Advances in CHEMICAL PHYSICSdownload.e-bookshelf.de/download/0000/5696/...EDITORIAL BOARD CARL BALLHAUSEN, Kobenhavns Universities Fysisk-Kemiske Institut, Kemisk BRUCE BERNE, Columbia

Advances in CHEMICAL PHYSICS

Edited by

I. PRIGOGINE

University of Brussels Brussels, Belgium

and University of Texas

Austin, Texas

and

STUART A. RICE

Department of Chemistry and

The James Franck Institute The University of Chicago

Chicago, Illinois

VOLUME LXXXIII

AN INTER SCIENCE^ PUBLICATION JOHN WILEY & SONS, INC.

NEW YORK CHICHESTER BRISBANE TORONTO SINGAPORE

ADVANCES IN CHEMICAL PHYSICS VOLUME LXXXIIl

EDITORIAL BOARD

CARL BALLHAUSEN, Kobenhavns Universities Fysisk-Kemiske Institut, Kemisk

BRUCE BERNE, Columbia University, Department of Chemistry, New York, New

RICHARD BERNSTEIN, University of California, Department of Chemistry, Los

G. CARERI, Istituto di Fisica “Guglielmo Marconi,” Universita delli Studi, Piassle

MORREL COHEN, Exxon Research and Engineering Company, Annandale, New

KARL F. FREED, The James Franck Institute, The University of Chicago, Chicago,

RAYMOND E. KAPRAL, University of Toronto, Toronto, Ontario, Canada WILLIAM KLEMPERER, Department of Chemistry, Harvard University, Cambridge,

Yu. L. KLIMONTOVICH, Department of Physics, Moscow State University, Mos-

V. KRINSKI, Institute of Biological Physics, U.S.S.R. Academy of Science Pus-

MICHEL MANDEL, Goriaeus Laboratories, University of Leiden, Department of

RUDY MARCUS, Department of Chemistry, California Institute of Technology,

PETER MAZUR, Instituut-Lorentz, voor Theoretische Natuurkunde, Leiden, The

GREGOIRE NICOLIS, UniversitC Libre de Bruxelles, Facult6 des Sciences,

A. PACAULT, Centre de Recherches Paul Pascal, Domaine Universitaire. Talence,

Y. POMEAU, Service de Physique Theorique, Centre d’Etudes Nucl6aires de

A. RAHMAN, Argonne National Laboratory, Argonne, Illinois, U.S.A. P. SCHUSTER, Institut fur Theoretische Chemie und Strahlenchemie, Universitat

ISAIAH SHAVITT, Ohio State University, Department of Chemistry, Columbus,

KAZUHISA TOMITA, Department of Physics, Faculty of Science, University of

Laboratorium IV, Kobenhaven, Denmark

York, U.S.A.

Angeles, California, U.S.A.

delle Scienze, Rome, Italy

Jersey, U.S.A.

Illinois, U.S.A.

Massachusetts, U . S. A.

cow, U.S.S.R.

chino, Moscow Region, U.S.S.R.

Chemistry, Leiden, The Netherlands

Pasadena, California, U.S.A.

Netherlands

Bruxelles, Belgium

France

Saclay, Gif-sur-Yvette, France

Wien, Wien, Austria

Ohio, U.S.A.

Kyoto, Kyoto, Japan

Advances in CHEMICAL PHYSICS

Edited by

I. PRIGOGINE

University of Brussels Brussels, Belgium

and University of Texas

Austin, Texas

and

STUART A. RICE

Department of Chemistry and

The James Franck Institute The University of Chicago

Chicago, Illinois

VOLUME LXXXIII

AN INTER SCIENCE^ PUBLICATION JOHN WILEY & SONS, INC.

NEW YORK CHICHESTER BRISBANE TORONTO SINGAPORE

In recognition of the importance of preserving what has been written, it is a policy of John Wiley & Sons, Inc. to have books of enduring value published in the United States printed on acid-free paper, and we exert our best efforts to that end.

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Library of Congress Cataloging Number: 58-9935

ISBN 0-471-54018-8

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

CONTRIBUTORS TO VOLUME LXXXIII

A. C. ALBRECHT, Department of Chemistry, Cornell University, Ithaca, New York

JEREMY K. BURDE~T, Department of Chemistry, The James Franck Insti- tute, and the NSF Center for Superconductivity, The University of Chicago, Chicago, Illinois

JEFFREY A. CINA, Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois

NOEL A. CLARK, Condensed Matter Laboratory, Department of Physics, University of Colorado, Boulder, Colorado

W. T. COFFEY, School of Engineering, Department of Microelectronics and Electrical Engineering, Trinity College, Dublin, Ireland

P. J. CREGG, School of Engineering, Department of Microelectronics and Electrical Engineering, Trinity College, Dublin, Ireland

JACK H. FREED, Baker Laboratory of Chemistry, Cornell University, Ithaca, New York

KARL F. FREED, The James Franck Institute and the Department of Chemistry, The University of Chicago, Chicago, Illinois

MAITHEW A. GLASER, Condensed Matter Laboratory, Department of Physics, University of Colorado, Boulder, Colorado

VINCENT HURTUBISE, The James Franck Institute and the Department of Chemistry, The University of Chicago, Chicago, Illinois

Yu. P. KALMYKOV, The Institute of Radioengineering and Electronics, Russian Academy of Sciences, MOSCOW, Russia

DUCKHWAN LEE, Department of Chemistry, Sogang University, Seoul, Korea

ANTONINO POLIMENO, Department of Physical Chemistry, University of Padua, Padua, Italy

V~CTOR ROMERO-ROCHiN, Instituto de Fisica, Universidad Nacional Autonoma de Mexico, Mexico

TIMOTHY J. SMITH, JR. , Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois

INTRODUCTION

Few of us can any longer keep up with the flood of scientific literature, even in specialized subfields. Any attempt to do more and be broadly educated with respect to a large domain of science has the appearance of tilting at windmills. Yet the synthesis of ideas drawn from different subjects into new, powerful, general concepts is as valuable as ever, and the desire to remain educated persists in all scientists. This series, Advances in Chemical Physics, is devoted to helping the reader obtain general information about a wide variety of topics in chemical physics, a field which we interpret very broadly. Our intent is to have experts present comprehensive analyses of subjects of interest and to encourage the expression of individual points of view. We hope that this approach to the presentation of an overview of a subject will both stimulate new research and serve as a personalized learning text for beginners in a field.

I . PRIGOGINE STUART A. RICE

vii

CONTENTS

TIME-RESOLVED OPTICAL TESTS FOR ELECTRONIC GEOMETRIC PHASE DEVELOPMENT

By Jeffrey A . Cina, Timothy J . Smith, J r . , and Victor Romero-Rochin

ON GLOBAL ENERGY CONSERVATION I N NONLINEAR LIGHT-MATTER

PASSIVE INTERACTION: THE NONLINEAR SPECTROSCOPIES, ACTIVE AND

By Duckhwan Lee and A . C. Albrecht

A MANY-BODY STOCHASTIC APPROACH TO ROTATIONAL MOTIONS IN

LIQUIDS

By Antonino Polimeno and Jack H. Freed

SOME STRUCTURAL-ELECTRONIC ASPECTS OF HIGH TEMPERATURE SUPERCONDUCTORS

By Jeremy K. Burdett

ON THE THEORY OF DEBYE A N D N ~ E L RELAXATION OF SINGLE DOMAIN FERROMAGNETIC PARTICLES

By W. T. Coffey, P, J . Cregg, and Yu. P. Kalmykov

THE ALGEBRA OF EFFECTIVE HAMILTONIANS AND OPERATORS: EXACT OPERATORS

By Vincent Hurtubise and Karl F. Freed

MELTING AND LIQUID STRUCTURE IN TWO DIMENSIONS

By Matthew A. Glaser and Noel A . Clark

AUTHOR INDEX

SUBJECT INDEX

1

43

89

207

263

465

543

71 1

723

ix

ADVANCES IN CHEMICAL PHYSICS VOLUME LXXXIII

TIME-RESOLVED OPTICAL TESTS FOR ELECTRONIC GEOMETRIC PHASE

DEVELOPMENT

JEFFREY A. CINA* and TIMOTHY J. SMITH. JR.

Department of Chemistry and the James Franck Institute, The University of Chicago, Chicago, Illinois

VICTOR ROMERO-ROCHIN

lnstituto de Fisica, Universidad Nacional Autonoma de Mexico, Mexico, D. F.

CONTENTS

I. Introduction 11. Hamiltonian and Induced Vector Potential

A. General Expression B. Particular Example C. Adiabatic Approximation D. Gaussian Wave Packets

A. B.

Appendix A:

Appendix B: Quadratic Expansion Coefficients Appendix C: Initial Conditions

111. Interference Signal

IV. Calculated Interferograms Weak Coupling, Large Amplitude Case Strong Coupling, Intermediate Amplitude Case

V. Concluding Remarks Dependence of Optical Impulsive Excitation of Pseudorotation on Molecular Orientation

Acknowledgments References

* Camille and Henw Drevfus Teacher-Scholar.

Advances in Chemical Physm, Volume L X X X / / I , Edited by 1. Prigogine and Stuart A. Rice. ISBN 0-471-54018-8 0 1993 John Wiley & Sons, Inc.

1

2 CINA, SMITH, AND ROMERO-ROCH~N

I. INTRODUCTION

The occurrence of electronic geometric phase factors in time-dependent states of molecules is a subtle consequence of the Born-Oppenheimer method. Based on the fortunate separation in timescales between electronic and nuclear motions, the Born-Oppenheimer treatment provides the familiar and useful vision of molecular dynamics in which the nuclear motion proceeds, sometimes in a more or less classical fashion, on adiabatic electronic potential energy surfaces. In the presence of a degeneracy or near degeneracy between electronic potential energy surfaces at some nuclear configuration, it is also possible for the Born- Oppenheimer separation to give rise to an induced adiabatic vector potential in the nuclear Hamiltonian [ l , 21. In some cases, the adiabatic vector potential is the source of an effective magnetic field which exerts a force on the nuclei. The adiabatic vector potential can also be the source of an electronic geometric Berry phase, which appears in the time- dependent molecular wave function as a line integral of the adiabatic vector potential along the path or paths of nuclear motion. As with any quantum mechanical phase effect, electronic geometric phase development can only be observed in an interference experiment. The purpose of this article is to summarize our own recent theoretical studies on the accessibility of electronic geometric phases to measurement in a certain class of time-resolved interference experiments, termed wave packet interferometry experiments. Geometric phases in general and molecular geometric phases in particular have both been the subject of recent review articles [3-61, so we do not attempt to survey the field as a whole.

The wave packet interferometry experiments proposed and investigated here depend on the newly developed capability of producing sequences of ultrashort optical pulses among which there are fixed and actively stabilized optical phase relationships [7,8]. When a molecular electronic transition is driven by a resonant pair of optical-phase- controlled light pulses, the excited state wave packets produced by the two pulses bear a definite quantum mechanical phase relationship and are therefore subject to interference with each other [8,9]. In the situation considered here, of small transition amplitude and pulse durations short on a vibrational timescale, the interference is between two copies of the ground state nuclear wave function, one of which has propagated for the inter-pulse delay time under the excited state nuclear Hamiltonian, the other of which has developed under the ground state Hamiltonian. The quantum interference can be constructive or destructive, leading respectively to increased or decreased excited state population relative to the separate effects of the two pulses. Since the magnitude and sign of the

ELECTRONIC GEOMETRIC PHASES 3

interference signal for a given inter-pulse delay depend on the time development of the electronically excited wave packet produced by the initial pulse and the reference (ground state) wave packet transferred by the delayed pulse, the signal is sensitive to the overall quantum phase development of the two packets, including any geometric contributions.

We can sketch the basic idea of the class of experiments considered by adopting an idealized model, due to Longuet-Higgins and co-workers, of a molecule exhibiting an adiabatic electronic sign change. The adiabatic electronic sign change is the simplest example of a molecular Berry phase. The model consists of three “electronic states” of a charged particle on a ring and the ring, representing the nuclear degrees of freedom, sustains elliptical deformations. A nodeless electronic ground state is energetically well separated from the other states. But the adiabatic electronic excited states, with nodal points connected by lines parallel or perpendicular to the instantaneous elliptical distortion, be- come degenerate at the circular ring configuration [lo].

A cyclic change in direction of the elliptical deformation, describing a closed path in the ( q , , q2) plane, is an example of molecular pseudorotation. In the excited adiabatic electronic states, a complete pseudo-


rotation leads to a sign change of the electronic wave function 1111, in the ground state it does not. To observe the sign change experimentally, one would use a sequence of nonresonant pulses to drive coherent pseudorotation of the molecule in its electronic ground state by an impulsive Raman process. The first pulse of a resonant in-phase-locked pulse pair transfers a small nuclear amplitude of the pseudorotating system to an excited state, say l e - ) , (this choice is governed by the polarization of the resonant pulses). The delayed pulse transfers to the excited state an additional amplitude which may interfere constructively, destructively, or not at all with the propagated initial amplitude. Let us assume for the sake of illustration that the pseudorotational period is the same in the ground and excited states and consider inter-pulse delays corresponding to whole numbers of the pseudorotational periods. The initial propagated amplitude carries the sign-changing excited electronic wave function, while the reference amplitude generated by the delayed pulse carries the nodeless electronic ground state. Thus, the sense of interference will depend on whether the system completes an odd or even number of complete molecular pseudorotations during the inter-pulse delay. The resulting dependence of the excited state population on odd versus even numbers of nuclear excursions around the point of electronic degeneracy is the evidence we seek for electronic Berry phase development in the excited state.

We should note that Loss, Goldbart, and Balatsky, and Loss and Goldbart have considered a system analogous to the Longuet-Higgins model of a Jahn-Teller molecule [12]. They have described a geometrical phase resulting from the quantum orbital motion of electrons in a mesoscopic normal metal ring subject to an inhomogeneous magnetic field and have shown that the Berry’s phase leads to persistent equilib- rium charge and spin currents. In addition, certain molecular manifesta- tions of the nonabelian generalizations of the geometric phase have been the subject of a recent investigation [13].

The experiments proposed here, and wave packet interferometry experiments in general, should be distinguished from recent suggestions for temporal control of chemical reaction dynamics. Tannor and Rice, and Tannor, Kosloff, and Rice have considered temporally separated optical pulses of different colors interacting with molecular systems in pump-dump experiments [14]. They have shown that by controlling the duration of propagation on the excited state electronic potential energy surface, corresponding to the delay between the two pulses, selectivity of products formed on the ground state electronic potential energy surface can in principle be achieved. In addition, a greater degree of control can be gained by modifying the phase structure of the individual pump and


dump pukes. The effect of temporally controlled transfer of nuclear probability density among different electronic potential energy surfaces on the chemical reaction rate has in fact been studied in a recent experiment [15].

Another strategy for optical control of chemical reaction dynamics has been investigated by Brumer and Shapiro. The simplest version of their scheme relies on quantum mechanical interference between one and three photon absorption pathways to a given final continuum state [16]. Thus a fixed relative optical phase between continuous wave light sources of different colors must be maintained. Some essential features of Brumer and Shapiro's proposed method have been implemented in recent experimental work [17].

This article is organized in the following way. Section I1 presents a model Hamiltonian consisting of three electronic states and a pair of nuclear modes. The forms of the adiabatic electronic states, their energies and the induced vector potential are specified. In Section 111, time- dependent perturbation theory is used to obtain an expression for the interference signal produced when a system executing coherent molecular pseudorotation in its ground electronic state undergoes an electronic transition driven by a resonant pair of in-phase-locked light pulses. In order to specify matrix elements of the electronic dipole moment operator, we adopt the Longuet-Higgins model values. We state the conditions under which excited state motion is adiabatic. As an aid to interpretation, we adapt the well studied locally quadratic Hamiltonian approach to Gaussian wave packet dynamics [18] to incorporate the effects of a locally linear vector potential. The equations of motion are found for the Gaussian parameters. A simple expression is obtained for the interference signal under the assumption that the locally quadratic Hamiltonian gives an adequate treatment of the nuclear motion in the excited state. The calculated signal in the locally quadratic theory is shown to be independent of the arbitrary choice of phases in the adiabatic electronic states (i.e., the approximate method is gauge invariant). In Section IV, interferograms are calculated and discussed. The interference signal is calculated in the case of weak electronic-nuclear coupling and large amplitude pseudorotation, using the locally quadratic Hamiltonian approach. The weak coupling-large amplitude case clearly manifests the adiabatic electronic sign change. We check the resiliency of this result by direct numerical calculation. We also investigate the spherical average necessary to predict the interference signal of an' ensemble of randomly oriented molecules. For fairly short delays, the interferogram in the strongly coupled system at intermediate amplitude also proves amenable to calculation with the locally quadratic theory, despite the instability of


the method in this case. The strong coupling interferogram for arbitrary time delay is obtained by direct numerical diagonalization of the Hamilto- nian and interpreted with numerically calculated excited state wave packets and by comparison to the locally quadratic result. There are three appendices. Appendix A discusses the preparation of coherent pseudorotation by nonresonant impulsive Raman excitation and its dependence on molecular orientation. Appendices B and C list some details pertain- ing to the locally quadratic theory.

11. HAMILTONIAN AND INDUCED VECTOR POTENTIAL

Our model system will be a molecule with three electronic levels. The electronic ground state is taken to be nondegenerate. The two excited electronic states are Jahn-Teller active, being degenerate at the symmetrical nuclear configuration and coupled by distortion of the molecule away from the symmetrical configuration. The symmetric shape corre- sponds to values q, = q2 = 0 of the two internal coordinates. We suppress all other vibrational modes. In the experiments suggested here, the most significant effects of spatial orientation will be in determining the nature of the time-dependent vibrational state prepared by a sequence of light pulses with specified spatial polarizations. We therefore treat the molecule as having a specific fixed orientation, as in a low temperature crystal. It may prove advantageous to perform the experiments in a jet, so we later discuss an orientational average.

In the electronic ground state, I g ) , the nuclear Hamiltonian,

is a two dimensional harmonic oscillator, with q = ( q l , q 2 ) and p =

(p l , p 2 ) . The Hamiltonian in the electronic excited manifold is given by

where the constant, K = k.\/hMw:, defines the strength of the electronic- nuclear coupling and k is the dimensionless vibronic coupling parameter introduced by Longuet-Higgins and co-workers [ 10, 19-21]. The Pauli matrices in Eq. (2.2) are constructed from the nuclear coordinate independent electronic states, I+) and I-), in the forms


An external magnetic field or spin-orbit coupling would add a term za3 , with z a constant, to the excited state Hamiltonian (2.2) [l(c), 221. The strategy outlined below for measuring the adiabatic electronic sign change will be equally well applicable to the complex Berry phase factors occurring in the presence of a a, term.

The development of electronic geometric phase factors is governed by an adiabatic vector potential induced in the nuclear kinetic energy when we extend the Born-Oppenheimer separation to the degenerate pair of states [ I , 23-25]. To see how the induced vector potential appears, we consider the family of transformations which diagonalize the excited state electronic coupling in the form

441a1 + 92%) = .4U(q)g3Ui(q) 7 (2.4)

so that it is expressed as a rotation of the diagonal Pauli matrix dependent on the nuclear coordinate operators. The unitary operator in Eq. (2.4) can be written as a product,

where

with q l = q cos 4 and qr = q sin 4, and

f(q) and g(q) in (2.7) are arbitrary single-valued functions. The eigenkets of the operator (2.4) are the adiabatic electronic states,

whose energies, including the harmonic term from Eq. (2.2), correspond to the electronic potential energy surfaces


The presence of a conical degeneracy at q = 0 between the upper and lower linear Jahn-Teller surfaces of Eq. (2.9) accounts for the occurrence of nontrivial electronic geometric phase factors in this system [ 1 , 2, 261.

Applying the unitary operator (2.5) to the molecular Hamiltonian for the excited state manifold allows an easy identification of the adiabatic and nonadiabatic parts. We write the Hamiltonian (2.2) as

where

1 MWS Xe = E + - (p + A(q))' + - q2 + K 4 U 3 2M 2

The vector potential in Eq. (2.11) is given by

h A = T U'(q)V,U(q)

A = A, + A , a , + A,a2 + A,a,

I

The coefficients are

and A, = A,, + with a transverse (divergence-free) part

and a longitudinal (curl-free) part

(2.11)

(2.12)

(2.13)

(2.14)

(2.15)

(2.16)

(2.17)

(see, for example, [27]). We can then decompose Ze into adiabatic (electronically diagonal) and


nonadiabatic (electronically off-diagonal) parts; i.e., Xe = Xad + Xnad, with

(2.19)

The transverse part of the induced adiabatic vector potential, (2.16), which appears in the kinetic energy of the adiabatic nuclear Hamiltonian, (2.18), governs electronic geometric phase development [25,28]. The two dimensional curl of A,, yields a solenoidal “magnetic field” of the form rhS( q , ) S ( ql ) (in the general case of z # 0, it yields a monopolar field). The A: + - f 2 term is an “electric gauge potential” giving rise to an inverse cubed force directed away from the point of degeneracy [29].

P P 2M 2M x n a d = (uIAI + aZA2) - + - * (VIA, + ~ 2 . 4 , )

111. INTERFERENCE SIGNAL

A. General Expression

The model Hamiltonian of Section I1 captures some of the essential features of the electronic and vibrational structure of polyatomic molecules, like benzene and sym-triazine, that have both nondegenerate and degenerate, Jahn-Teller active, electronic levels. In this section interference experiments are described which will be sensitive to the geometric phase development accompanying adiabatic nuclear motion on either of the electronic potential energy surfaces in the Jahn-Teller pair.

The system is to be prepared in a coherently pseudorotating nuclear wave packet in the ground electronic state. Specifically, the wave packet is a moving displaced vibrational ground state of the two-dimensional oscillator with the form

In Eq. (3.1), (Y = iMu,/2, and the other barred quantities are the usual time-dependent parameters. The formation of the coherent state (3.1) by nonresonant optical impulsive excitation [30] and the dependence of the initial values of the parameters on pulse sequence and molecular orientation are discussed in Appendix A.


The coherently pseudorotating molecule then interacts with a phase- locked pair of electronically resonant light pulses. The laser electric field, polarized in the I direction of a space-fixed Cartesian frame, gives an interaction of the form

V ( t ) = - f i - I (E( t ) + E(t - t d ) ) cos(Rt) (3.2)

where f i is the electronic dipole moment operator, connecting 1s) with I ? ) , and E( t ) is the field envelope. Since both pulses in (3.2) have the same carrier wave, they are said to be phase-locked at the frequency R. The pulses are short compared to the vibrational timescale and hence will transfer copies of the moving wave packet to one or both of the excited state electronic potential energy surfaces. It is now experimentally possible to produce ultrashort laser pulse-pairs phase-locked in phase (as in (3.2)), in quadrature, or out of phase at the carrier frequency or at any other component frequency [8].

A straightforward first order perturbation theory calculation gives the probability that the interaction (3.2) will leave the system electronically excited. The excited state population can be measured experimentally by monitoring fluorescence or photoionization. Of particular interest is the cross term coming from interaction with the fields of both pulses, which is given by

2 P n t ( t d ) = 7 Re /-: dt" n dt'E(t" - id) cos(flt")E(t') cos(fli')

The quantity (3.3) represents the effect of quantum mechanical interference between the two excited state amplitudes generated by the pulse pair from the moving ground state wave packet. The amplitude prepared by the initial pulse propagates under He during the interpulse delay; it can acquire geometric phases and, in general, undergo complicated nonadiabatic dynamics [31-331. A reference wave packet, which has undergone comparatively simple motion under H,, is transferred to the excited state by the delayed pulse. It is assumed that no radiative or material dephasing processes disrupt the interference of the wave packets on the timescale of the interpulse delay.

It is possible to evaluate Eq. (3.3) for arbitrary pulse shapes using the well known numerical diagonalization of the Jahn-Teller Hamiltonian [lo]. However, geometric phase development will be most clearly mani- fest under simplifying experimental conditions which make possible an

ELECTRONIC GEOMETRIC PHASES 1 1

illuminating approximate treatment of the interference signal. If we specialize to Gaussian pulses

which approach delta functions on a vibrational timescale, and make a rotating wave approximation, Eq. (3.3) reduces to

rrEir2 P'"'(t,) = ~ Re[e'"'"( $(0)le'Hg'd'*( glfi *Ie-'He'd'*fi -Ilg)l&(O))].

h2 (3.5)

The quantity in square brackets in (3.5) resembles the overlap kernel in the time-dependent theory of the continuous-wave absorption spectrum [34], but here involves the nonstationary ground state wave packet I&(O)) rather than a stationary vibrational wave function. The interference signal in the impulsive limit directly measures the overlap between pseudorotating wave packets propagated in the ground and excited states for a time t,.

B. Particular Example

We need the specific form of the matrix elements of the electronic dipole moment operator as functions of the nuclear coordinates in order to calculate interferograms using (3.3) or (3.5). To illustrate ideas, we adopt a simple realization of the Hamiltonian (2.1) and (2.2), due to Longuet- Higgins and co-workers [lo], which provides closed-form expressions for the q dependence of the matrix elements of p. Longuet-Higgins' realization of a Jahn-Teller active system treats the motion of a single electron on a continuous ring which is subject to elliptical distortions. It therefore mimics the Jahn-Teller effect in a monocyclic aromatic molecule.

We focus on the electronic ground state of the charged particle, ((1g) = (2rr)-"*, and the first, degenerate, pair of excited states, ( ( I+ ) = (2rr)-"*e"'. The dipole moment operator,

fi = p(i cos 6 + j sin 6) (3.6)

has transition elements [35]

(3.7) p . _ ..

( + / f i l g ) = 2 ( I + LJ)

The unit vectors in (3.6) and (3.7) belong to a molecule-fixed Cartesian frame ( i , j, k) in which k is normal to the molecular plane. The ring


distortions are all combinations of two orthogonal elliptical deformations, q1 and q2, having their major axes at a 45” angle. The major axis of the q1 mode defines the i direction in the molecule-fixed frame and the q2 distortion is along the (2)-I”(i + j) direction.

The adiabatic electronic states of Eq. (2.8) follow the ring distortion. Their transition elements with the ground state follow from (3.7) and are given by

Notice that the transitions to le+(q)) and le-(q)) are parallel and perpendicular, respectively, with respect to the major axis of the ellipse (for instance, a field along i prepares the state I + ) + I-) le+(d, = 0)) or le-(4 = n-))). To calculate the interferogram, we must also specify the relative orientations of the molecule-fixed (i, j, k) and space-fixed (I , J , K ) axes. As is explained further in Appendix A, we use the freedom in defining q1 and q2 to ensure that the ik-plane includes the polarization direction I of the phase-locked pulse pair.

C. Adiabatic Approximation

Since we are interested in electronic phase effects accompanying adiabatic nuclear excursions, and are considering motion near a conical intersec- tion, we must specify the conditions under which adiabaticity obtains. A wave packet moving on one of the excited state potential surfaces will not undergo nonadiabatic transition to the other provided it is further from q = 0 than its width and that

(3.9)

where q, and q, are the average position and velocity [25]. Since 6, is the instantaneous pseudorotational frequency, Eq. (3.9) is the usual require- ment that the local electronic splitting exceeds the frequency of nuclear motion. A ground state wave packet, IJ(O)), obeying (3.9) will initially evolve adiabatically following an impulsive transition to the excited state surfaces. Adiabatic motion will continue until the trajectory of the wave packet or its spreading give significant amplitude for q < h ~ , / 2 ~ .

To introduce the adiabatic approximation in the expression (3.5) for the interferogram, we use the excited state Hamiltonian in the form (2.10) and propagate under the adiabatic Hamiltonian (2.18) while


neglecting the nonadiabatic contribution (2.19). Thus, we take

in which X.:d are given by (2.18) with cr3 replaced by +1. The quantity (3.10) operates on the state

(3.11)

in the interferogram expression (we have used j . Z = 0). We may center the initial wave packet around a negative value of q , (i.e., 4 = T) so that (3.11) is a pseudorotating wave packet in the lower electronic excited state (cf. following (3.8)). The first term in square brackets in (3.10), which governs adiabatic motion o n the upper electronic potential energy surface, essentially vanishes in operation on (3.11); polarization perpendicular to the instantaneous long axis gives little amplitude on the upper surface. The Xl,, term would also be discriminated against if the light pulses had spectral width less than 2 ~ q / h and the center frequency were near resonance with the lower excited state surface. Combining the results of this paragraph with Eq. (3.5) gives an interference signal

(3.12)

in the adiabatic approximation. The initial condition on the location of the ground state wave packet assumed in (3.12) might appear difficult to produce except in an oriented sample, such as a low temperature crystal. In Appendix A, we show that in fact large amplitude excitation along the q,-mode can in principle be achieved for all but a small fraction of the molecules that will interact with the polarized resonant pulse pair in a randomly oriented sample.

D. Gaussian Wave Packets

We have found, by direct numerical propagation under H e , that initially Gaussian pseudorotating wave packets often remain fairly well localized over several periods of motion. Even a wave packet excited into the trough of a sym-triazine-like system with k - 2 executes more than a


complete pseudorotation before breaking up into several pieces (see further, below). The persistence of localized wave packets and the resulting prominent peaks in the calculated interference signal suggest that at least the important central portions of excited state wave packets may be treated as Gaussians with time-dependent width, momentum, coordinate and phase parameters. Such a treatment, which employs a simple analytic form for the wave function, will also be helpful in identifying geometric phase effects in the predicted interference signal.

We therefore adapt the locally quadratic Hamiltonian treatment of Gaussian wave packets, pioneered by Heller [18], to a system with an induced adiabatic vector potential. The locally quadratic theory replaces the anharmonic time-independent nuclear Hamiltonian by a time-dependent Hamiltonian which is taken to be of second order about the instantaneous center of the wave packet. Since the nuclear wave packet continually evolves under an effective harmonic Hamiltonian, an initially Gaussian wave form remains Gaussian. The treatment yields equations of motion for the wave function parameters that can be solved numerically 136-381. The locally quadratic Hamiltonian includes a second order expansion of the scalar potential, consisting of the last three terms in Eq. (2.18), which we write as

Expressions for the various coefficients are listed in Appendix B. The presence of an induced adiabatic vector potential in the kinetic

energy of (2.18) raises the question of how to properly handle the expansion of A,(q) + cr,A,(q) about the center of the wave packet. This question can be posed and resolved, as follows, by recalling the definition (2.12) of that quantity in terms of U ( q ) . The interferogram expression (3.12) calls for propagation of wave packets under %'id. But the Gaussian wave packet I $(O)) appears there preceded by exponential factors de- pending on q - q 1 to all orders. There is no need to approximate the factor exp(- ig(q) + if(q)); it merely cancels the longitudinal contributions (2.13) and (2.17) to the vector potential in (2.18). The other factor, (1 - exp(i4)), comes from the electronic matrix elements of U:(q) involved in the operation of (3.10) on (3.11). This factor accounts for a slight distortion of the wave packet upon excitation to the lower Jahn- Teller surface, which occurs despite the arbitrarily short pulse duration. To remain consistent with the Gaussian approximation, we must truncate the expansion of 4(q) in the exponent at second order, writing


In more general terms, consistent application of the Gaussian approximation requires use of the quadratic expansion (3.14) in the unitary operator U,(q) given by (2.6). Hence, we obtain a h e a r approximation to the transverse vector potential (2.16) of the form

A3L(q) ( a + aql( 41 - 411) f aq2(q2 - q z t ) , b + bql(9* - 91r)

+ by,( 42 - q?r)) (3.15)

which in turn entails a locally uniform approximation to the field derived from the induced vector potential. The coefficients in the expansions (3.14) and (3.15) are given for reference in Appendix B [39].

Determination of the time-dependent nuclear wave function now becomes a straightforward exercise in the locally quadratic theory. We need to propagate Gaussian wave packets of the form

under the locally quadratic version of the Hamiltonian Xid with the longitudinal parts of the vector potential removed from the kinetic energy operator (i.e., the Hamiltonian of Eq. (2.18) with a, replaced by -1 and both A, and A,II set to zero). Equations of motion for the parameters in (3.16) are obtained in the usual way, by substituting that function in the time-dependent Schrodinger equation and equating coefficients of the various powers of ( q1 - q,,) and ( q2 - q2r) . We write Q = a, + ia,, etcetera for the complex parameters and obtain the following equations of motion after some algebra:

1 4 1 r = ( P I - a) (3.17)

4 2 r = ( P ? - b ) (3.18) 1

e,,= - u + - 1 ( P I - a)aql + 1 ( P 2 - b)h71 M

(3.19)

(3.20)


= - M (&, + &) + ad1, + b&, - LJ - - h (a; + P ; ) 2 M

2 2 2 2 M M

ff,= - ( a ; - a ; + s j - 6 , ) + - (qp, + b,,6,) - i.,,,,

8; = - - 2 (a; + P;)6, - - 2 (a, + p,p; + 1 (aq2a, + b,,Pi) M M

(3.2 1 b)

(3.22)

(3.23)

(3.24)

(3.25)

(3.26)

(3.27)

To evaluate the interference signal (3.12) in the Gaussian approximation, one needs to solve the twelve equations of motion (3.17) through (3.28) twice. In one instance, the initial conditions are the parameters of the ground state wave packet (ql$(O)) given in (3.1). In the second instance, the initial values of some parameters are slightly modified by incorporating the coefficients of the expansion (3.14) in the multiplying factor exp(i+(q)). The two sets of initial conditions are given explicitly in Appendix C.

Equations (3.17)-(3.20) are the usual classical Hamilton’s equations for a particle in the vector potential (a(q), b(q)). Equation (3.21b) for the phase of the wave function involves the Lagrangian for such a particle plus a quantum correction. The equation of motion for the imaginary part of the overall phase, (3.22), is redundant in the sense that it merely ensures continued adherence to the normalization condition

p A = [4(a;P; - s;)/7r”2]1’4 (3.29)

In practice it is useful to solve (3.22), despite its redundancy, since it provides a check on the accuracy of numerical integration.

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Advances in CHEMICAL PHYSICSdownload.e-bookshelf.de/download/0000/5696/...EDITORIAL BOARD CARL BALLHAUSEN, Kobenhavns Universities Fysisk-Kemiske Institut, Kemisk BRUCE BERNE, Columbia