advances in multi-photon processes and spectroscopy. volume 15

ADVANCES IN MULTI-PHOTON PROCESSES AND SPECTROSCOPY

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S'*'-*

Volume 0 5 )

Edited by

SHLin Institute of Atomic and Molecular Sciences, TAIWAN & Arizona State University, USA

A A Villaeys Institut de Physique et Chimie des Materiaux de Strasbourg, FRANCE

Y Fujimura Graduate School of Science

Tohoku University, JAPAN

V f e World Scientific w b New Jersey • London • Singapore • Hong Kong

Published by

World Scientific Publishing Co. Pte. Ltd.

P O Box 128, Farrer Road, Singapore 912805

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British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

ADVANCES IN MULTI-PHOTON PROCESSES AND SPECTROSCOPY — Vol. 15

Copyright © 2003 by World Scientific Publishing Co. Pte. Ltd.

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Printed in Singapore by Mainland Press

V

Preface

In view of the rapid growth in both experimental and theoretical

studies of multiphoton process and multiphoton spectroscopy of atoms,

ions and molecules in chemistry, physics, biology, materials science, etc.,

it is desirable to publish an Advanced Series that contains review papers

readable not only by active researchers in these areas, but also by those

who are not experts in the field but who intend to enter the field. The

present series attempts to serve this purpose. Each review article is

written in a self-contained manner by the experts in the area so that the

readers can grasp the knowledge in the area without too much

preparation.

The topics covered in this volume are "Polarizabilities and

Hyperpolarizabilities of Dendritic Systems", "Molecules in Intense Laser

Fields: Nonlinear Multiphoton Spectroscopy and Near-Femtosecond To

Sub-Femtosecond (Attosecond) Dynamics", and "Ultrafast Dynamics and

non-Markovian Processes in Four-Photon Spectroscopy". The editors

wish to thank the authors for their important contributions. It is hoped

that the collection of topics in this volume will be useful not only to

active researchers but also to other scientists in biology, chemistry,

materials science and physics.

S. H. Lin

A. A. Villaeys

Y. Fujimura

vii

Contents

Preface v

Part One: Polarizabilities and Hyperpolarizabilities of

Dendritic Systems 1

Masayoshi Nakano and Kizashi Yamaguchi

Part Two: Molecules in Intense Laser Fields: Nonlinear

Multiphoton Spectroscopy and Near-Femtosecond

To Sub-Femtosecond (Attosecond) Dynamics 147

Andre D. Bandrauk and Hirohiko Kono

Part Three: Ultrafast Dynamics and non-Markovian

Processes in Four-Photon Spectroscopy 215

B. D. Fainberg

PART ONE

Polarizabilities and Hyperpolarizabilities of Dendritic Systems

3

Polarizabilities and Hyperpolarizabilities of Dendritic Systems

Masayoshi Nakano and Kizashi Yamaguchi

Department of Chemistry, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan

Contents

Abstract 6

1. Introduction 8

2. Polarizabilities and Hyperpolarizabilities of Dendritic Aggregate Systems 11 2.1. Aggregate Models 11 2.2. Density Matrix Formalism for Molecular Aggregate under Time-Dependent

Electric Field 14 2.3. Nonperturbative (Hyper)polarizabilities and Their Partition into the Contribution

of Exciton Generation 17 2.4. Off-Resonant Polarizabilities of Dendritic Aggregates 20

2.4.1. One-Exciton States and Their Spatial Contribution to a 21 2.4.2. Effects of Intermolecular Interaction and Relaxation on the Spatial

Contribution of One-Exciton Generation to a 24 2.4.3. Comparison of Polarizabilities of Dendritic Aggregates with Those of

One-Dimensional (Linear) and Two-Dimensional (Square-Lattice) Aggregates 26

2.5. Off-Resonant Second Hyperpolarizabilities of Dendritic Aggregates 28 2.5.1. Two Types of Dendritic Aggregates with and without a Fractal Structure 28 2.5.2. Spatial Contributions of One- and Two-Exciton Generations to y of D10 30 2.5.3. Effects of Intermolecular Interaction and Relaxation on the Spatial

Contributions on One- and Two-Exciton Generation to y of D10 31 2.5.4. Spatial Contributions on One- and Two-Exciton Generations to yof D25 33 2.5.5. Effects of Intermolecular Interaction and Relaxation on the Spatial

Contributions of One- and Two-Exciton Generations to yof D25 34 2.6. Near-Resonant Second Hyperpolarizabilities of Dendritic Aggregates 35 2.7. Summary 37

3. Polarizabilities and Hyperpolarizabilities of Dendrimers 41 3.1. Cayley-Tree-Type Dendrimers with ^-Conjugation 41 3.2. Finite-Field Approach to Static (Hyper)polarizabilities 43 3.3. Hyperpolarizability Density Analysis 47 3.4. Size Dependencies of a and yof Oligomer Models for Dendron Parts 53

3.4.1. Model Oligomers 53 3.4.2. Comparison of the a and y Values and Their Density Distributions of

Stilbene Calculated by the PPP CHF Method with Those by the B3LYP Method 55

3.4.3. Size Dependencies of a and y for Model Oligomers 57 3.4.4. a and yDensity Distributions for Model Oligomers 59

3.5. Second Hyperpolarizabilities of Cayley-Tree-Type Phenylacetylene Dendrimers 62 3.5.1. Calculation of yof D25 63 3.5.2. Comparison of the y Value and /Density Distribution of

Diphenylacetylene Calculated by the INDO/S CHF Method with Those by the B3L YP Method 63

3.5.3. y and y Densities of D25 64 3.6. Summary 67

4. Extensions of Models and Analysis 69 4.1. Master Equation Approach Involving Explicit Exciton-Phonon Coupling 69

4.1.1. Model Hamiltonian Involving Exciton-Phonon Coupling 70 4.1.2. Master Equation Approach 74

4.2. Analytical Expression of Hyperpolarizability Density 79 4.2.1. Analytical Formula of Hyperpolarizability Density 81 4.2.2. Imy Density of trans-Stilbene 89

4.3. Summary 91

5. Concluding Remarks 92

Acknowledgments 95

References 96

6

Abstract

Recently, a new class of polymeric systems, i.e., dendrimers, has attracted

much attention because of their high controllability of architecture and several

interesting properties, e.g., high light-harvesting and drug delivery properties.

"Dendrimers" are characterized by a large number of terminal groups originating in a

focal point (core) with at least one branch (linear-leg region) at each repeat unit. Such

attractive architecture is expected to have a close relation to various chemical and

physicsl functionalities of systems. In this review, we focus on the theoretical studies

on optical response properties such as polarizabilities and hyperpolarizabilities for these

systems from the view point of the relation among the unique architecture (dendritic

structure) and the optical response properties. As an example, we consider Cayley-

tree-type dendrimers, which are known to have fractal antenna architectures and to

exhibit high light-harvesting properties. Since exciton migration is pre imed to be

essential for the high light-harvesting properties for these dendrimers, we firstly

consider exciton models, i.e., molecular aggregate models with fractal antenna

structures composed of two-state monomers. We apply the numerical Liouville

approach to the exciton migration dynamics and investigate the influence of the multi-

step exciton states and relaxation processes among exciton states on the polarizabilities

(a) and second hyperpolarizabilities (f). It is found that the dominant spatial

contributions of excitons to a nd y are localized in linear-leg regions and reflect the

dendritic structures. The difference among the spatial dimensions of structures is also

found to remarkably affect the size dependency of intermolecular-interaction effects on

a. Second, the longitudinal a and y values of dendron parts (oligomers) involved in

7

Cayley-tree-type dendrimers composed of phenylene vinylene units and the /values of

a phenylacetylene dendrimer are examined using the molecular orbital (MO) method.

With the aid of (hyper)poarizability density analyses, such dendrimers with fractal

antenna structures are found to provide spatially localized contributions of electrons to

(hyper)polarizabilities similarly to the dendritic aggregate case and are predicted to have

the possibility of controlling the magnitude and sign of their (hyper)polarizabilities by

changing the size of generations and the connection forms between linear-leg regions.

These features are expected to be advantage to the molecular design of novel

(controllable) nonlinear optical materials.

8

1. Introduction

A new class of polymeric systems, i.e., dendrimers, has been synthesized and

has attracted much attention because of their unique chemical, transport and optical

properties [1-10]. In general, dendritic molecules are characterized by a large number

of terminal groups originating in a focal point (core) with at least one branch at each

repeat unit. The efficient excitation energy cascades to the core is predicted to be

caused by the fact that the molecular architecture provides an energy gradient as the

energy decreases as a function of position from the periphery to the core and has the

relaxation in the exciton states [11-19]. In particular, exciton funnels are expected to

exist along the antenna structures in Cayley-tree-type dendrimers, which possess a

fractal structure composed of linear-leg regions (para-connected units, e.g.,

phenylacetylenes) and meto-connected branching points, e.g., benzene rings. There

have been lots of theoretical and experimental investigations on such dendritic

aggregate models and dendrimers. Judging from these results, the efficient energy

transfer from the periphery to the core due to the energy gradient and the relaxation

among exciton states are expected to strongly depend on the architecture, the size and

the unit molecular species. On the other hand, there have been a small number of

studies on the optical response properties, e.g., nonlinear optical (NLO) properties, for

such systems [21-27] though the first-order optical processes, i.e., absorption and

emission of light, have been investigated actively.

For the past two decades, organic molecular systems with large NLO response

properties have attracted great attention both in scientific and technological fields due to

their large susceptibilities, fast responsibility and possibility of modification [28-32].

9

The organic NLO effects originate in the microscopic nonlinear polarization mainly

enhanced by ^-electron conjugation at the molecular level. Such microscopic

nonlinear polarization is characterized by the hyperpolarizabilities. A variety of

organic ^-conjugated molecular systems, e.g., donor-acceptor substituted polymeric

systems, polyaromatic systems and molecular crystal systems, have been intensely

investigated due to their low-excitation energies and large transition properties.

Considering the feature that the molecular (hyper)polarizabilities remarkably depend on

the slight variations in quantities (transition energies and transition moments) of excited

states, the optical response properties of such dendritic systems are predicted to

sensitively reflect the effects of the unique structure, i.e., dendritic structures with

fractal antenna shapes. In this review, we firstly consider dendritic molecular

aggregates modeled after phenylacetylene dendrimers: simple aggregate models are

composed of two-state monomers which interact with each other by the dipole-dipole

coupling [19,20,24,25,33,34]. The structure of exciton states and its dependency on

the configuration of monomers are elucidated. The polarizabilities (a) and second

hyperpolarizabilities j) are calculated using the definition of nonperturbative

(hyper)polarizabilities in the numerical Liouville approach (NLA) [35-39]. The spatial

contributions of one- and two-excition generations and the relaxation (among exciton

states) effects to a and y are elucidated in dendritic aggregate systems. The size

dependencies of intermolecular-interaction effects on a of dendritic aggregate systems

are also compared to those of aggregate systems with other spatial configurations, i.e.,

linear-chain (one-dimensional) and square-lattice (two-dimensional) systems. Second,

we consider supramolecular systems composed of ^-conjugated units (phenylacetylenes

or phenylene vinylenes) linked with each other by para- or meta-connection. In order

10

to elucidate the effects of fractal structures with meta- and para-connected benzene

rings on the a and y, we consider several oligomers modeled after dendron parts.

Further, the several components of y of a real Cayley-tree-type phenylacetylene

dendrimer (D25) composed of 24 units of phenylacetylenes are investigated using the

coupled-Hartree-Fock (CHF) calculations [40] at the INDO/S approximation level [41].

In the analysis of these (hyper)polarizabilities, the spatial distributions of

(hyper)polarizability densities [42-45], which are defined (in the static case) as the

derivatives of charge densities with respect to applied electric fields, are employed in

order to elucidate the structure-property relation of (hyper)polarizabilities for these

dendritic systems.

The present review is organized as follows. In Sec. 2, the calculation methods

of exciton dynamics and nonperturbative a and y are described in the numerical

Liouville approach [35-39]. The spatial contributions of exciton generations to a and y

are analyzed and the size and spatial-dimension dependencies of a and y are

investigated in the off-and near-resonant regions. In Sec. 3, the size dependencies of a

and yof phenylene vinylene oligomers and the /values of a phenylacetylene dendrimer

(D25) are investigated using (hyper)polarizability density analysis. Section 4

describes an extension of the procedure treating exciton dynamics, i.e., an explicit

treatment of relaxation originating in the exciton-phonon coupling, and presents an

analytical expression of dynamic (hyper)polarizability densities, which are useful for

investigating the imaginary y describing two-photon absorption (TPA) spectra of

dendritic systems. In conclusion (Sec. 5), the results obtained here are discussed from

the viewpoint of the relation between the (non)linear optical response properties and the

unique spatial structures of systems, i.e., dendritic structures with fractal-antenna

shapes.

11

2. Polarizabilities and Hyperpolarizabilities of Dendritic Aggregate Systems

2.1. Aggregate Models

The spatial architectures of fractal-antenna dendrimers, e.g., phenylacetylene

dendrimers, shown in Fig.l are composed of branchings at the meta positions of the

benzene nodes [11-19]. The fractal nature in the number of units involved in the

linear-leg regions is caused by the meta-position substitutions. Other types of

branchings, i.e., para- and ortfco-positions, will generate a linear chain and will

terminate the tree structure due to the steric hindrance. It is found from recent

calculations that ^-conjugation is well decoupled at the zneta-positions [16,26,27], and

that the meta-position branchings allow some distortions from a planar structure, the

feature of which enables us to synthesize larger dendritic systems by overcoming steric

hindrance.

As simple models of the fractal antenna dendrimers, we consider molecular

aggregates (D4, D10, D25, D58 and D127 shown in Fig.2) in which a monomer is

assumed to be a dipole unit (a two-state monomer, which is illustrated by an arrow)

arranged as modeled after the dendritic structure shown in Fig. 1 [19,24]. The klh

monomer possesses a transition energy, £*,(= E\ — E\~), and a transition moment, /if2.

This approximation is considered to be acceptable if the intermolecular distance (Rkl) is

larger than the size of a monomer. These aggregate models possess slight

intermolecular interactions between adjacent legs at the branching points since their

intermolecular distance (i?V3) is larger than that (/?) in the same leg regions. It is

12

noted that such decreases in the intermolecular interactions at the branching points are

similar to the situation in real dendrimers, in which the meta-branching points destroy

the TT-electronic conjugation between adjacent linear legs. For two dipoles k and /, the

angle between a dipole k(l) and a line drawn from the dipole site k to / is 9k(8l ). The

Hamiltonian for the aggregate model (composed of N monomers) is written by [19]

= ttEt<% + jtrft<;<;[(cos(0t, - ^ ) - 3 c o s V o s ^ ) / 4 K « f ; < « , ; ,

(1)

where the first (H0) and second (f/int) terms represent a noninteracting Hamiltonian and

a dipole-dipole interaction, respectively. N represents the number of dipole units.

E\ is an energy of the state ik for monomer k, and /if,, is a magnitude of a transition

matrix element between states ik and i'k for monomer k. The a* and aL represent

respectively the creation and annihilation operators for the ik state of monomer k. By

using the basis for the aggregate W]^ •••(pfN )h which is constructed by a direct

product of a state vector for each monomer (k',*/. the matrix elements of H0 i

represented by [19]

is

and the matrix element of Hint is expressed by [19]

13

(3)

where f(0ki,6,t,Ru) is

f(eki,eit,Rkl) = (cos(0t/ -6 l k ) -3cos0 t ; cos0, t ) / (4»e^) , (4)

By diagonalizing the Hamiltonian matrix H (Eq. (1)), we can obtain

eigenenergies E,"88 and eigenstates yf88) (/ = 1, ..., M), where M is the size of

the basis used. The M is l+NCt+NC2 since we consider one-and two-excitons since

they are at least necessary for reproducing reliable /which we focus on in this review.

The transition dipole matrix element (//j)?8) in the direction of applied field for this new

state model are also calculated by [19]

J\ Jn Ji.-J'n

(5)

where ^ .. represents a transition dipole matrix element (in the direction of

polarization vector of applied field) between jk and j'k of monomer k. It is noted that

the transition moments between the ground and one-exciton states, and those between

14

one- and two-exciton states, exist in the present model.

2.2. Density Matrix Formalism for Molecular Aggregate under Time-Dependent

Electric Field

The time evolution of a molecular aggregate model is treated by the following

density matrix formalism [19]:

^Ik P ( ° = [Ht)'p(t)] ~irpi0' (6)

where pt) indicates the molecular aggregate density matrix and the second term on the

right-hand side of Eq. (6) represents the relaxation processes in the Markoff

approximation. The total Hamiltonian H(t) is expressed by the sum of the aggregate

Hamiltonian, H , and aggregate-field interaction, V(t),

M M

Ut) = #age + V(t) = J^E^b^b, - I f t f ( m F c o s a ) i i > , (7) /=i ;,r=i

where F is an external field amplitude in the direction of x since the incident fields are

assumed to be plane waves with frequency ft) and wave vector Jfc travelling

perpendicular to the molecular aggregate plane and the polarization vector is parallel to

x axis. The b\ and b, represent respectively the creation and annihilation operators

for the / state of aggregate state-model obtained in the previous section. It is noted that

15

we take m = 1 for considering the case of a and m = 3 for considering /for the third-

order harmonic generation (THG) [28-32]. The matrix representation of Eq. (7) is

expressed as

p„,(0 = -i(l - Sir)E™Pll,(t) - «X(Mm(Opm,(0 - P„„(t)Vmr(t)) - (rP(0) r , (8) m

where Vu.(t) = -^(mF cos cor) (m = 1 or 3).

The relaxation term (-(Tp(OV) in Eq. (8) can be considered as the following

two types of mechanisms [46-48].

-(rP(0)„ = -r„p„(o+%mlpmmt) (9)

and

-(rp(ov=-r rp„,(o. (io)

Equations (9) and (10) describe the population and coherent-relaxation mechanisms,

respectively. The y r ( * yn) represents a feeding parameter. The off-diagonal

relaxation parameter is expressed as

r„, = ±(r„ + r,r)+r,;, dD

and

r =r 1 iv * n'

(12)

16

where r,'r is the pure dephasing factor. In this review, since we assume a closed

system, the factor yu is related to the decay rate as

M

Equation (8) is numerically solved by the fourth-order Runge-Kutta method.

The density matrix representation in the aggregate basis yp]^ "'"<?")) a t time ' is

calculated by

M

A„, IM , i(o=i«-<|^r)p f f.(^? iK'-<>. d4)

where pir(t) is calculated by Eq. (8). Using this density matrix, the polarization p(t)

is calculated by

M

P^) = 5X;2 IM rKPn.a r„-j,.k .•» W . d5) '1.12 'n. 'i.'i '"w

where the transition matrix element u, , ..... , is represented by ~'l-'2 'H'll''2 '/V r J

7=1 1=1 \nfl J

17

Here /i/J, represents a transition dipole matrix element (in the direction of polarization

vector of applied field) between i, and /,' of monomer /.

2.3. Nonperturbative (Hyper)polarizabilities and Their Partition into the

Contribution of Exciton Generation

In this section, the definitions of nonperturbative polarizability and the second

hyperpolarizability are explained [35]. The nonlinear optical response of systems to

the external polychromatic electric fields is represented by a polarization p(t), which is

the induced electric dipole moment. The polarization p(t) can be written with the

dipole moment operator fi and density matrix operator pit) as

p(t) = Tr[fip(t)]. (17)

The polarization pt) can be expanded as a Fourier series in the external frequencies in

the steady state as follows.

J»(0= X^^W^*"^-'"'"'*- (18) /ni,w, mL

Here, pm ,„ (ft)) is the Fourier component at frequency co = m,©, + m2a>2 + ... +mL(0L.

Although the definitions of nonperturbative (hyper)polarizabilities are given for various

18

(non)linear optical processes, we here consider a nonperturbative polarizability

(a"0"(-a);ft))) and the second hyperpolarizability (yno"(-3fi);ft),ft),ft))) in the third-

harmonic generation (THG). In the polarizability case, the polarization p(r) with a

frequency 0) is induced by incident beams with amplitudes £(ft)). The response

p(co) is expressed by the perturbative polarizability a(-co,co) and hyperpolarizabilities

such as y(-(o;o),(o,-(o):

p(m) = p,(ft>) + pn_1(co) +... =a(-(o;<D)e((D) + ... , (19)

where the Fourier components pm m (ft)) satisfy the condition w, +... + mL = 1. The

a"°"(-ft);fi)) can be defined as

am(-m;a» = ^ , (20)

where p(co) is obtained by the Fourier transformation of induced polarization time

series. In the THG process, the polarization with a frequency 3ft) is induced in the

system by incident beams with an amplitude £(»). The third-harmonic response

p(3(o) can be expressed by using the perturbative hyperpolarizabilities as follows.

p(co) = pm(3m) + />210(3ft)) + p120(3ft)) + p012(3ft)) +

p021(3ffl) + pwl(3<0) + p102(3o)) +...

= 27y(-3ft);ft),ft),ft))e3(ft)) +....

19

Here, the Fourier components pm m (o) satisfy the condition m, +... + mL = 3. The

y"on(-3ffl;©,fl),ffl) in THG can be defined as

ynon(-30);(0,(0,co) = />(^ft)) , (22) ' 27e3(ft»

where p(3o>) is obtained by the Fourier transformation of induced polarization time

series. As can be seen from Eqs. (19)-(22), for weak incident fields, the

nonperturbative (intensity-dependent) (hyper)polarizabilities can reduce to the

perturbative (intensity-independent) (hyper)polarizabilities. In contrast, for strong

incident fields, the contributions of intensity-dependent terms involving higher-order

hyperpolarizabilities will be important.

We apply this nonperturbative approach to the calculation and analysis of

(hyper)polarizabilities for aggregate systems [24,25] since the nonperturbative approach

has an advantage of easily treating relaxation effects, which originate in exciton-phonon

coupling, in the (near)resonant region. As mentioned in Sec. 1, the exciton migration

dynamics under the (near)resonant condition is a recent topical subject, so that we

employ our nonperturbative approach, which is applicable to the calculation of

(hyper)polarizabilities both in off- and on-resonant regions, though this approach is not

necessary for the case of weak off-resonant fields. From Eqs. (17), (20) and (22), the

a(-(o;(0)(= a"0"(-(o;(o)) and y(-3ft);(B,fi),fi))(=ynon(-3ffl;ft),ft),tt))) are also expressed

as

20

a^co;co) = ±aa_^±2^f(0\ (23) a>b a>b EV10)

and

y(-3co;co,co,co) = ±ya„h = l ^ S u T < <24>

where a and b indicate the aggregate basis V •••<)'")> a n d P*T'(fi)) a n d PfT'C3®)

are Fourier components of the real parts of density matrix element (p™\t)) in the

aggregate basis. For the case of a, we consider only one-exciton states, while for the

case of y, we consider one- and two-exciton states since their perturbation expressions

involve one-exciton states and one- and two-exciton states, respectively. Equation

(23) indicates that the total a can be partitioned into the virtual excitation contribution

(aa_A) between bases a and b. In the one-exciton case, either a or b is |11...1) (1: the

ground state of monomer), so that we can elucidate the spatial contribution of one-

exciton generation to aby showing the one-exciton distribution, e.g., |121...1) (2: the

excited state of monomer). Similarly, Eq. (24) indicates that the total y can be

partitioned into the virtual excitation contribution (ya_h) between bases a and b. In the

two-exciton model, the fiah exists between the |11--1) (1: the ground state of

monomer) and one-exciton states or between one- and two-exciton states, so that the

Ya-b represents the contribution of the virtual one- or two-exciton generation

represented by basis pair a-b. We can elucidate the spatial contribution of ya_h by

showing exciton distribution represented by bases a and b.

2.4. Off-Resonant Polarizabilities of Dendritic Aggregates

21

2.4.1. One-Exciton States and Their Spatial Contribution to a

Figure 2 shows dendritic molecular aggregate models (D4, D10, D25, D58 and

D127), which involve all the same dipole units. The transition energy and transition

moment of the dipole unit (monomer) are assumed to be 38000 cm"1 and 10 D,

respectively. The intermolecular distance in the linear-leg region is fixed to 15 a.u. It

is noted that the magnitudes of these parameters do not affect the qualitative results for

the relative size-dependency of intermolecular-interaction and relaxation effects on a

and their spatial contributions to a. Since these aggregate models possess slight

intermolecular interactions between adjacent legs at the branching points since their

intermolecular distance (15-73 a.u.) is larger than that (15 a.u.) in the same leg regions,

the reduction of the intermolecular interactions at the branching points is similar to the

situation in real phenylacetylene dendrimers, in which the meta-branching points

destroy the ^-electronic conjugation between adjacent linear legs [16,26,27]. The

one-exciton states / (/ = 2,..., M) and the magnitudes of transition moments between the

ground and the one-exciton states for these aggregate models are shown in Fig. 3.

There are found to be explicit multi-step energy states (with significant transition

moments) for these dendritic aggregates (except for a small dendritic aggregate (D4)) in

contrast to the case of linear aggregates which possess almost only a one-exciton energy

state with a significant transition moment [20]. It is also found that the number of such

multi-step energy states and the energy width of their distribution increase with the

increase in the dendritic-aggregate size, corresponding to the number of generations

involved in the dendritic structure. These multi-step energy structures can be

explained by the J- and //-aggregate-type interactions [49,50] involved in the dendritic

22

fractal-antenna structure, and are known to be important for the exciton migration from

the periphery to the core.

Sine the relaxation effects in one-exciton states are predicted to be essential for

the exciton migration from the periphery to the core of dendrimer, we consider the

relaxation terms in Eq. (6). The factor ya in Eq. (13) is determined by an energy-

dependent relation: yu =/(£ ,fsg-£',a8g) (f = 0.1, i(>/): one-exciton state). This

indicates that the population of higher energy state faster decreases and damps into the

lower energy states. The external single-mode laser with 100 MW/cm2 has a

frequency (3000 cm"1), which is sufficiently off-resonant with respect to one-exciton

states. It is expected from the perturbational formula (Eq. (25)) that the off-resonant a

can be qualitatively well described in the one-exciton model. The division number of

the one optical cycle of the external field used in the numerical calculation is 80, and the

a is calculated by using 100 optical cycles far after an initial non-stationary time

evolution (2000 cycles). In general, the feature of off-resonant a is described by the

following perturbational formula:

"^••m)'2l^fT^fy <25)

Namely, all the virtual excitation contributions are positive in the first off-resonant

region and are more enhanced in the case of smaller transition energies and larger

magnitudes of transition dipole Lu, J.

In the present case (off-resonant a of systems with weak intermolecular

interactions), the total a value is found to almost linearly enhance with the increase in

23

the number of monomers. This feature indicates that the magnitude of off-resonant a

for the present systems is primarily determined by the number of monomers and their

relative configurations with respect to the applied field. Therefore, we confine our

attention to the intermolecular-interaction and relaxation effects on a. The a value

including only intermolecular-interaction effects is referred to as aim, the a value

including both intermolecular interaction and relaxation effects is done to as aint+rel.

The a value including neither effects is referred to as anon. The monomer value of

anon is 89.97 a.u. Figures 4(a) and (b) show the size dependencies of intermolecular-

interaction (aint -a n o n ) and relaxation (aint+re] -« i m ) effects on a per unit dipole for

these aggregate models, respectively. It is found that the intermolecular-interaction

and relaxation effects provide positive and negative contributions to a, respectively (see

the legend of Fig. 4). According to the fitting procedure in previous studies [43,51],

Aa (= Gfint - anon or aint+re| - orint) per unit dipole are fitted by the least squares to

vJM\ = p + ± + J-, (26) \ N ) N N2

where the extrapolated values for infinite N are (\Aa\/N)N_>X =10''. The fitting

parameters p, q and r are listed in Table 1. In this plot, we use the data from D10 to

D127, while the data for D4 are not used since D4 possesses only one generation and

exhibits a distinct feature of structure compared with other dendritic aggregates

composed of multi-generations. In this model study, dendritic aggregates larger than

D127 are not considered since such larger systems are predicted to have non-planar

24

structures, which do not conform to the present models. The nonlinear size-

dependency of (.OCj^-a^/N suggests that the intermolecular-interaction effects on

transition energies and moments nonlinearly depend on the number of monomers in

linear-leg regions parallel to the polarization vector of the applied field. Since it is

well-known that the /-aggregate-type interaction decreases the dipole-allowed

excitation energies, the size-dependency of the intermolecular-interaction effect on aIN

is predicted to be determined by the number of ./-aggregate-type interaction pairs

involved in linear-leg regions in each model. For large-size aggregates, the relaxation

effects (negative contribution) are found to overcome the intermolecular-interaction

effects (positive contribution), so that the total a values for large-size dendritic

aggregates tend to slightly decrease.

The spatial one-exciton contributions to the off-resonant a for D10 (non-fractal

structure) and D58 (fractal structure) are shown in Fig. 5. All the contributions are

found to be positive in sign as expected from Eq. (25). In agreement with our

prediction, the dominant contributions for both systems are shown to be distributed in

linear-leg regions parallel to the polarization vector of the applied field. This feature

can be understood by the fact that these linear-leg regions possess dominant interaction

with the applied field since their dipole units are parallel to the polarization vector of

applied field.

2.4.2. Effects of Intermolecular Interaction and Relaxation on the Spatial

Contribution of One-Exciton Generation to a

25

We first consider the effects of intermolecular interaction on a. Figures

6(D10-a) and (D58-a) show the effects of the intermolecular interaction on the one-

exciton generation contributions to a. The total a values for both systems are found to

be slightly enhanced by the intermolecular interaction as compared to their amn,

respectively. The primarily enhanced contributions of one-exciton generation are

shown to be located in the linear leg regions parallel to the polarization vector of the

applied field. Although the contributions in these linear-leg regions of both systems

are found to be different among generations, D10 shows much smaller variation than

D58 (for example, compare regions a and e(e') in D10 with regions a, e(e'), and f(P) in

D58). This feature relates to the fact that D10 possesses smaller number of monomers

in its linear legs (only a monomer per each liner leg) as compared to D58. For D58,

the contributions in inner-leg regions are distinctly larger than those in outer-leg

regions: a > e(e') > f(P) and h(h') > g(g') > i(i'), j(j') and k(k'). Namely, the inner

linear-leg regions for D58 primarily contribute to lower one-exciton energy states due to

their larger number of ./-aggregate-type pairs as compared to outer-leg regions. Such

differences in the spatial contributions to a between D10 and D58 reflect their

architectures, i.e., fractal and non-fractal structures, respectively, since the number of

monomers in linear-leg regions for the fractal architecture increases as going from the

periphery to the core.

Next, the relaxation effects in one-exciton states on a are considered. In

contrast to the intermolecular-interaction effects, the contributions (negative in sign)

occur significantly both in inner and in outer linear-leg regions. The relaxation effects

in a large-size aggregate (D58) are also found to remarkably reduce the aim+re, as

26

compared to the small-size aggregate (D10) case. Judging from our introducing way

of relaxation terms, these reductions are presumed to originate in the decoherence

process, i.e., the decrease in the off-diagonal density matrices, due to the phase-

relaxation effects (see Eqs. (11) and (13)).

Although the total effects (intermolecular-interaction and relaxation effects) for

D10 are found to slightly enhance a, those for D58 are found to reduce a (see Figs. 5

(DIO-c) and (D58-c)). This feature implies that the relaxation effects (negative

contribution) are more significant for larger-size dendritic aggregates since larger-size

aggregates possess larger multi-step energy widths which lead to larger relaxation

factors.

2.4.3. Comparison of Polarizabilities of Dendritic Aggregates with Those of One-

Dimensional (Linear) and Two-Dimensional (Square-Lattice) Aggregates

In this section, we compare the difference in polarizabilities (a) of a fractal

antenna dendritic aggregates (D) and those of linear- (L) and square-lattice (S)

aggregates (see for example Fig. 7) [20]. The transition energy and transition moment

of the dipole unit (monomer) are assumed to be 38000 cm"1 and 5 D, respectively. The

intermolecular distance in the linear-leg region is fixed to 15 a.u. The polarization

vector is fixed to be parallel with the direction (x) of dipoles shown in Fig. 7(L25J),

(L25H) and (S25).

We first investigate the size dependencies of off-resonant a(=axjr) (under a

field with frequency co = 3000 cm1) for models (L), (S) and (D) (see Fig. 8(a)). It is

found that the size dependencies of a for (L) and (S) are nearly equal to each other,

27

while they are larger than those for (D). This feature can be explained by the fact that

all the dipoles involved in (L) and (S) are parallel to the polarization vector of the

applied field in contrast to the case of (D), where partial dipoles are parallel to the

polarization vector of the applied field. Namely, this size-dependency of off-resonant

a is predicted to be mainly determined by the relative configuration among dipoles and

applied field. The size dependencies of intermolecular-interaction effect on a IN,

i.e., (aint -«„„„)/ N, are shown in Fig. 8(b). Similarly to the case in Sec. 2.4.1, the

intermolecular-interaction effects on a/N for these systems are shown to have

nonlinear dependencies on their sizes. It is also noted that the size dependency of

intermolecular-interaction effects on a/N for (D) is different from those for (L) and

(S), which are similar to each other. Namely, (aint - anon) / N values for (L) and (S)

rapidly increase with increasing the number of units and then exhibit the saturation of

(aint -«„„„)/ N, while (aint -anon)/7V for (D) exhibits a relatively small increasing

rate in the small-size region but preserves a large size-dependency in the large-size

region. This unique feature can be interpreted as follows. Since it is well-known that

the /-aggregate-type interaction decreases the allowed excitation energies, these

intermolecular-interaction effects on a/N are nonlinearly related to the number of J-

aggregate-type interaction pairs involved in each model. Namely, since (L) and (S)

systems have a large number of /-aggregate-type interaction pairs in the linear-leg

regions and their numbers increase monotonously with the increase in the number of

units, these systems are predicted to exhibit a rapid increase and the subsequent

saturation of a/N. In contrast, for (D) systems the number of 7-aggregate-type

interaction pairs increases in a fractal manner as increasing the size of (D). Namely, in

28

the small-size region, only the short 7-aggregate regions are involved, while in the

larger-size region, longer 7-aggregate regions are involved. As a result, in the large-

size region, the size dependency of (ainl - otnoa)IN for (D) is predicted to be larger than

that for (L) and (S).

We next elucidate the spatial contributions of intermolecular interaction

(aint - anon) of each dipole unit in these aggregates to a (see Fig. 9) by the procedure

presented in Sec. 2.3. The /-aggregate-type interaction is known to reduce the

transition energy, while the //-aggregate-type interaction is known to enhance it.

Namely, the /-aggregate-type interaction tends to enhance a, while the //-aggregate-

type interaction tends to reduce it. This supports the feature that the enhancement

(reduction) of 7 (W)-aggregate model occurs except for both-end regions (see Figs.

9(L25J) and (L25H)). It is also found for (S25) that the contributions in the middle

region are more enhanced in the column direction, while those are somewhat more

reduced in the row direction. This spatial variation in the intermolecular-interaction

contribution can be well explained by considering the features observed in (L25J) and

(L25H). In contrast, the magnitudes of intermolecular-interaction contributions to a of

(D25) are found to enhance as going from the periphery to the core. This feature

originates in the fractal structure of (D25), where the each generation is well decoupled

and the lengths of /-aggregate-type linear legs more enlarge in the internal generations.

2.5. Off-Resonant Second Hyperpolarizabilities of Dendritic Aggregates

2.5.1. Two Types of Dendritic Aggregates with and without a Fractal Structure

Two types of dendritic molecular aggregates with different sizes (D10 and D25

29

shown in Fig. 2) are considered. They involve all the same dipole units. The

excitation energy and transition moment of the dipole unit (monomer) is assumed to be

38000 cm'1 and 5 D, respectively. D10 has a non-fractal structure in the sense that the

number of dipole units involved in regions c and a, which belong to adjacent different

generations, respectively, is equal to each other. In contrast, D25 has a fractal

structure since the number of dipole units involved in region a is larger than that in

region c. The one- and two-exciton states for these dendritic aggregates are shown in

Fig. 10. Both one- and two-exciton states are found to have multi-step energy

structures, which are distributed around E2l (excitation energy of monomer) and 2£21,

respectively. Such multi-step energy structures can be explained by the J- and H-

aggregate-type interactions involved in the dendritic structure. Apparently, the widths

of the multi-step energy states for D25 are larger than those for D10. This can be

understood by the fact that the number of intermolecular interaction pairs in D25 is

larger than that in D10. The relaxation factor yn in Eq. (13), which is essential for the

exciton migration from the periphery to the core, is determined by an energy-dependent

relation: yH = f(E,m -£fEg) (f = 0.02, /(>/) : one- or two-exciton state). The

external single-mode laser with 10 MW/cm2 has a frequency (3000 cm"1), which is

sufficiently off-resonant with respect to exciton states. The division number of the one

optical cycle of the external field used in the numerical calculation is 100, and the / i s

calculated by using 20 optical cycles after an initial non-stationary time evolution (300

cycles).

The static yis described by the following perturbational formula [38,53,54]:

30

y _ yQ) + y(H) + yW)

• y C ^ . ) 2 ^ ) 2 y (K. i ) 4 y (K.i)2(/Q2 (27) /|=2 ^nl n=2 ^ n l m,n=2 £'nl i-'ml

Here, /in, is the transition moment between the ground (1) and the nth excited states,

Hnn is the transition moment between the mth and the nth excited states, the difference

of dipole moments between the ground and the nth excited states, and Enl is the

transition energy given by ( £ „ - £ , ) . From Eq. (27), in the static case, the

contributions of types (I) (y(l)) and (ni)(y(III)) are positive in sign, whereas the

contribution of type (II) (y(11>) is negative. It is noted that the type (I) contribution of

the present dendritic aggregates vanish since the dipole moments of the ground and

excited states for the dendritic aggregates are zero due to their spatial symmetry.

2.5.2. Spatial Contributions of One- and Two-Exciton Generations to yof D10

In this section, we elucidate the exciton contribution to y of a dendritic

aggregate (D10) with a non-fractal structure. From Eq. (27), types (II) and (III)

virtual excitation contributions are found to be more enhanced in the case of smaller

transition energies and larger magnitudes of transition moments. Figure 11 shows the

sum of partitioned yu_b on each site for D10 including intermolecular-interaction and

relaxation effects. It is found that main parts of total, one-exciton and two-exciton

contributions are distributed in leg regions a, e and e'. This feature can be understood

by the fact that these regions possess dominant interaction with the applied field since

31

their dipole units are parallel to the polarization vector of the applied field. Each

exciton contribution (one-exciton contribution = -141330 a.u., two-exciton

contribution = 99000 a.u.) is found to have mutually opposite sign and to have much

larger magnitude than total y (-42330 a.u.). Sine the one-exciton contribution

(corresponding to type (II)) is found to be larger than the two-exciton contribution

(corresponding to type (III)), the total y value is found to be negative in sign. This

feature reflects the fact that the present dendritic aggregate is composed of two-state

monomers which provide only type (II) (negative) contribution. Namely, in the

present model, positive contribution (type (III)) originates in the two-exciton

generation composed of excitations on two different sites.


Contributions on One- and Two-Exciton Generation to yof D10

Similarly to Sec. 2.4.1, we use the following notation. The /value including

only intermolecular-interaction effects is referred to as yim. The 7 value including

both intermolecular-interaction and relaxation effects is referred to as yint+rei. The y

value including neither of effects is referred to as ynon.

Firstly, we only focus on the effects of intermolecular interaction on y so that

the relaxation terms are omitted. Figure 12(a) shows the effects of the intermolecular

interaction on the one- and two-exciton generation contributions to y. The total y value

is found to be slightly reduced by the intermolecular interaction compared to

7non(yint-7non= -761 a.u.). The total y value is found to be composed of the

32

negative contribution of one-exciton generation (type (II)) and the positive contribution

of two-exciton generation (type (III)). The primary contributions of one-exciton

generation appear on sites a, e and e', while their magnitudes are almost equal to each

other. This feature originates in the configuration of dipoles parallel to the applied

field. Namely, /-aggregate-type interaction effects, which enhance the magnitudes of

one-exciton contributions to y are more enhanced in such regions. In contrast, the

two-exciton contributions are shown to be distributed extensively in the whole region of

the aggregate compared to the one-exciton case. This feature reflects the fact that the

two-exciton generation (on two different sites) occurs in more spatially extensive region

than the one-exciton generation (on a single site). In the present case, the one-

exciton contribution is found to be larger than the two-exciton contribution. This is

explained as follows. Although the /-aggregate-type intermolecular interaction

decreases the exciton energies and then enhances one- and two-exciton contributions,

the reduction of exciton energies more contributes to type (II) compared with type (III)

in the present two-exciton models.

Next, we consider the relaxation effects in one- and two-exciton states on y.

It is found from Fig. 12(b) that the relaxation effects slightly enhance the total y

compared to yim (yinc+rel - yin,= 13 a.u.). The magnitudes of relaxation effects on one-

and two-exciton contributions to /are shown to be slightly reduced, respectively, and

the degree of the variation is more remarkable on the periphery regions (c, c', e and e')-

Judging from our introducing way of relaxation terms, these reductions are presumed

to originate in the decoherence process, i.e., the decrease in the off-diagonal

aggregate density matrices, due to the phase-relaxation effects. Since the relaxation

33

factor ytt in Eq. (13) is determined by an energy-dependent relation, this reduction

effect is expected to be more remarkable in higher exciton energy states (where the

exciton is distributed on periphery regions). This feature corresponds to the relaxation

effects on the spatial exciton contributions shown in Fig. 12(b).

It is found that the intermolecular interaction and relaxation in one-and two-

exciton states provide effects with mutually opposite sign on 7. Although the total

effects are found to slightly reduce / i n the present D10 case (see Fig. 12(c)), larger

relaxation factor or smaller intermolecular interaction may change the sign of total

effects on 7.

2.5.4. Spatial Contributions on One- and Two-Exciton Generations to yof D25

The spatial one-, two- and total exciton contributions to off-resonant 7 (-

135050 a.u.) of D25 (with a fractal structure) involving intermolecular-interaction and

relaxation effects are shown in Fig. 13. All the contributions of one-exciton

generations are found to be negative in sign (-930570 a.u.), while those of two-exciton

generations are positive in sign (795520 a.u.). In the present model, the one-exciton

contribution is larger than the two-exciton contributions, so that the total /value is

negative in sign. It is found from Fig. 13 that main contributions are distributed in leg

regions f, f, h, h \ e, e' and a. This feature can be understood by the fact that these

regions possess dominant interaction with the applied field since their dipole units are

parallel to the polarization vector of the applied field. Similar features are observed in

the two-exciton contributions. The features of these spatial contributions in D25 (with

a fractal structure) are equal to those in D10 (with a non-fractal structure).

34


Contributions of One- and Two-Exciton Generations to /of D25

We first investigate the effects of the intermolecular interaction on the one- and

two-exciton generation contributions to /(Fig. 14(a)). The total /value is found to be

slightly reduced by the intermolecular interaction (yint - ynon = -23968 a.u.) since the

one-exciton contributions (-30103 a.u.) are found to much exceed the two-exciton

contributions (6135 a.u.). Similarly to the D10 case, the enhancement of the

magnitudes of one- and two-exciton contributions is ascribed to the reduction of

excitation energies caused by the /-aggregate-type intermolecular interaction in linear-

leg regions f, P, h, h', e, e' and a. The relative magnitude of one- and two-exciton

contributions is determined by the relative magnitude of types (II) and (III)

contributions (Eq. (27)). In contrast to the D10 case, the contributions are put in the

magnitude order: (a), (e and e') and (f, f\ h and h'). This implies that the

coherency between the exciton and the ground states is more enhanced in more internal

linear legs parallel to the polarization vector of applied field. This feature can be

explained by the larger interaction of dipoles in such regions with the applied field and

the significant decrease in the exciton energies due to the larger /-aggregate-type

interactions as going from the periphery to the core. This increase in the spatial

contribution of exciton generation from the periphery to the core is found to originate in

the fractal structure of D25 with weta-branching points.

Next, we consider the relaxation effects in one- and two-exciton states on y.

It is found from Fig. 14(b) that the relaxation effects very slightly reduce the total y

35

(7int+rei ~7im= -4.9155 a.u.). The magnitudes of relaxation effects on one- and two-

exciton contributions to y are found to slightly decrease, respectively. These slight

changes seem to be caused by small values of yu (f= 0.02) in Sec. 2.5.1 in contrast to

those (f = 0.1) in Sec. 2.4.1. The dominant reductions occur in regions f, f, h, h \ e, e'

and a and their magnitudes slightly decrease as going from the periphery to the core.

This feature is caused by the fact that the decoherence process is more remarkable in

periphery regions (higher energy exciton states). Although the total relaxation effects

on yfor D10 and D25 are shown to have contributions with mutually opposite signs, the

magnitudes of one- and two-exciton contributions are reduced in both systems by

relaxation effects. In the present case, one- and two-exciton contributions have

mutually opposite signs and the relaxation parameters are small (yu (f= 0.02) in Sec.

2.5.1), so that the features of total relaxation effects on yfor D10 and D25 are more

complicated as compared to the case of a (Sec. 2.4.1).

It is found that the intermolecular interaction and relaxation in the one(two)-

exciton state provide effects with mutually opposite sign on y. Although both the

intermolecular-interaction and relaxation effects are found to slightly reduce y in the

present case (see Fig. 14(c)), the features of sign and magnitudes can be changed

depending on the relative magnitudes between the relaxation factor and the strength of

intermolecular interaction.

2.6. Near-Resonant Second Hyperpolarizabilities of Dendritic Aggregates

In the near-resonant region, the contributions of intermolecular interactions and

36

relaxation effects to response properties are expected to be remarkably enhanced.

Therefore, we investigate the near-resonant 7 (real part) for a dendritic-antenna

aggregate (D25) (Fig. 7) composed of two-state monomers (E2l = 38000 cm"' and [i2S -

5D) for example [23]. The frequency of applied electric fields polarized to x-axis is

taken to be 37950 cm"1, which is close to the highest exciton state with the largest

transition moment. We consider three types of models (A, B and C): (A) involves both

intermolecular-interaction and relaxation terms, (B) involves only the intermolecular-

interaction term, and (C) involves neither intermolecular-interaction nor relaxation term

(Yu = f(Eigg ~ Etgg) </ = 0 0 2> li>i) • one- or two-exciton state, see Eq. (13)). The

difference between (A) and (B) shows the relaxation effect and that between (B) and (C)

shows the intermolecular-interaction effect. Figure 15(a) shows the 7value (-3.7771 X

104 a.u) and the spatial contribution of exciton generation to y for model (A). Although

most of sites contribute to 7 negatively, there are some sites contributing positively to 7

in the core region. Figures 15(b) and (c) show the spatial contributions of

intermolecular-interaction and relaxation effects on 7, respectively. It is found that the

intermolecular-interaction effect decreases 7 at all sites. Especially, the contributions

of dipoles, which are parallel to x-axis, are shown to be large. As is well known, one-

and two-exciton energy states are split, respectively, by the intermolecular interaction

and then construct a multi-step energy structures of exciton states. Since the external

field frequency is higher than many allowed one-exciton states except for the highest

allowed one-exciton state, intermolecular-interaction effect on 7 is predicted to become

negative in sign. In contrast to the off-resonant case in Sec. 2.5, the difference in the

magnitude of the contribution among generations is hardly observed. This seems to be

37

ascribed to the fact that the external field is resonant with plural exciton states and their

contributions are more complicated. It is found that the relaxation contributions are

positive in sign and their distributions in the periphery region are larger than those in the

internal region. For the present non-relaxation model (B), the 7 value tends to

negatively diverge at the one-photon resonance point, while model (A) (involving both

intermolecular-interaction and relaxation effects) is generally predicted to exhibit a

broad spectrum of 7 in the resonant region. This feature supports the positive

contribution of the relaxation effects on 7 (Fig. 15(c)). In the present model, the

relaxation terms of high-lying exciton states, in which the excitons are mainly

distributed in the periphery region, are larger than those in lower-lying exciton states, in

which the excitons are mainly distributed in the internal region. This feature

corresponds to the feature of spatial distributions of relaxation effects.

2.7. Summary

In Sec. 2, we first investigated the off-resonant a for several sizes of model

dendritic aggregates (D4, D10, D25, D58 and D127) with planar structures and

elucidated the features of spatial contributions of one-exciton generation to a for fractal

and non-fractal dendritic aggregate systems. It was found that the size dependency of

intermolecular-interaction enhancement of oc/N is nonlinear and is closely related to the

number of ./-aggregate-type interaction pairs involved in each system. The

intermolecular interaction and relaxation in one-exciton states were shown to provide

slight effects with mutually opposite sign on a: the former enhances the a, while the

latter reduces that. The a values of the present fractal dendritic systems were found to

38

be dominantly contributed by the one-exciton generation distributed in linear-leg

regions, especially the central-leg region, parallel to the polarization vector of applied

field, while for non-fractal dendritic systems, the dominant intermolecular-interaction

contribution in each generation showed slight differences among generations. This

feature can be understood by the fact that the fractal structure provides a larger number

of 7-aggregate-type interaction pairs, which lead to the decrease in the one-exciton

energies, as going from the periphery to the core in contrast to the non-fractal structure.

In contrast to these intermolecular-interaction effects, the relaxation effects were shown

to provide mutually similar contributions both in inner and in outer linear-leg regions

for these systems. The reduction of a by the relaxation effects is presumed to be

caused by the phase-relaxation effects. In the present models, the reduction of a was

more significantly observed in larger-size aggregates than in smaller-size aggregates.

We also compared the a of fractal antenna dendritic aggregates (D) with those of linear-

(L) and square-lattice (S) aggregates. It was found that intermolecular-interaction

effects enhance the a/iVand their features are determined by the relative configurations

of dipoles with respect to apllied fields and the number of 7-aggregate-type interaction

pairs involved in each model. It was also found that the size dependency of

(aint - a n o n ) / N for (D) is different from that for (L) and (S): the am-anon)IN

values for (L) and (S) exhibit rapid increase and then become saturated, while

(ccint - amn) / N for (D) exhibits a relatively small increasing rate in the small-size

region but preserves a large increasing rate in the large-size region. This unique size

dependency of (aint - a n o n ) / N for (D) is predicted to originate in the fractal structure

of(D).

39

Second, we investigated the effects of intermolecular-interaction and relaxation

on the one- and two-exciton contributions to y for two different sizes of dendritic

molecular aggregates, D10 (with a non-fractal structure) and D25 (with a fractal

structure), by visualizing the spatial contributions of exciton generations to off-resonant

y. These aggregates were found to provide negative y, which are composed of the

negative contribution of one-exciton generation (type (II) in Eq. (27)) and the positive

contribution of two-exciton generation (type (HI) in Eq. (27)). The main contributions

of one- and two-exciton generations were found to be distributed in linear-leg regions

parallel to the polarization vector of the applied field since the interactions among their

dipole units and applied field are larger than others. It is also noted that there is a

distinct difference in the intermolecular-interaction effects on the spatial contribution of

exciton generation between fractal and non-fractal structures: for D10 (with a non-

fractal structure) the contributions of exciton generations distributed on such linear-leg

regions were found to be almost equal to each other, while for D25 (with a fractal

structure) those were found to be more enhanced as going from the periphery to the core

since more internal regions have more /-aggregate-type interaction pairs, which lead to

the more decrease in exciton energies and thus more enhance the contribution to y. In

contrast, the relaxation effects in the one- and two-exciton states were found to decrease

their contributions especially on the periphery regions. This is ascribed to the

decoherence of off-diagonal aggregate density matrices between the ground and higher-

lying exciton states (mainly distributed on periphery regions) caused by the relaxation

effects. In the case of small relaxation and large intermolecular-interaction effects

(present case), large-size fractal dendritic aggregates composed of two-state monomers

are predicted to exhibit significant enhancement of contributions of exciton generations

40

to y as going from the periphery to the core due to the larger number of 7-aggregate-

type-interaction pairs in internal generations. In the case of large relaxation and small

intermolecular-interaction effects, however, the reduction of \y\ for these dendritic

systems could be observed. Further, dendritic aggregates composed of three-state

monomers have the possibility of exhibiting positive total 7 since the type (III)

(positive) contribution exists even in a monomer.

Third, we investigated the characteristics of near-resonant 7 (real part) for the

dendritic molecular aggregate D25. In this case, the intermolecular-interaction and

relaxation effects were found to be negative and positive in sign, respectively. The

magnitudes of intermolecular-interaction effects were found to be much larger than

those of relaxation effects, so that the total 7 becomes negative in sign. The

intermolecular-interaction effects were found to be large at the dipole unit parallel to the

external field, while the relaxation effects were found to cause the larger contribution

especially in the periphery region. In contrast to the off-resonant case, both effects

were shown to dramatically change the near-resonant 7 values. This indicates that the

intermolecular-interaction and relaxation effects are extremely important to control the

near-resonant 7 value for dendritic systems. In particular, the relaxation effects among

exciton states are expected to be unique for dendritic systems with fractal structures due

to their multi-step allowed exciton states. We will not expect such complicated

changes in 7 for conventional one- and two-dimensional systems since the allowed

exciton state (which dominantly contributes to 7) is almost only one in such systems.

On the analogy of the present results, the fractal supramolecular systems with

electronically well decoupled generations are predicted to possess attractive features:

41

the contributions of one- and two-exciton generations to (hyper)polarizabilities vary for

each generation from the periphery to the core, and the relaxation effects largely

contribute to the (near)resonant (hyper)polarizabilities especially in the periphery

regions. Such features are also expected to be observed more remarkably and more

complicatedly in the (near)resonant higher-order response processes in dendrimers. In

the next section, we investigate the a and y for several supramolecules including

dendrimers by using molecular orbital (MO) calculations.

3. Polarizabilities and Hyperpolarizabilities of Dendrimers

3.1. Cayley-Tree-Type Dendrimers with ^-Conjugation

In this section, we consider supramolecular systems with fractal antenna

structures, i.e., Cayley-trees (or Bethe-lattices), shown in Fig. 16. As mentioned in Sec.

1, the efficient excitation energy cascades from the periphery to the core are observed

experimentally, and the existence of exciton-migration (energy-transfer) funnels along

ordered multi-step exciton states are predicted. These features are found to be closely

related to the fractal structure. As predicted in Sec. 2, the spatial contributions of a

and y for Cayley-tree-type dendritic model aggregates also reflect the fractal

architectures. For phenylacetylene dendrimer (Fig. 16(A)), there are ^-conjugations in

its linear-leg (para-substituted phenylacetylene) regions and their decoupling at the

meto-branching points (benzene rings) are predicted to be essential for generating multi-

step energy structures, each of which corresponds to the exciton distribution well

localized in each generation. Considering the fact that ^-conjugation sensitively

affects optical response properties, the above unique feature of ^-conjugation in fractal

42

dendrimers is expected to also provide a remarkable influence on (hyper)polarizabilities.

In particular, the size-dependencies of a and y for fractal dendrimers are expected to

sensitively reflect the change in the ^-conjugation lengths (linear-leg regions) from the

periphery to the core in a fractal manner.

Therefore, we first investigate the longitudinal a and y for several sizes of n-

conjugated model oligomers involved in the Cayley-tree-type fractal structure made of

phenylene vinylenes (Fig. 16(B)) (Sec. 3.4). For these fractal-structured meta-

oligomers, several different-length linear-leg regions (made of phenylene vinylene units

connected through para-substitutions of benzene rings) are mutually connected through

meta-substitutions of benzene rings. For comparison studies, non-fractal-structured

oligomers, i.e., para- and non-fractal-structured /rceta-oligomers, which only involve

para- and meta-substitutions, respectively, are also considered. Second, we consider

the y of an actual Cayley-tree-type phenylacetylene dendrimer D25 (see Fig. 16(A))

with a fractal dimension D (1 < D < 2), so that the structure-property relation in /for

such dendrimers is considered based on orientationally averaged values, ys, involving

several components of yijkl.

In Sec. 3.2, we explain the finite-field (FF) approach [55] using the coupled

Hartree-Fock (CHF) method, which is applied to the evaluation of a and /values. In

order to elucidate the structure-property relation in (hyper)polarizabilities, we perform

the hyperpolarizability density analysis (Sec. 3.3), which can visualize the contributions

of electrons in arbitrary spatial regions to (hyper)polarizabilities. The relations among

the fractal architecture, (hyper)polarizabilities and the spatial contributions of electrons

to 7 are explored. These structure-property relations are also useful for designing

43

novel nonlinear optical (NLO) materials.

3.2. Finite-Field Approach to Static (Hyper)polarizabilities

The molecular optical response can be described in a power series expansion of

the molecular polarization p' (the i component of polarization oscillating at angular

frequency ft)) by an external electric field:

p\(0) = X«,^y '(®,) + JJPiJkFi(o)l)F

k(co2) + ^riJklFJ(coi)F

k(co2)Fl((oi) + ... (28)

J jk jkl

Here, F'(ft),) represents the i component of the applied electric field oscillating at a>,.

a,j, PiJk and yijkl are the tensor components of the polarizability, the first

hyperpolarizability and the second hyperpolarizability, respectively. In the case of the

medium intense electric field, the right-hand side of Eq. (28) can be approximated by

the first-order term with respect to F, while in the presence of the strong electric field,

the higher-order terms cannot be ignored. These higher-order terms give rise to

various nonlinear optical properties. Real and imaginary parts of (hyperpolarizability

characterize the (non)linear polarization and (non)linear absorption of the system,

respectively. In the off-resonant (non)linear optical phenomena, the static

(hyper)polarizabilities are considered to be good approximations to the off-resonant

frequency dependent (hyper)polarizabilities.

There have been various theoretical approaches for calculating

(hyper)polarizabilities. The sum-over-state (SOS) approach [28-32] based on the time-

44

dependent perturbation theory (TDPT) is useful for elucidating the contribution of

transitions between electronic excited states to (hyper)polarizabilities (see Eqs. (25) and

(27)). However, the application of this approach at the ab initio MO level is limited to

the relatively small-size molecules since this needs transition properties (transition

moments and transition energies) over many excited states, which are hard to describe

precisely at the ab initio MO level at the present time. Alternatively, the approaches

using numerical or analytical derivatives of total energy with respect to the applied

electric field are widely employed for calculating static (hyper)polarizabilities.

Particularly, the FF approach using the CHF method at the semiempirical [56-63] and

ab initio [51] MO levels, various electron-correlated ab initio MO methods [61-63] and

the DF methods [64-68] has been widely employed for calculating static

(hyper)polarizabilities. Although most of the ab initio MO and DF methods using

extended basis sets cannot be applied to the calculation of (hyper)polarizabilities of

excessively large size of the target molecule such as D25 (see Fig. 16), the

semiempirical MO methods will be appropriate for the qualitative or semi-quantitative

description of (hyper)polarizabilities for such large size systems. Actually, these

computationally efficient semiempirical MO methods have been remarkably successful

at both predicting and explaining observed y for ^-conjugated organic molecules [56-

60]. We here employ the two types of semiempirical MO methods, i.e., PPP [69] and

INDO/S [70] methods, to obtain the total energies of a molecule under the applied

electric fields.

We explain the FF approach to the calculation of static a and y. The total

Hamiltonian in the presence of a uniform electric field F is expressed as

45

H = H0+J^F-r,-J^ZlF-Rl, (29)

where indices i and / signify electrons and nuclei, respectively. Z, is the atomic

number of the /th nucleus and H0 is the field-free Hamiltonian. The total energy E can

be obtained as the expectation values (XP|//|VF) for the wavefunctions *P in the

presence of the electric field. Similarly, the dipole moment \l is expressed as

M=Cr|Iz,*,-Xim (30)

The differentiation of total energy E with respect to F' gives

dE_ dF'

d*F dF'

H\V) + V\^-\Y) + V\H d*F dF •^7 • (3D

If f i s the true wavefunction, the first and third terms on the right-hand side of Eq. (31)

is equal to zero by the Hellmann-Feynman theorem [71]. The variational methods

such as CHF satisfy the theorem. If the Hellmann-Feynman theorem is satisfied, the

dipole moment defined by Eq. (30) can be expressed as

/ i ' dE_ dF'

(32)

The total energy and dipole moment can be expanded as the power series of the applied

field:

46

E = E0- 2 > i F - Wa^F1 - \Y,PiikFF'Fk -^y^F'F'FF' -..., (33)

i ^ ij J ijk 4 ijkl

and

A*' = Hi+J.a^ +YJPljkFJFk +^YiiklF

jFkF' -..., (34)

where ^ is the permanent dipole moment. The Hellmann-Feynman theorem asserts

that Eqs. (33) and (34) are compatible. We use the definition of (hyper)polarizability

based on Eq. (33). From this relation, the static atj can be expressed by

a,. d2E

dF'dF' (35)

and the static yijU can be expressed by

d*E lijkl ' dFdFjdFkdF'

(36)

To calculate afj, we use the following numerical differentiation formula:

au = (EF) + E(-F') - 2£(0))/(F f (37)

where E(F') represents the total energy in the presence of the field F. The

longitudinal components of au and /,,,, are sufficient for describing their optical

properties of one-dimensional systems, while for systems with larger spatial dimensions,

orientationally averaged (hyper)polarizabilities are considered. For example, the

orientationally averaged second hyperpolarizability, ys, in THG is expressed (in the

static case) by [28-32]

47

7S = - (Ynu + Yyyyy + Y + ty„y + ^Y yyzz + ^Y^ ) • (38)

Since the phenylacetylene dendrimer such as D25 has a planar architecture, we should

consider the averaged ys in Eq. (35), which includes two types of components of % i.e.,

YiUi and yUjj (i, j = x,y,z). These /values are numerically calculated by

and

ym = E(3F) - 12E(2F) + 39 E(F) - 56£(0)

+39E(-F)-l2E(-2F) + E(-3F)/36(F'f,

Ym = -E(F,FJ) + E(-F,FJ) + E(F,~Fj) + E(-F,-FJ)

+2[E(FJ) + E(-FJ ) + E(F') + E(-F')] /6(Fi )4.

(39)

(40)

In order to avoid numerical errors, we use several minimum field strengths. After

numerical differentiations using these fields (from 0.0005 a.u. to 0.002 a.u.), we adopt a

numerically stable a and y.

3.3. Hyperpolarizability Density Analysis

In the time-dependent perturbation theory, the hyperpolarizability is described

by the virtual excitation process involving ground and excited states [28-32]. The

appropriately approximated perturbational expression is well-known to be useful for

understanding the mechanism of the hyperpolarizability by partitioning it into the

48

contributions of virtual excitation processes [32] (see Eq. (27) in Sec. 2.5.1). On the

other hand, in the FF approach, we have proposed an alternative method for analysis of

hyperpolarizabilities, which is referred to as "hyperpolarizability density analysis".

This analysis method can elucidate the spatial contributions of electrons to

hyperpolarizabilities. Such spatial contributions of electrons to hyperpolarizabilities

provide a local view of hyperpolarizability though it is generally a global value. This

is expected to be useful for chemists to understand the mechanism of

hyperpolarizabilities intuitively and pictorially and to construct the design rule of

nonlinear optical molecules. Although the analysis method can be extended to treat

the dynamic (frequency-dependent) hyperpolarizabilities and to divide the spatial

contributions to each virtual excitation process [72], we here briefly explain the static

hyperpolarizability density analysis in the FF approach.

The charge density function p(r,F) can be expanded in powers of the electric

field F in the same manner as the expansions of energy and dipole moment (see Eqs.

(33) and (34)). The charge density function p(r,F) are expressed as

pr,F) = p^\r) + ^\r)F' +\-yPmr)F'Fk +\yjp

m(r)F'FkF' + ... .(41)

The dipole moment can be expanded as follows.

49

H'(F)s-jripr,F)d3r

= -\r'p*\r)d'r - £ Jr'p< "(rKVF —^pfir^rPF* i *•• jt

—Zj'pgWWF*?-.... 3> ill

(42)

Here, r' is the i component of the electron coordinate. From Eqs. (34) and (42), the

polarizability and second hyperpolarizability can be expressed by

and

where

and

av=-jr>pi;\r)d3r.

Yukl=--jriP$(r)dir

P<1)(r) = ^ V' dF>

0»'(r) = d'P Pjkl() dF>dFkdF<

(43)

(44)

(45)

(46)

These first and third derivatives, pf\r) and p'«(r), of charge density with respect to

external electric fields are referred to as the polarizability (a) and second

hyperpolarizability (f) densities, respectively. Other hyperpolarizability densities are

also defined in a similar manner [43]. These quantities can be calculated in a good

50

precision by discretizing the space and by using the efficient numerical differentiation

method. The numerically precise value of this density is useful for investigating the

effects of basis sets and electron correlations on the hyperpolarizabilities of three

dimensional systems in the ab initio MO method [32,66,67]. For example, we can use

the charge densities over a three-dimensional grid of points, which is obtained by the

routines involved in the GAUSSIAN 94/98 program package [73].

In the case of evaluating the characteristics of y of large-size molecules using

the semiempirical MO method, the hyperpolarizability density analysis can be

performed approximately using the Mulliken charge densities. We consider the charge

density (PS) partitioned into each atomic orbital s in the Mulliken population analysis:

P.M^lPWMr)* (47) s,t

where 0s(r) denotes the atomic orbital s. From this equation,

Jp(r)dV = £(«)» = £ Jp,(r)</V. (48) s s

Here, (S)sl and (P)a are overlap and bond-order matrix elements, respectively.

Equation (47) can be rewritten as follows using charge density function p(r) divided

into each atomic orbital s.

51

(49)

y-'jn

The following approximation is applied to the charge density function ps(r).

ps(r) = (PS)J(r-rs). (50)

This approximation implies that ps(r) is concentrated to the center rs of atomic orbital

s. Substituting Eq. (50) into Eq. (49), the following relation is obtained.

21'ik * V J

^r;(PS)ZklF>F*F'+..).

(51)

From this equation, we obtain approximate a and yas

and

«,~-X^PS)^> (52)

r^-ll^Ci. <53>

where

52

and

d\PS)sl K hsJkl dF>dFkdFl

(54)

(55)

These quantities represent the a and /densities in the Mulliken approximation. Here,

r's represents the ;' component of the coordinate of the atom located at the center of the

atomic orbital s. The quantities (PS)"',, and (PS)"',,,, which are respectively or,., and

ymi densities of atomic orbital s, are calculated by the following numerical

differentiation method:

and

(PS)% =(PSUF)-(PS)J-F')-2(PS)SS(0)/(F)2, (56)

(PS)% = (PSU2F) - (PS)J-2F)- 2((PSUF)- (PS)J-F))/2(Ff,

(57)

where (PS)SS(F) is the Mulliken charge density of atomic orbital s in the presence of

the electric field F'.

In order to explain a method for analysis employing the plots of

hyperpolarizability densities, we consider a pair of localized /density, p^(r), as an

example (see Fig. 17). The positive sign of p^\r) implies that the second derivative

of the charge density increases with the increase in the field. As can be seen from Eq.

53

(44), the arrow from positive to negative p~\r) shows the sign of the contribution to

yiiU determined by the relative spatial configuration between the two plf/ir). Namely,

the sign of the contribution to ym becomes positive when the direction of the thick

arrow coincides with the positive direction of the coordinate system. The contribution

to ym determined by p^\r) of the two points is more significant, when their distance

is larger.

3.4. Size Dependencies of a and yof Oligomer Models for Dendron Parts

3.4.1. Model Oligomers

We consider the size dependencies of longitudinal y for three types of

oligomers shown in Fig. 18 in order to elucidate the effects of fractal structure involved

in Cayley-tree-type dendrimers on y. For Pn oligomers (n = 1, 4, 7, 11 and 16)(Fig.

18(a)), all the repeat units (phenylene vinylenes) are connected at the para-positions of

benzene rings, while MNFn (Fig. 18(b)) and MFn (Fig. 18(c)) oligomers (n = 1, 4, 7, 11

and 16) involve connections of the repeat units through meta-substitutions. The

difference between MNFn and MFn oligomers appears in the number of repeat units

involved in the linear-leg (para-substituted) regions between benzene rings with meta-

substitution. MNFn oligomers involve the same number of repeat unit (one unit) in a

linear-leg region, while for MFn oligomers the number of repeat units involved in the

largest linear-leg region increases with the increase in the chain-length. This implies

that MNFn and MFn oligomers exhibit non-fractal (uniform) and fractal structures

concerning the number of repeat units in linear-leg region, respectively. Since we only

54

focus on the qualitative chain-length dependencies of longitudinal a O a n ) and y

(= y^) for these systems, we use the standard bond lengths (Rl = 1.396 A, R2 = 1.473

A and R3 = 1.336 A) and bond angles (0 = 120°) of stilbene (Fig. 18(a) PI) for all these

models. The z-axis is taken to be along the longitudinal direction of each model

oligomer.

It is well-known for polymeric systems that the derealization of ^-electrons is

primarily contribute to the longitudinal (hyper)polarizabilities [28-32]. In order to

obtain the qualitative chain-length dependencies of a and y, we need to calculate their

values for larger-size systems. Although it is well-known that ab initio calculations

using extended basis sets and high-order electron correlation methods can provide semi

quantitative (hyper)polarizabilities for small molecules such as hydrogen fluoride, such

rigorous calculations can hardly be performed for large-size molecules like the present

oligomers. Fortunately, it has been known that the basis-set dependence of a and y

become small when the lengths of oligomers are sufficiently large and the scaling

procedure using the FF semi-empirical MO calculations can reproduce the tendency of

the results obtained by ab initio large-scale calculations for the longitudinal a and /of

large oligomers [51,56]. Therefore, we apply the CHF method in the PPP

approximation, which treats only ^-electrons and needs much lower cost than the ab

initio approximation, to the investigation of the chain-length dependencies of model

oligomers involving large chain-length systems. Further, to check the reliability of the

PPP CHF results, we compare the a and /values for stilbene (PI) calculated by the PPP

CHF method with those by the B3LYP method [74,75], which is one of the hybrid

density functional approach and is known to be able to reproduce the semi-quantitative

(hyper)polarizabilities for relatively small organic molecules [64-68], though the

55

reliable y values for large-size systems such as linear-chain oligomers and the

qualitatively correct size dependency of a and y cannot be reproduced using the present

exchange-correlation functionals [68].

3.4.2. Comparison of the a and y Values and Their Density Distributions of

Stilbene Calculated by the PPP CHF Method with Those by the B3LYP

Method

Before discussing the a and /values and their density distributions calculated

by the PPP CHF method, we examine the qualitative reliability of those quantities for a

small-size system, i.e., stilbene (PI), involving a phenylene vinylene unit, by comparing

the PPP CHF results with the B3LYP results. In this study, we use the 6-31G**+d

(£rf =0.0523 on carbon atoms) basis set, which is known to be able to well reproduce at

least the longitudinal components of (hyper)polarizabilities for ^-conjugated organic

molecules [32,66,67]. The a and y values and their density distributions for stilbene

(PI) are shown in Fig. 19. The a density (p^\r)) and /density (p^!(r)) values are

drawn at the plane located at 1 a.u. above the molecular plane in order to elucidate the

contribution of ^-electrons. It is found that although the a (= 164.9 a.u.) by the PPP

CHF method (Fig. 19(b)) is smaller than that (= 303.9 a.u.) by the B3LYP method (Fig.

19(a)), the features (sign and relative magnitude) of the distribution of a densities

between these two methods are similar to each other. This difference between the a

value by the PPP CHF method and that by the B3LYP method are presumed to be

caused by the fact that the PPP CHF result does not include the effects of extended basis

sets, those of cr-electrons and electron-correlation effects. As mentioned above, the

effects of extended basis sets and the contribution of cr-electrons to a for flr-conjugated

56

linear chain molecules are known to become small in the large chain-length region, and

the electron-correlation effects on a are not considered to much affect the qualitative

tendencies of these three types of oligomers. Similarly, it is found that the features

(sign and relative magnitude) of the distribution of y densities between these two

methods are similar to each other though the y (= 67290 a.u.) by the PPP CHF method

(Fig. 19(d)) is smaller than that (= 106500 a.u.) by the B3LYP method (Fig. 19(c)).

The reason of this difference is considered to be the same as that in the case of a.

Similarly to the case of or, however, the effects of extended basis sets on y for K-

conjugated linear chain molecules are known to become small in the large chain-length

region, and the electron-correlation effects on y are not considered to much affect the

qualitative tendencies of these three types of oligomers calculated by the PPP CHF

method. Judging from these things, it is expected that the qualitative a and y values

and their density distributions for phenylene vinylene oligomers can be reproduced by

the PPP CHF calculations.

The primary contributions of a density distributions can be well divided into

benzene-ring regions (a-f and i-n) and vinylene region (g-h) (see Fig. 18(a) PI), all of

which are shown to provide positive contributions to a. This feature suggests that the

virtual excitation process for a of stilbene (PI) are well separated into the

contributions of benzene rings and those of vinylene regions. This also corresponds to

the fact that the second-order virtual excitation processes involving relatively low-lying

excited states primary contribute to a. In contrast, the primary contributions of y

density distributions are found to be caused by the both-ends benzene-ring regions (a-d-

f-e and j-i-k-n) with large positive contribution and vinylene region (g-h) with small

negative contribution (see Fig. 18(a)Pl). This is distinct from the case of a, in which

57

the contributions of a density are shown to be well separated into those of benzene rings

and those of vinylene regions. This feature suggests that the 7value of stilbene (PI) is

described by the virtual excitation processes (involving higher-lying excited states)

from one benzene ring to the other due to the ^-conjugation between both benzene rings

as compared to the lower-order virtual excitation processes (involving lower-lying

excited states) for a.

3.4.3. Size Dependencies of a and /for Model Oligomers

We first consider the chain-length dependency of a (longitudinal (z)

component) per repeat unit, i.e., a In (n: the number of phenylene vinylene unit), for

three types of oligomers (Pn, MNFn and MFn) from n = 4 to 16 to remove the effects of

end-capped benzene ring (see Fig. 20(a)). There are found to be distinct differences in

variations in aln with the increasing chain-length among these model oligomers.

Apparently, the aln values of Pn oligomers are larger than those of MNFn and MFn

oligomers. MNFn oligomers exhibit much smaller chain-length dependency than Pn

and MFn oligomers. Although MFn oligomers show intermediate aln values

between those of Pn and MNFn oligomers, the aln values of MFn oligomers exhibit

attractive increase behavior: the aln values of MFn oligomers are close to those of

MNFn oligomers in the small chain-length region, while those of MFn oligomers

approach those of Pn oligomers in large chain-length region. Namely, MFn oligomers

are found to preserve a high increasing rate even in the large chain-length region in

contrast to Pn oligomers. In the large chain-length region, the aln values tend to be

saturated as observed in other ^-conjugated oligomers [43,51]. According to the

58

extrapolation procedure in previous studies [51], the aln values are fitted by least

squares to

log(^) = a + + 4 , (58)

where the extrapolated values for infinite polymers are (a/n)„_>„ = 10°. As expected

from the chain-length dependency, the saturated aln (= 179.2 a.u.) of MFn oligomers

is found to be larger than that (155.3 a.u.) of MNFn and to be fairly approach that

(187.2 a.u.) of Pn oligomers. This result suggests that the number of phenylene

vinylene units involved in the linear-leg regions between meta-substituted benzene rings

closely relate to the chain-length dependencies of aln and the saturated aln for

these oligomers. These features are predicted to originate in the differences in the

lengths of ^-conjugated linear-leg regions (Pn segments) involved in each oligomer.

Namely, the unique increase behavior of aln for MFn oligomers is attributed to the

fact that the number of ^-conjugated linear-legs involved in chain and the length of the

largest ^-conjugated linear-leg for MFn oligomers gradually increase with the

increasing chain-length.

We next investigate the chain-length dependencies of y (longitudinal

component (z)) per repeat unit, i.e., y/n, (n: the number of phenylene vinylene unit)

for three types of model oligomers, Pn, MNFn and MFn (n = 1, 4, 7, 11 and 16) (see Fig.

20(b)). Similarly to the case of a, it is found that the log( y I n) values of Pn oligomers

increase most fast and dramatically with the increasing chain-length among these

oligomers, while MNFn oligomers exhibit slight chain-length dependencies. Although

59

MFn oligomers show intermediate log(y/n) values between those of Pn and MNFn

oligomers, the increasing rate of their y In values in large chain-length region is found

to exceed those of Pn and MNFn oligomers. Applying the extrapolation procedure (Eq.

(58)) to y/n, the extrapolated values for Pn, MNFn and MFn oligomers are found to

be 5.413 x 105 a.u., 1.278 x 105 a.u. and 3.419 x 105 a.u., respectively. Similarly to the

case of a, the saturated yIn of MFn oligomers actually takes an intermediate value

between those of Pn and MNFn oligomers, while the saturated y In of MFn oligomers

is found to be relatively close to that of Pn oligomers and to be much larger than that of

MNFn oligomers. This feature is also predicted to originate in the differences in the

lengths of ^-conjugated linear-leg regions (Pn segments) involved in each oligomer. It

is also found that the differences in saturated y/n and chain-length dependencies for

these three types of oligomers are remarkably enhanced compared to the case of a (see

Figs. 20(a) and (b)). It is noted that the differences in the size dependency of y I n

aIn) between Pn and MFn oligomers are similar to that of intermolecular-interaction

effect on a I n between linear chain (7-aggregate) and dendritic aggregate models (Sec.

2.4.3). These two tendencies are predicted to originate in the fractal structure of

dendritic systems.

3.4.4. a and /Density Distributions for Model Oligomers

Figure 21(a) shows the a density distributions for PI, PI6, MNF16 and MF16

oligomers, respectively. For P16 oligomer, there are similar a density distributions

(positive contribution to a) to the case of stilbene (PI) in the whole chain-length region

though their magnitude of a densities is more enhanced than that of PI. Also, a

60

densities distributed in the middle region of chain are shown to be somewhat enhanced

compared to those distributed on both-ends regions. These features are assumed to be

caused by the ^-conjugation over the entire region of chain. The a density distribution

on each vinylene region (g'-h') (see Fig. 18(b) MNFn) for MNF16 oligomer is similar

to that for PI, while the enhancement of a density distribution in the middle region of

MNF16 oligomer is not observed in contrast to the Pn case. It is further found that

there is little a density distribution on sites d', c \ 1' and k' (see Fig. 18(b) MNFn) in

benzene rings and the contributions of a densities (distributed on (a', b')-(e\ P) and (i',

j ' ) -(m\ n')) in benzene rings of MNFn oligomers are smaller than those (a-f and i-n) of

Pn oligomers. These features can be understood by the fact that all phenylene vinylene

units for MNFn oligomers are linked through weto-substitutions, which cause the

segmentation of ^-conjugation extended over the entire region of chain. Such feature

of a density distributions for MNFn oligomers well supports the smallest chain-length

dependence of a In for MNFn oligomers (Fig. 20(a)). In contrast to the results for Pn

and MNFn oligomers, the ^-electron contribution in MFn oligomers is found to possess

both features of Pn and MNFn oligomers (Figs. 21(a) MF16). It is found that the

contributions of /?ara-substituted phenylene vinylene units (linear-leg regions for MFn

oligomers) are similar to those of Pn oligomers with corresponding chain-lengths, while

the contributions of linear-leg regions for MFn oligomers are well segmented at the

meta-substituted benzene rings. Since larger linear-leg regions are involved in larger-

size MFn oligomers, the significant contribution of the linear segments to a of MFn

oligomer is predicted to enhance its chain-length dependency and to allow its saturated

a/n value to be closer to that of Pn oligomers.

Figure 21(b) shows the /density distributions for PI, P16, MNF16 and MF16

61

oligomers, respectively. For P16 oligomer, the enhanced y densities distributed in a

similar manner to the feature of stilbene (PI) are observed in both-ends benzene rings,

while in the intermediate chain region, positive and negative y density regions with

significantly enhanced magnitude alternately appear along the chain-length direction,

the feature of which reflects the ^-conjugation between benzene rings and double-bond

units in the entire region of chain. These distributions contribute to y positively in

both-ends regions though there is considerable cancellation between positive and

negative contributions to 7 in the intermediate region of chain. For MNF16 oligomer,

all phenylene vinylene units are linked through meta-substitutions, which cause the

segmentation of ^-conjugation extended over the entire region of chain. Actually, the

magnitude of y density distribution for MNF16 oligomer is similar to that for PI

oligomer: the enhancement and delocalization of /density distribution observed in P16

oligomer does not appear in MNF16 oligomer. For example, there are sites with little

/density distribution in benzene rings (see Fig. 21(b)MNF16). These features of y

density distributions for P16 and MNF16 oligomers well support the largest chain-

length dependence of yln with the largest saturated yln for Pn oligomers and the

smallest chain-length dependency of yln with the smallest saturated yln for MNFn

oligomers, respectively. In contrast to P16 and MNF16 oligomers, the ^-conjugation

contribution for MF16 oligomers exhibits behavior including both features of P16 and

MNF16 oligomers: the contributions of para-substituted phenylene vinylene units

(linear-leg regions) in MF16 oligomer are similar to those in P16 oligomer, while the

contributions of linear-leg regions for MF16 oligomer are well segmented at the mete-

substitution points of benzene rings. Judging from these results, for small-size MFn

62

oligomers, the /density distributions in the linear-leg regions are not enhanced similarly

to the MNFn case, while for large-size MFn oligomers, large linear-leg regions (which

provide enhanced 7 density distributions which are similar to those of large-size Pn

oligomers) are involved and thus their significant contributions to 7 are predicted to

enhance the chain-length dependency and to allow the saturated jln value to be closer

to that of Pn oligomers. Compared the results for /with that for a, such features in 7

are found to be more enhanced than the case of a.

These unique features observed in a and yare found to be realized by taking a

fractal structure (with meta-substitutions of benzene rings), in which the largest linear-

leg region (whose ^-conjugation is well decoupled at the meta-substituted benzene

rings) gradually extends for the increasing oligomer-size.

3.5. Second Hyperpolarizabilities of Cayley-Tree-Type Phenylacetylene

Dendrimers

In Sec. 3.4, we considered various oligomers involved in phenylene vinylene

dendrimers with fractal antenna (Cayley-tree) structure. This result indicates that the

fractal structure and meta-branching points (benzene rings) cause the localization of a

and 7 densities in the linear-leg (para-connected) regions and the unique size

dependencies of longitudinal a and y(see Fig. 20). However, those systems are linear-

like chains and actual dendrimers have a fractal dimension D (1 < D < 2). Therefore,

we consider a phenylacetylene dendrimer (D25, see Fig. 16) with an intermediate size.

The above unique features concerning ^-conjugation length are predicted to be closely

63

related to the size dependencies of NLO properties such as y Since Cayley-tree-type

phenylacetylene dendrimers have a fractal dimension, we need to consider

orientationally averaged values, ys, involving several components of yiJkl.

3.5.1. Calculation of yof D25

A dendrimer, D25, has a fractal structure: central generation involves longer

linear-leg regions as compared to the outer region. Since we focus on the qualitative

structure-property relation of yfor D25, we use the bond lengths (Rl = 1.3992 A, R2 =

1.4283 A, R3 = 1.216 A and R4 = 1.0854 A) and bond angles (0 = 120°) of

dipheylacetylene (calculated by the B3LYP method using 6-31G** basis sets) for D25.

The B3LYP method is known to well reproduce the experimental geometries of organic

molecular systems [64,74,75].

3.5.2. Comparison of the y Value and 7 Density Distribution of Diphenylacetylene

Calculated by the INDO/S CHF Method with Those by the B3LYP Method

The CHF method in the INDO/S approximation is known to provide

qualitatively or semi-quantitatively well results of y for large-size organic molecules.

To confirm the reliability of our calculations using the INDO/S method, we examine the

longitudinal y value, yzzzz (which is a dominant component of y), and y density of

relatively small-size system, i.e., diphenylacetylene, by comparing the INDO/S CHF

results with the B3LYP results. In this study, we use the 6-31G**+d (£, =0.0523 on

64

carbon atoms) basis set, which is known to be able to well reproduce the y of n-

conjugated organic molecules [32,51]. The /values and their density distributions of

diphenylacetylene (calculated by Eqs. (39) and (57)) are shown in Figure 22. The y

densities, p™(r), are drawn at the plane located at 1 a.u. ( = 0.52917 A) above the

molecular plane in order to mainly elucidate the contribution of ^-electrons. It is

found that the yHK (= 112400 a.u.) by the B3LYP method (Fig. 22(a)) is in good

agreement with that (= 113100 a.u.) by the INDO/S CHF method (Fig. 22(b)). It is

noted that 1 a.u. = 5.0366 xl0~40esu for y. Further, the features (sign and relative

magnitude) of the distribution of y densities between these two methods are similar to

each other though the INDO/S y densities are calculated in the Mulliken approximation.

Judging from these things, it is expected that the qualitative or semi-quantitative y

values and y density distributions for phenylacetylene oligomers can be reproduced by

the INDO/S CHF calculations. The primary contributions of /density distributions are

found to be caused by the virtual excitation from the left-hand-side benzene ring to the

right-hand-side one (positive contribution), while the contribution of central C-C triple

bond region to y is found to be much smaller with negative sign. As a result, total y

value becomes positive in sign. This dominant spatial contribution of electrons

between the one-end benzene ring to the other-end one suggests that the y of

diphenylacetylene is described well by the virtual excitation processes involving high-

lying excited states due to the ^-conjugation between benzene rings.

3.5.3. yand y Densities of D25

65

Each component of calculated yijkl in Eqs. (39) and (40) and the averaged

value, ys, (Eq. (38)) are given in Table 2. As expected from the planar structure, ys

is found to be dominantly determined by two components, i.e., yxxxx (= 2171000 a.u.)

and Yaa (= 2170000 a.u.), which are at least about one order larger than other

components. For comparison, we also examine the unit monomer, i.e.,

phenylacetylene (see Fig. 16(A)), the ys (= 2600 a.u.) of which is also dominantly

determined by the longitudinal component. Namely, the total ys of D25 is found to

be about 450 times larger than that of monomer (phenylacetylene). Since the present

dendrimer D25 is composed of 24 units of phenylacetylenes, the ys per unit molecule

(phenylacetylene) of D25 becomes 48750 a.u., which is shown to be about 19 times

larger than that of phenylacetylene. Judging from such remarkable enhancement of ys

per monomer, the ^-electron conjugation in the longitudinal direction (along the

benzene-ring and acetylene units) of phenylacetylene is expected to significantly

contribute to the enhancement of the component of y. In order to better elucidate the

features of the contribution of electrons to y, we next investigate the y density

distributions concerning two main components, yxxxx and yzuz.

Figures 23(a) and (b) show yxxxx and yaa density distributions of D25,

respectively. The spatial contributions of these components are found to be localized

in the linear-leg regions, A-G for yxxxx and A'-J' for yzm, parallel to the directions, x

and z, of the applied electric fields, respectively. For yxxxx density distributions (Fig.

23(a)), all the spatial contributions except for region C are found to be similar to those

of diphenylacetylene (see Fig. 22), while the contribution (positive in sign) in the

central linear-leg region C (composed of two phenylacetylene units) is found to be

66

caused by the virtual excitation from one-side phenylacetylene to the other-side one and

its magnitude is shown to be more enhanced compared to other contributions. These

features suggest that ^-conjugation is well localized in these linear-leg regions mutually

connected by wefa-substitutions and is enhanced in the central long para-connected

linear-leg region C since larger ^-conjugation decreases excitation energies and thus

leads to the enhancement of the contribution to y as expected by the perturbational

formula of y. On the other hand, for yzzzz density distributions (Fig. 23(b)) in regions

K'-L' and M'-N', considerable cancellation of positive and negative contributions is

observed at the weta-connected benzene rings, so that the primary contribution (positive

in sign) seems to be caused by the virtual excitation from the one-end benzene ring to

the other-end one. However, these features can be explained by the simple addition of

the /density distributions of diphenylacetylenes (see Fig. 22): /density distributions on

the central meta-connected benzene ring are similar to the sum of y densities on the

overlapped each end benzene ring of diphenylacetylenes arranged as shown in K'-L'

and M'-N', respectively. The well decoupling of the ^-conjugation at the meta-

connected benzene rings is predicted to cause such simple addition of y density

distributions of diphenylacetylenes and the non-enhancement of y densities. Similar

features are also observed in regions A'-B'-C'-D', E'-F' and I'-J'. It is noted that the

7 density distributions in regions G' and H', which involve two units of

phenylacetylenes, respectively, are more enhanced than those in other linear-leg regions.

The contributions of electrons in these regions are shown to be positive in sign and to be

caused by the virtual excitation from the one-side phenylacetylene unit to the other-side

one, while the contributions are shown to be well segmented at the central meta-

67

connected benzene ring. Similarly to yxxxx case, these features are predicted to relate

to the well decoupling of ^-conjugation at the central meta-connected benzene ring and

to the structural feature that ^-conjugation length is larger in central linear-leg regions

as compared to those in periphery regions.

3.6. Summary

In this section, we investigated the relation between the architecture of

supramolecules and (hyper)polarizabilities. From the investigation of longitudinal a

and 7 for three types of oligomers (Pn, MNFn and MFn) constructed from phenylene

vinylene units, the chain-length dependencies of longitudinal a(f) of relatively small-

size oligomers increase in the order : MNFn « MFn < Pn. The saturated value of

a(f) was also shown to increase in the order: MNFn « MFn < Pn. These features

were well elucidated by the differences in the a(y) density distributions along the chain-

length direction, the features of which are sensitively reflect the differences in the K-

conjugation of these oligomers. Namely, the destruction of ^-conjugation at the meta-

substitution points of benzene rings is assumed to reduce the enhancement of a(y)

density distribution in the whole chain region. For example, MFn oligomers involve

larger /?ara-substituted leg regions due to the fractal structure in contrast to MNFn

oligomers involving no para-substituted leg regions. As expected from this feature,

the saturated a and y values of MFn oligomers were found to much exceed those of

MNFn oligomers and approach those of Pn oligomers, respectively. We also found

that these structure-property dependencies for /are more remarkable than those for a.

68

We next investigated the contributions of electrons to y for a real

phenylacetylene dendrimer with a fractal antenna structure. The orientationally

averaged y value was found to be dominantly determined by the two components in

plane, yxxxx and yuu, which were shown to be contributed by the electrons in linear-

leg regions parallel to the applied electric fields, x and z, respectively. It was further

elucidated that such contributions are well segmented at the meta-connected benzene

rings and are more enhanced in internal regions than those in outer (periphery) regions.

Similarly to the results for the y of model oligomers, these localizations of the spatial

contributions of electrons to y are predicted to originate in the fact that ^-electron

conjugation in linear-leg regions primarily contribute to the corresponding component

of y and is well decoupled at the meta-connected benzene rings. The enhancement of

the contributions to y in internal linear-leg regions can be explained by the reduction of

the excitation energies in the internal linear-leg regions due to the longer ^-conjugation

lengths compared to the shorter ^-conjugation lengths in outer regions.

Judging from these results, the spatial contributions of electrons to

(hyper)polarizabilities for larger phenylacetylene dendrimers with fractal antenna

structures are expected to be localized in linear-leg regions and their contributions

increase as going from the periphery to the core. Such unique spatial contributions to

(hyper)polarizabilities reflect the fractal architecture of antenna dendrimers involving

weto-connected benzene rings. As a result, the novel architecture of fractal antenna

dendrimer involving meta-connected benzene rings closely relates to the unique feature

of spatial contributions to (hyper)polarizabilities. This result can then be employed to

adjustably design novel molecular systems exhibiting desired NLO characteristics by

69

constructing relevant fractal architecture composed of ^-conjugation linear-legs and

meta-connected benzene rings. Further, the dynamic (near-resonant)

hyperpolarizabilities for these systems are expected to be interesting since the exciton

migration is assumed to remarkably affect the (near)resonant nonlinear optical response

properties. Toward the understanding of such phenomena, the methods for dynamics

treating exciton migration in systems with explicit exciton-phonon coupling and the

analysis using an analytical expression of dynamic hyperpolarizability density are

presented in the next section.

4. Extensions of Models and Analysis

4.1. Master Equation Approach Involving Explicit Exciton-Phonon Coupling

Previous studies [18,19] elucidate that the relaxation effect is necessary for an

efficient exciton migration in addition to the multi-step energy structure. As for the

relaxation term, however, we used the phenomenological parameters yu (see Sec.2.4.1),

which only depend on the differences among exciton states. We also performed the

exciton dynamics of dendritic molecular aggregate model using the master equation

involving weak exciton-phonon coupling [76]. Recently, Takahata et al. [77] have

formulated the master equation involving general exciton-phonon coupling in the

Markoff approximation and discussed the relation among the relaxation parameters and

exciton wavefunctions. Applying our treatment of nonperturbative

(hyper)polarizabilities (Sec. 2.3) to such rigorous approach, we will be able to more

profoundly investigate the effects of exciton migration on (near)resonant nonlinear

optical properties in view of the spatial architecture dependency. We briefly explain

70

the formulation of this approach in this section.

4.1.1. Model Hamiltonian Involving Exciton-Phonon Coupling

Similarly to the case of Sec. 2.1, we consider a molecular aggregate composed

of two-state monomers, in which excitation energies are £, and the magnitudes of

their transition moments are /x,. The non-interacting part HQ for the molecular-

aggregate Hamiltonian Hs (see Eq. (61)) is written by

where |i) is the aggregate basis, in which the /th monomer is excited, and N is the

number of monomers. Intermolecular interaction is assumed to be the dipole-dipole

interaction. The interaction part Hjm for the molecular-aggregate Hamiltonian Hs is

given by

^n. = ^ i l ^ W M c o s ( 0 , . - 6 h ) - 3cos0,y cos0, )\i)(j\ = i £ 7 t f | , X / | . (60)

The intermolecular distance between dipoles i and j is R;j, and the angle between a

dipole /(/') and a line drawn from the dipole site ;' toy is 0,-.(0-.)• From Eqs. (59) and

(60), the Hamiltonian for molecular aggregate, Hs, is represented by

71

HS = H0+ Hml = X£,|«><«| + ^iUl<->(j| . (6D

The electronic eigenstates and eigenenergies are obtained by solving the following

eigenvalue problem:

H,\wk) = ^k\vk)> (62)

with

K>=Sl'X'1vOsEc«i'>- <63>

These electronic states are assumed to interact with phonon states composed of

many harmonic oscillators with the frequency ]Qq.. \, which represent nuclear

oscillations. qUJ) denotes the phonon state coupled with both the ith and y'th

monomers. The Hamiltonian for the phonon state, HR, is given by

».=XI°fcJ,C«w (64)

where c* and cq are creation and annihilation operators corresponding to phonon

q0J) in the reservoir coupled with the ith and jth monomers. ^ . means the

summation over all possible pairs of ;' and j . These operators satisfy the following

commutation relations:

72

[c ,c+, ] = 8,8 ,8 , (1-8 ) +8 ,8..8 , (65)

and

K,^;,,]=K,,'4,,J=°- <66>

The interaction Hamiltonian between the molecular-aggregate state and the phonon state

is assumed to be

"SR = I I O O K ^ C + ^ W V u , ) - (67)

where Kij)q represents a coupling constant between a pair of ;'th and y'th

monomers and phonon qiijy

The total Hamiltonian is given by

H = HS + HR+HSR+HE, (68)

where HE represents the Hamiltonian for the interaction between an aggregate system

and an electric field E. We consider a semiclassical electric field for HE in the

dipole approximation since we consider the situation, in which the initial exciton

populations are generated by a stable laser beam. The Hamiltonian HE is given by

HE = -fiE, (69)

73

where fi represents the total dipole moment operator of the aggregate system.

The time evolution of density operator (# ) for the total system (system +

reservoir) is described in terms of the von Neumann-Liouville equation:

X = ^r[H,xl (70) in

Using both Eqs. (68) and (70), we obtain

X-^[Hs + HR+HSR+HE,Xh^[Hs + HR+HSR,X] + ^[HE,x]- (71)

When we take the trace for the both sides of Eq. (71) over the phonon states (denoted by

t r R - /^ , where / , is the thermal-equilibrium density operator of phonon states), the

Hamiltonian in the second term on the right-hand side is not affected at all:

Tn^[^x] = Mx] = P \ (72)

where we define a reduced density operator p for the electronic state as

P = trRU]. (73)

Thus, we can only consider Hamiltonian Hs, HR and HSR in order to formulate the

master equation involving exciton-phonon coupling.

74

4.1.2. Master Equation Approach

We start with a master equation (in the interaction picture) in the Born-Markoff

approximation [78]:

~p = ~ fVtrR[#SR(0,[#SR(O,p(0/g], (74)

where p denotes the reduced density operator of the electronic state in the interaction

picture. We define the interaction Hamiltonian (HSR (?)) in the interaction picture as

HSR(t) = eW*x"s+"»»//sRe-<W"'s+»K». (75)

Using equations (64), (67) and (75), we have

••i-iu.it

(76)

where we use

em**<c+^e-mn« = cl/'"^"-"', (77)

and

e ( ; / * , / , R 'V < w "" = cqiue~°"')a<"<''. (78)

http://i-iu.it

75

The right-hand side of Eq. (74) has four terms as shown below.

p = -^j'odt'lvRHSR(t)HSR(trp(t)Ro ~ WSR(f')p(0«o^SR(0

-HSR(t)p(t)RoHSR(t') + p(O^WSR(f')«SR(0. (79)

Using Eq. (75), the first term on the right-hand side of Eq. (79) is rewritten as

-^ldt'trRHSR(t)HSR(trp(t)R,

= __L ^j'gdt'eiin)H^\i)(j\e~u'h)H^'-n\«X7y'/ft,"s('-'>~("S,"s'(l-5ff) i-i-ii.i

W7ft ,"s'|J-)0-K<'7s,"s""' , |7)(/k ( ,7e ,''s(M'V'' ,S)H5'

(80)

where we use

KK.C;,,^) = ixR(%icqi,y^) = o, (si)

fr«<C/v„,*»> - W«w*v„ v-s^+WMW "(Q*,> ' r ) ' ( 8 2 )

and

tr»(c,((J c ; , 3 ) = W W l < , , , < ! - 5*->+ W«,^t^^T) +1), (83)

76

with

nQqij,T) = [eia^>/k"T) -1]"1. (84)

n(fl.q ,T) denotes the mean phonon number for phonons with frequency £2 in

thermal equilibrium state at temperature T. Inserting the completeness relation with

respect to eigenstates of Hs:

X|V*)(V*| = 1 (85)

into Eq. (80), we obtain

1 V^. hjldt'tTRHSR(t)HSR(trp(t)Ro

= - i I ljyeUmHs'\i)(j\¥k)(vk\i)(j\¥m)(¥mWM^'"""d-*„) i,j,q(ij) k,m,n

^,'7fi,"l0(yV,(^|y)(#ra)(^|pk„)(^k-,'v"''i

Kanqu n f n(fl,i( yi, 7 > ; <"«'•;> '*— +"-)('-'' > +(«(fl,(( yi, T) + i y <-*«<"' ft-ffl*+ffl")('-'''].

(86)

Markoff approximation is used to evaluate the time integral on the right-hand

side of Eq. (86). This approximation is acceptable when the order of t is much larger

than that of t'. In this case, we can assume t to be infinity. Under this

approximation, Eq. (86) is expressed by

77

~JVtrR/JsR(0£SR('')p(0^

= - i X X^'^'IWI v*Xv* \i)(Av,M„ \PWM rilh),h'v - sv)

(87)

where

7(U)K -<»*)= XK7H,J2«H„J,':r)5(Q,(,J, -«>* +A>J

+(n(Q,((ji ,T) + D5(-n,((jl - a>k + »„,). (88)

Similarly, from the second, third and fourth terms on the right-hand side of Eq. (79), we

respectively obtain

- ^ J VtrR -tfSR (OP W / / S R (0

= - i l XR'^I^Xv* l«X;l v-Xv.. ipkXv. WVH""H*<\ - *«>

^('/"w1^X^I^>-X^M^J^J0O>-l''*,,',')r(<J,K-«'J.

(89)

-p-£*'trR-«SR(0p(0^HsR(O

78

**• i,j k,m,n

(90)

and

-p£*'trRp(0^sR(0ffw(O

=-ilIK/w,v|y.Xr.|p|y.XÎ'->(/V*XÎ'-)0>","JWa-*,)

+^('")H1^X^JpkJ(rJÔVO(^l0O1^(',*,Ws')r(,,)K-^)-

(91)

Using Eqs. (69), (72), (79), (87), (89), (90) and (91), we get the master equation (in the

matrix representation using the eigenstates of Hs) involving the exciton-phonon

coupling and exciton-field interaction in the Schrodinger picture:

P^ = -'(«a -^)P^-rrS r*m»A™ -rZtAWtf-P-A/i) ' (92)

" m,n " n

where the relaxation parameter rag.im is represented by

i.j.k'- l J

79

-I[(i - s^cXAe®+CQC;.^ x7, .,(<»,„ - *>„)+7(,,,,K - a,)].

(93)

For the exciton migration dynamics, we consider the diagonal part of Eq. (92):

raa ~ *2 * f aa;mmPmm -^^^anPna-Pm^na)' (94) " m " n

where

^;„„„ = X[2(1 - ) C X < A + |Qf|Q.|2r(,y)(ft),„ - fl)t)j i.J.*L

- i W d - 5,y)CCm.CQ + iQflQfly^ Cft),,, - a,0)l. (95) •j

We can numerically solve Eqs. (92) and (94). Although it is difficult to

nonempirically determine the coupling constant K in Eq. (67), we can apply some

appropriate approximations to K and /,,-., (Eq. (88)). Some examples of these

approximations are described in refs. [79] and [80].

4.2. Analytical Expression of Hyperpolarizability Density

In the perturbation theory, physical pictures of hyperpolarizability can be

understood by virtual excitation processes [28-32]. In Sec. 3.3, we explain a method

80

for analysis of hyperpolarizability using "hyperpolarizability density", which can

describe the spatial contributions of electrons to hyperpolarizability. This analysis can

be easily applied to arbitrary molecules calculated by the FF approach (Sec.3.2) at

various MO and density functional (DF) methods. However, this hyperpolarizability

density analysis based on the FF approach can be only applied to the static case.

Namely, this can only provide a real part of hyperpolarizability density in the static

region. In the dynamic case, frequency dependent hyperpolarizability densities can be

calculated using the Fourier transformed charge densities [72]. Such dynamic

hyperpolarizability density analysis is also useful for combing the physical picture

(virtual excitation process) with chemical picture (spatial contributions of electrons) of

hyperpolarizability. In order to treat near-resonance case, in which the imaginary parts

of hyperpolarizability is important, we have to perform the dynamics in the Liouville

(master equation) approach and obtain the time series of polarization. However, such

dynamics is very demanding and is hard to apply to larger systems (with a large number

of states) such as dendrimers. In this section, an alternative approach using analytical

expressions of hyperpolarizability densities is explained. This procedure has an

advantage of being applied to arbitrary response properties described by perturbational

expressions. As an example, we apply this procedure to obtaining the analytical

expression of the density for second hyperpolarizability, y(~co;co,co,-co), whose

imaginary part (Im?) is concerned with the two-photon absorption (TPA) phenomena

[81-83], which is now one of topics in chemical properties of dendrimers. It is noted

that such quantities (imaginary parts of hyperpolarizabilities) cannot be calculated by

the FF approach.

81

4.2.1. Analytical Formula of Hyperpolarizability Density

First of all, we briefly explain a definition of hyperpolarizability density using

an example of the static and dynamic y. In Eq. (44), p]k!r) represents a static y

density, which is a third derivative of charge density with respect to fields F', Fk and

F' (ij,k,l = x, v, z). The r' is the rth component of the electron coordinate. The

dynamic hyperpolarizability density also can be defined in a similar expression to Eq.

(44) though that is concerned with Fourier transformed charge density, e.g., p(3co,r)

(ft): frequency of incident field) in the case of third harmonic generation (THG). The

analysis method using the plots of these hyperpolarizability densities is explained in Sec.

3.3.

From the comparison of time-dependent perturbational formulae (sum-over

state (SOS) formulae) of hyperpolarizabilities, e.g., Orr-Ward formulae [84], with the

definition of Eq. (44), it is noted that the direction of electron coordinate r' in Eq. (44)

coincides with the direction of polarization (the fth component of yjjkl represents the

direction of polarization). Since the numerators in the perturbational formula of y~u

are composed of the product of transition moments with i,j, k and / components, we can

rewrite the perturbational formula of yijU to the same form as Eq. (44) by representing

the i components of transition moments (/i^,) between electronic states a and b with

transition density puh(r) and electron coordinate r'. To this end, we rewrite a

transition moment operator to the following form:

82

A' = II«)[-|^('-)^V]^I- (96> a,b

By substituting this transition moment operator into the perturbational

formulae of hyperpolarizabilities, we can obtain a similar form to Eq. (44). We can

extract the analytical expression of hyperpolarizability density from this rewritten

formula. For the following discussion of TPA, we consider the analytical expression

of y(-ft);co,co,-co) using the Orr-Ward formula of y(-co;co,co,-co) and Eq. (96).

The resultant complex y.Jkl(-co;co,co,-co) density, which is referred to as p*kl(co,r), is

expressed by

p]u(co,r) = p # W ) + pTi«>,r) + p*"1"1 W > + pj™"2 W ) , (97)

p * ' W ) = - T T X [ P , » 4 " L ( ^ L ; " ; : I + A&4 i)

x[(Qal-cor\nal-2cor, +(Q'a! +a»-2(flfll+2a>r' Hna]-coy2Q-j+(n'al +<»r2^V] +2(AnJ

aaAnln'al)[(X2a, -coy[Q-J(£2al +©)- '+(«; , + coy1 a.? (Q'al - coy1 ]

+Apaa(r)n']aAnkaaiiL + riaA/iLHai)

x[(f2al -fi))-'(i2al -2a>r ,<«a, -coy1 +(Qal +coyl(a:i +2coy\Q'a] +coyl

•HA.',+fl))-ifl;1'(flal+<»)-'-K^., -©r'fl.r'cfi.*. -o»"'] +2(///a^aX1)[(A:1 +coy'n-J(.i2a]-coyl + (Qal - © r ' ^ ' ^ , +©)"']],

(98)

pj2"W) = p-S[p,.(r)20xiX>*t.)

x[(fl„, - toy\aal + coy1 (flw - a))-' + (flal + coy' (o'hl + coy] (Qhl - coy' +n*ai + coy1 (Q'al - coy1 (o'bl +coy[+ (flal - coy1 (flw - «)-' (A;, +a»-' ] x[(X2al -coy2(Ohl -coy' +(Qal-coy\Q'bl-coy\£2hl -coy1

+(flal +coy\i2'bl+coy,+(Qal +coy\i2bl +coy\£2'bi +coy]

Hnal-coy2(f2bl + coy> + (£2ol -coy\Q'bl +coy\nb + coyx

83

+(fl.*i + (»r\n'M - coy[ + (Q'al + ft))"' (flw - a))"' (13,*, - fl))"

(99)

p£nM W ) = - ^ X[pla(r)4pL(A&A4. + /£/*».) + PxWuMMl + <4<) ] a.h

p l a ( r )M u „(^X, +Afliaril) + plh(r)Av'jvLvkub+»>L)]

(flal -fl))-Iflar'(flw -or1+(fla*.+fl))-IflBV(fl;1 +©)-']

x[(flal - a ) -'a.,-'(flw+o))-'+(«Bl +a»r,flav'(fl;l -&))-']

x[(Aw -flir'^-Ufl,,,+©)-'+(«;, +ft))-1r2;,"'(i2;, -a))-1]

x[(flBl - ft))"1 (flBl - 2ft))"1 (flw -&))-'+ (flBl + ft))"1 (flBl + 2ft))-'(£21 + ft))"1 ]

+Pabr)n'M(Ha^l + tia^i)

x[ (^ , -0))-'(flBl -2fl))-'(flBl -ft))"1 +(flBl +ft))-'(R*1 +2ft))-1(r2;i +0))-']

X[(A;,+ f l))- |flBl-|(flw -a))-1+(ABl - f t ) ) - 1 ^ ; , - ' ^ , +©)-']

+pah(r)An'00vkMnci+ vLlAa)

X[(A;, +(»riflBr1(flBi -®r'+(«». -©r'^V'c^;, +«r ' ]

x[(flB',+<»)-'flBl-'(flw+(«)-'+(AB1 -ft))-1^;,-'^;, -a))-1]

x[(r26*, +fl))-'flBl-1(flBl + &))-' +(«, , -o))-1 flBr'(flal - ») ' ' ] .

(100)

and

p*'"-2 W ) = -±- S[p la(r)^(^X + A*U*.) a.h.i Ui*b*c)

x[(t2a] -©r'cfl., -2a»r'(flrl -ft))-1+(A;, +fi»r,(fl;I+2a))-'(fl;1 +»)-'] +/40&/4+/&#.)

+ft1(Wt + M, . ) (AB1 - f t ) ) - 1 ^ , - 1 ^ ,+o) ) - ' + ( A ; , +fl))-,fi;r'(fl;1 -a))-1]

84

x[(£2'Bl - fl))"1 (flM - 2a))-1 (flcl - ft))"' + (fl., + to)'1 (fl;, + lay' Q'c, +©)-']

+ A 4 « ^ i + < M H ) K * , +a»-'flw(flel -a))"1 + (flal -or112;,(13;, +«)- ' ]

+/4(A'X+<Mt)[(^*, +fi»-|flM(flel +©r'+(R1 -fflr'fl;,^;, -*»)-•]],

(101) where

«.i =«». i - '^ .1 /2 . (102)

Here, zlp^, ^0Jal, 4/x^, rol and ft) represent the charge density difference (between

the excited state a and the ground state (1)), the transition energy (between the ground

(1) and excited (a) states), dipole moment difference (between the ground (1) and

excited (a) states), relaxation parameter of excited state a, and frequency of external

field, respectively. ^ indicates a summation over all states except the ground state

(1). The real and imaginary parts of Eq. (97) correspond to analytical expressions for

real and imaginary prjkl((o,r), respectively. We here partition pJH(<o,r) (or

y(-(D,(0,(0,-(o)) into three types of contributions, each of which corresponds to a

virtual excitation pathway described by an expression (\-a-b-c-\). The types (I)

( 1 - a - a - a - l ) and (III-1) (\-a-a-b-\) processes exist only in the case of

molecules with noncentrosymmetric charge distributions due to A\im = 0 for

centrosymmetric systems. The type (II) (1 - a -1 - b -1) process involves the ground

state in the middle of the excitation path. In contrast to type (ni-1), type (ffl-2)

(1 - a - b - c -1) process can contribute to pJH(fi),r) (or y(-ar,o),co,-a>)) of arbitrary

molecules. From these formulae, the feature of pjt; (©,/•) (or y(-a>;(0,a,-co)) is

shown to be closely related to that of each type of virtual excitation process. In fact,

various classification rules of y have been presented based on the analysis of such

85

excitation processes [52,85-88].

Since two-photon absorption (TPA) cross section is related to the imaginary

part of ym(-co;co,co,-co) (ImyijU(-co;co,co,-co)), the spatial Im yiJU(-co;co,co,-co)

density distributions are important for understanding the structure-property relation in

TPA. Recently, Fujita et al. have developed a procedure of obtaining compact forms

of real and imaginary parts of y(-co\co,co,-co) by using trigonometric functions and

have provided some approximate formulae of lmy(-co;co,co,-co) [89]. They have

been also applied this procedure to obtaining the compact expression of real and

imaginary /densities at various approximate levels. For example, the exact analytical

expressions of real and imaginary ym densities for three types of virtual excitation

processes are respectively expressed by

Re[<W)] = - p I Pla(0(4lO2^, X

[cos(2e;l+e); cos(-2e;,-C), cos(0-+g:,+g;,) f cos(-2e;,-e;,) 1 A-JA\~ (A+JK; y ^ X (yc,)2<,

| cos(2e;, +e^) cos(-e;, -eaN,-e;,)|

(Ai.)\N, A4?A, J

+Pa«(rWJ1AvL x

f cose.2,- -cosC cos(-e;,+0>e;,) cos(e--ej-e;,) [Mft (K,U\\ ^ A A , *AMA,

^ cose.'j -cosgaN,

(103)

86

" a,b

fcos(e;,+e;1+eM) | cos(e;,-e;,+eM) | cos(-e;l-e;,-e+) |

[ 4,4A, K\KIA~M K\K\K\

cos(-e;, + e;,-e+) | cos(2ea-,+eM) | cos(e;,) | cosfo-+0+) ^ 4A,4, A-JAM A-al(A-hlf A-JAI,

cos(e;,) | cos(-2e;,-e;,) | cos(-e;,) | cos(-2e;,-e-) | cos(-e;,)1 4,(4,)2 (4,)24, 4,(4,)2

(4,)2AW 4 , K ) 2 )

(104)

1 Re[PrW)] = --JrX

" »,*,rL

cosfc+ffr+e;,) cos(-fl;,-C-e;,), cos(o-+etN,+e;,)

A - *2- A- + A+ A2+ A+ +

A- M .+ +

4>,4,l4'l 4,Al4l 4 A A ,

cos(-e;,-el-a;,) cos(e;,+e» +e;,) cos(-e;,-<,-g-)] 4 A A , KAKi 4XA,

+Pdb(r)AaH'tcHcl X

[cos(-o;,+ot2r+^l) | co s ( e ; , -C -^ ) , cos(-fl;, + e>e f l ) |

4,4i4, 4AA, 4.4,4,

cos(e;,-efcN,-e;,), cos^+^.+e;,) cos(g- -efc

N.- ,)l A A A 4 A A , 4 . A A

(105)

Im[p^(«J)r)] = - ^ X Pi.(»0(4fO2X, x

[sin(20a-,+0oy) gin(-2g;,-^) sinfc, + 0aN, + O;,) sin(-20;, -0a

N,)

[ K , ) 2 ^ K. ) 2 ^r *ANi*« K . ) 2 ^

|sin(2e;, + 0aN,)|sin(-g;,-eo

N,-0a-)l

K > ) 2 ^ - ' ' J

[ s inC i -smdlt |SinK,+0oN,+0a-,) |sin(e;,-0a

N,-C1)

iK,)2^; K,)2^r ^ A ' A«>^>

sin0„N, .. , sinC

(4,)X K,K (106)

IratpffW)] = ^ Z p j ' - K M * ) 2 x

[Sin(e-+fl;,+e-) | sin(0;,-e;,+e-,) [ sin(-e;,-ea-,-e;,) |

A.lAlA.l ^ l^ I iA i

and

sin(-e; ,+^,-^,) | sin(20;,+0-) [ sinfo) < sinfo;, + 0+) |

sin(e-) | ^(-20;,-0+) | 8in(-o;,) | sin(-2e;,-e;,) | sin(-e;,)|

^,K)2 W\i A+M>f K$*u Kfa)2]'

(107)

« n h /• A»/4/4X,x

[sin(g- +et2,-+e;,) 8in(-e;,-et

I,+ -e;,) rinfo+ea+e;,) A" A^~ A- A* A2+ A+ A- M A+

\A'b\ \ \ A,lA,lA:l A.lA>lAl

88

sin(-ff,-etw,-o;,) sin(e-+C+g:,), sin(-e;,-e»w,-e;.)1

A:,<A;, ^ , A > ; , A:,A>;, I

+PAr)V'u,H'hcHU x

[sin(-e;1+^r+g;,); sin(fl,-et2r-ff,), sin(-e;,+et

w,+e;,) A;,4;A;, ^ X X ^ X A ,

, sin(e;,-etw,-e;,) | sin(-e;,+ot

N,+e;,) | s in(g; , -e» w , -%•, ) ] AaXA« KAlK> *aXAi

, (108)

where ]|T indicates a summation over all states except the ground state (1) and the

summations in Eqs. (105) and (108) exclude the case a = b = c. Diagonal transition

moment (//„,) and charge density (pm(r)) in Eqs. (103)-(108) mean the dipole moment

difference and charge density difference between state a and the ground state (1),

respectively. In the above equations, we use the definitions of energy terms as

follows:

A2ale

±iB'' = cotll - 2ft)± ira 12, (for two-photon rotating terms) (109)

A~ate±,e" = ft),, - ft) ± iral 12, (for one-photon rotating terms) (110)

ANale

±ie" = ft),,, ±iTal / 2, (for non-rotating terms) (111)

A*ale±,e°' = ft),,, + ft) ± JTO1 / 2, (for one-photon anti-rotating terms) (112)

and

89

A2*e±w" = (t>aX +2(0±iral/2, (for two-photon anti-rotating terms) (113)

where

A2' = -j(Oal-2cof+(ral/2f , tan 92aX~ = r „ / [2(ft>„, - 2ft))], (114)

A;. -A /K.-° )) 2 +( ra . / 2) 2- tane«~.s r . iy t 2 K -«)]. (115)

^ - ^M2 + ral/2)\ tan02 ^ Tal / (2fl)al), (116)

^ . • V K i + f l , ) 2 + ( r - i / 2 ) 2 ' tane;lSr(I1/[2o».1 + fl))], ( in)

and

A2* = A/(ft)ul+20))2+(ral/2)2, tan # EE r „ / [2(a>al + 2a>)]. (118)

The signs of the phase parts, e.g., ±i92aX in Eq. (109), correspond to the signs of

imaginary parts, e.g., ±iTa l/2 in Eq. (109). Superscripts (p±) of A^f and S f

indicate the coefficient (p) and the sign (±) of external field frequency added to coal,

while in the case of A*, and 0* external field frequency is not added to coaX. Such

compact formulae are useful for analyzing a mechanism of nonlinear optical processes

and constructing structure-property relations in hyperpolarizabilities for molecular

systems.

4.2.2. ImyDensity of frans-Stilbene

90

As a typical example of y density analysis, we consider the longitudinal

component of Iray (=lmy^(-(ow^-a)) of /rans-stilbene (see Fig. 18(a)Pl) at the

two-photon resonant point (hm = 2.89 eV) with respect to state 2Ag ( to M g l A g = 2hco

= 5.78 eV, see Fig. 24(a)). The state model is obtained by the Pariser-Parr-Pople

(PPP) - single-double excitation configuration interaction (SDGI) (using full 1275

configurations) calculation. The Imy value calculated using 117 states, which can

nearly reproduce Imy at the full SDCI level, is 155.7 x 10"36 cm7/esu2. It is found that

the dominant contribution of excitation process to Imy is type (ffl-2) (lAg-lBu-2Ag-

lBu-lAg) (185.4 x 10"36 cm7/esu2) and is much larger than type (II) contribution, which

is mainly described by excitation process (lAg-lBu-lAg-lBu-lAg) (-2.313 x 1036

cmVesu2) (see Fig. 24(b)). It is noted that types (I) and (ffl-1) vanish because of the

symmetric systems with zero dipole moment in the direction of z axis. The much

smaller type (II) contribution can be understood by the fact that type (II) process

involves only terms concerning one-photon processes though we focus on the two-

photon resonance region. As expected from Orr-Ward formula in the two-photon

resonant region, type (III-2) (lAg-lBu-2Ag-lBu-lAg) contribution to Imyis positive in

contrast to the type (II) (lAg-lBu-lAg-lBu-lAg) contribution (with a negative value).

The Imy density distributions (in the Mulliken approximation) of dominant processes

(types (ffl-2) (lAg-lBu-2Ag-lBu-lAg) and (II) (lAg-lBu-lAg-lBu-lAg)) are shown

in Fig. 24(b). The primary contributions of Imy density for type (ffl-2)(l Ag-lBu-2Ag-

lBu-lAg) are found to be caused by the both-ends benzene-ring regions (a-d-e and j-k-

n) with large positive contribution and vinylene region (g-h) with small positive

91

contribution (see Fig. 24(b)(III-2)). This feature indicates that the virtual excitation

process of type (III-2) is spatially described by the virtual charge transfer from one

benzene ring to the other due to the ^-conjugation between both benzene rings. It is

also found that the Imy density contribution of type (II) (lAg-lBu-lAg-lBu-lAg)

comes from the similar virtual charge transfer to the case of type (ffl-2)(lAg-lBu-2Ag-

lBu-lAg) though the sign of Imy density of type (II) on each site is opposite to that of

type (III-2) (so that the contribution is negative in sign) and the magnitude of Imy

density of type (II) on each site is much smaller than that of type (III-2).

4.3. Summary

In Sec. 4.1, we explained the master equation approach (involving a general

form of exciton-phonon coupling) to exciton migration dynamics for aggregate systems.

In contrast to the Liouville approach (Sec. 2.2) using phenomenological relaxation

parameters, the behavior of relaxation among exciton states (exciton migration) were

shown to closely depend on the features of exciton wavefunctions, which depend on the

structures of molecular systems. The relations among such exciton migration (photon-

energy transfer) and near-resonant (hyper)polarizabilities are very interesting from

scientific and technological viewpoints, e.g., a novel multi-functionality of substances.

The present master equation approach combined with the nonperturbative calculation

method of (hyper)polarizabilities (Sec. 2.3) will be useful for the investigation of such

dynamics.

In Sec. 4.2, we explained a procedure providing an analytical expression of

dynamic hyperpolarizability density and applied it to the case of Imy for TPA. The

92

dynamic hyperpolarizability density analysis has the advantage of elucidating local

spatial contributions of electrons to dynamic hyperpolarizability. Its analytical

expression is useful for partitioning of dynamic hyperpolarizability density into real and

imaginary parts and for analyzing the contribution of each virtual excitation process.

The procedure will be easily applied to obtaining analytical expressions of response

densities for arbitrary response properties described by perturbational formulae. The

analytical expressions of dynamic real and imaginary response properties will be useful

for investigating the chemical and physical pictures (spatial contributions of electrons

for each virtual excitation process) of (perturbationally treated) near-resonant response

properties for larger-size systems such as dendrimers since this does not require any

Liouville dynamics but only needs static quantities (transition energies, transition

moments and transition density matrices) concerning the ground and excited states.

5. Concluding Remarks

We investigated the features of (hyper)polarizabilities of dendritic

supramolecular systems using the fractal-antenna-structured aggregate and molecular

models which mimic the phenylacetylene dendrimers. The nonperturbative calculation

and analysis methods based on the numerical Liouville approach have been developed

and applied to the off- and near-resonant a and y for dendritic aggregate models. On

the other hand, the static a and y for three types of phenylene vinylene oligomers have

been investigated as well as several components of y of a phenylacetylene dendrimer by

the finite-field approach and hyperpolarizability density analysis using molecular orbital

and density functional methods.

93

For dendritic aggregate models, we elucidated that the segmentation of exciton

distributions in linear-leg regions causes the differences in the intermolecular-

interaction effects on the spatial contribution of exciton generation to a and 7 in each

generation. The spatial contributions of intermolecular-interaction effects in inner

generations are predicted to be larger than those in outer regions. This feature is

considered to reflect the fact that the excitation energies in inner regions are smaller

than those in outer regions due to the stronger intermolecular interactions originating in

longer ./-aggregate-type structures in inner regions. The effects of relaxation among

exciton states, which are predicted to be essential for the efficient exciton migration

from the periphery to the core, have been also shown to provide remarkable changes in

7 in the near-resonant region. The magnitudes of relaxation effects on the spatial

contribution of exciton generation to 7 tend to be larger in outer regions than in inner

regions. This can be understood by the larger relaxation factors of higher-lying

exciton states, which exist in outer regions. The magnitude and sign of

hyperpolarizabilities, particularly in the near-resonant region, were found to be

remarkably related to the configuration of monomers and the state models of monomers

in dendritic aggregates.

As examples of dendrimers, we investigated the static (hyper)polarizabilities

per unit for three types of phenylene vinylene oligomers and a phenylacetylene

dendrimer. We showed that the size dependencies of aln and 7/n (n: the number of

units) for all />ara-connected oligomers are much larger than those of all meta-

connected oligomers, while fractal-structured meta-connected oligomers exhibit an

intermediate feature: the size dependencies of aln and yln for fractal-structured meta-

connected oligomers in the small-size region are smaller than those for para-connected

94

oligomers, while they preserve a large increasing rate and become close to those for

para-connected oligomers in the large-size region. From the (hyper)polarizability

density analysis, such size dependencies were found to be caused by the fractal nature in

the structure and the segmentation (localization) of spatial ^--electron contribution to a

and 7 in linear-leg regions (mutually connected at m<?ta-positions of benzene rings).

This feature suggests that (hyper)polarizability remarkably reflects the segmentation of

^-electron conjugation predicted in these dendrimers. Actually, the y density

distributions for two main components of y for phenylacetylene dendrimer (D25) were

shown to be localized in the linear-leg regions parallel to the applied field and to be well

segmented at znefa-connections. Similar unique size dependency and spatial

contribution features in dendrimers with fractal structures were also observed in the

intermolecular-interaction effects on aJn for dendritic aggregate systems (Sees. 2.4 and

2.5).

In contrast to the present dendritic systems, there are a variety of dendrimers

with three-dimensional (strictly, with a fractal dimension D (2 < D < 3)) structures and

non-7T-conjugations. These systems are not expected to exhibit any segmentation of

electronic derealization. For example, a tree-structured dendrimer involving

chromophores were found to have the effects of controlling the configurations of

chromophores and to exhibit large second-order NLO susceptibilities at the

macroscopic level [21]. However, even in such systems, the relations among exciton

migration (energy transfer) and (near)resonant hyperpolarizabilities have not been

elucidated. To elucidate the nature of such multi-functionality, more profound

investigations based on the exciton migration dynamics involving exciton-phonon

coupling will have to be performed.

95

In this review, we also introduced two extensions of methodologies: one is a

treatment of exciton migration dynamics based on the master equation approach

involving explicit exciton-phonon coupling and the other is the dynamic

hyperpolarizability density analysis using analytical expressions. These methods will

be useful for elucidating the relations and cooperative effects among (near)resonant

hyperpolarizabilities and exciton migration. The fractal structure is considered to

affect the exciton state energies, transition properties and exciton wavefunctions, while

the relaxation among exciton states are also predicted to depend on the relative phase

and the spatial distribution of each exciton wavefunction. The investigation toward

understanding the structure dependency of (near)resonant (hyper)polarizabilities of

dendritic systems is expected to create a new class of multi-functional nonlinear optical

substances with other attractive properties, e.g., energy and electron transfer. Such

studies will be also interesting in view of the realization of attractive chemical and

physical properties of substances with a new spatial dimension, i.e., fractal dimension.

Acknowledgments

We wish to thank especially H. Fujita and M. Takahata for their collaboration

and fruitful and enlightening discussions relating to this subject. We also gratefully

acknowledge the support of the Grant-in-Aid for Scientific Research (Nos. 12042248

and 12740320) from Ministry of Education, Culture, Sports, Science and Technology,

Japan. The author also thanks Professor David M. Bishop for his useful suggestions

on the subject in Sec. 4.2.

96

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103

Table 1.

Fitting parameters p, q and r in Eq. (26) for the intermolecular-interaction

(«im_anon) a n d relaxation (aint+rel -aim) effects in dendritic aggregates

from D10 to D127 shown in Fig. 2.

p q r

Intermolecular-interaction effect 0.7056 -8.412 21.95

Relaxation effect 1.988 -51.22 205.0

104

Table 2. y^, ym, ym, y„„, 7^, 7^ and the orientationally averaged / ( y s

values [a.u.] (Eq. (38)) of D25 shown in Fig. 23(a).

Components

1 XXXX

>yyyy

' zzzz

7xxyy

fyyzz

1 ZZXX

Value [a.u.]

2171000

567.5

2170000

13430

13370

72800

y.. 1170000

105

Figure Captions

Figure 1. Structures of Cayley-tree-type phenylacetylene dendrimers (D4, D10, D25,

D58 and D127). D4 and D10 are referred to as compact Cayley-tree-type

dendrimers, while D25, D58 and D127 are referred to as extended Cayley-

tree-type dendrimers.

Figure 2. Structures of dendritic molecular aggregates (D4, D10, D25, D58 and

D127) which mimic skeletons of phenylacetylene dendrimers. N

represents the number of monomers. Each two-state monomer dipole

unit (with transition energy E2i = 38000cm~' and transition moment

/i21 =10D) is represented by an arrow. The intermolecular distance in

linear legs and the angle between neighboring linear legs at all branching-

points are assumed to be 15 a.u. and 120°, respectively.

Figure 3. Calculated one-exciton state energies £,fe [cm1] and the magnitude of

transition moments tffg [D] between the ground (1) and one-exciton (/)

states of the dendritic molecular aggregates shown in Fig. 2.

Size dependency of intermolecular-interaction effects (log( |orint - anon |

/(Nxla.u.)) vs. 1/AO (a) and the relaxation effects

(log(|aim+rd -aint|/(M<la.u.)) vs. 1/AO (b). The N represents the number

of monomers. (ajM - amJ/ N values for D10, D25, D58 and D127 are

1.100 a.u., 2.504 a.u., 3.546 a.u. and 4.418 a.u., respectively,

while (aim+Kl-aim)/N values for D10, D25, D58 and D127 are -

0.06667 a.u., -1.742 a.u., -11.62 a.u. and-41.93 a.u., respectively.

Calculated ai^a^) [a.u.] and its partitioned o^u yh (spatial

contribution) for the dendritic aggregates D10 and D58 (Fig. 2) involving

intermolecular interactions and relaxation effects. The size of circle at

each dipole site represents the magnitude of a^,..,,)_,,. The scale

factors of these systems are different from each other. The symbol b

indicates a one-exciton aggregate basis (see Eq. (14)). These a values

are the same as ain,.„, in Sec. 2.4.1.

107

Figure 6. (DIO-a) and (D58-a) show the differences: aim -ocm„ (see Sec. 2.4.1

for the notation of a) of D10 and D58 (Fig. 2), respectively.

Relaxation effects in one-exciton states are omitted in these cases.

(DIO-b) and (D58-b) show the difference: aint+rel - aim of D10 and D58,

respectively (Fig. 2). (DIO-c) and (D58-c) show the total effects:

aint+rei ~ non of D10 and D58, respectively. It is noted that D10 takes a

non-fractal structure, while D58 does a fractal structure. The white and

black circles represent positive and negative contributions, respectively,

and the size of the circle indicates the magnitude of contribution. The

scale factors of circles for these systems are different from each other.

Figure 7. Four types of molecular aggregate models: (D25) dendritic aggregate

model, (L25J) /-aggregate model, (L25H) ^/-aggregate model and (S25)

square-lattice aggregate model. Each arrow represents a transition

dipole unit, i.e., a two-state monomer with transition energy

E2[ = 38000 cm"' and transition moment /z21 = 5 D.

Figure 8. Size dependencies of cc^a^) [a.u.] (a) and their intermolecular-

interaction effects (ajnl-anon)/Af (b) for systems: /-aggregate (L),

square-lattice aggregate (S) and dendritic aggregate (D) models (see Fig.

7). N represents the number of monomers.

Figure 9. Intermolecular-interaction contribution (aim - anon) of each dipole unit to

a(= a J of (L25J), (L25H), (S25) and (D25) shown in Fig. 7.

Figure 10. Calculated one- and two-exciton states for D10 ((D10-o) and (D10-t)) and

for D25 ((D25-o) and (D25-t)) (see Fig. 2). The excitation energy and

transition moment of two-state monomers involved in these dendritic

aggregates are E2I = 38000 cm-1 and n2] = 5 D, respectively.

Figure 11. Calculated / ( ^ y ^ . ) [a.u.] and the one- and two-exciton contributions

(spatial contributions) of the dendritic aggregate D10 (Fig. 2) involving

intermolecular interactions and relaxation effects. The excitation energy

and transition moment of two-state monomers involved in these dendritic

aggregates are E21 = 38000 cm"1 and fi2l = 5 D, respectively. The

white and black circles represent positive and negative contributions,

respectively, and the size of the circle indicates the magnitude of

contribution. This /value is the same as yint+re, in Fig. 12.

109

Figure 12. (a) shows the difference (yint - ynon) between exciton contributions to

y(= Yxny) of D10 (Fig. 2) composed of interacting monomers and those

composed of noninteracting monomers. Relaxation effects in exciton

states are omitted in the case of (a), (b) shows the difference

(/im+rei ~ Y-™) between exciton contributions to y of D10 (Fig. 2) with

relaxation effects and those without relaxation effects, (c) shows the

total effects (yim+re, -ynon) of intermolecular interaction and relaxation:

(a)+(b). See Fig. 11 for further legends.

Figure 13. Calculated y(= y„„) [a.u.] and the one- and two-exciton contributions

(spatial contributions) for the dendritic aggregate D25 (Fig. 2) involving

intermolecular interactions and relaxation effects. See Fig. 11 for

further legends.

Figure 14. (a) shows the difference (yint - ynon) between exciton contributions to

y(= yjttti) of D25 (Fig. 2) composed of interacting monomers and those

composed of noninteracting monomers. Relaxation effects in exciton

states are omitted in the case of (a). (b) shows the difference

(yim+re, -y i m) between exciton contributions to y of D25 with relaxation

effects and those without relaxation effects, (c) shows the total effects

(yint+re, - ynon) of intermolecular interaction and relaxation: (a)+(b). See

Fig.l 1 for further legends.

110

Figure 15. Spatial contributions of exciton generation to a real part of y(= y ^ ) for

D25 (see Fig. 2) in the near-resonant region (electric field frequency

co = 37950 cm"'): total contribution (a), intermolecular interaction effects

(b) and relaxation effects (c). The scale factors of circles for these

systems are different from each other. See Fig. 11 for further legends.

Figure 16. Cayley-tree-type dendrimers (D25 and D58) composed of

phenylacetylene (A) and phenylene vinylene (B) units. For example, the

dendron part indicated by a thick line for D25(D58) represents a meta-

oligomer (made of phenylene vinylenes) with a fractal structure referred

to as MF4(MF11) shown in Fig. 17(c).

Figure 17. Schematic diagram of the second hyperpolarizability (y„„) densities

(P/ffV))- The white and black circles represent the positive and

negative p]?\r), respectively. The size of circle represents the

magnitude of p|?V) and the arrow shows the sign of p(^(,r)

determined by the relative spatial configuration between these two

111

Figure 18. Structures of para-oligomers ((a) Pn) and meta-oligomers ((b) MNFn and

(c) MFn) (n = 1, 4, 7, 11 and 16) made of phenylene vinylene units.

Only skeletons of carbon (C) atoms are illustrated. C-C bond-lengths

(Rl = 1.396 A, R2 = 1.473 A and R3 = 1.336 A) and C-C-C bond angle

(6 = 120°) are taken from the standard geometry data of stilbene (PI).

MFn oligomers have linear-leg regions with different lengths, in which

double-bond units are linked through para-substitutions of benzene rings,

while those different-length legs are linked at the meta-positions of

benzene rings. In contrast, for MNFn oligomers, all phenylene vinylene

units are linked through the meta-positions of benzene rings. These

features imply that MNFn and MFn oligomers have non-fractal and

fractal structures concerning the number of phenylene vinylene units

involved in linear-leg region, respectively.

Figure 19. aH and yuzL values and their density plots of stilbene (Fig. 18(a) PI)

calculated by the B3LYP with 6-31G**+d basis sets ((a) and (c)) and PPP

CHF ((b) and (d)) methods. For the PPP CHF results, azz and y....

densities in the Mulliken approximation are plotted, while for the B3LYP

results aa and yaa densities are plotted on the plane located at 1 a.u.

above the molecular plane. Lighter areas represent the spatial regions

with larger yzuz densities.

Figure 20. Size dependencies of longitudinal a(=a,z)/n ((a)) and y(=/zz22)/«

((b)) (n: the number of phenylene vinylene units) for Pn, MNFn and MFn

oligomers (Fig. 18) calculated by the PPP CHF method.

Figure 21. a(= a^) ((a)) and y(= yzzzz) ((b)) density plots of PI, PI6, MNF16 and

MF16 oligomers (Fig. 18) calculated by the PPP CHF method. The

white and black circles represent positive and negative contributions,

respectively, and the size of the circle indicates the magnitude of

contribution. The gray points indicate the positions of C atoms.

Figure 22. Longitudinal components, yzuz, and their density plots of

diphenylacetylene calculated by the B3LYP with 6-31G**+d basis sets

((a)) and INDO/S CHF ((b)) methods. For the INDO/S CHF result,

yzzzz densities in the Mulliken approximation are plotted, while for the

B3LYP result, yzzzz density is plotted on the plane located at 1 a.u. above

the molecular plane. Lighter areas represent the spatial regions with

larger y,^ densities.

Figure 23. yxxxx ((a)) and yzzzz ((b)) values and their density plots of D25 (Fig. 16)

calculated by the INDO/S CHF method. The white and black circles

represent positive and negative y densities, respectively, and the size of

the circle indicates the magnitude of y densities. The gray points

indicate the positions of carbon atoms.

113

Figure 24. (a) Three-state model, which mainly contributes to lmy(-(o;co,o),-(o)

(ftft)=2.89 eV) of /ra«s-stilbene (see Fig. 18(a)Pl), calculated by the

PPP-SDCI method, (b) lmy(-co;o),o),-co) tico=2.%9 eV) and their

density distributions (in the Mulliken approximation) for types (II) (lAg-

lBu-lAg-lBu-lAg) and (III-2) (!Ag-lBu-2Ag-lBu-lAg) processes.

115

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117

3 CO

c o

« L

lz

e D ) O

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

(a) Intermolecular-interaction effects

o.oo 0.12

(b) Relaxation effects

CO

S"

J2T

O

0.12

Fig. 4

118

(D10)a=819.2a.u.

o o

o o o o

(D58) a = 4668.8 a.u.

o o o o

o o o o o o o o

o o

o o ° o o

o o o o o

o o o o ° c > o c> o o o o o o o

° o o ° o o

o o o o o o o o

o o o o

Fig. 5

119

(D10-a) o o

o O O o

o o cciM-anon = 9.9 a.u.

(DIO-b)

• •

Cl ^ in t+re]

(DIO-c)

O

o Cl ^int+rel

i - #int = -0.6 a.u

O O

o O O o

o - ocnon = 9.3 a.u.

Fig. 6-1

120

(D58-a) o o o o

° o _ „ o o o o

• o

o o •

O

« o o o o •

o o o o

«int-«non = 202.1a.U.

(D58-b) • •

• • « 9 0 « • • # t #

# • " #

• • • . # »» »» # . • • •

* t t *

°Wei ~ am - -662.2 a.u.

(D58-C) • • • •

. . . . . . . . # t #

• • • • • • . • • * * • • . • • • • • • •

° W i -«non =-460.1 a.u.

Fig. 6-2

121

CM

X

lO

CM

K

CM

CM

Q

22.7

22.7

22.6

22.6

22.5

22.5

(b)

A"'

" A

......-A

,u-"' _-- — -—""

A (L)

-O-(D)

1 _

~T

-

0 10

46.6

46.4

46.2

46.0

45.8

45.6

45.4

45.2

3

05

o 3 effects

o

a/N

0)

c for

P § Q .

20 30 40 N

50 60 <y>

Fig. 8

123

o o

X ra o cs •<* vo oo m c5 d> d> d d> d CM I I

[•rve] snjBA.

in CM Q

3S CM _ l

a — a — , in

• S B 5 5 E E 5 B "bo S B S S B B S ^ B O N

n u-i o >o o xi ^ —I d d

[•tru] anpA

0 O O O 0 o OOO o o OOO o o OOO o oOOOo

(S25

)

124

CO

CD

••

-»

CO

4—»

CO

c o

•1—

»

o X

CD

O

5 H

^

M^

,

+-< o

T

™

Q

i

i

11 i i i

i i ~

i i i i

-!

--1

1 1

1 1 |

"

o •*

to

CO

OC

D

CO

*-

CO

CO

C\J

o CM

m

•»

-

CO

CD

+-» CO

*-» CO

c o

o X

CD

o 5

H

-1—

»

m

CM

Q

o o

o o

o o

o o

o o

o o

CO

•* C

M

O

CO

C

O

CD C

O

CO

C

O

W

LO

r—

r~—

r-—

I—

r~—

r~—

[.LU

O] A

6jeug [t.u

jo] A

6jeug

CO

CD

•*-» CO

CO

c o

-1—

»

o X

CD

CD

c o

^

-"V

o

o —

Q

i -

i i -

i i

i i

CO 0

)

15 55

o o C

O

CO

C

O

o o CO

0

0 C

O

o o •<d-0

0 0

0

o o CM

0

0 C

O

o o o CO

C

O

o o 00

1^

CO

[t.aio

] A6J9U

3

CO

CD

•+-« CO

*-» CO

c o

*-» o

X

CD

CD

c o

3" in

CVJ Q

i i

-i i

1 i i i i i i i i i

i

S1

CD

CO

•4—

•

,C0

o o o o •>*

o o w

OJ

CO

o o o en C

O

o o in 0

0 C

O

o o o 0

0 C

O

o o LO

r

CO

o o o i^ C

O

[^oio] A6jeug

Total = -42330 a.u.

One-exciton contribution = -141330 a.u.

Two-exciton contribution = 99000 a.u.

O O

o o o o

126

(a) Intermolecular-interaction effect 7m ~Ynan)

Total = -761 a.u.

o o



O O

o O O o

(b) Relaxation effect (7int+rei -7 im)

Total = 13a.u.

o o

o o

O O

One-exciton (type (II)) contribution = 194 a.u.

o o

o o

Two-exciton (type (III)) contribution = -181 a.u.

o • m o

28

(c) Intermolecular-interaction and relaxation effect (7iru+rel

Total = -748 a.u.

O O


o o


o o

O o o o o

o o

129

Total = -135050 a.u.


• • • • • •

• • • •


o o o o o o

o o ° o

o o o o o o o o

o o o o o o

Fig.13

130

(a) Intermolecular-interaction effect (7i„t - /„<>.>)

Total = -23968 a.u.

• • •

• • • • • •

• • • •


• • • •

* .

• • • •


o o o o

o o o

o o o o O o o o

o o o o

Fig. U(a)

131

(b) Relaxation effect rint+Kl -y i n t)

Total =-4.9155 a.u.

• • • • • •

• •

One-exciton (type (II)) contribution = 2.8508 a.u. O O

o o o o

o o ° o

o o o o o o o o

o o o o o o

Two-exciton (type (III)) contribution = -7.0463 a.u.

• • • •

• • • • • • • •

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PART TWO

Molecules in Intense Laser Fields: Nonlinear Multiphoton Spectroscopy and Near-Femtosecond To Sub-Femtosecond

(Attosecond) Dynamics

MOLECULES IN INTENSE LASER FIELDS: NONLINEAR MULTIPHOTON SPECTROSCOPY

AND NEAR-FEMTOSECOND TO SUB-FEMTOSECOND (ATTOSECOND) DYNAMICS

Andre D. Bandrauk Canada Research Chair

Laboratoire de Chimie Theorique Universite de Sherbrooke (Qc) J1K 2R1, CANADA

Hirohiko Kono Department of Chemistry

Graduate School of Science Tohoku University

Sendai 980-8578, JAPAN

150

1 Introduction

Current laser technology has opened up a new field of study, the nonlinear, nonperturbative interaction of matter with intense, ultrashort laser pulses [1-2]. Much of the earlier experimental and theoretical work in this nonperturbative regime of radiation-atom interaction has been summarized by Gavrila [3]. Thus in the atomic case many new nonperturbative optical phenomena and processes have been found such as above threshold ionization (ATI) [3-6], tunnelling ionization [7-9], and High-Order Harmonic Generation (HOHG) [10-13].

The rapid development of laser technologies allows for shaping and focussing laser pulses to higher intensities and even shorter times, thus inducing radiative coherences that are shorter than nonradiative (radiationless) relaxation times. The latter usually occur in molecules on sub-picosecond (femtosecond-fs) time scales [14]. We give in table I a brief "history" of this evolution (revolution) and the new terminology, which accompanies this evolution. Whereas at the end of the 20th

century the focus was on femtosecond (fs) photochemistry and photophysics culminating with a Nobel Prize to A.H. Zewail for "Femtochemistry", a major effort is now underway to develop single attosecond optical pulses [15-16] based on the physics of HOHG [10-13] using very intense short pulses [16]. Clearly we are entering a new area where the nonlinear dynamics induced by the nonperturbative laser-matter interaction regime is being exploited to generate high order laser-matter coherences on ultrashort (attosecond) time scales, shorter than the time scale of relaxation or decoherence which occurs in molecules on femtosecond scales [14].

Table I Evolution of Laser Parameters [1]

Time (s)

Nano 10"9

Pico 10"12

Femto 10-15

la.u.: 24x10 18

Atto 10"18

Zepto 10"21

Yocto 10-24

Intensity (Watts/cm2)

Giga 10+9

Tera 10+12

Peta 10+15

/0=3.5xl0+ 1 6

Exa 10+18

Zetta 10+21

Yotta 10+24

Year

1980

1990

1995

2001

2003

?

151

In spite of these highly nonlinear nonperturbative effects induced by current intense short laser pulses, it has been possible to explain ATI and HOHG spectra in atoms using simple "quasistatic" models of tunneling ionization. This simple model first proposed by Keldysh [7] has been used successfully by Corkum et al. to explain the mechanism of ATI and HOHG in atoms [9-11]. In the high intensity and low-frequency regime the internal Coulomb potential is distorted by a laser field to form a "quasistatic" barrier through which an electron can tunnel. The rate of tunneling ionization can be calculated by "quasistatic" theories i.e., using standard quantum mechanical expressions for tunneling through a barrier induced by the field. This method of calculating atomic ionization rates usually gives reasonable orders of magnitude for atomic ionization rates and is now called the ADK method [3-4, 17-18]. Further extension of this simple model to HOHG is made by assuming that the velocity of the electron after quasistatic tunneling is zero and the evolution of the ejected electron is then described by classical mechanics [7, 11, 19].

The interaction of molecules with intense laser pulses introduces new challenges due to the presence of nuclear motion, i.e., extra degrees of freedom, and thus additional time scales. As seen later, typical ionization rates at intensities of 1014 W/cm2 at wavelengths of 800-1064 nm in the near IR are of the order of 1014 s"1, i.e., ionization times of 10 fs. Laser cycle periods are 3.5 fs at 1064nm and 2.7 fs at 800nm. The shortest nuclear motion period is that of the proton, 10-15 fs, thus confirming that the motion of the most elusive particle (important in many biological and chemical processes) is a near-fs process, which needs to be considered on equal footing with radiative processes induced by intense laser fields. We can immediately conclude that the extra nuclear degrees of freedom in molecules will have a great influence on the photochemical dynamics in the nonperturbative multiphoton regime. At intensities / <1013 W/cm2 where ionization is negligible, a successful approach to treat nonperturbatively molecule-radiation interaction has been the dressed-molecule representation summarized in [20]. In this representation, the total system consisting of molecule and photon is described semiclassically (quantum molecular motion + semiclassical radiation field) and due to energy conservation between the molecular and photon subsystems, resonant processes are pictured as crossing of "dressed" molecular potentials [20-21]. At high intensities, laser-induced avoided crossings occur creating Laser-Induced Molecular Potentials (LIMP's) predicted as early as 1978 [22] and recently confirmed experimentally [23]. The dynamics of the photophysical processes which occur in the presence of LIMP's can be described successfully by nonadiabatic theories of nuclear transitions originally developed for nonperturbative molecular spectroscopy phenomena such as predissociation [24-25]. As we show further on, such nonadiabatic theories of nonperturbative (beyond

152

Fermi's golden rule) transitions need to be considered at high intensities in view of the nonseparability of electron and nuclear motion.

Intensities exceeding 1013W/cm2 imply considerable ionization. This can be concluded from table II where we tabulate the atomic units (a.u.) of physical parameters relevant to this present article:

Table II Atomic units

Potential energy

Electric field:

Intensity:

Distance:

Time:

: V0 = e2/a0 =1 Hartree = 27.2 eV,

E0=e/a02 =5xl09 V/cm,

I0= cE\ I%K = 3.5xl0'6 W/cm2,

a0 = 0.0529nm,

f0=24.2as.

(1)

(2)

(3)

(4)

(5)

The intensities discussed in the present article, 1014 < / < 1015 W/cm2, correspond to fields approaching the internal Coulomb potentials of atoms and molecules, Vo [cf. Eqs. (l)-(3)], thus inducing considerable distortions of intermolecular potentials. In the dressed state representation these radiatively induced distortions creating LIMP's as discussed above lead to bond softening via laser-induced avoided crossing of molecular potentials [26-27]. At such intensities, one needs to consider further ionization and the remaining molecular ion potentials become LIMP's in the presence of intense laser pulses. The molecular ions can also undergo Above-Threshold Dissociation (ATD) [20, 26-27], the equivalence of ATI in atoms. As shown below, the quasistatic picture of atomic tunnelling ionization needs to be modified for molecular ATI and ATD in view of the multi-centre nature of electron-potentials in molecules and the presence of large radiative transition moments, originally described by Mulliken as early as 1939 [28] as Charge Resonance (CR) transitions.

One of the fundamental differences between intense field nonperturbative ionization of atoms and molecules is "Enhanced Ionization." The first numerical evidence of this nonperturbative phenomenon was found in exact three dimensional (3-D) time-dependent Schrodinger equation (TDSE) simulations of the Born-

153

Oppenheimer (static nuclei) ionization of the simplest one-electron molecular ion H2

+ [29-31]. This was confirmed by subsequent 1-D calculations on single- and two- electron models of molecular ions [32-36]. This enhanced ionization was first interpreted in terms of laser-induced localization of the electron in molecules, akin to similar effects predicted for double quantum-wells [37]. This unusual phenomenon has been observed experimentally as a preponderant production of Coulomb Explosion molecular fragments at critical distances Rc greater than equilibrium distances Re for molecules exposed to intense short pulses [38-41]. The experimental critical distances R c for Coulomb Explosion were found to agree well with the calculated enhanced ionization maxima.

Quasistatic models of molecular ionization have been successful in predicting values for R c [42]. The main tenet of this model, which differs fundamentally from quasistatic atom models, is that laser induced charge resonance (CR) effects localize temporarily the electron by charge transfer in one [30-33] and two-electron [34-36] molecular systems. We have called this phenomenon Charge Resonance-Enhanced Ionization, CREI, in honor of Mulliken who predicted strong radiative effects in spectra of symmetric molecules due to the presence of large electronic transition moments as a result of electronic charge transfer across the whole length of the molecule [28]. CREI has been used to explain recent exact numerical simulations of ionization in one-electron systems such as H2

+ and other linear [34] and even nonlinear molecules [43]. There is now clear experimental confirmation of CREI in the diatomics I2 [38] and H2

+ [39-40] from detailed pulse-probe experiments. In addition CREI has been observed also in recent experiments for many-electron molecules such as CO2 [44]. As emphasized above, the extra degree of freedom arising from the nuclear motion necessitates the use of new concepts such as Laser-Induced Molecular Potentials, LIMP's [20, 22-27]. The complete study of the dynamics of dissociative ionization of even the simplest system such as H2

+, has been recently achieved by exact non-Born-Oppenheimer simulations using current state of the art supercomputers [45-47]. Such "numerical experiments" have confirmed that nonadiabatic transitions through nuclear as well as field-induced potential crossing points become essential in describing the ionization process of molecules at high intensities.

In the present paper, we will focus on exact numerical simulations of dissociative ionization of one and two-electron systems in order to establish the universality of CREI and its origin. Simple quasistatic models will be used to derive analytic expressions for the critical CREI distance R c. Models for nuclear and radiative nonadiabatic transitions will be presented in order to interpret quantitatively the exact numerical simulation results. The doorway states to ionization are identified by the

154

use of field-following time-dependent adiabatic states. Finally, possible future applications of CREI and intense laser field multiphoton processes will be discussed for small molecules and clusters and future research directions will be suggested.

2 Numerical Methods

We have solved previously the TDSE for one-electron, both linear and nonlinear, systems H„ ~1)+ interacting with intense laser pulses using an exact three-dimensional (3-D) Hamiltonian. We have also carried out 1-D exact non-Born-Oppenheimer simulations with moving nuclei. For two-electron systems, a 1-D Hamiltonian with a regularized (softened) Coulomb potential to remove singularities is commonly employed. In the 1-D models, an electron or electrons are allowed to move only along the molecular axis. The first 3-D static nuclei (Born-Oppenheimer) ionization calculation was performed for H2

+ [29-31] and later extended to include moving nuclei [47]. For linear systems with the molecule parallel to the laser field, i.e., the spatial configuration with maximum radiative coupling due to CR effects [20], the dimensionality of the TDSE (a parabolic partial differential equation) is 3«, where "«" is the number of electrons (3n-l for the electrons, +1 for symmetric nuclear motion). For the parallel linear configuration due to the cylindrical symmetry, one can reduce the dimensionality by one for one-electron systems and even two-electron systems such as H2 [48]. For circularly or elliptically polarized laser fields, one has to resort to full 3-D calculations [49]. We have recently generalized the latter to be able to go beyond the usual dipole approximation [50] which is only valid for electron dimension r much less than the field wavelength A, i . e . , r /A« l [20].

We describe here as an example the 1-D H2+ system with moving nuclei, which

we have used to calculate completely ATI and ATD spectra [45-47]. The advantage of the 1-D model is that in the near-IR regime (1064-880 nm) the electron is restricted to motion close to the internuclear axis for short pulses, thus giving very reliable results that are comparable to exact 3-D calculations performed earlier [47]. Owing to computational (memory storage) limitations, the 3-D calculations necessitate absorbing boundaries at the numerical grid edges which cause an elimination of high-energy components in both electron and nuclear spectra. In the case of 1-D models, due to the reduced dimensionality, we can extract complete information on the ATI, ATD and Coulomb Explosion (CE) spectra from exact simulations.

The 1-D TDSE for H2+ with both electronic and nuclear degrees of freedom

included is exactly given by (in atomic units, h = me = e = 1)

155

ldy/<<Zdt

R,t^[HNR) + Vcz,R) + Hez,R,t)]¥z,R,t), (6)

where "z" is the electron coordinate parallel to the molecular axis (with respect to the nuclear centre of mass), R is the internuclear distance. Here, the electronic Hamiltonian/ff(z,i?,/), the nuclear Hamiltonian/fA,(i?), and the regularized 1-D Coulomb potential Vc are given as follows:

Hez,R,t) = -71^T + XzEt), # „ ( * ) = - - L i l + I (7) dz mp oR R

K(z,R) = -? V^-f V^' (8)

\[z + RI2) +c j y(z-R/2) +c j

where mp = 1837 a.u. is the proton mass (me = \) and

(2m +l) 1 T>=L^-L> ^= 1 +^r^r- (9)

Am 2mp+1 The value of c can be obtained as the average value of the cylindrical coordinate yO = ( x 2 + y ) in full 3-D calculations, and this turns out to be c — \, giving very similar ionization potentials Ip as in the 3-D case [49]. The Hamiltonian used in Eq. (6) is the exact three-body Hamiltonian obtained after separation of the centre of mass motion in 1-D [45-47]. The field E(t) is expressed in the dipole approximation [50]:

Et) = EMfs(t)coso)t), (10)

where fs(t) is the shape of the field envelope, a> the frequency, and EM the maximum field amplitude related to the intensity / as in Eq. (3). We shall be using in our simulations: /i = 1064nm (<y = 0.043a.u.) for YAG lasers and/or /i = 800nm (ftj=0.057a.u.) for Ti/Sapphire lasers; the intensity range islO14 <7<10 l5W/cm2, below the atomic units of intensity I0 =3.5xl016W/cm2 [Eq. (3)] and the corresponding field E0 [Eq. (2)]. Such fields as we will show further on will induce complete single electron ionization on time scale of ~10 fs and less. The total electron-nuclear wave function y/(z,R,t) is advanced in a time step St by an exponential method, called the split- operator method [49-51],

156

^(z,Rj,+St) = exp(--StK)exp[-iSt(Vc+Val)]exJ--StK) + 0(Sti), (11)

where K = -rjd2 Idz2 -(l/mp)d2 IdR2, the kinetic energy terms only which can be

treated by FFT methods, and the middle exponential contains all Coulomb terms (electron and proton) plus the external field interaction Vat = %zE(t). The split-operator method can be extended to higher order to verify accuracy of the lowest three term symmetric scheme (11) [51] and is very efficient for regularized (singularity free) potentials. In 3-D calculations, singularities in Coulomb potentials and kinetic energy Hamiltonian, can be removed by expanding in Bessel functions and propagating with high-level split operators [29-31] or by using general Crank-Nicholson schemes [49]. Alternatively, we can use non-linear coordinate transformation [52-54] combined with the Crank-Nicholson integration scheme or the more general alternating-direction implicit scheme.

Two important physical parameters for laser-induced electronic processes need to be considered as those determine the size of the grid space required for computation. These are the ponderomotive energy Up and ponderomotive radius aM describing the oscillatory motion of an electron in a time-dependent field E(t) = EM cos cot. Solving the classical equations of motion [11, 19], gives

'z(t) = -EMcos(ot)t + <p); z (t) = -(EM/co)[sin (cot+ q>)-sin (<p)~\, (12)

z (t) = (EMI co2 )[cos(<w? + cp) + cot sin (cp)- cos (#>)], (13)

where q> is the initial laser phase at which moment the electron is ionized [ at t = 0 in Eqs. (12) and (13)] assuming an initial zero velocity z(0) = 0. These classical equations define the ponderomotive radius aM and the ponderomotive energy Up:

aM=EM/of, Up=±a2Ma)2=E2

M/4co2. (14)

The classical Eqs. (12)-(14) allow us to predict energies and displacements of electrons induced by the laser after ionization. As a first hypothesis we shall look at maximum acceleration, which in the classical limit should produce maximum radiation emitted by an electron [55]. From Eq. (12), z(max) occurs at phase cot + (p = n with resulting velocity z = (EMlai)sm.(p, with a maximum z(max) = £'M/CO at cp = njl. The resulting energy is £"(max) = z 2 /2 = E2

Ml2cd —2UP. The corresponding distance traveled by the electron, which can recollide with a neighbouring ion is zt) = aM [-1 + cot] = 0.57aM . Recombination of the electron with the parent ion gives rise to a maximum HOHG order [10-13, 19]

157

for maximum returning velocity at z = 0. The condition z = 0 yields the transcendental equation tan<p = (cosait-1)/(sincot-cot). Inserting <p as a function of cot into z(t) in Eq. (12), one finds that the rescattering time for which the returning velocity is maximized is cot — 1 .ZK . The corresponding phase is cp = 0. \K . The resulting maximum velocity is z(max) = l.26(EM I CO) and the maximum energy is £(max) = 3.17 Up. This simple classical model explains the theoretical HOHG maximum order in atoms by recollision [10-13, 19],

Nm=(Ip+3.l7UP)/co, (15)

where Ip is the ionization potential. However, as emphasized by us [19] in the molecular case, collision will occur also with neighbouring ions. The above maximum velocity condition (cot + cp) = "in 12 gives the velocity: z=(EMl'ft>)[l + sin(9>)l, i.e., z(max) = 2£'MIco at cp = n!2. Thus the maximum energy in this case is £ (max) = 2El, / co2 =8Up in agreement with numerical simulations for the corresponding distance traveled by the electron in a half cycle cot = n, i.e., z(n/co) = naM [19].

The simple classical model described above where one assumes an electron is ionized with initial zero velocity, the basis of tunnelling ionization models [7-9] allows us to deduce the laser induced dynamics of the electron after ionization. Thus, for A = 1064nm and intensity 7 = 1014W/cm2, one obtains aM =29 a.u. and Up = 9co~ lOeV. Then distances of TUXU = lOOa.u. and energies SUP ~ 80eV are readily achieved by ionized electrons. This implies the necessity of using large grids and high spatial and temporal resolutions in FFT's used in calculating the split-operators, Eq. (11). As we have shown above, the grid size for high intensity simulations are governed by the dimensions of the ponderomotive radius OTM and ponderomotive energy UP , Eq. (14). As mentioned above, the first recollision can occur at phase cot-\.l>n , i.e., in 0.65 cycle of the laser field [11,19,56]. This period corresponds to 2.3 fs at X = 1064 nm and 1.8 fs at 800 nm. Clearly, the laser-induced electron dynamics occur on near-femtosecond time scale and on attosecond time scale for shorter wavelengths [15].

In simulations, we generally set zmax = ± 500 a.u. corresponding to ~ 4000 grid points; in 3D simulations, the cylindrical coordinate pm3X = (x ,ax + y^) is set to be 125 a.u. (500 grid points). The step sizes in z and t are determined by the limits imposed by the uncertainty principle SzSp - SeSt ^ 1 (e.g., for £ = 5 a.u., i.e., p = lO^a.u., 8x < 0.3a.u. and St < 0.2a.u.). For ionization rate calculations, an absorbing potential is used at both z and p boundaries [49]. The ionization rate r(s~l) is obtained from the decrease of the time dependent norm N of the wave function,

158

dNldt = -rN , N(t)= fy/(t)\2dv. (16)

Calculations of complete ATI and ATD spectra can be readily obtained in 1-D models, i.e., one electron coordinate z and one nuclear coordinate R, thus reducing the numerical solution of the TDSE to a tractable 2-D problem with current computer technology. This allows for using ever larger numerical grids and circumventing the problem of loss of information of high-energy components of the total wavefunction y/(z,R,t) by projecting onto exact electron wavefunctions in the laser field, called Volkov states [4, 57]. This exact non-Born-Oppenheimer 1-D numerical procedure is described in detail for one-colour (single frequency) [45] and for two colour [46] dissociative ionization of H2

+. Such exact 1-D non-Born-Oppenheimer have allowed us to propose new methods of dynamic imaging of moving nuclear wave packets on near femtosecond time-scales for rapid proton motion [58-59].

3 Charge Resonance Enhanced Ionization and Quasistatic Models: One-Electron Systems

The first attempt to derive simple analytic formulae for atomic multiphoton transitions beyond usual perturbation (Fermi's golden rule) theory was undertaken by Keldysh [7] who showed that a parameter y, today called the Keldysh parameter [3-4], allows to separate the perturbative multiphoton regime from the nonperturbative tunnelling region. This parameter is obtained from the low frequency limit of the transition probability from an initial bound state to a Volkov state [4, 8] and is given by:

r = yllP/2Up, (17)

where Ip is the ionization potential and Up the ponderomotive energy defined in Eq. (14). This can also be interpreted as the ratio of the laser frequency co to the tunnelling frequency CO, through a barrier of width l = Ipl E where E is the electric field amplitude corresponding to the peak intensity I-cE2 I%K [7, 60]. Thus, provided the ionization proceeds faster than the laser induced tunnel barrier lifetime or alternatively the tunnelling frequency co, is larger than the laser frequency co, then the tunnelling ionization model should apply for y<\ where E is calculated at the peak of the field.

In this tunnelling regime of y <\, we can estimate the intensity at which ionization will occur completely by over-barrier passage for any given ionization potential Ip and charge q, using a simple electrostatic potential V(z) = -q/\z\-Ez. The barrier height of the total electrostatic potential V(z) takes the maximum value

159

at the position zm = jq/E . Equating —Ip to V(zm) gives the electric field Eb or intensity Ib at which complete over-barrier ionization should occur:

Eb=I2p/4q, Ib=I4

p/l6q2, (18)

For an H atom for which Ip = 0.5 a.u., the critical intensity Ib for over-barrier ionization can be readily estimated from the data in Eqs. (l)-(3) as 1.4xlOl4W/cm2. The corresponding Keldysh parameter is

rb=l6qa)/(2Ipf\ (19)

which now depends on frequency a>. Numerical comparisons of TDSE solutions for the H atom [18] with tunnelling rates obtained from quasistatic ADK theory [17] show that indeed for E field values below Eb, the tunnelling model gives satisfactory ionization rates but fails above Eb > since tunnelling no longer applies. It was further concluded that for noble gases and for laser wavelengths in the visible and the near infrared, the Keldysh parameter yb is ~ 1 , which means that in this frequency range multiphoton ionization goes over directly into above-barrier ionization and tunnel ionization plays a minor role.

Molecules in intense laser fields will undergo dissociative ionization upon multiphoton absorption. This is due to the competition between dissociation giving rise to ATD spectra [20, 27] and ionization giving rise to ATI spectra [45]. In particular for H2

+ for which a detailed comparison between experiment and theory has been performed [61-62], the extremely rapid motion of the protons on a time-scale of ~ 10 fs allows for dissociated H2

+ protons to travel to large distances R > Re

and Rc, where j^, is the equilibrium and Rc is the critical distance for Charge Resonance Enhanced Ionization (CREI). Enhanced ionization then occurs at a critical distance Rc> Re, which results in rapid creation of H+H+. Likewise, in the case of multielectron molecules, multiply charged fragments are created [38-41,44,61-66].

Following the procedure described in the previous section, i.e., Eq. (16), we have obtained ionization rates from numerical solutions of the TDSE for H2

+. In Fig. 1, we show 3-D ionization rates for H2

+ at intensity 7 = 1014 W/cm2 and the wavelength X = 800 nm. In Fig. 2, we give 1-D ionization rates at A = 1064nm (YAG) and 10.6 um(C02) . All rates are obtained for fixed internuclear distance R and have been performed in two different gauges, the length (E -r) gauge and a new space-translation representation in which Vc(r) is replaced with Vc(r — a(t)), where a(t) = [ dt'\ dt"E(t") is the ponderomotive radius as given in Eq. (14). The exact wave functions in the two representations are related to each other by a unitary transformation [ 49, 50 ] so that the ionization rates must be independent of these as

160

10 12

R (a.u.)

Figure 1 3-D H* ionization rates (units of lO'V ) as a function of R at A = 800nm : 0 - for circularly polarized light with / = 2 X10M w/cm2 ; + -linearly polarized light with/ = 1014 w/cm2, parallel to the molecular axis R : o - linearly polarized light with i" = 1014 w/cm2 , perpendicular to R.

1 I

1 c o

1 c o

&H-13

5e+13

4e+13

3e+13

2e+13

1e+13

0

a)

2 4 6 8 10 12 Irrtemuelear distance (a.u.)

~^1 b)

2 4 6 8 10 12 Internuclear distance (a.u.)

14

Figure 2 1-D Hj ionization rates as a function of R at intensity /

(a) X = 1064 nm (YAG); (b) X = 10.6 um (C0 2 ) .

:1014

162

found numerically. This serves as a validation of the numerical procedure. It is to be observed that in the near-infrared region, ionization rates are large in a window 6<i?<10a.u. and exceed the asymptotic atomic H rates by over one order of magnitude. Figure 1 where circularly and linearly polarized light calculation are compared at 800 nm illustrate that for radiation parallel to the internuclear axis the ionization rate is large in the range of 6 < R < lOa.u. and decreases rapidly for R >10 a.u. Figure 2b shows a single ionization maximum at R = 7 a.u. for the IR wavelength X (C02) = 10.6 urn. All three figures show considerable enhanced ionization for 6</?<10a.u. The 1064 and 800 nm results show some features of resonant absorption whereas the /l(C02) spectra have less resonance in the 6 < R < 10 a.u. range yet a wide ionization rate maximum exists around Rc=l a.u. which we call the critical distance Rc for CREI [30-32, 42-43]. We show next that a molecular quasistatic picture of electron tunnelling and above-barrier ionization via highest occupied (HOMO) and lowest unoccupied (LUMO) molecular orbitals can provide a simple interpretation of CREI.

In analogy with barrier suppression in atoms, Codling et al. proposed a similar effect in the tunnelling y < 1 ionization regime [65-66] for molecules, but now one has to take into account the multiwell structure of the Coulomb potential experienced by the electron. In the presence of the laser induced electrostatic potential +Ez, distortion of the double-well in Hj occurs. As illustrated in Fig.3, the instantaneous electronic potential at a nonzero field has a descending (lower) and an ascending (upper) well which yield the Stark-shifted HOMO and LUMO adiabatic states 1<7_ and 1<7+ , respectively. In addition to the middle internal barrier (inner barrier), when the field strength is nonzero, a barrier of finite width is formed outside the descending well (outer barrier). If the temporal change in the electric field E(t) is slow in comparison with the time scale of the electronic response, lsag and ls<Ju at zero fields are adiabatically connected to \a_ and lcr+, respectively. We plot in Fig. 3 the energies £_ and £+ of \a_ and \a+ as a function of R for a static field E corresponding t o / = 1014W/cm2. For large R, one can expect the quasidegenerate energies of hag and \sau to touch the maximum of the inner barrier at some critical distance Rc. An estimate oiRc~'ilIp=6 a.u. was proposed based on this model [33]. This simple one-electron model used for multi-electron molecular ions neglects the Stark-shifts of the MO's, which was shown previously to be indispensable as it can lead to possible charge localization [31] and maxima in HOHG [19]. Calculations of the energies of the HOMO-LUMO's in static fields E corresponding to the peak field intensities / confirmed the essential role of the Stark-shifts of these orbitals [30-32, 34]. We have also performed these calculations to investigate the influence of a magnetic field of field strength /? ( 1 a.u. = 107 Gauss) which acts as a

163

0.4

0 2

0.0

0 2

-0.4

-0.6

0.8

10

1.2

1.4

- R*4a.u. B=o.o

/--r>2.1x10"'aii. -

r_.2.9x10'Wu.

1°+

l ° - \

' -V -/) \

t\

I

/ -

I _

_ --

"R=9.5a.u. p . o.o

-r,.i.9xio r..i.exio

10

:t/l 1 -

/-/ W-

'4 l L

-

L

•R-14a.u. B-0.0 r.»3.5x10"*a.u

r -3.9x10sa.u i o t

-15-10-5 0 5 10 1515-10-5 0 5 10 15-15-10-5 0 5 10 15-15-10-5 0 5 10 1515-10-5 0 5 10 15

"R=6a.u. B = 0.1

-r,.9.5x10"'a.il

r..8.1x10''ai

1 0 . ^ _

-15-10-5 0 5 10 1515-10-5 0 5 10 15-15-10-5 0 5 10 15-15-10-5 0 5 10 1515-10-5 0 5 10 15

> • Z(a.u.)

Figure 3 3-D H2+ energies £_ and £+ of HOMO \a_ and LUMO lcr+ (horizontal

lines) and corresponding linewidths r_ 12 and r+12 (in a.u.) in a static field E = 0.053 a.u. (/ = 1014 W/cm2 j : (a) zero magnetic field/? = 0; (b) magnetic field /? = 0.1 a.u. = 106Gauss. The internuclear distance is fixed at various values of R = 4, 6, 7.5, 9.5, and 14 a.u. The electronic potential along the molecular axis at the nonzero field, - l / | / ? /2 + z| - l / | i ? / 2 - z | +Ez , is plotted as a function of the electron coordinate z. The 1 <7_ and 1 <7+ are localized in the descending and ascending wells, respectively.

164

magnetic bottle and confines (contracts) the electron density along the internuclear axis [54]. We give the linewidths T of these levels, which provide estimates of ionization rates x = \lT [ 1 a.u. = 24xl0~18s, See Eq. (5)]. One observes from Fig.3 that the LUMO (lcr+) is situated above the internal barrier in the range 5 < R < 10 a.u. in good agreement with the TDSE ionization rate maxima illustrated in Figs. 1-2. The effect of a magnetic field is to lower the energies of the LUMO, thus resulting in a narrower range of critical distances Rc [54]. The above correlation between exact TDSE (Figs. 1-2) and static (Fig. 3) ionization rates suggest a wide applicability of the quasistatic model not only to atoms but also to molecules to interpret intense laser-molecule interactions. Figure 4 gives a detailed variation of the ionization rates in the static field of intensity / = 1014 W/cm2 for zero magnetic field. Figure 5 gives the corresponding energies of the Stark-shifted HOMO and LUMO, i.e., £_ and £+, as a function of R for the same intensity as in Fig. 4. Vin is the maximum height of the inner (central) barrier of the instantaneous electronic potential and Vout is the maximum of the outer barrier (left barrier when E > 0, as shown in Fig. 3). One sees in Fig. 5 that both barriers are of equal height around R- 6a.u. where an ionization maximum appears in the TDSE calculation (Figs. 1-2) and in the static line widths /+ (Fig. 4). The equality of the two barrier heights causes an additional enhancement at R - 6 a.u. because around this distance the energy difference between the LUMO and the higher one of these barriers is maximized. Finally, one observes the gradual trapping of the LUMO in the ascending well for R>l0a.u., resulting in the disappearance of over-barrier ionization.

The above static over-barrier model involving Stark-shifted HOMO and LUMO is devoid of any time dependent dynamics. Thus, various time scales must be considered in order to understand the complete time dependent dynamics. Firstly, the energy separation A(i?) = f,CT„ (R)-£]r7g(R) can be taken to be a measure of the tunnelling time t, = \/A between the two Coulomb wells created by the two protons (Fig. 3) at zero fields. Another time scale is the laser photon cycle time tL = 1 / co. The maximum radiative coupling strength is determined by the maximum amplitude EM and the transition moment//g!/(^?) = < l<rg|z|lcr„> parallel to the internuclear axis. For one or odd electron symmetric systems this type of transition moment is due to Charge Resonance (CR) or charge transfer phenomena. In the case of H2

+, fi^ (R) is ~ R/2 corresponding to half an electron transfer from one end of the molecule to the other [28]. The relation //^ (i?) - R/2 can be easily proven by approximating \ag and \ou as |l<7g>B) = (±a + Z>)/v2 , where a and b are Is atomic orbitals located on the left and right protons, respectively. Then the maximum radiative coupling coR=EMRI2 is the Charge Resonance Rabi frequency with a time scale TR = 2n I coR . Finally, one has to consider the ionization rate r -1 /1, and the

165

ft (a.u.)

Figure 4 3-D H* linewidths T_ and r+ of HOMO and LUMO in a static field E -0.053 a.u. ( / = 1014 w/cm2) as a function of R. The case corresponds to Fig.3 with

166

Figure 5 HOMO (£_) and LUMO (£+) energies of 3-D H+2 as a function of

R. Vin and Vm, are the maxima of the inner and outer barriers (Fig. 3) at p = x2+y2) =la.u.

167

nuclear motion time scale ^ s l O - l S f s (the vibrational frequency 0)p of H2+ is

2000 cm"', i.e., tp — 15 fs). We now compare these various parameters at 7 = 1014 W/cm2 and A= 1064 nm [in atomic units, EM =0.053 a.u. and co = 0.043 a.u. ( ^ = 3.5 fs) ]. Thus for 6<i?<10a.u., we find that 0.16 < coR < 0.27 a.u. and 0.1 >A> 0.02 a.u. Thus, in the enhanced ionization range, the strong radiative-coupling limit holds; coR >A> coL, coP [67]. Clearly, the radiative coupling between the HOMO and LUMO dominates around the ionization maxima 6<JR<10a.u. so that these two electronic states can be considered as the essential states, i.e., the CR states of the system.

We develop next a two-level model to describe the time-dependent dynamics for this strongly driven system based on Bloch equations [68] which we used previously to describe HOHG in such a system [30, 67]. The wave function of a two-level system can be written as

\¥t))-Cgt)\¥g) + Cut)\¥u). (20)

where \y/g) and \y/u) are the wave functions of field-free time-independent two states g and u. The corresponding TDSE takes on the form:

''^H^te+rCOC,,, (2D idu= + Y2)Cu+Vt)Cg, (22)

where A = (su-£g is the field free transition energy and V(t) = jUguE(t). //^ is the transition dipole moment. Defining the three real functions:

xt) = CgC'u+C„C'g, (23)

yt) = iC«C]-CgC:), (24)

z ( 0 = | c „ | 2 - | C g | \ (25)

one obtains two important physical parameters : d(t) = Hmxt) is the induced dipole moment responsible for HOHG and z(t) is the population inversion. One derives easily the corresponding two level Bloch equations by differentiating Eqs. (23)-(25) and using Eqs. (21) and (22)

x = -Ay, y = Ax + 2V(t)z, (26)

z = -2V(t)y. (27)

168

These equations have the constant of motion, x2 + y2 + z2, i.e., the norm of the wave function, which is set equal to one. This means that negligible ionization is assumed. In the strong field limit which is valid in the enhanced ionization region 6 < R < 10 a.u. as discussed above, V (t)» A. Neglecting A in Eq. (26), one obtains

readily the strong field solutions:

y(t) = y0)cos[F(t)] + z0) sin [ > ( , ) ] , (28)

z[t) = z (0 )cos [F( / ) ] - y0)sin[F(t)], (29)

where

F(t) = 2J'oV(t')dt'. (30)

The quantity F(t) is therefore the total field area [67]. For a monochromatic field E(t) = EM cos cot) , then selecting the initial

conditions j (0 ) = 0 and z(0)=+l (as shown in [67] this gives only odd harmonics), and using the trigonometric relation [69],

cos[Ssm(a)t)]^J0(S) + 2YjJu(2kci)t), (31)

one obtains the expression for the population difference between the l<Tg and \au :

\cg(t)\2 -\Cu(t)\

2 =J0(2o)R/ o)) + 2^J2k(2o)R/ co)cos(2kcot). (32)

The Js are Bessel functions and coR = EMR12 is the Rabi frequency. In fact, AxJ0 (2coRl 0)) is the energy difference of the dressed (Floquet) states, i.e., the field free energy separation A is renormalized (dressed) by the time dependent field [70]. Clearly, at zeros of J0, there is a maximum charge asymmetry, i.e., Cg (t) = ±CU [t) which corresponds to localization in atomic electronic states, (lcr, ±\au) = b or a, the atomic orbitals to which the HOMO and LUMO dissociate. Thus, the electron is localized at one well or another [67]. This is known as laser-induced or dynamical electron localization due to tunnelling suppression [71-72]. The second time-dependent term in Eq. (32) corresponds to the electron following the field at multiple laser frequencies ka>, which in the long wavelength limit can be viewed as appearance of the upper state lcr+ or lower state \a_, i.e., the Stark-induced linear combinations of the field-free HOMO ( lag) and LUMO ( lau). At large EM>0, lcr and 1<7+ approach the localized atomic orbitals a and b, respectively; for EM <0,

169

vice versa. As an example, we take a case in which the distance R = 7a.u., intensity / = 1014 W/cm2, and X = 1064 nm. Then, the argument of J0 is 2caRl a> = %.l, which is very close to the third zero (x = 8.6) of J0(x) , thus suggesting dynamical localization. Such electron localization has been confirmed in full 3-D TDSE calculations of the ion H2

+ [31]. In general, due to the strong radiative coupling oR being larger than the essential

state energy separation A charge asymmetry due to the field induced charge transfer results in laser induced population of the upper state la+ as illustrated in Fig.3. This population inversion can be more quantitatively described in a nonadiabatic description [73] and is discussed in Sees. 7 and 8 where it is shown that the result (32) is an asymptotic, strong field limit. We next derive analytic formulae for the critical distance Rc where enhanced ionization is observed from numerical simulations. As this is due to charge resonance or charge transfer effects induced by the laser field as discussed above, we call this CREI-Charge Resonance Enhanced Ionization. The simplest one-electron system H2

+ may be viewed as a prototype of odd-electron molecules and general features of odd-electron molecular ionization can be elucidated by investigating the dynamics of H2

+ for which we have performed the first full non-Born-Oppenheimer calculations in 3-D [47] and also in 1-D [45-46].

The electronic dynamics in H2 prior to ionization is dominated by the strong radiative coupling between the essential frontier orbitals, i.e., the HOMO ( lc g ) and LUMO ( \a u ). The transition moment between these increases linearly with intermiclear distances R as R/2, a spectroscopic observation first made by Mulliken [28], which he called a charge resonance (CR) transition between a bonding and corresponding antibonding molecular orbital. The strong radiative coupling changes the electronic potential surfaces of the bound 2~L* and repulsive 2E* states into new distorted (dressed) potentials as a result of the creation of adiabatic HOMO (1<7_) and LUMO (lcr+) energies, £± (R) - -Ip ± EMR12, where /,, is the ionization potential of H atom and EM is the maximum field amplitude. This quasistatic Stark shifted orbital picture was earlier discussed with respect to intense static field dissociation of molecules, where the concept of barrier-suppression of ground state molecular potentials was inferred as an efficient mechanism of dissociation in intense fields [74-75]. This static picture has been shown to be very useful in describing thresholds for laser-induced photodissociation at high intensities and low frequencies [76-77] and also in controlling dissociative ionization in two colour excitation schemes [46, 78].

As shown in Fig. 5, while £_ is usually well below the barrier heights, £+ is higher than the heights of the inner and outer barriers, Vjn and Vm, , in the range 6<RC <10 a.u., where Rc is called the critical distance for Charge Resonance

170

Enhanced Ionization (CREI). In this critical range ofR, ionization proceeds rapidly from the upper adiabatic state | lc+) (when E(t) > 0, |lcr+) - b, i.e., the right atomic orbital of a dissociating H2

+). Assuming that at Rc, £+(Rc) is at the top of both barriers maxima Vin and Voul, one obtains simple analytic expressions for Rc first derived for highly charged one-electron ions A2

2,?+ [32] and further confirmed later [79]. The analytical expression for H2 is Rc-AIIp whereas it is Rc—5IIP for linear H3

2+ (where R is the total bond length between the two outer protons) [32, 34, 42]. The latter result was also confirmed numerically for highly charged ionsA2

?+

where q>2 [33-34, 80]. As we show next for H2+ and H3

2+ , these results are independent of charge, depend weakly on field strength, are further consistent with the exact TDSE numerical simulations described in the previous section, and confirm experimental results in Coulomb explosion of H2 and other molecules at high intensities [38-41,44, 61-66].

We follow here the original 1 -D derivation [32] for one-electron models of A22,+

systems in the one-electron approximation undergoing Coulomb explosion at high intensities and summarized for 1-D models [42]. Exact Born-Oppenheimer (static nuclei) TDSE simulations of such highly charged ions show ionization maxima for 6<i?<10a.u. evolving from ionization "windows" peaked around Rc = 8a.u. into plateaus starting at R-6a.u. at high intensities [32-33]. Following Chelkowski and Bandrauk [32] the necessary conditions for CREI to occur is that the upper electronic eigenstate £+(R) exceeds the two potential barriers, i.e.,

Vmt(R)<VinR)< £+R), (33) where Vin and Vmt are the barrier heights of the following electronic potential V(z,R) at the maximum field strength EM (the nucleus-nucleus repulsion is not included)

V(z,R) = -q/\R/2 + z\-q/\R/2-z\+Euz, (34)

and £+(R) is given by

£+(R) = -qIp-q/R + EMR/2. (35)

Ip is the ionization potential of a neutral atom A. In Eq. (35), we use the empirical formula for the ionization potential of electrons from the same shell, i.e., 7ion - qlp

for an ion of charge q-\. For the CI atom, this formula is exact within 2-3% for q>2 and 10% for q = 2 [32]. The potential (34) has maxima at z_ (left) and z+(right) for EM >0, i.e., Vin(R) = V(z+,R) and Vml (R) = V(z_,R):

z_=-R/2-E^>2, z^EqR?IZ2, (36)

171

where Eq=EMlq.

The approximate solution of the inequalities (33) gives the following condition for Rc, Rt < Rc < R2, where CREI is predicted to occur:

R =1.464£-"2 , R2 = l-(l-6^/V)" hIE,- (37)

For Eq «. I2P16, one obtains Rc-R2='i/Ip. This result used by Seideman et al. to

predict onset of electron localization [33] neglects the Stark-shift of the LUMO, which is an essential feature of CREI as discussed above. Defining the optimal condition for CREI by the equation RI=R2 [i.e., Vin(R) and Vm, (R) equal to £+(R)] gives

Ec= 0.12912pq ,RC=4.07/Ip. (38)

For fields EM >I2q/4, the CREI effect will disappear since the electron in the

separated ion of charge q - 1 is itself above the barrier. This was confirmed in Fig. 1

of [32] for the system Cl25+ -»Cl3+ +C13+, which shows a plateau instead of a CREI

peak for 7>2xl01 5 W/cm2. Formula (38) shows the independence of Rc from field strengths and charge q.

Such independence was originally discovered in Coulomb explosion kinetic energy

spectra of ion fragments [41], thus confirming the existence of a critical distance

Rc for enhanced ionization. More detailed experiments on Hj [61-62] and

comparison with full non-Born-Oppenheimer simulations [45, 62] confirm CREI as

the principle mechanism for the production of highly charged ions at critical

distances Rc predicted by Eq. (38). The Coulomb explosion measurements must

necessarily depend on the pulse length since the molecular ion must reach Rc from

its equilibrium Re in order to undergo CREI. Thus using a classical model of

evolution of highly charged ions from Re to Rc on a purely Coulombic potential

V = q21R, one obtains for the time tc in which dissociating fragment of charge

+^move from Re to Rcthe expression given by [32],

tc=(/l/2)"2j£(£0-q2R)-"2dR, (39)

= (M/2)V2ReV2[x/(x2-\)-0.5\n(x + l)/(x-l))]/q, (40)

where £0 is the initial potential energy q21 Re, |X the molecular reduced mass, and x = [Rc /(Rc -Re)] . Thus, times 17 > tc > 75 fs are obtained from molecules N2 to

172

I2 for fragments of charge q = +2 to reach Rc as predicted by Eq. (38). Recently, the effect of short pulses on Coulomb explosion of I2 has been observed [81-82] demonstrating indeed deviations from Eq. (38) as the heavy fragments F+do not reach Rc for ultrashort pulses (t he = 30fs).

Extending the quasistatic ionization model of H2 to one-electron triatomics such as Hj+ involves consideration now of at least three doorway states. Both for the linear [31, 34] and nonlinear H3

2+ [43] sharp ionization maxima are found to occur at Rc -lOa.u. (as shown in Fig. 6) and 7a.u., respectively, from TDSE Born-Oppenheimer (static nuclei) calculation. R is the total bond length of the linear molecule. For the nonlinear geometry, an ionization maximum occurs also at a critical bond angle 6C - 87° [43] which can be also explained as over-barrier ionization of the LUMO. Thus, the CREI mechanism is seen to be operative even in nonlinear molecules through the field induced displacement of the three doorway molecular orbitals (MO's), as we show next.

In the linear case of H32+ with equidistant atoms, the three MO's are given in the

LCAO approximation by [83]:

lc7g=r1,2[c + 2-V2(a + b)], (41)

lau=2-V2[b-a], (42)

2(Tg=2m[c-2-l'2(a + bj], (43)

where a, b, and c are the Is atomic orbitals on the left, right, and central protons, respectively. The transition moment /x = U<7g|z|lt7\ can readily be shown to be R/2in in contrast to H2

+ where ju = R/2. In both cases, such divergent moments are due to the general phenomenon of CR effects [28]. In the presence of a static field of amplitude EM (maximum amplitude of a laser field), one obtains three new field-induced states as shown in Figs. 7 and 8. The energies are approximately equal to

£l=-Ip-3/R-\EM\R/2 ;£i=-Ip-3/R + \EM\R/2, (44)

£2=-Ip-4/R, (45)

where £2 of the middle level localized in the central proton remains undisplaced with energy corresponding to the atomic ionization potential, - / , perturbed by the Coulomb potential 2—2IR) due to the two protons adjacent to the central one.

Efficient ionization from the state of energy £2, the LUMO for this system, will occur at critical distances Rc where £2(RC) equals the highest of the outermost

173

H32*, 3D

(a) l = 1014W/cm:

v ' X = 1064 nm .

Figure 6 Ionization rates of linear H32+ at / = 1014 w/cm2 and A = 1064nm (units of

1 0 ' V ) : (a)3-D;(b)l-D.

174

I 1

0.2

d

-0.2

-0.4

-0.6

-6.6

-1

-1.2

-1.4 h

H3a*, 3D

l = 1014W/em2

R = 12a.u.

0.0000

0.3962

0.5861

-16 -10

1

1

0.2

0

-0.2

-0.4

-0.6

-0.6

-1

-1.2

-1.4

10 15

i ' T " 1 1 1 T / 1

Hj4*, 3D / . i = id14w/ci«s /

fi = 19a.u. f

0.1196 e3 >^K

0.3831 ^K e2 j \ / \ /

• x 1 U ^ ^ \ 0 . 1 ? 3 2 / e . : W II

0.Z7X1O-, r 3 .

o.3?xibia,r2_

0.64x1 o - i , r t .

-20 -15 - 1 0 - 6 0 6 10 15 20 2 (a.u.)

Figure 7 Energies £t and linewidths r, (in a.u.) for linear Hj+ at/? = 12 and 19 a.u. in a static field of E = 0.053 a.u. ( / = 1014 w /cm 2 ) . The values on horizontal energy lines are the occupations of the corresponding three states. The energies Fj, V2, and V3

denote the barrier maxima of the electronic potential.

175

barrier, i.e., Vx shown in Fig. 7. The electronic potential Vz,R) can be expressed in terms of the shift x of the electron's position from the left proton (when EM > 0):

r(x,R) = -^-—L-~^--EM(x + R/2), (46) x x + R/2 x + R

where x - -[z-(-R/2)] = -z-R/2. Evaluating the height of the outermost barrier of V(x, R) and equating the obtained Vx to £2 gives the simple result [34,42]

X-5/V (4?) Since 7P (H) = 0.5a.u., this gives readily the value of the critical distance Rc = lOa.u. in agreement with numerical solutions of the TDSE for 3-D [34, 85] and 1-D H3

2+ [80, 86]. This result can also be obtained by considering the value of R at which the unperturbed middle level £2 of energy -IP-AIR is degenerate with the field free barriers at z = ±R IA, i.e., setting

Ip+l/Rc=2/(Rc/4) + l/(Rc/2 + RJ4) = 28/3Rc (48)

gives RC=16/3IP. Inserting Ip =0.5a.u. gives Rc =lla.u., which is in satisfactory agreement with the major ionization peak of H3

2+ shown in Fig. 6 [34]. The field-free value of Rc, Eq. (48), is again close to Rc obtained from a rigorous calculation with the field Stark displacements included in Eq. (47).

As Eq. (38) in the case of H j , R c in Eq. (47) is independent of field strength and agrees with the previous estimate Eq. (48) from a field free model. The coincidence of this last result confirms the field strength independence of Rc, which comes from the near equality of the classical energy shifts of the barrier maxima Vt and K3 in Figs. 7-8 (or Vin and Vmt in Fig. 5) with the charge resonance energy shifts of the quantum levels. Thus both classical and quantum energy shifts cancel each other rendering the values of the CREI distance Rc nearly independent of field strength EM as in H2

+. Similarly, the independence of Rc with charge q comes from the narrowness of the wells with increasing q, thus leading to potential maxima, for instance, Vin and Vm, of H2

+ shown in Fig. 5 [or Vx ,V2, and V3 in H32+ (Figs. 7-8)]

equal to the Stark shifts of the electronic levels. We end this section emphasizing that CREI occurs also in nonlinear, i.e., triangular H3

2+, with a critical distance Rc -7 a.u. and a critical angle 0C ~ 87° [43]. Thus in nonlinear molecules, CREI is expected to occur also at critical bond distances Rc and critical bond angles 6C. This is a general phenomenon of enhanced ionization via over-barrier ionization induced by Stark-displacements of LUMO's.

176

.1.6 L: 1 1 1 1 1 1 1 1 6 8 10 12 14 16 IB 20 22 24

R (a.u.)

Figure 8 Energy levels £t and barrier maxima V^, V2, and V3 plotted against R. The notations are the same as in Fig. 7.

177

4 Two-Electron Systems

The first detailed experimental study of ATI for molecules was performed in H2 by Verschuur et al. [84] where it was discovered that the photoelectrons obtained in intense fields showed anomalous vibrational structure of H2

+ later interpreted as due to bond softening [26] via LIMP'S dominated by laser induced avoided crossings between dressed molecular potentials [20, 21-23, 27]. 3-D ionization rates for H2

and H3+ in intense fields were first attempted using time-dependent Hartree-Fock

methods and finite-element techniques [85]. However, since such methods fail at large R due to configuration interaction (CI), exact 1 -D calculations were pursued in the long wavelength regime (A = 1064nm) where it is expected that 1-D models should be adequate and that CR effects should dominate. Thus, maxima in the ionization rate with respect to R have been again found for 1-D two-electron models such as H2, H3

+ [19, 34, 85], H42+ and H5

3+ [34]. These maxima have recently been interpreted as due to strong radiative interactions between essential doorway states resulting in charge transfer at the peak of the laser pulse. Ionic states thus created as ground states in intense fields play the role of main doorway states to ionization [35-36, 42]. The electron transfer and resulting enhanced ionization were confirmed by recent wave packet calculations of a 3-D model of H2 [86]. The results of static field calculations of H2 also agree with the interpretation of two-electron enhanced ionization [34, 87-88].

The existence of similar critical Rc 's as in one or odd-electron systems indicates that enhanced ionization is a universal phenomenon. The mechanism of enhanced ionization of even-electron molecules is however expected to differ nevertheless from that of one or odd-electron systems due to electron correlation. As an example, we have found that the HOHG atomic plateau which continues up to the maximum IP+3A7UP given by Eq. (15) can extend to IP+\2UP in two-electron extended molecular systems [19]. The two-electron diatomic prototype is H2 for which we have discussed the mechanism of enhanced ionization in [35, 86]. In this case, ionization is enhanced when the ionic state H+H~ is most efficiently created from the covalent ground state HH in the level dynamics prior to ionization. The importance of ionic states in the spectroscopy of symmetric molecules was emphasized by Mulliken [28] and recently such a state in 0 2 (0"0+) has been confirmed experimentally [89]. In H2, ab initio calculations have proven the existence of H+H" states with multiple crossings with valence (Rydberg) states at R = 6a.u.[90]. An analytic expression for the ionic and covalent crossing condition was obtained for H2 in terms of three doorway states [42, 86] which agrees well with the numerical results [34-35].

178

2\ , , , , , , 1 4 6 8 10 12 14 16 18

Internuclear distance R (a.u.)

Figure 9 Ionization rates (units of lO 'V ) for linear symmetric 1-D H* at A = 800 nm : (a) / = 3.2xl0'4 w/cm2 ; (b) / = 1014 w/cm2 ; (c) / = 1013 w/cm 2 . In (a), the result for the static field of the same peak intensity 7 = 3.2xl014 W/cm2 is presented as well as the ac field case: (i) ac field; (ii) dc field.

179

The two-electron triatomic system, H3+, is the simplest model to understand the

electron dynamics of extended systems. We already have reported calculations of enhanced ionization of both symmetric and non-symmetric 1-D H3

+ [34]. We examine here in detail the mechanism of enhanced ionization (see Fig. 9) of linear symmetric H3

+ by employing the adiabatic-fleld state method previously used for H2

+ (Sec. 3) and for H2 [35]. Simple Stark-shifted MO's will be again shown to give a clear interpretation of the numerical TDSE results and of the origin of the R dependence of the ionization rate. Figure 9 illustrates the ionization rate for fixed nuclei (Born-Oppenheimer) for H3

+ at three different intensities. A maximum occurs around Rc =9 a.u., close to Rc =10 a.u. for H3

2+ (Fig. 6). A static field calculation for the same static electric field amplitude as the maximum of the laser field essentially reproduces the same Rc. See the static field case (ii) in Fig. 9(a). We shall develop next a quasistatic model based on radiatively coupled doorway states to predict the occurrence of such critical distance for enhanced ionization as resulting of the crossing of the ground covalent state HH+H with the ionic state H+H+FT or H"H+H+, similar to H2 [34-35,42].

At large R the lowest three states of symmetric H3+ become degenerate

corresponding to the fact that H3+ dissociates into H+HH, HH+H and HHH+. The

electronic transition moments ju between states i and j are defined as

Mij =(^(z 1 ,z 2) | (z ,+z 2) |^ . (z 1 ,z 2)) . (49)

The first transition moment //,2 corresponding to the transition XlZg —¥ S'X„+ has a linear R dependence at large R (up to R ~ 4 a.u.) as well as the second moment //23

for BlZ„ —> Ex£g (up to R ~ 10 a.u.), confirming the CR character of these states [28], The linear R dependence of these moments can be estimated from the LCAO-MO method [34] which give in=R/2 and jU2i=R/23'2 for the states described next.

In order to incorporate the ionic character one now needs to incorporate six doorway states for H3

+ whereas for H2 only three states were essential [35]. Using the lowest three MO's of H3

2+ in Eqs. (41)-(43), we obtain the lowest six states of H3

+:

fl(l,2) = UT,(l)l<T,(2), (50)

% ( l , 2 ) = [l(7g(l)lcTli(2) + lC7g(2)l(Tu(l)]/V2, (51)

ft(l,2) = [l<r,(l)2ffg(2) + l<7,(2)2ag(l)]/>/2 , (52)

^ ( l , 2 ) = lff„(l)l<r.(2), (53)

180

<p5(l,2) = [lau(\)2<7g(2) + lau(2)2c7g(l)]/^2, (54)

%(l,2) = 2<7g(l)2c7g(2). (55)

If \<7g and 1<T„ are replaced with (±a + b)/yJ2 of H2+, q\, q>2, and (p4 become the

three essential states of H2. For H3 , the transition moments as discussed above are //,2 = //24 = -ju45 = -fiSi = R12 ; //23 = - / ^ = R12'1, where R is the total bond length of the molecule; other matrix elements are zero. The state /' and j are thus coupled with each other by the radiative matrix element EM(i\z\ j) at maximum field strength £ M , i.e., the peaks of the laser field. Assuming that these radiative matrix elements far exceed the zero-field energy separation between the six electronic states (^, -(p6) , we obtain the following adiabatic states p. and energies £ in the field EM by diagonalizing the 6x6 Hamiltonian matrix: for E„>0,

£,=-\EM\R, ^ , = a ( l ) a ( 2 ) , (56)

£2=-\EM\R/2, yr2=[a(l)c(2) + a(2)c(l)]/>/2, (57)

£3=0, j / 3 =c( l ) c (2 ) , (58)

£3' = 0, y/i=[a(\)b(2) + a2)b\)\l42 (59)

£A=\EM\RI2, i /4=[6(l)c(2) + A(2)c(l)]/>/2> (60)

e5=\EM\R, ¥5=b\)b2), (61)

where a, b, and c are the Is atomic orbitals on the left, right, and central protons, respectively. For EM <0 , a and b in Eqs. (56)-(61) should be interchanged with each other. The two electron states y/2 , \f/\ , and y/4 in a long distance case of R = 14 a.u. are illustrated in Fig. 10 as states (a), (b) and (c), respectively. The states \ffx , i//2 , and y/5 are clearly the ionic states H^H^Hj, H^H^H* and H^H^H* . Equation (56) shows that the lowest y/ state is ionic and the energy £x agrees with the electrostatic energy of a charge transferred through the distance R by the field EM-

As is illustrated by the two-electron distribution ^(z^Zj,?) in Fig.l 1 obtained from exact TDSE numerical solutions, the ionization dynamics via laser-induced charge transfer or charge resonance [36] is determined by three electronic configurations H^H^H^, H^H^H*, and H^H^Hj for EM > 0. Which doorway states (configurations) to ionization mainly appear depend on the characteristic parameters of laser-molecule interaction. The electronic response to the ac field can be classified

181

-15 -10 -5 0 5 10 15 z,(a.u.)

-15 -10 -5 0 5 10 15 z, (a.u.)

-15 -10 -5 0 5 10 15 z, (a.u.)

Figure 10 Two-electron probability densities of the three adiabatic field states of linear symmetric H* at E(t) =0.1 a.u.; (a) y/2 of Eq. (57); (b) y/[ of Eq. (59); (c) ^ of Eq. (60). z, and z2 are the coordinates of electrons 1 and 2. The internuclear distance is fixed at R = 14 a.u. The corresponding electronic spatial configurations are schematically illustrated in the right panels, where the closed circles denote the positions of two electrons and the open circles denote the protons. The dipole interaction with the field is represented by a slanted bar (at this moment, the field is positive).

182

30

20

Si 10 CVJ

N

0 •

-10 •

-10 0 10 20 30

10

~ 5

si N" o

-5

-10 -10 -5 0 5 10 15

z, (a.u.)

Figure 11 Two-electron probability densities for a pulse ofA = 800nm and the peak intensity / = 3.5xlO14 w/cm2 (EM=0Aa.u.) . Two example are shown: (a) a diabatic case where R is fixed at a large value of R = 14 a.u.; (b) an adiabatic case where R = 8 a.u. In (a), a snapshot at t =3.875 cycles is presented. At this moment, Et)=- 0.54a.u. In (b), a snapshot is taken at t= 3.75 cycles at which Et)= -0.72 a.u. The ionic site z=z2=R/2 of the lowest adiabatic state y/\ is denoted by the symbol I, while the covalent site z, = -z2 = R12 of y/[ is denoted by C. The site M (z, = R12 and z2 = 0) corresponds to the second lowest adiabatic state y/2. In order to obtain the wave functions of adiabatic states for E(t) < 0 , a and b in Eqs. (56)-(61) should be interchanged with each other. In (b), three types of ionization occur: from the ionic site I denoted by the solid line (4); from the covalent site C denoted by the line (1); from the site M of the second lowest adiabatic state yr2, denoted by the line (2).

183

into three different categories [67]: (i) in the adiabatic regime of R < 9 a.u., H~H*Hj (i.e., ionic \ffx) appears for EM > 0; (ii) in the diabatic regime of R>9a.u., HoH*H4

(covalent y/'z); (iii) in the mixed regime of R ~ 9a.u., H^HCH4 (y/4). The adiabatic energy of each electronic configuration can be estimated from electrostatic principles assuming negligible overlap between atomic orbitals on neighbouring sites:

£lR) = -Ip-Aa-6/R-\EM\R, (62)

fSR) = -2Ip-4/R, (63)

£4R) = -2Ip-3/R + \Eu\R/2, (64)

where Aa is the ionisation potential of H" (0.06 a.u.) and Ip is that of H (0.67 a.u.) in 1-D.

The adiabatic regime of R<Rc=9 a.u. corresponds to the regime where electrons follow nonresonantly the field; as shown in Fig. 11 (b), an ionic state is formed near the site I of the ionic configuration H*H*Hj. This is the quasistatic regime of strong-field atomic physics [7-9]. In the molecular case, this corresponds to efficient charge transfer via the creation of the ionic configuration ^, in the adiabatic regime (i). As in the case of H2 [35, 42], one can estimate the critical distance Re for enhanced ionization on the basis of the simple principle of most efficient charge transfer at the field-induced energy crossings of the ionic states H^H^H; for £ M > 0 o r H+H^H~ for EM < 0 [the ionic site I in Fig. 11 (b)] with the covalent state HaH*H4 (the covalent site C). Thus equating £\R) and E[R), we obtain an expression similar to the case of H2. The maximum field strength at which the two energies can be equalized is given by Emax = (lp - Aa) /8 = 0.045 a.u. At this strength, the energy gap between the two adiabatic states is maximized, indicating adiabatic electron transfer from the covalent state y/'^ to the ionic state ^, as a doorway state to ionization. At Emax , the two states cross each other in energy at the internuclear distance Rc

K = (/, - Aa )/2£max = 4/(/„ - A J . (65)

This gives Rc=6-1 a.u. by using Ip-Aa = 0.6 a.u., which is in agreement with peaks around R ~ 8 a.u. in Fig. 9 or the first high intensity peak in Fig. 9(c). It should be pointed out that in Fig. 11(b) the probability of ionization from the second lowest adiabatic state y/2 (site M) is fairly large. The state is regarded as a transition state from the covalent y/' to ionic state y/x . The ionization process via the intermediate state y/2 is not taken into account in the derivation of Eq. (65).

184

A snapshot of the wave packet in the diabatic regime (ii) is shown in Fig. 11(a) where if is large (14 a.u.). It is clearly demonstrated that ionization proceeds from the covalent configuration denoted by the symbol C in Fig. 11(a). There exist two routes to ionization. As shown in Fig. 10(b), one electron is localized in the lowest well and the other one is in the uppermost well. In the ionization process denoted by the solid line in Fig. 11(a), the electron in the lowest well is ionized; in the ionization process denoted by the broken line, the electron ejected from the uppermost well penetrates the middle and lowest wells.

The mixed regime around Rc=9 a.u.corresponds to laser-induced transitions mainly between the three field-free electronic states XXZ^, BxZt and E]Ug which are degenerate at large R as they dissociate symmetrically to the three degenerate asymptoties H+HH , HH+H and HHH+. As in the case of H2

+ , these three asymptotic configurations appear in the adiabatic states illustrated in Fig. 10. We therefore adopt again the quasistatic above-barrier method described in the previous section for the one-electron systems H2

+ and H32+. We write the energy of the

adiabatic state y/4 in the mixed regime (iii) as

£4(R) = £upR) + £midR), (66)

£upR) = -Ip-\/R + \EM\R/2, £mid(R) = -Ip-2/R, (67)

where one electron (z, = R12) is on the outer right proton H*, i.e., in the uppermost well (yielding £up) and the other one (z2 =0) is on the central atom Hc (yielding £mid). The total energy (66) is equal to Eq. (64). Thus the ionizing upper electron (zi > 0), i.e., leaving from the LUMO, only interacts with the bare proton Ho

+ and the electrostatic field EM since the central proton Hc is shielded by the other electron (z2 =0). The potential of the upper electron with z = z,> 0 at distance z from FL is then

V(z) = -- \ \ + EMz. (68) V ; (R/2) + z (R/2)-z

We note the similarity between the H2+ CREI model potential and the present H3

+

model, i.e., between Eqs. (34) and (68). Defining z = (R14) + y and searching for the maximum barrier Vm from the condition (dV/dz) = 0 gives readily y-EMRi /128-R/9. Setting next Vm(y) = £up(R) gives the equation for Rc

Ip^EMRj2 + U/4Rc. (69)

185

The maximum field strength that satisfies Eq. (69) is Em!iX -ill5. The internuclear distance for enhanced ionization is found to be i?c — 9 a.u. at Emax, which is in agreement with the dominant ionization maximum shown in Fig. 9.

We conclude that the quasistatic model of over-barrier ionization for molecules exposed to intense laser fields as originally proposed by Codling and Frasinski et al. [65-66] needs to be modified to take into account the important Stark or equivalently Charge Resonance [20, 28] energy shifts of the LUMO. By examining the heights of barriers for tunnelling and the field-induced energy shifts of adiabatic states, we explained qualitatively the main enhanced ionization critical distance Rc — 4/Ip - 8 a.u. for H2

+, Rc = 51 Ip =10 a.u. for H23

+, and Rc=9a.u. for H3+ calculated from

appropriate TDSE's. In this last case, enhanced ionization at Rc - 9 a.u. occurs from the laser-induced excitation of the configuration H^H^Hj with the electron situated on the uppermost quasistatic well at H4 contributing mainly to the ionization by over-barrier passage. We emphasize once again the similarity of this ionization regime to H2

+ where the electron trapped in the upper Stark-shifted LUMO state <J+

(see Fig. 3) contributes mainly to the ionization. This is the fundamental difference between atoms where only one well exists and no LUMO. Thus in the molecular case, at Rc the electron must cross the whole molecule in order to ionize via CREI. Electron repulsion at lower laser intensities as in Fig. 9 (a) and (b) suppresses the creation of the ionic state, H^H^Hj, i.e., electron correlation limits charge transfer. The ionic state begins to contribute to the ionization dynamics only at higher intensities I> 1015W/cm2 [as in Fig. 9 (c)] due to the high energy of these states. The critical distance Rc for this ionic mechanism, Eq. (65), is obtained under the condition that this ionic state and the ground covalent state cross each other at the maximum field strength for crossing, £max.

There are at least two mechanisms of enhanced ionization, that is, CREI and ionic mechanism. In the ionic mechanism, the ionization step proceeds mainly through the lowest adiabatic state (localized ionic state such as H+H") created by field-induced transfer of an electron; in the CREI mechanism, it proceeds through the upper LUMO or an upper adiabatic states. Upper LUMO's are populated by HOMO-LUMO crossings at near-zero fields, as we show next; ionic states are populated by crossings between covalent and ionic states at nonzero fields, as discussed above. We have introduced in Sec. 3 a strong-field excitation two-level model, Eqs. (20)-(32), which we have used previously to explain molecular HOHG [19, 67] and CREI in H2

+ [67] around the critical distance Rc. This strong field regime where coR> A is called the diabatic regime [see Eq. (32)]. The more precise definitions of adiabatic and diabatic regimes will be given in Sees. 5 and 7. In the next section, we extend this theory to describe the full nonadiabatic dynamics of

186

electron-nuclear-photon dynamics in intense laser fields [91] based on analytical solutions of the two-level model in strong fields [73].

5 Adiabatic State Formalism

5.1 Time-dependent adiabatic states

As has been shown in previous sections, bond stretching in a low-frequency intense laser field enhances the rate of tunnel ionization (known as enhanced ionization). The experimentally observed kinetic energies of ionized fragments of a molecule are large (> a few eV) and are consistent with Coulomb explosions of multiply charged cations at a specific internuclear distance Rc which is much longer than the equilibrium internuclear distance Rg. Other numerical simulations also indicate that the tunnel ionization rates around Rc exceed those near Re and those of dissociative fragments. In what follows, we present numerical simulations and analyses of above-mentioned nonperturbative phenomena in intense fields in terms of strictly defined field-following adiabatic states. The analyses are devoted to identifying doorway states to ionization [86, 91, 92].

To analyze laser-induced electronic and nuclear dynamics in intense fields, we first define time-dependent " field-following " adiabatic states \n) as eigenfunctions of the "instantaneous" electronic Hamiltonian HQ(t). The H0(t) is the sum of the Born-Oppenheimer (B-O) electronic Hamiltonian Hel at zero field strengths and the interaction with light, VEt) [92, 93]:

H0(t) = Hel +VE(t). (70)

To obtain !«), we diagonalize H0(t) by using bound eigenstates of Hel as a basis set [94]. The time t and the internuclear distance R are treated as adiabatic parameters.

13 2

In an intense field (7>10 W/cm ), an adiabatic state or adiabatic states are populated by laser-induced intramolecular electronic motion. Tunnel ionization to Volkov states (quantum states of a free electron in a laser field) proceeds through such an adiabatic state (adiabatic states). Intramolecular electronic motion also affects nuclear motion; e.g., after one-electron ionization from H2, the bond distance of the resultant H2

+ stretches on the lowest adiabatic potential surface [26, 62, 95]. Field-induced nonadiabatic transitions through avoided crossing points in time and internuclear coordinate space govern the electronic and nuclear dynamics in intense fields. As explained below, the dynamics of bound electrons and the subsequent ionization process, as well as laser-induced nuclear motion, can be understood by analyzing the time-dependent populations of adiabatic states.

187

We take a one-electron molecule H2+ as an example. In the intense and low-

frequency regime, H2+ is aligned by a linearly polarized laser field, parallel to the

polarization direction. Then, VE(t) in the dipole approximation is given by jeE(t) = zE(t) as in Eq. (7) [74]. Here, z is the electronic coordinate parallel to the molecular axis, and E(t) is the linearly polarized laser electric field. The electronic dynamics of H2

+ prior to tunnel ionization is determined by the large radiative dipole coupling of zE(t) between the lowest two B-0 electronic states, ls<7g and ls<7u, respectively (abbreviated as |g) and |u)). The dipole transition moment between them, parallel to the molecular axis, increases as RI2, where R is the internuclear distance. This large transition moment is characteristic of a charge resonance transition between a bonding and a corresponding antibonding molecular orbital, which was originally pointed out by Mulliken [28].

We diagonalize the instantaneous electronic Hamiltonian including the radiative interaction, H0(t)-Hel +VEt), by using the radiatively coupled |g) and |u) states as basis functions. The eigenvalues of Hel for ls(7g and \s(Ju, are denoted by E (R) and £U(R) , respectively. The resulting eigenfiinctions of H0(t) are given by [91, 94]

|l) = cos0|g)-sin<9|w) (71a)

|2) = cos0|w) + sin0|g) (71b)

where

~2(g|z|u)ff(/r

. Aeag(R) _

with the B-0 energy separation A£ug(R)

A£ug(R) = £u(R)-£g(R). (73)

The phases of the two wave functions |g) and |u) are chosen so that (g|z|u) is positive. The corresponding eigenvalues of H0(t) are given by

6 = — arctan 2

(72)

£2,(R,t)=Ueg(R) + £a(R)±jA£2ag+4\(g\z\u)E(t)\

2

~±[£,(R) + £u(R)±R\E(t)\] forlargetf. (74)

The strong radiative coupling of the charge resonance transition yields the "field-following" time-dependent adiabatic surfaces, £xR,t) and £2(R,t) , which are adiabatically connected with lscrg and ls(7u, respectively. The adiabatic energies £x 2(R,0 at E(t)=0.053 a.u. are shown in Fig. 12 as well as £„ and £u. If £ is

188

assumed to be equal to the field envelope / ( / ) , the field strength E in atomic units corresponds to the intensi ty/as/= 3.5x10 E W/cm .

The instantaneous electrostatic potential for the electron, V0(t), has two wells around the nuclei, i.e., z=±R/2. The dipole interaction energy for an electron is E(t)R/2 at the right nucleus and —E(t)R/2 at the left nucleus. As E(t) increases from zero, the potential well formed around the right nucleus ascends and the well formed around the left nucleus descends. Therefore, the ascending and descending wells yield the adiabatic energies £2 and £], respectively; |2) and |l) are localized near the ascending and descending and wells, respectively (irrespective of the sign of the electric field). In addition to the barrier between the two wells (inner barrier), when \Et)\^Q, there exists a barrier of finite width outside the descending well (outer barrier). The barrier heights of V0(t) evaluated along the molecular axis are also plotted in Fig. 12. It is clearly demonstrated that while £x is usually below both of the barrier heights, £2 can be higher than the barrier heights in a range around ^.=6 a.u. Except the atomic limit where R is much larger than the equilibrium internuclear distance, the upper adiabatic state |2) is easier to ionize than is |l).

In Fig. 13, the rate of ionization from |2) in a DC field, ri, is plotted as a function of R. The field strength is fixed at a constant E(t)= 0.053 a.u. In the numerical calculation, we use cylindrical coordinates z and p, where p is the coordinate perpendicular to z. Then, the z-component of the electronic angular momentum is conserved: only z and p are necessary to describe the electronic state. We solve the time-dependent Schrodinger equation for H2

+, with nuclei frozen at various values of R, by using a grid point method for Coulomb systems (the dual transformation method) [52,91]. To eliminate the outgoing ionizing flux and to calculate the ionization probability, we set absorbing boundaries for p and z: the ionization rate is obtained by fitting the norm within the space encircled by the boundary planes to an exponential decay form. Roughly speaking, owing to barrier suppressions favorable for tunnel ionization from |2), 7"2 is large in a wide range around R^,. However, a couple of peaks are observed. The maximum ionization rate from |2) is found at R ~9 a.u. For this field strength, at R ~9 a.u., £2t) is however only a little above the inner barrier, as shown in Fig. 12. The rapid ionization for the instantaneous potential is attributed to the following facts: at R~9 a.u., |2) is resonant with an adiabatic state in the descending well that is higher than |l); the outer barrier is thin and is much lower than £2 . Another peak due to resonance is also found at R~6 a.u. but it is lower than that at R ~9 a.u. The rate of ionization from |2) seems to be more sensitive to the outer barrier than the inner barrier.

In Fig. 13, we also show ionization rates r in an alternating intense field under the condition that the initial state is the ground state Is (J . The laser electric field is

189

-0.2-1— l

-0.3-

7 -0-4" 3-g5 -0.5-<D c

LU

Inner barrier height

1SOc . - - * *

Outer barrier .height

— i 1-

4 6 8 10

Internuclear Distance R (a.u.)

Figure 12. Potential energies of the lowest two field-following adiabatic states, £i and £2, of H2

+ (denoted by solid lines) and the heights of the inner and outer barriers for tunnel ionization at £ (0 = 0.053 a.u. i.e., 7 = 10 W / c m (dotted lines). The broken lines denote the Born-Oppenheimer potential surfaces of lscr and lsou . The internuclear repulsion l/R is included in the energies.

190

i i i i | i i i i | i i i i | i i i "'I | i i i i | i i U

5 6 7 8 9 10

Internuclear Distance R (a.u.)

Figure 13. Ionization rates of H2+ as functions of the internuclear distance R (1 a.u. in

ionization rate = 4.1x10 /s). The open squares denote ionization rates r2 of the upper adiabatic state \l) at a constant field strength Ei) = 0.0533a.u. (which corresponds to 7 = 10'4W/ cm ); the open circles denote ionization rates 7" in an alternating intense field of A=1064 nm and 1= 1014 W / cm2 under the condition that the initial state is the ground state IscT, ( the field envelope has a five-cycle linear ramp). The scales for r2 and r are marked on the right and left ordinates, respectively.

191

assumed to take the form E(t) = f(t)cos(cot), where a =0.0428 a.u. (Z = 1064 nm) and the envelope / ( / ) is linearly ramped with t so that after five optical cycles f(t) attains its peak value / 0 =0.0533 a.u. ( / = 1014 W/cm ). As clearly demonstrated in Fig. 13, the ionization rate r is correlated with 7 2 (e.g. both have peaks at R=6 and 9 a.u.), which indicates that ionization proceeds via the |2) created from the initial state lscr The mechanism of this correlation is revealed in the following sections.

5.2 Coupled equations for the two-adiabatic-state model

The initial state of Hj created from one-electron tunnel ionization of H2 is the lower adiabatic state |1) (equal to ls(Xg at zero field strengths). Using the two-adiabatic-state model, we next show how the upper adiabatic state 12) is populated from |l). The electronic and nuclear dynamics of H2

+ within the model is expressed as

\¥) = &R)\\) + X2Rp), (75)

where ^ ( - K ) and Xi^^ a r e m e nuclear wave functions associated with |1) and |2). We insert Eq. (75) into the time-dependent Schrodinger equation for the total Hamiltonian

1 d2

H(t) = -2mJ2)dR-

• + H0(t), (76)

where mp is the mass of a proton. The following coupled equations are obtained [91]:

j;xW-1 d2

-£l(R,t)+-wp dR m]

de_X dR)

ZlR)-A(R,t)Z2(R), (77a)

VtxM)-i mp dR mp

de_ dR

Z2(R) + A(R,t)zlR), (77b)

where the coupling that induces transitions between the two states (laser-induced nonadiabatic transitions) is

. , _ , dd 2 / dd d A(R,t)=— —-—

dt mvdR dR

d2e mpdR2

(78)

Here the small terms g\d2ldR2\g)/mp and (u\d2/dR':|w)/mp appearing in the diagonal and off-diagonal elements in Eq. (77) are ignored. The term ddjdt is the nonadiabatic coupl ing due to the change in the electric field, and the other coupling terms in A(R,t) are due to the joint effect o f the electric field and the nuclear motion.

192

It is expected from Eqs. (77) and (78) that nonadiabatic transitions occur if the main coupling term dOjdt is large in comparison with the gap between the two adiabatic potential surfaces, £2 - £\ • The term dd/dt is expressed as

d6ldtM2\U)dE«\COs28f. (79) 1 A£ag(R) dt

Since E(t) is a sinusoidal function of time, | dEt)ldt | and (cos2#) becomes largest when the field E(t) changes its sign, i.e., t„ = nn/a> (n is an integer) for E(t) = smcot. The gap £2 - £\ becomes A£ug, i.e., smallest when the two adiabatic potential surfaces come closest to each other, i.e., again, when the field E(t) changes its sign. Thus, a nonadiabatic transition between the two adiabatic states is expected to occur within a very short period rtr before and after the field Et) changes its sign [96]. In this case, a nonadiabatic transition between |l) and |2) corresponds to suppression of electron transfer between the two wells, i.e., two nuclei. We will later show the condition for temporally localized noadiabatic transition. It should be noted that (g\z\u)/A£ug(R) increases as R increases [(g|z|w) is an increasing function of R and A£ug(R) is a decreasing function of/?]. Larger internuclear distances are thus favorable for nonadiabatic transition. If the nuclear motion is slow in comparison with the time scale of rlr, the nonadiabatic transition probability per level crossing at R can be defined and it is given by exp(-2 nS), where

S = A£ug(R)2/sg\z\u)f(t)o). (80)

The meaning of 8 will be discussed later. The probability exp(-2;rJ) can be increased as the intensity, frequency, and R are increased.

Thachuk et al. [97] have developed a semiclassical formalism for treating time-dependent Hamiltonians ( nuclei are propagated classically on the surfaces) and applied it to the dissociation of diatomic ions. They have derived the nonadiabatic couplings dd/dt and vdOl dR, where v is the relative nuclear velocity. These two terms correspond to the first and second terms in Eq. (78). For homonuclear ions, dd/dt is much larger than vddldR , except when A£ug » ( g | z | w ) x (pulse envelope).

6 Adiabatic State Population Analysis

6.1 Numerical simulation of ionization of a dissociating H2*

The above qualitative analysis of the coupled equations is consistent with the actual numerical simulations of nuclear dynamics and ionization in the intense- and low-

193

frequency regime. In the case of H2+, it is possible beyond the two-state model. The

exact electronic wave packet 0(p,z,R;t) can be calculated [47, 91]. The components ;fr(i?) and z2($

a r e m u s obtained by projecting the exact electronic and nuclear wave packet onto |l) and |2). The mechanism of enhanced ionization has been directly proved using the populations of adiabatic states. We show that tunnel ionization proceeds in the region R>2Re through the |2) state populated from |l), where R,, is the equilibrium internuclear distance. Unlike the case of Fig. 13, here, the role of bond stretching is investigated by treating the internuclear distance quantum-mechanically. The molecular axis is again assumed to be fixed, parallel to the polarization direction, by the field E(t). For the following numerical calculation, the grid ends in z are ± 30 a.u. and the grid ends in pare 0 and 30 a.u. We assume that H2

+ interacts with the following laser pulse:

E(t) = f(t)sm(wt), (81)

where a =0.06 a.u. (A = 760 run) and the envelope ft) is linearly ramped with t so that after two optical cycles / ( / ) attains its peak value f0 = 0.12 a.u. ( / = 5.0xl014 W/cm2). The total pulse duration is 20 cycles (40 fs).

Ff2 is prepared by one-electron ionization from H2. The resultant state of H2+ is

|l). Since the creation of an ionizing electronic wave packet through tunneling begins and ends within a very short period, i.e., a half-optical cycle (characteristic of tunnel ionization), the vibrational wave packet of H2 created is nearly equal to the ground vibrational state of H2 (vertical transition). We thus assume that the initial total wave function of H2

+ at t = 0 has the following form [62, 95]:

0p,z,R;t = V) = X«^(.R)V), (82)

where %vjb=aR) is the lowest vibrational state of H2 in the ground electronic state X ILg . Here, we simply replace |l) at t-0 with the Lserg B-O electronic wavefunction of H2

+ , yf\sa (p,z;R). This replacement does not affect the following conclusions.

The dynamics of the nuclear motion is summarized as follows. The H2+ wave

packet starts from the equilibrium internuclear distance ^,=1.5 a.u. of H2. Since Rg is longer for H2

+ than for H2, the nuclear component of the H2+ wave packet moves

toward the outer turning point R ~ 4 a.u. on £ . Without an external field, the main component of the packet reflects at the outer turning point, coming back toward the inner turning point. On the other hand, under the condition of an intense pulse, dissociation takes place. When the instantaneous field strength \E(t)\ is large, the potentials £]R,t) and £2(R,t) given by Eq. (74) are distorted as shown in Fig. 12.

194

The nuclear wave packet is moved toward larger internuclear distances by the field-induced barrier suppression in £x(R,t) (bond softening due to a laser field).

We map the calculated &(p,z,R;t) onto |1) and |2) to obtain the projected vibrational components %l(R,t) = (l\0) and %2(R,t) = (2\0) [92, 93]. The overall populations /;(*)= j|;ft(/?,*)|2<#* a n d ^ ( 0 = ^ZiiÔfdR are shown in Fig. 14 together with the total norm in the grid space enclosed with the absorbing boundaries, PsM(t). In the early stage, in which the internucear distance is small (R<3 a.u.), the response to the field is adiabatic: the main component of the wave packet is still electronically |l). As the packet reaches internuclear distances around R-3.5 a.u., the adiabatic parameter 8 becomes as small as 0.2; exp(—2nd) = 0.23 . Nonadiabatic transitions between |1) and |2) occur when the field E(t) changes its sign, i.e., when the two adiabatic potential surfaces come closest to each other: around ? =210 a.u. (=47u/a>), a part of the population in |l) is transferred to |2), as shown in Fig.14. The increase in Pît) at a crossing point coincides with the decrease in Px (t).

During the next half cycle after a nonadiabatic transition (as the field strength approaches a local maximum), a reduction in the population of |2), Pi(t), is clearly observed, whereas P\t) changes very little. According to nonadiabatic transition theory, without ionization, Px(t) and Pitt) are constant between adjacent crossing points. The reduction in Pît) denoted by open circles is hence due to ionization. This is also confirmed by the correlation in reduction between PgM(t) and Pi(t) as indicated by the line with arrows at both ends. After the incident pulse has fully decayed, the relation P](t) + P2(t) = PgM(t) holds. We hence conclude that the state 12) is the main doorway state to tunnel ionization. Enhancement of ionization in H^ around R > 2Re is due to the joint effect of the following three factors: (i) the upper adiabatic state |2) is easier to ionize than is |l); (ii) nonadiabatic transitions from |l) to 12) occur at large R; (iii) the ionization probability of |2) has peaks around certain internuclear distances R>2Re.

The above mechanism of ionization is called charge resonance-enhanced ionization because nonadiabatic transitions occur mainly between the two adiabatic states |l) and |2) arising from a charge resonance pair such as lscr, and lsC7a. The analysis using the populations of the two adiabatic states is validated by the fact that in the high-intensity and low-frequency regime only these states are mainly populated before ionization [92]. We have diagonalized H0(t) by using the lowest six a -states. The total population of the resultant six adiabatic states is nearly equal to the sum of the populations of |l) and |2). The lifetimes of intermediate states other than |l) and |2) are shorter than a half optical cycle.

Using the right- and left-centered localized bases,

195

c o 12 Q. O Q_

1.0-

0.8-

0.6

0.4-

0.2-

0.0- r— r

' " w

I

>

*

ft N

, P, \ P

fSi 5 \

v\-\ \

^

1 I

grid

1

200 400 Time (a.u.)

600

Figure 14. Overall populations P(i) and P2 (t) on the lowest two adiabatic states, 11) and 12), of H2

+ . The populations P, (?) and P2 (t) integrated over R are denoted by a dotted line and a solid line, respectively. To eliminate the outgoing ionizing flux, we set absorbing boundaries for the electronic coordinates p and z. Ionization corresponds to the reduction of the norm of the wave function remaining in the grid space. The total norm in the grid space enclosed with the absorbing boundaries, Pgnd(t), is denoted by a broken line. Level crossing points for H2

+ , where Ei) = 0, are indicated by vertical dotted lines. Around t =210 a.u.(—4TT / ffl\ the wave packet approaches i?~3.5 a.u.; then, nonadiabatic transitions between 11) and 12) begin. Nonadiabatic transitions occur around level crossing points t = nn\ (O for n >4. The reduction in P2 (?) denoted by open circles is due to ionization. The upper adiabatic state 12) is regarded as the main doorway state to tunnel ionization. A reduction in P2 (t) corresponds to the subsequent reduction in Pgrid (?) as indicated by the line with arrows at both ends.

196

lL)=(ls>-lM»A/2' (83b) we can also describe the dynamics as

|^>= /!rR(^)|R> + jL( JR)|L), (84)

where the vibrational parts ZR and %h are related with Z\ an^ Xi a s

ZR(R) = [(COS 6 - sin 0)Zi (R) + (cos 6 + sin ff)z2 (R)]/yf2, (85a)

ZL (R) = [(cos 6 + sin ff)Xi (R) ~ (cos 6 - sin 0)Xl (R)]/J2. (85b)

The populations in the right and left wells associated with two nuclei, PR and PL , are expressed as follows [91]

PAR) = \zA^2 =^2[^+^)\MR)\2 +^d+^\Z2(R)\2

+ Re[z1\R)z2(R)cos28], (86a)

PdR) = \zd^ -^2^fj\x(R)\2 +^^^\z2(R)\2

-Re[z1'(R)Z2(R)™s28]. (86b)

The third term in both equations represents the interference between |1> and |2>. During a half cycle between two adjacent crossing points, the populations of |l)

and |2) are nearly constant. From Eq. (77), we notice that Zj(R) is expressed, in the period, as the product of the dynamical phase factor expj -i\ EjR,t') + ddldR)1 jmv \dt'\ and the modulus. The interference term thus takes the form of

ZxR)ZiR)^%26 -

*20 \z'R)Z2R)I e x p | - / J ' [ f 2 ( * / ) - £ , (*/)]<*']> (87) cos.

which oscillates with the period of the phase factor, 2nj\e1 [R,t)-£ (R,t)~\ • In intense fields, the period ~2K ^Keli+A\l^g\z\u)E[t)\2 ~7tj\ig\z\u)Et)\ can be much shorter than a half cycle, which results in ultrafast interwell electron transfer (whose period is neither intramolecular one 2#7A£ nor 2nl co) [91]. Field-induced ultrafast interwell transition is pronounced when both |l) and |2) are equally

197

populated and cos 28 is large. In the diabatic regime, \8\ is close to K/A except in the vicinities of crossing points; consequently, the interference term in Eq. (86) becomes negligible. This type of interference disappears when Z\(R) a nd %2^) do not overlap with each other in .ft-space (e.g. if the speeds of Z\(R) and %2(R) are extremely different from each other).

6.2 Adiabatic doorway states of H2

The tunnel ionization of H2 in an intense laser field (7=10 Wcm~ and A=760 nm) has been examined with accurate evaluation of two-electron dynamics by the dual transformation method [48, 86]. The molecular axis is assumed to be parallel to the polarization direction of the applied laser field. We have estimated the ionization probabilities at different values of/? to reveal the mechanism of enhanced ionization in a two-electron molecular system. An ionic component characterized by the electronic structure H H~ or H~H is created near the descending well owing to laser-induced electron transfer from the ascending well. As R increases, while the population of H~H decreases, a pure ionic state H~H+ becomes more unstable in an intense field because of the less attractive force of the distant nucleus. As a result, ionization is enhanced at the critical distance Rc — 4-6 a.u. Although H2 in the ground vibrational state does not expand to R^ [98], the peak of the ionization probability around Rv indicates that ionization would be strongly enhanced when H2

is vibrationally excited.

Ionization proceeds via the formation of a localized ionic component in the descending well, in contrast to the H2

+ case where the electron is ejected most easily from the ascending well. We have estimated that the change in R from 1.6 a.u. to 4 a.u. would enhance the ionization rate by a factor of 10-20. In the small R (<4 a.u.) region, the main doorway states to ionization are identical to the localized ionic states H H~ and ETH . Around R = 6 a.u., the ionization from the covalent state competes with that from the created localized ionic state. As R increases further, the electron density transferred between the nuclei is suppressed: the main character becomes covalent. Although the rate of direct ionization from a covalent state is much smaller than that from an ionic state, the ionization at large R (>8 a.u.) mainly proceeds from the remaining covalent component.

We have also investigated the intramolecular electronic dynamics that governs the ionization process by analyzing the populations of field-following adiabatic states defined as eigenfunctions of the instantaneous electronic Hamiltonian. In a high-intensity and low-frequency regime, only a limited number of adiabatic states participate in the intramolecular electronic dynamics, i.e., dynamics of bound

198

electrons. The effective instantaneous Hamiltonian for H2 is constructed from three main electronic states, X, B I a

+ , and EF [99] (At large R, the last two electronic states are replaced with the ungerade and gerade ionic states.)- We thus have three time-dependent adiabatic states |l) , |2), and |3). By solving the time-dependent Schrodinger equation for the 3 x 3 effective Hamiltonian, we have found that the difference in electronic and ionization dynamics between the small R case and the large R case originates in the character of the level crossing of the lowest two adiabatic states.

As the field strength increases, the lowering second lowest adiabatic state |2) comes closer to the lowest adiabatic state |l) starting from the X state. The transition period in which a nonadiabatic transition between |l) and |2) is completed is much smaller than the quarter optical cycle 7tj2co. Thus, a nonadiabatic transition is localized around the time tc when the field strength Et) reaches the value required for a crossing, Etc). In the case of R < 4 a.u., the energy gap at the avoided crossing between |l) and |2) is as large as that at zero field strengths. As a result, nonadiabtic transitions to upper adiabatic states hardly occur. When E(t) is larger than E(tc), |1) is ionic, while |2) is covalent. Therefore, ionization occurs from state |l) characterized by a localized ionic state H H~ or H~H directly to Volkov states. In the case of large R (>4 a.u.), nonadiabatic transitions occur from |l) to |2) when these two states cross each other. For R>6 a.u., ionization proceeds mainly through state 12), which is a covalent character-dominated state when E(t)> Etc). The three-state problem can be reduced to a two-state problem by prediagonalizing the 2x2 matrix constructed in terms of the upper two states B and EF. On the basis of the two-state model, an analytical expression of the field strength required for the crossing of |1) and |2) is derived; moreover, the probability of a nonadiabatic transition between |1) and |2) is expressed by the Landau-Zener formula. The results based on these simple formulas agree with those in the three-state treatment.

The characteristic features of electronic dynamics of H2 in an intense laser field leads to a simple electrostatic view that each atom in a molecule is charged by field-induced electron transfer and ionization proceeds via the most unstable (most negatively charged) atomic site. The success of the adiabatic state analysis for H2 means that the dynamics of bound electrons and the subsequent ionization process can be clarified in terms of a small number of "field-following" adiabatic states. The properties of adiabatic states of multi-electron molecules can be calculated by ab initio MO methods. While the time-dependent adiabatic potentials calculated by an MO method are used to evaluate the nuclear dynamics (such as bond stretching) till the next ionization process, the charge distributions on individual atomic sites are used to estimate the ionization probability. We have already applied this approach to

199

a multi-electron polyatomic molecule, CO2, in an intense field [100, 101] and revealed that in the C02

2+ stage the two C-0 bonds can be symmetrically (conceitedly) stretched while accompanied by a large-amplitude bending motion. This approach is simple but it has wide applicability in predicting the electronic and nuclear dynamics of polyatomic molecules in intense laser fields.

7 Transfer Matrix Formalism

We here treat R as a time-dependent parameter R(t) to solve the coupled equations (77). The nuclear motion can be estimated by semiclassical formalisms. Thachuk et al. [97] have proposed a criterion as to how classical trajectories should be hopped between the £x and £2 time-dependent surfaces. The conservation principle to apply during a hop depends upon its physical origin. The nonadiabatic coupling ddl dt mainly induces energy exchange between the electron and the field. When ddl dt is dominant, nuclear momentum conservation is appropriate. On the other hand, when vddl dt is dominant, energy exchange occurs between the electron and nuclei: total energy conservation is appropriate. The two limiting cases can be smoothly bridged with a physically justified conservation scheme. Since the interference between the two components of |l) and |2) vanish when Z\(R) a nd Zi(^) do n o t overlap with each other, we focus on the case where the difference in nuclear motion between 11) and 12) is negligible.

First, we derive a characteristic duration time of a nonadiabatic transition. Near the nth crossing point tn, the gap between the two adiabatic states is given by

2J = A£ug(Rn), (88)

where Rn = Rt =tn). The coupling strength near the crossing point, ddldt )t=t in Eq. (77) or (79), is

(J0/Jt)l=tn=(g\z\uR=Rn[ciE(t)/dt]t=t l2J = Aco/4J, (89)

where A is a slowly varying function

A = 2(g\z\u)R=Rnf(t„) . (90)

We have assumed that the internuclear distance hardly changes for the time duration Ttr of a nonadiabatic transition (the Zener transition time given below); i.e., (g|z|w) and A£ug(R) change only a little in the range between Rn±vrtr. The transition process is adiabatic or diabatic according to whether the inverse of the scaled nonadiabatic coupling (ddldt)t=,nlA£ug(R„) is larger or smaller than unity. The quantity A£ag(R„)l(ddldt)t=tn is on the order of

200

8 = J2/Aco. (91)

Although the Zener transition time ztr is defined by detailed analyses of nonadiabatic transitions in two-level systems [96], we can derive it from the following simple discussion. For 8 > 1 (adiabatic case), the time duration rlr that the system exists in the transition region is given by the temporal width of dOI dt around tn, i.e., JI ACQ , because the coupling dOldt is always smaller than the gap £2 - £\; for <5 < 1 (diabatic case), Ttr is given by the period of effective coupling where 901dt is larger than £2-£\, which leads to Ttr < 1/VAm. The condition for which leads to Ttr < \h isolated transition is determined from r,„ « nlco. Conditions are tabulated below.

Adiabatic parameter 8 = J 1 ACQ

Transition time xtr

Condition for isolated transition

Adiabatic case

8>\

J IACO

AIJ>\

Diabatic case

8<\

\I^~AM

(an overestimated value)

•JAICO>\

Combining the conditions for 8 and isolated transition, we obtain more specific conditions for isolated transition; Q)< J <A in the adiabatic case and max(ffl, J ) < A in the diabatic case.

Nonadiabatic transitions can be discussed by using a more familiar representation, namely, the diabatic representation. Inserting Eq. (84) into the time-dependent Schrodinger equation for the total Hamiltonian, we obtain the equations of motion for the localized diabatic bases, |R) and |L), as

dt

d_ dt

zM = -i

ZL(R) = -

1 wp

1

d1

dR2

d2

mp dR2

+ £R(R,t)

+ £L(R,*)

ZR(R)-i^4^^(R)> (92a)

ZL(R)-i^VXR(R), (92b)

where 1 1 d2

£R(R,t)=-[£AR)+£u(R)]-— <R|—|R> + (gNw>£(0, mp dR2 (93a)

201

£L(R,t)=\[£„(R)+£u(R)]-—(L\^-\L)-(g\z\u)E(t). (93b) 2 B mp dR2

In the above, we have neglect the small additional coupling terms (L|cr I dR |R)/wp

and ( R l ^ / e ^ l O / W p . Since (R |^ /c^ 2 |R) = Ud2ldR2\V) , the periodic energy difference A£RL(R,t) = £R(R,t)-£L(R,t) is given by 2g\iu)Et). If Ttr « KIco is satisfied (isolated transition), A£RL(Rn,t) can be linearized, in the range between t„ ± Ttr around the «th crossing point t„, as

A£RL(Rn,t)«2(gUu)R=RndE(t)/dt\l=ln(t-t„)

= Ao)(t-tn), (94)

where Aco is interpreted as the velocity of the change in the energy difference A£RL(Rn,t). Here, the field is assume to change from negative to positive at the nth crossing point. In the diabatic representation, the coupling A£u„(Rn)/2-J is independent of time. In the case of rtr « n I a>, the present curve crossing problem is represented by a linear potential model in the time domain with a constant coupling: the exact solution can be given by the Landau-Zener formula [102].

The transition amplitudes ( ^ R ^ L ) Jus t before t-tn-rtr and (Z'R,ZL) Jus t

after t = tn+ ttr is connected, aside from a common irrelevant phase factor, by the transfer matrix Md [73]

Md=( JJL , ^ e ~ * ) t (95)

where q is the asymptotic Zener transition probability and <j) is the so-called Stokes phase

q = e~1"\ (96)

<z> = ;r/4 + argr(l-;<S)+£(ln(5-l). (97)

The phase <p is a decreasing function of 5 which takes (f)5 —> 0) = n'I'4 and 0(<S—>°°) = 0. The above Md is obtained for the case where E(t) changes from negative to positive at the crossing point, i.e., |R) crosses |L> from the lower-energy side. In the opposite case, M^ should be replaced with its transpose Md.

The transfer matrix in the diabatic representation, Md, can be converted to the

adiabatic one Ma which connects (X\->Xi) Jus t before t„ - rtr and x'\,X'i) Jus t after

202

M„ -& q e"1

^ (98)

The outgoing state at the «th crossing and the incoming state at the (n+\)th crossing are connected by the propagator (dynamical phase factor) which is in the adiabatic representation expressed as

-ia.

G": e

V 0

0 -ia:

(99) •2)

where the phase a" arises from the terms [...] in Eq. (77), i.e., from the dynamics on they'th adiabatic potential

anj = \,

tn+lejRt'),t')dt'. (100)

If the state just after the wth crossing is given by %" -xl^x'i) > m e subsequent dynamics of the amplitudes, e.g., just after the («+2)th crossing, obeys the following operation

M tt+2 / - i n+\ jk/wn+\ g~in _ , n (101)

where M^ denotes the transfer matrix for the specific «th crossing event. The general extension to multiple crossings is straightforward. When nuclear motion is taken into account, A and J must be treated as functions of Rn.

In the case where the applied periodic field has a constant envelope and the internuclear distance is fixed, an elementary arithmetic of 2x2 matrices as in Eq. (101) yields very simple and physically intuitive formulas. The probabilities of |1) and 12) after the «th crossing, P" and P%, under the condition that the system exists in 11) before the first crossing are expressed as

i f =|^1"|2=l-g'sin[«(^ + / 2 ) ] / cos^ ] 2 , (102a)

and

where

Pi = b f =9sin[n(?7 + *72)]/cos772 (102b)

sin7]- VI - q sin(a+ </>). (103)

Here, a is the substantial dynamical phase, namely, the half of the difference between a2 and ax

CC: :a2-ax)l2= ^'l<0[E2t')-Ext')\l2 dt'

203

= _[* "\J4J2+ A2 sin2 at / 2 dt'. (104)

The maximum of the sequence P2" , P™x , is given by q I cos2 rj = q/[l-(\-q)sin2(a+<p)] . If sin(a+ (/>) = 0 , P™* = q (destructive hopping case, localization case). In this case, the condition ri=kn with an integer k holds ; hence, the sequence P2 simply takes the values of 0 (for even n) and q (for odd «): the time-averaged population \P2 = q/2. The transition paths to |2), |l) —>|2) —>|2) and |l) —>|l) —»|2) , interfere completely destructively every two crossings. The condition has a more general meaning. Whenever sin(a+^) = 0 holds, the same populations are fully recovered every two crossings; i.e., P" ,P2")- (P"+2 ,P2

n+2). If both components always exist, i.e., P" * 0 and P2 * 0 for any n, using the transfer matrix formalism, we find that \P2) in the destructive hopping case takes a value in the following range:

q/2 + (l-q)P2n±^iq(\-q)P2"(\-P2

n) . (105)

For sin(«+ <p) = ±1, P2maK = 1 (constructive hopping case). The sequence P2

n distributes uniformly and densely over the interval [0,1]: the average is \P2j =1/2 irrespective of the initial condition. This value of 1/2 can be achieved even if the nonadiabatic transition probability q per hopping is much smaller than unity, i.e., in an adiabatic case; complete inversion from |l) to |2) requires many crossings, at least,

. All the transition paths interfere so that the hopping probabilities from |1) to 12) (or from |2) to |1)) are accumulated every crossing most efficiently. The hopping pattern is governed by the interference originating from the sum of the dynamical phase a acquired during the propagation between adjacent crossing points and the Stokes phase <f> at the crossing point.

Since tunnel ionization proceeds via the upper adiabatic state 12), the ionization probability increases as the average population of |2) increases. If the system exists in |l) before ionization starts, ionization is enhanced when \P2j is maximized, i.e., in the constructive hopping case (when ionization occurs, (^2/ ' s normalized by setting the sum P" +P2 to be unity). This condition can be fulfilled even in the adiabatic case. If the state just before ionization is |2), the destructive hopping case is favorable for enhanced ionization: then, the hopping to |l) is suppressed. In the diabatic regime, however, the difference between the two hopping cases is very small; \P2")~l/2 in any hopping case. For example, in the case of large R, the value of q is very close to unity. Then, \P2J in the destructive hopping case is ~ql2 from Eq. (105), which is nearly equal to 1/2 of the constructive hopping case. For instance, in the case of Fig.13 (A = 1064 nm and I = 1014W/cm2), the situation at R-l a.u. is

204

nearly a destructive hopping case, where sin(0 + a) ~0.1. We then obtain \P2/-0.55 which is in fact close to 1/2. The difference :=0.05 mainly comes from the interference term in Eq. (105), ylq(l-q)P^(l-P2") ( in t h i s case, g=0.99). In the case of Fig. 13, the average [P?) is nearly independent of R. The tendency of the change in the AC ionization rate r is therefore governed by the change in the DC ionization rate of \2),r2. The ratio of r2 to r, however, peaks at R~6 and 9 a.u. For an instantaneous value of £(/) = 0.0533 a.u., F2 becomes larger around R=6 and 9 a.u owing to resonance with an adiabatic state in the descending well: the role of the ionization from |2) at the peak of the field strength is exaggerated.

In the diabatic case, \6\~7tlA at t = tn+Tlr . Thus, except in the isolated transition region, the adiabatic states are identical with the localized states; in the range tn+rlr <t <tn+l-rtr , |l)->|L) and 12) —> | R> for E(t„+Ttr)>0 and 11) —>|R) and 12) —»-jL) for E(tn+rlr)<0. See Eq. (71). Even in the adiabatic case, the condition for an isolated transition, AIJ»\, means that the adiabatic states become identical with the localized states around t — (t„+tn+l)/2. Around this moment, the field strength takes a local maximum value, for which |#|= ;r/4 is satisfied. Whenever the transfer matrix formalism is valid, except in the isolated transition region, an adiabatic state can be connected to a localized state by one-to-one mapping. Thus, we obtain the population of a localized state ( |R) or |L)) after the wth crossing, P", under the condition that the system starts from the other localized state

P2m~l =\-qcos[(2m~l)7]]/cosT]2, (106a)

and

P2m = qs'm(2mT]) /cos T]2. (106b)

For parameter values satisfying r\=kn with an integer k, the sequence Pn simply takes two values, namely, 0 (for even ri) and 1 - q (for odd ri): The transition paths to the other localized state interfere completely destructively every two crossings. This is nothing but the destructive hopping case. In the diabatic limit 8«\, the following replacements can be made in Eq. (106): q—¥\, \-q —>2xS , 0—>;r/4, and a-¥ Al co. We find

P"=sin2[«V2~^?sin04/ft>+;r/4)]. (107)

The rate of tunnelling or transition between the two localized states is reduced from the zero-field value J by the factor 426)17tAsm(AI (0+ x/4). The transition is completely suppressed when the total acquired phase Ala>+ nlA is kn, i.e., in the destructive hopping case. Equation (107) is valid in the diabatic isolated transition

205

case [ max(<w, J) < A ]. In the limit of AI co »1, we can use sin[( AI co) + n 14)] = \lnAI 2nco J0(A/Q)), where J0 is the zeroth order Bessel function. We thus have

P" = sin2 [(naJ/a>)J0(AIco)] . (108)

The condition for localization in the strong-field limit of AI co»1 is that J0(A/ co)= 0 [103], which is equivalent to the condition J0i2coR I co) = 0 stated in Sec. 3 because AI co = 2coRl co.

8 High-Frequency Limit

We next derive a formula that is valid in the rapid oscillation limit 0)» J and discuss the suppression of transitions between the two localized states [73]. It is convenient to use the coupled equations in a rotating frame as

d dt' —X*(RA = -U™9 2i(g\z\u) \ Et')dt' xdR,t),

-2i(g\z\u)^ E(t')dA^R,t),

where

ZR(R,t) = exp / j'eR(R',t')dt' ZR(R,t),

ZL(R,t) = exp ij'€L(R',t')dt'jzdR,t) •

(109a)

(109b)

(110a)

(110b)

In Eq. (109), the interwell transition rate / is modified by the exponential phase factors exp[+2('(g|z|w) \ E(t')dt'] . When the condition co»J is satisfied, the exponential phase factors change as fast as the time scale of llco while the state vector changes at most on order of 1 / / . Then, the effective interwell transition rate is determined by cycle-averages of the exponential phase factors. When the pulse envelope / ( / ) , i.e., A, is a slowly carrying function of time in comparison with the oscillation period 2KI co, the cycle-average over one period is given by

m r'\^\±2i(g\z\u)( E(t")dt"' In

dt'

CO rl+itla

exp Jt-jtlco 2x

• 2i(g\z\u) fit') cos cot'

CO dt'

• J0[2(g\z\u)fit)/co] , (111)

206

where the pulse is assumed to be turned on adiabatically

r E(Odt'=-fit)co&(0t. (H2) J ~ co

In the high-frequency limit, the tunnelling rate is reduced from the zero-field value J by the factor J0[2(g|z| u)f(t)/co], as in the strong field case.

Inserting Eq. ( I l l ) into Eq. (109), under the condition that it starts from |l_) at t = 0, we obtain

jL(?)=c o s jr jo[2(^iz iM)^') /^^'' ^ii3a>

XR(t) = -ismj^J,2(g\z\u)ft')lai\dt^ . (113b)

If / ( / ) has a constant envelope as f(t) = f0, JQ[---] m Eq. ( I l l ) should be multiplied by an additional phase (for the sinusoidal form of sin cot). By choosing proper phase factors for the localized states, we can eliminate the factor from the final formulas as

ZL(t)=cos[JtJ0(2(g\z\u)f0/a))], (114a)

ZRU) = -ism[JtJ0(2g\z\u)f0/co)] . (114b)

Tunnelling is completely suppressed if J(£2(g\z\u)f0/co) = J0(A/oo) = 0 is satisfied. Upon replacing tlnl co) with n, i.e.,. the number of crossings, we find that The square of Eq. (114b) is nothing but Eq. (108) which is valid in the asymptotic region of max(ft>, J)«A . The two equations should coincide with each other because the regions where Eq. (114b) and (108) are valid, i.e., J«co and max(<y, J)« A, have the common subregion of J « co « A [73].

Analytic formulas can be obtained for a specific pulse shape as / ( / ) = f0 sin(7i;t/Tp) for 0 < t < pulse duration T ; otherwise, zero. We can calculate the amplitudes at t - Tp, i.e., at the end of pulse

ZLTp) = cosjTp[j0((g\Z\u)f0/co)]2 , (115a)

ZR(Tp) = -ismjTp[j0((g\z\u)f0/co)]2 . (115b)

If Jd^g\z\u)folco) = 0, complete localization is resumed at t=Tp irrespective of the pulse length T. Complete localization is also resumed if JTp[Jo((g\z\u)fo I a>)] = kn ; complete transition to |R) occurs if

207

JTp[J0((g\4u)fo10))] = (k + \I2)JI . The present result is an example that shows a possibility of controlling molecular systems by using pulses of realistic shapes other than pulses of constant amplitude.

At larger internuclear distances, the transition between the two wells is suppressed. In the high-frequency limit, the transition rate for interwell tunnelling is given by the well known form Jx J 0[2(g\z\u)f(t)/ 0)] . Interwell tunnelling is further suppressed with increasing field strength. There also exist specific conditions for the Bessel function to be zero. For the zeros of the Bessel function, interwell transition is inhibited. The coherent destruction of tunnelling is due to interference between the two adiabatic (or diabatic) components at periodic crossing points tn = nxl a)(n=\,2,...).

9 Conclusion

We have summarized in this review the current status of our knowledge of the behaviour of molecules in intense laser fields. This is a new area of research, which will continue to expand due to the quickly evolving accessible laser technology [1-2]. Needless to say, we have entered already the attosecond era in time scale [ 16] with the foreseeable application to controlling electron dynamics in molecules [104]. Similarly in intensity, we have gone beyond Exawatts and are approaching the Zettawatt limit (see table I) with foreseeable applications to nuclear fusion induced by Coulomb explosion of molecular clusters [105].

Similarities of atomic and molecular processes have been emphasized through the unifying concepts of quasistatic models, which predict tunnelling ionization, barrier suppression mechanism of enhanced dissociation and ionization at high intensities. However, one fundamental difference exists between atomic and molecular multiphoton processes at high intensities: the existence of doorway states in molecular systems due to charge resonance effects predicted as early as by Mulliken in 1939 [28]. Thus as shown in [67], there are various regimes of radiative couplings as a function of internuclear distances and the multiphoton transitions in each regime can be adequately described by a nonadiabatic theory (Sees. 5 - 8). The most interesting regime is the nonadiabatic regime where the excitation energy between the LUMO and HOMO in one (odd) electron systems approaches the laser frequency CO and/or Rabi frequency a>R [Eq. (32)]. This leads to Charge Resonance-Enhanced Ionization (CREI) at critical distances and structures of a molecule where molecular ionization rates exceed that of the dissociative fragments by orders of magnitude. In the case of even-electron systems, the creation of ionic states by laser-induced crossing of these states with the ground state at critical

208

distances also leads to enhanced ionization. It is in these nonadiabatic regimes that field-induced population of excited states, be it LUMO's or ionic states, leads to enhanced ionization where an electron must cross the whole molecule in order to exit the molecule through tunnelling ionization.

The combination of the atomic quasistatic tunnelling ionization model and the laser-induced excitation, be it resonant or nonresonant with transitions between molecular essential (doorway) states as described in the nonadiabatic model, offers a consistent simple framework to understand the highly nonlinear nonperturbative response of molecules to intense laser pulses. We reemphasize this fundamental difference between molecules and atoms, i.e., in the former there exist essential or doorway states for enhanced ionization at critical radiative couplings in molecules. These nonperturbative radiative couplings depend on nuclear configurations such as internuclear distances. It is under these conditions that new Laser-Induced Molecular Potentials (LIMP's) are created [20-27]. It has been suggested recently that such doorway states affect ionization, fragmentation, and energetics in large molecules where charge resonance effects dominate [106]. Numerical simulations on the simple systems H2

+ and H2 have shown furthermore that it is in this enhanced ionization regime that laser control of ionization is most sensitive to laser parameters such as phase [78, 107].

One outstanding issue is the connection between excitations of the doorway states and plasmons, i.e., the multielectron collective excitations, which appear at critical electron densities. Recent numerical simulations of ionization and Coulomb explosion of large clusters emphasize the persistence of the CREI mechanism at a critical initial configuration as a precursor to the facile ionization and explosion of these large systems [108]. Earlier suggestions that plasmon resonances are the essential doorway states for multielectron ionization [109] require a unifying model, allowing us to link the CREI regime and the collective regime of plasmons. Thus many body effects in strong laser fields, specific to molecules and clusters, remain an issue. Laser-induced electron collisions with neighbours in molecules have been shown to enhance and extend HOHG plateaus [34] and lead to a distinct possibility of cluster heating with long-range electron-atom collisions [109]. The enhanced ionization mechanism discovered in our numerical simulations as described in the preceding sections shows that freed or hopping electrons are created in the high field regime and must traverse a whole molecule or cluster in order to ionize. Such effects have been shown to be very sensitive to electron correlation [86] and have not yet been considered in multielectron excitations of large systems. The extension of the simple models developed in the previous sections for small molecules to large molecules and clusters remains an open problem.

209

The emergence of intense ultrashort pulses (7>1014W/cm2and r/^!e<10fe) is ushering a new era of imaging and diagnostic of molecular dynamics with these new tools. We already have shown through non-Born-Oppenheimer simulations on the simple Hj system that Laser Coulomb Explosion Imaging (LCEI) can be a viable new method of measuring the time evolution of nuclear wave packets [58-59]. As discussed in Sec. 2, intense low frequency fields have a new interesting effect, i.e., such fields induce recollision of the electrons with the ionic core. This suggests the possibility of using such "recolliding" electrons to probe short time molecular dynamics via Laser-Induced Electron Diffraction (LIED) [110]. The first experiment based on this idea has been recently achieved [111] demonstrating the use of sub-laser-cycle electron pulses for probing molecular dynamics and thus opening a new area of science, measurement of molecular dynamics in a new time regime, sub-femto or attosecond time regime.

Finally, the spectroscopy of gases interacting with short intense laser pulses [112] offers a new method of detecting small concentrations due to the intense signals generated by such pulses [112]. These are new exciting applications of the new laser technology.

Acknowledgments

We thank various colleagues, P. B. Corkum, Y. Fujimura, Y. Yamanouchi, S. Chelkowski, Y. Kayanuma, Y. Ohtsuki and I. Kawata for valuable discussions concerning interaction between intense laser fields and molecules. We also thank the Japan Society for Promotion of Science for supporting collaborative projects leading to this review. This work was supported in part by grants-in-aid for scientific research from the Ministry of Education, Culture, Sports, Science, and Technology, Japan (12640484 and 14540463).

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PART THREE

Ultrafast Dynamics and non-Markovian Processes in Four-Photon Spectroscopy

217

Ultrafast Dynamics and non-Markovian

Processes in Four-Photon Spectroscopy

B.D.Fainberg

Raymond and Beverly Sackler Faculty of Exact Sciences,

School of Chemistry, Tel Aviv University, Tel Aviv 69978,

Israel

and Physics Department, Center for Technological Education

Holon, 52 Golomb St., Holon 58102, Israel

Contents

1 Introduction 221

2 Hamiltonian of chromofore molecule in solvent and basic

218

methods of the resonance four-photon spectroscopy 226

3 Calculation of nonlinear polarization 233

4 Stochastic models in transient R F P S 238

4.1 Non-Markovian relaxation effects in two-pulse RFPS with

Gaussian random modulation of optical transition frequency 238

4.2 Transient four-photon spectroscopy of near or overlapping

resonances in the presence of spectral exchange 246

4.3 Non-Markovian relaxation effects in three-pulse RFPS . . . . 252

5 Non-Markovian theory of steady-state R F P S 255

5.1 Introduction and the cubic susceptibility in the case of

Gaussian-Markovian random modulation of an electronic

transition 255

5.2 Model for frequency modulation of electronic transition of

complex molecule in solution 257

5.3 Cubic susceptibility for detunings larger than reciprocal

correlation time 262

5.3.1 Classical system of LF motions 262

5.3.2 Quantum system of LFOA vibrations 265

219

6 Four-photon spectroscopy of superconductors 271

7 Ultrafast spectroscopy with pulses longer than reciprocal

bandwidth of the absorption spectrum 276

7.1 Theory of transient RFPS with pulses long compared with

reversible electronic dephasing 279

7.1.1 Condon contributions to cubic susceptibility 282

7.1.2 Nonlinear polarization in a Condon case for non-

overlapping pump and probe pulses 286

7.1.3 Non-Condon terms 289

7.2 Nonlinear polarization and spectroscopy of vibronic

transitions in the field of intense ultrashort pulses 291

7.2.1 Classical nature of the LF vibration system and

the exponential correlation function 295

7.2.2 General case. Quantum nature of the LF vibration

system 299

8 Experimental study of ultrafast solvation dynamics 300

8.1 Introduction 302

8.2 Calculation of HOKE signal of i?800 in water and D20 . . . 303

220

8.3 Method of data analysis 309

8.4 Discussion 314

9 Prospect: Spectroscopy of nonlinear solvation 318

9.1 Four-time correlation functions related to definite electronic

states 321

9.2 Simulation of transient four-photon spectroscopy signals

for nonlinear solvation 324

9.3 Spectral moments of the non-equilibrium absorption and

luminescence of a molecule in solution 331

9.4 Broad and featureless electronic molecular spectra 334

9.5 Time resolved luminescence spectroscopy 336

9.6 Time resolved hole-burning study of nonlinear solvation . . . 337

9.7 Stochastic approach to transient spectroscopy of nonlinear

solvation dynamics 339

9.8 Summary 344

10 Acknowledgments 350

A Appendix 350

221

1 Introduction

Recent investigations with tuning lasers and lasers for ultrashort (up to a

few femtoseconds) light pulses have led to the revision of simple models for

the light-condensed matter interaction.

The first interpretations of the corresponding experiments with dye solu

tions were based on a consideration of a model of two-level (or three-level)

atoms characterized by certain constants of the energy T\ and phase Ti re

laxation and also by the spread of the transition frequencies modeling the

inhomogeneous broadening in the system [1-4]. Phenomenological terms tak

ing into account the processes of cross-relaxation T3 [1, 5-7] or vibrational

relaxation [8] were also introduced, with the goal of constructing a more re

alistic model in the case of condensed media. In these studies by comparing

the experimental data to the theory, times T2 and T3 were estimated to be

of the order of hundred femtoseconds.

However, such extremely fast relaxation phenomena, whose duration is of

the order of the correlation time rc of the thermal reservoir, can no longer be

analyzed by the conventional theory based on the relaxation time description.

In this ultrashort time region one should take into account effects of memory

222

in the relaxation (non-Markovian effects) [9, 10].

Another reason for the revision of the simple models is the impossibility

to divide a relaxation between closed states into an energetic and a phase

one [11-13].

At the beginning of the eighties a non-Markovian theory of the tran

sient and steady-state resonance four-photon spectroscopy (RFPS) began to

develop in this respect [14-21]. A non-Markovian character of the optical

transition broadening can be illustrated by a simple example of an oscillator

whose natural frequency ui' is randomly modulated [9].

The resonance absorption spectrum of such an oscillator at the frequency

u> is given by

1 r°° F(u - w0) = — / exp[-i(w - uQ)t\ft)dt

Z7T J - o o

where to0 is the time average of UJ',

f(t) = (eXp[i£u1(t')dt'))

is the relaxation function of oscillator, uii (t) — UJ (t) — LU0.

The resonance intensity distribution F (LJ — UJ0) is an observed quantity

and it is broadened around the center w0 by the random modulation LJ\ (<).

223

Consider the shape of this resonance spectrum and its relation to the nature

of the modulation. Let us suppose that the frequency modulation is charac

terized by a probability distribution -P(^'i). If we suppose that P (ui) has

only one peak at u>i = 0, the stochastic process u\ (t) can be described in

terms of two characteristic parameters.

(1) Amplitude of modulation a: a2 = J uifP (to) dijj\ = (u>f).

(2) Correlation time of modulation, rc.

The correlation function of modulation is defined by

5 ( r ) = 4 r ( W l ( i ) W l ( t + r)>.

Then the correlation time TC is given by rc = J"0°° S (t) dt, i.e. (wj (t) wi (i + r)) ~

0 when r ^> rc; rc thus measures the speed of modulation.

It can be demonstrated [9] that two typical situations are distinguished

by the relative magnitude of a and TC:

(a) Slow modulation.

a - T c » l , (1)

(b) Fast modulation.

a • rc < 1. (2)

In case (a) rc is large compared to 1/a. Then it turns out that F (u> — u>0) =

224

P(u> —u>o)- Thus the intensity distribution reflects directly the distribution

of the modulation. The width of the intensity distribution curve will be

about a and the response is dynamic and coherent. This case corresponds

to an inhomogeneous broadening of the transition under consideration. Thus

one can say that the inhomogeneous broadening of the optical transition

corresponds to an extreme case of the non-Markovian relaxation, when the

"memory" in a system is completely conserved.

In case (b), TC becomes small and any modulation UJ\ hardly lasts for any

significant time (~ 1/wi), so that the fluctuation is smoothed out and the

resonance line becomes sharp around the center. In this limit a • rc —> 0 the

line have a Lorentzian form in which the half-width 7 = a2 • rc. This case

corresponds to a homogeneous broadening of an optical transition when the

relaxation is Markovian with a very short "memory" determined by rc.

As will be seen in later sections, the broadening of electronic transitions in

dye solutions used in four-photon experiments corresponds to the case of slow

modulation (a) (or the intermediate one) rather than to case (6). Therefore,

we need to use a non-Markovian theory for the description of corresponding

experiments.

The literature on non-Markovian effects in nonlinear spectroscopy is cur-

225

rently quite voluminous, but the framework of the review article does not

enable us to cover all the issues concerning the subject. Therefore we will

confine the review mainly to our recent theoretical results related to an ex

periment. We do not discuss such interesting problems as the non-Markovian

theory of the Resonance Raman Scattering [22-29], spectroscopy of dimers

[30, 31], etc. At present, there are a number of review articles [32-34] and a

monograph [31] which cover questions which are not discussed here.

A large contribution to the theory of four-photon spectroscopy has been

made by Mukamel and coauthors (see [31] and references here). Experimen

tally, the RFPS has been developed in the works of Yajima's, Wiersma's,

Shank's and Fleming's groups, Vohringer and Scherer and others (see refer

ences here).

The outline of this chapter is as follows. In Sec.2 we describe the Hamil-

tonian of a chromofore molecule in a solvent and basic methods of RFPS.

In Sec.3 we present the corresponding theory. In Sec.4 we consider non-

Markovian relaxation effects in transient RFPS by the use of stochastic mod

els. A non-Markovian theory of steady-state RFPS is presented in Sec.5.

Sec.6 is devoted to the real time four-photon spectroscopy of superconduc

tors. In Sec.7 we present a theory for transient RFPS with pulses long com-

226

pared with the electronic dephasing and its generalization for strong light

fields not satisfying the four-photon approximation. In Sec.8 we describe our

experimental results obtained by the heterodyne optical Kerr effect (HOKE)

spectroscopy on ultrafast solvation dynamics study of rhodamine 800 (i?800)

and DTTCB in water and D2O. In Sec.9 we shall discuss a prospect of spec

troscopy with pulses longer than the reciprocal bandwidth of the absorption

spectrum: nonlinear solvation study. In the Appendix we carry out auxiliary

calculations.

2 Hamiltonian of chromofore molecule in sol

vent and basic methods of the resonance

four-photon spectroscopy

Let us consider a molecule with two electronic states n — 1 and 2 in a solvent

described by the Hamiltonian

Ho = J2 I") [#» - *'Hi + wnQ) H (3) n=\

where E2 > Ei,En and 2jn are the energy and the inverse lifetime of state

n,Wn(Q) is the adiabatic Hamiltonian of reservoir R (the vibrational sub-

227

systems of a molecule and a solvent interacting with the two-level electron

system under consideration in state n).

The molecule is affected by electromagnetic radiation of three beams

1 3 -E(r , t) = E+(r , t) + E"( r , t) = - ^ £m(t) exp[i(kmr - umt)] + c.c. (4)

^ m = l

Since we are interested in both the intramolecular and the solvent-solute in-

termolecular relaxation, we will single out the solvent contribution to Wn(Q):

Wn(Q) = WnM + Wns where Wns is the sum of the Hamiltonian governing

the nuclear degrees of freedom of the solvent in the absence of the solute,

and the part which describes interactions between the solute and the nuclear

degrees of freedom of the solvent; WnM is the Hamiltonian representing the

nuclear degrees of freedom of the solute molecule.

A signal in any method of nonlinear spectroscopy can be expressed by

the nonlinear polarization PNL. We will consider both the steady-state and

the transient methods of the RFPS.

The steady-state RFPS methods [35, 1, 36, 5, 2, 37, 3, 8, 6, 21, 38-50] are

based on an analysis of the frequency dependence of the cubic polarization

P ' 3 ' of the medium studied at the signal frequency u>s = u'm + LJ'^ — um. The

case Lo'm = UJ'^ = wx = const, u>m — u>2 — var corresponds to spectroscopy

228

based on a resonance mixing of the Rayleigh type [35, 1, 36, 5, 21, 39-50].

The optical scheme describing the principles of the method is shown in Fig.l.

Two laser beams with different frequencies u,'i ^ u>2 and wave vectors ki

3=2K,-K2

Figure 1: The scheme describing the principles of spectroscopy based on a

resonance mixing of the Rayleigh type.

and k2 produce a nonstationary intensity distribution in the medium under

investigation. This intensity exhibits a wavelike modulation with a grating

vector q = kj—k2 and a frequency ft — u>i— ui^. The wavelike modulated light

intensity changes the optical properties of the material in the interference

region, resulting in moving grating structure.

If this wavelike modulation is slow in comparison with the relaxation

time, rrei of the optical properties of a material (r re; <C fi_ 1) the latters

follow the intensity change and the grating amplitude does not decrease. If

229

0. > T~J, the optical properties of a material do not follow the intensity

modulation, and the grating structure becomes less contrast, and therefore

the grating amplitude decreases. The grating effectiveness measured by the

self-diffraction of waves u\ drops for this case. In the moving grating method

the relaxation velocity of the optical properties of a material is compared with

the motion velocity of the grating (which is proportional to the frequency

detuning).

When applied to a spectroscopy of inhomogeneously broadened transi

tions, the self-diffraction signal will change like ~ fl~2 for T2-1 S> \Cl\ S> Xi_1

and like ~ fi'4 for \Cl\ > T j - 1 , ^ - 1 [35].

Now let us consider transient methods of RFPS. In a three-pulse time-

dependent four-wave-mixing experiment (Fig.2a) [51-64], pump pulses prop

agate in the directions k% and k2 and induce grating in a medium. The

dependence of the grating efficiency on the delay time r between pulses ki

and k2 is recorded using the scattering of the probing pulse k3 , delayed by a

fixed time T with respect to pulse k2. The resonance transient grating spec

troscopy (see Fig.2b) is the particular case of a three-pulse time-dependent

four-wave-mixing when r = 0 and T is variable [65-67]. For k3 = k2 (T = 0)

we obtain the spatial parametric effect [4, 68-70] and the two pulse photon

230

b)

kg+lc^k!

k ! -Jc2*t 2 T * 3

Wki

^ - ^ ^ 3

Figure 2: Geometry for three-pulse time-delayed four-wave-mixing experi

ments: (a) grating on the basis of a polarization (r variable, T = const); (b)

population grating (r = 0, T variable).

231

echo case [71-74, 59, 75]. In these experiments, the signal power Is in the

direction ks = k3 ± (k2 — k t ) at time t, is proportional to the square of

the modulus of the corresponding positive frequency component of the cubic

polarization P ' 3 ) + : Is(t) ~ | P ' 3 ' + ( r , t) \ . In pulsed experiments, the depen

dence of the signal energy Js is usually measured on the delay time of the

probe pulse relative to the pump ones:

J , ~ /°° |P<3)+(r,*)|2di (5) J — oo

When the stimulated photon echo is time gated, for instance, by mixing

the echo signal with an ultrashort gating pulse in a nonlinear crystal [76, 77],

the signal is proportional to the echo profile at time tg of the arrival of the

gating pulse: Jgated(tg,T,T) ~ | P£P+

Etg,T,T) | .

Other methods of transient RFPS are: the transmission pump-probe ex

periment [78-80], the heterodyne optical Kerr effect (HOKE) spectroscopy

[81-84], and time resolved hole-burning experiments [78, 85-90].

In the transmission "pump-probe" experiment [78-80], a second pulse

(whose duration is the same as the pump pulse) probes the sample transmis

sion AT at a delay r . This dependence AT(r ) is given by [91]

/

oo S;rt-r)VNL+(t)dt (6)

-oo

232

where Spr and VNL+(t) are the amplitudes of the positive frequency compo

nent of the probe field and the nonlinear polarization, respectively.

In resonance HOKE spectroscopy [81, 83, 84], a linearly polarized pump

pulse at frequency w induces anisotropy in an isotropic sample. After the

passage of the pump pulse through the sample, a linearly polarized probe

pulse at IT/4 rad from the pump field polarization, is incident on the sample.

A polarization analyzer is placed after the sample oriented at approximately

~/2 (but not exactly) with respect to the probe pulse polarization. A small

portion of the probe pulse that is not related to the induced anisotropy plays

the role of a local oscillator (LO) with a controlled magnitude and phase.

The HOKE signal can be written in the form:

/

oo

£lo(t - r ) expirl>)VNL+(t)dt (7) -co

where if; is the phase of the LO. If i\> = 0, the resonance HOKE spectroscopy

provides information similar to that of the transmission pump-probe spec

troscopy (see Eq.(6)). If i\) — 7r/2, the resonance HOKE spectroscopy pro

vides information about the real part of the nonlinear susceptibility (the

change in the index of refraction).

In the time-resolved hole-burning experiment [78, 85, 86, 92-97], the sam-

233

pie is excited with a ~ 100 fs pump pulse, and the absorption spectrum is

measured with a 10 fs probe pulse that is delayed relative to the pump pulse

by a variable r. In another variant of such an experiment, a delayed pump

pulse, broadened up to a continuum, can play the role of a probe pulse. The

difference in the absorption spectrum at w' + u is determined by [91, 86]

^ Q ( W ' ) ~ -Im[VNLJ)iepr(u')\ (8)

where

/

oo VNL+t)exv(iLo't)dt (9)

-oo

is the Fourier transform of the nonlinear polarization, and

/

CO

£pr(t - r) exp(iu't)dt (10) -oo

is the Fourier transform of the probe field amplitude.

3 Calculation of nonlinear polarization

The electromagnetic field (4) induces an optical polarization in the medium

P(r , t) which can be expanded in powers of E(r , t) [98]. For cubic polarization

of the system under investigation, we obtain:

P(3>(r, t) = p(3>+(r, t) + c.c. = iVJL4(rrR(D12^3

1)(i)) + c.c.)or (11)

234

where N is the density of particles in the system; L is the Lorentz correction

factor of the local field; D is the dipole moment operator of a solute molecule;

(.. . )o r denotes averaging over the different orientations of solute molecules;

p'3 ' is the density matrix of the system calculated in the third approximation

with respect to E(r,£).

The equation for the density matrix of the system can be written in the

form

/> = - * ( I o + £i)p (12)

where Lo and L\ are the Liouville operators denned by the relationships

LQp = h~l[Ho,p] and Lip = &-1[—D • E(r , £),/?]. Using the interaction

representation (int) by means of the transformation pmt = exp(iLot)p and

£mt _ eXp(iLot)LiexTp(—iLot), solving the resultant equations by perturba

tion theory with respect to L™' in the third order, and using the resonance

approximation, we find p^z\ The diagram representation of p^ can be found

in monographs [98, 31].

The a-th component (a, b,c,d = x, y, z) of the amplitude of the positive

frequency component of the cubic polarization V^+(t) describing the gen

eration of a signal with a wave vector ks = km/ + km» — km and a frequency

235

o.'s (P(3>+(r,i) = VW+t)exp[ik3r-<jjst)) is given by the formula [99]

x Yl H dTidT2dT3exp-[i(u21 - ua) + J]TI mm'm" bed

-[iiom - um>) + T^]T2 - ~fT3exp[i(u;21 - wm)T3]Fla6crf(r1, r2, r3)

x£m' C (* - Ti - T 2 )£ '^ ( i ( i - Ti - T2 - T3)

+ e x p [ - l ( w 2 l - W r o ' )r3]F2a6cd(Tl, T2, T3)

x£m/c(< - TI - r2 - T3)£*md(t - n - T2)£m»bt - n ) (13)

where T\ = (27 2) - 1 = (27) - 1 is the lifetime of the excited state 2, w2i =

'ei — (W2 — Wi)/h is the frequency of the Franck-Condon transition 1 —> 2

(see the definition of W2 and Wi in Sec.2), we; = (£ 2 — Ei)/h is the frequency

of purely electronic transition with corrections from the electronic degrees of

freedom of the solvent [99, 86]. The summation in Eq.(13) is carried out

over all fields that satisfy the condition k s = km< + km» — k m . The functions

F\,2abcdTi-,'r2,Tz) are sums of four-time correlations functions corresponding

to the four photon character of light-matter interaction:

Flabcd(n, 1~2, T3) = Kdcab0, ^ 3 , n + T2 + T3, T2 + T3)

+A^6ac(0, T2 + r3, Ti+r2 + r3, r3) , (14)

236

F2abcdn,T2, T3) = K*dba(0, r3, r2 + r3, r : + r2 + r3)

+ ^ M ( O . T I + T2 + r3, r2 + r3, r3) , (15)

where

A ^ O , ^ , * , , ^ ) = (DltexpiiWih/tyD^expiiWifo-tJ/h)

xDc12exp(iW2(t3 - i 2 ) / ^ ) ^ 1 e x p ( - ? W 1 f 3 / a ) ) ) o r (16)

are the tensor generalizations of the four-time correlation functions A'(0, t\, t2, i3)

which were introduced in four-photon spectroscopy by Mukamel [16, 100].

Here (...) = T r # ( . . . pR) denotes the operation of taking a trace over the

reservoir variables, pn = exp[—Wi/(kT)]/Trnexp[— Wi/(kT)] is the density

matrix of the reservoir in the state 1, W2 = W2 — (W2 — W\) is the adia-

batic Hamiltonian in the excited state without the reservoir addition to the

frequency of the Franck-Condon transition (the term (W2 — Wi)).

It follows from Eqs. (13),(14),(15), (16) that the nuclear response of any

four-photon spectroscopy signal, generally speaking, depends on the polar

izations of the excited beams because of the tensor character of the values

Fl,2abcd a n d Kabcd0,ti,t2,t3).

We can represent the latter quantity as a product of the Condon (FC)

237

and non-Condon (NC) contributions [101, 99]:

Kabcd(0,h,t2,t3) = KFC(0,h,t2,t3) • K^d(0,h,t2,t3). (17)

The Condon factors KFC(0,ti,t2,t3) do not depend on the polarization

states of exciting beams, however the non-Condon ones Kavb^d(0,ti,t2,t3) de

pend on their polarizations. The origin of the non-Condon terms stems

from the dependence of the dipole moment of the electronic transition on

the nuclear coordinates Di 2 (Q) . Such a dependence is explained by the

Herzberg-Teller (HT) effect i.e., mixing different electronic molecular states

by nuclear motions. When D 1 2 does not depend on the nuclear coordi

nates (the Condon approximation), the non-Condon terms are constants:

K?£(0,tut3,t3) = Dl2Db21D\2D

d2l)or ~ D4 where D = \Dl2\.

Let us introduce the quantity u = W2 — W\ which determines the strength

of the bonding of the vibrational subsystem with the electronic transition,

and characterizes the Condon perturbations of the electronic transition (un

like non-Condon perturbations which are determined by the dependence

Di2(Q))- Thus, if the value u is a Gaussian one (intermolecular nonspe

cific interactions, linear electronic-vibrational coupling etc.), and also in the

case of a weak electronic-vibrational coupling, irrespective of the nature of

238

u, the Condon contribution can be represented in the form [100, 39, 31]:

A ' F C ( 0 , t i , t 2 , i 3 ) = e x p [ 5 ( i 3 - i 2 ) + ^ i ) + 5 ( i 2 - i i ) - ^ 2 ) - ^ 3 - i i ) + ^ 3 ) ]

(18)

where

g(t) = - / r 2 f dt\t - t')Kt') (19) Jo

is the logarithm of the characteristic function of the spectrum of single-photon

absorption after substraction of a term which is linear with respect to t and

determines the first moment of the spectrum, K(t) = (u(0)u(t)) — (u)2 is the

correlation function of the value u.

4 Stochastic models in transient RFPS

4.1 Non-Markovian relaxation effects in two-pulse RFPS

with Gaussian random modulation of optical tran

sition frequency

Let us consider the spatial parametric effect (SPE)[4, 68]. The SPE consists

in the following. When the medium under study is acted upon by two short

light pulses of frequency u> with wave vectors kx and k2 separated by a time

239

interval i2 ( s e e Eq.(4) for TO = 1,2 and wi = w2 = w), signals with wave

vectors k3 = 2k2 — kj and k4 = 2kx — k2 are generated in this medium. The

temporal characteristics of these signals provide information on the phase

relaxation time of the optical transition studied.

The problem of non-Markovian relaxation in the SPE was first discussed

independently in Refs. [14, 15]. Aihara[14] examined the transient SPE

in the case of a system with a linear and quadratic electron-phonon bond,

and obtained numerical results illustrating a non-Markovian behavior. In

Ref.[17] (see also Ref.[18]) a stochastic model was used to derive simple an

alytical relationships, which could serve as the basis for the spectroscopy of

non-Markovian relaxations based on SPE. The model interpolates in a contin

uous way between the inhomogeneous broadening case and the homogeneous

broadening case.

The cubic polarization of the medium, corresponding to wave k3, is given

by Eq.(13) for m = 1 and m' = TO" = 2. If pulses ki and k2 are well

separated in time, then only the first term in the curly brackets on the right

hand side of Eq.(13) (~ i*\) makes a contribution. Let us assume that u(t)

is a Gaussian-Markovian random process with correlation function K(t) =

h2a2 exp(—|i|/rc). Such a model of relaxation perturbation is highly realistic.

240

It corresponds to solvated systems in the case of the Debye spectrum of

dielectric losses [102], concentration-dependent dephasing in mixed molecular

crystals [103], Doppler-broadened lines in gases in weak collisions [104], and

phase modulation by phonons in crystals [25].

The quantity Fi in the case of the Gaussian-Markovian random modula

tion of an electronic transition has been calculated in Refs.[17, 18]. In the

Condon approximation, without taking into account tensor properties, the

quantities Fit2 have the following form:

^ , 2 ( r i , r 2 , r 3 ) = £ 4 e x p - p 2 [ e x p ( - ^ ) + ^ + e x p ( - ^ ) + ^ - 2

T e x p ( - ^ ) ( e x p ( - ^ ) - l ) ( e x p ( - ^ ) - 1)], (20)

where p = arc.

Considering the limit of short pulses and introducing the pulse areas

e m = DH~l /f^ £m(t')dt', we obtain the signal strength I3(t) ~ |p(3)+|2 by

means of Eqs.(13) and (20):

I3(t) = J B e x p - T 1 - 1 t - 2 p 2 [ - + 2 e x p ( - ^ ) - e x p ( - - ) + 2 e x p ( - ^ ^ ) - 3 ] ,

(21)

where t\ = 0 is the moment of the appearance of the first pulse ki , t > t2

- the delay time of the pulse k2 with respect k i , B = AA^O^O2, and A is

241

a proportionality factor. In the case of intense light fields, the quantity B

should be replaced by B' = 16AN2 s in 4 (0 2 /2)s in 2 (0 1 ) [21].

In a more general case when ut) is a Gaussian random process with an

arbitrary correlation function K(t), one can find K(t) on the basis of the

function Iz(t) [21]:

-ft -2r — In I3(t) = 2h~2[Kt) - 2K(t - t2). (22)

When £2 > Tc the second term on the right hand side of Eq.(22) makes

the primary contribution.

It follows directly from Eq.(21) that the maximum of signal h(t) corre

sponds to instant [17]

tmar = rcln[p2(2exp(^) - l ) / (p 2 + T^TC/2)\. (23)

The energy dependence of signal IC3 is equal to

/•CO in f„

Js(t2) = / ht)dt = TcBexp-Tf1*2 - 2 p 2 [ ^ + 2 e x p ( - ^ ) - 3]

x2p2[2 - e x p ( - ^ ) ] - ( T r 1 - . + ^ ) 7 [ T - i T c + 2p2,2p2(2 - e x p ( - % (24)

where 7(6,0) is an incomplete gamma function [105].

At the limit of fast modulation, which corresponds to a homogeneous

242

broadening (p < 1; t2,t — t2~> rc),

/3(t) = Bexp[-(T 1 - 1 + 270(,)i],

tmax _ , llad t2 ^2 — In ——i—;— ~r — ~ — i

To Iad + Tll2 TC T c '

Js(t2) = , B e x p K T f 1 + 2lad)t2], J-1 + ^7ati

where -jad = ap is the contribution of elastic (adiabatic) processes to the

phase relaxation.

At the limit of slow modulation, corresponding to inhomogeneous broad

ening (p > 1; t2,t - t2 < rc),

h(t) = 5 e x p ( - T f 4 )exp [ -a 2 ( i - 2t2)2],

Js(h) = ( S A / ( 2 a ) ) e x p ( ^ 7 - 2T~H2)

T - i

x[l + $(a i 2 1 "

2a "

As can readily be seen, a photon echo takes place in the latter case [71].

Fig.3 shows the signal I3(t) and Js(t2) in the case of intermediate modulation

( p = l ) .

Thus, as follows from the non-Markovian theory of SPE described in this

subsection, the methods of transient RFPS (together with the spectra of

single-photon absorption) make it possible to find the parameters of relax-

243

tf c

CM

- CJ y

/ / - / ^

^

/ 1 1

+

CJ -•—

1

t p / T ,

t ? / T r

Figure 3: Time response ^ ( i ) (a) and pulse energy J,(<2) for T j - 1 ^ -C p = 1;

£ — £2 = ro- The inset shows the instant of appearance of the maximum of

the signal /3(f) as a function of the delay time £2 of the second pulse for

TI1TC -C p2 [Eq. (23)]; the tangent 2t2JTc corresponds to the case of photon

echo.

244

ation perturbation, i.e., the modulation amplitude a and correlation time TC,

which, in the general case, determine the non-Markovian relaxation of the

system studied.

The model of the Gaussian-Markovian stochastic modulation for optical

dephasing has been used for the description of a non-Markovian relaxation

behavior in a number of two-pulse SPE experiments [72, 106, 59]. Fig.4

shows the result of the corresponding femtosecond experiment on resorufin in

dimethylsulfoxide (DMSO) by Wiersma et al. [72]. The solid line is a fit based

on the Gaussian-Markovian stochastic modulation model for optical dephas

ing. Wiersma et al. showed that optical dephasing of resorufin in DMSO

can be described by using a stochastic modulation model. With Gaussian-

Markovian statistics, both the femtosecond photon-echo experiment and the

steady-state absorption spectrum can be adequately simulated with the same

values for the stochastic parameters [72].

Saikan et al. [106] observed non-Markovian relaxation in photon echos of

iron-free myoglobin which was described by the Gaussian-Markovian stochas

tic modulation model for optical dephasing by using Eq.(20).

245

-40 0 40 80 120 pulse delay time (fs)

160

Figure 4: Photon-echo signal for resorufin dissolved in DMSO (dotted trace).

The solid line is a fit based on the Gaussian-Markovian stochastic modulation

model for dephasing, with parameters a = 41 THz and r - 1 = 27 THz [72].

246

4.2 Transient four-photon spectroscopy of near or over

lapping resonances in the presence of spectral

exchange

In Subsec.4.1 we considered non-Markovian relaxation effects in two-pulse

RFPS with the Gaussian random modulation of the optical transition fre

quency. In this case the four-time correlation functions describing a four-

photon light-matter interaction can be expressed in terms of the correlation

function K(t) (see Eqs.(18) and (19)). Another example of random modu

lation, which allows a detailed and general mathematical development, and

has important physical applications, is the Markovian modulation [9, 11].

Let us return to the oscillator whose natural frequency is randomly modu

lated (see See l ) . For the sake of simplicity we will assume that the oscillator

takes on any of two states a and b. The resonance frequency in states a and b

will be wo ±^t, respectively [107]. In this model, the frequency stays at some

value, say uia for a certain time T„, and changes suddenly to other value W&,

remains constant for time period TJ, returns to ioa and so on. The quantities

r~l and T^1 describe the phenomenon of spectral exchange in the system

which is related to a coupled damping of oscillators [9, 11-13, 107].

247

The spectral exchange strongly influences the shape of an absorption

spectrum of such an oscillator [9, 11,.108]. Let us introduce the time rc such

as T~X = T~1 + T^1. Then in the limit TC —y 0 the spectrum will be a sharp

line at the equilibrium average of the two frequencies coa and W&. As rc —> oo

two sharp peaks will appear at wa and w& [9].

The model under consideration is interesting in relation to a spectroscopy

of near or overlapping resonances [107]. The question of RFPS methods

establishing the presence or absence of spectral exchange in a system, is of

rather great current interest. Actually, one can distinguish two mechanisms

of formation of spectra from superimposed lines. In the first mechanism the

different lines belong to noninteracting transitions (for example, in different

centers) and each transition decays independently of the others. In the second

mechanism relaxation of transitions with different frequencies is connected

with transfer of excitation between them, due to which relaxation of the

transitions occurs in a coupled way. This is the case of spectral exchange. If,

for example, one turns to doublets in the spectra of polyatomic molecules,

the mechanism of line broadening due to spectral exchange is determined by

isomeric transitions occurring in one center [109-111], while the mechanism

of formation of doublets connected with the presence of different spatially

248

separated types of active centers, corresponds to the absence of spectral

exchange [112]. Thus, revealing the nature of the overlap of near lines is

of undoubted physical interest. To resolve the indicated question it has been

proposed to use two-pulse RFPS [107].

In the case being considered, the four-time correlation functions describ

ing a four-photon light-matter interaction can not be expressed in terms of

the correlation function Ki) (see Eqs.(18) and (19)), as it was for the Gaus

sian modulation. Therefore, for solving the problem we used Burshtein's

theory of sudden modulation [11]. We have calculated the intensity of the

SPE signal in the limit of short pulses [107]:

ht)~\P^+\2 = B'\R^\\ (25)

where in the case of equal times ra — T& = r0 the quantity |i?k3|2 c a n D e

represented in the form

l^ksl2 = ( 4 « T X e x p H T f 1 + 2r0'1)t)R2(t2)Sm

2(Kt + v(t2)), (26)

R(t2) = fifi2 + T~2 + 2r0-1[«;sin (2«i2) - T'1 cos (2/d2)]1 / 2 ,

s inp (t2) = Br1 (i2) [r0-2 - fx2 cos (2/ei2)], (27)

cos tp (i2) = -R-1 (t2) [KTO1 + /J,2 sin (2/rf2)], K = (A*2 - r 0- 2 ) 1 / 2 .

249

Consider, for the sake of comparison, two noninteracting resonances, the

frequency difference of which is 2A, decaying with the constant T1_1/2 (sys

tem without spectral exchange - wse). In this situation

IZ"t) ~ exp ( -Tf 4 ) sin2[Af + (TT/2 - 2At2). (28)

Eqs.(26),(27) for fj, > T^1 and Eq.(28) describe the beats of the intensity

of the signal k3, which can be used to reveal the hidden structure of single-

photon spectra. This aspect of the problem has been well studied in photon

echo spectroscopy [113]. Further, from a comparison of Eqs.(26),(27) and

Eq.(28) it is seen directly that they are characterized by different dependences

on t2 of the amplitude and phase of the corresponding signals. This can be

used to reveal the mechanism responsible for the overlap of near lines [107].

As a matter of fact, the zeroes of intensity of the signal IC3 are realized when

Kt + f(t2) = n7r for the case of the presence of spectral exchange, and when

At + 7r/2 — 2At2 = TITT in its absence (n is an integer). Let, for example, the

delay t2 be chosen in such a way that Is(t2) = ^3/se(^2) = 0 for t — t2. Denote

by t" the moment of time of the appearance of the next zero of intensity.

Then it is not difficult to show that in a system with spectral exchange

t" 1

— = l + 7r/arccos( ), (29) t2 \XTQ

250

while in the absence of spectral exchange is always t'^se/t2 = 3. When JJLTQ 3>

1 (the case of slow modulation), arccos( — l/(fiT0)) —>• n/2 and t"jti —> 3,

while for (J,T0 ~ 1 the ratio t"jti ^ 3. For example, if fir0 = \ /2 , then

t" / i 2 = 2 | . Thus, the nonstationary behavior of the SPE signal allows

distinguishing situations corresponding to the presence or absence of spectral

exchange in the system investigated.

It is worth noting that a manifestation of spectral exchange by a phase

shift between components is the charactestic feature of four-photon beats

spectroscopy. The proposed method of the establishment of the fact of the

presence of spectral exchange has been used in Ref.[114] devoted to transient

spectroscopy of coherent anti-Stokes Raman scattering (CARS) of thulium

(Tm) atoms in a buffer gas. In this experiment, a slowing of Doppler de-

phasing and a spectral exchange effect have been detected for the first time

in optical-range atomic spectroscopy. Ganikhalov et al. [114] observed the

quantum beats stemmed from the hyperfine splitting of the 4F5 /2 and 4F7/2

states of the thulium atoms. They found Tminlrmin = 2-6 ± 0.1 ^ 3 for

thulium in xenon (see Fig. 5), where rm'in are the delay times which deter

mine the positions of the first and second minima. Therefore, authors [114]

concluded that they were dealing with a manifestation of spectral exchange.

251

0

2 - 2

^ - 3

I"4

- 6

-

-

. 1

• Tmin

i

i i !

= 960ps

•

• • r * i n = 2500ps •

\ • •

3 r rn in=2880ps

• J

i i i

0 1000 2 0 0 0 3 0 0 0 4000 T.ps

Figure 5: Temporal response at a xenon pressure of 825 torr. Shown here are

the delay times r^'t2n, which determine the positions of the first and second

minima of beats of the hyperfine-structure components, and the delay time

3T1 • , which determines the position of the second minimum of the beats in

the absence of spectral exchange [114].

252

An examination of the non-Markovian effects on the relaxation processes

in the excited levels between which beating occurs has been also made in

Ref.[115].

4.3 Non-Markovian relaxation effects in three-pulse

R F P S

Let us consider three-pulse time-dependent four-wave-mixing experiments

(Fig.2a) [51-64]. A stochastic theory of these experiments for Gaussian ran

dom modulation of a frequency of an optical transition has been developed in

Ref.[116]. Let us consider the signal corresponding to wave k3 = k3 + k2 — kt

in the limit of short pulses. Then using Eqs.(13),(14),(17),(18) and (19),

one can show that the correlation function K(t) is obtained from the time

dependence of the signal power Is(t) [116]:

— In /.(<) = -2H-2K(t -T-T) + [K(t - r) - K(t). (30)

When T 3> TC, the first term on the right-hand side of Eq.(30) makes the

main contribution.

To carry out further calculations, we need to specify the form of the

correlation function K(t). We assumed that u(t) is a Gaussian-Markovian

253

random process [116]. Then using Eq.(20), one can show that the maximum

of signal Is(t) occurs at the instant time [116]

r T tmax = rclnexp(—)[1 +exp(—)] - 1. (31)

One can see that for T,T <C rc tmax = T + 2r, i. e. the stimulated photon

echo [51] is realized in this case.

Three pulse stimulated photon echo experiments [62-64] showed that the

echo peak shift, as a function of a delay between the second and the third

pulses, could give accurate information about solvation dynamics. This as

pect of the problem is covered in excellent review [34].

The energy dependence of signal k s is the following [116, 117]:

JS(T) ~ | / (r) |2exp(g)Q-2 p 27(2p2 , , ?) , (32)

where

q = 2p2l + exp( -T/ r c ) [ l - e x p ( - r / r c ) ] . (33)

Here / ( r ) = exp[gr(r)] is the characteristic function of the resonance absorp

tion spectrum F(ui — w2i) and is expressed by / ( r ) = /f^ F(LJ') exp(iw'r)rfw',

where o;2i is the frequency of the corresponding transition 1 —>• 2. It is well-

known that the characteristic function is the relaxation function that de-

254

scribes the relaxation of the response of a system after removal of the outer

disturbance [118, 9].

Interesting aspects of the influence of non-Markovian effects in a three-

pulse time-dependent four-wave-mixing experiment have been noted by Lavoine

and Villaeys [119]. They showed that the energy of the diffracted light can

be expressed as a square of the relaxation function / ( r ) :

Mr) ~ | / ( r ) | 2 , (34)

when the medium is excited by very short pulses. Their calculation does not

make assumptions about the analytical form of / ( r ) . For this reason it is

possible to consider result (34) as general, and this result is interesting from

the point of view of learning about the dynamics of the bath.

Eqs.(32) and (33) enable to define more precisely the conditions for the

correctness of Eq.(34) [117]. One can see that Eqs.(32) and (33) reduce to

Eq.(34) only for T » TC. That is to say, the delay time T of the probe pulse

must be much larger than the correlation time.

Further, in the slow modulation limit (inhomogeneously broadened tran

sition) the dependence JS(T) for T 3> rc is the following (Ref.[116], Eq.(17)):

JS(T) ~ exp(—a2r2). In other words, in this case, formula (34) does not

255

provide information about the correlation time of the interaction with the

surrounding bath. Such information can be obtained by the modification of

a three-pulse four-wave-mixing experiment to the population grating config

uration [Fig.2b] (see [120] and Subsec.7.1 below).

Thus, formula (34) is of value in the case of fast or intermediate modula

tion of the frequency of a transition under study when T 3> rc.

5 Non-Markov ian t h e o r y of s t e a d y - s t a t e R F P S

5.1 Introduction and the cubic susceptibility in the

case of Gaussian-Markovian random modulation

of an electronic transition

A non-Markovian theory of steady-state RFPS has been developed in Refs.[19,

16, 20, 21, 39, 121, 40-43]. In Ref.[19] only non-Markovian corrections to the

Markovian approximation were considered: a situation not appropriate to

the broad inhomogeneously broadened bands of dyes. In Refs.[16, 121] the

factorization approximation was proposed to calculate the cubic suscepti

bility. This approximation enables us to express the cross section for an

256

arbitrary multiphoton process in terms of ordinary single-photon line-shape

functions. It is exact in the Markovian limit, however it is not a good ap

proximation in the extreme non-Markovian case corresponding to the broad

inhomogeneously broadened dye bands.

A stochastic theory of steady-state RFPS describing in a continuous fash

ion, a transition from the Markovian limit (a homogeneously broadened op

tical spectrum) to the extreme non-Markovian case (an inhomogeneously

broadened optical transition), has been developed in Refs.[20, 21]. Eq.(13)

for the steady-state case can be written as

Vi3)+ = l E £xiLK)£m«>£m<c£w (35) mm'm" bed

where Xabidi^s) 1S ^ e c ubic susceptibility,

X$L(u.) = 2NL4h-3D:DbDeD-d)or £ Q(un,uj'm,^), (36) mm'tri"

the quantities Q (u>m, uj'm, w^) determine the frequency dependences of the

susceptibility. In the case of the Gaussian-Markovian random modulation

of an electronic transition, the quantities ( ^ ( w m , ^ , ^ ) have the following

form [20, 21]:

r\ I \ • 2 V~* P ft™ \xs) Q (u>m,um.,um.,) = -irc ^ — s^frp-i. , \

n=o n- i (wm - um>) + (7\ + n/Tc) x[i?n(a;m) + ( - l ) n

J R n ( a ; m 0 ] , (37)

257

where Xj - [T^1TC/2 + p2J + ircAwj; j = s, m, TO'; Aus = w2i - ws, Aum, =

u.'2i — wm/, Aa;m = wm — u>2i;

#„ (XJ) = n!$ (ra + l , i j + n + l;p2) [XJ (x3 + 1)... (XJ + n)]~\ $(n + l,Xj + n + l;p2)

is a confluent hypergeometric function [122].

Eq.(37) is convenient for calculations for the cases of fast (p <C 1), inter

mediate (p ~ 1) and also slow (p ^> 1) modulation [21]. In the last case its

convenience is confined to detunings

|fi| « («V) 1 / 3 (38)

(fi = wi — w2) as applied to spectroscopy based on a resonance mixing of

the Rayleigh type [35, 1, 36, 5]. For detunings T~1 < |fi| < (aVj"1)1/3,

\Q\ ~ Ifil-1/2. For larger detunings fi, the quantity Q can be calculated for

a more general and more realistic model of a complex molecule in a solution

than the model of Gaussian-Markovian modulation (see Subsec. 5.3).

5.2 Model for frequency modulation of electronic t ran

sition of complex molecule in solution

The effect of the vibrational subsystem of a molecule and a solvent on

the electronic transition can be represented as a modulation (a quantum

258

modulation, in the general case) of the frequency of the electronic transi

tion. According to Eqs.(18) and (19)), when the quantity u is Gaussian,

a four-photon light-matter interaction can be completely described by the

correlation function K(t) or the corresponding power spectrum $(w) =

(2w)~1 Jf^ A'(£)exp( —iu>t)dt, expressing the Fourier transform of K(t). It

is obvious that $(w) has maxima in the regions corresponding to the opti

cally active (OA) vibrations i.e., vibrations which change their equilibrium

positions when the electronic transition occurs (see Fig. 6). The molecu

lar electronic transition model under consideration includes two groups of

OA vibrations [39, 40, 101, 43, 120]: low-frequency (LF) (HUJS < 2kT) and

high-frequency (HF) (huh > kT). Accordingly, K[t) = Ks(t) + Kh(t) and

g(t) = ght) + 9st)- The corresponding contributions to the spectrum $(u>)

are $s(u>) and $A(U;): $ ( W ) = $s(w) + $/,(w). It follows from the relationship

$ ( - w ) = $(w) exp[-/»w/(fcT)] (39)

that the HF part of the spectrum $ji(w) is localized mainly in the region,

corresponding to the frequency of the HFOA vibrations: 1000 — 1500cm -1.

As to the OALF vibrations, the value of Ks(0) — /f^ $a(u)dw = h2a2s

is determined by the area included between the curve $4(w) and the LO axis

259

$saM

3>(OJ)

*sC | (w) <S>h(cu)

0 | q " ajh QJ

Figure 6: Approximate power spectrum of the vibrational perturbation for

the model of the electronic transition of a complex molecule in solution.

\q"\ = <?z,l<y2s = / ! t , - . -$ . (w)dw/ /^ 0 $, (w)du; . The methods of RFPS al

low us to find such parameters of the distribution $(w) as q1. T and q",

determining the vibronic relaxation of the optical transition. In the spe

cial case of shifted adiabatic potentials, cr^s — Hi Si coth[/iu;„72kT)u]2si,

\q"\=Y.iSi^Ja23.

260

(Fig.6). The quantity a2s is the contribution from the OALF vibrations to

the second central moment of the absorption spectrum. In the situation con

sidered, one can represent Ks(t) in the form of two contributions: a classical

Kaci(t) and a quantum Ksq(t): Ks(t) = Ksc\i) + Ksq(t) and, correspond

ingly, 4>s(w) = 3>sc/(w) + $ s?(w). The classical part (huis -C 2kT) corre

sponds to intermolecular motions and also to the quadratic electron-phonon

interaction [123, 39, 40]. The quantum contribution arises from the LF in

tramolecular vibrations [124, 43] and may stem from molecular librations,

for which huis ~ 2kT. It follows from Eq.(39) that the spectrum $SCI(LU) is

approximately symmetric relative to the frequency to — 0. An approximate

dependence $(w) is shown in Fig. 6 for the model being considered in this

part. Using the properties of the Fourier transform, one can show that the

correlation time for Kst) (which determines the characteristic decay time of

Ks(t)) is rs where rs_1 ~ max(q', \q"\) and \q"\ ~ LOS [43].

We consider that the condition of "strong heat generation" [123, 21, 39,

40, 43] (a2s S> ^) is realized for the LF system ws. If the system u)s

is purely classical, then the fulfillment of the latter condition is guaranteed

by the inequality hus <C 2kT. If the system ws is basically quantum,

then large shifts of the minima of the adiabatic potentials upon electronic

261

excitation (J2iSi 2> 1, where 5,- are the dimensionless parameters of the

shift) are necessary for fulfillment of this condition. Because of the inequality

<72aT* S> 1 in the case under consideration, there is a large parameter in the

exponents in Eq.(18). This makes it possible to limit the expansion of these

exponents to power series at the extremum points T\ = r3 = 0 with an

accuracy up to the second order terms with respect to T\ and r3 [21, 116, 39,

40, 101, 43, 120, 99]:

Kfc0, r3, n + T2 + T3, T2 + r3) = exp[G?(T1,T2)T3)], (40)

Kfc* (0, T3,T2 + r3, n + r2 + r3)

Kf c ( 0 , r2 + r3, T, + T2 + r3, T3) exp~i2Tjmgs(T2) + G?TUT2, r3)](41)

Kf c*(0, Tl + T2 + r3, T2 + r3, r3)

where

Gr(r1 ?T2 , r3) = - ^ [ r 2 + r 2 T 2r1r3(EeV's(r2) ± z / m ^ ( r 2 ) ) ] , (42)

gs(r2) = dgs/dT2 and T/>S(T2) = A's(r2)/A's(0) is the normalized correlation

function of the system ws. If the system HJS is classical, then the term

—2Imgs(T2) = ujst[l — T/>.J(T2)] describes the dynamical Stokes shift [120, 99]

where uost is the contribution of the LFOA vibrations to the Stokes shift

between steady-state absorption and luminescence spectra.

262

5.3 Cubic susceptibility for detunings larger than re

ciprocal correlation time

5.3.1 Classical system of LF motions

It follows from Eq.(13) that when detuning is \u>m — wm/| ^> rs_1, we need to

consider only the behavior of ij)a[T2) at small values of r2 . Then for the classi

cal system w3 (hua -C 2kT) we obtain an approximate analytic expression,

using Eqs.(40), (41),(42) [39, 40]:

Q(w r a ,w;,u;^)~-t/30-1 /(A)Mo), (43)

where /J,0 = [i(u>m - w3) + T^1]/'y/ff^, 01 = [i(um - wm ') + T^/q', q' =

—i?e0s(+O), Ref3o > 0; f(z) = ciz) sinz — si(z) cos z, ci(z) and si(z) are

the integral cosine and sine, respectively [122]. An approximate analytic

expression (43) practically coincides with the rigorous dependence in the

region important for comparison with experiment [43].

We will now consider the dependence |Q(w2,Wi,u>i)| of Eq.(43) (Fig. 7).

If

2|fi|3/2/(<72s<z')1/2 « 1, (44)

we find that \Q(W2,LO\,UI)\ OC Ifil -1/2 and reversal of the above inequality

263

[Q(aj2,cu,,ai, >[rel. unit*

5 10 I<JL»|—LU2I/C

Figure 7: Dependence of \Q(u2,u)i,Ui)\ (curve 1) on the detuning for the

LF system (s) compared with the experimental results of Ref.[5] (points) ob

tained for rhodamine B in ethanol when c = 182cm - 1 . The open circles do

not satisfy the condition \Cl\/q' > 1 and can not be compared with the theo

retical curve 1. Curve 2 is the theoretical approximation to the experimental

data of Ref.[5j obtained using two fitting parameters T[ and T2.

264

yields \Qu>2,uii,uJi)\ oc | f i | - 2 . The inequality of Eq.(44) is also the condition

of validity of the solution of Subsec.5.1 [see Eq.(38)], but the latter is not

limited by the condition \tom — iomi\Tc 3> 1. Therefore, in the range where

the solutions of Subsec.5.1 and this subsection are valid (this range is char

acterized by the dependence |Q(u;2,w1,a;1)[ oc | f i | - 1 ' 2 ) , we can match the

solutions. The fact that the characteristic scale of variation of |<5(oj'2, u;i, OJJ) |

with detuning wj — w2 is considerably greater than rc-1 can be explained

by the fact that, in addition to the Rayleigh scattering, a RFPS signal also

includes contributions from the processes of multiphonon Raman scattering

by the LF system (s) [21, 39, 40]. This follows directly from Eq.(37), which

represents expansions typical of the theory of multiphonon processes.

Experimental data [5] for a solution of rhodamine B in ethanol are in

satisfactory agreement with the present theory by using only one adjustable

parameter c = (iq'o^s)1 ' '3 = 182cm -1 (Fig.7), while the theoretical treatment

in Ref.[5] requires at least two fitting parameters T2 and T[ for the same

range1. Within the framework of the theory of this subsection, the ratio

T2/T ~ 1 reported in Ref.[5] (and also in the experiments on rhodamine 6G

xIn the case of rhodamine B one can take into account only the system (s) [39, 40].

265

reported in Ref.[6]) becomes understandable. The value of c just given and

an estimate of the half-width of the subband of rhodamine B in ethanol 5u> =

1070cm-1 (y/a^ = 454cm"1) of Ref.[125] yield q' « 60cm"1 (rs « 0.09psec).

The inclusion of the HFOA vibrations enables us to explain the increase,

observed in Ref.[5], of the frequency difference, for which \Q(U)2,LOI,UJI)\ OC

(UJI — w<i)~2, w i th increasing Ui towards the blue wavelengths [39, 40].

5.3.2 Quantum system of LFOA vibrations

When detuning is large (|wm — wm#| S> TS_ 1) , one can also obtain an analytic

expression for the quantum system u>s (hu>s ~ 2kT) [42, 43]:

Q^LJUUH) ~ att'^Kfro), (45)

where (32 = 9. + i2Y0)a^/3'', Reft > 0; a3s = id3gsq(0)/dt3 is the contribution

of the LFOA vibrations to the third central moment of the absorption spec

trum; I/Q = (T\ -1 — iCl)<r^s , To is the decay parameter of the zero-phonon

line [126]. Function f(z) has been determined in Subsec.5.3.1 and can be

represented in the following form for the case under consideration:

s-i ; /5 N • 2/3 r J. exp[(in - rr1)* P fP"o) = -io-3s / dx— —— —

Jo 21 o — iil + vzsx*)

= p-'iciipvo) s in(^o) - si(Pvo) cos(/3z/0)]. (46)

266

Eq.(45) is written for the special case 2r 0 — T^1 = 0 (see Refs. [42, 43] for

general case).

Let us discuss the dependence \QUI2,UI,UJ1)\, determined by Eq.(45).

For detunings (|H|3/|a-3s)1/'2 < 1, \Q(LJ2,U>I,UI)\ CC l^ l - 1 / 2 and reversal of

the above inequality yields \Q(u;2,u>i,iVi)\ oc | f i | - 2 for detunings fi > 0. On

the basis of the form of the integrand in Eq.(46), one should expect a more

rapid decrease of the quantity \Q(LO2,LOI,U>I)\ for positive detunings Q > 0

in comparison with the case of negative detunings fi < 0 since resonances

arise when Q, < 0, in particular when D, = —a3sx2. Such a behavior is

explained by the predominant contribution to the intensity of the us signal

from the multiphonon Raman processes when the excitation frequency is LO2

and scattering frequency is u>\ [43]. It is clear that for such processes the

corresponding probabilities will be larger for u>2 > U\. One can note the

related mechanisms leading to the discussed behavior of \Q(ui2,UJI,LOI)\ when

ui2 — u>i S> T3_1 and to the appearance of a red wing of the multiphonon

Stokes Raman scattering [24].

The aforesaid conclusions are illustrated in Fig. 8.

Experimentally, the effect has been observed for ethanol solutions of mala

chite green [44] (Fig. 9). Fig. 9 shows that the dependence \Q(u>2,u>i,u)i)\

I g j Q ( c a 2 , <JU ,, cu,) j

/ /

/

0

0 - I - I

i g d n i / c , )

\ \

\

0

267

Figure 8: Dependences of \Q(U2,(JJI,OJI)\ on the detuning for quantum LF

system (s) when Q > 0 (curve 1) and Q, < 0 (curve 2) for Toc^ = 0.25,

Tr1^3 = 0.1, d = <T2s\q"\)1/3 (Ref.[43]). |Q(wj,wi,wi)| is in arbitrary

units.

268

2 3

log ((lill/27rc)-cm)

Figure 9: The experimental behavior of |x^3 ' (n)| for a solution of malachite

green in ethanol (Ref.[44]). U1/2TTC) = 16077cm-1 for 1,2; wx/(2-c) =

17361cm -1 for 3,4; ft < 0 for 1,3 and ft > 0 for 2,4. |x ( 3 ) ( n ) l i s i n arbitrary

units.

269

is asymmetrical with respect to the sign of Q (the corresponding curves are

not parallel). The values of \Q(CJ2,^I,^I)\ for Q < 0 are larger than the cor

responding values of \Q(U>2,LOI,U)I)\ for fi > 0. This circumstance indicates

the presence of a nonclassical system of HFOA vibrations.

The structure observed near |fi|/(27rc) ss 230cm -1 corresponds to the vi

bration of 230cm -1 defining beats in the photoinduced changes in the trans

mission of the malachite green solution [79, 80, 101].

For a quantitative estimate of the vibronic relaxation parameters, the

experimental curve 2 (Fig. 9) was compared with theoretical dependence for

the classical LFOA system [43] (Fig. 10a) and Eq.(45) (Fig. 10b). With

the assumption about the greatest contribution to the RFPS signal of the

LF system (for excitation in the region of the 0 — 0 transition with respect

to the OAHF vibration) the following estimates were obtained:^' ~ 100cm -1

(rs ss 0.05psec), if one considers the LF system classical, and \q"\ = cr3s/a2s ~

50crn -1 (TS « O.lpsec), if one considers the LF system quantum. In these

estimates we used the value a2s » 500cm -1 , determined from the spectrum

of the single- photon absorption of malachite green in ethanol.

We note that the estimates obtained agree with the estimate TS ~ O.lpsec,

which was obtained by a treatment within the framework of the theory

270

> s X

S - 2

C3i O

- 3

"V

\

\ i

\

3 2

log[(|ft|/27Tc)-cnn]

Figure 10: Comparison of the theoretical dependences, for the classical LFOA

system [43]-a and Eq.(45)-b with the part of the experimental curve 2 of Fig.9

for detunings ft > 90cm - 1 . |x (3 )( f i)l is in arbitrary units.

271

Ref.[101] of the results of direct femtosecond experiments [79, 80].

6 Four-photon spectroscopy of super

conductors

Recently a number of papers have been published devoted to laser-induced

grating spectroscopy [45-49, 127, 50] and femtosecond ultrashort pulse spec

troscopy [128-135] of metallic and high- temperature superconductors. A

new method has been proposed in Refs.[45, 47, 48] for the investigation of

electron-phonon interaction in metals and superconductors on the basis of

laser-induced moving gratings. Shuvalov et al. observed a well-defined dip in

the nonlinear spectroscopy signal of superconducting Y-Ba-Cu-0 thin films

based on a biharmonic pumping technique [46, 49, 50] (see Sec.2). The upper

limit of the region of this dip corresponded to a value of 2A of the supercon

ducting energy gap.

It has been proposed in Refs.[128, 129] to use an impulsive stimulated

light scattering [136-139] in order to study the energy gap in superconductors,

since in the usual Raman spectrum the superconducting energy gap is very

272

weakly displayed. The corresponding method presents a real time optical

spectroscopy of superconducting-gap excitation, and therefore it is essentially

non-Markovian.

In Refs.[128, 129] the signal was been calculated in scheme corresponding

to an impulsive stimulated light scattering when ultrashort light pulses inter

act with a superconducting film (see Fig.2b; below we use r instead of T). In

this method the energy Js of the signal ks generated due to the four-photon

interaction of the type k s = k3 + kj — k2, is measured. The beats have been

predicted in the dependence of the energy Js on the delay time r of the probe

pulse k3 with respect to pump pulses ki and k2. The beats are due to oscilla

tions of the charge density, and their doubled period determines the value of

the superconducting band 2A. The examination was performed on the basis

of the phenomenological BCS model [140] for an isotropic superconductor

with a large correlation length.

The signal electromagnetic wave with vector potential As is generated by

a nonlinear current j^3 ' ( r , t) in the medium, created by waves Aj(j = 1,2,3):

A3r,t) = ( l /2)a$(t)exp-i[wt - k,(n + ib)r] + c.c, (47)

where |kj | = k = co/c. One can obtain for the positive frequency component

273

of the nonlinear current [128, 129]:

/ 3 > + (r, t) = j ( 3 ) + (t) V exp [iksr (n + *&)], (48)

where

J'(3)+ W = -JT^T^ / ^ [J" (* " a) + 7" (* + ^ ^ ( s A ) a3 (*) ^ P (-iut) 4 (27rfic) Jo

(49)

determines the time dependence of the current, the factor U describes the

nonlocal character of the interaction [128, 129],

Ip(t)=l-Re[a'1(t)a'2*(t)],

f (sA) = J0 (sA) Y0 (sA) - Jx (sA) Yx (sA),

Jn (sA) and Yn (sA) are Bessel functions of the first and second kind, respec

tively.

Solving the Maxwell equations for the signal wave with current (48) we

find the amplitude of the signal wave after the passage of the superconducting

sample of thickness /

A TI

as(l, t) = j(3)+(*)[/exp(«;/) - K'1 sinhUl), (50) CK

where « = — kb + ikn.

274

In the experiment it is convenient to record the energy Js of signal ks,

which is proportional to f dt\as(l,t)\2, as a function of delay r . In the case

of sufficiently short pump and probe pulses tp <C A - 1 , the time dependence

JS(T) is determined by the function / ' : JS(T) ~ / , 2 ( T A ) . This function for

large values of the argument has the asymptotic representation

/ ' ( r A ) ~ - [ 2 / ( T T T A ) ] COS(2TA).

The dependence / , 2 ( r A ) is shown in Fig.l l , in which the beats with fre

quency 4A are clearly seen.

Using Eqs.(49) and (50), one can obtain an estimate of the ratio of the

intensities of the signal and probe fields [128, 129]. For characteristic val

ues of the parameters n ~ b ~ 3, I ~ 10~5cm, A/UJ ~ 10 - 3 , this ratio is

\as(l,t)\2/\a'3(t)\

2 « l O - 1 5 ^ ^ , where Eit2 are the energy densities of the

pump pulses in terms of J/m2.

275

Figure 11: Plot of fn versus rA .

276

7 Ultrafast spectroscopy with pulses longer

than reciprocal bandwidth of the absorp

tion spectrum

The consideration of the transient RFPS up till now (see Sec.4) has been

based upon the assumption that exciting pulses are very short with respect to

all relaxation times in a system. In the case of inhomogeneously broadened

transitions, one must distinguish two dephasing times [98]: reversible (~

&2 ) a n d irreversible T" (T" > <rj ' ) where a2 is the second central moment

of the absorption spectrum. Therefore, for the electronic spectra of complex

organic molecules in solutions the shortest relaxation time corresponds to the

reversible dephasing of the electronic transition which is about equal to the

reciprocal bandwidth of the absorption spectrum (~ 10/sec). At present,

pulses of such durations are used in ultrafast spectroscopy [55, 60, 61, 76, 75,

62, 63, 59], and they provide unique information concerning ultrafast intra-

and intermolecular processes.

However, there are situations when the pulses long compared with a re

ciprocal bandwidth of the absorption spectrum have decisive advantages [81,

277

141, 142]. In the four-photon spectroscopy methods with pulses long com

pared with reciprocal bandwidth of the solvent contribution to the absorption

spectrum ( tv 3> a2a where <723 is the solvent contribution to the central

second moment of the absorption spectrum) [65-67, 81, 120], pump pulses of

the frequency u create light-induced changes in the sample under investiga

tion, which are measured with a time delayed probe pulse. Due to condition

— 1/2

tp S> (T2s , pump pulses have a relatively narrow bandwidth and therefore

create a narrow hole in the initial thermal distribution with respect to a

generalized solvation coordinate in the ground electronic state (Fig. 12) and,

simultaneously, a narrow spike in the excited electronic state. These distri

butions tend to the equilibrium point of the corresponding potentials over

time.

By varying the excitation frequency w, one can change the spike and the

hole position on the corresponding potential. The rates of the spike and

the hole movements depend on their position. The changes related to the

spike and the hole are measured at the same or another frequency u>\ by the

delayed probe pulse. Therefore, one can control relative contribution of the

ground state (a hole) and the excited state (a spike) to an observed signal. i In

This property of the spectroscopy with pulses tv ^> a2s can be used for

278

U

U2(QS)

U|(QS)

Solvat ion Cordinate •Q.

Figure 12: Potential surfaces of the ground and the excited electronic states

of a solute molecule in a liquid: one dimensional potential surfaces as a

function of a generalized solvent polarization coordinate.

279

the nonlinear solvation study, when the breakdown of linear response for

solvation dynamics occurs [81, 141, 142] (see Sec.9).

A theoretical description of the interaction of pulses comparable with

relaxation times is an essentially more complex problem than that of pulses

short with respect to all relaxations in a system. However, it is possible to

develop a method of solving of such problems even in strong electromagnetic

field (without using four-photon approximation) for pulses long compared

with the electronic dephasing [143, 91]. This issue will be considered in

Subsec.7.2.

We will also present recent experimental results obtained by pulses long

compared with the electronic dephasing [83] (Sec.8).

7.1 Theory of transient RFPS with pulses long com

pared with reversible electronic dephasing

We will use the general theory of Sec.3 for the model described in Subsec.5.2.

The transient nonlinear optical response strongly depends on the relations

between the intramolecular chromophore relaxation and solvation dynam

ics. Numerous experiments [144-147, 65, 148, 62] show that the Franck-

280

Condon molecular state achieved by an optical excitation, relaxes very fast

and the relaxed intramolecular spectrum forms within 0.1 ps. Therefore,

we shall consider that the intramolecular relaxation takes place within the

pump pulse duration. Such a picture corresponds to a rather universal dy

namical behavior of large polar chromophores in polar solvents, which may

be represented by four well-separated time scales [62]: an intramolecular

vibrational component, and intermolecular relaxation which consists of an

ultrafast (~ 100/s),l ~ 4ps, and 10 ~ lOOps decay components.

As to the interactions with the solvent, they satisfy the slow modulation

limit [99, 66, 120, 117, 86] in the spirit of Kubo's theory of the stochastic

modulation [9] (see See l ) :

T> 2 5 » 1 (51)

As a consequence of condition (51), times Ti and T3 (see Eq.(13)) become

fast [21, 39, 116, 40, 43]. Therefore, we can use Eqs.(40),(41),(42) and in

tegrate the right-hand side of Eq.(13) with respect to them if the exciting

pulses are Gaussian of frequency w [120]:

Smt) = S0exp[-52/2)t - tmf + iutm]

281

with pulse duration of

tp = 1.665/6 » <72~1/2 (52)

As a result, Eq.(13) is strongly simplified [120, 99]:

mm'm" oca

(53)

where xlfcL^) *> r2) *s the cubic susceptibility. It can be represented as a sum

of products of "Condon" X^Cc,*^^,^) a n < i a "non-Condon" Babcd'v(T2)

parts

x i L ( " , ' , r2) = ^ x j & a > , *, T2)B»bT

crr2) (54)

where indices a, v? of XFC a n d B%>Id show that the corresponding values are

related to nonequilibrium processes in the absorption (a) or emission ip) (for

more details see below). The "Condon" factors XFCa,v(wi *> r2) depend on the

excitation frequency u,t and r2, but they do not depend on the polarization

states of exciting beams. The "non-Condon" terms Babc^'vr2) do not depend

on to, but depend on r2 and the polarizations of the exciting beams. The

origin of the "non-Condon" terms Babc£,ip stems from the dependence of the

dipole moment of the electronic transition on the nuclear coordinates D i 2 (Q)

that is explained by the Herzberg-Teller (HT) effect i.e., mixing different

282

electronic molecular states by nuclear motions (see Sec.3).

We are interested mainly in non-Condon effects in solvation. Therefore,

we shall consider for simplicity high-frequency "intramolecular" vibrations

as Condon ones.

We consider the translational and the rotational motions of liquid molecules

as nearly classical at room temperatures, since their characteristic frequencies

are smaller than the thermal energy kT.

Here we do not consider the rotational motion of a solute molecule as a

whole. The corresponding times are in the range of several hundreds picosec

onds for complex molecules and are not important for ultrafast investigations

(< 10 — lOOps). In the ultrafast range such effects are only important for

small molecules. One can take into account the influence of the rotational

motion of an impurity molecule on P ' 3 ' by using approach [149].

7.1.1 Condon contributions to cubic susceptibil ity

At first, let us consider a case of one optically active (OA) intramolecular

vibration of frequency wo- Then the Condon contributions XFCa^i^i ^ ri) t °

the cubic susceptibility (54) can be written in the form [120]:

283

OO

x J2 In(So/s'mh6o)IkSo/s\nhdo)exp[-2Socoth.Oo + (n + k)60 n,k=—oo

Here So is the dimensionless parameter of the shift of the equilibrium point for

the intramolecular vibration u0 under electronic excitation, 0o = hui0/(2kT),

In(x) is the modified Bessel function of first kind [122], F^ (u — nui0 —

we/) = (2ira2s)~1/2exp[— (w — nw0 — ioei — us)/h)2/(2a2s) is the equilibrium

absorption spectrum of a chromophore corresponding to a n-th member of a

progression with respect to the vibration co0, us — W2s — Wis,

w(z) = exp-z2)[l + (2i/y/n) T exp(t2)dt] Jo

is the error function of the complex argument [122],

za,v = iS2[r2(2 + S(T2)) - *(3 + 5(r2)) + tm„ + tm, + tm(l + S(r2))] +

+u - wa ,v(r2) + Uo(Tk + n5(r2))/(2<T(r2))1/2 (56)

<r(r2) = a2s\ - S2(r2) + — [ 3 + 25(r2) + 52(r2)] (57)

is the time-dependent central second moment of the changes related to nonequi-

librium processes in the absorption and the emission spectra , at the active

pulse frequency u>,

uwfo) = ^ ± 'Y + S(T2)[U - (We/ ± ^ ) ] (58)

284

are the first moments related to the solvent contribution to transient ab

sorption (a) and emission (<f) spectra, respectively, uat — 2(us) is the sol

vent contribution to the Stokes shift between the equilibrium absorption and

emission spectra, fi2a2sSt) = (us(0)us(t)) — (u s)2 , S(t) is the normalized

solute-solvent correlation function, <T2S = h~2(ul(0) — (us)2) is the solvent

contribution to the second central moment of both the absorption and the

luminescence spectra. The terms w(za^) on the right-hand side of Eq.(55)

describe contributions to the cubic polarizations of the nonequilibrium ab

sorption and emission processes, respectively.

The third term on the right-hand side of Eq.(57) which is proportional to

S2/cr2s, plays the role of the pulse width correction to the hole or spike width.

This term is important immediately after the optical excitation when r2 w 0

and, therefore, Sfa) ss 1. The first term on the right-hand side of Eq.(56)

which is proportional to 52 ~ l/t2, takes into account the contribution of the

electronic transition coherence.

It is worth noting that Eqs.(53), (55),(56),(57),(58) describe in a contin

uous fashion a transition from the time frame in which coherent effects like

photon echo exist to the time range where reversible dephasing disappears

285

[120, 117]. For acting pulse durations tv satisfying the condition

<72~1/2 « tP « K/<x2*)1/3 = T' (59)

Eqs.(53), (55),(56),(57),(58) describe the effects of two-pulse and three-pulse

(stimulated) photon echo [120, 117]. For example, ignoring the vibration wo,

one can obtain from these equations at the specified conditions for two-pulse

excitation [120]: V^+ ~ exp[— (S2/6)(t — 2 T ) 2 ] , i.e., a photon echo appears

in the system. Here r is the delay time between the first and the second

pulses.

When

tv » V (60)

and the pump and the probe pulses do not overlap in time, one can ig

nore terms ~ S1 in Eqs.(56) and (57) [120, 117]. In the last case, Eq.(53)

can be used for any pulse shape, and the cubic susceptibilities Xabcd(UJ^^T'^)

and XFCa,<p(LJ,t,T2) in Eqs.(53),(54),(55) do not depend on time t, i.e. they

convert to usual steady-state susceptibilities. In this case, the signal ks =

km/ + km» — km only exists when pulses £mi and Sm overlap in time. In

other words, coherence effects associated with the reversibility of dephasing

disappear.

286

Thus, the quantity T" = (r5/cr23)1/'3 plays the role of the irreversible de-

phasing time in the system under consideration [120, 117, 91]. Such an inter

pretation of T" is consistent with the behavior of the four-photon scattering

signal excited by biharmonic pumping (see Eq.(44)).

7.1.2 Nonlinear polarization in a Condon case for nonoverlapping

pump and probe pulses

The consideration of Subsec. 7.1.1 is confined by the Gaussian character of

the value us = W2s — WXs. For nonoverlapping pump and probe pulses when

condition (60) is satisfied, a nonlinear polarization in a Condon case can be

expressed by the formula [150, 67]:

PNL+(r,t) =^ND12(-D21ET,t))i[Fa(LJ,u,t) - Fv(u>,u,t)]

+[$ a (w' ,w, t ) -$„(<" ," ' ,*) ] (61)

for any us. Here 3

E(r,t) = J2 Smt)expikmr), 771 = 1

/

OO

^ ' F Q , V M ( W ' ) ^ C , I ¥ . « ( W I - ujei ~ w', w, t) (62) -OO

287

are the spectra of the non-equilibrium absorption (a) or luminescence (ip) of

a molecule in solution,

1 r00

Fa>lfi3(ij',u,t) = — dT1fc,tV,s(Ti,t)exp(-iuj'T1) (63) Z7T J-oo

and

1 f°° Fa,vM(u') = — / ^ri/a,¥,M(T-i)exp(-2u;'T1) (64)

ZlT J-oo

the corresponding "intermolecular" (s) and "intramolecular" (M) spectra;

$a ,v(wi,u;,i) = 7r XP / du'-^j '- (6o) J — oo ( J — CJj

are the non-equilibrium spectra of the refraction index which are connected

to the corresponding spectra Fa^(u>i,ui,t) by the Kramers-Kronig formula,

P is the symbol of the principal value.

fa,VM(Ti) = TrM[exp(±(i /h)W2iiMTi) exp(^(i /h)W1>2MTi)pi,2M] (66)

are the characteristic functions (the Fourier transforms) of the "intramolec

ular" absorption (a) or emission (<p) spectrum [151],

Pi,2M = exp(-/?Wi,2M) /TrM exp ( - /W 1 ) 2 M )

is the equilibrium density matrix of the solute molecule,

fa,Vs(ri,t) = Tra[exp((i/h)u3Ti)pias(t)] (67)

288

are the characteristic functions of the "intermolecular" absorption (a) or the

emission (<p) spectra, pit2s(t) is the field-dependent density matrix of the

system describing the evolution of the solvent nuclear degrees of freedom in

the ground (1) or in the excited (2) electronic states. The latter magnitude

can be calculated by using the method of successive approximations with

respect to the light intensity [91, 150, 142].

The signals in the pump-probe and in the time-resolved hole-burning ex

periments are determined only by the non-equilibrium absorption and emis

sion spectra [150, 141, 142]:

AT(r ) ~ -u>[Fa(u>,u,T) - F^(UJ,U,T)] (68)

and

Aa(u')~-[Fa(u + Lj',u,T)-Fvu> + u]',u,T)] (69)

Eqs.(68) and (69) have been obtained for pump pulse duration shorter than

the solute-solvent relaxation time.

The formulae of this subsection are not limited by the four-photon ap

proximation because they are based on the approach of Refs. [143, 91] (see

Subsec. 7.2), which has been developed for solving problems related to the

interaction of vibronic transitions with strong fields.

289

7.1.3 Non-Condon terms

Let us consider the non-Condon terms in Eq.(54) for Xabidi^^i^)- They

have the following forms [99]:

BlT\T2) = J Jdpdv(aab(v)adc(t))orexp-2j2[(Ql3(0)) 3

x(^ 2 + v) + 2fijisjysjT2)) + i5mtfidsju3l - * „ - ( T 2 ) ) ] (70)

where m — a,<p; 5mvis the Kronecker delta,

aabv) = . M J dQsaab(Qs)exp(-i/Qs) (71)

is the Fourier-transformation of the tensor

<r«6(Q.) = Da12(Q3/2)Db

21(Qs/2), (72)

M i s the dimensionality of the vector Qs, $ s j(r2) = (Qsj0)QsjT2))/(Q2sj0))is

the correlation function, corresponding to coordinate Qsj- If this vibration

is an OA one, then the solvation correlation function S(T2) is related to the

correlation functions ^sj(r2). In the classical case this relation is [83]

S(T2) = J2"st,3ysj(T2)/ust (73) 3

where uistj is the contribution of the j-th. intermolecular motion to the whole

"intermolecular" Stokes shift wsi (iost = Ylj^st^)- S(T2) can be considered

290

as an average of the values ^ / (T 2 ) distributed with the density w s ij/u; s t . If

the non-Condon contribution is due to a non-OA vibration which does not

contribute to the Stokes shift wst, then \P(r2) is not related to S(T2).

The second addend in the square brackets in Eq.(70) describes the inter

ference of the Franck-Condon and Herzberg-Teller contributions. The value

of the parameter dsj can be expressed by the following equation [83]:

\dtj\ = (aw^)1 '2 /^ (74)

For freely orientating molecules, the orientational averages d'abv)<JdciJ'))or

can be expressed by the tensor invariants a0, hs and ha [99] (see Appendix

A). In the last case the values Babcj ;can be expressed by the values [99, 83]

-\-i8mifidsjVj(l - * s j(r2))] h.fav) (75)

ha(&, v)

related to the tensor invariants.

291

7.2 Nonlinear polarization and spectroscopy of vi-

bronic transitions in the field of intense ultrashort

pulses

The four-photon approximation used up till now is inadequate in a number of

cases. These are the application of intense ultrashort pulses to femtosecond

spectroscopy [152], the transmission of strong pulses through a saturable

absorber and an amplifier of a femtosecond laser and so on. In Refs.[143, 91]

the problem of calculating the non-linear polarization of electronic transitions

in a strongly broadened vibronic system in a field of intense ultrashort pulses

of finite duration, has been solved. This problem is of interest as it involves

two types of nonperturbative interactions: light-matter and relaxation (non-

Markovian) ones.

This problem is similar to that of calculating chemical reactions under

strong interaction [102, 153]. Let us consider a molecule with two electronic

states (Eq.(3)) which is affected by electromagnetic radiation of frequency u:

E(r, t) = -E(r, t) exp(-zu;i) + c.c.

One can describe an electronic optical transition as an electron-transfer re-

292

action between photonic 'replication' 1' of state 1 and state 22 (or between

state 1 and photonic 'replication' 2' of state 2) induced by the disturbance

V(t) = — r>2i-E(t)/2. The problem of electron transfer for strong interaction

has been solved by the contact approximation [102, 153], according to which

the transition probability is taken as proportional to S(Q — Qo) where Q0 is

the intersection of terms. The contact approximation enables one to reduce

the problem to balance equations.

A similar approximation can be used in the problem under consideration.

One can describe the influence of the vibrational subsystems of a molecule

and a solvent on the electronic transition within the range of definite vi-

bronic transition 0 —>• k related to HFOA vibration (f« 1000 — 1500cm -1) as

a modulation of this transition by LFOA vibrations ws (see Subsec.5.2). In

accordance with the Franck-Condon principle, an optical electronic transition

takes place at a fixed nuclear configuration. Therefore, the highest probabil

ity of optical transition is near the intersection Qo of 'photonic replication'

2 The wave function of the system can be expanded in Fourier series due to the periodic

dependence of the disturbance on time: ^x,t) = Y^aa Vnx,t) exp[—i(s + nu)t], where

<p„(x,t) is a slowly varying function. Photonic 'replication' 1' corresponds to the ground

state wave function for n = 1.

293

Figure 13: The adiabatic potentials corresponding to electronic states 1,2

and their photonic 'replications' 1', 2'.

294

and the corresponding term (Fig. 13) and rapidly decreases as \Q — Q0\ in

creases. The quantity u„(Q) = W-23(Q) — Wi3(Q) is the disturbance of nuclear

motion under electronic transition. Electronic transition relaxation stimu

lated by LFOA vibrations is described by the correlation Ks(t) = (us(0)us(t))

of the corresponding vibrational disturbance with characteristic attenuation

time TS (see Sec.3 and Subsec.5.2). <J2srs2 3> 1 for broad vibronic spectra

satisfying the 'slow modulation' limit, where <72s = Ks(0)h~2 is the LFOA

vibration contribution to a second central moment of an absorption spec

trum. According to Ref. [120], the following times are characteristic for

the time evolution of the system under consideration: a2s < X" -C rs,

where cr^ and X" = (T^/O^S)1 /3 are the times of reversible and irreversible

dephasing of the electronic transition, respectively (Subsec. 7.1.1). Their

characteristic values are a^s1/2 s» 10_145, V « 2.2 x 10 - 1 45, rs sa 10 - 1 3s for

complex molecules in solutions. The inequality rs 3> T" implies that optical

transition is instantaneous and the contact approximation is correct. Thus,

it is possible to describe vibrationally non-equilibrium populations in elec

tronic states 1 and 2 by balance equations for the intense pulse excitation

(pulse duration tp > T"). This procedure enables us to solve the problem for

strong fields.

295

7.2.1 Classical nature of the LF vibration sys t em and the expo

nential correlation function

We suppose that Huis <§C kT. Thus uis is an almost classical system

and operators Wns are assumed to be stochastic functions of time in the

Heisenberg representation. ua can be considered as a stochastic Gaussian

variable. We consider the case of the Gaussian-Markovian process when

Ks(t)/Ks(0) = S(t) = exp(—\t\/rs). Using Burshtein's theory of sudden

modulation [11, 102, 153], one can obtain the approximate balance equa

tions for the density matrix of this system when it is excited with pulses of

duration tp » (rs/<723)1/3 [143, 91]:

j f P j j (a',t) = (-l)jh-2 (ir/2)8(u21 -u> -a') \V21E(t)\2A(a>,t)+L33p33 (a',t)

(76)

where j = 1,2; a' = —u/H, u>2\ is the frequency of Franck-Condon transition

1 —y 2, A (a', t) = p\\ (a', t) — p22 (<*', t). The operator L33 is determined by

the equation:

Ljj — Ts \ , / r d d2

1 + (a - Sj2u.t) a/ysJ t i i N + ^2*^7-; r , 2 , (77)

da'-532ust) 'sda'-5j2Lost)2

5ij is the Kronecker delta, wst is the Stokes shift of the equilibrium absorption

and luminescence spectra. The partial density matrix of the system p33 (a', t)

296

describes the system distribution in states 1 and 2 with a given value at time

t. Eq.(76) corresponds to the contact approximation [153].

The complete density matrix averaged over the stochastic process which

modulates the system energy levels, is obtained by integration of pjj (a', t)

over a':

Pn(t) = Jft,K<) da'. (78)

The positive frequency component of the nonlinear polarization is expressed

in terms of A = pu — p22 [143, 91]:

P ^ + (t) = -±-ND12aa (u;21) (2rr<r2s)1/2 f drl (i - r ) A (u;21 - LO, t - r )

in Jo

x f drlD2l .E(t-n) £ exp-i<7 (r) r\ - i [Uj ( r) - w] n,(79)

where a (r) = a2s [1 — S2 (r)] is the time-dependent central second moment

of spectra, aa (w2i) is the cross section at the maximum of the absorption

band, / (t) is the power density of the exciting radiation, uij (T ) = w21 —

$jv>ust + (ui — w2i + $j<p<jjst)S (T) are the first moments of transient absorption

(j — a) and emission (j = ip) spectra. The quantity A (u;2i — u>, t — r ) is the

solution of the integral equation:

A' (t) = 1 - aa (OJ21 ) f drI(r)A'(T)R(t-r), (80) Jo

297

where A' (t) = A (w21 - to, t) / A (w2i - w, 0),

R (t) = [a (t) /<72s]-1 /2 Ej=a,<p e x p - [w - Uj (t)2 / [2a (*)] describes the con

tributions from induced absorption (j = a) and induced emission (j = if) to

A ' ( i ) .

Eq.(80) is the main result of the section devoted to intense pulses. As it

follows from Eq.(79), the distinction from four-photon calculations consists

in substituting the solution of Eq.(80) A (w21 — u>,t — T) for the equilibrium

value A (w2i - w, 0) = (27r<r25)~1/2 exp [- (w21 - w)2 / (2<r2s)] .

The solution to Eq.(80) by Pade approximant [0/1] [154] is [143, 91, 155]:

A' (t) = [l + <ra (u>21) f drl (T) R(t- r ) ] " 1 . (81) Jo

This solution does not practically differ from the exact one, even at a compar

atively large saturation parameter <ra (w2J) Imaxtp ~ 1, when the perturbation

theory does not hold [155].

Formulae (79), (80), (81) solve the problem of calculating a nonlinear

polarization of the system under study in the field of sufficiently intense

ultrashort pulses whose intensity is confined by the condition aa (tu21) 7max <C

(T ' )" 1 .

Using Eq.(81), one can find criteria for the necessity of taking into ac-

298

count the saturation effect. For long pulses 2tp S> rs it has the usual

form: aa (W21) 4ax ~ tp1- However, for sufficiently short pulses (2tp <

rs) saturation is realized for essentially smaller intensities: aa (W21) /max ~

(2<pTs)-1'2. In the latter case, due to inhomogeneous broadening, the sat

uration is reached in a range narrower than the width of the equilibrium

absorption spectrum.

Eq. (79), which is linear with respect to E(t — T\), is correct for any

duration of the pulse corresponding to this field. If the weak probe pulse is

merely a copy of the pump pulse shifted in time, as it was in the transmission

pump-probe experiments [79, 80] (see Sec. 2) then the imaginary part of

the positive frequency component of the total polarization (not only of its

nonlinear part) has the following form [91]:

ImP+ (t) = Im[ND12p2i (<)exp(tut)] = -^NT)12IT [ D 2 I • E(<)] A(w21 -u,t).

(82)

This quantity defines an absorption of the field E(t).

Using the developed theory we have generalized the four-photon approx

imation theories [86, 156] of the time-resolved hole-burning experiment for

the case of sufficiently intense pump pulses [91].

299

7.2.2 General case. Quantum nature of the LF vibration system

In the general (non-classical and non-Gaussian) case, one can also reduce

the problem under consideration to the solution of equations (operator ones)

for the populations of electronic states [91, 150, 142]. Using these equations

strongly simplifies the problem, because they may be solved to any order n

with respect to the quantity |D2i • E\2, which is proportional to the light

intensity. As a result, the polarization P + ( i ) may be calculated to any order

2n + 1 with respect to the acting field. For example, the cubic polarization

can already be calculated by solving the population equations only to the first

order with respect to ID21 • E\2. However, one can not obtain in the general

case a closed equation for the averaged population difference like Eq.(80). It

was possible only due to the Markovian character of the modulating pertur

bation in Subsec. 7.2.1.

For a chromofore molecule in a solvent a nonlinear polarization can be

expressed by the formulae of Subsec. 7.1.2.

300

8 Experimental study of ultrafast solvation

dynamics

Recently, time resolved luminescence (TRL) and four-photon spectroscopy

have been applied to probe the dynamics of electronic spectra of molecules

in solutions (solvation dynamics) [157-163, 64, 164, 31, 55, 60, 61, 76, 75,

62, 63, 65-67, 81]. In TRL spectroscopy a fluorescent probe molecule is elec

tronically excited and the fluorescence spectrum is monitored as a function of

time. Relaxation of the solvent polarization around the newly created excited

molecular state led to a time-dependent Stokes shift of the luminescence spec

trum. Such investigations are aimed at studying the mechanism of solvation

effects on electron transfer processes, proton transfer, etc. [157-159, 162-164].

In this regard it is worth noting the works by the Fleming's and Barbara's

groups on observation of ultrafast (subpicosecond) components in the sol

vation process [162-164, 157, 144, 165] and systematic studies of solvation

dynamics by Maroncelli and others [166, 148]. The experimental efforts were

supplemented by results of molecular dynamics simulations and the theory

by Maroncelli and Fleming [158, 167], Neria and Nitzan [168], Fonseca and

Ladanyi [169], Perera and Berkowitz[170] and Bagchi and others [171-173].

301

The four-photon experiments were carried out with both very short pump

pulses (pulse duration tv ~ 10/s) [55, 60, 61, 76, 75, 62, 63, 59] , and

pulses long compared with reciprocal band%vidth of the absorption spectrum

and irreversible electronic dephasing T' (tp ~ 100/s) [65-67, 81] (see also

[174]). Photon echo measurements which were conducted with former pulses

in Shank's, Wiersma's, Fleming's groups and by Vohringer and Scherer, pro

vided important information on solvation in the condensed phase [55, 60, 76,

61, 75, 62, 63]. For example, three pulse stimulated photon echo experiments

[62-64] showed that the echo peak shift, as a function of a delay between the

second and the third pulses, could give accurate information about solvation

dynamics.

Recently an excellent review [34] has been published devoted to photon

echo and fluorescence Stokes shift experiments. Therefore, we will concern

ourselves here with four-photon spectroscopy with pulses tv 3> cr2s • We

have already discussed the potentials of this spectroscopy in Sec. 7.

As an example we will consider the resonance heterodyne optical Kerr-

effect spectroscopy of solvation dynamics in water and D20 [83].

302

8.1 Introduction

Recently, interesting results have been obtained concerning the ultrafast sol

vation dynamics in liquid water [175, 144, 164, 176, 173, 177]. It was found,

experimentally [164], by use of molecular dynamical simulations and theory

[176, 173, 177] that the solvation of a solute molecule (or ion) in water is

bimodal. The solvation correlation function is Gaussian at short times and

exponential at long times. Solvation studies are of great importance, since

the time response of solvent molecules to the electronic rearrangement of a

solute has an essential influence on the rates of chemical reactions in liquid

[164, 178] and, particularly in liquid water.

A question arises when and if the solvation dynamics of a solute in deuter-

ated water is similar to water [173]. The Debye relaxation time, measured

by the dielectric relaxation technique for D20 is slower than that for H20 at

the same temperature [179]. Deuterated water is a more ordered liquid with

a stronger hydrogen bond compared to normal water [180]. It was predicted

that a significant isotope effect may be observed in ion solvation of normal

and deuterated water in a (sub)picosecond range [173]. It was reported in

Ref.[165] (see also Ref.[144]), that a small isotope effect exists in water for

303

the longitudinal relaxation time.

Using the resonance heterodyne optical Kerr-effect technique [82, 81, 181]

we studied the solvation dynamics of two organic molecules: rhodamine 800

(.R800) and 3,3-diethylthiatricarbocyanine bromide (DTTCB) in normal and

deuterated water in femto - and picosecond ranges [83, 84]. We found a

rather significant isotope effect in the picosecond range for i?800, but not for

DTTCB. We attribute the i?800 results to a specific solvation in rhodamine

800 due to the formation (breaking) of an intermolecular solute-solvent hy

drogen bond. Another important aspect of this study is that the solvation

correlation function is bimodal with an ultrafast femtosecond component

< 100/5.

8.2 Calculation of H O K E signal of #800 in water and

D20

Here we will apply the general theory described in Subsec. 7.1 to the calcu

lation of the HOKE signal of RS00 in water and D2O.

Let us consider the spectra of 72800 in water and other solvents (Fig. 14).

304

1.0-

-0.5-

I 0.0-1.2x10* 1.5x10*

cm -1 1.8x10*

1.0

1

CO

0.0

• • c ^

1.2x10* 1.5x10* cm -1

1.8x10*

Figure 14: Absorption (1) and emission (2) spectra of i?800 in water (a),

D20 (b), ethanol (c), acetone (d), propylene carbonate (e), and dimethyl

sulfoxide ( / ) .

305

This molecule has a well structured spectra which can be considered as

a progression with respect to an OA high frequency vibration ~ 1500cm -1

[125]. The members of this progression are well separated, and their am

plitudes rapidly attenuate when the number of the progression member in

creases (as one can see from Fig. 14, the amplitude of the third component

is rather small). Such behavior provides evidence of a small change of the

molecular nuclear configuration on an electronic excitation. In other words,

the Franck-Condon electron-vibrational interactions in rhodamine molecules

are small. The resonance Raman scattering studies of rhodamine dyes [182,

183] display intense lines in the range of ~ 1200 — 1600cm -1 and the lowest-

frequency one at 600cm -1 in both alcohol and water solutions. Therefore,

one can assume that the intramolecular vibrational contribution to the line

broadening of the i?800 in water in the range between the electronic tran

sition frequency u>e[ and the first maximum is minimal. In our experiments

the excitation frequency corresponds to this range to = 13986cm -1).

Let us discuss the interactions with the solvent. Bearing in mind our

comments concerning the role of the intra - and inter-molecular interactions,

we can assume that criterion (51) is correct for the first maxima in both the

absorption and luminescence spectra of the i?800 in water. In the last case,

306

a2s is the central second moment of the first maximum.

The criterion (52) is also well realized in our experiments, since tp ~ 100/s

(in the first series of our measurements tv w 150/s) and a2s ~ 14/s.

Let us discuss the role of non-Condon effects for i?800 in H20 and D20.

The absorption spectra of i?800 in water and D20 differ from the correspond

ing spectra in other solvents (Fig. 14). Solvents like H20 and D20 influence

the relative intensities of spectral components in the absorption band. Such

behavior can be described by the dependence of the dipole moment of the

electronic transition D12 on a solvent coordinate Di2(Q s) [99], i.e. by the

non-Condon effect. Thus, the electronic dipole moment dependence on a sol

vent coordinate must be a necessary component of our consideration. More

over, the i?800 absorption spectrum in D20 differs from that of H20. The

substitution of H by D influences the absorption spectrum shape. Therefore,

one can assume that the dependence Di2(Q s) is determined by the solute-

solvent H-bond in water. The analytical form of the Di 2 (Q s ) dependence is

determined by invoking a specific model for the interaction.

Let us consider the HOKE signal for the LO phase ip = 0 (Eq.(7)).

Bearing in mind Eq.(55), we can write the imaginary part of the Condon

307

contributions XFCa,v(UJ^^T2) m the form:

IrnX$Lju,t,T2) = - (2 7 r 3 ) 1 / 2 iVL 4 r 3 e^( - r 2 /T 1 ) (<T(r 2 ) ) - 1 / 2

xFlsa(Lo-uel)Rewza^) (83)

where we insert tm = 0 and im» + tmi = r in Eq.(56) for zatV.

The last equation corresponds to a case where only the first maxima of the

absorption and the emission spectra are taken into consideration (n = k = 0).

This simplification is justified due to the specific relative position of the

excitation frequency OJ with respect to the rhodamine's spectra.

The cubic polarization for the HOKE experiment (Y is the signal polar

ization axis, the probe pulse polarization is along the X axis and the pump

pulse is at 45° with respect to both X and Y) can be written in the form (see

Eqs.(13),(54),(70),(75) and Appendix A):

^ 3 ) + C ) = IT, fy^xPcaJ^t,r2)i\B^(r2)\£2x(t-r2)\2

o a<v JO o

x£3x(t - r ) + [BS«(TI) + ^Bf«(T2) - \BZ«(T2)]

x£ix(0£te(* - r2 - r)£*2xt - T2) (84)

For subsequent calculations we ought to choose a concrete dependence of

D(Q. ) .

308

When the dipole moment Di2(Q s) changes its direction only but preserves

its modulus [99, 66] (see Eqs. (125),(126),(127),(128) below), the values B^a

are given by the following equations:

B» = Bt = D40/% Ba

a = B* = 0; (85)

Bim)(r2) = Dt/2)\ + e x p [ - £ r ;2 ( l - *sj(r2))]

i

x cos[5mv £ rAl ~ ^s3r2))l3nojsttJfl2) (86)

3

where ry = 2ajJQlj(0) are constants characterizing the correlations of the

vector D2 i with the j'-th intermolecular vibration, D0 = |£>2i|, ^st,j is the

contribution of the j'-th intermolecular motion to the total "intermolecular"

Stokes shift ujat (iost = Ylj^atJ )> ^sjfa) is the normalized correlation func

tion, corresponding to the j'-th intermolecular vibration which is related to

the solvation correlation function S(T2) by Eq.(73). It is worth noting that

the cosine term on the right-hand side of Eq.(86) for Bf describes the interfer

ence of the Franck-Condon (dynamical Stokes shift) and the Herzberg-Teller

relaxation dynamics.

309

8.3 Method of data analysis

Our aim is to determine the solvation correlation function by resonance

HOKE spectroscopy. According to Eqs.(7),(56),(57), (58),(83),(84) we need

to know, for this purpose, the following characteristics of the steady-state

spectra: uei and the solvent contribution to the Stokes shift between the

equilibrium absorption and emission spectra Lost. The latter is related to the

solvent's contribution to the second moment a2s by the relation: u>st = /J/3<J2S.

One can determine uei as the crossing point in the frequency scale of the equi

librium absorption and emission spectra of R800 (wej = 14235cm -1 for water

and is about the same for D2O).

The solvent contribution to the central moment <T2S can be determined

by the relation SO, — 2\/2<72s In 2 where SO, is the half-width of the first

absorption maximum. In order to exclude from our consideration the con

tribution of the second maximum and the optically active vibration of the

frequency ~ 600cm -1 , we determined 5Q as twice the distance (in the

frequency domain) between the luminescence maximum and the right-hand

side half maximum of the first luminescence maximum. Using this method,

we obtain <r2s = 115416cm-2 for the water solution and therefore ujst =

310

H3a2s = 550cm -1 , which conforms with the experimentally measure value.

For D20 the relation uist = hj3cr2s is an approximate one, and we suppose in

this case that a2s = 123900cm-2 .

Bearing this in mind, we fit our experimental data by Eqs.(7),(56),(57),(58),

(73),(74),(83),(84), (85),(86). We present the correlation function S(r2) in

the form of a sum of a Gaussian and one or two exponentials:

2

S(T2) = af exp[ - ( r 2 / r / )2 ] + J2 «i e x p ( - r 2 / r e i ) , (87)

f=i

where aj + J2iai = 1; Te2 is the decay time of the slow (picosecond) expo

nential. We relate it to the solute-solvent H-bond , and therefore connect

the correlation function for the "non-Condon" intermolecular motion on the

right hand side of Eq.(87) with this exponential:

<Js,(r2) - exp( - r 2 / r e 2 ) (88)

Comparing Eqs.(87) and (73), we can express the value u>stj in Eq.(73) by

parameters a/,a2 and u>st :

wst,j = (1 - af - a^Ust (89)

Correspondingly, the fitting parameters are a,f,ai,Tf,Tei,Te2 and r2 = r2.

311

The pulse duration tp in our experiments is tp « 70 —150/5, depending on

the laser excitation wavelength. In the case of ultrafast OKE experiments,

the decay time 7\ in Eq.(83) is replaced by the orientation relaxation time

ror of the solute molecules, if the latter is shorter than T\ . For rhodamine

dyes Ti ~ 1 - 2ns > ror ~ 150ps. We multiplied the experimental data

by the factor exp(r/ro r) and compared the theoretical and the experimental

data for delay times r < Tor « 150ps. Fig. 15 shows the computer fit results

of the experimental data of i?800 in H20 and D20. The fit of the theoretical

calculations to the experimental curves is good. The insert in Fig. 15 shows

the solvation correlation functions S(t) of RS00 for H20 and D20 found by

the computer fitting procedure.

We also carried out the corresponding measurements for i?800 in water

at different excitation frequencies u> (Fig. 16a). Fig. 16b shows theoretical

spectra for different excitation conditions, i.e. w and tp for rhodamine 800

in water. We used the same parameter values of the previous fit (Fig. 15)

for curves of Fig. 16b. One can see that the theoretical curves reproduce all

the fine details observed in the experiment (in particular, the decrease in the

amplitude of the slower signal component for "blue" excitations).

312

1.0

^0.8-

.N

o

'I 0.4 CD

0.6

I 0.2

0.0

-

•

~

™*

•

-

*

-

•

.

'

'

• I

\ 1

-

r

5 0.4-

: 0.2-

\ °-°;

V ^

^ " S .

1

\ ^ \ _ ^ 2

1 ^ ^ ~ ^ ~ _

3 5

Time [psj

2

*K- * ^ ^ * ^ ^ f c _ _

1

10

1

0 5 Time [ps]

10

Figure 15: HOKE signals for #800 in water (1) and D20 (2).

Dots and diamonds - experiment, solid lines - computer fit using

Eqs. (7)>(56),(57),(58),(73),(74),(83),(84),(85),(86),(87),(88),(89) for t, =

150/5, r2 = 2.5, Tf = 85/5; a = 0.6 (1) and 0.44 (2), aj = 0 (1) and

0.156 (2), rei = 146/a (2), re2 = 6.8ps (1) and lOps (2). Insert - solvation

correlation functions for H20 (1) and D20 (2).

313

1.0

0.8

0.6 "en

CO

S 0-4

CO

§i 0.2 CO

0.0

2

0 7

7//77e [ps]

Figure 16: Experimental (a) and calculated (b) HOKE signals for #800 in

water for different excitation frequencies ui and pulse duration tp.

a: u = 13755cm"1. tp = 125/3 (1); u = 13550cm"1, tp = 100/a (2).

b: us = 13831cm"1. tv = 130/5 (1); u = 13441cm"1, tp = 90/s (2).

314

8.4 Discussion

The correlation solvation functions for R800 in water and D2O consist of two

main components: an ultrafast Gaussian one with TJ ~ 85/s < 100/s, and a

slow one with an exponential decay of a few picoseconds. Only a small part

of the fast signal component can be explained by the coherent spike. The

main contribution to it is due to the hole burning effect.

The amplitude of the Gaussian component is about 60% for water, and

the sum of a Gaussian and a fast exponential for D20 is also 60%. This

value is close to that observed by Fleming et al. (~ 50%) for coumarin 343

solvation in liquid water [164]. Its duration (85fs) is about 1.7 times longer

than that observed in Ref. [164]. The large difference can be explained as

follows. The solvation, observed in Ref. [164], has been interpreted as an ion

one [173]. According to Ref. [173], the dipole solvation is slower than the ion

one. Therefore, if in the case of #800, the solvation is due to dipole or higher

multipole interactions, its fast component is slower than that of ion solvation.

The fast exponential of 146/s for D2O corresponds to that observed for a

water solvation in Refs.[144, 164].

Let us consider the slow components of the correlation functions for H^O

315

and D20 (Eq.(87)) (re2 = 6.8ps for H20 and re2 = lOps for D20). They

are close to the Debye relaxation times TQ for these solvents (8.27ps and

10.37/JS, respectively [179]). Such long components have not been observed

in recent studies of solvation dynamics of other solutes in water [144, 164].

We interpret our observations as a specific solvation related to formation (or

breaking) of an intermolecular solute-solvent hydrogen bond between i?800

and water molecules. The situation is similar to that observed by Berg and

coauthors [184, 185] on specific solvation dynamics of resorufin in alcohol

solutions. In hydrogen-bonding solvents, the longest component of the Debye

dielectric relaxation is assumed to be related to the rate of hydrogen-bond

reorganization of the solvent [185-188]. According to Ref.[186], time TQ may

reflect translation in water. In computer simulations the autocorrelation time

of hydrogen bonds in water is 5 — 7ps [185, 189]. Thus, the assumption that

the slowest solvation is related to the reorganization of a hydrogen bond,

seems rather plausible.The experimental data for R800 show a significant

isotope effect in water (~ 32% for times r e 2) , in contrast to study [184] in

which an isotope effect in deuterated ethanol was not observed. It would be

expected in view of the larger number of H-bonds that water makes [190].

The hydrogen-bond formation (or breaking) assumption correlates with

316

occurrence of non-Condon effects. The dependence D(Q) is essential for a

large change in Q. This is the case of hydrogen-bond formation (or breaking)

where a large Q is accompanied by a large hopping distance (3.3A for water

[186]) and a small activation energy.

In Fig. 17 the HOKE data for DTTCB solution in water and D20 are

shown. These data reflect only fast dynamics of solvation (the non-specific

one) and do not show any significant isotope effect.

In conclusion, using the time resolved HOKE technique, we have studied

the ultrafast solvation dynamics of i?800 and DTTCB in water and DiO.

According to our findings, the time dependence of the HOKE signal for

i?800 at the frequency domain under consideration, is determined mainly

by solute-solvent interactions. The significant change in the HOKE signal

during the first ~ 100/s is determined largely by the transient hole-burning

effect. A biphasic behavior of the solvation correlation function is essential

for obtaining a good fit with the experimental data. The fast component

of solvation dynamics for both #800 and DTTCB is determined by the

non-specific solvation. The slowest component for i?800 (which is close to

the Debye relaxation time) is determined by a specific solvation related to

formation (or breaking) of an intermolecular solute-solvent hydrogen bond.

317

. "O a: . N

*^5 03

t O ^ ^

"55 0

*+«*.

^

^^, Ct5

CO

I.U

0.8

0.6

0.4

0.2

n n

\

-

•J

\ \ \ \

\ \

\ \ ^* .

* v .

^ - ^ _ ^ ^

1 • • ,

0 5

Time [ps]

10

Figure 17: HOKE data for DTTCB solutions in water (solid line) and DoO

(dotted line); tp = 70/3, w = 13330cm-1.

318

Correspondingly, we observe a significant isotope effect for the R800 solution,

and do not observe an isotope effect for DTTCB, which does not seem to

form a solute-solvent hydrogen bond.

9 Prospect: Spectroscopy of nonlinear

solvation

In this section we will discuss the advantages of transient four-photon spec

troscopy with pulses longer than the electronic transition dephasing. We will

show that it can be used for the nonlinear solvation study, i.e., when the

linear response for the solvation dynamics breaks down.

As has already been intimated in the beginning of Sec.7, one can control

relative contribution of the ground state (a hole) and the excited state (a

spike) to an observed signal by changing excitation frequency ui. This prop

erty of the spectroscopy with pulses long compared with electronic dephasing

can be utilized for the nonlinear solvation study.

In the last few years, much attention has been given to the problem of

nonlinear solvation [191-193, 169, 194-198]. In the case of linear solvation

319

the spike and the hole motions can differ only by initial conditions of the

excitation. However, in the case of nonlinear solvation when the field created

by a solute strongly changes during electronic excitation, relaxation of the

solvent polarization occurs under conditions which are essentially different

from those of the initial electronic state, and the spike and hole motions

strongly differ irrespective of the initial conditions of excitation.

A molecular dynamics simulation study of solvation dynamics in methanol

[169] and polyethers [195] showed the breakdown of the linear response theory

for this process. Solvation dynamics processes in the ground state of a solute

differ from those of the excited electronic state [169]. In this section we

will show that spectroscopy methods considered in Sec. 7, allow to obtain

separate information concerning solvation dynamics of a solute in the ground

and in the excited electronic states, and therefore, enable the study nonlinear

solvation.

The aim of the theory is to relate the signal obtained in transient spec

troscopy measurements to the solvation characteristics. There are four-time

correlation functions in nonlinear spectroscopy where the system evolution

is determined in the excited electronic state, and the thermal averaging is

carried out in the ground electronic state [31, 67, 199, 78, 32, 150]. Apply-

320

ing stochastic models to the calculation of such correlation functions, results

in missing any effects connecting with the situation when the chromophore

affects the bath, in particular, the dynamical Stokes shift (see, for exam

ple, review [32]). Indeed, the electronic excitation of a molecule results in a

situation where the solvent configuration does not correspond to the upper

electronic molecular state. The solvent (bath) is forced to relax to a new equi

librium configuration. It is precisely this reverse influence of the molecule

on the solvent (bath) that cannot be taken into account in the limits of

the stochastic approach. The latter makes the use of stochastic models in

nonlinear spectroscopy of solvation dynamics meaningless, because the main

effect of solvation is the Stokes shift. This is unfortunate since the stochastic

models enable taking into account, in a simple way, many features of nuclear

dynamics.

However, it is possible to overcome this difficulty if one takes into account

the change in molecular electronic states before using the stochastic approach

[141, 142]. One can express four-time correlation functions by the ones in

which the thermal averaging is carried out in the same electronic state as

the system evolution (equilibrium averages). In the last case, there is no

change in the electronic molecular states while using the stochastic approach,

321

and therefore applying the stochastic models will not result in missing the

dynamical Stokes shift. We shall use such an approach in this section and

investigate the influence of the intramolecular spectrum on the transient

spectroscopy signal (TRL and RTFPS with pulses long compared with the

electronic transition dephasing).

9.1 Four-time correlation functions related to defi

nite electronic states

In the case of nonlinear solvation, us = W2s — W\, is not a Gaussian quantity

due to different solvation dynamics in the ground and in the excited electronic

states. Therefore, we will use a non-Gaussian formulation of the nonlinear

polarization of Subsec.7.1.2. We can represent the formula for the charac

teristic functions of the "intermolecular" spectra fa,va(Ti,t) (see Eq.(67)) in

the four-photon approximation in the following form [141, 142]:

fj.(ri,t) = ^ f dr2\B21E(v,t - r 2 ) | 2

4/z Jo

/

oo dT3fa,M(T3) exp[- r 2 /Ti + ir3(w - u^Mjfa, r2, r 3 ) , (90)

-OO

322

where j = 1,2; a = 1, ip = 2;

M ^ n , r2, r3) = Tr s exp[ i ( U i ( r 2 ) r 1 - « j(0)r3)]/>L, (91)

the index V denotes the equilibrium state,

i i uATi) = exp ( -Wj r 2 )u s exp( - -W^T 2 ) .

Averages (Eq.(91)) can be found using either classical or stochastic ap

proaches. In the classical analysis [32] we can calculate ui,2(r2), if we find

the classical trajectories Q I , 2 ( T 2 ) in the ground (Qi) or in the excited (Q2)

electronic states such that UI,2(T2) = u s(Qi,2(r2)). In the last case, the values

U1I2(T-2) = W S (QI , 2 (T 2 ) ) are C-numbers. Apparently, the value ui(r2) is deter

mined by a motion in the ground electronic state, and U2(T2) - by a motion

in the excited electronic state. The thermal averaging in Eq.(91) is carried

out with respect to the ground electronic state. However, applying stochas

tic models to the calculation of M2(r1 ,T2,r3), where the system evolution

is determined in the excited electronic state 2, and the thermal averaging

is carried out in the ground electronic state 1, results in missing any bath

effects on the chromophore, in particular, the dynamical Stokes shift. We

can overcome such a difficulty if we will express M2(TI,T2, T 3 ) , by a four-time

323

correlation function related to a definite electronic state [141, 142]:

h i MjTU 72, T3) = ^ eXP[-<U,)i(Ti - T3 - i ^ p j ^ . T j , T3) (92)

where 6JS = Tr,exp(—/?Wjs),

Mj(ri ,T2 ,r3) = <exp^[iii(T2)Ti - Uj(0)(r3 + i ^ f i ) ] ) , - , (93)

are the central four-time correlation functions, UJ(T) = Uj(r) — (u(0))j is the

central value of WJ(T2); (• • -)j = Trs- • • pejs denotes the average with respect

to electronic state j ; J2j is the Kronecker delta.

In Eq.(93) the averaging is carried out in the same electronic state as

the classical trajectory calculations (equilibrium average), and the stochastic

model can be used for the calculation of Eq.(93) and all the expressions

resulting from it.

We expand the four-time correlation function by cumulants [200] (see

explanation of cumulant averages in Ref.[31], chapter 8)

00 i 1 A ^ n ^ T a ) = e x p X : ( ^ ) n ^ ( f e ( r 2 ) r 1 - ^ (0 ) ( r 3 + i82jph)]n)cj (94)

71=2

Here suffix 'c ' means that (• • -)cj is defined as a cumulant average with respect

to state j .

324

9.2 Simulation of transient four-photon spectroscopy

signals for nonlinear solvation

In this subsection we will simulate by using Eqs.(5),(61),(62),(63),(64),(65),(66),(68),(69),

(90), (92),(94) and (96) (see below) the signal for various methods of transient

spectroscopy with pulses long compared with electronic dephasing. We shall

use the solvation correlation functions of a nonlinear solvation calculated by

Fonseca and Ladanyi for a dipole solute in methanol [169] (Fig. 18). Unfortu

nately, their simulations are limited by normalized correlations functions of

the second order. Therefore, in our simulations we can use expansion Eq.(94)

only up to the first term, thus confining our consideration up to the second

order cumulants. The corresponding formula for the cubic polarization dif

fers from the one for. linear solvation (Subsec.7.1) by the presence of different

solvation correlation functions describing the dynamics in the ground or in

the excited electronic states.

Fig. 19a shows the calculation results for RTGS. For comparison we also

show the corresponding signals when both ground and excited states corre

lation functions coincide (linear case) and are equal either to the correlation

function of the excited state Se3(t) (curve 2) , or to the ground state Sgs(t)

325

0.4 0.6 T lME(ps)

0.8

Figure 18: Solvation correlation functions for the ground (gs) and the excited

(es) electronic states calculated in Ref. [169].

326

(curve 3). The equilibrium spectra of the molecule in solution F£(u) (curve 4)

and F£(u) (curve 5), and the shapes of the 'intramolecular' spectra FVM(W)

(curve 6) and the FaMw) (curve 7) (when the solvent contribution is absent)

are shown in the inserts to Figs. 19,20,21. The arrows show the relative posi

tions of the excitation frequency w. One can see that for the excitation at the

frequency of the purely electronic transition, the signal provides combined

information concerning the solvation dynamics in both states. But for the

excitation at the maximum of the absorption band, the signal mainly reflects

the solvation dynamics in the excited electronic state (Fig. 19b).

Fig. 20 shows the calculation results using Eq.(68). It can be seen that

for the excitation on the blue side of the absorption spectrum, the transmis

sion pump-probe experiment provides information concerning the solvation

dynamics in the ground electronic state.

The same is true for the HOKE signal at ifr = 0 (Eq.(7)), since the right

hand side of Eq.(68) also describes a signal for the latter case (see above).

Such behavior can be understood if we compare the contributions from the

transient absorption FC(UJ,UJ,T) and emission Fv(u,u, T) spectra related to

the dynamics in the ground and in the excited electronic states, respectively.

Let us first consider only the contribution of the intermolecular motion to

327

TIME DELAY(ps) TIME DELAY(ps)

Figure 19: The RTGS signal for the case of nonlinear solvation (1) calculated

by the correlation functions of Ref. [169] (see Fig. 18); other parameters of

the system are identical to the parameters used in the numerical calculations

in Refs. [65, 99]. Curves 2 and 3 in Figs. 19,20,21 correspond to signals when

both ground and excited states correlation functions coincide (linear case)

and are equal either to the correlation function of the excited state (2) or

to the ground state (3). Insertions to Figs. 19,20,21: equilibrium spectra of

the molecule in solution (4,5), and the shapes of the 'intramolecular' spectra

(6,7); the arrows show the relative positions of the excitation frequency u>.

Figure 20: The transmission of the probe signal in pump-probe experiments

(and the HOKE signal at ip = 0) in the case of nonlinear solvation. The rest

is the same as in Fig. 19.

329

Fa,¥>(w,u;,r). Then, using the four-photon approximation with respect to

light-matter interaction, we obtain [142]

\FV(UJ,UJ,T)\ - = exp \Fa(u,u,T)

ftp Lu'ei — UT (95) l + 5(r)

One can see that for the delay time r = 0 (5(0) = 1), the ratio (95) is

equal to one. If u> > u>ei, this ratio diminishes when r increases (S'(oo) =

0) and approaches to exp[—hf3(u — we/)] which is much smaller than 1 for

ft 3 (w — u>ei) 3> 1. This is explained by the fact that the spike relaxes much

faster than the hole for H0 (u> — wei) 3> 1. It is related to the Franck-Condon

principle: the sublevels of the excited electronic state achieved upon vertical

optical transition, correspond to a higher excitation level than the ones in

the ground electronic states and therefore the former relax faster [120].

Such a picture is qualitatively held for the case which includes intramolec

ular vibrations (see Ref.[142]).

Now let us consider the HOKE signal when the LO phase i\> — 90°. Using

Eqs.(7) and (61), we obtain:

JHET^ = 90°) ~ -[$ a(w,o; ,T) - $ V ( W , W , T ) ] (96)

We can see that for the excitation, as it is shown in Fig. 21, the HOKE signal

reflects mainly the solvation in the excited electronic state.

330

0.4 0 0.2 0.4 0.6

TIME DELAY (ps) 0.8

Figure 21: The HOKE signal at ip = 90° for nonlinear solvation. The rest is

the same as in Fig. 19.

331

Thus, using different methods of spectroscopy with pulses long compared

with the electronic transition dephasing at different excitation frequencies,

we can separately investigate the solvation dynamics in the ground electronic

state or in the excited state, and this enables us to study the nonlinear

solvation.

9.3 Spectral moments of the non-equilibrium

absorption and luminescence of a molecule in

solution

The previous subsection's consideration was of a preliminary nature since it

was limited by the second order cumulants. Here and in the following sub

sections, we shall show how to correctly characterize the nonlinear solvation

(non-Gaussian) case.

The nonlinear polarization and the transient spectroscopy signal can be

expressed by the spectra of non-equilibrium absorption (a) and luminescence

(<p) of a molecule in solution [Eqs.(61),(62),(63),(64),(65),(66),(90), (68),(69)

and (96)] which are the convolutions of the "inter" - and "intra"-molecular

spectra (Eq.(62)). Therefore, we calculate the normalized spectral moments

332

i n (T\\

/a'o \ of the "intermolecular" spectra F a ¥ , s(w' ,w,r) . By using them, one

can easily find the spectral moments of the whole spectra FatV(uii,u),T).

For simplicity, we consider that the pump pulses are shorter than the

solute-solvent relaxation time and do not overlap with the probe ones. The

rc-th noncentral moment of non-equilibrium "intermolecular" spectra Fa^s is

determined by

/oo

( ^ F ^ K ^ r ) ^ ! (97) -oo

Using Eqs.(90),(92) and (94), one can obtain for the first moment [141, 142]:

«V,M> = (ujj/h + al3 (r) (98)

where

1 ^ ( - 1 ) " / „ - n ^ - , \ r 1 ^ ^ ) °\i r E W r < « " ( 0 ) f i i ( r ) > f l -

^ -h(3)k ( f - ^ H 23 h kl(n-k)\ du*-* ( 9 9 )

The values <rn;- (r) are the "partial" central moments:

an] (r) = I" (<* - (u^/hT F ] s ^ T ) ^ (100)

where FJS(U>I,U;,T) are the "intermolecular" spectra of the non-equilibrium

absorption (j = a) or luminescence [j = <p) determined by Eq.(63).

333

Eq.(99) relates the first moments of the nonequilibrium absorption (j = 1)

and emission (j = 2) spectra to the correlation functions (u"(0)w ;(r))cj of

the solute-solvent interaction us = W2S — W\a . The coefficients of expansion

(99) are determined by the experimentally measurable values: derivatives of

the equilibrium absorption spectrum F£(u) of a solute molecule in a solu

tion. Dependence of the first moments on F*(UJ) reflects the fact that in the

nonlinear case, the spectral dynamics depends on the excitation conditions.

Eq.(99) can be considered as a generalization of the fluctuation-dissipation

theorem for the nonlinear solvation case [142].

In the particular case of the linear solvation when the magnitude ua is

Gaussian, the cumulants of the order higher than the second are equal to

zero, and expansion (99) comes abruptly to an end after the first term. In

the last case Eq.(99) reflects the fluctuation-dissipation theorem [142].

Eq.(99) expresses the first moments of nonequilibrium spectra by the

derivatives of the equilibrium absorption spectrum of a solute molecule in

solution FQ(U>) which can be experimentally measured. The second par

tial central moment can be also presented in the form similar to Eq.(99).

However, the formula for this moment is more complex and therefore is not

presented here. In general, the corresponding formulae become complicated

334

when the order of a moment increases. However, formulae for high order

moments can be written in a more compact form if one refrains from using

F » [142]:

0W(T) = fittjreft^fcfffit7")exp (5VPU°)

F°>M (<" - we; - «./»)>; (101)

Eq.(lOl) for j = 2 can also be presented in the form of a nonequilibrium

average [142]:

°n2 (r) = hnFe^un2(r)FaM(w - We/ - u./ft))i (102)

Eqs.(lOl) and (102) express the partial central moments of nonequilibrium

spectra by the "intramolecular" absorption spectrum FaM- It cannot be

measured directly for a molecule which is in a polar solvent. However, the

intramolecular spectrum FC,MU) can be determined as the spectrum of the

same solute in a nonpolar solvent [201].

9.4 Broad and featureless electronic molecular spec

tra

Let us consider the particular but very important and widely-distributed case

of very broad and featureless electronic spectra of solute organic molecules

335

in solutions. The examples are LDS-750 [65], phtalimides [202] and many

others. For such molecules the square root of the second central moment of

the equilibrium absorption spectrum is rather large y/a^ ~ 1700cm -1 . As

a result, the formulae for the "partial" central moments for the spectra of

such molecules are strongly simplified [141, 142]:

<Tn2 (r) = ^ i / r n ( ^ ( r ) exp (f3us))2 (103)

Eq.(103) can also be considered as a generalization of the fluctuation-dissipation

theorem for the nonlinear solvation case [142].

We can also rewrite Eq.(103) in the form of the nonequilibrium averages

[141, 142]:

Tn2(r) = h-n(un2(r))1 (104)

where U2(j) is determined by the motion in the excited electronic state 2,

however, the averaging is carried out with respect to the ground electronic

state 1.

336

9.5 Time resolved luminescence spectroscopy

The time shift of the first moment of the luminescence spectrum is charac

terized by the equation [169, 144]:

c^ = r"Wrr^ (105)

(uv (0)) - uv (oo))

where (wa,v (r)) = fl (oji,u>,T)duii is the first moment of the ab

sorption (a) or emission (ip) spectrum.

The quantity CVT) can be presented in the form [141, 142]:

CV(T) = <X12(T)/<T12(0) (106)

where the normalized first moment of the TRL spectrum <7i2 (r) is determined

in particular by Eq.(99) for j=2 . One can see that for the case of nonlinear

solvation, the first moment of the TRL spectra <T12 (T) is determined not

only by the correlation function of the second order, but also by cumulant

averages (U%(Q)U2T))C2 higher than the second (n > 1).

Computer simulations [169] calculated the normalized correlation func

tion in the ground electronic state ((u1(0)ui(r))1 /(uj(0))i) , in the excited

electronic state ( ( U 2 ( 0 ) U 2 ( T ) ) 2 / ( W 2 ( 0 ) ) 2 ) , and the nonequilibrium response

function C(T) ( ~ (U2(T))I) which was assumed to correspond to the exper-

337

imental measurements of a12 (r) . One can see that the first normalized mo

ment <7i2 (r) depends on the equilibrium absorption spectrum of the solute

molecule in solution, F*(u>) and in general is not reduced to the nonequi-

librium average (U2(T) ) I . However, for the case of broad and featureless

electronic spectra and for excitation near the Franck-Condon frequency of

the transition 1 —> 2 [141, 142], the "partial" central moments of the emis

sion band <Xi2 (r) almost do not depend on F^(CJ) , and are expressed by the

nonequilibrium averages (Eq.(104)). Thus, the nonequilibrium average ap

proximately describes the first moment of the TRL spectrum only in the case

of broad and featureless electronic spectra of a solute molecule in solution

for the excitation near the frequency of the Franck-Condon transition.

9.6 Time resolved hole-burning s tudy of nonlinear

solvation

Let us consider the time resolved hole-burning experiment [85]. Similar to

TRL studies [169, 144] (see Eq.(105)), one can characterize the time shift of

the first moment of the difference absorption spectrum (Eq.(69))

/»oo

(wAa (r)) = / LOI Aa (ux - u) dux (107) Jo

338

by the equation:

r (r\ - (^° (r)) ~ K " (°°)) nns^ (WAo (0 ) ) - (WAo (OOJ)

where u>i = w + u). The quantity («4 a (r)) can be expressed by the first mo

ments of the non-equilibrium absorption (a) and luminescence (<p) spectra:

( ^ A c ( r ) ) = ( w v ( T ) ) - ( w a ( T ) ) .

The quantity C&a (r) can be presented by aXj (r) [141, 142]:

CAQ (r) = [<r12 (r) + <7n (r)]/[a12 (0) + au (0)] (109)

According to Eqs.(99) and (106), the term CVT) provides information on

solvation dynamics in the excited electronic state, while C^a (r) provides

both the solvation dynamics in the ground and in the excited electronic states.

We will show how the solvation dynamics in the ground electronic state can

be found by the time resolved hole-burning spectroscopy. Let us assume

that we have determined both <x12 (r) and <7i2(0) by TRL spectroscopy. By

measuring the dependence C&a (T ) , we can determine the function crn (T ) ,

describing the dynamics in the ground electronic state. Using Eq.(109), one

can show [141, 142] that

an (r) = CAa (r) [2<r12 (0) - cjst] - ax% (0) Cv (r) (110)

339

where ust = h~l ((us)i — us)2) is the solvent contribution to the Stokes shift

between the equilibrium absorption and emission spectra.

9.7 Stochastic approach to transient spectroscopy of

nonlinear solvation dynamics

In this subsection we show how to use a stochastic approach to the calculation

of the spectral moments of the non-equilibrium absorption and luminescence

of a solvating molecule [142].

We consider Uj (r) as a random function of a parameter r . Equilib

rium averages in formulae (99), (101) and (103) have the following form:

(01(u:7'(r))^2(wi(O)))j where ^>1)2 (UJ) are given functions of Uj . Let us de

note UJ(T) = uT and Uj(0) = u0. Then the equilibrium averages under

discussion can be presented in the form [203, 204]:

(lpl(uT)lp2(u0))j =11 ^O^TWJ(WO)U ; (UT |T ,UO)V' 1 (UT )V'2( '"O) ( H I )

where Wj (UQ) describes a law of probability in electronic state j , and VJ(UT\T, UO)

is the density of the conditional probability that u takes the value uT at time

r if it takes the value uo at time 0.

It is worth noting that u in Eq . ( l l l ) is a stationary random function, and

340

therefore, Wj(u0) does not depend on r , and Vj depends only on one time

variable. This is due to the fact that in our formulae the average is carried

out with respect to the same electronic state as the determination of value

u.

Thus, in order to calculate averages (Eq.( l l l ) ) , we must know the corre

sponding conditional probability VJ(UT\T,U0). It has been calculated for the

rotational diffusion model in the case of nonlinear solvation [142].

Let us consider the nonequilibrium averages (Eq.(104)) which appear in

the theory of broad and featureless electronic molecular spectra (Subsec.9.4).

Using permutations under the trace operation, we can present Eq.(104)) in

the form:

an2 (r) = h-^u^T))! = h-nTrs(us - (us)2)n p2s ( r ) (112)

where p2s (T) = exp (— W2T) P\a e x P ( s ^ 7 " ) is the density matrix describ

ing the evolution of the solvent nuclear degrees of freedom in the excited

electronic state 2 for the specific initial condition: p2s (0) = p\a, i.e. it coin

cides with p\s for r = 0.

The classical analog of p2s (r) is a one-dimensional distribution w2 (u„ r )

341

for a nonstationary random process for the initial condition:

w2(us,0) = wi(u3) (113)

where w\ (us) describes the stationary probability in the ground electronic

states.

Thus we can write the value <rn2 (r) corresponding to Eq.(104), in the

form:

cr„2 (T) - H~n J (u. - us)2)n w2 (u„ T) dus (114)

where w2 (u3,t) must be determined for nonstationary conditions which cor

respond to the ground state for t < 0, and the excited one for t > 0.

Sometimes finding w2 (us, t) for suitable initial conditions is an easier task

than finding the conditional density V2(UT\T,U0) for arbitrary u0 [142].

The general formulae presented in Subsections 9.3, 9.4 and 9.7, have been

applied to the calculation of the spectral moments of a molecule in a model

solvent [142]. According to Debye [205-207], the solvent was considered to

be composed of point dipoles d. Each dipole undergoes rotational Brownian

motion as a result of interactions with a bath. The quantity us that is the

difference between interactions of the solvent with the excited state solute

and with the ground-state solute, can be represented in the form: us =

342

Hn(^2s — Wis ) where W 2 / (or W2™ ) denotes the interaction between the

solvent molecule labeled n and the excited-state (or ground-state) solute.

The value us for the interaction of the solute with a single solvent molecule

is u' = —d- ( E ( 2 ' — E^M, where E^ is the electrical field created by a solute

in the electronic state j .

In order to avoid nonprincipal complications, we considered that the elec

tric field created by a solute in both electronic states 1 and 2, is directed along

the same straight line, but can differ by its value or the direction with re

spect to this line. In this case, one can write: u' = —d • (E^ — E^n cos 9 =

—dE<i\ cos 0, where 6 is the angle between the dipole and the direction of the

field E^2' , and E^ is the value of the field E ^ with a sign plus or minus de

pending on the orientation of E ^ with respect to E*2', and E2\ = -E*2' — E^.

A long-time solution for the model under consideration in a strong elec

trical field has been obtained in Ref.[142].

Fig. 22 illustrates nonlinear solvation behavior for the solute which does

not create a field in the ground electronic state (E^ = 0, £21 = E^ ).

Fig. 23 shows the time-dependent first moment of the difference absorption

spectrum for time resolved hole-burning (HB) experiments. We can see that

343

Figure 22: TRL signal as a function of x =• DT for y = 1 and different

c = 0dE(2' where D is the rotational diffusion coefficient for the solvent and

y = h(ujei -u>)/(dE2i).

344

in the general case the signal is an intermediate one with respect to the

dynamics in the ground and in the excited electronic states (Fig. 23a). How

ever, we can obtain a signal that mimics either the excited or the ground

state dynamics by tuning the excitation frequency (Figs. 23b and 23c).

9.8 Summary

In this section we have shown that by using different methods of spectroscopy

with pulses long compared with electronic dephasing and tuning the excita

tion frequencies, one can investigate separately solvation dynamics in the

ground or in the excited electronic states, and consequently, study nonlinear

solvation.

We used the theory [67, 150] developed for pulses long compared with

electronic dephasing (tp ~^> T"). In this regard, the question arises whether

the condition tp 3> T' is necessary for a separate study of solvation dynamics

in the ground or in the excited electronic states. Following the considerations

of this section, the pump pulses must be sufficiently long in order to provide

a definite position of a spike and a hole on the potentials of the excited or

ground electronic state, respectively. One can formulate the corresponding

345

2 0J

b~

1.0

n °-8

b~

T 0.6

b"

~ 0.4 a <

0.2

A

- \ \

\ V b

\ ^»£

0.5 1.0 2.0 2.5 2:o

Figure 23: The first moment of the difference absorption spectrum for time-

resolved HB experiments (1) for c = 3 and y = 1 (a), y = 0.75 (6), y = 0.01

(c). Transient first moments for the absorption (2) and the emission (3) are

also shown for comparison.

346

condition as follows: tv 3> a2s where Gis is the contribution of solvation

processes to the second central moment of an absorption (or emission) band

(u^s ~ 10_143 [66, 67, 99]). For some cases the criterion tp 3> crjs is

weaker than tp 3> T'. However, the delay time r between the pump and the

probe pulses must be larger than irreversible dephasing time T". In reality,

short pump pulses tp < T" induce a polarization grating whose attenuation

is determined by T", i.e. the relaxation of the non-diagonal (with respect to

electronic indices) density matrix ^21 • The relaxation of pi\ is determined

by the evolution of the system both in the excited and ground electronic

states. Therefore, the creation of a polarization grating is an unfavorable

situation which interferes with the separate study of dynamics in the ground

and in the excited electronic states. The polarization grating relaxes for

delay times r > T". Thus, the analysis, conducted in this section, is limited

to experiments performed with pulses tv ~^> 10_14s, and delay times between

the pump and the probe pulses r > T".

We have used a new approach for the calculation of the four-time cor

relation functions in nonlinear spectroscopy of nonlinear solvation when the

breakdown of the linear response of the solvation dynamics occurs. In this

347

approach the thermal averaging is carried out in the same electronic state as

the system evolution calculations (equilibrium averages).

This approach has a number of advantages. First, stochastic models can

be used for the calculation of the corresponding averages in Eqs.(93),(94),(99),

(101) and (103), while the information concerning the time resolved Stokes

shift is preserved. In contrast, the application of stochastic models to the

calculation of M2(T1,r2,r3) (Eq.(91)), where classical trajectories are deter

mined in the excited electronic state, and thermal averaging is carried out

in the ground electronic state, cannot be used to model the time-resolved

Stokes shift. Secondly, the computer calculation of the equilibrium aver

ages consumes less computing time than that of the averages when classical

trajectory calculations and the averaging are carried out in different elec

tronic states [208]. Finally, Eqs.(93),(94) and (99) provide simple analytical

expressions for the important particular case (Subsec.9.4).

We would particularly like to note Eq.(lOl) which seems to us most per

spective in terms of practical usage. It was used in Ref.[142].

We have investigated the correctness of the description of the TRL spec

trum first moment by the nonequilibrium average which is commonly used

for nonlinear solvation studies [169] (Subsec.9.5). We have shown how sol-

348

vation dynamics in the ground electronic state is obtained by time resolved

hole-burning spectroscopy (Subsec.9.6).

We demonstrated the use of stochastic models for calculating the spectral

moments of the non-equilibrium absorption and luminescence of a solvating

molecule (Subsec.9.7). We have formulated two approaches. The first is

more general and corresponds to a model of a stationary random process.

It can be realized with the four-time correlation functions related to def

inite electronic states. The second approach corresponds to a model of a

nonstationary random process, and can be used for the calculation of the

nonequilibrium averages which appear in the theory of broad and featureless

electronic molecular spectra. We applied our results to the Debye model of

a rotation diffusion for the case of nonlinear solvation in Ref.[142].

Let us discuss an experimental study of nonlinear solvation effects. Large

nonlinear effects have been found in many kinds of solvents with hydrogen

bonds by using molecular dynamics simulations [169, 194, 195]. The detailed

studies of Ladanyi and coworkers [169,194] have shown that the breakdown of

the linear response occurs when substantial differences exist between the pat

tern of solute-solvent hydrogen bonding in the initial and final solute states.

Kumar and Maroncelli have found nonlinear effects for the solvation of rela-

349

tively small solutes of the size of benzene in methanol [209]. Their finding was

consistent with measurements of the solute dependence of solvation dynamics

in 1-propanol [210]. Simple aromatic amines (aniline, 1-aminonaphthalene,

2-amino-anthracene, 1-aminopyrene and dimehtylaniline) showed behavior

which was inconsistent with expectations based on non-specific theories of

solvation dynamics [210]. The agreement between theory and experiment is

improved by taking into account the effects of the solute self-motion [211].

According to Ref.[210], the key features of these solutes that differentiate

them, are: 1) that the hydrogen-bonding effect is localized to a single inter

action and 2) that partly as a result of this localization, the perturbation

caused by the So —> 5"i transition causes the response 'driven' in a nonlinear

fashion.

Thus, the simple aromatic amines under discussion can show the nonlinear

solvation behavior, therefore, they can be used in experiments concerning

the nonlinear solvation study. Diatomic molecules also show large nonlinear

effects due to solute motion [212].

350

10 Acknowledgments

I am very grateful to Professor D. Huppert for stimulating discussions and

for his support. This work was supported by grants from the United States-

Israel Binational Science Foundation (BSF), and the James Franck Binational

German-Israel Program in Laser Matter Interaction.

A Appendix

Let us consider the case of freely orientating molecules. In order to calculate

(5"o6(i')5'dc(M))or, we shall expand the tensor aabV) (or <70&(Qa)) by irreducible

parts (i.e. parts that transform only by themselves at any coordinate trans

formations):

aab(P) = a\V)5ab + fjp) + crlbu) (115)

where

*V) = | £*..(*) = v* (lie)

is a scalar,

351

is a symmetrical tensor, and

*«V*) = |(M*)-*>.(*)) (us)

is an antisymmetrical tensor.

One can show that the following values: d-°(P),

M*>fl=£^(*K.(/Z), (H9) ab

and

M*/?) = £*«V*K,(£), (120) ab

are invariants of the tensor crabu)(\.e. values that are constants for ten

sor in any coordinate system). We can express any orientation average

(&abv)<7dcfi))orby the tensor invariants a°,hs and ha :

<*«« (*)*». (£)>»• = A?)*°(P) + &(?,?) (121)

15

(craaji)abbv))or = cr\jl)cr\V) - ^hs(t, u) (a # b) (122)

(°ab(il)vba(v))or = -^hsfl, v) + -hafl, V) (a ^ b) (123)

VabifiZabityor = Jfi/».(& v) ~ g &„(£, V) (a ^ b) (124)

All the other averages are equal to zero.

352

As an example, let us first consider a molecule, where the direction of its

dipole moment depends on the excitation of some (intermolecular) motions

[99]. In the "molecular" frame of references (x'y'z1)

DX,QS) = D0cos(aQs) = D0cosJ2ajQsj), (125) 3

Dy,(Qs) = Dosin(aQs),Dz, = 0 (126)

We obtain for this model

^ V ) = ( 5 o / 3 ) ¥ ) , ^ » 0 , (127)

h.(W = (D*/4)[8(a-ji)6($ + v)

+8a + j:)Sa-i;) + ls(p)SP)] (128)

where 8(v) is the ^-function of Dirac.

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353

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Volume 15 In view of the rapid growth in both experimental and

theoretical studies of multi-photon processes and

multi-photon spectroscopy of atoms, ions and

molecules in chemistry, physics, biology, materials

science, etc., it is desirable to publish an advanced

series of volumes containing review papers that can

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areas, but also by those who are not experts but

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