a qua ntu m -cla ss ical investiga tion of envi r o nment ... · 4.1 1 ground stat e p opul atio n...

A Quantum-Classical Investigation of EnvironmentalEffects on Electronic Dynamics at Conical Intersections

by

Aaron Kelly

A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Chemistry

University of Toronto

Copyright c! 2010 by Aaron Kelly

Abstract

A Quantum-Classical Investigation of Environmental E!ects on Electronic Dynamics at

Conical Intersections

Aaron Kelly

Doctor of Philosophy

Graduate Department of Chemistry

University of Toronto

2010

In this thesis we employ the methods of quantum classical Liouville theory in order to

explore the e!ects of an external environment on electronic dynamics, in systems contain-

ing conical intersections. In studying a simple, yet nontrivial, model in the gas phase we

find that the short-time quantum dynamics are well approximated by a hybrid MC/MD

solution to the QCL equation. Upon including an external environment the popula-

tion transfer profile changes, based on the various parameter values chosen. Electronic

decoherence and the partial/complete destruction of geometric phase e!ects are also ob-

served. Based on earlier work on master equation dynamics in QCL theory, we find that

a Markovian approximation to the dynamics for systems containing conical intersections

is not easily justified. In order to access longer time-scale phenomena another, less com-

putationally demanding, solution method was also derived in the mapping basis. This

method involves more severe approximations to the QCL equation than the MC/MD

method, and as such is is not as physically robust, it still manages to reproduce some

important aspects of the dynamics over a picosecond timescale.

ii

Dedication

To my teachers, my family and my friends.

iii

Contents

1 Scope and Focus 1

1.1 The Born - Oppenheimer Approximation . . . . . . . . . . . . . . . . . . 1

1.2 Landau - Zener Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Conical Intersections and the Geometric Phase . . . . . . . . . . . . . . . 5

1.4 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Overview of Quantum-Classical Theory 9

2.1 Quantum-Classical Liouville Dynamics . . . . . . . . . . . . . . . . . . . 10

2.2 Representations and Solutions . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 The subsystem basis . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.2 The adiabatic basis . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.3 The force basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.4 The mapping basis . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Other Quantum-Classical Approximations . . . . . . . . . . . . . . . . . 24

2.3.1 Mean field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.2 Surface-hopping dynamics . . . . . . . . . . . . . . . . . . . . . . 26

2.4 Observables and Correlation Functions . . . . . . . . . . . . . . . . . . . 31

2.5 Reaction Rate Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5.1 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.6 Comments and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

iv

3 Molecular Models 42

3.1 A Two State -Two Mode Conical Intersection: The FLV Model . . . . . 43

3.1.1 Avoided Crossing Model . . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Photoisomerization of a Protonated Schi! Base . . . . . . . . . . . . . . 50

3.3 Another Two State -Three Mode Model . . . . . . . . . . . . . . . . . . 54

4 Molecular Dynamics 57

4.1 Trotter-Based Propagation Algorithm . . . . . . . . . . . . . . . . . . . . 57

4.2 Initial state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3 Numerical Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.4 Population and Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.4.1 BCH Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5 E!ects of the Environment 79

5.1 Population Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2 Electronic Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.3 Geometric Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.4 On the Markovian Approximation . . . . . . . . . . . . . . . . . . . . . 89

6 Another Solution: The Mapping Basis 94

6.1 Approximate QCL equation . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.2 Solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7 Conclusions 104

Bibliography 106

v

List of Figures

1.1 Schematic view of two adiabatic (non-crossing) and the corresponding dia-

batic (crossing) potential energy surfaces in one dimension spatial dimen-

sion, X. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Nonadiabatic vs adiabatic reaction rate for ! = 1.0. The blue curve is the

adiabatic result while the green curve is the nonadiabatic QCL rate. . . . 39

2.2 Forward rate coe"cient kAB(t) as a function of time for ! = 1.0. The

upper (blue) curve is the adiabatic rate, the purple curve is the result

obtained by Tully’s surface-hopping algorithm, the middle (black) curve

is the quantum master equation result, the green curve is the QCL result,

and the lowest dashed line (grey) is the result using mean-field dynamics. 40

3.1 Plot of the FLV diabatic electronic surfaces V00(X, Y ) and V11(X, Y ). The

parameter values used in this figure are summarized in Eq.(3.8). . . . . . 43

3.2 Plot of the diabatic electronic coupling V01(X, Y ) for " = 0.01. Brighter

areas correspond to positive coupling, and darker regions correspond to

negative coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3 Plot of the FLV adiabatic electronic surfaces for " = 0.01. . . . . . . . . 46

3.4 Contour plot of the X-component of the nonadiabatic coupling vector

d01(X, Y ) · X for " = 0.01. Bright regions correspond to strong positive

coupling, and dark regions correspond to near zero coupling. . . . . . . . 47

vi

3.5 Contour plot of the Y -component of the nonadiabatic coupling vector

d01(X, Y ) · Y for " = 0.01. Bright areas correspond to positive coupling,

dark areas correspond to negative coupling strength, and grey regions to

near zero coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.6 Seams of conical intersections prescribed by the BCH and XYZ models. . 56

4.1 Propagation and branching of the wavepacket along the X coordinate.

Results from QCL simulations of the original FLV model in the adiabatic

basis with " = 0.01au. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2 Average number of quantum transitions per trajectory as a function of

time for the QCL and Tully surface hopping simulations of the original

FLV model in the adiabatic basis, with " = 0.01. . . . . . . . . . . . . . 65

4.3 Distribution of the number quantum transitions per trajectory for the

QCL and Tully surface hopping simulations performed of the original FLV

model in the adiabatic basis, with " = 0.01. . . . . . . . . . . . . . . . . 66

4.4 Distribution (normalized) of Monte Carlo weights recorded at t = 70fs

for QCL simulations of the original FLV model performed in the adiabatic

basis, for a range of coupling strengths. . . . . . . . . . . . . . . . . . . . 68

4.5 Distribution of quantum transitions as a function of the X and Y coor-

dinates for QCL and Tully surface hopping simulations performed in the

adiabatic basis, with " = 0.01. . . . . . . . . . . . . . . . . . . . . . . . . 69

4.6 Distribution of quantum transitions as a function of the torsional coordi-

nate #, obtained for QCL simulations of the BCH model performed in the

adiabatic basis, with c! = 2.5eV . . . . . . . . . . . . . . . . . . . . . . . 70

4.7 Probability density of transition energies for the surface hopping imple-

mentation of the QCL equation of the original FLV model, as well as the

generalized model with a set of di!erent characteristic bath frequencies.

For each case " = 0.01. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

vii

4.8 Evolution of the ground state adiabatic population PS0(t) for the gas phase

model. The quantum results are taken from [1], and the quantum-classical

results refer to those generated from the QCL equation using the Trotter-

based algorithm in the adiabatic basis, with " = 0.01, and # = 2000, using

107 trajectories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.9 Real part of the o!-diagonal element of the subsystem density matrix as a

function of time, for " = 0.01. Note that the scale on the ordinate changes

from the right panel to the left panel. . . . . . . . . . . . . . . . . . . . 73

4.10 Ground adiabatic state population at t = 50fs as a function of electronic

coupling strength in the FLV model, as predicted by various approximate

nonadiabatic methods and exact quantum mechanics (QM). . . . . . . . 74

4.11 Ground state population as a function of time in the BCH model, for

c! = 0.25eV . In the case with the bath present $c = 0.05a.u., and % = 1. 77

4.12 Electronic coherence as a function of time in the BCH model, for c! =

0.25eV . In the case with the bath present $c = 0.05a.u., and % = 1. . . . 78

5.1 Ground adiabatic state population PS0(t) versus time, for " = 0.01, %=0.1,

and a range of di!erent bath characteristic frequencies, $c. . . . . . . . . 80

5.2 Ground adiabatic state populations PS0(t = 50fs) versus " with % = 1, for

a range of di!erent bath characteristic frequencies, $c. For reference, recall

that the frequencies of the nuclear vibrational modes are $X = 219cm!1

and $Y = 850cm!1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.3 Di!erence in the ground adiabatic state populations with the bath on as

compared to the gas phase model as a function of the Kondo parameter

%, for a range of di!erent bath characteristic frequencies, $c, and " = 0.01. 83

5.4 Ground state population at t = 50fs, for " = 0.01 , & = 1, in the avoided

crossing model. In the case with the bath present % = 1. . . . . . . . . . 84

viii

5.5 Ground state population as a function of time in the XYZ model, for

" = 0.01. In cases with the bath present % = 1. . . . . . . . . . . . . . . . 85

5.6 Evolution of the electronic coherence as a function of time, for a selection

of characteristic bath frequencies, for " = 0.01. . . . . . . . . . . . . . . 86

5.7 Purity of the quantum subsystem as a function of time, for a selection of

characteristic bath frequencies, for " = 0.01. . . . . . . . . . . . . . . . . 87

5.8 Electronic coherence as a function of time in the ACM with & = 1 (regular

width), and & = 12 (half-width), for " = 0.01. In cases with the bath

present % = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.9 Electronic coherence as a function of time in the three mode XYZ model,

for " = 0.01. In cases with the bath present % = 1. . . . . . . . . . . . . . 89

5.10 Branching of the wavepacket along the Y coordinate, results from QCL

simulations in the adiabatic basis. Emergence of Berry’s phase is seen as

a node develops on the excited state surface, S1, after the packet leaves

the region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.11 Destruction of the node in the subsystem reduced density along the Y

coordinate, as a function of $c from QCL simulations in the adiabatic basis. 91

5.12 Subsystem density in the excited state as a function of the Y coordinate,

for QCL simulations of the ACM model with & = 1 (regular width), and

& = 12 (half-width), and the XYZ model. . . . . . . . . . . . . . . . . . . 92

5.13 Plot of five di!erent realizations (corresponding to five di!erent initial

phase points (X, PX , Y, PY )) of the bath averaged memory function "M2121 (X, t)#e

versus time, for a selection of di!erent bath frequencies $c, at " = 0.01. 93

6.1 Ground adiabatic state populations PS0(t = 50fs) versus ". The quantum

results are taken from [2], and the force basis results are from [3]. . . . . 101

ix

6.2 Ground adiabatic state populations PS0 as a function of time (in picosec-

onds) for " = 0.01. Here we compare the original FLV conical intersection

model with the ACM for & = 1 (Regular width), and & = 12 (Half-width). 102

6.3 Ground adiabatic state populations PS0 as a function of time (in picosec-

onds) for " = 0.01. Comparison of the mapping basis data: FLV model

results (2D; no bath), XYZ model results (3D; no bath), and generalized

FLV model results (2D; $c = 500cm!1). Only the original FLV model

results remain physical (0 $ PS0 $ 1) for the entire silmulation. . . . . . 103

x

Chapter 1

Scope and Focus

In order to accurately yet practically describe the dynamics of an N -body quantum

system we are faced with a task that becomes exponentially more di"cult as N increases.

For small systems, with roughly ten to one hundred particles, full quantum simulations

can be carried out but at extreme computational cost. The dynamical description of

larger systems thus necessitates employing some approximation to quantum mechanics.

Indeed, in many instances a fully quantum mechanical treatment is probably unnecessary,

and numerous approximate quantum theories have allowed deep insights into a broad

range of physical and chemical problems [4].

1.1 The Born - Oppenheimer Approximation

One of the earliest and most important quantum approximations known in chemistry

is the Born-Oppenheimer approximation [5] (BOA). The BOA is a cornerstone of mod-

ern molecular intuition, enabling the interpretation of many spectroscopic and chemical

phenomena[6]. In brief, it exploits the fact that the nuclei and electrons have disparate

masses, which leads to the approximate separability of their coupled motion. As the two

di!erent types of particles have the same magnitude of electrostatic charge, the forces

they feel will be of similar magnitude. Thus in a stable structure, such as a molecule,

1

Chapter 1. Scope and Focus 2

we may assume that the nuclei and electrons have comparable momenta. However, since

nuclei have are much more massive than electrons (by a factor of about 103), we imagine

that the nuclei move relatively slowly while the electrons tend to zip about with much

higher velocities. Consequently, we may also think of the nuclei having a smaller total

kinetic energy than the electrons; in the context of the Schrodinger equation this amounts

to the assertion that electronic and nuclear motions are uncoupled. More formally, the

time-independant Schrodinger equation for the composite system (in the position repre-

sentation) is

H(r, R, p, P )|$(r, R)# = E(r, R)|$(r, R)#. (1.1)

The multi-dimensional electronic coordinates and momentum operators are(r, p), while

(R, P ) denote the corresponding quantities for the nuclei. The total Hamiltonian, H,

is a sum of kinetic and electrostatic potential energy terms for the collection of charged

particles,

H = Te + TN + Ve + VN + Ve!N . (1.2)

The kinetic energy terms are Te = p2

2m and TN = P 2

2M , and the potential energy terms refer

to the isolated electric potentials Ve and VN , and the static coupling between these two

sets of particles Ve!N . Invoking the BOA involves first solving a Schrodinger equation for

the electrons, in a fixed field of the nuclei. By temporarily neglecting the nuclear kinetic

energy, the electronic Hamiltonian becomes He = H % TN . The total wave-function

may then be written approximately as a product of two separate functions describing the

electrons and nuclei, |$(r, R)# = |'(r; R)#|((R)#, and the electronic Schrodinger equation

is

He(r, R)|'(r; R)# = Ee(R)|'(r; R)#. (1.3)

The electronic wave-functions, '(r; R), and energies, Ee(R), thus depend parametically

on the coordinates of the nuclei. By repeatedly solving (1.3) for a suitable grid in R,

one may obtain the set of electronic energy surfaces. The adiabatic theorem [7] ensures

that this procedure is possible, as it guarantees that the eigenstate does not change for


infinitesimal changes in R. The adiabatic surfaces obtained from the repeated solution

of (1.3), also called Born-Oppenheimer surfaces, may then be used to solve the e!ective

nuclear Schrodinger equation,

[TN + Ee(R)]|((R)# = E|((R)#. (1.4)

Since the electronic energy levels actually do depend on the kinetic energy of the

nuclei, there are instances where the BOA breaks down. The neglect of the nuclear

kinetic energy term in the electronic Schrodinger equation is often a good approxima-

tion due to the mass di!erence between electrons and nuclei, yet the e"cacy of the

BOA also intimately depends on the validity of the adiabatic approximation. When two

Born-Oppenheimer surfaces intersect the electronic and nuclear systems may become

strongly coupled, resulting in sudden changes in the electronic eigenstate and correspond-

ing changes in the momenta of the nuclear system. In situations of this type the BOA

is inapplicable and the dynamics are may referred to as non-adiabatic. Non-adiabatic

dynamics comprises a vast field of processes in physical, chemical, and biological systems

[4].

1.2 Landau - Zener Theory

Two of the earliest attempts to describe quantum dynamics beyond the Born-Oppenheimer

approximation are independently due to L. Landau and C. Zener. In the BOA picture

the adiabatic states and potential energy surfaces which are obtained simply correspond

to the energy eigenstates and eigenvalues of the electronic Hamiltonian (neglecting the

nuclear kinetic term, of course). Another basis representation, known as the diabatic

basis, is (nonuniquely [6]) defined such that nuclear kinetic coupling terms vanish. That

is,

dn,diaij (R) = "'dia

i (r; R)| )n

)Rn|'dia

j (r; R)# = 0, n = 1, 2. (1.5)


In one dimension the diabatic potential energy surfaces are quite similar to the adiabatic

picture, except for regions near avoided crossings. An avoided crossing in one dimension

is depicted in Fig.(1.1). These regions are characterized by a degeneracy in the diabatic

picture, corresponding to a small adiabatic energy gap. In the semiclassical model con-

5 6 7 8

50

100

150

200

Ferretti_model.nb 1

X

E

Figure 1.1: Schematic view of two adiabatic (non-crossing) and the corresponding dia-

batic (crossing) potential energy surfaces in one dimension spatial dimension, X.

sidered by Landau and Zener a two level quantum system is coupled to a one dimensional

classical degree of freedom. Similar expressions were separately formulated for the quan-

tum transition probability involving two intersecting diabatic potential energy surfaces;

by Landau working in the perturbative limit [8] and by Zener in a general case [9]. The

quantum system is prepared at some time in the infinite past, in one of the diabatic states,

well outside the crossing region. The electronic transition probability is then defined as

the asymptotic (t & ') probability of finding the system in the other diabatic state.

The following set of approximations were made in order to employ analytical techniques


to the full quantum mechanical equations of motion. First, the form of the diabatic

electronic coupling (Vc) is assumed to be constant. Second, the slopes of the upper and

lower diabatic energy surfaces are assumed to be constant, (i.e) the diabatic potentials

are linear. Finally, it is assumed that the nuclear dynamics are purely classical, and the

nuclear motion only enters the problem parametrically. Given these assumptions Zener

showed that the electronic Schrodinger equation may be cast into Weber form, which

corresponds to family of di!erential equations with known asymptotic properties [10] .

The Landau-Zener (LZ) formula for the transition probability is

PLZ = exp[%2*|Vc|2

hv%F], (1.6)

where v is the velocity of the classical coordinate as the system passes through the

crossing point, and %F is the di!erence between the slopes of the diabatic surfaces at

the crossing point.

The LZ theory has enabled the understanding many quantum phenomena which can

be modeled using a one dimensional curve-cossing model [11], and many attempts have

been made to extend its regime of validity [11]. However, no successful generalizations

have been obtained which can adequately describe multidimensional curve-crossing dy-

namics that is not well represented by a reduction to a one dimensional model [11].

1.3 Conical Intersections and the Geometric Phase

Indeed, the possibility of exact degeneracies in multidimensional potential energy sur-

faces has been long known by theorists [12]; electronic degeneracies may be required by

symmetry (as in the case of the Jahn-Teller e!ect [12]), and can arise in certain regions

of the space of nuclear configurations. An early paper by Teller [12] concisely notes that

exact degeneracies can occur in the electronic manifold of all molecules containing at least

three atoms; a fact which could have important implications in photochemical processes.

In the simplest case of a two dimensional potential energy surface, Teller showed that the


local geometry of the degeneracy resembles that of a double cone, hence the term conical

intersection has been used to classify multidimensional electronic degeneracies. These

structures were long thought to be very rare occurrences by most chemists. However,

with the advent of modern (ab initio) electronic structure methods, conical intersections

have been found to be quite common in many polyatomic systems [6]. In fact, the exis-

tence of conical intersections is thought to be essential in a host of important processes

including many biologically relevant photo-induced chemical reactions. However, since

the equations of quantum mechanics are prohibitively hard to solve for these systems,

and because the time and length scale of the coupled electronic dynamics and nuclear

motion is very small, insight into these systems was very di"cult to obtain.

An important theoretical work in the sixties by Herzberg and Longuet-Higgins [13]

rekindled some interest into these issues. They pointed out that a serious problem that

arises concerning the structure of the total wave-function; if one considers the change

in the phase of the electronic wave-function when the nuclei complete some adiabatic

circuit, C, in coordinate space. When C encloses a region which contains a degenerate

point, such as a conical intersection, the electronic system accrues a phase of *, which

results in a sign change of electronic wave-function. In order for the total wave-function

to be single-valued, an opposite phase change is required in the nuclear wave-function.

Under the adiabatic approximation this phase change is neglected, and thus it must be

somehow re-introduced into any theory based on adiabatic passage.

Twenty years later Berry formalized these ideas mathematically, and showed how

an evolving quantum system retains a memory of its past motion upon returning to its

initial state. This memory emerges in the form of a geometric phase factor, which is also

known as Berry’s phase. In the special case of adiabatic evolution on some particular

quantum state |'n(R)#, which is returned to its initial state after some time T via some

closed path C, the wave-function may be written as

|'(T )# = exp[% ı

h

! T

0dt"En(R(t"))] exp[%ı"n(C)]|'n(R(0))#. (1.7)


The first factor is the more familiar dynamical phase, which dominates at long times.

Berry showed that the geometric phase, "(C), is a non-integrable quantity which depends

only on the geometry of the problem.

"n(C) = ı"

c"'n(R)| )

)R|'n(R)# · dR (1.8)

This factor takes an especially simple form for any trajectory encircling a conical in-

tersection, exp[%ı"n(C)] = %1, [14] in accordance with the result of Herzberg and

Longuet-Higgins. The geometric phase, like the dynamical phase, can produce inter-

ference phenomena. Indeed, the presence of geometric phase e!ects have been recently

been observed in chemical systems containing a conical intersection [15, 16]. However the

extent to which this e!ect may influence chemical dynamics remains an open question.

With the advent of high performance computing and advanced femtosecond spec-

troscopic techniques [17, 18], interest has been stimulated in studying the electronic

dynamics of various molecular systems (some of which contain conical intersections, e.g.

[19]). Calculations have been performed using exact and approximate quantum dynam-

ics methods for various model molecular systems containing conical intersections. Exact

methodologies include direct wave-packet propagation methods [2], and the MCTDH

method [20]. Approximate theories such as the semiclassical initial value representa-

tion of the Feynman path integral (SC-IVR) [21, 22], and Markovian master equation

descriptions (e.g. Redfield [23, 24], Lindblad [25, 26]) have seen use in attempting to

gain insight into the dynamics induced by conical intersections. Generally speaking the

subsystem dynamics, corresponding to the isolated molecule, have been relatively well

investigated and understood. The e!ects of an external environment on this process are

not yet well-known, and very few theoretical/numerical approaches exist which allow one

to obtain dynamical information.


1.4 Synopsis

In this thesis we shall apply a recently formulated approximation to full quantum me-

chanics, known as the quantum-classical Liouville (QCL) equation, in order to investi-

gate the molecular models involving conical intersections. This approximate dynamical

scheme is useful if one is interested in a small subset of system degrees of freedom whose

quantum mechanical character is important, while the remainder of the system (i.e. the

environment) may be approximated by classical mechanics [27, 28, 29]. For example,

a decomposition of this type is appropriate for a subsystem composed of light particles

(e.g. electrons or protons), interacting with a bath of heavy particles (other molecules).

An overview of QCL theory is o!ered in Chapter 2. In Chapter 3 a selection of molecular

models, involving conical intersections in two and three dimensions, are described. These

models attempt to incorporate the e!ects of electronic and vibrational dephasing and re-

laxation, which have been reported to be important in these systems [25]. In Chapter

4 we discuss a hybrid Monte-Carlo / Molecular Dynamics (MC/MD) algorithm used to

solve the QCL equation. As well, we outline some of the numerical details associated

with the implementation of this algorithm, in order to evaluate its e"cacy. In Chapter

5 we present the MC/MD simulation results on the various conical intersection models,

and attempt to evaluate environmental e!ects where possible. As the MC/MD method

is rather restrictive in terms of total simulation time, we develop another approximate

QCL solution in Chapter 6 in order to attempt to capture longer time-scale behavior.

Finally, Chapter 7 o!ers a summary of the results obtained in this work and a perspective

on future research.

Chapter 2

Overview of Quantum-Classical

Theory

The quantum-classical Liouville equation describes the dynamics of a quantum subsystem

coupled to an almost classical environment. The term “almost” used here refers to the fact

that while the environment evolves by the classical equations of motion in the absence of

coupling to the quantum subsystem, in the presence of coupling a description in terms of

single classical trajectories is no longer possible. After discussing the QCL equation and

its properties, we outline a number of ways one may construct numerical solutions to this

equation by projecting it onto di!erent bases. In certain limits, making approximations,

quantum-classical Liouville dynamics may be reduced to some commonly-used mixed

quantum-classical approaches; in particular, mean field and surface hopping schemes, as

well as the Wigner-Liouville approach. Quantum time correlation functions, which are

related to transport properties, are then discussed. The computation of a reaction rate

for a model quantum chemical system is then described as an instructive example.

9

Chapter 2. Overview of Quantum-Classical Theory 10

2.1 Quantum-Classical Liouville Dynamics

The time evolution of a quantum mechanical system is governed by the quantum Liouville-

von Neumann equation,

)

)t+(t) = % i

h[H, +(t)], (2.1)

where +(t) is the density matrix, H is the total Hamiltonian, and the square brackets

denote the commutator. Quantum-classical Liouville dynamics is an approximation to

this equation that is appropriate for situations where the full quantum system may be

partitioned into a quantum subsystem, and a classical environment. This partition is

motivated by the observation that for many condensed phase processes the quantum

mechanical character of only a few degrees of freedom need be taken into account to

accurately describe the system’s overall dynamics. To this end, we let q = {qi}, i =

1, ..., n be a set of coordinate operators for the n subsystem degrees of freedom with

mass m, while the remaining N environmental degrees of freedom with mass M have

coordinate operators Q = {Qi}, i = 1, ..., N . The total Hamiltonian can then be written

as

H =P 2

2M+

p2

2m+ V (q, Q), (2.2)

where we have written the momentum operators for the subsystem and environment

as p and P , respectively. In keeping with this partition scheme, the potential energy

operator, V (q, Q), V (q, Q) = Vs(q)+Ve(Q)+Vc(q, Q), may be decomposed into subsystem,

environment, and coupling terms respectively.

By performing a partial Wigner transform with respect to the coordinates of the

environment, we obtain a classical-like phase space representation of those degrees of

freedom. The subsystem coordinate operators are left untransformed, thus, retaining the

operator character of the density matrix and Hamiltonian in the subsystem Hilbert space

[30]. In order to take the partial Wigner transform of Eq.(2.1) explicitly, we express the


Liouville-von Neumann equation in the {Q} representation,

)"Q|+(t)|Q"#)t

= % i

h

!dQ""

#"Q|H|Q""#"Q""|+(t)|Q"# % "Q|+(t)|Q""#"Q""|H|Q"#

$. (2.3)

Using the definition of the Wigner transform [31] of the density matrix,

+W (R,P ) = (2*h)!3N!

dZ eiP ·Z/h"R% Z

2|+|R +

Z

2#, (2.4)

and the formula for the partial Wigner transform of a product of two operators [32]

#AB

$

W(R,P ) = AW (R,P )eh!/2iBW (R,P ), (2.5)

Eq.(2.3) becomes

)+W (R,P, t)

)t= % i

h

#HW (R,P )eh!/2i+W (R, P, t)

%+W (R,P, t)eh!/2iHW (R,P )$. (2.6)

The operator & =(%)P ·

%&)R%(%)R ·

%&)P is the negative of the Poisson bracket operator, and

the subscript W indicates the partial Wigner transform. The partial Wigner transform

of the total Hamiltonian is written as,

HW (R,P ) =P 2

2M+

p2

2m+ VW (q, R). (2.7)

The quantum-classical Liouville equation can be derived by formally expanding the

operator on the right side of Eq.(2.6) to O(h). Alternatively, one may justify [30] such an

expansion for systems where the masses of particles in the environment are much greater

than those of the subsystem, M * m. In this case the small parameter in the theory is

µ = (m/M)1/2. This factor emerges in the equation of motion quite naturally through

a scaling of the variables motivated by the classical theory of Brownian motion. One

expands the total system propagator to first order in µ, and truncates the remainder of

the series. Through such an analysis [30], one obtains the quantum-classical Liouville

equation [29, 30, 33, 34, 35, 36, 37, 38, 39, 40, 41]

)+W (R,P, t)

)t= % i

h

%HW (R,P ), +W (R,P, t)

&


+1

2({HW (R,P ), +W (R,P, t)}% {+W (R, P, t), HW (R,P )})

= %iL+W (R, P, t). (2.8)

The last line defines the QCL (super)operator L.

The QCL superoperator has many desirable features required to produce physical

dynamics; it conserves total mass, energy, momentum and phase space volumes [30, 42,

43]. However, it does not provide a fully consistent treatment of mixed quantum-classical

dynamics. The quantum-classical bracket defined by the right side of Eq.(2.8) does not

possess a Lie algebraic structure since it fails to satisfy the Jacobi identity [42, 44]:

[[AW , BW ], CW ] + [[AW , [BW , CW ]] + [[BW , [AW CW ]] = O(h), (2.9)

where the right hand side must be zero for the Jacobi identity to be satisfied.

A detailed discussion of the consequences of this lack of a Lie algebraic structure

can be found in Ref. [42]. There have been attempts to construct quantum-classical

brackets that possess a Lie algebraic structure [45, 46] although these constructions have

been shown to have di"culties [47, 48]. In addition to these attempts, there have been

more recent formulations of quantum-classical dynamics based on di!erent premises that

have a Lie algebraic structure [49]. In spite of these limitations the quantum-classical

Liouville description is one of the most accurate, computationally tractable methods for

the study of the quantum dynamics of large complex systems. For example, we observe

that it is equivalent to the full quantum dynamics described by Eq. (2.1) for arbitrary

quantum subsystems bilinearly coupled to harmonic baths. In fact, the QCL equation is

equivalent to a whole set linearized quantum dynamics schemes derived from the Feynman

path integral [50]. Additionally, we shall show how approximations to QCL dynamics

yield the mean field and surface-hopping schemes.

In the next section we describe how the QCL equation may be expressed in any basis

that spans the subsystem Hilbert space. Here we observe that the subsystem may also

be Wigner transformed to obtain a (basis free) phase-space-like representation of the


subsystem variables as well as those of the environment. Taking the Wigner transform of

Eq. (2.8) over the subsystem, we obtain the quantum-classical Wigner-Liouville equation

[51],

')

)t+ iL(0)

" + iL(0)h

(

+W (p, P, r, R, t)

=2

h(*h)n

!ds

)!dr V (r % r, R) sin(

2sr

h)*+W (p% s, P, r, R, t)

+!

ds %F (R, s))

)P+W (p% s, P, r, R, t), (2.10)

where the force %F (R, s) is defined as

%F (R, s) =1

(*h)n

)

)R

)!dr cos(2sr/h)V (r % r, R)

*. (2.11)

The classical free streaming Liouville operators are iL(0)" = p

m##r and iL(0)

h = PM

##R for the

light (,) subsystem particles and (h) heavy environmental particles, respectively. The

quantum-classical Wigner-Liouville equation (2.10) can be written in a more compact

form,

')

)t+ iL" + iLh

(

+W (p, P, r, R, t) =!

ds K(s, P, r, R)+W (p% s, P, r, R, t) (2.12)

where iL" = pm

##r + Fs(r)

##p is the full classical Liouville operator for the subsystem and

iLh = PM

##R + Fe(R) #

#P is the full classical Liouville operator for the environment. Here

Fs(r) = %)Vs(r)/)r and Fe(R) = %)Ve(R)/)R. The kernel K(s, P, r, R) is the sum of

two contributions, K = K" + Kh with

Kh(s, P, r, R) =1

(*h)n

+)

)R

!dr cos(2sr/h)Vc(r % r, R)

,)

)P,

K"(s, P, r, R) = %)Vs(r)

)r

d-(s)

ds

+2

h(*h)n

!dr [Vs(r % r) + Vc(r % r, R)] sin(

2sr

h). (2.13)

This equation gives the dynamics of the quantum-classical system in terms of phase

space variables (R,P ) for the bath and the Wigner transform variables (r, p) for the

quantum subsystem. This equation cannot be simulated easily but can be used when


a representation in a discrete basis is not appropriate. It is easy to recover a classical

description of the entire system by expanding the potential energy terms in a Taylor

series to linear order in r. Such classical approximations, in conjunction with quan-

tum equilibrium sampling, are often used to estimate quantum correlation functions and

expectation values. Classical evolution in this full Wigner representation is exact for

harmonic systems since the Taylor expansion truncates.

2.2 Representations and Solutions

In many cases, in order to compute the dynamics of condensed phase systems, one invokes

a basis representation for the quantum degrees of freedom in the system. Typically, one

computes the dynamics of these systems in order to obtain quantities of interest, such

as an average value, A(t) = Tr [A+(t)], or a correlation function, as will be discussed

below. Since such averages are basis independent one may project Eq. (2.8) onto any

convenient basis. This is in principle a nice feature, and one that is often exploited

to aid in calculations. However, it is important to note that the basis onto which one

chooses to project the QCLE has important implications on how one goes about solving

the resulting equations of motion. Ultimately the time-dependent average value of an

observable is expressed as a trace over quantum subsystem states and a phase space

average over classical-like coordinates; this feature is intimately linked to constructing a

solution using statistical mechanics. Trajectory-based simulation methods for comput-

ing phase-space averages are often sought once the system Hamiltonian is known in a

given basis. However, other schemes have been proposed that do not rely on computing

ensembles of trajectories [52, 53].

In this section we present the major basis representations that have been used in the

literature to solve the QCL equation. For each representation - subsystem, adiabatic,

force, and mapping, respectively - we present the QCL equation in the particular repre-


sentation, and briefly describe the schemes used to solve the equation of motion in that

particular basis.

2.2.1 The subsystem basis

The Hamiltonian one obtains when partitioning a system into a subsystem and its envi-

ronment, Eq. (2.2), is composed of subsystem, environment, and coupling parts. Thus,

representing the QCL equation in the subsystem basis is a natural starting point.

Let us consider a simple partitioning of the total Hamiltonian into two parts; one

containing terms corresponding to the isolated quantum subsystem only, hs, and a re-

mainder that contains all the bath and coupling terms. The subsystem basis is then

defined by the following eigenvalue problem, hs|.# = /$|.#, where hs = p2/2m + Vs(q).

These basis states, and the associated energy eigenvalues, are independent of the coordi-

nates of the environment. The quantum-classical Liouville superoperator when written

in the subsystem basis is given by,

%iLs$$!,%%! = %i($s

$$! + Ls$$!)-$%-$!%! +

i

h(-$%V %!$!

c % V $%c -$!%!)

+1

2

-

-$!%!)V $%

c

)R+ -$%

)V %!$!c

)R

.

· )

)P, (2.14)

where we have used the following notation for subsystem quantities: $s$$! = (/$% /$!)/h,

V $$!c = ".|Vc|."#, iLs

$$! = PM · #

#R +Fe(R) · ##P , and Fe(R) = %)Ve/)R is the force exerted

by the environment. Also, this equation of motion has been derived from the linearized

influence functional in a path integral representation expressed in this basis [41] prior to

the demonstration that the two descriptions are equivalent in basis free form [50].

Donoso and Martens [39, 54] have developed a method for simulating the dynamics

prescribed in this representation in the spirit of classical molecular dynamics. The algo-

rithm is based on writing each element of the density matrix in the subsystem basis as a

weighted sum over an ensemble of classical trajectories,

+$$!

s (X, t) =N!!!/

k=1

a$$!

k (t)-(X %X$$!

k (t)). (2.15)


The ensemble contains N$$! classical trajectories of the type X$$!k (t) = (R$$!

k (t), P$$!k (t)),

with weight a$$!k (t). Population transfer and phase oscillations in the subsystem occur

via the time variation of the weights attributed to the ensemble. However, in this con-

struction the density is not a smooth function of the phase space coordinates [39], so

a smoothing process is implemented to obtain appropriately scaled gaussian wavepack-

ets. A short time approximation of the propagator is then performed and the resulting

equations of motion for the weights are numerically integrated, whilst the ensemble un-

dergoes Hamiltonian dynamics. The results from simulations using this algorithm (called

the semiclassical Liouville method) are generally in excellent agreement with exact quan-

tum mechanical solutions for model problems. The method has also been applied to the

computation of vibrational dephasing rates of the I2 molecule in a low temperature Kr

matrix and the results are in good agreement with experiment [55].

2.2.2 The adiabatic basis

In contrast to the subsystem representation, the adiabatic basis depends on the environ-

mental coordinates. As such, one obtains a physically intuitive description in terms of

classical trajectories along Born-Oppenheimer surfaces. A variety of systems have been

studied using QCL dynamics in this basis. These include: the reaction rate and the

kinetic isotope e!ect of proton transfer in a polar condensed phase solvent and a cluster

[56, 57, 58, 59, 60], vibrational energy relaxation of a hydrogen bonded complex in a polar

liquid [61], photodissociation of F2 [62], dynamical analysis of vibrational frequency shifts

in a Xe fluid [63], and the spin-boson model [64, 65], which is of particular importance

as exact quantum results are available for comparison.

The adiabatic basis is defined by the eigenvalue problem, hW (R)|.; R# = E$(R)|.; R#,

where hW (R) = HW (R,P )%P 2/2M is the Hamiltonian for the subsystem in a static en-

vironment, and the adiabatic energies, E$(R), depend parametrically on the coordinates

of the environment, R. In this representation, the time evolution of an element of the


density matrix, ".; R|+W (R,P, t)|."; R# = +$$!W (R,P, t), is given by

)+$$!W (R, P, t)

)t= %i

/

%%!L$$!,%%!+

%%!

W (R,P, t), (2.16)

where the evolution operator is now [30]

iL$$!,%%! = iL(0)$$!-$%-$!%! % J$$!,%%!

= (i$$$! + iL$$!)-$%-$!%! % J$$!,%%! . (2.17)

The structure of L given above, consists of two distinct components; (i) classical prop-

agation on mean surfaces accompanied by quantum mechanical phase oscillations with

frequency $$$! = (E$%E$!)/h, and (ii) nonadiabatic transitions accompanied by changes

in the momentum of the environment in order to conserve energy. The classical Liouville

operator

iL$$! =P

M· )

)R+

1

2

#F$

W + F$!

W

$· )

)P, (2.18)

specifies the propagation of the environmental coordinates via Hellmann-Feynman forces,

F$W = %".; R|#VW (q,R)

#R |.; R#. The operator, J , responsible for nonadiabatic transitions

may be written as follows,

J$$!,%%! = % P

M· d$%

-

1 +1

2S$% ·

)

)P

.

-$!%!

% P

M· d#$!%!

-

1 +1

2S#$!%! ·

)

)P

.

-$% . (2.19)

where the quantity S$% is defined as S$% = F$W -$%%F$%

W ( PM ·d$%)!1 = E$%d$%( P

M ·d$%)!1,

where F$%W are the o!-diagonal matrix elements of the force and d$% is the nonadiabatic

coupling matrix element, d$% = ".; R| ##R |!; R#. The presence of the nonadiabatic cou-

pling matrix elements in this operator accounts for nonadiabatic transitions, which change

the quantum state of the subsystem from a diagonal to an o!-diagonal state or vice versa.

The environment momentum derivative accounts for the energy transfer involved in the

subsystem state change.

Shi and Geva [41] have also derived the QCL equation in the adiabatic basis start-

ing from the full path integral expression for the quantum mechanical problem. In this


representation their particular derivation starts with the partial Wigner transform of the

environmental degrees of freedom. The final form of the equation is then obtained by

expressing the adiabatic matrix elements of the partially Wigner transformed density,

+$$!W (R,P ; t + /) in terms of +$$!

W (R,P ; t), and linearizing the resulting system propaga-

tors. It is interesting to note that the choice of basis has consequences on the order of

operations between the linearization and partial Wigner transform in the derivation of

these equations. In the subsystem basis, the partial Wigner transform is a consequence

of the linearization of the forward-backward action, however in the adiabatic case, the

partial Wigner transform is applied to the equation of motion, and the propagators are

subsequently linearized in order to obtain the QCL equation. Nevertheless, the work

on general linearization approaches [50] establishes the equivalence of these techniques

despite some of the seemingly noncommutative mathematical operations employed in the

projections onto particular bases.

One can show how a trajectory picture of the dynamics emerges by applying the

Dyson identity to the formal solution of Eq. (2.16):

+$$!

W (R,P, t) = e!ı(&!!!+L!!! )t+$$!

W (R,P, 0) (2.20)

+/

%%!

!dt"e!ı(&!!!+L!!! )(t!t!)J$$!%%!

#e!ı(Lt

$

$$!%%!+%%!

W (R,P, 0)

The iterative solution of the Dyson equation (2.20) then results in a series of trajectory

segments.

+$$!

W (R,P, t) = e!ı(&!!!+L!!! )t+$$!

W (R,P, 0) (2.21)

+/

%%!

!dt"e!ı(&!!!+L!!! )(t!t!)J$$!%%!e

!ı(&""!+L""! )t+%%!

W (R,P, 0)

+/

%%!µµ!

!dt"dt""e!ı(&!!!+L!!! )(t!t!)J$$!µµ!e

!ı(&µµ!+Lµµ! )(t!!t!!)

+Jµµ!%%!e!ı(&""!+L""! )t+%%!

W (R,P, 0) + . . .

These segments consist of evolution along the surface (.."), which may be adiabatic

(. = .") or the arithmetic mean of two adiabatic surfaces, (. ,= ."), governed by the


propagator, e!i(&!!!+L!!! )t. Each subsequent term in the series includes this type of prop-

agation interrupted by the nonadiabatic coupling operator, J , an incremental number of

times. Since, the operator J accounts for nonadiabatic transitions, these contributions

represent trajectory segments that evolve along some surface, undergo a transition to a

new surface, followed by subsequent evolution on this surface and so on.

The presence of the momentum derivatives in J makes the action of this operator

di"cult to simulate, because it acts on all functions to its right. This will generate

a branching tree of trajectories. This di"culty is avoided by making the momentum-

jump approximation. To see how this approximation is obtained, the following change

of variables is made:

1 +1

2S$% ·

)

)P= 1 + %E$%M

)

)(P · d$%)2. (2.22)

For small %E$%M this expression corresponds to the linear expansion of the exponential

translation operator e"E!"M · #

#(P ·d!")2 , whose action on a function f(P ) is a translation

of the square of the environment momentum by %E$%M along the component of the

momentum that is parallel to the nonadiabatic coupling matrix element:

e"E!"M · #

#(P ·d!")2 f(P ) (2.23)

= e"E!"M · #

#(P ·d!")2 f0(P · d$$%)d$$% + sgn(P · d$%)

1(P · d$%)2)d$%

2

= f0(P · d$%)d$$% + sgn(P · d$%)

1(P · d$%)2 + %E$%Md$%

2.

The approximations surrounding the definition of the J operator comprise the momentum-

jump approximation. This translation or shift of the momentum corresponds precisely

to the amount of energy transferred during a transition and thus satisfies energy conser-

vation. In situations where there is insu"cient kinetic energy available from the environ-

ment for the subsystem to make the transition, %E$%M/(P · d$%)2 > 1, the transition is

simply not allowed, and the evolution continues evolving in its given state.

Several algorithms exist to simulate this evolution equation [40, 65, 66, 67, 68]. The

computation of observables in this approach is accomplished by Monte Carlo sampling


configurations from the initial quantum equilibrium distribution followed by propagation

of the observable by some algorithm [65, 67]. The sequential short time propagation

algorithm [67] is one such algorithm where the propagator is divided into N propagators

that act for a short time interval:

(eiLt)$$!,$N$!N=

/

($1$!1)...($N"1$!N"1)

N3

j=1

(eiL"tj)$j"1$!j"1,$j$!j. (2.24)

The short-time propagators can be solved through application of the Dyson identity

truncated to first order. The subsequent dynamics of the quantity of interest are obtained

by propagating the classical variables along a surface that corresponds to the quantum

state (..") followed by Monte Carlo sampling of the nonadiabatic transition events:

(eiL"tj)$j"1$!j"1,$j$!j- W$j"1$!j"1

(tj!1, tj)eiL!j"1!!

j"1"tj

+#-$j"1$j-$!j"1$!j

+ %tJ$j"1$!j"1,$j$!j

$. (2.25)

where W$%(t, 0) = e!i4 t

0d'&!" [R(')] is a phase factor. Simulations using this algorithm

[67] and a Trotter-based scheme [65], which will be described in detail later, are able to

reproduce the exact quantum results for the spin-boson model, verifying its utility.

2.2.3 The force basis

When the quantum-classical Liouville equation is expressed in the adiabatic basis, the

most di"cult terms to simulate come from the o!-diagonal force matrix elements, which

give rise to the nonadiabatic coupling matrix elements. As described above, contributions

coming from this term were computed using the momentum-jump approximation in the

context of a surface-hopping scheme.

One way to simplify this term in the evolution equation is to make use of a basis that

diagonalizes the force contribution [69]; i.e., we represent the quantum-classical Liouville

equation in a basis |i; R# such that

%"i; R|)V (R)

)R|j; R# = F i

F (R)-ij, (2.26)


where the subscript F is used to denote the force basis. Taking the matrix elements of

Eq. (2.8) in this basis, we obtain

)+ijF (X, t)

)t= % i

h

/

k

#Hik

F +kjF (t)% +ik

F (t)HkjF

$

%-

P

M· )

)R+

1

2(F i

F + F jF ) · )

)P

.

+ijF , (2.27)

where HijF = H ij

F + ih PM · dF

ij. Here dFij is the nonadiabatic coupling matrix element in the

force basis. It can be related to the usual nonadiabatic coupling matrix element in the

adiabatic basis by inserting complete sets of adiabatic states:

dFij =

/

$%

"i; R|.; R#d$%"!; R|j; R#. (2.28)

Evolution governed by the last term in Eq. (2.27) is simple and can be solved in char-

acteristics. The first term is responsible for coupling among the elements of the density

matrix and its inclusion makes the computation of the dynamics a di"cult task.

The quantum-classical Liouville equation in the force basis has been solved for low-

dimensional systems using the multithreads algorithm [69, 3]. Assuming that the density

matrix is localized within a small volume of the classical phase space, it is written as

linear combination of matrices located at L discrete phase space points as

+W (R,P, t) =L/

k=1

+(k)(t)-(R%Rk(t))-(P % Pk(t)). (2.29)

The evolution equations for the quantities entering the right side of this equation are

obtained by substitution into the quantum-classical Liouville equation. For a variety of

one- and two-dimensional systems for which exact results are known, excellent agreement

was found.

2.2.4 The mapping basis

The quantum-classical Liouville equation was expressed in the subsystem basis in Sec.

(2.2.1). Based on this representation, it is possible to recast the equations of motion in a


form where the discrete quantum degrees of freedom are described by continuous position

and momentum variables [70, 71, 72, 73, 74, 75]. In the mapping basis the eigenfunctions

of the n-state subsystem can be replaced with eigenfunctions of n fictitious harmonic

oscillators with occupation numbers limited to 0 or 1: |0# & |m(# = |01, · · · , 1(, · · · 0n#.

This mapping basis representation then makes use of the fact that the matrix element of

an operator BW (X) in the subsystem basis, B((!W (X), can be written in mapping form as

B((!W (X) = "0|BW (X)|0"# = "m(|Bm(X)|m(!#, where

Bm(X) =/

((!B((!

W (X)a†(a(! . (2.30)

The mapping annihilation and creation operators are given by

a( =

51

2h(q( + ip(), a†( =

51

2h(q( % ip(). (2.31)

The mapping basis has been exploited in quantum-classical calculations based on a lin-

earization of the path integral formulation of quantum correlation functions in the LAND-

map method [76, 77, 78].

Given this correspondence between the matrix elements of a partially Wigner trans-

formed operator in the subsystem and mapping bases, we can express the quantum-

classical Liouville equation in the continuous mapping coordinates [79]. The first step in

this calculation is to introduce an n-dimensional coordinate space representation of the

mapping basis,

"m(|Bm(X)|m(!# =!

dqdq""m(|q#"m(!|q"#"q|Bm(X, t)|q"#, (2.32)

and then write the coordinate space matrix elements in terms of Wigner transforms in

the mapping space to obtain

"r % z

2|Bm(X, t)|r +

z

2# =

1

(2*h)n

!dp e!ipz/hBm(x, X, t). (2.33)

Carrying out this change of representation on the quantum-classical Liouville equation

and using the product rule formula for the Wigner transform of a product of operators,


we obtain

d

dt+m(x, X, t) = %2

hHm sin(

h&m

2)+m(t) (2.34)

+)Hm

)Rcos(

h&m

2) · )Bm(t)

)P% P

M· )+m(t)

)R,

where the negative of the Poisson bracket operator on the mapping phase space coor-

dinates is defined as &m =(%)p ·

%&)r %(%)r ·

%&)p. The Hamiltonian in the mapping basis

is

Hm(x, X) =P 2

2M+ Ve(R) +

1

2h

/

((!h((!(R)(r(r(! + p(p(! % h-((!), (2.35)

where h((!(R) = "0|p2/2m + Vs(q) + Vc(q, R)|0"# = /(-((! + V ((!s (R). Explicitly comput-

ing the exponential Poisson bracket operators, we find the quantum-classical Liouville

equation in the mapping basis,

d

dtBm(x, X, t) = %{Hm, Bm(t)}x,X (2.36)

+h

8

/

((!

)h((!

)R(

)

)r(!

)

)r(+

)

)p(!

)

)p() · )

)PBm(t)

. iLmBm(t),

where {Am, Bm(t)}x,X denotes a Poisson bracket in the full mapping-bath phase space

of the system.

The first term in the evolution operator has the form of a Poisson bracket and evo-

lution under this part of the operator can be expressed in terms of characteristics. The

corresponding set of ordinary di!erential equations is

dr((t)

dt=

1

h

/

(!h((!(R(t))p(!(t),

dp((t)

dt= %1

h

/

(!h((!(R(t))r(!(t),

dR(t)

dt=

P (t)

M;

dP (t)

dt= % )Hm

)R(t). (2.37)

The last term involves derivatives with respect to both mapping and environmental

variables. Essentially it contains part of the influence of the quantum subsystem on


the e!ective force felt by the bath. Its contribution is di"cult to compute, and is a

matter under current investigation [79, 80]. Calculations on the spin-boson model have

shown that even if the last term is neglected, excellent agreement with the exact results

for a wide range of system parameters is obtained [79].

2.3 Other Quantum-Classical Approximations

Other approximate quantum-classical theories have been in use for some time now. In

particular, Ehrenfest’s mean field theory and variants of Tully’s surface-hopping scheme

are commonly employed to treat the same class of problems as the QCL equation. These

methods are attractive due to their computational simplicity; however, quantum coher-

ence and correlations are handled somewhat severely in these approaches. Nevertheless,

as the QCL equation is an approximation in its own right, examining connections with

these well-known methods can shed light on its utility.

2.3.1 Mean field theory

Mean field theories of mixed quantum-classical systems are based on approximations that

neglect correlations in Ehrenfest’s equations of motion for the evolution of the position

and momentum operators of the heavy-mass nuclear degrees of freedom. The approxi-

mate evolution equations take the form of Newton’s equations of motion where the forces

that the nuclear degrees of freedom experience involve mean forces determined from the

time-evolving wave function of the system.

We now show how the mean field equations of motion can be derived as an approx-

imation to the quantum-classical Liouville equation (2.8) [35]. The Hamiltonian can be

written again as the sum of environmental, subsystem and interaction contributions,

HW =P 2

2M+ Ve(R) +

p2

2m+ Vs(q) + Vc(q, R) = He(X) + Hs(q) + Vc(q, R).


In order to study the e!ects of neglecting correlations in this description of the dynamics,

we define a general phase space coordinate X = (R, P ) for the classical degrees of freedom,

and the reduced density matrices for the environment and subsystem, respectively, as

+e(X, t) = Tr"+W (X, t), +s(t) =!

dX +W (X, t), (2.38)

where Tr" is the partial trace. The reduced distributions are normalized so that4

dX +e(X, t) =

1 and Tr"+s(t) = 1. We also define the correlation density operator +cor(X, t) by +W (X, t) .

+s(t)+e(X, t) + +cor(X, t). Given that the density operator satisfies the normalization

Tr"4

dX +W (X, t) = 1, we have Tr"4

dX +cor(X, t) = 0.

If we substitute the above expression for +W (X, t) into the quantum-classical Liouville

equation we find

+s(t))+e(X, t)

)t+ +e(X, t)

)+s(t)

)t+

)+cor(X, t)

)t=

%iL+s(t)+e(X, t)% iL+cor(X, t). (2.39)

To obtain the mean field equations, we make the assumption that all terms in this

equation containing +cor(X, t) can be neglected. Then, integration over X and use of the

normalization conditions yields

)+s(t)

)t= % i

h

%Hs +

!dX Vc+e(X, t), +s(t)

&, (2.40)

while the trace over the quantum degrees of freedom gives

)+e(X, t)

)t=

6He + Tr"Vc+s(t), +e(X, t)

7=

6He# , +e(X, t)

7, (2.41)

where He# = P 2/2M +Ve#(R) and the e!ective potential is defined as Ve#(R) = Ve(R)+

Tr"Vc+s(t). This equation can be solved in terms of characteristics. The density function

takes the form +e(X, t) = -(X % X(t)), where X(t) = (R(t), P (t)) satisfy Newtonian

equations of motion,

R(t) =P (t)

M, P (t) = %)Ve#(R(t))

)R(t). (2.42)


Using +e(X, t) = -(X %X(t)), we may write

!dX Vc(R)+e(X, t) = Vc(R(t)). (2.43)

As a result, Eq.(2.40) is equivalent to the pair of Schrodinger equations,

ih)|'(R(t), t)#

)t=

#Hs + Vc(R(t))

$|'(R(t), t)#, (2.44)

and its adjoint. Equations (2.42) and (2.44) are the standard mean field equations of

motion for a mixed quantum-classical system.

Thus, we see that in order to obtain the mean field equations of motion, the den-

sity matrix of the entire system is assumed to factor into a product of subsystem and

environmental contributions with neglect of correlations. The quantum dynamics then

evolves as a pure state wave function depending on the coordinates evolving in the mean

field generated by the quantum density. As we have seen in the previous sections, these

approximations are not valid and no simple representation of the quantum-classical dy-

namics is possible in terms of single e!ective trajectories. Consequently, in contrast to

claims made in the literature [81], quantum-classical Liouville dynamics is not equivalent

to mean field dynamics.

2.3.2 Surface-hopping dynamics

With an aim to provide nonadiabatic corrections compatible with classical molecular dy-

namics techniques, surface-hopping schemes [82, 28, 83] generally start with the ansatz

that the heavy particles follow some, not necessarily classical, trajectory R(t). From

this trajectory ansatz, Tully constructed an ad hoc method to solve the time depen-

dant Schrodinger equation for the quantum subsystem known as surface hopping. This

method, and other variants, consist of concatenating a number of continuous evolution

segments which are interspersed with instantaneous probabilistic hopping between quan-

tum states.


Assuming knowledge of the trajectory, R(t), one expands the wavefunction for the

quantum subsystem in the adiabatic basis,

|'(R, t)# =/

$

C$(t)|.; R(t)#, (2.45)

where C$(t) are complex valued expansion coe"cients. This form for the wavefunc-

tion is then directly substituted into the time dependent Schrodinger equation, and the

coe"cients are evolved accordingly:

ıh)C$

)t=

/

$!C$!(V$$! % ıh

P

Md$$!), (2.46)

where V$$! = ".; R(t)|H % P 2

2M |.; R(t)# are Born-Oppenheimer potential energy surfaces.

These BO potential surfaces that the classical trajectories evolve along correspond to one

of the adiabatic surfaces used in the expansion of the subsystem wavefunction, while the

subsystem evolution is carried out coherently and may develop into linear combinations

of these states.

This coupled set of equations for the subsystem coe"cients is integrated simulta-

neously with Newton’s equations for the positions and momenta of the heavy parti-

cles. Quantum transitions are then implemented as discrete, instantaneous hops between

Born-Oppenheimer surfaces, that conserve the total energy of the system. Tully’s fewest

switches algorithm prescribes a hopping probability that ensures the correct distribution

of trajectories on each surface, |C$|2, using a minimal number of hops. At any given

time, the probability of switching from state . to state .", during the interval (t, t+ -) is

P$%$! =2Im(C#

$(t + -)C$!(t + -)V$$!)% 2Re(C$(t + -)C#$!(t + -) P

M d$$!)

|C$(t + -)|2 . (2.47)

In these schemes, the environment does not experience the force associated with the

true quantum state of the subsystem, and the e!ect of the environment on the evolution

of coherences in the subsystem is not properly taken into account. Nonetheless, these


methods have provided computationally tractable, and under some conditions accurate,

descriptions of nonadiabatic dynamics.

Decoherence in the quantum subsystem of condensed phase systems is a well estab-

lished phenomenon [84, 85] and should be accounted for in surface-hopping schemes.

We note that various phenomenologically motivated prescriptions have been proposed to

incorporate decoherence into the dynamics of the subsystem [83, 86, 87, 88, 89].

A connection between surface-hopping schemes and the dynamics prescribed by the

QCL equation may be established by considering the conditions under which it is rea-

sonable to express the dynamics given by the QCL equation in terms of evolution along

single adiabatic surfaces. Since coherence is described by the o!-diagonal elements of the

density matrix, the problem reduces to the examination of the conditions under which

decoherence leads to rapid enough decay of the o!-diagonal elements to justify the use

of a quantum-classical master equation. The master equation approach indeed produces

surface-hopping-like trajectories, however, as further approximations to the QCL equa-

tion are required in its derivation it must be applied with care [90].

Starting from the QCL equation expressed in the adiabatic basis,

)

)t+$$!

W (X, t) = %iL$$!%%!+$$W (X, t), (2.48)

one may indeed derive a generalized master equation for the diagonal elements of the

density matrix. Next, we separate the above equation into diagonal and o!-diagonal

parts,

+d(X, t) = +$$!

W (X, t)-$$! , (2.49)

+o(X, t) = +$$!

W (X, t)(1% -$$!),

which yields two coupled di!erential equations describing the evolution of the density

matrix.

)

)t+d = %ıLd+d(X, t)% ıLd,o+o(X, t), (2.50)

)

)t+o = %ıLo+o(X, t)% ıLo,d+d(X, t)


Here we have adopted the use of diagonal and o!-diagonal superscripts to define di!erent

parts of the Liouvillian operator as follows:

Ld = L$$!%%!-$%-$$! , -%%! = ıL$$!-$%-$$!-%%! , (2.51)

Lo = L$$!%%!(1% -$$!)(1% -%%!),

Lo,d = %J$$!%%!(1% -$$!)-%%! ,

Ld,o = %J$$!%%!-$$!(1% -%%! .

The solution to the o!-diagonal evolution is obtained formally and used to solve for the

diagonal part. Assuming the initial state is pure, (i.e.) +o(X, 0) = 0, one obtains the

following expression,

)

)t+d(X, t) = %iLd+d(X, t) +

! t

0dt"iLd,oe!iLo(t!t!)iLo,d+d(X, t), (2.52)

which has the form of a generalized master equation. Returning to the original notation,

we have

)

)t+$$(X, t) = %iL$$+$$(X, t) +

! t

0dt"

/

%

M$%(t")+%%(X, t% t"), (2.53)

where the memory kernel operator M$%(t) is defined as,

M$%(t) =/

))!,µµ!J$$,µµ!

#e!iL(X)(t)

$

µµ!,))!J))!,%%. (2.54)

The memory kernel acts on all phase space functions that appear to its right, and by

considering the action of the operators given in Eq. (2.54), the memory kernel operator

may be reduced to a memory function defined through its action on functions of X.

M$%(X, t)f(X) = (1% -$%)M$%$% (X, t)f(X$%

$%,t) + -$%

/

)

M$))$ (X, t)f(X)$

$),t)(2.55)

M$%$% (X, t) = 2Re [W$%(t, 0)] D$%(X)D$%(X$%,t), (2.56)

where W$%(t, 0) = e!i4 t

0d'&!" [R(')] is a phase factor, D$% = P · d$%, and the subscripts

and superscripts on the memory function label the indices on the first and second D


functions respectively. The bar on the phase space variable X indicates the action of a

momentum shift by the J operator, the subscript notation indicates that X has been

evolved along the surface (.!) for a time t, and the superscripts (if indicated) denote

a momentum shift operation, corresponding to the action of the second factor of J in

equation (2.54). In this form, the memory function accounts for all coherent evolution.

Decoherence, arising from interaction of the subsystem with an environment, may be

incorporated into the evolution expression by averaging over the environmental degrees

of freedom through the application of projection operator techniques to the generalized

master equation. Hence, let us denote the degrees of freedom of the subsystem and the

environment as Xe and Xs, respectively, and consider the following projection operator,

P = +eq(Xe; Rs)!

dXe·, (2.57)

which invokes an average over an equilibrium environment, conditional on the coordinates

(Rs) of the system. Upon applying P to the generalized master equation, (2.53), the

projected evolution of the subsystem density is obtained [90]. The projected is evolution

has a complicated form, yet can be greatly simplified through the use of a Markovian

approximation [90, 91].

The average over the environment (denoted "· · ·#e),

"M$%$% (X, t)#e =

!dXeM

$%$% (X, t)+eq(Xe; Xs), (2.58)

provides a mechanism for decoherence and eventually leads to decay of the kernel involv-

ing the memory function. If this decay occurs faster than all other relevant relaxation

processes then a Markovian approximation, which asserts that the projected subsystem

has an infinitely short memory, may be justified. In this context the explicit form of

the Markovian approximation M$%$% (X, t) is replaced with the value of its infinite time

integral,

"M$%$% (X, t")#e = 2

! &

0dt"M$%

$% (X, t)#e-(t") = 2m$%(Xs)-(t"). (2.59)


The resulting projected subsystem evolution equation is then lifted back into the full

phase space of the system, yielding a Markovian master equation description,

)

)t+$$(X, t) = %ıL$$+$$(X, t)%m$$(Xs)+

$$(X, t) +/

%

m$%(Xs)j$%%+%%(X, t), (2.60)

where the jump operators j$%% prescribe momentum shifts corresponding to population

transfer directly between adiabatic states [90].

2.4 Observables and Correlation Functions

Thus far we have focussed on the dynamics of quantum-classical systems. In practice, we

are primarily interested instead in computing observables that can be compared eventu-

ally to experimentally obtainable quantities. To this end, consider the general quantum

mechanical expression for the expectation value of an observable,

A(t) = Tr(+(t)A) = Tr(+A(t)) (2.61)

= Tr"!

dQ1dQ2"Q1|A(t)|Q2#"Q2|+(0)|Q1#

= Tr"!

dRdP AW (R,P, t)+W (R,P ) = Tr"!

dXAW (X, t)+W (X). (2.62)

In the above expressions we have introduced the (primed) partial trace over the Hilbert

space of the susbsystem, Tr"+W (R, P ) = +e(R,P ), and the symbols Tr" refers to taking

the partial trace over the subsystem.

If we expand Eq.(2.61) in the coordinate {Q}-representation for the environmental

degrees of freedom only we obtain the second line. Taking the Wigner representation for

these degrees of freedom and finally, defining the general coordinate X = (R,P ) gives

Eq. (2.62).

For a quantum mechanical system in thermal equilibrium a transport coe"cient 0AB

may be determined from the time integral of a quantum flux-flux correlation function

[92]. The quantum mechanical form of the correlation function is:

CAB(t) = " ˙B(t), ˙A#Q =1

ZQTr( ˙B(t) ˙A+eq), (2.63)


where, for example ˙B = (i/h)[H, B] = jB(t) is the flux of B. The equilibrium quantum

canonical average is "· · ·#Q = Z!1Q Tr · · · e!%H where ZQ is the partition function.

The quantum mechanical expressions for transport coe"cients involving correlation

functions are well known and may be derived by invoking linear response theory [92] or the

Mori-Zwanzig projection operator formalism [93, 94]. In the linear response formalism,

for example, the response function of the system,

(AB(t) = " ih

[jB(t), A†]#Q = !"jA; jB(t)#, (2.64)

is related to 0AB(t) through the use of the Kubo transform, (2.65):

" ihjA; jB(t)# =

1

!

! %

0d0Tr(jA(%ıh0)jB+eq), (2.65)

Here we have introduced the notation, "·; ·#, to indicate the Kubo transformed-correlation

function. The transport coe"cient may then be obtained from the plateau value of 0AB(t)

[95],

0AB(t) =! t

0dt""jA; jB(t")# =

1

!

! t

0dt"(AB(t"). (2.66)

However, we would like to evaluate transport properties for quantum-classical sys-

tems. We thus take the quantum mechanical expression for a transport coe"cient as

a starting point and then consider a limit where the dynamics are approximated by

quantum-classical dynamics [96, 97, 98]. The advantage of this approach is that the full

quantum equilibrium structure can be retained.

In simulations it is convenient to obtain the transport coe"cient from the plateau

value of 0AB(t). Writing Eq. (2.66) in detail, we can express the time-dependent trans-

port coe"cient 0AB(t) as,

0AB(t) =1

!

! t

0dt"" ˙B(t); ˙A#

=1

!ZQ

! %

0d0Tr

0˙Ae

ih H(ih()B(t)e!

ih H(ih()!%H

2. (2.67)


Rewriting the expression in the coordinate representation for the full system, {Q} =

{q}{Q} (calligraphic symbols are used to denote variables for the entire system, subsys-

tem plus bath),

0AB(t) =1

!ZQ

! %

0d0

!dQ1dQ"1dQ2dQ"2"Q1| ˙A|Q"1#"Q"1|e

ih H(t+ih()|Q2#

+"Q2|B|Q"2#"Q"2|e!%H! ih H(t+ih()|Q1#. (2.68)

Making a change of variables, Q1 = R1%Z1/2, Q"1 = R1+Z1/2, etc., and then expressing

the matrix elements in terms of the Wigner transforms of the operators, we have [97]

0AB(t) =1

!

! %

0d0

!dX1dX2(A)W (X1)BW (X2)

1

(2*h)2) ZQ

+!

dZ1dZ2e! i

h (P1·Z1+P2·Z2)8R1 +

Z1

2

999eih H(t+ih()

999R2 %Z2

2

:

+8R2 +

Z2

2

999e!%H! ih H(t+ih()

999R1 %Z1

2

:. (2.69)

Here we used the fact that the matrix element of an operator A can be expressed in terms

of its Wigner transform AW (X ) as

8R% Z

2

999A999R+

Z2

:=

1

(2*h))

!dPe!

ihP·ZAW (X ), (2.70)

where 1 is the coordinate space dimension.

If we define the spectral density by,

W (X1,X2, t) =1

(2*h)2) ZQ

!dZ1dZ2e

! ih (P1·Z1+P2·Z2)

+8R1 +

Z1

2

999eih Ht

999R2 %Z2

2

: 8R2 +

Z2

2

999e!%H! ih Ht

999R1 %Z1

2

:, (2.71)

we can write the transport coe"cient as

0AB(t) =!

dX1dX2(A)W (X1)BW (X2)W (X1,X2, t), (2.72)

where

W (X1,X2, t) =1

!

! %

0d0W (X1,X2, t + ih0) . (2.73)


To take the quantum-classical limit of this general expression for the transport coe"-

cient we partition the system into a subsystem and bath and use the notation R = (r, R),

P = (p, P ) and X = (r, R, p, P ) where the lower case symbols refer to the subsystem

and the upper case symbols refer to the bath. To make connection with surface-hopping

representations of the quantum-classical Liouville equation [30], we first observe that

AW (X1) can be written as

AW (X1) =!

dz1 eih p1·z1"r1 %

z1

2|AW (X1)|r1 +

z1

2#, (2.74)

where AW (X1) is the partial Wigner transform of A over the bath degrees of freedom.

We may now express the subsystem operators in the adiabatic basis to obtain,

AW (X1) =/

$1$!1

!dz1 e

ih p1·z1"r1 %

z1

2|.1; R1#A

$1$!1W (X1)"."1; R1|r1 +

z1

2#, (2.75)

where A$1$!1W (X1) = ".1; R1|AW (X1)|."1; R1#. Inserting this expression, and its analog for

BW (X2), into Eq.(2.69) we have

0AB(t) = %/

$1,$!1,$2,$!2

! 23

i=1

dXi A$1$!1W (X1)B

$2$!2W (X2)

)

)tW

$!1$1$!2$2(X1, X2, t), (2.76)

where the matrix elements of W are given by

W $!1$1$!2$2(X1, X2, t) =! 23

i=1

dZie! i

h (P1·Z1+P2·Z2) 1

ZQ

1

(2*h)2)h

+"."1; R1|"R1 +Z1

2|e i

h Ht|R2 %Z2

2#|.2; R2#

+"."2; R2|"R2 +Z2

2|e! i

h Ht!!|R1 %Z1

2#|.1; R1# , (2.77)

with t"" = t% i!h.

The quantum-classical limit of the transport coe"cient is obtained by evaluating the

evolution equation for the matrix elements of W in the quantum-classical limit. This

limit was taken in Ref. [96] and the result is

)

)tW

$!1$1$!2$2(X1, X2, t) =

1

2

/

%!1%1%!2%2

#iL$!1$1,%!1%1(X1)-$!2%!2

-$2%2 % iL$!2$2,%!2%2(X2)-$!1%!1-$1%1

$

+W%!1%1%!2%2(X1, X2, t) . (2.78)


We use the first equality in Eq.(2.78), insert this into Eq.(2.76), and move the evo-

lution operator iL(X1) onto the AW (X1) dynamical variable. Next, we use the sec-

ond equality in Eq.(2.78) and formally solve the equation to obtain W (X1, X2, t) =

e!iL(X2)tW (X1, X2, 0). Finally we substitute this form for W (X1, X2, t) into Eq. (2.76)

and move the evolution operator to the dynamical variable BW (X2). In the adiabatic

basis, the action of the propagator e!iL(X2)t on BW (X2) is

B$2$!2W (X2, t) =

/

%2%!2

#e!iL(X2)t

$

$2$!2,%2%!2B

%2%!2W (X2) . (2.79)

The result of these operations is

0AB(t) =/

$1,$!1,$2,$!2

! 23

i=1

dXi (iL(X1)AW (X1))$1$!1

+B$2$!2W (X2, t)W

$!1$1$!2$2(X1, X2, 0). (2.80)

This equation can serve as the basis for the computation of transport properties for

quantum-classical systems. Note that full quantum equilibrium e!ects are described by

the initial value of W .

2.5 Reaction Rate Calculation

The rate coe"cient of a reactive process is a transport coe"cient of interest in chemical

physics. It has been shown from linear response theory that this coe"cient can be ob-

tained from the reactive flux correlation function of the system of interest. This quantity

has been computed extensively in the literature for systems such as proton and electron

transfer in solvents as well as clusters [56, 59, 99, 83, 100, 101, 102, 103, 104, 105], where

the use of the QCL formalism has allowed one to consider quantum phenomena such

as the kinetic isotope e!ect in proton transfer [59]. Here, we outline the problem of

formulating an expression for a reactive rate coe"cient in the framework of the QCL

theory. Results from a model calculation will be presented including comparisons with

the approximate methods previously described.


Consider a simple reactive process A 23 B. The quantum mechanical expression for

the time dependent forward rate coe"cient for such a process is given by,

kAB(t) = % 1

neqA

! t

0dt""- ˙NB(t"); - ˙NA#, (2.81)

where the -Ni are the species operators representing the deviations of each species from

their equilibrium number densities, and the angled brackets denote the Kubo transformed

correlation function as defined in Eq. (2.67). This expression was derived previously by

following the projection operator methods of Mori and Zwanzig in the linear response

regime [106]. Following the discussion of the previous section, the quantum classical limit

of this expression, (2.80) can be obtained and is given by [98]

kAB(t) =1

neqA

/

$

/

$!'$

(2% -$!$) (2.82)

+!

dXRe

'

N$$!

B (X, t)W$!$A (X,

ih!

2)

(

.

Here, the integration over X1 was performed in Eq. (2.80) to define W$$!A (X, ih%

2 ) which

is the integrated value of the combination of the spectral density function with the time

independent operator. This spectral density function contains the quantum equilibrium

structure of the system. N$$!B (X, t) is the time evolved matrix element of the number

operator for the product state B. Thus, to calculate the rate, one samples initial con-

figurations from the quantum equilibrium distribution, and then computes the evolution

of the number operator for product state B. The QCL evolution of the species operator

is accomplished using one of the algorithms discussed in Sec. (2.2.2). Alternative ap-

proaches to the dynamics may also be used such as the further approximations to the

QCL equation discussed in Sec. (2.3).

The sampling of initial configurations from the spectral density function remains

a challenging task as the structure of this function is complicated. By factoring this

quantity into a subsystem and conditional environment distribution, W $!$A (X, ih%

2 ) =

+$!$A (X0)+c

e(Xe; R0), the problem simplifies. In particular, if the environment consists of


harmonic oscillators, as is the case here, the exact form of the spectral density is known

for the environment. For the subsystem, one can appeal to the fact that typically the

barrier region, where the largest contribution to the dynamics in this problem take place,

is approximately harmonic. Doing so, one is able to obtain an approximate analytic form

of this function given by [60],

+$!$A (X0) =

ia

ZQ*2

!dZ0e

! ih (P0·Z0)

/

$1$2

A$!$$1$2

B$1$2(0) (2.83)

+%-$1$2e

!a(4R20+Z2

0 )u!12aM0Z0u21

%d$1$2(0)e!2a

%(R0+

Z02 )

2u!1+(R0!Z0

2 )2u!2

&/

u1u2

(

,

where u"i = uicothui, ui = uicschui, ui = !h$i/2, $i is the barrier region frequency

for the state .i which may be real or imaginary, and a = M0/(2!h2). By numerically

integrating over Z0 we obtain the quantum mechanical equilibrium structure for the

subsystem, +0(X0). Sampling from this distribution is performed from the harmonic part

and subsequently reweighted by the remaining term. With this quantity, in conjunction

with that for the environment and Eq. (2.82), the quantum mechanical rate-coe"cient

can be computed via computer simulation.

We consider the same reaction model used in previous studies as a simple model for

a proton transfer reaction. [59, 90, 91] The subsystem consists of a two-level quantum

system bilinearly coupled to a quartic oscillator, Vq(R), and the bath consists of 1 % 1 =

300 harmonic oscillators bilinearly coupled to the non-linear oscillator but not directly

to the two-level quantum system. In the subsystem representation, the partially Wigner

transformed Hamiltonian for this system is,

HW

=

;

<<=Vq(R0) + h' h"0R0

h"0R0 Vq(R0)% h'

>

??@ (2.84)

+

;

= P 20

2M0+

)!1/

j=1

P 2j

2Mj+

Mj$2j

2

-

Rj %cj

Mj$2j

R0

.2>

@ I.

The solution of the eigenvalue problem, hW (R)|.; R0# = E$(R)|.; R0#, yields the adia-


batic eigenstates, |.; R0#, and eigenvalues E$(R) = Vq(R0)+Ve(Re; R0)0h1

'2 + ("0R0)2,

where 2' is the adiabatic energy gap. The parameters of this model characterize a

harmonic environment with ohmic spectral density, [107] the details of which can be

found elsewhere. [59, 96, 91] The reaction coordinate R0 undergoes dynamics char-

acteristic of a well defined barrier crossing process and the product species operator

N$$!B (R0) = #(R0)-$$! is initially diagonal in the adiabatic basis. Here #(R0) is the

Heaviside function.

2.5.1 Simulation results

The plot in Fig. (2.1) shows the reaction rate computed via adiabatic versus nonadiabatic

dynamics. The rate constant, given by the plateau value of the correlation function, is

lower when nonadiabatic dynamics is considered. This reduction is due to enhanced

barrier crossing as a result of motion on either the excited state or mean surfaces. When

the system is propagating on either the mean or excited state surface it is confined close

to the barrier region such that it crosses the barrier more frequently. On the other hand,

once the system jumps down to the ground state, it stabilizes in one of the wells such

that recrossings are less frequent. The increase of recrossings due to dynamics on these

surfaces leads to decay of the correlation function.

In Fig. (2.2), the simulation results for the same model problem are presented using

the QCL equation, the master equation, Tully’s surface-hopping approach, the mean

field approach, and adiabatic dynamics. The algorithmic details of each approach can be

found elsewhere [28, 67, 91]. From the figure we see that the surface-hopping result using

Tully’s algorithm is almost identical to the adiabatic rate for the parameters given here.

By comparison to both the QCL equation simulation and the master equation simulation,

one can conclude that for this set of parameters this overestimates the reaction rate. The

primary reason for the discrepancy is the manner in which coherence and decoherence

is treated in the theory. Although the master equation also restricts motion to single


Figure 2.1: Nonadiabatic vs adiabatic reaction rate for ! = 1.0. The blue curve is the

adiabatic result while the green curve is the nonadiabatic QCL rate.

adiabatic surfaces, the probability of hopping is obtained from a calculation that accounts

for decoherence in a di!erent manner. The prediction of the rate obtained by the mean-

field approach, shown in the figure, underestimates the rate. This can be attributed to

the neglect of correlations in the equations of motion as discussed in Sec. (2.3.1).

2.6 Comments and Outlook

So far we have reviewed some of the most recent developments in the computation and

modeling of quantum phenomena in condensed phased systems in terms of the quantum-

classical Liouville equation. Although the theory involves several levels of approximation,

QCL dynamics performs extremely well when compared to exact quantum calculations

for some important benchmark models such as the spin-boson system. Consequently,

QCL dynamics is an accurate theory to explore the dynamics of many other quantum

condensed phase systems.

In practice, one’s ability to simulate any model system via the QCL equation depends


0 5 100

0.5

1

1.5

2

t

10

2 k

AB(t

)

Figure 2.2: Forward rate coe"cient kAB(t) as a function of time for ! = 1.0. The upper

(blue) curve is the adiabatic rate, the purple curve is the result obtained by Tully’s

surface-hopping algorithm, the middle (black) curve is the quantum master equation

result, the green curve is the QCL result, and the lowest dashed line (grey) is the result

using mean-field dynamics.

on the appropriate choice of representation of the quantum system. As we have men-

tioned, and shall see later, each representation brings with it a unique set of challenges

and limitations in the development of e"cient algorithms for computing the dynamics

as well as sampling initial configurations. For these reasons it is interesting and instruc-

tive to consider some of the earlier more approximate schemes such as the mean field

and surface-hopping approaches. We have shown here that there are conditions under

which the approximations underlying these methods are reasonable and thus one can

take advantage of the computational simplicity involved in these schemes to obtain a

computationally cheap solution to the problem. One, however, must keep in mind that


QCL theory is already an approximate approach, and that these schemes involve further,

often severe, approximations to the QCL equation. Thus, the QCL equation continues

to emerge as an analytic tool that one can use as a theoretical basis to simulate, and

evaluate alternative approaches to, the dynamics of open quantum systems.

Chapter 3

Molecular Models

We seek to study quantum relaxation processes through conical intersections, which

exist in d 1 2 dimensions, and correspond to a d % 1 dimensional degeneracy in the

potential energy hypersurface. Thus, the minimum requirements for a nontrivial physical

model that can capture this feature are two quantum states which are coupled to two

linearly independent environmental degrees of freedom. With molecular systems in mind,

we roughly imagine the electronic states of single atom or molecule as comprising the

quantum subsystem, and approximate the nuclei and other surrounding molecules as

the environment. In systems of this type the initial application of quantum-classical

Liouville theory is justified by the mass ratio argument [30]. Thus far only a few attempts

have been made to study model conical intersection systems using quantum-classical

methods [108, 3], and thus a host of open questions pertaining to the application of mixed

quantum-classical methods exist. Additionally, QCL theory allows one to incorporate the

e!ects of dissipation and other external fields on molecular dynamics in a straightforward

manner. This theory can serve as a starting point to investigate the e!ects of dissipation

in condensed phase processes by involving (e.g.) coupling to an external heat bath. In

this chapter we describe a selection of physically motivated mathematical models which

aim to represent a condensed phase molecular system containing a conical intersection.

42

Chapter 3. Molecular Models 43

3.1 A Two State -Two Mode Conical Intersection:

The FLV Model

The key model for a quantum subsystem containing a conical intersection chosen for use

in this study is due mainly to Ferretti, Lami, and Villiani [2, 1]. For convenience, we

shall refer to it as the FLV model. This two level - two mode quantum model represents

the coupled vibronic states of a linear triatomic of the type ABA. The nuclei of this

symmetric linear molecule are described using two vibrational degrees of fredom (Jacobi

coordinates, A % B distances): a symmetric stretch, the so-called ’tuning coordinate’

coordinate, X, and an anti-symmmetric stretch Y , known as the coupling coordinate.

The (Wigner transformed) electronic Hamiltonian matrix in the diabatic basis is given

Figure 3.1: Plot of the FLV diabatic electronic surfaces V00(X, Y ) and V11(X, Y ). The

parameter values used in this figure are summarized in Eq.(3.8).

by

hs,W

(X, Y ) = (P 2

X

2MX+

P 2Y

2MY)I +

;

<<=V00(X, Y ) V01(X, Y )

V01(X, Y ) V11(X, Y )

>

??@ , (3.1)


where

V00(X, Y ) =1

2MX$2

X(X %X1)2 +

1

2MY $2

Y Y 2, (3.2)

V11(X, Y ) =1

2MX$2

X(X %X2)2 +

1

2MY $2

Y Y 2 + %,

V01(X, Y ) = "Y e!$(X!X3)2e!%Y 2.

The above model prescribes two shifted 2D parabolic electronic surfaces which are

depicted in Fig(3.1), with an antisymmetric localized coupling as shown in Fig(3.2).

The two diabatic surfaces exhibit minima at (X1, 0) and (X2, 0) respectively, and are

degenerate at a single point (X3, 0). The electronic coupling is localized to the region of

the intersection by a Gaussian cuto! function, such that it vanishes at large distances

from the coupling region, and is antisymmetric in the coupling coordinate. The conical

intersection coincides with the minimum of the upper diabatic potential surface, and

under to the classification proposed by Atchity et al. is an intermediate case between a

sloped and a peaked crossing [109].

This model represents a challenge to the QCL theory since the Gaussian cuto! on

V01 produces an infinite set of higher-order terms in the full quantum propagator which

are treated approximately in the QCL theory. An algebraically simpler model exists

that is highly similar to the FLV model, known as the the linear vibronic model (LVM)

[110, 111]. With the exception of the linear diabatic coupling terms that are present in

the LVM, it is identical in structure to the FLV model. In fact, the QCL equation is

exact for the LVM, and the short time dynamics have been studied using QCL methods

[108]. However, the diabatic coupling term in the FLV model is localized in the region of

the degeneracy, whereas in the linear case this coupling is global. Studies by electronic

structure theorists, (e.g.) [112, 113], and dynamics theorists [25, 26], suggest that the

localized form of the coupling is more physically relevant in general, as well as for the

particular case of charge transfer processes. Although the FLV model was originally

posited with a small molecule in mind, it is quite generally appealing; for example, this


Figure 3.2: Plot of the diabatic electronic coupling V01(X, Y ) for " = 0.01. Brighter areas

correspond to positive coupling, and darker regions correspond to negative coupling.

model could possibly also be used to study charge transfer processes between identical

moiteties in (e.g.) biphenyl-type compounds, in which the two rings are held at 90 degrees

by steric interactions [114].

Of particular interest are the subsystem adiabatic energies (Born-Oppenheimer sur-

faces) and nonadiabatic coupling matrix elements. We shall find use for these quanti-

ties in carrying out the numerical solution to Eq. (2.62), which shall be described in

the next chapter. Recall that the adiabatic basis is defined by the eigenvalue problem,

hs,W (R)|.; (R)# = E$(R)|.; (R)#. The adiabatic energies, E$(R), depend parametrically


on the coordinates of the environment,

E0,1(R) =V00(R) + V11(R)

20 1

2

1(V00(R)% V11(R))2 + 4V01(R)2. (3.3)

The adiabatic surfaces are depicted in Fig.(3.3), displaying the (conical) intersection

between the two surfaces. The elements of the nonadiabatic coupling matrix are,

Figure 3.3: Plot of the FLV adiabatic electronic surfaces for " = 0.01.

d01(R) =h

Nd%a(R)$01(R)

%)R(V11(R)% V00(R)) +

#V00(R)% V11(R)

V01(R)

$)RV01(R)

&. (3.4)

In the last expression Nd%a(R) is a normalization factor for the transformation from the

diabatic basis to the adiabatic basis. These functions have a significantly more compli-

cated structure than the diabatic coupling, and are depicted in Fig.(3.4) and Fig.(3.5).

The X component is a symmetric structure mainly that is focused along the line of

diabatic crossings (X = 3), while the Y -component reflects the anti-symmetry of the

coupling mode focused mainly about the conical intersection.

We now proceed to generalize the FLV model to include coupling to a heat bath by

imagining that each of the stretching coordinates experience a further bilinear coupling to

identical, but independent, sets of harmonic oscillators. Thus we may include the e!ect

of dissipation on the system in our QCL description. The (partially Wigner transformed)


Figure 3.4: Contour plot of the X-component of the nonadiabatic coupling vector

d01(X,Y ) · X for " = 0.01. Bright regions correspond to strong positive coupling, and

dark regions correspond to near zero coupling.

total Hamiltonian is

HW

(X, Y, R, P ) = hs,W

(X, Y ) +# /

j

P 2j

2Mj+

/

l

P 2l

2Ml(3.5)

+# /

j

1

2Mj$

2j (Rj %

cj

Mj$2j

X)2 +/

l

1

2Ml$

2l (Rl %

cl

Ml$2l

Y )2$$

I.

The coordinates and momenta of each bath oscillator with mass Mj are (e.g) (Rj, Pj),

and the coupling constants and frequencies (cj, $j) are chosen to correspond with those


Figure 3.5: Contour plot of the Y -component of the nonadiabatic coupling vector

d01(X,Y ) · Y for " = 0.01. Bright areas correspond to positive coupling, dark areas

correspond to negative coupling strength, and grey regions to near zero coupling.

of a infinite harmonic bath with an Ohmic spectral density [107],

J($) = */

c2j/(2Mj$j)-($ % $j), (3.6)

where cj = (%h$0Mj)1/2$j, $j = %$c ln (1% j$0/$c) and $0 = $c

#1% e!&max/&c

$/NB.

The subsystem free energy surfaces are defined as

W$(X, Y ) =%1

!ln

1

Z

!(jdRjPue

!%E!(R), (3.7)

Z =!

dXdY (jdRje!%(E0+E1),

where Pu is a uniform probability distribution and Z is the subsystem partition function.


We note here that for this generalized model, the subsystem free energies correspond to

the adiabatic potential surfaces of the original FLV model.

A set of parameter values for the subsystem has been employed that is relevant to

molecular systems [1], where (e.g.) the frequencies of the oscillators are $X = 219cm!1,

and $Y = 850cm!1. A summary of the values chosen for the model parameters is provided

(in atomic units) below.

MX = 20000 MY = 6667 $X = 0.001 $Y = 0.00387

X1 = 4 X2 = X3 = 3 . = 3 ! =3

2% = 0.01 " = {0, 0.08} (3.8)

In numerical simulations one normally works with a model in terms of dimensionless

variables so that the results may be scaled to any relevant regime. For example a dimen-

sionless energy scale corresponds to the characteristic energy of a bath oscillator, h$c,

hence HW = HW /h$c. A set of coordinate transformations that render the above model

dimensionless to aid in further calculations is as follows:

X =#MX$c

h

$ 12 X, Y =

#MY $c

h

$ 12 Y, PX = (hMX$c)

! 12 PX , PY = (hMY $c)

! 12 PY

Rj =#Mj$c

h

$ 12 Rj, Pj = (hMj$c)

! 12 Pj, %X =

h%

MX$c, %Y =

h%

MY $c, $j = $j/$c

cXj = (%X$0)

12 $j, cY

l = (%Y $0)12 $l, $X = $X/$c, $X = $Y /$c

. =h

MX$c., ! =

h

MY $c!, " =

"

h$c

# h

MY $c

$ 12 , % =

%

h$c. (3.9)

3.1.1 Avoided Crossing Model

By dropping the linear factor in the diabatic coupling term in FLV model we return to

the case of an avoided crossing, where the degeneracy in the diabatic basis is lifted by the

transformation to the adiabatic basis. Although this type of model does not necessarily

pose the same steep theoretical challenges as in the case of the conical intersection, we

shall later find it useful for the purpose of comparison. A two dimensional avoided

crossing model (ACM) may then be defined in the diabatic basis exactly as per the


Ferreti model with the exception of the diabatic coupling, where we make the substitution

V ACM01 (X, Y ) = "e!$(X!X3)2/*2

e!%Y 2/*2. The new parameter & is a measure of the size of

the localized coupling region.

3.2 Photoisomerization of a Protonated Schi! Base

In order to capture further e!ects of an environment on the molecular conical intersec-

tion a model system due to Burghardt, Cederbaum, and Hynes (BCH) was chosen to be

included in this study. This model represents a protonated Schi! base (PSB) molecule in

a polar medium, which may undergo ultrafast cis % trans isomerization upon photoex-

citation. The model consists of two coupled, charge localized, vibronic states which are

subsequently coupled to a continuous dielectric medium. The electronic states of the PSB

molecule are modeled based on an earlier description the structure of the conical intersec-

tion, by Bonacic-Koutecky, Koutecky and Michl (BKKM) [115]. In this particular case

the conical intersection arises in the PSB electronic structure as the torsional coordinate

about a carbon-carbon double bond reaches a twisted geometry, corresponding to a con-

figuration known as a biradicaloid structure. The e!ect of the environment is included

in the model through the use of dielectric continuum theory and free energy arguments.

Upon combining these e!ects, a two-state three-mode model for the environmental e!ects

on a system containing a conical intersection is prescribed. [116, 117, 118, 119, 120]

Let us first review the BKKM model for the electronic states. The structure of the

PSB molecule of interest is%CH3 % CAH = CBH % CH = NH2

&+. For simplicity we

invoke a Valence Bond type picture where the two *-bonding electrons are idealized as

localized p orbitals, which are associated with the two carbon nuclei labeled CA and CB.

The S1 % S0 conical intersection model may then be constructed in terms of the two

singlet electronic states:

|AB# =1/2[A(1)B(2) + B(1)A(2)]2 |)#, (3.10)


|B2# = [B(1)B(2)]2 |)#,

where (A(i), B(i)) refer to the occupation of a p-orbital on CA,B by electron i, and |)#

is the singlet spin configuration for two electrons, (i.e.) |)# = 1(2[.1!2 % !1.2]. The

corresponding |A2# configuration is shifted to higher energies and is neglected in this

treatment. The two electronic configurations correspond to di!erent localized charge

distributions. In this case the |AB# (”dot-dot”) configuration corresponds to the excess

positive charge being localized at the nitrogen end of the molecule, (i.e.) to the right

of the CACB double bond, and the |B2# (”hole-pair”) configuration corresponds to the

positive charge localized at the hydrocarbon end of the molecule, to the left of the CACB

double bond. Thus an electronic transition between these two states constitutes a charge

transfer process. At the twisted, biradicaloid, geometry the two states decouple due

to symmetry requirements; as the torsional angle twists away from this configuration

the two states exhibit a non-zero overlap integral which is reflected in the * bonding

character these states exhibit. The torsional angle, #, thus emerges naturally here as

a coupling coordinate in the model. The C = C double bond stretching coordinate,

denoted R0 here, is chosen to play the role of a tuning mode, and does not contribute to

the interstate coupling.

The polarization field of a polar solvent is expected to be strongly coupled to the

charge transfer process described above. Within the dielectric continuum picture the

general nonequilibrium polarization of the solvent (a vector field in coordinate space)

may be broken into two major collective contributions; a fast electronic part, and a

slower orientational component,

P = Pel + Por. (3.11)

The electronic part corresponds to the response of solvent electronic dipoles, while the

orientational part refers the orientational dipole response. As a first order approxima-

tion, assuming an appropriate separation of time-scales between these two responses, the

electronic component may be neglected and the orientational component is represented


as a linear combination in terms of a single solvent coordinate Z and the equilibrium

polarization field contributions corresponding to each charge-localized state,

Por(Z; R0) = ZP eqor,AB(R0, #) + (1% Z)P eq

or,B2(R0, #). (3.12)

Under a polarization field of this type the free energy surfaces are parabolic along the

solvent coordinate Z [116],

W sAB(R0, #, Z) = W eq

AB(R0, #) +1

2ksZ

2, (3.13)

W sB2(R0, #, Z) = W eq

B2(R0, #) +1

2ks(1% Z)2,

which motivate the use of displaced parabolic potentials in the model to describe the

solvent interaction.

Based upon these considerations the solution phase diabatic free energy model was

constructed to resemble electronic surfaces obtained from electronic structure calcula-

tions, and has been refined in a series of articles [116, 117, 118, 119]. The (partial

Wigner transformed) electronic Hamiltonian matrix, in the diabatic basis, is

hs,W

(R0, #, Z) = (P 2

R0

2µR0

+P 2

+

2µ++

P 2Z

2µZ)I +

;

<<=VAB(R0, #, Z) "(#)

"(#) VB2(R0, #, Z)

>

??@ . (3.14)

The diagonal elements of the matrix above correspond to the energies of the |AB# and

|B2# states in solution, and the o! diagonal element describes the interstate coupling:

VAB(R0, #, Z) = V MorseAB (R0) + V Shift

AB (R0, #) + V +AB(#) + V solv

AB (Z), (3.15)

VB2(R0, #, Z) = V MorseB2 (R0) + V Shift

B2 (R0, #) + V +B2(#) + V solv

B2 (Z) + V FCB2 (R0, #) + %EB2 ,

"(#) = c! cos(#).

The electronic coupling term reflects the overlap integral between the localized electronic

functions on neighboring centers, and was chosen in accordance with existing models

for ethylene [121]. As an example of the parametrization chosen for this model, the


individual potential functions are as follows:

V MorseB2 (R0) = DB2(1% exp(%.B2(R0 %Req

0,B2)))2, (3.16)

V ShiftB2 (R0, #) = cShift

B2 (R0 %Req0,B2) sin2(#),

V +B2(#) = c+

B2 cos2(#),

V FCB2 (R0, #) =

1

2kFC sin2(#) exp(%(R0 %R0,FC)2/&R0FC) exp(% sin2(#)/&+FC),

V MorseAB (R0) = DAB(1% exp(%.AB(R0 %Req

0,AB)))2,

V ShiftAB (R0, #) = cShift

AB (R0 %Req0,AB) sin2(#),

V +AB(#) = c+

AB cos2(#).

The V Morse functions are chosen to reflect the vibrational character of the R0 coordinate,

while the V Shift potentials displace the minimum of the Morse curves for each state as

the torsional angle increases. The form of the V + potential was chosen based on the

results of the work of Tennyson and Murrell [121] on the electronic structure of ethylene.

The function V FCB2 is a correction factor for the local structure of of the Franck-Condon

(FC) region. This overall parametrization was chosen so as to match available ab initio

data as well as possible [112, 118]. The solvent contribution to the diabatic potentials

mimic the free energy picture,

V solvAB (Z) =

1

2ksZ

2, (3.17)

V solvB2 (Z) =

1

2ks(1% Z)2.

To complete our description of this open system, we simply imagine that the solvent

coordinate experiences a further bilinear coupling to a heat bath composed of independent

harmonic oscillators. The (partially Wigner transformed) total Hamiltonian is

HW

(R0, #, Z, R, P ) = hs,W

(R0, #, Z) +/

j

# P 2j

2Mj+

1

2Mj$

2j (Rj %

cj

Mj$2j

Z2)$I. (3.18)

The coordinates and momenta of each bath oscillator with mass Mj are (Rj, Pj), and

the coupling constants and frequencies (cj, $j) are, again, chosen to correspond with those


of a infinite harmonic bath with an Ohmic spectral density [107]. We have constructed

the bath coupling for this model such that the subsystem free energy surfaces correspond

to the original diabatic free energy model potential energy surfaces of the BCH model.

A summary of the parameter values chosen for the model parameters can be found in

reference [53, 118].

In this three-mode model the conical intersection manifests as a seam of points in

the (R0, Z)-plane, at # = 0, where the diabatic coupling vanishes and the two electronic

states are degenerate. This seam of conical intersection points is the solution to

VB2(R0, 0, Z)% VAB(R0, 0, Z) = 0, (3.19)

and is depicted (for a particular set of parameter values) in figure (3.6). In this way we

clearly see how the presence of a solvent can a!ect the position, and even the existence,

of the the conical intersection.

3.3 Another Two State -Three Mode Model

The utility of the three dimensional BCH model is partially reduced by its algebraic

complexity which spawns some numerical di"culties associated with the QCL solution

techniques, which will be discussed later. However, motivated by the general physical ap-

peal of the environmental e!ects included in the BCH model, and the algebraic concision

of the FLV model, we propose the following (partially Wigner transformed) subsystem

Hamiltonian:

hs,W

(X,Y, Z) = (P 2

X

2MX+

P 2Y

2MY+

P 2Z

2MZ)I +

;

<<=V00(X, Y, Z) V01(X, Y )

V01(X, Y ) V11(X, Y, Z)

>

??@ , (3.20)

where

V00(X, Y, Z) =1

2MX$2

X(X %X1)2 +

1

2MY $2

Y Y 2 +1

2MZ$2

ZZ2, (3.21)

V11(X, Y, Z) =1

2MX$2

X(X %X2)2 +

1

2MY $2

Y Y 2 +1

2MZ$2

Z(1% Z)2 + %,

V01(X, Y ) = "Y e!$(X!X3)2e!%Y 2.


Clearly, we have appropriated the FLV vibronic system, and coupled it with the solvent

polarization coordinate from the BCH model; for convenience it shall be referred to as

the XYZ model. We further imagine immersing this system in an infinite heat bath,

modeled in the following familiar fashion,

HW

(X, Y, Z, R, P ) = hs,W

(X,Y, Z) +# /

j

P 2j

2Mj+

/

k

P 2k

2Mk+

/

l

P 2l

2Ml(3.22)

+/

j

1

2Mj$

2j (Rj %

cj

Mj$2j

X)2

+/

l

1

2Mk$

2k(Rk %

ck

Mk$2k

Y )2

+/

l

1

2Ml$

2l (Rl %

cl

Ml$2l

Z)2$I.

The XYZ model prescribed above thus aims to describe a host of environmental e!ects

on a molecular subsystem containing a conical intersection; the polarization field of the

solvent induces fluctuations in the adiabatic energy gap giving rise to electronic dephas-

ing, and the presence of the heat bath provides vibrational dissipation and dephasing.

The addition of the solvent coordinate to the 2-dimensional model generates a new nona-

diabatic coupling term, d12 · Z, and the conical intersection corresponds to a linear seam

of points in the (X, Z)-plane at Y = 0, shown below in Fig. (3.6).


0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3

R0 , X

Z

BCH Model

XYZ Model

Figure 3.6: Seams of conical intersections prescribed by the BCH and XYZ models.

Chapter 4

Molecular Dynamics

Among the major goals of this work is to explore how mixed quantum-classical meth-

ods, specifically those based on the QCL equation, can be used to treat problems in

condensed phase quantum dynamics. The aforementioned conical intersection models

represent systems for which the QCL equations of motion are not exact due to the form

of the electronic coupling matrix element in the diabatic basis. As such they provide new

challenges that test the utility of the QCL equation itself, as well as the various algorithms

that are used in its solution. In this chapter we outline an approximate dynamical scheme

based on the quantum-classical Liouville equation which allows a non-perturbative, non-

Markovian, yet numerically tractable solution involving relatively minimal approxima-

tions. The details of these molecular dynamics simulations shall then be discussed. While

exact numerical quantum results are known for the original FLV model [2, 1], little is

currently known about the generalized models which include the environment. Before

going too far afield, details of the numerical method are put under scrutiny.

4.1 Trotter-Based Propagation Algorithm

As an example let us consider the computation of a quantity of interest such as an

electronic state population as a function of time; the ground and excited adiabatic state

57

Chapter 4. Molecular Dynamics 58

populations are PS0 and PS1 respectively,

PS!(t) = Tr"!

dXPS!(X, t)+W (X). (4.1)

The projector onto the adiabatic state . is PS!(X, t) = |.; (X(t)#".; (X(t))|, and +W (X) =

+s+W (X) is the (joint uncoupled) initial density of the subsystem and environment. We

shall also consider other quantities such as the reduced densities, +$$s (X, Y ), and the

electronic coherence, +01s (t).

As stated previously, the evaluation of this expression hinges on computing the time

evolution of some quantity, inevitably involving the nonadiabatic system propagator, and

sampling from some initial distribution. The bulk of the calculations performed for this

work were based on an algorithm developed by MacKernan et al., [122]. The starting

point of this Trotter-based solution is to write the Liouvillian in the adiabatic basis, and

write it as a sum of diagonal and o!-diagonal parts,

iL$$!,%%! = (i$$$! + iL$$!)-$%-$!%! % J$$!,%%! . (4.2)

Since L commutes with itself (due to its time-independence) one may divide eiLt into a

string of N = t/- short time propagators without loss of generality. Denoting pairs of

quantum states (..") by a single index, s = 2.+.", the total propagator may be written

as

(eiLt)s0,sN =/

s1...sN"1

N3

j=1

(eiL,)sj"1,sj . (4.3)

The Trotter factorization for each short time propagator is now employed on each

term in the product (4.3),

(eiL,)sj"1,sj = (eiL(0),/2)sj"1(e!J ,)sj"1sj(e

iL(0),/2)sj +O(-3). (4.4)

Through the use of a further Trotter decomposition on the short time nonadiabatic

propagator, one can show that

(e!J ,)ss! = (Q1)ss!

#1 + Css!

)

)P

$+O(-2). (4.5)


The derivation of the matrices Q1 and C is discussed in [122]. These matrices are generally

defined for any N -level quantum subsystem, and for a two level subsystem are

Q1 =

;

<<<<<<<<<<=

cos2(a) % cos(a) sin(a) % cos(s) sin(a) sin2(a)

cos(a) sin(a) cos2(a) % sin2(a) % cos(a) sin(a)

cos(a) sin(a) % sin2(a) cos2(a) % cos(a) sin(a)

sin2(a) cos(s) sin(a) cos(s) sin(a) cos2(a)

>

??????????@

, (4.6)

C =

;

<<<<<<<<<<=

0 S01 S01 2S01

S10 0 0 S01

S10 0 0 S01

2S10 S10 S10 0

>

??????????@

, (4.7)

where a = PM · d$%- and S01 = h$01d01/(2(P/M)d01).

Now the well-known ’momentum-jump’ approximation is utilized to facilitate compu-

tation of the nonadiabatic propagator. Within this approximation the continuous action

of e!J is approximated by discrete changes in the environmental momenta which accom-

pany the subsystem transitions. Employing this approximation here essentially entails

exchanging 1 + Css!#

#P for the exponential eCss!#

#P [122], which is valid when the changes

in environmental momenta are small. The nonadiabatic propagator obtained in this case

is

(e!J ,)ss! - (Q1)ss!eCss!

##P +O(-2) = M(-) +O(-2). (4.8)

The MacKernan matrix M(-), in the case of a two level subsystem, has the explicit

form

M(-) =

;

<<<<<<<<<<=

cos2(a) % cos(s) sin(a)j01 % cos(s) sin(a)j01 sin2(a)j0%1

cos(a) sin(a)j10 cos2(a) % sin2(a) % cos(a) sin(a)j01

cos(a) sin(a)j10 % sin2(a) cos2(a) % cos(a) sin(a)j01

sin2(a)j1%0 cos(s) sin(a)j10 cos(s) sin(a)j10 cos2(a)

>

??????????@

.

(4.9)


The operators j$% and j$%% give rise to the momentum changes which accompany nona-

diabatic transitions. Their action on an arbitrary function f(P ) yields f(P +%P ), where

%P$% = d$%sgn(P · d$%)1

(P · d$%)2 + hM$$% % (P · d$%),

%P$%% = d$%sgn(P · d$%)1

(P · d$%)2 + 2hM$$% % (P · d$%). (4.10)

If we now write the propagator (eiL(0),)sj as the product of an accumulating phase

factorWsj(tj!1, tj) = ei4!tj

d'&sj (Rsj ('))and a classical evolution propagator on the surface

sj, the following expression for a single short time segment is obtained,

(eiL,)sj"1,sj -W(tj!1, tj % -/2)(eiL,/2)sj"1Msj"1sj(-)W(tj % -/2, tj)(eiL,/2)sj . (4.11)

Formulating the dynamics in this way may be interpreted as a type of surface hopping

representation. Each sequence of short time propagators corresponds to a trajectory com-

posed of classical evolution segments on a single, or the mean of two, Born-Oppenheimer

surfaces interspersed with energy conserving quantum transitions. Collecting an ensem-

ble of trajectories enables one to reconstruct the expression for the average value (2.62),

B(t) =/

s0

!dXBs0

W (X, t)+s!0W (X) =

/

s0

!dX+

s!0W (R,P )

/

s1...sN"1

% N3

j=1

W(tj!1, tj % -/2)(eiL(0),/2)sj"1Msj"1sj(-)W(tj % -/2, tj)(eiL(0),/2)sj

&

+Bs0W (R,P ). (4.12)

where s"0 is obtained from s0 by interchanging . and .". Evaluation of the above expres-

sion is carried out via a Monte Carlo procedure described in detail in [122, 123]. Briefly,

one performs importance sampling on the phase space density of the environment and

Monte Carlo sampling on the initial subsystem states as dictated by +s!0W (X). This en-

semble of initial conditions is then evolved in time via (4.11) employing a Monte Carlo

sampling on the matrix M. Each segment of forward propagation in time is carried out

by evaluating the action of the propagators, in order from left to right, on the given state

at that time [122, 123].


Incorporating the Monte Carlo sampling of quantum transitions introduces weight

factors based on the initial density and the elements matrix Q1 at each transition. The

Monte Carlo average value has the form

B(t) =N 2

K

K/

-=1

+s!$0W (R-, P -)

|+s!$0W (R-, P -)|

% N3

j=1

#Ws$

j"1(tj!1, tj % -/2)(eiL(0),/2)s$

j"1+ (4.13)

Ms$j"1s$

j(-)Ws$

j(tj % -/2, tj)(e

iL(0),/2)s$j

$

s$j"1,s$

j

&B

s$N

W (R-, P -),

where the index 4 refers to the sampling of the elementary event (R-, P -, s-0 , s

-1 , ..., s

-N),

and average is assembled over K of such events. The factor N 2 comes from the uniform

sampling for the sum on initial states s0.

A final and important part of the algorithm is the use of a filter on the square bracketed

term in equation (4.14). The average value is dominated by large fluctuations in the

summand due to this term; owing its origin to the quantum-mechanical sign-problem

these fluctuations occur as the increasing statistical weights amplify the rotating phase

factors in the evolution. For the studies performed in this work the filter was employed

as a strict upper-bound (or cuto!), #, on the magnitude of the square-bracketed term.

When, at step j in the evaluation of a trajectory, the magnitude of the summand exceeds

#, nonadiabatic transitions are no longer sampled and the value of the running product

is set to unity. In practice the appropriate value of # depends on the nonadiabaticity

of the system and the length of the simulation. In the simulations reported here values

chosen for # typically ranged between 103 and 104. Imposing this numerical cuto! on the

observable in the MC-MD simulations is an uncontrolled approximation, which introduces

a systematic error in the average values reported. In practice, the value for # is chosen

to be as large as possible, in order to minimize this error.

4.2 Initial state

In order to model the initial state of the system we imagine that the electronic subsys-

tem has been prepared in a Franck-Condon type state, which is modeled as a Gaussian


wavepacket on the excited adiabatic state surface (perhaps via photoexcitation from some

lower-lying state), +s(t = 0) = |1; (R)#"1; (R)|.

Invoking the well-known, albeit approximate, Feynman-Vernon type initial conditions

the initial density may be decomposed into electronic subsystem and environmental (vi-

brational and bath) components, (e.g.) +W (X) = +s+(vib)W (X, Y )+(b)

W (R,P ). The initial

vibrational states are chosen for the various models as Gaussian packets of the following

form:

'(vib)(X, Y ) =1/

*%X%Yexp

#% (X %X0)2

2%2X

% Y 2

2%2Y

$, (4.14)

'(vib)(R0, #, Z) =1

1*3/2%R0%+%Z

exp#% (R0 %R0,FC)2

2%2R0

% #2

2%2+

% (Z % ZFC)2

2%2Z

)$,

'(vib)(X, Y, Z) =1

1*3/2%X%Y %Z

exp#% (X %X0)2

2%2X

)$

exp#% Y 2

2%2Y

)$

exp#% (Z % ZFC)2

2%2Z

)$.

These initial wavepackets are then Wigner transformed,

+vibW (R,P ) = (2*h)!d

!dZ eiP ·Z/h'#(vib)(R% Z

2)'(vib)(R +

Z

2), (4.15)

where d is the dimensionality of the space, to obtain the initial densities,

+(vib)W (X,Y, PX , PY ) =

# 1

2*

$2exp

%% (

P 2X%2

X

h2 +(X %X0)2

%2X

)&+ (4.16)

exp%% (

P 2Y %2

Y

h2 +Y 2

%2Y

)&.

The oscillator baths are assumed to initally be in thermal equilibrium, and we adopt

the quantum bath distribution function [96],

+(b)W (R,P ) =

3

j

tanh(!$j/2)

*exp

%% 2 tanh(!$j/2)

$j

#P "2j

2+

$2j R

"2j

2

$&, (4.17)

where (e.g.) (R"j, P

"j) = (Rj% cj

&2jX, Pj). The bath is characterized by the cut-o! frequency

$c, and the Kondo parameter, %, [124]. In the simulations presented here, NB = 100 for

each set of oscillators, and $max = 3$c.


4.3 Numerical Details

The molecular dynamics algorithm prescribed in the adiabatic basis is a surface-hopping

type scheme consisting of segments of classical coordinate propagation, interspersed with

the sampling of energy conserving quantum transitions. As such, an analysis of the details

of the quantum transitions that occur in the simulations can reveal various important

aspects of the models and solution method. Some comparisons with Tully’s simple surface

hopping method are provided as a reference point from which to shed light on the nature

of this algorithm.

As a prototypical case we shall mainly focus on the FLV model, which was developed

initially to study the quantum dynamics of a wave-packet as it makes a single passage

through a conical intersection. The e!ects of an external environment such as decoherence

and dissipation are neglected. From the initial state prescribed above the short time

dynamics of this model consist of propagation along the positive X direction, concurrently

with slightly faster oscillations in Y coordinate. As the system enters the strong coupling

region nonadiabatic transitions begin to take place. This is reflected in Fig. (4.1) where

a branching of the (wave-packet) population from excited state to ground state occurs

between t = 15fs and t = 35fs, as the packet travels through the strong coupling

region. After this time the state populations stabilize as the wave-packet leaves the

strong coupling region and approaches the classical turning point on each surface.

First we observe in the simulations when, on average, the system undergoes nonadia-

batic transitions. The average number of hops per trajectory, as shown in Fig. (4.2), rises

from zero to some finite value over a period of time corresponding to the interval during

which the system passes through the region of the intersection. As the nonadiabatic

coupling is essentially zero outside this region, this behavior is expected.

Of course, one expects the number of transitions in the QCL simulations to be higher

than in the case of Tully’s method since in Tully’s scheme population is transfered directly

from one adiabatic state to another, whereas in the QCL scheme quantum transitions


0.01

0.02t = 20 fs

t = 30 fs

t = 40 fs

2 3 4 5

X (a.u.)

0

0.01

0.02

re

du

ce

d d

en

sity

S1

S0

Figure 4.1: Propagation and branching of the wavepacket along the X coordinate.

Results from QCL simulations of the original FLV model in the adiabatic basis with

" = 0.01au.

typically occur between an adiabatic surface and the mean of two adiabatic surfaces. Thus

it typically takes at least two hops in QCL trajectory in order to transfer population from

one state to another. This is not strictly the case however, since a double-jump term

does exist in the transition matrix. However, in practice the double jump transitions

only account for around one percent of the total number surface hops that take place in

the ensemble. As a result we may naively expect the QCL trajectories to show at least

twice as many surface hops, on average, as the Tully trajectories. Furthermore, each

single-hop term contributes to the creation or destruction of coherent evolution in the

system.

In fact, we see that the QCL trajectories experience approximately three times, to

about once per trajectory for the Tully results. This three fold di!erence is due to the

fraction of trajectory segments that evolve on mean surfaces during the QCL calculation.


10 20 30 40 50 60time (fs)

0

1

2

3

NJU

MP

(t)

QCL

Tully

Figure 4.2: Average number of quantum transitions per trajectory as a function of time

for the QCL and Tully surface hopping simulations of the original FLV model in the

adiabatic basis, with " = 0.01.

These segments of evolution on the mean surface are a major di!erence between the

two surface hopping algorithms, and they are essential (in the QCL implementation) to

properly treat the evolution of coherence in the system.

The distribution of the number of jumps per trajectory for the QCL and Tully sur-

face hopping simulations is depicted in Fig. (4.3). We see that the for the gas phase

model the QCL simulations have a rather wide distribution indicating many instances

of population transfer and the creation and destruction of coherence. In the case of the

generalized model, as one increases $c the distribution does not significantly change until

the characteristic frequency surpasses approximately 500cm!1. For small $c the jump

distribution is essentially coincident with the gas phase result, as evidenced by the over-

lap with the $c = 50cm!1 curve shown in Fig. (4.3). For larger $c the distribution begins


2 4 6 8 10N

JUMP

0.2

0.4

0.6

0.8

frequency

Tully (gas phase)

QCL (gas phase)

!c = 50 cm

-1

!c = 500 cm

-1

!c = 850 cm

-1

!c = 5000 cm

-1

Figure 4.3: Distribution of the number quantum transitions per trajectory for the QCL

and Tully surface hopping simulations performed of the original FLV model in the adia-

batic basis, with " = 0.01.

to narrow and the average/peak value reduces, and the distribution approaches the jump

distribution corresponding to Tully’s algorithm. This trend may reflect decoherence in

subsystem as a result of the presence of the bath. A reduction in the number of overall

jumps per trajectory indicates that there may be less coherent evolution present in the

dynamics, and that a master equation approach may be justified in this limit. Another

a consequence of the suppression of jumps (due to the presence of a fast bath) is a slight

alleviation in the overall accumulation of weight factors from the Monte Carlo sampling.

As the number of quantum transitions increases, so too does the cumulative weight

factor that comes from the Monte Carlo sampling thereof. In performing QCL simula-

tions the cumulative weight factor imposes an added challenge in terms of the numerical

e!ort (number of trajectories) required as compared to Tully’s algorithm. Using Tully’s

algorithm, which does not incorporate weight factors, one may obtain high precision


results (error - 1%) by using 2000 trajectories for " = 0.01. The Trotter-based QCL

algorithm requires approximately 105 trajectories to solve the corresponding problem,

however the precision of the observable is lower (error - 10%). Hence, in order to re-

solve delicate (quantum) features of the dynamical problem an extremely large number

of trajectories were sampled, in most cases - 107. This is an unfortunate feature of

the error scaling (error 3 1/Ntraj

) in the Monte Carlo algorithm; in order to extend the

QCL method to larger systems involving multiple di!erent CI’s, a di!erent solution al-

gorithm shall likely be required. A detailed comparison of the numerical performance

of the Trotter-based QCL method is compared with other trajectory based simulation

methods for the linearized path integral [94].

As mentioned earlier, a cuto! is imposed on the cumulative weight factor in order to

relieve some of the computational e!ort and smooth the data. For a given set of system

parameters, the distribution of weight factors for an ensemble of trajectories exhibits an

initially exponential-type decay that flattens out into a very long tail that extends for

several orders of magnitude, as depicted in Fig. (4.4). The fraction of weights in the tail

hamper the fidelity of the QCL simulations for finite sample sizes. However, the influence

of this fraction of trajectories is thought to be quite small due to the oscillating phase

factors involved in the simulations. As such the cuto! value, #, is chosen such that as

much of the distribution as possible is captured. The breadth of the weight distribution

changes depending on the nonadiabaticity of the system, which in this case is controlled

by the value chosen for ". The weight distribution is narrow for small ", and becomes

wider and wider as one increases ". Thus the chosen value of # increases with ", and

typically ranges from values around 103 for the weak coupling up to around 104 for strong

coupling. In a sense the original gas phase model is the worst-case scenario for a Monte

Carlo scheme attempting to sample quantum transitions, and one expects that adding

the bath will partially alleviate the phase oscillations due to the quantum subsystem.

It is also interesting to examine where the system undergoes quantum transitions


1000 2000 3000 4000 5000weight factor

0

5e-06

1e-05

1.5e-05

2e-05

2.5e-05

frequency

!!"!#$#%

!!"!#$#&

!!"!#$#'

!!"!#$#(

!!"!#$#)

!!"!#$#*

Figure 4.4: Distribution (normalized) of Monte Carlo weights recorded at t = 70fs for

QCL simulations of the original FLV model performed in the adiabatic basis, for a range

of coupling strengths.

during the simulations. Since the nonadiabatic coupling is strongest near the conical

intersection, we expect most of the transitions to occur close to (X = 3, Y = 0). Indeed

this is the case for both the QCL simulations in the adiabatic basis, as well as for

the simulations realized using Tully’s method. The jump distributions, shown in Fig.

(4.5), exhibit a large degree of overlap and are rather sharply focussed about the conical

intersection. The main di!erence is that the peak of the X distribution in the case of

Tully dynamics is shifted to the left of the intersection. This is most likely due to the

well-known problem of over coherence present in Tully’s method.

In all the model cases studied we find that simulating the dynamics of the system

plus bath leads to a broadening of the jump distributions, and a slight (but favourable)

shift in the observable distributions. This is expected as the bath tends to ’smear’


-1 -0.5 0 0.5 1Y (a.u.)

0

0.5

1

1.5

2

2.5J

um

p D

istr

ibu

tio

n

QCL

Tully

2.5 3 3.5 4X (a.u.)

0

1

2

3

4

5

Ju

mp

Dis

trib

uti

on

QCL

Tully

Figure 4.5: Distribution of quantum transitions as a function of the X and Y coordinates

for QCL and Tully surface hopping simulations performed in the adiabatic basis, with

" = 0.01.


the dyanmics of the classical coordinates, which leads to decoherence so that the sign

problem is partially mitigated. As an example we show the torsional coordinate jump

distributions for the original and generalized BCH model in Fig. (4.6). When the bath is

not present quantum jumps are very sharply peaked about the odd half-integer multiples

of *, corresponding to the subspace of conical intersections. When the bath is turned on

this distribution become somewhat less sharp. It is important to note that in executing

-10 0 10 20θ (rad)

0.1

0.2

0.3

0.4

0.5

0.6

Jum

p Di

strib

utio

n

no bathωc = 0.01

Figure 4.6: Distribution of quantum transitions as a function of the torsional coordinate

#, obtained for QCL simulations of the BCH model performed in the adiabatic basis,

with c! = 2.5eV .

the Trotter-based surface hopping algorithm to solve the QCL equation in the adiabatic

basis we have invoked the momentum-jump approximation (MJA). Recall that the MJA

is valid when the transition energy is small compared to the classical kinetic energy, (i.e.)

%E$% << P 2

2M . In order to check the validity of the MJA we may examine the distribution

of transition energies, for example, which are depicted in Fig. (4.7). We see that for the

gas phase FLV model the distibution is rather broad, exhibits a shoulder-like feature and


then decays to zero well before unity (in dimensionless units). Hence we see that most

of the transitions do indeed occur at relatively low energies and the approximation is

(loosely) justified. When we consider the generalized model, these distributions become

narrower, and the shoulder sharpens to a peak that moves toward the origin as the

characteristic frequency of a bath oscillator increases. Thus we see that the e!ect of

adding a dissipative bath strengthens the validity of the momentum jump approximation

for the conical intersection model.

5

10

15

no bath !c = 50 cm

-1!

c = 220 cm

-1

0 0.50

10

20

P

[ "

E |

jum

p]

!c = 500 cm

-1

0.5"E (arb. units)

!c = 850 cm

-1

0.5 1

!c = 5000 cm

-1

Figure 4.7: Probability density of transition energies for the surface hopping implemen-

tation of the QCL equation of the original FLV model, as well as the generalized model

with a set of di!erent characteristic bath frequencies. For each case " = 0.01.


4.4 Population and Coherence

An important test of the QCL equation (and the associated solution algorithm) is a

comparison with the exact numerical quantum results. These data are provided in [1]

and [3]. Of particular interest are the average values of quantities such as the electronic

populations PS0 and PS1 . It is crucial that the QCL methods are able to reproduce

these results to high accuracy in order to justify further investigation of the more general

models. Indeed, we see in Fig. (4.8) that for low coupling strengths the agreement

0 10 20 30 40 50time (fs)

0

0.2

0.4

0.6

0.8

P S 0 (t)

QCL - adiabatic basisexact quantum

Figure 4.8: Evolution of the ground state adiabatic population PS0(t) for the gas phase

model. The quantum results are taken from [1], and the quantum-classical results refer

to those generated from the QCL equation using the Trotter-based algorithm in the

adiabatic basis, with " = 0.01, and # = 2000, using 107 trajectories.

between the exact results and the QCL results is excellent.

While the data for population decay are acceptable at low coupling, Tully’s algorithm


fails when one attempts to compute the coherence in the system. This problem with

Tully’s method is commonly known as over-coherence; an e!ect which can be seen by

examining the value of the o!-diagonal element of the subsystem density matrix, as shown

in Fig. (4.9). One expects that some coherence should be created in the subsystem whilst

it passes through the region of the conical intersection, as population is being transferred

from the excited state to the ground state. Once the system has left the strong coupling

region, however, the coherence that has built up will subsequently be destroyed, i.e.

decoherence will quickly ensue. While the QCL results do indeed reflect this scenario

quite well, Tully’s method yields a result that explodes to unphysical values shortly after

the first pass through the region of the intersection.

0 10 20 30 40 50 t (fs)

-0.2

-0.1

0

0.1

0.2

0.3

Re{ ρ s01

(t)}

QCLTully

50 60 70 80

-100

0

100

Figure 4.9: Real part of the o!-diagonal element of the subsystem density matrix as a

function of time, for " = 0.01. Note that the scale on the ordinate changes from the right

panel to the left panel.

In order to evaluate QCL theory as a function of the electronic coupling strength the


plateau value for the population transfer to the ground adiabatic state, S0, was taken as

the value of PS0 at t = 50fs. This quantity is shown for a range of coupling strengths in

Fig. (4.10).

0 0.02 0.04 0.06 0.08

!!(a.u.)

0.6

0.7

0.8

0.9

1P

S0 (t

= 5

0 f

s)

QM

QCL - Adiabtic Basis

Tully

Figure 4.10: Ground adiabatic state population at t = 50fs as a function of electronic

coupling strength in the FLV model, as predicted by various approximate nonadiabatic

methods and exact quantum mechanics (QM).

The QCL results given in Fig. (4.10) tend to underestimate the exact transition

probabilities, by approximately five to ten percent over the entire coupling range. As

well they exhibit excellent agreement with the shape of the transition probability curve

as a function of coupling strength. Results generated using Tully’s method also show

excellent agreement at low coupling strengths, but as the coupling increases this method

begins to fail as it underestimates the total probability of decay by around twenty percent.

The shape of the quantum curve, as a function of coupling strength, can be understood

through semiclassical arguments encompassed in a Landau-Zener based vibronic model

(LZVM). This model is based on a quasi-monodimensional approximation to the full

dynamics. Ferretti and co-workers constructed this semiclassical treatment [1] via an

explicitly classical treatment of the motion along the X coordinate. This corresponds


to a Born-Oppenheimer separation of the the X oscillator by neglecting the X kinetic

energy term from the total Hamiltonian,

H = hs + TY . (4.18)

Then a linear o!-diagonal electronic coupling is assumed in accordance with the limits

of the Landau-Zener approximation. V01 = "Y . A product-state diabatic basis is then

invoked, |I, 1# = |I#2 |1#, where |I# is the I-th diabatic electronic state and |1# is the 1th

state of the Y harmonic oscillator. A key feature of this treatment is that only the first

three vibrational levels are included. In this basis the diagonal elements of the electronic

Hamiltonian are,

"1, 1|hs|1, 1# =1

2MX$2

X(X %X1)2 + h$Y (1 +

1

2), (4.19)

"2, 1|hs|2, 1# =1

2MX$2

X(X %X2)2 + h$Y (1 +

1

2) + %.

The only non-zero o!-diagonal elements are,

"1, 1|hs|2, 1 % 1# = "

51

2MY $Y, (4.20)

"2, 1|hs|1, 1 + 1# = "

51 + 1

2MY $Y.

The diagonal energies are taken to be the manifold of diabatic electronic surfaces for this

approximate model, which indeed exhibit many level crossings [1]. For the purposes of

this particular model three possible vibrational states, 1 = {0, 1, 2}, are considered. If

one imagines that the system is initiated on the |1, 0#, state, the only allowed transition is

|1, 0# & |2, 1#. If this transition is successful then there is one possible allowed transition

for the resultant state, |2, 1# & |1, 2#. Employing the Landau-Zener approximation [8, 9]

to these possible transitions yields the following set of occupation probabilities

P1,0 = e!(, (4.21)

P1,2 = 1% e!( % e!2( + e!3(,

P2,1 = e!2( % e!3(.


where 0 = .!2

XKX |X1!X2|MY &Y, and X is the speed of the X-coordinate at the intersection

point.

The total probability of being in electronic state 1 is PLZ1 = P1,0 + P1,2. Since the

diabatic and the adiabatic states asymptotically coincide, we have

PS0(t&') = 1% exp!2( + exp!3( . (4.22)

Although this model involves many approximations it indeed shows that the population

transfer should decrease for small ", exhibit a minimum, and then continue to increase

with increasing coupling. While providing a good representation of the quantum result

a low couplings the model fails as " increases (see Fig. (4.10)), as the dynamics exhibit

true two-dimensional character and the anharmonicity of the potential energy surfaces

begins to play an important role in the transfer process.

4.4.1 BCH Model

The results obtained for the BCH model correspond to a lower electronic coupling

strength than originally prescribed in the model, in order to mitigate the statistical

weight problem. Unfortunately, even with a lower coupling strength the total possible

simulation time is still not su"cient to observe a full passage through the CI region, due

to the Monte Carlo weight problem a!ecting the uncertainty in the observable. This

di"culty stems from the particular Trotter-based simulation method employed here, and

is not associated with the QCL equation itself. On a short time-scale, before the weights

become too cumbersome, we see in Fig. (4.11) that the population transfer ratio and

the evolution of the electronic coherence (Fig. (4.12)) are somewhat insensitive to the

presence of the (slow) bath.


0 20 40 60 80 100 120

time (fs)

0.2

0.4

0.6

0.8

PS

0

(t)

no bath!

c = 0.05 a.u.

Figure 4.11: Ground state population as a function of time in the BCH model, for

c! = 0.25eV . In the case with the bath present $c = 0.05a.u., and % = 1.

4.5 Summary

Although the number of molecular dynamics trajectories required to simulate the QCL

equation becomes substantial at long times and large coupling strengths, the data pro-

duced represent a marked improvement over previous quantum-classical theories such as

Tully’s simple surface hopping technique and the Landau-Zener theory. Furthermore, we

see (in accordance with [3]) that the QCL theory is indeed an accurate approximation to

full quantum mechanics for a system containing a conical intersection, over a wide range

of coupling strengths. In the strong coupling regime the accumulation of Monte Carlo

weights from the sampling of quantum transitions renders low precision averages unless

one samples a very large number of trajectories. In the case of the BCH model, it seems


0 20 40 60 80 100 120time (fs)

0.05

0.1

0.15

0.2

0.25

0.3

Re {

ρs01

(t) }

no bathωc = 0.05 a.u.

Figure 4.12: Electronic coherence as a function of time in the BCH model, for c! =

0.25eV . In the case with the bath present $c = 0.05a.u., and % = 1.

that the Trotter-based solution algorithm is not well suited to further investigations as

a result of the Monte Carlo sampling e!ort that would be required. Nevertheless, for

problems where the accumulation of weights is not an insurmountable obstacle, the QCL

equation is a useful theoretical tool. In particular it could be useful in forming the basis

for open quantum system calculations where the addition of environmental degrees of

freedom tend to dull some of the numerical barbs that are characteristic of the isolated

quantum problem. Clearly, these results motivate the need for better solution methods

and simulation algorithms for the QCL equation.

Chapter 5

E!ects of the Environment

We would now like to explore further the e!ects of an external environment on the

subsystem dynamics. To this end the influence of the bath on subsystem population

transfer is assessed in the various models outlined earlier. As well, we examine how the

environment changes the evolution of the electronic coherence, including a discussion of

aspects the geometric phase in these model problems. In connection with the evolution

of electronic coherence in the presence of an environment, we examine the conditions

under which a Markovian master equation approach (based on the QCL equation) would

be applicable.

5.1 Population Dynamics

We first consider QCL dynamics simulations of the generalized two-dimensional coni-

cal intersection model. As mentioned previously, this model contains the original FLV

conical intersection model, plus two independent heat baths (composed of harmonic os-

cillators) that are bilinearly coupled to the vibrational coordinates. The classical heat

bath is completely characterized by three parameters; the characteristic frequency $c, the

dimensionless Kondo parameter %, and the reciprocal temperature ! = h&ckBT . The charac-

teristic frequency determines the peak and width of the (Ohmic) bath spectral density,

79

Chapter 5. Effects of the Environment 80

and defines the characteristic time-scale of the bath dynamics. The Kondo parameter de-

termines the amplitude of the spectral density, and the coupling strengths to the nuclear

coordinates. The e!ects of finite temperature in this model arise in the widths of the ini-

tial distributions for the coordinates and momenta of the bath. In all simulations of the

generalized model we have chosen ! at room temperature, corresponding to T = 300K.

10 20 30 40 50t (fs)

0

0.2

0.4

0.6

0.8

1

P S 0 (t)

exact gas phaseQCL gas phaseωc = 50 cm -1

ωc = 220 cm-1

ωc = 500 cm-1

ωc = 850 cm-1

ωc = 1500 cm-1

ωc = 5000 cm-1

Figure 5.1: Ground adiabatic state population PS0(t) versus time, for " = 0.01, %=0.1,

and a range of di!erent bath characteristic frequencies, $c.

Figure (5.1) depicts the population of the adiabatic ground state as a function of time

for a range of bath frequencies, $c at a fixed value of the Kondo parameter. Interestingly,

we see that at a low electronic coupling strength, and weak damping, the bath can

promote the population transfer to the ground state in comparison to the gas phase

model result. This is seen for all but the highest characteristic frequency reported in

Fig. (5.1). Thus we see that in the case of weak coupling and weak damping, that the


population transfer to the ground state can be enhanced for a wide range of characteristic

bath frequencies, but is suppressed at bath frequencies that are significantly higher than

the nuclear vibrations.

0 0.01 0.02 0.03 0.04

!!(a.u.)

0

0.2

0.4

0.6

0.8

1

PS

0 (

t =

50 f

s) no bath

"c = 50 cm

-1

"c = 220 cm

-1

"c = 500 cm

-1

"c = 850 cm

-1

"c = 5000 cm

-1

Figure 5.2: Ground adiabatic state populations PS0(t = 50fs) versus " with % = 1, for

a range of di!erent bath characteristic frequencies, $c. For reference, recall that the

frequencies of the nuclear vibrational modes are $X = 219cm!1 and $Y = 850cm!1.

The enhancement of the population transfer at low coupling strength can be under-

stood in terms of earlier results on the FLV model [125], where it was found that slowing

down the wave-packet (reducing X via reducing the slope of the potential curves) leads to

more population transfer to the ground state. This e!ect does not follow the predictions

of the semiclassical Landau-Zener vibronic model discussed earlier.

The population transfer to the ground state is enhanced for a range of low coupling

strengths and for bath frequencies from 50cm!1 up to 850cm!1. However, the population


transfer is a decreasing function of " for these bath frequencies, which follows the shape

of the bath free curve. At moderate coupling strengths (" > 0.02) in Fig.(5.2) we see

that the population transfer is suppressed for all but the slowest bath simulated. When

the bath becomes very fast relative to the subsystem ($c = 5000cm!1) the population

transfer is heavily suppressed for all (", %) studied. In this case the transfer ratio is shown

to be relatively insensitive to the electronic coupling strength.

Figure (5.3) shows the change in population transfer relative to the gas phase model

at low coupling strength for a range of di!erent ($c, %). For cases where the bath is at

most as fast as the nuclear dynamics the value of the Kondo parameter has no e!ect on

the population transfer over the range of values studied. When the bath is faster than

the gas phase nuclear vibrations however, there is a decrease in the transfer ratio as % is

increased. As the coupling strength increases one also expects that the transfer would

be suppressed at higher frequencies due to the localizing e!ect of the bath. This e!ect is

also generally known as the Zeno e!ect, which refers to the inhibitive e!ect that external

interactions may have on the evolution of a quantum system [126] which is subsequently

subjected to repeated measurements (or interactions) at a given characteristic frequency

by its environment. As the interaction frequency increases, so too may the probability

of finding the system in the initial state. In the (Zeno) limit of continuous measurement,

the quantum system becomes ”frozen” in its initial state. These localizing e!ects of the

high frequency bath are only partial in this case however, as even the fastest (finite $c)

bath permits some population transfer.

In the case of the avoided crossing model the population transfer to the ground state

monotonically decreases as a function of the electronic coupling strength. Clearly this

behaviour is di!erent than in the conical intersection model, where the transfer ratio

increases for higher coupling strengths. The transfer ratio also decays as a function of "

in the case of a moderately fast bath, as shown in Fig.(5.4). However, the extent to which

the transfer ratio decreses as a function of " is less severe than in the gas phase scenario.


0 0.5 1 1.5 2

!

-0.6

-0.4

-0.2

0

0.2

!"P

S0

(t =

50 f

s)

#c = 50 cm

-1

#c = 220 cm

-1

#c = 500 cm

-1

#c = 1500 cm

-1

#c = 5000 cm

-1

Figure 5.3: Di!erence in the ground adiabatic state populations with the bath on as

compared to the gas phase model as a function of the Kondo parameter %, for a range of

di!erent bath characteristic frequencies, $c, and " = 0.01.

This e!ect is similar to that seen in the generalized FLV conical intersection model, where

for low coupling strength the transfer ratio is higher than the gas phase case for moderate

bath frequencies. In the three mode models the Monte Carlo e!ort required to obtain

converged averages is increased in comparison with the two-dimensional case. First, we

focus on a case for the XYZ model where the solvent coordinate has a larger e!ective

inertial mass than the vibrational modes (MZ = 152 MX), and the solvent frequency

is less than vibrational frequencies, $z - 65cm!1. This solvent mass and frequency

loosely correspond to those found using classical molecular dynamics simulations of the

interactions of a CCl4 solvent interacting with a low frequency molecular vibrational

mode [127]. Interestingly, we see in Fig.(5.5) that for the particular choice of subsystem


0 0.005 0.01 0.015 0.02γ (a.u.)

0

0.2

0.4

0.6

0.8

1

P S 0 (t

=50

fs)

no bathωc = 500 cm-1

Figure 5.4: Ground state population at t = 50fs, for " = 0.01 , & = 1, in the avoided

crossing model. In the case with the bath present % = 1.

parameters, the population transfer in the three mode model is rather insensitive to the

presence of the dissipative bath.

5.2 Electronic Coherence

The e!ects of decoherence by the bath can be seen by examining the o!-diagonal element

of the subsystem density matrix +12s (t), or the purty, Tr(+2

s). These quantities are depicted

for the generalized FLV model in Fig. (5.6) and Fig. (5.7). It is plainly evident here

that by increasing the characteristic frequency of the bath the envelope of coherence

oscillations, that are induced during motion through the region of the conical intersection,

is progressively destroyed. As one might expect from the population transfer results


0 20 40 60 80time (fs)

0

0.2

0.4

0.6

0.8

1P S 0

(t)


ωc = 1500 cm-1

Figure 5.5: Ground state population as a function of time in the XYZ model, for " = 0.01.

In cases with the bath present % = 1.

there is an increase in the purity after a single pass through the intersection at most

characteristic bath frequencies in the case of weak electronic coupling due to the enhanced

transfer to the ground state. However, once the bath becomes much faster than the

subsystem, the purity decays monotonically with time.

It has been shown in other model studies [25, 26] that a narrow avoided crossing may

cause the system to evolve in a similar manner to the case of a true conical intersection.

This situation may obtained by a narrowing of the diabatic coupling function, i.e. reduc-

ing the value of &. In the case of the conical intersection & = 1. We explore this point by

examining the evolution of the electronic coherence in the avoided crossing model (ACM)

for two di!erent values of &, shown in Fig.(5.8). One can see from Fig.(5.8) that in the

bath-free case and the original coupling width (& = 1, regular width) that the evolution


Figure 5.6: Evolution of the electronic coherence as a function of time, for a selection of

characteristic bath frequencies, for " = 0.01.

of the electronic coherence does not resemble the FLV case. However, it is evident that

when a bath is added or the width of the coupling is reduced (& = 12 , half-width), the

coherence indeed resembles the FLV result. The avoided crossing system is more sensitive

to the presence of the bath than its conical counterpart however, cf. the 500cm!1 data

for the FLV coherence in Fig.(5.6). In accordance with the insensitivity of the population

transfer data for the three mode models, we see in Fig.(5.9) and Fig.(4.12) that, for these

particular parameter choices, the electronic coherence is also rather insensitive to the

presence of the dissipative bath.


10 20 30 40 50time (fs)

0.5

0.6

0.7

0.8

0.9

1

Tr ρ

s2

no bathω c = 220 cm-1

ωc = 500 cm-1

ωc = 1500 cm-1

ωc = 5000 cm-1

Figure 5.7: Purity of the quantum subsystem as a function of time, for a selection of

characteristic bath frequencies, for " = 0.01.

5.3 Geometric Phase

One important consequence of the presence of a conical intersection on the potential

energy landscape is the emergence of Berry’s (geometric) phase. In the FLV model the

evidence of a Berry phase in the system arises as a nodal structure in the subsystem

excited state density projected along the Y -coordinate [2, 15]. The reproduction of this

delicate quantum phenomenon provides an important challenge to any approximate the-

ory and its accompanying simulation algorithm. At low coupling strengths, this e!ect

manifests as a single node in the excited state density at Y = 0, a feature is well re-

produced by the QCL simulations in the adiabatic basis. For larger values of " this

interference pattern becomes more complicated. The semiclassical methods mentioned


20 40 60 t (fs)

-0.1

-0.05

0

0.05

0.1

Re{ ρ s01

(t)}

no bathωc = 500cm-1

0 20 40 60 80

-0.2

-0.1

0

0.1

0.2

no bathωc = 500cm-1

Regular Width Half - Width

Figure 5.8: Electronic coherence as a function of time in the ACM with & = 1 (regular

width), and & = 12 (half-width), for " = 0.01. In cases with the bath present % = 1.

earlier (Tully, Landau-Zener) are incapable of reproducing this structure[2]. Earlier QCL

calculations in the force basis, however, faired somewhat better at capturing this delicate

feature [3]. As shown in Fig(5.10), the MC-MD QCL simulations are indeed able to re-

produce this feature. In additon, Fig(5.11) shows the gradual destruction of this node as

we speed up the bath in the generalized FLV model. It is also interesting to see if nodal

structure, or other interference phenomena, arise as consequence of the geometric phase

in the the models studied. Particularly, in the three-mode case the system experiences

a spatial continuum of conical intersections associated with the seam of intersections

depicted in Fig. (3.6), as opposed to a single intersection, and thus some shift in the

interference pattern is expected. Indeed, we see in Fig. (5.12), that there is no node at

Y = 0 in the subsystem excited state coordinate distribution. In the case of the avoided


10 20 30 40 50 60 70time (fs)

-0.1

0

0.1

0.2

Re{ ρ s01

(t) }


ωc = 1500cm-1

Figure 5.9: Electronic coherence as a function of time in the three mode XYZ model, for

" = 0.01. In cases with the bath present % = 1.

crossing model, while no geometric phase e!ects are expected, we find that by narrowing

the electronic coupling region, what could appear to be a hint of a node results in the

excited state coordinate distribution.

5.4 On the Markovian Approximation

In order to investigate the applicability of a Markovian master equation based on the QCl

equation, we present selected realizations of the bath averaged memory function. In par-

ticular we display selected realizations of the memory function for downward transitions

as a function of time, for a range of characteristic bath frequencies in Fig. (5.13). The

di!erent curves correspond to di!erent initial nuclear phase points, Xs = (X, Y, PX , PY ).


-0.5 0 0.5 1 Y (Bohr)

0

0.5

1

1.5

0.5

1

ρ s(Y,t)

0

0.5

1

1.5

2

S0

-0.5 0 0.5 1

S1

x 5

x 5

20 fs

30 fs

40 fs

Figure 5.10: Branching of the wavepacket along the Y coordinate, results from QCL

simulations in the adiabatic basis. Emergence of Berry’s phase is seen as a node develops

on the excited state surface, S1, after the packet leaves the region.

It it evident that the breadth of the oscillations in "M2121 (X, t)#e does not decrease signifi-

cantly as a function of the characteristic frequency of the bath. Consequently, the conical

intersection models presented do not readily permit a Markovian type description within

the context of the QCL equation.


-1 0 1 Y (Bohr)

0

1

gas phase

1

2

ρ s11(Y

, t =

40

fs)

ωc = 50 cm-1

1

2ωc = 220 cm-1

0 1

ωc = 850cm-1

ωc = 1500 cm-1

ωc = 5000 cm-1

Figure 5.11: Destruction of the node in the subsystem reduced density along the Y

coordinate, as a function of $c from QCL simulations in the adiabatic basis.


0

0.03

0.06 ACM - Half Width

0

0.3

ρ s11(Y

, t =

40

fs)

ACM - Regular Width

-0.5 0 0.5 1Y (Bohr)

0

0.3 XYZ Model

Figure 5.12: Subsystem density in the excited state as a function of the Y coordinate, for

QCL simulations of the ACM model with & = 1 (regular width), and & = 12 (half-width),

and the XYZ model.


10 20 30 40 50 60 time (fs)

-2

0

2

4

<

M10

10 (t

) >e

50 cm-1

10 20 30 40 50

500 cm-1

1500 cm-1

0

1

2

850 cm-1

Figure 5.13: Plot of five di!erent realizations (corresponding to five di!erent initial phase

points (X,PX , Y, PY )) of the bath averaged memory function "M2121 (X, t)#e versus time,

for a selection of di!erent bath frequencies $c, at " = 0.01.

Chapter 6

Another Solution: The Mapping

Basis

There exists a characteristic time scale after which the computational task associated

with the of Monte Carlo sampling of quantum transitions becomes too cumbersome, and

the QCL solution scheme (in the adiabatic basis) breaks down. For the conical inter-

section models under study here this time scale typically turns out to be approximately

150fs . This particular limitation restricts the Trotter-based QCL simulations to one or

two periods of vibrational motion at the most. Thus in order to study relaxation process

on a longer time scale we must appeal to some other method of solving the QCL equation,

which scales more favorably with the simulation length. In chapter 2 the QCL equation

was expressed in various basis representations, each yielding a di!erent (approximate)

numerical solution algorithm. We present a description and implementation of another

solution method based on the QCL equation in the mapping basis (equation (2.36)). In

the mapping basis an approximate form of the QCL equation may be obtained which

requires a numerical e!ort that scales linearly with time. This approach also has lim-

itations, associated with the approximate form of the Liouvillian, and the form of the

integration scheme implemented in the solution.

94

Chapter 6. Another Solution: The Mapping Basis 95

6.1 Approximate QCL equation

In this section an approximate version of the QCL equation, expressed in the Meyer-Miller

mapping basis, is developed intended for use in calculations on the aforementioned conical

intersection models. Closely following the work of Kim, Nassimi and Kapral [79], we write

the expression for an average value in the mapping basis. Briefly, this is accomplished by

writing the expression in the subsystem basis, transforming to the mapping basis, and

then taking the Wigner transform over the mapping degrees of freedom.

Recalling from chapter 2 that the QCL equation (2.8) may be written in the mapping

basis using the coordinate representation, we rewrite the expression for an observable as

follows,

B(t) =/

(,(!

!dX"m(|Bm(X, t)|m(!#"m(!|+m(X)|m(#

=/

(,(!

!dX

!dqdq"dq""dq""""m(|q#"q|Bm(X, t)|q"#

+ "q"|m(!#"m(!|q""#"q""|+m(X)|q"""#"q"""|m(#. (6.1)

where the subscript m refers to the mapping basis representation of the quantities shown.

The Wigner transforms of the coordinate space matrix elements of the mapping vari-

ables are,

"r % z

2|Bm(X, t)|r +

z

2# =

1

(2*h)N

!dp e!ipz/hBm(x, X, t),

"r" % z"

2|+m(X)|r" + z"

2# =

!dp"e!ip!z!/h+m(x", X), (6.2)

where x = (r, p) are the phase space coordinates of the mapping variables. Using these

definitions Eq. (6.1) can be written as

B(t) =!

dXdxdx" Bm(x, X, t)f(x, x")+m(x", X)

=!

dXdx Bm(x, X, t)+m(x, X), (6.3)

where +m(x, X) =4

dx"f(x, x")+m(x", X) and

f(x, x") =1

(2*h)N

/

((!

!dzdz""m(|r %

z

2#"r +

z

2|m(!# + (6.4)


"m(!|r" %z"

2#"r" + z"

2|m(#e!i(p·z+p!·z!)/h.

Again here, we see that the task of computing an average consists of two separate parts;

sampling from +m(x, X), and evaluating the dynamics dictated by eıLmt).

The time evolution of a general operator in the mapping basis, equation (2.36), is

d

dtBm(x, X, t) = %{Hm, Bm(t)}x,X (6.5)

+h

8

/

((!

)h((!

)R(

)

)r(!

)

)r(+

)

)p(!

)

)p() · )

)PBm(t)

. iLmBm(t),

where {Am, Bm(t)}x,X denotes a Poisson bracket in the full mapping-bath phase space

of the system.

As previously mentioned, the complicated last term involves derivatives with respect

to both mapping and environmental variables, and its contribution to the dynamics is

di"cult to compute. If we neglect this then then we obtain an approximate QCL in this

basis,

d

dtBm(x, X, t) = %{Hm, Bm}x,X = ıL0

mBm(x, X, t). (6.6)

6.2 Solution Algorithm

It was found that simple numerical integration methods, (e.g.) previously used to study

the spin-boson model [79], were not appropriate to integrate the system of equations for

this model, due to the timescale separation of the subsystem and environental degrees

of freedom. Another numerical solution using MD-type trajectories that are generated

by using a short time decomposition of the propagator was constructed in an attempt to

alleviate this problem [128]. We shall start with the formal solution to equation (6.6),

Bm(x, X, t) = eiL0mtBm(x, X, 0), (6.7)


and write a short time decomposition of the approximate mapping propagator,

eiL0mt =

N3

k=1

eiL0m(tk!tk"1), (6.8)

where t = tN % t0 = N%t.

The total Hamiltonian shall be written as a sum of two parts,

Hm = H1m + H2

m. (6.9)

The first part of the Hamiltonian is chosen to be the kinetic energy of the environment,

H1m = P 2

2M , and the second part contains the remainder of the Hamiltonian (eq.(2.35)),

H2m = VB(R)+ 1

2h

A((! h((!(R)(r(r(! + p(p(! % -((!). Partitioning the Hamiltonian in this

way generates new Liouville operators,

ıLim = %{H i

m, ·}x,X , (6.10)

such that

ıL0m = ı(L1

m + L2m). (6.11)

We then express the short-time propagators using the symmetric Trotter decomposi-

tion,

eı(L1m+L2

m)"t = eıL1m(!t

2 )eıL2m"teıL1

m(!t2 ) +O(%t3). (6.12)

This yields the following overall expression for the time-dependant observable,

Bm(x, X, t) = limn%&,"t%0

n3

k=1

(eıL1m(!t

2 )eıL2m"teıL1

m(!t2 ))kBm(x, X, 0). (6.13)

Thus the evolution under ıL1m gives rise to a system propagator on the environmental

coordinates alone. The following set of equations is prescribed:

dr(

dt= 0,

dp(

dt= 0,

dP

dt= 0,

dR

dt=

P

M. (6.14)


Evolution under ıL2m is somewhat more complicated, however it may evaluated exactly

since R(t) is stationary under this portion of the dynamics. The equations of motion are

as follows:

dr(

dt=

1

h

/

(!h(,(!(R)p(! , (6.15)

dp(

dt=

%1

h

/

(!h(,(!(R)r(! , (6.16)

dP

dt= %)VB(R)

)R% 1

2h

/

(,(!

)h(,(!(R)

)R(r(r(! + p(p(! % h-(,(!). (6.17)

The system of equations for the evolution of the x-coordinates, (6.15-6), has a closed

form and can be solved analytically, and subsequently used to integrate the expression

for the environmental momenta.

Consider the spectral decomposition of the mapping Hamiltonian,

h((!(R) =/

),)!c()(R)(E)(R)-))!)(c

!1))!(!(R) (6.18)

=/

)

c()(R)E)(R)(c!1))(!(R),

where E)(R) are the eigenvalues (adiabatic energies) of h. The columns of the matrix c

correspond to the eigenvectors of h. Using the spectral decomposition of h we perform

the following transformation,

Br) =/

(

(c!1))(r(, Bp) =/

(

(c!1))(p(, (6.19)

which simplifies the evolution equations for the mapping variables;

dBr(

dt=

E((R)

hBp(,

dBp(

dt=%E((R)

hBr(. (6.20)

The above system may be expressed as the matrix equation,

du

dt= M · u. (6.21)

At present we are simply interested in the case of a two-level quantum subsystem, as


such the vector u may be written

u =

;

<<<<<<<<<<=

Br1

Bp1

Br2

Bp2

>

??????????@

, (6.22)

such that the matrix M has a simple (block diagonal) form,

M =1

h

;

<<<<<<<<<<=

0 E1 0 0

%E1 0 0 0

0 0 0 E2

0 0 %E2 0

>

??????????@

. (6.23)

The general solution to Eq. (6.21) is

u(t) = eMtu(0), (6.24)

where, in this particular case, the matrix exponential has the form

eMt =

;

<<<<<<<<<<=

cos($1t) sin($1t) 0 0

% sin($1t) cos($1t) 0 0

0 0 cos($2t) sin($2t)

0 0 % sin($2t) cos($2t)

>

??????????@

, (6.25)

with $)(R) = E)(R)/h. The time evolved tilde variables are obtained via

ui(t) =/

j

(eMt)ijuj(0). (6.26)

These results can then be back-transformed to the original (untilded) variables, and

used to perform the integrals for the momenta of the environment given by equation

(6.17). A potential problem with this method has been noted by Stock and co-workers

in their work on the mapping basis formalism [108]. This problem lies in the fact that

the classical degrees of freedom may experience the e!ect of inverted potential energy


surfaces during the dynamics. This scenario occurs in equation (6.17) for instances where

(r(r(! + p(p(! % h-(,(!) < 0. The set of these cases comprises a region of nonzero measure

in phase space since the subsystem coordinates and momenta, (r(, p(), are sampled from

independent Gaussian distributions. This sort of instability poses such problems as the

system coordinates diverging in finite time for certain types of potential energy functions;

it thus precludes the treatment of models that incorporate real exponential potential

functions (e.g. Morse oscillators) in the subsystem Hamiltonian. As such we cannot use

the aforementioned method to simulate the dynamics of the BCH model.

6.3 Numerical Results

In order to test the approximate QCL mapping approach outlined above, we first compare

the approximate mapping basis results for ground adiabatic state populations at t = 50fs

as a function of coupling strength, ", with the exact quantum results and results generated

from other QCL methods. We see in Fig.(6.1) that all the QCL methods shown reproduce

the shape of the FLV population transfer curve over the low coupling range quite well.

The QCL equation in the force basis was solved using the ’multiple threads’ algorithm

by Wan and Schofield [69, 3]. All three methods shown agree extremely well with the

quantum result at low coupling, with only the force basis calculation slightly missing

the position of the minimum. At higher coupling strengths the adiabatic basis is quite

capable of matching the shape of the curve, whereas the other two results show slightly

less agreement as " increases. The Trotter-based QCL results seem to agree best with the

exact numerical data, however overall there seems to be a constant underestimation of the

population transfer for all strengths in this particular data set. This di!erence is likely

due to the use of a filter; (i.e.) the choice to place a cuto! on the observable in the Monte

Carlo sampling of quantum jumps. Of course, this underestimation may also result from

the momentum jump approximation, or could even stem from the approximate nature of


0 0.02 0.04 0.06 0.08γ (a.u.)

0.8

0.85

0.9

0.95

P S 0 (t

= 5

0 fs

)

Exact quantumQCL - adiabatic basisQCL - force basisQCL - mapping basis

Figure 6.1: Ground adiabatic state populations PS0(t = 50fs) versus ". The quantum

results are taken from [2], and the force basis results are from [3].

the QCL equation itself. The mapping and force basis results exhibit excellent agreement

at low coupling strengths, with both methods just slightly overestimating the population

transfer to the ground state. At the minimum of the curve the mapping basis, however,

is well over the quantum result. While the force basis calculation remains relatively

accurate at higher coupling strengths, the mapping basis results begin to underestimate

the ground state population. Deviations from the exact result are not entirely surprising

for the mapping basis calculations, as the approximate system propagator used in the

solution algorithm neglects one of the terms giving rise to coupling between the nuclear

and electronic degrees of freedom in the QCL system propagator, which is already an

approximation to the full system propagator. The dynamics depicted in Figs. (6.2)

and (6.3) show the population transfer to the ground state in the case of model conical


0.5 1 1.5 2time (ps)

0.2

0.4

0.6

0.8P S 0

FLV - CIACM ; σ = 1ACM ; σ = 1/2

Figure 6.2: Ground adiabatic state populations PS0 as a function of time (in picoseconds)

for " = 0.01. Here we compare the original FLV conical intersection model with the ACM

for & = 1 (Regular width), and & = 12 (Half-width).

intersections and avoided crossings. In the case of the systems with a CI, the initial fast

electronic decay over the first 80 fs is followed by oscillatory decay which dissipates on a

sub-picosecond time-scale. The avoided crossing models exhibit a somewhat simpler and

less rapid decay profile. However, when the interaction width of the crossing is reduced,

the decay time also moves into the sub-picosecod range. These general observations are

in accord with the MCTDH [20], Redfield [24] and Lindblad [25, 26] equation results

that exist for these systems. It is obvious from Fig. (6.3) that there are problems with

the implementation of the approximate mapping basis calculation in the three mode

model, and in the two mode model in the presence of the bath, as the average ground

(and excited) state population attains unphysical values. This is not entirely suprising,


0.5 1 1.5time (ps)

0.2

0.4

0.6

0.8

1

1.2P S 0

2D ; no bath3D ; no bath2D ; ωc = 500 cm-1

Figure 6.3: Ground adiabatic state populations PS0 as a function of time (in picosec-

onds) for " = 0.01. Comparison of the mapping basis data: FLV model results (2D;

no bath), XYZ model results (3D; no bath), and generalized FLV model results (2D;

$c = 500cm!1). Only the original FLV model results remain physical (0 $ PS0 $ 1) for

the entire silmulation.

as total population is conserved, and there are no constraints on the individual states

in this approximate scheme. Overall the surface-hopping algorithm in the adiabatic

basis, although computationally much more expensive than the mapping and force basis

calculations, is more adept at reproducing the short time quantum results over the entire

range of coupling strengths. That said, the mapping basis calculations may provide an

alternative computationally tractable solution allowing access to longer time-sclaes, when

the coupling strength is appropriate.

Chapter 7

Conclusions

Clearly our understanding of conical intersections and chemical dynamics has grown

vastly since the seminal work of Teller [12]. However, the quantum coherence generated

in the CI quantum dynamics problem makes it a paradigmatic case for our fundamental

understanding of other coherently coupled systems.

In this thesis we have explored the various e!ects of an external environment on

the electronic dynamics in a selection of model molecular systems containing conical

intersections and avoided crossings. In doing so we have mainly utilized the methods

of quantum-classical Liouville dynamics, attempting to test the limits of validity of the

QCL equation and the associated computational algorithms used in its solution. As well,

we have tried to set the QCL theory into perspective alongside many of the other major

approximate (nonadiabatic) quantum theories.

We found that the short-time quantum dynamics were well approximated by a hy-

brid MC/MD surface-hopping type solution to the QCL equation, which was confirmed

by comparison with exact results. Adding a bilinear coupling to an external environ-

ment caused the population transfer to be enhanced or suppressed, based on the relative

values of the characteristic frequencies of the subsystem and environment. Electronic

decoherence and the destruction of geometric phase e!ects were observed as the external

104

Chapter 7. Conclusions 105

degrees of freedom became much faster than the molecular vibrations. However, this

MC/MD approach su!ers serious problems associated with the accumulation of statis-

tical weights, and as such is not appropriate for the investigation of longer time-scale

relaxation processes involving CI models.

In accordance with earlier work [26], we also found that the dynamics at a narrow

avoided crossing somewhat mimics a corresponding scenario involving a conical inter-

section. However, the dynamics though a CI was shown to be much more robust to

the presence of an environment when compared to the avoided crossing model. Further,

the dynamics at higher dimensional CI’s, in the BCH and the XYZ models, were shown

to be even less sensitive to the presence of the environment. This observation is quite

interesting, and may be a hint as to how large chromophoric systems sustain electronic

coherence over large length scales in the condensed phase. We also showed that, based

on earlier work on master equation dynamics in QCL theory [90], that a Markovian ap-

proximation to the dynamics for systems containing conical intersections is not easily

justified. This is yet another reflection of the robustness of the coherence in the system

due to the presence of the CI.

A more computationally facile solution was also derived in the mapping basis, which

allowed for a perspective of the dynamics on a longer time-scale. This method, also

based on a Trotter-factorization, involved more severe approximations to the original

QCL equation than the MC/MD method and was shown to be not as physically robust.

Despite its approximate nature, this solution in the mapping was able to reproduce

important aspects of the vibronic relaxation problem on a picosecond time-scale.

The QCL theory is certainly emerging as a formidable theoretical tool in condensed

phase chemical problems. Future research into quantum dynamics problems in which

electronic coherence plays an important role could certainly benefit from making use of

this theoretical framework. Of course, new solution algorithms must also be sought out

in order to facilitate the description a broader range of model physical systems.

Bibliography

[1] A. Ferretti, A. Lami, and G. Villani. Quantum dynamics of a model system with

a conical intersection. J. Chem. Phys., 106:934–941, 1996.

[2] A. Ferretti, G. Granucci, A. Lami, M. Persico, and G. Villani. Quantum and

semiclassical dynamics at a conical intersection. J. Chem. Phys., 104:5517–5527,

1996.

[3] C. C. Wan and J. Schofield. Solutions of mixed quantum-classical dynamics in

multiple dimensions using classical trajectories. J. Chem. Phys., 116:494–506, 2002.

[4] A. Kelly R. Grunwald and R. Kapral. Quantum dynamics in almost classical

environments. In I. Burghardt, editor, Energy Transfer Dynamics in Biomaterial

Systems, pages 383–413. Springer, Berlin, 2009.

[5] M. Born and I. Oppenheimer. On the quantum theory of molecules. Ann. der

Physik, 84:457–484, 1927.

[6] D. R.Yarkony. Diabolical conical intersections. Rev. Mod. Phys., 68:985–1013,

1996.

[7] M. Born. The adiabatic principle in the quantum mechanics. Z. Phys., 40:167–192,

1926.

[8] L. D. Landau. Zur theorie der energieubertragung. II. Phys. Z., 2:46, 1932.

106

Bibliography 107

[9] C. Zener. Nonadiabatic crossing of energy levels. Proc. R. Soc. London Series A,

137:692–702, 1932.

[10] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions. Dover,

New York, 1972.

[11] C. Wittig. The Landau-Zener approximation. J. Phys. Chem. B., 109:8428–8430,

2005.

[12] E. Teller. The crossing of potential surfaces. J. Phys. Chem., 41:109–116, 1937.

[13] G. Herzberg and H.C. Longuet-Higgins. Intersection of potential energy surfaces

in polyatomic molecules. Discuss. Faraday Soc., 35:77–82, 1963.

[14] M.V. Berry. Quantal phase factors accompanying adiabatic changes. Proc. R. Soc.

London Series A, 392:45–57, 1984.

[15] J. Schon and H. Koppel. Femtosecond time-resolved ionization spectroscopy of

Na3(B) and the question of the geometric phase. Chem. Phys. Lett., 231:55–63,

1994.

[16] J. Anandan. The geometric phase. Nature, 360:307–313, 1992.

[17] S. K. Kim, J. K. Wang, and A. H. Zewail. Femtosecond pH jump - dynamics of

acid-base reactions in solvent cages. Chem. Phys. Lett., 228:369–378, 1994.

[18] S. K. Kim, J. J. Breen, D. M. Willberg, L. W. Peng, A. Heikal, J. A. Syage,

and A. H. Zewail. Solvation ultrafast dynamics of reactions: Acid-base reactions

in finite-sized clusters of naphthol in ammonia, water, and piperidine. J. Phys.

Chem., 99:7421–7435, 1995.

[19] V. I. Prokhorenko, A. M. Nagy, S. A. Waschuk, L. S. Brown, R. R. Birge, and

R. J. Dwayne Miller. Coherent control of retinal isomerization in bacteriorhodopsin.

Science, 313:1257–1261, 2006.

Bibliography 108

[20] H.D. Meyer A. Raab, G. Worth and L.S. Cederbaum. J. Chem. Phys., 110:936,

1999.

[21] X. Sun, H. Wang, and W. H. Miller. Semiclassical theory of electronically nonadi-

abatic dynamics: Results of a linearized approximation to the initial value repre-

sentation. J. Chem. Phys., 109:7064–7074, 1998.

[22] W.H. Miller M. Thoss and G. Stock. Semiclassical description of nonadiabatic

quantum dynamics: application to the S1-S2 conical intersection in pyrazine. J.

Chem. Phys., 112:10282, 2000.

[23] W. Domcke and G. Stock. Theory of ultrafast nonadiabatic excited-state processes

and their spectroscopic detection in real time. Adv. Chem. Phys., 100:1, 1997.

[24] A. Kuhl and W. Domcke. E!fect of a dissipative environment on the dynamics at

a conical intersection. Chem. Phys., 259:227–236, 2000.

[25] R. Koslo! G. Katz and M. Ratner. Conical intersections: Relaxation, dephasing

and dynamics in a simple model. Israel J. Chem., 44:53–64, 2004.

[26] R. Koslo! D. Gelman, G. Katz and M.A. Ratner. Dissipative dynamics of a system

passing through a conical intersection: Ultrafast pump-probe observables. J. Chem.

Phys., 123:134112, 2005.

[27] P. Pechukas. Time-dependent semiclassical scattering theory .i. potential scattering.

Phys. Rev., 181:166–174, 1969.

[28] J. C. Tully. Molecular-dynamics with electronic-transitions. J. Chem. Phys.,

93:1061–1071, 1990.

[29] R. Kapral. Progress in the theory of mixed quantum-classical dynamics. Ann. Rev.

Phys. Chem, 57:129–157, 2006.

Bibliography 109

[30] R. Kapral and G. Ciccotti. Mixed quantum-classical dynamics. J. Chem. Phys.,

110:8919–8929, 1999.

[31] E. Wigner. On the quantum correction for thermodynamic equilibrium. Phys. Rev.,

40:749, 1932.

[32] K. Imre, E. Ozizmir, Rosenbaum, and P. F. Zweifel. Wigner method in quantum

statistical mechanics. J. Math. Phys., 8:1097, 1967.

[33] I. V. Aleksandrov. The statistical dynamics of a system consisting of a classical

and a quantum subsystem. Z. Naturforsch., 36:902–850, 1981.

[34] V. I. Gerasimenko. Correlation-less equations of motion of quantum-classical sys-

tems. Rep. Acad. Sci. Ukr. SSR., 10:65–68, 1981.

[35] V. I. Gerasimenko. Dynamical equations of quantum-classical systems. Theor.

Math. Phys., 50:49–55, 1982.

[36] W. Boucher and J. Traschen. Semiclassical physics and quantum fluctuations. Phys.

Rev. D, 37:3522–3532, 1988.

[37] W. Y. Zhang and R. Balescu. Statistical-mechanics of a spin-polarized plasma. J.

Plasma Phys., 40:199–213, 1988.

[38] Y. Tanimura and S. Mukamel. Multistate quantum Fokker–Planck approach to

nonadiabatic wave packet dynamics in pump–probe spectroscopy. J. Chem. Phys.,

101:3049, 1994.

[39] C. C. Martens and J. Y. Fang. Semiclassical-limit molecular dynamics on multiple

electronic surfaces. J. Chem. Phys., 106:4918–4930, 1997.

[40] I. Horenko, C. Salzmann, B. Schmidt, and C. Schutte. Quantum-classical Liouville

approach to molecular dynamics: Surface hopping Gaussian phase-space packets.

J. Chem. Phys., 117:11075–11088, 2002.

Bibliography 110

[41] Q. Shi and E. Geva. A semiclassical generalized quantum master equation for an

arbitrary system-bathcoupling. J. Chem. Phys., 120:10647–10658, 2004.

[42] S. Nielsen, R. Kapral, and G. Ciccotti. Statistical mechanics of quantum-classical

systems. J. Chem. Phys., 115:5805–5815, 2001.

[43] A. Sergi. Non-hamiltonian commutators in quantum mechanics. Phys. Rev. E,

72(6):066125, 2005.

[44] L.L. Salcedo. Absence of classical and quantum mixing. Phys. Rev. A, 54:3657–

3660, 1996.

[45] V. V. Kisil. A quantum-classical bracket from p-mechanics. EPL, 72(6):873–879,

2005.

[46] O.V. Prezhdo. A quantum-classical bracket that satisfies the jacobi identity. J.

Chem. Phys., 124:201104.

[47] S. Caprara F. Agostini and G. Ciccotti. Do we have a consistent non-adiabatic

quantum-classical mechanics? EPL, 78:30001, 2007.

[48] L.L. Salcedo. Comment on ”a quantum-classical bracket that satisfies the jacobi

identity. J. Chem. Phys., 126, 2007.

[49] M.J.W. Hall and M. Reginatto. Interacting classical and quantum ensembles. Phys.

Rev. A, 72(6), 2005.

[50] G. Ciccotti S. Bonella and R. Kapral. Linearization approximation and Liouville

quantum-classical dynamics. Chem. Phys. Lett., 484:399–404, 2010.

[51] R. Kapral and A. Sergi. Dynamics of condensed phase proton and electron transfer

processes. In Handbook of Theoretical and Computational Nanotechnology, page 92.

American Scientific Publishers, 2005.

Bibliography 111

[52] A. Sergi. Quantum-classical dynamics of wave fields. J. Chem. Phys., 126:074109,

2007.

[53] G. Parlant L. S. Cederbaum I. Burghardt, K. B. Moller and E. R. Bittner. Quan-

tum hydrodynamics: Mixed states, dissipation and new hybrid quantum-classical

approach. Int. J. Quantum Chem., 100:1153–1162, 2004.

[54] A. Donoso and C. C. Martens. Simulation of coherent nonadiabatic dynamics using

classical trajectories. J. Phys. Chem. A, 102:4291–4300, 1998.

[55] J. M. Riga, E. Fredj, and C. C. Martens. Simulation of vibrational dephasing of I-2

in solid Kr using the semiclassical Liouville method. J .Chem. Phys., 124:064506,

2006.

[56] G. Hanna and R. Kapral. Quantum-classical Liouville dynamics of nonadiabatic

proton transfer. J. Chem. Phys., 122:244505, 2005.

[57] G. Hanna and R. Kapral. Nonadiabatic dynamics of condensed phase rate pro-

cesses. Acc. Chem. Res., 39:21–27, 2006.

[58] H. Kim, G. Hanna, and R. Kapral. Analysis of kinetic isotope e!ects for nonadia-

batic reactions. J. Chem. Phys., 125:084509, 2006.

[59] H. Kim and R. Kapral. Quantum bath e!ects on nonadiabatic reaction rates.

Chem. Phys. Lett., 423:76–80, 2006.

[60] H. Kim and K. J. Shin. Di!usion influence on michaelis-menten kinetics. ii. the

low substrate concentration limit. J. Phys.: Condens. Matter, 19:065137, 2007.

[61] G. Hanna and E. Geva. Vibrational energy relaxation of a hydrogen-bonded com-

plex dissolved in a polar liquid via the mixed quantum-classical Liouville method.

J. Phys. Chem. B, 112:4048–4058, 2008.

Bibliography 112

[62] B. Schmidt I. Horenko and C. Schutte. A theoretical model for molecules interacting

with intense laser pulses: The floquet-based quantum-classical Liouville equation.

J. Chem. Phys., 115:5733–5743, 2001.

[63] C. M. Morales and W. H. Thompson. Mixed quantum-classical molecular dynamics

analysis of the molecular-level mechanisms of vibrational frequency shifts. J.Phys.

Chem. A, 111:5422–5433, 2007.

[64] D. MacKernan, G. Ciccotti, and R. Kapral. Surface-hopping dynamics of a spin-

boson system. J. Chem. Phys., 116:2346–2353, 2002.

[65] G. Ciccotti D. MacKernan and R. Kapral. Trotter-based simulation of quantum-

classical dynamics. J. Phys. Chem. B, 112:424, 2008.

[66] U. Manthe M. Santer and G. Stock. Quantum-classical Liouville description of

multidimensional nonadiabatic molecular dynamics. J. Chem. Phys., 114:2001–

2012, 2001.

[67] R. Kapral D. Mac Kernan and G. Ciccotti. Sequential short-time propagation

of quantum–classical dynamics sequential short-time propagation of quantum-

classical dynamics. J. Phys.: Condensed Matter, 14:9069, 2002.

[68] G. Ciccotti A. Sergi, D. MacKernan and R. Kapral. Simulating quantum dynamics

in classical environments. Theor. Chem. Acc., 110:49, 2003.

[69] C. C. Wan and J. Schofield. Mixed quantum-classical molecular dynamics: Aspects

of the multithreads algorithm. J. Chem. Phys., 113:7047–7054, 2000.

[70] J. Schwinger. On angular momentum. In L. C. Biedenharn and H. Van Dam,

editors, Quantum Theory of Angular Momentum, page 229. Academic Press, New

York, 1965.

Bibliography 113

[71] H. D. Meyer and W. H. Miller. Classical analog for electronic degrees of freedom

in non-adiabatic collision processes. J. Chem. Phys., 70:3214–3223, 1979.

[72] G. Stock. A semiclassical self-consistent-field approach to dissipative dynamics -

the spin-boson problem. J. Chem. Phys., 103:1561–1573, 1995.

[73] G. Stock and M. Thoss. Semiclassical description of nonadiabatic quantum dynam-

ics. Phys. Rev. Lett., 78:578–581, 1997.

[74] U. Muller and G. Stock. Consistent treatment of quantum-mechanical and clas-

sical degrees of freedom in mixed quantum-classical simulations. J. Chem. Phys.,

108:7516–7526, 1998.

[75] M. Thoss and G. Stock. Mapping approach to the semiclassical description of

nonadiabatic quantum dynamics. Phys. Rev. A, 59:64–79, 1999.

[76] S. Bonella and D. F. Coker. Semiclassical implementation of the mapping Hamil-

tonian approach for nonadiabatic dynamics using focused initial distribution sam-

pling. J. Chem. Phys., 118:4370–4385, 2003.

[77] S. Bonella and D. F. Coker. Land-map, a linearized approach to nonadiabatic

dynamics using the mapping formalism. J. Chem. Phys., 122:194102, 2005.

[78] S. Bonella, D. Montemayor, and D. F. Coker. Linearized path integral approach

for calculating nonadiabatic time correlation functions. Proc. Natl. Acad. Sci.,

102:6715–6719, 2005.

[79] A. Nassimi H. Kim and R. Kapral. Quantum-classical Liouville dynamics in the

mapping basis. J. Chem. Phys., 129:084102, 2008.

[80] A. Nassimi and R. Kapral. Liouville quantum-classical dynamics in the mapping

basis. unpublished, 2010.

Bibliography 114

[81] Y. Lin F. Zhan and B. Wu. Equivalence of two approaches for quantum-classical

hybrid systems. J. Chem. Phys., 128(20):0204104, 2008.

[82] J. C. Tully and R. K. Preston. Trajectory surface hopping approach to nonadiabatic

molecular collisions: The reaction of H+ with D2. J. Chem. Phys., 55(2):562–572,

1971.

[83] S. Hammes-Schi!er and J. C. Tully. Proton-transfer in solution - molecular-

dynamics with quantum transitions. J. Chem. Phys., 101:4657–4667, 1994.

[84] W. H. Zurek. Decoherence and the transition from quantum to classical. Physics

Today, 44(10):36–44, 1991.

[85] D. F. Coker, H. S. Mei, and J.-P. Ryckaert. Thermal average time correlation

functions from nonadiabatic md: application to rate constants for condensed phase

non-adiabatic reactions. In Bruce J. Berne, Giovanni Ciccotti, and David F. Coker,

editors, Classical and Quantum Dynamics in Condensed Phase Simulations, pages

539–582. World Scientific, Singapore, 1998.

[86] P. J. Rossky F. Webster, E. T. Wang and R. A. Friesner. Stationary phase sur-

face hopping for nonadiabatic dynamics: Two-state systems. J. Chem. Phys.,

100(7):4835–4847, 1994.

[87] D. F. Coker and L. Xiao. Methods for molecular-dynamics with nonadiabatic

transitions. J. Chem. Phys., 102:496–510, 1995.

[88] E. R. Bittner and P. J. Rossky. Quantum decoherence in mixed quantum-classical

systems: Nonadiabatic processes. J. Chem. Phys., 103:8130–8143, 1995.

[89] O. V. Prezhdo and P. J. Rossky. Mean-field molecular dynamics with surface

hopping. J. Chem. Phys., 107(3):825–834, 1997.

Bibliography 115

[90] R. Grunwald and R. Kapral. Decoherence and quantum-classical master equation

dynamics. J. Chem. Phys., 126:114109, 2007.

[91] R. Grunwald, H. Kim, and Raymond Kapral. Quantum equilibrium structure and

decoherence in quantum-classical dynamics. J. Chem. Phys., 2008.

[92] R. Kubo. Fluctuation-dissipation theorem. Rep. Prog. Phys., 29:255, 1966.

[93] R. W. Zwanzig. Statistical Mechanics of Irreversibility. Wiley-Interscience, 1961.

[94] H. Mori. Transport, collective motion, and Brownian motion. Prog. Theor. Phys.,

33(3):423, 1965.

[95] S. Consta R. Kapral and L. McWhirter. Chemical rate laws and rate constants.

In In B. J. Berne, G. Ciccotti, and D. F. Coker, editors, Classical and Quantum

Dynamics in Condensed Phase Simulations, pages 583–616. World Scientific, 1998.

[96] A. Sergi and R. Kapral. Quantum-classical dynamics of nonadiabatic chemical

reactions. J. Chem. Phys., 118:8566–8575, 2003.

[97] H. Kim and R. Kapral. Transport properties of quantum-classical systems. J.

Chem. Phys., 122:214105, 2005.

[98] H. Kim and R. Kapral. Nonadiabatic quantum-classical reaction rates with quan-

tum equilibrium structure. J. Chem. Phys., 123:194108, 2005.

[99] H. Kim and R. Kapral. Proton and deuteron transfer reactions in molecular nan-

oclusters. ChemPhysChem, 9(3):470–474, 2008.

[100] A. Warshel. Dynamics of reactions in polar-solvents - semi-classical trajectory stud-

ies of electron-transfer and proton-transfer reactions. J. Phys. Chem., 86(12):2218–

2224, 1982.

Bibliography 116

[101] R. A. Marcus and N. Sutin. Electron transfers in chemistry and biology. Biochim.

Biophys. Acta, 811(3):265–322, 1985.

[102] D. Laria, G. Ciccotti, M. Ferrario, and R. Kapral. Molecular-dynamics study of

adiabatic proton-transfer reactions in solution. J. Chem. Phys., 97:378–388, 1992.

[103] S. Hammes-Schi!er and J. C. Tully. Vibrationally enhanced proton transfer. J.

Phys. Chem., 99:5793–5797, 1995.

[104] S. Consta and R. Kapral. Dynamics of proton transfer in mesoscopic clusters. J.

Chem. Phys., 104:4581–4590, 1996.

[105] M Tosi A. A. Kornyshev and editors. J Ulstrup. Electron and Ion transfer in

Condensed Media. World Scientific, 1997.

[106] D. Laria G. Ciccotti, M. Ferrario and R. Kapral. Simulation of classical and quan-

tum activated process in the condensed phase. In L Reatto and F Manghi, editors,

Progress of Computational Physics of Matter: Methods, Software and Applications.

World Scientific, New Jersey, 1995.

[107] K. Thompson and N. Makri. Influence functionals with semiclassical propagators

in combined forward-backwardtime. J. Chem. Phys., 110:1343–1353, 1999.

[108] G. Stock and M. Thoss. Classical description of nonadiabatic quantum dynamics.

Adv. Chem. Phys., 131:243–375, 2005.

[109] S. S. Xantheas G. J. Atchity and K. Ruedenberg. Potential energy surfaces near

intersections. J. Chem. Phys., 95:1862–1876, 1991.

[110] H. Koppel L. S. Cederbaum, W. Domcke and W. von Niessen. Vibronic coupling

e!ects in the photoelectron spectrum of ethylene. Chem. Phys., 26:169, 1977.

Bibliography 117

[111] L. S. Cederbaum and W. Domcke. Theoretical aspects of ionization potentials

and photoelectron spectroscopy: a green’s function approach. Adv. Chem. Phys.,

36:205, 1977.

[112] F. Bernardi M.A. Robb M. Garavelli, P. Celani and M. Olivucci. The C5H6NH+2

protonated Schi! base: An ab initio minimal model for retinal photoisomerization.

J. Am. Chem. Soc.

[113] L. Noodleman M. J. Thompson, D. Bashford and E. D. Getzo!. Photoisomerization

and proton transfer in photoactive yellow protein. J. Am. Chem. Soc., 125:8186–

8194, 2003.

[114] A. Ferretti, A. Lami, and G. Villani. A model study of the wavepacket dynamics

around a Jahn-Teller conical intersection in a symmetric charge-transfer system.

Chem. Phys., 259:201–210, 2000.

[115] J. Koutecky V. Bonacic-Koutecky and J. Michl. Neutral and charged biradicals,

zwitterions, funnels in S1, photochemistry and proton translocation: Their role in

photochemistry, photophysics and vision. Angew. Chem. Intl. Edn. Engl., 26:170,

1987.

[116] L. S. Cederbaum I. Burghardt and J. T. Hynes. Environmental e!ects on a conical

intersection: A model study. Faraday Discuss., 127:395–411, 2004.

[117] L. S. Cederbaum I. Burghardt and J. T. Hynes. Ultrafast excited-state charge

transfer at a conical intersection: e!ects of an environment. Comp. Phys. Comm.,

169:95–98, 2005.

[118] I. Burghardt and J. T. Hynes. Excited state charge transfer at a conical intersection:

e!ects of an environment. J. Phys. Chem A., 110:11411–11423, 2006.

Bibliography 118

[119] E. Gindensperger I. Burghardt, J.T. Hynes and L. S. Cederbaum. Ultrafast excited-

state charge transfer at a conical intersection: the role of environmental e!ects.

Phys. Scr., 73:C42–C46, 2006.

[120] I. Burghardt R. Spezia and J. T. Hynes. Conical intersections in solution: nonequi-

librium versus equilibrium solvation. Mol. Phys., 104:903–914, 2006.

[121] J. Tennyson and J.N. Murrell. Analysis of a three state model for the non-adiabatic

processes accompanying the internal rotation of ethylene. Nouv. J. Chim., 1981.

[122] D. MacKernan, G. Ciccotti, and R. Kapral. Trotter-based simulation of quantum-

classical dynamics. J. Phys. Chem. B, 112:424–432, 2008.

[123] D. MacKernan R. Kapral S. Bonella, D. Coker and G. Ciccotti. Trajectory based

simulations of quantum-classical systems. In I. Burghardt, editor, Energy Transfer

Dynamics in Biomaterial System, pages 415–436. Springer, Berlin, 2009.

[124] N. Makri. The linear response approximation and its lowest order corrections: An

influence functional approach. J. Phys. Chem. B, 103:2823–2829, 1999.

[125] A. Ferretti, A. Lami, and G. Villani. Transition probability due to a conical inter-

section: On the role of the initial conditions and the geometric setup of the crossing

surfaces. J. Chem. Phys., 111:916–922, 1999.

[126] B. Misra and E. C. G. Sudarshan. The Zeno’s paradox in quantum theory. J. Math.

Phys., 18:756–763, 1977.

[127] O. Kuhn and H. Naundorf. Dissipative wave-packet dynamics of the intramolecular

hydrogen bond in o-phthalic acid monomethylester. PCCP, 2003.

[128] J. Schofield A. Kelly, R. van Zon and R. Kapral. On the solution to the quantum-

classical Liouville equation in the mapping basis. unpublished, 2010.

a qua ntu m -cla ss ical investiga tion of envi r o nment ... · 4.1 1 ground stat e p opul atio n...

Documents