Advances in multi photon processes and spectroscopy volume 21

www.pdfgrip.com

8851hc.9789814518338.tp.indd 1

28/10/13 10:16 AM


November 5, 2013

13:17

9in x 6in

Advances in Multi-Photon Processes and Spectroscopy Volume 21

www.pdfgrip.com

This page intentionally left blank

b1629-fm


www.pdfgrip.com

Volume

21

World Scientific
NEW JERSEY

•

LONDON

8851hc.9789814518338.tp.indd 2

•

SINGAPORE

•

BEIJING

•

SHANGHAI

•

HONG KONG

•

TA I P E I

•

CHENNAI

28/10/13 10:16 AM


www.pdfgrip.com

Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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

Library of Congress Control Number: 86643116

Advances in Multi-Photon Processes and Spectroscopy — Vol. 21
ADVANCES IN MULTI-PHOTON PROCESSES AND SPECTROSCOPY
(Volume 21)
Copyright © 2014 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance
Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy
is not required from the publisher.

ISBN 978-981-4518-33-8

Typeset by Stallion Press
Email:

Printed in Singapore


November 5, 2013

13:17

9in x 6in

Advances in Multi-Photon Processes and Spectroscopy Volume 21

b1629-fm

www.pdfgrip.com

PREFACE

In view of the rapid growth in both experimental and theoretical studies
of multi-photon processes and multi-photon spectroscopy of molecules,
it is desirable to publish an advanced series that contains review articles
readable not only by active researchers, but also by those who are not yet
experts and intend to enter the field. The present series attempts to serve this
purpose. Each chapter is written in a self-contained manner by experts in
the area so that readers can grasp the content without too much preparation.
This volume consists of six chapters. The first chapter presents the
results of both theoretical and experimental studies of “Vibrational and

Electronic Wavepackets Driven by Strong Field Multi-photon Ionization”.
First, basic theoretical ideas essential to understanding multiphoton ionization and laser control of molecules are described. Secondly, experimental
techniques for molecular control such as phase-dependent dissociation,
photon locking and spatial hole burning are explained by taking halogenated
methanes (CH2 BrI, CH2 I2 ) as a model system.
The second chapter deals with the results of experimental studies
on “Orientation-selective Molecular Tunneling by Phase-controlled Laser
Fields”. After the basic properties of tunneling ionization (TI) of atoms
and molecules are introduced, the experimental results of directionally
asymmetric TI of CO, OCS, iodohexane, and bromochloroethane, which
are induced by ω + 2ω laser pulses, are presented.
The third chapter presents experimental and theoretical results of
“Reaction and Ionization of Polyatomic Molecules Induced by Intense
Laser Pulses”. The emphasis is on ionization rates, resonance effects,
dissociative ionization and Coulomb explosion of polyatomic molecules
v


November 5, 2013

13:17

9in x 6in

Advances in Multi-Photon Processes and Spectroscopy Volume 21

b1629-fm

www.pdfgrip.com

vi

Advances in Multi-Photon Processes and Spectroscopy

such as cyclopentanone (C5 H8 O), which are induced by intense fs-laser
fields.
The fourth chapter presents the reviews of experimental studies on
“Ultrafast Internal Conversion of Pyrazine via Conical Intersection”.
Pyrazine is one of the typical azabenzenes undergoing ultrafast S2 –S1 internal conversion through conical intersection. In this chapter, experimental
results of femtosecond internal conversion of pyrazine, that are observed
in real time using a time-resolved photoelectron imaging method with a
time resolution of 22 fs are presented. The method enables us to obtain a
time–energy map of the photoelectron angular anisotropy as well.
The fifth chapter deals with the theoretical studies of “Quantum
Dynamics in Dissipative Molecular Systems”. Dissipation is essential
in condensed phase systems. Femtosecond time-resolved spectroscopy
applied to photosynthetic antenna in proteins manifests as quantum beats,
which indicates the quantum nature of the system. The timescale of the
protein environment memory is found to be comparable to that of the energy
transfer. For such a system, traditional perturbative Markovian quantum
dissipation theories are inadequate. The reviews of theoretical studies in
the nonperturbative and non Markovian treatments are presented on the
basis of the hierarchical equation of motion approach.
The sixth chapter presents the results of the theoretical and computational studies of “First-principle Calculations for Laser Induced Electron
Dynamics in Solids”. Electron dynamics in a crystalline solid induced by
strong ultrashort laser pulses is totally different from that observed in atoms
and molecules. The basic principles and restrictions for treating electrons
in crystalline solids are described. Time-dependent Kohn-Sham equation
in a unit cell is solved based on the time-dependent density functional
theory. The present theory and computational method provide the most

comprehensive description for the interactions of strong and ultrashort laser
pulses with solids.
The editors wish to thank all the authors for their important contributions toAdvances in Multi-photon Processes and Multiphoton Spectroscopy
Vol. 21. It is hoped that the collection of topics in this volume will be
useful not only to active researchers but also to other scientists and graduate
students in scientific research fields such as chemistry, physics, and material
science.


November 5, 2013

14:2

9in x 6in

Advances in Multi-Photon Processes and Spectroscopy Volume 21

b1629-fm

www.pdfgrip.com

CONTENTS

Preface

v

1. Vibrational and Electronic Wavepackets Driven
by Strong Field Multiphoton Ionization

1

P. Marquetand, T. Weinacht, T. Rozgonyi,
J. González-Vázquez, D. Geißler and L. González
1.1 Introduction . . . . . . . . . . . . . . . . . . . . .
1.2 Theoretical Concepts . . . . . . . . . . . . . . . .
1.2.1 The time-independent Schrödinger equation
and its implications on dynamics . . . . . .
1.2.2 Spin-orbit coupling and diabatic vs.
adiabatic states . . . . . . . . . . . . . . . .
1.2.3 Nuclear time-dependent Schrödinger
equation . . . . . . . . . . . . . . . . . . .
1.2.3.1 Second-order differentiator . . . . .
1.2.3.2 Split-operator method . . . . . . . .
1.2.4 Stark shifts . . . . . . . . . . . . . . . . . .
1.2.5 Multi- vs. single-photon transitions . . . . .
1.2.6 Laser-dressed states . . . . . . . . . . . . .
1.2.7 Photon locking . . . . . . . . . . . . . . . .
1.2.8 Hole burning . . . . . . . . . . . . . . . . .
1.2.9 Strong-field ionization . . . . . . . . . . . .
1.3 Computational and experimental details . . . . . .

vii

. .
. .

1
3

. .

3

. .

7

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

9
11
11
13
15
19
20
21
23
27


November 5, 2013

13:17

9in x 6in

Advances in Multi-Photon Processes and Spectroscopy Volume 21

b1629-fm

www.pdfgrip.com

viii

Advances in Multi-Photon Processes and Spectroscopy

1.4 Vibrational Wavepackets Created by Multiphoton
Ionization . . . . . . . . . . . . . . . . . . . . . .

1.4.1 Phase-dependent dissociation . . . . . . . .
1.4.1.1 Photon locking . . . . . . . . . . .
1.4.1.2 Hole burning . . . . . . . . . . . .
1.4.2 Ionization to different ionic states . . . . . .
1.4.2.1 Preparing electronic wavepackets
via SFI . . . . . . . . . . . . . . .
1.4.2.2 VMI measurements to identify
dissociation pathways following SFI
1.5 Conclusion and Outlook . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . .
2.

.
.
.
.
.

.
.
.
.
.

29
29
35
36
38

. .

38

. .
. .
. .

44
49
50

Orientation-Selective Molecular Tunneling Ionization
by Phase-Controlled Laser Fields

55

H. Ohmura
1
2

3

4
5

Introduction . . . . . . . . . . . . . . . . . . . . . . .
Photoionization Induced by Intense Laser Fields . . . .
2.1
MPI in standard perturbation theory . . . . . . .

2.2
Keldysh theory: From MPI to TI . . . . . . . .
2.3
Characteristics of TI . . . . . . . . . . . . . . .
2.4
Molecular TI . . . . . . . . . . . . . . . . . . .
Directionally Asymmetric TI Induced by Phase-controlled
Laser Fields . . . . . . . . . . . . . . . . . . . . . . .
3.1
Phase-controlled laser fields . . . . . . . . . . .
3.2
Directionally asymmetric TI (atoms) . . . . . .
3.3
Directionally asymmetric TI (molecules) . . . .
Experimental . . . . . . . . . . . . . . . . . . . . . .
Results and Discussion . . . . . . . . . . . . . . . . .
5.1
Diatomic molecule: CO . . . . . . . . . . . . .
5.1.1 Photofragment detection . . . . . . . .
5.1.2 Photoelectron detection . . . . . . . . .
5.2
Other molecules . . . . . . . . . . . . . . . . .
5.2.1 Nonpolar molecule with asymmetric
structure: Br(CH2 )2 Cl . . . . . . . . .

55
58
58
59
62

66
68
68
71
75
77
83
83
83
87
92
92


November 5, 2013

13:17

9in x 6in

Advances in Multi-Photon Processes and Spectroscopy Volume 21

b1629-fm

www.pdfgrip.com

Contents

ix

Large molecule: C6 H13 I . . . . . . . .
Systematically changing molecular
system: CH3 X(X=F, Cl, Br, I) . . . . .
5.2.4 OCS molecule investigated by nanosecond
ω + 2ω laser fields . . . . . . . . . . .
6 Summary . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgments . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . .

94

97
98
99
100

Reaction and Ionization of Polyatomic Molecules
Induced by Intense Laser Pulses

105

5.2.2
5.2.3

3.

96

D. Ding, C. Wang, D. Zhang, Q. Wang, D. Wu and S. Luo
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .

1.2 Ionization Rate of Molecules in Intense
Laser Fields . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Theoretical approaches for ionization rates of
molecules in intense laser fields . . . . . . . . .
1.2.2 Experimental measurements of ionization rates
of molecules and comparations with theory . . .
1.3 Fragmentation of Molecules in Intense Laser Fields . .
1.3.1 Ionization-dissociation of molecules in intense
laser fields and statistical
theoretical description . . . . . . . . . . . . . .
1.3.2 Effects of cation absorption on molecular
dissociation . . . . . . . . . . . . . . . . . . .
1.4 Dissociative Ionization and Coulombic Explosion
of Molecules in Intense Laser Fields . . . . . . . . . .
1.4.1 Dissociative ionization of formic acid molecules
1.4.2 Coulombic explosion of CH3 I . . . . . . . . . .
1.5 Summary and Perspectives . . . . . . . . . . . . . . .
Acknowledgments . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . .

105
110
110
117
124

124
125
127
128

131
133
135
135


November 5, 2013

13:17

9in x 6in

Advances in Multi-Photon Processes and Spectroscopy Volume 21

b1629-fm

www.pdfgrip.com

x

4.

Advances in Multi-Photon Processes and Spectroscopy

Ultrafast Internal Conversion of Pyrazine Via Conical
Intersection

139

T. Suzuki and Y. I. Suzuki

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
1.2 Pyrazine: Ultrafast S2 (1 B2u , ππ∗ ) — S1 (1 B3u , nπ∗ )
Internal Conversion Via Conical Intersection . . . . .
1.3 Sub-20 fs Deep UV Laser for TRPEI of Pyrazine . .
1.4 Time-Resolved Photoelectron Imaging . . . . . . . .
1.4.1 TRPEI of Ultrafast S2 –S1 internal conversion
in pyrazine . . . . . . . . . . . . . . . . . . .
1.4.2 Analysis of PAD . . . . . . . . . . . . . . . .
1.5 Conical Intersections in Cation and Rydberg
States of Pyrazine . . . . . . . . . . . . . . . . . . .
1.6 Toward Sub-30 fs TRPEI in VUV Region . . . . . .
1.7 Summary . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgments . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . .
5.

.

139

.
.
.

141
144
147

.
.

149
155

.
.
.
.
.

161
166
170
171
171

Quantum Dynamics in Dissipative Molecular Systems

175

Hou-Dao Zhang, J. Xu, Rui-Xue Xu and Y. J. Yan
1
2

3

Introduction . . . . . . . . . . . . . . . . . . . . .
HEOM versus Path Integral Formalism:
Background . . . . . . . . . . . . . . . . . . . . .
2.1

Generic form and terminology of HEOM . .
2.2
Statistical mechanics description
of bath influence . . . . . . . . . . . . . . .
2.3
Feynman–Vernon influence functional
formalism . . . . . . . . . . . . . . . . . .
2.4
General comments . . . . . . . . . . . . . .
Memory-Frequency Decomposition of Bath
Correlation Functions . . . . . . . . . . . . . . . .
3.1
PSD of Bose function . . . . . . . . . . . .
3.2
Brownian oscillators decomposition of bath
spectral density function . . . . . . . . . . .

. .

175

. .
. .

177
177

. .

177

. .
. .

179
180

. .
. .

182
182

. .

185


November 5, 2013

13:17

9in x 6in

Advances in Multi-Photon Processes and Spectroscopy Volume 21

b1629-fm

www.pdfgrip.com

Contents

xi

4

Optimized HEOM Theory With Accuracy Control . . .
4.1
Construction of HEOM via path integral
formalism . . . . . . . . . . . . . . . . . . . .
4.2 Accuracy control on white-noise
residue ansatz . . . . . . . . . . . . . . . . . .
4.3
Efficient HEOM propagator: Numerical filtering
and indexing algorithm . . . . . . . . . . . . .
5 HEOM in Quantum Mechanics for Open Systems . . .
5.1
The HEOM space and the Schrödinger
picture . . . . . . . . . . . . . . . . . . . . . .
5.2
HEOM in the Heisenberg picture . . . . . . . .
5.3
Mixed Heisenberg–Schrödinger block-matrix
dynamics in nonlinear optical response
functions . . . . . . . . . . . . . . . . . . . . .
6 Two-Dimensional Spectroscopy: Model
Calculations . . . . . . . . . . . . . . . . . . . . . . .
7 Concluding Remarks . . . . . . . . . . . . . . . . . .
Acknowledgments . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . .

6.

First-Principles Calculations for Laser Induced Electron
Dynamics in Solids

187
187
190
192
193
193
195

196
201
205
206
206

209

K. Yabana, Y. Shinohara, T. Otobe, Jun-Ichi Iwata
and George F. Bertsch
1
2

3

Introduction . . . . . . . . . . . . . . . . . . .
Formalism . . . . . . . . . . . . . . . . . . . .

2.1 A time-dependent Kohn-Sham equation
in periodic systems . . . . . . . . . . . .
2.2
Polarization field . . . . . . . . . . . . .
2.3
Derivation from a Lagrangian . . . . . .
2.4
Computational method . . . . . . . . . .
Real-Time Calculation for Dielectric Function .
3.1
Linear response calculation in transverse
geometry . . . . . . . . . . . . . . . . .

. . . .
. . . .

210
212

.
.
.
.
.

.
.
.
.
.

212
213
215
218
219

. . . .

219

.
.
.
.
.

.
.
.
.
.


November 5, 2013

13:17

9in x 6in

Advances in Multi-Photon Processes and Spectroscopy Volume 21

b1629-fm

www.pdfgrip.com

xii

Advances in Multi-Photon Processes and Spectroscopy

3.2

Linear response calculation in longitudinal
geometry . . . . . . . . . . . . . . . . . . . .
3.3
Example: Dielectric function of bulk Si . . . .
4 Coherent Phonon Generation . . . . . . . . . . . . .
4.1
Physical description . . . . . . . . . . . . . .
4.2
TDDFT calculation for Si . . . . . . . . . . .
5 Optical Breakdown . . . . . . . . . . . . . . . . . .
5.1
Incident, external, and internal electric fields .
5.2
Intense laser pulse on diamond . . . . . . . .
6 Coupled Dynamics of Electrons and Electromagnetic
Fields . . . . . . . . . . . . . . . . . . . . . . . . .
6.1
Maxwell + TDDFT multiscale simulation . .

6.2
Example: Laser pulse irradiation on Si surface
7 Summary . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.
.
.

220
221
223
223
225
230
230
232

.
.
.
.
.

237

237
240
242
243


November 5, 2013

13:17

9in x 6in

Advances in Multi-Photon Processes and Spectroscopy Volume 21

b1629-ch01

www.pdfgrip.com

CHAPTER 1

VIBRATIONAL AND ELECTRONIC WAVEPACKETS
DRIVEN BY STRONG FIELD
MULTIPHOTON IONIZATION
P. Marquetand,ả , T. Weinacht , T. Rozgonyi ,
J. Gonzỏlez-VỏzquezĐ , D. Geißler† and L. González∗
We present basic theoretical ideas underlying multiphoton ionization and laser
control of molecules. Approaches to describe molecular electronic structure, spinorbit coupling, dynamic Stark shifts, dressed states, and multiphoton excitations
are shortly reviewed. Control techniques such as phase-dependent dissociation,
photon locking, and spatial hole burning are explained and illustrated exemplarily
using halogenated methanes (CH2 BrI, CH2 I2 ) as model systems. Theoretical

approaches are compared with experiments and the complex signals resulting from
phenomena like electronic wavepackets are elucidated and understood. Hence,
we show how strong-field control concepts developed for simple systems can be
transferred to more complex ones and advance our ability to control molecular
dynamics.

1.1. Introduction
The development of intense ultrafast lasers over the past two decades has
led to dramatic advances in our ability to follow molecular dynamics on
femtosecond and attosecond timescales.1–5 Furthermore, intense ultrafast
lasers not only provide the means to study electronic and nuclear dynamics,
∗ Institute of Theoretical Chemistry, University of Vienna, Währinger Straße 17, 1090 Vienna, Austria
† Department of Physics, Stony Brook University, Stony Brook, New York 11794, USA
‡ Institute of Materials and Environmental Chemistry, Research Centre for Natural Sciences, Hungarian

Academy of Sciences, Pusztaszeri út 59-67, Budapest, HU-1025, Hungary
§ Instituto de Qmica Física Rocasolano, CSIC, C/Serrano 119, 28006 Madrid, Spain
¶ Email:

1


November 5, 2013

13:17

9in x 6in

Advances in Multi-Photon Processes and Spectroscopy Volume 21

b1629-ch01

www.pdfgrip.com

2

Advances in Multi-Photon Processes and Spectroscopy

but also allow for influencing their evolution. While several control schemes
have been described and implemented in diatomic molecules,6–16 this
chapter focuses on following and controlling vibrational dynamics in
a family of small polyatomic molecules — the halogenated methanes
CH2 XY (X, Y = I, Br, Cl . . .). Being small enough to allow for
high-level ab initio electron-structure calculations, but offering sufficient
complexity for chemical relevance (e.g. atmospheric chemistry, bond
selective dissociation, conical intersections), and presenting a homologous
series for laser selective chemistry, these molecules are ideal for testing
different control schemes, characterizing electronic wavepackets generated
via strong-field ionization (SFI), and for implementing strong field control
over bond breaking.
In this chapter, we outline many of the basic physical and computational
principles underlying the dynamics and control, and discuss several
measurements and calculations which illustrate them. The first few sections
deal with solving the time-independent and time-dependent Schrödinger
equation (TISE and TDSE, respectively) for polyatomic molecular systems
via ab initio electronic structure theory and wavepacket propagations. The
following sections give a brief and simple discussion of basic principles
required to understand strong field control, including AC Stark shifts,
multiphoton transitions, dressed states and SFI. After these basic ideas are
introduced, we discuss the ideas and implementation of photon locking,

spatial hole burning, and phase-dependent dissociation. The term “photon
locking” (or “optical paralysis”),17–23 is used to describe the mixing (or
dressing) of two potential energy surfaces in order to lock a vibrational
wavepacket in position. “Hole burning” (also termed r-dependent excitation
or “Lochfrass”)24, 25 uses strong field excitation to reshape a vibrational
wavepacket by population transfer in a spatially narrow window. Similar
approaches, using position dependent ionization or strong field driven AC
Stark shifts, have been used to create or reshape molecular wavepackets
in diatomic molecules.26–31 Other works using strong fields focused on
using light-dressed states to control the branching ratio in dissociation.32–36
Finally, we show how pump-probe spectroscopy of vibrational dynamics in
conjunction with electronic structure and quantum dynamics can be used
to characterize electronic wavepackets generated via strong-field molecular
ionization. We conclude with a discussion of future perspectives.


November 5, 2013

13:17

9in x 6in

Advances in Multi-Photon Processes and Spectroscopy Volume 21

b1629-ch01

www.pdfgrip.com

Wavepackets Driven by Strong-Field Multiphoton Ionization

3

1.2. Theoretical Concepts
1.2.1. The time-independent Schrödinger equation and its
implications on dynamics
Understanding a chemical reaction induced and/or controlled by strong
laser pulses at the molecular level, requires the simulation of motion
of nuclei under the influence of an external electric field. This can be
done either classically or quantum-mechanically. In both cases the forces
governing the motion of the nuclei must be determined either a priori or
on-the-fly for all the relevant configurations. A fundamental approximation
here is the Born–Oppenheimer (BO) approximation which — based on the
huge difference between masses of electrons and nuclei — assumes that
the motion of nuclei and that of the electrons are separable, i.e., electrons
adjust to a nuclear configuration abruptly and the nuclei move in an effective
field of the electrons, expressed by the electronic ground- or excited-state
potential, V(R). (R represents the coordinates of the nuclei and accordingly,
we will denote a vector as a and a matrix as A in the following.) Apart from
the most simple cases (when one can use some analytic functions for V(R)
fitted to spectroscopic data), the forces acting on the nuclei — being usually
simply the gradient of V(R) — are obtained by solving the time-independent
Schrödinger equation (TISE) for the electronic system. Treating the motions
of both the electrons and the nuclei quantum-mechanically, the system is
described by the total wavefunction, (r, R), written as
(r, R, t) =

ψn (R, t)φn (r, R),

(1.1)

n

where φn (r, R) are the eigenfunctions of the electronic TISE,
ˆ e φn (r, R) = Vn (R)φn (r, R)
H

(1.2)

and ψn (R) are the nuclear wavefunctions in electronic states n. In Eq. (1.2),
ˆ e is the Hamilton operator of the whole system for fixed nuclei. In
H
the semiclassical dipole approximation the motion of the nuclei in the
presence of an external electric field, ε(t), is governed by the time-dependent
Schrödinger equation (TDSE). In matrix form,
∂
ih¯ ψ = (T + V − µε(t))ψ,
∂t

(1.3)


November 5, 2013

13:17

9in x 6in

Advances in Multi-Photon Processes and Spectroscopy Volume 21

b1629-ch01

www.pdfgrip.com

4

Advances in Multi-Photon Processes and Spectroscopy

where T is the kinetic energy operator for the nuclei and µ is the dipole
matrix with elements µnm defined as
µnm = φn |er|φm ,

(1.4)

with e being the electron charge and the elements of the ψ vector are the
ψn (R) wavefunctions. In the BO approximation V is a diagonal matrix
with elements being the Vn (R) solutions of Eq. (1.2). In the following,
we assume the laser field polarization and the dipole moment vector to be
aligned and hence, neglect their vectorial properties.
Depending on the size of the system and the required accuracy, solving
the electronic TISE can be very time-consuming so that this is the bottleneck
from the point of view of the simulation time. Since quantum-dynamical
simulations require the solution of the electronic TISE for several nuclear
configurations and also the solution of the nuclear TDSE can become very
costly, such computations can only be performed in reduced dimensionality.
Therefore, the first step is to choose coordinates appropriate to the process
under investigation (e.g., bond length in case of a dissociation). Using
normal-mode coordinates (e.g., in case of a bending motion) can simplify
the numerical treatment of the nuclear TDSE considerably. Normal-mode
coordinates are determined by diagonalizing the mass-weighted Hessian
matrix, the elements of which are the second derivatives of the potential

energy, V , with respect to Cartesian displacement coordinates of the nuclei
from their equilibrium configuration. Having determined the V(R) on a
grid in the space of the selected coordinates, the eigenfunctions belonging
to V(R) can be determined by solving the TISE for the nuclei, e.g., by
the Fourier-Grid-Hamiltonian method.37 In most cases, the lowest-energy
vibrational eigenfunction represents the initial nuclear wavefunction for the
quantum-dynamical simulations.
In the following, we consider two different approaches to solve the
electronic TISE: (i) the wavefunction-based (ab initio) methods and (ii)
the density-based (Density Functional Theory, DFT) methods. Ab initio
methods start from the Hartree–Fock (HF) wavefunction, which is an antisymmetrized product (a Slater-determinant) of one-electron spin-orbitals
(molecular orbitals, MO).38 These orbitals are products of a spatial part and
the spin-eigenfunction. In practice, the spatial orbitals are constructed by
linear combinations of atomic orbitals (LCAO), the so-called basis set. At


November 5, 2013

13:17

9in x 6in

Advances in Multi-Photon Processes and Spectroscopy Volume 21

b1629-ch01

www.pdfgrip.com

Wavepackets Driven by Strong-Field Multiphoton Ionization

5

the HF level of theory, the electronic Hamiltonian is a sum of one-particle
operators, the so-called Fock-operators. In this theory, the expansion coefficients in the LCAO are determined by solving the TISE in a self-consistent
iterative procedure (called self-consistent field (SCF)) which — according
to the variational principle — results in the lowest-energy electronic
eigenfunction. Such a wavefunction fulfills the Pauli exclusion principle
for fermions, it accounts for the correlation between electrons of the same
spin. However, methods based on one Slater determinant, as HF, cannot
describe the correlated motion of electrons completely and are generally
not appropriate to describe excited electronic states. The correlation effects
missing from HF-theory can be classified as static and dynamic correlations.
The former arises e.g., in bond dissociations or when different electronic
excited states get close in energy. Description of such situations requires
multiconfigurational wavefunctions, which are linear combinations of
different Slater determinants, obtained by promoting one or more electrons
from occupied MOs of the reference Slater determinant to unoccupied
ones. Typical multiconfigurational wavefunctions include only the most
important determinants. In the complete active space self-consistent field
(CASSCF) method,39 these configurations are constructed by all possible
arrangements of electrons within a properly selected small set of orbitals,
the so-called active orbitals, and the coefficients of these configurations (CI
coefficients) are optimized together with the MO coefficients in the SCF
procedure. In this framework, excited electronic states are computed in
the state-averaged CASSCF (SA-CASSCF) procedure, where the average
energy of a prescribed number of electronic states is minimized in the SCF.
While multiconfigurational procedures like SA-CASSCF account for
static or long range electron correlation effects, they are usually not
good enough to obtain spectroscopic accuracy, since they do not include
enough dynamic correlation. This type of correlation is the result of the

instantaneous repulsion of electrons, i.e., the fact that they avoid each other
during their motion. The multi-reference configuration interaction (MRCI)
method40 offers a solution to this problem. It relies on a multiconfigurational
wavefunction (typically a CASSCF wavefunction) as a reference function
and includes further single, double, etc. CI excitations on top of it. This
highly accurate method suffers however from two shortcomings: First, it is
applicable only to relatively small molecules due to its huge computational


November 5, 2013

13:17

9in x 6in

Advances in Multi-Photon Processes and Spectroscopy Volume 21

b1629-ch01

www.pdfgrip.com

6

Advances in Multi-Photon Processes and Spectroscopy

cost and second, it is not size-consistent and therefore requires further
corrections, such as Davidson correction.41
A popular alternative to MRCI is the CASPT2 method in which secondorder perturbation theory is applied to a SA-CASSCF reference wavefunction.42, 43 This method, whose success still strongly depends on the adequate
choice of the active space, is also able to provide good estimates of electronic
energies, while it is — due to its considerably lower computational costs —

applicable to larger systems than MRCI. Computing different electronic
states separately by CASPT2 can however result in nonorthogonal electronic wavefunctions. This is an unphysical solution of the nondegenerate
eigenvalue problem and can cause inaccurate results when the electronic
states are close in energy and their wavefunctions are mixed with one
another at the SA-CASSCF level. A solution to this problem is offered
by the multistate version of the CASPT2 method44 in which an effective
Hamiltonian is constructed from the single-state solutions and diagonalized
producing new wavefunctions and accurate excitation energies.
In addition to methods based on multiconfigurational wavefunctions,
there are several other approaches based on a single reference description
of the ground state, which are used to compute excited electronic states.
Such methods are e.g., the configuration interaction singles (CIS)45 which is
simple and fast but often cannot even provide qualitatively correct results46
or the equation of motion coupled cluster (EOM-CC) methods47 which can
produce accurate excitation energies but only at a high excitation level and
therefore for an extraordinary computational cost.
Among methods based on the single-reference ground-state description, the most popular for computing excited states is an extension of
DFT: the time-dependent density functional theory (TDDFT). The original
DFT is based on the finding, that all molecular electronic properties
(including energy and wavefunction) are uniquely determined by the
electronic ground-state electron density.48, 49 The energy of the electronic
ground state is a functional of the ground-state electron density and the true
density minimizes this energy functional. The form of this functional is,
however, unknown. Plenty of high quality functionals have been developed,
the difference among them being the way they construct the so-called
exchange-correlation part of the functional. One of the most widely used
functional is the B3LYP.50–52


November 5, 2013

13:17

9in x 6in

Advances in Multi-Photon Processes and Spectroscopy Volume 21

b1629-ch01

www.pdfgrip.com

Wavepackets Driven by Strong-Field Multiphoton Ionization

7

Solving the frequency-dependent polarizability equations,53, 54
TDDFT is able to determine the excitation energies and transition dipole
moments (TDFs) without explicitly determining the electronic states. In
contrast to CASSCF-based methods, TDDFT is much more simple to use
as it does not require the — sometimes tedious — construction of a proper
active space. In addition, it is much faster and applicable to much larger
systems than multiconfigurational methods. The main disadvantage of the
method is that the single configuration for the ground state does not allow
a correct description of double or higher excitations and the method is
unable to treat degenerate situations correctly. Furthermore, in contrast to
multiconfigurational methods, where the accuracy of the computations can
be systematically improved by increasing the number of configurations
(e.g., increasing the active space in CASSCF), in case of DFT there is no
universal functional equally good for any system and there is no way to
systematically improve the accuracy.

1.2.2. Spin-orbit coupling and diabatic vs. adiabatic states
The electron spin, which cannot be classically understood, is an intrinsic
angular momentum of the electron. It gives rise to a magnetic moment,
which can interact with the magnetic field that is created when the electron
orbits the nucleus. This interaction is consequently termed spin-orbit
coupling (SOC). The spin arises naturally from a relativistic description of
the electron, as in Dirac’s theory.55 However, the Dirac equation is a singleparticle equation and a many-body equation has not yet been derived.56
Thus, approximate Hamiltonians are used for the electron system, e.g., the
Dirac–Coulomb–Breit (index DCB) operator:
ˆ DCB =
H

n
i=1

hˆ D (i) +

n

n

i=1 i
1
− Bˆ ij ,
rij

(1.5)

which — besides the well-known Coulomb interaction r1ij — contains the

ˆ 57 The latter
Dirac single-particle Hamiltonian hˆ D and the Breit operator B.
accounts mainly for SOC while the former accommodates predominantly
other, scalar relativistic effects.


November 5, 2013

13:17

9in x 6in

Advances in Multi-Photon Processes and Spectroscopy Volume 21

b1629-ch01

www.pdfgrip.com

8

Advances in Multi-Photon Processes and Spectroscopy

The Breit operator can be transformed to the so-called Breit–Pauli
operator,58, 59 which in principle can be solved numerically but contains
many two-electron integrals. The latter can be approximated in the spinorbit mean-field operator approach, where a single particle is treated in
a mean-field of all the others (similar to HF theory).60 The numerical
implementation is called atomic mean field integrals (AMFI).61
The Dirac Hamiltonian hˆ D contains the so-called Dirac matrices, which
are of size 4 × 4, and consequently, the corresponding wavefunction has to
be a 4-component vector (called a 4-spinor).57 They contain contributions

of electronic as well as positronic type. Note that the latter are not related
to positrons but rather negative energies and thus, unphysical artifacts.62
According to Douglas–Kroll theory, these electronic and positronic states
can be decoupled by a unitary transformation of the Dirac Hamiltonian,
where the latter is then in a block-diagonal form.63 The method was later
adapted for numerical implementation by Hess.64 The Douglas–Kroll–Hess
method is nowadays used in many quantum chemistry packages and
provides scalar relativistic corrections at low computational cost.62
The relativistic corrections change the potential shape while the SOCs
introduce off-diagonal elements in the Hamiltonian matrix. In cases when
the potentials get close in energy, these off-diagonal elements have the effect
that population is transferred between the different electronic states. The
same effect can also be introduced by other nonadiabatic couplings, e.g., the
commonly evaluated kinetic couplings (also called derivative couplings),
which can be transformed to potential couplings. In all these cases, the BO
approximation breaks down and a single potential is not enough to describe
the dynamics of the system.21
As indicated above, the off-diagonal elements can be in the potential
part as well as in the kinetic part of the Hamiltonian. Different representations exist, where these couplings are transformed in order to ease their
application in different methods. First, we focus in a representation that
makes use of adiabatic potentials. In this so-called adiabatic picture,
the potential matrix is diagonal and the eigenvectors of this matrix are
the adiabatic eigenfunctions of the system, which form the basis for the
expansion of the total wavefunction. Note, that the term “adiabatic” is
sometimes used in a sloppy way to describe simply the output of electronic
structure calculations based on the BO approximation. Typical programs


November 5, 2013

13:17

9in x 6in

Advances in Multi-Photon Processes and Spectroscopy Volume 21

b1629-ch01

www.pdfgrip.com

Wavepackets Driven by Strong-Field Multiphoton Ionization

9

yield adiabatic potentials as long as SOC is not considered. As soon as
SOCs are computed, they are usually given as potential couplings and
the term “adiabatic” for the corresponding potentials (i.e., the diagonal
elements of the nondiagonal matrix) is not appropriate anymore. Only after
a diagonalization of the potential matrix, the adiabatic picture is obtained.
Also the laser interaction can be regarded as a potential coupling. If the
matrix including these dipole couplings is diagonalized, the result are the
so-called field-dressed states (see Sec. 1.2.6).
If the potential matrix is not diagonal, we speak of a diabatic
representation. However, there is no unique definition of a diabatic picture
and great care has to be taken in order to avoid misunderstandings. Here,
we shortly concentrate on a special case. Each diabatic representation
has its respective basis functions. If the wavefunction character of every
eigenfunction is retained and thus, the basis functions are time-independent,
we speak of a spectroscopic representation since spectroscopic properties
very much relate to the wavefunction character. Sometimes, this special case

is also termed as “the” diabatic representation. Note that all representations
can in principle be interconverted by similarity transformations (although
often difficult in practice).
1.2.3. Nuclear time-dependent Schrödinger equation
In this section, we describe the possible ways to solve the TDSE, see
Eq. (1.3), focusing on vibrational one-dimensional systems (i.e., we use
R instead of R), where the kinetic operator can be described as a diagonal
matrix within the BO approximation with elements:65
h¯ 2 ∂ ∂
pg
ˆ pˆ
Tˆ = −
g
=
,
2 ∂R ∂R
2

(1.6)

where g represents the inverse of the moved mass in the coordinate R.
This coordinate can be an internal coordinate (e.g., a bond distance) or a
collective coordinate (e.g., a normal mode). Depending on the definition
of R, g can be a function (e.g., the bending angle), or a constant (e.g., the
reduced mass belonging to some normal vibrational mode of a polyatomic
molecule),65 in which case the kinetic operator is just Tˆ = gpˆ 2 /2. Applying
this definition of the kinetic operator to the TDSE, we obtain a series of


November 5, 2013

13:17

9in x 6in

Advances in Multi-Photon Processes and Spectroscopy Volume 21

b1629-ch01

www.pdfgrip.com

10

Advances in Multi-Photon Processes and Spectroscopy

equations coupled by the electric field,
ih¯

∂
pg
ˆ pˆ
ψm (R, t) = (Vm +
)ψm (R, t) −
∂t
2

µnm ε(t)ψm (R, t).

(1.7)

n

This set of equations can be solved by applying the Hamiltonian to the
wavefunction, where the procedure mainly depends on the chosen basis.
On the one hand, it is possible to choose the i vibrational eigenfunctions of
the field-free Hamiltonian for the m electronic potentials, Vm , as a basis,
ψm (R, t) =

ci,m (t)φi,m (R),

(1.8)

i

where (Vm + Tˆ )φi,m (R) = Ei,m φi,m (R) and ci,m (t) are the amplitudes at
every time. By inserting Eq. (1.8) in Eq. (1.7) and projecting on φj,n , we
obtain
ih¯

∂
cj,n (t) =
∂t

j,n

j,n

Ei,m − µi,m ε(t) ci,m (t),

(1.9)

i

where the integrals of the time-independent Hamiltonian elements are
j,n
∗ (V
ˆ + Tˆ )φi,m dR, which in case of orthogonal eigenfunctions
Ei,m = φj,n
j,n

j,n

are just Ei,m = Ei,m δij δmn . Similarly, µi,m are the matrix elements of the
dipole moment, which can be related to the electronic dipole moment µnm
j,n
∗ µ φ dR. If there is no electric field, this equation can
as µi,m = φj,n
nm i,m
be analytically solved and the time evolution of the coefficients is just
i
ci,m (t) = ci,m (0) exp − Ei,m t .
h¯

(1.10)

However, neither the calculation of the eigenfunctions nor the solution
including an electric field are analytic beyond the harmonic model and the
numerical approximation requires many basis functions to solve the above
equation.21
On the other hand, we can work directly in the one-dimensional grid

R, where Vm and µmn are directly defined. In this case, the problem is the
definition of the kinetic operator, which is readily applied in the momentum
space but not in the coordinate space. However, the nuclear wavefunction
can be easily transformed to the momentum space by a Fourier transform.


November 5, 2013

13:17

9in x 6in

Advances in Multi-Photon Processes and Spectroscopy Volume 21

b1629-ch01

www.pdfgrip.com

Wavepackets Driven by Strong-Field Multiphoton Ionization

11

In the momentum representation, the kinetic operator is diagonal,
FT

ci,m (t) |ri −→ ψm (p, t) =

ψm (R, t) =
i

ci,m (t) |pi ,

(1.11)

i

where |ri and |pi are basis sets with zeros in all the grid points and 1 when
R = ri and p = pi , respectively. Using this definition, the application of
the kinetic operator to the wavefunction is simple. For example,
1
− gpˆ 2
2

ci,m (t) |pi = −
i

i

g
ci,m (t) p2i |pi
2

and similarly in the potential part, where Vm |ri
µmn |ri = µmn (ri ) |ri .

(1.12)

= Vm (ri ) |ri and

1.2.3.1. Second-order differentiator

One of the simplest methods to solve the TDSE on a grid is the second-order
differentiator (SOD) method.66 In this approach, the wavefunction at time
t + t is expanded in a second-order Taylor expansion:
ψ(t +

t) = ψ(t) +

t

t 2 ∂2
∂
ψ(t),
ψ(t) +
∂t
2 ∂t 2

(1.13)

where we can obtain the temporal derivative of ψ using Eq. (1.7). To
avoid the application of the Hamiltonian twice, it is possible to modify
the propagator, so that21
ψ(t +

t) = ψ(t −

t) + 2 t

∂
ψ(t).
∂t

(1.14)

The main problem of this propagator is the numerical instability. Since
the propagator is not unitary, the time-step t should be very small to assure
the conservation of the norm.
1.2.3.2. Split-operator method
A more elaborated propagator is the so-called split-operator (SO) technique.67–69 In this method, we integrate the TDSE from Eq. (1.10) and


November 5, 2013

13:17

9in x 6in

Advances in Multi-Photon Processes and Spectroscopy Volume 21

b1629-ch01

www.pdfgrip.com

12

Advances in Multi-Photon Processes and Spectroscopy

arrive at a solution as:
ψ(t +

i

i
ˆ t ψ(t) = exp − (W
ˆ + Tˆ ) t ψ(t),
t) = exp − H
h¯
h¯
(1.15)

ˆ = Vˆ − µE represents the potential part of the Hamiltonian,
where W
including the field interaction, which is represented in coordinate space,
and Tˆ is the kinetic part that should be applied in momentum space. As
they are not represented on the same grid, it is not possible to apply the
exponential including both potential and kinetic parts at the same time and
ˆ and Tˆ do
they have to be split into two terms. The problem is that W
not commute, and it is not exact to describe the exponential term as the
ˆ = exp(W
ˆ ) exp(Tˆ ).
multiplication of two noncommuting ones, i.e., exp(H)
This problem is solved in the SO by splitting one of the parts in two. For
example, splitting the potential part,
ψ(t +

i
t
i
i ˆ t
ˆ
exp − Tˆ t exp − W

ψ(t).
t) ≈ exp − W
h¯
h¯
h¯
2
2
(1.16)

The application of the potential part is very simple even if it is not
diagonal, e.g., when the electric field couples two electronic states. In that
case, the operators are 2 × 2 matrices, where every matrix element depends
on R represented by the grid points ri . The potential part of the propagation
can be easily carried out after a diagonalization of the W matrix at
every ri ,
t
t
i
i
ψ = Z exp − Z† W Z
Z† ψ
exp − W
h¯
h¯
2
2
t
i
= Z exp − D
Z† ψ,

h¯ 2

(1.17)

where D is a diagonal matrix containing the eigenvalues of W and Z is the
unitary transformation matrix containing the corresponding eigenvectors.
In contrast to the SOD, the SO is unitary and very stable. However, the
kinetic operator in the exponent cannot be applied using Fourier transform
when g depends on the coordinate.