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 Quantum Dynamics of Finite Atomic and Molecular Systems Through Density Matrix Methods
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 THORNDYKE, BRIAN ( Author, Primary )
 Copyright Date:
 2008
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 Subjects / Keywords:
 Atoms ( jstor )
Cauldrons ( jstor ) Electronics ( jstor ) Electrons ( jstor ) Helium ( jstor ) Lithium ( jstor ) Physics ( jstor ) Simulations ( jstor ) Sine function ( jstor ) Trajectories ( jstor )
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 University of Florida
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 University of Florida
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 4/30/2004
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QUANTUM DYNAMICS OF FINITE ATOMIC AND MOLECULAR SYSTEMS
THROUGH DENSITY MATRIX METHODS
By
BRIAN THORNDYKE
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2004
Copyright 2004
by
Brian Thorndyke
To my father,
Gerry Thorndyke
ACKNOWLEDGMENTS
First of all, I would like to thank my parents, Gerry and Carol Thorndyke,
whose unwavering support and love my entire life have allowed me to pursue my
dreams.
On the academic side, I would like to thank my advisor, Dr. David Micha, for his
excellent guidance and encouragement throughout my doctoral work. I would also
like to thank the following colleagues in Quantum Theory Project and the Physics
department for their friendship and insights during my stay in Gainesville: Jim
Cooney, Herbert DaCosta, Alex Pacheco, Dave Red, Andres Reyes, Akbar Salam,
Alberto Santana and Zhigang Yi.
On a personal level beyond the Physics department, I would be remiss if I
didn't express my love and gratitude to Natasha Lepor6. She has been my ..11i
sister" for over a decade, and I hope our lives will continue to run with fascinating
parallels and intertwine for many decades to come! I'd also like to recognize Albert
Vernon who, since our early days in the Computer Science department, has been
my partner in our relentless pursuit of aesthetically pleasing code. His friendship
has both contributed to some of the best of times and helped me through some of
the worst over the last 8 years. Finally, I'd like to express my appreciation to Mike
Kuban and Rob Thorndyke for being with me in spirit throughout my Ph.D., and
for always providing wonderful avenues of escape and adventure year after year!
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ................... ....... iv
LIST OF TABLES ........... ...... ........ ...... ix
LIST OF FIGURES ................................ x
ABSTRACT ................... ............. xiii
CHAPTER
1 INTRODUCTION .................. ........ 1
1.1 Overview of Classical and Quantum Dynamics ........... 2
1.2 Approximations to Quantum Dynamics ...... ........ 5
1.2.1 WavefunctionBased Approaches ....... ..... ... 5
1.2.2 Density Operator Approaches ...... .......... 7
1.3 QuantumClassical Liouville Equation ... . 8
1.4 Our Approach ....... ......... ... ...... 10
1.5 Simple OneDimensional TwoState Models . ... 11
1.6 LithiumHelium Clusters ................ .... .. 11
1.7 Outline of the Dissertation ................. .. 12
2 QUANTUMCLASSICAL LIOUVILLE EQUATION: FORMULATION 14
2.1 Introduction .................. ........... .. 14
2.2 Quantum Liouville Equation ............ .. .. 14
2.3 W igner Representation .................. ... 15
2.4 QuantumClassical Liouville Equation . 18
2.5 Effective Potential .................. ........ .. 20
3 QUANTUM CLASSICAL LIOUVILLE EQUATION: COMPUTATIONAL
ASPECTS ................ .. ........ ..... 22
3.1 Introduction .................. ........... .. 22
3.2 Trajectory Solution .................. .. .... .. .. 22
3.3 Electronic Basis Set .................. ..... 23
3.4 Nuclear Phase Space Grid ................... 24
3.5 RelaxandDrive Algorithm ................. .. 25
3.5.1 W Independent of Time .... .. 26
3.5.2 W Dependent on Time ....... . ... 26
3.5.3 Velocity Verlet for the Classical Evolution ... 31
3.5.4 Algorithm Details .. ...................
3.6 Computing Observables .. ....................
3.6.1 Operators in an Orthonormal Basis .. ..........
3.6.2 Population Analysis .. ...................
3.6.3 Expectation Values .. ..................
3.6.4 Hamiltonian Eigenstates and Eigenvalues .. ........
3.7 Programming Details .. .....................
3.7.1 Orthogonality of Code Development .. ...........
3.7.2 Extensibility . . . .
4 ONEDIMENSIONAL TWOSTATE MODELS .. ...........
4.1 Introduction . . . .
4.2 EffectivePotential QCLE in the Diabatic Representation .....
4.3 NearResonant Electron Transfer Between an Alkali
Metal Surface ..... ..............
4.3.1 M odel Details ... .. .. .. .. ..
4.3.2 Properties of Interest .........
4.3.3 Results. . .
4.4 Binary Collision Involving Two Avoided Crossings .
4.4.1 M odel Details .............
4.4.2 Properties of Interest .........
4.4.3 Results... . .
4.5 Photoinduced Dissociation of a Diatomic System .
4.5.1 M odel Details ... .. .. .. .. ..
4.5.2 Properties of Interest .........
4.5.3 Results...... . .
4.6 Comparison Using Variable and Constant Timesteps .
4.7 C conclusion . . .
Atom and
. 41
. 41
. 42
. 47
. 59
. 59
. 60
. 62
. 63
. 63
. 68
. 70
. 72
. 72
5 ALKALI ATOMRARE GAS CLUSTERS: GENERAL FORMULATION 80
Introduction . . .
Physical System .. ... .. .. .. .. ..
Properties of Interest ...........
Hamiltonian for AlkaliRare Gas Pairs ...
Hamiltonian for the AlkaliRare Gas Cluster .
Electronic Spectral Calculations ......
Electronic Basis of Gaussian Atomic Functions .
5.7.1 Equations of Motion .........
5.7.2 Overlap Matrix Elements ......
5.7.3 Kinetic Energy Matrix Elements ..
5.7.4 Coulomb Matrix Elements ......
5.7.5 Momentum Coupling Matrix Elements .
5.7.6 Dipole Matrix Elements .......
5.7.7 Pseudopotential Matrix Elements ..
. 80
. 80
. 81
. 81
. 84
. 86
. 87
. 87
. 89
. 90
. 90
. 91
. 91
. 92
5.8 Computing the Quasiclassical Trajectory .
5.9 Computational Details ...........
5.10 C conclusion . . .
6 LITHIUMHELIUM CLUSTERS .........
6.1 Introduction . . .
6.2 Description of the System ..........
6.3 Properties to be Investigated ........
6.4 Preparation of LithiumHelium Clusters ..
6.4.1 Bulk Helium ... .. .. .. .. ..
6.4.2 Liquid Helium Droplets .......
6.4.3 LithiumHelium Interactions ....
6.5 Results: Lithium Inside the Helium Cluster .
6.6 Results: Lithium on the Helium Cluster Surface .
6.6.1 Dynamics of Li(2pa) ..
6.6.2 Dynamics of Li(2p7) ..
6.7 C conclusion . . .
. 93
. 93
. 97
. 99
. 99
. 99
. 101
. 102
. 102
. 109
. 109
. 119
. 121
. 124
. 130
. 137
7 CONCLUSION .. .............
Effective Potential QuantumClassical Liouville Equation 140
OneDimensional TwoState Models ... . 141
AlkaliRare Gas Clusters .... . 142
Software Development .................. ...... 144
Future Work ....... ...................... 145
APPENDIX
A THE CAULDRON PROGRAM .................. ..... 146
A.1 Overview ........................ ....... 146
A.2 Component Descriptions ................ .... 147
A.2.1 Read Input File ... . ..... 147
A.2.2 System: Get Differential Equation Coefficients 147
A.2.3 Propagation: Evolve Single Timestep . ... 149
A.2.4 Properties: Output Properties . 149
A.3 Subroutine Details ................... 150
B SPLIT OPERATORFAST FOURIER TRANSFORM METHOD .
C THE QUALDRON PROGRAM. . .....
C 1 O verview . . .. ..
C.2 Component Descriptions . .....
C.2.1 Read Input File . .
C.2.2 System: Get Hamiltonian Matrix Elements .
C.2.3 Propagation: Evolve Single Timestep . .
. 151
. 154
. 154
. 155
. 155
. 155
. 155
C.2.4 Properties: Output Properties ... . 157
C.3 Subroutine Details .................. ..... 157
REFERENCE LIST .................. ............. 158
BIOGRAPHICAL SKETCH .................. ......... 168
viii
LIST OF TABLES
Table page
41 Parameters used in the N.. n f. .:e and Lisurface models. ...... ..42
42 Parameters used in the dual avoided crossing collision model. 60
43 Parameters used in the Nal complex model. ............ ..68
51 Pseudopotential rotation for dfunction mixing. ..... 98
61 Parameters for the HeHe interaction from Aziz (VA). ... 106
62 Parameters for the correction to the HeHe interaction (V) 106
63 Parameters for the eLi interaction. ................. ..113
64 Parameters for the eHe interaction. ................ ..113
65 Parameters for the LiHe core interaction. .. . ..... 114
LIST OF FIGURES
Figure page
41 Potential curves for Hamiltonian I: Na incident upon a metal surface. 43
42 Potential curves for Hamiltonian II: Li incident upon a metal surface. 44
43 T(R) at t = 0 au, for the N..ii if.,.e model. This wavefunction is
evolved through the SOFFT algorithm. ............. ..48
44 F11 at t = 0 au, for the N..II. f.i.:e model. This PWTDM is evolved
through the EPQCLE method. ................. 49
45 T(R) at t = 14000 au, for the N..11. f.,.!e model. . 50
46 Phase space grid points at t = 14000 au, for the N..iii f.,.e model. 51
47 Na populations qi and 7q2 vs. time. ................. 52
48 Li populations mq and 7q2 vs. time. ................. 53
49 Coherence described by Re(iq12) vs. time, for the .. iii f.i:e system. 54
410 Coherence described by Re(M12) vs. time, for the Lisurface system. 55
411 Expectation of position and dispersion for the N.. ,i f.,.:e system. .56
412 Expectation of momentum and dispersion for the N..i if.i.:e system. 57
413 Density function p(R) for the N.. ii f.,:e system. . ... 58
414 Potential curves for the dual avoided crossing collision. ... 61
415 Populations piq and 7q2 vs. time for the dual crossing collision model. 64
416 Coherence described by Re(r12) vs. time, for the dual crossing collision
m odel. ........... ...... ........ ..... 65
417 Grid deformation at t = 1400 au, for the dual crossing collision model. 66
418 Probability of transmission in the ground state, for the dual crossing
collision model. .................. ..... 67
419 Potential curves for the Nal complex. ................. 69
420 Ionic and neutral populations over time, for the Nal complex. 73
421 Expectation of position and its deviance, for the Nal complex. 74
422 Coherence as a function of time, for the Nal complex. ... 75
423 Phase space grid at the end of the simulation, for the Nal complex. 76
424 Number of steps required by the relaxanddrive algorithm, compared
to an estimated number required for a fixed timestep version. 77
61 Schematic of Li(2p) above a He surface. A) Li(2pw). B) Li(2p(r). 101
62 Radial distribution functions for bulk liquid helium .... 106
63 Comparison of the Aziz potential with the effective form. ...... .107
64 Effective HeHe potential. ........ . ..... 108
65 Constraining potential used to keep He atoms from evaporating. 110
66 Temperature fluctuations of the He droplet over time .... 111
67 Helium density profile from the centerofmass of the cluster. 112
68 Adiabatic energy for Li and He as a function of internuclear distance. 114
69 Adiabatic energies for Li and one or more He along the zaxis. 116
610 Adiabatic energy for Li and one or more He along the yaxis. 117
611 Adiabatic energy for Li and a surface of He atoms parallel to the xy
plane. . . .. .. .......118
612 Evolution of ground state Li embedded in the center of a He cluster.
A) Initial time t = 0 au. B) Final time t = 10,000 au. ...... .120
613 Comparison of Li and He motion within a He cluster. The time scale
has been reduced by a factor of 100 for the He curve. ..... ..122
614 Electronic population of Li as it emerges from the He cluster. 123
615 Evolution of Li(2pa) as it recedes from the He cluster surface. A)
Initial time t = 0 au. B) Final time t = 33, 000 au. ... 125
616 Mixing of the Li(2pa) and Li(2pr) states at distances where Li(2p) is
triply degenerate. ................ ........ 126
617 Electronic population of Li with and without a perturbing electro
magnetic field, resonant to the D line. . 128
618 Dipole emission spectra of Li(2pa) as it recedes from the He cluster
surface. .... .. .. .... ..... ...... 129
619 Snapshot of Li(2pr) as it interacts with the He cluster surface. A)
Initial time t = 0 au. B) Final time t = 67, 000 au. ... 131
620 Electronic population of Li(2p7) as it interacts with the He cluster
surface. . .. .. .. .. 132
621 Dipole emission spectrum of Li(2pr) during the first 3000 au. .... 134
622 Dipole emission spectrum of Li(2pr) during the final 3000 au. .... 135
623 Adiabatic curves of Li surrounded by a cubic lattice of He atoms. The
parameter R refers to the halflength of the lattice edge. 136
624 Decay of Li(2pr) surrounded by surface He atoms, induced by an EM
field with frequency resonant to the Li(2pr < 2sra) transition. .138
A1 Flowchart describing the cauldron program. . 148
C1 Flowchart describing the qualdron program. . 156
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
QUANTUM DYNAMICS OF FINITE ATOMIC AND MOLECULAR SYSTEMS
THROUGH DENSITY MATRIX METHODS
By
Brian Thorndyke
M.I. 2004
Chair: David A. Micha
M., i r Department: Physics
We develop a mixed quantumclassical formulation to describe the dynamics of
few and hm..iii body atomic systems by applying a partial Wigner transform over
the quantum Liouville equation of motion. In this approach, the density operator
becomes a function in quasiclassical phase space, while remaining an operator over
a subset of quantal variables. By taking appropriate limits and introducing an
effective potential, we derive equations of motion describing quasiclassical nuclear
trajectories coupled to quantal electronic evolution. We also introduce a variable
timestep procedure to account for the disparity between slow nuclear motion and
fast electronic fluctuations.
Our mixed quantumclassical method is applied to the study of three simple
onedimensional twostate models. The first model represents the photoinduced des
orption of an alkali atom from a metal surface, where nearresonant electron transfer
is important. A second model explores a binary collision under conditions where two
avoided crossings are present. The third model follows the photoinduced dissocia
tion of the sodium iodide complex, whose longrange attractive surface results in
oscillations of internuclear distance. Quantities such as state populations and quan
tum coherence are computed, and found to be in excellent agreement with precise
quantal results obtained through fast Fourier transform grid methods.
Having validated our approach, we turn to the study of alkali atoms embedded
in rare gas clusters, treating the alkali atomrare gas interactions with idependent
semilocal pseudopotentials. Light emission from the electronic motion of the alkali
atom is derived in the semiclassical limit, and computational methods to render
the simulation feasible for a manyatom cluster are discussed. The formalism is
applied to lithium atoms in helium clusters, where the cluster configuration and the
electronic population dynamics of the lithium atom are monitored over time. We
study both the ground and first excited states of lithium, and introduce a resonant
electromagnetic field to induce electronic transitions. Our results correlate well with
other experimental and theoretical studies on doped helium droplets, and provide
insight into the dynamics of an excited lithium atom near a helium cluster surface.
CHAPTER 1
INTRODUCTION
This work is part of a broader effort to bring new insights into the time depen
dence of few and manybody molecular systems. In order to study these systems,
we are particularly interested in mixed quantumclassical methods. There are many
approaches to combining classical and quantum mechanics,1'2'3 but the underlying
theme is to use classical mechanics where quantal descriptions are not essential to
the dynamics of the system. By doing so, we can save a tremendous amount of
computational time with hopefully minimal expense in accuracy.
In our study, we focus on the density operator treatment, which allows for a
general introduction of semiclassical and classical limits for some degrees of freedom.4
In the applications being considered, electronic variables are described quantally and
the nuclear variables are propagated quasiclassically, but our theoretical treatment
of quantumclassical coupling is more general. We are able to validate our methods
through extensive study of small test models. These models can be evolved through
a fully quantal propagation using fast Fourier grid methods, permitting a rigorous
assessment of the accuracy of our approach.
As a realistic application, we turn to the study of ground and excited alkali
atoms embedded in clusters of rare gas atoms.5'6'7'8'9 Rare gas clusters provide an
interesting bridge between fewatom systems and bulk matter. A mixed quantum
classical approach allows us to follow the nuclear motion and population state
dynamics as the alkali atom interacts with the cluster. By tracking the electronic and
induced dipole, we are able to compute the electronic spectra of the alkali atom; and
by explicitly introducing an electromagnetic field, we are able to induce electronic
transitions.
While our formalism is applicable to arbitrary alkalirare gas combinations, we
have concentrated on the dynamics of lithium embedded in helium clusters, for which
there has been a surge of recent experimental and theoretical activity. 10,11,12,13,9,14
Stable helium clusters doped with lithium atoms have been produced at ultralow
temperatures, where the ground lithium atom has been shown to preferentially reside
on the surface of the helium cluster. Furthermore, the behavior of the lithium atom
subsequent to electronic excitation depends heavily on the orientation of the excited
state. Our mixed quantumclassical approach corroborates these findings, and leads
to additional insight into the dynamics of these interactions.
1.1 Overview of Classical and Quantum Dynamics
The vast majority of chemical and biological processes can be described, in
principle, by nonrelativistic quantum mechanics. Within this context, the state
of a system of nuclei and electrons is represented entirely by a wavefunction or
density operator.15 A Hamiltonian operator describes all interactions between the
particles, and can be extended to include environmental components (for example,
a boundary or electromagnetic field). When the Hamiltonian does not depend on
time, its eigenstates are stationary (up to a phase), and represent timeinvariant
configurations of the system. When the Hamiltonian contains time dependence, or
the initial state is nonstationary, the molecular system evolves through the action
of the Hamiltonian.
The solutions to the time independent Schr6dinger equation (TISE) are the
eigenstates of the Hamiltonian at any given time. The corresponding eigenvalues
are the energy levels available to the system. While conceptually compact, the ana
lytical solution of the TISE is not possible for more than two particles in the general
case. An enormous body of computational work in chemical and molecular ]1li,i, ,
is devoted to the numerical solution of the TISE,16,17,18 and accurate ground state
energies have been computed for molecules involving hundreds of atoms.19 Knowl
edge of the full spectrum of eigenstates would allow one to follow the dynamics of
the system as well, but unfortunately it is very difficult to obtain accurate results for
excited states, and in any case, the number of states required for accurate computa
tions would be prohibitively expensive for most dynamics problems. An alternative
is to follow the dynamics directly.
Quantum dynamics (QD) follows the evolution of a system in real time. The
wavefunction evolves according to the time dependent Schridinger equation (TDSE),20
while the density operator (DOp) evolves through the quantum Liouville equation
(QLE).21 These formalisms are equivalent, although statistical ensembles are more
naturally described by the density operator. Quantum dynamical calculations are
important in understanding the pathways between initial and final states, for exam
ple in chemical reactions or molecular collisions, and have become an important
complement to modern experiments that use pico or femtosecond light pulses to
probe ultrafast dynamics.22 The computational complexity of numerical solutions
to full QD, however, severely limits the number of degrees of freedom that can
be studied, and incorporation of classical concepts is necessary for most realistic
applications.
Full QD solutions can be readily implemented by discretizing the wavefunc
tion along a multidimensional grid. Since the Hamiltonian contains nonlocal opera
tors like kinetic energy, methods such as finite dill. n 1,25,26 or Fourier trans
form27,28 are needed to evaluate the action of the Hamiltonian on the wavefunction.
In the general case, these techniques require a dense grid to obtain accurate results.
Furthermore, their nonlocal nature can result in significant shifts of the wavefunction
over time, so that even if the initial state is spatially compact, a large grid is required
to accommodate translation. Sparse grid methods29 can alleviate some of the com
putational burden of using large grids, and dynamically changing grids30 can better
follow the distortion and translation of the wavefunction over time. Alternatively,
discretization on a basis set instead of a grid can simplify derivative calculations.31
Ultimately, however, the exponential scaling of the number of grid points or basis
functions with the system size renders full QD solutions intractable for more than
a few degrees of freedom.
Classical molecular dynamics (\!1)) can be derived from the QLE in the classical
limit (h  0).32 In the context of molecular simulations, the most basic MD treats
nuclei as point particles in phase space, and follows trajectories according to the
Hamilton equations of motion.33 Internuclear potentials are derived from ab initio,
experimental and empirical results, and the forces on the nuclei are obtained by
summing over the partial derivatives of the pair potentials. These classical force
calculations are now the bottleneck, and straightforward evolution of N classical
degrees of freedom has only O(N2) time complexity.34 Tree methods based on octree
spatial partitions35 or multiple potential .::p.ini. .I '. 37 reduce this complexity to
O(N log N), with a proportionality constant dependent on the desired accuracy of
the simulation. Because of its intuitive nature and computational efficiency, MD is
routinely used to study molecular systems with 103 106 atoms.38 The problem with
MD is its inability to adequately describe quantum mechanical phenomena, such as
charge transfer, electronic excitation, tunneling and zeropoint motion. These effects
are ubiquitous in chemical and biological processes at thermal or lower energies, and
cannot be completely neglected in most cases.3
An attractive alternative is to construct an approximate QD model that ideally
combines the accuracy of QD with the computational efficiency and intuitive sim
plicity of MD. In general, this can be done by augmenting MD methods to include
quantum features, or by simplifying QD models to incorporate classical or quasiclas
sical trajectories.1 There are many variations of this theme, and question remains as
to which approach is the most suitable under arbitrary conditions. In Section 1.2, we
survey some of the most common approximation schemes, differentiating between
methods based on the wavefunction and those centered on the density operator.
1.2 Approximations to Quantum Dynamics
1.2.1 WavefunctionBased Approaches
Wavefunctionbased approaches are most useful for the propagation of pure
states in closed environments, and have been studied theoretically and numeri
cally since the dawn of quantum mechanics. We limit our survey to some prin
cipal methods that incorporate some classical concepts to solve the TSDE, includ
ing time dependent selfconsistent field (TDSCF) calculations,39,40,41'42 Gaussian
wavepacket (GWP) propagation,43'44 surface hopping methods,45 electron nuclear
dynamics (END),46'47 and path integral methods.48'49
The time dependent selfconsistent field approximation begins with the wave
function written as a product of nuclear and electronic wavefunctions, and uses
time dependent variational principles to evolve each wavefunction along a potential
averaged over all other wavefunctions. When the nuclear variables are expressed in
the classical limit, the TDSCF approximation reduces to a set of electronic wave
functions evolving over nuclear trajectories, the trajectories propagating accord
ing to effective forces from the quantal system. Many mixed quantumclassical
schemes retain this meanfield flavor, where classical trajectories propagate along
precomputed or simultaneously evolving electronic states. On the other hand,
quantum dynamics has a very different character than classical propagation, and
mixed quantumclassical methods vary tremendously in their treatment of quantum
classical interactions, initial conditions, and measurement of system observables.
Gaussian wavepacket propagation, in its original form, takes the system to be
a Gaussian wavepacket in nuclear space, that propagates along a single electronic
surface. By locally expanding the potential up to harmonic contributions, equations
of motion for the Gaussian wavepacket parameters are derived, which lead to shifts
and distortions of the Gaussian over time. The Gaussian center follows classical
equations of motion, while its distorting shape results from quantal corrections to
the classical motion. The GWP method has been extended to describe nonadiabatic
processes with the multiple spawning approach,50 where several wavepackets prop
agate on multiple electronic surfaces, and proliferate into additional wavepackets at
crossing points.
Surface hopping begins with nuclear trajectories evolving on one or more elec
tronic surfaces, and reproduces nonadiabatic events through statistically based jumps
between the surfaces. In its earliest version, these transitions take place near avoided
crossings, but later generalizations allow hops to occur at any time during the sim
ulation.51 One drawback to the surface hopping approach is the need to rescale
velocities after stochastic transitions, in order to conserve energy. Although accu
rate state transitions are often achieved, the dynamics are clearly not representative
of the true system evolution, except in a statistical sense.
Electron nuclear dynamics is an entirely different approach, that derives equa
tions of motion by minimizing the MaxwellSchrodinger Lagrangian density and
treating the nuclear degrees of freedom as coherent states52 in the semiclassical
limit. By expanding the electronic wavefunction in a basis of Slater determinants,
nuclear and electronic motion are combined in a computationally accessible scheme.
Path integral methods solve the TDSE for the quantum time evolution opera
tor, exp(iHt/h), in the coordinate representation.53 Path integral equations are
equivalent to full QD, but unfortunately they are computationally intractable in
the general case. Harmonic approximations to the intermolecular potentials sub
stantially reduce the complexity of the calculations, making it possible to reproduce
the dynamics of a quantum system embedded within an harmonic bath.54'55 For
example, variations such as the initial value representation,56'57'58 have successfully
described the spinboson model of coupled electronic states in a condensed phase
environment. While not applicable to arbitrary systems, these methods are quite
useful when the system and its interactions can be cast into the appropriate forms.
1.2.2 Density Operator Approaches
There are a number of advantages to using the density operator. First, it
provides a convenient representation of mixed rather than pure states.59 Second,
many systems can be naturally partitioned into a primary i, 1ii IiI of interest and
its surrounding bath. By taking the trace over the bath degrees of freedom, the
QLE provides a reduced density description of the primary system interacting with
a bath.21'59 Finally, it is possible to include terms directly in the QLE that represent
energy dissipation into the environment.60 For these reasons, the QLE is often used
where initial conditions are specified as statistical averages, or when departure from
a full quantum treatment is necessary due to the size of the system.
Among mixed quantumclassical solutions to the QLE, an early approach splits
the density operator into a product of Gaussian wavepackets, and derives equations
of motions for the Gaussian parameters using selfconsistent field approximations.61
In the semiclassical limit, this method has similarities to generalized forms of GWP
propagation, but because it is based on the density matrix, it can naturally treat
both open and closed systems.
Another method is similar in spirit to END, but uses density matrices rather
than Slater determinants to represent the electronic state.62 The quantum degrees
of freedom are expanded in some basis set, while the classical degrees of freedom
are written as coherent states. The Lagrangian is minimized through the Dirac
Frenkel variational principle, and one arrives at equations of motion for a reduced
density matrix, coupled to semiclassical equations of motion for the coherent state
parameters.
The density matrix evolution (DME) method63,64'65 divides the system into a
quantum and classical space. The quantum 1i,I 1. iIl is expanded over a nonlocal,
orthogonal basis set, while the classical coordinates evolve along trajectories accord
ing to the HellmannFeynman force. This method has worked well for simple models
involving analytically accessible matrix elements, but is not designed for arbitrary
I' II with nonlocal and possibly nonorthogonal basis sets.
Eikonal methods66'67'68 exploit the limit of small nuclear wavelengths to sepa
rate quantal and classical motion. Through this formalism, the electronic density
matrix propagates simultaneously with nuclear trajectories, the latter guided by
effective forces from the quantum density. Combined with variable timestep meth
ods and travelling atomic function basis sets, the eikonal approach is particularly
well suited for the study of binary collision problems.
A recent branch of QLE methods uses the Wigner transform (WT) to cast the
operatorbased QLE into an equation of motion over phase space variables. The
classical Liouville equation for the density function emerges in the classical limit,
while judicious application of the WT to a subset of the system variables leads to
well defined quantumclassical schemes. In Section 1.3, we outline some of the major
approaches found in the literature.
1.3 QuantumClassical Liouville Equation
It is fascinating that a phase space representation of quantum mechanics can be
developed, that is fully equivalent to the Hilbert space representation. By applying
the Wigner transform to both sides of the QLE, an equation of motion is derived for
a Wigner function, which is a function of classical position and momentum variables.
The Wigner function does not evolve through simple classical equations of motion,
however, but involves nonlocal operators in phase space.4 One class of solutions
exploits the similarity of these equations of motion with hydrodynamics found in
classical li, i. Socalled hydrodynamic quantum dynamics was formulated by
Bohm in 1952, and recent developments in the field have revitalized interest in phase
space trajectory methods for solving QD.69'70'71'72'73 Unfortunately, these methods,
like those involving basis set or grid solutions to the TDSE or QLE, suffer from
exponential growth with system size.
A mixed quantumclassical scheme arises by dividing the quantum system into
quantal and quasiclassical variables, the quasiclassical variables associated with ener
gies or masses much greater than the quantal variables. Taking the partial Wigner
transform (PWT) of the QLE over only the quasiclassical variables produces an
equation of motion for a partially transformed Wigner operator (PTWDOp), which
is a function of phase space in the quasiclassical variables, but remains a quantal
operator in the space of the quantal variables. After some appropriate approxi
mations relating to the different masses of the variables, we arrive at a differential
operator equation referred to as the quantumclassical Liouville equation (QCLE).74
While the quantal evolution must still be tackled in Hilbert space, the propagation
of the quasiclassical phase space acquires a classical character. If the majority of
the system can be described with quasiclasical variables, the computation savings
may be considerable.
One numerical approach represents the PWTDOp in phase space with a fixed
number of delta functions.75,76'77 Specifically, the PWTDop is projected on a basis,
resulting in a set of diagonal and offdiagonal functions in phase space. Each function
is approximated by a set of delta peaks, that evolve along surfaces constructed
from the densities and coherences. Nonadiabatic coupling is implemented through a
modified surface hopping procedure, where hops occur between trajectories at curve
crossings.
An alternative solution uses the same delta peak representation, but rather
than use stochastic jumps between trajectories, new trajectories are generated at
curve crossings.78 A variation on this approach, the multithread method, spawns
new trajectory points at every timestep, but combines the newly generated pools
of trajectory points into smaller numbers through energy and other conservation
considerations.79,80,81
Recent efforts have been made to combine Gaussian wavepackets in phase space
(GPPs) with the trajectory solutions along diagonal and offdiagonal surfaces.82
Rather than represent quasiclassical functions through delta peaks, the functions
are expanded as a linear combination of GPPs, thereby more effectively covering
quasiclassical phase space. Stochastic jumps between surfaces at crossing points
reproduce nonadiabatic behavior, and avoid the need to spawn new GPPs.
1.4 Our Approach
In our approach, we introduce an effective potential to the QCLE, to provide
a simple numerical procedure for solving the equation.83 Our effective potential
quantumclassical Liouville equation (EPQCLE) permits solution in terms of full
quantal evolution of the quantal variables along quasiclassical trajectories guided
by the quantal state. We introduce a quantal basis and a quasiclassical grid to
render the equations suitable for numerical propagation, and implement a scheme to
efficiently account for the different time scales of quasiclassical and quantal motion.
Our approach shares several positive features with other QCLE methods of solu
tion. The QCLE rigorously combines quantum and classical motion, and provides
1,1.I'., i.il ways to incorporate higherorder approximations and quantumclassical
coupling. The formalism is based on the density operator, making it attractive for
incorporating thermodynamical features or dissipative environments. The QCLE
also lends itself to trajectorybased numerical solutions that reduce to the classi
cal Liouville equation in the classical limit, providing an intuitive computational
framework.
Our EPQCLE solution differs from other QCLE primarily in its generality and
its scalability. The effective potential can take a variety of forms, and while we
have found the HellmannFeynman force to provide the best results for our model
'1i I.. it is possible that other forms would provide even better results under
different circumstances. Furthermore, an electronic basis is only introduced once
the EPQCLE has been developed in terms of partial Wigner operators, and no
assumptions are made on the form of the basis set. This flexibility has allowed a
uniform treatment of problems ranging from onedimensional models expanded in
a twostate diabetic basis, to fully threedimensional atomic clusters in a basis of
Gaussian atomic functions. Finally, while the majority of QCLE solutions rely on
interactions between the propagating trajectories, our use of an effective potential
results in completely independent trajectories whose only connection follows from
initial conditions. Such independent trajectory evolution maps optimally to certain
parallel architectures, and substantially reduces the cost of propagation, that would
otherwise be required to compute trajectory interactions at each timestep.
1.5 Simple OneDimensional TwoState Models
We test our methods on a set of onedimensional twostate models, which are
simple enough to be evaluated precisely through fast Fourier grid methods. Among
these methods, we study a model of an alkali atom approaching a metal surface,
where nearresonant electronic transfer is important.84s85 Secondly, we consider a
system representing a molecular collision with two avoided crossings, where impor
tant interference effects arise as one varies the collision energy.79 Finally, we study
a system representing the predissociation of the sodium iodide complex, where the
longrange attraction of the excited state results in oscillatory nuclear motion.86
1.6 LithiumHelium Clusters
Clusters are .: regates of atoms containing between three and a few thousand
atoms. The smallest clusters, or microclusters, have between 3 and 10 atoms, and
are just small enough that molecular concepts still apply to some degree. The
next range of clusters, or small clusters, have between 10 and 100 atoms, a range
where molecular concepts break down and the clusters form shells with nonempty
interiors. Large clusters, between 100 and 1,000 atoms, provide the final transition
from isolated molecules to the bulk condensed state.
Rare gas clusters can be probed by doping them with a chromophore and follow
ing the chromophore through various laser detection methods.9'87'88'89 Our study
used a lithium atom as the dopant in a pure helium cluster. Semilocal idependent
pseudopotentials are used to describe the lithiumhelium interactions, because they
are found to accurately reproduce adiabatic energy curves for the lithiumhelium
pair for s, p and d symmetries.90'91'92'93'94 By introducing the pseudopotential for
malism into the EPQCLE, we are able to follow the nuclear configuration over time.
In addition, we are able to probe the evolving electronic energy surface by monitor
ing the electronic and induced dipole, and computing the resulting dipole emission
spectrum. By applying a resonant electromagnetic field, we are able to stimulate
the emission of light by excited lithium atoms near the cluster surface. Both the
nuclear configuration of the cluster and the spectral properties of its dopants have
been under intense investigation over the recent years, and our contribution gives
insight into the dynamics of these interactions.
1.7 Outline of the Dissertation
Chapter 2 presents the formalism we have used to explore the dynamics of
mixed quantumclassical systems. The ideas are presented for the general case of
quantal and quasiclassical variables, and it is shown how the EPQCLE naturally
emerges from the partial Wigner transform over the quasiclassical variables.
Chapter 3 explores the computational aspects of the EPQCLE. In particular,
the application of an electronic basis and a nuclear grid render the equations of
motion suitable to numerical solution. We also show how observable quantities can
be calculated within this framework.
Chapter 4 applies our formalism to three different onedimensional twostate
,1 in,. where the results can be compared to exact quantal simulations based on
the Schridinger equation and numerically precise grid methods. Along the way,
valuable insights into the accuracy and limitations of our methods are obtained.
Chapter 5 presents the formalism for general alkalirare gas clusters. We describe
the Hamiltonian for general alkalirare gas interactions, and derive matrix elements
for a basis set of Gaussian atomic functions. We further discuss the dipole of the
cluster and its matrix elements in detail. Finally, we present computational meth
ods used to render the numerical propagation of the equations of motion feasible for
small to large clusters.
Chapter 6 presents specific results for lithiumhelium clusters. We consider
the thermodynamic equilibration of the helium cluster in detail, followed by the
evolution of the lithium atom through the cluster and near its surface once the cluster
has reached equilibrium. We follow the nuclear motion and discuss the interaction
of the excited lithium atom near surface helium atoms, in particular with regard to
the nuclear configuration and the dipole spectrum. Finally, we introduce a resonant
electromagnetic field to stimulate photon emission after the excited lithium atom
embeds itself within the surface.
Chapter 7 summarizes the main conclusions obtained in this dissertation.
Appendix A discusses the program cauldron developed over the course of this
dissertation, to simulate the EPQCLE for the test model systems and the full
lithiumhelium cluster.
Appendix B derives the numerical evolution of the Schridinger equation through
the split operator fast Fourier transform method. The solution obtained by this
approach is exact to within numerical precision, but is computationally expensive
and thus prohibitive for more than a few degrees of freedom.
Appendix C presents the qualdron code, developed alongside cauldron to com
pare the results of the test models using mixed quantumclassical dynamics with the
full quantal treatment.
CHAPTER 2
QUANTUMCLASSICAL LIOUVILLE EQUATION: FORMULATION
2.1 Introduction
Rather than focus on the wavefunction of a molecular system, we construct its
density operator and proceed from there. The DOp is more general than the wave
function, in that it naturally describes a statistical ensemble of quantum states.
This is particularly useful when the molecular system interacts with its environ
ment, and one does not have a complete knowledge of that environment. The DOp
also provides a convenient starting point for deriving mixed quantumclassical meth
ods, because the classical limit of the DOp is the classical density function. One
particular route to a mixed quantumclassical description of a molecular system is
through the the Wigner transform. By applying a partial Wigner transform to the
DOp and its equation of motion, one obtains a new representation that is a function
in the phase space of the Wigner transformed variables, but remains an operator in
the remaining variables. If the PWT is judiciously applied to classicallike (or qua
siclassical) variables, then approximations can be made to the equations of motion
that provide a classical evolution of the quasiclassical variables coupled to a quantal
evolution of the quantal variables. In the case of molecular systems, the quasiclas
sical variables are typically the nuclear coordinates, while the remaining variables
describe the electronic state. In this chapter, the PWT is described in detail, and
its application to molecular systems is outlined.
2.2 Quantum Liouville Equation
A system of atoms or molecules is represented by nonrelativistic quantum
mechanics as a state vector IT) in Hilbert space. This state vector evolves in time
according to Schridinger's equation,
ih Ot = HI), (2.1)
where H is the Hamiltonian of the full system. If, however, we are studying an
ensemble of states {li)} with statistical weights {', }, it is convenient to construct
the density operator,
= IQ') . (2.2)
Taking its time derivative and using Eq. 2.1, we find the density operator to evolve
in time according to the quantum Liouville equation of motion,
ih = [H, F]. (2.3)
There are a number of advantages to using the density operator, but in our case
the primary advantage is that it leads directly to the classical density function in
the limit h  0. We first discuss the full Wigner transform and its application to
the QLE, and then show how the partial Wigner transform can be used to derive a
mixed quantumclassical representation in the appropriate limits.
2.3 Wigner Representation
The Wigner transform provides a phase space representation of the DOp (the
Wigner function) and other quantum mechanical operators. It is defined as the
Fourier transform of an operator projected on coordinate space,4'95'96'97
Fw(r, p) (2Ih) dzexp(ip z/)(r z/2r + z/2), (2.4)
Aw(r,p) = d'zexp(ip z/h)(r /2AIr + /2), (2.5)
where A is arbitrary. The integration generates functions that are local in both
coordinate and momentum space, which is important for the emergence of classical
features in the development of our mixed quantumclassical method. The prefactors
are defined differently for the density operator than for other operators, in order to
provide convenient parallels between the Wigner function and the classical Liouville
density.
Classically, the probability distribution function is well defined. For an Nbody
v1, in with coordinates r = (ri, r,..., r3N) and moment p = (pl, I_,... ,P3N), the
classical Liouville density p = p(r, p) generates the expectation value of any function
A A(r, p),32
(A) drdpp(r,p)A(r,p). (2.6)
Quantum mechanically, even the notion of phase space is problematic, as Heisen
berg's uncert.,iiii' principle prohibits the simultaneous measurement of position
and momentum for a given degree of freedom. However, quasidistribution functions
like the Wigner function provide an analogous form of the quantum mechanical
expectation value. The expectation value of an arbitrary operator is well known,59
(A) Tr(FA), (2.7)
which can readily be seen by expanding the trace over eigenstates of A. However,
by Wigner transforming both P and A, we arrive at a form for the expectation value
similar to the classical case,
(A) drdpfw(r,p)Aw(r,p). (2.8)
This can be seen by expanding Fw and Aw in Eq. 2.8,
I drdpFw(r, p)Aw(r, p)
S(2h) Ndrd I dzexp(ip z/h)(r +z/2,r /2)
x f dz' exp(ip z'/h)A(r + z'/2, r z'/2)
(2h)3N / drdpdzdz' exp[ip (z +z')/h]
xF(r + z/2, r z/2)A(r + z'/2, r z'/2)
/ drdzF(r + z/2, r z/2)A(r z/2, r + z/2).
(2.9)
Transforming variables q
value,
I drdpFw(
r+z/2, q' = rz/2, we retrieve the quantum expectation
r, p)Aw(r,p) = dqdq'F(q, q')A(q', q)
= dqdq'(qPfq')(q'A q'q)
f dq(qfPA q)
Tr(FA).
(2.10)
Although Fw is not a probability distribution, various properties justify its
classification as a quasiprobability function:
I(r)2 dpFw(r,p).
(p)2 'drFw(r,p).
For any function f(r), (f(r)) drdpFw(r,p)f(r).
For any function g(p), (g(p)) = fdrdpw(r, p)g(p).
One can also derive the WT of products of operators in terms of the WT of the
operators themselves.4 For a general operator P = AB, we find that
Aw exp(hA /2i)Bw,
(2.11)
where A is the bidirectional operator,
A  (2.12)
Op Or Or Op
Expanding the commutator directly, we have
[A, B]w = (AB)w (BA)w
S Awexp(h A/2i)Bw Bw exp( A/2i)Aw. (2.13)
2.4 QuantumClassical Liouville Equation
We have seen that the density operator evolves according to the QLE (Eq. 2.3).
Rather than study the solution to the QLE, we will examine the evolution of the
Wigner function. When the density operator is transformed over all its variables, we
arrive at equations of motion that, in the limit that h 0, reduce to the classical
Liouville equation. Here we are interested in performing a partial Wigner trans
form of the density operator over a subset of variables (those termed quasiclassical),
while leaving the remaining variables (those termed quantal) in their original rep
resentation. By taking appropriate limits, we derive equations of motion for the
quasiclassical variables, coupled to quantal equations of motion for the remainder.
To be specific, we divide the degrees of freedom into N quasiclassical variables
Q (Qi,Q2, .* QN) and n quantal electronic variables q = (q, q2,..., ). The
Wigner transform is performed over quasiclassical variables only,
w(R,P) (27h)N / dZexp(iP Z/h)(R Z/21IR+ Z/2), (2.14)
Aw(R, P) dZexp(iP Z/h)(R Z/2PR+ Z/2). (2.15)
These are distinguished from fully Wigner transformed operators by virtue of remain
ing operators in the quantal space. By taking the PWT of both sides of the QLE, we
find the equation of motion for the partially Wigner transformed density operator
Sw(R, P),98,99, 100
ih Hw exp(hAf/2i)w Fw exp(hA f/2i)Hw, (2.16)
where
Ac (2.17)
OP OR OR OP
To proceed further, we approximate Eq. 2.16 to O(h A cl),
1Fw I I
S [I^, F] + ({Hw, Fw} {Fw, Hw}), (2.18)
Ot ih 2
where {A, B} is the Poisson bracket,
{A, B} A A ciB. (2.19)
This is an approximation within the quasiclassical space, and reduces quantal motion
of the quasiclassical degrees of freedom, but keeps dependence of the quasiclassical
variables on the quantal state through the interaction potential. It is an appropriate
truncation when the quasiclassical variables are associated with a much greater mass
than the quantal variables.75'101'80'79
For a specific example, consider a Hamiltonian for a molecular system composed
of kinetic, potential and interaction terms,
p2 2
H +V(Q)+ ()+ '(4, + Q), (2.20)
2M 2m
where P is the nuclear momentum operator, V(Q) the nuclear potential, p the elec
tronic momentum, v(q) the electronic potential, and V'(q, R) the electronicnuclear
coupling. By interpreting the nuclear variables Q as quasiclassical and taking the
PWT over the nuclear variables, we find the partially Wigner transformed Hamil
tonian,
2 + V() + v() +V'(,).
(2.21)
Defining
fH +p + ) + V'(4, R),
2m
V V(R) + V'(,R),
(2.22)
(2.23)
we get the quantumclassical Liouville equation of motion for Fw(R, P),
OFw
Ot
[I q p Fw] I+ + (2.24)
ih M 2 OOR OP aP aR
2.5 Effective Potential
The third term on the RHS of Eq. 2.24 presents a challenging obstacle to solving
the equation numerically. One problem common among many proposed schemes is
the requirement of computationally demanding algorithms to evaluate the partial
derivatives of the PWTDOp. To avoid this problem, we introduce an effective
potential V(R, P) in Eq. 2.24,
9Pw 1 P avP w ap 1V OE w
[H i, Fw ] +
Ot ih M OR OR OP
{ ( R aR a _R aR (2.25)
2 OR OR OP OP iR ORf v
The introduction of V becomes computationally useful insofar as we can neglect
the fourth term in Eq. 2.25. While the choice of V is clearly arbitrary, we have found
optimal results by setting the expectation value over quantal variables of (aRVaRv)
to zero,
Trqu 1W ( RV a VI
OR OR
(2.26)
In this way, we retrieve the HellmannFeynman force,
9 Trq, q['wwv/9Rj
0 Trq u[P (2.27)
OR Trq. [fW]
FHF(R,P). (2.28)
The denominator in Eq. 2.27 is itself a function of quasiclassical phase space, which
differs from effective potential approaches involving a single normalized density oper
ator. Neglecting the fourth term in Eq. 2.25, we find an approximated but compu
tationally advantageous effectivepotential QCLE (EPQCLE),
0f1w 1 P 80w 0 aw
Ot i[H F M OR F P
The approximations used to derive the EPQCLE substantially reduce the
quantal character of the quasiclassical solution space. In contrast, the best adia
batic methods, based on the BornOppenheimer separation of nuclear and electronic
motion, retain full quantum nuclear dynamics along adiabatic curves.102 A princi
pal advantage to the EPQCLE is its nonadiabatic character, which is capable of
describing nonadiabatic events when the BornOppenheimer limit no longer applies.
A further benefit is its suitability for solution using trajectory methods, greatly
increasing the size of problems amenable to numerical analysis. The theoretical and
computational aspects of the trajectory solution to the EPQCLE are discussed in
the next chapter.
CHAPTER 3
QUANTUM CLASSICAL LIOUVILLE EQUATION: COMPUTATIONAL
ASPECTS
3.1 Introduction
The EPQCLE is a partial differential operator equation in the quasiclassical
variables and time. One way of solving this kind of problem is to represent the
operators as matrices on a large grid, and evolve the matrices using finite difference
or spectral methods. The drawback to this approach is that a very dense grid is
required for numerical accuracy, and a very large grid is necessary if the quasiclassi
cal density shifts location appreciably as it evolves in time. Since the grid dimension
varies directly with the classical degrees of freedom, a multiparticle system presents
very serious numerical difficulties. Moreover, finite difference and spectral grid solu
tions are inherently difficult to parallelize, as substantial communication is required
between processors regardless of the division of computational labor. An alterna
tive approach, applicable to the class of partial differential equations to which the
EPQCLE belongs, is to follow trajectories in phase space as the system evolves.
Only the important trajectory points are represented, so that the (moving) grid
maintains a minimal size. In this chapter, we explore the trajectory approach and
see how the EPQCLE can be solved in an efficient and even parallel manner.
3.2 Trajectory Solution
We can formally solve Eq. 2.29 by following trajectories in classical phase space,
with R and P becoming functions of time,
dR P
(3.1)
dt M'
dP
d FHF(R,P). (3.2)
dt
One could follow any paths in phase space, but by using those II'... . .1 in Eq. 3.2
we are able to transform Eq. 2.29 into an ordinary differential equation in time.
Inserting Eq. 3.2 in Eq. 2.29 and moving the partial derivatives to the LHS, we
derive the change of the PWTDOp along the quasiclassical trajectories,
dF 1
Sfi[HF] (3.3)
dtih
Note that we have omitted the subscript on the PWTDOp for notational conve
nience, and from here onward will continue to label all PWT operators without this
subscript.
Eq. 3.3 remains a formal solution, and before solving it numerically we must
discretize the equations in both quantal and quasiclassical space. This is the subject
of the next two sections.
3.3 Electronic Basis Set
Let us introduce an arbitrary basis, {
tions, a Gaussian basis is used, but for now we consider the basis to be general, and
not necessarily orthogonal or normalized. Converting to matrix notation, we let 1I4)
be the row matrix,
14) (1 I) 1 ). (3.4)
Then we can expand our operators,
PF I4) Sl (FIIl4)S1 (4,, (3.5)
F
A 4) S1 (4 AI4) S I(4 (3.6)
A
where S is the overlap (44). Projecting Eq. 3.3 on this basis set, and setting
h 1, we obtain
dT
r(iH q _t)S1 Sl(iH' + )F (3.7)
dt
where we have used the notation,
dR dP
S (fl ( //&R4) + (4d/&P4). (3.8)
dt dt
3.4 Nuclear Phase Space Grid
Although we've transformed the EPQCLE into a discrete representation in
electronic space and are now dealing with the partially Wigner transformed density
matrix (PWTDM) instead of operator, we must still discretize the quasiclassical
phase space. To this end, we choose a set of initial grid points {(Ri, Pi)}. Their
distribution should approximately cover the domain of F(R, P), and should be suf
ficiently dense to well represent the evolution of the PWTDM. In practice, the grid
should be adjusted until convergence is achieved.
Once a grid is chosen, the grid points follow nuclear trajectories {(Rj(t), Pj(t))}
according to Eqs. 3.1 and 3.2:
dRj P1
S (3.9)
dt M'
dP
FHF(RJ, P). (3.10)
dt
At the same time, Eq. 3.7 becomes a set of uncoupled equations, one for each
trajectory:
dr F(iHq Q)S S 1 (iH + (3.11)
dt r^H s H (3.11)
where
r r((Rj(t),Pj(t)), (3.12)
Hj H(Rj(t),Pj(t)), (3.13)
Qfj F(R,(t),Pj(t)). (3.14)
Each trajectory follows the evolution of the PWTDM along that path, independent
of the other trajectories.
While one cannot expect coherence between the classical degrees of freedom to
be represented by this approach, there are some substantial computational advan
tages. In particular, the scheme can be optimally ported to a parallel processor,
whereby each processor independently evolves a single trajectory; communication
between the processors is unnecessary.
3.5 RelaxandDrive Algorithm
Before the trajectory solution can be implemented, a detailed propagation
scheme needs to be specified. One would expect that with a sufficiently small
timestep At, all propagation methods would converge to the same results, provided
roundoff error were not significant. However, the paths to convergence will certainly
differ, in that methods with higher accuracy can use larger timesteps. The relax
anddrive method, developed originally by Micha and Runge,66'67'68 incorporates
the rapid electronic oscillatory behavior with the relatively slowly evolving nuclear
variables in an accurate, variable timestep scheme. The relaxanddrive procedure
has been shown to give excellent results for a wide variety of twobody collision
problems. For completeness, we review its details next.
First of all, since all trajectories are propagated analagously, it suffices to look
at the evolution of a single trajectory. Accordingly, let us rewrite the EPQCLE for
a single trajectory,
S =. W(t)F(t) F(t)Wt(t), (3.15)
dt
where we have defined
W S1(H + i+).
(3.16)
We wish to propagate from initial conditions Wo = W(to), Fo = r(to). If W
is independent of time, we find a numerical solution that is exact up to machine
precision. If W depends on time, but is slowly varying with respect to the timescale
of F, we can propagate to a high degree of accuracy by linearizing Eq. 3.15 in time
and incrementing in small timesteps At = tl to.
3.5.1 W Independent of Time
When W is independent of time, W(t) = Wo, we can formally solve Eq. 3.15,
F(t) Uo(t,to)roUt(t, to), (3.17)
where
U(t, to) = exp[iWo(tto)]. (3.18)
If we diagonalize Wo,
Wo = TAT1, (3.19)
we can rewrite Eq. 3.17 as,
F(t) T[TUo(t, to)T]T o(Tt)1[TtUt(t, to)(Tt)1]Tt. (3.20)
The exponential matrices can be formed analytically, since
T1Uot, to)T exp[iA(tto)], (3.21)
and F(t) can be computed in the time complexity of the diagonalization of Wo.
3.5.2 W Dependent on Time
When W depends on time, we separate F into a reference F and correction Q
term,
r(t) r(t) + Q(t).
(3.22)
The reference density is propagated by the time independent Wo of Section 3.5.1,
dro(t)
dt
The evolution of Q(t) is formed by inserting Eq. 3.22 into Eq. 3.15,
(3.23)
.d
i(ro + Q)
dt
(Wo + AW)(r Q)
(r + Q)(Wo + AW)t,
(3.24)
where
AW
W Wo.
(3.25)
Using Eq. 3.23 we obtain
AWro + WoQ + AWQ
FOAWt QWt QAWt. (3.26)
Transforming to the local interaction picture, where
A UoALU ,
UoAWLUoUoFrLU
+UoAWLUoUoQLUt
UoFiUoUoAW Uo
UoQLUoUoAWLUo. (3.28)
We can simplify Eq. 3.28 by multiplying on the left by U 1 and on the right by
(U )1 to obtain
DL + [AWLU UoQL
QLUoUoAWj],
(3.29)
where
AWLUoUoFL FLUoUoAWL.
dQ
dt
we get
iU dQL
iUo Uo
dt 0
(3.27)
dQL
dt
WoF(t) ro(t)Wt0
DL(t)
(3.30)
Formally solving for QL,
Q(t) (t) + dt'[AWL(t')U(t', to)Uo(t', to)QL(t')
to
QL(t')Uo(t', to)Uo(t', to)AWt(t', to)], (3.31)
where
t
Qf (t) f dt'[AWL(t')U(t', to)Uo to)F (t')
to
ro t')U (t', to)Uo(t', to)AW (t', to)]. (3.32)
Solving by iteration, Eq. 3.31 becomes
t
Q t) Qf() f + dt'[AWL(t')Utt',to)Uo(t', to) Qf(t)
to
Qf (t')U (t', to)Uo(t', to)AW (t', to)] (3.33)
Neglecting the second term and higher for small timesteps At = ti to,
tl
QL(tf) dt'[AWL(t')Ut(t', to)Uo(t1', 10) (tl')
to
ro (t')Uto(t', to)Uo(t', to) AW (t', to)]. (3.34)
Converting back to the original representation,
AL = UoA(U)1, (3.35)
we have
t1
Uo(tl, to)Q(t1)Uo(t1, to)1 dt'[Uo l(t, to)AW(t')ro(t')Ut(t', to)1
to
Uo1 (t', to)Fo(t')AW (t')U (t', to)1].
(3.36)
Multiplying Eq. 3.36 on the left by Uo(ti, to) and on the right by Ut(ti, to), and
noting
Uo(t, to)Uo (t', to)
exp[iWo(ti
exp[iWo(ti
Uo(tl, t'),
exp[iWV(t'
exp[iW(ti 
to)] exp[iWo(t'
(3.37)
U (ti, t'),
(3.38)
we get an approximate correction term,
Q(ti)
to
dt'Uog(t t')D(t')U (tl, t'),
to
where
D(t') AW(t')F(t') F(t')AWt(t').
We can compute Q(tl) by quadrature if we assume
D(t) = D(t1/2), to t < ti
where t1/2
to + At/2. We then have
Q(ti)
tl
S dt'Uo(t, t')D(tl/2)U (t t'),
to
(3.39)
(3.40)
(3.41)
(3.42)
Ut (t, to)I u (ti, to)
to)] exp [^iWo (ti
which we can rewrite as
Q(ti)
T Idt'exp[iA(ti
to
t')]DT exp[iAt(ti
T D(tl/2)(Tt)1
Examining the elements of Q,
t')] T k
Sdt'exp[iAo(tl
oi
f dt' exp[
to
A )(ti t')]DL Ttk
1 exp[i(Ai A )At] T
(A m A )k
where we have used the notation Ai Aii. Reverting back to matrix form, we have
Q(ti)
TXTt,
exp[i(A A )At] 1 T
A, (TAm
T D(tl/2) )1.
The full density matrix at tl is then simply,
(3.48)
tl
dtT[T Uo(t, t')T]DT[TU(Tt)1]T
to
where
t')] Tt,
(3.43)
Qjk(tl)
(3.44)
Til
Tm
Im
i
Ti1
l ZTn
i(Al
(3.45)
where
(3.46)
XDm
DT
(3.47)
t')]DI exp[iA (t,
F(ti) r(ti) + Q(ti).
3.5.3 Velocity Verlet for the Classical Evolution
In order to complete the relaxanddrive algorithm, we need to account explicit
for the propagation of the classical variables. We do this by assuming that during
each timestep, R and P are advanced by the reference density F, and that the
correction term contributes negligibly to the classical propagation. The precise
nature of the classical propagation is independent of the relaxanddrive algorithm,
although to keep accuracy to O(At2) it is necessary to integrate using an algorithm
like velocity Verlet or RungeKutta. We choose to use the velocity Verlet method,
which is accurate to O(At2) and is selfstarting. We proceed by advancing the
classical positions,
P (t) 1 dP(t) 2
R(t + At) = R(t) + At + t2. (3.49)
M 2M dt
The last term of Eq. 3.49 is the acceleration term, which comes from the effective
potential. Having advanced the positions, we calculate the acceleration at the new
location, and advance the moment,
P(t + At) P(t) + At. (3.50)
In the context of the relaxanddrive procedure, the classical coordinates are advanced
in two steps of At/2, in order to compute the correction term Q.
Variable timestep. As the propagation proceeds, the initial timestep may no
longer be appropriate. This often occurs when the system enters a region where the
magnitude of potential interactions changes so that the reference density fluctuates
at a different rate. An example is the collision of two atoms, where large timesteps
can be taken at large distances, but small timesteps are required at close range. We
can monitor this fluctuation by observing the correction term Q. As Q becomes
too small (large), we need to increase (decrease) the timestep to maintain efficiency
(accuracy). To this end, we define a correction measurement,
if 2
e i max (3.51)
At the end of each timestep, we evaluate e. If c is less than some threshold, say cl,
we discard the step and use a new c 3 x c. If c is greater than some threshold,
.1,v c, we discard the step and use a new  e/2. By either multiplying by 3 or
dividing by 2, we avoid any oscillation (and thus infinite loop) between a pair of
timesteps.
3.5.4 Algorithm Details
The algorithm can be divided into an outer piece (say, Main), which calls an
inner piece (say, Propagate). They are described in point form as follows:
Main: To propagate F from tA to tB, given At = Ato and {fl, ej},
1. At min{At, tB tA}.
2. Propagate, with to = tA.
3. Compute = n.i:: ; Qj/  l2.
4. Is e < I and tA At < tB?
YES: Reset variables (matrices and functions return to their values at
to), set At  3 x At and return to step 1.
5. Is e > c,?
YES: Reset variables to to, set At + At/2 and return to step 1.
6. IstA At
YES: Set tA ` tA + At, and return to step 1.
Propagate: To propagate F from to to tl = to + At,
1. Initialize variables: Wo = W(to), Fo = ro(to) = r(to), R(to), P(to), At.
2. Diagonalize Wo = TAT.
3. Advance classical variables by half timestep, {R(tl/2), P(tl/2)}, where t1/2
t + At/2, using initial reference density Fo(to) and Wo.
Compute W(tl/2) using R(ti/2) and P(tl/2).
Compute or(tl/2) Texp[iA(At/2)]Tlro(Tt)1 exp[iAt(At/2)]Tt.
Compute Q(ti) by assuming D(t) = D(tl/2), to < t < ti.
(a) Calculate AW(tl/2) = W(t/2) Wo.
(b) Calculate D(tl/2) AW(tl/2)Fr(tl/2) F(tl/2)AWt(tl/2).
(c) Compute DT = T1D(tl/2)(Tt)1
(d) Compute X, where
XIm.
exp[i(A A )At] 
tA DlT
A, Amtn
(e) Compute Q(ti) = TXTt.
7. Advance classical variables to full timestep, {R(ti), P(ti)}, using Fo(tl/2) and
W(ti/2).
8. Compute W(ti) using {R(ti), P(ti)}.
9. Compute Fr(ti) Texp(iAAt)Tlro(Tt)1 exp(iAtAt)Tt.
10. Compute F(ti) F (ti) + Q(ti).
3.6 Computing Observables
3.6.1 Operators in an Orthonormal Basis
Up to this point, we have considered the general basis 4) without any condition
of orthogonality or normality. We can transform this basis to an orthonormal one,
V'), through a L6wdin transformation,
I4)S1/2
(3.52)
Since quantal traces are naturally formulated in orthogonal bases, it is useful to
express the relationship between matrix representations of operators in both bases.
For the density operator,
r I\'}r'{^'
I=)S1/2r'S1/2(4
)  F)r 1. (3.53)
Thus we equate
S S1/2F'S1/2. (3.54)
Since the density matrix has been defined differently than the matrix representations
for general operators, we also consider the representations for a general operator A,
A ')A'({'
= 1)S1/2A'S1/2(
I4)S'AS (). (3.55)
Thus
A S1/2A'S1/2. (3.56)
3.6.2 Population Analysis
The population is naturally defined in an orthonormal basis, such that the
population of state i is the ith diagonal element of the orthonormal representation
of the density matrix,
ri [Sii
[S1/2rsl/2]ii. (3.57)
In the case of the PWT representation we must integrate over nuclear phase space
as well, so that Eq. 3.57 becomes
S= dRdP[S1/2(R, P)F(R, P)S1/2 R, P) (3.58)
3.6.3 Expectation Values
The expectation value of a general operator Aw is found by taking both the
quantal and classical trace of the product of the operator with the PWTDM,
(Aw) = Tr[AwFw]
STrciTrqu[AwFw]
dRdPTrqu[Aww]. (3.59)
The quantal trace is naturally computed in the orthonormal basis, but as we
now show, the nonorthonormal basis representation can also be used:
Trqu[AwFw] = Trqu[A' F']
STrqu[S1/2AwS1/2S1/2FrS1/2]
STrqu[AwFw]. (3.60)
3.6.4 Hamiltonian Eigenstates and Eigenvalues
Using the orthonormal representation of the Hamiltonian,
H' = S1/2HS1/2, (3.61)
we can compute its eigenvalues by diagonalizing the matrix,
L1H'L = H',
(3.62)
where H' is the diagonalized Hamiltonian and contains the energy eigenvalues along
its diagonal. Since the Hamiltonian is Hermitian, its eigenvalues are real, so that
LtH'(L')t = H, (3.63)
and the diagonalizing matrix L is seen to be unitary,
Lt = L. (3.64)
The columns of a unitary matrix are orthogonal, and since the columns of L
are the eigenstates of the Hamiltonian, we see that the eigenstates produced by
diagonalizing the orthonormal representation of the Hamiltonian are also orthonor
mal. This is useful in the case of degenerate eigenstates, as they are automatically
orthogonal and no additional procedures are needed to ensure orthogonality in the
degenerate subspace.
3.7 Programming Details
When designing a computational package, it is desirable to ensure the code
remains orthogonal and extensible throughout the design and implementation. Mod
ern programming languages use objectoriented concepts to achieve these goals,103
but unfortunately a great deal of computational work is built on older procedural
languages and cannot be readily incorporated into an objectoriented scheme with
out substantial effort. Indeed, as much of this legacy code has been developed over
many years and has undergone extensive t. l iii. it is enticing to adhere to the older
languages in which they were written and incorporate them directly. In the devel
opment of code for this research, a compromise was found by using many advanced
features found in Fortran 90, but maintaining a coding style which permit straight
forward integration of legacy Fortran 77 code. Details of the package (cauldron)
are found in Appendix A, and in the remainder of this chapter we briefly overview
the programming principles used in the development of the package.
3.7.1 Orthogonality of Code Development
By orthogonality of code development, we mean that different aspects of the
code can be developed independently. Thus one may decide to build a completely dif
ferent algorithm than relaxanddrive, for example, to propagate the mixed quantum
classical system, but be able to do so without changing aspects of the code which
define the system, compute its properties, read configuration files, generate output
files, and so forth. This helps ensure that once a version of the code works well,
changes to its components will be less likely to introduce errors. This aspect of
program development is crucial for software designed in a team environment, but
also very useful for the solitary designer when the problems and their solutions may
rapidly change. Orthogonality has been kept within cauldron through judicious use
of variable and subroutine naming conventions, and by building a solid hierarchy of
directories and subroutines from the beginning.
3.7.2 Extensibility
In scientific work, the systems studied and the solutions used are constantly
(1I.ii,il,. as progress is made in understanding the solutions, and new problems
arise. One way to help maintain flexibility, which has been used throughout cauldron,
is to ensure that systems are represented as generically and as dynamically as pos
sible. Generic code attempts to represent the fundamental aspects of all molecular
 1. ii, . for example, by the same set of variables and .iili.,. When new com
ponents (e.g., new kinds or numbers of nuclei) are added to the system, the same
variables are used, and it is only the interpretation of the results that differs from
,i1. I, to system. By dynamic representations, we refer to the defining of variable
size at runtime rather than fixing the size at compilation. The major benefit in this
comes from being able to implement models of varying sizes without recompiling
the code and creating a new executable for each system studied. Systems can then
38
be defined completely within input files, for example, preserving the polished exe
cutable without modification. Fortran 90 encourages dynamic memory allocation,
which has been used to great advantage in cauldron to provide very extensible code.
CHAPTER 4
ONEDIMENSIONAL TWOSTATE MODELS
4.1 Introduction
While our ultimate goal is to study realistic threedimensional models of alkali
atoms embedded in rare gas clusters, the complexity of these systems places full
quantal solutions out of computational reach. On the other hand, simple models
can sometimes capture elements of larger and more realistic systems, and provide
a rigorous basis for validating approximate numerical methods. In this chapter,
we study the dynamics of three simple models involving two electronic states and
one nuclear coordinate. The first represents photoinduced desorption of an alkali
atom from a metal surface, where nearresonant electron transfer is important. The
second models the collision between two nuclei in a framework involving two avoided
crossings. The third models the photoinduced dissociation of the Nal complex,
where oscillatory motion between neutral and ionic states is observed. Because of
the limited size of these models, in each case we are able to propagate a grid solution
to the TDSE, and thus compare our EPQCLE approach to the dynamics of the full
quantal system. For all three models, we will see that the mixed quantumclassical
methods provide qualitatively, and often quantitatively similar results to the full
quantal evolution.
4.2 EffectivePotential QCLE in the Diabatic Representation
In Chapter 3, we derived the EPQCLE for an arbitrary basis. Here, we consider
the specific case where the system is described in an orthonormal diabetic basis.
There are many varieties of diabetic bases,104'105 but here we refer to the strictly
diabetic representation {()d} where the momentum coupling vanishes,106
({Qd =/Ol d) = 0. (4.1)
We also assume that the basis does not explicitly depend on P or t, so that Q = 0.
Since the basis is orthonormal, the overlap is unity, and Eq. 3.7 reduces to the simple
form,
dF
S [Hqu, ]. (4.2)
In the diabetic representation, the effective force is also simplified, since the opera
tors in the quantal trace can be replaced directly with their matrix representations,
Trqu [FOV/OR]
FHF (4.3)
Trqu [r]
For our test models, the partial derivatives of the potential can be calculated ana
lytically at each grid point in phase space. Since the PWTDM is propagated along
these grid points, the product in the numerator of Eq. 4.3 is computed through
matrix multiplication, while the quantal trace is calculated by summing over the
diagonal components of this matrix product. The quantal trace in the denominator,
on the other hand, is simply the sum over the diagonal components of the PWTDM.
Had we used a different basis, we would not have been able to simplify the
EPQCLE in this way. However, the real advantage to the diabetic representation
is that for very small systems, it lends itself to a fully quantal numerical solution
through the propagation of the TDSE on a grid. One scheme, the split operator
fast Fourier transform (SOFFT) method, splits the Hamiltonian into its kinetic
and potential components, and uses the fast Fourier transform to compute the evo
lution due to the kinetic terms.107'108 While this method is very accurate, it is also
intractable for systems with more than a few degrees of freedom. However, because
our models are simple, the SOFFT procedure provides an excellent test of the accu
racy of the EPQCLE. A complete description of the SOFFT is given in Appendix
B, and the code implementing the SOFFT (qualdron) is described in Appendix C.
4.3 NearResonant Electron Transfer Between an Alkali Atom and
Metal Surface
4.3.1 Model Details
In the first of our test systems, we consider a model describing the nearresonant
electron transfer between an alkali atom (Ak) and a metal surface (I\) at thermal
energies. The model consists of two diabetic surfaces corresponding to a ground state
of neutral components Ak + M (state 1) and an excited state for ionic components
Ak+ + M (state 2), which cross at short distance. The surfaces and interaction
term are given by,85,84
H11(R) = Uo{exp[2a(R Ro)]+ 2exp[a(R Ro)]} /2, (4.4)
H2(R) Uo{exp[2a(R Ro)] 2exp[o(R Ro)]} + /2, (4.5)
H12(R) = cexp[a2(R 1) 2]. (4.6)
Here, R is the distance between the metal surface and the nuclear center of the Ak
atom, and is the quasiclassical variable over which we take the PWT. The ionic
curve, H22(R), is a Morse potential with a binding energy Uo. The repulsive neutral
curve, H11(R), is offset relative to the ionic curve to give an excitation potential Ac.
The strength of the coupling term, H12(R), is characterized by 3 and peaks at the
crossing R = R, between H11 and H22.
The initial state is formed by approximating the ionic surface as a harmonic
potential around its minimum R = Ro, and finding the lowest bound vibrational
state within that (harmonic) well,
(R) = ( ) exp R2 exp[iPo(R Ro)]. (4.7)
7ff2 $U
Table 41. Parameters used in the N..U1 f...:e and Lisurface models.
Hamiltonian I Hamiltonian II
Parameter value (au) value (au)
Uo 0.025 0.184
a 0.4 0.4
e 0.005 0.147
Ro 5.0 5.0
R, 12.5 9.0
Po 0.0 0.0
a 0.233153 0.1i
f 0.15 0.15
M 42,300 12,800
The PWTDOp is the PWT of Eq. 4.7 over R, giving a Gaussian density in (P,R),
F(P,R) I RoR 2( _Po)2 (4.8)
At t = 0, the electronic state is promoted by a sudden optical excitation to the
repulsive neutral potential, so that the PWTDM becomes
l(P, R) = 1 (R Ro 2 U(P PO P (4.9)
with F12 F21 = 22 0. The simulation follows the spontaneous decay of this
state.
The parameters used in the calculation are shown in Table 41, where we con
sider two model Hamiltonians: (I) N..II[ f.,:e and (II) Lisurface. The diabetic
potentials for Hamiltonians I and II are shown in Figures 41 and 42.
4.3.2 Properties of Interest
Populations. We can follow the populations over time by taking the full
classical trace over either diagonal element of the PWTDM,
r] J = fdRdPri(R, P). (4.10)
(7
43
0.02
H22
22 ro 
0.015 \H12
0.01
0.005
s 7           
H0    " < ^ 
CT 0
C 0.005
LU
0.01 
0.015
0.02 
0.025
0 5 10 15 20 25 30
R (au)
Figure 41. Potential curves for Hamiltonian I: Na incident upon a metal
surface.
44
0.15
H11
\H22
H12
0.1
0.05
5 'i  "
) 0
0.05
0.1
0.15
0 5 10 15 20
R (au)
Figure 42. Potential curves for Hamiltonian II: Li incident upon a metal
surface.
Since the system begins in the repulsive (neutral) state, one expects a certain per
centage of the population to fall to the attractive (ionic) state as the system passes
through the region of nonnegligible potential interaction.
Coherences. The coherence between the neutral and ionic state is described
by the real and imaginary components of the offdiagonal terms of the PWTDM,
for example
i= j dRdPiFj(R, P). (4.11)
Coherence is a purely quantum phenomenon, and one measure of the 11,.,lil of a
mixed quantumclassical method is the degree to which it maintains coherence.
Position expectation values. One observable we can study is the expecta
tion of position,
(R) =Tr[F(R,P)R]
I dRdPTrqu[F(R, P)]R. (4.12)
We can also measure the dispersion,
aR [((R (R))2)]1/2
[f dRdPTrqu[F(R, P)](R (R))2 (4.13)
Momentum expectation values. Similarly, we can compare the expecta
tion and dispersion of moment,
(P) = Tr[F(R, P)P]
f dRdPTrqu[F(R, P)]P, (4.14)
p = [((p (P))2)]1/2
/ dRdPTrqu[F(R, P)](P (P))2 (4.15)
/ 1/
Probability density. We can compute the probability density p(R) from the
PWTDM by taking the trace over quantum variables and moment,
00
p(R) = dPTrq[F(R, P)]. (4.16)
00
In practice, since the grid in phase space quickly deforms as the system evolves, we
must find a way of approximating this integral. One procedure is to determine the
support of the PWTDM in quasiclassical phase space, [Rmin, Rmx] x [Pmin, PFia].
We then divide this space into NR x Np equisized bins {bi}, such that bin bij spans
the rectangular region,
Rn (i 1 Rmax Rmin Rmax Rmin
[Rmin + (i 1) Rmin + i ]
NR NR
x [PTi + (j ) Pmin + J ]. (4.17)
Np Np
We then assign a value to each bin, pij, which is the weighted sum of all Nij trajec
tory points which fall within that bin,
Nij
r F(Rk, Pk)
P 1 (4.18)
We can determine the matrix probability density from pai by summing over all bins
containing a given position R,
p(R) = VpR, VR b. (4.19)
Finally, we compute the probability density by taking the quantal trace over the
matrix probability density,
p(R) = Trqu[p(R)].
(4.20)
This probability density can be compared to the density function obtained from the
SOFFT simulation,
p(R) = I(R) 2 + 2( R)2. (4.21)
Wavefunction and PWTDM. Of course, the evolution of the quantum
wavefunction can be contrasted directly with the evolution of the PWTDM. How
ever, we can also observe the distortion in the phase space grid used by the EPQCLE
method. While the grid is initially uniform, it changes shape in an interesting way
because of the action of the effective potential.
4.3.3 Results
Figures 43 and 44 show the initial wavefunction and its PWT, respectively.
Note that the PWTDM formed from a Gaussian wavepacket is a Gaussian function
itself, albeit in twodimensional phase space. Figure 45 shows the wavefunction at
t = 14000 au, having been propagated through the SOFFT method. The PWTDM
evolves in phase space through the EPQCLE, and in Figure 46 we show the grid in
phase space at the final time. While substantially distorted from its initial uniform
distribution, we notice that the points are globally positioned along a straight line
in phase space. This reflects the .ii. ,,,.1, . i' state, where each point is subject to a
vanishing HellmannFeynman force, and thus propagates at constant velocity.
The observables are presented in Figures 47 to 413. There are a number of
interesting things we can glean from these plots. From Figures 47 and 48 we see
that as the atom moves away from the metal surface, much of its population shifts
from the neutral to ionic state. In the case of Na, approximately 2/3 of the neutral
population shifts, while for Li the transfer is total. This may reflect the stronger
interaction coupling involved with Li.
Figures 49 and 410 describe the coherence between the states. For the
N. ii f.i.e system, the coherence remains large for long times, while for the Lisurface
3 3.5 4 4.5 5 5.5 6 6.5 7
R (au)
Figure 43. T(R) at t = 0 au, for the N..i i f..e model. This wavefunction
is evolved through the SOFFT algorithm.
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
Figure 44. F1n at t = 0 au, for the N..II. f.i.e model. This PWTDM is
evolved through the EPQCLE method.
I I I
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
20
Figure 45. T/(R) at t = 14000 au, for the N..11. f.,.e model.
25 30 35
R (au)
60 65 70 75 80
P (au)
85 90 95
Figure 46.
Phase space grid points at t
model.
14000 au, for the N..II. f.T.e
100
0 2000 4000
6000
8000 10000 12000 14000
time (au)
Figure 47. Na populations p]q and rq2 vs. time.
1 I ii SOFFT: rii
/ EPQCLE: 2
SOFFT: q2
0.8
"0
S0.6
.5 0.4
o
0.2
2000
nm1 .500
0 1000
0 time (au)
Figure 48. Li popul:lltin r and 'l2 vs. time.
0.4
EPQCLE
SOFFT
0.3 
0.2
0.1
0
0.1
0.2
0.3
0.4
0 2000 4000 6000 8000 10000 12000 14000
time (au)
Figure 4 9. Coherence described by Re(r/12) vs. time, for the N.. u1 f.,.:e sys
tem.
0.45 uI I
0.4
0.35
0 0.3
S 0.25
S 0.2
o 0.15
0.1
0.05
0
0.05
0 500 1000 1500 2000
time (au)
Figure 410. Coherence described by Re(lq12) vs. time, for the Lisurface sys
tem.
0 2000
4000
6000 8000 10000 12000 14000
time (au)
Figure 411. Expectation of position and dispersion for the N..i f.,.:e sys
tem.
EPQCLE:
SOFFT:
EPQCLE: Gp
SOFFT: op
0 2000 4000
6000
8000 10000 12000 14000
time (au)
Figure 412. Expectation of momentum and dispersion for the N..i fi .e
system.
58
0.16
EPQCLE
SOFFT
0.14
0.12
0.1
 0.08
0.06
0.04 
0.02 
0
0 5 10 15 20 25 30 35 40
R (au)
Figure 413. Density function p(R) for the N..U1 f[i.:e system.
i,I1 iii. the coherence rapidly diminishes once the interaction potential is crossed.
This can be explained by the size of the energy gap at ..1i.. .1l i' distances, as
larger ionization energies lead to more rapidly vanishing coherences.
Figure 411 shows the expectation of the position of Na steadily increases
from its initial average, while the dispersion in position initially decreases and then
increases again. On the other hand, Figure 412 shows a marked difference in the
behavior of the expectation of the momentum, where it begins at (P) = 0 au, rapidly
increases and then becomes stationary around (P) = 78 au. We also find that the
momentum dispersion first increases, and then decreases.
Finally, Figure 413 presents the density function at the end of the simulation.
Because of the distortion of the phase space grid, the density function obtained
from the EPQCLE had to be calculated using bins and through approximations
within these bins, and thus is subject to some noise. Nevertheless, we find excellent
agreement between the EPQCLE and SOFFT results.
For all observables, the EPQCLE results are quantitatively similar to the SO
FFT results to visual resolution. Having studied both Na and Li approaching a
metal surface, we see that the EPQCLE can be expected to yield very accurate
results for these kinds of systems, even when coherence is maintained over long
periods.
4.4 Binary Collision Involving Two Avoided Crossings
4.4.1 Model Details
For the second system, we look at a twostate collision model where the diabetic
surfaces intersect twice. Because of the dual crossing and the coupling in this region,
quantum interference and effects such as tunnelling play a substantial role in the
dynamics of the quasiclassical variable. As such, this model is quite demanding
for mixed quantumclassical methods, where one can expect deviations at lower
energies.
Table 42. Parameters used in the dual avoided crossing collision model.
Parameter Value (au)
Ro 8
Po [10, 30]
a 2.5176
M 2000
The Hamiltonian elements are79
H1 (R) = 0, (4.22)
H22(R) = 0.1exp(0.28R2) + 0.05, (4.23)
H12(R) = 0.015 exp(0.06R2). (4.24)
In the above, R is the nuclearnuclear separation. The initial state is a ground state
Gaussian wavepacket which begins in the .i.,mptotic region R = 8 au. Its PWT
over R gives the Gaussian density,
ll(P, R) = 1 [ R) 2p 2 (4.25)
with F12 F21 F22 = 0. At t 0, the wavepacket propagates toward the
region of coupling, where it is partially transmitted and partially reflected, now
with populations in both the ground and excited state. The parameters used in the
calculation are shown in Table 42. The diabetic potentials used in the model are
shown in Figure 414.
4.4.2 Properties of Interest
As in the alkali atomsurface model, we can follow the populations over time.
The system begins in the ground state, and as it passes through the collision region,
we expect some of the population to transfer to the excited state. The amount of
transfer depends on the collision energy. We can also follow the coherence once the
H 
H22
H12

10 5 0 5 10
R (au)
Figure 414. Potential curves for the dual avoided crossing collision.
0.08
0.06
0.04
0.02
0
0.02
0.04
0.06
collision has ended and the system reaches ..i mptotic values. We also consider the
transmission of the ground state as the wavepacket passes through the interaction.
Transmission. As the wavepacket passes through the collision, part of it
continues forward while the remainder is reflected back. We can study this by
computing the probability of transmission of the ground state,
0
^ = dRdPTn{RP). (4.26)
Although not included, we could also compute the probability of reflection of the
ground state, as well as the .1 i.l.1 .tic populations of the excited state.
4.4.3 Results
In Figure 415 we see the total population transfer at energy Po = 30 au. In
this energy region, the EPQCLE matches the quantum results to within a cou
ple percent. For both the EPQCLE and SOFFT simulations, the reflection of
the wavefunction (or PWTDM) was negligible, so that we need only compare the
transmission. This negligible reflection was found throughout the energies studied.
In Figure 416 we see the coherence as a function of time. The variations are
large initially and diminish .i ,'iiil' .tically to regular oscillations. The EPQCLE
captures this behavior qualitatively, and to within 5% quantitatively.
We show the deformation of the phase space grid in Figure 417. We see
that it is primarily the inner part of the grid that undergoes deformation, while
the enclosing points maintain a structured order. This is likely due to only a small
range of values in phase space which are seriously affected by the potentials. Outside
the interaction region, all HellmannFeynman forces are zero, so one would expect
deformations only for points whose time in the interaction region was significant.
In Figure 418 we display the transmission probability for a wide range of ener
gies. As expected, the EPQCLE performs better for higher energies, and while the
doublewell is qualitatively reproduced, the EPQCLE fails to give quantitatively
accurate results for energies lower than (log E = 2). This is likely due to the quan
tum tunnelling effects described earlier, which are not expected to be reproduced
well by the EPQCLE. We also compare transmission probabilities obtained through
surface hopping methods by Tully and coworkers.51 These surface hopping calcu
lations show deviations from the quantal results that are similar to the EPQCLE
probabilities at lower energies; at higher energies, the EPQCLE is slightly superior
to the surface hopping scheme.
4.5 Photoinduced Dissociation of a Diatomic System
4.5.1 Model Details
For the third test system, we explore a model of the Nal complex. As in
the previous models, it involves two diabetic surfaces and an interaction around
the avoided crossing. A substantially different feature, however, is a longrange
Coulombic attraction in an ionic state. As we shall see, this attractive potential
results in the the complex oscillating between a neutral and ionic state as the sodium
and iodine separate and come back together, partially dissociating at each crossing
into an .i, ,iii'l.itically neutral state. This oscillatory motion is a good test for the
EPQCLE at long times in cases where ..i mptotic states are not reached quickly.
The Hamiltonian elements .i "''
H11(R) = Aiexp[/3(R Ro)], (4.27)
H22R) [A2 (B2/R)] exp(R/p) /R (+ )/2R4
C2/R6 2+/R7 + AEo, (4.28)
H12(R) A12 exp[ 12(R R)2]. (4.29)
To form the initial state, we Taylor expand (to first order) the ionic potential H22
about its minimum R = Ro, and find the lowest energy state of this harmonic well.
At t = 0, the wavepacket undergoes a sudden optical promotion to the neutral curve,
EPQCLE: rl
SOFFT: l
EPQCLE: 12
SOFFT: 12
0 200 400 600 800 1000 1200 1400
time (au)
Figure 415. Populations qrl and rl2 vs. time for the dual crossing collision
model.
0.1
0.05
0
0.05
0.1
0.15
0.2
0 200 400
Figure 416.
EPQCLE
SOFFT
600 800 1000 1200 1400
time (au)
Coherence described by Re(riq2) vs. time, for the dual crossing
collision model.
ue 41
(rid ,, rMu/ 3) 34
S400 a, for le dualoss
0coli,..i,
67
S" / \ EPQCLE
S SOFFT 
S0.9 Tully et al. 
0.8
0.7
I \\
F
0.6 
0.5
0.4
0.3
4 3.5 3 2.5 2 1.5 1
loge(E) (au)
Figure 418. Probability of transmission in the ground state, for the dual
crossing collision model.
Table 43. Parameters used in the Nal complex model.
Parameter Value (au)
Ro 5.047
Po 0.0
a 0.12462
A1 0 i111 i ',
A2 101.43
A12 0.00202
B2 3.000
C2 18.950
A+ 2.756
A 12.179
p 0.660
AEo 0.07626
/1 2.158
/12 0.194
R, 13.24
M 35,482
so that the PWTDM becomes,
rl(P, R) 1 R RO 2 2(p o)2 (4.30)
with F12 F21 F22 = 0. The simulation follows the resulting motion of this
state. The model parameters are shown in Table 43. The diabetic potentials are
displayed in Figure 419.
4.5.2 Properties of Interest
In addition to observing the coherence, expectation value of the position and its
dispersion, and the phase space grid, we also consider bound and free neutral and
ionic populations as the Nal complex oscillates from its primarily ionic to primarily
covalent state.
Bound and free populations. It is interesting to follow the populations of
the ionic and neutral states as the system evolves, giving insight into the nature of
the dissociation into an ..I.i.l.1 i ..lly free neutral system. For this, we define three
5 10 15 20 25
R (au)
Figure 419. Potential curves for the Nal complex.
0.05
0
0.05
populations: the ionic, the bound neutral and the free neutral. The ionic popula
tion is the probability of finding the system in the ionic state at any internuclear
separation,
00
/2 = dRdP22(R, P). (4.31)
00
We define the bound neutral population is the probability of finding the Nal complex
in the neutral state at nuclear separation up to the crossing point R = R,,
R
f = fdRdPI (R, P), (4.32)
0
while the free ionic population is defined as the probability of the neutral state from
the crossing point beyond,
J{ = dRdPFi(R, P). (4.33)
R
This division between bound and free is motivated by the observation that the
majority of the transfer between ionic and neutral potentials occurs at the avoided
crossing, and that any part of the wavepacket which ends up in the neutral state
but propagating toward infinite separation has a negligible probability of shifting to
the ionic state much beyond the avoided crossing.
4.5.3 Results
The ionic and covalent populations are displayed in Figure 420. We see the
oscillations in the populations between ionic and covalent, repeating approximately
every 40 000 au. This pattern can be compared to the expectation values of position
see in Figure 421. The position oscillates with the same frequency as the change
in population, showing that each time the wavepacket heads across the avoided
crossing from its bound covalent state, it converts almost completely into the ionic
state, with a small amount escaping into the free neutral state. Over time, the
free neutral population gradually increases at these crossings, and the Nal slowly
dissociates. The EPQCLE is quantitatively similar to the exact results for the first
half of the simulation, and maintains qualitative accuracy for the remainder.
The dispersion in Figure 421 shows an interesting difference between the exact
and the quantumclassical algorithm. While the dispersion continues to rise in the
SOFFT simulation, it reaches its first peak and then begins to decline somewhat in
the EPQCLE simulation. This reflects the nature of the effective potential, where
each point is guided by a combination of excited and ground state forces. Because
the ionic curve does not permit escape, what would normally be ..i .mptotically free
wavepackets tend to be pulled back toward the crossing because of the attractive
ionic potential. Consequently, the free neutral population is always lower using the
EPQCLE equation than the SOFFT, an observation supported by Figure 420.
Since the majority of the population remains in the ionic or bound neutral state,
and these populations are well matched between the exact and quantumclassical
simulations, it is not surprising that the expectation value of the position is quan
titatively in the beginning, and qualitatively for the i. is,.iinii.. similar for both
methods. However, the dispersion is much more sensitive to the increased ..Ii.mp
totically free wavepackets in the SOFFT simulation, and for the reasons discussed,
we find significant divergence between the EPQCLE and SOFFT results.
The coherence, shown in Figure 422 initially peaks through the first crossing,
but through subsequent crossings it is substantially diminished. However, the EP
QCLE shows quantitatively similar results to the SOFFT calculations.
The deformation of the phase space grid, plotted in Figure 423, has char
acteristics not seen in the other two models. One line of points emits from the
center of the cluster, quickly straightening and reflecting the negligible force on the
points. These points correspond to the .,i ,,iiilitically free neutral components of
the PWTDM. The second group circles around, gaining velocity and position, then
turning. These ellipses are characteristic of the phase space of classical particles in
a well, and indeed reflect the quasiclassical motion under the HellmannFeynman
force of the PWTDM points as they follow the ionic and bound covalent curves.
4.6 Comparison Using Variable and Constant Timesteps
In this Section, we evaluate the usefulness of the variable timestep aspect of
the relaxanddrive algorithm. To do this, we consider the N..II f.,:e algorithm of
Section 4.3, and simulate using varying upper and lower tolerances. The number
of steps taken in each case is compared with the number that would be required of
the same algorithm, but keeping the timestep fixed. The fixed timestep would nec
essarily advance by steps no greater than the smallest timestep used in the variable
timestep algorithm, and it is based on this timestep that we estimate the corre
sponding steps required for the fixed timestep approach.
The results are shown in Figure 424. We see that as the tolerance decreases
(and thus the accuracy increases), the fraction of steps saved by the introduction of
the variable timestep increases superlinearly. One concludes that while the variable
timestep may not be important for low accuracy simulations, considerable compu
tational savings can be had for high accuracy propagation.
4.7 Conclusion
By examining three simple twostate models, we were able to compare the
EPQCLE method with the exact SOFFT quantum mechanical solution. For all
models, we found very good agreement between the EPQCLE and the SOFFT
results. The agreement was at least qualitative, and in many cases quantitative to
visual precision. We also saw conditions under which the quantumclassical model
deviated from exact quantal results. Finally, we compared a fixed timestep variant
of the relaxanddrive algorithm with the variable timestep version.
For the model of an alkali atom approaching a metal surface, we examined prob
ability transitions, position expectation and its deviation, momentum expectation
73
2
EPQCLE: Ionic
SOFFT: Ionic
EPQCLE: Neutral Bound
SOFFT: Neutral Bound
EPQCLE: Neutral Free
C 1.5 SOFFT: Neutral Free
0
0
0
0z
." 
"oz    
o 0.5 .
   
0 T  T  I I'^ 'T '''
0 20000 40000 60000 80000 100000 120000
time (au)
Figure 420. Ionic and neutral populations over time, for the Nal complex.
74
25
EPQCLE:
SOFFT: :i
EPQCLE: GR
SOEFT: o :
20
15
A
v
10 / ."
5
0 I I I I I I I I
0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000
time (au)
Figure 421. Expectation of position and its deviance, for the Nal complex.
0.35
EPQCLE
SOFFT
0.3
0.25
(1)
0 0.2
rr
o
 0.15
0
t
0.1
0.05 
0
0 20000 40000 60000 80000 100000 120000
time (au)
Figure 422. Coherence as a function of time, for the Nal complex.
140
120
100
80
rr
60
40
20 ..
0
80 60 40 20 0 20 40 60 80
P (au)
Figure 423. Phase space grid at the end of the simulation, for the Nal
complex.
3500
Variable
Fixed x
3000
2500
0
0)
a 2000
E
0
5 1500 
E
1000
500 
103101 105103 10 5 109 107
tolerance
Figure 424. Number of steps required by the relaxanddrive algorithm,
compared to an estimated number required for a fixed timestep
version.
and its deviation, density function and phase space grid evolution. We found that
the EPQCLE method reproduced quantitatively the exact values found through the
quantum model for all observables. We also found the phase space grid to deform
substantially as the system evolved, initially distorting but eventually lining up in
a straight line as the effective force vanished.
For the dual crossing diabetic surface collision, in addition to probability tran
sitions for a given energy, we calculated the transmission probability of the ground
state for a wide range of energies. This showed the EPQCLE to deviate at the lower
energies, where nuclear interference effects were important. At higher energies, the
correspondence between the mixed quantumclassical results and the exact quantum
results was good. We also noted that the phase space grid deformed only for points
which spent a significant amount of time in the region of strong interaction. This is
reasonable, as the deformation comes from the effective force, which is nonvarying
(and in fact zero) outside the interaction.
The Nal model showed interesting backandforth transitions between the ionic
and covalent state, as the internuclear distance oscillated due to the longrange
attractive ionic potential. We also saw that each passing through the avoided cross
ing led to a small amount of dissociation into the ..i1.mptotically free neutral state.
Ionic, bound neutral and free neutral populations were quantitatively similar for the
EPQCLE and SOFFT algorithms for the beginning of the simulations, and quali
tatively similar for the remainder. Coherences were quantitatively similar, showing
an initial peak after the first pass through the avoided crossing, and much smaller
magnitudes thereafter. Finally, the grid deformation showed the grid to split into
two groups. The first represented the .i 1'mptotically free neutral state, where the
points form a straight line as the HellmannFeynman force vanishes and the parti
cles propagate to infinite distances. The second group circled in phase space, as the
79
points oscillated back and forth in the potential well formed by the ionic attraction
at large distances and the neutral repulsion at small distances.
The EPQCLE method was shown to be very robust under a wide variety of
conditions. Comparing its variable timestep version (which was used in all the simu
lations) to the fixed timestep alternative, we found that not only was the EPQCLE
method accurate, but use of the relaxanddrive propagation generated much more
efficient simulations than could have been obtained without dynamically varying the
timestep.
CHAPTER 5
ALKALI ATOMRARE GAS CLUSTERS: GENERAL FORMULATION
5.1 Introduction
In this chapter, we describe the interactions of an alkali atom (Ak) embedded in
a cluster of rare gas atoms (RgN). Cluster dynamics are interesting insofar as they
present a bridge between isolated atoms and the (effectively) infinite bulk liquid, and
can provide insight into the dynamics of molecules on extended surfaces. The Ak
RgN clusters are among the simplest cluster systems to study because of the presence
of a single valence electron in the alkali atom, and the closed shell structure of the
rare gas atoms. Pseudopotential descriptions of the interactions between the valence
electron (e), the Ak core and the Rg atoms can greatly simplify the description of
the cluster with very modest penalties in accuracy. In the context of mixed quantum
classical models, these clusters are well suited to simulation through the EPQCLE
by treating the valence electron quantally and the nuclear cores quasiclassically. For
the majority of the chapter, we consider the general case of an alkali atom embedded
in a rare gas cluster, although our simulations in Chapter 6 focus on the specific
case of the lithium atom (Li) embedded in a helium cluster (HeN).
5.2 Physical System
We consider a cluster initially at thermal equilibrium, and introduce a ground
or excited alkali atom to the center of the cluster. We do not concern ourselves
with the means by which the alkali atom is excited or embedded within the cluster,
although typically the excitation is due to a laser pulse.
From this initial setup, we follow the dynamics of the alkali atom as it deexcites
and moves within the cluster. We further assume the cluster is isolated from any
environmental effects, so that spectral and configuration measurements represent
those of an isolated AkRgN cluster. As we see later, we introduce a containing
potential to keep the Rg atoms from dispersing. This potential is used strictly to
maintain a cluster formation, and does not represent interaction with an environ
ment.
5.3 Properties of Interest
First of all, we are interested in the structure of the Rg cluster in equilibrium,
before the introduction of the Ak atom. We need to reproduce known density profiles
and pairpair correlation functions to ensure the cluster is representative of a real
ll,i, .1i system. Secondly, we are interested in following the dynamics of the Ak
atom as it moves within the cluster. In particular, we are looking for migration from
the center of the cluster to its surface, as other experimental and theoretical studies
indicate that the Ak atom tends to reside on the cluster surface. Finally, we wish
to compute the time dependent emission spectra resulting from the deexcitation of
the Ak atom,
Ak(n'l') +RgN Ak(nl) + RgN + (5.1)
where 0 is a photon with energy corresponding to the electronic decay.
5.4 Hamiltonian for AlkaliRare Gas Pairs
Following the dynamics of an AkRgN cluster through a full quantal treatment
would be an extremely demanding problem computationally for more than a few Rg
atoms. Instead, we reduce the AkRg interaction to a threebody problem by using
pseudopotential interactions between the nuclear cores and the electron. The pseu
dopotential treatment is explored extensively for the single AkRg pair in Reyes.109
In this section, we summarize this approach.
In order to describe the threebody interaction, we consider a fixed nuclear
configuration with RAB the position vector from the alkali core (A) to the rare gas
atom (B), and rA (rB) the position vector from the alkali core (rare gas atom) to
the electron e. With this notation, we can write the Hamiltonian in five distinct
components,
pair 2 r Ak(r)
f = VA A A)+VBgrB)+VA cro A, AB)
+V1ore(RAB). (5.2)
We have implicitly taken the PWT over nuclear variables but not electronic variables.
Because the potentials can all be expressed as polynomial functions of Q, the PWT
amounts to replacing Q by R throughout the Hamiltonian. In what follows, we
will assume we are working with partially Wigner transformed operators, but drop
the subscript 'W' for notational simplicity. Also note that the electronic variable
remains quantal, although we will use the notation r rather than q for consistency
with common usage in the literature.
The first term on the RHS of Eq. 5.2 is, of course, the kinetic energy operator
of the valence electron. The second and third terms are the potentials arising from
the interaction of the electron with the Ak core and the Rg atom, respectively. The
VA'O term is a crossterm stemming from the polarization of the Rg atom by both
the valence electron and Ak core. Finally, the last term Vo" is the interaction
between the Ak core and the Rg atom. We examine each of these potentials in
detail for the general AkRg pair, and provide specific parameters for the LiHe
interaction in Chapter 6.
The eAk core potential can be divided into three components,
V k (AZA ^tAk
k(rAk ) Z VA (A) V+T(rA), (5.3)
TA
where the first term is the Coulomb interaction between the Ak core of charge ZA
and the electron. The second contribution arises from the dipole polarization of the
Ak core by the valence electron,
d
VT^'j A A) A w(A, 6A)2, (5.4)
2rA
with a$ the dipole polarizability of the core and w a cutoff function of the distance
6,
w(r, 6) = exp(6r2). (5.5)
The final contribution to VAk is an idependent shortrange pseudopotential,
r (rx) B1, exp( / )P1x, (5.6)
l,i
where B,,i and /3,i are pseudopotential parameters adjusted to fit experimental data,
Pix is the projection operator on angular symmetry 1,
Pix = Y Yim(rx))(Yim(x)I, (5.7)
and the states { Yim(x))} are the spherical harmonic functions centered on the core
X. This pseudopotential simulates repulsion due to the effects of the Pauli exclusion
principle when the valence electron approaches the core electrons, as well as the
attraction due to the incomplete screening of the nuclear charge.
The third term in Eq. 5.2 is the eRg potential, and can be divided into two
terms,
V""(rB) pol(r) + *(rB). (5.8)
v g^rB VToTM (5.8)
The first component stems from the polarization of the Rg atom by the electron,
d q4
tR(r) (rB, 6)4 (r, 6) (5.9)
VB ()r
where aC a/ 6/1, a' being the quadrupole polarizability of the Rg atom
and 6/1 the dynamical correction to the static polarizability. The shortrange
pseudopotential VJ' is defined in Eq. 5.6.
The fourth term in Eq. 5.2 is a crossterm arising from the polarization of the
Rg atom by both the Ak core and the valence electron,
Ir Pi (cos) q P2 PCos0)
VAo(rA, RAB) ZA 2 2 WB, 6)2 ZAa 2 2 B 6)3. (5.10)
IAB B 1)ABrB
Here, P1 and P2 are the Legendre polynomials and 0 is the angle between the vectors
RAB and rB.
Finally, the last term in Eq. 5.2 is the interaction between the alkali core and
the rare gas atom. It is assumed to involve shortrange, dipole and quadrupole
contributions,
I1 1 '1
VA re(RAB) r V(RAB) 2 B 2 2 BB 3 (5.11)
where ac' = a + 2a' d2. The shortrange term VjI' has been determined by
assuming it to have the form,
VI (RAB) = Aexp(bRAB), (5.12)
and adjusting the parameters A and b so that VAI'e(RAB) in Eq. 5.11 fits the most
accurate curves in the literature. This procedure has been shown to accurately
reproduce AkRg energy curves for a variety of alkali atom and rare gas combina
tions.109
5.5 Hamiltonian for the AlkaliRare Gas Cluster
The cluster is similar to the pair, in that each AkRg pair in the cluster is
treated as a threebody problem involving alkali core, valence electron and rare
gas atom, and described using the same Hamiltonian elements. In the case of the
cluster, however, we also require that the Rg atoms maintain a cohesive structure
rather than drift apart. To ensure this, we impose a constraining potential on the
; 1 I III
Defining RABi as the position vector from the alkali core to the Rg atom i, and
RB, the position vector of the Rg atom from the origin, we can write the cluster
Hamiltonian as,
icluster 1 2 Ak NR B
i=1
2 A (rA) + vY, rB)
N
i=1
N
+ VCore(R RB, ). (5.13)
j>i=1
The final term in Eq. 5.13 is the interaction between the Rg atoms for the
quasiclassical motion. Following Aziz and coworkers,110 we take the RgRg potential
to have the form, for internuclear distance R,
Veore(R) = V*(x), (5.14)
where
V*(x) A* exp(a*x + *2) F(x) 2 2 (5.15)
2 ;+ '6
j=0
F (x) = exp[(D/x 1)2], x < D (5.16)
F(x) (5.16)
1, x >D
This potential has been written in terms of the dimensionless distance x = R/RM.
The constraining potential, Vhold is taken to be a sigmoidal function centered
along the boundary of a sphere of radius Rhold,
hold (RB ) a ex(5.17)
1+ exp [b(RB, Rhold)]'
This potential shows .'.mptotes Vhold(oo) = 0 and Vzold(+oo) = a, with midpoint
at the holding radius Vhod(Rhold) = a/2. The steepness of the function is controlled
by the parameter b, and determines the strength and range of the holding force. It
acts on the Rg atoms only, keeping them bound roughly within a sphere of radius
Rhold, while permitting the alkali atom free motion within the cluster.
5.6 Electronic Spectral Calculations
When a molecular system undergoes electronic motion, the accelerating charges
emit electromagnetic radiation. At distances large compared to the electronic motion,
the flux of this radiation at point r from the center of the system is given by the
Poynting vector,111
1
S(t) (Ex B)
Pto
to 2sin2 2
Ii[t)2 r (5.18)
where D(t) is the dipole moment of the system, [to is the permittivity constant and
c is the speed of light. By integrating over all angles, we obtain the power emited
by the source,
P(t) S(t) da
I A (t)2 (5.19)
where
A(t) (t). (5.20)
From Eq. 5.20, we can see that the computation of the dipole moment is essential
to the spectral calculation. In a mixed quantumclassical system, the dipole moment
is obtained by calculating the expectation value of the dipole operator D,
D(t) Tr[F(t)D]/Tr[P(t)].
(5.21)

Full Text 
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QUANTUM D YNAMICS OF FINITE A TOMIC AND MOLECULAR SYSTEMS THR OUGH DENSITY MA TRIX METHODS By BRIAN THORND YKE A DISSER T A TION PRESENTED TO THE GRADUA TE SCHOOL OF THE UNIVERSITY OF FLORID A IN P AR TIAL FULFILLMENT OF THE REQUIREMENTS F OR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORID A 2004
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Cop yrigh t 2004 b y Brian Thorndyk e
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T o m y father, Gerry Thorndyk e
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A CKNO WLEDGMENTS First of all, I w ould lik e to thank m y paren ts, Gerry and Carol Thorndyk e, whose un w a v ering supp ort and lo v e m y en tire life ha v e allo w ed me to pursue m y dreams. On the academic side, I w ould lik e to thank m y advisor, Dr. Da vid Mic ha, for his excellen t guidance and encouragemen t throughout m y do ctoral w ork. I w ould also lik e to thank the follo wing colleagues in Quan tum Theory Pro ject and the Ph ysics departmen t for their friendship and insigh ts during m y sta y in Gainesville: Jim Co oney Herb ert DaCosta, Alex P ac heco, Da v e Red, Andres Rey es, Akbar Salam, Alb erto San tana and Zhigang Yi. On a p ersonal lev el b ey ond the Ph ysics departmen t, I w ould b e remiss if I didn't express m y lo v e and gratitude to Natasha Lep or e. She has b een m y \soul sister" for o v er a decade, and I hop e our liv es will con tin ue to run with fascinating parallels and in tert wine for man y decades to come! I'd also lik e to recognize Alb ert V ernon who, since our early da ys in the Computer Science departmen t, has b een m y partner in our relen tless pursuit of aesthetically pleasing co de. His friendship has b oth con tributed to some of the b est of times and help ed me through some of the w orst o v er the last 8 y ears. Finally I'd lik e to express m y appreciation to Mik e Kuban and Rob Thorndyk e for b eing with me in spirit throughout m y Ph.D., and for alw a ys pro viding w onderful a v en ues of escap e and adv en ture y ear after y ear! iv
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T ABLE OF CONTENTS P age A CKNO WLEDGMENTS . . . . . . . . . . . . . . iv LIST OF T ABLES . . . . . . . . . . . . . . . . ix LIST OF FIGURES . . . . . . . . . . . . . . . . x ABSTRA CT . . . . . . . . . . . . . . . . . . xiii CHAPTER1 INTR ODUCTION . . . . . . . . . . . . . . . 1 1.1 Ov erview of Classical and Quan tum Dynamics . . . . . 2 1.2 Appro ximations to Quan tum Dynamics . . . . . . . 5 1.2.1 W a v efunctionBased Approac hes . . . . . . . 5 1.2.2 Densit y Op erator Approac hes . . . . . . . . 7 1.3 Quan tumClassical Liouville Equation . . . . . . . . 8 1.4 Our Approac h . . . . . . . . . . . . . . 10 1.5 Simple OneDimensional Tw oState Mo dels . . . . . . 11 1.6 LithiumHelium Clusters . . . . . . . . . . . 11 1.7 Outline of the Dissertation . . . . . . . . . . . 12 2 QUANTUMCLASSICAL LIOUVILLE EQUA TION: F ORMULA TION 14 2.1 In tro duction . . . . . . . . . . . . . . . 14 2.2 Quan tum Liouville Equation . . . . . . . . . . 14 2.3 Wigner Represen tation . . . . . . . . . . . . 15 2.4 Quan tumClassical Liouville Equation . . . . . . . . 18 2.5 Eectiv e P oten tial . . . . . . . . . . . . . 20 3 QUANTUM CLASSICAL LIOUVILLE EQUA TION: COMPUT A TIONAL ASPECTS . . . . . . . . . . . . . . . . 22 3.1 In tro duction . . . . . . . . . . . . . . . 22 3.2 T ra jectory Solution . . . . . . . . . . . . . 22 3.3 Electronic Basis Set . . . . . . . . . . . . . 23 3.4 Nuclear Phase Space Grid . . . . . . . . . . . 24 3.5 RelaxandDriv e Algorithm . . . . . . . . . . . 25 3.5.1 W Indep enden t of Time . . . . . . . . . 26 3.5.2 W Dep enden t on Time . . . . . . . . . . 26 3.5.3 V elo cit y V erlet for the Classical Ev olution . . . . . 31 v
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3.5.4 Algorithm Details . . . . . . . . . . . 32 3.6 Computing Observ ables . . . . . . . . . . . . 33 3.6.1 Op erators in an Orthonormal Basis . . . . . . 33 3.6.2 P opulation Analysis . . . . . . . . . . . 34 3.6.3 Exp ectation V alues . . . . . . . . . . . 35 3.6.4 Hamiltonian Eigenstates and Eigen v alues . . . . . 35 3.7 Programming Details . . . . . . . . . . . . 36 3.7.1 Orthogonalit y of Co de Dev elopmen t . . . . . . 37 3.7.2 Extensibilit y . . . . . . . . . . . . . 37 4 ONEDIMENSIONAL TW OST A TE MODELS . . . . . . . 39 4.1 In tro duction . . . . . . . . . . . . . . . 39 4.2 Eectiv eP oten tial QCLE in the Diabatic Represen tation . . 39 4.3 NearResonan t Electron T ransfer Bet w een an Alk ali A tom and Metal Surface . . . . . . . . . . . . . . 41 4.3.1 Mo del Details . . . . . . . . . . . . 41 4.3.2 Prop erties of In terest . . . . . . . . . . 42 4.3.3 Results . . . . . . . . . . . . . . 47 4.4 Binary Collision In v olving Tw o Av oided Crossings . . . . 59 4.4.1 Mo del Details . . . . . . . . . . . . 59 4.4.2 Prop erties of In terest . . . . . . . . . . 60 4.4.3 Results . . . . . . . . . . . . . . 62 4.5 Photoinduced Disso ciation of a Diatomic System . . . . . 63 4.5.1 Mo del Details . . . . . . . . . . . . 63 4.5.2 Prop erties of In terest . . . . . . . . . . 68 4.5.3 Results . . . . . . . . . . . . . . 70 4.6 Comparison Using V ariable and Constan t Timesteps . . . . 72 4.7 Conclusion . . . . . . . . . . . . . . . 72 5 ALKALI A TOMRARE GAS CLUSTERS: GENERAL F ORMULA TION 80 5.1 In tro duction . . . . . . . . . . . . . . . 80 5.2 Ph ysical System . . . . . . . . . . . . . . 80 5.3 Prop erties of In terest . . . . . . . . . . . . 81 5.4 Hamiltonian for Alk aliRare Gas P airs . . . . . . . 81 5.5 Hamiltonian for the Alk aliRare Gas Cluster . . . . . . 84 5.6 Electronic Sp ectral Calculations . . . . . . . . . 86 5.7 Electronic Basis of Gaussian A tomic F unctions . . . . . 87 5.7.1 Equations of Motion . . . . . . . . . . . 87 5.7.2 Ov erlap Matrix Elemen ts . . . . . . . . . 89 5.7.3 Kinetic Energy Matrix Elemen ts . . . . . . . 90 5.7.4 Coulom b Matrix Elemen ts . . . . . . . . . 90 5.7.5 Momen tum Coupling Matrix Elemen ts . . . . . 91 5.7.6 Dip ole Matrix Elemen ts . . . . . . . . . . 91 5.7.7 Pseudop oten tial Matrix Elemen ts . . . . . . . 92 vi
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5.8 Computing the Quasiclassical T ra jectory . . . . . . . 93 5.9 Computational Details . . . . . . . . . . . . 93 5.10 Conclusion . . . . . . . . . . . . . . . 97 6 LITHIUMHELIUM CLUSTERS . . . . . . . . . . . 99 6.1 In tro duction . . . . . . . . . . . . . . . 99 6.2 Description of the System . . . . . . . . . . . 99 6.3 Prop erties to b e In v estigated . . . . . . . . . . 101 6.4 Preparation of LithiumHelium Clusters . . . . . . . 102 6.4.1 Bulk Helium . . . . . . . . . . . . . 102 6.4.2 Liquid Helium Droplets . . . . . . . . . . 109 6.4.3 LithiumHelium In teractions . . . . . . . . 109 6.5 Results: Lithium Inside the Helium Cluster . . . . . . 119 6.6 Results: Lithium on the Helium Cluster Surface . . . . . 121 6.6.1 Dynamics of Li(2 p ) . . . . . . . . . . 124 6.6.2 Dynamics of Li(2 p ) . . . . . . . . . . 130 6.7 Conclusion . . . . . . . . . . . . . . . 137 7 CONCLUSION . . . . . . . . . . . . . . . . 140 7.1 Eectiv e P oten tial Quan tumClassical Liouville Equation . . 140 7.2 OneDimensional Tw oState Mo dels . . . . . . . . 141 7.3 Alk aliRare Gas Clusters . . . . . . . . . . . 142 7.4 Soft w are Dev elopmen t . . . . . . . . . . . . 144 7.5 F uture W ork . . . . . . . . . . . . . . . 145 APPENDIXA THE CAULDRON PR OGRAM . . . . . . . . . . . . 146 A.1 Ov erview . . . . . . . . . . . . . . . . 146 A.2 Comp onen t Descriptions . . . . . . . . . . . 147 A.2.1 Read Input File . . . . . . . . . . . . 147 A.2.2 System: Get Dieren tial Equation Co ecien ts . . . 147 A.2.3 Propagation: Ev olv e Single Timestep . . . . . . 149 A.2.4 Prop erties: Output Prop erties . . . . . . . . 149 A.3 Subroutine Details . . . . . . . . . . . . . 150 B SPLIT OPERA TORF AST F OURIER TRANSF ORM METHOD . . 151 C THE QUALDRON PR OGRAM . . . . . . . . . . . . 154 C.1 Ov erview . . . . . . . . . . . . . . . . 154 C.2 Comp onen t Descriptions . . . . . . . . . . . 155 C.2.1 Read Input File . . . . . . . . . . . . 155 C.2.2 System: Get Hamiltonian Matrix Elemen ts . . . . 155 C.2.3 Propagation: Ev olv e Single Timestep . . . . . . 155 vii
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C.2.4 Prop erties: Output Prop erties . . . . . . . . 157 C.3 Subroutine Details . . . . . . . . . . . . . 157 REFERENCE LIST . . . . . . . . . . . . . . . . 158 BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . 168 viii
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LIST OF T ABLES T able page 4{1 P arameters used in the Nasurface and Lisurface mo dels. . . . 42 4{2 P arameters used in the dual a v oided crossing collision mo del. . . 60 4{3 P arameters used in the NaI complex mo del. . . . . . . . 68 5{1 Pseudop oten tial rotation for d function mixing. . . . . . . 98 6{1 P arameters for the HeHe in teraction from Aziz ( V Az iz ). . . . . 106 6{2 P arameters for the correction to the HeHe in teraction ( V 0 ) . . . 106 6{3 P arameters for the e Li in teraction. . . . . . . . . . 113 6{4 P arameters for the e He in teraction. . . . . . . . . . 113 6{5 P arameters for the LiHe core in teraction. . . . . . . . . 114 ix
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LIST OF FIGURES Figure page 4{1 P oten tial curv es for Hamiltonian I: Na inciden t up on a metal surface. 43 4{2 P oten tial curv es for Hamiltonian I I: Li inciden t up on a metal surface. 44 4{3 ( R ) at t = 0 au, for the Nasurface mo del. This w a v efunction is ev olv ed through the SOFFT algorithm. . . . . . . . 48 4{4 11 at t = 0 au, for the Nasurface mo del. This PWTDM is ev olv ed through the EPQCLE metho d. . . . . . . . . . 49 4{5 ( R ) at t = 14000 au, for the Nasurface mo del. . . . . . 50 4{6 Phase space grid p oin ts at t = 14000 au, for the Nasurface mo del. 51 4{7 Na p opulations 1 and 2 vs. time. . . . . . . . . . 52 4{8 Li p opulations 1 and 2 vs. time. . . . . . . . . . 53 4{9 Coherence describ ed b y Re( 12 ) vs. time, for the Nasurface system. 54 4{10 Coherence describ ed b y Re( 12 ) vs. time, for the Lisurface system. 55 4{11 Exp ectation of p osition and disp ersion for the Nasurface system. . 56 4{12 Exp ectation of momen tum and disp ersion for the Nasurface system. 57 4{13 Densit y function ( R ) for the Nasurface system. . . . . . 58 4{14 P oten tial curv es for the dual a v oided crossing collision. . . . . 61 4{15 P opulations 1 and 2 vs. time for the dual crossing collision mo del. 64 4{16 Coherence describ ed b y R e ( 12 ) vs. time, for the dual crossing collision mo del. . . . . . . . . . . . . . . . . 65 4{17 Grid deformation at t = 1400 au, for the dual crossing collision mo del. 66 4{18 Probabilit y of transmission in the ground state, for the dual crossing collision mo del. . . . . . . . . . . . . . . 67 4{19 P oten tial curv es for the NaI complex. . . . . . . . . . 69 4{20 Ionic and neutral p opulations o v er time, for the NaI complex. . . 73 x
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4{21 Exp ectation of p osition and its deviance, for the NaI complex. . . 74 4{22 Coherence as a function of time, for the NaI complex. . . . . 75 4{23 Phase space grid at the end of the sim ulation, for the NaI complex. 76 4{24 Num b er of steps required b y the relaxanddriv e algorithm, compared to an estimated n um b er required for a xed timestep v ersion. . 77 6{1 Sc hematic of Li(2 p ) ab o v e a He surface. A) Li(2 p ). B) Li(2 p ). . 101 6{2 Radial distribution functions for bulk liquid helium. . . . . . 106 6{3 Comparison of the Aziz p oten tial with the eectiv e form. . . . 107 6{4 Eectiv e HeHe p oten tial. . . . . . . . . . . . . 108 6{5 Constraining p oten tial used to k eep He atoms from ev ap orating. . 110 6{6 T emp erature ructations of the He droplet o v er time. . . . . . 111 6{7 Helium densit y prole from the cen terofmass of the cluster. . . 112 6{8 Adiabatic energy for Li and He as a function of in tern uclear distance. 114 6{9 Adiabatic energies for Li and one or more He along the zaxis. . . 116 6{10 Adiabatic energy for Li and one or more He along the yaxis. . . 117 6{11 Adiabatic energy for Li and a surface of He atoms parallel to the xy plane. . . . . . . . . . . . . . . . . . 118 6{12 Ev olution of ground state Li em b edded in the cen ter of a He cluster. A) Initial time t = 0 au. B) Final time t = 10,000 au. . . . 120 6{13 Comparison of Li and He motion within a He cluster. The time scale has b een reduced b y a factor of 100 for the He curv e. . . . 122 6{14 Electronic p opulation of Li as it emerges from the He cluster. . . 123 6{15 Ev olution of Li(2 p ) as it recedes from the He cluster surface. A) Initial time t = 0 au. B) Final time t = 33 ; 000 au. . . . . 125 6{16 Mixing of the Li(2 p ) and Li(2 p ) states at distances where Li(2 p ) is triply degenerate. . . . . . . . . . . . . . 126 6{17 Electronic p opulation of Li with and without a p erturbing electromagnetic eld, resonan t to the D line. . . . . . . . . 128 6{18 Dip ole emission sp ectra of Li(2 p ) as it recedes from the He cluster surface. . . . . . . . . . . . . . . . . 129 xi
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6{19 Snapshot of Li(2 p ) as it in teracts with the He cluster surface. A) Initial time t = 0 au. B) Final time t = 67 ; 000 au. . . . . 131 6{20 Electronic p opulation of Li(2 p ) as it in teracts with the He cluster surface. . . . . . . . . . . . . . . . . 132 6{21 Dip ole emission sp ectrum of Li(2 p ) during the rst 3000 au. . . 134 6{22 Dip ole emission sp ectrum of Li(2 p ) during the nal 3000 au. . . 135 6{23 Adiabatic curv es of Li surrounded b y a cubic lattice of He atoms. The parameter R refers to the halflength of the lattice edge. . . . 136 6{24 Deca y of Li(2 p ) surrounded b y surface He atoms, induced b y an EM eld with frequency resonan t to the Li(2 p 2 s ) transition. . 138 A{1 Flo w c hart describing the cauldron program. . . . . . . . 148 C{1 Flo w c hart describing the qualdron program. . . . . . . . 156 xii
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Abstract of Dissertation Presen ted to the Graduate Sc ho ol of the Univ ersit y of Florida in P artial F ulllmen t of the Requiremen ts for the Degree of Do ctor of Philosoph y QUANTUM D YNAMICS OF FINITE A TOMIC AND MOLECULAR SYSTEMS THR OUGH DENSITY MA TRIX METHODS By Brian Thorndyk e Ma y 2004 Chair: Da vid A. Mic ha Ma jor Departmen t: Ph ysics W e dev elop a mixed quan tumclassical form ulation to describ e the dynamics of fewand man yb o dy atomic systems b y applying a partial Wigner transform o v er the quan tum Liouville equation of motion. In this approac h, the densit y op erator b ecomes a function in quasiclassical phase space, while remaining an op erator o v er a subset of quan tal v ariables. By taking appropriate limits and in tro ducing an eectiv e p oten tial, w e deriv e equations of motion describing quasiclassical n uclear tra jectories coupled to quan tal electronic ev olution. W e also in tro duce a v ariable timestep pro cedure to accoun t for the disparit y b et w een slo w n uclear motion and fast electronic ructuations. Our mixed quan tumclassical metho d is applied to the study of three simple onedimensional t w ostate mo dels. The rst mo del represen ts the photoinduced desorption of an alk ali atom from a metal surface, where nearresonan t electron transfer is imp ortan t. A second mo del explores a binary collision under conditions where t w o a v oided crossings are presen t. The third mo del follo ws the photoinduced disso ciation of the so dium io dide complex, whose longrange attractiv e surface results in xiii
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oscillations of in tern uclear distance. Quan tities suc h as state p opulations and quantum coherence are computed, and found to b e in excellen t agreemen t with precise quan tal results obtained through fast F ourier transform grid metho ds. Ha ving v alidated our approac h, w e turn to the study of alk ali atoms em b edded in rare gas clusters, treating the alk ali atomrare gas in teractions with l dep enden t semilo cal pseudop oten tials. Ligh t emission from the electronic motion of the alk ali atom is deriv ed in the semiclassical limit, and computational metho ds to render the sim ulation feasible for a man yatom cluster are discussed. The formalism is applied to lithium atoms in helium clusters, where the cluster conguration and the electronic p opulation dynamics of the lithium atom are monitored o v er time. W e study b oth the ground and rst excited states of lithium, and in tro duce a resonan t electromagnetic eld to induce electronic transitions. Our results correlate w ell with other exp erimen tal and theoretical studies on dop ed helium droplets, and pro vide insigh t in to the dynamics of an excited lithium atom near a helium cluster surface. xiv
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CHAPTER 1 INTR ODUCTION This w ork is part of a broader eort to bring new insigh ts in to the time dep endence of fewand man yb o dy molecular systems. In order to study these systems, w e are particularly in terested in mixed quan tumclassical metho ds. There are man y approac hes to com bining classical and quan tum mec hanics, 1 2 3 but the underlying theme is to use classical mec hanics where quan tal descriptions are not essen tial to the dynamics of the system. By doing so, w e can sa v e a tremendous amoun t of computational time with hop efully minimal exp ense in accuracy In our study w e fo cus on the densit y op erator treatmen t, whic h allo ws for a general in tro duction of semiclassical and classical limits for some degrees of freedom. 4 In the applications b eing considered, electronic v ariables are describ ed quan tally and the n uclear v ariables are propagated quasiclassically but our theoretical treatmen t of quan tumclassical coupling is more general. W e are able to v alidate our metho ds through extensiv e study of small test mo dels. These mo dels can b e ev olv ed through a fully quan tal propagation using fast F ourier grid metho ds, p ermitting a rigorous assessmen t of the accuracy of our approac h. As a realistic application, w e turn to the study of ground and excited alk ali atoms em b edded in clusters of rare gas atoms. 5 6 7 8 9 Rare gas clusters pro vide an in teresting bridge b et w een fewatom systems and bulk matter. A mixed quan tumclassical approac h allo ws us to follo w the n uclear motion and p opulation state dynamics as the alk ali atom in teracts with the cluster. By trac king the electronic and induced dip ole, w e are able to compute the electronic sp ectra of the alk ali atom; and b y explicitly in tro ducing an electromagnetic eld, w e are able to induce electronic transitions. 1
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2 While our formalism is applicable to arbitrary alk alirare gas com binations, w e ha v e concen trated on the dynamics of lithium em b edded in helium clusters, for whic h there has b een a surge of recen t exp erimen tal and theoretical activit y 10 11 12 13 9 14 Stable helium clusters dop ed with lithium atoms ha v e b een pro duced at ultralo w temp eratures, where the ground lithium atom has b een sho wn to preferen tially reside on the surface of the helium cluster. F urthermore, the b eha vior of the lithium atom subsequen t to electronic excitation dep ends hea vily on the orien tation of the excited state. Our mixed quan tumclassical approac h corrob orates these ndings, and leads to additional insigh t in to the dynamics of these in teractions. 1.1 Ov erview of Classical and Quan tum Dynamics The v ast ma jorit y of c hemical and biological pro cesses can b e describ ed, in principle, b y nonrelativistic quan tum mec hanics. Within this con text, the state of a system of n uclei and electrons is represen ted en tirely b y a w a v efunction or densit y op erator. 15 A Hamiltonian op erator describ es all in teractions b et w een the particles, and can b e extended to include en vironmen tal comp onen ts (for example, a b oundary or electromagnetic eld). When the Hamiltonian do es not dep end on time, its eigenstates are stationary (up to a phase), and represen t timein v arian t congurations of the system. When the Hamiltonian con tains time dep endence, or the initial state is nonstationary the molecular system ev olv es through the action of the Hamiltonian. The solutions to the time indep enden t Sc hr odinger equation (TISE) are the eigenstates of the Hamiltonian at an y giv en time. The corresp onding eigen v alues are the energy lev els a v ailable to the system. While conceptually compact, the analytical solution of the TISE is not p ossible for more than t w o particles in the general case. An enormous b o dy of computational w ork in c hemical and molecular ph ysics is dev oted to the n umerical solution of the TISE, 16 17 18 and accurate ground state
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3 energies ha v e b een computed for molecules in v olving h undreds of atoms. 19 Kno wledge of the full sp ectrum of eigenstates w ould allo w one to follo w the dynamics of the system as w ell, but unfortunately it is v ery dicult to obtain accurate results for excited states, and in an y case, the n um b er of states required for accurate computations w ould b e prohibitiv ely exp ensiv e for most dynamics problems. An alternativ e is to follo w the dynamics directly Quan tum dynamics (QD) follo ws the ev olution of a system in real time. The w a v efunction ev olv es according to the time dep enden t Sc hr odinger equation (TDSE), 20 while the densit y op erator (DOp) ev olv es through the quan tum Liouville equation (QLE). 21 These formalisms are equiv alen t, although statistical ensem bles are more naturally describ ed b y the densit y op erator. Quan tum dynamical calculations are imp ortan t in understanding the path w a ys b et w een initial and nal states, for example in c hemical reactions or molecular collisions, and ha v e b ecome an imp ortan t complemen t to mo dern exp erimen ts that use picoor fem tosecond ligh t pulses to prob e ultrafast dynamics. 22 The computational complexit y of n umerical solutions to full QD, ho w ev er, sev erely limits the n um b er of degrees of freedom that can b e studied, and incorp oration of classical concepts is necessary for most realistic applications. F ull QD solutions can b e readily implemen ted b y discretizing the w a v efunction along a m ultidimensional grid. Since the Hamiltonian con tains nonlo cal op erators lik e kinetic energy metho ds suc h as nite dierence 23 24 25 26 or F ourier transform 27 28 are needed to ev aluate the action of the Hamiltonian on the w a v efunction. In the general case, these tec hniques require a dense grid to obtain accurate results. F urthermore, their nonlo cal nature can result in signican t shifts of the w a v efunction o v er time, so that ev en if the initial state is spatially compact, a large grid is required to accommo date translation. Sparse grid metho ds 29 can alleviate some of the computational burden of using large grids, and dynamically c hanging grids 30 can b etter
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4 follo w the distortion and translation of the w a v efunction o v er time. Alternativ ely discretization on a basis set instead of a grid can simplify deriv ativ e calculations. 31 Ultimately ho w ev er, the exp onen tial scaling of the n um b er of grid p oin ts or basis functions with the system size renders full QD solutions in tractable for more than a few degrees of freedom. Classical molecular dynamics (MD) can b e deriv ed from the QLE in the classical limit ( ~ 0). 32 In the con text of molecular sim ulations, the most basic MD treats n uclei as p oin t particles in phase space, and follo ws tra jectories according to the Hamilton equations of motion. 33 In tern uclear p oten tials are deriv ed from ab initio exp erimen tal and empirical results, and the forces on the n uclei are obtained b y summing o v er the partial deriv ativ es of the pair p oten tials. These classical force calculations are no w the b ottlenec k, and straigh tforw ard ev olution of N classical degrees of freedom has only O ( N 2 ) time complexit y 34 T ree metho ds based on o ctree spatial partitions 35 or m ultip ole p oten tial expansions 36 37 reduce this complexit y to O ( N log N ), with a prop ortionalit y constan t dep enden t on the desired accuracy of the sim ulation. Because of its in tuitiv e nature and computational eciency MD is routinely used to study molecular systems with 10 3 {10 6 atoms. 38 The problem with MD is its inabilit y to adequately describ e quan tum mec hanical phenomena, suc h as c harge transfer, electronic excitation, tunneling and zerop oin t motion. These eects are ubiquitous in c hemical and biological pro cesses at thermal or lo w er energies, and cannot b e completely neglected in most cases. 3 An attractiv e alternativ e is to construct an appro ximate QD mo del that ideally com bines the accuracy of QD with the computational eciency and in tuitiv e simplicit y of MD. In general, this can b e done b y augmen ting MD metho ds to include quan tum features, or b y simplifying QD mo dels to incorp orate classical or quasiclassical tra jectories. 1 There are man y v ariations of this theme, and question remains as to whic h approac h is the most suitable under arbitrary conditions. In Section 1.2 w e
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5 surv ey some of the most common appro ximation sc hemes, dieren tiating b et w een metho ds based on the w a v efunction and those cen tered on the densit y op erator. 1.2 Appro ximations to Quan tum Dynamics 1.2.1 W a v efunctionBased Approac hes W a v efunctionbased approac hes are most useful for the propagation of pure states in closed en vironmen ts, and ha v e b een studied theoretically and n umerically since the da wn of quan tum mec hanics. W e limit our surv ey to some principal metho ds that incorp orate some classical concepts to solv e the TSDE, including time dep enden t selfconsisten t eld (TDSCF) calculations, 39 40 41 42 Gaussian w a v epac k et (GWP) propagation, 43 44 surface hopping metho ds, 45 electron n uclear dynamics (END), 46 47 and path in tegral metho ds. 48 49 The time dep enden t selfconsisten t eld appro ximation b egins with the w a v efunction written as a pro duct of n uclear and electronic w a v efunctions, and uses time dep enden t v ariational principles to ev olv e eac h w a v efunction along a p oten tial a v eraged o v er all other w a v efunctions. When the n uclear v ariables are expressed in the classical limit, the TDSCF appro ximation reduces to a set of electronic w a v efunctions ev olving o v er n uclear tra jectories, the tra jectories propagating according to eectiv e forces from the quan tal system. Man y mixed quan tumclassical sc hemes retain this meaneld ra v or, where classical tra jectories propagate along precomputed or sim ultaneously ev olving electronic states. On the other hand, quan tum dynamics has a v ery dieren t c haracter than classical propagation, and mixed quan tumclassical metho ds v ary tremendously in their treatmen t of quan tumclassical in teractions, initial conditions, and measuremen t of system observ ables. Gaussian w a v epac k et propagation, in its original form, tak es the system to b e a Gaussian w a v epac k et in n uclear space, that propagates along a single electronic surface. By lo cally expanding the p oten tial up to harmonic con tributions, equations of motion for the Gaussian w a v epac k et parameters are deriv ed, whic h lead to shifts
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6 and distortions of the Gaussian o v er time. The Gaussian cen ter follo ws classical equations of motion, while its distorting shap e results from quan tal corrections to the classical motion. The GWP metho d has b een extended to describ e nonadiabatic pro cesses with the m ultiple spa wning approac h, 50 where sev eral w a v epac k ets propagate on m ultiple electronic surfaces, and proliferate in to additional w a v epac k ets at crossing p oin ts. Surface hopping b egins with n uclear tra jectories ev olving on one or more electronic surfaces, and repro duces nonadiabatic ev en ts through statistically based jumps b et w een the surfaces. In its earliest v ersion, these transitions tak e place near a v oided crossings, but later generalizations allo w hops to o ccur at an y time during the simulation. 51 One dra wbac k to the surface hopping approac h is the need to rescale v elo cities after sto c hastic transitions, in order to conserv e energy Although accurate state transitions are often ac hiev ed, the dynamics are clearly not represen tativ e of the true system ev olution, except in a statistical sense. Electron n uclear dynamics is an en tirely dieren t approac h, that deriv es equations of motion b y minimizing the Maxw ellSc hr odinger Lagrangian densit y and treating the n uclear degrees of freedom as coheren t states 52 in the semiclassical limit. By expanding the electronic w a v efunction in a basis of Slater determinan ts, n uclear and electronic motion are com bined in a computationally accessible sc heme. P ath in tegral metho ds solv e the TDSE for the quan tum time ev olution op erator, exp ( i ^ H t= ~ ), in the co ordinate represen tation. 53 P ath in tegral equations are equiv alen t to full QD, but unfortunately they are computationally in tractable in the general case. Harmonic appro ximations to the in termolecular p oten tials substan tially reduce the complexit y of the calculations, making it p ossible to repro duce the dynamics of a quan tum system em b edded within an harmonic bath. 54 55 F or example, v ariations suc h as the initial v alue represen tation, 56 57 58 ha v e successfully describ ed the spinb oson mo del of coupled electronic states in a condensed phase
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7 en vironmen t. While not applicable to arbitrary systems, these metho ds are quite useful when the system and its in teractions can b e cast in to the appropriate forms. 1.2.2 Densit y Op erator Approac hes There are a n um b er of adv an tages to using the densit y op erator. First, it pro vides a con v enien t represen tation of mixed rather than pure states. 59 Second, man y systems can b e naturally partitioned in to a primary subsystem of in terest and its surrounding bath. By taking the trace o v er the bath degrees of freedom, the QLE pro vides a reduced densit y description of the primary system in teracting with a bath. 21 59 Finally it is p ossible to include terms directly in the QLE that represen t energy dissipation in to the en vironmen t. 60 F or these reasons, the QLE is often used where initial conditions are sp ecied as statistical a v erages, or when departure from a full quan tum treatmen t is necessary due to the size of the system. Among mixed quan tumclassical solutions to the QLE, an early approac h splits the densit y op erator in to a pro duct of Gaussian w a v epac k ets, and deriv es equations of motions for the Gaussian parameters using selfconsisten t eld appro ximations. 61 In the semiclassical limit, this metho d has similarities to generalized forms of GWP propagation, but b ecause it is based on the densit y matrix, it can naturally treat b oth op en and closed systems. Another metho d is similar in spirit to END, but uses densit y matrices rather than Slater determinan ts to represen t the electronic state. 62 The quan tum degrees of freedom are expanded in some basis set, while the classical degrees of freedom are written as coheren t states. The Lagrangian is minimized through the DiracF renk el v ariational principle, and one arriv es at equations of motion for a reduced densit y matrix, coupled to semiclassical equations of motion for the coheren t state parameters. The densit y matrix ev olution (DME) metho d 63 64 65 divides the system in to a quan tum and classical space. The quan tum subsystem is expanded o v er a nonlo cal,
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8 orthogonal basis set, while the classical co ordinates ev olv e along tra jectories according to the HellmannF eynman force. This metho d has w ork ed w ell for simple mo dels in v olving analytically accessible matrix elemen ts, but is not designed for arbitrary systems with nonlo cal and p ossibly nonorthogonal basis sets. Eik onal metho ds 66 67 68 exploit the limit of small n uclear w a v elengths to separate quan tal and classical motion. Through this formalism, the electronic densit y matrix propagates sim ultaneously with n uclear tra jectories, the latter guided b y eectiv e forces from the quan tum densit y Com bined with v ariable timestep metho ds and tra v elling atomic function basis sets, the eik onal approac h is particularly w ell suited for the study of binary collision problems. A recen t branc h of QLE metho ds uses the Wigner transform (WT) to cast the op eratorbased QLE in to an equation of motion o v er phase space v ariables. The classical Liouville equation for the densit y function emerges in the classical limit, while judicious application of the WT to a subset of the system v ariables leads to w ell dened quan tumclassical sc hemes. In Section 1.3 w e outline some of the ma jor approac hes found in the literature. 1.3 Quan tumClassical Liouville Equation It is fascinating that a phase space represen tation of quan tum mec hanics can b e dev elop ed, that is fully equiv alen t to the Hilb ert space represen tation. By applying the Wigner transform to b oth sides of the QLE, an equation of motion is deriv ed for a Wigner function, whic h is a function of classical p osition and momen tum v ariables. The Wigner function do es not ev olv e through simple classical equations of motion, ho w ev er, but in v olv es nonlo cal op erators in phase space. 4 One class of solutions exploits the similarit y of these equations of motion with h ydro dynamics found in classical ph ysics. Socalled h ydro dynamic quan tum dynamics w as form ulated b y Bohm in 1952, and recen t dev elopmen ts in the eld ha v e revitalized in terest in phase space tra jectory metho ds for solving QD. 69 70 71 72 73 Unfortunately these metho ds,
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9 lik e those in v olving basis set or grid solutions to the TDSE or QLE, suer from exp onen tial gro wth with system size. A mixed quan tumclassical sc heme arises b y dividing the quan tum system in to quan tal and quasiclassical v ariables, the quasiclassical v ariables asso ciated with energies or masses m uc h greater than the quan tal v ariables. T aking the partial Wigner transform (PWT) of the QLE o v er only the quasiclassical v ariables pro duces an equation of motion for a partially transformed Wigner op erator (PTWDOp), whic h is a function of phase space in the quasiclassical v ariables, but remains a quan tal op erator in the space of the quan tal v ariables. After some appropriate appro ximations relating to the dieren t masses of the v ariables, w e arriv e at a dieren tial op erator equation referred to as the quan tumclassical Liouville equation (QCLE). 74 While the quan tal ev olution m ust still b e tac kled in Hilb ert space, the propagation of the quasiclassical phase space acquires a classical c haracter. If the ma jorit y of the system can b e describ ed with quasiclasical v ariables, the computation sa vings ma y b e considerable. One n umerical approac h represen ts the PWTDOp in phase space with a xed n um b er of delta functions. 75 76 77 Sp ecically the PWTDop is pro jected on a basis, resulting in a set of diagonal and odiagonal functions in phase space. Eac h function is appro ximated b y a set of delta p eaks, that ev olv e along surfaces constructed from the densities and coherences. Nonadiabatic coupling is implemen ted through a mo died surface hopping pro cedure, where hops o ccur b et w een tra jectories at curv e crossings. An alternativ e solution uses the same delta p eak represen tation, but rather than use sto c hastic jumps b et w een tra jectories, new tra jectories are generated at curv e crossings. 78 A v ariation on this approac h, the m ultithread metho d, spa wns new tra jectory p oin ts at ev ery timestep, but com bines the newly generated p o ols
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10 of tra jectory p oin ts in to smaller n um b ers through energy and other conserv ation considerations. 79 80 81 Recen t eorts ha v e b een made to com bine Gaussian w a v epac k ets in phase space (GPPs) with the tra jectory solutions along diagonal and odiagonal surfaces. 82 Rather than represen t quasiclassical functions through delta p eaks, the functions are expanded as a linear com bination of GPPs, thereb y more eectiv ely co v ering quasiclassical phase space. Sto c hastic jumps b et w een surfaces at crossing p oin ts repro duce nonadiabatic b eha vior, and a v oid the need to spa wn new GPPs. 1.4 Our Approac h In our approac h, w e in tro duce an eectiv e p oten tial to the QCLE, to pro vide a simple n umerical pro cedure for solving the equation. 83 Our eectiv e p oten tial quan tumclassical Liouville equation (EPQCLE) p ermits solution in terms of full quan tal ev olution of the quan tal v ariables along quasiclassical tra jectories guided b y the quan tal state. W e in tro duce a quan tal basis and a quasiclassical grid to render the equations suitable for n umerical propagation, and implemen t a sc heme to ecien tly accoun t for the dieren t time scales of quasiclassical and quan tal motion. Our approac h shares sev eral p ositiv e features with other QCLE metho ds of solution. The QCLE rigorously com bines quan tum and classical motion, and pro vides systematic w a ys to incorp orate higherorder appro ximations and quan tumclassical coupling. The formalism is based on the densit y op erator, making it attractiv e for incorp orating thermo dynamical features or dissipativ e en vironmen ts. The QCLE also lends itself to tra jectorybased n umerical solutions that reduce to the classical Liouville equation in the classical limit, pro viding an in tuitiv e computational framew ork. Our EPQCLE solution diers from other QCLE primarily in its generalit y and its scalabilit y The eectiv e p oten tial can tak e a v ariet y of forms, and while w e ha v e found the HellmannF eynman force to pro vide the b est results for our mo del
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11 systems, it is p ossible that other forms w ould pro vide ev en b etter results under dieren t circumstances. F urthermore, an electronic basis is only in tro duced once the EPQCLE has b een dev elop ed in terms of partial Wigner op erators, and no assumptions are made on the form of the basis set. This rexibilit y has allo w ed a uniform treatmen t of problems ranging from onedimensional mo dels expanded in a t w ostate diabatic basis, to fully threedimensional atomic clusters in a basis of Gaussian atomic functions. Finally while the ma jorit y of QCLE solutions rely on in teractions b et w een the propagating tra jectories, our use of an eectiv e p oten tial results in completely indep enden t tra jectories whose only connection follo ws from initial conditions. Suc h indep enden t tra jectory ev olution maps optimally to certain parallel arc hitectures, and substan tially reduces the cost of propagation, that w ould otherwise b e required to compute tra jectory in teractions at eac h timestep. 1.5 Simple OneDimensional Tw oState Mo dels W e test our metho ds on a set of onedimensional t w ostate mo dels, whic h are simple enough to b e ev aluated precisely through fast F ourier grid metho ds. Among these metho ds, w e study a mo del of an alk ali atom approac hing a metal surface, where nearresonan t electronic transfer is imp ortan t. 84 85 Secondly w e consider a system represen ting a molecular collision with t w o a v oided crossings, where imp ortan t in terference eects arise as one v aries the collision energy 79 Finally w e study a system represen ting the predisso ciation of the so dium io dide complex, where the longrange attraction of the excited state results in oscillatory n uclear motion. 86 1.6 LithiumHelium Clusters Clusters are aggregates of atoms con taining b et w een three and a few thousand atoms. The smallest clusters, or micr o clusters ha v e b et w een 3 and 10 atoms, and are just small enough that molecular concepts still apply to some degree. The next range of clusters, or smal l clusters ha v e b et w een 10 and 100 atoms, a range where molecular concepts break do wn and the clusters form shells with nonempt y
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12 in teriors. L ar ge clusters b et w een 100 and 1,000 atoms, pro vide the nal transition from isolated molecules to the bulk condensed state. Rare gas clusters can b e prob ed b y doping them with a c hromophore and follo wing the c hromophore through v arious laser detection metho ds. 9 87 88 89 Our study used a lithium atom as the dopan t in a pure helium cluster. Semilo cal l dep enden t pseudop oten tials are used to describ e the lithiumhelium in teractions, b ecause they are found to accurately repro duce adiabatic energy curv es for the lithiumhelium pair for s p and d symmetries. 90 91 92 93 94 By in tro ducing the pseudop oten tial formalism in to the EPQCLE, w e are able to follo w the n uclear conguration o v er time. In addition, w e are able to prob e the ev olving electronic energy surface b y monitoring the electronic and induced dip ole, and computing the resulting dip ole emission sp ectrum. By applying a resonan t electromagnetic eld, w e are able to stim ulate the emission of ligh t b y excited lithium atoms near the cluster surface. Both the n uclear conguration of the cluster and the sp ectral prop erties of its dopan ts ha v e b een under in tense in v estigation o v er the recen t y ears, and our con tribution giv es insigh t in to the dynamics of these in teractions. 1.7 Outline of the Dissertation Chapter 2 presen ts the formalism w e ha v e used to explore the dynamics of mixed quan tumclassical systems. The ideas are presen ted for the general case of quan tal and quasiclassical v ariables, and it is sho wn ho w the EPQCLE naturally emerges from the partial Wigner transform o v er the quasiclassical v ariables. Chapter 3 explores the computational asp ects of the EPQCLE. In particular, the application of an electronic basis and a n uclear grid render the equations of motion suitable to n umerical solution. W e also sho w ho w observ able quan tities can b e calculated within this framew ork. Chapter 4 applies our formalism to three dieren t onedimensional t w ostate systems, where the results can b e compared to exact quan tal sim ulations based on
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13 the Sc hr odinger equation and n umerically precise grid metho ds. Along the w a y v aluable insigh ts in to the accuracy and limitations of our metho ds are obtained. Chapter 5 presen ts the formalism for general alk alirare gas clusters. W e describ e the Hamiltonian for general alk alirare gas in teractions, and deriv e matrix elemen ts for a basis set of Gaussian atomic functions. W e further discuss the dip ole of the cluster and its matrix elemen ts in detail. Finally w e presen t computational metho ds used to render the n umerical propagation of the equations of motion feasible for small to large clusters. Chapter 6 presen ts sp ecic results for lithiumhelium clusters. W e consider the thermo dynamic equilibration of the helium cluster in detail, follo w ed b y the ev olution of the lithium atom through the cluster and near its surface once the cluster has reac hed equilibrium. W e follo w the n uclear motion and discuss the in teraction of the excited lithium atom near surface helium atoms, in particular with regard to the n uclear conguration and the dip ole sp ectrum. Finally w e in tro duce a resonan t electromagnetic eld to stim ulate photon emission after the excited lithium atom em b eds itself within the surface. Chapter 7 summarizes the main conclusions obtained in this dissertation. App endix A discusses the program cauldron dev elop ed o v er the course of this dissertation, to sim ulate the EPQCLE for the test mo del systems and the full lithiumhelium cluster. App endix B deriv es the n umerical ev olution of the Sc hr odinger equation through the split op erator fast F ourier transform metho d. The solution obtained b y this approac h is exact to within n umerical precision, but is computationally exp ensiv e and th us prohibitiv e for more than a few degrees of freedom. App endix C presen ts the qualdron co de, dev elop ed alongside cauldron to compare the results of the test mo dels using mixed quan tumclassical dynamics with the full quan tal treatmen t.
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CHAPTER 2 QUANTUMCLASSICAL LIOUVILLE EQUA TION: F ORMULA TION 2.1 In tro duction Rather than fo cus on the w a v efunction of a molecular system, w e construct its densit y op erator and pro ceed from there. The DOp is more general than the w a v efunction, in that it naturally describ es a statistical ensem ble of quan tum states. This is particularly useful when the molecular system in teracts with its en vironmen t, and one do es not ha v e a complete kno wledge of that en vironmen t. The DOp also pro vides a con v enien t starting p oin t for deriving mixed quan tumclassical metho ds, b ecause the classical limit of the DOp is the classical densit y function. One particular route to a mixed quan tumclassical description of a molecular system is through the the Wigner transform. By applying a partial Wigner transform to the DOp and its equation of motion, one obtains a new represen tation that is a function in the phase space of the Wigner transformed v ariables, but remains an op erator in the remaining v ariables. If the PWT is judiciously applied to classicallik e (or quasiclassic al ) v ariables, then appro ximations can b e made to the equations of motion that pro vide a classical ev olution of the quasiclassical v ariables coupled to a quan tal ev olution of the quan tal v ariables. In the case of molecular systems, the quasiclassical v ariables are t ypically the n uclear co ordinates, while the remaining v ariables describ e the electronic state. In this c hapter, the PWT is describ ed in detail, and its application to molecular systems is outlined. 2.2 Quan tum Liouville Equation A system of atoms or molecules is represen ted b y nonrelativistic quan tum mec hanics as a state v ector j i in Hilb ert space. This state v ector ev olv es in time 14
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15 according to Sc hr odinger's equation, i ~ @ j i @ t = ^ H j i ; (2.1) where ^ H is the Hamiltonian of the full system. If, ho w ev er, w e are studying an ensem ble of states fj i ig with statistical w eigh ts f w i g it is con v enien t to construct the densit y op erator, ^ X i w i j i ih i j : (2.2) T aking its time deriv ativ e and using Eq. 2.1 w e nd the densit y op erator to ev olv e in time according to the quan tum Liouville equation of motion, i ~ @ ^ @ t = [ ^ H ; ^ ] : (2.3) There are a n um b er of adv an tages to using the densit y op erator, but in our case the primary adv an tage is that it leads directly to the classical densit y function in the limit ~ 0. W e rst discuss the full Wigner transform and its application to the QLE, and then sho w ho w the partial Wigner transform can b e used to deriv e a mixed quan tumclassical represen tation in the appropriate limits. 2.3 Wigner Represen tation The Wigner transform pro vides a phase space represen tation of the DOp (the Wigner function ) and other quan tum mec hanical op erators. It is dened as the F ourier transform of an op erator pro jected on co ordinate space, 4 95 96 97 W ( r ; p ) = 1 (2 ~ ) n Z d n z exp ( ip z = ~ ) h r z = 2 j ^ j r + z = 2 i ; (2.4) A W ( r ; p ) = Z d n z exp ( ip z = ~ ) h r z = 2 j ^ A j r + z = 2 i ; (2.5) where ^ A is arbitrary The in tegration generates functions that are lo cal in b oth co ordinate and momen tum space, whic h is imp ortan t for the emergence of classical features in the dev elopmen t of our mixed quan tumclassical metho d. The prefactors
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16 are dened dieren tly for the densit y op erator than for other op erators, in order to pro vide con v enien t parallels b et w een the Wigner function and the classical Liouville densit y Classically the probabilit y distribution function is w ell dened. F or an N b o dy system with co ordinates r ( r 1 ; r 2 ; : : : ; r 3 N ) and momen ta p ( p 1 ; p 2 ; : : : ; p 3 N ), the classical Liouville densit y = ( r ; p ) generates the exp ectation v alue of an y function A = A ( r ; p ), 32 h A i = Z dr dp ( r ; p ) A ( r ; p ) : (2.6) Quan tum mec hanically ev en the notion of phase space is problematic, as Heisenb erg's uncertain t y principle prohibits the sim ultan teous measuremen t of p osition and momen tum for a giv en degree of freedom. Ho w ev er, quasidistribution functions lik e the Wigner function pro vide an analogous form of the quan tum mec hanical exp ectation v alue. The exp ectation v alue of an arbitrary op erator is w ell kno wn, 59 h ^ A i = T r( ^ ^ A ) ; (2.7) whic h can readily b e seen b y expanding the trace o v er eigenstates of ^ A Ho w ev er, b y Wigner transforming b oth ^ and ^ A w e arriv e at a form for the exp ectation v alue similar to the classical case, h ^ A i = Z dr dp W ( r ; p ) A W ( r ; p ) : (2.8)
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17 This can b e seen b y expanding W and A W in Eq. 2.8 Z dr dp W ( r ; p ) A W ( r ; p ) = 1 (2 ~ ) 3 N Z dr dp Z dz exp ( ip z = ~ )( r + z = 2 ; r z = 2) Z dz 0 exp ( ip z 0 = ~ ) A ( r + z 0 = 2 ; r z 0 = 2) = 1 (2 ~ ) 3 N Z dr dpdz dz 0 exp [ ip ( z + z 0 ) = ~ ] ( r + z = 2 ; r z = 2) A ( r + z 0 = 2 ; r z 0 = 2) = Z dr dz ( r + z = 2 ; r z = 2) A ( r z = 2 ; r + z = 2) : (2.9) T ransforming v ariables q = r + z = 2, q 0 = r z = 2, w e retriev e the quan tum exp ectation v alue, Z dr dp W ( r ; p ) A W ( r ; p ) = Z dq dq 0 ( q ; q 0 ) A ( q 0 ; q ) = Z dq dq 0 h q j ^ j q 0 ih q 0 j ^ A j q i = Z dq h q j ^ ^ A j q i = T r( ^ ^ A ) : (2.10) Although W is not a probabilit y distribution, v arious prop erties justify its classication as a quasiprobabilit y function: j ( r ) j 2 = R dp W ( r ; p ). j ~ ( p ) j 2 = R dr W ( r ; p ). F or an y function f ( r ), h f ( r ) i = R dr dp W ( r ; p ) f ( r ) : F or an y function g ( p ), h g ( p ) i = R dr dp W ( r ; p ) g ( p ) : One can also deriv e the WT of pro ducts of op erators in terms of the WT of the op erators themselv es. 4 F or a general op erator ^ F = ^ A ^ B w e nd that F W = A W exp ( ~ = 2 i ) B W ; (2.11)
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18 where is the bidirectional op erator, @ @ p @ @ r @ @ r @ @ p : (2.12) Expanding the comm utator directly w e ha v e [ A; B ] W = ( AB ) W ( B A ) W = A W exp ( ~ = 2 i ) B W B W exp ( ~ = 2 i ) A W : (2.13) 2.4 Quan tumClassical Liouville Equation W e ha v e seen that the densit y op erator ev olv es according to the QLE (Eq. 2.3 ). Rather than study the solution to the QLE, w e will examine the ev olution of the Wigner function. When the densit y op erator is transformed o v er all its v ariables, w e arriv e at equations of motion that, in the limit that ~ 0, reduce to the classical Liouville equation. Here w e are in terested in p erforming a partial Wigner transform of the densit y op erator o v er a subset of v ariables (those termed quasiclassic al ), while lea ving the remaining v ariables (those termed quantal ) in their original represen tation. By taking appropriate limits, w e deriv e equations of motion for the quasiclassical v ariables, coupled to quan tal equations of motion for the remainder. T o b e sp ecic, w e divide the degrees of freedom in to N quasiclassical v ariables Q ( Q 1 ; Q 2 ; : : : ; Q N ) and n quan tal electronic v ariables ^ q = ( ^ q 1 ; ^ q 2 ; : : : ; ^ q n ). The Wigner transform is p erformed o v er quasiclassical v ariables only ^ W ( R ; P ) = (2 ~ ) N Z d Z exp ( iP Z = ~ ) h R Z = 2 j ^ j R + Z = 2 i ; (2.14) ^ A W ( R ; P ) = Z d Z exp ( iP Z = ~ ) h R Z = 2 j ^ j R + Z = 2 i : (2.15) These are distinguished from fully Wigner transformed op erators b y virtue of remaining op erators in the quan tal space. By taking the PWT of b oth sides of the QLE, w e nd the equation of motion for the partially Wigner transformed densit y op erator
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19 ^ W ( R ; P ), 98 99 100 i ~ @ ^ W @ t = ^ H W exp ( ~ cl = 2 i ) ^ W ^ W exp ( ~ cl = 2 i ) ^ H W ; (2.16) where cl @ @ P @ @ R @ @ R @ @ P : (2.17) T o pro ceed further, w e appro ximate Eq. 2.16 to O ( ~ cl ), @ ^ W @ t = 1 i ~ [ ^ H W ; ^ W ] + 1 2 ( f ^ H W ; ^ W g f ^ W ; ^ H W g ) ; (2.18) where f A; B g is the P oisson brac k et, f A; B g A cl B : (2.19) This is an appro ximation within the quasiclassical space, and reduces quan tal motion of the quasiclassical degrees of freedom, but k eeps dep endence of the quasiclassical v ariables on the quan tal state through the in teraction p oten tial. It is an appropriate truncation when the quasiclassical v ariables are asso ciated with a m uc h greater mass than the quan tal v ariables. 75 101 80 79 F or a sp ecic example, consider a Hamiltonian for a molecular system comp osed of kinetic, p oten tial and in teraction terms, ^ H = ^ P 2 2 M + ^ V ( ^ Q ) + ^ p 2 2 m + ^ v ( ^ q ) + ^ V 0 ( ^ q ; ^ Q ) ; (2.20) where ^ P is the n uclear momen tum op erator, ^ V ( ^ Q ) the n uclear p oten tial, ^ p the electronic momen tum, ^ v ( ^ q ) the electronic p oten tial, and ^ V 0 ( ^ q ; R ) the electronicn uclear coupling. By in terpreting the n uclear v ariables ^ Q as quasiclassical and taking the
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20 PWT o v er the n uclear v ariables, w e nd the partially Wigner transformed Hamiltonian, ^ H W = P 2 2 M + V ( R ) + ^ p 2 2 m + ^ v ( ^ q ) + ^ V 0 ( ^ q ; R ) : (2.21) Dening ^ H q ^ p 2 2 m + ^ v ( ^ q ) + ^ V 0 ( ^ q ; R ) ; (2.22) ^ V V ( R ) + ^ V 0 ( ^ q ; R ) ; (2.23) w e get the quan tumclassical Liouville equation of motion for ^ W ( R ; P ), @ ^ W @ t = 1 i ~ [ ^ H q ; ^ W ] P M @ ^ W @ R + 1 2 ( @ ^ V @ R @ ^ W @ P + @ ^ W @ P @ ^ V @ R ) : (2.24) 2.5 Eectiv e P oten tial The third term on the RHS of Eq. 2.24 presen ts a c hallenging obstacle to solving the equation n umerically One problem common among man y prop osed sc hemes is the requiremen t of computationally demanding algorithms to ev aluate the partial deriv ativ es of the PWTDOp. T o a v oid this problem, w e in tro duce an eectiv e p oten tial V ( R ; P ) in Eq. 2.24 @ ^ W @ t = 1 i ~ [ ^ H q ; ^ W ] P M @ ^ W @ R + @ V @ R @ ^ W @ P + 1 2 ( @ ^ V @ R @ V @ R @ ^ W @ P + @ ^ W @ P @ ^ V @ R @ V @ R !) : (2.25) The in tro duction of V b ecomes computationally useful insofar as w e can neglect the fourth term in Eq. 2.25 While the c hoice of V is clearly arbitrary w e ha v e found optimal results b y setting the exp ectation v alue o v er quan tal v ariables of ( @ R ^ V @ R V ) to zero, T r qu ^ W @ ^ V @ R @ V @ R !# = 0 : (2.26)
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21 In this w a y w e retriev e the HellmannF eynman force, @ V @ R = T r qu h ^ W @ ^ V =@ R i T r qu [ ^ W ] (2.27) = F H F ( R ; P ) : (2.28) The denominator in Eq. 2.27 is itself a function of quasiclassical phase space, whic h diers from eectiv e p oten tial approac hes in v olving a single normalized densit y op erator. Neglecting the fourth term in Eq. 2.25 w e nd an appro ximated but computationally adv an tageous eectiv ep oten tial QCLE (EPQCLE), @ ^ W @ t = 1 i ~ [ ^ H q ; ^ W ] P M @ ^ W @ R F H F @ ^ W @ P : (2.29) The appro ximations used to deriv e the EPQCLE substan tially reduce the quan tal c haracter of the quasiclassical solution space. In con trast, the b est adiabatic metho ds, based on the BornOpp enheimer separation of n uclear and electronic motion, retain full quan tum n uclear dynamics along adiabatic curv es. 102 A principal adv an tage to the EPQCLE is its nonadiabatic c haracter, whic h is capable of describing nonadiabatic ev en ts when the BornOpp enheimer limit no longer applies. A further b enet is its suitabilit y for solution using tra jectory metho ds, greatly increasing the size of problems amenable to n umerical analysis. The theoretical and computational asp ects of the tra jectory solution to the EPQCLE are discussed in the next c hapter.
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CHAPTER 3 QUANTUM CLASSICAL LIOUVILLE EQUA TION: COMPUT A TIONAL ASPECTS 3.1 In tro duction The EPQCLE is a partial dieren tial op erator equation in the quasiclassical v ariables and time. One w a y of solving this kind of problem is to represen t the op erators as matrices on a large grid, and ev olv e the matrices using nite dierence or sp ectral metho ds. The dra wbac k to this approac h is that a v ery dense grid is required for n umerical accuracy and a v ery large grid is necessary if the quasiclassical densit y shifts lo cation appreciably as it ev olv es in time. Since the grid dimension v aries directly with the classical degrees of freedom, a m ultiparticle system presen ts v ery serious n umerical diculties. Moreo v er, nite dierence and sp ectral grid solutions are inheren tly dicult to parallelize, as substan tial comm unication is required b et w een pro cessors regardless of the division of computational lab or. An alternativ e approac h, applicable to the class of partial dieren tial equations to whic h the EPQCLE b elongs, is to follo w tra jectories in phase space as the system ev olv es. Only the imp ortan t tra jectory p oin ts are represen ted, so that the (mo ving) grid main tains a minimal size. In this c hapter, w e explore the tra jectory approac h and see ho w the EPQCLE can b e solv ed in an ecien t and ev en parallel manner. 3.2 T ra jectory Solution W e can formally solv e Eq. 2.29 b y follo wing tra jectories in classical phase space, with R and P b ecoming functions of time, dR dt = P M ; (3.1) dP dt = F H F ( R ; P ) : (3.2) 22
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23 One could follo w any paths in phase space, but b y using those suggested in Eq. 3.2 w e are able to transform Eq. 2.29 in to an ordinary dieren tial equation in time. Inserting Eq. 3.2 in Eq. 2.29 and mo ving the partial deriv ativ es to the LHS, w e deriv e the c hange of the PWTDOp along the quasiclassical tra jectories, d ^ dt = 1 i ~ [ ^ H q ; ^ ] : (3.3) Note that w e ha v e omitted the subscript on the PWTDOp for notational con v enience, and from here on w ard will con tin ue to lab el all PWT op erators without this subscript. Eq. 3.3 remains a formal solution, and b efore solving it n umerically w e m ust discretize the equations in b oth quan tal and quasiclassical space. This is the sub ject of the next t w o sections. 3.3 Electronic Basis Set Let us in tro duce an arbitrary basis, fj i ig In man y c hemical ph ysics applications, a Gaussian basis is used, but for no w w e consider the basis to b e general, and not necessarily orthogonal or normalized. Con v erting to matrix notation, w e let j i b e the ro w matrix, j i ( j 1 i j 2 i j 3 i ) : (3.4) Then w e can expand our op erators, ^ = j i S 1 h j ^ j i S 1  {z } h j ; (3.5) ^ A = j i S 1 h j ^ A j i  {z } A S 1 h j ; (3.6) where S is the o v erlap h j i Pro jecting Eq. 3.3 on this basis set, and setting ~ = 1, w e obtain d dt = ( i H q n y ) S 1 S 1 ( i H q + n ) ; (3.7)
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24 where w e ha v e used the notation, n h j @ =@ t j i + dR dt h j @ =@ R j i + dP dt h j @ =@ P j i : (3.8) 3.4 Nuclear Phase Space Grid Although w e'v e transformed the EPQCLE in to a discrete represen tation in electronic space and are no w dealing with the partially Wigner transformed densit y matrix (PWTDM) instead of op erator, w e m ust still discretize the quasiclassical phase space. T o this end, w e c ho ose a set of initial grid p oin ts f ( R i ; P i ) g Their distribution should appro ximately co v er the domain of ( R ; P ), and should b e sufcien tly dense to w ell represen t the ev olution of the PWTDM. In practice, the grid should b e adjusted un til con v ergence is ac hiev ed. Once a grid is c hosen, the grid p oin ts follo w n uclear tra jectories f ( R j ( t ) ; P j ( t )) g according to Eqs. 3.1 and 3.2 : dR j dt = P j M ; (3.9) dP j dt = F H F ( R j ; P j ) : (3.10) A t the same time, Eq. 3.7 b ecomes a set of uncoupled equations, one for eac h tra jectory: d j dt = j ( i H qj n yj ) S 1 j S 1 j ( i H qj + n j ) j ; (3.11) where j ( R j ( t ) ; P j ( t )) ; (3.12) H j H ( R j ( t ) ; P j ( t )) ; (3.13) n j n ( R j ( t ) ; P j ( t )) : (3.14)
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25 Eac h tra jectory follo ws the ev olution of the PWTDM along that p ath indep enden t of the other tra jectories. While one cannot exp ect coherence b et w een the classical degrees of freedom to b e represen ted b y this approac h, there are some substan tial computational adv antages. In particular, the sc heme can b e optimally p orted to a parallel pro cessor, whereb y eac h pro cessor indep enden tly ev olv es a single tra jectory; comm unication b et w een the pro cessors is unnecessary 3.5 RelaxandDriv e Algorithm Before the tra jectory solution can b e implemen ted, a detailed propagation sc heme needs to b e sp ecied. One w ould exp ect that with a sucien tly small timestep t all propagation metho ds w ould con v erge to the same results, pro vided roundo error w ere not signican t. Ho w ev er, the paths to con v ergence will certainly dier, in that metho ds with higher accuracy can use larger timesteps. The relaxanddriv e metho d, dev elop ed originally b y Mic ha and Runge, 66 67 68 incorp orates the rapid electronic oscillatory b eha vior with the relativ ely slo wly ev olving n uclear v ariables in an accurate, v ariable timestep sc heme. The relaxanddriv e pro cedure has b een sho wn to giv e excellen t results for a wide v ariet y of t w ob o dy collision problems. F or completeness, w e review its details next. First of all, since all tra jectories are propagated analagously it suces to lo ok at the ev olution of a single tra jectory Accordingly let us rewrite the EPQCLE for a single tra jectory i d ( t ) dt = W ( t ) ( t ) ( t ) W y ( t ) ; (3.15) where w e ha v e dened W S 1 ( H + i n ) : (3.16)
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26 W e wish to propagate from initial conditions W 0 W ( t 0 ), 0 ( t 0 ). If W is indep enden t of time, w e nd a n umerical solution that is exact up to mac hine precision. If W dep ends on time, but is slo wly v arying with resp ect to the timescale of w e can propagate to a high degree of accuracy b y linearizing Eq. 3.15 in time and incremen ting in small timesteps t = t 1 t 0 3.5.1 W Indep enden t of Time When W is indep enden t of time, W ( t ) = W 0 w e can formally solv e Eq. 3.15 ( t ) = U 0 ( t; t 0 ) 0 U y0 ( t; t 0 ) ; (3.17) where U ( t; t 0 ) exp [ i W 0 ( t t 0 )] : (3.18) If w e diagonalize W 0 W 0 = TT 1 ; (3.19) w e can rewrite Eq. 3.17 as, ( t ) = T [ T 1 U 0 ( t; t 0 ) T ] T 1 0 ( T y ) 1 [ T y U y0 ( t; t 0 )( T y ) 1 ] T y : (3.20) The exp onen tial matrices can b e formed analytically since T 1 U 0 ( t; t 0 ) T = exp [ i ( t t 0 )] ; (3.21) and ( t ) can b e computed in the time complexit y of the diagonalization of W 0 3.5.2 W Dep enden t on Time When W dep ends on time, w e separate in to a reference 0 and correction Q term, ( t ) = 0 ( t ) + Q ( t ) : (3.22)
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27 The reference densit y is propagated b y the time indep enden t W 0 of Section 3.5.1 i d 0 ( t ) dt = W 0 0 ( t ) 0 ( t ) W y 0 : (3.23) The ev olution of Q ( t ) is formed b y inserting Eq. 3.22 in to Eq. 3.15 i d dt ( 0 + Q ) = ( W 0 + W )( 0 + Q ) ( 0 + Q )( W 0 + W ) y ; (3.24) where W = W W 0 : (3.25) Using Eq. 3.23 w e obtain i d Q dt = W 0 + W 0 Q + W Q 0 W y QW y 0 Q W y : (3.26) T ransforming to the lo cal in teraction picture, where A U 0 A L U y0 ; (3.27) w e get i U 0 d Q L dt U y0 = U 0 W L U y0 U 0 0L U y0 U 0 0L U y0 U 0 W y L U y0 + U 0 W L U y0 U 0 Q L U y0 U 0 Q L U y0 U 0 W y L U 0 : (3.28) W e can simplify Eq. 3.28 b y m ultiplying on the left b y U 1 0 and on the righ t b y ( U y0 ) 1 to obtain i d Q L dt = D L + [ W L U y0 U 0 Q L Q L U y0 U 0 W y L ] ; (3.29) where D L ( t ) W L U y0 U 0 0L 0L U y0 U 0 W y L : (3.30)
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28 F ormally solving for Q L Q L ( t ) = Q DL ( t ) + 1 i t Z t 0 dt 0 [ W L ( t 0 ) U y0 ( t 0 ; t 0 ) U 0 ( t 0 ; t 0 ) Q L ( t 0 ) Q L ( t 0 ) U y0 ( t 0 ; t 0 ) U 0 ( t 0 ; t 0 ) W y L ( t 0 ; t 0 )] ; (3.31) where Q DL ( t ) 1 i t Z t 0 dt 0 [ W L ( t 0 ) U y0 ( t 0 ; t 0 ) U 0 ( t 0 ; t 0 ) 0L ( t 0 ) 0L ( t 0 ) U y0 ( t 0 ; t 0 ) U 0 ( t 0 ; t 0 ) W y L ( t 0 ; t 0 )] : (3.32) Solving b y iteration, Eq. 3.31 b ecomes Q L ( t ) = Q DL ( t ) + 1 i t Z t 0 dt 0 [ W L ( t 0 ) U y0 ( t 0 ; t 0 ) U 0 ( t 0 ; t 0 ) Q DL ( t 0 ) Q DL ( t 0 ) U y0 ( t 0 ; t 0 ) U 0 ( t 0 ; t 0 ) W y L ( t 0 ; t 0 )] + : (3.33) Neglecting the second term and higher for small timesteps t = t 1 t 0 Q L ( t 1 ) 1 i t 1 Z t 0 dt 0 [ W L ( t 0 ) U y0 ( t 0 ; t 0 ) U 0 ( t 0 ; t 0 ) 0L ( t 0 ) 0L ( t 0 ) U y0 ( t 0 ; t 0 ) U 0 ( t 0 ; t 0 ) W y L ( t 0 ; t 0 )] : (3.34) Con v erting bac k to the original represen tation, A L = U 1 0 A ( U y0 ) 1 ; (3.35) w e ha v e U 1 0 ( t 1 ; t 0 ) Q ( t 1 ) U y0 ( t 1 ; t 0 ) 1 = 1 i t 1 Z t 0 dt 0 [ U 1 0 ( t 0 ; t 0 ) W ( t 0 ) 0 ( t 0 ) U y0 ( t 0 ; t 0 ) 1 U 1 0 ( t 0 ; t 0 ) 0 ( t 0 ) W y L ( t 0 ) U y0 ( t 0 ; t 0 ) 1 ] : (3.36)
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29 Multiplying Eq. 3.36 on the left b y U 0 ( t 1 ; t 0 ) and on the righ t b y U y0 ( t 1 ; t 0 ), and noting U 0 ( t 1 ; t 0 ) U 1 0 ( t 0 ; t 0 ) = exp [ i W 0 ( t 1 t 0 )] exp [ i W 0 ( t 0 t 0 )] = exp [ i W 0 ( t 1 t 0 )] = U 0 ( t 1 ; t 0 ) ; (3.37) U y0 ( t 0 ; t 0 ) 1 U y0 ( t 1 ; t 0 ) = exp [ i W y 0 ( t 0 t 0 )] exp [ i W y 0 ( t 1 t 0 )] = exp [ i W y 0 ( t 1 t 0 )] = U y0 ( t 1 ; t 0 ) ; (3.38) w e get an appro ximate correction term, Q ( t 1 ) = 1 i t 1 Z t 0 dt 0 U 0 ( t 1 ; t 0 ) D ( t 0 ) U y0 ( t 1 ; t 0 ) ; (3.39) where D ( t 0 ) = W ( t 0 ) 0 ( t 0 ) 0 ( t 0 ) W y ( t 0 ) : (3.40) W e can compute Q ( t 1 ) b y quadrature if w e assume D ( t ) = D ( t 1 = 2 ) ; t 0 t t 1 ; (3.41) where t 1 = 2 = t 0 + t= 2. W e then ha v e Q ( t 1 ) 1 i t 1 Z t 0 dt 0 U 0 ( t 1 ; t 0 ) D ( t 1 = 2 ) U y0 ( t 1 ; t 0 ) ; (3.42)
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30 whic h w e can rewrite as Q ( t 1 ) = 1 i t 1 Z t 0 dt 0 T [ T 1 U 0 ( t 1 ; t 0 ) T ] D T [ T y U y0 ( T y ) 1 ] T y = 1 i T 24 t 1 Z t 0 dt 0 exp [ i ( t 1 t 0 )] D T exp [ i y ( t 1 t 0 )] 35 T y ; (3.43) where D T T 1 D ( t 1 = 2 )( T y ) 1 : (3.44) Examining the elemen ts of Q Q j k ( t 1 ) = X l m 1 i T j l 24 t 1 Z t 0 dt 0 exp [ i l ( t 1 t 0 )] D T l m exp [ i ym ( t 1 t 0 )] 35 T y mk = X l m 1 i T j l 24 t 1 Z t 0 dt 0 exp [ i ( l ym )( t 1 t 0 )] D T l m 35 T y mk = X l m 1 i T j l 1 exp [ i ( l ym ) t ] i ( l ym ) D T l m T y mk ; (3.45) where w e ha v e used the notation i ii Rev erting bac k to matrix form, w e ha v e Q ( t 1 ) = TXT y ; (3.46) where X l m = exp [ i ( l ym ) t ] 1 l ym D T l m ; D T = T 1 D ( t 1 = 2 )( T y ) 1 : (3.47) The full densit y matrix at t 1 is then simply ( t 1 ) = 0 ( t 1 ) + Q ( t 1 ) : (3.48)
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31 3.5.3 V elo cit y V erlet for the Classical Ev olution In order to complete the relaxanddriv e algorithm, w e need to accoun t explicit y for the propagation of the classical v ariables. W e do this b y assuming that during eac h timestep, R and P are adv anced b y the reference densit y 0 and that the correction term con tributes negligibly to the classical propagation. The precise nature of the classical propagation is indep enden t of the relaxanddriv e algorithm, although to k eep accuracy to O ( t 2 ) it is necessary to in tegrate using an algorithm lik e v elo cit y V erlet or RungeKutta. W e c ho ose to use the v elo cit y V erlet metho d, whic h is accurate to O ( t 2 ) and is selfstarting. W e pro ceed b y adv ancing the classical p ositions, R ( t + t ) = R ( t ) + P ( t ) M t + 1 2 M dP ( t ) dt t 2 : (3.49) The last term of Eq. 3.49 is the acceleration term, whic h comes from the eectiv e p oten tial. Ha ving adv anced the p ositions, w e calculate the acceleration at the new lo cation, and adv ance the momen ta, P ( t + t ) = P ( t ) + 1 2 M dP ( t ) dt + dP ( t + t ) dt t: (3.50) In the con text of the relaxanddriv e pro cedure, the classical co ordinates are adv anced in t w o steps of t= 2, in order to compute the correction term Q V ariable timestep. As the propagation pro ceeds, the initial timestep ma y no longer b e appropriate. This often o ccurs when the system en ters a region where the magnitude of p oten tial in teractions c hanges so that the reference densit y ructuates at a dieren t rate. An example is the collision of t w o atoms, where large timesteps can b e tak en at large distances, but small timesteps are required at close range. W e can monitor this ructuation b y observing the correction term Q As Q b ecomes to o small (large), w e need to increase (decrease) the timestep to main tain eciency
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32 (accuracy). T o this end, w e dene a correction measuremen t, max i;j Q ij 0ij 2 : (3.51) A t the end of eac h timestep, w e ev aluate If is less than some threshold, sa y l w e discard the step and use a new 3 If is greater than some threshold, sa y u w e discard the step and use a new = 2. By either m ultiplying b y 3 or dividing b y 2, w e a v oid an y oscillation (and th us innite lo op) b et w een a pair of timesteps.3.5.4 Algorithm Details The algorithm can b e divided in to an outer piece (sa y Main ), whic h calls an inner piece (sa y Propagate ). They are describ ed in p oin t form as follo ws: Main : T o propagate from t A to t B giv en t = t 0 and f l ; u g 1. t = min f t; t B t A g 2. Propagate with t 0 = t A 3. Compute = max i;j j Q ij = ij j 2 4. Is < l and t A + t < t B ? YES: Reset v ariables (matrices and functions return to their v alues at t 0 ), set t 3 t and return to step 1 5. Is > u ? YES: Reset v ariables to t 0 set t t= 2 and return to step 1 6. Is t A + t < t B ? YES: Set t A t A + t and return to step 1 Propagate : T o propagate from t 0 to t 1 = t 0 + t 1. Initialize v ariables: W 0 = W ( t 0 ), 0 = 0 ( t 0 ) = ( t 0 ), R ( t 0 ), P ( t 0 ), t 2. Diagonalize W 0 = T T 1 3. Adv ance classical v ariables b y half timestep, f R ( t 1 = 2 ) ; P ( t 1 = 2 ) g where t 1 = 2 = t + t= 2, using initial reference densit y 0 ( t 0 ) and W 0
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33 4. Compute W ( t 1 = 2 ) using R ( t 1 = 2 ) and P ( t 1 = 2 ). 5. Compute 0 ( t 1 = 2 ) = T exp[ i ( t= 2)] T 1 0 ( T y ) 1 exp [ i y ( t= 2)] T y 6. Compute Q ( t 1 ) b y assuming D ( t ) = D ( t 1 = 2 ), t 0 t t 1 (a) Calculate W ( t 1 = 2 ) = W ( t 1 = 2 ) W 0 (b) Calculate D ( t 1 = 2 ) = W ( t 1 = 2 ) 0 ( t 1 = 2 ) 0 ( t 1 = 2 ) W y ( t 1 = 2 ). (c) Compute D T = T 1 D ( t 1 = 2 )( T y ) 1 (d) Compute X where X l m = exp [ i ( l ym ) t ] 1 l ym D T l m : (e) Compute Q ( t 1 ) = TXT y 7. Adv ance classical v ariables to full timestep, f R ( t 1 ) ; P ( t 1 ) g using 0 ( t 1 = 2 ) and W ( t 1 = 2 ). 8. Compute W ( t 1 ) using f R ( t 1 ) ; P ( t 1 ) g 9. Compute 0 ( t 1 ) = T exp ( i t ) T 1 0 ( T y ) 1 exp ( i y t ) T y 10. Compute ( t 1 ) = 0 ( t 1 ) + Q ( t 1 ). 3.6 Computing Observ ables 3.6.1 Op erators in an Orthonormal Basis Up to this p oin t, w e ha v e considered the general basis j i without an y condition of orthogonalit y or normalit y W e can transform this basis to an orthonormal one, j 0 i through a L owdin transformation, j 0 i = j i S 1 = 2 : (3.52) Since quan tal traces are naturally form ulated in orthogonal bases, it is useful to express the relationship b et w een matrix represen tations of op erators in b oth bases.
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34 F or the densit y op erator, ^ = j 0 i 0 h 0 j = j i S 1 = 2 0 S 1 = 2 h j = j i h j : (3.53) Th us w e equate = S 1 = 2 0 S 1 = 2 : (3.54) Since the densit y matrix has b een dened dieren tly than the matrix represen tations for general op erators, w e also consider the represen tations for a general op erator ^ A ^ A = j 0 i A 0 h 0 j = j i S 1 = 2 A 0 S 1 = 2 h j = j i S 1 AS 1 h j : (3.55) Th us A = S 1 = 2 A 0 S 1 = 2 : (3.56) 3.6.2 P opulation Analysis The p opulation is naturally dened in an orthonormal basis, suc h that the p opulation of state i is the i th diagonal elemen t of the orthonormal represen tation of the densit y matrix, i = 0ii = [ S 1 = 2 S 1 = 2 ] ii : (3.57)
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35 In the case of the PWT represen tation w e m ust in tegrate o v er n uclear phase space as w ell, so that Eq. 3.57 b ecomes i = Z dR dP [ S 1 = 2 ( R ; P ) ( R ; P ) S 1 = 2 ( R ; P )] ii : (3.58) 3.6.3 Exp ectation V alues The exp ectation v alue of a general op erator ^ A W is found b y taking b oth the quan tal and classical trace of the pro duct of the op erator with the PWTDM, h ^ A W i = T r[ ^ A W ^ W ] = T r cl T r qu [ ^ A W ^ W ] = Z dR dP T r qu [ ^ A W ^ W ] : (3.59) The quan tal trace is naturally computed in the orthonormal basis, but as w e no w sho w, the nonorthonormal basis represen tation can also b e used: T r qu [ ^ A W ^ W ] = T r qu [ A 0W 0W ] = T r qu [ S 1 = 2 A W S 1 = 2 S 1 = 2 W S 1 = 2 ] = T r qu [ A W W ] : (3.60) 3.6.4 Hamiltonian Eigenstates and Eigen v alues Using the orthonormal represen tation of the Hamiltonian, H 0 = S 1 = 2 HS 1 = 2 ; (3.61) w e can compute its eigen v alues b y diagonalizing the matrix, L 1 H 0 L = H 0D ; (3.62)
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36 where H 0D is the diagonalized Hamiltonian and con tains the energy eigen v alues along its diagonal. Since the Hamiltonian is Hermitian, its eigen v alues are real, so that L y H 0 ( L 1 ) y = H 0D ; (3.63) and the diagonalizing matrix L is seen to b e unitary L y = L 1 : (3.64) The columns of a unitary matrix are orthogonal, and since the columns of L are the eigenstates of the Hamiltonian, w e see that the eigenstates pro duced b y diagonalizing the orthonormal represen tation of the Hamiltonian are also orthonormal. This is useful in the case of degenerate eigenstates, as they are automatically orthogonal and no additional pro cedures are needed to ensure orthogonalit y in the degenerate subspace. 3.7 Programming Details When designing a computational pac k age, it is desirable to ensure the co de remains orthogonal and extensible throughout the design and implemen tation. Mo dern programming languages use ob jectorien ted concepts to ac hiev e these goals, 103 but unfortunately a great deal of computational w ork is built on older pro cedural languages and cannot b e readily incorp orated in to an ob jectorien ted sc heme without substan tial eort. Indeed, as m uc h of this legacy co de has b een dev elop ed o v er man y y ears and has undergone extensiv e testing, it is en ticing to adhere to the older languages in whic h they w ere written and incorp orate them directly In the dev elopmen t of co de for this researc h, a compromise w as found b y using man y adv anced features found in F ortran 90, but main taining a co ding st yle whic h p ermit straigh tforw ard in tegration of legacy F ortran 77 co de. Details of the pac k age ( cauldron ) are found in App endix A and in the remainder of this c hapter w e briery o v erview the programming principles used in the dev elopmen t of the pac k age.
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37 3.7.1 Orthogonalit y of Co de Dev elopmen t By orthogonalit y of co de dev elopmen t, w e mean that dieren t asp ects of the co de can b e dev elop ed indep enden tly Th us one ma y decide to build a completely differen t algorithm than relaxanddriv e, for example, to propagate the mixed quan tumclassical system, but b e able to do so without c hanging asp ects of the co de whic h dene the system, compute its prop erties, read conguration les, generate output les, and so forth. This helps ensure that once a v ersion of the co de w orks w ell, c hanges to its comp onen ts will b e less lik ely to in tro duce errors. This asp ect of program dev elopmen t is crucial for soft w are designed in a team en vironmen t, but also v ery useful for the solitary designer when the problems and their solutions ma y rapidly c hange. Orthogonalit y has b een k ept within cauldron through judicious use of v ariable and subroutine naming con v en tions, and b y building a solid hierarc h y of directories and subroutines from the b eginning. 3.7.2 Extensibilit y In scien tic w ork, the systems studied and the solutions used are constan tly c hanging, as progress is made in understanding the solutions, and new problems arise. One w a y to help main tain rexibilit y whic h has b een used throughout cauldron is to ensure that systems are represen ted as generically and as dynamically as p ossible. Generic co de attempts to represen t the fundamen tal asp ects of all molecular systems, for example, b y the same set of v ariables and arra ys. When new comp onen ts (e.g., new kinds or n um b ers of n uclei) are added to the system, the same v ariables are used, and it is only the in terpretation of the results that diers from system to system. By dynamic represen tations, w e refer to the dening of v ariable size at run time rather than xing the size at compilation. The ma jor b enet in this comes from b eing able to implemen t mo dels of v arying sizes without recompiling the co de and creating a new executable for eac h system studied. Systems can then
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38 b e dened completely within input les, for example, preserving the p olished executable without mo dication. F ortran 90 encourages dynamic memory allo cation, whic h has b een used to great adv an tage in cauldron to pro vide v ery extensible co de.
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CHAPTER 4 ONEDIMENSIONAL TW OST A TE MODELS 4.1 In tro duction While our ultimate goal is to study realistic threedimensional mo dels of alk ali atoms em b edded in rare gas clusters, the complexit y of these systems places full quan tal solutions out of computational reac h. On the other hand, simple mo dels can sometimes capture elemen ts of larger and more realistic systems, and pro vide a rigorous basis for v alidating appro ximate n umerical metho ds. In this c hapter, w e study the dynamics of three simple mo dels in v olving t w o electronic states and one n uclear co ordinate. The rst represen ts photoinduced desorption of an alk ali atom from a metal surface, where nearresonan t electron transfer is imp ortan t. The second mo dels the collision b et w een t w o n uclei in a framew ork in v olving t w o a v oided crossings. The third mo dels the photoinduced disso ciation of the NaI complex, where oscillatory motion b et w een neutral and ionic states is observ ed. Because of the limited size of these mo dels, in eac h case w e are able to propagate a grid solution to the TDSE, and th us compare our EPQCLE approac h to the dynamics of the full quan tal system. F or all three mo dels, w e will see that the mixed quan tumclassical metho ds pro vide qualitativ ely and often quan titativ ely similar results to the full quan tal ev olution. 4.2 Eectiv eP oten tial QCLE in the Diabatic Represen tation In Chapter 3 w e deriv ed the EPQCLE for an arbitrary basis. Here, w e consider the sp ecic case where the system is describ ed in an orthonormal diab atic basis. There are man y v arieties of diabatic bases, 104 105 but here w e refer to the strictly 39
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40 diab atic represen tation f di g where the momen tum coupling v anishes, 106 h d j @ =@ R j d i = 0 : (4.1) W e also assume that the basis do es not explicitly dep end on P or t so that n = 0. Since the basis is orthonormal, the o v erlap is unit y and Eq. 3.7 reduces to the simple form, d dt = i [ H qu ; ] : (4.2) In the diabatic represen tation, the eectiv e force is also simplied, since the op erators in the quan tal trace can b e replaced directly with their matrix represen tations, F H F = T r qu [ @ V =@ R ] T r qu [ ] : (4.3) F or our test mo dels, the partial deriv ativ es of the p oten tial can b e calculated analytically at eac h grid p oin t in phase space. Since the PWTDM is propagated along these grid p oin ts, the pro duct in the n umerator of Eq. 4.3 is computed through matrix m ultiplication, while the quan tal trace is calculated b y summing o v er the diagonal comp onen ts of this matrix pro duct. The quan tal trace in the denominator, on the other hand, is simply the sum o v er the diagonal comp onen ts of the PWTDM. Had w e used a dieren t basis, w e w ould not ha v e b een able to simplify the EPQCLE in this w a y Ho w ev er, the real adv an tage to the diabatic represen tation is that for v ery small systems, it lends itself to a fully quan tal n umerical solution through the propagation of the TDSE on a grid. One sc heme, the split op erator fast F ourier transform (SOFFT) metho d, splits the Hamiltonian in to its kinetic and p oten tial comp onen ts, and uses the fast F ourier transform to compute the ev olution due to the kinetic terms. 107 108 While this metho d is v ery accurate, it is also in tractable for systems with more than a few degrees of freedom. Ho w ev er, b ecause
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41 our mo dels are simple, the SOFFT pro cedure pro vides an excellen t test of the accuracy of the EPQCLE. A complete description of the SOFFT is giv en in App endix B and the co de implemen ting the SOFFT ( qualdron ) is describ ed in App endix C 4.3 NearResonan t Electron T ransfer Bet w een an Alk ali A tom and Metal Surface 4.3.1 Mo del Details In the rst of our test systems, w e consider a mo del describing the nearresonan t electron transfer b et w een an alk ali atom (Ak) and a metal surface (M) at thermal energies. The mo del consists of t w o diabatic surfaces corresp onding to a ground state of neutral comp onen ts Ak + M (state 1) and an excited state for ionic comp onen ts Ak + + M (state 2), whic h cross at short distance. The surfaces and in teraction term are giv en b y 85 84 H 11 ( R ) = U 0 f exp [ 2 ( R R 0 )] + 2 exp[ ( R R 0 )] g = 2 ; (4.4) H 22 ( R ) = U 0 f exp [ 2 ( R R 0 )] 2 exp[ ( R R 0 )] g + = 2 ; (4.5) H 12 ( R ) = exp[ 2 ( R R x ) 2 ] : (4.6) Here, R is the distance b et w een the metal surface and the n uclear cen ter of the Ak atom, and is the quasiclassical v ariable o v er whic h w e tak e the PWT. The ionic curv e, H 22 ( R ), is a Morse p oten tial with a binding energy U 0 The repulsiv e neutral curv e, H 11 ( R ), is oset relativ e to the ionic curv e to giv e an excitation p oten tial The strength of the coupling term, H 12 ( R ), is c haracterized b y and p eaks at the crossing R = R x b et w een H 11 and H 22 The initial state is formed b y appro ximating the ionic surface as a harmonic p oten tial around its minim um R = R 0 and nding the lo w est b ound vibrational state within that (harmonic) w ell, ( R ) = 1 2 1 = 4 exp ( R R 0 ) 2 2 2 exp [ iP 0 ( R R 0 )] : (4.7)
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42 T able 4{1. P arameters used in the Nasurface and Lisurface mo dels. Hamiltonian I Hamiltonian I I P arameter v alue (au) v alue (au) U 0 0.025 0. 184 0.4 0. 4 0.005 0. 147 R 0 5.0 5. 0 R x 12.5 9. 0 P 0 0.0 0. 0 0.233153 0. 1908559 0.15 0. 15 M 42,300 12,800 The PWTDOp is the PWT of Eq. 4.7 o v er R giving a Gaussian densit y in (P ,R), ^ ( P ; R ) = 1 R R 0 2 2 ( P P 0 ) 2 # : (4.8) A t t = 0, the electronic state is promoted b y a sudden optical excitation to the repulsiv e neutral p oten tial, so that the PWTDM b ecomes 11 ( P ; R ) = 1 R R 0 2 2 ( P P 0 ) 2 # ; (4.9) with 12 = 21 = 22 = 0. The sim ulation follo ws the sp on taneous deca y of this state. The parameters used in the calculation are sho wn in T able 4{1 where w e consider t w o mo del Hamiltonians: (I) Nasurface and (I I) Lisurface. The diabatic p oten tials for Hamiltonians I and I I are sho wn in Figures 4{1 and 4{2 4.3.2 Prop erties of In terest P opulations. W e can follo w the p opulations o v er time b y taking the full classical trace o v er either diagonal elemen t of the PWTDM, i = Z dR dP ii ( R ; P ) : (4.10)
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43 0.025 0.02 0.015 0.01 0.005 0 0.005 0.01 0.015 0.02 0 5 10 15 20 25 30 Energy (au)R (au) H 11 H 22 H 12 Figure 4{1. P oten tial curv es for Hamiltonian I: Na inciden t up on a metal surface.
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44 0.15 0.1 0.05 0 0.05 0.1 0.15 0 5 10 15 20 Energy (au)R (au) H 11 H 22 H 12 Figure 4{2. P oten tial curv es for Hamiltonian I I: Li inciden t up on a metal surface.
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45 Since the system b egins in the repulsiv e (neutral) state, one exp ects a certain p ercen tage of the p opulation to fall to the attractiv e (ionic) state as the system passes through the region of nonnegligible p oten tial in teraction. Coherences. The coherence b et w een the neutral and ionic state is describ ed b y the real and imaginary comp onen ts of the odiagonal terms of the PWTDM, for example ij = Z dR dP ij ( R ; P ) : (4.11) Coherence is a purely quan tum phenomenon, and one measure of the qualit y of a mixed quan tumclassical metho d is the degree to whic h it main tains coherence. P osition exp ectation v alues. One observ able w e can study is the exp ectation of p osition, h R i = T r [( R ; P ) R ] = Z dR dP T r qu [ ( R ; P )] R : (4.12) W e can also measure the disp ersion, R = [ h ( R h R i ) 2 i ] 1 = 2 = Z dR dP T r qu [ ( R ; P )]( R h R i ) 2 1 = 2 : (4.13) Momen tum exp ectation v alues. Similarly w e can compare the exp ectation and disp ersion of momen ta, h P i = T r[( R ; P ) P ] = Z dR dP T r qu [ ( R ; P )] P ; (4.14) P = [ h ( P h P i ) 2 i ] 1 = 2 = Z dR dP T r qu [ ( R ; P )]( P h P i ) 2 1 = 2 : (4.15)
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46 Probabilit y densit y W e can compute the probabilit y densit y ( R ) from the PWTDM b y taking the trace o v er quan tum v ariables and momen ta, ( R ) = 1 Z 1 dP T r qu [ ( R ; P )] : (4.16) In practice, since the grid in phase space quic kly deforms as the system ev olv es, w e m ust nd a w a y of appro ximating this in tegral. One pro cedure is to determine the supp ort of the PWTDM in quasiclassical phase space, [ R min ; R max ] [ P min ; P max ]. W e then divide this space in to N R N P equisized bins f b ij g suc h that bin b ij spans the rectangular region, [ R min + ( i 1) R max R min N R ; R min + i R max R min N R ] [ P min + ( j 1) P max P min N P ; P min + j P max P min N P ] : (4.17) W e then assign a v alue to eac h bin, ij whic h is the w eigh ted sum of all N ij tra jectory p oin ts whic h fall within that bin, ij = N ij P k =1 k ( R k ; P k ) N ij : (4.18) W e can determine the matrix probabilit y densit y from ij b y summing o v er all bins con taining a giv en p osition R ( R ) = X j ij ; 8 R 2 b ij : (4.19) Finally w e compute the probabilit y densit y b y taking the quan tal trace o v er the matrix probabilit y densit y ( R ) = T r qu [ ( R )] : (4.20)
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47 This probabilit y densit y can b e compared to the densit y function obtained from the SOFFT sim ulation, ( R ) = j 1 ( R ) j 2 + j 2 ( R ) j 2 : (4.21) W a v efunction and PWTDM. Of course, the ev olution of the quan tum w a v efunction can b e con trasted directly with the ev olution of the PWTDM. Ho wev er, w e can also observ e the distortion in the phase space grid used b y the EPQCLE metho d. While the grid is initially uniform, it c hanges shap e in an in teresting w a y b ecause of the action of the eectiv e p oten tial. 4.3.3 Results Figures 4{3 and 4{4 sho w the initial w a v efunction and its PWT, resp ectiv ely Note that the PWTDM formed from a Gaussian w a v epac k et is a Gaussian function itself, alb eit in t w odimensional phase space. Figure 4{5 sho ws the w a v efunction at t = 14000 au, ha ving b een propagated through the SOFFT metho d. The PWTDM ev olv es in phase space through the EPQCLE, and in Figure 4{6 w e sho w the grid in phase space at the nal time. While substan tially distorted from its initial uniform distribution, w e notice that the p oin ts are globally p ositioned along a straigh t line in phase space. This rerects the asymptotic state, where eac h p oin t is sub ject to a v anishing HellmannF eynman force, and th us propagates at constan t v elo cit y The observ ables are presen ted in Figures 4{7 to 4{13 There are a n um b er of in teresting things w e can glean from these plots. F rom Figures 4{7 and 4{8 w e see that as the atom mo v es a w a y from the metal surface, m uc h of its p opulation shifts from the neutral to ionic state. In the case of Na, appro ximately 2/3 of the neutral p opulation shifts, while for Li the transfer is total. This ma y rerect the stronger in teraction coupling in v olv ed with Li. Figures 4{9 and 4{10 describ e the coherence b et w een the states. F or the Nasurface system, the coherence remains large for long times, while for the Lisurface
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48 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 3 3.5 4 4.5 5 5.5 6 6.5 7 Re( Y1)R (au) Figure 4{3. ( R ) at t = 0 au, for the Nasurface mo del. This w a v efunction is ev olv ed through the SOFFT algorithm.
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49 15 10 5 0 5 10 15 P (au) 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 R (au) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 G 11 Figure 4{4. 11 at t = 0 au, for the Nasurface mo del. This PWTDM is ev olv ed through the EPQCLE metho d.
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50 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 20 25 30 35 40 Real( Y1)R (au) Figure 4{5. ( R ) at t = 14000 au, for the Nasurface mo del.
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51 22 24 26 28 30 32 34 36 60 65 70 75 80 85 90 95 100 R (au)P (au) Figure 4{6. Phase space grid p oin ts at t = 14000 au, for the Nasurface mo del.
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52 0 0.2 0.4 0.6 0.8 1 0 2000 4000 6000 8000 10000 12000 14000 Populations h1 and h2time (au) EPQCLE: h 1 SOFFT: h 1 EPQCLE: h 2 SOFFT: h 2 Figure 4{7. Na p opulations 1 and 2 vs. time.
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53 0 0.2 0.4 0.6 0.8 1 0 500 1000 1500 2000 Populations h1 and h2time (au) EPQCLE: h 1 SOFFT: h 1 EPQCLE: h 2 SOFFT: h 2 Figure 4{8. Li p opulations 1 and 2 vs. time.
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54 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0 2000 4000 6000 8000 10000 12000 14000 Coherence: Re( h12)time (au) EPQCLE SOFFT Figure 4{9. Coherence describ ed b y Re( 12 ) vs. time, for the Nasurface system.
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55 0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 500 1000 1500 2000 Coherence: Re( h12)time (au) EPQCLE SOFFT Figure 4{10. Coherence describ ed b y Re( 12 ) vs. time, for the Lisurface system.
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56 0 5 10 15 20 25 30 35 0 2000 4000 6000 8000 10000 12000 14000 Position and dispersion (au)time (au) EPQCLE: SOFFT: EPQCLE: s R SOFFT: s R Figure 4{11. Exp ectation of p osition and disp ersion for the Nasurface system.
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57 0 20 40 60 80 100 0 2000 4000 6000 8000 10000 12000 14000 Momentum and dispersion (au)time (au) EPQCLE: SOFFT: EPQCLE: s P SOFFT: s P Figure 4{12. Exp ectation of momen tum and disp ersion for the Nasurface system.
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58 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0 5 10 15 20 25 30 35 40 r (R)R (au) EPQCLE SOFFT Figure 4{13. Densit y function ( R ) for the Nasurface system.
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59 system, the coherence rapidly diminishes once the in teraction p oten tial is crossed. This can b e explained b y the size of the energy gap at asymptotic distances, as larger ionization energies lead to more rapidly v anishing coherences. Figure 4{11 sho ws the exp ectation of the p osition of Na steadily increases from its initial a v erage, while the disp ersion in p osition initially decreases and then increases again. On the other hand, Figure 4{12 sho ws a mark ed dierence in the b eha vior of the exp ectation of the momen tum, where it b egins at h P i = 0 au, rapidly increases and then b ecomes stationary around h P i = 78 au. W e also nd that the momen tum disp ersion rst increases, and then decreases. Finally Figure 4{13 presen ts the densit y function at the end of the sim ulation. Because of the distortion of the phase space grid, the densit y function obtained from the EPQCLE had to b e calculated using bins and through appro ximations within these bins, and th us is sub ject to some noise. Nev ertheless, w e nd excellen t agreemen t b et w een the EPQCLE and SOFFT results. F or all observ ables, the EPQCLE results are quan titativ ely similar to the SOFFT results to visual resolution. Ha ving studied b oth Na and Li approac hing a metal surface, w e see that the EPQCLE can b e exp ected to yield v ery accurate results for these kinds of systems, ev en when coherence is main tained o v er long p erio ds. 4.4 Binary Collision In v olving Tw o Av oided Crossings 4.4.1 Mo del Details F or the second system, w e lo ok at a t w ostate collision mo del where the diabatic surfaces in tersect t wice. Because of the dual crossing and the coupling in this region, quan tum in terference and eects suc h as tunnelling pla y a substan tial role in the dynamics of the quasiclassical v ariable. As suc h, this mo del is quite demanding for mixed quan tumclassical metho ds, where one can exp ect deviations at lo w er energies.
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60 T able 4{2. P arameters used in the dual a v oided crossing collision mo del. P arameter V alue (au) R 0 8 P 0 [10, 30] 2.5176 M 2000 The Hamiltonian elemen ts are 79 H 11 ( R ) = 0 ; (4.22) H 22 ( R ) = 0 : 1 exp ( 0 : 28 R 2 ) + 0 : 05 ; (4.23) H 12 ( R ) = 0 : 015 exp( 0 : 06 R 2 ) : (4.24) In the ab o v e, R is the n uclearn uclear separation. The initial state is a ground state Gaussian w a v epac k et whic h b egins in the asymptotic region R 0 = 8 au. Its PWT o v er R giv es the Gaussian densit y 11 ( P ; R ) = 1 R R 0 2 2 ( P P 0 ) 2 # ; (4.25) with 12 = 21 = 22 = 0. A t t = 0, the w a v epac k et propagates to w ard the region of coupling, where it is partially transmitted and partially rerected, no w with p opulations in b oth the ground and excited state. The parameters used in the calculation are sho wn in T able 4{2 The diabatic p oten tials used in the mo del are sho wn in Figure 4{14 4.4.2 Prop erties of In terest As in the alk ali atomsurface mo del, w e can follo w the p opulations o v er time. The system b egins in the ground state, and as it passes through the collision region, w e exp ect some of the p opulation to transfer to the excited state. The amoun t of transfer dep ends on the collision energy W e can also follo w the coherence once the
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61 0.06 0.04 0.02 0 0.02 0.04 0.06 0.08 10 5 0 5 10 Energy (au)R (au) H 11 H 22 H 12 Figure 4{14. P oten tial curv es for the dual a v oided crossing collision.
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62 collision has ended and the system reac hes asymptotic v alues. W e also consider the transmission of the ground state as the w a v epac k et passes through the in teraction. T ransmission. As the w a v epac k et passes through the collision, part of it con tin ues forw ard while the remainder is rerected bac k. W e can study this b y computing the probabilit y of transmission of the ground state, T 1 = 1 Z 0 dR dP 11 ( R ; P ) : (4.26) Although not included, w e could also compute the probabilit y of rerection of the ground state, as w ell as the asymptotic p opulations of the excited state. 4.4.3 Results In Figure 4{15 w e see the total p opulation transfer at energy P 0 = 30 au. In this energy region, the EPQCLE matc hes the quan tum results to within a couple p ercen t. F or b oth the EPQCLE and SOFFT sim ulations, the rerection of the w a v efunction (or PWTDM) w as negligible, so that w e need only compare the transmission. This negligible rerection w as found throughout the energies studied. In Figure 4{16 w e see the coherence as a function of time. The v ariations are large initially and diminish asymptotically to regular oscillations. The EPQCLE captures this b eha vior qualitativ ely and to within 5% quan titativ ely W e sho w the deformation of the phase space grid in Figure 4{17 W e see that it is primarily the inner part of the grid that undergo es deformation, while the enclosing p oin ts main tain a structured order. This is lik ely due to only a small range of v alues in phase space whic h are seriously aected b y the p oten tials. Outside the in teraction region, all HellmannF eynman forces are zero, so one w ould exp ect deformations only for p oin ts whose time in the in teraction region w as signican t. In Figure 4{18 w e displa y the transmission probabilit y for a wide range of energies. As exp ected, the EPQCLE p erforms b etter for higher energies, and while the doublew ell is qualitativ ely repro duced, the EPQCLE fails to giv e quan titativ ely
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63 accurate results for energies lo w er than (log E = 2). This is lik ely due to the quantum tunnelling eects describ ed earlier, whic h are not exp ected to b e repro duced w ell b y the EPQCLE. W e also compare transmission probabilities obtained through surface hopping metho ds b y T ully and co w ork ers. 51 These surface hopping calculations sho w deviations from the quan tal results that are similar to the EPQCLE probabilities at lo w er energies; at higher energies, the EPQCLE is sligh tly sup erior to the surface hopping sc heme. 4.5 Photoinduced Disso ciation of a Diatomic System 4.5.1 Mo del Details F or the third test system, w e explore a mo del of the NaI complex. As in the previous mo dels, it in v olv es t w o diabatic surfaces and an in teraction around the a v oided crossing. A substan tially dieren t feature, ho w ev er, is a longrange Coulom bic attraction in an ionic state. As w e shall see, this attractiv e p oten tial results in the the complex oscillating b et w een a neutral and ionic state as the so dium and io dine separate and come bac k together, partially disso ciating at eac h crossing in to an asymptotically neutral state. This oscillatory motion is a go o d test for the EPQCLE at long times in cases where asymptotic states are not reac hed quic kly The Hamiltonian elemen ts are 86 H 11 ( R ) = A 1 exp [ 1 ( R R 0 )] ; (4.27) H 22 ( R ) = [ A 2 + ( B 2 =R ) 8 ] exp ( R = ) 1 =R ( + + ) = 2 R 4 C 2 =R 6 2 + =R 7 + E 0 ; (4.28) H 12 ( R ) = A 12 exp [ 12 ( R R x ) 2 ] : (4.29) T o form the initial state, w e T a ylor expand (to rst order) the ionic p oten tial H 22 ab out its minim um R = R 0 and nd the lo w est energy state of this harmonic w ell. A t t = 0, the w a v epac k et undergo es a sudden optical promotion to the neutral curv e,
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64 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 200 400 600 800 1000 1200 1400 Populations h1 and h2time (au) EPQCLE: h 1 SOFFT: h 1 EPQCLE: h 2 SOFFT: h 2 Figure 4{15. P opulations 1 and 2 vs. time for the dual crossing collision mo del.
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65 0.2 0.15 0.1 0.05 0 0.05 0.1 0 200 400 600 800 1000 1200 1400 Coherence: Re( h12)time (au) EPQCLE SOFFT Figure 4{16. Coherence describ ed b y R e ( 12 ) vs. time, for the dual crossing collision mo del.
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66 0 5 10 15 20 25 30 35 40 45 20 22 24 26 28 30 32 34 36 R (au)P (au) Figure 4{17. Grid deformation at t = 1400 au, for the dual crossing collision mo del.
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67 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 4 3.5 3 2.5 2 1.5 1 h1 Tlog e (E) (au) EPQCLE SOFFT Tully et al. Figure 4{18. Probabilit y of transmission in the ground state, for the dual crossing collision mo del.
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68 T able 4{3. P arameters used in the NaI complex mo del. P arameter V alue (au) R 0 5.047 P 0 0.0 0.12462 A 1 0.02988 A 2 101. 43 A 12 0.00202 B 2 3.000 C 2 18. 950 + 2.756 12. 179 0.660 E 0 0.07626 1 2.158 12 0.194 R x 13. 24 M 35,482 so that the PWTDM b ecomes, 11 ( P ; R ) = 1 R R 0 2 2 ( P P 0 ) 2 # ; (4.30) with 12 = 21 = 22 = 0. The sim ulation follo ws the resulting motion of this state. The mo del parameters are sho wn in T able 4{3 The diabatic p oten tials are displa y ed in Figure 4{19 4.5.2 Prop erties of In terest In addition to observing the coherence, exp ectation v alue of the p osition and its disp ersion, and the phase space grid, w e also consider b ound and free neutral and ionic p opulations as the NaI complex oscillates from its primarily ionic to primarily co v alen t state. Bound and free p opulations. It is in teresting to follo w the p opulations of the ionic and neutral states as the system ev olv es, giving insigh t in to the nature of the disso ciation in to an asymptotically free neutral system. F or this, w e dene three
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69 0.1 0.05 0 0.05 0 5 10 15 20 25 30 Energy (au)R (au) H 11 H 22 H 12 Figure 4{19. P oten tial curv es for the NaI complex.
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70 p opulations: the ionic, the b ound neutral and the free neutral. The ionic p opulation is the probabilit y of nding the system in the ionic state at an y in tern uclear separation, 2 = 1 Z 1 dR dP 22 ( R ; P ) : (4.31) W e dene the b ound neutral p opulation is the probabilit y of nding the NaI complex in the neutral state at n uclear separation up to the crossing p oin t R = R x b 1 = R x Z 0 dR dP 11 ( R ; P ) ; (4.32) while the free ionic p opulation is dened as the probabilit y of the neutral state from the crossing p oin t b ey ond, f 1 = 1 Z R x dR dP 11 ( R ; P ) : (4.33) This division b et w een b ound and free is motiv ated b y the observ ation that the ma jorit y of the transfer b et w een ionic and neutral p oten tials o ccurs at the a v oided crossing, and that an y part of the w a v epac k et whic h ends up in the neutral state but propagating to w ard innite separation has a negligible probabilit y of shifting to the ionic state m uc h b ey ond the a v oided crossing. 4.5.3 Results The ionic and co v alen t p opulations are displa y ed in Figure 4{20 W e see the oscillations in the p opulations b et w een ionic and co v alen t, rep eating appro ximately ev ery 40 000 au. This pattern can b e compared to the exp ectation v alues of p osition see in Figure 4{21 The p osition oscillates with the same frequency as the c hange in p opulation, sho wing that eac h time the w a v epac k et heads across the a v oided crossing from its b ound co v alen t state, it con v erts almost completely in to the ionic state, with a small amoun t escaping in to the free neutral state. Ov er time, the
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71 free neutral p opulation gradually increases at these crossings, and the NaI slo wly disso ciates. The EPQCLE is quan titativ ely similar to the exact results for the rst half of the sim ulation, and main tains qualitativ e accuracy for the remainder. The disp ersion in Figure 4{21 sho ws an in teresting dierence b et w een the exact and the quan tumclassical algorithm. While the disp ersion con tin ues to rise in the SOFFT sim ulation, it reac hes its rst p eak and then b egins to decline somewhat in the EPQCLE sim ulation. This rerects the nature of the eectiv e p oten tial, where eac h p oin t is guided b y a com bination of excited and ground state forces. Because the ionic curv e do es not p ermit escap e, what w ould normally b e asymptotically free w a v epac k ets tend to b e pulled bac k to w ard the crossing b ecause of the attractiv e ionic p oten tial. Consequen tly the free neutral p opulation is alw a ys lo w er using the EPQCLE equation than the SOFFT, an observ ation supp orted b y Figure 4{20 Since the ma jorit y of the p opulation remains in the ionic or b ound neutral state, and these p opulations are w ell matc hed b et w een the exact and quan tumclassical sim ulations, it is not surprising that the exp ectation v alue of the p osition is quantitativ ely in the b eginning, and qualitativ ely for the remaining, similar for b oth metho ds. Ho w ev er, the disp ersion is m uc h more sensitiv e to the increased asymptotically free w a v epac k ets in the SOFFT sim ulation, and for the reasons discussed, w e nd signican t div ergence b et w een the EPQCLE and SOFFT results. The coherence, sho wn in Figure 4{22 initially p eaks through the rst crossing, but through subsequen t crossings it is substan tially diminished. Ho w ev er, the EPQCLE sho ws quan titativ ely similar results to the SOFFT calculations. The deformation of the phase space grid, plotted in Figure 4{23 has c haracteristics not seen in the other t w o mo dels. One line of p oin ts emits from the cen ter of the cluster, quic kly straigh tening and rerecting the negligible force on the p oin ts. These p oin ts corresp ond to the asymptotically free neutral comp onen ts of the PWTDM. The second group circles around, gaining v elo cit y and p osition, then
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72 turning. These ellipses are c haracteristic of the phase space of classical particles in a w ell, and indeed rerect the quasiclassical motion under the HellmannF eynman force of the PWTDM p oin ts as they follo w the ionic and b ound co v alen t curv es. 4.6 Comparison Using V ariable and Constan t Timesteps In this Section, w e ev aluate the usefulness of the v ariable timestep asp ect of the relaxanddriv e algorithm. T o do this, w e consider the Nasurface algorithm of Section 4.3 and sim ulate using v arying upp er and lo w er tolerances. The n um b er of steps tak en in eac h case is compared with the n um b er that w ould b e required of the same algorithm, but k eeping the timestep xed. The xed timestep w ould necessarily adv ance b y steps no greater than the smallest timestep used in the v ariable timestep algorithm, and it is based on this timestep that w e estimate the corresp onding steps required for the xed timestep approac h. The results are sho wn in Figure 4{24 W e see that as the tolerance decreases (and th us the accuracy increases), the fraction of steps sa v ed b y the in tro duction of the v ariable timestep increases sup erlinearly One concludes that while the v ariable timestep ma y not b e imp ortan t for lo w accuracy sim ulations, considerable computational sa vings can b e had for high accuracy propagation. 4.7 Conclusion By examining three simple t w ostate mo dels, w e w ere able to compare the EPQCLE metho d with the exact SOFFT quan tum mec hanical solution. F or all mo dels, w e found v ery go o d agreemen t b et w een the EPQCLE and the SOFFT results. The agreemen t w as at least qualitativ e, and in man y cases quan titativ e to visual precision. W e also sa w conditions under whic h the quan tumclassical mo del deviated from exact quan tal results. Finally w e compared a xed timestep v arian t of the relaxanddriv e algorithm with the v ariable timestep v ersion. F or the mo del of an alk ali atom approac hing a metal surface, w e examined probabilit y transitions, p osition exp ectation and its deviation, momen tum exp ectation
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73 0 0.5 1 1.5 2 0 20000 40000 60000 80000 100000 120000 Ionic and Neutral Populationstime (au) EPQCLE: Ionic SOFFT: Ionic EPQCLE: Neutral Bound SOFFT: Neutral Bound EPQCLE: Neutral Free SOFFT: Neutral Free Figure 4{20. Ionic and neutral p opulations o v er time, for the NaI complex.
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74 0 5 10 15 20 25 0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000 time (au) EPQCLE: SOFFT: EPQCLE: s R SOFFT: s R Figure 4{21. Exp ectation of p osition and its deviance, for the NaI complex.
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75 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 20000 40000 60000 80000 100000 120000 Coherence: Real( G12)time (au) EPQCLE SOFFT Figure 4{22. Coherence as a function of time, for the NaI complex.
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76 0 20 40 60 80 100 120 140 80 60 40 20 0 20 40 60 80 R (au)P (au) Figure 4{23. Phase space grid at the end of the sim ulation, for the NaI complex.
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77 0 500 1000 1500 2000 2500 3000 3500 10 3 10 1 10 5 10 3 10 7 10 5 10 9 10 7 number of timestepstolerance Variable Fixed Figure 4{24. Num b er of steps required b y the relaxanddriv e algorithm, compared to an estimated n um b er required for a xed timestep v ersion.
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78 and its deviation, densit y function and phase space grid ev olution. W e found that the EPQCLE metho d repro duced quan titativ ely the exact v alues found through the quan tum mo del for all observ ables. W e also found the phase space grid to deform substan tially as the system ev olv ed, initially distorting but ev en tually lining up in a straigh t line as the eectiv e force v anished. F or the dual crossing diabatic surface collision, in addition to probabilit y transitions for a giv en energy w e calculated the transmission probabilit y of the ground state for a wide range of energies. This sho w ed the EPQCLE to deviate at the lo w er energies, where n uclear in terference eects w ere imp ortan t. A t higher energies, the corresp ondence b et w een the mixed quan tumclassical results and the exact quan tum results w as go o d. W e also noted that the phase space grid deformed only for p oin ts whic h sp en t a signican t amoun t of time in the region of strong in teraction. This is reasonable, as the deformation comes from the eectiv e force, whic h is nonv arying (and in fact zero) outside the in teraction. The NaI mo del sho w ed in teresting bac kandforth transitions b et w een the ionic and co v alen t state, as the in tern uclear distance oscillated due to the longrange attractiv e ionic p oten tial. W e also sa w that eac h passing through the a v oided crossing led to a small amoun t of disso ciation in to the asymptotically free neutral state. Ionic, b ound neutral and free neutral p opulations w ere quan titativ ely similar for the EPQCLE and SOFFT algorithms for the b eginning of the sim ulations, and qualitativ ely similar for the remainder. Coherences w ere quan titativ ely similar, sho wing an initial p eak after the rst pass through the a v oided crossing, and m uc h smaller magnitudes thereafter. Finally the grid deformation sho w ed the grid to split in to t w o groups. The rst represen ted the asymptotically free neutral state, where the p oin ts form a straigh t line as the HellmannF eynman force v anishes and the particles propagate to innite distances. The second group circled in phase space, as the
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79 p oin ts oscillated bac k and forth in the p oten tial w ell formed b y the ionic attraction at large distances and the neutral repulsion at small distances. The EPQCLE metho d w as sho wn to b e v ery robust under a wide v ariet y of conditions. Comparing its v ariable timestep v ersion (whic h w as used in all the sim ulations) to the xed timestep alternativ e, w e found that not only w as the EPQCLE metho d accurate, but use of the relaxanddriv e propagation generated m uc h more ecien t sim ulations than could ha v e b een obtained without dynamically v arying the timestep.
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CHAPTER 5 ALKALI A TOMRARE GAS CLUSTERS: GENERAL F ORMULA TION 5.1 In tro duction In this c hapter, w e describ e the in teractions of an alk ali atom (Ak) em b edded in a cluster of rare gas atoms (Rg N ). Cluster dynamics are in teresting insofar as they presen t a bridge b et w een isolated atoms and the (eectiv ely) innite bulk liquid, and can pro vide insigh t in to the dynamics of molecules on extended surfaces. The AkRg N clusters are among the simplest cluster systems to study b ecause of the presence of a single v alence electron in the alk ali atom, and the closed shell structure of the rare gas atoms. Pseudop oten tial descriptions of the in teractions b et w een the v alence electron ( e ), the Ak core and the Rg atoms can greatly simplify the description of the cluster with v ery mo dest p enalties in accuracy In the con text of mixed quan tumclassical mo dels, these clusters are w ell suited to sim ulation through the EPQCLE b y treating the v alence electron quan tally and the n uclear cores quasiclassically F or the ma jorit y of the c hapter, w e consider the general case of an alk ali atom em b edded in a rare gas cluster, although our sim ulations in Chapter 6 fo cus on the sp ecic case of the lithium atom (Li) em b edded in a helium cluster (He N ). 5.2 Ph ysical System W e consider a cluster initially at thermal equilibrium, and in tro duce a ground or excited alk ali atom to the cen ter of the cluster. W e do not concern ourselv es with the means b y whic h the alk ali atom is excited or em b edded within the cluster, although t ypically the excitation is due to a laser pulse. F rom this initial setup, w e follo w the dynamics of the alk ali atom as it deexcites and mo v es within the cluster. W e further assume the cluster is isolated from an y en vironmen tal eects, so that sp ectral and conguration measuremen ts represen t 80
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81 those of an isolated AkRg N cluster. As w e see later, w e in tro duce a con taining p oten tial to k eep the Rg atoms from disp ersing. This p oten tial is used strictly to main tain a cluster formation, and do es not represen t in teraction with an en vironmen t. 5.3 Prop erties of In terest First of all, w e are in terested in the structure of the Rg cluster in equilibrium, b efore the in tro duction of the Ak atom. W e need to repro duce kno wn densit y proles and pairpair correlation functions to ensure the cluster is represen tativ e of a real ph ysical system. Secondly w e are in terested in follo wing the dynamics of the Ak atom as it mo v es within the cluster. In particular, w e are lo oking for migration from the cen ter of the cluster to its surface, as other exp erimen tal and theoretical studies indicate that the Ak atom tends to reside on the cluster surface. Finally w e wish to compute the time dep enden t emission sp ectra resulting from the deexcitation of the Ak atom, Ak ( n 0 l 0 ) + R g N Ak ( nl ) + R g N + ; (5.1) where is a photon with energy corresp onding to the electronic deca y 5.4 Hamiltonian for Alk aliRare Gas P airs F ollo wing the dynamics of an AkRg N cluster through a full quan tal treatmen t w ould b e an extremely demanding problem computationally for more than a few Rg atoms. Instead, w e reduce the AkRg in teraction to a threeb o dy problem b y using pseudop oten tial in teractions b et w een the n uclear cores and the electron. The pseudop oten tial treatmen t is explored extensiv ely for the single AkRg pair in Rey es. 109 In this section, w e summarize this approac h. In order to describ e the threeb o dy in teraction, w e consider a xed n uclear conguration with R AB the p osition v ector from the alk ali core (A) to the rare gas atom (B), and r A ( r B ) the p osition v ector from the alk ali core (rare gas atom) to
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82 the electron e With this notation, w e can write the Hamiltonian in v e distinct comp onen ts, ^ H pair = 1 2 r 2r A + ^ V Ak A ( r A ) + ^ V Rg B ( r B ) + ^ V cr oss AB ( r A ; R AB ) + ^ V cor e AB ( R AB ) : (5.2) W e ha v e implicitly tak en the PWT o v er n uclear v ariables but not electronic v ariables. Because the p oten tials can all b e expressed as p olynomial functions of ^ Q the PWT amoun ts to replacing ^ Q b y R throughout the Hamiltonian. In what follo ws, w e will assume w e are w orking with partially Wigner transformed op erators, but drop the subscript 'W' for notational simplicit y Also note that the electronic v ariable remains quan tal, although w e will use the notation r rather than ^ q for consistency with common usage in the literature. The rst term on the RHS of Eq. 5.2 is, of course, the kinetic energy op erator of the v alence electron. The second and third terms are the p oten tials arising from the in teraction of the electron with the Ak core and the Rg atom, resp ectiv ely The ^ V cr oss AB term is a crossterm stemming from the p olarization of the Rg atom b y b oth the v alence electron and Ak core. Finally the last term ^ V cor e AB is the in teraction b et w een the Ak core and the Rg atom. W e examine eac h of these p oten tials in detail for the general AkRg pair, and pro vide sp ecic parameters for the LiHe in teraction in Chapter 6. The e Ak core p oten tial can b e divided in to three comp onen ts, ^ V Ak A ( r A ) = Z A r A + ^ V pol Ak A ( r A ) + ^ V sr A ( r A ) ; (5.3) where the rst term is the Coulom b in teraction b et w een the Ak core of c harge Z A and the electron. The second con tribution arises from the dip ole p olarization of the
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83 Ak core b y the v alence electron, ^ V pol Ak A ( r A ) = d A 2 r 4 A w ( r A ; A ) 2 ; (5.4) with d A the dip ole p olarizabilit y of the core and w a cuto function of the distance w ( r ; ) = 1 exp ( r 2 ) : (5.5) The nal con tribution to ^ V Ak A is an l dep enden t shortrange pseudop oten tial, ^ V sr X ( r X ) = X l ;i B l ;i exp ( l ;i r 2 X ) P l ;X ; (5.6) where B l ;i and l ;i are pseudop oten tial parameters adjusted to t exp erimen tal data, P l ;X is the pro jection op erator on angular symmetry l P l ;X = X m j Y l m ( ^ r X ) ih Y l m ( ^ r X ) j ; (5.7) and the states fj Y l m ( ^ r X ) ig are the spherical harmonic functions cen tered on the core X. This pseudop oten tial sim ulates repulsion due to the eects of the P auli exclusion principle when the v alence electron approac hes the core electrons, as w ell as the attraction due to the incomplete screening of the n uclear c harge. The third term in Eq. 5.2 is the e Rg p oten tial, and can b e divided in to t w o terms, ^ V Rg B ( r B ) = ^ V pol Rg B ( r B ) + ^ V sr B ( r B ) : (5.8) The rst comp onen t stems from the p olarization of the Rg atom b y the electron, ^ V pol Rg B ( r B ) = d B 2 r 4 B w ( r B ; ) 4 q 0 B 2 r 6 B w ( r B ; ) 6 ; (5.9)
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84 where q 0 B = q B 6 1 q B b eing the quadrup ole p olarizabilit y of the Rg atom and 6 1 the dynamical correction to the static p olarizabilit y The shortrange pseudop oten tial ^ V sr B is dened in Eq. 5.6 The fourth term in Eq. 5.2 is a crossterm arising from the p olarization of the Rg atom b y b oth the Ak core and the v alence electron, ^ V cr oss AB ( r A ; R AB ) = Z A d B P 1 ( cos ) R 2 AB r 2 B w ( r B ; ) 2 + Z A q B P 2 ( cos ) R 2 AB r 2 B w ( r B ; ) 3 : (5.10) Here, P 1 and P 2 are the Legendre p olynomials and is the angle b et w een the v ectors R AB and r B Finally the last term in Eq. 5.2 is the in teraction b et w een the alk ali core and the rare gas atom. It is assumed to in v olv e shortrange, dip ole and quadrup ole con tributions, ^ V cor e AB ( R AB ) = ^ V sr AB ( R AB ) 1 2 d B ( R 2 AB + d 2B ) 2 1 2 q 00 B ( R 2 AB + d 2B ) 3 ; (5.11) where q 00 B = q B + 2 d B d 2B The shortrange term ^ V sr AB has b een determined b y assuming it to ha v e the form, ^ V sr AB ( R AB ) = A exp ( bR AB ) ; (5.12) and adjusting the parameters A and b so that ^ V cor e AB ( R AB ) in Eq. 5.11 ts the most accurate curv es in the literature. This pro cedure has b een sho wn to accurately repro duce AkRg energy curv es for a v ariet y of alk ali atom and rare gas com binations. 109 5.5 Hamiltonian for the Alk aliRare Gas Cluster The cluster is similar to the pair, in that eac h AkRg pair in the cluster is treated as a threeb o dy problem in v olving alk ali core, v alence electron and rare gas atom, and describ ed using the same Hamiltonian elemen ts. In the case of the cluster, ho w ev er, w e also require that the Rg atoms main tain a cohesiv e structure
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85 rather than drift apart. T o ensure this, w e imp ose a constraining p oten tial on the system. Dening R AB i as the p osition v ector from the alk ali core to the Rg atom i and R B i the p osition v ector of the Rg atom from the origin, w e can write the cluster Hamiltonian as, ^ H cl uster = 1 2 r 2r A + ^ V Ak A ( r A ) + N X i =1 ^ V Rg B i ( r B ) + N X i =1 h ^ V cr oss AB i ( r A ; R AB i ) + ^ V cor e AB i ( R AB i ) + ^ V hol d i ( R B i ) i + N X j >i =1 ^ V cor e ( j R B i R B j j ) : (5.13) The nal term in Eq. 5.13 is the in teraction b et w een the Rg atoms for the quasiclassical motion. F ollo wing Aziz and co w ork ers, 110 w e tak e the RgRg p oten tial to ha v e the form, for in tern uclear distance R ^ V cor e ( R ) = V ( x ) ; (5.14) where V ( x ) = A exp ( x + x 2 ) F ( x ) X j =0 2 C 2 j +6 x 2 j +6 ; (5.15) F ( x ) = 8><>: exp [ ( D =x 1) 2 ] ; x D 1 ; x > D : (5.16) This p oten tial has b een written in terms of the dimensionless distance x = R =R M The constraining p oten tial, ^ V hol d i is tak en to b e a sigmoidal function cen tered along the b oundary of a sphere of radius R hol d ^ V hol d i ( R B i ) = a 1 + exp [ b ( R B i R hol d )] : (5.17)
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86 This p oten tial sho ws asymptotes ^ V hol d i ( 1 ) = 0 and ^ V hol d i (+ 1 ) = a with midp oin t at the holding radius ^ V hol d i ( R hol d ) = a= 2. The steepness of the function is con trolled b y the parameter b and determines the strength and range of the holding force. It acts on the Rg atoms only k eeping them b ound roughly within a sphere of radius R hol d while p ermitting the alk ali atom free motion within the cluster. 5.6 Electronic Sp ectral Calculations When a molecular system undergo es electronic motion, the accelerating c harges emit electromagnetic radiation. A t distances large compared to the electronic motion, the rux of this radiation at p oin t r from the cen ter of the system is giv en b y the P o yn ting v ector, 111 S ( t ) = 1 0 ( E B ) = 0 16 2 c [ D ( t )] 2 sin 2 r 2 ^ r ; (5.18) where D ( t ) is the dip ole momen t of the system, 0 is the p ermittivit y constan t and c is the sp eed of ligh t. By in tegrating o v er all angles, w e obtain the p o w er emited b y the source, P ( t ) = Z n S ( t ) d a = j A ( t ) j 2 (5.19) where A ( t ) = r 1 6 0 c 3 D ( t ) : (5.20) F rom Eq. 5.20 w e can see that the computation of the dip ole momen t is essen tial to the sp ectral calculation. In a mixed quan tumclassical system, the dip ole momen t is obtained b y calculating the exp ectation v alue of the dip ole op erator ^ D D ( t ) = T r [ ^ ( t ) ^ D ] = T r[( t )] : (5.21)
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87 F or the AkRg N cluster, the dip ole op erator can b e written with the origin at the Ak core, so that it b ecomes, ^ D = r + d A r A r 3A w ( r A ; A ) + d B N X i =1 r B i r 3B i w ( r B i ; B ) + Z A R AB R 3AB w ( R AB ; B ) # : (5.22) The rst term is the dip ole from the v alence electron. The second is the induced dip ole in the Ak core b y the v alence electron. The third and fourth terms sum o v er all Rg atoms, and represen t the induced dip ole in the Rg core b y the v alence electron and Ak core, resp ectiv ely In practice, the induced dip oles are m uc h smaller than the v alence electron dip ole. W e can calculate the emission sp ectrum b y taking the F ourier transform of Eq. 5.19 I ( ) = 1 p 2 Z P ( t ) exp( i! t ) dt = 1 p 2 Z j A ( t ) j 2 exp ( i! t ) dt: (5.23) If the emission sp ectrum c hanges o v er time, one can use a windo w ed F ourier transform cen tered ab out the time of in terest. In the case of AkRg N clusters, this allo ws us to trace the ev olution of the emission sp ectrum as the alk ali atom mo v es from the cen ter of the cluster to its surface. 5.7 Electronic Basis of Gaussian A tomic F unctions 5.7.1 Equations of Motion W e recall the EPQCLE in an arbitrary basis, Eq. 3.7 repro duced here for con v enience: d dt = ( i H q n y ) S 1 S 1 ( i H q + n ) ; (5.24)
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88 where w e ha v e used the notation, n h j @ =@ t j i + dR dt h j @ =@ R j i + dP dt h j @ =@ P j i : (5.25) F or the cluster, H q = H cl uster A con v enien t basis in v olv es Gaussian functions cen tered on the Ak core, and leads to matrix elemen ts whic h can b e ev aluated analytically T o this end, w e in tro duce primitiv e cartesian Gaussian states, dened b y an in teger triplet a = ( a x ; a y ; a z ), a Gaussian cen ter A = ( A x ; A y ; A z ) and an exp onen tial co ecien t a suc h that their pro jection on the electronic co ordinate r is h r j a ; A ; a i = ( r x A x ) a x ( r y A y ) a y ( r z A z ) a z exp [ a ( r A ) 2 ] : (5.26) F rom these primitiv es, w e can form Gaussian atomic functions (GAFs) whic h resemble the h ydrogenic orbitals. The lo w est three symmetries are j s ; R ; i j a = (0 ; 0 ; 0); R ; i = j 000; R ; i ; (5.27) j p ; R ; i 2 fj 100; R ; i ; j 010; R i ; j 001; R ; ig ; (5.28) j d ; R ; i 2 fj 110; R ; i ; j 011; R ; i ; j 101 ; R ; i ; j 200 ; R ; i j 020; R ; i ; 2 j 002; R ; i j 200; R ; i j 020; R ; ig : (5.29) Finally w e construct the basis elemen ts b y com bining linear com binations of these GAFs in segmen ted con tractions, j i = N X j c j j ; R ; j i ; (5.30) where 2 f s; p; d g f c j g are the con traction co ecien ts and N is a normalization factor. W e can write the full M dimensional basis as a ro w v ector of these segmen ted
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89 con tractions, j i = ( j 1 i j 2 i : : : j M i ) ; (5.31) corresp onding to the notation used to deriv e Eq. 3.7 Since j i do es not dep end explicitly on time or n uclear momen ta, n reduces to a momen tum coupling term, n = dR dt h j @ @ R j i : (5.32) A non trivial matter is the analytical computation of these matrix elemen ts, atten tion to whic h w e turn presen tly 5.7.2 Ov erlap Matrix Elemen ts In computing matrix elemen ts, w e fo cus on the essen tial part of the computation, whic h is the matrix elemen t b et w een primitiv e cartesian Gaussians. F or the o v erlap S w e need to compute h a j b i whic h is p ossible using the recursion form ulae from Obara, 112 113 h a + 1 i j b i = ( P i A i ) h a j b i + 1 2 a i h a 1 i j b i + 1 2 b i h a j b 1 i i ; (5.33) h a j b + 1 i i = ( P i B i ) h a j b i + 1 2 h a j b 1 i i + 1 2 a i h a 1 i j b i ; (5.34) whhere 1 i is an in teger triplet ( ix ; iy ; iz ), = a + b and P = ( a A + b B ) = Recursion con tin ues to the base case, h s ; A ; a j s ; B ; b i = 2 exp [ ~ ( A B ) 2 ] ; (5.35) where ~ = a b = If computational eciency is a concern, some of the lo w er recursion relations can b e rolled out explicitly rather than computed recursiv ely
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90 5.7.3 Kinetic Energy Matrix Elemen ts F or the kinetic energy w e wish to compute h a j ^ K j b i = h a j 1 2 r 2r j b i W e can do this b y using the recursion relations, h a + 1 i j ^ K j b i = ( P i A i ) h a j ^ K j b i + 1 2 a i h a 1 i j ^ K j b i + 1 2 b i h a j ^ K j b 1 i i = 2 ~ h a + 1 i j b i 1 2 a a i h a 1 i j b i ; (5.36) h a j ^ K j b + 1 i i = ( P i B i ) h a j ^ K j b i + 1 2 b i h a j ^ K j b 1 i i + 1 2 a i h a 1 i j ^ K j b i = 2 ~ h a j b + 1 i i 1 2 b b i h a j b 1 i i : (5.37) The recursion con tin ues to the base case, h s ; A ; a j ^ K j s ; B ; b i = ~ (3 2 ~ ( A B ) 2 ) h s ; A ; a j s ; B ; b i = ~ (3 2 ~ ( A B ) 2 ) 2 exp [ ~ ( A B ) 2 ] : (5.38) 5.7.4 Coulom b Matrix Elemen ts The n uclear attraction in tegral is a threecen ter in tegral, h a j ^ V C j b i = Z d r ( A ; a ) 1 j r C j ( B ; b ) : (5.39) F ollo wing Obara, 113 this can b e ev aluated using the recursion relations, h a + 1 i j ^ V C j b i ( m ) = ( P i A i ) h a j ^ V C j b i ( m ) ( P i C i ) h a j ^ V C j b i ( m +1) + 1 2 a i h h a 1 i j ^ V C j b i ( m ) h a 1 i j ^ V C j b i ( m +1) i + 1 2 b i h h a j ^ V C b 1 i i ( m ) hj ^ V C j b 1 i i ( m +1) i ; (5.40) and similarly for h a j ^ V C j b + 1 i i The recursion base is found to b e, h s ; A ; a j ^ V C j s ; B ; b i ( m ) = 2 ~ 1 = 2 h s ; A ; a j s ; B ; b i F m ( U ) ; (5.41)
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91 where U = ( P C ) 2 and F m is the sp ecial function, F m ( T ) = 1 Z 0 dt t 2 m exp ( T t 2 ) : (5.42) 5.7.5 Momen tum Coupling Matrix Elemen ts The momen tum coupling elemen ts h j @ =@ R j i are ev aluated in terms of the o v erlap elemen ts, h a j @ =@ B x j b i = b h a j b + 1 x i b 1 h a j b 1 x i ; (5.43) h a j @ =@ B y j b i = b h a j b + 1 y i b 2 h a j b 1 y i ; (5.44) h a j @ =@ B z j b i = b h a j b + 1 z i b 3 h a j b 1 z i : (5.45) Eac h elemen t on the RHS of Eqs. 5.43 to 5.45 is a straigh tforw ard o v erlap in tegral, whose ev aluation w e ha v e already deriv ed previously 5.7.6 Dip ole Matrix Elemen ts Recalling the dip ole for the AkRg system, w e see that w e require the follo wing matrix elemen ts, h a j r A j b i ; (5.46) h a j r A r 3A (1 exp [ r A A ]) j b i ; (5.47) h a j r B r 3B (1 exp [ r B B ]) j b i ; (5.48) h a j b i : (5.49) The rst in tegral is a momen t in tegral, whic h can b e computed using the recursiv e pro cedure from Obara. 113 Consider the general momen t in tegral, h a jM ( ) j b i ; (5.50)
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92 where M ( ) = x x y y z z : (5.51) These in tegrals can b e found through the recursiv e form ula, h a + 1 i jM ( ) j b ) = ( P i A i ) h a jM ( ) j b i + 1 2 a i h a 1 i jM ( ) j b i + 1 2 b i h a jM ( ) j b 1 i i + 1 2 i h a jM ( 1 i ) j b i : (5.52) The recursion con tin ues to the base case, h s ; A ; a jM ( ~ 0 ) j s ; B ; b i = h s ; A ; a j s ; B ; b i = 2 exp [ ~ ( A B ) 2 ] : (5.53) The second and third in tegrals can b e calculated using a pro cedure presen ted b y McMurc hie and Da vidson, 114 and summarized in detail in Rey es. 109 The nal in tegral is a simple o v erlap in tegral, b ecause the op erator do es not dep end on electronic co ordinates. W e ha v e presen ted the calculation of the o v erlap in tegral previously 5.7.7 Pseudop oten tial Matrix Elemen ts In order to calculate the pseudop oten tial matrix, w e need the additional elemen ts, h a j 1 r 4 [1 exp ( c r 2 )] 2 j b i ; (5.54) h a j 1 r 4 [1 exp ( c r 2 )] 4 j b i ; (5.55) h a j 1 r 6 [1 exp ( c r 2 )] 6 j b i ; (5.56) h a j l X m = l j l m ih l m j j b i : (5.57)
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93 The rst three elemen ts are computed using a pro cedure from McMurc hie and Da vidson, 114 while the fourth elemen t is calculated based on metho ds in Sc h w erdtfeger and Silb erbac h. 115 These calculations are summarized in detail in Rey es. 109 5.8 Computing the Quasiclassical T ra jectory The c hange in p osition of core i (Ak or Rg) is simply calculated from the momen tum, d R i dt = P i M i : (5.58) Ho w ev er, the c hange in momen tum is substan tially more in v olv ed. Eac h quasiclassical v ariable k is sub jected to the HellmannF eynman force, F k = T r ^ @ ^ H @ R k # = T r[] ; (5.59) where w e ha v e used the abbreviated notation ^ H = ^ H cl uster Using our nonorthogonal basis j i this expression b ecomes, F k = T r h j @ ^ H @ R k j i # = T r[ S ] = 1 T r [ S ] T r[ @ H @ R k ] T r[ n yR k S 1 H ] T r [ HS 1 n R k ] : (5.60) The rst term on the RHS of Eq. 5.60 in v olv es the partial deriv ativ e of the cluster Hamiltonian matrix, and can b e ev aluated n umerically b y computing H at a p erturb ed co ordinate R k + @ H @ R k = H ( R k + ) H ( R k ) : (5.61) 5.9 Computational Details Examining the EPQCLE, w e see that a n um b er of matrices can b e computed a single time at the b eginning of the sim ulation, suc h as the o v erlap, momen tum, electronic dip ole and kinetic energy Ho w ev er, the pseudop oten tial is dep enden t
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94 on the relativ e AkRg congurations, and m ust b e computed at eac h timestep. In addition, the n umerical ev aluation of the HellmannF eynman force requires that the Hamiltonian for sev eral p erturb ed congurations b e ev aluated for eac h force calculation. Consequen tly it is imp ortan t that the pseudop oten tial calculations run quic kly This can b e done b y precomputing the pseudop oten tial matrix for sev eral in teratomic distances along the zaxis, extrap olating to the desired distance, and then rotating to the actual orien tation. In this section, w e demonstrate that the rotation is a simple matter of m ultiplication b y rotation matrices. Consider a singlecen ter basis j i and general op erator ^ O in the threeb o dy AkRg system, where the Ak atom is cen tered at the origin, R is the p osition v ector from this core to the Rg atom, and r is the p osition v ector of the v alence electron to the Ak core. Supp ose that w e ha v e precomputed O = h j ^ O j i for sev eral v alues of R along the zaxis. Our task is to compute the new matrix O 0 for the general case where the Rg atom has b een rotated b y n = ( ; ), where and are the p olar and azim uthal angles, resp ectiv ely If the op erator is of the form, ^ O = ^ O ( j R j ; j r j ; j R r j ) ; (5.62) whic h is the case for the cluster Hamiltonian op erator, w e can compute O 0 through a rotation D (n), j 0 i = j i D (n) : (5.63) Algebraically w e can see this b y examining a single matrix elemen t with R along the zaxis, O = h j ^ O j i = Z d r ( r ) ^ O ( j R j ; j r j ; j R r j ) ( r ) : (5.64)
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95 When R is in the rotated p osition R 0 = D (n) R O 0 = Z dr ( r ) ^ O ( jD (n) R j ; j r j ; jD (n) R r j ) ( r ) : (5.65) Substituting r D (n) r = D r O 0 = Z dr ( D r ) ^ O ( jD R j ; jD r j ; jD R D r j ) ( D r ) : (5.66) Since a rotation of a v ector do es not c hange its magnitude, O 0 = Z dr ( D r ) ^ O ( j R j ; j r j ; j R r j ) ( D r ) (5.67) ) O 0 = Z dr ( D r ) ^ O ( j R j ; j r j ; j R r j ) ( D r ) = D y h j ^ O j i D : (5.68) T urning our atten tion to the construction of D w e use the fact that our basis in v olv es segmen ted con tractions of Gaussian atomic functions. In this case, basis elemen ts of same symmetry mix, while dieren t symmetries do not. W e will sho w the full deriv ation for s t yp es and p t yp es. Denoting the s t yp e con traction, s ( r ) = X i exp ( i r 2 ) ; (5.69) w e w an t to calculate s ( D r ). Since D r implies the follo wing co ordinate transformations, x x cos cos + y sin sin z sin ; (5.70) y x sin + y cos ; (5.71) z x cos sin + y sin sin + z cos : (5.72)
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96 Then, s ( D r ) = X i c i exp ( i r 2 ) = s ( r ) : (5.73) Similarly consider the p t yp e con tractions, p x ( r ) = X i c i x exp ( i r 2 ) ; (5.74) p y ( r ) = X i c i y exp ( i r 2 ) ; (5.75) p z ( r ) = X i c i z exp ( i r 2 ) : (5.76) Then p x ( D r ) = X i c i ( x cos cos + y sin cos z sin ) exp( i r 2 ) = p x ( r ) cos cos + p y ( r ) sin cos + p z ( r )( sin ) ; (5.77) p y ( D r ) = p x ( r )( sin ) + p y ( r ) cos ; (5.78) p z ( D r ) = p x ( r ) cos sin + p y ( r ) sin sin + p z ( r ) cos : (5.79) Supp ose our basis is ( r ) = [ s ( r ) p x ( r ) p y ( r ) p z ( r )] : (5.80) Then, ( D r ) = [ s ( D r ) p x ( D r ) p y ( D r ) p z ( D r )] = [ s ( r ) p x ( r ) p y ( r ) p z ( r )] D (n) = ( r ) D sp (n) ; (5.81)
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97 where D sp (n) = 0BBBBBBB@ 1 0 0 0 0 cos cos sin cos sin 0 sin cos cos sin sin 0 sin 0 cos 1CCCCCCCA : (5.82) W e can w ork out the mixing of d functions similarly the results of whic h w e presen t in T able 5.9 5.10 Conclusion W e ha v e thoroughly describ ed the metho ds to ev olv e the AkRg N cluster, starting with the Hamiltonian for the AkRg pair and generalizing to a full cluster in three dimensions. W e ha v e treated the Ak v alence electron explicitly and ha v e computed the electron in teractions using semilo cal l dep enden t pseudop oten tials. In the context of the EPQCLE w e ha v e treated the electron as a quan tal v ariable, and the Ak and Rg cores as quasiclassical. By computing the dip ole of the molecular system, w e ha v e sho wn ho w the electronic emission sp ectra of an initially excited Ak atom can b e computed. In order to carry out the ev olution of the system n umerically w e ha v e in tro duced a basis of segmen ted con tractions of Gaussian atomic functions, and ha v e ev aluated all required matrix elemen ts explicitly The n umerical computation of the HellmannF eynman force guiding the quasiclassical motion of the n uclear cores has b een deriv ed, and the greatly accelerated computation of the pseudop oten tial matrices through table lo okup, n umerical in terp olation and rotation has b een discussed in detail.
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98 T able 5{1. Pseudop oten tial rotation for d function mixing. D ij (n) elemen t v alue D xy ;xy (2 cos 2 1) cos D xy ;y z (2 cos 2 1) sin D xy ;xz 2 sin cos sin cos D xy ;x 2 y 2 2 sin cos cos 2 + sin cos sin 2 D xy ; 2 z 2 x 2 y 2 p 3 sin cos sin 2 D y z ;xy cos sin D y z ;y z cos cos D y z ;xz 2(cos 2 1) sin D y z ;x 2 y 2 sin sin cos D y z ; 2 z 2 x 2 y 2 p 3 sin sin cos D xz ;xy sin sin D xz ;y z sin cos D xz ;xz (2 cos 2 1) cos D xz ;x 2 y 2 cos sin cos D xz ; 2 z 2 x 2 y 2 p 3 cos sin cos D x 2 y 2 ;xy 2 sin cos cos D x 2 y 2 ;y z 2 sin cos sin D x 2 y 2 ;xz (2 cos 2 1) sin cos D x 2 y 2 ;x 2 y 2 (2 cos 2 1) cos 2 + 1 2 (2 cos 2 1) sin 2 D x 2 y 2 ; 2 z 2 x 2 y 2 q 3 4 (2 cos 2 1) sin 2 D 2 z 2 x 2 y 2 ;xy 0 D 2 z 2 x 2 y 2 ;y z 0 D 2 z 2 x 2 y 2 ;xz p 3 sin cos D 2 z 2 x 2 y 2 ;x 2 y 2 q 3 4 sin 2 D 2 z 2 x 2 y 2 ; 2 z 2 x 2 y 2 cos 2 1 2 sin 2
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CHAPTER 6 LITHIUMHELIUM CLUSTERS 6.1 In tro duction In Chapter 5, w e explored all asp ects of the sim ulation of alk alirare gas clusters, in the con text of the EPQCLE. In this c hapter, w e rene our description to the sp ecic case of a lithium atom em b edded in a cluster of helium atoms. The formalism is the same, in that the helium atoms and lithium core are treated quasiclassically while the lithium v alence electron is describ ed quan tally These clusters are only stable at v ery lo w temp eratures (a few degrees Kelvin), where core quan tal eects b ecome imp ortan t. Ho w ev er, b y appro ximating the heliumhelium in teraction with an eectiv e p oten tial, w e are able to mo del bulk liquid helium classically suc h that its temp erature, densit y and radial distribution function matc h exp erimen tal and path in tegral results. F urthermore, b y extracting a sphere of helium atoms from this bulk, and imp osing an appropriate constraining p oten tial, w e are able to repro duce a liquid helium droplet whose densit y prole matc hes quan tum Mon te Carlo sim ulations. With an adequate classical description of the cluster, w e in tro duce a ground state lithium atom in to its cen ter and monitor the conguration and electronic p opulation dynamics o v er time. W e also study the ev olution of the excited lithium atom on a mo del of the cluster surface, in tro ducing electromagnetic elds to induce electronic transitions and ligh t emission. 6.2 Description of the System There are t w o stable isotop es of liquid helium, fermionic 3 He and b osonic 4 He. 116 W e are fo cused on the 4 He isotop e (whic h w e will designate as simply He), although in the classical appro ximation the only dierence resides in their mass. A t temp eratures b elo w 4.2 K, helium v ap or condenses in to the liquid state, 117 while b elo w 99
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100 2.17 K the liquid exhibits sup erruid prop erties. 118 117 Indeed, the thermal w a v elength at these energies exceeds the t ypical corecore separation, and quan tal eects suc h as zerop oin t motion b ecome imp ortan t. 119 Ho w ev er, the excited electronic state of the He atom has an energy of 2 : 3 10 5 K ab o v e the ground state, making it v ery reasonable to assume the He atoms remain in their ground electronic conguration. 118 A t these temp eratures, the lo w kinetic energy p ermits the formation of stable clusters of helium con taining sev eral tens to sev eral h undreds of He atoms. These liquid droplets exhibit the bulk densit y and structure near their cen ter, with a decreasing densit y to w ard their surface. 120 When lithium atoms are in tro duced in to the cluster, they tend to reside on the surface, b ound b y a v ery shallo w w ell in the ground electronic state. 13 When the surface Li atom is excited, its subsequen t b eha vior dep ends hea vily on the excited state. The dierence in excited states is pictorally describ ed in Figure 6{1 Near a rat helium surface, the rst t w o excited states are degenerate, and rerect the p orbital aligned parallel to the surface. This Li(2 p ) conguration minimizes the electronic o v erlap of the helium with the lithium, and results in an attraction to w ard the surface. The third excited state, 2 p has the p orbital aligned orthogonally to the surface, where the electronic o v erlap induces repulsion b et w een the atoms. 121 While the Li(2 p ) tends to mo v e a w a y from the surface with minimal distortion of the He distribution, the Li(2 p ) mo v es to w ard the surface, where the He atoms resp ond b y clustering around the attractiv e Li atom. In exp erimen tal studies with excited Na, excimers of Na and He are found to desorb from the He surface within 70 to 700 ps, dep ending on the initial attractiv e state. 121 The desorb ed excimer is then found to emit to the red of the gas phase Na(3 p 3 s ) transition. Indeed, Li(2 p ) surrounded b y surface He atoms also emits to the red of the corresp onding
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101 A B Figure 6{1. Sc hematic of Li(2 p ) ab o v e a He surface. A) Li(2 p ). B) Li(2 p ). gas phase transition, but in terestingly no ligh t emission is observ ed when the Li(2 s ) is excited in the He bulk. 122 6.3 Prop erties to b e In v estigated First of all, w e are in terested in a classical description of the He atoms. While quan tal n uclear eects are clearly imp ortan t for ab initio descriptions of liquid helium, w e are fo cused on the in teraction of Li with the He, and seek only to reproduce the quan tum exp ectation of the He conguration. T o this end, it is imp ortan t to nd an eectiv e p oten tial for the HeHe in teraction that generates an appro ximately correct He distribution through classical sim ulation, for a giv en He densit y and temp erature. Second, w e wish to mo del the liquid He droplet, where the He atoms hold together and exhibit a densit y distribution that approac hes the bulk helium densit y in the cen ter and tap ers o to w ard the edge. This can b e done b y using an appropriate constraining p oten tial whic h k eeps the atoms in a cluster. Ha ving formed a classical mo del of the He droplet, w e wish to examine the dynamics of a Li atom in tro duced in to the cluster. Sp ecically w e are in terested in
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102 follo wing the n uclear motion and electronic p opulations of ground state Li initially em b edded in the cen ter of the cluster, to b etter understand the mec hanism b y whic h Li atoms tend to reside on the surface of the He cluster. Once w e ha v e clearly established that the Li atom is not miscible with the He liquid, w e shift fo cus to the in teraction b et w een Li and the surface He atoms. In part, w e are in terested in the congurational ev olution of the He atoms and Li core once the Li atom is elev ated to an excited electronic state. Dep ending on whether the or state is p opulated, the dynamics are exp ected to b e quite dieren t, and these dierences are in v estigated. W e are also in terested in the c hange in electronic p opulations and energy lev els as the excited Li atom either approac hes or recedes from the He surface. By in troducing a classical electromagnetic eld resonan t to electronic transitions, w e are able to induce p opulation transitions in Li(2 p ), and stim ulate ligh t emission that can b e seen in the dip ole emission sp ectrum. In addition to pro viding sp ectral results for the LiHe N cluster, w e gain in teresting insigh ts in to the inclusion of an external electromagnetic eld within the mixed quan tumclassical con text. 6.4 Preparation of LithiumHelium Clusters 6.4.1 Bulk Helium Our rst task is to generate a reasonable classical sim ulation of liquid He at ultralo w temp eratures. The four principal elemen ts in v olv ed in this kind of sim ulation are the system b oundaries, the propagation sc heme, the equilibration to attain the correct temp erature, and the in teratomic p oten tials. W e will discuss eac h of these in turn. In order to capture the eectiv ely innite spatial exten t of the liquid on the atomic scale, an attractiv e approac h is to use p erio dic b oundary conditions. 34 In this sc heme, a cen tral b o x is (imagined to b e) replicated at its sides and corners, so that there are 26 additional b o xes surrounding the cen tral v olume. Eac h of these
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103 b o xes con tains precisely the same conguration of He atoms as the cen tral b o x, and an y mo v emen t of an atom in the cen tral b o x (considered a r e al atom) is replicated b y the atoms in the surrounding b o xes (considered virtual atoms). If the in teratomic p oten tial is short range (as is the case for He atoms), then the minimum image con v en tion can b e used, whereb y a cuto distance of half the b o x length is used in the force calculations. This means that for a giv en He atom, only the neigh b oring atoms within a half b o xlength con tribute to the force on that He atom. In this w a y ev ery atom in the cen tral b o x con tributes to the force on ev ery other atom precisely once. If t w o atoms are within a half b o xlength of one another, then their in teraction is computed b et w een the corresp onding real atoms. If they are more than a half b o xlength from one another, then their in teraction o ccurs b et w een the real atom and the corresp onding virtual atom. In our sim ulations, w e b egan with a cen tral square b o x of length L = 40 : 46 au. This giv es a densit y = 0 : 00326 au 3 corresp onding to the kno wn bulk liquid He densit y 118 13 As for the initial state, w e w ould lik e to sim ulate the bulk He liquid in the canonical ensem ble, k eeping the n um b er of atoms, v olume and temp erature constan t. W e can pro vide a reasonable starting conguration b y randomly p ositioning the atoms within the cen tral b o x so that they are some minimal distance from one another. W e can also imp ose an initial temp erature b y assigning their momen ta acccording to the Maxw ellian distribution, p / exp p 2 2 mk B T ; (6.1) where k B is the Boltzmann constan t. W e p opulated our cen tral b o x using the minimal distance R min = 5 : 6 au for the conguration and temp erature T = 0 : 5 K for the momen tum distribution.
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104 There are man y metho ds of classical propagation, although w e found it useful to emplo y the v elo cit y V erlet algorithm, whic h is accurate to O ( t 2 ) and is selfstarting. In our case, it has the additional adv an tage of corresp onding directly to the propagation sc heme used for the quasiclassical v ariables in the relaxanddriv e pro cedure. While the c hoice of initial conditions ma y b e reasonable, it is necessary to equilibrate the system b y ev olving the atoms, and p erio dically rescaling the momen ta to obtain the desired temp erature, p p r T desir ed T actual : (6.2) There are other approac hes to temp erature calibration, but this rescaling sc heme a v oids c hanging the tra jectory directions so that the dynamics are b etter preserv ed. 34 Another imp ortan t issue in the equilibration is the timestep. T o k eep n umerical stabilit y w e imp osed a maxim um distance d max b y whic h an y atom could mo v e in a giv en timestep. This w as done b y nding the particle with maxim um sp eed v max and setting the (dynamically c hanging) timestep, t = d max v max : (6.3) Of course, this ignores the second order con tribution to the step in the v elo cit y V erlet algorithm, whic h dep ends on the particle accelerations. Ho w ev er, b y k eeping d max sucien tly small w e can neglect these higher order con tributions. W e found d max = 0 : 01 au to deliv er stable results without unnecessarily prolonging the sim ulation. Finally w e need to determine the eectiv e HeHe in teratomic p oten tial to generate the correct particle distribution from classical sim ulations. A t higher temp eratures, the HeHe p oten tial from Aziz and co w ork ers, 110 V Az iz ( R ), pro vides a
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105 v ery accurate description of the HeHe in teraction. Its form w as describ ed in Chapter 5, and its parameters for helium are sho wn in T able 6{1 Ho w ev er, b ecause of the quan tal eects at lo w temp eratures, w e exp ect an eectiv e p oten tial V ef f ( R ) substan tially dieren t from V Az iz ( R ) to b e necessary for the classical sim ulation to describ e the bulk He distribution. This can b e seen in Figure 6{2 where w e plot the radial distribution function obtained using V Az iz in our classical sim ulation, and compare it to path in tegral results. 118 Not only is the true radial distribution substan tially rattened b y appro ximately an order of magnitude, but it is also shifted to the righ t b y roughly 1 au. W e found that b y adding a correction, V 0 ( R ), to the Aziz p oten tial, V ef f ( R ) = V Az iz ( R ) + V 0 ( R ) ; (6.4) w e w ere able to generate a similar radial distribution function to results from pathin tegral calculations. 118 The eectiv e p oten tial con v erges to the Aziz p oten tial at short distances, but has an attractiv e w ell whic h is shifted to the righ t and is signican tly more shallo w than V Az iz The correction term w as constructed b y taking the dierence b et w een a scaled, shifted Aziz p oten tial and the original Aziz p otential, and m ultiplying b y a sigmoidal function that is asymptotically zero at short distances, V 0 ( R ) = ( R )[ V Az iz ( R R M =R s ) V Az iz ( R )] ; (6.5) ( R ) = 1 1 + exp [ a ( R R )] : (6.6) The parameters for V 0 are sho wn in T able 6{2 The p oten tials are graphed in Figure 6{3 and the eectiv e p oten tial is sho wn with expanded axes in Figure 6{4 Note that while the eectiv e p oten tial is substan tially ratter than the original Aziz p oten tial, it nonetheless retains an attractiv e w ell. Sim ulations rev ealed that the radial distribution function is v ery sensitiv e to the heigh t and p osition of this w ell.
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106 T able 6{1. P arameters for the HeHe in teraction from Aziz ( V Az iz ). R m A D C 6 C 8 C 10 5.6125 0.34648 1.922 10.735 1.893 1.414 1.349 0.414 0.171 T able 6{2. P arameters for the correction to the HeHe in teraction ( V 0 ) a R R S 4.0 4.0 0.003 6.62 0 2 4 6 8 10 12 14 2 4 6 8 10 12 14 16 g(R)R (au) V Aziz classical V Aziz path integral V eff classical 0 0.5 1 1.5 2 4 6 8 10 12 14 Figure 6{2. Radial distribution functions for bulk liquid helium.
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107 0.0006 0.0004 0.0002 0 0.0002 0.0004 0.0006 0.0008 0.001 2 3 4 5 6 7 8 9 10 Energy (au)R (au) V Aziz V' V eff = V Aziz + V' Figure 6{3. Comparison of the Aziz p oten tial with the eectiv e form.
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108 2e07 1.5e07 1e07 5e08 0 5e08 1e07 1.5e07 2e07 4 5 6 7 8 9 10 Energy (au)R (au) V eff Figure 6{4. Eectiv e HeHe p oten tial.
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109 6.4.2 Liquid Helium Droplets Ha ving formed an acceptable mo del of liquid He, w e are no w in a p osition to generate a mo del of a He droplet. W e b egin with a w ellequilibrated p erio dic b o x of liquid He at = 0 : 00326 au 3 and T = 0 : 5 K, and extract a sphere of 100 He atoms. Although there are w eak v an der W aals forces holding these atoms together, one m ust remem b er that w e ha v e used an eectiv e p oten tial with a v ery shallo w w ell, so that ev en at these ultralo w temp eratures, the He atoms will ultimately escap e in to the gas phase. T o coun ter this, w e in tro duce the constraining p oten tial discussed in Chapter 5, whic h consists of a sigmoidal function near the edge of the cluster. The parameters of the sigmoidal function w ere adjusted un til w e obtained a stable cluster whose densit y distribution w as consisten t with path in tegral and diusion Mon te Carlo calculations. 118 13 F or the constraining function, V hol d ( R ) = a 1 + exp [ b ( R R hol d )] ; (6.7) w e w ere able to generate an acceptable pure He droplet using a = 5 10 4 b = 0 : 08 and R hol d = 80 : 0. In Figure 6{5 w e graph the constraining p oten tial. Figure 6{6 sho ws that the temp erature of the drop is prop erly calibrated to T = 0 : 5 K, and that ructuations are small. Finally Figure 6{7 displa ys the densit y prole from the cen terofmass of the cluster, comparing our results with path in tegral and diusion Mon te Carlo calculations.6.4.3 LithiumHelium In teractions The LiHe p oten tials are describ ed using the pseudop oten tial formalism discussed in the previous c hapter. The parameters are dened in T ables 6{3 to 6{5 F or the electronic basis, w e used the 5s5p4d basis set of Cartesian Gaussian atomic functions tak en from Czuc ha j and co w ork ers. 90 In the w ork b y Rey es, 109 adiabatic p oten tial curv es including the d symmetries w ere ev aluated as a function of basis
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110 0 5e05 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004 0.00045 0.0005 0 20 40 60 80 100 120 140 160 Energy (au)R (au) V hold Figure 6{5. Constraining p oten tial used to k eep He atoms from ev ap orating.
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111 0.485 0.49 0.495 0.5 0.505 0.51 0.515 0.52 0 2e+06 4e+06 6e+06 8e+06 1e+07 Temperature (K)time (au) Figure 6{6. T emp erature ructations of the He droplet o v er time.
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112 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0 5 10 15 20 25 30 35 40 r (R)R (au) V eff classical V Aziz PIMC V Aziz DMC bulk 4 He Figure 6{7. Helium densit y prole from the cen terofmass of the cluster.
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113 T able 6{3. P arameters for the e Li in teraction. c c l B l l 0.1915 0.831 0 5.786 1.276 1 1.065 1.607 T able 6{4. P arameters for the e He in teraction. d B q B 1 l i B l l 1.3843 2.3265 0.706 0.79 0 1 0.83 1.30 0 2 2.27 0.50 0 2 2.27 0.50 1 1 0.12 0.75 1 2 1.87 1.00 size, and con v ergence w as found with the 5s5p4d basis. This basis pro vides an excellen t description of the lo w er adiabatic energies for the LiHe pair, as sho wn in Figure 6{8 F or this reason, w e emplo y ed this basis set for the remainder of our LiHe cluster sim ulations. F rom Figure 6{8 w e see that the 2 s ground state is essen tially repulsiv e. In fact, it con tains a v ery shallo w w ell near 11 au (not sho wn), whic h is resp onsible for the lo ose binding of Li atoms to the surface of He clusters. Ho w ev er, its depth is only 2 K, so that it will easily ev ap orate from the cluster. W e also see that the rst t w o excited states of lithium, 2 p and 2 p are substan tially dieren t in c haracter. In fact, the 2 p is a degenerate state of p orbitals in the plane p erp endicular to the LiHe axis, while 2 p con tains the p orbital parallel to the LiHe axis. The electronic o v erlap in the 2 p results in the curv e sho wing repulsion at all distances, while the 2 p curv e has an attractiv e w ell with a minim um around 4 : 5 au. These dierences result in v ery dieren t LiHe dynamics for dieren t electronic states.
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114 T able 6{5. P arameters for the LiHe core in teraction. A b R D 27.26 2.29 1.93 0.2 0.15 0.1 0.05 0 3 4 5 6 7 8 9 10 11 12 Energy (au)R (au) 2s s 2p p 2p s 3s s Figure 6{8. Adiabatic energy for Li and He as a function of in tern uclear distance.
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115 W e w ere also in terested in the eect of additional He atoms on the adiabatic energy giv en our in ten t to in tro duce the Li atom in to a cluster of He atoms. T o explore this eect, w e placed a single He atom at the origin, and then observ ed the c hange in adiabatic curv es as w e placed additional He atoms in the vicinit y of the origin. F or eac h conguration of helium atoms, w e mo v ed the lithium atom to w ard the origin along the zaxis, plotting the adiabatic energy as a function of the distance from the lithium atom to the origin. Initially w e added a single He atom at (0 ; 0 ; 6) au, and then another at (0 ; 0 ; 12) au. W e used a spacing of 6 au, since the radial distribution function p eak ed around this distance. The resulting adiabatic curv es are sho wn in Figure 6{9 where it is clear that the addition of the nal He atom aects the energy curv es negligibly In a second trial, w e b egan with a single He atom at the origin, and then added t w o along the yaxis: (0 ; 6 ; 0) au and (0 ; 6 ; 0) au. W e then added t w o more along the same line, (0 ; 12 ; 0) au and (0 ; 12 ; 0) au. The adiabatic curv es are plotted in Figure 6{10 where w e see the eects of additional He atoms are greater, but nonetheless the addition of the second set con tributes negligibly Finally w e lo ok ed at appro ximations to a cluster surface b y studying a 3 3 2 lattice of He atoms spanning [ 6 ; 6] [ 6 ; 6] [ 6 ; 0] au 3 with spacing of 6 au along eac h axis. The resulting adiabatic curv e w as compared to a larger 5 5 3 lattice of He atoms spanning [ 12 ; 12] [ 12 ; 12] [ 12 ; 0] au 3 again with 6 au spacing. The curv es are compared to the LiHe pair in Figure 6{11 The results rerect our exp ectations from the earlier curv es, in that there is v ery little dierence b et w een the smaller and larger surface in terms of adiabatic energies. These results giv e us condence that one can safely use a cuto in the order of 24 au (or ev en less) when computing the LiHe in teractions. This is esp ecially imp ortan t in larger cluster sim ulations, where the dominan t computational time is sp en t on matrix calculations.
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116 0.2 0.15 0.1 0.05 0 3 4 5 6 7 8 9 10 11 12 Energy (au)R (au) 2s s 2p p 2p s One He Two He Three He 0.13176 0.13168 0.1316 4.2 4.3 4.4 4.5 4.6 4.7 4.8 2p p 2s s 2p p 2p s Figure 6{9. Adiabatic energies for Li and one or more He along the zaxis.
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117 0.2 0.15 0.1 0.05 0 3 4 5 6 7 8 9 10 11 12 Energy (au)R (au) 2s s 2p p 2p s One He Three He Five He 0.1322 0.13204 0.13188 0.13172 4.2 4.3 4.4 4.5 4.6 4.7 4.8 2p p 2s s 2p p 2p s Figure 6{10. Adiabatic energy for Li and one or more He along the yaxis.
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118 0.2 0.15 0.1 0.05 0 3 4 5 6 7 8 9 10 11 12 Energy (au)R (au) 2s s 2p p 2p s One He 3x3x2 He 5x5x3 He 0.198115 0.198095 0.198075 9.5 10 10.5 11 11.5 12 12.5 13 2s s 2s s 2p p 2p s Figure 6{11. Adiabatic energy for Li and a surface of He atoms parallel to the xy plane.
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119 6.5 Results: Lithium Inside the Helium Cluster Exp erimen tal and quan tum mec hanical structure calculations sho w that the Li atom in v ariably resides on the surface of the He cluster, rather than within its bulk. W e examined this phenomenon b y b eginning with a liquid helium droplet in thermal equilibrium at T = 0 : 5 K. Our droplet con tained 100 He atoms, and w as brough t to equilibrium through the v elo cit y rescaling describ ed earlier. A t this p oin t, the He atom closest to the cen terofmass of the cluster w as replaced with a Li atom in the molecular ground state. This Li atom w as tak en to ha v e no momen tum initially so that all subsequen t dynamics w ere a result of in teractions with the surrounding He atoms. W e sho w the ev olution of the LiHe 99 cluster in Figure 6{12 where snapshots of the xy plane are compared at initial and nal times. As exp ected, the Li atom mo v es to w ard the cluster surface. What is in teresting, ho w ev er, is the sp eed at whic h it is ejected from the cluster's in terior. The path is direct, and motion from the cen ter to the surface (corresp onding to the t w o snapshots) tak es only 10,000 au (241 fs). This is a far greater sp eed than w ould b e exp ected through thermal motion at 0 : 5 K, and indicates that not only is the Li atom ejected from the cluster, but is ejected violen tly The con v erse is seen in exp erimen ts where Li can only b e forced in to liquid He through laser ablation. This is explained b y the strong repulsion of the ground state, esp ecially at t ypical distances separating He atoms, and sho ws that while the ground Li atom migh t b e pic k ed up on the cluster surface, it is extremely unlik ely to p enetrate this surface in to the cluster in terior. T o v erify that the motion of the Li atom is drastically dieren t from the thermal motion of the He atoms, w e p erformed the same sim ulation, but without replacing the He atom b y the Li atom. W e then follo w the ev olution of the cen tral He atom as it slo wly w anders through the cluster, and compare the motion to the Li atom in Figure 6{13 There are t w o ma jor dierences in their motion, as seen from the
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120 40 30 20 10 0 10 20 30 40 40 30 20 10 0 10 20 30 40 Y (au)X (au) A Initial 40 30 20 10 0 10 20 30 40 40 30 20 10 0 10 20 30 40 Y (au)X (au) B Initial Final Figure 6{12. Ev olution of ground state Li em b edded in the cen ter of a He cluster. A) Initial time t = 0 au. B) Final time t = 10,000 au.
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121 gure. The rst is that the He atom mak es essen tially a random w alk through the cluster, and sta ys within a cen tral v olume of radius 11 au. The second is that the He atom mo v es m uc h more slo wly than the Li atom. In fact, on the time scale of the Li motion, the He atom app ears nearly stationary whic h is wh y w e ha v e had to scale the He time axis b y t w o orders of magnitude to sho w an y appreciable mo v emen t. In Figure 6{14 w e trac k the electronic p opulations of the Li as it emerges from the cluster. W e see that the p opulation remains almost exclusiv ely in the ground state, with a ground p opulation loss of less than 0 : 5% b y the time the Li atom reac hes the surface. This small loss is lik ely attributable to the higher collision energies a v ailable to the Li atom as it gains momen tum, so that close approac hes to He atoms resulting in p opulation transfer are p ossible. One should note that at times earlier than 2000 au, b efore the Li atom pic ks up m uc h momen tum, p opulation transfer is not visibly detectable on the scale of the graph. This is a recurren t theme that w e will see again as w e ev olv e excited Li atoms, in that the v ery lo w kinetic energies are insucien t to bring the Li and He atoms close enough where curv e crossing, and therefore state mixing, are appreciable. 6.6 Results: Lithium on the Helium Cluster Surface Ha ving established that the Li atom do es not rest within the cluster, w e wish to in v estigate its b eha vior on the cluster surface. Sp ecically w e are in terested in follo wing the dynamics once the Li has b een excited to either the 2 p or 2 p state. Ho w ev er, since our cluster has a rather diuse surface, w e instead mo del the surface as a square lattice of He atoms. Reviewing the adiabatic curv es computed earlier, w e nd a lattice of 5 5 3 He atoms to b e sucien tly large to mo del a droplet surface. Using a square lattice also p ermits us to b etter dene the inital electronic state of the Li atom (2 p or 2 p ), and ev aluate the resp onse of the He atoms to the presence and motion of the attac hed Li atom.
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122 5 10 15 20 25 30 0 2000 4000 6000 8000 10000 distance from c.o.m. (au)time (au) Li(2s s ) He Figure 6{13. Comparison of Li and He motion within a He cluster. The time scale has b een reduced b y a factor of 100 for the He curv e.
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123 0.98 1 1.02 1.04 1.06 1.08 1.1 0 2000 4000 6000 8000 10000 h (2s s )time (au) 0.997 0.9975 0.998 0.9985 0.999 0.9995 1 1.0005 1.001 0 2000 4000 6000 8000 10000 Figure 6{14. Electronic p opulation of Li as it emerges from the He cluster.
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124 F or this lattice, w e nd a minim um energy along the ground state adiabatic curv e at R = 10 : 7 au, and th us place the Li atom initially at (0 ; 0 ; 10 : 7) au. Similar to the last section, w e b egin with the Li atom motionless, so that all n uclear motion comes from the LiHe in teractions. In the next t w o sections, w e follo w the dynamics of the Li atom once it has b een electronically excited. 6.6.1 Dynamics of Li( 2 p ) W e assume the Li atom has b een excited to 2 p and follo w its dynamics from this initial state. Snapshots of its ev olution are displa y ed in Figure 6{15 where w e see the motion of the He atoms and Li atom along the yz plane. Since this is a repulsiv e state, the Li atom mo v es a w a y from the He surface. W e also see the He atoms rep el in w ard from the Li atom. The snapshots are tak en at the initial time t = 0 au, and at the nal time t = 33 ; 000 au. Similar desorption from the surface is seen in exp erimen ts where either Na or K is excited to the repulsiv e state, and then immediately lea v es the surface. 11 The excited state deca ys slo wly and in the time frame of our sim ulation the Li atom remains in the 2 p state throughout. Ho w ev er, as sho wn in Figure 6{16 w e notice that while the total 2 p p opulation remains unit y there is almost complete transfer b et w een the 2 p and 2 p p opulations at around t = 2000 au. This is explained b y the fact that the 2 p state b ecomes triply degenerate at large distances, so that the 2 p and 2 p p opulations mix completely Ho w ev er, their degeneracy means that v ery little ligh t emission can b e exp ected. On the other hand, w e can induce dip ole emission b y p erturbing the system with an electromagnetic (EM) eld, ^ H 0 = ^ H ^ D E 0 cos ( t + ) ; (6.8) where ^ D is the system dip ole. F or an isolated Li atom, w e nd an energy dierence E = 0 : 068 au b et w een the ground and rst excited state. An EM eld with
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125 20 15 10 5 0 5 10 15 20 20 15 10 5 0 5 10 15 20 25 30 Y (au)Z (au) A Li 20 15 10 5 0 5 10 15 20 20 15 10 5 0 5 10 15 20 25 30 Y (au)Z (au) B Li Figure 6{15. Ev olution of Li(2 p ) as it recedes from the He cluster surface. A) Initial time t = 0 au. B) Final time t = 33 ; 000 au.
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126 0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 5000 10000 15000 20000 25000 30000 35000 htime (au) 2p p 2p s 2p p + 2p s Figure 6{16. Mixing of the Li(2 p ) and Li(2 p ) states at distances where Li(2 p ) is triply degenerate.
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127 frequency = 0 : 068 au 1 will stim ulate absorption and emission of ligh t b y the gas phase Li, as the Li atom exp eriences Li(2 p 2 s ) (called here the D ) transitions. As the EM frequency shifts from the electronic transition energy ligh t absorption and emission are still presen t, but with less magnitude. F or Li(2 p ) near the He surface, w e in tro duced an EM eld resonan t to the D line, with amplitude E 0 = 1 : 0 10 5 In addition, w e added the phase = 3 = 2 so that the electric eld rises from zero amplitude, rather than b eginning abruptly Finally w e c hose = (1 ; 1 ; 1) = p 3 so that there are comp onen ts along eac h axis, and the p erturbation has an eect regardless of the orien tation of the Li atom. In Figure 6{17 w e see the eect of in tro ducing the eld at t = 30 ; 000 au. By t = 32 ; 000 au, appro ximately half the total 2 p p opulation has deca y ed to the ground state. A t this p oin t, the 2 p p opulation b egins to rise again, corresp onding to ligh t absorption. This pattern of electronic deca y and reco v ery results from our semiclassical treatmen t of ligh tmatter in teractions. The eect of the EM eld is to induce the emission of dip ole radiation, whic h is measured from the second time deriv ativ e of the dip ole. Figure 6{18 sho ws the resulting dip ole sp ectrum for t w o time p erio ds, one when the p erturbing eld is applied during the initial 3000 au, and the second when the eld is applied during the nal 3000 au. W e also sho w the gas phase Li(2 p 2 s ) emission line, whic h w e compute from our asymptotic LiHe energies as 14903 cm 1 (and is extremely close to the kno wn v alue, 14904 cm 1 ). W e see that the Li(2 p ) emission p eak is initially blue shifted b y ab out 50 cm 1 ; at nal times, the p eak is substan tially broadened. This broadening can b e explained b y the motion of the He atoms, whic h ha v e acquired additional thermal energy from the repulsiv e in teraction with the Li atom.
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128 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 29000 30000 31000 32000 33000 h2ptime (au) without field with field Figure 6{17. Electronic p opulation of Li with and without a p erturbing electromagnetic eld, resonan t to the D line.
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129 13000 13500 14000 14500 15000 15500 16000 16500 17000 Intensity (arbitrary units)n (cm 1 ) initial time final time gas phase Li(2p <2s) Figure 6{18. Dip ole emission sp ectra of Li(2 p ) as it recedes from the He cluster surface.
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130 6.6.2 Dynamics of Li( 2 p ) The ev olution of the Li(2 p ) atom presen ts substan tially more in teresting features than the Li(2 p ) atom, as it is dra wn in to w ard the cluster surface. As suc h, it is not a matter of simply lea ving the cluster, but rather in v olv es non trivial in teractions with the He atoms. Snapshots are sho wn in Figure 6{19 again displa y ed as cross sections along the yz plane. Because of the attractiv e nature of the 2 p surface, w e see the Li atom indeed mo v es in to w ard the cluster surface. The Li atom ev en tually em b eds itself in the surface, and b y t = 67 ; 000 au it is completely surrounded b y He atoms. The He densit y around the Li atom eviden tly increases, although the HeHe repulsion coun ters this tendency once a certain He densit y is reac hed. Exp erimen tal results on excited Na atoms sho w the formation of NaHe N excimers, with N t ypically 3 or 4. According to the exp erimen tal results, once the excimers form, they desorb from the surface in 70 to 700 ps. 121 Our dynamics indicate that LiHe N excimers form on a m uc h shorter timescale (1.66 ps), alb eit w e are considering a dieren t alk ali atom. Ho w ev er, w e w ere not able to repro duce subsequen t desorption in longer sim ulation runs, partly b ecause of n umerical instabilities whic h ev en tually accum ulate. In Figure 6{20 w e sho w the p opulation of the 2 p state as the Li atom en ters the surface. P opulation transfer is exceptionally small, sho wing less than 0 : 01% o v er t = 67 ; 000 au, and once again reinforcing the conclusion that at these ultralo w temp eratures, collisioninduced p opulation transfer is negligible. Ho w ev er, similar to the study of Li(2 p ), w e are able to prob e the electronic state b y in tro ducing an electromagnetic eld and observing the dip ole emission sp ectrum. In Figure 6{21 w e sho w the dip ole sp ectrum resulting from an EM eld resonan t to the D line, in tro duced during the rst 3000 au of the sim ulation. The p eak of this sp ectrum o ccurs at = 14895 cm 1 whic h is shifted to the red of the gas phase emission b y 8 cm 1 Laserinduced ruorescence exp erimen ts sho w that a
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131 20 15 10 5 0 5 10 15 20 20 15 10 5 0 5 10 15 20 25 30 Y (au)Z (au) A Li 20 15 10 5 0 5 10 15 20 20 15 10 5 0 5 10 15 20 25 30 Y (au)Z (au) B Li Figure 6{19. Snapshot of Li(2 p ) as it in teracts with the He cluster surface. A) Initial time t = 0 au. B) Final time t = 67 ; 000 au.
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132 0.999 0.9992 0.9994 0.9996 0.9998 1 1.0002 1.0004 0 10000 20000 30000 40000 50000 60000 70000 h2p ptime (au) Figure 6{20. Electronic p opulation of Li(2 p ) as it in teracts with the He cluster surface.
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133 Li(2 s ) attac hed to a He cluster surface emits at a frequency whic h is red shifted b y ab out 10 cm 1 13 agreeing with our results. A m uc h more substan tial shift, ho w ev er, is found when the same EM eld is in tro duced in the nal 3000 au of the sim ulation, while the Li atom is surrounded b y the He atoms. Sho wn in Figure 6{22 the dip ole sp ectrum is shifted to the red b y appro ximately 4000 cm 1 While this v ery large shift initially app ears susp ect, it is explained b y examining the adiabatic energy curv es for a Li atom surr ounde d b y He atoms, rather than approac hing a surface. In this calculation, w e ha v e k ept the Li atom at the origin, and surrounded it with a square lattice of 8 He atoms (one at eac h corner of the lattice). W e ha v e then computed the adiabatic energy as a function of the distance the He atoms are lo cated along eac h axis. The curv es for the ground and rst excited state are sho wn in Figure 6{23 W e nd that ev en substan tially b efore the minim um of the w ell is reac hed, the distance b et w een the t w o curv es corresonds to the energy p eak of the dip ole emission sp ectrum. It is also in teresting to note ho w close the t w o curv es come to one another at small distances. This ma y explain wh y radiativ e deca y is not observ ed for Li in bulk He, where the n uclear congurations are forcibly suc h that the Li atom is alw a ys surrounded b y He atoms. In this case, it ma y b e energetically p ossible for excited Li atoms to reac h the left side of the adiabatic curv e, and deca y through curv e mixing in to the ground state. If this transfer w ere to o ccur b efore the time scale of sp on taneous emission, there w ould b e v ery little ligh t emission, and certainly none in the frequency range of the D emission line. In order to v alidate these sp ectral and adiabatic curv e results for Li(2 p ), w e in tro duced an electromagnetic eld resonan t with the supp osed sp ectral p eak, = 10 ; 730 cm 1 W e k ept the amplitude and phase the same as b efore, and turned on the eld at t = 64 ; 000 au. The results are sho wn in Figure 6{24 One can see that the 2 p p opulation deca ys steadily to ab out 25%, while the 2 s p opulation rises
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134 13000 13500 14000 14500 15000 15500 16000 16500 17000 Intensity (arbitrary units)n (cm 1 ) initial time gas phase Li(2p <2s) Figure 6{21. Dip ole emission sp ectrum of Li(2 p ) during the rst 3000 au.
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135 7000 8000 9000 10000 11000 12000 13000 14000 Intensity (arbitrary units)n (cm 1 ) Figure 6{22. Dip ole emission sp ectrum of Li(2 p ) during the nal 3000 au.
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136 0.2 0.18 0.16 0.14 0.12 0.1 2 2.5 3 3.5 4 4.5 5 5.5 6 Energy (au)R (au) D E = 10730 cm 1 2s s 2p p Figure 6{23. Adiabatic curv es of Li surrounded b y a cubic lattice of He atoms. The parameter R refers to the halflength of the lattice edge.
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137 corresp ondingly The EM eld frequency is rerected in the small ondulations, but the net p opulation transfer o v er time can b e in terpreted as induced ligh t emission. This sho ws that the classical electromagnetic eld can indeed b e used to induce b oth absorption and emission of ligh t. 6.7 Conclusion W e found that w e could repro duce the distribution of bulk liquid He at ultralo w temp eratures through a classical sim ulation b y using an eectiv e p oten tial whic h w as signican tly scaled do wn as w ell as shifted from the p oten tial prop osed b y Aziz and co w ork ers. Sp ecically w e w ere able to qualitativ ely repro duce the radial distribution function in the bulk. F urthermore, b y extracting a sphere of He atoms from this bulk, and imp osing a sigmoidal con tainmen t p oten tial on the system, w e w ere able to pro duce a liquid helium droplet whose densit y prole matc hed path in tegral and diusion Mon te Carlo results. Ha ving sucien tly mo deled the He droplet, w e in tro duced a ground state Li atom in to its cen ter, and follo w ed its dynamics. W e found that it w as rapidly exp elled from the cluster, and con trasted this motion from the slo w random w alk tak en b y a t ypical He atom. W e also found that the p opulation of the Li atom remained predominan tly in the ground state. W e then examined the dynamics of an excited Li(2 p ) atom near a mo del He surface. W e found that it mo v ed a w a y from the surface, while the He atoms recoiled somewhat in to the cluster. As with the Li(2 s ), the p opulation remained primarily in the original 2 p state. In order to prob e the electronic structure, w e in tro duced an electromagnetic eld to the system, resonan t to the gas phase Li(2 p 2 s ) line. W e computed the resulting dip ole sp ectrum, and found it to b e initially shifted appro ximately 50 cm 1 to the blue of the D line. A t later times, the p eak broadened, o wing to the increased thermal motion of the He atoms.
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138 0 0.2 0.4 0.6 0.8 1 64000 64500 65000 65500 66000 66500 67000 htime (au) Li(2p p ) Li(2s s ) Figure 6{24. Deca y of Li(2 p ) surrounded b y surface He atoms, induced b y an EM eld with frequency resonan t to the Li(2 p 2 s ) transition.
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139 W e also studied the dynamics of the excited Li(2 p ) atom near the mo del surface. In con trast to the 2 p state, the Li(2 p ) atom mo v ed to w ard the surface, and ev en tually found itself surrounded b y a high densit y of He atoms. These excimers ha v e b een sho wn to form for Na atoms near He clusters, follo w ed b y desorption of the excimer from the cluster. W e w ere able to sho w the formation of the excimers, but could not repro duce the subsequen t desorption, p ossibly due to the v ery long times required for the desorption to tak e place. An external electromagnetic eld ga v e rise to a dip ole emission sp ectrum initially shifted to the blue of the gas phase transition for Li(2 p ), and shifted to the red for Li(2 p ). These shifts are in agreemen t with exp erimen tal and other computational results. When the Li w as surrounded b y surface He atoms, the dip ole sp ectrum w as found to b e shifted to the red b y ab out 4000 cm 1 This v ery large shift w as explained b y lo oking at the adiabatic curv es of Li surrounded b y a matrix of He atoms, whic h rev ealed that the ground and rst excited states came sucien tly close to one another to corresp ond to the dip ole sp ectrum p eak. The adiabatic curv es also rev ealed ho w excited Li atoms in bulk He migh t b e able to deca y to the ground state with minimal ligh t emission, giv en the pro ximit y of the curv es. Finally w e in tro duced an electromagnetic eld resonan t with the dip ole emission p eak for Li(2 p ) surrounded b y He atoms. This electromagnetic eld induced deca y in to the ground state, un til the 2 s p opulation reac hed appro ximately 75%. This v alidated the sp ectral results, and demonstrated that the classical electromagnetic eld can b e used in mixed quan tumclassical sim ulations to induce b oth electronic excitation and deexcitation.
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CHAPTER 7 CONCLUSION The purp ose of this study has b een to dev elop a computationally feasible time dep enden t formalism to describ e the dynamics of fewand man yatom systems. W e ha v e dev elop ed a mixed quan tumclassical approac h whic h sho ws excellen t agreemen t with full quan tal calculations on small mo del systems, and is also capable of treating the ev olution of a realistic threedimensional mo del of an alk ali atom em b edded within a cluster of rare gas atoms. The general pro cedure used for the tests mo dels in tro duces an eectiv e p oten tial to the mixed quan tumclassical formalism, and uses classical tra jectories coupled to quan tal ev olution to generate the dynamics. The eciency of the calculations w as impro v ed b y in tro ducing a propagation algorithm whic h tak es in to accoun t the dieren t time scales of the n uclear and electronic motion, and pro vides a means to ev olv e the system with v arying timesteps, maximizing eciency while ensuring a prescrib ed lev el of accuracy In order to describ e the alk alirare gas clusters, the general formalism w as extended to incorp orate pseudop oten tial in teractions b et w een the alk ali v alence electron and the n uclear cores, and n umerical in terp olation metho ds w ere used to render the calculation of the Hamiltonian matrix elemen ts computationally feasible. In this conclusion, w e review the principal results of eac h c hapter. 7.1 Eectiv e P oten tial Quan tumClassical Liouville Equation Beginning with the quan tum Liouville equation of motion for the densit y op erator, w e divided the (general) system in to quan tal and quasiclassical degrees of freedom. By taking the partial Wigner transform o v er the quasiclassical degrees of freedom only and appro ximating for a large quasiclassical/quan tal mass ratio, 140
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141 w e deriv ed a mixed quan tumclassical equation of motion for the partially Wigner transformed densit y op erator. The transformed densit y op erator b ecomes a function of the quasiclassical phase space, but remains an op erator in quan tal space. By in tro ducing an eectiv e p oten tial to the mixed quan tumclassical equations of motion, w e generate a simplied v ersion, the eectiv e p oten tial quan tumclassical Liouville equation (EPQCLE), whic h can b e solv ed b y follo wing tra jectories through the quasiclassical phase space. This sc heme is rendered n umerically accessible b y in tro ducing a general basis set for the quan tal space, and b y discretizing the quasiclassical space on to a grid. The grid p oin ts ev olv e through quasiclassical tra jectories whic h are coupled to the quan tal state through the eectiv e force. The quan tal state ev olv es sim ultaneously using the relaxanddriv e metho d, whic h divides the quan tal motion in to a relaxation term com bined with a driving con tribution from the ev olving quasiclassical v ariables. The relaxanddriv e sc heme w as mo died to pro vide a robust v ariable timestep whic h accoun ted for the t ypically fast quan tal motion compared to the slo w er quasiclassical tra jectory ev olution. In the EPQCLE sc heme, measured quan tities w ere computed through an in tegration o v er quasiclassical phase space of the trace o v er quan tal v ariables. In this w a y exp ectation v alues of a general op erator, as w ell as p opulations of quan tal states, could b e computed analogously to the full quan tal calculations. 7.2 OneDimensional Tw oState Mo dels By sim ulationg three simple t w ostate mo dels, w e w ere able to compare the EPQCLE metho d with the exact quan tum mec hanical solution, obtained through the splitop erator fast F ourier transform grid metho d. W e found the EPQCLE to sho w excellen t agreemen t with the full quan tal results for man y observ ables, and at least qualitativ e agreemen t for all mo dels and energies. W e follo w ed a mo del of an alk ali atom aproac hing a metal surface, and ev aluated probabilit y transitions, p osition exp ectation and its deviation, momen tum
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142 exp ectation and its deviation, the densit y function and phase space grid ev olution. W e found the EPQCLE metho d repro duced qualitativ ely the exact v alues obtained through the quan tal treatmen t for all observ ables, and sho w ed the phase space grid to deform due to the action of the eectiv e p oten tial. W e examined a collision mo del in v olving t w o a v oided crossings, and found the transmission probabilit y obtained b y the EPQCLE to deviate from the quan tal results at lo w er energies. This problem w as observ ed in other theoretical mo dels whic h did not incorp orate n uclear coherence, and w as exp ected to arise b ecause of the imp ortance of n uclear coherence in the dual crossing mo del. Ho w ev er, w e found excellen t transmission probabilit y agreemen t at higher energies. Finally w e sim ulated a mo del of so dium in teracting with io dine, where the NaI complex oscillated b et w een a long range attractiv e ionic state and an asymptotically free neutral state. Disso ciation w as seen to o ccur at the diabatic crossing, as some of the ionic state w ould cross in to the neutral state and propagate to w ard free Na and I atoms. The oscillatory b eha vior w as repro duced qualitativ ely b y the EPQCLE mo del, and coherence b et w een the states sho w ed agreemen t with the quan tal results for long sim ulation times. Ev aluation of the phase space grid rev ealed b oth the oscillatory motion of the b ound NaI complex, and the disso ciated neutral state of the free atoms. 7.3 Alk aliRare Gas Clusters W e ha v e applied the EPQCLE formalism to the study of alk ali atoms em b edded within rare gas clusters. This has b een done b y treating the AkRg in teraction as a threeb o dy problem comp osed of the alk ali core, its v alence electrons and the rare gas core. In the con text of the EPQCLE, w e ha v e treated the electron as a quan tal v ariable and the atomic cores as quasiclassical. The electroncore in teractions ha v e b een describ ed using semilo cal l dep enden t pseudop oten tials, while the AkRg core in teractions ha v e b een describ ed with a tted parametric curv e. The
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143 RgRg in teraction w as mo deled b y a parametric curv e, and the threeb o dy system w as generalized to the case of one Ak atom and sev eral Rg atoms. In order to propagate the system, w e in tro duced a basis of segmen ted contractions of Gaussian atomic functions cen tered on the alk ali core. W e ev aluated all required matrix elemen ts explicitly and describ ed in detail the n umerical ev aluation of the HellmannF eynman force guiding the atomic cores. In addition, w e ha v e implemen ted a sc heme in v olving lo okup tables, in terp olation and rotation to ecien tly compute the pseudop oten tial matrix elemen ts, a ma jor b ottlenec k in the n umerical propagation of the cluster. W e ha v e studied the dynamics and sp ectra of a lithium atom em b edded in a cluster of helium atoms at v ery lo w temp eratures ( 0 : 5 K). W e follo w ed the path of the ground state lithium atom, and found it to rapidly mo v e from the cen ter of the cluster to its exterior. This is in agreemen t with exp erimen tal results whic h indicate that alk ali atoms reside on the surface of rare gas clusters and not within their in terior. W e ha v e also examined the dynamics of excited lithium atoms near a mo del helium surface, and found that the b eha vior is strongly dep enden t on the symmetry of the excited state. In the case of Li(2 p ), the atom is immediately rep elled from the surface, while the He atoms recoil sligh tly On the other hand, Li(2 p ) is attracted to the surface, and ev en tually em b eds itself completely within the He atoms. F or b oth excited states, w e induce dip ole emission b y in tro ducing a p erturbing electromagnetic eld resonan t to the Li(2 p 2 s ) line. The computed dip ole sp ectra ha v e b een calculated for initial and nal times, where w e nd sp ectral shifts to generally matc h other theoretical and exp erimen tal results. In the case of Li(2 p ) surrounded b y surface He atoms, w e observ e a v ery high red shift in the sp ectrum, whic h can b e explained b y examining the pro ximit y of the ground and rst excited state adiabatic curv es for a lithium atom approac hed b y a surrounding lattice of
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144 He atoms. This curv e pro ximit y ma y also explain the absence of radiation detected from excited Li in bulk liquid He, with radiationless deca y b eing p ossible through state mixing at sucien tly lo w LiHe distances. Finally w e w ere able to induce deca y and reco v ery of Li(2 p ) b y applying an electromagnetic eld resonan t to the Li(2 p 2 s ) transition, conrming our sp ectral results as w ell as demonstrating the abilit y of a classical electromagnetic eld to induce these electronic state c hanges in the con text of a mixed quan tumclassical sim ulation. 7.4 Soft w are Dev elopmen t Ov er the course of this pro ject, w e ha v e dev elop ed a general program, cauldron from scratc h in F ortran 90, to sim ulate the dynamics of arbitrary systems whic h can b e cast in to the EPQCLE formalism. Cauldron mak es no assumptions ab out and puts no limitations on the system size, and has b een dev elop ed in a mo dular fashion with the system, propagation and prop ert y calculations designed orthogonally Indeed, all three simple onedimensional mo dels, as w ell as the full LiHe N cluster, w ere sim ulated with precisely the same relaxanddriv e propagation co de. This orthogonal approac h has enabled cauldron to b e extended as new questions arose and new systems w ere studied, without the need to mo dify already w orking co de. W e ha v e also dev elop ed a second program, qualdron analgous to cauldron but for sim ulating onedimensional t w ostate systems using the splitop erator fast F ourier transform formalism. Qualdron has b een used primarily to generate precise results for comparison with the ( cauldron ) EPQCLE results. Qualdron has also b een dev elop ed with orthogonal mo dules for the system, propagation and prop erties, but is limited to onedimensional t w ostate systems describ ed in a diabatic basis. Ho w ev er, the extension to m ultistate systems w ould b e a trivial task, requiring the mo dication of a couple of subroutines in the propagation mo dule. Both cauldron and qualdron are large programs, in v olving o v er 35 ; 000 lines of co de and 200 subroutines. Both their use and in ternal functioning ha v e b een
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145 extensiv ely describ ed in t w o accompan ying man uals, where sp ecic proto cols for extending the co des to new problems ha v e b een suggested. 7.5 F uture W ork While w e ha v e fo cused on the application of the EPQCLE to three small mo dels as w ell as the realistic threedimensional LiHe N cluster, it w ould b e in teresting to ev aluate the p erformance and accuracy of the EPQCLE for additional test mo dels where precise quan tal results can b e computed, in order to further assess the b enets and limitations of the metho d. It w ould also b e in teresting to examine the dynamics of AkRg N clusters in v olving hea vier alk ali or rare gas atoms. Finally it w ould b e instructiv e to study the dynamics and sp ectra of collisions of Ak atoms with Rg atoms at higher temp eratures, where the Rg clusters v ap orize in to the gas phase. AkRg collisions at thermal and h yp erthermal energies ha v e b een studied recen tly b y Rey es, and cauldron pro vides a means to study collisions at these energies where m ultiple rare gas atoms are in v olv ed.
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APPENDIX A THE CAULDRON PR OGRAM A.1 Ov erview Cauldron has b een written to n umerically solv e the EPQCLE for an arbitrary n um b er of quan tal and classical degrees of freedom. F rom the programmer's p oin t of view, cauldron solv es the follo wing partial dieren tial matrix equation: @ @ t = a + b + ~ c @ @ ~ R + ~ d @ @ ~ P ; (A.1) where the matrices a b and ha v e dimensions of the electronic basis set, and the v ectors ~ c ~ d ~ R and ~ P ha v e dimensions rerecting the classical degrees of freedom. is a function of R P and t while the co ecien t v ectors and matrices can b e functions of these v ariables as w ell as itself (for example, when the HellmannF eynman eectiv e force is used). P osed this w a y Eq. A.1 can represen t man y systems and can b e solv ed using a v ariet y of propagation metho ds. T o this end, cauldron w as designed along three orthogonal branc hes, reminiscen t of classes used in ob jectorien ted design. Ho wev er, while v ery mo dular, cauldron is not truly ob jectorien ted in either design or implemen tation, and in the in terest of a v oiding confusion and misrepresen tation, ob jectorien ted terminology will not b e used. The three orthogonal branc hes of cauldron are: 1. System: The system branc h pro vides the time dep enden t co ecien ts for Eq. A.1 completely dening the mo del under study All Hamiltonian terms (quan tal, classical and coupling), as w ell as the eectiv e p oten tial, are con tained within the co ecien ts and ev aluated en tirely within the system branc h. New mo dels 146
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147 are implemen ted b y dev eloping a system branc h sp ecic to the mo del, without reference to an y other branc h. 2. Propagation: The propagation branc h uses the co ecien ts computed b y the system branc h to ev olv e the system. This branc h is general, and a giv en metho d (e.g., relaxanddriv e) ma y b e applied to all systems without mo dication, regardless of system dimension. 3. Prop erties: A t eac h timestep, the prop erties branc h p erforms calculations on the system (e.g. exp ectation v alues of observ ables) and sends the results to appropriate output les for p ostpro cessing. The main cauldron executable calls the appropriate co de from eac h branc h, as sho wn in Figure A{1 Since these co de branc hes are orthogonal to one another, it is a simple matter to add a new system, propagation metho d or prop ert y calculation of arbitrary complexit y without aecting the rest of the co de. A.2 Comp onen t Descriptions A.2.1 Read Input File Eac h sim ulation is describ ed b y a single input le, cauldron.init comp osed of three namelists of data: one for the system, one for the propagation, and one for the prop ert y calculations. T o run a sim ulation, cauldron m ust b e executed in a directory con taining the desired cauldron.init le. A.2.2 System: Get Dieren tial Equation Co ecien ts A t this p oin t, the system branc h pro vides the dieren tial equation co ecien ts at a giv en time. Curren tly implemen ted systems are: Alk ali atommetal surface. This is a onedimensional t w ostate mo del of an alk ali atom approac hing a metal surface. Dual a v oided crossing. This system mo dels a collision b et w een t w o n uclei in v olving t w o electronic states and t w o a v oided crossings.
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148 Evolve single timestep Propagation: yes no end begin finished? Properties: Output properties Get diffeq coefficients System: read input file Figure A{1. Flo w c hart describing the cauldron program.
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149 NaI complex. Here w e mo del the NaI complex with the n uclear separation and t w o diabatic electronic surfaces. Bulk liquid helium. This is a threedimensional mo del of an arbitrary n um b er of classical helium atoms in a p erio dically replicating cen tral v olume. Helium droplet. Here w e mo del a liquid droplet of classically in teracting helium atoms. Lithiumhelium cluster. This system mo dels the in teraction of a lithium atom within a cluster of helium atoms in the framew ork of the EPQCLE, treating the atomic cores quasiclassically and the lithium v alence electron quan tally Their in teraction is describ ed through l dep enden t semilo cal pseudop oten tials and the eectiv e HellmannF eynman force. A.2.3 Propagation: Ev olv e Single Timestep The propagation branc h uses the the dieren tial equation co ecien ts to adv ance the system a single timestep. Curren tly implemen ted branc hes are: Relaxanddriv e. This is the mixed quan tumclassical relaxanddriv e algorithm, with the capabilit y to use either a xed or v ariable timestep. The propagation can also b e split b et w een the relaxation and the driving terms, omitting the driving term if desired. V elo cit y V erlet with p erio dic b oundaries. This is a classical v elo cit y V erlet propagation, but using p erio dic b oundary conditions so that particles lea ving the cen tral v olume reapp ear on the opp osite side. V elo cit y V erlet without p erio dic b oundaries. This is the classical v elo cit y V erlet algorithm with innite b oundaries, useful for propagating isolated clusters of helium. A.2.4 Prop erties: Output Prop erties The prop erties branc h outputs v arious quan tities of in terest to sp ecic and predened output les. Curren tly implemen ted prop erties are:
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150 Exp ectation v alue of p osition and momen tum. P osition and momen tum disp ersion. T race (b oth quan tum and classical) of the partially Wigner transformed densit y matrix. State probabilities. Nuclear conguration, for b oth original and smo othed surfaces. T emp erature, kinetic and p oten tial energy Nuclear densit y prole. P air distribution function. Dip ole exp ectation v alues. A.3 Subroutine Details Cauldron has b een written en tirely from scratc h in F ortran 90, and in v olv es nearly 10,000 lines of co de in o v er 100 les. It w ould b e a hop eless task to describ e the details in the space of an app endix. Ho w ev er, a detailed reference man ual 123 has b een written to accompan y cauldron whic h guides the programmer through all asp ects of b oth use and extension of the co de.
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APPENDIX B SPLIT OPERA TORF AST F OURIER TRANSF ORM METHOD Consider a quan tal system describ ed b y ( r ; R ; t ). The BornOpp enheimer separation builds this w a v efunction in terms of a complete and orthonormal set f k ( r ; R ) g of electronic functions, ( r ; R ; t ) = X l l ( r ; R ) ( R ; t ) : (B.1) Beginning with the time dep enden t Sc hr odinger equation, i ~ @ @ t = ^ H ; (B.2) w e can insert a general Hamiltonian in v olving n uclear (N) and electronic (e) co ordinates, ^ H = ^ K N + ^ T e + ^ V N N + ^ V N e + ^ V ee  {z } ^ H e (B.3) so that i ~ @ @ t = [ ^ K N + ^ H e ] : (B.4) Switc hing to Dirac notation for con v enience, and setting ~ = 1, w e m ultiply on the left b y h k j h k j i @ @ t X l j l ij l i = h k j ^ K N + ^ H e ( j l ij l i ) (B.5) so that i @ @ t j k i = h k j ^ K N ( j l ij l i ) + h k j ^ H e j l i  {z } ^ V k l j l i : (B.6) 151
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152 Expanding the rst term on the RHS, h k j ^ K N ( j l ij l i ) = h k j 1 2 M r 2R ( j l ij l i ) = h k j 1 2 M r 2R j l ij l i + j l ir 2R j l i : (B.7) In the diabatic basis, the momen tum coupling h k jr R j l i disapp ears, so that i @ @ t k = 1 2 M r 2R k + ^ V k l l : (B.8) F or a t w ostate system, denoting the column v ector ( 1 ; 2 ), w e can write @ @ t = i 0B@ 1 2 M r 2R 0 0 1 2 M r 2R 1CA  {z } ^ K i 0B@ ^ V 11 ^ V 12 ^ V 21 ^ V 22 1CA  {z } ^ V : (B.9) F ormally w e can ev olv e a timestep t using the timeordered op erator ^ T : ( t ) = ^ T exp 24 t Z 0 ( i ^ K ( t 0 ) i ^ V ( t 0 )) dt 0 35 (0) : (B.10) If the v ariation of the Hamiltonian is small on the timescale of t w e can write Eq. B.10 as ( t ) = exp h ( i ^ K (0) i ^ V (0)) t i (0) : (B.11) The split op erator metho d breaks up the exp onen tial, ( t ) = exp i t 2 ^ K (0) exp h i t ^ V (0) i exp i t 2 ^ K (0) (0) + O ( t 3 ) : (B.12)
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153 Since ^ K is diagonal but nonlo cal in the n uclear co ordinate space, w e can ev alute its exp onen tial b y computing the deriv ativ es in F ourier space, ~ ( k ) = r 1 2 1 Z 1 ( R ) e ik R dR ) @ 2 @ R 2 ~ ( k ) = k 2 ~ ( k ) On the other hand, ^ V is lo cal but nondiagonal. T o ev aluate its exp onen tial, w e need to diagonalize ^ V : ^ V = D 1 D ) exp [ i t ^ V ] = D exp[ i t ] D 1 = D 0B@ exp ( i t 1 ) 0 0 exp ( i t 2 ) 1CA D 1 : Th us, the split op erator fast F ourier transform metho d ev olv es the quan tal system a single timestep as follo ws: 1. Ev aluate exp [ i t= 2 ^ K ] ( t ) b y computing the second spatial deriv ativ e in F ourier space. 2. Apply exp [ i t ^ V ( t )] b y diagonalizing ^ V ( t ) and then con v erting the exp onential of the matrix in to a matrix of exp onen tials. 3. Apply exp [ i t= 2 ^ K ] once again using the fast F ourier transform. In practice, when sev eral timesteps are computed in a ro w, adjoining exp onen tials of ^ K (i.e., step 1/ follo wing step 3/) can b e com bined in to a single op eration with double the timestep.
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APPENDIX C THE QUALDRON PR OGRAM C.1 Ov erview Qualdron has b een written to n umerically solv e the time dep enden t Sc hr odinger equation for a t w ostate electronic system with a single classical degree of freedom. F rom the programmer's p oin t of view, qualdron solv es the follo wing partial dieren tial equation: i @ @ t = ( ^ K + ^ V ) ; (C.1) where is the column v ector of t w o electronic states, = ( 1 ; 2 ). The basis is assumed to b e diabatic, so that ^ K is diagonal and ^ V con tains the coupling b et w een states. While qualdron has b een written for the t w ostate case, expansion to an arbitrary n um b er of states w ould b e straigh tforw ard, and require the replacemen t of a single subroutine to rerect the m ultidimensional problem. P osed this w a y Eq. C.1 can represen t man y dieren t systems b y c hanging the v alue of the p oten tial coupling elemen ts ^ V in the Hamiltonian. T o this end, qualdron w as designed along three orthogonal branc hes, similar to the design of cauldron These branc hes are: 1. System: The system branc h pro vides the initial v alues and Hamiltonian for Eq. C.1 This section of co de denes the system, and can represen t an y onedimensional t w ostate mo del that can b e describ ed b y a Hamiltonian in the diabatic represen tation. 2. Propagation: The propagation branc h uses the Hamiltonian computed b y the system branc h to ev olv e the system. The propagation co de is general, and 154
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155 a giv en metho d (e.g., split op erator) ma y b e applied to all dened systems without mo dication. 3. Prop erties: A t eac h timestep, the propagation co de calls the prop erties co de, whic h mak es system calculations and outputs appropriate les. The main qualdron executable calls the appropriate co de from eac h area to create the sim ulation, as sho wn in Figure C{1 Since these co de branc hes are for the most part orthogonal to one another, it is a simple matter to add a new system, a new propagation metho d, or new prop ert y calculations without aecting the rest of the co de. C.2 Comp onen t Descriptions C.2.1 Read Input File Eac h sim ulation is describ ed b y a single input le, qualdron.init comp osed of three namelists of data: one for the system, one for the propagator, and one for the prop erties. T o run a sim ulation, qualdron m ust b e executed in a directory con taining the desired qualdron.init le. C.2.2 System: Get Hamiltonian Matrix Elemen ts A system pro vides the Hamiltonian matrix elemen ts at a giv en time. Curren tly implemen ted systems are: Alk ali atommetal surface. Dual a v oided crossing. NaI complex. C.2.3 Propagation: Ev olv e Single Timestep The propagation co de uses the Hamiltonian elemen ts to adv ance the system a single timestep. Curren tly the only implemen ted propagation is the split op erator fast F ourier transform metho d.
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156 Propagation: Evolve single trajectory Get Hamiltonian System: yes no end begin finished? Properties: Output properties read input file Figure C{1. Flo w c hart describing the qualdron program.
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157 C.2.4 Prop erties: Output Prop erties The prop erties co de outputs v arious quan tities of in terest to sp ecic and predened output les. Curren tly implemen ted prop erties are: Exp ectation v alue of p osition and momen tum. P osition and momen tum disp ersion. State probabilities. Nuclear congurations. C.3 Subroutine Details Qualdron has b een written en tirely from scratc h in F ortran 90, and in v olv es nearly 2,500 lines of co de in o v er 100 les. Since these details could not b e adequately describ ed b y an app endix, a detailed reference man ual 124 has b een written to accompan y qualdron whic h guides the programmer through all asp ects of b oth use and extension of the co de. In particular, instructions for c hanging qualdron from a t w ostate to a m ultistate system are presen ted in detail.
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BIOGRAPHICAL SKETCH Brian Thorndyk e w as b orn on Octob er 17 th 1969, in Calgary Alb erta, Canada. As he grew up, his mother, Carol Thorndyk e, w as an elemen tary sc ho ol teac her while his father, Gerry Thorndyk e, taugh t highsc ho ol biology and general science. Their p ositiv e academic inruence w as notable, as Brian ended up graduating high sc ho ol with an adv anced In ternational Baccalaureate diploma, and indeed w as oered a full y ear's credit at the Univ ersit y of T oron to in the Departmen t of Ph ysics. Rather than mo v e to T oron to, Brian c hose to remain an additional y ear in Calgary immerse himself in v arious F renc h programs at the Univ ersit y of Calgary and then en ter in to a bac helor's program in ph ysics at an allF renc h univ ersit y in Mon treal. He remained in Mon treal for b oth his bac helor's and master's degrees. After his time in Mon treal, he sp en t a y ear at the Univ ersit y of British Colum bia, w orking on pro jects in the Computer Science and Electrical Engineering Departmen ts. He w ould ha v e remained in computer science in V ancouv er had he not b een en ticed to mo v e to Florida, and complete a master's degree in computer science with Dr. P aul Fish wic k. Tw o y ears later, ho w ev er, Brian decided he had to return to his rst lo v e, ph ysics, and transferred to the Ph ysics Departmen t in the Quan tum Theory Pro ject with Dr. Da vid Mic ha. Tw o y ears b efore graduation, Brian's father w as diagnosed with terminal cancer, and Brian to ok a y ear of semilea v e to sp end time with his paren ts and supp ort them emotionally When his father passed a w a y in Jan uary of 2003, Brian returned to nish his Ph.D., and has since accepted a p ostdo ctorate p osition in the Departmen t of Radiation Oncology Division of Radiation Ph ysics at Stanford Univ ersit y 168

