm t , where t is a time variable.1)   m1 m2 To verify that the coefficients I2 in the series (3.43) are for each pair of n1 n2 levels ðm1 ; m2 Þ and values of a distribution for several sets of choices ðn1 ; n2 Þ and vice versa, we first note the simple property of G2 :   A1 ðw1 ; w2 ; z1 ; z2 Þ exp
B1 ðw1 ; w2 ; z1 ; z2 Þ 1=2 1=2 G2 ðw1 ; w2 ; z1 ; z2 Þ ¼ 4b1 b2 ; ð3:45Þ ½B1 ðw1 ; w2 ; z1 ; z2 ÞB2 ðw1 ; w2 ; z1 ; z2 Þ1=2 which follows directly from Equations 3.31–3.34. In other words, the complex conjugate of G2 is obtained simply by replacing the variables wm and zm with their with conjugates wm and zm ðm ¼ 1; 2Þ. In conjunction   (3.43), it follows that the m1 m2 coefficients  of the series (3.43), I2 n1 n2 , are real. In particular, m1 m2 0 for all pairs of levels (m1 ; m2 ) or ðn1 ; n2 Þ by definition. (This I2 n1 n2 point emerges clearly from the fourfold integration approach to evaluation of the generating function G2 and Equation 3.44). Finally, it is a simple matter to verify that for each pair of levels ðm1 ; m2 Þ,   1 X m1 m2 I2 ¼ 1: ð3:46Þ n1 n2 n ;n ¼0 1

2

For the proof of (3.46), it is convenient to express bilinear form [109]:   d00;00  d  10;00 A1 ðw1 ; w2 ; z1 ; z2 Þ ¼ k1; w1 ; w2 ; w1 w2 k    d01;00  d 11;00

A1 ðw1 ; w2 ; z1 ; z2 Þ in terms of a d00;10 d10;10

d00;01 d10;01

d01;10 d11;10

d01;01 d11;01

  d00;11   1    d10;11    z1   d01;11    z2  z z d 11;11

1 2

        

¼ wt kdm;n k41 z; ð3:47Þ

where W and z are column matrices col(1, w1 ; w2 ; w1 w2 ) and col(1; z1 ; z2 ; z1 z2 ), respectively, m ¼ ðm1 ; m2 Þ; and n ¼ ðn1 ; n2 Þ. The sign t in (3.47) denotes transposition. This matrix approach is very convenient for generalization of three, four, or more dimensions. If the number of the vibrational modes becomes very large, that is, N, then the order of the square matrix in (3.47) is 2N . It follows from (3.39) that the

1) We note a certain ambiguity in the definition of the variables wm and zm. The ambiguity is, however, not genuine and therefore the definition can not be arbitrarity. This becomes evident in the applications. The simplest case is, if we assume wm and zm to be dummy variables.

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44

sum of the elements of each row, each column, and any main diagonals is the same, namely, zero (compare with a magic square). Similarly, representing B1 ðw1 ; w2 ; z1 ; z2 Þ in the form of a product of three matrices, a row, a square, and a column matrix,  4 B1 ðw1 ; w2 ; z1 ; z2 Þ ¼ wt am;n 1 z; ð3:48Þ where, according to (3.40), the sum of elements in the first, second, third, and fourth row of the square matrix in (3.48) are 4b1 b2 ;
4b1 b2 ;
4b1 b2 , and 4b1 b2 , respectively. Correspondingly, the sum of elements in the first, second, third, and fourth columns are 4,
4,
4, and 4, respectively. Finally, we note that  4 B2 ðw1 ; w2 ; z1 ; z2 Þ ¼ wt bm;n 1 z: ð3:49Þ  4   Here the sum of the elements  in each 4 row of bm;n 1 is the same as the sum of the   elements in the columns of am;n 1 and vice versa. By direct substitution in Equations z1 ¼ z2 ¼ 1 and taking into  4 3.47–3.49 4 4 account the above properties of dm;n 1 ; am;n 1 , and bm;n 1 , we have after matrix multiplication A1 ðw1 ; w2 ; 1; 1Þ ¼ 0 B1 ðw1 ; w2 ; 1; 1Þ ¼ 4b1 b2 ð1
w1 Þð1
w2 Þ B2 ðw1 ; w2 ; 1; 1Þ ¼ 4ð1
w1 Þð1
w2 Þ:

ð3:50Þ

Note that the right-hand side of (3.43) converges as z1 ! 1 and z2 ! 1. Substituting these expressions in (3.31) gives G2 ðw1 ; w2 ; 1; 1Þ ¼

1 X 1 ¼ w m1 w m2 : ð1
w1 Þð1
w2 Þ m1 ;m2 ¼0 1 2

ð3:51Þ

Comparing this result with (3.43), we have  m1 m2 G2 ðw1 ; w2 ; 1; 1Þ ¼ I2 w1m1 w2m2 n1 n2 m1 ;m2 ¼0 n1 ;n2 ¼0  ! 1 1 X X m1 m2 w1m1 w2m2 ; ¼ I2 n n 1 2 m1 ;m2 ¼0 n1 ;n2 ¼0 1 X

1 X



ð3:52Þ

which completes the proof of (3.46).  Equation  3.46 underlines the correctness and m1 m2 . generality of our definition of I2 n1 n2
4 ð0; 1Þ [|zm | < 1, Similarly, if we consider the function (3.31) in the polydisc D |wm | 1; m ¼ 1; 2], we have for each pair of levels (n1 ; n2 ) 1 X m1 ;m2 ¼0

 I2

m1 n1

m2 n2

 ¼ 1:

ð3:53Þ

3.2 The Generating Function for Intramolecular Distributions I1 and I2

Proof Analogous to the foregoing treatment, we have A1 ð1; 1; z1 ; z2 Þ ¼ 0 B1 ð1; 1; z1 ; z2 Þ ¼ 4ð1
z1 Þð1
z2 Þ B2 ð1; 1; z1 ; z2 Þ ¼ 4b1 b2 ð1
z1 Þð1
z2 Þ

ð3:54Þ

and this gives the series for the generating function (3.31) G2 ð1; 1; z1 ; z2 Þ ¼

1 X 1 zn1 zn2 : ¼ ð1
z1 Þð1
z2 Þ n1 ;n2 ¼0 1 2

Expressing (3.43) in a different manner, we obtain  ! 1 1 X X m1 m2 zn11 zn22 : G2 ð1; 1; z1 ; z2 Þ ¼ I2 n n 1 2 n ;n ¼0 m ;m ¼0 1

2

1

ð3:55Þ

ð3:56Þ

2

On comparison with the series (3.55) follows Equation 3.53.  3.2.5 m1 Symmetry Properties of I2 n1

m2 n2



The matrix representation of A1 ðw1 ; w2 ; z1 ; z2 Þ (and similarly of B1 and B2 ) introduced above allows us to investigate the question of how the distribution I2 is affected by exchange of parameters. To be explicit, we will investigate the behavior of I2 under the following change of parameters: ð1Þ

ð12Þ

ð2Þ

ð12Þ


1
1 D12 to D1 ; D12 to D2 ; b1 to b
1 1 ; b2 to b2 ; b12 to b21 :

ð3:57Þ

Using formula (3.35), one can prove that utilizing the change (3.57), the coefficient matrix of the form A1 ðw1 ; w2 ; z1 ; z2 Þ in (3.47) transforms to     dm;n 4 ! b
1 b
1 dn;m 4 ; 1 2 1 1

ð3:58Þ

where the matrix on the right-hand side of (3.58) is the transposition of the original in (3.47). Analogously, it follows from Equations 3.36 and 3.37 that by changing the parameters as (3.57),  4   am;n  ! b
1 b
1 an;m 4 1 2 1 1

ð3:59Þ

and similarly,  4   bm;n  ! b
1 b
1 bn;m 4 ; 1 2 1 1

ð3:60Þ

where use has been made of the fact that the matrix elements of (3.21) obey w11 ¼ w22 ¼ cos w and w12 ¼
w21 ¼ sin w after parametrization of the rotation matrix W.

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46

Combining the above change of parameters (3.57) with the interchange of the variables ðw1 ; w2 Þ $ ðz1 ; z2 Þ, the following relations hold:2)  4  4  4
1 t 
1
1 t    A1 ðw1 ; w2 ; z1 ; z2 Þ ¼ wt dm;n 1 z ! b
1 1 b2 z dn;m 1 w ¼ b1 b2 w dm;n 1 z; ð3:61Þ       4 4 4
1
1
1
1 B1 ðw1 ; w2 z1 ; z2 Þ ¼ wt am;n 1 z ! b1 b2 zt an;m 1 w ¼ b1 b2 wt am;n 1 z; ð3:62Þ

and

 4  4  4
1 t 
1
1 t    B2 ðw1 ; w2 z1 ; z2 Þ ¼ wt bm;n 1 z ! b
1 1 b2 z bn;m 1 w ¼ b1 b2 w bm;n 1 z: ð3:63Þ

Combining Equations 3.61 through 3.63, the exchange of parameters (3.57) produces the result: A1 ðw1 ; w2 ; z1 ; z2 Þ A1 ðz1 ; z2 ; w1 ; w2 Þ ¼ ; B1 ðw1 ; w2 ; z1 ; z2 Þ B1 ðz1 ; z2 ; w1 ; w2 Þ

and correspondingly    !  Dð1Þ Dð2Þ  b b   Dð12Þ Dð12Þ  b
1 b
1  12  1  2 12  1 2 ¼ G2 z1 ; z2 ; w1 ; w2  ð1Þ ð2Þ ;  1
1 2
1 : G2 w1 ; w2 ; z1 ; z2  ð12Þ ð12Þ ;  D D D  b21 b12 b12 b21 D2 1 12 12 ð3:64Þ

If we now expand both sides of the identity (3.64) in the polydisc D4 ð0; rÞ according to (3.43) in a power series and equate the terms of w1m1 w2m2 zn11 zn22 , we have the following: Corollary 3.1 The distribution I2 is unaltered by the exchange of parameters (3.57), provided that the integer variables mm and nm are simultaneously exchanged, mm $ nm   ! !  ð1Þ ð2Þ  ð12Þ ð12Þ b
1 b
1 m1 m2  D12 D12 b1 b2 n1 n2  D1 D2 1 2 ¼ I2 : I2 ; ;   n1 n2  Dð12Þ Dð12Þ b12 b21 m1 m2  Dð1Þ Dð2Þ b
1 b
1 21 12 2 1 12 12 ð3:65Þ This says that due to the presence of the cross-frequency parameters b12 and b21 ðmÞ in (3.65), as well as the fact that Dð12Þ 6¼ D12 , there exists no mirror images in respect m to ðm1 ; m2 Þ $ ðn1 ; n2 Þ (i.e., in emission and absorption spectra). This holds only if the rotation angle w ¼ 0 and the frequency changes b1 ¼ b2 ¼ 1 (see below). Similarly, since the indices 1 and 2 in (3.43) are indistinguishable, they can exchange roles:   ! !  ð1Þ ð2Þ  ð2Þ ð1Þ b b m1 m2  D12 D12 b1 b2 m2 m1  D12 D12 2 1 ¼ I2 : I2 ; ;   n1 n2  Dð12Þ Dð12Þ b12 b21 n2 n1  Dð12Þ Dð12Þ b21 b12 2 2 1 1 ð3:66Þ 2) In the expressions 3.61–3.63 the variables w and z are treated as dummy variables.

3.2 The Generating Function for Intramolecular Distributions I1 and I2

The proof of this identity results from the fact that the coefficients dm1 m2 ;n1 n2 and am1 m2 ;n1 n2 appearing in the generating function G2 are invariant under the operation F (“flip”), which follows simply by changing m1 to m2 and n1 to n2 dm1 m2 ;n1 n2 ¼ dm2 m1 ;n2 n1 ¼ Fdm1 m2 ;n1 n2 am1 m2 ;n1 n2 ¼ am2 m;n2 n1 ¼ Fam1 m2 ;n1 n2

as well as the parameters of I2 as displaced in (3.66) explicitly. Equations 3.65 and 3.66 are very valuable for calculating I2 . 3.2.6 Case w ¼ 0 ð12Þ

ðmÞ

If w ¼ 0, it 2follows from (3.21), (3.29), and (3.30) that km ¼ k12 and 2 ðmÞ ¼ bm D12 ðm ¼ 1; 2Þ. In this special case, Equations 3.32–3.37 can be written Dð12Þ m as follows: A1 ðw1 ; w2 ; z1 ; z2 Þ ¼

ð1Þ2

b1 D12 ð1
w1Þ Þð1
z1 Þ½ð1 þ w2 Þð1
z2 Þ þ b2 ð1
w2 Þð1 þ z2 Þ ð2Þ2

þ b2 D12 ð1
w2 Þð1
z2 Þ½ð1 þ w1 Þð1
z1 Þ þ b1 ð1
w1 Þð1 þ z1Þ Þ; 2 Y B1 ðw1 ; w2 ; z1 ; z2 Þ ¼ ½bm ð1
wm Þð1 þ zm Þ þ ð1 þ wm Þð1
zm Þ; m¼1

and B2 ðw1 ; w2 ; z1 ; z2 Þ ¼

2 Y

½bm ð1 þ wm Þð1
zm Þ þ ð1
wm Þð1 þ zm Þ:

m¼1

Dividing A1 by B1, we find that ð1Þ2

A1 ðw1 ; w2 ; z1 ; z2 Þ b1 D12 ð1
w1 Þð1
z1 Þ ¼ B1 ðw1 ; w2 ; z1 ; z2 Þ b1 ð1
w1 Þð1 þ z1 Þ þ ð1 þ w1 Þð1
z1 Þ ð2Þ2

b2 D12 ð1
w2 Þð1
z2 Þ þ ; b2 ð1
w2 Þð1 þ z2 Þ þ ð1 þ w2 Þð1
z2 Þ

where the right-hand side breaks down into two terms, each of which depends on ðmÞ variables wm ; zm and parameters D12 and bm of only one component m (the crossparameters b12 and b21 disappear). Thus, in the case of w ¼ 0, the two-dimensional GF can simply be written as a product of two single-mode GF: G2 ðw1 ; w2 ; z1 ; z2 Þ ¼ G1 ðw1 ; z1 ÞG1 ðw2 ; z2 Þ;

ð3:67Þ

where  exp




bD2 ð1
wÞð1
zÞ bð1
wÞð1 þ zÞ þ ð1 þ wÞð1
zÞ

G1 ðw; zÞ ¼ 2b1=2

1=2 ð1 þ b2 Þð1
w 2 Þð1
z2 Þ þ 2b½ð1 þ w 2 Þð1 þ z2 Þ
4wz ð3:68Þ

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48

is clearly an one-dimensional GF (see later). The result obtained can be stated as follows: Corollary 3.2 If the angle w ¼ 0, the two-dimensional GF G2 ðw1 ; w2 ; z1 ; z2 Þ factors into a product of one-dimensional terms G1 , each of which depends on variables and parameters belonging to one component only.

3.3 Derivation of the Promoting Mode Factors Kg(t) and Ig(t)

Before we examine further the behavior of the generating function G2 ðtÞ, let us return to Equation 3.16 for the transition probability. To this end, we have yet to obtain the expression for the factor Kg ðtÞ, which includes the q=qqg overlap integral arising from the promoting mode (3.14). Proceeding as in the previous manner and writing Equation 3.14 explicitly with the help of Equation 3.18, we find that


1=2

 ð1 ð bsg blg 2 2
1 i
q 1 h
2

g þ qsg

qsg w
g Kg ðtÞ¼
dqlg d
qlg i s exp
bsg qsg þ
qsg w 
 1=2 h qqg 4 p 1
wg2 1
z2g


1 h
2 2 i q 1
i l exp
blg qlg þ
qlg
zg þ qlg

qlg
z
1 g 4 q
qg
1=2

 ð1 ð
bsg blg
s s 2
1 i 1 s h
s s 2 2 l l


q w w ¼
h dq d
q þ
q þ q

q b exp
g 
 1=2 g g g g g 4 g g g p 1
wg2 1
z2g
1


2 2 i 1 h
exp
blg qlg þ
qlg
zg þ qlg

qlg
z
1 g 4



 h  
1 i   i

1
1 h

g þ qsg

qsg w
g
blg qlg þ
qlg
zg
qlg

qlg
z
1 : 
bsg qsg þ
qsg w g 2 2 ð3:69Þ

Introducing now, as above, new coordinates for the gth promoting mode
 qg ¼ 2
1=2 qlg þ
qlg

qg ¼ 2
1=2 qlg

qlg


 q0g ¼ 2
1=2 qsg þ
qsg

q0g ¼ 2
1=2 qsg

qsg

ð3:70Þ

according to (3.23), the relations q0g ¼ qg þ 2
1=2 kg
q0g ¼
qg

ð3:71Þ

are obtained. Substituting (3.70) and (3.71) into Equation 3.69 and integrating, we obtain

3.3 Derivation of the Promoting Mode Factors Kg(t) and Ig(t)

 Kg ðtÞ ¼

8 9 > > = 2bsg blg k2g 1 2 s l < 1 1 ðgÞ
þ
h bg bg h i 2 G1 ðtÞ s l l
1 > 2 s
1 l
1 >
g bsg
z
1
g :bg
zg þ bg w ; g þ bg w
g bg
zg þ bg w ðgÞ

¼ Kg0 ðtÞG1 ðtÞ;









ð3:72Þ



where bsg ¼ vsg =
h and blg ¼ vlg =
h are the frequency factors of the promoting mode g in the states s and l and kg is the geometrical displacement between the potential surfaces in the configuration space along the promoting mode coordinate qg . If the promoting mode is nontotally symmetric, one can assume that kg ¼ 0 and bg ¼ bsg =blg  1

and it is clearly sufficient to reduce Equation 3.72 to the much simpler form in conformity with [110]  



1 Kgð0Þ ðtÞ ¼ h hvg =2kB TÞ
1 e
ivg t ;
vg cothð
hvg =2kB TÞ þ 1 eivg t þ cothð
4 ð3:73Þ

where the substitutions (3.17) have been made for the variables wg and zg . Analogously, one can prove from Equation 3.15 that Ig ðtÞ ¼ i
h

bsg blg kg l
1
g bsg
z
1 g þ bg w

g

G1 ðtÞ:

ð3:74Þ

This vanishes if, as in this case, the promoting mode g is nontotally symmetric, giving kg ¼ 0. Having calculated the generating functions Kg ðtÞ and GN
1 ðtÞ, we are now in a position to obtain the final expression for the transition probability (3.16). Substituting the expression for Kg ðtÞ (Equation 3.73) into Equation 3.16 yields ( 1 ð X  1 vg  g 2 R  ½cothð
hvg =2kB TÞ þ 1 dt hwnonrad iT ¼ Z sl 4
h g
1


1 exp
itðV
vg Þ
C
h jtj GN ðtÞ ) 1 ð


1 þ ½cothð
hvg =2kB TÞ
1 dt exp
itðV þ vg Þ
C
h jtj GN ðtÞ :
1


1

ð3:75Þ

In writing Equation 3.75, we have included the single-mode generating function g G1 ðtÞ from (3.72) into GN ðtÞ. The former is reduced in the approximation used above g to G1 ðtÞ ¼ ð1
expð

hvg =kB TÞÞ
1 and its contribution to GN ðtÞ is approximately unity. From Equation 3.75, we deduce that the effect of the promoting mode is to decrease or increase the effective energy gap to
hðV vg Þ. For transitions that terminate in the first vibrational excited level ng ¼ 1 of the l electronic state, the

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50

energy gap is decreased by an amount
hvg . It is increased at about the same vibrational energy if the transition take place from the first vibrational excited level mg ¼ 1 of the s electronic state (see also Appendix K, case b). Since the second term in Equation 3.75 appears only at high temperatures, at low temperatures extending to room temperatures, it is negligibly small. This development bears a strong resemblance to vibronic induced optical transitions (see Section 7.1). The latter are characterized, as the nonradiative transitions, by a “false origin” at the energy
hvg off the 0–0 line and can be formally regarded as a symmetry-forbidden emission process, where the nontotally symmetry modes play the roles of promoting modes that induce the optical transition. We now return to the calculation of (3.75). Using Equations 3.16 and 3.43 and the transformation 1 ð 1 2c expð
cjxjÞ ¼ dy expð
ixyÞ 2 ; ð3:76Þ 2p y þ c2
1

by setting c ¼ C=
h for the decay time, the integrals in Equation 3.75 may now be represented in terms of the following lineshape function:
I 2 ðV vg Þ ¼

1 ð



dt exp
itðV vg Þ
cjtj GN ðtÞ


1

!   2 X 2c X X m1 m2 s exp

h mm vm =kB T dy 2 I2 n1 n2 y þ c2 m¼1 m n f g f g m m
1 " !# 1 ð 2 2 X X mm vsm
nm vlm þ y  dt exp
it V vg þ 1 ¼ 2p

1 ð

m¼1


1

¼

P P fmm g fnm g

I2

m1 n1

m¼1

2 X 2c exp

h mm vsm =kB T

!

m¼1

2 2 X X mm vsm
nm vlm V vg þ m¼1 m¼1  !  ð1Þ ð2Þ m2  D12 D12 b1 b2 : ;  n2  Dð12Þ Dð12Þ b12 b21 1

ð3:77Þ

!2 þ c2

2

To evaluate the last result we use the representations 1 ð

dteiVt ¼ 2pdðVÞ

ð3:78Þ

f ðxÞdðx
yÞdx ¼ f ðyÞ

ð3:79Þ


1

and 1 ð


1

with the latter being valid for each continuous function f ðxÞ.

3.3 Derivation of the Promoting Mode Factors Kg(t) and Ig(t)

Equation 3.75 in conjunction with Equation 3.77 provides an explicit expression for the nonradiative transition probability, where   the width c is assumed to be independent of the particular vibronic levels lnm . This expression is general, being applicable for both the statistical limit and for the small molecule cases. The Lorentzian function in (3.77) exhibits a sharp peak around P P V vg ¼
m mm vm þ m nm vm . The height of this peak is 2=c, while its width is given by c. The sharp peak of the Lorentzian function around P V ¼ vg
m ðmm vsm
nm vlm Þ very strongly favors the transitions toward those final resonance levels nm of the l electronic state, the so-called close coupled levels, for P which the quantity V vg þ m ðmm vsm
nm vlm Þ does not deviate from zero by an amount larger than c. Thus, the nonradiative transitions occur mainly toward those P final states, the energies of which El0 þ m nm
hvlm do not differ from the initial energy P
hðV vg þ m mm vsm Þ by more than DE þ Esm
Eln  c
h. Hence, the transition results in an approximate conservation of energy, the spread in energy being given by   c. Since the manifold of final states lnm becomes more densely spaced with the increase in the energy gap DE ¼ Es0
El0 between the two electronic states s and l in view of the large number of possible   combinations of fundamental vibrational modes, the background of states lnm consists of a dense distribution of vibronic states. In this case, for the so-called statistical limit
hc r
1 l , where rl is the density of the final states, the overlapping Lorentzian peaks in the lineshape function (3.77) yield a smooth function of energy, which is practically independent of c [111]. The situation is quite different in the small molecule case, where only a few or a single intramolecular vibrational mode acts as an accepting mode. This situation ðmÞ prevails in diatomic and triatomic molecules. For D12 < 1, it is possible that the largest contribution to the decay rate (originating for instance from the lowest vibrational level of the excited state) will not originate from the close lying levels of the final electronic state, but rather from off-resonance low-lying states (with smaller nm values), which contribute via the tail of the Lorentzian distribution. In the P case of accidental degeneracy, when ðV
m nm vlm Þ  c and provided that the ðmÞ displacements of the potential surfaces D12 are not too small, it is usually justified to simply drop the frequency term in the denominator of (3.77) to write (in the zero-temperature limit) w

 vg  g 2 0 Rsl I2 n1
hc

 0 : n2

ð3:80Þ

The main feature of this result is the dependence of w on the reciprocal of c. This situation corresponds to the Robinson–Frosch formula, applicable for the case of near-degeneracy in a small molecule embedded in an “inert” medium (Shpolskii matrix [112]). If the major contribution to (3.77) originates from the off-resonance P low-lying levels, that is, ðV
m nm vlm Þ c, the transition decay rate will vary as
hcð
hVÞ
2 I2 , so that the nonradiative transition probability is linear with the coupling to the medium, expressed in terms of the width
hc. The energy gap law for the nonradiative decay of a small molecule in a dense medium is not expected to be of general validity, as in the case for large molecules. The energy gap law w / I1 ðnÞ,

j51

j 3 Description of Radiationless Processes in Statistical Large Molecules

52

where n ffi V=vlm , applies only when the major contribution will originate from nearresonance coupling.

3.4 Radiationless Decay Rates of Initially Selected Vibronic States in Polyatomic Molecules

Until now the discussion of electronic relaxation processes in large molecules embedded in an inert medium implicitly assumed that the nonradiative decay occurs from a manifold of thermally equilibrated levels of the initial electronic state. In the present section, we present an expression for the nonradiative decay probability of a single vibronic level of a large isolated molecule.  We consider a molecule that has a set of zero-order “selectable” vibronic states ysma , belonging to electronic state s, which are optically accessible and well separated from each other. These states can undergo radiationless transitions to some other zero-order set of vibronic states n o ylnm that belong to a lower electronic state l of any multiplicity but of a density sufficiently large in the region of ysmm so that we can safely assume that the largemolecule statistical limit ensures (Figure 2.1). The crucial questions that now arise are (1) is the initial step in the vibrational relaxation dominated by an intramolecular process or by direct energy transfer to the host and (2) if it is intramolecular, what is the role of the host. In this respect it is important to mention the results of Amirav et al. [113] in the relaxation of isolated large molecules (tetracene, pentacene) cooled in a supersonic jet. They observed predominantly unrelaxed fluorescence from levels with vibrational energy smaller than 1200 cm
1 . This prove that a free molecule as large as pentacene with 102 vibrations does not have enough degrees of freedom to act as its own heat bath in the low vibrational energy range. Electronic relaxation in different excited vibronic levels corresponding to the same electronic configuration can be experimentally studied, provided that, as mentioned above, (1) single vibrational levels within the initial electronic state are populated and (2) the excited molecule decays nonradiatively on a timescale much shorter than the mean time between deactivating collisions or by other means such as infrared fluorescence [115]. For typical polyatomic molecules in the gas phase, a narrow-band  optical excitation pulse (as small as 1 A) and shorter relative to the genuine decay times c
1 s will result in the selection of a single vibronic state. Under these conditions, the emission lifetimes and quantum yields of individual vibronic levels have been measured for several organic molecules [115–121]. We assume further that upon optical excitation to the state ysma from the ground electronic state, only the ath vibrational mode (the optical mode) is excited with ma quanta. The theory can be easily generalized to include cases in which more than one vibrational mode is excited in the optical selected state ysmm . The rate for nonradiative decay of the initially prepared state can be written in the “golden rule” form 2 X  g 2 C wsl ðma Þ ¼ ; ð3:81Þ Vsm;ln 
h n ðEsma
Eln Þ2 þ C2

3.4 Radiationless Decay Rates of Initially Selected Vibronic States in Polyatomic Molecules g

where Vsm ln is the electronic factor that includes the q=qqg overlap integral arising from the promoting mode g (see Equation 3.3) and Esma
Eln ¼ DE þ
hma vsa

hna vla


X m


hnm vlm ;

with DE ¼ Es0
El0

being the energy gap between the lowest vibronic levels in the s and l electronic states. The summation on n includes the set of vibronic levels of the final electronic states jlna i and that of the remaining modes that are not finally in the optical mode in jli. At first sight, it appears that much of the formal derivation can be taken over s directly from the theory in Section 3.3 by specifying the complex variable w ¼ e
iva t (i.e., omitting the Boltzmann factor
hvsa =kB T in the exponent). We must dispel any apprehensions that Equation 3.81 differs from the thermally averaged rate expression (3.16) since thermal averaging must be avoided here. At this point we have to depart from the previous derivation of the thermal averaging of the generating function Kg ðtÞ for the promoting mode(s) g. The corresponding factor now takes the form ~ g ðtÞ ¼
h2 K

1 Xð ð ng


1

dqlg d
qlg i


s1=2 s 
l1=2 l 
s1=2 s  q b q x b q x b
q x qqsg smg g g lng g g smg g g


1=2  m n q  i l xlng blg
qlg wg g zgg ; q
qg

ð3:82Þ

s

where now wg ¼ e
ivg t and zg are still given by Equation 3.17. Interchanging orders of summation and integration in (3.82) and noting that qxav ¼ qq

) rffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffi(rffiffiffi va v vþ1 x
x 2 av
1 2 av þ 1
h

ð3:83Þ

(where use has been made of the fact that ba ¼ ðva =
hÞ1=2 ), Equation 3.82 can be considerably simplified to Equation 3.84 by assuming as above that for nontotally symmetric promoting modes Dg ¼ 0 and bg ¼ 1, ~ g ðtÞ ¼ K

 


hvg ðmg þ 1Þeivg t þ mg e
ivg t : 2

ð3:84Þ

The expression for the nonradiative decay rate of the optically prepared state jsma i now becomes wsl ðma Þ ¼

v     g  g 2 Rsl ðmg þ 1Þ~I 2 ðV
vg Þ þ mg~I 2 ðV þ vg Þ ; 2
h

ð3:85Þ

j53

j 3 Description of Radiationless Processes in Statistical Large Molecules

54

where for  gthe  present we have assumed that there is only one (promoting) mode for which Rsl  6¼ 0. The expression ~I2 ðV vg Þ in Equation 3.85 is written as ~I 2 ðV vg Þ ¼

X n

 I2

2c ðV vg þ ma vsa
na vla
nm vlm Þ2 þ c2  ðaÞ ðmÞ ! ma 0m  D12 D12 ba bm  ð12Þ ð12Þ ; bam bma na nm  Da Dm

ð3:86Þ

and differs from Equation 3.77 in that now the (normalized) Boltzmann weights (3.7) assigned to each state of energy Es;ma are omitted and the summation is restricted only to the final manifold of vibronic levels n ¼ ðna ; nm Þ. If the optical mode is parallel to those due to the rest of the molecule, it is natural to employ the partitioning technique to separate the optical mode from the other modes. Thus, the two-dimensional ID in Equation 3.86 decomposes in a convolution of two onedimensional IDs, each for the individual modes a and m. As is shown in Chapter 4, the theory is easily generalized to include cases in which more than one vibrational mode is optically selected and more than two accepting modes ða; mÞ are involved in the transition s ! l. Several comments should be made at this point. (a) The effective energy gap is modified by the energy of the promoting mode
h vg . The propensity rule for the promoting mode (previously derived for molecular nonradiative processes under thermally equilibrium conditions) is also satisfied for the decay of any single vibronic state [122]. (b) Furthermore, it is instructive to see how that analysis emerges in the limit ma ! 0. The nonradiative decay rate from the lowest vibronic level of the excited electronic state, whereupon mm ¼ 0 for all modes m, is obtained from Equation 3.85 in the form v   g  g 2
Rsl I 2 ðV
vg Þ; wsl ðma ¼ 0Þ ¼ ð3:87Þ 2
h since in this case ~I 2 ðV
vg Þ ¼
I 2 ðV
vg Þ. This result coincides, as expected, with the zero-temperature case of Equations 3.75 and 3.77. (c) Equations 3.85 and 3.86 provide some insight into the dependence of the nonradiative decay rate of a single vibronic state on the excess of electronic energy above the true electronic origin Es0 ,
hma vsa , or generally, X Ev ¼
hma vsa ð3:88Þ a

(if more than one mode is excited in the prepared initial state), which is converted into vibrational energy in the electronic state jli. Depending on the energy gap
hV ¼ DE, the nonradiative decay probability can either increase or decrease with increasing excess electronic energy. Precisely speaking, the expressions ~I2 ðV vg Þ in Equation 3.85 describe the density of states weighted with the intramolecular hV. The distribution I2 , which in the statistical limit increases with increasing


3.4 Radiationless Decay Rates of Initially Selected Vibronic States in Polyatomic Molecules

series in Equation 3.86 must be summed over all resonance levels na and nm that give the appreciable contributions to the transition rate wsl . In spite of this, it is also important to consider the influence of the geometry change of the excited state relative to the ground state on the properties of the intramolecular distribution. We shall state this more precisely in Section 4.2, where numerical results for the IDs and g multidimensional IDs are presented. The coupling element Rsl decreases  matrix  simultaneously with the density of states, rl in the lnm manifold increases with the growing excess of electronic energy Ev. Detailed calculations of the relative nonradiative rates for benzene and perdeuterobenzene have been given by Heller et al. [114]. Using numerical estimates for the double optical progression 6m 1n , they found that the nonradiative decay rate wsl ðm1 ; m6 ; n1 ; n6 Þ increases moderately with the increasing vibrational excitation within the first excited singlet state of benzene ðC6 H6 and C6 D6 Þ for the S1 ! T1 ð1 B2u ! 3 B1u Þ nonradiative transition. This is in agreement with the increase observed experimentally [115–120]. The spectroscopic data for both the optical ðC
CÞa1g mode a of benzene ðva ¼ v1 Þ and the nontotally symmetric bending mode ðv6 Þ used for the calculations, as well as the energy gap are v1 ¼ 923 cm
1 ; v6 ¼ 521 cm
1 ;
hV ¼ 8200 cm
1 ;

Dv1 ¼ vTa1
vSa1 ¼ 25 cm
1 ; Dv6 ¼ 50 cm
1 ;

X1 ¼ XC
Cðv1 Þ ¼ 0:025; X6 ¼ 0;

vg ¼ 1500 cm
1 ;

with D2a ¼ 2Xa . The calculations have accordingly been performed in the parallel mode approximation (the effect of Duschinsky rotation on the vibrational overlap for benzene is small), taking proper account of the effect of geometry ðD1 Þ and frequency changes ðDv1 ; Dv6 Þ. For large effective energy gaps, the increase in rl dominates and the nonradiative decay rate increases with increasing Esm [122]. For extremely large energy gaps ð
hV ffi 30; 000 cm
1 Þ, the increase in rl leads to sizable alternations of the nonradiative decay rates that may exhibit an exponential dependence on the excess vibrational energy [121, 122].

j55

j57

4 Calculational Methods for Intramolecular Distributions I1, I2, and IN

The principal subject of this chapter is the calculation of the intramolecular distribution (ID) [109]. The computational procedure utilizes the generating functions derived in the previous chapter. We begin in Section 4.1 with the derivation of expressions (in closed form) of the one-dimensional ID in the special (zero temperature) case w ¼ 0. Then, on this basis we discuss some important properties that we will formulate in terms of addition theorems. Likewise, we show in the next section how the analysis can again be generalized to deal with the finite temperature case ðw 6¼ 0Þ and derive a sequence of further intramolecular distributions. We conclude this section by listing the principal aims of the method described for the special case, where one of the parameters, for example, bm becomes unity. Finally, we treat more complicated case of mode mixing and show how the generating function technique can be used to calculate multidimensional intramolecular distributions (MID). We will see this idea exploited in some routines given in this chapter. They are realized in terms of recurrence equations. As an illustration of the practical utility of this calculation, we discuss mode mixing in an unusual distribution behavior. Our discussion of the symmetry or invariance properties of the MID in respect to the exchange of parameters may appear to be very long and circuitous to people familiar only with the customary “hit and run” treatment of transition processes given in many elementary quantum mechanics texts. Indeed, we shall find ultimately the standard results that are usually qualified by the assurance that they can be justified by the consideration of algebraic manipulation. Our aim is to provide such assurance.

4.1 The One-Dimensional Distribution I1(0, n; a, b)

The previous discussion will be concluded by considering the simpler case of the one-dimensional generating function, Equation 3.68, derived in Section 3.2.6. A convenient representation of the generating function (or GF), from which the

j 4 Calculational Methods for Intramolecular Distributions I , I , and I

58

1

2

N

distributions are obtained, is obtained by writing the latter as a product of two functions, which have many of the same structural features (see below) 2   3 1
z 2 6ð1þbÞ a 1
bz w 7 7 6   exp 6
 7 5 4 z
b
að1
zÞ 1
w exp 1
bz 1
bz G1 ðw;z;a;bÞ ¼ ð1
b2 Þ1=2
  1=2 : ½ð1
bzÞ ð1þbzÞ1=2 z
b zþb 1
w 1
w 1
bz 1þbz ð4:1Þ Here z and w are generally complex variables in the bidisc D1(0,1)  D1(0,1). b 1
b In writing (4.1), we have set in Equation 3.68 D2 ¼ a and ¼ b; so that a 1þb 1þb and b are real parameters within the intervals a  0 and
1 < b < 1. The obvious interpretation of the parameter a and b is that they represent the linear and quadratic interaction terms of a vibrational mode that is active in transitions between an excited electronic state with excited vibrational levels of a lower electronic state. The subscript 1 of the function G1 , denotes the first-order or the one-dimensional case and is associated with the power of the denominator of G1 . The function (4.1) is regular in the bidisc and its infinite series representation is G1 ðw; z : a; bÞ ¼

1 X 1 X

I1 ðm; n : a; bÞw m zn :

ð4:2Þ

m¼0 n¼0

It will be shown that for each integral values m  0, I1 ðm; n; a; bÞ is an integer valued probability distribution of n and vice versa. Hence, I1  0 and the sum 1 X

I1 ðm; n; a; bÞ ¼ 1;

ð4:3Þ

n¼0

holding for each value of m ¼ 0; 1; 2; 3 . . . . To investigate this point more clearly, we shall first examine the special case w ¼ 0, before studying it at w 6¼ 0. Physically, this case corresponds to the limit of zero temperature, where only the lowest (vibrational) level m ¼ 0 of the initial electronic state is occupied. For this case, (4.1) together with (4.2) can be written as

að1
zÞ exp 1 X 1
bz G1 ð0; z; a; bÞ ¼ ð1
b2 Þ1=2 ¼ I1 ð0; n; a; bÞzn : ð4:4Þ 1=2 ½ð1
bzÞ ð1 þ bzÞ n¼0 The function in the exponent of (4.4) constitutes a homographic transformation, which describes a mapping of the unit circle |z| ¼ 1 in a circle lying in the left z-half plane and that passes tangential to the point z ¼ 0. Therefore, G1 is regular over the unit circle D1 ð0; 1Þ and univalent if a=ð1 þ bÞ  p. For larger values of a, the function G1 ð0; z : a; bÞ becomes polyvalent. In general the mapping has a fix point, which is 1 at z ¼ 1; G1 ð0; 1; a; bÞ ¼ 1 and where max jG1 ð0; z; a; bÞj ¼ G1 ð0; 1; a; bÞ ¼ 1. 1 z2D ð0;1Þ

4.1 The One-Dimensional Distribution I1(0, n; a, b)

We expand the exponential term in (4.4) in a power series of z over the unit circle D1 ð0; 1Þ

 1 X
að1
zÞ að1
bÞ exp ck zk ; ð4:5Þ ¼ exp ð
aÞ exp z ¼ exp ð
aÞ 1
bz 1
bz k¼0 where c0 ¼ 1

and ck ¼

  k X 1 k
1 i¼1

i!

i
1

ai ð1
bÞi bk
i ;

k  1:

ð4:6Þ

Representing the denominator of G1 (or strictly, its regular branch of positive function value at z ¼ 0) in terms of binomial series, we obtain by forming the Chauchy product of the two series after rearranging and collecting terms of an , (    n X
1=2
1=2 2 1=2 I1 ð0; n; a; bÞ ¼ ð1
b Þ exp ð
aÞ ð
1Þn bn ð
1Þi n
i i i¼0 þ ð
1Þn
1 þ ð
1Þn
2 þ ð
1Þn
3

   n
a n
1 X
3=2
1=2 b ð
1Þi
1 n
i i
1 1! i¼1

   n
a2 n
2 X
1=2
5=2 ð
1Þi
2 b i
2 n
i 2! i¼2    n
a3 n
3 X
1=2
7=2 b ð
1Þi
3 i
3 n
i 3! i¼3

þ 

   n X
an
1
1=2 i
n þ 1
n þ 1=2 þ ð
1Þ b ð
1Þ i
n þ 1 n
i ðn
1Þ! i¼n
1 )   
an
n
1=2
1=2 ; ð4:7aÞ þ 0 0 n! 1

which satisfies the recurrence relation ðn þ 1Þ I1 ð0; n þ 1; a; bÞ ¼ ð
a þ nbÞ I1 ð0; n; a; bÞ þ b ð
a þ nbÞ I1 ð0; n
1; a; bÞ
ðn
1Þ b3 I1 ð0; n
2; a; bÞ;

ð4:8Þ

where
a ¼ a ð1
bÞ has been set. Equation 4.8 represents a preferred method for calculating the values of I1 ð0; n; a; bÞ, starting from the initial value I1 ð0; 0; a; bÞ ¼ ð1
b2 Þ1=2 exp ð
aÞ. The term in curly brackets in (4.7a) is a square, so that I1 ð0; n : a; bÞ can be written as a square of a polynomial of degree n in
a1=2 . Thus, we have

j59

j 4 Calculational Methods for Intramolecular Distributions I , I , and I

60

1

2

N

" ½n=2 #2 1 X n! 2 1=2 r n=2
r a I1 ð0; n; a; bÞ ¼ ð1
b Þ exp ð
aÞ : b
n! r¼0 2r ðn
2rÞ!r!

ð4:7bÞ

The right-hand series of (4.4) converges as z ! 1. Hence, by a simple substitution of z ¼ 1 it follows immediately from (4.4) that 1 X

I1 ð0; n; a; bÞ ¼ 1

ð4:9Þ

n¼0

as was claimed. The relation (4.9) of I1 ð0; n; a; bÞ can be also verified directly using (4.7a). 4.1.1 The Addition Theorem

If G1 ð0; z; a1 ; bÞ and G1 ð0; z; a2 ; bÞ are generating functions described by (4.4), the product G1 ð0; z; a1 ; bÞ G1 ð0; z; a2 ; bÞ ¼ G2 ð0; z; a1 þ a2 ; bÞ

is a generating function of the same kind, but of order (dimensionality) two. At the same time, as a consequence of (4.4), we also have ! ! 1 1 X X G2 ð0; z; a1 þ a2 ; bÞ ¼ I1 ð0; n1 ; a1 ; bÞzn1 I1 ð0; n2 ; a2 ; bÞzn2 n1 ¼0

¼

1 X

I2 ð0; n; a1 þ a2 ; bÞzn

n2 ¼0

ð4:10Þ

n¼0

Equating the coefficients of equal powers of z on both sides, we find the combined (joined) distribution of order two n X I2 ð0; n; a1 þ a2 ; bÞ ¼ I1 ð0; n1 ; a1 ; bÞ I1 ð0; n
n1 ; a2 ; bÞ ð4:11Þ n1 ¼0

as a convolution of two probability distributions of dimensionality one. Equation 4.11 expresses the so-called addition theorem in respect to the parameter a. The twodimensional probability distribution on the left side of (4.11) can simply be written as a convolution of one-dimensional probability distributions, the a parameter of which is the sum a ¼ a1 þ a2 . It must be emphasized that the one-dimensional distributions in (4.11) have the same b parameter (for arguments). If we define

að1
zÞ exp 1 X 1
bz Gi ð0; z; a; bÞ ¼ ð1
b2 Þi=2 ¼ Ii ð0; n; a; bÞzn ; ð4:12Þ ½ð1
bzÞ ð1 þ bzÞi=2 n¼0 where i is an arbitrary positive integer, the following addition theorem can be considered and similarly calculated:

4.2 The Distributions I1(m, n; a, b) n X

Ii ð0; n1 ; a1 ; bÞ Ij ð0; n
n1 ; a2 ; bÞ ¼ Ii þ j ð0; n; a1 þ a2 ; bÞ:

ð4:13Þ

n1 ¼0

Because of the convolution form of (4.13), it is obvious that the sum of Ii þ j ð0; n; a; bÞ over n equals 1. As before, the distribution addition theorem (4.13) is obtained by setting a1 þ a2 ¼ a and by summing the orders i þ j to obtain the order of the convoluted distribution. 4.2 The Distributions I1(m, n; a, b) 4.2.1 Derivation of I1(m, n; a, b)

Having determined the probability distribution for m ¼ 0, I1 ð0; n; a; bÞ, we proceed to the general case m 6¼ 0. For this purpose, the w-dependent factor of (4.1) must be expanded in power of wm . This can be done by analogy to the expansion of G1 ð0; z; a; bÞ, taking into account the following assignment: z ! w; a ð1
zÞ ; 1
bz   z
b b! : 1
bz a!

Hence

 1
b ! 1


ð4:14Þ

   z
b 1
z ¼ ð1 þ bÞ : 1
bz 1
bz

ð4:15Þ

With these substitutions, the kernel of G1 ð0; z : a; bÞ can be brought to the form G1 ðw; z; a; bÞ 2   3 1
z 2
 6að1 þ bÞ 1
bz w 7 a ð1
bÞ 7 6   ð4:16Þ exp z ! exp 6 7; z
b 5 4 1
bz 1
w 1
bz or after expansion in power series of z and w, respectively, as !   1 n 1 X X X 1 n
1 k n
k k a ð1
bÞ b zn ! 1 þ 1þ Am ðzÞw m ; k
1 k! n¼1 m¼1 k¼1

ð4:17Þ

where    2k   m X 1 m
1 k z
b m
k k 1
z a ð1 þ bÞ Am ðzÞ ¼ ; k! k
1 1
bz 1
bz

m  1:

k¼1

ð4:18Þ

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j 4 Calculational Methods for Intramolecular Distributions I , I , and I

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1

2

N

Note that by the assignment (4.14), the parameter a becomes complex in the   að1
zÞ z
b z-plane, but with Re  0, as it should be. Accordingly,
1  Re 1 1
bz 1
bz for jzj  1, similar to
1 < b < 1. Expanding the regular branch of



1=2 X 1 zþb ¼ Bn ðzÞ w n ; ð4:19Þ w 1 þ bz n¼0   zb  1 assume positive function values) in terms of (which for
1  Re 1  bz binomial series, where 1


z
b w 1
bz



Bn ðzÞ ¼ ð
1Þn

1


     n  X z
b k z þ b n
k
1=2
1=2 n
k

k

k¼0

1 þ bz

1
bz

;

ð4:20Þ

by forming the Chauchy product of the two series above, we obtain   3 1
z 2 6ð1 þ bÞa 1
bz w 7 7 6   exp6 7 5 4 z
b 1
w 1
bz
  1=2 ¼ z
b zþb 1
w 1
w 1
bz 1 þ bz 2

¼

1þ

1 X

! n

An ðzÞw Þ

n¼1

1 X m¼0

1 X

! Bn ðzÞw

n

n¼0

ðmÞ

C1 ðz; a; bÞw m ; jw j < 1;

ð4:21Þ

where ðmÞ

C1 ðz; a; bÞ ¼ Bm ðzÞ þ

m X

Ak ðzÞBm
k ðzÞ:

ð4:22Þ

k¼1

After substitution of (4.18) and (4.20) in (4.22) and subsequent lengthy derivation, we have ðmÞ C1 ðz; a; bÞ

¼ ð
1Þ

m

     m  X z
b m
i z þ b i
1=2
1=2 m
i

i¼0

m
1

þ ð
1Þ  

i

1
bz

1 þ bz

    m  a 1
z 2 X
1=2
3=2 i
1 1! 1
bz i¼1 m
i

_

   z
b m
i z þ b i
1 1
bz 1 þ bz

4.2 The Distributions I1(m, n; a, b)

   m     _2  a 1
z 4 X z
b m
i z þ b i
2
1=2
5=2 þ ð
1Þm
2 i
2 2! 1
bz i¼2 m
i 1
bz 1 þ bz þ ð
1Þm
3

_3

a 3!



   m     1
z 6 X z
b m
i z þ b i
3
1=2
7=2 i
3 1
bz i¼3 m
i 1
bz 1 þ bz

þ       m  a 1
z 2m
2 X z
b m
i
1=2
m þ 1=2 þ ð
1Þ i
m þ 1 m
i 1
bz ðm
1Þ! 1
bz i¼m
1 1

 

_ m
1

  _m    z þ b i
m þ 1 1
z 2m
m
1=2
1=2 a þ : 0 0 m! 1
bz 1 þ bz

ð4:23Þ

where _ a ¼ að1 þ bÞ. Substituting (4.4) and (4.21) in (4.1) and equating terms of w m in the two resulting series gives exp ð1
b2 Þ1=2



að1
zÞ 1
bz

ð1
b2 z2 Þ1=2

ðmÞ

C1 ðz; a; bÞ ¼

1 X

I1 ðm; n; a; bÞzn :

ð4:24Þ

n¼0

Using Leibnitz’s formula for the nth derivative of a product, we obtain "   # n X 1 d k ðmÞ I1 ðm; n; a; bÞ ¼ I1 ð0; n
k; a; bÞ C1 ð0; a; bÞ ; k! dz k¼0

ð4:25Þ

in which (4.4) has been used. Note that (4.24) and (4.25) are valid for m ¼ 0, since ð0Þ C1 ðz; a; bÞ ¼ 1. The left-hand side of (4.24) can be regarded as a generating function of I1 ðm; n; a; bÞ for each integer m  0. Summing both sides of (4.25) over n gives !   1 1 n X X X 1 d k ðmÞ I1 ðm; n; a; bÞ ¼ I1 ð0; n
k; a; bÞ C1 ð0; a; bÞ k! dz n¼0 n¼0 k¼0 ! !   1 1 X X 1 d n ðmÞ I1 ð0; n; a; bÞ C1 ð0; a; bÞ ¼ n! dz n¼0 n¼0   1 X 1 d n ðmÞ C1 ð0; a; bÞ; ð4:26Þ ¼ n! dz n¼0 where we have used the fact that I1 ð0; n; a; bÞ satisfies (4.9). On the other hand, after simply substituting z ¼ 1 in (4.23) and (4.24), the interpretation yields 1 X n¼0

ðmÞ

I1 ðm; n; a; bÞ ¼ C1 ð1; a; bÞ ¼ 1;

m0

ð4:27Þ

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In view of the formulas (4.26) and (4.27), it is natural to expect simple relations ðmÞ among the developing coefficients of C1 ðzÞ appearing in the sum (4.26). To ðmÞ demonstrate this, it suffices to note that the functions C1 ðzÞ appearing in (4.24) are linearly connected with coefficients dependent of z, that is, "        1
z 2 1 z
b 1 ðm þ 1Þ ðm þ 1Þ C1 ðzÞ ¼ _ a þ 2m þ þ mþ 1
bz 2 1
bz 2 "      zþb 1
z 2 z þ b ðmÞ _ C1 ðzÞ
a 1 þ bz 1
bz 1 þ bz

 

      1 z
b 2 1 z
b þ 2m
þ m
2 1
bz 2 1
bz    zþb z
b 2 ðm
1Þ  C1 ðzÞ þ ðm
1Þ 1 þ bz 1
bz 

 

 zþb ðm
2Þ ðzÞ: C 1 þ bz 1

ð4:28Þ

This may be formulated more compactly as ðm þ 1Þ

ðm þ 1Þ C1

ðmÞ

ðmÞ

ðmÞ

ðm
1Þ

ðzÞ ¼ a0 ðz; a; bÞ C1 ðzÞ þ a1 ðz; a; bÞ C1 ðmÞ

ðm
2Þ

þ a2 ðz; bÞC1

ðzÞ:

ðzÞ ð4:29Þ ðmÞ

We assume here and throughout of the rest of this section that C1 ðzÞ ¼ The relation (4.29) provides a convenient starting point in the determination of the distributions I1 ðm:n; a; bÞ (see below). One can obtain not only the ðmÞ functions C1 ðzÞ and their derivatives in succession starting with the lowest, ð0Þ specifically C1 ðzÞ ¼ 1, but also the other quantities of interest from them. The ðmÞ essential formulas are found in Ref. [109] where the derivatives of ai ðzÞ (i ¼ 0, 1, 2) at z ¼ 0 are also given. ðmÞ C1 ðz; a; bÞ.

Corollary Comparing (4.7a) with (4.23), we have the relation ðnÞ

I1 ð0; n; a; bÞ ¼ ð1
b2 Þ1=2 exp ð
aÞC1 ð0; a;
bÞ ¼ I1 ðn; 0; a;
bÞ:

ð4:30Þ

It may be shown that the general symmetry property I1 ðm; n; a; bÞ ¼ I1 ðn; m; a;
bÞ

ð4:31Þ

holds. This follows immediately from the symmetry property of G1 ðw; z; a; bÞ ¼ G1 ðz; w; a;
bÞ.

4.2 The Distributions I1(m, n; a, b)

This is an important conclusion. It represents a precise statement of the related conjecture made following Equation 3.65. The meaning of Equation 4.31 is essentially that a transition in emission is equivalent to a transition in absorption when b ¼ 0. Otherwise, there exists no mirror image between transition in emission and absorption. 4.2.2 The Addition Theorem for I1(m, n; a, b)

For completeness, we use (4.1) and (4.2) to derive the addition theorem for I1 ðm; n; a; bÞ. In close analogy to the case (m ¼ 0), we have G1 ðw; z; a1 ; bÞ G1 ðw; z; a2 ; bÞ ¼ G2 ðw; z; a1 þ a2 ; bÞ

ð4:32Þ

and referring to (4.2), we see that 1 X 1 X

G2 ðw; z; a1 þ a2 ; bÞ ¼

! I1 ðm1 ; n1 ; a1 ; bÞ w m1 zn1

m1 ¼0 n1 ¼0 1 X 1 X



! m 2 n2

I1 ðm2 ; n2 ; a2 ; bÞ w z

m2 ¼0 n2 ¼0

¼

1 X 1 X I2 ðm; n; a1 þ a2 ; bÞ w m zn ;

ð4:33Þ

m¼0 n¼0

where X

X

I2 ðm; n; a1 þ a2 ; bÞ ¼

I1 ðm1 ; n1 ; a1 ; bÞ I1 ðm2 ; n2 ; a2 ; bÞ

ð4:34Þ

m1 þ m2 ¼m n1 þ n2 ¼n

is again the convolution I1  I1 . As before, the parameters a1 and a2 and the orders or dimensionalities of the distributions on the right side are summed to give the a parameter and order of the convoluted distribution. The successive application of (4.34) gives IN ¼ I1  I1  . . .  I 1 :

ð4:35Þ

They have the norm 1 X n¼0

IN m; n;

N X k¼1

! ak ; b

 ¼

 N þ m
1 : m

ð4:36Þ

4.2.3 The Recurrence Formula

As already mentioned, the optimum strategy for most simply finding the values of I1 ð0; n; a; bÞ is given by applying the recurrence (4.8). The same pertains to the

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distribution I1 ðm; n; a; bÞ. The corresponding recurrence may be derived by using (4.25) and (4.28). Indeed, starting from the relation (4.29), we first determine the nth ðm þ 1Þ ðzÞ for z ¼ 0 as derivative of C1  n  n
k 2 n  k X X d d d ðm þ 1Þ ðmÞ ðm þ 1Þ C1 ð0Þ=n! ¼ ai ð0Þ dz dz dz i¼0 k¼0 ! ðm
iÞ

C1

ð0Þ=k!ðn
kÞ!

ð4:37Þ

and  substitute the result in (4.25). After collecting terms containing the same factor  d k ðmÞ ai ð0Þ=k!, this yields dz   d ðmÞ ðmÞ ðm þ 1Þ I1 ðm þ 1; nÞ ¼ a0 ð0Þ I1 ðm; nÞ þ a ð0Þ I1 ðm; n
1Þ dz 0   1 d 2 ðmÞ þ a0 ð0Þ I1 ðm; n
2Þ þ    2! dz þ

  1 d n ðmÞ ðmÞ a0 ð0Þ I1 ðm; 0Þ þ a1 ð0Þ I1 ðm
1; nÞ n! dz 

þ

   d ðmÞ 1 d 2 a1 ð0ÞI1 ðm
1; n
1Þ þ dz 2! dz ðmÞ

 a1 ð0Þ I1 ðm
1; n
2Þ þ    þ

  1 d n ðmÞ a1 ð0Þ n! dz 

 d ðmÞ a ð0Þ dz 2

ðmÞ  I1 ðm
1; 0Þ þ a2 ð0Þ I1 ðm
2; nÞ þ

 I1 ðm
2; n
1Þ þ

þ  þ

  1 d 2 ðmÞ a2 ð0Þ I1 ðm
2; n
2Þ 2! dz

  1 d n ðmÞ a2 ð0Þ I1 ðm
2; 0Þ; n! dz



ð4:38Þ

 d k ðmÞ ai ð0Þ=k!, dz ði ¼ 0; 1; 2Þ are given explicitly in Ref. [109]. Equation 4.38 enables us to determine I1 ðm; nÞ completely for all values m and n, provided that the values of I1 ð0; nÞ are already available.

where we have assumed I1 ðm; n; a; bÞ ¼ I1 ðm; nÞ. The coefficients

4.2 The Distributions I1(m, n; a, b)

4.2.4 Case b ¼ 0

When b ¼ 0; a special case of the formulas of the preceding section exists. In this case, the problem simplifies considerably, since an n!

I1 ð0; n; a; 0Þ ¼ exp ð
aÞ

ð4:39Þ

reduces to the Poisson distribution of probability theory with mean a. In comparison with the latter, I1 ðm; n; a; bÞ gives skew line shapes, with a skewness to lower n or to higher n values, depending on whether b < 0 or b > 0 (see Figure 4.2). We can use the exact expression (4.23), but for b ¼ 0 ðmÞ

C1 ðz; a; 0Þ ¼ zm þ  þ



m m

m 1





a ð1
zÞ2 zm
1 þ 1!



m 2



a2 ð1
zÞ4 zm
2 þ    2!

am ð1
zÞ2m ; m!

ð4:40Þ

   x þ k
1
x for x > 0. ¼ ð
1Þk k k Differentiating in respect to z and substituting z ¼ 0, we have 

where use has been made of the fact that

     1 d n ðmÞ 2 m a C1 ð0; a; 0Þ ¼ ð
1Þn
m þ 1 þð
1Þn
m þ 2 n
m þ 1 1 1! n! dz    

4 n
m þ2



   m a2 2m
2 þ     þð
1Þn
1 2 2! n
1

 m
1    m a m m a 2m þ ð
1Þn þ dmn ; m
1 ðm
1Þ! m m! n



n

ð4:41Þ

d ðmÞ C1 ð0; a;0Þ are given in dz Ref. [109] for several of the values of m in terms of polynomials of a. It may be where dmn is the Kronecker delta. The derivatives n!1

seen that the right side of Equation 4.41 is symmetrically distributed with respect ðmÞ to n ¼ m (where n denotes the nth derivative of C1 ðz;a:0Þ) and the sum over all derivatives for a given m equals 1, as Equation 4.26 clearly shows. Moreover, the ðmÞ number of the derivatives of C1 ðz; a;0Þ for each m is finite (see also Table 1 in Ref. [109]). Substitution of (4.39) and (4.41) into (4.25) leads after rearrangement to   n! m
n  m
n 2 I1 ðm; n; a;0Þ ¼ exp ð
aÞ Ln ðaÞ ; a m!

ð4:42Þ

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where Lan ðaÞ ¼

n X Cða þ n þ 1Þ ð
aÞl ; Cðaþ l þ 1Þ l!ðn
lÞ! l¼0

a >
1

ð4:43Þ

is the Laguerre polynomial.1) In expression (4.42), it is assumed that m  n. If n > m, simply exchange m and n. 4.2.5 Case b 6¼ 0

As with the result (4.42), an explicit representation of I1 ðm; n; a; bÞ is obtained for the general case b 6¼ 0. The result is by similar arguments leading to Equation 4.42 " ½m=2 ½n=2 m!n! X X ð
1Þi bi þ r 2 1=2 I1 ðm; n; a; bÞ ¼ ð1
b Þ exp ð
aÞ m þ n ðm
2iÞ!i!ðn
2rÞ!r! 2 i¼0 r¼0 #2 m
n
2ði
rÞ n
2r m
n
2ði
rÞ
C Ln
2r ðaÞ ; ðn
2rÞ!A

ð4:44aÞ

where _ 1=2
¼ ½2að1 þ bÞ1=2 ¼ ð2a Þ ; A


¼
½2að1
bÞ1=2 ¼
ð2
B aÞ1=2 ;

C ¼ 2½ð1
bÞð1 þ bÞ1=2

ð4:45Þ

m
n
2ði
rÞ Ln
2r ðaÞ

and are again Laguerre polynomials. This is computationally useful formula. As before, in deriving expression (4.44a), it is assumed that m  n and m
2i  n
2r (there is a case in which one can have m
2i < n
2r for one or finitely limited values of i). In this case, replace the corresponding factor(s) on the right side of (4.44a) ð*Þ

m
n
2ði
rÞ


ðn
2rÞ!A

m
n
2ði
rÞ

C n
2r Ln
2r

ðaÞ

by
ðm
2iÞ!B

n
m
2ðr
iÞ

n
m
2ðr
iÞ

Cm
2i Lm
2i

ðaÞ:

Analogously if m < n, 2 1=2

I1 ðm; n; a; bÞ ¼ ð1
b Þ

m!n! exp ð
aÞ m þ n 2

" ½m=2 ½n=2 XX i¼0 r¼0

ð
1Þi bi þ r ðm
2iÞ!i!ðn
2rÞ!r! #2


n
m
2ðr
iÞ

ðm
2iÞ!B

n
m
2ðr
iÞ Cm
2i Lm
2i ðaÞ

ð4:44bÞ

provided n
2r  m
2i holds for all nonnegative integers i and r. Otherwise, perform an exchange of the appropriate factors appearing in (4.44b) according to R1 1) It is easily to verify from Equation 4.42 that I1 ðm; n; a 0Þda ¼ 1. 0

4.2 The Distributions I1(m, n; a, b)

( ), but in the reverse order. The terms within the square brackets in (4.44) are polynomials in a, the coefficients of which are polynomials of b. We have thus shown m is a square that each member of the sequence of I1 ðm; n; a; bÞ for different integers  m!n! 1=2 of a polynomial multiplied by a positive factor ð1
b2 Þ exp ð
aÞ m þ n . As 2 mentioned earlier, the result (4.44)is a generalization of Equation 4.42, to which it reduces if we set b equal to zero. The properties (4.27) and (4.44) satisfy the requirements for a definition of I1 ðm; n; a; bÞ. 4.2.6 Numerical Results

Finally, we illustrate the role of the parameters a (or D2 ) and b (or b) on the distribution I1 ðm; n; a; bÞ in the following figures. We remark that I1 ðm; n; a; bÞ is an integer-valued function, but it is often convenient to have a representation of I1 ðm; n; a; bÞ transcribed to the form appropriate when n is continuous. This can be carried out by analytical continuation. Thus, the graphic presentation of I1 ðm; n; a; bÞ in such continuous form for preselected m levels and a large a parameter looks as shown in Figure 4.1. The curves are smooth functions of the variable n and exhibit m þ 1 peaks along the n-axis. This behavior is usual for the case of strongly coupled states, that is, those states with a large difference in their geometrical equilibrium positions a. The b (or b) dependence is very slight. The b parameter solely influences the skewness of the curves (see Figure 4.2). The situation for weakly coupled states, where the displacement parameter a is small, that is, a < 1, is quite different. If we plot I1 ðm; n; a; bÞ versus n, we find the result shown in Figure 4.3. The distribution I1 ðm; n; a; bÞ varies from a value of nearly one by a factor of 1010 or more, as n increases from zero to n ¼ 10 and the analytical continuation of I1 ðm; n; a; bÞ exhibits discontinuities (jumbs) at even integrals n. The surprise comes in Figure 4.4, which indicates that the effect of b (or b) on the distribution I1 ðm; n; a; bÞ is quite remarkable. The distribution I1 ð0; n; a; bÞ is relative to the Poisson distribution I1 ð0; n; a; 0Þ typically 10 000 times (ore more) greater, when the parameter a continuously drops. Moreover, the quotient between these distributions becomes greater still as the frequency parameter b ¼ vs =vl drops or deviates from unity to a slight extend. This of course should come as no surprise and may be seen in somewhat greater detail as follow. Since the a (or D) and b (or b) parameters are related to the linear and quadratic interaction terms of the accepting mode that is active in the electronic transition, this means that with a large linear interaction term the influence of the quadratic term is only slight. On the other hand, the latter frequency effect becomes very strong for weakly coupled states having a small a parameter. Considerable insight has been obtained into the properties of the IDs in Figures 4.5–4.8, where the multidimensional distribution Ii ðm; n; a; bÞ of i accepting modes is represented, respectively, for the cases of moderately strong-coupled states ! i X aj and weakly coupled states with a ¼ 0:3. The i vibrational having a ¼ 2 a ¼ j¼0

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Figure 4.1 The intramolecular distribution I1 ðm; n; a; bÞ as function of n for preselected levels m of the electronic excited state (strongly coupled states a ¼ 12; b ¼ 0:05).

modes belong either to an i-fold degenerate mode or to modes of different frequencies, but having the same frequency factor b ðbÞ. In the first case of i-fold degenerate mode, the norm, Equation from the fact that the degeneracy of the mth  4.36, results  i þ m
1 vibrational level is just . The enormous rise of the area under the curves m in Figures 4.7 and 4.8 is confirmed by the fact that the norm of the ID and hence the density of states increases with i. Thus, the multidimensional ID expresses the interplay of vibrational overlap and density of states. This, in turn, reflects the dependence of the multidimensionless ID on the density of (vibrational) states. This behavior is essential for our calculations; it simplifies the discussion about the irreversibility. The multidimensional distribution Ii ðm; n; a; bÞ for weakly coupled electronic states will also increase with deviating b from zero (or when b deviates from unity), but this effect (typically described by a power law) is quite overwhelmed by the more rapid increase in the density of states with increasing i.

4.3 Calculation of the Multidimensional Distribution

Figure 4.2 The intramolecular distribution for the lowest m levels ðm ¼ 0; 1; 2Þ and a value of a ¼ 12:5. When the parameter a is large, the b-dependence of I1 ðm; n; a; bÞ is weak.

4.3 Calculation of the Multidimensional Distribution  0 1 ð2Þ b1 b2 m1 ; m2  Dð1Þ 12 D12  A ; I2 @  n1 ; n2  Dð12Þ Dð12Þ b12 b21 1 2 4.3.1 Preliminary Consideration

The multidimensional distribution IN ðm; n; a; bÞ, which  we described in the previous section, is appropriate for a set of separable modes nm , especially for components  of a degenerate vibration. Here nm represents the set of choices of the occupation numbers n1 ; n2 ; . . . ; nN , (and correspondingly for the excited state), where each nm can assume all integer values between zero and n, and for which n1 þ n2 þ    þ nN ¼ n. The individual nm can be treated as the occupation number of the mth

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Figure 4.3 Same as Figure 4.1, for weakly coupled states ða ¼ 0:15; b ¼ 0:025Þ.

separate vibrational mode. The characteristic behavior of such multidimensional distributions is that they arise via a convolution of one-dimensional distributions, each of which is associated with an individual vibrational mode. Quite often we are dealing with systems involving many mixed (nonseparable) vibrational components, with these components in a state vector characterized by a set of choices of n1 ; n2 ; . . . ; nN , where each nm can take all integral values  0. The formalism just outlined is applied to this case, where many ðNÞ vibrational modes of a molecule are not separable modes from each other in the sense that some or all of them are connected by a linear relationship, that is, by the Duschinsky rotation. The multidimensional distributions of this kind differ in an essential manner from those of separable vibrational modes. The latter are highly complex compared to those for separable modes. This results, as we have seen, from the increasing number of parameters appearing in the generating function and, as mentioned above, the parameters becoming dependent. Therefore, in this case, we cannot hope to obtain solutions in closed form for a large number of vibrational degrees of freedom as in the one-dimensional case and it is necessary to develop suitable methods. These can be formulated in terms of recurrence equations (there is one for each occupation

4.3 Calculation of the Multidimensional Distribution

Figure 4.4 The ratio I1 ð0; n; a; bÞ=I1 ð0; n; a; 0Þ (of the distribution (4.7b) to the Poisson distribution (4.39)) versus displacement parameter D at b ¼ 0:18; 0:11; 0:05 for a preselected levels n ¼ 10. Note the strong

dependence of I1 ð0; n; a; bÞ on the frequency 1
b vl
vs for weakly coupled change b ¼ ¼ l 1 þ b v þ vs b states, when the shift D or a ¼ D2 is small. 1þb

Figure 4.5 The multidimensional distribution Ii ð0; n; a; bÞ (in the parallel-mode approximation) for moderately strongly coupled states a ¼ 2; b ¼
0:2 and different orders of degeneracy i.

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Figure 4.6 Same as Figure 4.5 for weakly coupled electronic states a ¼ 0:045 and b ¼ 0:025.

number nm and correspondingly mm ). This and the related problems will be discussed in the next section for the case of N ¼ 2 [123]. In the last section, the result of Section 4.3 will be extended to the case of N-vibrational components, where N is any integral value [123].

Figure 4.7 The multidimensional distribution Ii ð1; n; a; bÞ in the parallel-mode approximation for moderately strongly coupled electronic states a ¼ 2 and b ¼
0:2. Note how the degeneracy i of the modes influences the distribution extensively.

4.3 Calculation of the Multidimensional Distribution

Figure 4.8 Same as Figure 4.7, for weakly coupled states a ¼ 0:045 and b ¼ 0:025. For even numbers n the analytical continuation of the ID has a discontinuity (jumb in Ii , as n crosses an even integer).

4.3.2 Derivation of Recurrence Equations

We begin by recalling the two-dimensional generating function of two nonseparable accepting modes given by Equations 3.31–3.37 namely, G2

 ð1Þ ! ð2Þ D  12 D12 b1 b2 w1 ; w2 ; z1 ; z2  ;  Dð12Þ Dð2Þ b12 b21 1 12

1=2 1=2

¼ 4b1 b2


 Aðw1 ; w2 ; z1 ; z2 Þ exp
B1 ðw1 ; w2 ; z1 ; z2 Þ ½B1 ðw1 ; w2 ; z1 ; z2 Þ B2 ðw1 ; w2 ; z1 ; z2 Þ1=2

;

ð4:45Þ

and consider a function of four complex variables w1 ; w2 ; z1 ; z2 and the eight real parameters written explicitly on the left side of Equation 4.45. Based on this equation and the following Equations 3.32–3.37 in Chapter 3, we try to obtain a solution of our problem in terms of recurrence equations. If we differentiate Equation 4.45 with

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respect to one of the four complex variables, for example, w1 , we obtain

 qG2 qB1 qA q 2 2B1 B2 ¼ G2 2AB2
2B1 B2
B1 ðB1 B2 Þ : qw1 qw1 qw1 qw1

ð4:46Þ

We proceed with an expansion in powers of w1 w2 z1 z2 of the functions contained in Equation 4.46 and find (see Appendix B) 3 X

j

cij;kl w1i w2 zk1 zl2 ;

ð4:47Þ

2 X 3 X q ð1Þ j ðB1 B2 Þ ¼ gij;kl w1i w2 zk1 zl2 ; qw1 i¼0 j;k;l¼0

ð4:48Þ

B21 B2 ¼

i;j;k;l¼0

B1

and AB2

1 X 3 X qB1 qA ð1Þ j
B1 B2 ¼ dij;kl w1i w2 zk1 zl2 : qw1 qw1 i¼0 j;k;l¼0

ð4:49Þ

Substituting Equation 4.47 through (4.49) and the expansion of the GF G2 (see Equation 3.43) into Equation 4.46 and equating terms of the same power in w1 w2 z1 z2 , we obtain the following equation: (     3 3 X X m1 þ 1; m2
j m1 ; m2
j ð1Þ ð1Þ ð2doj;kl
g0j;kl Þ I2 2ðm1 þ 1Þ c0j;kl I2 ¼ n1
k; n2
l n1
k; n2
l j;k;l¼0 j;k;l¼0 
2m1 c1j;kl I2

m1 ; m2
j n1
k; n2
l 


2ðm1
1Þc2j;kl I2



ð1Þ

ð1Þ

þ ð2d1j;kl
g1j;kl Þ I2



m1
1; m2
j n1
k; n2
l



  ) m1
1; m2
j m1
2; m2
j ;
2ðm1
1Þc3j;kl I2 n1
k; n2
l n1
k; n2
l

ð4:50Þ

where in the last row, the identity g2j;kl ¼ 2c3j;kl has been used (see Appendix B). For simplicity of notation, we have set I2

m1 ;

m2

n1 ;

n2

! ¼ I2

 ! ð2Þ b1 b2 m1 ; m2  Dð1Þ 12 D12 : ;  n1 ; n2  Dð12Þ Dð12Þ b12 b21 2 1

For computational convenience, we have arranged the term in Equation 4.50 by   m1 þ 1; m02 , then those that contain collecting together all those that contain I2 n01 ; n02   0 m1 ; m 2 , and so on. Equation 4.50 constitutes a four-term recurrence relation I1 n01 ; n02 (in each of the quantum numbers m1 ; m2 ; n1 ; n2 ), that is, it allows one to express the     m1 ; m2 0; m2
j arbitrary value I2 in terms of all preceding values I2 n1 ; n2 n1
k; n2
l

4.3 Calculation of the Multidimensional Distribution

where j; k; l ¼ 0; 1; 2, and 3. Therefore, in order to solve the problem fully, three additional recurrence equations are necessary. In particular, we need a recurrence equation that raises the number m2 to m2 þ 1. In order to derive the latter equation, we can proceed as we did previously with the exception that the GF (4.45), as well as its expansion, must be differentiated with respect to w2 . Alternatively, it is simpler to recall the invariance of I2 with respect to the interchange of modes discussed in Section 3.2.5 and the symmetry properties of the coefficients involved in Equation 4.50 (see Appendix B). In view of these symmetries, the desired recurrence equation can be obtained directly from Equation 4.50 as follows: i. Apply the operation F (flip) to both sides of Equation 4.50 taking consideration of the properties of the coefficients appearing in Equation 4.50 (see Appendix B); ii. Invoking the symmetry of I2 , reverse the order of the integer variables there and ð2Þ ð1Þ ð12Þ ð12Þ simultaneously change D12 to D12 ; D2 to D1 and b2 to b1 , and so on iii. Finally, replace m1 by m2 ; n1 by n2 ; i by j; k by l, and vice versa. Thus, ! ! ( 3 3 X X m1
i; m2 þ 1 m1
i; m2 ð2Þ ð2Þ ¼ ð2di0;kl
gi0;kl ÞI2 2ðm2 þ 1Þ ci0;kl I2 n1
k; n2
l n1
k; n2
l i;k;l¼0 i;k;l¼0 ! ! m1
i; m2 m1
i; m2
1 ð2Þ ð2Þ
2m2 ci1;kl I2 þ ð2di1;kl
gi1;kl ÞI2 n1
k; n2
l n1
k; n2
l ! m1
i; m2
1
2ðm2
1Þci2;kl I2 n1
k; n2
l !) m1
i; m2
2 ð4:51Þ
2ðm2
1Þci3;kl I2 n1
k; n2
l   m 1 ; m2 Equations 4.50 and 4.51 allow us to determine I2 for all occupation n1 ; n2   0; 0 are already available. In numbers m1 ; m2 ; n1 ; n2 , provided the values of I2 n1 ; n2 this manner, we have reached the point in our procedure where only values of I2 for quantum numbers m1 ¼ m2 ¼ 0 remain to be calculated. Physically, such a situation is realized, for example, at low temperatures, where only the vibrationless level m1 ¼ m2 ¼ 0 of the initial electronic state is populated. In this limit, the GF (4.45) can be rewritten (by taking w1 ¼ w2 ¼ 0) in a more explicit manner (see Equation 3.42)
 d00;00 þ d00;10 z1 þ d00;01 z2 þ d00;11 z1 z2 exp
a00;00 þ a00;10 z1 þ a00;01 z2 þ d00;11 z1 z2 ð0Þ 1=2 1=2 G2 ¼ 4b1 b2 h ð4:52Þ i1=2 : ða00;00 þ a00;11 z1 z2 Þ2
ða00;10 z1 þ a00;01 z2 Þ2 Consequently, Equation 3.43 reduces to  ! ð2Þ X   b1 b2 0; 0  Dð1Þ ð0Þ 12 D12 zn11 zn22 zm   1 ðm ¼ 1; 2Þ: G2 ¼ I2  ð12Þ ð12Þ ;  b b n1 ; n 2 D D 12 21 n ;n ¼0 1

2

1

2

ð4:53Þ

j77

j 4 Calculational Methods for Intramolecular Distributions I , I , and I

78

1

2

N

In Equation 4.53, we note the presence of dm1 m2 ;n1 n2 and am1 m2 ;n1 n2 coefficients for which the first two indices m1 and m2 are zero. This greatly simplifies the calculation effort, since the multiple sums that appear in the recurrence Equations 4.50 and 4.51 reduce to only one. Differentiating both sides of Equation 4.53 with respect to z1 , and proceeding as before, we obtain 3 3 n X X ð1Þ ð1Þ 2ðn1 þ 1Þ c00;0l I2 ðn1 þ 1; n2
lÞ ¼ ð2d00;0l
g00;0l Þ I2 ðn1 ; n2
lÞ l¼0

l¼0

ð1Þ ð1Þ
2n1 c00;1l I2 ðn1 ; n2
lÞ þ ð2d00;1l
g00;1l Þ I2 ðn1
1; n2
lÞ

o
2ðn1
1Þc00;2l I2 ðn1
1; n2
lÞ
2ðn1
1Þc00;3l I2 ðn1
2; n2
lÞ ;

ð4:54Þ

where again for notational convenience, we have set  ! ð2Þ b1 b2 0; 0  Dð1Þ 12 D12 : I2 ðn1 ; n2 Þ ¼ I2 ;  n1 ; n2  Dð12Þ Dð12Þ b12 b21 1

2

The last recurrence equation that we need to complete our procedure can be obtained from Equation 4.54 by applying the operation F and invoking the symmetry of I2 in respect to the interchange n1 , n2 . This yields 2ðn2 þ 1Þ

3 3 n X X ð2Þ ð2Þ c00;k0 I2 ðn1
k; n2 þ 1Þ ¼ ð2d00;k0
g00;k0 Þ I2 ðn1
k; n2 Þ k¼0

k¼0

ð2Þ ð2Þ
2n2 c00;k1 I2 ðn1
k; n2 Þ þ 2d00;k1
g00;k1 I2 ðn1
k; n2
1Þ o
2ðn2
1Þ c00;k2 I2 ðn1
k; n2
1Þ
2ðn2
1Þc00;k3 I2 ðn1
k; n2
2Þ :

ð4:55Þ

Equations 4.54 and 4.55 have the same analytic structure as the corresponding Equations 4.50 and 4.51. Both are four-term recurrence equations that express the arbitrary element I2 ðn1 ; n2 Þ in terms of I2 ðn1
k; n2
lÞ, where k; l ¼ 0; 1; 2; 3; k þ l > 0. Similarly, the coefficients contained in Equations 4.54 and 4.55 have their counterparts in Equations 4.50 and 4.51 (see Appendix B). 4.3.3 The Calculation Procedure



 m1 ; m 2 As may be seen from the preceding section, the calculation of I2 n1 ; n2   0; 0 necessarily requires two steps. In the first step, we calculate I2 for all n1 ; n2 pairs ðn1 ; n2 Þ by successive use of Equations 4.54 and 4.55. Knowledge of   0; 0 I2 then permits us, by using the two remaining Equations 4.50 and 4.51 n1 ; n2   m1 ; m2 for all numbers m1 ; m2 ; n1 and n2 of in a second step, to obtain I2 n1 ; n2

4.3 Calculation of the Multidimensional Distribution

vibrational quanta. In detail, the procedure is as follows: We start with the initial value I2

0; 0 0; 0

! ¼ G2

 ð1Þ !   ð2Þ 1=2 1=2 D 4b b d00;00  12 D12 b1 b2 ¼ 1 2 exp
0; 0; 0; 0 ;  Dð12Þ Dð12Þ b12 b21 a00;00 a00;00 2 1

ð4:56Þ  0; 0 and calculate in the first step the sequence I2 , n1 ¼ 0; 1; 2; . . . by successive n1 ; 0 utilizations of Equation 4.54 n1 times. After the evaluation of these quantities, it is possible to progress from any ðn1 ; 0Þ to any finite ðn1 ; n2 Þ by n2 repeated utilization of Equation 4.55. Should one of the integer numbers n1
k or n2
l become negative, the corresponding values of I2 ðn1
k; n2
lÞ contained in Equation 4.54 or 4.55 are set to zero. After obtaining all values of I2 ðn1 ; n2 Þ, we are in position to raise the numbers m1 and m2 , employing Equations 4.50 and 4.51 (second step). Here, the procedure is precisely the same as that applied before: We calculate first the sequence m1 ; 0 I2 , m1 ¼ 1; 2; 3; . . . applying Equation 4.50 m1 times, and subsequently, n1 ; n2 taking these as our initial values, we repeat this procedure m2 times with the help of Equation 4.51.



4.3.3.1 Some Numerical Results With the aid of the recurrence procedure described in the previous section, the w ð12Þ dependence of I2 incorporated in the interactive displacement parameters D1 and ð12Þ D2 as well as in the cross-frequency parameters b12 and b21 has been investigated. Note that if w 6¼ 0, the two vibrational components are interdependent, which is manifested in an unusual behavior of the distribution I 2 . This is illustrated in  m1 ; m2 Figure 4.9a–c, which represent relief plots of I2 over the plane ðn1 ; n2 Þ n1 ; n2 ð1Þ ð2Þ for various pairs ðm1 ; m2 Þ and moderately large parameters D12 ¼ D12 ¼ 4. In the patterns shown in Figure 4.9a for m1 ¼ 1; m2 ¼ 0, the distribution I2 is bell shaped ð1Þ ð2Þ and exhibits two maxima. One can verify that when the parameters D12 and D12 are sufficiently large, then the number of maxima increases as m1 þ m2 increases (see Figure 4.9b–c). Generally, m1 and m2 coincide with the number of valleys, which for w ¼ 0 run, respectively, perpendicularly to the n1 or the n2 axis. Furthermore, if the angle w changes, a renormalization among the modes occurs and the maximum (maxima) of I2 moves in the ðn1 ; n2 Þ plane, running, for special values of w, close to the n1 axis or close to the n2 axis. In these special cases (i.e., w ¼ 30 and w ¼ 120 in Figure 4.9), the distribution I2 becomes more complex and behaves as a onedimensional distribution with a remarkably complicated course. Finally, note that I2 is periodic in w with the period p. These effects on the spectroscopy will be discussed in chapter 7. ð1Þ ð2Þ ðmÞ When D12 and D12 are small ðD12  1Þ, I2 is not so simply described graphically, since I2 decreases nearly exponentially as the numbers n1 and n2 increase. In this case, the maximum of I2 lies in the vicinity of n1 ¼ m1 ; n2 ¼ m2 . As the angle w

j79

Figure 4.9  (a) Relief plot of ! ð2Þ b b m1 ; m2  Dð1Þ 12 D12 for m1 ¼ 1; I2 ; 1 2  ð12Þ  n1 ; n2 D Dð12Þ b12 b21 1 2 m2 ¼ 0. The spectroscopic parameters are ð1Þ ð2Þ D12 ¼ D12 ¼ 4 and b1 ¼ 0:9; b2 ¼ 1:18;

b12 ¼ 0:53 and b21 ¼ 2:0. The values of the angle of rotation j are indicated in the figure. (b) Same as (a) but for m1 ¼ 0; m2 ¼ 1. (c) Same as (a) but for m1 ¼ 2; m2 ¼ 0. (d) Same as (a) but for m1 ¼ 0; m2 ¼ 2.

4.3 Calculation of the Multidimensional Distribution

Figure 4.9 (Continued ).

j81

j 4 Calculational Methods for Intramolecular Distributions I , I , and I

82

1

2

N

varies, this maximum displaces, as before, toward the n1 axis, then from the n1 axis to the n2 axis, returning at w ¼ 180 to the initial position for w ¼ 0 .

4.4 General Case of N-Coupled Modes 4.4.1 The Generating Function GN

With this machinery in hand, we look at the general Nd problem, where N may be identified with some or all molecular modes. Since the molecule in the sth electronic state generally does not share the symmetries of the lth state, we must consider the largest subgroup common to both the s and l state conformations to classify the molecular normal modes. As we have already noted the mass-weighted normal coordinates of the l electronic state manifold and of an arbitrary secondary manifold s are linearly related     ð1Þ   s   w11 w12   w1N   q1    ql1   k12...N               s    l w21 w22   w2N   q   kð2Þ   q2       2 12...N         ¼  þ ð4:57Þ     ;                                     qs    ql      kðNÞ  w N w w N N1

N2

NN

12...N

where W is a rotation matrix defined in Section 1.3. If the molecule has elements of symmetry, the matrix W will be block diagonal and Equation 4.57 decomposes into a set of several simultaneous equations for the irreducible representations of the point ðmÞ group under consideration. k12...N of Equation 4.57 is the component of a vector k12...N , which describes the difference of the equilibrium positions of the s and the l electronic states in a coordinate system with base vectors qlm . Of these components, only those that belong to the totally symmetric modes in the largest subgroup common to both the s and l state are nonzero. In the notation of Section 3.2.1, the multidimensional GF associated with the s ! l transition is given by Z Z GN ðtÞ ¼

1


1



Z Y N m¼1

N

N

rs ðqsm ;
qsm ; wm Þrl ð
qlm ; qlm ; zm Þdql d
ql ;

ð4:58Þ

where rs and rl are the trace functions defined by (3.17)  (3.18).   Here, wm and zm  and are complex variables lying in the polydisc D2N ð0; 1Þ wm 1; zm 1; m ¼ 1; 2; . . . N . The overlap integral, Equation 4.58, may be evaluated according to the procedure used in Section 3.2. The only difference in the treatment in this section is that the generating function has a higher dimension and depends on the above 2N complex variables wm and zm ðm ¼ 1; 2; . . . NÞ. As before, to facilitate integration we define a new set of coordinates qm and q0m

4.4 General Case of N-Coupled Modes

qm ¼ 2
1=2 ðqlm þ
qlm Þ;
qm ¼ 2
1=2 ðqlm

qlm Þ;

j83

ð4:59Þ

m ¼ 1; 2; . . . N

and correspondingly for the s state q0m ¼ 2
1=2 ðqsm þ qsm Þ;
q0m ¼ 2
1=2 ðqsm
qm Þ;

ð4:60Þ

m ¼ 1; 2; . . . N

Equations 4.59 and 4.60 guarantee that the transformation from qm to q0m and from
qm to
q0m are of the same form as the transformation (4.57) from qlm to qsm . The only distinction is that the displacement vector in this transformation is 21=2 k12  N and 0, respectively (see, for example, Equation 3.23). On substitution of Equations 4.59 and 4.60 into Equation 4.58 and considering the transformation between qm and q0m (and also between
qm and
q0m ) discussed above, we obtain 0 1 Z þ1 Z N ðbsm blm Þ1=2 1 BY C GN ¼ N @ h  dq1 d
q1 dq2 d
q2 ...dqN d
qN i1=2 A p
1 m¼1 ð1
z2 Þð1
w 2 Þ m m   8 N  X X X >   > l s s 2 s >



w b þb ; w w b ;  ; w w b w z w w   > m 1 m1 m2 m m1 mN m m1 m 1 m m >   > > m m m¼1   > >   N > >   X > X X < 1 s l s   2 s



w w z w w w b ; w b þb ;  ; w w b m2 m1 m m 2 m2 mN m   m2 m m 2 m exp
kq1 ;q2 ;...;qN k  m m > m¼1 2   > >  >      > >   N > >   X X X > > s s s l   2 >



w w b ; w w b ;  w b þb w w w z > mN m1 m m mN m2 m m  mN m m N N :   m¼1 m m    q1     q2     1      
k
q1 ;
q2 ; 
qN k   2        qN   N X X  X 2 s
1  
m þbs1
z
1

1

1 wm1 bm w wm1 wm2 bsm w  ; wm1 wmN bsm w   1 ; m ; m ;   m¼1 m m   X X   X s
1 l
1 s
1 2 s
1 
m ;
m þb2
z2 ;  ;
m ;  wm2 wm1 bm w wm2 bm w wm2 wmN bm w     m m m           X N X X  s
1 s
1 s
1 l
1  2 
m ;
m ;  ;
m þbN
zN ;  wmN wm1 bm w wmN wm2 bm w wmN bm w   m¼1

m

m

)

  
q1       q1         
q2     q2  X   X N N N X 2       ðmÞ ðmÞ
m ;...;
m    
bsm kðmÞ
m ;    
21=2  wm1 bsm k12N w wmN bsm k12N w 12N w       m¼1 m¼1       m¼1        qN  
q  N

ð4:61Þ

j 4 Calculational Methods for Intramolecular Distributions I , I , and I

84

1

2

N

where
m ¼ w

1
wm 1
zm ;
zm ¼ ; m ¼ 1;2;...;N: 1þwm 1þzm

ð4:62Þ

This result should be compared with Equation 3.24. It is a natural extension of the integral (3.24) to N dimensions. As before, it is a product of the two multiple Gaussian integrals, one over q1 ; q2 ; . . . ; qN and the other over
q1 ;
q2 ; . . . ;
qN as variables, respectively. Each of these integrals can be evaluated explicitly by carrying out the integration using (cf. Equation 3.25) Z

1

 


1

Z




   1 t 1
1 N=2
1=2 t exp
q xq þ y q dq1 dq2 . . . dqN ¼ ð2pÞ ðdet xÞ exp yx y ; 2 2 ð4:63Þ

where q and y are N-dimensional column vectors and x stands for the N  N matrix in Equation 4.61. Without repeating the details, we follow once the steps that led to Equation 3.31 and after a lot of algebra, we obtain n on o  ði i2 ...ip Þ s l ; b GN w1 ;w2 ;...;wN ;z1 ;z2 ;...zN  kj1 1j2 ...j ;b m m q
 A1 ðw1 ;w2 ;...;wN ;z1 ;z2 ;...;zN Þ exp
N Y B1 ðw1 ;w2 ;...;wN ;z1 ;z2 ;...;zN Þ ¼ 2N ðbsm blm Þ1=2 ½B1 ðw1 ;w2 ;...;wN ;z1 ;z2 ;...;zN ÞB2 ðw1 ;w2 ;...;wN ;z1 ;z2 ;...;zN Þ1=2 m¼1 ð4:64Þ

where A1 ðw1 ;w2 ;...;wN ;z1 ;z2 ;...;zN Þ ¼

1 X

1 X

m1 ;m2 ;...;mN ¼0 n1 n2 ...nN ¼0

m

m

m

dm1 m2 ...mN ;n1 n2 ...nN w1 1 w2 2 ...wNN zn11 zn22 ...znNN ;

ð4:65Þ

B1 ðw1 ;w2 ;...;wN ;z1 ;z2 ;...;zN Þ ¼

1 X

1 X

m1 ;m2 ...;mN ¼0 n1 ;n2 ;...;nN ¼0

m

m

m

ð4:66Þ

m

m

m

ð4:67Þ

am1 m2 ...mN ;n1 n2 ...nN w1 1 w2 2 ...wNN zn11 zn22 ...znNN

B2 ðw1 ;w2 ;...;wN ;z1 ;z2 ;...;zN Þ ¼

1 X

1 X

m1 ;m2 ;...;mN ¼0 n1 ;n2 ;...;nN ¼0

bm1 m2 ...mN ;n1 n2 ...nN w1 1 w2 2 ...wNN zn11 zn22 ...znNN :

With this economical notation, it will be possible to discuss this type of problem in exactly the same manner as in the two-dimensional case. The coefficient dm1 m2 ...mN ;n1 n2 ...nN ; am1 m2 ...mN ;n1 n2 ...nN and bm1 m2 ...mN ;n1 n2 ...nN , which can be regarded as elements of matrices of order 2N are real and given by

4.4 General Case of N-Coupled Modes

dm1 m2 ...mN ;n1 n2 ...nN ¼

X

j85

ð12...NÞ2

1 jp N

ð
1Þm1 þ m2 þ  mN þ njp bs1 bs2 . . . bsN bljp kjp X

X

þ

ð
1Þmi1 þ mi2 ...miN
1 þ njp þ njq

1i1 hi2 ...iN
1 N 1 jp h jq N ði i ...i

Þ2

 bsi1 bsi2 . . . bsiN
1 bljp bljq kip 1jq2 N
1 X ði1 Þ2 þ  þ ð
1Þmi1 þ n1 þ n2 þ  nN bsi1 bl1 bl2 . . . blN k12...N ; 1i1 N

ð4:68Þ

where i1 hi2 h. . . hir form a complete system of r indices as do j1 h j2 h. . . h js , both taken from among indices 1; 2; . . . ; N and where r þ s ¼ N þ 1. For convenience of ði i ...i Þ notation, we use here the dimensioned quantities kj1 1j22...jqp and bsm and blm instead ði1 i2 ...ip Þ of dimensionless Dj1 j2 ...jq and bm and bmn of the preceding section. In Section 8.1, it ði i2 ...ip Þ will often be convenient to justify our notation. For example, the parameters kj1 1j2 ...j q and those of the frequency factors bm and bmn can be normalized to dimensionless ði i ...i Þ ones in a condensed form. The interactive parameters kj1 1j22...jqp appearing in (4.68) are given by ! ! ! 23...N ð1Þ 13...N ð2Þ 12...N
1 ðNÞ ð12...NÞ N þ1 k12...N
W k12...N  þð
1Þ k12...N ; k1 ¼W W 23...N 23...N 23...N ! ! ! 23...N ð1Þ 13...N ð2Þ 12...N
1 ðNÞ ð12...NÞ N þ2 k2 ¼
W k12...N þW k12...N  þð
1Þ W k12...N ; 13...N 13...N 13...N .. . ð12...NÞ kN ¼ð
1ÞN þ1 W

23...N

! ð1Þ k12...N  þð
1Þ2N W

12...N
1

12...N
1 12...N
1

! ðNÞ

k12...N : ð4:69aÞ

ði i ...i Þ k121 2 N
1 ¼W

i2 i3 ...iN
1

!

34...N

þð
1ÞN W ði i ...i Þ k131 2 N
1 ¼
W

þð
1Þ

i1 i2 ...iN
2

24...N W

ði

i1 i2 ...iN
2

Þ

i1 i3 ...iN
1 24...N

! ði

! ði Þ

2 k12...N 

Þ

N
1 ; k12...N

.. . ði1 i2 ...iN
1 Þ ¼
ð
Þ1=2NðN
1Þ W kN
1;N

ði Þ

2  k12...N

N
1 ; k12...N

ði1 Þ k12...N þW

24...N

!

34...N

!

34...N !

i2 i3 ...iN
1

N þ1

i1 i3 ...iN
1

ði1 Þ
W k12...N

i2 i3 ...iN
1 12...N
2

! ði2 Þ þ 
ð
Þ1=2NðN þ1Þ W k12...N

i1 i2 ...iN
2 12...N
2

! ði

Þ

N
1 : k12...N

ð4:69bÞ

j 4 Calculational Methods for Intramolecular Distributions I , I , and I

86

1

2

N

for all combinations of N
1, indices 1i1 hi2 h...hiN
1 N selected from among the indices 1;2;...;N and arranged in lexicographic order,  and finally ði1 i2 Þ k12...N
1 ¼W

i2 N

! ði1 Þ k12...N
W

!

i2

ði i Þ

1 2 k1...N
2;N ¼
W

ði Þ

N
1

.. . ði1 i2 Þ ¼ð
1ÞN þ1 W k23...N

i2 1

i1

!

N

1 k12...N þW

ði Þ

2 k12...N ;

i1 N
1

! ði1 Þ k12...N þð
1ÞN W

! ði Þ

2 k12...N ;

i1 1

ð4:69cÞ

! ði Þ

2 k12...N ð1i1 hi2 NÞ:



 i 1 i2 . . . i p are minors of order p of the N-dimensional orthogonal   j1 j2 . . . j p i1 i 2 . . . i p refer to the rows and the subindices to matrix W. The upper indices of W j1 j2 . . . j i the column of the matrix W. As can be seen, the construction of the scheme (4.69) for ði i2 ...ip Þ due to mixing of components is simple. These the interactive parameters kj1 1j2 ...j q parameters are derivable from the components of the displacement vector k12...N in combination with minors of W of decreasing order. Moreover, as the order of the minors is decreased, the number of the components of k12...N on the right side of Equations 4.69 is successively diminished by 1. In the language of matrix theory [124], the coefficients in the system of (4.69a) are, apart from the sign , elements of the ðN
1Þth compound matrix of W. Those of the system (4.69b) are elements of the ðN
2Þth compound matrix and so on, until the first compounds matrix (Equation 4.69c) is achieved. Next, we observe that the upper and lower indices of Here, W

ði i ...i Þ

kj1 1j22...jqp in (4.69) have a close relationship with the indices of the minors   i 1 i2 . . . i p W . Thus, the additional parameters on the left side of (4.69) are j 1 j 2 . . . jp uniquely determined by the properties of the matrix W and the vector k12...N. The   N  X N N , which evidently intotal number of these parameters is k N
k þ 1 k¼1 creases rapidly with the number of components N. From the viewpoint of the parallelmode approximation this is quite a surprising result, for we are describing the ðmÞ displacements of the normal modes m in terms of k12...N only. From the relation (4.69), it can also be inferred that a reversion of the vector k12...N !
k12...N leaves the ðmÞ by the GF (4.64) invariant, since all components k12...N , as well asthose generated  i1 i 2 . . . ip effect of W enter the expression (4.68) as squares. Since W are minors j1 j2 . . . j p of an orthogonal matrix, relation (4.69a) can be written more compactly as kð12...NÞ ¼ W
1 k12...N :

Equation 4.70 is a generalization of (3.30) to N dimensions.

ð4:70Þ

4.4 General Case of N-Coupled Modes

Similarly, we have m1 þ m2 þ  þ mN

am1 m2 ...mN ;n1 n2 ...nN ¼ ð
1Þ

X

þ

W X

12 . . . N 12 . . . N

þ

bs1 bs2 . . . bsN

ð
1Þmi1 þ mi2 þ  þ miN
1 þ njp

1i1 hi2 h...hiN
1 N 1 jp N

W

!2

i1 i2 . . . iN
1 j1 . . . jp
1 jp þ 1 . . . jN X X

!2 bsi1 bsi2 . . . bsiN
1 bljp ð
1Þm1 þ m2 þ  mN
2 þ njp þ njq

1i1 hi2 h...hiN
2 N 1 jp h jq N

W

i1 i2 . . . iN
2

!2

j1 . . . jp
1 jp þ 1 . . . jq
1 jq þ 1 . . . jN

bsi1 bsi2 . . . bsiN
2 bljp bljq

þ    þ ð
1Þn1 þ n2 þ  þ nN bl1 bl2 . . . blN ; ð4:71Þ

and finally bm1 m2 ...mN ;n1 n2 ...nN ¼ ð
1Þm1 þ m2 þ  mN þ n1 þ n2 þ  þ nN am1 m2 . . .mN ;n1 n2 ...nN :

ð4:72Þ

4.4.2 Properties of dm ,n , am ,n , and bm ,n

Representing the expression (4.65) in a bilinear form, analogously to Equation (3.61), we have  2N A1 ðw1 ; w2 ; . . . ; wN ; z1 ; . . . ; zN Þ ¼ wt dm;n 1 z; ð4:73Þ where the first subscript m ¼ ðm1 ; m2 ; . . . ; mN Þ denotes the row and the second subscript n ¼ ðn1 ; n2 ; . . . ; nN Þ the column of the matrix element dm;n . It is convenient here to consider w and z as column symbols with the elements arranged in some definite, for example, lexicographic order     1     1     z     1 w1         :     :     :         z N wN         z z 1 2 w w 1 2     ; z ¼  : w¼ :         : :      wN
1 wN   zN
1 zN       w1 w2 w3   z 1 z2 z 3          :     :     :       w1 w2 . . . wN   z 1 z2 . . . zN

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2

N

  2N In this representation, the matrix dm;n 1 behaves like a magic square, since the sum of elements in each row and column and any main diagonal is the same, namely, zero 1 X n1 ;n2 ;...;nN ¼0

dm1 m2 ...mN ;n1 ;n2 ...nN ¼ 0;

ð4:74aÞ

for each row with the subscript m ¼ ðm1 ; m2 ; . . . mN Þ and 1 X m1;m2 ;...;mN ¼0

dm1 m2 ...mN ;n1 n2 ...nN ¼ 0;

ð4:74bÞ

for each column with the subscript n ¼ ðn1 ; n2 ; . . . nN Þ. The properties (4.74) follow directly from (4.68)and are 2N analogous to the relation (3.39). Note that the explicit matrix realization of dm;n 1 is effected by a matrix of 2N rows and columns; consequently, w and z have 2N components. Now, with the same treatment regarding the form  2N B1 ðw1 ; w2 ; . . . ; wN Þ ¼ wt am;n 1 z;

ð4:75Þ  2 N we establish some properties of the matrix am;n  . Starting from the fundamental 1

formula (4.71), we have for the sum of the elements of the row with the subscript m ¼ ðm1; m2 ; . . . ; mN Þ 1 X n1 ;n2 ;...;nN ¼0

am1 m2 ...mN ;n1 n2 ...nN ¼ ð
1Þm1 þ m2 þ m3 þ  þ mN 2N bs1 bs2 . . . bsN

ð4:76aÞ

am1 m2 ...mN ;n1 n2 ...nN ¼ ð
1Þn1 þ n2 þ ... þ nN 2N bl1 bl2 . . . blN ;

ð4:76bÞ

and analogously 1 X m1;m2 ;...;mN ¼0

for the sum of elements of the column with the subscript n ¼ ðn1 ; n2 ; . . . ; nN Þ. To complete the development thus far we attained, we write  2N B2 ðw1 ; w2 ; . . . ; wN Þ ¼ wt bm;n 1 z:

ð4:77Þ

On the basis of formulas (4.71) and (4.72), we find that 1 X n1 ;n2 ;...;nN ¼0

bm1 m2 ...mN ;n1 n2 ...nN ¼ ð
1Þm1 þ m2 þ  þ mN 2N bl1 bl2 . . . blN ;

ð4:78aÞ

where m ¼ ðm1 ; m2 ; . . . ; mN Þ denotes the subscript of the row being considered and 1 X m1 ;m2 ;...;mN ¼0

bm1 m2 ;...mN ;n1 n2 ...nN ¼ ð
1Þn1 þ n2 þ  þ nN 2N bs1 bs2 . . . bsN ;

where n ¼ ðn1 ; n2 ; . . . ; nN Þ; denotes the subscript of the selected column.

ð4:78bÞ

4.4 General Case of N-Coupled Modes

  4.4.3 m1 ; m2 ; . . . ; mN The Distribution IN and its Properties n1 ; n2 ; . . . ; nN

The sum rules Before proceeding, we note that the generating function (4.64) is holomorphic in the polydisc D2N ð0; 1Þ and can be developed in following power series 1 X

GN ðw1 ; w2 ; . . . ; wN ; z1 ; z2 ; . . . ; zN Þ ¼  IN

m1 ; m2 ; . . . ; mN n1 ; n2 ; . . . ; nN

!

1 X

m1 ;m2 ;...;mN ¼0 n1 ;n2 ;...;nN ¼0

w1m1 w2m2 . . . wNmN zn11 zn22 . . . znNN

ð4:79Þ

where IN in (4.79) is, as will be shown, for each choice of integers ðm1 ; m2 ; . . . ; mN Þ, an N-dimensional probability distribution of n1 ; n2 ; . . . ; nN and vice versa. We will show that the distribution so obtained obeys conditions similar to those of (3.46) and (3.53) but generalized to 2N integer variables. Simultaneously, we show how the symmetry properties of the distribution, analogously to Equation 3.65 for N ¼ 2, can be extended to the general case of 2N arguments. In the rest of this section, m1 ; m2 ; . . . ; mN and n1 ; n2 ; . . . ; nN signify any integers  0. With the help of the expressions (4.73), (4.75) and (4.77), we can establish the following theorem. Theorem 4.1 For each choice of integers ðm1 ; m2 ; . . . ; mN Þ 1 X n1 ;n2 ;...nN ¼0

 IN

m1 ; m2 ; . . . ; mN n1 ; n2 ; . . . ; nN

 ¼ 1:

ð4:80Þ

Proof The proof of this remarkable theorem offers no difficulty, since it closely follows the lines of the proof of the corresponding identity for N ¼ 2. By direct substitution in (4.73), (4.75), and (4.77) z1 ¼ z2 ¼    ¼ zN ¼ 1, we have, according to (4.74a), (4.76a), and (4.78a) A1 ðw1 ; w2 ;. . .; wN ; 1; 1;. . .; 1Þ¼ 0; B1 ðw1 ; w2 ; .. . ;wN ;1;1; .. . ;1Þ¼ 2N bs1 bs2 .. .bsN

1 X m1;m2 ;...;mN ¼0

m

m

m

m

m

m

ð
1Þm1 þm2 þ  þmN w1 1 w2 2 . .. wNN

¼ 2N bs1 bs2 .. .bsN ð1
w1 Þð1
w2 Þ . .. ð1
wN Þ; and B2 ðw1 ; w2 ; .. . ;wN ;1;1; .. . ;1Þ ¼ 2N bl1 bl2 .. .blN

1 X m1 ;m2 ;...;mN ¼0

ð
1Þm1 þ m2 þ  þmN w1 1 w2 2 . . .wNN

¼ 2N bl1 bl2 .. .blN ð1
w1 Þð1
w2 Þ . .. ð1
wN Þ:

ð4:81Þ

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2

N

Substituting these expressions in (4.64), we obtain GN ðw1 ; w2 ; . . . ; wN ; 1; 1; . . . ; 1Þ ¼

 N  1 Y X 1 w1m1 w2m2 . . . wNmN : ¼ 1
w m m¼1 m1 ;m2 ;...;mN ¼0 ð4:82Þ

while, according to (4.79), 1 X

GN ðw1 ; w2 ; . . . ; wN ; 1; 1; . . . ; 1Þ ¼ 

1 X



IN

n1 ;n2 ;...;nN ¼0

m1 ;m2 ;...;mN ¼0

m1 ; m2 ; . . . ; mN

!

n 1 ; n 2 ; . . . ; nN

w1m1 w2m2 . . . wNmN ;

ð4:83Þ

which proves the theorem. Equation 4.80 appears as natural generalization of the corresponding Equation 3.46.    
2N ð0; 1Þ wm   1; zm  < 1; Regarding the function (4.64) in the polydisc D m ¼ 1; 2; . . . ; N, the same procedure as before enables us to prove the following supplementary Theorem 4.2 1 X m1 ;m2 ;...;mN ¼0

 IN

m1 ; m2 ; . . . ; mN n1 ; n2 ; . . . ; nN

 ¼ 1;

ð4:84Þ

which is valid for each choice of integers ðn1 ; n2 ; . . . ; nN Þ. Proof The deduction is precisely like that of theorem 1. Based on (4.73), (4.75), and (4.77) and relations (4.74b), (4.76b), and (4.78b), we have after direct substitution of w1 ¼ w2 ¼    ¼ w N ¼ 1 A1 ð1; 1; . . . ; 1; z1 ; z2 ; . . . ; zN Þ ¼ 0 B1 ð1; 1; . . . ; 1; z1 ; z2 ; . . . ; zN Þ ¼ 2N bl1 bl2 . . . blN

1 X n1 ;n2 ;...;nN ¼0

ð
1Þn1 þ n2 þ  þ nN zn11 zn22 . . . znNN

¼ 2N bl1 bl2 . . . blN ð1
z1 Þð1
z2 Þ . . . ð1
zN Þ; 1 X ð
1Þn1 þ n2 þ  þ nN zn11 zn22 . . . znNN B2 ð1; 1; . . . ; 1; z1 ; z2 ; . . . ; zN Þ ¼ 2N bs1 bs2 . . . bsN ¼

n1 ;n2 ;...;nN ¼0 s N s s 2 b1 b2 . . . bN ð1
z1 Þð1
z2 Þ . . . ð1
zN Þ:

ð4:85Þ

Substituting these expressions in (4.64), we obtain  N  1 Y X 1 GN ð1; 1; . . . ; 1; z1 ; z2 ; . . . ; zN Þ ¼ zn11 zn22 . . . znNN : ¼ 1
zn n¼1 n1 ;n2 ;...;nN ¼0 ð4:86Þ

4.4 General Case of N-Coupled Modes

On the other hand, (4.79) gives GN ð1; 1; . . . ; 1; z1 ; z2 ; . . . ; zN Þ ¼

1 X n1 ;n2 ;...;nN ¼0

1 X



IN

m1 ;m2 ;...;mN ¼0

m1 ; m2 ; . . . ; mN

!!

n1 ; n2 ; . . . ; nN

zn11 zn22 . . . znNN ;

ð4:87Þ

which leads to (4.84).  4.4.3.1 Symmetry Property of IN

m1 ; m2 ; . . . ; mN n1 ; n2 ; . . . ; ; nN



In a manner similar to the treatment in Section 3.2.5, we investigate the invariance of IN under the exchange of parameters below bsm $ blm ;

m ¼ 1; 2; . . . ; N ð4:88Þ

and ð12...NÞ

k12...N $ k

:

First, we note that the replacement k12...N by kð12...NÞ requires, according to relation (4.70), that W ! W
1 . Thus, starting with the vector kð12...NÞ and the inverse matrix W
1 instead of k12...N and W, we have the following lemma [109]: Lemma 4.1 If we replace the vector k12...N by kð12...N Þ and the orthogonal matrix W by its inverse W
1 in Equations , the interactive displacement parameters convert into ði i ...i Þ kj1 1j22...jqp

!

ð j j2 ...jq Þ ki1 i12 ...i : p

1

i1 hi2 h. . . hip j1 h j2 h. . . h jq

! N

ð4:89Þ

In other words, the transformation (4.89) consists of exchanging the lower and ði i ...i Þ upper indices of kj1 1j22...jqp with each other. The proof of this lemma is provided in detail in Ref. [109]. The relation (4.89) can be transferred to the expression of the matrix elements dm;n , which, with the exchange of parameters (4.88) transforms into dn;m , or more precisely  N  N dm;n 2 ! dn;m 2 : 1 1

ð4:90aÞ

In the same manner, we find for the remaining coefficient matrices appearing in the GF  N  N am;n 2 ! an;m 2 1 1

ð4:90bÞ

 2N  2N bm;n  ! bn;m  : 1 1

ð4:90cÞ

and

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2

N

Finally, if we simultaneously exchange the role of the dummy variables zm $ wm ; ðm ¼ 1; 2; . . . ; NÞ, as in the Section 3.2.5, we write     zt dn;m w ¼ wt dm;n z ¼ A1 ðw1 ; w2 ; . . . ; wN ; z1 ; . . . ; zN Þ and similarly for the expressions B1 and B2 . This, of course, implies that GN ðw1 ; . . . ; wN ; z1 ; . . . ; zN

hn

ði i ...i Þ

2 p kj1 1j2 ...j q

o n oi ; bsm ; blm

n o n o  ð j j2 ...jq Þ ; blm ; bsm ¼ GN z1 ; . . . ; zN ; w1 ; . . . ; wN  ki1 i12 ...i p

ð4:91Þ

and after expanding both sides of (4.91) in power series in the polydisc D2N ð0; 1Þ and mN 1 m2 equating terms of w1m1 w2m2 . . . wNmN zm 1 z2 . . . zN , our conclusions are embodied in the following Theorem 4.3 The distribution IN is left invariant by the exchange of parameters (4.88), provided the integer variables mm and nm ðm ¼ 1; 2: . . . ; NÞ are simultaneously exchanged mm $ nm  IN

n n  on o on o m1 ;m2 ;...;mN  ði1 i2 ...ip Þ n1 ;n2 ;...;nN  ðj1 j2 ...jq Þ s l ¼ IN kj1 j2 ...jq ; bm ;bm ki1 i2 ...ip ; blm ;bsm :   n1 ;n2 ;...;nN m1 ;m2 ;...;mN ð4:92Þ

Equation 4.92 appears as natural generalization of the corresponding Equation 3.65 to an arbitrarily number of vibrational modes. 4.4.4 A Special Case

We have derived all of the general properties of the multidimensional distributions that have been noted in Sections 3.2.4 and 3.2.5 (for the case N ¼ 2), except that of the special case  N W ¼ dij 1 ;

ð4:93Þ

where dij is the Kronecker delta. In this case, all minors in the system of Equations 4.69 and 4.71 vanish, except those of the principal minors, which are 1. Hence, the system of Equations 4.69 simplifies considerably, since k12...N ¼ k12...N ð4:94Þ

and ði i ...i ...i Þ ki11j22...jqr p

¼

ðir Þ k12...N

4.4 General Case of N-Coupled Modes

if j1 ; j2 ; . . . ; jq completed by i1 ; i2 ; . . . ; ip form a complete sequence 1; 2; . . . ; N and where ir ¼ (rest of is), or ði i ;...ip Þ

2 kj1 1j2 ...j q

¼ 0;

otherwise:

From this, it follows that 2

ðmÞ N X bsm blm k12...N ð1
wm Þð1
zm Þ A1 ðw1 ; w2 ; . . . wN ; z1 ; . . . ; zN Þ ¼ s l B1 ðw1 ; w2 ; . . . ; wN ; z1 ; . . . ; zN Þ m¼1 bm ð1
wm Þð1 þ zm Þ þ bm ð1 þ wm Þð1
zm Þ

B1 ðw1 ; w2 ; . . . ; wN ; z1 ; . . . ; zN Þ ¼

N h Y m¼1

B2 ðw1 ; w2 ; . . . ; wN ; z1 ; . . . ; zN Þ ¼

N h Y m¼1

i

bsm ð1
wm Þð1 þ zm Þ þ blm ð1 þ wm Þð1
zm Þ

i

bsm ð1 þ wm Þð1
zm Þ þ blm ð1
wm Þð1 þ zm Þ

and lastly GN ðw1 ; w2 ; . . . ; wN ; z1 ; . . . ; zN Þ ¼

N Y

G1 ðwm ; zm Þ;

ð4:95Þ

m¼1

where G1 ðw:zÞ is again the one-dimensional GF considered in Sections 3.2 and 4.1. Thus, we have the following Theorem 4.4 If the matrix (4.93) is a unit matrix, the multidimensional GF GN factors into a simple product of N one-dimensional GF, each of which depends on variables and parameters of one vibrational component only. In addition, if the parameters bsm =blm ¼ bm for all components m are equal, the corresponding distribution IN generated by (4.95) coincides with formula (4.35). 4.4.5 Concluding Remarks and Examples

The explicit separation of vibrations in the multidimensional GF of Theorem 4.4 allows an effective reduction of the number of degrees of freedom and facilitates the calculation of transition rates. In this regard, it cannot be too strongly emphasized that the calculation of transition rates in the parallel-mode approximation has a fundamental inadequacy that is evident from the derivation above. The defect emerges, if we return to the case of N-vibrational modes some of which are not parallel to each other. This situation occurs if the latter have the same symmetry, especially if they are totally symmetric in the molecular group. In this case, the complexity introduced by the reciprocal (4.69a) and interactive displacement parameters (Equations 4.69b and 4.69c) is considerable and this fact cannot be overlooked in any “parallel-mode estimate” of the transition probability. Even with considerable effort, it is impossible to factor the exact GF into a product of one-dimensional GF.

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2

N

Note that the interactive displacement parameters are not quantities with physical significance. They merely represent the error incurred in making a crude approximation, assuming that all modes are parallel. Another remarkable feature of mode mixing is the dependence of the transition rate on the cross-frequency parameters (see Equation 4.71). (Note that in the parallel-mode approximation the transition rate depends solely upon km and bm ¼ vsm =vlm per mode). Both of these findings are manifested in the molecular spectra: (i) A striking energy narrowing or broadening of the absorption and fluorescence spectra does occur [123] and (ii) regularly structured emission bands (in form of single-mode progressions) occur only at special values of w, especially when the (orthogonal) matrix W is a unit matrix. Under all other conditions, vibrational components in the spectra are more or less scrambled and irregularly distributed. 4.4.6 Recurrence Relations

We raise the important question of finding the values of the MID. Based on our analysis pertaining to the one-and two-dimensional IDs, this problem will again be solved by constructing 2N four-term recurrence equations. There is one relation for each of the vibrational quantum numbers m1 ; m2 ; . . . ; mN ; n1 ; n2 ; . . ., nN contained in IN . Invoking the methods used above in Section 4.3.1, the recurrence equations are obtained by differentiating the GF (4.64) with respect to wi ði ¼ 1; 2; . . . ; NÞ and by substituting the series solution (4.79) for GN in this equation. Using the notation introduced in Section 4.3.2 and extending it to N dimension, we get 3 X

2ðmi þ1Þ

cm1 ...mi
1 0miþ1 ...mN ;n1 ...nN IN

m1 ;...;mi
1 ;miþ1 ;...;mN ¼0

...;mi þ1;...

!

...;ni
ni ;...

n1 ;n2 ;...;nN ¼0 3 X

¼

(

ðiÞ ðiÞ 2dm1 ...mi
1 0miþ1 ...mN ;n1 ...nN
gm1 m2 ...mi
1 0miþ1 ...mN ;n1 ...nN

m1 ;...;mi
1 ;miþ1 ;...;mN ¼0 n1 ;n2 ;...;nN ¼0


2mi cm1 ...mi
1 1miþ1 ...mN ;n1 ...nN IN ðiÞ

...;mi ;...

þð2dm1 ...mi
1 1miþ1 ...mN n1 ...nN
gm1 ...mi
1 1miþ1 ...mN n1 ...nN ÞIN
2ðmi
1Þcm1 ...mi
1 2miþ1 ...mN ;n1 ...nN IN
2ðmi
1Þcm1 ...mi
1 3miþ1 ...mN ;n1 ...nN IN ði ¼ 1;2;...;NÞ;

IN

...;mi ;...

!

...;ni
ni ;...

!

...;ni
ni ;...

ðiÞ

...;mi
1;...

!

...;ni
ni ;... ! ...;mi
1;...

...;ni
ni ... !) ...;mi
2;... ; ...;ni
ni ;... ð4:96Þ

j95

4.4 General Case of N-Coupled Modes

where IN

...;mi þr;...

!

...;ni
ni ;...

¼ IN

m1
m1 ; ...; mi
1
mi
1 ; mi þr; miþ1
miþ1 ; ...; mN
mN n1
n1 ; ... ...;

ni
ni ; ...

!

...; nN
nN

ðr ¼ 1;0;
1;
2Þ:

To obtain the second set of equations, we insert w1 ¼ w2    ¼ wN ¼ 0 into Equations 4.64 and 4.79 and m1 ¼ m2    ¼ mN ¼ 0 in the latter. We proceed in the above manner with the exception that the GF as well as its expansion must be differentiated with respect to zi . This gives ! 3 0:; . . . ; 0 X 2ðni þ 1Þ c0...0;n1 ...ni
1 0ni þ 1 ...nN IN . . . ; ni þ 1; . . . n1 ;...;ni
1 ;ni þ 1 ;...;nN ¼0 ( ! 3 0:; . . . ; 0 X ðiÞ ðiÞ ð2d0...0;n1 ...ni
1 0ni þ 1 ...nN
g0...0;n1 ...ni
1 0ni þ 1 ...nN ÞIN ¼ ni n1 ;...;ni
1 ;ni þ 1 ;...;nN ¼0 ! 0:; . . . ; 0 ðiÞ ðiÞ
2ni c0...0;n1 ...ni
1 1ni þ 1 ...nN IN þ ð2d0...0;n1 ...ni
1 1ni þ 1 ...nN
g0...0;n1 ...ni
1 1ni þ 1 ...nN Þ . . . ; ni ; . . . ! ! 0:; . . . ; 0 0; . . . ; 0 IN
2ðni
1Þc0...0;n1 ...ni
1 2ni þ 1 ...nN IN . . . ; ni
1; . . . . . . ; ni
1; . . . !) 0; . . . ; 0 ;
2ðni
1Þc0...0;n1 ...ni
1 3ni þ 1 ...nN IN . . . ; ni
2; . . . i ¼ 1; 2; . . . ; N:

ð4:97Þ

In both sets of equations, our notation suppresses the parametric dependence of IN on the spectroscopic parameters. The coefficients cm;n ; gm;n , and dm;n appearing in these equations are given in the Appendix C. Notice the upper indices on dm;n and gm;n in Equations 4.96 and 4.97. These are chosen deliberately to distinguish between different dm;n and gm;n in the system of recurrence (Equations 4.96 and 4.97). In order to complete the final step in the derivation, we insert w1 ¼ w2    ¼ wN ¼ 0 and z1 ¼ z2    ¼ zN ¼ 0 into Equation 4.79 and obtain   n o n o 0; . . . ; 0  ði i2 ...ip Þ ; bsm ; blm ¼ GN 0; . . . ; 0; 0; . . . ; 0 kj1 1j2 ...j IN q 0; . . . ; 0   ; ð4:98Þ N Y
d0...0;0...0 a0...0;0...0 ¼ 2N ðbsm blm Þ1=2 exp a0...0;0...0 m¼1 which is the starting point in our recurrence procedure. Equations 4.96 and 4.97 are four-term recurrenceequations and solving them is equivalent to the evaluation of  m 1 ; m2 ; . . . ; m N IN for different values of the vibrational quantum numbers n1 ; n2 ; . . . ; nN m1 ; m2 ; . . . ; mN and n1 ; n2 ; . . . ; nN . This procedure offers a very convenient and efficient manner of calculating the MIDs. The calculation proceeds as already

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2

N

described in  Section 4.3.2 forN ¼ 2 in two steps. In the first step, the numerical 0; 0; . . . ; 0 for all systems ðn1 ; n2 ; . . . ; nN Þ are calculated by values of IN n1 ; n2 ; . . . ; nN successive use of the N recurrence Equation 4.97: The ith equation employed to raise the quantum number ni. Taking these values as our initial values in Equation 4.96, we proceed in a similar manner as above and raise the numbers m1 ; m2 ; . . . ; mN , again by successive use of each of the Equation 4.96. 4.4.7 The Three-Dimensional Case

For illustrative purposes, we will present some results for the case N ¼ 3. In this case, we use the three Euler angles to describe the rotation matrix W in the (z,x,z) convention    cos j cos y
sin j cos q sin y sin j cos y þ cos j cos q sin y sin q sin y     W¼ 
cos j sin y
sin j cos q cos y
sin j sin y þ cos j cos q cos y sin q cos y ;  sin j sin q
cos j sin q cos q 

where 0  j  2p; 0  y  2p; 0  q  p

and  det W ¼ W

1 2 3 1 2 3

 ¼1

The influence of W is essentially attributable to the displacement parameters. According to Equations 4.69, there are a total of 15 parameters: Three “direct”  ð1Þ ð2Þ ð3Þ  geometrical displacements defined by the vector k123 ¼ col k123 ; k123 ; k123 , three reciprocal displacements specified in Equation 4.70 kð123Þ ¼ W
1 k123 ;  ð123Þ ð123Þ ð123Þ  where kð123Þ ¼ col k1 ; k2 ; k3 , and nine further (interactive) displacements generated by the first compound matrix of W, Equation 4.69c:     2 ð1Þ 1 ð2Þ ð12Þ k123
W k123 k12 ¼ W 3 3     2 ð1Þ 1 ð2Þ ð12Þ k13 ¼
W k123 þ W k123 2 2     2 ð1Þ 1 ð2Þ ð12Þ k23 ¼ W k123
W k123 1 1     3 ð1Þ 1 ð3Þ ð13Þ k12 ¼ W k123
W k123 3 3

4.4 General Case of N-Coupled Modes ð13Þ

k13

ð13Þ

k23

ð23Þ

k12

ð23Þ

k13

ð23Þ

k23

    3 ð1Þ 1 ð3Þ ¼
W k123 þ W k 2 2 123     3 ð1Þ 1 ð3Þ ¼ W k
W k 1 123 1 123     3 ð2Þ 2 ð3Þ ¼ W k123
W k 3 3 123     3 ð2Þ 2 ð3Þ ¼
W k þW k : 2 123 2 123     3 ð2Þ 2 ð3Þ ¼ W k123
W k 1 1 123

Given values for all these displacements together with the frequencies factors bsm ¼ vsm =
h and blm ¼ vlm =
h ðm ¼ 1; 2; 3Þ (the latter, for instance, may be determined from spectroscopic data), the coefficients dm1 m2 m3 ;n1 n2 n and am1 m2 m3 ;n1 n2 n3 (see Equations 4.68 and 4.71) of the homogeneous functions A1 ðw1 ; w2 ; w3 ; z1 ; z2 ; z3 Þ and B1 ðw1 ; w2 ; w3 ; z1 ; z2 ; z3 Þ, respectively, can be calculated. This may be performed ðmÞ either in dimensioned quantities of k123 ; bsm ; and blm or in dimensionless quantities: i1 i2 ...ip ðmÞ D123 and aj1 j2 ...jq , normalized according to the procedure given in Section 8.1 (see Equations 8.14 and 8.15. The latter quantities are quite convenient because the numerical values have an order of magnitude of 1. Next, starting from dm;n and am;n , we obtain cm;n ; gm;n ; and dm;n via Equations C1–C8. With all of these coefficients and the initial value of I3 , Equation 4.98, the system of recurrence Equations 4.96 and 4.97 may be solved. 4.4.8 Some Numerical Results

We present results of such calculations to illustrate some of the effects of normalmode mixing. There are obviously a large number of possible examples and a few of them will be included here. The calculations are performed for m1 ¼ m2 ¼ m3 ¼ 0, that is, for the transition from the lowest vibrational level of the optically excited initial s state into the vibrational manifold of the lower electronic state (the zero-temperature limit). The results that we shall display are the final-state vibrational distribution (as given by the multidimensional surface of I3 ) and the relative nonradiative decay probability, both represented as functions of the normal coordinate rotation. Since the multidimensional surface of IN for dimensions N > 2 is hard to visualize, Figure 4.10 represents cross sections of I3 as functions of the lower-state occupation numbers n1 ; n2 at n3 ¼ 10. These cross sections are displayed for various values of the first Euler angle j in order to illustrate the influence of some mixed quadratic interaction terms of the excited-state force field on the vibrational distribution. In all of the examples presented in Figure 4.10, the geometry distortions along ð1Þ ð2Þ the first and second coordinates qs1 and qs2 ; k123 and k123 , have been chosen to be ð3Þ strong, whereas the remaining displacement k123 was moderately strong. In this figure, some of the effects of normal-mode mixing and of the interactions responsible

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1

Figure spatial variation of  4.10 Characteristic  0 0 0 with the Euler angle j I3 n1 n2 n3 depicted for a high sheet of I3 at n3 ¼ 10. The values of the Euler angle j are indicated in the figure. The remaining angles are constant and equal q ¼ y ¼ 30 . The dimensionless normal-

2

N

ð1Þ

ð2Þ

mode displacements are D123 ¼ D123 ¼ 4:0,

ð3Þ D123

¼ 1:0 and the normal-mode frequencies of the excited initial state s and the final states l are vs1 ¼ 200; vl1 ¼ 240; vs2 ¼ 400; vl2 ¼ 600; vs3 ¼ 900 and vl3 ¼ 800 cm
1 .

for them become apparent. If the angle j varies (while keeping the remaining angles q and y constant), the coordinate system ðqs1 ; qs2 ; qs3 Þ rotates about the ql3 axis (or in general, about an axis that is parallel to the ql3 ). When the above rotation is carried out for any fixed value of the occupation number n3, the surface of I3 moves in the ðn1 ; n2 Þ

4.4 General Case of N-Coupled Modes

plane toward the n2 axis. This is demonstrated in Figure 4.10 for the highest cross section occurring (for any value of j) at n3 ¼ 10. As can be seen, for an angle of j ffi 30 the ID is distributed practically in the ðn2 ; n3 Þ plane (assuming in the remaining space spanned by the n1 ; n2 , and n3 axes approximate values of zero). If the angle j is further increased, the surface of I3 moves away from the n2 axis, running at approximately j ¼ 120 close to the n1 axis. Clearly, these special values of j are affected by the values of spectroscopic parameters. However, they are primarily ðmÞ dependent on the normal coordinate displacements k12...N ðm ¼ 1; 2; . . . ; NÞ. At l j ¼ p, that is, after a C2 ðq3 Þ rotation, the surface again reaches the initial position for j ¼ 0 . Therefore, the period of j is p. In this context, the three-dimensional and two-dimensional IDs behave similarly. The variation of I3 with q and y is more complicated and cannot be fully represented by one cross section as above. Another interesting feature of mode mixing, not shown in Figure 4.10, is the variation of the distribution along the n3 axis. Since the distortion along the third ð3Þ acceptor mode was chosen to be moderately strong, D123 ¼ 1, only low vibrational l levels of the q3 normal mode should be occupied (cf. Figure 4.3). However, from Figure 4.10, we conclude that this effect does not generally occur. The ql3 mode can be highly occupied if the coordinate axes are rotated due to mode mixing. Thus, the maximum decay in the examples illustrated in Figure 4.10 takes place at n3 ¼ 10. This unusual spread in the population of vibrational levels, which has no counterpart in the parallel-mode approximation ðj ¼ q ¼ y ¼ 0Þ,is attributed solely to mode mixing. As reported previously, due to normal coordinate rotation the direct ðmÞ displacements k12...N measured along the base vectors qlm are inadequate to describe ðmÞ spectral line broadening or narrowing. More suitable than the k12...N are the dm;n coefficients, which according to Equations 4.68 and 4.69 are complicated functions of ðmÞ the k12...N ; bsm ; blm ðm ¼ 1; 2; . . . ; NÞ and the coefficients of the rotation matrix W (or the Euler angles). Certainly, the dm;n coefficients may be large even though all ðmÞ k12...N ðm ¼ 1; 2; . . . ; NÞ are small. This behavior of the ID is directly related to experimental data such as optical spectra of polyatomic molecules or nonradiative decay rates. For example, the spectral line shape of a spectrum may be deduced by convoluting the ID with Lorentzian functions centered around V
vg
n1 vl1
n2 vl2
n3 vl3 , where V corresponds to the energy gap between the two molecular (electronic) states and vlm are the vibrational frequencies of the final state. This point will be addressed in greater detail in Chapter 7. Here, we merely note that the effect shown in Figure 4.10 leads to a certain mode selection. If  the angle j is equal to 30 , or lies within a certain range  0; 0; 0 ffi 0 for all occupation numbers n1 6¼ 0. The spectroaround 30 , I3 n1 ; n2 ; n3 scopic consequence is an increase in the intensities for the vibronically assisted (electronic) transitions involving the vl2 and vl3 vibrations at the expense of the vl1 vibration, the latter being completely eliminated. The same holds at j ¼ 120 with regard to the second vibration. Mode selection can also occur (in our case of three accepting modes) with the complete elimination of two modes and the dominance of the third remaining mode. Such situation arises if the molecular geometry change upon electronic excitation l ! s locates the initial-state coordinate axes ðqs1 ; qs2 ; qs3 Þ

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1

2

N

relative to the axes ðql1 ; ql2 ; ql3 Þ in a position as described by the displacement vector ðmÞ k123 (or D123 ) and the three Euler angle j; q; and y using the values given in the caption of Figure 4.11. It should be emphasized that such extreme cases occur at those values of j; q, and y which lie in certain finite interval, determined by the spectroscopic constants. In each of the three extreme cases depicted in Figure 4.11, the three-dimensional ID behaves one-dimensionally, being distributed along one of the n1 ; n2 , or n3 axes, respectively. This implies that instead of a level density typical of a polyatomic molecular system, a discrete level structure in the form of a single-mode progression can be resolved from the vibronically assisted molecular spectra. The net result is a proportionate increase in the intensity of these resolved levels with respect to those remaining. The line shape of such spectra or their envelopes are directly given by the IDs. The shape of the plots shown in Figure 4.11, although being functions of one quantum number nm only, are far more complicated than those expressed by one-dimensional IDs. The main difference is that in the latter the

Figure 4.11 Selective properties of I3 showing the reduction of the three-dimensional ID into one-dimensional IDs: (a) distributed along the n1 axis if j ¼ 60 ; q ¼ 30 and y ¼ 270 ; (b) distributed along the n2 axis if j ¼ 30 ;

q ¼ 30 and y ¼ 210 ; and (c) distributed along the n3 axis if j ¼ 30 ; q ¼ 60 and y ¼ 60 . The normal-mode displacements are ð1Þ ð2Þ ð3Þ D123 ¼ D123 ¼ D123 ¼ 3:0 and the vibrational frequencies are the same as in Figure 4.10.

4.4 General Case of N-Coupled Modes ðmÞ

number of progressional members is related to the electronic origin shift k123 upon the electronic excitation, whereas in the former it is governed by the dm;n coefficients ðmÞ which are complicated functions of the k123 and the Euler angles, as mentioned above. Therefore, single-mode progressions such as in Figure 4.11 are either very short (Figure 4.11a) or can be much longer (Figure 4.11b and c) than in the case of a onedimensional ID. These model examples reflect the spectroscopic implications of mode mixing and indicate that in practice special care must be exercised in the analysis of spectral line shapes. Fitting calculated to observed spectral intensities, although conceptually simple, can be difficult to perform, since for every particular molecular system one must find the appropriate coordinate transformation (4.57) between the normal coordinates of the states in question. This requires extensive input information on configurational and frequency changes for all molecular normal modes or the knowledge of the potential energy surfaces of both the states s and l [45]. Let us next investigate the role of normal-mode rotations on the nonradiative decay probability. To do this, one can ordinarily evaluate the quotient wnr ðj; q; yÞ=wnr ð0; 0; 0Þ, using Equation 3.75 together with Equation 3.77 for the nonradiative decay rate, but extended to N ¼ 3 accepting modes. Furthermore, we assume that the initial electronic state s is “vibrationless,” that is, has mm ¼ 0 quanta in all the normal modes. In this case, the sum in Equation 3.77 extends over all  possible final states lnm i whose energies  do not differ from the initial energy by more than c, V
vg
n1 vl1
n2 vl2
n3 vl3  < c. For the width, we have assumed that ~ N is the largest common integer divider of all frequencies vm . ~ N , where v c v Such a situation is encountered in the nonradiative decay of (nearly degenerate) levels in a small molecule embedded in a medium. Under these circumstances the probability per unit time wnr is proportional to c
1 [111] (see Equation 3.80) and the quotient wnr ðj; q; yÞ=wnr ð0; 0; 0Þ becomes independent of c. In the other limit ~ N , which is not considered here, the probability wnr is not dependent on c. This c v situation corresponds to the statistical limit. The results of such model calculations, some of which are portrayed in Figure 4.12, show a considerable change of wnr ðj; q; yÞ on increase of j; q, and y. The increase (decrease) of the transition probability due to rotation is a consequence of the increase (decrease) of the vibrational overlap when the overall shape of I3 “moves” in the space spanned by the occupation numbers n1 ; n2 and n3 . Note that the electronic coupling factor cancel in the expression wnr ðj; q; yÞ=wnr ð0; 0; 0Þ (if there is a single-promoting mode). Due to variation of the Euler angles the position of the maximum of I3 displaces in the direction of those occupation numbers nm having higher energy vibrations vm , thus leading to an increase of the nonradiative transition probability. On the contrary, if the peak in I3 moves toward an occupation number associated with a low-frequency vibration, wnr decreases. This is clearly seen from the curve for q ¼ y ¼ 0 , where the shift of the peak in I3 from the position ð0; 4; 0Þ to ð2; 0; 0Þ changes wnr by five orders of magnitude. Note that in this example vl2 > vl1 . The selectivity of I3 at different positions ðn1 ; n2 ; n3 Þ (see Figure 4.11) also comes into play here. A selection due to rotation of a progression in a high (or low)-frequency mode results almost always in an increase (decrease) of nonradiative transition probability. Such a selection

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1

Figure 4.12 Relative nonradiative decay probability as function of the rotational angles j; q, and y for a model molecule characterized ð1Þ by three accepting modes with D123 ¼ 2:0;

ð2Þ

2

N

ð3Þ

D123 ¼ 1:5; D123 ¼ 0:8 and a single-promoting mode. The effective energy gap V was taken to be 6000 cm
1 and the normal-mode frequencies are the same as in Figure 4.10.

becomes apparent in the curve for q ¼ y ¼ 30 , where the third vibration with the highest frequency chosen is involved. The magnitude and type of the variation of wnr thus depend on the spectroscopic constants describing the effect of frequency distortion and potential surface displacement.

4.5 Displaced Potential Surfaces 4.5.1 The Strong Coupling Limit

In this section, we provide an alternative derivation of the nonradiative decay rate for a statistically large molecule under the simplifying assumption that the normal modes are parallel and their frequencies are the same in the two electronic states

4.5 Displaced Potential Surfaces

j103

under consideration, except for displacements in the origins of the normal coordiðmÞ nates k12...N in the two electronic states under consideration [125–127]. For this case, we can set vsm ¼ vlm or bm ¼ 1 ðbm ¼ 0Þ for all the normal modes m and the singlemode generating function (4.1) simplifies to " # D2m ð1
wm Þð1
zm Þ exp
2ð1
wm zm Þ ðmÞ G1 ðwm ; zm ; am ; 0Þ ¼ : ð4:99Þ 1
wm zm Taking into account Equation 4.99 becomes

zm ¼ eivm t

that

and




wm ¼ e
bhvm
ivm t ðb ¼ 1=kB TÞ,

(

) D2m  
ivm t ivm t
G1 ðwm ; zm ; am ; 0Þ ¼ Zm exp
nm e
ð
nm þ 1Þe coth ðb hvm =2Þ

; 2 ð4:100Þ






m ¼ ðebhvm
1Þ
1 corresponds to the number of where Zm ¼ ð1
e
bhvm Þ
1 and n excited vibrations with frequency vm at thermal equilibrium. We then see that (on using the above substitution for the variables wm and zm ) the generating function becomes temperature dependent. The multidimensional generating function for all N
1 accepting modes is obtained as product of terms (4.100) ( ) N
1 Y X D2m
GN
1 ðtÞ ¼ Zm exp
ð4:101Þ coth ðb hvm =2Þ þ g
ðtÞ þ g þ ðtÞ ; 2 m m¼1 where for convenience we have introduced the relation g þ ðtÞ ¼

1X 2 D ð
nm þ 1Þeivm t ; 2 m m

ð4:102Þ

g
ðtÞ ¼

1X 2
m e
ivm t : D n 2 m m

ð4:103Þ

Taking the explicit expression (3.75) for the single-promoting mode generating function, the expression for the nonradiative decay rate in the simple displaced potential surface model can be written as ( Z1 v     g  g 2
hwnr iT ¼
Rsl dt coth ðb hvg =2Þ þ 1 4h
1 " # 1X 2
exp
itðV
vg Þ
D coth ðb hvm =2Þ þ g þ ðtÞ þ g
ðtÞ 2 m m " #) Z1 X   1 hvg =2Þ
1 dt exp
itðV þ vg Þ
D2 coth ðb
hvm =2Þ þ g þ ðtÞ þ g
ðtÞ þ cothðb
2 m m
1

ð4:104Þ

j 4 Calculational Methods for Intramolecular Distributions I , I , and I

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1

2

N

Under conditions that will be given later, Equation 4.104 is essentially independent of the vibrational relaxation width c and therefore valid when c ¼ 0. Therefore, in writing Equation 4.104 we have made use of the integral representation of the d-function (3.78) to insure conservation of energy. As follows, we will treat Equation 4.101 in the strong coupling limit gð0Þ ¼ g þ ð0Þ þ g
ð0Þ 1

ð4:105Þ

that is, the displacements Dm of the origins between the two electronic states considerably exceed the root mean square vibrational displacements ð
h=vm Þ1=2 , for at least some vibrational modes. This strong coupling limit is generally found for spin-allowed transitions in transition metal ion systems and several organic molecules. The weak coupling limit, which is encountered when gð0Þ  1, behaves differently than the strong coupling limit, so that the relative displacement for each normal mode is small. When gð0Þ 1, the functions g þ ðtÞ and g
ðtÞ in Equations 4.102 and 4.103 can be expanded in a power series of t around t ¼ 0 retaining only terms up to t2,2) 1 X 2 1 gðtÞ ¼ g þ ðtÞ þ g
ðtÞ ¼ gð0Þ þ it D vm
D2 t2 þ Oðt3 Þ; 2 m m 2

ð4:106Þ

where D2 ¼

1X 2 2 v D ð2
nm þ 1Þ; 2 m m m

ð4:107Þ

and gð0Þ ¼

1X 2 1X 2 Dm ð2
nm þ 1Þ ¼ D coth ðb
hvm =2Þ: 2 m 2 m m

ð4:108Þ



1
m ¼ ehvm =kB T
1 The quantity n in Equations 4.107 and 4.108 is the thermal equilibrium vibrational occupation number. Introducing the so-called molecular rearrangement energy in the excited state, EM ¼

1X
hvm D2m ; 2 m

ð4:109Þ

which corresponds half of the Stokes shift of the two electronic states under consideration, Equation 4.104 can be written as 2) If g þ ðtÞ þ g
ðtÞ is large, the integrand in function (4.104) fall off roughly as exp ð
gð0Þ hvi2 t2 =2Þ, so that at sufficient large time t, exp ð
gð0Þ hvi2 t2 =2Þ is already small that the decay is for practical purposes completed before the truncated power expansion of g þ ðtÞ þ g
ðtÞ becomes invalid.

4.5 Displaced Potential Surfaces



 Z1 v       1 22 g  g 2

hwnr iT ¼ coth ðb hv =2Þ þ 1 dt exp
it ðV
v Þ
E = h
t D R g g M sl 2 4
h
1

  þ coth ðb
hvg =2Þ
1



Z1



dt exp
it ðV þ vg Þ
EM

=
h


1

 1 22
D t 2

 : ð4:110Þ

Subsequent integration over t leads to the following result  2  pffiffiffiffiffiffi  (   g 2 )
Rsl vg 2p   hðV
vg Þ
EM
hwnr iT ¼ cothðb hv =2Þ þ 1 exp
g hD 4
2D2
h2 (  2 )
  hðV þ vg Þ
EM
; ð4:111Þ þ cothðb hvg =2Þ
1 exp
2D2
h2 which exhibits a Gaussian dependence on the energy parameter
hðV  vg Þ
EM. Using a kind of coarse graining of the vibrational frequencies permits further simplification of Equation 4.111. Introducing an average frequency hvi, we can write (see Equation 4.107)
h2 D 2

ffi
h hvi EM coth ð
hhvi=2kB T Þ  2EM kB T ;

where T is an effective temperature defined by [128] kB T ¼

  1
h hvi coth
hvg =2kB T : 2

ð4:112Þ

We next assume that the promoting-mode frequencies are small enough to be neglected in the exponents (4.111) and introduce an activation energy EA, given by the height of the point intersection of the two potential surfaces as measured from the bottom of the upper surface [125] E A ¼ ð
hV
EM Þ2 =4EM :

ð4:113Þ

The expression (4.111) for the decay rate assumes the form hwnr iT ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  g 2 p R  vg coth ð
hvg =2kB TÞexp ð
EA =kB T Þ; 4EM kB T sl

ð4:114Þ

At high temperatures, T coincides with the ambient temperature T, whereas at energy. In the lowlow temperatures kB T 
hhvi=2, the mean zero  point motion  temperature limit b
hvg 1 and therefore coth b
hvg =2  1. This permits us to rewrite Equation 4.111 in a form ( ) ffi v   rffiffiffiffiffi ðV
vg
vs Þ2 2p g  g 2 exp
; ð4:115Þ hwnr i ¼
Rsl 2h 2s2 s2

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1

2

N

where vs ¼

1X 2 D vm ; 2 m m

ð4:116Þ

s2 ¼

1X 2 2 D v ; 2 m m m

ð4:117Þ

and

which is in agreement with the result of [129] as was derived previously by an alternative way, using in a formal sense lim

c

c ! 0 ðEa
Ea0 Þ2

þ c2

¼ pd ðEa
Ea0 Þ;

which has meaning when the quantity appears under an integral sign. A quite general feature of the result obtained in the displaced potential energy surfaces approximation is the relation between the nonradiative transition rate and the energy gap. If the energy gap
hV is small, the exponential functions in Equation 4.115 are correspondingly large. As a rule of thumb, the less the energy gap, the larger the transition probability; conversely, the transition probability decreases exponentially as the energy gap is continuously increased. This results in the well-known “exponential energy gap law” that is common to several different types of multiphonon radiationless and vibrational relaxation processes. 4.5.2 The Weak Coupling Limit

The weak coupling limit is encountered as we have stated above, when

X m

D2m  1 or

EM 
hhvi. Thus, the relative displacement for each mode is small. This complicates the discussion of the correctness of our procedure. It merely confirms our earlier observation that if the displacement parameters are small, than the effect of the frequency distortion bm ¼ vsm =vlm on the values of the ID is appreciable. This conclusion can also be reached by a seemingly different argument recalling that the displacement parameters Dm refer to linear interaction terms (which produce a horizontal shift of the B–O potential energy surfaces), whereas the change of the vibrational frequencies bm refer to the quadratic interaction terms (the latter are derivable from the curvature of the potential energy surfaces). Thus, when Dm becomes large, the effect of bm is small and conversely, at small Dm along the qm coordinate, the effect of bm dominates.3) Therefore, there is no point in providing a derivation of the expression for the nonradiative decay rate in the weak coupling limit in the approximation of displaced potential surface model. Instead, in the weak coupling limit, the result of Section 4.3 is correct and must be used. 3) In fact, even if the linear coupling is weak, it is not correct to assume that the quadratic coupling is also weak.

4.6 The Contribution of Medium Modes

4.6 The Contribution of Medium Modes

Before leaving the subject of radiationless processes in an isolated molecule, we mention briefly the role of vibrational relaxation involved in the electronic relaxation process of a small molecule in a medium. To describe the influence of the medium, we shall use an approximate method involving the introduction of a so-called supermolecule [111] with a single intramolecular promoting mode and a single accepting mode and N intermolecular modes characterized by frequencies va and ðaÞ reduced displacements D12...N ; a ¼ 1:2; . . . ; N. We shall accept the experimental ðaÞ fact that the molecule medium coupling is weak and defined by the parameters D12...N and that the frequency vm considerably exceeds the medium frequencies va .4) The intramolecular modes are specified by the frequencies vg and vm , respectively, ðmÞ ðgÞ and the reduced displacement D1 and D1 ¼ 0. Treating the intermolecular vibrations analogously to the intramolecular vibrations, the physical situation is reminiscent of a statistical limit (whereupon the nonradiative decay rate is independent of the relaxation width c). Separating the intramolecular and the intermolecular contributions and using Equations 4.100–4.104 for moderately low temperatures b
hv 1; coth ðb
hv=2Þ  1, the transition probability takes the form wnr

" # v   1 ðmÞ2 1 X ðaÞ2 g  g 2 ¼
Rsl exp
D1
D 2 2 a 12...N 2h Z1 
1

!     1 ðmÞ2 ivm t 1 X ðaÞ2 iva t dt exp
itðV
vg Þ exp D1 e D e exp : 2 2 a 12...N ð4:118Þ

Equation 4.118 can be expressed in terms of separate contributions of the intramolecular high-frequency mode and intermolecular low-frequency modes. Using the expansion exp

n   X ðmÞ2 1 ðmÞ2 ivm t ðD1 =2Þ exp ðinvm tÞ; ¼ D1 e 2 n! n

ð4:119Þ

Equation 4.118 can be reformulated as   ðmÞ2 n o v   X 1 ðmÞ2 ðD1 =2Þn ~ g  g 2 ðaÞ wnr ¼
Rsl exp
D1 GN V
vg
nvm ; D12...N ; 2 2h n! n ð4:120Þ 4) To pursue this matter a little further, we remark that we may define a model Hamiltonian X for ðaÞ vibrational relaxation of the molecule in a dense medium in the form Hm ¼ Cqm exp
D12...N qa a [130].

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j 4 Calculational Methods for Intramolecular Distributions I , I , and I

108

1

2

N

where

~ N V
vg
nvm ; G Z1 
1

n

ðaÞ D12...N

o

1 X ðaÞ2 ¼ exp
D 2 a 12...N

!

  1 X ðaÞ2 iva t dt exp
itðV
vg
nvm Þ exp D e 2 a 12...N

! ð4:121Þ

plays the role of the medium line shape function. Its appearance here does not, of course, violate energy conservation since the energy gap
hV is bridged by intramolecular vibrations and as will be shown below by medium induced Stokes shift. Proceeding as in Section 4.5, expression (4.121) can be simplified further in the limit of strong coupling to the medium modes (phonons) X ðaÞ2 ðD12...N =2Þ 1 ð4:122Þ a

writing (  2 ) n o  2p 1=2 ðV
vg
nvm Þ
EM =
h ðaÞ ~ ¼ ; exp
GN V
vg
nvm ; D12...N D2M 2D2M ð4:123Þ

where D2M ¼

1 X 2 ðaÞ2 v D ; 2 a a 12...N

ð4:124Þ

is responsible for the medium broadening, while EM ¼

1X
ðaÞ2 hva D12...N ; 2 a

ð4:125Þ

is responsible for the Stokes shift due to the medium modes. We substitute Equation 4.121 into Equation 4.118 to obtain

n   DðmÞ2 =2 p 1=2  v   X 2 1 1 ðmÞ g  Rg  2 wnr ¼ exp
D1
sl hDM n! 2 2 n " # ðV
vg
nvm
EM =
hÞ2 :  exp
2 2DM

ð4:126Þ

Equation 4.126 describes the nonradiative decay of a small molecule strongly coupled to the medium in terms of a superposition of an intramolecular (Poisson) distribution and Gaussian. The summation over the occupation numbers n of the intramolecular accepting mode in the final electronic state is implied in Equation 6.126. The contribution of the medium modes in terms of the parameters ðmÞ2 EM and DM is contained in the Gaussian distribution. For D1 =2  1, it may happen

4.6 The Contribution of Medium Modes

that the largest contribution to this decay rate will occur from low-lying off-resonance levels in the final manifold, which contribute via the tails of the distributions. The methods we have described for studying electronic relaxation and decay of electronic states may be applied to a variety of problems. These include, for example, the study of vibrational relaxations (VR) which will be illustrated with a simple application to the decay of an initially prepared harmonic molecular oscillator state Q Q ji >¼ j1; a na > of a macroscopic system to the final state jf >¼ j0; a n0a >. The system is supposed to contain a guest, represented by a harmonic molecular oscillator of frequency v, coupled to a very large number of medium (phonon) states jfna g > of considerable lower frequencies va . Then, for the VR rate wvr , we have [130]   XXY 2p
h wvr ¼
hvÞ Za
1 expð
b
hna va Þ ½1
expð
b
h 2v 0 fn g fn g a a

a

 Y  Y 2  X  0  0

 n jFðfq gÞj n d ðn
n Þ hv
hv a a a a a a   a

a

a

ð4:127Þ

where the coupling of the guest molecule to the medium phonons is given by the Hamiltonian

X Hm ¼ Cq exp
a Da ðva =
hÞ1=2 qa ¼ q Fðfqa gÞ; ð4:128Þ P

which involves the Debye-Waller factor exp
a Da ðva =
hÞ1=2 qa due to the thermal vibrations of the medium modes about the equilibrium positions. In one of the case of greatest physical interest, the coupling varies linearly with the normal coordinate of the molecular oscillator q. We might calculate wvr directly. It is simpler, however, to evaluate it using the generating function technique of Chapter 3. By following the same steps that led to representation (4.99) one can represent the generating function of eq. (4.127) in the form XX exp½
na ðit va þ b
hva Þ expðitn0a va Þ fna g fn0a g

 Y   X  Y 2     na  exp
Da ðva =
hÞ1=2 qa  n0a   a

a

a

 Y  1X 2 ¼ Ga1
wa ;
za ;
Da ; 0 2 a a

ð4:129aÞ

where h 2 i ð1þwa Þð1þza Þ  exp D a 2ð1
wa za Þ 1 2 Ga1
wa ;
za ;
Da ; 0 ¼ 1
wa za 2
 1 2 ¼ Za exp Da ð2na þ 1Þ þ gþa ðtÞ þ g
a ðtÞ : 2 

ð4:129bÞ

j109

j 4 Calculational Methods for Intramolecular Distributions I , I , and I

110

1

2

N

Eq. (4.129b) has the form of the expression (4.99) for the generating functions for electronic relaxation between two vertically displaced potential surfaces, except with wm 2 replaced by
wa, zm by
za and D2m by
Da and exhibits the same t dependence as the 2 latter. We emphasize, however, that the effect of replacing the D2m by
Da changes the temperature dependence of wvr manifested through na dramatically in comparison to the expected for electronic relaxation processes [130]. Now, expression (4.129a) is expanded in terms of the associated MID for different vibrational occupation numbers n1 ; n2 ; . . . ; nN ; n01 ; n02 ; . . . ; n0N according to (4.79). This may be substituted into eq. (4.127), to give an expression for the rate constant wvr which has the form of a Fourier integral Z1 wvr ¼ D
1 Z1

¼D



Y X dt exp
ivt
a ca jtj Ga ðtÞ a 1


  X  X dt exp
it v þ ðna va
n0a va Þ
ca jtj a


1



XX

  X 0 0 0 ð
1Þn1 þn2 ...þnN þn1 þn2 þ...þnN exp
b
h na va

fna g fn0a g



Y

a

  1 2 0 I n ; n ;
D ; 0 1 a a a 2 a

a

ð4:130Þ

Q where D ¼ ð2
hvÞ
1 C 2 ½1
expð
b
hvÞ a Za
1 . In obtaining this result we have taken proper account of the conservation of energy P P manifested through the delta function dðv þ a na va
a n0a va Þ. If we wish the rate at which phonons are emitted into frequency range of the lattice including resonance and local impurity modes, we adopt the validity of the limit of fast P vibrational relaxation, defined in terms of the relation a ca t 1 with t being the time scale. The method

of carrying out the calculation of (4.130) and the MID Q 1 2 0 I n ; n ;
D ; 0 has been described before quite explicitly. a a a 1 2 a

j111

5 The Nuclear Coordinate Dependence of Matrix Elements With the conclusion of Chapter 4 we have completed our formal development of electronic relaxation processes in polyatomic molecules and the remaining portion of the book will be largely devoted to applying the results thus far obtained to special situations. We will also consider the nuclear coordinate dependence of electronic matrix elements that occurs in the rate expression for the nonradiative transitions. Besides, topics such as radiative decaying states, excitation energy transfer in molecular crystals, and other applications will also be taken up subsequently. Section 5.1 deals with the nuclear dependence of the electronic matrix elements for radiationless transitions by the use of a q-centroid approximation. The latter is obtained as an average in which the density of states is weighted by means of vibrational overlap factor using the ID procedure developed in earlier chapters. Such studies considerably enhance the understanding and accuracy of transition probability calculation and also underscore previous warnings as to the enormous errors incurred by using the Condon approximation for the nuclear coordinate-dependent energy denominators that appear in the electronic matrix elements.

5.1 The q-Centroid Approximation

As remarked in Chapter 3, simply formulating the non-Born–Oppenheimer (BO) coupling element Vsm;ln ¼ ðhjs ðr; qÞxsm ðqÞjH 0 BO jjl ðr; q Þxln ðq ÞiÞ     i  1
  2 P h
 ¼

h2 js xsm qjl =qqg qxln =qqg þ xsm js q =qq2g jl xln 2 g ð5:1Þ

as a product of an electronic transition matrix element and a density of states weighted vibrational overlap factor results in an enormous simplification of the actual problem of calculating nonradiative transitions between the electronic states s and l: In this chapter, we shall specify the nuclear configuration q0 at which the integration

j 5 The Nuclear Coordinate Dependence of Matrix Elements

112

over the electronic coordinates r of the integral in (5.1) is evaluated. From this, it consequently follows that the matrix element (5.1) for calculation of the thermally averaged internal conversion rates is written as a product of electronic factor and vibrational factor. As one should expect from the terminology, the earliest application of the q-centroid, strictly speaking, the r-centroid method was made in diatomic spectroscopy [131], although in a completely different physical sense. The q-centroid approximation presents a nontrivial generalization of the r-centroid approach of diatomic spectroscopy to the case of the nonradiative decay of polyatomic molecules. The importance of studying the q-dependence of the nonadiabatic coupling element was emphasized by Freed and Gelbart [132] and Freed and Lin [27]. The customary approach to (5.1) involves the expansion of the electronic wavefunctions appearing in (5.1) in a Taylor series about some reference configuration q0 [133, 134]. Often this reference configuration is chosen as the equilibrium configuration qe of either the ground electronic state or the excited electronic state. This is the approach that is almost universally used in the case of radiation transitions and corresponds to using the crude BO wavefunctions. In the case of nonradiative transitions when using adiabatic electronic wavefunctions, considerable errors are incurred, which are incorrect by orders of magnitude if the usual Condon approximation is invoked [136, 137]. On the other hand, the choice of q0 is purely ad hoc, since nothing has singled out q0 as natural, so there is no unique choice of q0 . If, for instance, terms in all orders in the Taylor expansion of the electronic wavefunction about q0 are retained, then any reasonable choice of q0 , which makes the series converge, is adequate. However, this is not necessarily true if only some leading terms in the expansion are retained. Thus, it is necessary to find an optimal q0 to ensure the best rapid convergence of the expansion. In Section 3.3, we saw that the energy conservation imposes a stringent selection rule on nonradiative transition. However, the energy conservation is often quite unfavorable with the vibrational overlap factor and this favors a q-centroid, which may considerably differ from that of the equilibrium position of either of the two electronic states. Therefore, Freed and Lin have shown that the proper treatment of this problem is obtained from a thermally averaged decay rate with density of states weighted vibrational overlap factor. This average is not restricted to geometries near the equilibrium positions of the electronic states under consideration [27]. The basic idea of the q-centroid approach rests on the following assumption. The vibrational overlap factor, as we saw in Chapter 4, is a sharply peaked function of nuclear coordinates, while the electronic matrix element (apart from particular instances) is often a slowly varying function of q. Thus, we assume that it is reasonable to expand the electronic matrix element about a point in the vicinity of the peak of the vibrational overlap factor. An optimal expansion point is chosen to guarantee the most rapid convergence of series expansion or as that that makes the leading correction term for the thermally averaged nonradiative decay rate vanish. This point is called the q-centroid and corresponds to a nuclear configuration very often of lower symmetry than the equilibrium configuration.

5.1 The q-Centroid Approximation

In this chapter, we shall extend the description given in Ref. [27]. This will include the effect of Duschinsky mixing for some (totally symmetric) modes. We shall also give an extensive discussion of the dependence of the electronic matrix element on the nuclear coordinates. In this section, we will justify the use of such favored nuclear configuration by means of which the electronic matrix element should be determined. We shall do this by considering the expansion of the electronic wavefunctions js and jl derived in Section 1.3: ð0Þ

ð1Þ

ð2Þ

ð0Þ

ð1Þ

ð2Þ

js ¼ js þ ljs þ l2 js þ    jl ¼ jl þ ljl þ l2 jl þ    ;

ð5:2Þ

where the order of perturbation have now been displaced explicitly in terms of ðiÞ ðiÞ a parameter l.1) The functions js and ðjl Þ in (5.2) are corrections of ith orders, obtained through the Rayleigh–Schr€odinger expansion of the adiabatic wavefunction js ðjl Þ in terms of crude BO wavefunctions about arbitrary reference values q0 . The latter may be determined by the generalized average q-centroid approximation. If we take ðqU=qqm Þ0 to be of the first order, ðq2 U=qqm qqg Þ0 of the second order, and so ð1Þ ð2Þ on, the functions jb ðb ¼ s; lÞ are linear in qm , jb are quadratic in qm , and so on. Substituting the expansion (5.2) of electronic wavefunction js and jl into the matrix element (5.1) and collecting terms of the same order in l after squaring the matrix element, we obtain
D E
D E   ð1Þ ð0Þ ð1Þ 0 Vsm;ln 2 ¼l2 jð0Þ jl xln jH0 BO jjs xsm s xsm jH BO jjl xln
D E

D E 
D E ð0Þ ð1Þ ð0Þ ð2Þ ð1Þ ð1Þ jl xln jH0 BO jjs xsm þ jl xln jH 0 BO jjs xsm þl3 js xsm jH 0 BO jjl xln

D E
D E
D E ð0Þ ð2Þ ð1Þ ð1Þ ð0Þ ð1Þ jl xln jH 0 BO jjs xsm þl3 js xsm jH 0 BO jjl xln þ js xsm jH 0 BO jjl xln ð2Þ

ð3Þ

¼Csmln þCsm;ln ; ð5:3Þ

where we have truncated the expansion after the third-order term of jVsm;ln j2 (in the expressions Csm;ln , we have set l¼1). Using the expansion (5.2), the electronic matrix elements appearing in (5.3) can be evaluated to give in lowest order of perturbation,   D E   * +  ð0Þ ð0Þ   q  js ðqU=qqlg Þ0 jl   ð1Þ g ; ð5:4aÞ ¼ i
h Rsl ¼
h jð0Þ j s i  qqlg  l El ðq0 Þ
Es ðq0 Þ   D E   + ð0Þ  s  ð0Þ  q  j ðqU=qq Þ j  s g 0 l  ð0Þ  ; ¼ i
h ¼h
jl i s jð1Þ  qqg  s Es ðq0 Þ
El ðq0 Þ *

g

Rsl

ð5:4bÞ

1) The purpose of introducing the parameter l is simply to identify the various orders in the perturbation. The parameter l is then set equal to 1 when this identification has been made.

j113

j 5 The Nuclear Coordinate Dependence of Matrix Elements

114

and in the next (third) order of perturbation 0 1  * +   2  ð0Þ  q Uðr; qÞ ð0Þ @  A   * + " jl  qqs qqs js     n g X ð0Þ  q  ð2Þ 0 ¼ i
h ðqsn
q0n Þ
h jl i s js ðq Þ
E ðq Þ E s l  qqg  0 0 n þ

  D ED  ð0Þ E  ð0Þ ð0Þ  ð0Þ  jc ðqU=qqsn Þ0 js X jl ðqU=qqsg Þ0 jc c6¼s

þ

 D E  ð0Þ ED ð0Þ   ð0Þ ð0Þ  jc ðqU=qqsg Þ0 js X jl ðqU=qqsn Þ0 jc c6¼s

¼

ðEs ðq0 Þ
El ðq0 ÞÞðEs ðq0 Þ
Ec ðq0 ÞÞ

X n

ðEs ðq0 Þ
El ðq0 ÞÞðEs ðq0 Þ
Ec ðq0 ÞÞ

ðqsn
q0n Þ # ðqsn
q0n Þ

Mgn ðslÞðqsn
q0n Þ;

ð5:5aÞ

  D ED E    + ð0Þ  ð0Þ    l  ð0Þ s  ð0Þ j j j ðqU=qq Þ ðqU=qq Þ  X X c c   n 0 g 0 js l ð1Þ  q  ð1Þ ¼ i
h ðqln
q0n Þ
h jl i s js ðq Þ
E ðq ÞÞðE ðq Þ
E ðq ÞÞ ðE s c c l  qqg  0 0 0 0 n c6¼s X ¼ Lgn ðslÞðqln
q0n Þ; ð5:6aÞ *

n

where the summation over n in Equations 5.5 and 5.6 includes all (promoting and accepting) modes and *
2 h

 E  2D ð0Þ  + 2  ð0Þ  jl ð1=2Þðq2 U=qqsg Þ0 js q2  ð2Þ j ¼ 2
h2 4 s2  s Es ðq0 Þ
El ðq0 Þ qqg   

ð0Þ  jl 

    D ED E  ð0Þ  ð0Þ 3 ð0Þ  ð0Þ  jc ðqU=qqsg Þ0 js X jl ðqU=qqsg Þ0 jc 5: þ ðEs ðq0 Þ
El ðq0 ÞÞðEs ðq0 Þ
Ec ðq0 ÞÞ c6¼s

ð5:7Þ

In Equations 5.4–5.7, the energy of the zero-order states, Eb ðq0 Þ ðb ¼ s; l; cÞ, is taken at the q0 -centroid. The third-order (in l) electronic matrix elements appearing on the left-hand (bra) side of (5.3) are obtained from the derivations (5.5a), (5.6a), and (5.7) with indices s and l interchanged:   * +   X ð0Þ  q  ð2Þ ¼
h js i l jl Mgn ðlsÞðqln
q0n Þ ð5:5bÞ  qqg  n and *
h

  +  q  X  ð1Þ ¼ Lgn ðlsÞðqsn
q0n Þ: jl l qqg  n

 jð1Þ s i

ð5:6bÞ

5.1 The q-Centroid Approximation

j115

With these expressions we have given a complete set of matrix elements to evaluate the thermally averaged rate constants for the radiationless decay between states s and l [138 –143] in successive order of approximation with respect to l. Following the concept of Section 3.3, we have in second order 2p X 1 C ð2Þ hwnr iT ¼ pðm; TÞCsm;ln ¼ w ð2Þ ðslÞc þ wð2Þ ðslÞcc ;
h fm g;fn g p ðEsm
Eln Þ2 þ C2 m m ð5:8Þ

where wð2Þ ðslÞc represents just the expression (3.75) for the nonradiative decay rate considered in Section 3.3. The second term wð2Þ ðslÞcc in Equation 5.8 contains mixedtype single promoting mode generating functions Ig ðtÞ and Ig0 ðtÞ (see Equations 3.10 and 3.74) and vanish, if the promoting modes are nontotally symmetric, as in this case kg ¼ kg0 ¼ 0. The q-centroid is introduced at the next level of approximation. Taking first the third-order terms occurring on the right (ket) side of (5.3), we have w ð3Þ ðslÞ ¼ w ð3Þ ðslÞ1 þ w ð3Þ ðslÞ2 ;

ð5:9Þ

where

  0 1    q   2p X X ð2Þ @ g s  l w ðslÞ1 ¼ pðm; TÞRsl  Rsl xsm ð
q Þi
h l xln ð
q ÞA xln ðql Þjxsm ðqs ÞÞ
h g;g0 fm g;fn g  q
qg  ð3Þ

n



1 C ; p ðEsm
Eln Þ2 þ C2

with ð2Þ Rsl

n


2 h ¼
2

*

ð5:10aÞ

  +  q2   ð2Þ ; 2 j qqsg0  s

ð0Þ  jl  

ð5:11Þ

and  1 0    XX X   2p q g ð3Þ  @ w ðslÞ2 ¼ pðm; TÞRsl xsm i
h l xln A
h g;g0 m fm g;fn g q
q  g n

n

 1 0  13    s  qx qx C A þLg0 m ðslÞ@xln ðql
q0m Þi
A5 1 h sm : 4Mg0 m ðslÞ@xln ðqm
q0m Þi
h sm m  s s qq qq p ðEsm
Eln Þ2 þ C2   g0 g0 2

0

ð5:12Þ

Introducing integral representation of the Lorentzian function (see Equation 3.9) into (5.10) and proceeding as in Section 3.3, we obtain w ð3Þ ðslÞ1 ¼ ð
h
2 ÞZ
1

X g;g0

g

ð2Þ

1 ð

Rsl Rsl

dt exp½
iV
cjtjIg ðtÞGN
1 ðtÞ;
1

ð5:10bÞ

j 5 The Nuclear Coordinate Dependence of Matrix Elements

116

where Equations 3.12, 3.15 and 3.74 have been used. Notice that wð3Þ ðslÞ1 ¼ 0, if the promoting modes g are nontotally symmetric. To continue, we consider w ð3Þ ðslÞ2. The appearance of qsm (and qlm ) coordinates in the expression of (5.12) must be considered in detail, because the summation over m covers normal modes that have nonvanishing normal coordinate displacements ðmÞ k23;...;N and mix under a perturbation of the full symmetry of the molecular group with corresponding partners of identically transforming symmetry representation. Thus, the modes constitute a block separated from the remaining (promoting) modes discussed so far (see the shaded area in the matrix W of Section 1.2). If for convenience in order to clearly portray the physical situation, we regard the presence of only two accepting modes, this may be done using Equation 3.21. Although our discussion deals with the case of two accepting modes, the principal results obtained can easily be generalized. To represent Equation 5.12 in a particular transparent manner, it is convenient to decompose (split) it into classes or prototype parts, depending on the nature of the triple indices g; g0 ; and m representing modes of different symmetries, w ð3Þ ðslÞ2 ¼ w ð3Þ ðslÞ21 þ w ð3Þ ðslÞ22 þ wð3Þ ðslÞ23 þ w ð3Þ ðslÞ24 þ w ð3Þ ðslÞ25 þ w ð3Þ ðslÞ26 ; ð5:13Þ

where the individual components are obtained directly from Equation 5.12. These are as under: 1( 0   X X    2p qx g ln w ð3Þ ðslÞ21 ¼ pðm; TÞRsl @xsm i
h l A Mgg ðslÞ þ Lgg ðslÞ
h g fm g;fn g  q
qg n

n

 1 0  19   =  l  qx qx sm A A þ kg Mgg ðslÞ@xln i
dðEsm
Eln Þ; h @xln ðqg
q0g Þi
h sm  s qqsg   qqg ; 0

ð5:14Þ

where for the promoting mode, we write qsg ¼ qlg þ kg . For convenience, we have reinserted in Equation 5.14 the energy conserving d-function in place of the Lorentzian distribution of Equations 5.10 or 5.12 (but with the understanding that smooth representations of the Dirac delta function are to be taken). For the second term in Equation 5.13, we may write w ð3Þ ðslÞ22

1( 0    xln 2p X X X g@  ¼ pðm; TÞRsl xsm i
h l A Mgm ðslÞ
h g m6¼g fm g;fn g  q
qg n

n

 1 0  19   =  s  qx qx sm A A þ Lgm ðslÞ@xln ðql
q0m Þi
dðEsm
Eln Þ: @xln ðqm
q0m Þi
h sm h m  qqsg qqsg ;   0

ð5:15Þ

The relationship here between the totally symmetry coordinates qsm and qlm ðm ¼ 1; 2Þ are given by Equation 3.21. Similarly,

5.1 The q-Centroid Approximation ð3Þ

w ðslÞ23

1( 0    qxln   2p X X X g@ ¼ pðm; TÞRsl xsm i
h l A Mmg ðslÞ þ Lmg ðslÞ
h g m6¼g fm g;fn g  q
qg n

n

 1 0  19    l  qxsm = qx sm @xln ðqg
q0g Þi
h s A þ kg Mmg ðslÞ@xln i
h s A dðEsm
Eln Þ; qqm   qqm ; 0

ð5:16Þ

where the summation over m covers totally symmetric modes. In the following expressions, g and g0 refer to promoting modes that are distinct: 1( 0    qxln   2p X X X g@ ð3Þ w ðslÞ24 ¼ pðm; TÞRsl xsm i
h l A Mg0 g0 ðslÞ þ Lg0 g0 ðslÞ
h g g0 6¼g fm g;fn g  q
qg n n m

 1 0  19    l  qxsm = qx sm @xln ðqg0
q0g0 Þi
h s Aþ kg0 Mg0 g0 ðslÞ@xln i
h s A dðEsm
Eln Þ; qqg0   qqg0 ; 0

ð3Þ

w ðslÞ25

ð5:17Þ

1( 0    qxln 2p X X X g@ ¼ pðm; TÞRsl xsm i
h l A Mg0 m ðslÞ
h g6¼g0 m6¼g fm g;fn g  q
qg n

n

 1 0  19   =  s  l qx x sm sm h s A dðEsm
Eln Þ: @xln ðqm
q0m Þi
h s A þ Lg0 m ðslÞ@xln ðqm
q0m Þi
qqg0 qqg0 ;   0

ð5:18Þ

Finally, we consider ð3Þ

w ðslÞ26

1( 0    qxln   2p X X X g@ ¼ pðm; TÞRsl xsm i
h l A Mgg0 ðslÞ þ Lgg0 ðslÞ
h g g0 6¼g fm g;fn g  q
qg n

n

 1 0  19    l  qxsm = qx sm   @xln ðqg0
q0g0 Þi
h s A þ kg0 Mgg0 ðslÞ@xln i
h s A dðEsm
Eln Þ: qqg   qqg ; 0

ð5:19Þ

This last expression may be regarded as a supplement to the expression w ð3Þ ðslÞ21 , if g and g0 are assigned to distinct promoting modes. For further discussion of these equations, it is useful to transform the w ð3Þ ðslÞ2i of Equations 5.14–5.19 into a more convenient form. Proceeding as in Chapter 3, the expression for w ð3Þ ðslÞ21 can be written as w ð3Þ ðslÞ21 ¼ ð
h
2 ÞZ
1

X g

g

1 ð

Rsl

dt exp½
iVt
cjtj
1

f½Mgg ðslÞ þ Lgg ðslÞ½Jgð0Þ ðtÞ
q0g Kgð0Þ ðtÞ þ kg Mgg ðslÞKgð0Þ ðtÞgGN ðtÞ;

ð5:20Þ

where Kg ðtÞ corresponds to the single-mode generating function (which involves the nuclear momentum operator for the promoting mode qg ) defined in Equation 3.72,

j117

j 5 The Nuclear Coordinate Dependence of Matrix Elements

118

and Jg ðtÞ is defined by X Jg ðtÞ ¼ exp½
mg ðit vsg þ
hvsg =kB TÞexpðitng vlg Þ mg ;ng

1 ð


h2

ð

qxlng ðblg
qlg Þ 1=2

dqlg d
qlg qlg xsmg ðbsg
qsg Þi 1=2

q
qlg


1

¼
h2

ÐÐ

qxsmg ðbsg qsg Þ 1=2

xlng ðblg qlg Þi 1=2

qqsg

q s s s q r ðq ;
q ; wg Þi l rlg ð
qlg ; qlg ; zg Þ: qqsg g g g q
qg

dqlg d
qlg qlg i

ð5:21Þ

Jg ðtÞ is calculated in the Appendix D and can be expressed as 0 1 2
g bsg kg 2bsg blg k2g w
g w
h2 s l A @ 4 Jg ðtÞ ¼
bg bg
3 s 2 s l s l
1
1
1 2
g þ bg
zg Þ ðbg
zg þ bg w
g Þ

1 ðbg w ðbg
zg þ blg w g Þ þ

#


z
1 g

g

g

G1 ðtÞ ¼ Jgð0Þ ðtÞG1 ðtÞ:

l
1

1 zg Þ ðbsg w g þ bg


ð5:22Þ

ð0Þ

The part of w ð3Þ ðslÞ21 involving Jg ðtÞ and kg vanishes if the promoting mode g is nontotally symmetric. Similarly, after a sequence of steps, the next expression can be brought to the form wð3Þ ðslÞ22 ¼ ð
h
2 ÞZ
1

XX g

m6¼g

g

1 ð

Rsl

h

ð0Þ ðtÞ
q0m dt exp½
itV
cjtj Mgm ðslÞ H m


1


i þ Lgm ðslÞ Hmð0Þ ðtÞ
q0m  Kgð0Þ ðtÞGN ðtÞ

ð5:23Þ

where ð Hm ðtÞ ¼

1 ð



dql1 dql2 d
ql1 d
ql2 qlm rs1 ðqs1 ;
qs1 ; w1 Þrs2 ðqs2 ;
qs2 ; w2 Þrl1 ð
ql1 ; ql1 ; z1 Þrl2 ð
ql2 ; ql2 ; z2 Þ;


1

ð5:24Þ

and correspondingly ðm ¼ 1:2Þ
m ðtÞ ¼ H

ð

ð1 

dql1 dql2 d
ql1 d
ql2 qsm rs1 ðqs1 ;
qs1 ; w1 Þrs2 ðqs2 ;
qs2 ; w2 Þrl1 ð
ql1 ; ql1 ; z1 Þrl2 ð
ql2 ; ql2 ; z2 Þ;


1

ð5:25Þ

with qsm and qlm given by Equation 3.21. The straightforward, but somewhat tedious, evaluation of these integrals is made in Appendix D. The results are Hm ðtÞ ¼ Hmð0Þ ðtÞG2 ðtÞ;

m ¼ 1; 2;

ð5:26Þ

5.1 The q-Centroid Approximation

with ð0Þ

H1 ðtÞ ¼

n o
1 ð12Þ
1w
2 þ w11 b1 kð1Þ
1
z2 þ w21 b21 kð2Þ
2
z2 ; b1 b2 k1 w 12 w 12 w B1 ðw1 ; w2 ; z1 ; z2 Þ ð5:27Þ

and ð0Þ

H2 ðtÞ ¼

n o
1 ð12Þ
1w
2 þ w12 b12 kð1Þ
1
z1 þ w22 b2 kð2Þ
2
z1 : b1 b2 k2 w 12 w 12 w B1 ðw1 ; w2 ; z1 ; z2 Þ ð5:28Þ

The quantity B1 was previously derived and given by Equation 3.27: 2 2
1w
2 þ w11
1
z2 þ w12
1
z1 B1 ðw1 ; w2 ; z1 ; z2 Þ ¼ b1 b2 w b1 w b12 w 2 2
2
z2 þ w22
2
z1 þ
z1
z2 : þ w21 b21 w b2 w

ð5:29Þ

Without repeating the details, we follow once again the steps that led to Equations 5.26–5.28. With somewhat greater effort and upon the results given in Appendix D, we find
ð0Þ ðtÞG2 ðtÞ;
m ðtÞ ¼ H H m

m ¼ 1; 2;

ð5:30Þ

with
ð0Þ H 1 ðtÞ ¼

n o 1 ð12Þ
2
z2 þ w22 b2 kð12Þ
2
z1 þ kð1Þ
w21 b21 k2 w z1
z2 ; 1 w 12
B1 ðw1 ; w2 ; z1 ; z2 Þ ð5:31Þ

and
ð0Þ ðtÞ ¼ H 2

n o 1 ð12Þ
1
z1 þ w11 b1 kð12Þ
1
z2 þ kð2Þ
w12 b12 k1 w z1
z2 : 2 w 12
B1 ðw1 ; w2 ; z1 ; z2 Þ ð5:32Þ

The complexity introduced by the normal coordinate mixing (3.21) is considerable. ð1Þ ð2Þ Apart from the displacements k12 and k12 (referred to as the direct terms) and frequency parameters b1 and b2 of the individual modes, expressions 5.27–5.32 ð12Þ ð12Þ involve reciprocal displacements k1 and k2 as well as cross-frequency parameters b12 and b21 . This implication will be subsequently overcome by approximations that make the problem more manageable. Let us now focus our attention on w ð3Þ ðslÞ23 . It is worth noting that the summation over m in (5.16) covers totally symmetric modes that have nonvanishing normal coordinate displacements. These modes are supposed to be mixed with each other as we have already shown above and separated from the remaining nontotally symmetric vibrational modes. Then in terms of Fourier integral, we may obviously

j119

j 5 The Nuclear Coordinate Dependence of Matrix Elements

120

rewrite Equation 5.16 as wð3Þ ðslÞ23 ¼ ð
h
2 ÞZ
1

XX g

m6¼g

g

1 ð

dt exp½
itV
cjtj ½Mmg ðslÞ

Rsl
1

ð0Þ

ð0Þ ð0Þ
þ Lmg ðslÞ½Dð0Þ g ðtÞ
q0g Ig ðtÞ þ kg Mmg ðslÞIg ðtÞgN m ðtÞGN ðtÞ;

ð5:33Þ

where again qsg ¼ qlg þ kg have been set. The function Dg ðtÞ for the promoting mode is defined by 1 ð

ð

Dg ðtÞ ¼
h
1


 q
 dqlg d
qlg qlg rsg qsg ;
qsg ; wg i l rlg
qlg ; qlg ; zg ; q
qg

ð0Þ

ð5:34Þ

ð0Þ


m ðtÞ ðm ¼ 1; 2Þ are related to H
m ðtÞ (Equation 5.25), with qsm being replaced and N there by the nuclear momentum operator i
hq=qqsm. Then (see Appendix D), # 2

"
z
1 2bsg k2g
z
1
zg 1 g g g l
s
s G1 ðtÞ Dg ðtÞ ¼ i
hbg s
1 l
1 l l
1 2
1 2





ðbg w g þ bg zg Þ ðbg w g þ bg zg Þ ðbg zg þ bg w g Þ g

¼ Dð0Þ g ðtÞG1 ðtÞ

ð5:35Þ

and
ð0Þ
ð0Þ
mH N hbsm w m ðtÞ ¼
i
m ðtÞ;

m ¼ 1; 2:

ð5:36Þ

ð0Þ

By retaining only the dominating term of Dg ðtÞ when vsg ¼ vlg and kg ¼ 0, we then find Dð0Þ g ðtÞ ¼

 i
h zg
wg i
h 
g expðit vg þ
¼ n hvg =kB TÞ
expð
itvg Þ ; 2 1
wg zg 2

ð5:37Þ


g ¼ ðeb
hvg
1Þ
1 is the number of excited vibrations of the promoting mode where n at thermal equilibrium. Likewise, w ð3Þ ðslÞ24 is rewritten as
2

ð3Þ

w ðslÞ24 ¼ ð
h ÞZ


1

P P

g g0 6¼g Rsl

g

1 ð

dt exp½
itV
cjtjf½Mg0 g0 ðslÞ
1

ð0Þ

ð0Þ

ð0Þ

þ Lg0 g0 ðslÞ½Bg0 ðtÞ þ q0g0 Ig0 ðtÞ þ kg0 Mg0 g0 ðslÞIg0 ðtÞgIgð0Þ ðtÞGN ðtÞ; ð5:38Þ

where 1 ð

Bg0 ðtÞ ¼
h
1

ð

dqlg0 d
qlg0 qlg0 i

q s s s r 0 ðq 0 ;
q 0 ; wg0 Þrlg0 ð
qlg0 ; qlg0 ; zg0 Þ: qqsg0 g g g

ð5:39Þ

5.1 The q-Centroid Approximation

Analogous to Dg ðtÞ, this is found to be 2



1 w
g0 w 1 g0 þ
 Bg0 ðtÞ ¼
i
hbsg0 4
s l s
1 2
g0 þ bg0
zg0
g0 þ blg0
z
1 bg0 w bg0 w g0 # 2bsg blg0 k2g0
z
1 g0 ð0Þ g0 g0

2 G1 ðtÞ ¼ Bg0 ðtÞG1 ðtÞ s
1 l
1
g0 bg0
zg0 þ bg0 w

ð5:40Þ

If the promoting modes g are nontotally symmetric, then w ð3Þ ðslÞ24 vanishes. Let us now return to Equation 5.18. Remembering that the summation over m covers nonseparable totally symmetric modes, we have, similar to as in the case of Equation 5.23, w ð3Þ ðslÞ25 ¼ ð
h
2 ÞZ
1

1 ð 2 n
 XXX g
ð0Þ ðtÞ
q0m Rsl dt exp½
itV
cjtj Mg0 m ðslÞ H m g g0 6¼g m¼1


1


o ð0Þ  Igð0Þ ðtÞIg0 ðtÞGN ðtÞ; þ Lg0 m ðslÞ Hmð0Þ ðtÞ
q0m

ð5:41Þ

from which follows that wð3Þ ðslÞ25 vanishes if the promoting modes g are nontotally symmetric. On completing the evaluation of Equation 5.19, we verify that w ð3Þ ðslÞ26 ¼ ð
h
2 ÞZ
1

XX g g0 6¼g

g

1 ð

Rsl

dt exp½
itV
cjtjf½Mgg0 ðslÞ
1

ð0Þ

þ Lgg0 ðslÞðCg0 ðtÞ
q0g0 Þ þ kg0 Mgg0 ðslÞg  Kgð0Þ ðtÞGN ðtÞ;

ð5:42Þ

with 1 ð

Cg0 ðtÞ ¼

ð

dqlg0 d
qlg0 qlg0 rsg0 ðqsg0 ;
qsg0 ; wg0 Þrlg0 ð
qlg0 ; qlg0 ; zg0 Þ ¼



1


g0 bsg0 kg0 w
g0 ðbsg0 w

þ blg0
zg0 Þ

g0

G1 ðtÞ: ð5:43Þ ð0Þ

For g0 being a promoting mode, only the part of (5.42) that contains q0g0  Kg ðtÞ contributes to the q-centroid. A discussion similar to that carried out above leads for the second part of the squared matrix elements (5.3), which comes from the left (bra) position, to further nonvanishing nonradiative rate constants. These are
ð3Þ ðslÞ21 ¼ ð
h
2 ÞZ
1 w

X

ð0Þ

g

g

1 ð

dt exp½
itV
cjtjf½Mgg ðlsÞ

Rsl


1

þ Lgg ðlsÞ½
J g ðtÞ
q0g Kgð0Þ 
kg Mgg ðlsÞKgð0Þ ðtÞgGN ðtÞ;

ð5:44Þ

j121

j 5 The Nuclear Coordinate Dependence of Matrix Elements

122

ð0Þ

where Kg ðtÞ is given by Equation 3.72 and the single-mode generating function
J g ðtÞ is now defined by ðð q q
J ðtÞ ¼
h2 d qlg d
qlg qsg i s rsg ðqsg ;
qsg ; wg Þi l rlg ð
qlg ; qlg ; zg Þ: ð5:45Þ g qqg q
qg Without repeating the details, we follow once again the steps that led to Equation 5.22 ð0Þ g and obtain
J g ðtÞ ¼
J g ðtÞG1 ðtÞ, where "

2  blg kg 2bsg blg k2g
zg h
ð0Þ s l
J ðtÞ ¼
bg bg g l
1 2 2
g þ blg
zg Þ ðbsg
z
1
g Þ ðbsg w g þ bg w
3


zg ðbsg
z
1 g



1 þ blg w g Þ

þ



1 w g

1 ðbsg w g

þ blg
z
1 g Þ

:

ð5:46Þ

We may also arrive at this result by considering the symmetry of the originally derived generating function Jg ðtÞ, interchanging bsg by blg and simultaneously wg by zg. Note g that incidentally the single-mode generating functions Kg ðtÞ and G1 ðtÞ are invariant under such transformation. We recall that the interchange s , l relates also the electronic matrix elements Mgg ðlsÞ and Lgg ðlsÞ appearing in (5.44) to those given in
ð3Þ ðslÞ21 Equations 5.5a and 5.6a. As a closing remark, we note that the part of w ð0Þ
involving J g ðtÞ and kg vanishes if the promoting modes g are nontotally symmetric. In precisely the same manner as we have done above, the reader can verify by elementary method that
2


ð3Þ

w ðslÞ22 ¼ ð
h ÞZ þ Lgm ðlsÞ


1




XX g

m6¼g

g Rsl


ð0Þ H m ðtÞ
q0m

1 ð

h
 dt exp½
itV
cjtj Mgm ðlsÞ Hmð0Þ ðtÞ
q0m


1 i  Kgð0Þ ðtÞGN ðtÞ;

ð5:47Þ

ð0Þ
ð0Þ ðtÞ ðm ¼ 1; 2Þ are where the two-dimensional generating functions Hm ðtÞ and H m given by Equations 5.27–5.32. Without repeating the details, we follow once again the steps that led to Equation 5.33 and may write


ð3Þ ðslÞ23 ¼ ð
h
2 ÞZ
1 w

XX g

m6¼g

g

1 ð

Rsl

dt exp½
itV
cjtjf½Mmg ðlsÞ
1

ð0Þ ð0Þ ð0Þ
ð0Þ þ Lmg ðlsÞ½D g ðtÞ
q0g Ig ðtÞ
kg Mmg ðlsÞIg ðtÞgNm ðtÞGN ðtÞ;

ð5:48Þ

where m covers nonseparable totally symmetric modes. The single-mode generating function appearing in Equation 5.48 is defined as
g ðtÞ ¼
h D

1 ð

ð

dqlg d
qlg
qsg i

q s s s r ðq ;
q ; wg Þrlg ð
qlg ; qlg ; zg Þ qqsg g g g

0
1 12 3 2

1

1 2blg k2g w w
w 1 g g g s 5Gg ðtÞ
s
s ¼ @ i
hbg A4 s 1 l
1 l
1 þ bl w
1 Þ2 2

1




z w z z ðbg w þ b Þ ðb þ b Þ ðb g g g g g g g g g g g ð0Þ


ðtÞG ðtÞ: ¼D g 1 g

ð5:49Þ

5.2 Determination of the q-Centroid ð0Þ


g ðtÞ Again, the interchange bsg , blg and wg , zg relates the generating function D ð0Þ
ð0Þ ðtÞ is to Dg ðtÞ. The dominating term of D g
ð0Þ ðtÞ ffi D g



   i
h wg
zg
i
h
g expðitvg þ
¼ hvg =kB TÞ
expð
itvg Þ : n 2 1
wg zg 2 ð5:50Þ ð0Þ

The functions Nm ðtÞ appearing in Equation 5.48 are Nmð0Þ ðtÞ ¼
i
hblm
zm Hmð0Þ ðtÞ;

m ¼ 1; 2:

ð5:51Þ

ð0Þ

ð0Þ


ð3Þ ðslÞ23 involving Ig ðtÞ and Since Ig ðtÞ ¼ 0 if qg is a promoting mode, the part of w ð3Þ
ðslÞ24 ; w
ð3Þ ðslÞ25 ; and that that correkg vanishes. The remaining expressions w ð3Þ sponds to w ðslÞ1 do not contribute to the derivation of the q-centroid (as do w ð3Þ ðslÞ24 ; w ð3Þ ðslÞ25 , and w ð3Þ ðslÞ1 above) and are not given here. Finally, analogous to Equation 5.42, we obtain the expression
ð3Þ ðslÞ26 ¼ ð
h
2 ÞZ
1 w

XX g g0 6¼g

g

Rsl

1 ð

n dt exp½
itV
cjtj ½Mgg0 ðlsÞ


1 o ð0Þ
þ Lgg0 ðlsÞ Cg0 ðtÞ
q0g0
kg0 Mgg0 ðslÞ  Kgð0Þ ðtÞGN ðtÞ;


ð5:52Þ

where
ð0Þ C g0 ðtÞ ¼

blg0 kg0
zg0
g0 þ blg0
zg0 Þ ðbsg0 w

:

ð5:53Þ

5.2 Determination of the q-Centroid

Our discussion up to this point has been restricted to determination of the decay rates of nonradiative transitions up to third order of perturbation theory. We have established how the transition probabilities may be reduced by the knowledge of the average of the density of states weighted vibrational overlap factor and its symmetry properties. Assuming that the promoting modes are nontotally symmetric (ns) and separated from the totally symmetric accepting modes (ts), we have found that
ð3Þ ðslÞ1 ¼ w ð3Þ ðslÞ24 ¼ w
ð3Þ ðslÞ24 ¼ w ð3Þ ðslÞ25 ¼ w
ð3Þ ðslÞ25 ¼ 0: w ð2Þ ðslÞcc ¼ w ð3Þ ðslÞ1 ¼ w ð5:54Þ

The nonradiative decay rate constant becomes hwnr iT ¼ w ð2Þ ðslÞc þ

3
 X
ð3Þ ðsl Þ2i þ w ð3Þ ðsl Þ26 þ w
ð3Þ ðsl Þ26 ; w ð3Þ ðsl Þ2i þ w i¼1

ð5:55Þ

j123

j 5 The Nuclear Coordinate Dependence of Matrix Elements

124

where w ð2Þ ðslÞc , the leading term, is given by (5.8) and w ð3Þ ðslÞ21 ; w ð3Þ ðslÞ22 ;w ð3Þ ðslÞ23 ; and w ð3Þ ðslÞ26 are given by the following:
2

ð3Þ

w ðslÞ21 ¼ ð
h ÞZ


1

X g

Rgsl

1 ð

dt exp½
iVt
cjtj½Mgg ðslÞ þ Lgg ðslÞKgð0Þ ðtÞGN ðtÞð
q0g Þ;


1

ð5:56aÞ

wð3Þ ðslÞ22 ¼ ð
h
2 ÞZ
1 þ Lgm ðslÞ




XX g

m6¼g

XX g

h

ð0Þ ðtÞ
q0m dt exp½
itV
cjtj Mgm ðslÞ H m


1

Hmð0Þ ðtÞ
q0m

w ð3Þ ðslÞ23 ¼ ð
h
2 ÞZ
1

1 ð

g

Rsl

m6¼g

i

g

 Kgð0Þ ðtÞGN ðtÞ; 1 ð

ð5:57aÞ

 dt exp½
itV
cjtj Mmg ðslÞ

Rsl
1


ð0Þ þ Lmg ðtÞ Dð0Þ g ðtÞN m ðtÞGN ðtÞ;

m ¼ 1; 2;

ð5:58aÞ

and w ð3Þ ðslÞ26 ¼ ð
h
2 ÞZ
1

XX

1 ð

g

g g0 6¼g

Rsl

dt exp½
itV
cjtj½Mgg0 ðslÞ
1

þ Lgg0 ðslÞKgð0Þ ðtÞGN ðtÞð
q0g0 Þ:

ð5:59aÞ

Similarly, we find ð3Þ


2


ðslÞ21 ¼ ð
h ÞZ w


1

X g

Rg sl

1 ð

  dt exp
itV
cjtj Mgg ðlsÞ


1

 þ Lgg ðlsÞ Kgð0Þ ðtÞGN ðtÞð
q0g Þ;


ð3Þ ðslÞ22 ¼ ð
h
2 ÞZ
1 w þ Lgm ðlsÞ




XX g

m6¼g


ð0Þ ðtÞ
q0m H m


ð3Þ ðslÞ23 ¼ ð
h
2 ÞZ
1 w

XX g

m6¼g

1 ð

g

Rsl



ð5:56bÞ

h
 dt exp½
itV
cjtj Mgm ðlsÞ Hmð0Þ ðtÞ
q0m


1

  Kgð0Þ ðtÞGN ðtÞ;

g

ð5:57bÞ

1 ð

Rsl

dt exp½
itV
cjtj½Mmg ðlsÞ
1

ð0Þ
ð0Þ þ Lmg ðlsÞD g Nm ðtÞGN ðtÞ;

ð5:58bÞ

5.2 Determination of the q-Centroid

and finally,
ð3Þ ðslÞ26 ¼ ð
h
2 ÞZ
1 w þ Lgg0 ðlsÞ



XX

g

Rsl

1 ð

j125

  dt exp
itV
cjtj Mgg0 ðlsÞ

g g0 6¼g
1 ð0Þ Kg ðtÞGN ðtÞð
q0g0 Þ:

ð5:59bÞ

After these preparations, the q-centroid may now be determined by those q0g and q0m appearing in (5.56)–(5.59), which likewise cause the sum in (5.55) to vanish. First, we considerthenontotallysymmetric (ns)vibrations. SinceEquations 5.56 and5.59 contain directly the nontotally symmetric mode q0g and Equations 5.58 are nonzero if g is ns and m is totally symmetric (ts), the q-centroid condition for the ns modes is given by 8 1 ð X <X g0  ð0Þ  Rsl ½Mg0 g ðslÞ þ Lg0 g ðslÞ dt exp
itV
cjtj Kg0 ðtÞGN ðtÞ : 0 g g
1 ) 1 ð   ð0Þ 0 g þ Rsl ½Mg0 g ðlsÞ þ Lg0 g ðlsÞ dt exp
itV
cjtj Kg0 ðtÞGN ðtÞ ð
q0g Þ þ

8 X <X g

:

m


1 1 ð

g

 
ð0Þ dt exp
itV
cjtj Dð0Þ g ðtÞN m ðtÞGN ðtÞ

Rsl ½Mmg ðslÞ þ Lmg ðslÞ

g þ Rsl ½Mmg ðlsÞ þ Lmg ðlsÞ


1 1 ð

 ð0Þ 
g ðtÞNmð0Þ ðtÞGN ðtÞ dt exp
itV
cjtj D

) ¼ 0;


1

ð5:60Þ

where we have combined Equations 5.56 and 5.59 and rewritten in a compact form. Then, g

Rsl

ð1 X  
ð0Þ ½Mmg ðslÞ þ Lmg ðslÞ dt exp
itV
cjtj Dð0Þ g ðtÞN m ðtÞGN ðtÞ m

g þ Rst


1

X m

½Mmg ðlsÞ þ Lmg ðlsÞ

ð1
1

  ð0Þ
ðtÞN ð0Þ ðtÞGN ðtÞ dt exp
itV
cjtj D m g

; q0g ¼ X n o ð1   ð0Þ g0 g0  Rsl ½Mg0 g ðslÞ þ Lg0 g ðslÞ þ Rsl ½Mg0 g ðlsÞ þ Lg0 g ðlsÞ dt exp
itV
cjtj Kg0 ðtÞGN ðtÞ
1

g0

g ¼ ns; m ¼ ts: ð5:61Þ

For totally symmetric vibrations, we need only Equations 5.57. The q-centroid is then
ð3Þ ðslÞ22 ¼ 0. This gives determined by setting w ð3Þ ðslÞ22 þ w ð1 X g   ð0Þ g
m ðtÞKgð0Þ ðtÞGN ðtÞ ½Rsl Mgm ðslÞ þ Rsl Lgm ðlsÞ dt exp
itV
cjtj H
1

g

ð1 X g   ½Rsl Lgm ðslÞ þ Rg M ðlsÞ dt exp
itV
cjtj Hmð0Þ ðtÞKgð0Þ ðtÞGN ðtÞ þ gm sl g


1

q0m ¼ X n ; o ð1 g g Rsl ½Mgm ðslÞ þ Lgm ðslÞ þ Rsl ½Mgm ðlsÞ þ Lgm ðlsÞ dt exp½
itV
cjtjKgð0Þ ðtÞGN ðtÞ g


1

m ¼ ts: ð5:62Þ

j 5 The Nuclear Coordinate Dependence of Matrix Elements

126

Several features of Equations 5.61 and 5.62 deserve comment: Equations 5.61 and 5.62 are quite general and have a clear physical significance. They provide us with the correct expression for the q-centroid configuration. Equation 5.61 as well as Equation 5.62 may be understood as an existential theorem, or more correctly, as a fix point theorem, since it may be written in the form q0 ¼ Jðq0 Þ, where the function J is given by the righthand side of Equation 5.61 or 5.62. The calculation of both Equations 5.61 and 5.62 then requires (i) knowledge of this function over the region of nuclear coordinates between the equilibrium position and the q-centroid and (ii) the density of states weighted vibrational overlap factor, which involves all vibrational (promoting and accepting) modes contributing to the radiationless decay rate. The evaluation of the former requires the knowledge of electronic matrix elements appearing in Equations 5.61 and 5.62, generally an involved numerical task. The latter is usually beyond our ability to accurately calculate. To further pursue this matter, we note that the q-centroid is temperature dependent. The latter is extracted by making the change of variables wm ¼ expð
ivsm tÞexpð

hvsm =kB TÞ as seen in Section 3.3. Later we will see from the discussion below that the q-centroid depends on the rotation of the normal coordinates upon electronic excitation. Let us now return to the basic relations (5.27) through (5.32), which play the role of generalized displacement parameters. To gain some insight into the significant feature of these quantities, we consider the zero-temperature limit. We assume that only the vibrationless level of the electronic excited state is occupied with an
m ¼ 1 ðm ¼ 1; 2Þ. appreciable probability in thermal equilibrium, and therefore w Furthermore, we make the special assumption by setting vsm ¼ vlm . Under these circumstances, Equation 5.27 reduces to the simpler form ð0Þ H1 ðtÞ

¼


1 ð0Þ

B1

(

h i h ð1Þ ð2Þ ð1Þ ðw11 þ w22 Þk12 þ ðw21 b21
w12 Þk12 þ ðw11 þ w22 Þk12

i ð2Þ ð2Þ þ ðw21 b21
w12 Þk12 expðiv1 tÞ
ðw21 b21 þ w12 Þk12 expðiv2 tÞ ) ð2Þ
ðw21 b21
w12 Þk12 exp½iðv1

þ v2 Þt ;

ð5:63Þ

where ð0Þ

2 2 2 2 B1 ¼ 2 þ w11 þ w22 þ w12 b12 þ w21 b21 :

ð5:64Þ

In deducing this result we have used Equation 3.30 to express the reciprocal ð12Þ ð12Þ ð1Þ ð2Þ parameters k1 and k2 by k12 and k12 and by setting z1 ¼ expðiv1 tÞ and ð0Þ z2 ¼ expðiv2 tÞ. Furthermore, the z-dependence of the denominator of H1 ðtÞ has been ignored, since it consists of oscillations about zero with an amplitude considerably smaller than the leading term (5.64). This approximation is made entirely for practical reasons to render a difficult computational problem more manageable. The ð0Þ corresponding expression H2 ðtÞ is obtained from Equation 5.63 by interchanging 1 , 2 (see Equations 5.27 and 5.28).

j127

5.2 Determination of the q-Centroid ð0Þ

ð0Þ


1 ðtÞ ðm ¼ 1; 2Þ. In complete analogy with H ðtÞ, To continue, we consider next H 1 we may recast Equation 5.31 into the form ( h i h 1 ð0Þ ð12Þ ð12Þ ð12Þ
1 ðtÞ ¼ ðw11 þ w22 Þk1 þ ðw12
w21 b21 Þk2
ðw11 þ w22 Þk1 H ð0Þ B1 ð12Þ

þ ðw12 þ w21 b21 Þk2

i

ð12Þ

expðiv1 tÞ
ðw12
w21 b21 Þk2 expðiv2 tÞ )

ð12Þ þ ðw12 þ w21 b21 Þk2 exp½iðv1

þ v2 Þt :

ð5:65Þ


ð0Þ ðtÞ is obtained from (5.65) by interchanging the indices 1 , 2. Similar to above, H 2 ð 0Þ We recall that w11 ¼ w22 ¼ cosj and w12 ¼
w21 ¼ sinj. Consequently, Hm ðtÞ and ð0Þ
m ðtÞ ðm ¼ 1; 2Þ become strongly j dependent, even for attainable changes of j. H This is among other things due to the presence of the cross-frequency parameters b12 and b21 that can differ appreciably from unity for different accepting modes. Thus, the entire integrand in Equation 5.62 is j dependent (recall that GN ðtÞ is also a function of j). For parallel accepting modes ðj ¼ 0Þ, Equations 5.63 and 5.65 reveal ðmÞ ð0Þ
ð0Þ ðtÞ reduce to that they depend only on k12 , that is, Hm and H m 1 ðmÞ  ð0Þ Hm ðtÞ ¼
k12 1 þ expðivm tÞ 2 1 ðmÞ  1 ð12Þ 
ð0Þ 1
expðivm tÞ ¼ k12 1
expðivm tÞ ; H m ðtÞ ¼ km 2 2

ð5:66Þ

m ¼ 1; 2:

ð0Þ
ð0Þ ðtÞ, we are faced with the Having extracted the significant features of Hm ðtÞ and H m problem of evaluating the Fourier integrals appearing in Equation 5.62 for the q0m -centroid. Formally, this problem can be solved by the same steps that led to Equations 3.75 and 3.77 for the nonradiative transition probability. A very general g analysis of how many promoting modes of nonvanishing coupling terms Rsl occur in Equation 5.62 is fairly complex due to our ignorance of molecular interstate coupling terms. For further discussion of this equation, let us now examine how the q-centroid ð0Þ
ð0Þ ðtÞ. It may first be is affected by the several contributions appearing in Hm ðtÞ and H m ð0Þ expected that the q-centroid arises from the constant (nonoscillatory) terms of Hm ðtÞ ð0Þ ð0Þ ð0Þ
ð0Þ) either directly or less directly through the electronic
ðtÞ (say Hm ð0Þ; H and H m m coupling terms. Indeed, to a first-order approximation, we can write

ð1 X g g
ð0Þ ð0Þ þ ½Rg Lgm ðslÞ þ Rg Mgm ðlsÞH ð0Þ ð0Þg f½Rsl Mgm ðslÞ þ Rsl Lgm ðlsÞH dtð    Þ m m sl sl q0m 

g

X g

g g fRsl ½Lgm ðslÞ þ Mgm ðslÞ þ Rsl ½Lgm ðlsÞ þ Mgm ðlsÞg

ð1
1


1

;

dtð    Þ ð5:67Þ

where the ratio of third-order electronic factors evaluated at q0 could be very large or ð0Þ
ð0Þ ðtÞ (not small in given instances. The next terms in the numerator of Hm ðtÞ and H m taken into account in (5.67)) contribute among other things due to the time factors

j 5 The Nuclear Coordinate Dependence of Matrix Elements

128

expðiv1 tÞ, expðiv2 tÞ, and exp½iðv1 þ v2 Þt that decrease the energy gap of
hV hv2 , and
hðv1 þ v2 Þ, respectively. The appearing in Equation 5.62 by sizes of
hv1 ,
latter can be appreciable for good accepting modes.2) Incidentally, this phenomenon is quite analogous to situation that we have already illustrated in Chapter 3 by including the function Kg ðtÞ in the expression for the nonradiative transition rate. There we concluded that the energy gap
hV was modified by the energy
hvg of the promoting mode. Although our estimate is very crude, it suffices to show that expression (5.67) leads to a q0m -centroid, which order of magnitude is of the ðmÞ displacement parameters k12 . The procedure in the case of nontotally symmetry modes is similar and can be handled on the same footing. The relevant displacement parameters appearing in
ð0Þ
ð0Þ ðtÞNmð0Þ ðtÞ and Dð0Þ Equation 5.61 are now given by the products D g ðtÞN m ðtÞ, where g (see Equation 5.51) ( i iv1 h ð0Þ ð0Þ ð1Þ ð2Þ l
N1 ðtÞ ¼
iv1 z1 H1 ðtÞ  ð0Þ ðw11 þ w22 Þk12
ðw12
w21 b21 Þk12 B1 ð2Þ

ð2Þ


ðw12 þ w21 b21 Þk12 expðiv2 tÞ þ 2w12 k12 exp½iðv1 þ v2 Þt ð1Þ ð1Þ ð2Þ þ ½ðw11 þ w22 Þk12
ðw12
w21 b21 Þk12 expð2iv1 tÞ
ðw12
w21 b21 Þk12 exp½ið2v1

) þ v2 Þt : ð5:68Þ

ð0Þ N2 ðtÞ

The second component is quite comparable to the first and is obtained from
ð0Þ ðtÞ ðm ¼ 1; 2Þ are them by interchanging the indices 1 , 2. The expressions N m ð0Þ
ðtÞ and (in the zero-temperature limit) given by closely related to H m ð0Þ
ð0Þ
ð0Þ
ð0Þ ðtÞ ¼
ivs H N m m m ðtÞ. The corresponding prefactors Dg ðtÞ and Dg ðtÞ reduce on ð0Þ the same level of approximation to the results Dg ðtÞ ¼ ð1=2Þi
h expðivg tÞ and
ð0Þ ðtÞ the
ð0Þ ðtÞ ¼
ð1=2Þi
h expðivg tÞ and cause apart of the factors Nmð0Þ ðtÞ and N D g m energy gap of
hV to decrease additionally of about
hvg . A comparison of expressions ð0Þ ð0Þ ð0Þ ð0Þ 0
ð0Þ
ð 0Þ
ð0Þ Hm ðtÞKg ðtÞ and H m ðtÞKg ðtÞ on the one side with Dg ðtÞNm ðtÞ and Dg ðtÞN m ðtÞ on the other indicates that the nontotally symmetric q0g can be as large as those for totally symmetric modes q0m . If the q-centroid for nontotally symmetric modes is large, the molecule is distorted by displacing the ion of the skeleton from the equilibrium positions (of high symmetry) to the q-centroid at considerably lower symmetry. At this unsymmetrized and distorted q-centroid configuration, the electronic wavefunctions will be deformed as well. Consequently, the electronic coupling terms in (5.61) and (5.62) enhance in a manner that can be quite difficult to deduce with any precision. With reference to the leading contribution of the nonradiative decay rate (Equation 5.8), one can expect an appreciable enhancement of the electronic matrix element evaluated at the q-centroid compared to the customary approach at qe resulting from a small (or rapidly varying) energy denominator Es ðq0 Þ
El ðq0 Þ of the electronic matrix elements (5.4). 2) Roughly speaking, the nonradiative decay rate increases as the energy gap decreases as one might expect from formula (4.111) for nonradiative transition in the displaced potential model.

j129

6 Time-Resolved Spectroscopy We have thus far analyzed the radiationless decay of a single excited state of a polyatomic molecule into a dense manifold of Born–Oppenheimer (BO) states, corresponding to another electronic configuration. Since the prepared zero-order excited state is induced by an electromagnetic field (i.e., by a broadband excitation), the decay by radiative transition of the excited state of the molecule should also be considered. This chapter mainly deals with a unified treatment of such a decay process, which includes the radiative transition to the lower electronic state as well. In this context, we follow here the treatment of Goldberger and Watson [81], Mower [80], and Freed and Jortner [144], which is based on the Green’s function formalism for the transition amplitude. In this formalism, the states of interest are selected by suitable projection operators, which we have already used in Chapter 2 in the case of nonradiative transitions, and the matrix elements can be displayed in an energy representation, which involves either the BO or the molecular eigenstate (ME) basis set. In this chapter we shall also address ourselves to a theoretical study of coherent optical effects such as optical free induction decay and photon echo from an assembly of collision-free molecules. From the mathematical point of view, many of the formal manipulations are given in Appendix A.2.

6.1 Formal Consideration

Before discussing the interaction of a molecule with an electromagnetic field in detail, it is useful to review some formal facts about electromagnetic radiation. When radiative decay processes are considered, the molecular Hamiltonian of the system molecule plus radiation field is given by H ¼ Hel þ Hrad þ Hint ;

ð6:1Þ

Hel ¼ HBO þ Hv :

ð6:2Þ

where

The operator Hel or HBO is the Hamiltonian for the internal electronic states of the molecule (which include electronic, vibrational, and rotational states). If the initial

j 6 Time-Resolved Spectroscopy

130

basis set is chosen in terms of BO wavefunctions ðysv ; ylv0 Þ, then Hv is the intramolecular perturbation, the so-called Born–Oppenheimer coupling, which promotes transitions between (see the discussion in Section 1.1) potential energy surfaces. Alternatively, if we choose the molecular eigenstate yn as basis set (see Appendix F), then Hel yn ¼ En yn

ð6:3Þ

and the perturbation consists of Hint . Hrad in Equation 6.1 is the Hamiltonian of the free radiation field (see Appendix E):
þ  1X þ Hrad ¼ ðkcÞ ak^ ð6:4Þ
hvk^e ¼ ðckÞ ðc ¼ 3
1010 cm=sÞ; e ak^e þ ak^e ak^e ; 2 k;^e þ where ak^e and ak^ e are the respective operators for annihilation and creation of photons in the state ðk;^eÞ. The sum on k and ^e in Equation 6.4 extends over all plane wave momentum states k, and respectively over the two allowed directions of polarization of a photon having momentum k. The eigenfunctions of (6.4) in the Dirac notation are

j ¼ jnk1^e1 ; nk2^e2 ; . . . ; nke ; . . .i;

ð6:5Þ

where nk1 ;^e1 ; nk2 ;^e2 ; . . . represent a complete set of photon numbers that specify how many photons of each type are present. Hint is the radiation–matter interaction term taken as Hint ¼ 

Z X e pj Aðrj Þ; mc j¼1

ð6:6Þ

where e and m are the charge and mass of an electron. The summation in (6.6) extends over the coordinates rj and momentum pj of the Z electrons in the molecule. The quantity Aðrj Þ is the vector potential of the radiation field evaluated at rj . This vector potential may be expanded in a series of plane waves [145] (see Appendix E): AðrÞ ¼

i X 2p
h2 c 1=2 h þ iðkrÞ=
h ^e ak;^e eiðkrÞ=
h þ ak;^ : ee Vk k;^e

ð6:7Þ

The sum on k here extends over all plane wave momentum states k in the cavity of þ volume V. The only nonvanishing matrix elements of ak;^e and ak;^ e are     pffiffiffiffiffiffiffi nk1 ;^e1 ; nk2^e2 ; . . . ; nk;^e 1; . . . ak;^e nk1 ;^e1 ; nk2^e2 ; . . . ; nk;^e ; . . . ¼ nk;^e    pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ð6:8Þ nk ;^e ; nk ^e ; . . . ; nk;^e þ 1; . . . a þ nk ;^e ; nk ^e ; . . . ; nk;^e ; . . . ¼ nk;^e þ 1: 1

1

2 2

k;^e

1

1

2 2

In discussing the electromagnetic interaction of a charged particle with the radiation field, the zero-order states are taken as eigenstates of the Hamiltonian: H0 ¼ HBO þ Hrad :

ð6:9Þ

They include the vibronic state ys ¼ jys ; vaci, the zero-order approximation of the electronically excited state, which carries oscillator strength to the ground state and the vibronic manifold fyl g ¼ fjyl ; vacig. The vibronic manifold corresponds to a lower electronic state that are quasi-degenerate with ys and does not carry oscillator

6.2 Evaluation of the Radiative Decay Probability of a Prepared State

strength to the ground state yk;^e ¼ jy0 ; k;^ei, (jvaci is the zero photon state) (we revert here to the simple notations s for ðsvÞ and l for ðlv0 Þ). Using Equations (6.6–6.8), we find 0 11=2 * +   2 D E  pe  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p
h c y ; vac A nk;^e þ 1 yk;^e  yk;^e jHint jys ; vac ¼ e@ Vk mc  s 0

11=2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p
h c A nk;^e þ 1 ¼ e@ Vk 0

1 +    p^e  s s   j ðr; q Þ xsv ðq ÞA ;
@x0v0 ðq Þ j0 ðr; q Þ mc  s *

0

0

ð6:10Þ

q

PZ

where nk;^e ¼ 0 and p ¼ j¼1 pj . For optically allowed transitions, the electronic matrix element hj jir in (6.10) is usually a slowly varying function of the nuclear coordinates over the range where xs  x0 significantly differs from zero. Therefore, it may be removed from the inner element of the nuclear integration by using the q-centroid-type approximation. To obtain (6.10), we have assumed that the electric dipole transition is allowed for the decay from state ys to the ground state y0 and set eikrj =
h ¼ 1 in Equation 6.7. This is an excellent approximation for atomic transitions, since called long-wave approximation. 6.2 Evaluation of the Radiative Decay Probability of a Prepared State

Let us now consider a system consisting of a molecule plus an electromagnetic radiation field. The molecule is prepared by optical excitation at time t ¼ 0 in a nonstationary state of H, which may be expressed as a superposition of either the BO states fys ; yl g or the molecular eigenstates yn [144]: X X yð0Þ ¼ bs ð0Þjys ; vaci þ bl ð0Þjyl : vaci ¼ an ð0Þjyn : vaci: ð6:11Þ l6¼s

n

In many cases of physical interest, the initial excited state of the system can be prepared by coherent excitation, by broadband source, or by a square light pulse [146–149] (having a pulse length chosen so that t  v1 , where v ¼ ck is the photon frequency),1) whereupon an ð0Þ ¼ hys jyn i; bl ð0Þ ¼ dls . In this case, the completeness 1) It is well known that by the uncertainty principle, a finite pulse length t gives rise to an energy spread DE 
h=t in the incident light. In this case, it is suitable to employ a wave packet formulation to provide the correct analysis of the system. The importance of using photon wave packets to provide a physically realistic model for the excitation process has been emphasized by Jortner and Mukamel [146] and

Freed [149]. Furthermore, Jortner and Mukamel discussed how no extra molecular information can be gained by altering the nature of the incident wave packet, so the reader is referred to their work for more details. The idealized square pulse employed here omits such energy spread and its simplicity enables the elucidation of the fundamental molecular phenomena we wish to illustrate.

j131

j 6 Time-Resolved Spectroscopy

132

of the molecular basis set for the excited states and the fact that ys is the only state that carries oscillator strength immediately imply that yð0Þ ¼ jys ; vaci. When narrow-band excitation is used (i.e., by using nanosecond lasers), the bl ð0Þ are not necessarily nonzero and bs ð0Þ can be greatly reduced, so the contributions of bl ð0Þ to the radiative decay should also be considered. After a time t has evolved, the state of the system is determined by the Hamiltonian (6.1) according to the expression2) yðtÞ ¼ expðitHÞyð0Þ:

ð6:12Þ

The probability P of finding the system in the one-photon ground state yk;^e ¼ jy0 : k; ^ei is Pk;^e ðtÞ ¼ jhyk;^e jjyðtÞij2 ¼ jhyk;^e jexpðitHÞyð0Þij2 :

ð6:13Þ

Using Equation 6.11, Equation 6.13 can be written as X Pk;^e ðtÞ ¼ hyk;^e jexpðiHtÞjyj ; vacibj ð0Þhyk;^e jexpðiHtÞjyj0 ; vaci bj0 ð0Þ j;j0 ¼s;l

X ¼ hyk;^e jexpðiHtÞjyn ;vacian ð0Þhyk;^e jexpðiHtÞjyn0 ; vaci an0 ð0Þ; n;n0

ð6:14Þ

where for the case of broadband excitation, the coefficients may simply be taken as bj ð0Þ ¼ djs : Otherwise, there is possibility of interference between the different states, ys and yl , as many of them come close together and may be strongly coupled. The nonzero matrix elements of H in the basis set ðys ;yl Þ and yk;^e ¼ jy0 ;k;^ei are given by hyk;^e jH jyk0 ;^e0 i ¼ ðE0 þ kcÞdkk0 d^e^e0 ;

ð6:15aÞ

hys jHjyl i ¼ hys jHv jyl i ¼ Vsl ;

ð6:15bÞ

hyk;^e jH jys i ¼ hyk;^e jHint jys i ¼ Ws;k^e ;

ð6:15cÞ

hys jHjys i ¼ Es ;

ð6:15dÞ

hyl jHjyl0 i ¼ El dll0 ;

ð6:15eÞ

where Ws;k;^e is given by Equation 6.10 and E0 ; Es , and El are respectively the energies of y0 ; ys ; and yl in the absence of the external radiation field. Using the technique of Chapter 2 and considering for simplicity the classic case of broadband excitation for which yð0Þ ¼ jys ;vaci, we define the projection operator P ¼ jys ; vacihys ;vacj 2) In this chapter we return to our convention by which we set
h ¼ 1.

ð6:16Þ

6.2 Evaluation of the Radiative Decay Probability of a Prepared State

and the projection operator on all other states separated from the state jys ;vaci Q ¼ 1P:

ð6:17Þ

Thus, to find the transition probability Pk;^e ðtÞ, we need the matrix element for the time evolution of the P ! Q excitation, hyk;^e jGðEÞjys : vaci ¼ hyk;^e jQGðEÞP jys ; vaci;

ð6:18Þ

or more explicitly the amplitude for the transition, Is;k;^e ðtÞ ¼ hyk;^e jexpðiHtÞjys ; vaci ð 1 ¼ hyk;^e jQGðEÞPjys ; vaciexpðiEtÞdE; 2pi

ð6:19Þ

c

where the contour c runs from þ ¥ to ¥ over the singularities of GðEÞ, above the real axis. As given in Equation A16b, QGðEÞP ¼ ðEQH0 QÞ1 QRðEÞP½EPH0 PPRðEÞP1 ;

ð6:20Þ

where H ¼ H0 þ V;

ð6:21aÞ

H0 ¼ HBO þ Hrad ;

ð6:21bÞ

with

and the perturbation V ¼ Hv þHint ;

ð6:21cÞ

responsible for the decay. RðEÞ in Equation 6.20 is the level shift operator given by Equation A15. Now, the electromagnetic interaction of a molecule with the radiation field is “weak” in the sense that virtual transitions involving the emission of photons contribute very little to the decay rates. This means that we may evaluate the matrix element in Equation 6.19 in the lowest order perturbation theory. Then according to Equation 6.20, hyk;^e jGðEÞjys ; vaci ¼ hyk;^e jQGðEÞP jys : vaci

1 1 1
¼ ðEE0 kcÞ hyk;^e jHint jys : vaci EE s þ i ðCs ðEÞ þ Ds ðEÞÞ ; 2

ð6:22Þ

where we have set Rss ðEÞ ¼ Ds ðEÞi½CðEÞ þDs ðEÞ=2 for the matrix element of the level shift operator. The real level shifts of the matrix element Rss , which represents
s, have the shift between the “unperturbed” energy Es and the “perturbed” energy E

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j 6 Time-Resolved Spectroscopy

134


s ¼ Es þDI þ DII .3) The latter represents the difference between the been included in E s s actual eigenvalues of H and the unperturbed eigenvalue Es . The term Ds ðEÞ is the damping of jys : vaci due to vibronic coupling (6.15b) with the manifold fjyl ;vacig4) and is given (see Appendix A.2) by a superposition of Lorentzians (each characterized by the strength jVsl j2 and the width Cl ) of yl : Ds ðEÞ ¼ 2

X

jVsl j2 Cl ðEÞ=2

l

ðEEl Þ2 þ ½Cl ðEÞ=22

:

ð6:23aÞ

Note that the widths Cl are very small as the levels of fyl g do not carry oscillator strength. In the statistical limit when the density rl of vibronic states fyl g is sufficiently large to exceed the reciprocal of the vibronic terms Vsl between the BO states, rl V  1, considerable configuration mixing Ð occurs. In this limit, we may P replace the sum over l by an integral using l ! dEl rl ðEl Þ. Thus, (6.23a) becomes valid in the limit of small Cl ðEÞ ! 0: ð ðCl ðEÞ=2Þ Ds ðEs Þ ¼ 2 dEl jVsl j2 rl ðEl Þ: ð6:23bÞ ðEs El Þ2 þ½Cl ðEÞ=22 Since Vsl ðEÞ and the density of states of fyl g, rl ðEl Þ, are rather slowly varying in large molecules for E ¼ Es, the width Ds ðEÞ is a smooth function of energy. Alternatively, a general condition for the smoothness of Ds ðEÞ is that the widths of successive Lorentzians considerably exceed their spacing. The term Cs in Equation 6.22 is the radiation damping associated with the decay of the initial state ys . That is, as the system radiates outgoing waves into the decay channel of yk;^e , the original state becomes depleted and this leads to a self-damping of the radiative process given by the imaginary part of the operator Rss (see Appendix A.2): Xð    2 dVk  yk;^e Hint ys  rph ðvph Þ; Cs ¼ 2p ð6:24Þ ^e

P Ð where rðvph Þ is the density of photon states of a given polarization ^e and ^e dVk corresponds to the integration over all propagation direction in k space directed into the solid angle Vk and summation over the two polarizations of each k. We have seen from Equation 6.22 that the matrix element of the Green’s operator  yk;^e jGðEÞjys ;vaci is analytic everywhere in the complex plane, except on the complex pole
s i 1 ðCs þ Ds Þ; E ¼E 2

ð6:25Þ

which is situated on the second Riemann sheet Im E < 0 and along the cut extending from E0 to plus infinity on the real axis. Provided that the width ðCs ðEÞ þ Ds ðEÞÞ is a 3) The quantities DIs and DII s are numerically much smaller than Es . 4) We write the damping of the state ys due to the vibronic coupling now by Ds rather than Cs as before in chapter 2. Here Cs is the damping of ys due to the interaction with the radiation field and the remaining notation is that of the previous sections.

6.2 Evaluation of the Radiative Decay Probability of a Prepared State

smooth function of E for E ffi Es ffi E0 þkc, Equation 6.25 is the only complex pole. Making the following partial fraction decomposition 1 1
s þ iðCs þ Ds Þ=2 ¼ ½E0 þkcE
s þ iðCs þ Ds Þ=2 ðEE0 kcÞ½EE   1 1

s þ iðCs þ Ds Þ=2 EE0 kc EE ð6:26Þ

and ignoring the contribution from the branch cut E > E0 , the transition amplitude is found simply by completing the contour in the lower half-plane and evaluating the integral in ð eiEt hyk;^e jHint jys ;vaci 1 Is;k;^e ðtÞ ¼ dE
s þ iðCs þ Ds Þ=2 ; 2pi c ðEE0 kcÞ½EE

ð6:27aÞ

in terms of residue at the pole given by Equation 6.25. Thus, we find Is;k;^e ðtÞ ¼ 



s iðCs þ Ds Þ=2tg hyk;^e jHint jys ; vaci exp½iðE0 þ kcÞtexpfi½E ;
s þ iðCs þ Ds Þ=2 ½E0 þ kcE ð6:27bÞ

and by taking the absolute value squared      y Hint y ; vac 2 k;^e s Pk;^e ðtÞ ¼
s Þ2 þ 1 ½Cs ðEs Þ þDs ðEs Þ2 ðE0 þ kcE 4 

1
s Þ t þ exp½ðCs þ Ds Þt ;
12exp  ðCs þDs Þt cosðE0 þkcE 2 ð6:28Þ

where Pk;^e ðtÞ is the probability of finding the system in the state yk;^e at time t. On integrating the integral (6.27) by contour analysis, it is necessary to continue analytically the matrix element of the level shift operator Rs ðEÞ from the first Riemann sheet through the cut onto the second sheet. This may be accomplished by defining the value of the level shift operator on the second sheet as I RII s ðEieÞ ¼ Rs ðE þ ieÞ ¼ Ds ðEÞi½Cs ðEÞ þ Ds ðEÞ=2;

where the complex energy near the cut E > E0 is E ie ðe > 0Þ. The superscripts on Rs denote the sheet on which the level shift operator is to be evaluated. With this definition in mind, the matrix element of the Green’s function in (6.27a) is analytic everywhere in the complex plane except on the cut along the real axis E > E0 and the pole (6.25) that is situated on the second Riemann sheet. Having performed the analytical continuation of Rs , we may now deform the contour of integration c (that

j135

j 6 Time-Resolved Spectroscopy

136

runs from þ ¥ to ¥ above the real axis along which the singularities of RðEÞ lie) in the integral (6.27a) by bending it to the left of E0 down along the first sheet to E ¼ E0 i¥. The contour to the right of E ¼ E0 may be deformed in a similar manner, provided that we pass through the cut onto the second sheet. In addition, a small
circular contour c0 encloses the poles of the singularities at E ¼ EiðC s þ Ds Þ=2 on the second sheet. The resulting expression for Is;k^e consists of two contour integrals, one about the new deformed cut c1 (extending from E0 to E ¼ E0 i¥) as well as the contour encompassing the singularities of the integrand, ð ð  1 ð    ÞdE: Is;k^e ¼ þ 2pi c0 c1 Following the procedure of Goldberger and Watson (see also Mower [80]), it can be shown that the contribution of the integral around c1 is of negligible magnitude and can therefore ordinarily be neglected. Let us now return to our problem of evaluating the probability of emission. Integrating Equation 6.28 over frequencies kc, all propagation directions in k space, and summation over the polarizations ^e, we obtain [144] Pemission ðtÞ ¼

Cs ðEs Þ f1expð½Cs ðEs Þ þ Ds ðEs ÞtÞg; ½Cs ðEs Þ þ Ds ðEs Þ

where we have used the familar relations ð dðkcÞ 2p ¼
s Þ2 þ 1 ðCs þ Ds Þ2 Cs þ Ds ðE0 þ kcE 4

ð6:29Þ

ð6:30aÞ

and ð




s Þt dðkcÞcos½ðE0 þ kcE 2p 1 ¼ exp  þ D Þt : ðC s s
s Þ2 þ 1 ðCs þ Ds Þ2 Cs þ Ds 2 ðE0 þ kcE 4

ð6:30bÞ

The last integral is evaluated in terms of residue at the pole E0 þ kc ¼
s i=2ðCs þ Ds Þ. With the formula (6.29), we have established a relation between E the decay rate of a prepared unstable state and its lifetime in the statistical limit. Equation 6.29 describes an exponential decay with quantum yield Pð¥Þ ¼ Cs =ðCs þ Ds Þ. The lifetime of the state is the reciprocal of ðCs þ Ds Þ, which consists of independent contributions of radiative and nonradiative components. Thus, the initially prepared resonance state ys subjected to a coupling scheme represented by Figure 2.1 decays exponentially by radiative and nonradiative transihr. tions on the timescale that is appreciably shorter than the recurrence time trec
The photon continuum allows the irreversible radiative decay, the states fyl g act as an effective continuum that enables irreversible decay into this manifold. The probability of finding the molecule in the fyl g manifold at time t, complementary to (6.29), is given by Pl ðtÞ ¼

Ds f1exp½ðCs þ Ds Þtg ðCs þ Ds Þ

ð6:31Þ

6.3 The Sparse Intermediate Case

and the probability that the molecule remains in its initial state jys ; vaci is 1Pemission ðtÞPl ðtÞ ¼ exp½ðCs þ Ds Þt:

ð6:32Þ

We conclude that a simple exponential decay is an adequate description of the time dependence of Pemission ðtÞ provided that coupling between all resonance levels (or optically active zero-order states) undergoing transitions may be ignored. The radiative damping matrix C is of greater importance for discussing radiative transitions and is closely related to the Fermi “Golden Rule.” It is defined in a manner that accounts for some type of interference effects (i.e., anticrossing-type interference effects): XXð dVk hajHint jy0 ; k; ^eihy0 ; k; ^ejHint ja0 irph ðckÞ: Caa0 ðEÞ ¼ 2p ð6:33Þ a

^e

In Equation 6.33, there is a sum over the zero-photon excited states a (after radiative decay) as well as a sum over the polarization^e and an integration over the direction Vk of the emitted photon. rph ðckÞ is the density of photon states. CðEÞ is generally nondiagonal, as will be illustrated in the next section. In the fys ; yl g representation however, it is easy to verify from Equation 6.15 that Css ¼ Cs 6¼ 0 Csl ¼ Cll0 ¼ 0

ð6:34Þ

for all l; l0 . Hence, C is diagonal in this basis. However, in another basis set, that is, a superposition of the zero-order states fys ; yl g, the C undergoes a unitary transformation and (6.33) is no longer diagonal. Molecular eigenstates (see Appendix F) provide an important illustration of this. For a single resonance, the resonance width is given by Caa. The off-diagonal matrix element of C represents coupling between the states ja; vaci via the one-photon states: ja; vaci ! jy0 ; k; ^ei ! ja0 ; vaci:

The off-diagonal contribution will be important only in the case of near-degeneracy when these terms are comparable to the energy spacing between the energy levels, that is, Caa0  jEa Ea0 j

ð6:35Þ 0

In such a case, it is known [80, 144] that the two states jai and ja i do not decay independently and the radiation coupling leads to a mixing of states.

6.3 The Sparse Intermediate Case 6.3.1 Preliminary Consideration

As we have already mentioned, in the statistical limit where the density of states is very large relative to the infrared or radiative decay width of the background states fyl g,

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138

Cl rl  1;

the function Ds ðEÞ is smooth and independent of the widths fCl g. In such a case, the adiabatic scheme is a good approximation for describing the bound-level structure of molecules. Large organic molecules (e.g., naphthalene, anthracene, tetracene [150], phenantren [150–152], therphenyl [154], and phentacene [155]) and a large group of inorganic transition ion complexes [156–162] fall in this category. For small molecules and also in the so-called sparse intermediate case, the vibronic coupling matrix element Vsl is sufficiently large (due to favorable Franck–Condon factor) and/or the level density rl is rather small. Consequently, the separation of the states may exceed their widths 1 jEl El0 j > ðCl Cl0 Þ 2

for adjacent levels in the fyl g manifold. In this case, DðEÞ is no longer expected to be a smooth function of energy E in the range of interest and the relevant matrix element of the Green’s function may be characterized by more than a single complex pole, resulting in an extensive mixing of states. In addition, the typical recurrence times trec ¼
hrl [11] for the occurrence of relaxation processes are now very short. In such a case, due to rapid recurrence between the states ys and fyl g, the adiabatic picture is physically meaningless and the experimental timescale for the fluorescence detection considerably exceeds the conveniently called Poincare recurrence time trec tmax . The molecule may resonate many times between fyl g and ys within a timescale characterized by the recurrence time. Typical molecules in this group are SO2 ; NO2 , and CS2 . The density of vibronic states (corresponding to the ground state and the first triplet state), which are quasi-degenerate with excited singlet state, is very low, being of the order of 0:1 cm1 . It has been experimentally demonstrated that the first excited singlet of SO2 ; NO2 , and CS2 exhibit anomalously long radiative lifetimes [163, 164], which are considerably longer than those expected on the basis of integrated oscillator strength. Furthermore, it was observed that the absorption spectra of these molecules are very complex, consisting of a large number of lines that are spread out over more than 150 cm1 and that could not be assigned to the vibrational–rotational manifold of a single electronic state. It was suggested by Douglas that this situation is characteristic of intramolecular vibronic coupling in these molecules. This results in the redistribution of the intensity of a zero-order BO state (corresponding to the excited singlet) among a large number of zero-order levels. These are quasi-degenerate with the former level and do not carry oscillator strength. From the spectroscopic point of view, the redistribution of the intensity of the zero-order component ys induces the appearance of many new well-resolved lines corresponding to all the molecular eigenstates yn in the optical spectrum (Figure 6.1). With this observation in mind, in this case it is more convenient to use the molecular eigenstate basis fyn g in the following section (see also Appendix F).

6.3 The Sparse Intermediate Case

V {|l >}

|s>

hω

Radiative Coupling

{|n>}

hω

|0>

|0>

(a) BO Zero-Order States

(b) Molecular Eigenstates

Figure 6.1 The Bixon and Jortner model [11] to describe the intrastate coupling and intramolecular vibrational redistribution. The zero-order molecular levels j0i, jsi, and fjlig are BO states for intrastate dynamics. They correspond, respectively, to the ground state j0i, the (one-photon) optically accessible doorway state jsi, and the background manifold

Radiative Coupling

fjlig. The wave lines represent the intramolecular interstate and intrastate coupling. The molecular eigenstates jni diagonalize the molecular Hamiltonian and are all radiatively coupled to the ground state. They are generally unevenly spaced and exhibit irregular variation in the radiative coupling with the ground state j0i.

6.3.2 The Molecular Eigenstates

As mentioned previously for the case in which the set of coupled states is a finite small number, the projection P takes the form (in terms of diagonalized projections) X P¼ ð6:36Þ jyn : vacihyn ; vacj; n

where Hel jyn i ¼ En jyn i:

The variable n as well as its range depends on the type of problem to be considered. The projection operator on the other states is Q ¼ 1P:

ð6:37Þ

The actual matrix elements are now D E yk;^e jGðEÞjyn ; vac ¼ hy0 ; k; ^ejQGðEÞP jyn ; vaci;

ð6:38Þ

and, as previously discussed in Section 6.1, the Hamiltonian of the system being considered is H ¼ H0 þ V; where now

H0 ¼ Hel þ Hrad ;

ð6:39aÞ ð6:39bÞ

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140

Hel ¼ HBO þ Hv ;

with

ð6:39cÞ

and the interaction V ¼ Hint :

ð6:39dÞ

The projection QGðEÞP is again given by (6.20). However, now hy0 ; k; ^ejQRðEÞPjyn i ¼ hy0 ; k; ^ejHint jyn ; vaci ¼ hy0 ; k; ^ejHint jys ; vacihys jyn i

ð6:40Þ

and PRðEÞP  iPCP=2;

ð6:41Þ

where PCP is the radiative damping matrix in the fyn g basis, which is nondiagonal as discussed above. Using Equations 6.40 and 6.41 and the expression for QGðEÞP given by Equation 6.22 leads to P hy0 ; k; ^ejQGðEÞPjyn ; vaci  ðEE0 kcÞ1 n0 hy0 ; k; ^ejHint jyn0 ; vaci   
yn0 ; vacj½EPH0 P þ iPCP=21 yn ; vac ; ð6:42Þ P where PH0 P ¼ n jyn ; vaciEn hyn ; vacj. Note that the last term in (6.42) is the inverse of the effective Hamiltonian (EH) Heff ¼ Hel iC=2:

ð6:43Þ

In the matrix form, Equation 6.43 becomes

Heff ¼

k

E1 iC1 =2 iC12 =2 .. . .. .

k

.. .. . . .. .. E2 iC2 =2 . . .. .. . . ; .. .. . . iC12 =2

so Hel is diagonal while the damping matrix is nondiagonal. On the contrary, in the BO basis, the effective Hamiltonian is

Heff

k

Es iCs =2 Vs1 Vs2 .. ¼ .

Vs1 E1

Vs2 E2



k

:

Now Hel is off-diagonal, while the damping matrix is diagonal [165]. For the case in which the set of closely coupled states fyn g is a finite number, the matrix elements of the inverse of the operator ½EPH0 P þ ði=2ÞPCP1 may be written formally as ½EPH0 P þ iPCP=21 ¼

cofactor½EPH0 P þ iPCP=2nn0 : Det½EPH0 P þ iPCP=2

ð6:44Þ

6.3 The Sparse Intermediate Case

If the number of closely coupled states is two, then the above matrix element reduces to hy0 ; k;^ejQGðEÞP jyn ; vaci ¼

hy0 ; k;^ejHint jyn ; vaci½EEn0 þ iCn0 n0 =2 þ hy0 ; k;^ejHint jyn0 ; vaciðiCnn0 =2Þ : ðEE0 kcÞf½EEn þ iCnn =2½EEn0 þ iCn0 n0 =2 þ C2nn0 =4g ð6:45Þ

It often happens that in studying a given transition, we find a single predominant contribution. This takes place if we stipulate that the molecular eigenstates are very sparse in the sense that En En0  Cnn0 :

ð6:46aÞ

The off-diagonal parts of CðEÞ in (6.45) can then be neglected and we obtain the simple result hy0 ; k;^ejQGðEÞPjyn ; vaci ¼

hy0 ; k;^ejHint jyn ; vaci ; ðEE0 kcÞ½EEn þ iCnn =2

ð6:47Þ

and correspondingly for hy0 ; k;^ejQGðEÞPjyn0 ; vaci. In this case, the radiative decay rate of the system is characterized by a superposition of exponentials with the widths Cnn and Cn0 n0 , provided that Cnn and Cn0 n0 are slowly varying for E  E0 þ kc. Furthermore, in this case, if the condition 1 En En0  ðCnn Cn0 n0 Þ 2

ð6:46bÞ

holds sufficiently, we can expect that the level crossing-type interference terms in (6.45) for ðn 6¼ n0 Þ are negligible and the molecular eigenstates decay independently with the widths Cnn and Cn0 n0 . A more general case for finite numbers n will be discussed in Section 7.5. Since the trace of the matrix C is invariant with respect to the basis set (Tr C in fys ; yl g basis ¼ Tr C in fyn g basis), the width Cs is given by Cs ¼

X n

Cnn ;

ð6:48Þ

so Cnn < Cs

for all n:

ð6:49Þ

Since the lifetime of the molecular eigenstates tn ¼ 1=Cnn , tn ¼ 1=Cnn > 1=Cs ¼ ts :

ð6:50Þ

This is the explanation for the anomalously long lifetime of small triatomic molecules observed by Douglas [163, 164]. Ordinarily, when conditions (6.46) are not satisfied

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142

for some of the states fyn g (for example, for closely spaced states), these states exhibit level anticrossing-type interference [166–168]. However, as shown by Bixon and Jortner [169], in the statistical limit, when the density of states rl is sufficiently high to exceed the reciprocal of the vibronic coupling term V between the zero-order BO states, that is, rl V  1, the result (6.29) is still obtained for broadband excitation even though the fyn g interfere with each other.

6.4 Radiative Decay in Internal Conversion by Introduction of Decay Rates for {c1}

In Section 6.2, we have described the decay of a single resonance that is coupled (apart from a radiation field) to a dense manifold of BO states fyl g that do not carry oscillator strength to the ground state y0 in view of spin selection rules (when the manifold fyl g is a triplet). However, when the BO states fyl g are singlets, they do carry oscillator strength to high vibrational levels of the ground electronic state yv0 . This will in general mean that we are now concerned with a situation where both ys and fyl g carry oscillator strength. Now, just as was done in Section 6.2, the initial state is taken as jys ; vaci, resulting from broadband excitation. Again, using Equation 6.18, the matrix elements of D

E D E yvk;^e jGðEÞjys ; vac ¼ yvk;^e jQGðEÞP jys ; vac

ð6:51Þ

 E    must be considered, since emission may also occur to yv0 , where yvk;^e ¼ yv0 ; k; ^e and P is again given as in Equation 6.16. We shall choose the label v to distinguish between transitions in the state yv0 and y0 . For the case in which fyl g form a dense manifold (see Equation 6.23), ð DðEÞ ¼ 2 dEl jVsl j2

Cl =2 ðEEl Þ2 þ ðCl =2Þ2

rl ðEl Þ;

ð6:52Þ

where rl is again the density of BO states fyl g. Cl ðEÞ are radiative widths due to the spontaneous emission given by Cl ðEÞ ¼ 2p

Xð ^e

D E2   dVk  yvk;^e jHint jyl  rph ðEE0v Þ;

ð6:53Þ

where it has been assumed that the damping matrix Cll0 is diagonal in the basis fyl g (see Equation 6.15). DðEÞ; Cl ðEÞ and Cs ðEÞ, as defined in Section 6.2, can all be considered to be smooth functions of E. Let us now turn to the problem of resolving the matrix element (6.51), which as in Section 6.2 can be written as D E ðEE0v kcÞ1 yvk;^e jQRðEÞP jys ; vac ðEEs þ i½Cs ðEÞ þ DðEÞ=2Þ1 :

ð6:54Þ

6.4 Radiative Decay in Internal Conversion by Introduction of Decay Rates for {y1}

But now since fyl g carry oscillator strength, the matrix element (6.54) may be decomposed into matrix elements for states jys i and jyl i: D

E v yvk;^e jHint jyl ; vac ¼ Wk;^ e ð1dv0 Þ;

ð6:55aÞ

D

E yvk;^e jHint jys ; vac ¼ Wk;^e dv0 ;

ð6:55bÞ

where dv0 is the Kronecker delta defined by  dv0 ¼

0; 1;

if v 6¼ 0; if v ¼ 0:

With the aid of the identity RðEÞ ¼ V þ VQðEQHQÞ1 QV, the matrix element of QRðEÞP that describes the emission to both yv0 and y0 states may be written in compact form D

E X yvk;^e jQRðEÞP jys ; vac ¼ Wk;^e dv0 þ W v V ð1dv0 Þ½EEl þ iCl ðEÞ=21 ; l k;^e ls ð6:56Þ

where Vls ¼ hyl jHjys i. Combining Equations 6.56 and 6.54, we obtain by counterintegration (neglecting as in Section 6.2 the contribution from the branch cut E > E0 ) D

E exp½iðE0 þ kcÞtexpfi½Es iðCs þ DÞ=2tg yk;^e jexpðiHtÞjys ; vac ¼ Wk;^e E0 þ kcEs þ iðCs þ DÞ=2 ð6:57aÞ

and [144] D

E P v yvk;^e jexpðiHtÞjys ; vac ¼ l Wk;^ e Vls

exp½iðE0 þ kcÞt Es El þ iðCl Cs DÞ=2

0 1expfi½Es E0n kciðCs þ DÞ=2tg
@ E0v þ kcEs þ iðCs þ DÞ=2 

! 1expfi½El E0n kciCl =2tg ; E0v þ kcEl þ iCl =2

ð6:57bÞ

where Cs ; D, and Cl can all be taken as CðE0 þ kcÞ in (6.57a) and CðE0v þ kcÞ in Equation 6.57b. Equation 6.57a is, of course, exactly equivalent to Equation 6.27b. In analogy with Equation 6.31, the fluorescence probabilities are obtained from Equation 6.57, by taking the absolute value squared, integrating over all energies, all directions in the angle Vk , and polarizations ^e, giving finally P0 ðtÞ ¼

 Cs
1exp½ðCs þ DÞt Cs þ D

ð6:58aÞ

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144

and Pv ðtÞ ¼

2 X 

1exp½iðCs þ DÞt X    hy jy ij2 1expðCl tÞ ij C þ  hy y l l s l s l l Cs þ D 0 1  1exp½ðC þ D þ C Þt=2 þ iðE E Þt s s l l 2 A; C hyl jys ij Re@ l l ðCs þ D þ Cl Þ=2 þ iðEl Es Þ

X

þ2

ð6:58bÞ

where (compare Equation F17) jhyl jys ij2 ¼ jVsl j2 =½ðEs El Þ2 þ ðCs þ DCl Þ2 =4

ð6:59Þ

is the component of ys in the molecular eigenstate yl . An important simplification of Equation 6.58b results in the statistical limit, where Ð P ! dE r ðE Þ and if we make the simple approximation that Cl is a constant. In l l l l this case, we may use Equation 6.52 to write Equation 6.58b in the form Pv ðtÞ ¼ Cl D

1exp½ðCs þ DÞt 1expðCl tÞ 1exp½ðCs þ DÞt þD þ 2Cl D : ðCs þ DÞðCs þ DCl Þ ðCs þ DCl Þ ðCs þ DCl Þ2 ð6:60Þ

The decay probability to the lowest vibronic level of the ground electronic state is determined by the branching ratio Cs =ðCs þ DÞ and by the lifetime ðCs þ DÞ1 . The close similarity between Equations 6.58a and 6.29 should be noted. The resonance fluorescence is expected to be very weak in view of the small branching ratio Cs =ðCs þ DÞ 104 . Reference to Equation 6.58b (or to Equation 6.60) shows that the fluorescence probability to higher vibronic levels consists of three contributions that involve two “direct decay” terms and an interference term: (a) The first term in Equation 6.60 describes an initial decay rate (for t ! 0) that is proportional to Cl . For a long time, this term exhibits a fast decay with a lifetime of ðCs þ DÞ1 , which arises from the fact that the molecular eigenstate yn that is primarily ys and has a width ðCs þ DÞ contains some fyl g. For this long timescale, the contribution from this term is negligible and the fluorescence is expected to be of order Cl =D, as for internal conversion between singlet states Cs D and Cl D. (b) The second term in Equations 6.58b and 6.60 corresponds to the “direct” radiation decay of the manifold fyl g, which is determined by the lifetimes fCl g of the fyl g states. For longer times, this term dominates the direct decay term (a). The fluorescence yield to the highly vibrationally excited ground state is close to unity. (c) The third contribution to the decay probability arises from level-crossing terms, which in the statistical limit gives a small contribution that is similar in form to (a). The radiative decay of the manifold fyl g exhibits interference effects in the intermediate case, where some of the states yl in the manifold fyl g couple to ys with different efficiency [144]. Equations 6.58b, 5.59b, and 6.60 have been of considerable importance in studying the radiative decay to higher vibronic components of the ground electronic state, which do not exactly overlap the fluorescence spectrum of the first excited

6.5 Dephasing and Relaxation in Molecular Systems

singlet state. This decay is also expected to reveal the interference effects between closely spaced states fyl g. The detailed features of this interesting new effect are of course determined by the details of the excitation process (i.e., narrow versus broadband excitation).

6.5 Dephasing and Relaxation in Molecular Systems 6.5.1 Introduction

We shall see in Chapter 7 that well-resolved sharp lines spectra of molecules can be obtained by incorporating them in suitable crystals. From such spectra, information about the interactions (vibronic coupling and spin–orbit coupling) and geometries (distortion) in the excited electronic state may be obtained. The results of these studies apply generally to static properties of the molecules, though the spectra also yield some information about relaxation phenomena. New developments in studying the dynamic interactions in such systems have occurred since the mid-eighties. These were mainly due to the use of tunable dye lasers, yielding time-resolved spectra. These were specifically important in the understanding of the dephasing processes in molecular mixed crystals, in particular those occurring on an ultrafast (picosecond) timescale. It is well known that by the uncertainty principle, the time-dependent processes give rise to the finite linewidth of an optical transition [170]. This homogeneous linewidth, however, can seldom be observed since crystal strain induces a spread in resonance frequencies, which exceeds in most cases the homogeneous width. The spectral line is then inhomogeneously broadened. The relaxation processes responsible for the homogeneous linewidth can be classified into two categories (in analogy with the longitudinal and transversal relaxation in magnetic resonance spectroscopy, see below). The processes with transfer of energy from the molecule to the surroundings (the bath) are called T1 processes. Fluorescence, internal conversion, and intersystem crossing fall in this category. The other types of relaxation processes are the pure dephasing T2 processes. Here the bath induces a random fluctuation of the transition frequency, but no energy is lost or transferred. By studying T2 -type processes, information about the interaction between the molecule and its environment (e.g., the phonons and transfer of excitation energy) can be obtained and the homogeneous from the inhomogeneous contribution of the fluorescence line can be separated. Dephasing can be studied either in the time domain (photon echo PE [171, 172], optical free induction decay (OFID) [173, 174] or in the frequency domain (hole burning) [175–177]. With these techniques, the homogeneous broadening can be circumvented and the pure homogeneous width can be measured. Dephasing processes were intensively studied by investigating the dynamics of the excited S1 state of pentacene (or strictly of the S1 S0 transition of pentacene)

j145

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146

in p-terphenyl and other molecular mixed crystals such as pentacene in naphthalene and zinc porphin in n-octane [178–184]. The system pentacene in p-terphenyl and pentacene in naphthalene proved to be good candidates for a detailed study, since the local phonons, responsible for the dephasing, can be observed directly in the electronic spectrum [181, 155]. 6.5.2 Interaction of a Large Molecule with a Light Pulse

As mentioned above, inhomogeneous broadening in molecular solids originates from the difference in the local crystal field acting on the molecules, while in the gas phase, differences in the velocity are responsible (Doppler effect). One of the methods for separating the homogeneous from the inhomogeneous contributions in the time domain is the OFID method. In this technique, the homogeneous broadening is removed by exciting an optically selected Born–Oppenheimer state with a laser pulse. For the optical phenomena of such type, Jortner and Kommandeur [185] have presented a theoretical study based on a generalized effective Hamiltonian (GEH) to account for the temporal characteristic of the macroscopic polarization of an assembly of “isolated” large molecules exposed to a short optical pulse. In the following, a brief description of this theory is given. Before describing the OFID method further, it is perhaps worth to consider a more systematic treatment for the interaction of a molecule with an electromagnetic field. The physical approximations underlying this analysis are no more rigorous or sophisticated than those used in the chapter before, merely more precisely stated. We consider a level scheme consisting of the ground state j g i ¼ j0i and a single doorway state jsi, which is coupled via nonadiabatic intramolecular interactions to a manifold of background states fjlig and where both the state jsi and the manifold fjlig are characterized by the radiative and nonradiative decay widths cs and cl , respectively (see Figure 6.2). The corresponding molecular eigenstates j ji for the problem of interest are obtained from the diagonalization of the effective Hamiltonian

Heff ¼

k

Eg 0 0 0 .. . .. .

.. . .. . Es ics =2 Vsl Vsl0 .. Vsl El icl =2 0 . .. Vsl0 0 El0 icl0 =2 . .. .. .. .. . . . . .. .. .. .. . . . . 0

0

0

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

k

ð6:61Þ

for which the levels in Figure 6.2 are the eigenfunctions j ji. They are in general unevenly spaced and exhibit irregular variations in the radiative coupling with the ground state j g i, so at first sight it does not seem worthwhile to observe phase coherence effects in such system.

6.5 Dephasing and Relaxation in Molecular Systems

γl

γj

{|l >}

{|j >}

Vsl |s > γs

γj Radiative Coupling

γl hω

Radiative Coupling

hω Diagonalize Heff

|0>

|0>

(a) BO Zero-Order States

(b) Generalized Molecular Eigenstates

Figure 6.2 Schematic picture of the effective Hamiltonian. The zero-order states jsi and fjlig are characterized by the energies Es and fEl g, respectively, and by the decay widths cs and fcl g. Vsl represents the intramolecular coupling between the doorway state jsi and the fjlig

manifold. Diagonalization of the effective Hamiltonian results in a set of independently decaying generalized molecular eigenstates fj jig, characterized by energies fEj g and decay widths fcj g.

To consider the response of an assembly of “isolated” molecules to a short optical pulse of duration Dt, the Hamiltonian for the system must be modified as follows:
cos vt; H ¼ Hm ~ m gs E H ¼ Hm ;

0  t  Dt; t  0; t Dt;

ð6:62Þ

where Hm is the molecular Hamiltonian, ~ E is the local electromagnetic field, characterized by the frequency v, while ~ m denotes the dipole operator. The time evolution of the levels j g i; jsi, and fjlig during the pulse 0  t  Dt is X
g ðtÞexpðivtÞj g i þ C
s ðtÞjsi þ
l ðtÞjli; YðtÞ ¼ C ð6:63Þ C where

l

D E
a ðtÞ ¼ ajUðt;
0ÞjYð0Þ ; C

a ¼ g; s; l:

ð6:64Þ


0Þ in Equation 6.64 is the time evolution operator during the pulse Uðt;
eff tÞ;
0Þ ¼ expðiH Uðt;

ð6:65Þ

determined by the generalized effective Hamiltonian


eff ¼ H

k

Eg þ v

vR =2

0

vR =2

Es ics =2

Vsl

0

Vsl

El icl

0 .. . .. .

Vsl0 .. . .. .

0 .. . .. .

k

.. .. . . .. .. Vsl0 . . .. .. 0 . . ; . . El0 icl0 =2 .. .. .. .. .. . . . .. .. .. . . . 0

ð6:66Þ

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148

which also contains the “dressed” ground state characterized by the energy Eg þ v and the coupling between the ground state j g i and the doorway state jsi in terms of the m gs~ mjsi. A closed formal solution of the E, where ~ m gs ¼ hgj
Rabi frequency vR ¼ ~ problem can be obtained by diagonalization of the GEH via a nonunitary transfor
0Þ in terms of the diagonalized projections: mation and by expansion of Uðt; X 
0Þ ¼ Uðt; ð6:67Þ j jiexpðiEj tcj t=2Þ
jj; j


eff and j ji the (complex) eigenwhere Ej icj =2 are the complex eigenvalues of H  P 
vectors, which satisfy the relation j j ji jj ¼ 1 with 
ji being the vector conjugate to j ji. Readers may refer to Appendix G for more details. We may use Equation 6.67 to write the probability amplitudes (Equation 6.64) in the form E XD
a ðtÞ ¼ aj j > expðiEj tcj t=2Þh
jjYð0Þ ; a ¼ g; s; l; 0  t  Dt; C j

ð6:68Þ

and the dipole moment induced in each molecule at t during the pulse  
g ðtÞexpðivtÞ þ cc:
 ðtÞC pðtÞ ¼ hYðtÞjmgs YðtÞi ¼ mgs C s

ð6:69Þ

Substituting Equation 6.63 into Equation 6.69 gives D E XXD   0 pðtÞ ¼ mgs Yð0Þj ji
jjs hgjj0 i
j jYð0Þ exp½iðEj Ej0 Þtðcj þ cj0 Þt=2: j0

j

ð6:70Þ

As the spacing of the real part of the eigenvalues ðEj Ej0 Þ is uncorrelated, one expects severe destructive interference effects for pðtÞ, when more than a single ME is driven by the field. Thus, coherent optical effects such as photon echoes or free induction decay experiments can be conducted only under special excitation conditions as follows: (a) Short-time excitation: This spans the entire congested excited spectrum of the molecule. This excitation mode, sometimes called as “coherent” molecular excitation [146–149, 186], corresponds to optical selection of the Born–Oppenheimer doorway state jsi. This excitation mode is specified by the condition Dt D1 sl ;

where Dsl ¼ 2p

X

jVsl j2 dðEs El Þ

ð6:71Þ

ð6:72Þ

l

is the width of the MEs. This excitation mode is feasible for an excited state that corresponds to a statistical large molecule or to a molecule with an intermediate-level structure. These assumptions are fairly justified in the case of picosecond excitation pulses as the coherent bandwidth of the pulses (typically 2--3 cm1 ) exceeds the

6.5 Dephasing and Relaxation in Molecular Systems

inhomogeneous linewidth (1 cm1 ). Under this special condition, the EH of Equation 6.61 can be considered as a scalar Heff ¼ Es Ds iðcs þ Dsl Þ=2, where Ds is the level shift. Then, the GEH that describes the interaction of the molecule with the pulse reduces to a 2
2 matrix, which represents a familiar two-level system [172],

k

k

Eg þ v vR =2 Heff ¼ vR =2 Es Ds iðc þ Dsl Þ=2 : s

If the pulse is sufficiently intense so that   vR  Eg þ vEs Ds iðcs þ Dsl Þ=2;

ð6:73Þ

ð6:74Þ

the time evolution during the pulse is simply
0Þ ¼ j g i cosðvR t=2Þ expðivtÞhgj þ jsi cosðvR =2Þhsj þ ij g i Uðt;
sinðvR t=2Þ expðivtÞhsj þ ijsi sinðvR t=2Þhgj:

ð6:75Þ

(b) Narrow-band excitation: In the small molecule limit or in a molecule that corresponds to the intermediate-level structure, the coupling Vsl between the Born–Oppenheimer states is large while the density of states in the background manifold is low. The states jli are coarsely spaced relative to their widths and the manifold fjlig cannot act as a dissipative channel. In this case, the ME’s basis j ji that diagonalizes Heff is of great utility. The level distribution of j ji is sufficiently sparse so that for the widths we have cj < r1  r1 j l and an optical excitation by a narrow-band pulse characterized by Dt < c1 j for all j;

Dt  D1 sl

ð6:76Þ

will result in the selection of a single ME in each molecule (if the inhomogeneous broadening will not bring different MEs of different molecules into resonance with the narrow-band excitation source). 6.5.3 Free Induction Decay of a Large Molecule

In what follows, we consider the time evolution within a subset of discrete levels jsi of an assembly of “collision-free” polyatomic molecules, which interact on the timescale 0  t  Dt with a short laser pulse and where the background manifolds fjlig do not carry oscillator strength to j g i. These time evolutions during the pulse 0  t  Dt are
eff tÞ, while after the pulse t ¼ Dt þ t are determined by
ðt; 0Þ ¼ expðiH given by U the time evolution operator in the diagonalized form as X  Uðt þ Dt; DtÞ ¼ j g iexpðiEg tÞhg j þ ð6:77Þ j jiexpðiEj tcj t=2Þ
jj; j

where the eigenvalues of EH (Equation 6.61) are Eg and ðEj icj =2Þ, while the eigenvectors are j g i and j ji and the corresponding left eigenvectors are denoted  by 
j . As the initial state of each molecule at t ¼ 0 is Yð0Þ ¼ j g i, the state of the

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system at time t ¼ Dt þ t can now straightforwardly be calculated by multiplication of the time evolution operators
ðDt; 0Þjg i: YðtÞ ¼ Uðt þ Dt; DtÞU

ð6:78Þ

Considering the radiative coupling with a p=2 pulse, that is, vR Dt ¼ p=2, Equation 6.78 together with Equations 6.75 and 6.77 results in X YðtÞ ¼ expðivDtÞexpðiEg tÞjg i þ iCss ðtÞjsi þ i Cls ðtÞjli; ð6:79Þ l

where the probability amplitudes after the termination of the pulse are given by X Cab ðxÞ ¼ hajjih
jjbiexpðiEj xcj x=2Þ; a; b ¼ s; l: ð6:80Þ j

The polarization per molecule after the pulse is obtained from Equation 6.69 is pðtÞ ¼ imgs expðivDtÞexpðiEsg tÞCss ðtÞ þ cc;

ð6:81aÞ

which may further be simplified to the form pðtÞ ¼ imgs expðivDtÞexpðiDEsg tÞf ðtÞ þ cc; where

DEsg ¼ Es Eg ;

ð6:81bÞ ð6:82Þ

mgs is the projection of the dipole operator in the direction of the field, and X f ðtÞ ¼ hsj
jih jjsiexp½iðEj Es Þtcj t=2; ð6:83Þ j

with ðEj Es Þ being the detuning of a particular molecule. Since f ðtÞ is expressed in terms of the intrinsic property of a single molecule, we shall refer to f ðtÞ as the intramolecular dephasing term to distinguish it from the portion of inhomogeneous broadening. In evaluating the macroscopic polarization, we must use an appropriate distribution for the inhomogeneous broadening. The latter can be described by a Gaussian distribution of DEsg
2 

ð6:84Þ WðEsg Þ ¼ ðbp1=2 Þ1 exp b2 DEsg DEsg0 ; which peaks at DEsg0 and is characterized by the width b. The macroscopic polarization PðtÞ of the molecular assembly is then ð PðtÞ ¼ N dðDEsg ÞWðDEsg ÞpðtÞ; which results in the expression o
n

 PðtÞ ¼ Nmgs exp b2 t2 =4 sin DEsg0 t Ref ðtÞ þ cos DEsg0 t Im f ðtÞ ;

with N in Equation 6.85 being the molecular number density.

ð6:85Þ

6.5 Dephasing and Relaxation in Molecular Systems

For some application to molecular physics, we may wish to generalize (6.85) to include the reaction of the detector to determine the OFID signal. This may be obtained by squaring of Equation 6.85 and performing an optical cycle averaging ðsi
n2 ðDEsg0 tÞ ¼ 1=2 . . .Þ to find IOFID ¼ A expðb2 t2 =2Þj f ðtÞj2 ;

ð6:86Þ

where the constant A refers to the time-independent parameters, such as frequency of the radiative field v, projection of the dipole operator mgs, molecular density N, and parameters of the detection system. Equation 6.86 provides us with an expression for the macroscopic polarization and the following remarks are worthy of note: 1) Inhomogeneous broadening: This contribution originates from the oversimplified assumption of light power and short-time excitation and is given by the Gaussian in Equation 6.86. 2) Intramolecular dephasing: This contribution is given by the function j f ðtÞj2 ¼ jCss ðtÞj2 . 1 For timescales relatively shorter than all genuine times c1 j , that is, t cj , Css ðtÞ ¼ expðDsl t=2ÞexpðiEs tÞ;

ð6:87Þ

so IOFID exhibits an exponential intramolecular dephasing process of the optically selected Born–Oppenheimer state with the lifetime ðDsl Þ1 . For longer times, t  c1 j , destructive interference effects erode all the contributions to the double sum of jCss ðtÞj2 , where j 6¼ j0 , so jCss ðtÞj2 

X  hsjjih
jjsi2 expðc tÞ; j

j

t c1 j ;

ð6:88Þ

and the IOFID exhibits a sum of exponentials [146–149, 186]. The decay of the OFID in such long timescale can be realized in real life for the case of the intramolecular structure, where the decay widths cj correspond to the genuine T1 relaxation rates of the (nonoverlapping) MEs. These remarks offer only the barest indication of why the following discussion has extensive practical application. 6.5.4 Photon Echoes from Large Molecules

A photon echo is a pulse of radiation emitted by the sample after irradiation with two or more coherent laser pulses, characterized by the durations Dt1 and Dt2 and separated by the time t12 . The first excitation pulse creates a macroscopic polarization in the sample, which vanishes rapidly due to differences in resonance frequency of

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the absorbing molecules (inhomogeneous dephasing). One or more subsequent pulses reverse the dephasing process, leading to a recovery of the macroscopic polarization and corresponding radiation (the echo). The photon echo is the optical analogue of the spin technique (of a spin one-half system) in magnetic resonance spectroscopy. This analogue was demonstrated by Dicke [187]. The corresponding equations for optical transitions were later derived by Feynman, Vernon, and Hellwarth [188]. The dynamics of the coherently prepared state of the system is described by a pseudo-spin vector, which during the system development with the resonant frequency of the optical transition rotates about a z-axis (the direction of the B field for magnetic transitions). In the course of this process, the z-component of this pseudo-spin describes the state of the inversion of the ensemble, while the projection in the complex x–y plane macroscopically describes the polarization p (the x-axis is the direction of an adjacent external field). Both components are subject to exponential decay. Correspondingly, both the longitudinal relaxation time T1 , which is identical with the lifetime of the excited state, and the transversal relaxation time T2 , which is also known as the total dephasing time, exist [189], since they describe the decay of the phase correlation between the excited ensemble molecules. Under certain conditions [189], the line shape is Lorentzian with a width Dnhom ¼

1 ; pT2

ð6:89Þ

where

1 1 1 ¼ þ : T2 T2 2T1

ð6:90Þ

T1 is the level depletion lifetime and T2 is the pure dephasing time associated with the phase destructive events (e.g., by phonons and intermolecular interactions). This point will be taken up in more detail in Chapter 7. Here we merely note that the pure dephasing time T2 is temperature dependent. Let us now return to the time development of the system, described in terms of the macroscopic polarization in spirit of the preceding section. In accordance with the above discussion, we shall now again consider short-time excitation conditions (6.71) and (6.74). The initial state of the system is Yð0Þ ¼ j g i, whereupon the state of the system at time t ¼ Dt1 þ t12 þ Dt2 þ t is
ðDt2 þ t12 YðtÞ ¼ Uðt þ Dt2 þ t12 þ Dt1 ; Dt1 þ t12 þ Dt2 ÞU ;
þ Dt1 ; t12 þ Dt1 ÞUðt12 þ Dt1 ; Dt1 ÞUðDt1 ; 0Þj g i

ð6:91Þ


during the pulses are given by Equation 6.65, where the time evolution operators U while the time evolution operators U, when the field is switched off, are given by Equation 6.77. The evaluation of Equation 6.91 is identical with that following Equations 6.63–6.69, which we need not repeat. Utilizing the conventional sequence of p=2 pulse, that is, vR Dt1 ¼ p=2, followed by a p pulse for which vR Dt2 ¼ p, the polarization per molecule at time t þ Dt2 þ t12 þ Dt1 assume the form

6.5 Dephasing and Relaxation in Molecular Systems

pðtÞ ¼ imgs fCss ðt12 ÞCss ðtÞexp½ivðDt1 Dt2 Þexp½iEg ðt12 tÞ þ ½Css ðt þ t12 ÞCss ðtÞCss ðt12 ÞexpðivDt2 ÞexpðiEg tÞg þ cc;

ð6:92Þ

where the probability amplitude Css ðxÞ for the time evolution is given by Equation 6.80. Following the discussion of the properties of this function given in the preceding section, it is apparent from Equation 6.80 that for the timescale t c1 j (for all j), the second term on the right-hand side of Equation 6.92 vanishes identically. Furthermore, it can also be shown that for longer times, this term will not contribute to the photon echo. Thus, Equation 6.92 may be simplified by removing this term and we will easily obtain pðtÞ ¼ imgs exp½ivðDt1 Dt2 Þexp½iDEsg ðt12 tÞFðt12 ; tÞ; þ cc

ð6:93Þ

where DEsg is given by Equation 6.82 and Fðt12 ; tÞ ¼ f ðtÞf  ðt12 Þ;

ð6:94Þ

with f ðxÞ given by Equation 6.83. Furthermore, as t12  Dt1 ; Dt2 and v ¼ DEsg , the exponential function exp½ivðDt1 Dt2 Þ in Equation 6.93 can be omitted in the subsequent discussion. Performing now the averaging over the inhomogeneous distribution (Equation (6.84), we find for the macroscopic polarization

  PðtÞ ¼ Nmgs exp b2 ðtt12 Þ2 =4 sin DEsg0 ðt12 tÞ Re Fðt12 ; tÞ  þ cos½DEsg ðt12 tÞIm Fðt12 ; tÞ :

ð6:95Þ

The final step in our procedure is to relate this result to the expression for the radiative intensity of the photon echo signal IPE at time t ¼ Dt1 þ t12 þ Dt2 þ t. After performing the optical cycle averaging of Equation 6.95 and following the method given in the section above, the PE intensity assumes the final form

 IPE ðtÞ ¼ A exp b2 ðtt12 Þ2 =2 jFðt; t12 Þj2 :

ð6:96Þ

Equation 6.96 has a clear physical significance. The temporal behavior of the photon echo from an assembly of large molecules, which are excited by two short onresonance laser pulses, is determined by the inhomogeneous dephasing contribution exp½b2 ðtt12 Þ2 =2 peaking at t ¼ t12 , while the amplitude aPE of the echo at t ¼ t12 is determined by the contribution of the intramolecular dephasing aPE ¼ jFðt; tÞj2 ¼ jCss ðtÞj4 , where Css ðtÞ being, of course, similar to that of the OFID signal. Now, following the above discussion given in connection with the function Css ðtÞ, we note that 1) For sufficiently short times t12 cj 1 (for all j), the relevant echo decay amplitude is given by Equation 6.80, so that at t ¼ t12 aPE / expð2Dsl t12 Þ;

exhibiting an exponential decay [172] with a lifetime ð2Dsl Þ1 .

ð6:97Þ

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2) For longer times t12 > c1 j , the decay amplitude is given by a sum of exponentials (compare with Equation 6.88) and the amplitude of the photon echo is X aPE / aj expð2cj t12 Þ; ð6:98Þ j

where aj  1=n2 and hcj i  cs =n þ cl with n being the dilution factor of the doorway state [146–149, 186]. This physical situation can be accomplished for the intermediate-level structure [146–149, 186, 190], where the long-time depletion rate T1 will be amenable to experimental observations.

j155

7 Miscellaneous Applications In this chapter, a number of rather disconnected topics are discussed to illustrate the application of the general technique developed in earlier chapters. We shall begin our discussion with the derivation of line shape functions for optically allowed and vibronically induced transitions. This will permit us to demonstrate the influence of the Duschinsky rotation on optical spectra. As an example, we shall describe the spectrum of the p-terphenyl crystal from the same point of view and of the ½CoðCNÞ2 ðtnÞ2
Cl3 H2 O complex in its crystal structure. These examples offer little more than a glimpse into the difficult, subtle, fascinating questions encountered in almost any attempt to interpret optical transitions. The discussion is extended in Section 7.2 to the analysis of phosphorescence spectra and the description of radiationless transition of aromatic molecules. Particular emphasis is placed on the mechanism of singlet–triplet relaxation in these molecules with nonbonding electrons. In Sections 7.3 and 7.4, the temperature dependence of radiationless transitions and the effect of deuteration on the lifetimes of excited electronic states are examined. In Section 7.5, a contribution to time-resolved spectroscopy is presented. In that section, we will discuss a problem dealing with transport phenomena of electronic excitations in doped molecular crystals. The theory of singlet excitation energy transfer uses an effective Hamiltonian to account for intramolecular excited-state depopulation and energy transfer by multistep migration among guest molecules. Finally, in Section 7.6 we shall illustrate our discussion of Section 1.4 with an application to the predissociation of the triatomic molecule H2 O þ .

7.1 The Line Shape Function for Radiative Transitions 7.1.1 Derivation

In addition to the study of nonradiative decay probability (of excited electronic states), the consideration of the decay of excited electronic states has many additional

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applications. This includes such familiar examples as decay by radiative transition. This can be best described in terms of a line shape that provides a very useful basis for discussing resonance and decay phenomena. The line shape function is given for transition in emission from excited state jsi to a ground state j0i in the form [92] XX

2 f ðEÞ ¼ pðm : TÞ
Vsm;0n
dðDE þ Esm Eln EÞ; ð7:1Þ fmm g fnm g

where the sums are extended over all final states n ¼ ðn1 ; n2 ; . . . ; nN Þ and averaged over the initial states m ¼ ðm1 ; m2 ; . . . ; mN Þ, being in thermal equilibrium at temperature T. pðm; TÞ is the probability that the system will be initially in the P zeroth-order states jsmi, so that m pðm; TÞ ¼ 1. Provided that vibrational relaxations are faster than electronic transitions, thermal equilibrium prevails and pðm; TÞ ¼ Z 1 expðbEsm Þ;

ð7:2Þ 1

where Z1 is the partition function defined in Equation 3.8 and b ¼ ðkB TÞ . DE corresponds to the energy gap between the lowest vibrational components of the two electronic states DE ¼ Es;0 E0;0 ¼
hV:

ð7:3Þ

The energies Esm and Eln of the vibrational levels in each electronic manifold are measured from the zeroth level of that manifold. Thus, X X Esm ¼
hmm vsm and E0n ¼ hnm v0m :
ð7:4Þ fmm g

fnm g

Rather than use (7.1), we consider its Fourier transform 1 ð

FðtÞ ¼ 1

  ðEDEÞt exp i f ðEÞ dE;
h

which can be expressed in the form X FðtÞ ¼ Z1 Vsm;0n expðiE0n t=
hÞV0n:sm expðiEsm t=
hbEsm Þ:

ð7:5Þ

ð7:6Þ

m;n

The function FðtÞ may be related to the generating function introduced in Chapter 3 for calculating the probability of radiationless transitions. If for the matrix elements in Equation 7.6 we write Vsm;ln ¼ ðxsm ðqs ÞjVs0 ðqÞjx0n ðq0 ÞÞ;

ð7:7Þ

where Vs0 ðqÞ ¼ hjs ðr; qs ÞjV jj0 ðr; q0 Þi

ð7:8aÞ

is the operator responsible for the radiation matter interaction. Using the Green’s functions for the nuclear motion in the electronic states |si and j0i given by

7.1 The Line Shape Function for Radiative Transitions

Equation 3.18, the summation in (7.6) over collections of vibrational quantum numbers m ¼ ðm1 ; m2 ; . . . ; mN Þ and n ¼ ðn1 ; n2 ; . . . ; nN Þ gives FðtÞ ¼ Z1

1 ð



ðY m

1

   0 0  dq0m d
q0m Vs0 ð
qÞrs qsm ;
qsm ; wm V0s ðqÞr0 ~ q m ; qm ; zm : ð7:9Þ

The operator Vs0 ðqÞ for the radiation matter interaction may be expanded about some reference configuration q0 along some nontotally coordinates q0g of the electronic ground state ! X qVs0 X g Vs0 ðqÞ ¼ Vs0 ðq0 Þ þ q0g þ    þ ¼ Vs0 ðq0 Þ þ Vs0 ðq0 Þq0g    ; 0 qq g g g 0

ð7:8bÞ

where the first term is responsible for electric dipole allowed transition (see Equation g 6.10). We shall not discuss here the detailed construction of the operator Vs0 ðq0 Þq0g for the vibronically induced transitions, however, since we require only the property that it is associated with the coordinate qg . This is done, for example, in Section 7.2. Since the vibrational amplitudes of qg ’s remain small, we terminate the expansion on terms linear in qg . This permits us to write Equation 7.9 as FðtÞ ¼ F 1 ðtÞ þ F 2 ðtÞ;

where F 1 ðtÞ ¼ Z 1 jVs0 ðq0 Þj2

ð7:10Þ N Y m¼1

and F 2 ðtÞ ¼ Z1

ðmÞ

G1 ðtÞ;

ð7:11Þ

Y ðmÞ X
g X g
g0 
V ðq0 Þ
2 K ~ g ðtÞ G ðtÞ þ Z1 V ðq0 ÞV ðq0 Þ s0

g

 ~I g ðtÞ~I g0 ðtÞ

Y m6¼g;g0

m6¼g

1

g6¼g0

ðmÞ

G1 ðtÞ:

s0

s0

ð7:12Þ

ðmÞ

The terms G1 ðtÞ correspond to the single-mode generating function defined in ~ g ðtÞ corresponds to a single-mode generating function Equation 3.13. The factor K that involves the nuclear coordinate qg of the promoting mode. It induces the radiative transition ~ g ðtÞ ¼ K

ð

1 ð

    dq0g d
q0g rs qsg ;
qsg ; wg
q0g r0
q0g ; q0g ; zg q0g :

ð7:13Þ

1

ðgÞ

The latter integral is related to the integral G1 ðtÞ analogous to relation (3.72), namely, " #
2g 2bs2 k2g w 1 1 1 g ðgÞ ~ g ðtÞ ¼ K G1 ðtÞ  þ s 2 0 1 0 2 bsg w
g þ b0g
zg bsg w
1


z þ b ðb þ b Þ w z g g g g g g g ð0Þ

ðgÞ

~ ðtÞG ðtÞ: ¼K 1 g

ð7:14Þ

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Considering the case with undisplaced and undistorted harmonic potential surface for the promoting mode, Equation 7.14 may be put into a simpler form analogous to the propensity rule for the promoting mode (3.73). This gives         
hvg h vg

h 1 1 ivg t ivg t ~ ð0Þ ðtÞ ¼ þ ; coth coth K þ 1 e 1 e g 2vg 2 2 2kB T 2kB T ð7:15aÞ

which for low temperatures goes to
ivg t h ~ ð0Þ e : K g ðtÞ ¼ 2vg

ð7:15bÞ

Finally, some mixed-type single-mode generating functions appear in Equation 7.12; these are governed by ~I g ðtÞ ¼

1 ð

ð

dq0g d
q0g rs ðqsg ;
qsg : wg Þ
q0g r0 ð
q0g ; q0g ; zg Þ:

ð7:16Þ

1

As discussed above in the case of radiationless transitions, in the harmonic approximation, where kg ¼ 0, the latter term vanishes (see the discussion in connection with Equation 3.74). Thus, the generating function (7.12) takes the more convenient form F 2 ðtÞ ¼ Z1

Y ðmÞ X
g

V ðq0 Þ
2 K ~ g ðtÞ G1 ðtÞ: s0 g

ð7:17Þ

m6¼g

To complete our derivation, we have yet to take the inversion formula (to (7.5)) f ðEÞ ¼ ð2p
hÞ1

1 ð

FðtÞ exp½iðEDEÞt=
h
dt:

ð7:18Þ

1

This permits us to expresses the line shape function f ðEÞ in terms of the generating function FðtÞ. This is a useful conclusion, since FðtÞ is ordinarily easier to evaluate directly than is f ðEÞ. The reader should be warned, however, that the Fourier transformations (7.5) and (7.18) hold in general only for generalized functions. In ordinary function theory, f ðEÞ and FðtÞ must fulfill some conditions for Equations 7.5 and 7.18.1) Therefore, we have chosen to modify Equation 7.1 to incorporate a finite width by using a Lorentzian distribution in Equation 7.1 to conserve energy, instead of the d-function of Dirac. This leads to a modification of Equation 7.18 by introducing a real factor C
h1 jtj in the exponent of the integrand of Equation 7.18. Using this modification, then inserting Equations 7.3, 7.4 and 7.11 in Equation 7.18, and finally carrying out 1) For example, Equation 7.18 does exist for any FðtÞ that falls off sufficiently rapidly as t !  1 that the integral (7.18) of its square is finite.

7.1 The Line Shape Function for Radiative Transitions

the integration by applying the procedure of Section 7.3, we find for the direct electric dipole allowed process  X  2cexp 
h mm vsm =kB T
2 X X 1 1

m f1 ðvÞ ¼ Z Vs0 ðq0 Þ
2  X X 2p fmm g fnm g mm vsm þ nm v0m þ c2 vV  IN

fmm g

 m1 ; m2 ; . . . ; mN : n1 ; n2 ; . . . ; nN

fnm g

ð7:19Þ

Similarly, for the vibronically induced transition with F 2 ðtÞ as generating function, we have 0 1 (  1 1 X @
h A

g

2

Vs0 coth ð
hvg =2kB TÞ þ 1 f2 ðvÞ ¼ Z 2p 4v g g  X  s 2cexp 
h mm vm =kB T XX m  2  X X fmm g fnm g mm vsm þ nm v0m þ c2 vV þ vg   IN

m1 ; m2 ; . . . ; mN n1 ; n2 ; . . . ; nN

XX þ coth ð
hvg =2kB TÞ1  fmm g fnm g

 IN

m1 ; m2 ; . . . ; mN n1 ; n2 ; . . . ; nN

)



fmm g

fnm g

 X  s 2cexp 
h mm vm =kB T

vVvg 

m

X fmm g

mm vsm þ

X

2 nm v0m

þ c2

fnm g

ð7:20aÞ

where we have introduced the notation c ¼ C
h1 and E ¼
hv is the energy of the emitted light. Despite the fact that the promoting modes g enter the ID IN , they make a negligible contribution to IN , since according to Equation 7.14 and the discussion in ðgÞ Section 3.3, G1 ðtÞ ffi 1. As in the case of radiationless transitions, we see that the net effect of the promoting mode is to decrease or increase the energy gap to
hV 
hvg with temperature dependence of ½coth ð
hvg =2kB TÞ  1
, respectively. Near v V; ðmm ¼ nm ¼ 0Þ;f2 ðvÞ is proportional to 0 1 (  1 1 X @
h A

g

2

2c f0 ðvÞ ¼ Vs0 cothð
hvg =2kB TÞ þ 1 Z 2p 4v ðvV þ vg Þ2 þ c2 g g )

 2c ; þ coth ð
hvg =2kB TÞ1 ðvVvg Þ2 þ c2 ð7:21aÞ

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j 7 Miscellaneous Applications

160

where the summation is restricted only to the promoting modes g. It is therefore possible to write Equation 7.20a as 0 1 XX X X s 0 f2 ðvÞ ¼ f0 @v mm vm þ nm vm Aexp ðmm
hvsm =kB TÞ  fmm g fnm g



IN

fmm g

 m1 ; m2 ; . . . ; mN : n1 ; n2 ; . . . ; nN

fnm g

ð7:20bÞ

This is a useful conclusion since f0 ðvÞ is ordinarily easier to estimate from an optical emission spectrum than f2 ðvÞ. s If we were to follow the discussion that led to Equation 7.20b by setting wn ¼ eivn t iv0n t
hv0n =kB T ivsn t
hvsn =kB T iv0n t and zn ¼ e (instead of wn ¼ e and zn ¼ e as in the previous derivations), we would obtain an expression for the line shape function in absorption in the same convenient form 0 1 X X ðabÞ X X ðabÞ f2 ðvÞ ¼ f0 @v mm vsm þ nm v0m Aexp ðnm
hv0m =kB TÞ fmm g fnm g

 IN

fmm g

fnm g

 m1 ; m2 ; . . . ; mN ; n1 ; n2 ; . . . ; nN

ð7:20cÞ

where ðabÞ f0 ðvÞ ¼

8 0 1  1 1 X @
h A

g

2 2 K, the dephasing of the electronic origin by a T1 -type scattering to local phonons and intermolecular interactions varies with temperature as   DE T2 ðTÞ ¼ T2 ð1Þexp ; kT

where DE is an activation energy and it is probably safe to consider the values for T2 ð1Þ between 2 and 5 ps (depending on the concentration cG ) [258]. This asserts that at T ¼ 10 K and values of DE lying somewhere in the range of phonons with energies of 30 cm1 , T2 reduces to about 200 ps. For the line shift dT , a similar expression holds, but with a different meaning of the preexponential factor. The description of the

7.6 Electronic Predissociation of the 2B2 State of H2O þ

excitation motion outlined above assumes that the spectral overlap of sensitizer and activator is constant. This is true if the temperature is kept constant. However, as described above, a temperature increase affects the spectral overlap mainly due to the redshift in fluorescence and absorption lines [257]. This leads to a decrease of the spectral overlap and thus to an energy transfer rate that varies with temperature.

7.6 Electronic Predissociation of the 2B2 State of H2O þ 7.6.1 Evaluation of the Nonadiabatic Coupling Factor

In this section, we limit our attention to a special case of the results obtained in ~ 2A0 states of H2 O þ ~ 2A0 and A Section 1.4 for the conical intersection connecting the B 2 to consider the vibronically induced predissociation of the B2 state. According to the interpretation in [260–262] and the calculations of the nonadiabatic couplings in Section 1.4, after preparing the molecule H2 O þ in the metastable state 2B2 by excitation with photon energy hn in the ranges between 18.0 and 20.2 eV, the latter plays an important role in producing OH þ ions. This nonradiative conical intersection seems to be at least competitive with the electronic predissociation of the 2B2 state of H2 O þ by two repulsive states 4A00 and 2A00 [262] for production of, respectively, H þ and OH þ . ~ 2A =2A 0 Þ ~ 1 A1 Þ þ hn ! H2 O þ ðB~ 2B2 = 2A 0 Þ ! H2 O þ ðA H2 OðX 1 ! OH þ ð3 S Þ þ H ! H þ þ OHð2 PÞ;

The nonradiative transition for the above process will be examined here. The required probability is then w /¼

2p X X



2 Cif ;
h i f

ð7:70aÞ

where Cif is the probability amplitude ~ jq=qRjBiqx ~ i =qRJ 1=2 Þ: Cif ¼ ðxf hA

ð7:70bÞ

~ jq=qRjBi ~ is the nonadiabatic electronic coupling matrix element and R is the gR ¼ hA reaction or dissociation coordinate, which will be defined below. The braket (    ) appearing in Equation 7.70b denotes integration over nuclear coordinates. Hence, determination of the matrix element (7.70) requires the evaluation of a multidimensional FC-type factor as well as the electronic matrix element hj ji. The Jacobian J in (7.70) insures that initial- and final-state wavefunctions are normalized with respect to the same volume element (in a sense to be made more precise below). The integrand in Equation 7.70 is a sharply peaked function over the nuclear coordinates, so that the integration may be confined to a restricted region of the

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nuclear coordinates around the apex of the conical intersection. This becomes particularly clear later when we estimate the electronic matrix element gR . Evaluation of gR is not trivial, however, it can be obtained from the nonradiative coupling factors derived in Section 1.4 (see formulas (1.68)). Since the dissociation process is represented as Y1 XY2 ! Y1 þ XY2 , a suitable coordinate system is given by the body-fixed (BF) reference frame [263–266] as shown in Figure 7.22, the Z-axis of which lies along the vector R from atom Y1 to the center of mass (c.m.) of the diatomic molecule XY2 . The angle c is the polar angle of the diatomic bond axis XY2 (the length of which is rd ) in the body-fixed system. The coordinates ðR; c; rd Þ, which we use below, are denoted as the Jacobi coordinates. In the same Figure 7.22, the angle w between the body-fixed Z-axis and the space-fixed (SF) x-axis describes the rotation of the entire system about the center of mass (C.M.). Since we are only interested in the motion in the xy plane, the azimuthal angles of rd in the body-fixed system are assumed to be zero. The coordinates ðR; c; rd Þ, which are suitable for describing the receding motion of the molecule on the potential surface, are now used to evaluate the matrix element (7.70). For this reason, we need a transformation between the normal coordinates of the triatomic molecule H2 O þ in its bound state and the Jacobi 3

mx

(a)

r2

α my

y

αeq

req

2

r1 my

1 x

(b) 3 γ

(c.m.) rd (C.M.)

y 2

R φ 1

x Figure 7.22 (a) Internal coordinates of a bent triatomic molecule. (b) Space-fixed and body-fixed coordinate systems for a 123 triatomic molecule.

7.6 Electronic Predissociation of the 2B2 State of H2O þ

mx

S1 αeq

re my

S1

S1

my

2my m x S2

S2 αeq S2

S2

my

my

2my sin αeq m x S3

S3 αeq my

S1

S1

my

Figure 7.23 Symmetry coordinates of a bent symmetric triatomic molecule. After Ref. [75].

coordinates ðR; c; rd Þ. Since the normal coordinates are related to the symmetry coordinates linearly (Figure 7.23), we start with the presentation of the latter in terms of ðR; c; rd Þ. The result is [267] 1 S1 ¼  ½R sin w þ ðmXY =mY Þrd sin ðw þ cÞ
þ re sin aeq ; 2 S2 ¼ 

1 ½R cos wð1 þ mXY =mX Þrd cos ðw þ cÞ
þ re cos aeq =p; 2p

1 S3 ¼ cos aeq ½R cos w þ ðmXY =mY Þ rd cosðw þ cÞ
; 2

ð7:71Þ

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where p ¼ 1 þ 2mY =mX , with mX and mY being the mass of atom XðX ¼ OÞ and atom YðY ¼ HÞ, respectively. re and aeq are the bond length and the half valence angle in the equilibrium geometry of the triatomic YXY in the initial bound state. mXY ¼ mX mY =ðmX þ mY Þ is the reduced mass of the diatomic fragment XY. The angle w describes the orientation of the molecular framework YXY during the ~ It is given by [267] ~ , A. transition B w ¼ wðR; rd ; cÞ ¼ arctan

  Rp tan aeq þ ðmXY =mY Þrd p tan aeq cos c þ ð1 þ mXY =mX Þrd sin c : Rð1 þ mXY =mX Þrd cos c þ ðmXY =mY Þrd p tan aeq sinc

ð7:72Þ

Over a restricted range of R, the transformation (7.71) is quasilinear in respect to R and rd but nonlinear in respect to c and w. With the aid of the nonadiabatic coupling terms along the internal coordinates a and r ¼ 12 ðR1 R2 Þ, the relevant nonadiabatic coupling between the potential surfaces of states 2B2 and 2A1 calculated along the reaction coordinate R can now be expressed as ~ jq=qRjBi ~ ¼ gR ðR; rd ; cÞ ¼ ga hA

qa qr þ gr : qR qR

ð7:73Þ

In order to calculate gR , we need the dependence of a on R and r on R. These relations can be obtained by recalling (see Ref. [268], p. 149, eq. (II. 122)) that the symmetry lowering coordinate r can be expressed in terms of the symmetry coordinate S3 , r ¼ 2tS3 ;

ð7:74Þ

where t ¼ 1 þ 2ðmY =mX Þsin2 aeq . Similarly, for the change of the valence angle a around its value ac at the conical intersection between states 2B2 and 2A1 (when r ¼ 0; ac ¼ 2aeq ¼ 71:6 ), we have (Figure 7.23) aac ¼ 2ðpS2 sin aeq S1 cos aeq Þ=re :

ð7:75Þ

Substituting Equation 1.68 in Equation 7.73 and using the relations (7.74) and (7.75), we write Equation 7.73 in the form gR ðR; rd ; cÞ ¼

1 tFa Fr =re 2 ðFr tS3 Þ2 þ ðFa =re Þ2 ðpS2 sin aeq S2 cos aeq Þ2 0 1 3 2 qS3 @ qS2 qS1 A 5 4  ðpS2 sin aeq S1 cos aeq Þ  p sin aeq cos aeq S3 ; qR qR qR ð7:76Þ

where according to (7.71) and (7.72)  qS1 1 qw ¼ sin w þ ½R cos w þ ðmXY =mY Þrd cosðw þ cÞ
; qR 2 qR

ð7:77Þ

7.6 Electronic Predissociation of the 2B2 State of H2O þ

 qS2 1 qw ¼ cos w þ ½R sin w þ ð1 þ mXY =mX Þrd sin ðw þ cÞ
; qR 2p qR

ð7:78Þ

 qS3 1 qw : ¼ cos aeq cos w½R sin w þ ðmXY =mY Þrd sinðw þ cÞ
qR qR 2

ð7:79Þ

and qw qR ¼

sin wp tan aeq cos w : R cos wð1 þ mXY =mX Þrd cosðw þ cÞ þ p tan aeq ½R sin w þ ðmXY =mY Þrd sinðw þ cÞ
ð7:80Þ

Figure 7.24 shows the nonadiabtic electronic matrix element gR plotted as a ˚ and fixed values of mY ¼ function of R and c for constant rd ¼ 1:15 A 1836:151 amu and mX ¼ 16mY . Figure 7.24 clearly demonstrates that the coupling factor gR has considerable variation with R and c, which may greatly affect the critical relaxation rate by orders of magnitude. One of the interesting results of this analysis is the extent to which the coupling factor gR resembles a generalized function. Viewing the c-axis, gR resembles a function d0 ðccc Þ, where d0 is the derivative of the delta Dirac distribution centered at cc ¼ 105:5 . Along the R-axis, we have a function ˚ The coupling gR is found to be positive for resembling dðRRc Þ, where Rc ¼ 1:14 A.

c smaller than cc ¼ 115:5 and negative for c larger than cc . A dynamic study by means of classical trajectory calculations [70] shows that the rate constant describing

Figure 7.24 Nonadiabatic electronic coupling matrix element gR as a function of R and c; (a) view against the c-axis. (b) A second view down the c-axis. The diatomic internuclear OH distance rd is

kept fixed at a value of 1.15 A.

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the internal conversion from the upper cone to the lower adiabatic surface is of the order of 1014 s1 . Note the difference of this reasoning to that employed at the end of Section 7.2. There, however, we concluded that the matrix element responsible for the nonradiative transition is determined in higher order by motion of the nuclei and by abrupt change in the spin configuration. Here, we conclude that the energies of the adiabatic states 2 B2 and 2 A1 are very close and the coupling matrix element gR ðR; rd ; cÞ between these states evolves the derivatives of the electronic wavefunction with respect to the nuclear coordinates (Equations 1.15 and 1.16). This term is expected to be large in the vicinity of the conical intersection (in the cusp intersection), where the wavefunction changes rapidly. If, for example, the bending frequency v2 of H2 O þ is taken as 1600 cm1 , it is seen that the m2 ¼ 1 level will lie 2400 cm1 above the 2 B2 minimum, which is very close to the energy of the cups of intersection ( 1300 cm1 ). Perturbations due to the E2 E1 coupling between the upper cone and the lower adiabatic surface may be very significant for levels in that energy region (see Figure 1.1) and lead to a ultrafast intramolecular relaxation with a lifetime of 1014 s. As far as the photodecomposition of H2 O þ is concerned, the nonadiabatic passing through the conical intersection between the ~ states of H2 O þ plays an important role in producing the surprisingly large ~ and A B amount of H þ . Finally we note that the rotation angle w is a slowly varying function of R and depends parametrically on c [267]. The change of the diatomic distance rd has little or no effect on the strength of gR and can be safely assumed rigid allowing rd to be replaced by a suitable equilibrium value r0 . When this is substituted into Equation 7.76, gR is then set to be gR ðR; r0 ; cÞ. The stretching coordinate rd becomes important in calculating the vibrational frequencies of the diatom and must be fully incorporated into the nuclear functions of both electronic states. 7.6.2 The Basis State Functions

We now turn to the vibrational wavefunctions appearing in the matrix element (7.70a). 7.6.2.1 The Initial-State Wavefunction xi To evaluate the integral (7.70a), it is necessary to obtain expressions for the nuclear wavefunctions xi and xf . The upper adiabatic state can often be described in the harmonic approximation as a product of three oscillator functions xi ¼ xi;m1 m2 m3 ðqm ðR; rd ; cÞÞ ¼ xi;m1 ðq1 Þxi;m2 ðq2 Þxi;m3 ðq3 Þ:

ð7:80aÞ

The normal coordinates q1 and q2 for the symmetric stretching and bending vibrations are then linear combinations of the symmetry coordinates S1 and S2 used above for calculating the electronic coupling factor gR and q3 ¼ c33 S3 is the asymmetric stretching coordinate [70]

7.6 Electronic Predissociation of the 2B2 State of H2O þ

q1 ¼ c11 S1 þ c12 S2 ; q2 ¼ c21 S1 þ c22 S2 ; q3 ¼ c33 S3 :

ð7:81Þ

After utilizing (7.71), we eliminate ðq1 ; q2 ; q3 Þ in favor of ðR; rd ; cÞ to give xi ¼ xi;m1 ;m2 ;m3 ðfqm ðR; rd ; cÞgÞ:

ð7:80bÞ

The transformation coefficients of Equation 7.81 are determined by solving the usual secular equation in terms of the masses, force constants, and frequencies [267]. The numerical values of the coefficients cij for the ground state of the neutral molecule are [70] c11 ¼ 49:343 aum1=2 ; c21 ¼ 35:772 aum1=2 ;

c12 ¼ 37:310 aum1=2 ; c22 ¼ 52:337 aum1=2 ;

c33 ¼ 62:922 aum1=2 :

Figure 7.23 provides a graphical representation of the results expressed by Equation 7.81. The Jacobian for that transformation is given by [267] J¼

Dðq1 q2 q3 Þ Rrd ¼C ; 2pD cos aeq DðR; rd ; cÞ

where C ¼ ðc11 c22 c12 c21 Þc33 and D ¼ R cos wð1 þ mXY =mX Þrd cosðw þ cÞ þ p tan aeq ½R sin w þ ðmXY =mY Þrd sin ðw þ cÞ
: 7.6.2.2 The Final Vibrational Wavefunction xf : The Closed Coupled Equations One of the most difficult aspects in treating polyatomic predissociation is the proper description of the xf wavefunction. The complication arises because the wavefunctions for the translational motion are obtained as a solution of coupled differential equations. In deriving this, we may proceed as follows: When the bent triatomic molecule is initially nonrotating (J ¼ 0) and the diatom is assumed to be bound for all energies considered, the nuclear Hamiltonian for motion of the nonlinear triatomic molecule Y1 XY2 on the final state of H2 O þ can be written as [263–266]  2
h2 q L2 HN ¼ Hd  þ VðR; r0 ; cÞ; ð7:82Þ R þ 2m1 R qR2 2m1 R2

where L is the angular momentum operator for the relative motion of atom Y1 about the diatomic center of mass (with zero projection on the space-fixed x-axis) and 1 1 is the interfragment reduced mass. That is, m1 1 ¼ mY1 þ ðmX þ mY2 Þ   h2
q q L2 ¼  sin c : ð7:83Þ sin c qc qc

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Hd in Equation 7.82 is the Hamiltonian for the bound component of the motion over the final surface  2
h2 q l2 Hd ¼  þ V þ ; ð7:84Þ r d d 2m2 rd qrd2 2m2 rd2 2 1 1 2 with m1 2 ¼ mX þ mY2 being the reduced mass and l ðcÞ ¼ L ðcÞ, where I is the rotational angular momentum operator associated with the diatomic fragment XY2 . VðR; r0 ; cÞ is the lower state adiabatic energy potential surface E1 , where the instantaneous interatomic distance of the diatomic fragment is replaced by its suitable “equilibrium” value r0 (the diatomic bond in the interaction region, at the apex of the cone). The potential VðR; r0 ; cÞ is not known analytically, but numerically ˚ in the region of strong for various values of R and c (at constant r0 ¼ 1:15 A) nonadiabatic interaction around the apex of the double cone (Figure 7.25). As a ˚ the potential VðR; r0 ; cÞ is attractive over the function of R, for R extending to 1:8 A, interval 0 c < 150 . For c outside this interval, VðR; r0 ; cÞ is repulsive. (The origin of the energy scale for VðR; r0 ; cÞ in Equation 7.82 is taken at the energy of the completely separated products OH þ þ H.) The diatomic potential Vd in Equation 7.84 is taken to be a simple harmonic oscillator of the form

1 Vd ¼ m2 v2d ðrd r0 Þ2 De 2

ð7:85Þ

with vd being the vibrational frequency.

Figure 7.25 Potential energy surface of the lower adiabatic surface E1 of H2 O þ in the region of strong nonadiabatic interaction around the apex of the double cone plotted as

function of Jacobi coordinates R and c. The insert shows the adiabatic potential surfaces E1 and E2 calculated in Jacobi coordinates. (Note: 103 hartree is equivalent to 219 cm1.)

7.6 Electronic Predissociation of the 2B2 State of H2O þ

As is usually done in atom–diatom scattering theory [263–266], we expand the final-state wavefunction in terms of diatomic basis functions. We choose these functions to be harmonic oscillator rigid rotor functions wn;l;0 ðrd ; c; 0Þ ¼ xn ðrd ÞYl;0 ðc; 0Þ;

ð7:86Þ

where xn ðrÞ ¼ gn ðjÞ=r is the diatomic vibrational wavefunction with gn ðjÞ ¼ ðb1=2 =p1=2 2n n!Þ1=2 exp ðbj2 =2ÞHn ðb1=2 jÞ;

j ¼ rd r0

ð7:87Þ

and Hn is the nth Hermite polynomial. These functions are, to good approximation solutions of the equation,  2  2    
h q 1 1 2 2   v ðr r Þ ðr Þ ¼ h
v n þ ð7:88Þ r x m x ðrd Þ: 0 d d d n 2 2m2 rd qrd 2 2 2 n It should be noted, however, that the diatomic basis functions of Equation 7.86 are not true eigenfunctions of the diatomic Hamiltonian Hd . This becomes evident if we calculate the matrix elements of the diatomic Hamiltonian between the basis functions (7.86) ðwnl0 jHd jwn0 l0 0 Þ ¼ enl dnn0 dll0 þ



2 lðl þ 1Þ

2 h ðxn 1=rd 1=r02
xn0 Þdll0 ; 2m2

ð7:89Þ

where   1 þ Blðl þ 1ÞDe ; en;l ¼
hvd n þ 2

ð7:90Þ

with vd and B ¼
h2 =2m2 r02 ¼
h2 =2I being, respectively, the vibrational frequency and the rotational constant of the diatomic fragment. The centrifugal coupling term in Equation 7.89 couple the rotational and the vibrational motion of the diatomic. The frequency factor in the oscillator function, Equation 7.87, is defined as b ¼ m2 vd =
h. The spherical harmonics in Equation 7.87 reduce to ordinary Legendre polynomials rffiffiffiffiffiffiffiffiffiffiffiffi 2l þ 1 Yl;0 ðc; 0Þ ¼ ð7:91Þ Pl ðcos cÞ: 4p The total wavefunction of the system must be an eigenfunction of the total angular momentum J and its z-component (M) in the space-fixed frame xy [264]. Since J ¼ 0, the total wavefunction, specified by the Hamiltonian (7.82), may therefore be written as a sum of products X xf ðR; rd ; cÞ ¼ xn ðrd ÞYl;0 ðc; 0ÞWnl;E ðRÞ=R; ð7:92Þ n;l

where Wnl;E are expansion coefficients to be determined numerically. All that remains is to determine these functions subjected to the boundary condition corresponding to those of the bound vibrational–rotational states of the system. To do this, let the Hamiltonian (7.82) operate on the function of this type and

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take matrix elements of (7.82) with respect to the vibrational and angular function. This gives 2 3 2 2 2 h
d h
4 þ lðl þ 1ÞðEenl Þ5Wnl;E ðRÞ þ 2m1 dR2 2m1 R2 X
  
xn Yl;0
V R; rd ; c
xn0 Yl0 ;0 Wn0 l0 ;E ðRÞ n0 ;l0

þ

X


2 h lðl þ 1Þ ðxn
1=rd2 1=r02
xn0 ÞWn0 l;E ðRÞ ¼ 0: 2m2 n0

The Equation 7.93 may be written in a more concise form X ½Hnl ðEenl Þ
Wnl:E þ Vnl;n0 l0 Wn0 l0 ;E ¼ 0

ð7:93Þ

ð7:94Þ

n0 l0 6¼nl

by introducing the notation
  
 Vnl;n0 l0 ¼ Ll;l0 dll0 þ xn Yl;0
V R; rd ; c
xn0 Yl0 ;0 dnn0 þ
 

2 h lðl þ 1Þ xn
1=rd2 1=r02
xn0 dll0 ; 2m2

ð7:95Þ

with Ll;l ¼


2 h lðl þ 1Þ 2m1 R2

ð7:96Þ

and Hnl ¼ 


2 d2 h þ Vnl;nl : 2m1 dR2

ð7:97Þ

In arriving at Equation 7.94, we have used the fact that the spherical harmonic Yl0 ðc; 0Þ is an eigenfunction of L2 and l2 L2 Yl0 ¼ l2 Yl0 ¼
h2 lðl þ 1ÞYl0 ;

ð7:98Þ

since L2 and l2 are diagonal in l, respectively. Equation 7.94 are identical to that that arises in the atom-diatomic scattering problem. They are a set of coupled differential equations. Each equation, or channel, is labeled by the set of quantum numbers ðnlÞ. The solution of these equations with the appropriate boundary conditions Wnl;E ðRÞ  !0 R!0 !0 Wnl;E ðRÞ  R!1

ð7:99Þ

corresponds to bound states at energies Enl;i , in particular, if ðEnl enl ÞVnl;nl < 0 for large R. This boundary conditions differ from those of scattering theory in that the wavefunction for the scattering coordinate R must go to zero at both R ¼ 0 and

7.6 Electronic Predissociation of the 2B2 State of H2O þ

R ¼ 1 rather than just at R ¼ 0 as in the scattering case. These boundary conditions are in fact identical to those of closed channels in scattering theory. We now utilize this fact and set up an artificial scattering problem in which the bound-state problem of interest forms the closed channels of a scattering problem [264]. This in turn enables us to use with a modification the standard techniques of molecular scattering theory [276–279]. In the following, it will often be convenient to condense our notation. For example, a single label a ¼ ðnlÞ may be employed to indicate state and channel. The label a0 will mean in general ðn0 l0 Þ a different specific channel and state of that channel. Then we may write quite generally X ½Ha ðEea Þ
Wa þ Vaa0 Wa0 ¼ 0: ð7:100Þ a0 6¼a

We have commented on the fact that the set of Equation 7.100 are identical to that arising in the atom-diatomic scattering problem. To utilize the well-developed theory of molecular scattering processes, the boundary conditions, as have been stated above, must correspond to continuum scattering states and not to the bound states as given by Equation 7.99. This can be accomplished by adding two additional equations, or channels denoted by b and c, to the set of coupled equations (7.100). Both these extra channels are open at large R and correspond asymptotically to scattering states that are oscillatory, that is, ðEeb ÞVbb > 0 for large R. The augmented set of differential equations has the form [264] ½Ha ðEea Þ
Ya þ

X

Vaa0 Ya0 ¼ Vac Yc ;

a ¼ 1; 2; . . . ; n;

ð7:101aÞ

a0 6¼a

½Hb ðEeb Þ
Yb ¼ 

X

Vba0 Ya0 ;

ð7:100bÞ

a0

½Hc ðEec Þ
Yc ¼ 0;

ð7:101cÞ

where Vab and Vac are the potential matrix elements coupling the artificial channels b and c to the bound-state channels a, respectively. Note that the two scattering channels are not coupled to each other directly. The boundary condition for the additional channel c that the wavefunction vanish at infinity is replaced by Yc ðRÞ ! sinðkc R þ jc Þ: R!1

ð7:102Þ

This condition restricts the form of Yc ðRÞ for large R. Here, k2c ¼ ð2m1 =
h2 ÞðEec Þ specifies the c channel wave number of the outgoing wave, with Eec being the asymptotic fragment energy. The real number jc is known as the phase shift, since it specifies the change of the phase of the outgoing wave due to the influence of the potential Vac . The Equation 7.101 appearing here have been used in Ref. [264] to calculate the vibrational–rotational energy levels of H2 O and the transition ampli-

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tudes or Tb ! c matrix element for an inelastic collision from channel b to channel c.4) The calculation of the vibrational–rotational levels has been carried out using a basis set of 47 (or 46) coupled channels (45 or 44 for the bound-state manifold and two for the extra channels). Considering the right-hand side of Equation 7.101 as an inhomogeneous term, the radial wavefunctions Ya of Equation 7.101 may be expressed as [264] 1 Xð Ya ðRÞ ¼ Gaa0 ðR; R0 ; EÞVa0 c ðR0 ÞYc ðR0 ÞdR0 ; ð7:103Þ a0

0

where Gaa0 is the Green’s function corresponding to the left-hand hand side of Equation 7.101a and Yc are solutions of the uncoupled Equation 7.101c with the boundary condition of Equation 7.102. These solutions, regular at R ¼ 0, are given by the following integral equation [269, 270]: 1 ð 2 Yc ðRÞ=R ¼ jl ðkc RÞ þ ð2m1 kc =
h2 Þ jl ðkc R< Þnl ðkc R> ÞVcc ðR0 ÞYc ðR0 ÞR0 dR0 ; 0

ð7:104Þ 0

where gl ðR; R Þ ¼ kc jl ðkc R< Þnl ðkc R> Þ is the Green function corresponding to the operator Hc with R< being the lesser and R> the greater of R and R0 . Here, jl ðkc RÞ and nl ðkc RÞ are, respectively, the spherical Bessel and the spherical Neumann functions of the first kind [271]. Both are particular solutions of the homogeneous counterpart of Equation 7.101c (when Vcc ¼ 0). Using the spectral resolution of a Green’s function, the Green’s function that appears inside the integrand (7.103) can be expressed (in the R-representation) in terms of eigenstates of the closed-state channels (7.94) 0

Gaa0 ðR; R ; EÞ ¼

X Wai ðRÞWa0 i ðR0 Þ i

EEi

1 ð

þ 0

Wa ðE 0 ; RÞWa0 ðE 0 ; R0 Þ dE 0 : E þ E 0

ð7:105Þ

The first term in Equation 7.104 is a summation over all the bound states Wai ðRÞ corresponding to the energies Ei of the original problem (Equation 7.94) and the second term is an integral over the continuum solution (E 0 ) of the same problem. We have so far not dealt with the potential energy matrix elements involving the extra two channels b and c. The theory leaves us here a more or less free choice. The only constraint on these matrix elements is that they must vanish at large R. (A diagonal potential matrix element of realistic strength is Vcc ¼ AexpðaRÞ and correspondingly for the off-diagonal ones Vac ¼ A0 expða0 RÞ). The kinetic energy E in the extra channels should also be chosen to be small so that the radial wavefunction should not be too oscillatory.5)

4) The T matrix element or precisely its energy dependence is applicable, for example, for the problem of determining the exact bound state energies Ei. 5) Wc oscillates very rapidly as kl becomes large.

7.6 Electronic Predissociation of the 2B2 State of H2O þ

The above calculations are quite extensive and highly specialized. To explore this problem further would take us well beyond the scope of this book, for we should have to review the entire area of quantum calculations of the three-dimensional reactive scattering problem. The reader who wishes to acquire even an elementary working knowledge to this methodology will find an extensive literature (of dedicated journals and books) on all aspects of this subject [272–280]. Thus, the present treatment of determining the radial function Wnl;E must be viewed as a brief introduction to this subject; this introduction may help the reader to be acquainted with the formalism.

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8 Multidimensional Franck–Condon Factor The consideration, which we will now in Chapter 8 devote to the rather old-fashioned Franck–Condon (FC) factor or the FC integral, may be greeted by the reader with some initial skepticism. These reservations and skepticism will be dispelled, at the latest, once we have explored a few applications in which we employ the FC integral. To obtain a complete description for even the simplest molecules, we wish to generalize our treatment to include several nonseparable normal modes. This means that the generating function, by which the factors will be derived, is a multidimensional function that depends on several “dummy variables.” The generating function incorporates the transformation (rotation) of both the normal mode coordinates between the initial and final electronic states and their frequency distortions. Both of them scramble the occupations of the normal modes, leading to unusual distributions. At certain values of the angle of rotation, the distribution may reduce to only a few modes or even to a single mode. Mathematically, this selectivity can be described in terms of normal mode displacement parameters generated by the rotation matrix. These normal mode displacements, as well as the set of crossed frequency parameters due to mode mixing, have no counterparts in the parallel mode approximation and are the reason that polyatomic FC integrals are of formidable analytical complexity. This complexity is analyzed with particular attention to the resemblance between the FC factors and the intramolecular distributions (IDs) used in our previous chapters. As an illustration of this complexity, calculations of the scattered intensities in the resonance Raman process as well as the study of sequential two-photon processes are presented. These show a strong dependence on the rotation angles.

8.1 Multidimensional Franck–Condon Factors and Duschinsky Mixing Effects 8.1.1 General Aspects

In Chapters 3 and 4, a multidimensional ID was derived that most often describes the complex problems of molecular spectroscopy such as nonradiative and any radiative

j 8 Multidimensional Franck–Condon Factor

226

(allowed and vibronically induced) transitions. The ID enters into the emission or absorption spectral line shape (SLS) directly as the square of the vibrational overlap integral convoluted with a Lorentzian profile. This convolution that creates a band shape function is weighted by electronic coupling factor. The Lorentzian, being a function of the variable frequency v, is centered around the individual vibronic components involved in the transition. The relative intensity distribution of these components is then governed by the values of the ID. In the case of radiationless transitions, the multidimensional ID appears in the decay rate expression. Here, it is coupled with a Lorentzian (to ensure energy conservation in the transition), but now v is fixed and equal to the energy gap V. The latter must be bridged by final-state vibrational quanta. This leads to a weighted sum (the weighting factors being the square of the electronic matrix elements) of density of state weighted vibrational overlaps (DSWVO), each of which includes all ground- and excited-state vibrational levels subject to the energy conservation constraint. Although the ID, as a vibrational overlap factor, is related to the Franck–Condon factor, a closer examination reveals several reasons for performing the former, which we enumerate below. First, the ID for N (separable or nonseparable) vibrational modes is derived from the knowledge of generating function GN . The Fourier transformation of GN is a DSWVO (or a line shape function), the latter of which appears in the decay rate expression (or in the SLS for any radiative transition). Thus, by expanding the generating function in a multiple power series (the coefficients of which are given by the values of an ID for several occupation number sets), the Fourier transformation of GN and therefore the aforementioned spectroscopic quantities can be calculated. This is what makes the ID so manageable. Second, since the ID for an arbitrary set of vibrational occupation numbers, m1 ; m2 ; . . . ; mN ; n1 ; n2 ; . . . ; nN , can be obtained by equating coefficients of the 2N expansion variables in the infinite multiple power series of GN , the IDs can be easily convoluted with themselves through the use of simple rules for multiplication of power series. In this way, some addition theorems concerning the normal mode shifts and the order (dimension) of the IDs can be derived (see Sections 4.1.1 and 4.2.2). So the convolution of two or more IDs of given orders leads to a joined ID, the order of which is equal to the sum of the orders of the factors. This important aspect of the ID is of practical use in treating dense manifolds of vibronic states that occur in the statistical limit. Third, as a consequence of the foregoing, the use of IDs is suitable in investigating the dependence of the electronic matrix element for radiationless transition on the nuclear coordinates. This problem can be solved, as has been shown in Chapter 5, by considering the matrix element as one that of an operator that depends upon both electronic and nuclear displacements and by introducing a q-centroid approximation for the electronic factor. The latter is obtained as an average with DSWVO factor. The familiar Condon approximation can be so improved as to write the whole matrix element as a product of a vibrational overlap integral and an electronic factor, the latter being evaluated at some q-centroid for the nuclear positions. It is not clear what the most useful applications of such procedure will turn out to be. We have already mentioned the determination of the q-centroid for evaluating

8.1 Multidimensional Franck–Condon Factors and Duschinsky Mixing Effects

matrix elements. There are applications to the theory of radiative and nonradiative transitions. In addition, a number of applications have been given in Chapter 7. A similar study of the effect of anharmonicity in the vibrational motion is not too far behind. For historical reasons, there is a great deal of satisfaction in solving the multimode problem (with high vibrational degrees of freedom) analytically or at least in reducing it to lower dimensionality in a manner that the problem becomes qualitatively understood. Despite the advantages of using the IDs in a wide range of spectroscopic phenomena, the determination of polyatomic FC factors currently has great appeal [281]. The FC integrals prominently appear in the so-called vibronic theories of Raman and resonance Raman intensities and in studies of time-delayed two-photon processes, the cross sections of which are obtained by adding amplitudes and then squaring (see below). In addition, the FC factors find the widest application in calculations of matrix elements between several molecular eigenstates. Calculations of such factors present some difficulties or problems (as also appears in calculating multidimensional IDs). In polyatomic molecules, this difficulty stems from the high vibrational degrees of freedom. This is because the nuclear motion is inextricably coupled to the motion of the valence electrons. Thus, upon excitation the molecular skeleton is deformed, displacing the nuclei from the ground-state equilibrium positions. Hence, the vibrational wavefunction will be deformed as well in a manner that can be quite difficult to deduce with any precision. One copes with this problem by making the so-called adiabatic approximation. In this approximation, the normal modes of vibrations in an electronic excited state (after separation from the electronic motion) are displaced and rotated relative to those of the electronic ground state. Since this normal mode rotation (and consequently mixing) cannot always be ignored, we must deal from the start with a multidimensional FC integral that is of formidable analytical complexity and that cannot be represented in many cases as a product of integrals of lower dimensionality that are easier to calculate. One of the earliest attempts at a quantitative analysis of the problem was put forward by Coon, DeWames, and Loyd [282] in an approximate method for calculating two-dimensional integrals specifically for nonlinear triatomic systems. Later, Sharp and Rosenstock [283] developed a more general approach in which a generating function was derived and the FC factors are obtained as coefficients in the expansion of this function in a multiple power series of dummy variables. In this way, they were able to determine relative probabilities of transitions starting from the vibrationless level (i.e., having zero vibrational quanta in all of the normal modes) of the ground (initial) electronic state of linear, symmetric, triatomic molecules to overtones as well as to some low combination levels of the final electronic state. The generating function technique was also used by Karplus and Warshel [284, 285] in their study of vibronically assisted electronic spectra of conjugated hydrocarbons and by Doktorov et al. [286]. The latter authors have employed the coherent-state method of Glauber to derive a generating function and recurrence equations to compute the vibrational overlap integral. However, the coefficients in these recurrence equations are given for only N ¼ 2. The authors

j227

j 8 Multidimensional Franck–Condon Factor

228

applied their method to analyzing the intensity distribution of a single mode progression in the 1 A1g ! 1 B2u electronic transition in benzene. Another approach for calculating polyatomic FC factors has been reported by Faulkner and Richardson [287]. As an alternative to the previous methods, they used contact transformation perturbation operator of the form U ¼ expðiSðp; qÞÞ to construct the vibrational wavefunctions of an excited electronic state in terms of the vibrational wavefunctions of the ground state. Thus, the calculation of polyatomic FC factors is reduced to the evaluation of vibrational matrix elements exclusively within the ground-state vibrational manifold. In the opinion of the authors, this method has only limited applicability due to the slow convergence of the perturbation expansion of Sðp; qÞ in power series. Another method developed by the same authors used a linear transformation of the normal coordinates to remove the Duschinsky rotation. The multidimensional FC integrals are then written as sums of factorizable integrals in the new intermediate nuclear coordinates. This method having markedly better computational efficiency than the perturbation method is restricted to the zero-temperature limit (i.e., when one of the vibrational wavefunctions in the FC integral is vibrationless). Recently, Kikuchi et al. based on Sharp–Rosenstock’s method have developed a computational algorithm to calculate MFC factors including mode mixing. As an application, the authors calculate the fluorescence spectrum of SO2 , originating from the lowest vibrational level of the excited electronic state, taking as input parameters the familiar G and F matrices of Wilson [288]. Another approach given in Ref. [289] is quite different. Here, one seeks closed-form solutions. In the sections that follow, we present a more elaborate analysis of mode mixing and give explicitly general expressions for calculating multidimensional FC integrals, from which precise qualitative and quantitative results can be extracted. The analysis is similar to that in Section 4.4. We shall take from the beginning an affine (not necessarily orthogonal) relation between the normal modes of the excited- and ground-electronic states and derive a multidimensional generating function FN . The latter quantity includes all mode mixing effects and frequency distortions. At this stage, we pause in our derivation to explain the physical phenomena governed by mode mixing. Subsequently, with the help of this generating function, a set of 2N recurrence equations (i.e., one recurrence equation for each of the excited- and ground-state occupation numbers m1 ; m2 ; . . . ; mN ; n1 ; n2 ; . . . ; nN ) is derived that offers a very convenient and efficient way of calculating polyatomic FC integrals. 8.1.2 Derivation

We consider two electronic states, l (lower or ground) and s (excited), and a set of N vibrational modes associated with these states. The harmonic vibrational wavefunctions in these states are denoted as before by xln ðql Þ and xsm ðqs Þ, respectively, where n ¼ fnm g represents the collection of vibrational occupation numbers in the normal coordinates, ql , of the lower electronic state, and m ¼ fmm g represents the collection of the vibrational occupation numbers in the normal coordinates, qs , of the excited

8.1 Multidimensional Franck–Condon Factors and Duschinsky Mixing Effects

electronic state. The relation between both sets of normal coordinates qs and ql is given by qs ¼ Wql þ k1 2


N ;

ð8:1Þ

or explicitly by Equation 4.57. The vector k1 2


N describes the shift of the coordinates qs measured in a coordinate system with base vectors qlm ðm ¼ 1; 2; . . . ; NÞ. In the notation established above, the FC integral associated with an electronic transition, jsmi , jlni, is given by
JN

m1 ; m2 ; . . . ; mN n1 ; n2 ; . . . ; nN



1 ð

¼

xsm ðqs Þxln ðql Þdql ;

ð8:2Þ

1

where the vibrational wavefunctions xsm ðqs Þ and xln ðql Þ are expressed explicitly as an N-fold product of one-dimensional harmonic oscillator functions (Equation 3.1). If we assume that the transformation (8.1) is affine, then the right-hand side of Equation 8.2 should be multiplied by the normalization constant of xsm ðqs Þ, ðdet WÞ1=2 .1) To derive the generating function for JN, we start from the well-known generating function for the Hermite polynomials X1 k

k!

  Hk ðxÞuk ¼ exp u2 þ 2ux ;

ð8:3Þ

where u is a dummy variable. With the help of Equations 1.49 and 8.3, we can write for every mode m in the excited electronic state   1  1=2 mm  1=4 X  1=2  1=2 1 2 wm xsmm bsm qsm ¼ exp wm2 þ 2wm bsm qsm  bsm qsm p=bsm 2mm =mm ! 2 mm ð8:4aÞ

and analogous expressions for the electronic state jli   1   1=4 X  1=2  1=2 nm 1=2 1 2 p=blm 2nm =nm ! zm xsnm blm qlm ¼ exp z2m þ 2zm blm qlm  blm qlm : 2 nm ð8:4bÞ

Now, multiplication of all 2N equations (8.4) with themselves and subsequent integration with respect to ql yields 1 X

1 X

w1m1 w2m2


wNmN zn11 zn22


znNN

m1 ;m2 ;...;mN ¼0n1 ;n2 ;...;nN ¼0

 ð2ðm1 þ m2 þ


þ mN þ n1 þ n2 þ


þ nN Þ =m1 !m2 !


mN !n1 !n2 !


nN !Þ1=2 1) TheÐ normalization constant C is determined from the relation (8.1) by the requirement that C 2 jxsm ðqs Þj2 dN ql ¼ 1.

j229

j 8 Multidimensional Franck–Condon Factor

230


 JN

m1 ; m2 ; . . . ; mN n1 ; n2 ; . . . ; nN



0 1N=2 ð ð N  Y 1=4 1 1 bsm blm


dN ql ¼@ A p 1 m¼1

2

3  1 s t s s 1 l t l l  t s1=2 s 1=2 l t l  exp4ðw w þ z zÞ þ 2 w b q þ z b q  ðq Þ b q  ðq Þ b q 5: 2 2 t

t

ð8:5Þ s

l

Here, w and z are N-dimensional column vectors (as are q and q ) of the dummy variables and bs and bl are diagonal matrices with elements ðbs Þmm ¼ bsm and ðbl Þmm ¼ blm , respectively. In writing Equation 8.5, we suppressed the parametric dependence of JN on the spectroscopic parameters. The integrand, as it is written in Equation 8.5, contains different sets of coordinates qs and ql . Therefore, to evaluate the integral, we utilize Equation 8.1 and rewrite the right-hand side of Equation 8.5 in the form " ð1 ð 1  Y  1 N=2 s l 1=4 N l bm bm FN ¼ ð1=pÞ


d q exp wt w þ zt z ðql Þt xql 2 1 m¼1 þ



yt1

þ yt2

þ yt3

where

x ¼ WT bs W þ bl ¼



k

# N N X 2 1X s ðmÞ s1=2 ðmÞ q b k þ 2 bm k12


N wm ; 2 m¼1 m 12


N m¼1 l

N X 2 s wm1 bm þ bl1 m¼1 N X m¼1

wm2 wm1 bsm .. .

N X m¼1

wmN wm1 bsm

P

s m wm1 wm2 bm






l 2 s m wm2 bm þ b2






P

P

s m wm1 wmN bm

P

s m wm2 wmN bm

.. .

P

s m wmN wm2 bm

ð8:6Þ

P

.. .

s l 2 m wmN bm þ bN






k

ð8:7Þ

and yt1

yt2

¼

N X m¼1

¼2

ðmÞ wm1 bsm k12


N ; 

N X m¼

1=2 wm1 bsm wm ;

X m

X m

ðmÞ wm2 bsm k12


N ; . . . ;

1=2 wm2 bsm wm ; . . . ;

1=2

1=2 1=2 yt3 ¼ 2 bl1 z1 ; bl2 z2 ; . . . ; blN zN :

X m

X m

! ðmÞ wmN bsm k12


N

;

ð8:8aÞ

! 1=2 wmN bsm wm

;

ð8:8bÞ

ð8:8cÞ

8.1 Multidimensional Franck–Condon Factors and Duschinsky Mixing Effects

Note that the matrix x, Equation 8.7, resembles those in Section 4.4 with the exception that now the dummy variables disappear in the matrix (8.7). The integral (8.6) is a multiple Gaussian integral over ql that can be carried out using the general formula (4.63). Thus, we find FN ðw1 ; w2 ; . . . ; wN ; z1 ; z2 ; . . . ; zN Þ ¼ FN ð0; . . . ; 0; 0; . . . ; 0Þ

ð8:9Þ

 exp½D11 þ D22 þ D12 þ D1 þ D2 þ wt wzt z;

where D11 ¼

1  t 1  y x y2 2wt w; 2 2

ð8:10aÞ

D22 ¼

1  t 1  y x y3 ; 2 3

ð8:10bÞ

D12 ¼

1  t 1  1  t 1  y x y3 þ y3 x y2 ; 2 2 2

ð8:10cÞ

D1 ¼

N X 1=2 ðmÞ 1  t 1  1  t 1  bsm k12


N wm ; y1 x y2 þ y2 x y1 þ 2 2 2 m¼1

ð8:10dÞ

D2 ¼

1  t 1  1  t 1  y x y3 þ y3 x y1 ; 2 1 2

ð8:10eÞ

and N=2

FN ð0; . . . ; 0; 0; . . . ; 0Þ ¼ 2

N  Y m¼1

¼ 2N=2

N  Y m¼1

2

1=4  1 1 bsm blm exp4 y1t x 1 y1  h

bsm =blm

2

1=4 exp 12 A=B

2 m¼1

i

ðBÞ1=2

3 N X 2 s ðmÞ bm k12


N 5

; ð8:11Þ

with A¼

X 1 j1 N

þ

ð1 2


NÞ2

Dj1

X

þ

X

X

ði i


iN1 Þ2

1i1 m, then
J1

! 
1=2

 2 Dm nm nm Dm m! 1 2 m p ffiffi ffi ; ;D ;0 ¼ exp  Dm Lm  n m 2 n! 4 2

where Dm ¼ bsm 1=2 km (see Equation 8.14c).

ð8:57bÞ

j259

Appendices

j261

Appendix A: Some Identities Related to Green’s Function

A.1 The Green’s Function Technique

To calculate the transition probability, we need to know the energy dependence of the resolvent operator or Green’s function in a manifold of states or in the representation appropriate to a set of wavefunctions of the zero-order Hamiltonian H0 jsi ¼ Es jsi;

ðA1Þ

where the behavior of the states jsi depends on the type of problem to be studied. The resolvent of the operator H ¼ H0 þ V (which we imagine to be decomposed into an “unperturbed part H0 and a “perturbation” V) is given by the object GðzÞ ¼

1 ; z
H

ðA2Þ

considered as a function of the complex variable z. It is a bound operator in the Hilbert space1 for every value of the variable z except at the eigenvalues of H. When the eigenvalues of H are discrete, Hjai ¼ Ea jai, the singularities of Ga ¼ hajGðEÞjai are poles on the E-axis at the position of these eigenvalues. Then Ga is analytic everywhere in the complex E-plane, except the poles on the real E-axis. When H has a continuum spectrum of eigenvalues, Hjli ¼ El jli, the situation is quite different. Here, let us suppose that the eigenvalues El of H are continuous over the range EM < E < 1. Then, Ga ðEÞ is analytic everywhere in the complex E-plane, except along the cut extending from EM to plus infinity on the real axis. This is illustrated in Figure A.1. The Green’s operator is thus defined as limit of the resolvent when z approaches to real axis: G ðEÞ ¼ limþ e!0

1 : E
H  ie

ðA3Þ

The quantity e is a very small (and positive) number that determines the position of the poles of GðEÞ with respect to the real axis, but which we always take to zero in the 1 That is, jj Gg jj  M jj g jj for all vectors g belonging to the Hilbert space. Here jj g jj ¼ ðg; gÞ1=2 is the norm of g and M is a constant.

j Appendix A: Some Identities Related to Green’s Function

262

ε

z-plane

EM Single poles of G(z)

E

Branch cut of G(z)

Figure A.1 Analytic behavior of GðzÞ ¼ ðz
HÞ
1 . The singular points or lines are on the real z-axis (when H is Hermitian) and provide information about the eigenvalues and eigenfunctions of H.

end. These conclusions concerning the domain of analyticity of Ga ðEÞ are important and provide a very useful basis for discussing decay phenomena. Let us now suppose the state jsi to be represented by the approximate Hamiltonian H0 of Equation A1. If, for example, the state jsi and another state jbi of the manifold (A1) between the transitions are isolated from the remaining states of H0 , the problem is relatively simple. However, if the states are not isolated, the calculation of the transition probability is not simple. To reduce the complexity and to make clear the states of the system to be included in the calculation of the matrix elements of the Green’s function, we use the Feshbach projection operator formalism. In this formalism, a projection operator associated with a model function or a set of states, the so-called close coupled states, is introduced X P¼ jsihsj; ðA4Þ s

where the sum over s depends on the problem to be considered. Since P is a projection operator, it is such that P 2 ¼ P:

ðA5Þ

Let us now introduce a second operator defined by the relation Q ¼ 1
P

ðA6Þ

for the complementary part of the functional space spanned by P and which we can call the Q space. Q is also a projection operator, since Q 2 ¼ ð1
PÞ2 ¼ I
2P þ P 2 ¼ I
P ¼ Q:

ðA7Þ

From Equations A5 and A7, we also deduce that the operators P and Q are orthogonal, so Q  P ¼ P  Q ¼ 0:

ðA8Þ

In the language of these projection operators, it is clear that for the calculations of the above-mentioned matrix elements, we need to know the projections of the Green’s function QGP and PGP. The first projection gives us the matrix element of G

Appendix A

between a state outside the set of states and a state inside the set of states, which we wish to treat on an equal basis. The second projection operator connects states that we treat on an equal footing. To derive the above projections of G, we start with the identity (see Equation A2) ðE
HÞGðEÞ ¼ 1:

ðA9Þ

Multiplying this identity from the right by P and introducing in (A9) the relation P þ Q ¼ 1, we get ðE
HÞPGðEÞP þ ðE
HÞQGðEÞP ¼ P

ðA10Þ

and then taking the left projection of this equation onto first P and then Q, we have PðE
HÞPðPGPÞ
PHQðQGPÞ ¼ P;

ðA11aÞ


QHPðPGPÞ þ QðE
HÞQðQGPÞ ¼ 0:

ðA11bÞ

Here we have introduced the identities P2 ¼ P and Q 2 ¼ Q. Assuming that the operator QðE
HÞQ possesses an inverse in the space spanned by Q, Equations A11a and A11b may be formally decoupled by first solving (A11b) for QGP, QGP ¼ ðE
QHQÞ
1 QHPðPGPÞ;

ðA12Þ

and inserting this result into Equation A11a, so that ½ðE
PHPÞ
PHQðE
QHQÞ
1 QHPPGP ¼ P:

ðA13Þ

Equation A13 may be rewritten as ½E
PH0 P
PRPPGP ¼ P;

ðA14aÞ

PGP ¼ ½E
PH0 P
PRP
1 P;

ðA14bÞ

or

for the time evolution of P states. The operator R introduced in Equations A14 is the so-called level shift operator given by [80,81] R ¼ V þ VQðE
QHQÞ
1 QV;

ðA15Þ

which is of real interest. The operator has complex diagonal matrix elements, the imaginary parts of which are of prime interest to nonradiative decay processes and will play an important role in our consideration. Similarly, from Equation A12, we have for the time evolution of the P ! Q excitation QGðEÞP ¼ ðE
QHQÞ
1 QHPðE
PH0 P
PRPÞ
1 :

ðA16aÞ

Equation A16a may be written in a more convenient form (for calculating transition probabilities) as QGðEÞP ¼ ðE
QH0 QÞ
1 QRðEÞPðE
PH0 P
PRPÞ
1 :

ðA16bÞ

j263

j Appendix A: Some Identities Related to Green’s Function

264

This can be verified by using the relationship ðE
QH0 QÞ
1 QRP ¼ ðE
QHQÞ
1 QVP ¼ ðE
QHQÞ
1 QHP;

ðA17Þ

which follows from the definition of R, QRP ¼ QVP þ QVQðE
OHQÞ
1 QVP ¼ ðE
QH0 QÞðE
QHQÞ
1 QVP: ðA18Þ

It is interesting to note that the operator appearing in Equations A14 and A16, Heff ¼ PH0 P þ PRP;

ðA19Þ

is the effective Hamiltonian for the states of P. A.2 Evaluation of the Diagonal Matrix Element of Gss

To understand how the result (6.24) and (6.25) is achieved, we turn to the derivation of the last term appearing in Equation 6.21 for a system, as was the case in Section 2.1: D



E ys
ðE
PH0 P
PRðEÞPÞ
1
ys ¼

1
s
Rss ; E
E

ðA20Þ

where Rss is understood that R is to operate on the state ys . It is easy to see that Rss satisfies the equation X X ~ ll0 Vl0 s þ ~ k^e;k0^e0 Wk0^e0 ;s ¼ RI þ RII ; Rss ðEÞ ¼ Vsl G Ws;k^e G ðA21Þ ss ss l;l0

k;^e

with ~ ¼ ðE
QHQÞ
1 : G

ðA22Þ

Making use of the Dyson equation, we have ~ ~ ¼ ðE
QHQÞ
1 ¼ ðE
H0
QVQÞ
1 ¼ G0 þ G0 QVQ G; G

ðA23Þ

where G0 ¼ ðE
H0 Þ
1 ;

and the interaction responsible for the transition ys ! yl ; y0;k;^e is V ¼ Hv þ Hint . Let us first consider the first term of (A21). For this case, we write ~ ll0 ¼ G

X dll0 1 ~ k^e;l0 þ Wl;k^e G E
El þ ie E
El þ ie k;^e

ðA24aÞ

and ~ k^e;l0 ¼ G

1 ~ ll0 ; Wk^e;l G E
Ek^e þ ie

ðA24bÞ

Appendix A

j265

where we have shifted the pole of GðEÞ below the real E-axis by an infinitesimal positive quantity e. Substituting (A24b) into (A24a) gives ~ ll0 ¼ G

X dll0 1 1 ~ ll0 þ Wl;k^e Wk^e;l G E
El þ ie E
El þ ie k;^e E
Ek^e þ ie

and after little manipulation dll0

~ ll0 ¼ G

E
El þ ie




2 ; P
Wl;k^e
k^e E
Ek^e þ ie

ðA25Þ

where Ek^e ¼ E0 þ kc. Going back to the first term in (A21) and letting e ! 0 þ , the expression for RIss as a function of E becomes "

# X X X
Wl;k^e
2 2 I ¼ Rss ðEÞ ¼ jVsl j = E
El
jVsl j2 =½E
El
Dl þ iCl ðEÞ=2; E
E k^e l k;^e l ðA26aÞ

or RIss ðEÞ ¼

X l


lÞ iX i jVsl j2 ðE
E jVsl j2 Cl ðEÞ I
2 2 2 2 ¼ Ds
2 Ds ðEÞ;

2 ðE
E l Þ þ ½Cl ðEÞ=2 ðE
E l Þ þ ½Cl ðEÞ=2 l

ðA26bÞ

where Re RIss ¼ DIs ðEÞ and Im RIss ðEÞ ¼
ð1=2ÞDs ðEÞ [81]. In Equation A26b, the
l ¼ El þ Dl . In deriving RI , we have tacitly level shift term Dl has been included in E ss assumed that the states yl have been allowed to have very small widths Cl due to highly forbidden radiative transition to the ground state y0 . In practice, the width Cl may be modified from intermolecular perturbation with the medium. The radiation matter interaction Hint of the molecule with the radiation field is “weak” in the sense that the virtual transition involving the emission of photons contribute very little to the decay rate. This means that we may evaluate the matrix II element RII ss in lowest order perturbation theory, to give Rss ¼
iCs =2, where X
Xð



Ws;k^e
2 dðEs
E0
kcÞ ¼ 2p
Ws;k^e
2 d r ðkcÞ; Cs ¼ 2p ðA27Þ ph ^e

^e

Es
E0 ¼ hns0  kc;

ðA28Þ

is the energy of the emitted photon and d rph ðkcÞ ¼

k2 ð2pÞ3 c

VdVk

ðA29Þ

is the number of photon states of a given polarization in the volume V per unit energy directed into the solid angle dVk . The real level shift DII s gives exceedingly small

j Appendix A: Some Identities Related to Green’s Function

266

contribution to the energy Es and may ordinarily be neglected [81]. This permits us to adopt Equation A20 to Equation A.12, which is equivalent to Gs ðEÞ ¼

1
s þ iðDs þ Cs Þ=2 ; E
E

ðA30Þ

where the real parts of Re Rss ¼ DIs þ DII s have been absorbed in the term
s ¼ Es þ DI þ DII . In this case, Gs ðEÞ is analytic everywhere in the complex EE s s plane, except the pole (poles) on the half plane of complex E. The question of the number of poles depends on whether Ds ðEÞ þ Cs ðEÞ (which itself is a function of E) is a smooth function of E or not.

j267

Appendix B: The Coefficients of the Recurrence
ð1Þ ! ð2Þ m1 ; m2

D12 D12 b1 b2 Equation for I2 ;
n1 ; n2
Dð12Þ Dð12Þ b12 b21 1

2

The expressions of coefficients of the recurrence Equations 4.50, 4.51, 4.54 and 4.55 may be obtained in a straightforward manner by expansion of the functions (4.47) through (4.49). We begin with the function 1)

B21 B2 Defining a0 pq;rs ¼

X

am1 m2 ;n1 n2 bm0 1 m0 2 ;n0 1 n0 2 ;

p; q; r; s ¼ 0; 1; 2;

ðB1Þ

m1 þ m0 1 ¼p m2 þ m0 2 ¼q n1 þ n0 1 ¼r n2 þ n0 2 ¼s

we may obtain the coefficients cij;kl in the expansion of B21 B2 by summing X cij;kl ¼ a0 m1 m2 ;n1 n2 am0 1 m0 2 ;n0 1 n0 2 ; i; j; k; l ¼ 0; 1; 2; 3: ðB2Þ m1 þ m0 1 ¼i m2 þ m0 2 ¼j n1 þ n0 1 ¼k n2 þ n0 2 ¼l

2)

B1 ðq=qw1 ÞðB1 B2 Þ Substituting Equations 3.33 and 3.34 and the expansion of Equation 4.47 to the identity B1

q q qB1 ðB1 B2 Þ ¼ ðB2 B2 Þ
B1 B2 qw1 qw1 1 qw1

and differentiating, rearranging, and comparing terms of the same powers in w1 w2 z1 z2 , we obtain X ð1Þ gij;kl ¼ ði þ 1Þci þ 1;j;kl
a0 i m2 ;n1 n2 a1m0 2 ;n0 1 n0 2 ; i ¼ 0; 1; 2; ðB3Þ m2 þ m0 2 ¼j n1 þ n0 1 ¼k n2 þ n0 2 ¼l

j Appendix B: The Coefficients of the Recurrence Equation

268

or written explicitly ð1Þ

g0j;kl ¼ c1j;kl


X m;n

ð1Þ

g1j;kl ¼ 2c2j;kl
ð1Þ

g2j;kl ¼ 3c3j;kl


a0 0m2 ;n1 n2 a1m0 2 ;n0 1 n0 2 ;

X m;n

X m;n

ðB3aÞ

a0 1m2 ;n1 n2 a1m0 2 ;n0 1 n0 2 ;

ðB3bÞ

a0 2m2 ;n1 n2 a1m0 2 ;n0 1 n0 2 ¼ 2c3j;kl ;

ðB3cÞ

where the indices ðm; nÞ run over a set of m2 ; n1 ; n2 ; m0 2 ; n0 1 ; n0 2 values as indicated in (B3).   qB1 qA1 3) A1
B1 B2 qw1 qw1 If we multiply the expansion 2 X qB1 qA1 ð1Þ q A1
B1 ¼ s w zr zs ; qw1 qw1 q;r;s¼0 0q;rs 2 1 2

q; r; s ¼ 0; 1; 2;

where ð1Þ

s0q;rs ¼

X

ða1m2 ;n1 n2 d0m0 2 ;n0 1 n0 2
a0m2 ;n1 n2 d1m0 2 ;n0 1 n0 2 Þ;

ðB4Þ

m2 þ m0 2 ¼q n1 þ n0 1 ¼r n2 þ n0 2 ¼s

by the function (3.34), we obtain the expression (4.49) with ð1Þ

dij;kl ¼

X

s0m2 ;n1 n2 bim0 2 ;n0 1 n0 2 ;

i ¼ 0; 1:

ðB5Þ

0

m2 þ m 2 ¼j n1 þ n0 1 ¼k n2 þ n0 2 ¼l

The coefficients appearing in Equation 4.51 can be treated similarly by differentiating the function in 1, 2 and 3 in respect to w2 . Alternatively, it is simpler to invoke the symmetry of I2 in respect to the interchange 1 , 2 (Equation 3.66), which yields the relationships Fcij;kl ¼ cji;lk ¼ cij;kl ð1Þ

Fgij;kl ¼ ði þ 1Þcj;i þ 1;lk


X 0

m2 þ m 2 ¼j 0

n1 þ n 1 ¼k n2 þ n0 2 ¼l

and by renaming i , j and k , l,

a0 m2 i;n2 n1 am0 2 1;n0 2 n0 1

ðB6Þ

Appendix B: The Coefficients of the Recurrence Equation

X

ð1Þ

Fgij;kl ¼ ð j þ 1Þcij þ 1;kl


ð2Þ

a0 m2 j;n2 n1 am0 2 1;n0 2 n0 1 ¼ gij;kl ;

j ¼ 0; 1; 2;

ðB7Þ

m2 þ m0 2 ¼i n1 þ n0 1 ¼l n2 þ n0 2 ¼k

ð1Þ

that is, the only distinction to gij;kl is that now the second index j plays the same role as ð1Þ the first index i of gij;kl . Similarly, we have X ð2Þ ð2Þ dij;kl ¼ sm1 0;n1 n2 bm0 1 j;n0 1 n0 2 ; j ¼ 0; 1; ðB8Þ m1 þ m0 1 ¼i n1 þ n0 1 ¼k n2 þ n0 2 ¼l

where

X

ð2Þ

sp0;rs ¼

ðam1 1;n1 n2 dm0 1 0;n0 1 n0 2
am1 0;n1 n2 dm0 1 1;n0 1 n0 2 Þ:

ðB9Þ

m1 þ m0 1 ¼p n1 þ n0 1 ¼r n2 þ n0 2 ¼s

The counterparts of (B3) and (B5) for the zero-temperature limit (see Equations 4.54 and 4.55) are similarly defined: X ð1Þ g00;kl ¼ ðk þ 1Þc00;k þ 1;l
a0 00;kn2 a00;1n0 2 ; k ¼ 0; 1; 2; l ¼ 0; 1; 2; 3; n2 þ n0 2 ¼l

ðB10Þ X

ð1Þ

d00;kl ¼

n2

þ n0

2 ¼l

ð1Þ

s00;0n2 b00;kn0 2 ;

k ¼ 0; 1;

ðB11Þ

where X

ð1Þ

s00;0s ¼

ða00;1n2 d00;0n0 2
a00;0n2 d00;1n0 2 Þ;

ðB12Þ

n2 þ n0 2 ¼s

and finally ð2Þ

g00;kl ¼ ðl þ 1Þc00;k;l þ 1
X

ð2Þ

d00;kl ¼

with ð2Þ

s00;r0 ¼

n1

þ n0

1 ¼k

X n1 þ n0 1 ¼r

ð2Þ

X

a0 00;n1 l a00;n0 1 1 ;

ðB13Þ

l ¼ 0; 1;

ðB14Þ

n1 þ n0 1 ¼k

s00;n1 0 b00;n0 1 l ;

ða00;n1 1 d00;n0 1 0
a00;n1 0 d00;n0 1 1 Þ:

ðB15Þ

j269

j271

Appendix C: The Coefficients of the Recurrence
!
m1 ; m2 ; . . . ; mN
i1 i2


ip Equations for IN g; fbsm ; blm g
fk n1 ; n2 ; . . . ; nN
j1 j2


jq Now we express the various c 0 s; g 0 s; and d0 s appearing in the recurrence equations in Section 4.4.6 in terms of coefficients dm;n and am;n given by Equations 4.65–4.67. For this reason, it seems appropriate to refer to Equations B1–B15 of Appendix B. Based on these equations and employing the corresponding equations involving derivatives with respect to the remaining complex variables wi , we have generally X

cm1 m2


mN ;n1 n2


nN ¼

a0 m01 m02


m0N ;n01 n02


n0N am001 m002


m00N ;n001 n002


n00N ;

m01 þ m001 ¼m1 ;...;m0N þ m00N ¼mN n01 þ n001 ¼n1 ;...;n0N þ n00N ¼nN

mi ; ni ¼ 0; 1; 2; 3;

ðC1Þ

where a0

X

m01 m02


m0N ;n01 n02


n0N ¼

am001 m002


m00N ;n001 n01


n00N bm0001 m0002


m000N ;n0001 n0002


n000N ;

0 00 000 0 m001 þ m000 1 ¼m1 ;...;mN þ mN ¼mN 0 00 000 0 n001 þ n000 1 ¼n1 ;...;nN þ n1 ¼nN

m0i ; n0i ¼ 0; 1; 2:

ðC2Þ

Further, ðiÞ

gm

1




mi1 kmi þ 1


mN ;n1


nN

 m01

X

þ m001 ¼m1 ;...;m0i1

¼

ðk þ 1Þcm1


mi1 k þ 1mi þ 1


mN ;n1


nN

a0 m01


m0i1 km0i þ 1


m0N ;n01 n02


n0N am001


m00i1 1m00i þ 1


m00N ;n001 n002


n00N ;

þ m00i1 ¼mi1

m0i þ 1 þ m00i þ 1 ¼mi þ 1 ;...;m0N þ m00N ¼mN n01 þ n001 ¼n1 ;...;n0N þ n00N ¼nN

k ¼ 0; 1; 2;

and

Transitions in Molecular Systems. Hans J. Kupka Copyright Ó 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-41013-2

ðC3Þ

j Appendix C: The Coefficients of the Recurrence Equations

272

ðiÞ

dm

1




mi1 kmi þ 1


mN ;n1


nN

X

¼

sm0


m0i1 0m0i þ 1


m0N ;n01


n0N ðiÞ bm001


m00i1 km00i þ 1


m00N ;n001


n00N ;

k ¼ 0; 1;

ðC4Þ

m0 þ m00 ¼m n0 þ n00 ¼n

where ðiÞ

sm1


mi1 0mi þ 1


mN ;n1


nN ¼ X ðam01


m0i1 1m0i þ 1


m0N ;n01


n0N dm001


m00i1 0m00i þ 1


m00N ;n001


n00N m0 þ m00 ¼m n0 þ n00 ¼n

am01


m0i1 0m0i þ 1


m0N ;n01


n0N dm001


m00i1 1m00i þ 1


m00N ;n001


n00N Þ

X

and

ðC5Þ

is an abbreviation for the summation written out explicitly in

m0 þ m00 ¼m n0 þ n00 ¼n

Equation C3. ðiÞ ðiÞ A similar derivation gives for the coefficients g0


0;n1


nN and d0


0;n1


nN appearing in Equation 4.97 for the zero-temperature limit. This is defined by the equations ðiÞ

g0


0;n1


ni1 kni þ 1


nN ¼ ðk þ 1Þc0


0;n1


ni1 k þ 1ni þ 1


nN X



a0 0


0;n01


n0i1 kn0i þ 1


n0N a0


0;n001


n00i1 1n00i þ 1


n00N

ðC6Þ

n01 þ n001 ¼n1
;...;n0i1 þ n00i1 ¼ni1 n0i þ 1 þ n00i þ 1 ¼ni þ 1 ;...;n0N þ n00N ¼nN

and X

ðiÞ

d0


0;n1


ni1 kni þ 1


nN ¼

n0

ðiÞ

þ n00 ¼n

s0


0;n0


n0 1

iþ1

0n0i1


n0N b0


0;n1


ni1 kni þ 1


nN ; 00

00

00

00

ðC7Þ

where ðiÞ

s0


0;n1


ni1 0ni þ 1


nN

¼

X n0 þ n00 ¼n

ða0


0;n01


n0i1 1n0i þ 1


n0N d0


0;n001


n00i1 0n00i þ 1


n00N

ðC8Þ

a0


0;n01


n0i1 0n0i þ 1


n0N d0


0;n001


n00i1 1n00i þ 1


n00N Þ:

These results come from Equations 4.64–4.67 by setting w1 ¼ w2 ¼


¼ wN ¼ 0 and differentiating the GF (4.64) with respect to zi , where i ¼ 1; 2; . . . ; N.

j273

Appendix D: Solution of a Class of Integrals In Chapter 5, we encountered a series of integrals for which the solution was simply stated. In this appendix, we shall derive the solutions. We begin with the following: a) The integral Jg ðtÞ. We use Equation 3.18 to write out Equation 5.21 in detail:

Jg ðtÞ

¼

ðbsg blg Þ1=2 ð

h2 Þ p½ð1
wg2 Þð1
z2g Þ1=2 1 ð


1

ð

8 9 < 1 h i=
g þ ðqsg

qsg Þ2 w

1 dqlg d
qlg exp
bsg ðqsg þ
qsg Þ2 w : 4 ;

9 8 < 1 h i= wg þ ðqsg

qsg Þ
w
1 
bsg ðqsg þ
qsg Þ
g ; : 2 8 9 < 1 h i= 2 2  exp
blg ðqlg þ
qlg Þ
zg þ ðqlg

qlg Þ
z
1 g : 4 ; 9 8 < 1 h i = l l 
bg ðqlg þ
qlg Þ
zg
ðqlg

qlg Þ
z
1 g qg : ; : 2

ðD1Þ

Let us now specialize (D1) to the coordinate system given by Equations 3.70 and 3.71. This gives us Jg ðtÞ ¼

ðbsg blg Þ3=2 ð

h2 Þ s 2 e
bg kg w
g pffiffiffi 2 2p½ð1
wg2 Þð1
z2g Þ1=2 1 ð


1

2

3 pffiffiffi s 1 s l 2
g þ bg
zg Þqg
2bg kg w
g qg 5 dqg exp4
ðbg w 2

Transitions in Molecular Systems. Hans J. Kupka Copyright Ó 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-41013-2

j Appendix D: Solution of a Class of Integrals

274

1 ð


1

2

3 1 s l

1 z
1 d
qg exp4
ðbg w q2g 5 g þ bg
g Þ
2

2


g
zg Þq2g
qg
g
zg q3 þ ð
w
1 w g
z
1 zg

w g
g þw p ffiffi ffi 6 pffiffiffi g 6 þ 2kg w
g
zg q2g þ 2kg ð
qg w g
zg

w g
z
1 g Þqg
6
1

1
1

1 2 6 þ ð


w

w

w Þq z z z q g g g g g g 4 pffiffiffi g g
g
z
1
2kg w w
1 q2g

z
1 q3g : g
g
g


# ðD2Þ

Equation (D2) presents a sum of products of Gaussian integrals of the type 1 ð

Pn ¼

xn e
ax

2

þ bx

ðD3Þ

dx;


1

s
g þ blg
zg Þ where pffiffiffi s a ¼ ð1=2Þðbg w
g.
2bg kg w In particular,   b P1 ¼ G; 2a

or

l
1

1 a ¼ ð1=2Þðbsg w zg Þ g þ bg


and

b¼

ðD4aÞ

 P2 ¼

 1 b2 þ 2 G; 2a 4a

" P3 ¼

ðD4bÞ

 #  b 3 3 b G; þ 2a 2 2a2

ðD4cÞ

and so on, where rffiffiffi p G¼ expðb2 =4aÞ: a

ðD5Þ

In such terms, we easily obtain Equation 5.22. Some of the terms in Equation (D2) containing
qng vanishes, if n is an odd number. b) The integral Dg ðtÞ Similarly, we may write Dg ðtÞ ¼

ðbsg blg3 Þ1=2 ði
hÞ 2p½ð1
wg2 Þð1
z2g Þ1=2

s 2

e
bg kg w
g

3 pffiffiffi s 1 s l 2
g þ bg
zg Þqg
2bg kg qg 5  dqg exp4
ðbg w 2
1 2 3 1 ð 1 s l
1
1 2
g þ bg
zg Þ
qg 5  d
qg exp4
ðbg w 2
1 h i qg þ
z
1 q2g : 

zg q2g þ ð
zg

z
1 g Þqg
g
1 ð

2

ðD6Þ

Appendix D: Solution of a Class of Integrals

Only the first and third terms in the bracket of Equation D6 contribute. Then, carrying out the integration using the formula (D4b) twice, we obtain Equation 5.35. c) The integral Bg ðtÞ In a similar fashion and by dropping the vanishing terms, we found Bg ðtÞ

¼




l 1=2 ðbs3 ði
hÞ g bg Þ

s 2

2p½ð1
wg2 Þð1
z2g Þ1=2 1 ð


1 1 ð


1

e
bg kg w
g

2

3 p ffiffi ffi 1 s l s
g þ bg
zg Þq2g
2bg kg w
g qg 5 dqg exp4
ðbg w 2 2

3 1 s l
1
1 2
g þ bg
zg Þ
qg 5 d
qg exp4
ðbg w 2

h i pffiffiffi

1
g qg þ w
g q2g þ 2kg w q2g :  w g


ðD7Þ

This is a relation similar to Equation D6. Carrying out the integration over qg and
qg leads to the result (5.40). ~ m ðtÞ d) The expressions Hm ðtÞ and H To evaluate these integrals, it is convenient to begin with the integral of the form     ð 1 1 dq1 dq2 exp
qt xq þ yt q q ¼ 2pðdet xÞ
1=2 exp yt x
1 y x
1 y: ðD8Þ 2 2 Without bothering to define new symbols, we may use for x ¼ jxij j21 the 2  2 symmetric matrices such as those occurring in the integrand of Equation 3.24. The others q ¼ colðq1 ; q2 Þ and y ¼ colðy1 ; y2 Þ are column vectors, as we have already used in Section 3.2 and in the integral of (3.24). We observe incidentally that Equation D8 is a rather straightforward generalization of Equation 3.25, which we have previously used to derive the generating function G2 ; we shall subsequently see that there is indeed an intimate relationship between G2 ðw1 ; w2 ; z1 ; z2 Þ and Hm ðw1 ; w2 ; z1 ; z2 Þ. To show this, we write according to (3.18) through (3.24) Hm ðw1 ; w2 ; z1 ; z2 Þ ¼

ðbs1 bs2 bl1 bl2 Þ1=2 p2 ½ð1
w12 Þð1
w22 Þð1
z21 Þð1
z22 Þ1=2 h i ð1Þ2 ð2Þ2
1 þ bs2 k12
2Þ w  exp
ðbs1 k12 w 8 2 9 3 1 ð < = 1 1  dq1 dq2 exp4
qt x1 q þ yt q5 pffiffiffi ðqm þ
qm Þ : ; 2 2
1

ð 

2

3 1
5;
t x2 q d
q1 d
q2 exp4
q 2

ðD9Þ

j275

j Appendix D: Solution of a Class of Integrals

276


where x1 and x2 denote the two 2  2 matrices appearing in Equation 3.24. q and q
¼ colð
q1 ;
q2 Þ obtained for the original are the column vectors q ¼ colðq1 ; q2 Þ and q (3.22). With the underqlm ’s and
qlm ’s, m ¼ 1; 2, according to the transformation pffiffiffi standing that in our decomposition of q0 m ¼ ð1= 2Þðqm þ
qm Þ it is only the qm term that gets a contribution, we can carry out the integration by using the formula (D8) twice. This leads to ! ! ð1Þ
1 þ w21 bs2 kð2Þ
2 H1 ðw1 ;w2 ; z1 ; z2 Þ w11 bs1 k12 w 12 w ¼ ð
1Þx1
1 G2 ðw1 ;w2 ;z1 ; z2 Þ ð1Þ H2 ðw1 ;w2 ; z1 ; z2 Þ
1 þ w22 bs2 kð2Þ
w12 bs1 k12 w w 2 12 ðD10Þ

and after a cumbersome algebra, to Hm ðw1 ;w2 ;z1 ; z2 Þ ¼ Hmð0Þ ðw1 ; w2 ; z1 ;z2 ÞG2 ðw1 ;w2 ; z1 ; z2 Þ; m ¼ 1; 2;

ðD11Þ

ð0Þ

with Hm given by Equations 5.27 and 5.28. ~ m ðtÞ ðm ¼ 1; 2Þ, recalling In a quite similar manner, we obtain the expression for H that (see Equations 3.22b and 3.23) !  s    ð1Þ 1 w11 w12 q1 q1 þ
q1 k12 p ffiffi ffi ðD12Þ ¼ þ ð2Þ : qs2 q2 þ
q2 2 w21 w22 k12 We may therefore substitute qsm in Equation D9 instead of qlm and use Equation D12 to obtain
1 ðw1 ; w2 ; z1 ; z2 Þ H

¼

ð0Þ

ð0Þ

½w11 H1 ðw1 ; w2 ; z1 ; z2 Þ þ w12 H2 ðw1 ; w2 ; z1 ; z2 Þ ð1Þ

þ k12 G2 ðw1 ; w2 ; z1 ; z2 Þ; and
2 ðw1 ; w2 ; z1 ; z2 Þ ¼ ½w21 H ð0Þ ðw1 ; w2 ; z1 ; z2 Þ þ w22 H2ð0Þ ðw1 ; w2 ; z1 ; z2 Þ H 1 ð2Þ

þ k12 G2 ðw1 ; w2 ; z1 ; z2 Þ;

ðD13Þ

omitting, as before, terms of
qm that do not contribute. After a little algebra and the use of Equations 5.27 and 5.28, we are led to expressions (5.31) and (5.32). This completes our derivations.

j277

Appendix E: Quantization of the Radiation Field If we view the field in empty space, then by means of the Coulomb gauge we can eliminate the scalar potential from the calculations. Aðr; tÞ is the vector potential, Eðr; tÞ is the electric field, and Bðr; tÞ is the magnetic induction. The fields are given in terms of vector potential by [145] r2 A


1 q2 A ¼ 0; c2 qt2

E¼


1 qA ; c qt

div A ¼ 0;

B ¼ r  A:

ðE1Þ

ðE2Þ

The initial and boundary conditions for A define themselves, from those to which E and B are subordinated. Since the equations that satisfy the vector potential A are linear, A can be expanded in a series of plane waves. This expression is [145] Aðr; tÞ ¼

 X
h 
1=2  ak^e Ak^e ðrÞe
iv^eðkÞt þ ak^e Ak^e ðrÞeiv^eðkÞt : 2vk^e k;^e

ðE3Þ

This is called development by modes. ak^e and ak^e are dimensionless complex numbers. The sum on k here extends over all plane wave momentum states k in the box V. The sum on ^e extends over the two allowed directions of polarization of a photon having a momentum k. The functions Ak^e satisfy the equations r2 Ak^e ðrÞ þ

v^2e ðkÞ c2

Ak^e ðrÞ ¼ 0 ð
hve^ðkÞ ¼ ckÞ;

ðE4Þ

r  Ak^e ðrÞ ¼ 0

and form a complete orthogonal system. Considering the case of radiation in a cube V of length L, imposing periodicity requirements instead of boundary conditions, one obtains rffiffiffiffiffiffiffiffiffiffiffiffi 4p
hc2 Ak^e ðrÞ ¼ ^e expðikr=
hÞ; ðE5Þ L3 Transitions in Molecular Systems. Hans J. Kupka Copyright
2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-41013-2

j Appendix E: Quantization of the Radiation Field

278

where ^e is the polarization unity vector. The normalization of the expression (E5) is determined from an evaluation of the Poynting flux S ¼ ðc=4pÞE  B. The field quantization can be performed by regarding the amplitudes ak^e and ak^e as þ operators ak^e and ak^ e , which satisfy the (canonical) commutation relations ½ak^e ak0^e0 

¼

þ þ ½ak^ e ak0^e0  ¼ 0

½ak^e akþ0^e0 

¼

dkk0 d^e^e0

The energy of the electromagnetic field is expressed as ð X 1 1X þ þ H ¼ ðE 2 þ B2 Þdr ¼ Hk^e ¼ hv^e ðkÞðak^
e ak^e þ ak^e ak^e Þ: 2 2 k;^e k;^e

ðE6Þ

ðE7Þ

This is nothing more than the sum of N independent oscillator Hamiltonians, one for each wave vector k and polarization ^e. Every mode then corresponds to a harmonic oscillator of which is known that the quantization leads to the following results: Hk^e jnk^e i ¼


 1 nk^e þ
hv^e ðkÞjnk^e i; 2

ak^e j0k^e i ¼ 0;

ðE8Þ ðE9Þ

ak^e jnk^e i ¼ ðnk^e Þ1=2 jnk^e
1i;

ðE10Þ

1=2 þ jnk^e þ 1i; ak^ e jnk^e i ¼ ðnk^e þ 1Þ

ðE11Þ

þ ak^ e ak^e jnk^e i ¼ nk^e jnk^e i;

ðE12Þ

jnk^e i ¼

þ nk^e ðak^ eÞ

ðnk^e !Þ1=2

j0k^e i;

ðE13Þ

where the ground state of the energy Hk^e is denoted by j0k^e i and nk^e is the number of photons in a momentum state k having polarization parallel to ^e (and, of course, k  ^e ¼ 0 since the field is transverse). Those eigenstates of the energy are stationary and are known as state vectors. They form a complete orthonormal system that spans the space of photon occupation numbers or Fock space. When a Hamiltonian H separates into a sum of commuting sub-Hamiltonians Hk , its eigenstates are simply products of eigenstates of each of the separate subHamiltonian, and the corresponding eigenvalues are the sum of the individual eigenvalues of the sub-Hamiltonians. We can therefore specify an eigenstate of H by giving a set of quantum numbers nk^e , one for each of the N independent oscillator Hamiltonians Hk^e , or for each of the photons nk^e of type ^e with wave vector k and frequency v^e ðkÞ: j ¼ jnk1^e1 ; nk2^e2 ; . . . ; nk^e ; . . .i:

ðE14Þ

Appendix E: Quantization of the Radiation Field þ Equations E10 and E11 lead to the interpretation of ak^e and ak^ e as the respective operators for absorption and creation of photons in the state jk; ^ei with the energy
hv^e ðkÞ. The development of the field is then determined by the equation

i
h

q jji ¼ Hjji: qt

ðE15Þ

The initial conditions are introduced by specification of a vector jj0 i at a time t0 . For a realistic description of an actual excitation process, Jortner and Mukamel [146] introduced on the basis of Equation E14 a photon wave packet X Xk^eA ¼ ank^e1 nk^e2  nk^e  jnk1^e1 nk2^e2    nk^e    i; ðE16Þ fnki ^ei g

which is characterized by the energies Ei ¼ ki (where we have used the units c;
h ¼ 1). For moderately weak fields, nki^ei ¼ 0 or 1 for all i and under these conditions, we may consider an initial state of the field j ð0Þ consisting of a wave packet of one-photon states X jð0Þ ¼ ak jk^ei; ðE17Þ k

P where ak is the initial amplitude of the state jk^ei, while the summation k represents the integration over photon energies over special directions and summation over all polarization directions.

j279

j281

Appendix F: The Molecular Eigenstates

Following here the treatment of Ref. [11], the molecular eigenstates are defined as a superposition of zero-order BO states by the Ansatz yn ¼

m X s¼1

ans js þ

1 X i¼
1

bni ji ;

ðF1Þ

where the states of energy Es ðs ¼ 1; 2; . . . ; mÞ carry oscillator strength to the ground state. These states are superimposed on a dense manifold of vibronic states fji g that do not carry oscillator strength. For simplicity, it will be assumed that the latter states are equally spaced, and let e be the spacing between these levels. The matrix elements between the zero-order states are hjs jHel jjt i ¼ Es dst ;

ðF2Þ

hji jHel jjj i ¼ Ei dij ;

ðF3Þ

hjs jHel jji i ¼ hjs jHv jji i ¼ vsi ¼ vs ;

ðF4Þ

where we have again assumed that all the vibronic coupling matrix element vsi are equal. The secular equations that determine the values of the expansion coefficients in (F1) are X ans Es þ bni vs ¼ En ans ; ðF5Þ i

X s

ans vs þ bni Ei ¼ En bni ;

where En are the new eigenvalues. The last equation can be solved for bni and one gets X bni ¼ ðEn
Ei Þ
1 ans vs : s

Transitions in Molecular Systems. Hans J. Kupka Copyright
2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-41013-2

ðF6Þ

ðF7Þ

j Appendix F: The Molecular Eigenstates

282

Substituting the result in Equation F5 yields P n a vt X ans þ t t vs ðEn
Ei Þ
1 ¼ 0; Es
En i

ðF8aÞ

or ½vs =ðEn
Es Þ

X t

ant vt

X i

ðEn
Ei Þ
1 ¼ ans :

ðF8bÞ

Multiplication by vs and summation over s leads to the result X
X X n X at vt ðEn
Ei Þ
1 ¼ ans vs ; v2s =ðEn
Es Þ s

t

ðF9Þ

s

i

or alternatively, X
s

" #
1 X  v2s =ðEn
Es Þ ¼ ðEn
Ei Þ
1 ;

ðF10Þ

i

which is a equation for En. The sum on the right-hand side of the equation is known ½11, so one gets !
1 X X v2 s ¼ ðEn
Ei Þ
1 ¼
ðp=eÞ cot g½ðp=eÞðEn
E0 Þ: ðF11Þ E
E n s s i The values of the coefficients ans are determined from the normalization condition hyn jyn i ¼ 1 ¼ " 

X i

X   2 X  2 an  þ bn  ¼ s i s

1 En
Ei

i

!2

X t

!2 ant vt

# X  ns 2 X  1 2 : þ En
Es En
Ei s i

ðF12Þ

The last sum on the right-hand side of Equation F12 can be expressed in the form p  p2 h p i X  1 2 p2 ¼ 2 cos ec2 ðEn
E0 Þ ¼ 2 1 þ cot g 2 ðEn
E0 Þ : En
Ei e e e e i ðF13Þ

Using the result of Equation F11, we get X i

1 En
Ei

2

X v2 p2 s ¼ 2 þ e En
Es s

!
2 :

Substitution of Equations F10 and F14 into Equation F12 leads to

ðF14Þ

Appendix F: The Molecular Eigenstates

X t

!2 ant vt

" ¼

X  vt 2 p2 þ1þ 2 En
Et e t

X t

v2s En
Et

!2 #
1

X t

v2t En
Et

!2 : ðF15Þ

Using Equations F11 and F15, one gets from (F8b) the following expression for the coefficient ans :  n 2 as ¼



vs En
Es

2 "X  t

vt En
Et

2 þ1þ

p2 e2

In the case of one resonance ðm ¼ 1Þ, we get  2 
1  n 2
: as ¼ a2n ¼ v2s ðEn
Es Þ2 þ v2s þ pv2s =e

X t

v2t En
Et

!2 #
1 :

ðF16Þ

ðF17Þ

These equations describe the probability of finding the vibronic states js in the new state yn . Furthermore, Equation F17 describes a Lorentzian as a function of En and a half width given by
 2 1=2 Ds ¼ v2s þ pv2s =e :

ðF18Þ

j283

j285

Appendix G: The Effective Hamiltonian and Its Properties

Consider a physically acceptable system of the complex-level structure that involves the Born–Oppenheimer basis, consisting of the ground electronic state j g i ¼ j0i and a single doorway state jsi that is coupled via nonadiabatic intramolecular interactions Hvib to a background of fjlig states, and where both the states jsi and the manifold fjlig are characterized by the radiative and nonradiative decay widths cs and cl , respectively (see Figure 6.2). The effective Hamiltonian responsible for this system is


Es
ics =2 Vsl Vsl0 



Vsl El
icl =2 0 


: ðG1Þ Heff ¼
Vsl0 0 El0
icl0 =2   




. . . . ..
.. .. ..
In the above basis, the Hamiltonian Heff is off-diagonal, while the damping matrix is diagonal. From the basis definition (see Equation A19), it is evident that the effective Hamiltonian is non-Hermitian. This can be rationalized by noting that we consider only a subspace of the Hilbert space, consisting of the above discrete zero-order basis set. In fact, the effective Hamiltonian of Equation G1 can be written as a sum Heff ¼ Hel
iC=2;

ðG2Þ

where Hel is a Hermitian matrix and
iC=2 is an anti-Hermitian matrix. Furthermore, when the effective Hamiltonian is nondiagonal within the basis of zero-order states, these states cannot be considered to decay independently. Thus, the time evolution operator expð
iHeff tÞ in the P subspace will contain off-diagonal contributions. We shall now transform the Heff matrix to a representation in which it is diagonal (as was done, for instance, in Section 7.5.5). This leads to a basis fj jig obtained by the transformation

where

D
1 Heff D ¼ L;

ðG3Þ






j1

s






l
¼ D

j2
:
.

.

..

..


ðG4Þ

Transitions in Molecular Systems. Hans J. Kupka Copyright Ó 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-41013-2

j Appendix G: The Effective Hamiltonian and Its Properties

286

D in Equation G3 is a nonunitary matrix. When a basis set is chosen such that Hel is a real matrix (as in the case above), Heff is a complex symmetry matrix, which can be diagonalized by an orthogonal transformation D. The matrix elements of the diagonal matrix L are in general complex Ljj0 ¼ ðEj
icj =2Þdjj0 ;

ðG5Þ

with a negative imaginary part cj =2 and in general we cannot find any orthogonal set of eigenvectors. While the eigenvectors are not orthogonal, they have an orthogonality relation with a different set of vectors, which must be defined. In the most cases, our eigenvectors have been column vectors x, which are multiplied to the right of a matrix A (Heff in our case), that is, A  x ¼ l  x;

ðG6Þ

where l are the eigenvalues determined by detjA
l1j ¼ 0:

ðG7Þ

These are termed right eigenvectors. We could also, however, try to find row vectors
x, which multiply A to the left and satisfy
x  A ¼ l 
x:

ðG8Þ

These are called left eigenvectors. By taking the transpose of this equation, we can see that every left eigenvector is the transpose of a right eigenvector of the transpose of A. Now by taking into account that the determinant of A in Equation G7 equals the determinant of the transpose of A, we also see that the left and right eigenvalues of A are identical. To pursue this matter a little further, we arrive to the conclusion that if the eigenvalues l are nondegenerate, each left eigenvector is orthogonal to all right eigenvectors, and vice versa. To return now to the question of the orthogonality relation of our eigenvectors j ji, we may quite generally write 
jj ji ¼ djj0 ; ðG9Þ    where
jj are the defined left eigenvectors. This complementary basis 
j is defined by the transformation   X þ 
j ¼ ðD Þjn jni; n ¼ s; l; l0 ; . . . : ðG10Þ n

  The corresponding projection operator P can be written in terms of j ji and 
j as X  P¼ ðG11Þ j ji
jj j

and the time evolution operator X X  P expð
iHeff tÞP ¼ j jiexpð
iLjj tÞh
jj ¼ j jiexpð
iEj t
cj t=2Þ
jj j

ðG12Þ

j

is expressed as a sum of independently decaying levels characterizing the molecular system. A rigorous discussion to this subject may be found in Refs [146, 149].

j287

Appendix H: The Mechanism of Nonradiative Energy Transfer

H.1 Single-Step Resonance Energy Transfer

To proceed with our discussion about the host-sensitized energy transfer in the system of pentacene in p-terphenyl, we review the simplest of energy transfers. This is the transfer in a system that consists of a molecule or ion S (sensitizer) and a molecule (ion) A (activator). If any possible interaction (i.e., exchange interaction) between S and A is disregarded, the system may be represented by the product of the wavefunction of S and the wavefunction of A. In particular, two states of the system are of interest:
 a) State
y
s ya with molecule S in the excited state and molecule A in the ground state.
 b) State
y
a ys with molecule S in the ground state and molecule A in the excited state. Initially the sensitizer S is in an excited state and the activator A is in the ground state, while after energy transfer the sensitizer is in the ground state and the activator in the excited state (see Figure H.1). The transfer rate between these states is described by

2 wsa ¼ ð2p=
hÞ
Mif
rf ðEi Ef Þ;

ðH1Þ

where Mif is the matrix element between initial and final states of the system and rf is the density of final states. The transfer matrix element is related to the Coulomb interaction of the electrons and nuclei and can be expanded in a series of multipole–multipole interactions about the sensitizer–activator separation Rsa . For neutral molecules, the leading term in this expansion is the electric dipole–dipole interaction first described by F€orster [302–305]: DD Hint ¼

  1 ðms Rsa Þðma Rsa Þ ; m  m 3 eR3sa s a R2sa

Transitions in Molecular Systems. Hans J. Kupka Copyright Ó 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-41013-2

ðH2Þ

j Appendix H: The Mechanism of Nonradiative Energy Transfer

288

|Ψs*>

|Ψa*>

Wsa

|Ψ s >

| Ψa >

Figure H.1 Resonant energy transfer between an excited sensitizer and unexcited activator molecule (ion).

where e is the dielectric constant of the medium and ms and ma are the two electric dipoles of S and A. Later Dexter expanded the treatment to include higher order electromagnetic and exchange interactions [306]. In the case of resonance electric dipole–dipole transfer, the relevant matrix element Mif is
DD

   
w w x 0 x 0 Mif ¼ xsm xan w
s wa
Hint a s am sn    1 
ws jms jws tsa wa jma jw
a ðxsm xan jxam0 xsn0 Þ ¼ ðH3Þ 3 eRsa ¼ JðRsa Þðxsm xan jxam0 xsn0 Þ;

where for the special case of a homogeneous mixed p-terphenyl crystal, the dipoles ms ¼ ma ¼ m are parallel to the short M-axis of pentacene and Rsa is taken along the b0
sa Þ, where R
sa is the unit vector crystallographic axis. Hence, tsa ¼ 13 cos2 ðm  R along the Rsa distance. The transition energies are not sharp but rather have finite widths due to thermal vibrations in the crystal. Thus, the density of states can be treated as function of energy, rðEi Ef Þ ¼

1 C ; p ðEi Ef Þ2 þ C2

ðH4Þ

where C ¼
hc;

Ei ¼
hVs þ

Ef ¼
hVa þ

X mm

X mm


hmm0 vam þ


hmm vsm þ X nm

X nm

hnm vam


and ðH5Þ


hn0m vsm :

Substituting (H3) through (H5) into (H1) and summing over the vibrational quantum numbers, the probability of resonant energy transfer can be written as wsa ¼

2p 2 1 X X J
h2 p mm ;nm mm0 ;n0m  INs

c X X ðmm vsm þ nm vam Þ ðmm0 vam þ n0m vsm Þ Vs Va þ m

 m1 ; m2 ; . . . ; mN a m10 ; m20 ; . . . ; mN0 IN ; n01 ; n02 ; . . . ; n0N n1 ; n2 ; . . . ; nN

!2 þ c2

m

ðH6Þ

H.2 Phonon-Assisted Energy Transfer

or in compact form ð 2p wsa ¼ 2 J 2 dv Fs ðvVs ÞFa ðvVa Þ;
h

ðH7Þ

where Fs and Fa are the spectral line shape functions for the sensitizer and activator, respectively.1) From (H7) we see that a nonzero overlap is essential for resonance F€orster energy transfer. The transfer rate (H7) can be expressed in more conventional units wsa ¼ ðtF Þ1 ðR0 =Rsa Þ6 ;

ðH8Þ

where tF is the fluorescence decay time (intrinsic lifetime in the absence of activator) of the sensitizer and R0 is a “critical transfer distance” given by [302, 303] R60 ¼

ð 9000 gt2sa ln 10 f ð~nÞeð~nÞ d~n 128p6 n4 NA ~n4



~n ¼

v  : 2pc

ðH9Þ

Here g is the quantum efficiency of the sensitizer (g ¼ tF =tsF ¼ 1=3 for pentacene) in the O4 site of p-terphenyl at 4 K,2) n is the index of refraction (n ¼ 1:7 for the p-terphenyl crystal), and NA is the Avogadro’s number. The integral in (H9) is calculated from the normalized fluorescence spectrum f ð~ nÞ and the decadic molar extinction coefficient eð~nÞ of pentacene at O4 site. The critical interaction distance is the sensitizer–activator separation for which the transfer rate is equal to the intrinsic decay time. Although derived for low temperatures, Equation H9 is also valid for arbitrary temperatures. In fact, the temperature dependence of the resonant energy transfer rate is contained in the spectral overlap integral.

H.2 Phonon-Assisted Energy Transfer

The rate of resonant energy transfer described above critically depends on the energy match between the sensitizer and activator transitions. This is reflected in the dependence of wsa on the spectral overlap integral. Lattice vibrations affect the widths and position of spectral transitions and, thus, the line shape function appearing in the overlap integral. If there is significant energy mismatch DEsa ¼
h ðVs Va Þ between sensitizer and activator transitions as shown in Figure H.2, the spectral overlap and, thus, the resonant energy transfer rate can 1) In writing (H7), we have made use of the fact that a convolution of two Lorentzians of widths cs and ca centered at V0s and V0a , respectively, is 1 cs þ ca a Lorentzian of the p ðV0s V0a Þ2 þ ðcs þ ca Þ2 width c ¼ cs þ ca . 2) The quantum yield g for pentacene in the O4 site of p-terphenyl is given by g ¼ tF =tsF ¼ 10=30, where tsF ¼ 30 ns is the

natural fluorescence lifetime in the absence of relevant radiationless or vibronic processes. It must be emphasized that the fluorescence lifetimes tF of pentacene are different in the (inequivalent) substitutional guest sites of the host crystal. The spectra of the four sites differ also in their intensity distributions and in their phonon sideband structures.

j289

j Appendix H: The Mechanism of Nonradiative Energy Transfer

290

|Ψs*>

∆Esa

|Ψa*> Wsa

|Ψ s >

| Ψa >

Figure H.2 Phonon-assisted energy transfer for molecules (ions) with a transition energy mismatch of DEsa .

be negligibly small. In this circumstance, the energy transfer can occur through phonon-assisted processes [307–312]. The interaction Hamiltonian Hint consists of DD two parts, one describing the dipole–dipole interaction Hint as discussed above and one describing the vibronic interaction Hv ¼

X

Vk ðiÞqk ;

ðH10Þ

i¼s;a

where Vk ðiÞ 6¼ 0 only in a small region located mainly at the positions of the sensitizer S and of the activator A and qk is the promoting mode displacement. The wavefunctions for the initial and final states must now include the vibrational factor for the promoting mode (phonon) that is emitted or absorbed:
 jyi i ¼
w
s wa xsm xan ;

ðH11Þ


E


yf ¼
ws w
a xs
n xam
:

ðH12Þ

Since the energy of the sensitizer (excited state) is greater than the energy of the activator (excited state), the transition takes place only in the direction S ! A with the emission of a phonon. By including Hv , the phonon-assisted energy transfer is a DD DD second-order process in respect to H0 ¼ Hint þ Hv (first order in Hint and first order in Hv ), so the relevant matrix element can now be expressed by [310] Mifk ¼

D E
DD

y ; nk þ 1 hy ; nk þ 1jHv ð jÞjy ; nk i X yf ; nk þ 1
Hint i i i Es ðEs 
h vk Þ ED E
DD

y ; nk X yf ; nk þ 1jHv ð jÞjyf ; nk yf ; nk
Hint i

j¼s;a

þ

j¼s;a

ðH13Þ

D

Es Ea

;

where jyi ; nk þ 1i is an intermediate state. Using the sum in equation H13 accounts for the possibility of the phonon process occuring at either the sensitizer or activator site

H.2 Phonon-Assisted Energy Transfer

Mifk ¼


DD

 
w w    ws w
a
Hint s a w
s jVk ðsÞjw
s xs;nk þ 1 jqk jxs;nk Es ðEs 
hvk Þ     þ hwa jVk ðaÞjwa i xa;nk þ 1 jqk jxa;nk xs;
n0 xa;m
0 jxs;m0 xa;n0

    w w

HDD
w
w þ s a int s a hws jVk ðsÞjws i xs;nk þ 1 jqk jxs;nk Es;
0 Ea;0    þ w
a jVk ðaÞjw
a xa;nk þ 1 jqk jxa;nk
   xs;
n0 xa;m
0
xs;m0 xa;n0 ; ðH14Þ

where the plus or minus sign refers to the case of phonon emitted or phonon absorbed, respectively. Setting DEsa ¼ Es;0 Ea;0 ¼
hvk for conservation of energy, Equation H14 simplifies to
DD

 
w w       ws w
a
Hint s a k Mif ¼ hws jVk jws i w
s jVk jw
s  hwa jVk jwa i w
a jVk jw
a
hvk 0 11=2
h ðn þ 1Þ k A ðxs
n0 xam
0 jxsm0 xan0 Þ @ 2vk 0 11=2 J
h ðn þ 1Þ k A ðxs
n0 xam
0 jxsm0 xan0 Þ: ¼ ð f gÞ@
hvk 2vk ðH15Þ

Here again we find that the promoting mode contribute to the transition via hws jVk jws i and hwa jVk jwa iand that the contribution to the vibrational overlap is given by ð
hðnk þ 1Þ=2vk Þ1=2 ðxs
n0 xam
0 jxsm0 xan0 Þ with nk ¼ 0. The comparison with Equation H3 that applies to the case of resonant energy transfer and which Ei and Ef are given by Equation H5 should be noted. The difference that arises from the additional vibronic interaction is manifested not only in the electronic part of the matrix element Mifk but also in a diminished initial energy gap
hðVs vk Þ in the density of states function (H4). If we make these modifications in Equation H1, we will immediately recover the appropriate expression for the phonon-assisted energy transfer:  ð p J 2
sa ¼ w ð f gÞ2 FS ðvVs ÞFA ðvVa vk Þdv; ðH16Þ
hvk
hvk where the spectral line shape functions for emission and absorption FS ðvÞ and FA ðvÞ are centered at VS and Va þ vk , respectively.

j291

j293

Appendix I: Evaluation of the Coefficients bmn , cmn , and bm in the Recurrence Equations 8.28 and 8.29 One must evaluate the derivatives q2 D11 =qwm qwn ; q2 D22 =qzm qzn , and qD1 =qwm by noting that they are the coefficients of the quadratic and linear terms in w and z in the forms (8.16), (8.18), and (8.22), respectively. By differentiating of Equation 8.16 with respect to wm and wg , we obtain with the help of Equation 8.17 for the case m 6¼ g: bmn ¼ bnm ¼ 2 ¼
B

(

1 q2 D11 2 qwm qwn X

ð
1Þm þ n W




1pN

1  m
1;m þ 1  N

 a1  p
1;p þ 1  N W þ

1    m
1; m þ 1    N 1    p
1; p þ 1    N






 1    n
1; n þ 1    N n
1;n þ 1  N a11   p
1;p þ 1  N 1    p
1; p þ 1    N X X ð
1Þr þ s

1i1 <  ir
1 <m