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 Greens Function The Greens 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 Greens 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 Greens 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 3K6. 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 M1=2 Aq;
where R and R0 are ð3K6Þ-dimensional column vectors of the instantaneous and equilibrium Cartesian coordinates, respectively, associated with the nonzero frequency normal modes. M is the ð3K6Þ ð3K6Þ 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 iE 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 iE xav ¼
X b6¼a
Xab xbv ;
ð1:8bÞ
where Xab ¼ hja ½TN ; jb ir and ½A; B ¼ ABBA. 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 ðqÞEa ðqÞ ja q=qqm jb ¼ ja qU=qqm jb ð1:13Þ and E D E D ½Eb ðqÞEb 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 ðqÞEa 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 lHospitals 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 ðqÞEb ðqÞ1 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; qÞUðr; 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:
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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 Þ1 q0 ðl0 Þt ðA0 Þ1 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 Þ1 q0 þ ðq0 Þt A0 F0 ðA0 Þ1 q0 ðq0 Þt A0 F0 ðA0 Þ1 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 Þ1 ðk0 Þt A0 F0 ðA0 Þ1 Þq0 þ ðq0 Þt A0 F0 ðA0 Þ1 q0 2 1 ðl0 Þt ðA0 Þ1 k0 þ ðk0 Þt A0 F0 ðA0 Þ1 k0 : ð1:39Þ 2 þ
In deriving (1.39), we have made use of the relation A1 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 Þ1 ðk0 Þt A0 F0 ðA0 Þ1 ¼ 0
and hence ðl0 Þt ðA0 Þ1 k0 ¼ ðk0 Þt A0 F0 ðA0 Þ1 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 Þ1 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 Þ1 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 Þ1 q0 þ ka Aa ðA0 Þ1 k0 ;
which we abbreviate to qa ¼ Wq0 þ k0a ;
ð1:44Þ
where W ¼ Aa ðA0 Þ1 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
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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
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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 ð1Þk n! ð2xÞn2k k!ðn2kÞ! 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]:
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X rw i ¼ ðgðqÞrqÞw i þ ½cos q hja jrj1 isin 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 ðgrqÞ 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 HOH 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
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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 ðaac 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 ðaac 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 ðaac Þ2 and gr ðrÞ ¼
qq Fr =½2Fa ðaac Þ ¼ : qr q;a 1 þ ½Fr =Fa ðaac 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 4:5 103 hartree/deg, compared to the value of 4:2 103 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].
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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 Greens 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 Greens 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 Greens 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 Lorentzians. 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 ðh=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 Greens 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 Greens 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 expðEsm =kB TÞ: expðEsm =kB TÞ pðm; TÞ ¼ P : m expðEsm =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 Zm1 ; Zm ¼ 1exp hvm =kB T ; ð3:8Þ m
Q where kB is the Boltzmann constant. The product Z1 ¼ m Zm1 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 ith1 ðDE þ Eln Esm ÞC h1 jtj
1 1 ð
Y
¼ h 1 dt exp iVtC h1 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 VCh1 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 itVC h1 jtj Kg ðtÞGN1 ðtÞ h g 1 ð3:10Þ ) 1 X g g0 ð
1 þ Rsl Rsl dt exp itVCh jtj Ig ðtÞIg0 ðtÞGN2 ðtÞ ; g;g0
where
1
Rgsl ¼ ih js q=qqg jl q0 ; Y m Y m G1 ðtÞ; GN2 ðtÞ ¼ G1 ðtÞ; GN1 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 dqlm 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 dqlg xsmg bsg qsg i l xlng blg qlg xlng blg qlg i s xsmg bsg qsg : qqg qqg 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 dqlg xsmg bsg qsg i l xlng blg qlg xsmg bs1 g qg xlng bg qg : qqg 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 itVC h1 jtj Kg ðtÞGN1 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,theoccurrenceofexpðC 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 h1 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 N1 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 expðivsm thvsm =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 1wm2 4 p ð3:18aÞ
where m ¼ w
1wm ; 1 þ wm
1 w m ¼
1 þ wm ; 1wm
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 1z2m ¼ @ A exp blm qlm þ qlm zm þ qlm qlm z1 ; m 4 p ð3:18bÞ
with zm given by zm ¼
1zm : 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 N1 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 1z2 1z2 1w 2 1w 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 z1 bs2 qs2 þ qs2 w 1 4 4 4 4 2 2 1 1 bl2 ql2 þ ql2 z2 bl2 ql2 ql2 z1 dql1 dql1 dql2 dql2 : 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 21=2 qlm þ qlm ð3:22aÞ qm ¼ 21=2 qlm qlm ; m ¼ 1; 2; and correspondingly qm0 ¼ 21=2 qsm þ qsm qm0 ¼ 21=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 dq1 dq2 dq2 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 z1 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 xÞ1=2 exp yt x1 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 ; 21=2 w12 bs1 kð1Þ 1 þ w22 bs2 kð2Þ 2 y ¼ col 21=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 colðq1 ; 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 z1 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 ðÞm1 þ n1 þ n2 b1 D12 þ ð1Þm2 þ n1 þ n2 b2 D12 ð12Þ2
þ ð1Þm1 þ m2 þ n1 b2 D1
ð12Þ2
þ ð1Þm1 þ m2 þ n2 b1 D2
;
2 2 b1 þ ð1Þm1 þ n1 w12 b12 am1 m2 ;n1 ;n2 ¼ ð1Þm1 þ m2 b1 b2 þ ð1Þm1 þ n2 w11 2 2 þ ð1Þm2 þ n1 w22 b2 þ ð1Þm2 þ n2 w21 b21 þ ð1Þn1 þ n2
ð3:35Þ
ð3:36Þ
and bm1 m2 ;n1 n2 ¼ ð1Þm1 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 d1m1 ;1m2 ;1n1 ;1n2 :
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 ! 1m1 ; m2 ! 1m2 ; n1 ! 1n1 ; and n2 ! 1n2 . 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
j41
j 3 Description of Radiationless Processes in Statistical Large Molecules
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 exp½b1 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 dq1 dq2 dq2 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=Tivsm t ðm ¼ 1; 2Þ, with a being a positive number, especially a ¼ hvsm =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 ð1w1 Þð1w2 Þ B2 ðw1 ; w2 ; 1; 1Þ ¼ 4ð1w1 Þð1w2 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 : ð1w1 Þð1w2 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 4ð1z1 Þð1z2 Þ B2 ð1; 1; z1 ; z2 Þ ¼ 4b1 b2 ð1z1 Þð1z2 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 ð1z1 Þð1z2 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 b1 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 ! b1 b1 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 ! b1 b1 an;m 4 1 2 1 1
ð3:59Þ
and similarly, 4 bm;n ! b1 b1 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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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 ! b1 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 ! b1 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 b1 b1 12 1 2 12 1 2 ¼ G2 z1 ; z2 ; w1 ; w2 ð1Þ ð2Þ ; 11 21 : 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 b1 b1 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 b1 b1 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 ð1w1Þ Þð1z1 Þ½ð1 þ w2 Þð1z2 Þ þ b2 ð1w2 Þð1 þ z2 Þ ð2Þ2
þ b2 D12 ð1w2 Þð1z2 Þ½ð1 þ w1 Þð1z1 Þ þ b1 ð1w1 Þð1 þ z1Þ Þ; 2 Y B1 ðw1 ; w2 ; z1 ; z2 Þ ¼ ½bm ð1wm Þð1 þ zm Þ þ ð1 þ wm Þð1zm Þ; m¼1
and B2 ðw1 ; w2 ; z1 ; z2 Þ ¼
2 Y
½bm ð1 þ wm Þð1zm Þ þ ð1wm Þð1 þ zm Þ:
m¼1
Dividing A1 by B1, we find that ð1Þ2
A1 ðw1 ; w2 ; z1 ; z2 Þ b1 D12 ð1w1 Þð1z1 Þ ¼ B1 ðw1 ; w2 ; z1 ; z2 Þ b1 ð1w1 Þð1 þ z1 Þ þ ð1 þ w1 Þð1z1 Þ ð2Þ2
b2 D12 ð1w2 Þð1z2 Þ þ ; b2 ð1w2 Þð1 þ z2 Þ þ ð1 þ w2 Þð1z2 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 ð1wÞð1zÞ bð1wÞð1 þ zÞ þ ð1 þ wÞð1zÞ
G1 ðw; zÞ ¼ 2b1=2
1=2 ð1 þ b2 Þð1w 2 Þð1z2 Þ þ 2b½ð1 þ w 2 Þð1 þ z2 Þ4wz ð3:68Þ
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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 dqlg i s exp bsg qsg þqsg w 1=2 h qqg 4 p 1wg2 1z2g
1 h 2 2 i q 1 i l exp blg qlg þqlg zg þ qlg qlg z1 g 4 qqg 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 1wg2 1z2g 1
2 2 i 1 h exp blg qlg þqlg zg þ qlg qlg z1 g 4
h 1 i i 1 1 h g þ qsg qsg w g blg qlg þqlg zg qlg qlg z1 : bsg qsg þqsg w g 2 2 ð3:69Þ
Introducing now, as above, new coordinates for the gth promoting mode qg ¼ 21=2 qlg þ qlg qg ¼ 21=2 qlg qlg
q0g ¼ 21=2 qsg þ qsg q0g ¼ 21=2 qsg qsg
ð3:70Þ
according to (3.23), the relations q0g ¼ qg þ 21=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 z1 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 TÞ1 eivg t ; vg cothðhvg =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 ih
bsg blg kg l 1 g bsg z1 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 GN1 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 ½cothðhvg =2kB TÞ þ 1 dt hwnonrad iT ¼ Z sl 4 h g 1
1 exp itðVvg ÞCh jtj GN ðtÞ ) 1 ð
1 þ ½cothðhvg =2kB TÞ1 dt exp itðV þ vg ÞC 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 ð1expðhvg =kB TÞÞ1 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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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 expðcjxjÞ ¼ dy expðixyÞ 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 Þcjtj 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 ðxÞdðxyÞdx 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 r1 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 hcðhVÞ2 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,
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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 cm1 . 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 c1 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 eiva 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 dqlg 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 ; qqg
ð3:82Þ
s
where now wg ¼ eivg 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 av1 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 eivg 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 ðVvg Þ þ mg~I 2 ðV þ vg Þ ; 2h
ð3:85Þ
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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 ðVvg Þ; wsl ðma ¼ 0Þ ¼ ð3:87Þ 2h since in this case ~I 2 ðVvg Þ ¼ I 2 ðVvg 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 ðCCÞa1g 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 cm1 ; v6 ¼ 521 cm1 ; hV ¼ 8200 cm1 ;
Dv1 ¼ vTa1 vSa1 ¼ 25 cm1 ; Dv6 ¼ 50 cm1 ;
X1 ¼ XCCðv1 Þ ¼ 0:025; X6 ¼ 0;
vg ¼ 1500 cm1 ;
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 ðhV ffi 30; 000 cm1 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].
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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
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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 1z 2 6ð1þbÞ a 1bz w 7 7 6 exp 6 7 5 4 zb að1zÞ 1 w exp 1bz 1bz G1 ðw;z;a;bÞ ¼ ð1b2 Þ1=2 1=2 : ½ð1bzÞ ð1þbzÞ1=2 zb zþb 1 w 1 w 1bz 1þbz ð4:1Þ Here z and w are generally complex variables in the bidisc D1(0,1) D1(0,1). b 1b 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 að1zÞ exp 1 X 1bz G1 ð0; z; a; bÞ ¼ ð1b2 Þ1=2 ¼ I1 ð0; n; a; bÞzn : ð4:4Þ 1=2 ½ð1bzÞ 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 að1zÞ að1bÞ exp ck zk ; ð4:5Þ ¼ exp ðaÞ exp z ¼ exp ðaÞ 1bz 1bz k¼0 where c0 ¼ 1
and ck ¼
k X 1 k1 i¼1
i!
i1
ai ð1bÞi bki ;
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 ð1b Þ exp ðaÞ ð1Þn bn ð1Þi ni i i¼0 þ ð1Þn1 þ ð1Þn2 þ ð1Þn3
n a n1 X 3=2 1=2 b ð1Þi1 ni i1 1! i¼1
n a2 n2 X 1=2 5=2 ð1Þi2 b i2 ni 2! i¼2 n a3 n3 X 1=2 7=2 b ð1Þi3 i3 ni 3! i¼3
þ
n X an1 1=2 in þ 1 n þ 1=2 þ ð1Þ b ð1Þ in þ 1 ni ðn1Þ! i¼n1 ) an n1=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 ða þ nbÞ I1 ð0; n; a; bÞ þ b ð a þ nbÞ I1 ð0; n1; a; bÞ ðn1Þ b3 I1 ð0; n2; a; bÞ;
ð4:8Þ
where a ¼ a ð1bÞ 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 ð1b2 Þ1=2 exp ðaÞ. 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
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" ½n=2 #2 1 X n! 2 1=2 r n=2r a I1 ð0; n; a; bÞ ¼ ð1b Þ exp ðaÞ : b n! r¼0 2r ðn2rÞ!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; nn1 ; 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 að1zÞ exp 1 X 1bz Gi ð0; z; a; bÞ ¼ ð1b2 Þi=2 ¼ Ii ð0; n; a; bÞzn ; ð4:12Þ ½ð1bzÞ 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; nn1 ; 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 ð1zÞ ; 1bz zb b! : 1bz a!
Hence
1b ! 1
ð4:14Þ
zb 1z ¼ ð1 þ bÞ : 1bz 1bz
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 1z 2 6að1 þ bÞ 1bz w 7 a ð1bÞ 7 6 ð4:16Þ exp z ! exp 6 7; zb 5 4 1bz 1 w 1bz or after expansion in power series of z and w, respectively, as ! 1 n 1 X X X 1 n1 k nk k a ð1bÞ b zn ! 1 þ 1þ Am ðzÞw m ; k1 k! n¼1 m¼1 k¼1
ð4:17Þ
where 2k m X 1 m1 k zb mk k 1z a ð1 þ bÞ Am ðzÞ ¼ ; k! k1 1bz 1bz
m 1:
k¼1
ð4:18Þ
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Note that by the assignment (4.14), the parameter a becomes complex in the að1zÞ zb z-plane, but with Re 0, as it should be. Accordingly, 1 Re 1 1bz 1bz 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
zb w 1bz
Bn ðzÞ ¼ ð1Þn
1
n X zb k z þ b nk 1=2 1=2 nk
k
k¼0
1 þ bz
1bz
;
ð4:20Þ
by forming the Chauchy product of the two series above, we obtain 3 1z 2 6ð1 þ bÞa 1bz w 7 7 6 exp6 7 5 4 zb 1 w 1bz 1=2 ¼ zb zþb 1 w 1 w 1bz 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 ðzÞBmk 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 ð1Þ
m
m X zb mi z þ b i 1=2 1=2 mi
i¼0
m1
þ ð1Þ
i
1bz
1 þ bz
m a 1z 2 X 1=2 3=2 i1 1! 1bz i¼1 mi
_
zb mi z þ b i1 1bz 1 þ bz
4.2 The Distributions I1(m, n; a, b)
m _2 a 1z 4 X zb mi z þ b i2 1=2 5=2 þ ð1Þm2 i2 2! 1bz i¼2 mi 1bz 1 þ bz þ ð1Þm3
_3
a 3!
m 1z 6 X zb mi z þ b i3 1=2 7=2 i3 1bz i¼3 mi 1bz 1 þ bz
þ m a 1z 2m2 X zb mi 1=2 m þ 1=2 þ ð1Þ im þ 1 mi 1bz ðm1Þ! 1bz i¼m1 1
_ m1
_m z þ b im þ 1 1z 2m m1=2 1=2 a þ : 0 0 m! 1bz 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 ð1b2 Þ1=2
að1zÞ 1bz
ð1b2 z2 Þ1=2
ðmÞ
C1 ðz; a; bÞ ¼
1 X
I1 ðm; n; a; bÞzn :
ð4:24Þ
n¼0
Using Leibnitzs formula for the nth derivative of a product, we obtain " # n X 1 d k ðmÞ I1 ðm; n; a; bÞ ¼ I1 ð0; nk; 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; nk; 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, " 1z 2 1 zb 1 ðm þ 1Þ ðm þ 1Þ C1 ðzÞ ¼ _ a þ 2m þ þ mþ 1bz 2 1bz 2 " zþb 1z 2 z þ b ðmÞ _ C1 ðzÞ a 1 þ bz 1bz 1 þ bz
1 zb 2 1 zb þ 2m þ m 2 1bz 2 1bz zþb zb 2 ðm1Þ C1 ðzÞ þ ðm1Þ 1 þ bz 1bz
zþb ðm2Þ 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
ðm1Þ
ðzÞ ¼ a0 ðz; a; bÞ C1 ðzÞ þ a1 ðz; a; bÞ C1 ðmÞ
ðm2Þ
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 ð1b2 Þ1=2 exp ðaÞC1 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 m1 : 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 nk 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 ! ðmiÞ
C1
ð0Þ=k!ðnkÞ!
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; n1Þ dz 0 1 d 2 ðmÞ þ a0 ð0Þ I1 ðm; n2Þ þ 2! dz þ
1 d n ðmÞ ðmÞ a0 ð0Þ I1 ðm; 0Þ þ a1 ð0Þ I1 ðm1; nÞ n! dz
þ
d ðmÞ 1 d 2 a1 ð0ÞI1 ðm1; n1Þ þ dz 2! dz ðmÞ
a1 ð0Þ I1 ðm1; n2Þ þ þ
1 d n ðmÞ a1 ð0Þ n! dz
d ðmÞ a ð0Þ dz 2
ðmÞ I1 ðm1; 0Þ þ a2 ð0Þ I1 ðm2; nÞ þ
I1 ðm2; n1Þ þ
þ þ
1 d 2 ðmÞ a2 ð0Þ I1 ðm2; n2Þ 2! dz
1 d n ðmÞ a2 ð0Þ I1 ðm2; 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 ðaÞ
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 ð1zÞ2 zm1 þ 1!
m 2
a2 ð1zÞ4 zm2 þ 2!
am ð1zÞ2m ; m!
ð4:40Þ
x þ k1 x for x > 0. ¼ ð1Þk 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 ð1Þnm þ 1 þð1Þnm þ 2 nm þ 1 1 1! n! dz
4 nm þ2
m a2 2m2 þ þð1Þn1 2 2! n1
m1 m a m m a 2m þ ð1Þn þ dmn ; m1 ðm1Þ! 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! mn mn 2 I1 ðm; n; a;0Þ ¼ exp ðaÞ Ln ðaÞ ; a m!
ð4:42Þ
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where Lan ðaÞ ¼
n X Cða þ n þ 1Þ ðaÞl ; Cðaþ l þ 1Þ l!ðnlÞ! 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 ð1Þi bi þ r 2 1=2 I1 ðm; n; a; bÞ ¼ ð1b Þ exp ðaÞ m þ n ðm2iÞ!i!ðn2rÞ!r! 2 i¼0 r¼0 #2 mn2ðirÞ n2r mn2ðirÞ C Ln2r ðaÞ ; ðn2rÞ!A
ð4:44aÞ
where _ 1=2 ¼ ½2að1 þ bÞ1=2 ¼ ð2a Þ ; A
¼ ½2að1bÞ1=2 ¼ ð2 B aÞ1=2 ;
C ¼ 2½ð1bÞð1 þ bÞ1=2
ð4:45Þ
mn2ðirÞ Ln2r 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 m2i n2r (there is a case in which one can have m2i < n2r 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
mn2ðirÞ
ðn2rÞ!A
mn2ðirÞ
C n2r Ln2r
ðaÞ
by ðm2iÞ!B
nm2ðriÞ
nm2ðriÞ
Cm2i Lm2i
ðaÞ:
Analogously if m < n, 2 1=2
I1 ðm; n; a; bÞ ¼ ð1b Þ
m!n! exp ðaÞ m þ n 2
" ½m=2 ½n=2 XX i¼0 r¼0
ð1Þi bi þ r ðm2iÞ!i!ðn2rÞ!r! #2
nm2ðriÞ
ðm2iÞ!B
nm2ðriÞ Cm2i Lm2i ðaÞ
ð4:44bÞ
provided n2r m2i 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 ð1b2 Þ exp ðaÞ 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 m1 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 1b 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
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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 lÞ2ðn1 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 1Þ2ðn2 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
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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 ).
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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 ðqlm ; qlm ; zm Þdql dql ;
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 21=2 ðqlm þ qlm Þ; qm ¼ 21=2 ðqlm qlm Þ;
j83
ð4:59Þ
m ¼ 1; 2; . . . N
and correspondingly for the s state q0m ¼ 21=2 ðqsm þ qsm Þ; q0m ¼ 21=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 dq1 dq2 dq2 ...dqN dqN i1=2 A p 1 m¼1 ð1z2 Þð1w 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 kq1 ;q2 ; qN k 2 qN N X X X 2 s 1 m þbs1 z1 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
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1
2
N
where m ¼ w
1wm 1zm ; 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
ð1Þm1 þ m2 þ mN þ njp bs1 bs2 . . . bsN bljp kjp X
X
þ
ð1Þmi1 þ mi2 ...miN1 þ njp þ njq
1i1 hi2 ...iN1 N 1 jp h jq N ði i ...i
Þ2
bsi1 bsi2 . . . bsiN1 bljp bljq kip 1jq2 N1 X ði1 Þ2 þ þ ð1Þmi1 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...N1 ðNÞ ð12...NÞ N þ1 k12...N W k12...N þð1Þ k12...N ; k1 ¼W W 23...N 23...N 23...N ! ! ! 23...N ð1Þ 13...N ð2Þ 12...N1 ðNÞ ð12...NÞ N þ2 k2 ¼W k12...N þW k12...N þð1Þ W k12...N ; 13...N 13...N 13...N .. . ð12...NÞ kN ¼ð1ÞN þ1 W
23...N
! ð1Þ k12...N þð1Þ2N W
12...N1
12...N1 12...N1
! ðNÞ
k12...N : ð4:69aÞ
ði i ...i Þ k121 2 N1 ¼W
i2 i3 ...iN1
!
34...N
þð1ÞN W ði i ...i Þ k131 2 N1 ¼W
þð1Þ
i1 i2 ...iN2
24...N W
ði
i1 i2 ...iN2
Þ
i1 i3 ...iN1 24...N
! ði
! ði Þ
2 k12...N
Þ
N1 ; k12...N
.. . ði1 i2 ...iN1 Þ ¼ðÞ1=2NðN1Þ W kN1;N
ði Þ
2 k12...N
N1 ; k12...N
ði1 Þ k12...N þW
24...N
!
34...N
!
34...N !
i2 i3 ...iN1
N þ1
i1 i3 ...iN1
ði1 Þ W k12...N
i2 i3 ...iN1 12...N2
! ði2 Þ þ ðÞ1=2NðN þ1Þ W k12...N
i1 i2 ...iN2 12...N2
! ði
Þ
N1 : k12...N
ð4:69bÞ
j 4 Calculational Methods for Intramolecular Distributions I , I , and I
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1
2
N
for all combinations of N1, indices 1i1 hi2 h...hiN1 N selected from among the indices 1;2;...;N and arranged in lexicographic order, and finally ði1 i2 Þ k12...N1 ¼W
i2 N
! ði1 Þ k12...N W
!
i2
ði i Þ
1 2 k1...N2;N ¼W
ði Þ
N1
.. . ði1 i2 Þ ¼ð1ÞN þ1 W k23...N
i2 1
i1
!
N
1 k12...N þW
ði Þ
2 k12...N ;
i1 N1
! ði1 Þ k12...N þð1ÞN 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 ðN1Þth compound matrix of W. Those of the system (4.69b) are elements of the ðN2Þth 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 Nk 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 W1 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 ð1Þ
X
þ
W X
12 . . . N 12 . . . N
þ
bs1 bs2 . . . bsN
ð1Þmi1 þ mi2 þ þ miN1 þ njp
1i1 hi2 h...hiN1 N 1 jp N
W
!2
i1 i2 . . . iN1 j1 . . . jp1 jp þ 1 . . . jN X X
!2 bsi1 bsi2 . . . bsiN1 bljp ð1Þm1 þ m2 þ mN2 þ njp þ njq
1i1 hi2 h...hiN2 N 1 jp h jq N
W
i1 i2 . . . iN2
!2
j1 . . . jp1 jp þ 1 . . . jq1 jq þ 1 . . . jN
bsi1 bsi2 . . . bsiN2 bljp bljq
þ þ ð1Þn1 þ n2 þ þ nN bl1 bl2 . . . blN ; ð4:71Þ
and finally bm1 m2 ...mN ;n1 n2 ...nN ¼ ð1Þm1 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 : : : wN1 wN zN1 zN w1 w2 w3 z 1 z2 z 3 : : : w1 w2 . . . wN z 1 z2 . . . zN
j87
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1
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 ð1Þm1 þ m2 þ m3 þ þ mN 2N bs1 bs2 . . . bsN
ð4:76aÞ
am1 m2 ...mN ;n1 n2 ...nN ¼ ð1Þn1 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 ð1Þm1 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 ð1Þn1 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
ð1Þm1 þm2 þ þmN w1 1 w2 2 . .. wNN
¼ 2N bs1 bs2 .. .bsN ð1w1 Þð1w2 Þ . .. ð1wN Þ; and B2 ðw1 ; w2 ; .. . ;wN ;1;1; .. . ;1Þ ¼ 2N bl1 bl2 .. .blN
1 X m1 ;m2 ;...;mN ¼0
ð1Þm1 þ m2 þ þmN w1 1 w2 2 . . .wNN
¼ 2N bl1 bl2 .. .blN ð1w1 Þð1w2 Þ . .. ð1wN Þ:
ð4:81Þ
j89
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90
1
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 1w 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
ð1Þn1 þ n2 þ þ nN zn11 zn22 . . . znNN
¼ 2N bl1 bl2 . . . blN ð1z1 Þð1z2 Þ . . . ð1zN Þ; 1 X ð1Þn1 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 ð1z1 Þð1z2 Þ . . . ð1zN 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 1zn 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 ! W1 . 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 W1 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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92
1
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 ð1wm Þð1zm Þ A1 ðw1 ; w2 ; . . . wN ; z1 ; . . . ; zN Þ ¼ s l B1 ðw1 ; w2 ; . . . ; wN ; z1 ; . . . ; zN Þ m¼1 bm ð1wm Þð1 þ zm Þ þ bm ð1 þ wm Þð1zm 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 ð1wm Þð1 þ zm Þ þ blm ð1 þ wm Þð1zm Þ
i
bsm ð1 þ wm Þð1zm Þ þ blm ð1wm 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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1
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 ...mi1 0miþ1 ...mN ;n1 ...nN IN
m1 ;...;mi1 ;miþ1 ;...;mN ¼0
...;mi þ1;...
!
...;ni ni ;...
n1 ;n2 ;...;nN ¼0 3 X
¼
(
ðiÞ ðiÞ 2dm1 ...mi1 0miþ1 ...mN ;n1 ...nN gm1 m2 ...mi1 0miþ1 ...mN ;n1 ...nN
m1 ;...;mi1 ;miþ1 ;...;mN ¼0 n1 ;n2 ;...;nN ¼0
2mi cm1 ...mi1 1miþ1 ...mN ;n1 ...nN IN ðiÞ
...;mi ;...
þð2dm1 ...mi1 1miþ1 ...mN n1 ...nN gm1 ...mi1 1miþ1 ...mN n1 ...nN ÞIN 2ðmi 1Þcm1 ...mi1 2miþ1 ...mN ;n1 ...nN IN 2ðmi 1Þcm1 ...mi1 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 ; ...; mi1 mi1 ; 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 ...ni1 0ni þ 1 ...nN IN . . . ; ni þ 1; . . . n1 ;...;ni1 ;ni þ 1 ;...;nN ¼0 ( ! 3 0:; . . . ; 0 X ðiÞ ðiÞ ð2d0...0;n1 ...ni1 0ni þ 1 ...nN g0...0;n1 ...ni1 0ni þ 1 ...nN ÞIN ¼ ni n1 ;...;ni1 ;ni þ 1 ;...;nN ¼0 ! 0:; . . . ; 0 ðiÞ ðiÞ 2ni c0...0;n1 ...ni1 1ni þ 1 ...nN IN þ ð2d0...0;n1 ...ni1 1ni þ 1 ...nN g0...0;n1 ...ni1 1ni þ 1 ...nN Þ . . . ; ni ; . . . ! ! 0:; . . . ; 0 0; . . . ; 0 IN 2ðni 1Þc0...0;n1 ...ni1 2ni þ 1 ...nN IN . . . ; ni 1; . . . . . . ; ni 1; . . . !) 0; . . . ; 0 ; 2ðni 1Þc0...0;n1 ...ni1 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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1
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 ysin j cos q sin y sin j cos y þ cos j cos q sin y sin q sin y W¼ cos j sin ysin 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 W1 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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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 cm1 .
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 Vvg 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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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, Vvg 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 c1 [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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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 cm1 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 ð1wm Þð1zm Þ exp 2ð1wm zm Þ ðmÞ G1 ðwm ; zm ; am ; 0Þ ¼ : ð4:99Þ 1wm zm Taking into account Equation 4.99 becomes
zm ¼ eivm t
that
and
wm ¼ ebhvm 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 1Þ1 corresponds to the number of where Zm ¼ ð1ebhvm Þ1 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 N1 accepting modes is obtained as product of terms (4.100) ( ) N1 Y X D2m GN1 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 eivm 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 itðVvg Þ D coth ðb hvm =2Þ þ g þ ðtÞ þ g ðtÞ 2 m m " #) Z1 X 1 hvg =2Þ1 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
104
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 ðh=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 ðgð0Þ hvi2 t2 =2Þ, so that at sufficient large time t, exp ðgð0Þ 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 ðVv ÞE = h t D R g g M sl 2 4 h 1
þ coth ðb hvg =2Þ1
Z1
dt exp it ðV þ vg ÞEM
= 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 hðVvg ÞEM hwnr iT ¼ cothðb hv =2Þ þ 1 exp g hD 4 2D2 h2 ( 2 ) hðV þ vg ÞEM ; ð4:111Þ þ cothðb hvg =2Þ1 exp 2D2 h2 which exhibits a Gaussian dependence on the energy parameter hðV vg ÞEM. 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 hVEM 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 ðEA =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 bhvg 1 and therefore coth b hvg =2 1. This permits us to rewrite Equation 4.111 in a form ( ) ffi v rffiffiffiffiffi ðVvg vs Þ2 2p g g 2 exp ; ð4:115Þ hwnr i ¼ Rsl 2h 2s2 s2
j105
j 4 Calculational Methods for Intramolecular Distributions I , I , and I
106
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 bhv 1; coth ðbhv=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 itðVvg 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 Vvg 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].
j107
j 4 Calculational Methods for Intramolecular Distributions I , I , and I
108
1
2
N
where
~ N Vvg 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 itðVvg 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 ðVvg nvm ÞEM = h ðaÞ ~ ¼ ; exp GN Vvg 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 " # ðVvg 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 Za1 expðb hna va Þ ½1 expðb 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 exp½na 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 2ð1wa za Þ 1 2 Ga1 wa ; za ; Da ; 0 ¼ 1wa za 2 1 2 ¼ Za exp Da ð2na þ 1Þ þ gþa ðtÞ þ ga 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 ð1Þn1 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 ð2hvÞ1 C 2 ½1 expðbhvÞ a Za1 . 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 ED E ð1Þ ð0Þ ð1Þ 0 Vsm;ln 2 ¼l2 jð0Þ jl xln jH0 BO jjs xsm s xsm jH BO jjl xln D ED 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 ED 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 ÞEs 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 ÞEl 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 ih ðqsn q0n Þ h jl i s js ðq ÞE 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 ÞEl ðq0 ÞÞðEs ðq0 ÞEc ðq0 ÞÞ
X n
ðEs ðq0 ÞEl ðq0 ÞÞðEs ðq0 ÞEc 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 ÞE ðq ÞÞðE ðq ÞE 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 2h2 4 s2 s Es ðq0 ÞEl 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 ÞEl ðq0 ÞÞðEs ðq0 ÞEc 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 qqg 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 ih 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 Þih 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 ðh2 ÞZ 1
X g;g0
g
ð2Þ
1 ð
Rsl Rsl
dt exp½iVcjtjIg ðtÞGN1 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 qqg n
n
1 0 19 = l qx qx sm A A þ kg Mgg ðslÞ@xln i dðEsm Eln Þ; h @xln ðqg q0g Þih 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 qqg n
n
1 0 19 = s qx qx sm A A þ Lgm ðslÞ@xln ðql q0m Þi dðEsm Eln Þ: @xln ðqm q0m Þih 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 qqg n
n
1 0 19 l qxsm = qx sm @xln ðqg q0g Þih 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 ih l A Mg0 g0 ðslÞ þ Lg0 g0 ðslÞ h g g0 6¼g fm g;fn g qqg n n m
1 0 19 l qxsm = qx sm @xln ðqg0 q0g0 Þih 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 ih l A Mg0 m ðslÞ h g6¼g0 m6¼g fm g;fn g qqg n
n
1 0 19 = s l qx x sm sm h s A dðEsm Eln Þ: @xln ðqm q0m Þih 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 ih l A Mgg0 ðslÞ þ Lgg0 ðslÞ h g g0 6¼g fm g;fn g qqg n
n
1 0 19 l qxsm = qx sm @xln ðqg0 q0g0 Þih 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 ðh2 ÞZ1
X g
g
1 ð
Rsl
dt exp½iVtcjtj 1
f½Mgg ðslÞ þ Lgg ðslÞ½Jgð0Þ ðtÞq0g 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,
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118
and Jg ðtÞ is defined by X Jg ðtÞ ¼ exp½mg ðit vsg þ hvsg =kB TÞexpðitng vlg Þ mg ;ng
1 ð
h2
ð
qxlng ðblg qlg Þ 1=2
dqlg dqlg qlg xsmg ðbsg qsg Þi 1=2
qqlg
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 ðqlg ; qlg ; zg Þ: qqsg g g g qqg
dqlg dqlg 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
#
z1 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 ðh2 ÞZ1
XX g
m6¼g
g
1 ð
Rsl
h ð0Þ ðtÞq0m dt exp½itVcjtj Mgm ðslÞ H m
1
i þ Lgm ðslÞ Hmð0Þ ðtÞq0m Kgð0Þ ðtÞGN ðtÞ
ð5:23Þ
where ð Hm ðtÞ ¼
1 ð
dql1 dql2 dql1 dql2 qlm rs1 ðqs1 ; qs1 ; w1 Þrs2 ðqs2 ; qs2 ; w2 Þrl1 ðql1 ; ql1 ; z1 Þrl2 ðql2 ; ql2 ; z2 Þ;
1
ð5:24Þ
and correspondingly ðm ¼ 1:2Þ m ðtÞ ¼ H
ð
ð1
dql1 dql2 dql1 dql2 qsm rs1 ðqs1 ; qs1 ; w1 Þrs2 ðqs2 ; qs2 ; w2 Þrl1 ðql1 ; ql1 ; z1 Þrl2 ðql2 ; 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
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rewrite Equation 5.16 as wð3Þ ðslÞ23 ¼ ðh2 ÞZ1
XX g
m6¼g
g
1 ð
dt exp½itVcjtj ½Mmg ðslÞ
Rsl 1
ð0Þ
ð0Þ ð0Þ þ Lmg ðslÞ½Dð0Þ g ðtÞq0g 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 dqlg qlg rsg qsg ; qsg ; wg i l rlg qlg ; qlg ; zg ; qqg
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
" z1 2bsg k2g z1 zg 1 g g g l s s G1 ðtÞ Dg ðtÞ ¼ ihbg 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
ih zg wg ih g expðit vg þ ¼ n hvg =kB TÞexpðitvg Þ ; 2 1wg zg 2
ð5:37Þ
g ¼ ðebhvg 1Þ1 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 ðh ÞZ
1
P P
g g0 6¼g Rsl
g
1 ð
dt exp½itVcjtjf½Mg0 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 dqlg0 qlg0 i
q s s s r 0 ðq 0 ; q 0 ; wg0 Þrlg0 ðqlg0 ; 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 ihbsg0 4 s l s 1 2 g0 þ bg0 zg0 g0 þ blg0 z1 bg0 w bg0 w g0 # 2bsg blg0 k2g0 z1 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 ðh2 ÞZ 1
1 ð 2 n XXX g ð0Þ ðtÞq0m Rsl dt exp½itVcjtj Mg0 m ðslÞ H m g g0 6¼g m¼1
1
o ð0Þ Igð0Þ ðtÞIg0 ðtÞGN ðtÞ; þ Lg0 m ðslÞ Hmð0Þ ðtÞq0m
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 ðh2 ÞZ1
XX g g0 6¼g
g
1 ð
Rsl
dt exp½itVcjtjf½Mgg0 ðslÞ 1
ð0Þ
þ Lgg0 ðslÞðCg0 ðtÞq0g0 Þ þ kg0 Mgg0 ðslÞg Kgð0Þ ðtÞGN ðtÞ;
ð5:42Þ
with 1 ð
Cg0 ðtÞ ¼
ð
dqlg0 dqlg0 qlg0 rsg0 ðqsg0 ; qsg0 ; wg0 Þrlg0 ðqlg0 ; 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 ðh2 ÞZ 1 w
X
ð0Þ
g
g
1 ð
dt exp½itVcjtjf½Mgg ðlsÞ
Rsl
1
þ Lgg ðlsÞ½ J g ðtÞq0g Kgð0Þ kg Mgg ðlsÞKgð0Þ ðtÞgGN ðtÞ;
ð5:44Þ
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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 dqlg qsg i s rsg ðqsg ; qsg ; wg Þi l rlg ðqlg ; qlg ; zg Þ: ð5:45Þ g qqg qqg 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 z1 g Þ ðbsg w g þ bg w 3
zg ðbsg z1 g
1 þ blg w g Þ
þ
1 w g 1 ðbsg w g
þ blg z1 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 ðh ÞZ þ Lgm ðlsÞ
1
XX g
m6¼g
g Rsl
ð0Þ H m ðtÞq0m
1 ð
h dt exp½itVcjtj Mgm ðlsÞ Hmð0Þ ðtÞq0m
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 ðh2 ÞZ1 w
XX g
m6¼g
g
1 ð
Rsl
dt exp½itVcjtjf½Mmg ðlsÞ 1
ð0Þ ð0Þ ð0Þ ð0Þ þ Lmg ðlsÞ½D g ðtÞq0g Ig ðtÞkg 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 dqlg qsg i
q s s s r ðq ; q ; wg Þrlg ðqlg ; qlg ; zg Þ qqsg g g g
01 12 3 2 1 1 2blg k2g w w w 1 g g g s 5Gg ðtÞ s s ¼ @ ihbg 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
ih wg zg ih g expðitvg þ ¼ hvg =kB TÞexpðitvg Þ : n 2 1wg zg 2 ð5:50Þ ð0Þ
The functions Nm ðtÞ appearing in Equation 5.48 are Nmð0Þ ðtÞ ¼ ihblm 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 ðh2 ÞZ1 w
XX g g0 6¼g
g
Rsl
1 ð
n dt exp½itVcjtj ½Mgg0 ðlsÞ
1 o ð0Þ þ Lgg0 ðlsÞ Cg0 ðtÞq0g0 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
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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 ðh ÞZ
1
X g
Rgsl
1 ð
dt exp½iVtcjtj½Mgg ðslÞ þ Lgg ðslÞKgð0Þ ðtÞGN ðtÞðq0g Þ;
1
ð5:56aÞ
wð3Þ ðslÞ22 ¼ ðh2 ÞZ1 þ Lgm ðslÞ
XX g
m6¼g
XX g
h ð0Þ ðtÞq0m dt exp½itVcjtj Mgm ðslÞ H m
1
Hmð0Þ ðtÞq0m
w ð3Þ ðslÞ23 ¼ ðh2 ÞZ1
1 ð
g
Rsl
m6¼g
i
g
Kgð0Þ ðtÞGN ðtÞ; 1 ð
ð5:57aÞ
dt exp½itVcjtj 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 ðh2 ÞZ1
XX
1 ð
g
g g0 6¼g
Rsl
dt exp½itVcjtj½Mgg0 ðslÞ 1
þ Lgg0 ðslÞKgð0Þ ðtÞGN ðtÞðq0g0 Þ:
ð5:59aÞ
Similarly, we find ð3Þ
2
ðslÞ21 ¼ ðh ÞZ w
1
X g
Rg sl
1 ð
dt exp itVcjtj Mgg ðlsÞ
1
þ Lgg ðlsÞ Kgð0Þ ðtÞGN ðtÞðq0g Þ;
ð3Þ ðslÞ22 ¼ ðh2 ÞZ1 w þ Lgm ðlsÞ
XX g
m6¼g
ð0Þ ðtÞq0m H m
ð3Þ ðslÞ23 ¼ ðh2 ÞZ1 w
XX g
m6¼g
1 ð
g
Rsl
ð5:56bÞ
h dt exp½itVcjtj Mgm ðlsÞ Hmð0Þ ðtÞq0m
1
Kgð0Þ ðtÞGN ðtÞ;
g
ð5:57bÞ
1 ð
Rsl
dt exp½itVcjtj½Mmg 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 ðh2 ÞZ1 w þ Lgg0 ðlsÞ
XX
g
Rsl
1 ð
j125
dt exp itVcjtj Mgg0 ðlsÞ
g g0 6¼g 1 ð0Þ Kg ðtÞGN ðtÞðq0g0 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 itVcjtj Kg0 ðtÞGN ðtÞ : 0 g g 1 ) 1 ð ð0Þ 0 g þ Rsl ½Mg0 g ðlsÞ þ Lg0 g ðlsÞ dt exp itVcjtj Kg0 ðtÞGN ðtÞ ðq0g Þ þ
8 X <X g
:
m
1 1 ð
g
ð0Þ dt exp itVcjtj 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 itVcjtj 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 itVcjtj 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 itVcjtj 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 itVcjtj 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 itVcjtj H 1
g
ð1 X g ½Rsl Lgm ðslÞ þ Rg M ðlsÞ dt exp itVcjtj 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 exp½itVcjtjKgð0Þ 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 expðivsm tÞexpðhvsm =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 tÞðw21 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 tÞðw12 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 1expðivm tÞ ¼ k12 1expðivm 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
