timescales of large amplitude motion - classical and
TRANSCRIPT
Timescales of Large Amplitude Motion - Classical and QuantumConsiderations
By
Amber Jain
A dissertation submitted in partial fulfillment of
the requirements for the degree of
Doctor of Philosophy
(Chemistry)
at the
UNIVERSITY OF WISCONSIN-MADISON
2014
Date of final oral examination: 10/30/14
The dissertation is approved by the following members of theFinal Oral Committee:
Edwin L. Sibert III, Professor, Theoretical Chemistry
James L. Skinner, Professor, Theoretical Chemistry
Robert C. Woods, Professor, Experimental Chemistry
Jordan R. Schmidt, Professor, Theoretical Chemistry
Qiang Cui, Professor, Theoretical Chemistry
i
Timsescales of Large Amplitude Motion - Classical and QuantumConsiderations
Amber Jain
Under the supervision of Professor Edwin L. Sibert III
At the University of Wisconsin – Madison
This thesis comprises two projects that investigate vibrational energy relaxation (VER), and
symmetric proton tunneling.
In the first project, we investigate the pathways of VER in themolecule iso-
chloroiodomethane CH2Cl-I embedded in a matrix of argon at 12K, motivated by the exper-
imental studies of Crim and coworkers. We study this relaxation theoretically using molecular
dynamics by considering two and three dimensional models. Multiple decay rate constants of
the same order of magnitude as the experiment are observed. These decay rate constants are in-
terpreted within the context of the Landau-Teller theory. Sensitivity of the decay rate constants
on the bath and system parameters shed more light into the mechanism of VER.
The second project focuses on proton tunneling, which playsa central role in many biolog-
ical reactions. We investigate a three dimensional model Hamiltonian coupled to a harmonic
bath that describes concerted proton transfer in formic acid dimer. The three modes provide
a paradigm for the symmetric and anti-symmetric coupled tunneling pathways. The effects of
temperature and coupling to the bath on the rates are presented. We compare three methods
that have been shown previously to provide good results for the tunneling dynamics – sur-
face hopping, ring polymer molecular dynamics, and the Makri-Miller method. We find that
surface hopping and ring polymer molecular dynamics do not describe some aspects of the
dynamics in the deep tunneling regime due to neglect of the coherence effects. Certain modi-
fications in the surface hopping algorithm are suggested to partially include these coherences.
The Makri-Miller method predicts the correct trends for thetunneling splittings, which govern
the dynamics.
In addition, we also develop a new method to compute tunneling splittings for highly excited
states. This method is based on making an adiabatic approximation to the Herring estimate, and
is in excellent agreement with the exact results.
ii
To my parents
iii
Published work and work in preparation
[1] A. Jain and E. L. Sibert, “Vibrational relaxation of chloroiodomethane in cold argon” J.
Chem. Phys.139, 144312 (2013).
[2] A. Jain and E. L. Sibert, “Rates of symmetric proton tunneling using semiclassical meth-
ods” J. Chem. Phys. submitted.
iv
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Published work and work in preparation . . . . . . . . . . . . . . . . . .. . . . . . iii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .x
1 Introduction 1
1.1 Vibrational energy relaxation . . . . . . . . . . . . . . . . . . . . .. . . . . . 2
1.2 Symmetric proton tunneling . . . . . . . . . . . . . . . . . . . . . . . .. . . 3
2 Vibrational relaxation of chloroiodomethane in cold argon 6
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Electronic structure calculations . . . . . . . . . . . . . . .. . . . . . 10
2.2.2 Potential energy surface . . . . . . . . . . . . . . . . . . . . . . . .. 11
2.3 Theoretical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 13
2.3.1 Non-equilibrium molecular dynamics . . . . . . . . . . . . . .. . . . 16
2.3.2 Landau-Teller theory: equilibrium molecular dynamics . . . . . . . . . 18
2.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19
2.4.1 2-D model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.2 3-D model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.3 Variation of the standard parameters . . . . . . . . . . . . . .. . . . . 28
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Rates of symmetric proton tunneling - surface hopping and ring polymer molecu-
lar dynamics 33
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34
3.2 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37
3.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.1 Exact quantum dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.2 Surface hopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
v
3.3.3 Ring polymer molecular dynamics . . . . . . . . . . . . . . . . . . .. 49
3.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50
3.4.1 Exact quantum dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4.2 Surface hopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4.3 Ring polymer molecular dynamics . . . . . . . . . . . . . . . . . . .. 56
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 Tunneling splittings - the Makri-Miller method 62
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63
4.2 Makri-Miller method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 64
4.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 67
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5 Tunneling splittings - the adiabatic Herring method 71
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72
5.2 Adiabatic Herring estimate . . . . . . . . . . . . . . . . . . . . . . . .. . . . 73
5.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 77
5.4 Additional methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 80
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6 Conclusions 84
6.1 Vibrational energy relaxation . . . . . . . . . . . . . . . . . . . . .. . . . . . 84
6.2 Symmetric proton tunneling . . . . . . . . . . . . . . . . . . . . . . . .. . . 85
Appendices 88
A Introduction for broader audience 89
B 2D Landau-Zener formalism 94
C Fewest switches surface hopping 99
D Invariant instanton theory 101
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
vi
List of Figures
Chapter 2
2.1 Schematic of the isomerization of chloroiodomethane . .. . . . . . . . . . . . 8
2.2 Equi-potential energy contour lines for chloroiodomethane . . . . . . . . . . . 14
2.3 Cartoon showing return of the shock wave energy back to themolecule . . . . . 15
2.4 A schematic comparing decay of the experimentally measured spectral intensity
with the computed energy decay rate . . . . . . . . . . . . . . . . . . . . .. . 18
2.5 Representative reactive trajectory . . . . . . . . . . . . . . . . .. . . . . . . . 20
2.6 Collective early time dynamics . . . . . . . . . . . . . . . . . . . . . .. . . . 21
2.7 Decay of the molecular energy . . . . . . . . . . . . . . . . . . . . . . .. . . 22
2.8 The average short-time Fourier transform . . . . . . . . . . . .. . . . . . . . 22
2.9 Number of trajectories in the different energy bins . . . .. . . . . . . . . . . . 23
2.10 Kinetic energy of different bath shells as a function oftime . . . . . . . . . . . 24
2.11 Decay of energy after quenching of the bath . . . . . . . . . . .. . . . . . . . 25
2.12 Decay of vibrational energy with trajectories initialized in the isomer well . . . 26
2.13 Landau-Teller rates as a function of normal mode frequency . . . . . . . . . . 27
2.14 Decay of the energy for the 3-D model . . . . . . . . . . . . . . . . .. . . . . 29
2.15 Comparison of energy decay for different normal mode frequencies . . . . . . 30
2.16 Comparison of energy decay for different equilibrium isomer configurations . . 30
2.17 Comparison of energy decay for different bath densities. . . . . . . . . . . . . 31
Chapter 3
3.1 Equipotential contour plots of the potential energy surface modelling the formic
acid dimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Slice of the adiabatic surface alongq4 at q6 = 0.6 A . . . . . . . . . . . . . . . 44
3.3 The full adiabatic surface as a function ofq4 andq6 for gas phase FAD . . . . . 44
3.4 Comparison of this diabatic surface with the adiabatic surface withq4 = 0 . . . 46
3.5 The first 8 tunneling splittings as a function of the coupling strength for the
formic acid dimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.6 The decay ofPR computed using exact quantum dynamics . . . . . . . . . . . 53
vii
3.7 Comparison of decay ofPR computed using LZ formalism and the fewest
switching criterion of surface hopping . . . . . . . . . . . . . . . . .. . . . . 54
3.8 The decay ofPR computed using NVE and NVT calcultions of surface hopping 54
3.9 The decay ofPR computed using a hybrid calculation of surface hopping . . . 55
3.10 The decay ofPR computed with the restriction on the intrinsic bath coordinate
applied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.11 Comparison of the decay ofPR computed exactly and assuming first order kinetics 58
3.12 The ratio of the rates at 300 K to 200 K computed using RPMD and exact
calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Chapter 4
4.1 An illustration of the straight line tunneling path for the Makri-Miller method . 65
4.2 Comparison of the instanton trajectory, the minimum energy path and the tun-
neling path chosen for Makri-Miller method . . . . . . . . . . . . . .. . . . . 67
4.3 Comparison of exact tunneling splittings with the tunneling splittings obtained
through the Makri-Miller method . . . . . . . . . . . . . . . . . . . . . . .. . 68
4.4 Comparison of decay ofPR computed using exact dynamics and the Makri-
Miller method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Chapter 5
5.1 Comparison of exact tunneling splittings with the tunneling splittings obtained
through the adiabatic Herring method . . . . . . . . . . . . . . . . . . .. . . 78
5.2 Comparison of decay ofPR computed using exact dynamics and adiabatic Her-
ring estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3 Comparison of tunneling splittings obtained through thevarious levels of ap-
proximation made in the adiabatic Herring method . . . . . . . . .. . . . . . 79
5.4 The diabatic wavefunctions and the contributions to theground state tunneling
splitting from various regions of the coordinate space given by the adiabatic
Herring theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.5 The diabatic wavefunctions and the contributions to an excited state tunneling
splitting from various regions of the coordinate space given by the adiabatic
Herring theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
viii
5.6 The instanton trajectory . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 81
Chapter B
B.1 Plot of∆0 and∆0d012 with respect toq6 . . . . . . . . . . . . . . . . . . . . . 96
B.2 Comparison of transition probabilities obtained throughthe 2D LZ formalism
with the exact transition probabilities . . . . . . . . . . . . . . . .. . . . . . . 98
ix
List of Tables
Chapter 2
2.1 The lowest two frequencies of CH2Cl-I computed using several levels of meth-
ods and basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Standard parameters chosen for the diabatic potential energy surfaces of
chloroiodomethane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Parameters for Lennard-Jones potential describing interactions between the
bath and CH2ClI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 VER decay rate constants computed using various methods. . . . . . . . . . . 28
Chapter 3
3.1 Parameters for the potential energy surface of FAD . . . . .. . . . . . . . . . 39
3.2 The parameters for the DVR basis for gas-phase FAD . . . . . .. . . . . . . . 41
3.3 The parameters for the DVR basis for gas-phase FAD coupled to a harmonic bath 42
3.4 Tunneling splittings for gas-phase FAD . . . . . . . . . . . . . .. . . . . . . 51
x
Acknowledgments
First and foremost, I will like to acknowledge my parents, who are my first teachers. They
always allowed me the freedom of choosing my career path, andhave been in constant support
in all of the decisions I have made. Their patience and understanding in my continued absence
is remarkable.
In my undergraduate studies, I was heavily influenced by Prof. Srihari and Prof. Gupta-
Bhaya. Prof. Srihari was my master’s thesis mentor. His novelway of thinking, and highly
inspiring nature heavily impacted my scientific thinking. The general chemistry course he
taught is one of the primary reasons of my interest in physical chemistry. I attended several
courses taught by Prof. Gupta-Bhaya. His honest, humble and hard-working way of life is truly
inspiring, and he is one of my role-models. I also want to mention two of my undergraduate
friends – Mithilesh and Raghav, with whom I had plenty of quality philosophical and scientific
discussions.
I have been blessed with several friends in my graduate studies. The former group members
of the Sibert group, Jayashree and Roumou, greatly helped me in my starting years as a graduate
student. Danny and Britta, the current group members, provide a very friendly environment to
work in. Constant sharing of ideas with them have greatly developed me as a scientist. I will
further like to acknowledge Jesse and Kuang, with whom I shared an office in my first year, and
had discussions over a very broad range of topics. Sriteja has served as a great friend, and I owe
him a great debt of gratitude. Nilay, who has been my roommatefor the last five years, have
supported me constantly. I am lucky to have him as one my friends.
Finally I will like to thank Prof. Sibert, my current advisor, without whom this journey
would have been incomplete. His deep physical insights continously amazes me, and pushes
the horizons of my scientific knowledge. I am indebted to the great academic freedom that I
have enjoyed in his group, which gave me the pleasure of finding things out. I hope to be able
to emaluate him as a scientist and a teachor.
1
Chapter 1
Introduction
Contents
1.1 Vibrational energy relaxation . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Symmetric proton tunneling . . . . . . . . . . . . . . . . . . . . . . . . . 3
Chemical reactions can be broadly viewed to proceed in two steps. The first step is bond
formation and breaking, leading to rearrangement of atoms to yield the products. These newly
formed products have excess thermal energy, that is lost to the environment in the second step by
the process of vibrational energy relaxation (VER). Both these steps are crucial in determining
the rates, pathways and efficiencies of the reaction.
In this thesis, we study both these steps from the perspective of dynamics and computation
of the relevant timescales. In the first part we unravel the mechanism and the underlying features
of the potential energy surface that dictate the timescalesof VER. The second part focuses on
the transfer of proton from one moiety to another. The aim of this part is to understand the
quantum mechanical effects associated with the proton tunneling, and be able to include these
effects into realistic simulations.
In both these projects we choose simple model Hamiltonians that are qualitatively based on
a real molecule or chemical reaction. The ability to comparethe results obtained by varying the
parameters of these model Hamiltonians allows one to generalize the observations to a broader
class of reactions, as well as gain predictive powers. The underlying ideas and the motivation
for both the projects are described next.
2
1.1 Vibrational energy relaxation
Fermi, Pasta, Ulam, and Tsingou (FPUT) in 1955 published theresults of perhaps the first
computer experiment to study ‘the rate of approach to equipartition of energy among the vari-
ous degrees of freedom of the system’.1 Surprisingly, they observed non-ergodic energy flow
for a model non-linear system, with the energy returning back periodically to its initial non-
equilibrium state! This led to numerous debates and furtherdevelopment of the theories of
solitons and chaos. Fortunately, the energy flow in most molecules can be described by statis-
tical mechanics at long times. A large number of studies havefocussed on computation of the
rates of VER. These rates of VER play a crucial role in determining the reaction mechanism
as the relative rate of VER compared to the rate of the transfer of the energy into the reaction
coordinate can influence the stability of the products.2
The numerical simulations require experimental validation. The pioneering works of Zewail
and coworkers on femtosecond resolved spectroscopy allowed following chemical reactions
in real time.3 With advances in this field over the last three decades, the rates of a host of
chemical reactions have been studied in great detail experimentally. The information obtained
through these experiments remain incomplete though, as they lack the mechanistic insights or a
fundamental understanding of what drives the reaction. Combining these experimental studies
with numerical simulations and theoretical formulations provide a fuller picture, with deep
insights into the mechanisms of the energy flow and effects ofvarious factors such as normal
mode frequencies, density, caging and shock-wave dynamicson the decay rate constants.
Here we focus on the time-resolved spectroscopic study of isomerization of
chloroiodomethane in an inert argon matrix at 12 K temperature performed by Crim and
coworkers.4 In this reaction, the product is iso-chloroiodomethane which features a chlorine
iodine bond and is approximately 10000 cm−1 higher in energy than the parent molecule. Apart
from the important role that these isomerization reactionsplay in atmospheric chemistry, they
also form a playground to test the theoretical methods. Withthe aim to understand the funda-
mental factors that govern energy flow, we build a reduced dimensional functional form of the
potential energy surface whose parameters are motivated byelectronic structure calculations
and experimental values.
We employ molecular dynamics simulations to follow the longtime dynamics of the isomer-
ization. Molecular dynamics simulations have a long and rich history starting with the FPUT
3
simulation, and have proved useful in providing qualitative understanding, as well as quantita-
tive rates in certain cases, of energy flow. To simulate the isomerization of chloroiodomethane,
the reaction is started out as the reactant, and the excess electronic excitation energy provided
experimentally to the C-I bond to initiate the isomerizationis modelled by putting the appro-
priate amount of kinetic energy into this bond. This leads tolarge amplitude motions of the
molecule and shock-wave like dynamics in the bath. A fraction of these trajectories are reac-
tive and following their subsequent dynamics gives the desired rates and the mechanism of the
energy flow. The true power of the numerical simulations can be realized by examining the
sensitivity of the decay rate constant on variation of several parameters, such as the temperature
and the density of the bath, and the parameters related to thepotential energy surface. This
leads to a generalized picture of VER not only for the system of interest, but for a broader class
of systems.
The rates obtained using these simulations are in qualitative agreement with the experimen-
tal work. We further elucidate the mechanisms of energy flow in the normal mode picture.
These rates and mechanisms are interpreted within the framework of Landau-Teller theory,5,6
which is predominant among theories that describe the vibrational relaxation in a solvent. This
formalism computes the decay rate constant in terms of the time correlation function of the
interactions between the system and the bath. The Landau-Teller theory is pivotal in explaining
the reasons for the different computed rates of energy flow from the different normal modes.
1.2 Symmetric proton tunneling
In this part of the thesis, we investigate proton tunneling.The phenomenon of tunneling allowed
one to explain the mysterious radioactive decay within a fewyears of the birth of quantum
mechanics.7 A simple one dimensional potential was used to model the radioactive decay,
and theoretical considerations explained all the observedfeatures qualitatively. Soon it was
realized that tunneling plays an important role in various other systems and processes including
proton tunneling which is central in many chemical and biological reactions. Yet, more than
seven decades later, neither do we have a quantitative theory to predict the timescales of proton
tunneling for a multidimensional system, nor can we incorporate the tunneling effects efficiently
into semiclassical (or quantum-classical) simulations, which are essential for simulating large
4
systems.
In this work, we first benchmark the performance of various popular semiclassical and
quantum-classical methods. For this purpose, we choose a symmetric double well system that
qualitatively models the symmetric proton tunneling in theformic acid dimer, and its further
coupling to a bath. This model Hamiltonian represents the deep tunneling regime where the
barrier height is much larger than the average thermal energy. Hence, this serves as a severe test
for these methods due to the inherent importance of the quantum coherences.
The methods we investigate are surface hopping and ring polymer molecular dynamics
(RPMD), both of which have shown promise to compute the tunneling rates. The RPMD
formalism, based on the path integral formalism of quantum mechanics, was developed by
Manolopoulos and coworkers to compute accurately the thermal rate constants using a flux-side
correlation function.8 It was later shown to have connections with the semiclassical instanton
theory in the deep tunneling regime, hinting that this theory can obtain dynamical information
in this deep tunneling regime.9 The surface hopping method, developed by Tully,10,11 is a mixed
quantum-classical method that incorporates non-adiabatic effects by appropriately allowing the
trajectories to hop between the adiabatic surfaces and has been employed extensively to obtain
the timescales of tunneling for a wide range of systems.12–14
Our findings indicate that these methods do not perform well for the model Hamiltonian
under consideration. It is well known that these theories can perform poorly for systems where
quantum coherences are important. We propose certain modifications to the surface hopping
method to partially include some of the coherence effects.
We explore alternative methods that can include these quantum coherence effects. The tun-
neling splittings, which is the difference between the energies corresponding to anti-symmetric
and symmetric eigen-functions, dictates the dynamics.15 There are numerous approaches
present in the literature to compute the ground state tunneling splitting. But computing the
excited state tunneling splittings is far more challenging. Makri and Miller developed a method
that extends the WKB formalism to compute tunneling splittings for multidimensional sys-
tems.16 Although the original work limits itself to ground state tunneling splitting, this formal-
ism readily extends to excited state tunneling splittings as well. This is well demonstrated by the
works of Kuhn and coworkers17 and Thompson and coworkers.18,19 Application of this methods
leads to qualitative agreement with the exact results. It further highlights the importance of the
5
effects of the path length in computation of tunneling timescales.
One of the issues with the Makri-Miller approach, and several other approaches, is the
dependence on the ‘tunneling path’. The tunneling path is a one-dimensional path, which con-
tributes the most in computation of the tunneling splittings. We develop a theory of our own
that can compute excited state tunneling splittings, without resorting to the tunneling path. This
new approach makes vibrational adiabatic approximation tothe well known Herring estimate.20
We obtain quantitative accuracy for the tunneling splittings for the model Hamiltonian. The
shortcomings of this theory include dependence of the results on the diabatic potential, and will
be further discussed.
6
Chapter 2
Vibrational relaxation of
chloroiodomethane in cold argon
Contents
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Electronic structure calculations . . . . . . . . . . . . . . . . . . . 10
2.2.2 Potential energy surface . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Theoretical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Non-equilibrium molecular dynamics . . . . . . . . . . . . . . . . 16
2.3.2 Landau-Teller theory: equilibrium molecular dynamics . . . . . . 18
2.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.1 2-D model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.2 3-D model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.3 Variation of the standard parameters . . . . . . . . . . . . . . . . 28
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Figures
2.1 Schematic of the isomerization of chloroiodomethane . . . . . . . . . . . . . 8
2.2 Equi-potential energy contour lines for chloroiodomethane . . . . . . . .. . 14
2.3 Cartoon showing return of the shock wave energy back to the molecule .. . . 15
7
2.4 A schematic comparing decay of the experimentally measured spectral inten-
sity with the computed energy decay rate . . . . . . . . . . . . . . . . . . . . 18
2.5 Representative reactive trajectory . . . . . . . . . . . . . . . . . . . . . . .. 20
2.6 Collective early time dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.7 Decay of the molecular energy . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.8 The average short-time Fourier transform . . . . . . . . . . . . . . . . . . .22
2.9 Number of trajectories in the different energy bins . . . . . . . . . . . . . .. 23
2.10 Kinetic energy of different bath shells as a function of time . . . . . . . . .. 24
2.11 Decay of energy after quenching of the bath . . . . . . . . . . . . . . . .. . 25
2.12 Decay of vibrational energy with trajectories initialized in the isomer well . .26
2.13 Landau-Teller rates as a function of normal mode frequency . . . . .. . . . 27
2.14 Decay of the energy for the 3-D model . . . . . . . . . . . . . . . . . . . . .29
2.15 Comparison of energy decay for different normal mode frequencies . . . . . 30
2.16 Comparison of energy decay for different equilibrium isomer configurations . 30
2.17 Comparison of energy decay for different bath densities . . . . . . .. . . . . 31
Tables
2.1 The lowest two frequencies of CH2Cl-I computed using several levels of
methods and basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Standard parameters chosen for the diabatic potential energy surfaces of
chloroiodomethane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Parameters for Lennard-Jones potential describing interactions between the
bath and CH2ClI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 VER decay rate constants computed using various methods . . . . . . . . . .28
2.1 Introduction
Vibrational energy relaxation (VER) is the process that leads to the thermal equilibration of the
excited vibrational modes in a solute molecule via transferof the excess energy to the various
solvent modes. Understanding VER is of fundamental importance in understanding chemical
reactions in condensed phase.2,21–23 Hence it has received significant interest in literature.24–28
Studying VER in small molecules is particularly insightfulbecause of the reduced complexity of
the problem. An interesting system where VER can be studied in detail is choloroiodomethane.
8
Ar
hv
Cl
I
H
H
Cl+
I
H
H
Cl
I
H
H
Ar
Ar
Ar
Ar Ar
Ar
Ar Ar
Ar
Ar
Ar
Ar Ar
Ar
Ar Ar
Ar
Ar
Ar
ArAr
Ar
Ar
Figure 2.1: Schematic of the isomerization of chloroiodomethane. A 267 nm pulse excites and
breaks the C-I bond. In presence of the matrix, the two fragments can recombine to produce the
isomer which can then relax vibrationally.
Halomethanes have been observed to generate isomers upon electronic excitation.29–32
Chloroiodomethane (CH2ClI) has an isomer iso-chloroiodomethane (CH2Cl-I) which features
a chlorine iodine bond and is approximately 10000 cm−1 higher in energy than the parent
molecule. In a recent paper,4 the C-I bond of CH2ClI, while embedded in an inert matrix,
was excited electronically using a 267 nm ultrafast laser pulse, leading to the dissociation of
the C-I bond. The two fragments so produced, unable to escape due to the inert matrix, can
recombine giving CH2Cl-I. The schematic of the process is shown in Fig. 2.1.
The formation and subsequent relaxation of the isomer has been studied by probing the
molecule with 435 nm and 485 nm pulses. The 485 nm pulse, having a lower energy, can
excite only the excited isomer molecules. The growth rates of 435 and 485 nm pulses give
the timescales of formation of the isomer, while the decay rate constants of the 485 nm pulse
gives the timescale of VER of the isomer. Multiple time scales on the order of picoseconds are
observed for the relaxation of the isomer.
Despite the experimental information, the mechanism of VERremains unclear in the exper-
iment. The aim of this work is to elucidate these pathways andtheir corresponding time scales.
The normal mode frequencies, anharmonicities in the electronic potential energy surface, and
the solvent structure and its reorganization as the molecule isomerizes are some of the important
factors that affect VER. Theoretical simulations allow us tostudy the effect of change in each
of the above factors in isolation leading to a better understanding of VER.
Experimentally the isomer appears following the excitation to the excited electronic state.
The subsequent short time dynamics in which the molecule evolves on the excited surface before
9
returning to the ground energy surface, while of significantinterest,33–35 is beyond the scope of
this work, which focuses on the long time VER of the isomer. Instead of building expensive
multi-dimensional excited and ground state potential energy surfaces, we start with 2 and 3
dimensional models of the ground electronic potential energy surface that can give a qualitative
understanding of the process.
Non equilibrium molecular dynamics (NEMD) has been used extensively in the past to in-
vestigate VER in halogens36–42and halomethanes.43–45 These studies elucidate the mechanisms
of the energy flow and demonstrate the effects of various factors such as density, caging, shock-
wave dynamics and solvent on decay rate constants.
The importance of frequency of the excited oscillator has been recognized in the literature.
For example, the decay rate constant of oxygen in an argon matrix (frequency=1556 cm−1)
is on milliseconds timescales,46 while the decay rate constant for iodine in the argon matrix
(frequency=215 cm−1) is on sub-nanosecond timescales. Dipole-dipole interactions between
the solvent and the bath can also lead to several order of magnitude change in decay rate con-
stants.43 The molecule of interest in this work, CH2Cl-I has time scales of a few picoseconds in
argon (a non-polar solvent), which suggests the importanceof low frequency modes.
The computed decay rate constants can be better understood in the framework of Landau
Teller (LT) theory.5,6 LT theory is predominant among theories that describes the vibrational
relaxation in a solvent. It has been primarily used to model VER in liquids, but the formalism
should hold true for solids as well. LT theory can (in principle) describe the collisional processes
that lead to the relaxation through the force-force correlation functions. Also it can be shown
that the classical LT rate formula is same as that of a completely quantum harmonic system
(quantum solute and quantum solvent).47 Corrections to LT theory have been proposed in the
past,48,49 but since our aim in this work is to develop a qualitative understanding of the process,
we employ the traditional LT approach.
The structure of this chapter is as follows: In section 2.2, we describe the model and the
potential energy surface. The theoretical methods are described in section 2.3, followed by
results and discussion in section 2.4. The conclusion of thechapter are given in section 2.5.
10
2.2 Model
We study the relaxation process using a reduced dimensionalHamiltonian. The isomerization
of the molecule entails excitation of the C-I stretch (a low frequency mode). The following
rearrangement involves large amplitude motion along Cl-I stretch (r), C-Cl-I bend (θ), and C-
Cl stretch. Since the coordinatesr andθ are the lowest two frequency modes, an energy surface
is developed with respect to these two coordinates. The effect of C-Cl stretch is included by
modelling it as a harmonic oscillator with the experimentalfrequency of 880 cm−1. Its coupling
to r andθ is purely kinetic in nature.
The focus of the simulations is to understand the long time energy decay in the isomer well.
The isomerization in the experiment entails excitation to the excited surface, leading to the pre-
dissociation of iodine and collisions of the molecule with the argon atoms. This would change
the solvent structure and temperature in the neighbouring shells, affecting the long time energy
decay. To include these effects in the simulations, the molecule is equilibrated in the parent
well and given energy in the C-I stretch which leads to the collisions of the molecule with the
argon atoms. The dependence of the decay rate constants on the amount of energy given may
incorporate some of the effects of the dynamics on the excited state potential energy surface.
Argon, nitrogen and methane are the three cryogenic matrices studied in Ref. 4. We choose
to use argon as the bath because it is relatively simple to model, and has no vibrational degrees
of freedom.
2.2.1 Electronic structure calculations
The frequencies and the minimum energy structure for the parent molecule (CH2ClI) and the
isomer (CH2Cl-I) are computed with several basis sets and different levels of theory. The fre-
quencies of the isomer are listed in the Table 2.1. The frequencies of the parent molecule are
207 cm−1 and 586 cm−1 (at MP2 level of theory, cc-pVTZ basis set for all atoms except iodine
and LANL2DZ basis set and LANL2DZ effective core potential50 for the iodine atom). The
frequencies for the parent molecule are relatively independent of the choice of basis set and the
level of theory.
The sensitivity of the frequencies of the isomer to different level of theory indicate the multi-
reference nature of the system. The focus of this paper is to understand qualitative features
11
Method C-Cl-I bend Cl-I stretch
MP2/LANL2DZ (LANL2DZ)50 165 243
MP2/ECP28MDF (ECP28MDF)51 165 252
MP2/Pross (ECP46MWB)52 170 263
CCSD/LANL2DZ (LANL2DZ) 125 182
CCSD/ECP28MDF (ECP28MDF) 137 200
CCSD/Pross (ECP46MWB) 145 212
Table 2.1: The frequencies of the first two normal modes of theisomer in cm−1. The basis on all
the atoms except iodine is cc-pVTZ. The basis and the effective core potential (ECP) on iodine
is listed in the Method column.
of VER and its dependence on the various bath and system parameters. Given the inherent
complexity of calculating an accurate potential energy surface, we adopted a broader view point.
The potential energy surface is written in a simple form withparameters that can be adjusted in
order to determine how the features of the potential energy surface influence the corresponding
relaxation.
2.2.2 Potential energy surface
The potential energy surface is constructed through a two step process, following the empirical
valence bond (EVB) approach.53 First diabatic potential energy surfaces are constructed that
describe the isomer region and the parent region. The full adiabatic surface is obtained by
solving for the lowest eigenvalue of the 2× 2 matrix
V ad =
(
V d1 V12
V12 V d2
)
, (2.1)
whereV d1 andV d
2 are the two diabatic potentials andV12 is an interaction matrix element in-
troduced to couple the diabatic surfaces. Next we describe the details of the diabatic surface,
12
followed by a discussion on the choice ofV12.
The functional form used for the isomer as well as the parent diabatic potential is
V d1/2 = a1(1 − e−a2∆r)2 +
(
a3∆θ2 + a4∆r∆θ)
e−a5∆r + E, (2.2)
whereE represents the energy of the isomer with respect to the parent molecule, and∆r and
∆θ are the changes inr andθ, respectively from their equilibrium values:
∆r = r − req, (2.3)
∆θ = θ − θeq. (2.4)
The values ofai are different for the isomer and the parent configurations and are listed in Table
2.2. All terms havingθ dependence are multiplied withe−a5∆r to ensure that at larger, there is
no unphysical dependence of the potential onθ.
For the parent molecule, the parameters for the potential are obtained by fitting the electronic
energy surface computed near the bottom of the parent well using the MP2/LANL2DZ method,
and the experimental dissociation energy of 18000 cm−1.54
For the isomer molecule, the parameters chosen are motivated by the results of the electronic
structure calculations, and correspond to normal mode frequencies of 147 cm−1 and 244 cm−1.
These parameters are given in Table 2.2.
The traditional approach to obtain the interaction matrix elementV12 is to fit it to a Gaussian
that exactly reproduces the transition state geometry, energy and the force constant matrix. This
approach assumes non-negligible values ofV12 only in regions near the transition state. We
deviate from this approach, as it leads to spikes in the adiabatic potential in the regions where
the two diabatic surfaces cross and are also far from the transition state geometry.
We instead use the functional formV12 = Ae−(V d
1−V d
2)2/B2
whereA = 107 cm−1, and
B = 10−4 cm−1. This form has non-vanishing values along the whole contourwhere the two
diabatic surfaces meet, which leads to a smooth adiabatic energy surface. The contour plot of
the adiabatic surface, corresponding to the parameters listed in Table 2.2, is shown in Fig. 2.2.
The direction of the normal modes around the isomer minima are also shown in this figure. The
evaluation of normal modes are described in the next section.
The interactions between the argon atoms and the argon with CH2ClI are described using
13
Parameter Parent Molecule Isomer Molecule
a1 18000 cm−1 8000 cm−1
a2 1.62A−1 1.72A−1
a3 195123 cm−1 10109 cm−1
a4 146217 cm−1A−1 1000 cm−1A−1
a5 1 A−1 1 A−1
E 0 cm−1 10000 cm−1
req 3.25A 2.7 A
θeq 35.46◦ 122◦
rC-Cl 1.78A 1.78A
ωC-Cl 880 cm−1 880 cm−1
Table 2.2: Standard parameters chosen for the diabatic potential energy surfaces. The parameter
rC-Cl refers to a fixed distance for 2-D model and equilibrium bond length for 3-D model. The
frequencyωC-Cl is applicable only for 3-D model.
Lennard Jones potentials. Parameters between different atoms are computed using Lorentz-
Berthelot rules. CH2 has been coarse-grained to one particle, as the two coordinatesr andθ do
not involve any hydrogen motion. The parameters are given inTable 2.3.
2.3 Theoretical methods
NEMD is used to study the isomerization. The molecule is placed inside a face-centered argon
lattice, with its initial condition near the parent’s ground state minima. Chlorine and iodine each
occupy a lattice site, while CH2 occupies a hole. A density of 1.71 g/cm3 is chosen for argon.58
Periodic boundary conditions are applied to simulate an infinite cell. Because of the large
amount of energy given to the molecule (37500 cm−1), the early time collisions of the molecule
with argon atoms generate a shock wave in the argon bath. The highly ordered nature of the
argon lattice causes most of the shock wave energy to travel along a straight line joining CH2 and
14
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
60 100 140 180
r Cl−
I(A
)
θ
30
27
2421 18
15
12
96
3
q1
q2
Figure 2.2: Equi-potential energy contour lines are plotted with respect to Cl-I distance (r) and
C-Cl-I angle (θ). The contours are drawn after every 3000 cm−1. The arrows show the direction
of the two normal modesq1 andq2, having frequencies of 167 cm−1 and 247 cm−1, respectively.
Atom σ(A) ǫ(K) Ref.
Ar 3.405 120 55
CH2 3.905 59 56
Cl 3.480 188 57
I 3.780 472 36
Table 2.3: Parameters for Lennard-Jones potential. Parameters between different atoms are
computed using Lorentz-Berthelot rules.
iodine. This leads to the unphysical return of the energy dueto periodic boundary conditions. To
overcome this problem, a cuboid box is chosen with five face-centered cells along two directions
and seven face-centered cells along the third direction (total of 701 atoms). This ensures that the
direction along which the shock wave is travelling misses the molecule by two shells of argon
15
(a) Cubic Cell (b) Cuboid Cell
Figure 2.3: Cartoon showing return of the shock wave energy back to the molecule in the cubic
cell, and displacement of the direction of the shock wave vector in the cuboid cell.
atoms as shown in Fig. 2.3.
The experimental temperature is 12 K, which is substantially less than the Debye tempera-
ture of argonTD = hωD/kB = 117K.59 This implies that the zero point energy of the argon
bath plays an important role. It has been reported in literature that using a higher effective tem-
perature60,61 in NEMD simulations can incorporate the zero point energy effects. This approach
assumes the bath particles behave as simple harmonic oscillators in equilibrium with a heat
bath at temperature T. By comparing the quantum and the classical density matrices, a scaled
temperatureT ′ is obtained:
T ′ =TD
2 tanh(
TD
2T
) (2.5)
This approach has been applied previously with reasonably good results.60,61 A value of the
Debye frequencyωD = 81.374 cm−1 gives simulation temperature ofT ′ = 63K for the exper-
imental temperatureT = 12K. As will be shown later, the temperature of the bath does not
have significant impact on the results.
16
2.3.1 Non-equilibrium molecular dynamics
Monte-Carlo simulations are equilibrated in the canonical ensemble over the course of 500,000
steps. To equilibrate the velocities, molecular dynamics simulations are run for 80 ps with a
time step of 4 fs. The velocity-Verlet62 scheme is used to integrate the equation of motion,
while SHAKE62 maintains the C-Cl bond at a constant length (for the 2-D model). The initial
conditions for the below described simulations are generated by saving the positions and the
velocities of all the particles after approximately every 500 fs.
A kinetic energy of 37500 cm−1 is given to the C-I bond, in such a way that conserves the
linear and the angular momentum of the center of mass. The trajectories that fail to reach the
isomer well (or re-cross back) are discarded. Each trajectory is simulated for 100 ps. The abrupt
nature of the initial conditions requires small time steps,but as the energy relaxation proceeds,
the time step size are increased while maintaining the energy conservation (fluctuations< 2
cm−1). The integration times are 0.1 for the first 100 fs, 0.5 fs forthe next 500 fs, 2 fs for the
next 10 ps, and 4 fs for the rest of the trajectory. The internal coordinatesr, θ and the energy of
the molecule are saved after every 10 time steps.
The resulting long time decay rate constants are interpreted in the normal mode representa-
tion. The transformation from internal (s) to normal coordinates (q) is given by
s − seq = Lq, (2.6)
whereseq is the equilibrium configuration (in internal coordinates). The transformation matrix
L is computed using the standard FG method described in Wilson, Decius and Cross.63 In
order to properly account for the constraint forces in the 2Dmodel, all terms having Cl-I stretch
dependence in theG−1 matrix are set to zero, whereG is the kinetic energy matrix in internal
coordinates. The direction of normal modes (shown in Fig. 2.2) are given by the columns ofL.
A substantial kinetic coupling is observed between the internal coordinates, making the
normal mode representation useful. The computed gas phase normal mode frequencies are 147
cm−1 and 244 cm−1. The gas phase frequencies are shifted in the solvent as are the equilibrium
configurations.48 The equilibrium configurationreq andθeq shift from 2.7A and 122◦ to 2.693
A and 122.766◦, and the normal mode frequencies shift from 147 cm−1 and 244 cm−1 to 166
cm−1 and 247 cm−1. The normal modes corresponding to 166 cm−1 and 247 cm−1 will be
17
referred as mode 1 and mode 2, respectively for the rest of thechapter. The total energy of
the molecule and the normal mode energies are averaged over 1000 trajectories (including the
discarded trajectories).
In addition to the temporal relaxation of the normal modes, we also perform the short-time
Fourier transform of the coordinates (Fourier transformation as a function of time). In this
technique, the internal coordinates (r(t) andθ(t)) are multiplied by a window functionW (τ)
which is nonzero only in some period of time. The Fourier transformation of the resulting signal
as the window slides over time gives a time-frequency representation of the internal coordinates.
Mathematically, it is given by:64
If (t, ω) =
∫ +∞
−∞dτ eiωτf(τ)W (τ − t), (2.7)
wheref(t) is eitherr(t) or θ(t). The window function we use is a Gaussian
W (τ) =1
σ√
2πe−τ2/2σ2
, (2.8)
whereσ is chosen to be 256 fs (which gives a width of approximately 1.5 ps). The most intense
peaks correspond to the normal modes of the molecule, and their decay of intensity can give
information of the decay of the normal mode energies.
The experimentally observed quantity is not the decay of thevibrational energy, but the de-
cay of the spectral intensity. The time evolution of this intensity is proportional to the product
of Franck-Condon factors and the population of vibrational energy levels. Without knowledge
of the electronically excited surface, these factors cannot be computed. However, the difference
in the energy of the two probe pulses 435 nm and 485 nm (2370 cm−1) suggests that the Franck-
Condon factors for the higher wavelength pulse change most rapidly for states with energies less
than 2370 cm−1. Therefore we compute the decay of the population over this energy range, and
compare the results to the spectral intensity decay rate constants. Figure 2.4 shows a schematic,
where the width of the transition between the two electronicenergy surfaces indicates the spec-
tral intensity, while the width of the vibrational transition indicates the VER rate. The decay of
the spectral intensity as well as the vibrational energy is dependent on the population relaxation
of the vibrational energy levels, and hence are correlated.The time evolution of the population
is determined by calculating the number of trajectoriesNi in an energy window ranging from
18
Figure 2.4: A schematic comparing decay of the experimentally measured spectral intensity
with the computed energy decay rate. Width of the arrow between electronic energy surfaces
indicates spectral intensity, while width of the arrow between the vibrational energy levels in-
dicate the VER.
Ei−1 to Ei, where
Ebini = iEbin (2.9)
andEbin = 500 cm−1.
2.3.2 Landau-Teller theory: equilibrium molecular dynamics
We use LT theory to calculate the decay rate constant of a classical excited harmonic oscillator
immersed in a classical bath at temperature T. Under the assumptions of linear coupling of the
19
bath and first order perturbation theory (Fermi’s golden rule), it can be shown5 that
〈E〉 = − 1
T1
(〈E〉 − 〈Eeq〉), (2.10)
1
T1
=1
2µkBTcδF (ω0), (2.11)
cδF (ω) =
∫ +∞
−∞dt cos(ωt)〈δF (0)δF (t)〉. (2.12)
Here〈E〉 is the energy of the harmonic oscillator averaged over bath configurations,〈Eeq〉 is
the thermal equilibrium energy,T1 is the vibrational relaxation time, andcδF (ω) is the Fourier
transform of the correlation function of the friction forceF (t) exerted by the bath on the har-
monic oscillator. This formalism has been applied previously on a number of systems in liquids,
but to our knowledge this is the first application of the theory to VER in a solid.
In the LT description, the Hamiltonian is partitioned into two parts. The system Hamiltonian
comprises the normal modes, that are defined with respect to abody fixed frame. We choose
this frame such that the molecule lies in the xy-plane with its origin at the center of the mass
and the x-axis parallel to the C-Cl vector. The friction forceF (t) along the 2 normal modes is
computed using the chain rule. The bath Hamiltonian includes hindered rotation and translation
of the molecule, together with all the argon modes.
To determine the friction forces, the molecule is fixed at thesolvent-shifted geometry, and
molecular dynamics simulations are run for 20 ps with an integration time step of 4 fs. The
correlation functioncδF is computed by averaging over 1000 such trajectories. Changing the
integration time step to 1 fs, or the total run time to 40 ps does not change the results appreciably.
The initial conditions are generated in a similar fashion asin NEMD calculations.
2.4 Results and discussion
This section is divided into three parts. In the first part we present the energy decay results for
the 2-D model Hamiltonian described in Sec. 2.2. The resultsobtained for the 3-D model and
by varying the parameters mentioned in Table 2.2 in the subsequent parts.
20
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
60 100 140 180
r Cl−
I(A
)
θ
0
100
200
300
400
500
600
700
800
900
1000
30
27
2421 18
15
12
96
3
15 fs
30 fs45 fs60 fs
120 fs
180 fs
300 fs
400 fs
Figure 2.5: A representative trajectory. The color bar represents time in fs. Time labels are
marked on the trajectory.
2.4.1 2-D model
A representative trajectory is shown in Fig. 2.5. Some of thetime values are shown on the tra-
jectory. The first turning point represents the first collision of the molecule with argon (around
60 fs).
The collective early time dynamics are shown in Fig. 2.6. Theintensity is proportional
to the number of trajectories that visit that region of configuration space during the first 400
fs. The trajectories follow similar paths up to the first collision after which the paths become
sensitive to the details of the collision. The trajectoriesthat do not have sufficient energy to
cross the barrier again stay in the isomer well.
The energy of the molecule is defined as the sum of the kinetic energy of the molecule and
the adiabatic potential energy given by Eq. 2.1. Figure 2.7 shows the average of this energy
over 1000 trajectories. There is a very fast decay for the first 100 fs, during which the molecules
loses∼ 15,000 cm−1 of energy due to collisions with argon atoms, as can been be seen from
the inset of Fig. 2.7.
After about 1 ps, we treat the VER in a framework of normal modes coupled to the bath.
Figure 2.7 compares the time evolution of the total energy ofthe molecule to that of the normal
21
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
60 100 140 180
r Cl−
I(A
)
θ
0
50
100
150
200
250
300
350
30
27
2421 18
15
12
96
3
q1
q2
Figure 2.6: Intensity at each point represents the number oftrajectories that visit that point for
the first 250 fs.
mode energies (see Fig. 2.2 caption). The energy of mode 1 (E1) decays faster than does the
energy of mode 2 (E2). The decay rate constant obtained by fittingE1 to an exponential depends
upon the starting time of the fit, and varies from 1 ps−1 to 0.5 ps−1. The long time decay rate
constants ofE2 and the total energy are similar; they range from 0.08 ps−1 to 0.1 ps−1. These
decay rate constants compare qualitatively to the experimental decay rate constants of 0.45 ps−1
and 0.07 ps−1.
The short-time Fourier transformIf (t, ω) of Eq. (2.7) provides both frequency and time
information. TheIθ(t, ω) result of figure 2.8 shows the decay of both the normal modes. In
addition to the slower decay of mode 2, the anharmonicity in this mode for the first 5 ps can be
seen. The decay of the two local maxima of the[Iθ(t, ω)]2 as a function ofω is consistent with
the decay of the normal mode energies as should be expected. We have not shownIr(t, ω) as it
does not resolve mode 1 clearly.
To get a better comparison with the experimental results, one can bin the energy [see Eq.
2.9] at each instant of time, obtaining a fuller picture of the relaxation process. Figure 2.9
shows the energy bins as a function of time, where each energybin corresponds to a 500 cm−1
energy difference. It also shows the experimental spectralresults for comparison. They-axis of
22
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5 10 15 20 25 30
E(×
103cm
−1)
time(ps)
0
5
10
15
20
25
30
0 0.5 1 1.5 2
Total EnergyE1E2
E2
Figure 2.7: Comparison of the ensemble average of the total molecular energy with the average
normal mode energies from 1 ps to 30 ps. The inset shows the average energy for the first 2 ps.
2
4
6
8
10
12
14
16
18
20
100 150 200 250 300
tim(p
s)
freq (cm−1)
0
1e−15
2e−15
3e−15
4e−15
5e−15
6e−15
Figure 2.8: The average short-time Fourier transform [Iθ(t, ω), see Eq. (2.7) in text] plotted as
a function of time and frequency.
the experimental results have arbitrary units. The energy bin corresponding to 500-1000 cm−1
matches with the experimental curve qualitatively. To understand its significance, we need to
23
0
100
200
300
400
500
600
700
800
0 20 40 60 80 100
Nu
mb
ero
ftra
js.
time(ps)
250cm−1
750cm−1
1250cm −1
Experiment
Figure 2.9: Number of trajectories in the different energy ranges as a function of time. The
average energy of each bin is labelled for clarity.
understand the temporal behavior of energy bins.
After about 10 ps, the excess energy exclusively resides in mode 2. Hence it is no surprise
that fitting the long time tail of the 500-1000 cm−1 energy bin to an exponential yields decay
rate constants that vary from of 0.07 ps−1 to 0.09 ps−1 which compare well with the decay rate
constant of the total energy (or that of mode 2). Furthermoreit tells us that the initial energy
deposited in mode 2 is primarily in the 500-1000 cm−1 range. This initial partitioning of energy,
as we will show later, is dependent on the early time dynamics.
Having looked at the VER of the molecule, we now consider how this energy is distributed
in the argon bath by dividing the argon atoms into shells. An argon atom is considered in the
first shell if its distance from any of the atoms of CH2ClI is less than 4.5A. Similarly, those
atoms whose distance lies between 4.5A and 9A are binned in the second shell. Beyond the
first shell, the radial distribution function is not able to resolve the shells distinctly. Hence this
partitioning of the argon atoms into shells is somewhat artificial, but it still provides insight on
how energy is flowing through the argon bath. Small variations in this distance (± 0.5 A) do
not change the result appreciably.
The kinetic energy of the first three shells is plotted as a function of time in Fig. 2.10. The
jump in the first shell results from the collision of the molecule with one of the argon atoms
24
0
2
4
6
8
10
12
14
16
18
20
0 1 2 3 4 5
K.E
.(×
103cm
−1)
time(ps)
Shell 1
Shell 2
Shell 3
Figure 2.10: Kinetic energy of different shells as a function of time. The dotted horizontal lines
represent the thermal equilibrium kinetic energy for the three shells. Time 450 fs represents the
initial excitation time.
in this shell. One might expect that the energy transfer is primarily due to collisions between
chlorine and argon atoms due to the small difference in theirmasses. But the direction of the
initial momentum (along the C-I bond), and the arrangement ofthe argon atoms with respect
to the atoms in the molecule lead to the CH2 collision being more significant. These collisions
create shock waves in the lattice38 along the CH2 iodine direction. There are corresponding
jumps in the second and the third shells immediately after this collision. All the shells are in
thermal equilibrium by about 3 ps. This indicates rapid distribution of the energy in the argon
shells.
It is interesting to consider how the detailed structure of the argon atoms around the solute
molecule affects the decay rate constants.65,66 To show the effect of the solvent structure, a
second simulation is run where the argon atoms are initiallyequilibrated at 300 K (in the parent
well), and then the temperature of the bath is decreased to 63K by rescaling the velocities by
the factor67
χ =
(
1 +δt
tT
(
T
Γ− 1
))1/2
, (2.13)
where T = 63 K,Γ is the current temperature,δt is the integration time step, andtT is a preset
25
10
11
12
13
14
15
16
17
1 2 3 4 5 6 7 8 9 10
E(×
103cm
−1)
time(ps)
Quench
Standard
Figure 2.11: Energy as a function of time when the bath temperature is quenched compared to
the standard case.
time constant that determines the rate of the temperature change. By choosing a smalltT value
(5 fs), the solvent does not have enough time to relax to its equilibrium structure. Figure 2.11
shows the different energy decay obtained as a result of thisquenching. A least-square bi-
exponential fit gives an initial decay rate constant which isapproximately two times more than
computed without quenching. The long time decay rate constants are similar.
To further explore the effects of the solvent structure, trajectories are equilibrated in the
isomer well (rather than the parent well) and are given approximately 100-200 cm−1 of kinetic
energy to each normal mode. The decay rate constants so obtained are about a factor of 1.5
smaller compared to the decay rate constant calculated by equilibrating the molecule in the
parent geometry.
Another interesting observation is the dependence of the decay rate constants on the initial
excitation energy, shown in Fig. 2.12. For example, the decay rate constant of mode 1 changes
from 0.32 ps−1 to 0.46 ps−1 when the initial excitation energy is changed from 75 cm−1 to 175
cm−1. For a harmonic oscillator linearly coupled to a thermally equilibrated bath, LT theory
predicts an exponential decay of the energy. In this limit the decay rate constant is independent
of the initial excitation energy. Clearly we are not in this limit. But as will be shown next,
comparing these results to the LT results helps elucidate the decay mechanisms.
26
0
100
200
300
400
10 20 30 40 50 60 70 80 90 100
E(×
103cm
−1)
time(ps)
(b) Normal Mode 20
50
100
150
200
E(×
103cm
−1)
(a) Normal Mode 1
Decay rate=0.040 ps−1
Decay rate=0.054 ps−1
Decay rate=0.32 ps−1
Decay rate=0.46 ps−1
Figure 2.12: Energy decay of (a)E1, (b) E2 for different initial excitation energies.
Figure 2.13 shows the decay rate constants, predicted by LT theory (given by Eq. 2.11), as a
function of the frequency of the two normal modes. Both the normal modes have the maxima of
cδF (ω) at about 60 cm−1 (not shown for clarity). At the solvent-shifted frequencies of 166 cm−1
and 247 cm−1, the decay rate constants obtained are 0.58 ps−1 and 0.018 ps−1, respectively.
The LT results indicate that mode 2 decays more slowly than mode 1 for two reasons. The
most obvious reason is that mode 2 has a higher frequency. However, even if their frequencies
were comparable, mode 2 would still decay more slowly as a result of the weaker interaction
with the bath compared to mode 1.
The most important cause of deviations in the decay rate constants obtained through LT
and NEMD results from neglect of the couplings between the two normal modes in LT theory.
We can demonstrate this by decoupling the two modes in the NEMD calculations. We do this
by significantly increasing the diagonal force constants ofone of the internal modes. This
decoupling produces good agreement between the decay rate constants computed through LT
and NEMD as shown in Table 2.4. For example, increasing the gas phase stretch frequency
to 565 cm−1 (with bend frequency remaining relatively unperturbed at 154 cm−1), yield decay
rate constants of 0.48 ps−1 computed through LT theory, and 0.44 ps−1 ± 0.02 ps−1 obtained
27
0.2
0.4
0.6
0.8
160 180 200 220 240 260 280 300
c δF/2
µk
bT
(ps−
1)
ω(cm−1)
(166,0.58)
(247,0.018)
Normal Mode 1
Normal Mode 2
Figure 2.13: Fourier transform of the time correlation function cδF (t)/2µkbT for the two normal
modes [see Eq. (2.11)]. The decay rate constants at the normal mode frequencies are 0.58 ps−1
and 0.018 ps−1 respectively, and are marked by•.
through NEMD.
The decay rate constants mentioned above are summarized in Table 2.4. NEMD (parent)
and NEMD (isomer) refer to the different starting configuration used to equilibrate with the
solvent. The decay rate constants for NEMD (uncoupled) refer to the computation done after
decoupling the two modes as descried above. If there is dependence of the decay rate constant
on the energy, we have reported an average value. The uncertainties reflect the variation in the
decay rate constants with the initial time for the exponential fit. In all of the cases, the decay rate
constants of mode 1 and mode 2 are of the order of 0.5 ps−1 and 0.05 ps−1, respectively. The
qualitative agreement of the decay rate constants obtainedby LT theory and NEMD indicates
that the system can be interpreted using linear response theory after 1 ps.
2.4.2 3-D model
We incorporate the effects of C-Cl stretch by modelling the C-Cl bond with a harmonic oscilla-
tor having 880 cm−1 frequency. Following the same procedure for initialization as is used for
the 2D model, we re-calculate the time evolution of the totalenergy of the CH2ClI molecule.
28
Mode 1 Mode 2
Experiment4 0.45±0.1 0.074±0.003
NEMD (parent) 0.6±0.3 0.081±0.006
NEMD (isomer) 0.43±0.05 0.047±0.005
LT 0.58 0.018
NEMD (uncoupled) 0.44±0.02 0.036±0.001
LT (uncoupled) 0.48 0.036
Table 2.4: Summary of calculated decay rate constants. All decay rate constants are in ps−1.
The energy of the molecule plotted in Fig. 2.14 shows a somewhat slower initial decay com-
pared to the 2D model. The largest difference is an upward shift of the 3D result. The early
time large amplitude motion, leading to the isomerization of the molecule, causes large kinetic
coupling imparting the energy to all the three vibrational modes. On the timescales of interest,
the energy deposited in the C-Cl stretch has been sequestered,hence the shift. Since the decay
rate constants ofr andθ degrees of freedom are similar for both models, we return to the 2D
model.
2.4.3 Variation of the standard parameters
Having examined VER for the model Hamiltonian, we now consider the sensitivity of our re-
sults to the parameters of this Hamiltonian. To obtain a better understanding of this sensitivity,
we vary several parameters and recompute the average energyfor 10 ps. The goal is to change
only one parameter at a time, but in some instances this is notpossible, for example changing
the density of the argon also changes the solvent-shifted normal mode frequencies.
The parameters have been divided into three categories. These are the parameters related to
a) the potential energy surface of the isomer reported in Table 2.2, b) the bath parameters which
include the density and the temperature of argon, and c) the initial conditions which include the
amount of the energy provided and the placement of the molecule in the bath. Most of these
29
0
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8 9 10
E(×
103cm
−1)
time(ps)
3D Model (includes C-Cl stretch)
2D Model
Figure 2.14: Average energy, in wavenumbers, as a function of time for the 3-D model.
parameters do not change the results appreciably. The parameters that change the VER decay
rate constants the most are the isomer frequency, the isomerposition and the density of argon,
which are discussed next.
The normal mode frequencies cause the largest change in the decay rate constants as shown
in Fig. 2.15. As is expected from the LT results, decreasing the frequency of both modes or only
of mode 2 increases the decay rate constant. Interestingly,decreasing the frequency of mode
1 leads to slower VER. This can be explained by the dependence of the initial partitioning
of the energy between the two normal modes on the frequenciesof the normal modes. This
dependence arises because the initial collision of the molecule with the argon atoms is sensitive
to the details of the potential energy surface, leading to different distribution of energies amongst
the different modes. When the mode 1 frequency decreases, mode 2 acquires more energy after
the collision with the argon bath, slowing down the overall energy decay.
The isomer equilibrium position also determines the distribution of the energy into the two
normal modes, as well as the anharmonicities which play an important role in determining the
initial decay rate constants as shown in Fig. 2.16. The long time energy decay remains the same
as the mode 2 frequency, which dictates the long time decay, remains approximately unchanged.
Figure 2.17 shows that changing the bath density affects thedecay rate constants, which is
in agreement with the results obtained by Brown, Harris and Tully.36 Increasing the density
30
11
13
15
E(×
103cm
−1)
11
13
15
1 3 5 7 9
E(×
103cm
−1)
time(ps)1 3 5 7 9
time(ps)
89 cm−1,166 cm−1
standard
124 cm−1,238 cm−1
standard
147 cm−1,211 cm−1
standard
171 cm−1,277 cm−1
standard
Figure 2.15: Energy as a function of time for different normal mode frequencies compared to
the standard case (147 cm−1, 244 cm−1). The frequencies listed are the gas phase normal mode
frequencies.
11
13
15
1 3 5 7 9
E(×
103cm
−1)
time(ps)
11
13
15
E(×
103cm
−1)
(req, θeq)
(req + δr, θeq)
(req + 2δr, θeq)
(req, θeq)
(req, θeq − δθ)
(req, θeq + δθ)
Figure 2.16: The energy as a function of the time for the different equilibrium isomer config-
urations compared to the standard case (2.7A and122◦). δθ andδr are set to10◦ and0.3A
respectively.
31
10
11
12
13
14
15
16
1 2 3 4 5 6 7 8 9 10
E(×
103cm
−1)
time(ps)
standard
1.62 g/cm3
1.77 g/cm3
Figure 2.17: Energy as a function of time for different bath densities compared to the standard
case (1.71 g/cm3).
leads to stronger interactions of the normal modes with the solvent modes, hence increasing the
decay rate constants.
2.5 Conclusions
NEMD calculations using two and three dimensional models allow us to determine the pathways
of VER in CH2Cl-I. The molecule loses about 15000 cm−1 of energy via impulsive collisions
with argon during the first 100 fs. This energy flows rapidly among the bath modes. After
1 ps, we find that VER can be described in a normal mode framework. These normal modes
roughly correspond to C-Cl-I bend (frequency∼ 167 cm−1) and Cl-I stretch (frequency∼ 247
cm−1), and are referred to as mode 1 and mode 2, respectively. Mode2 decays an order of
magnitude slower than does mode 1. The effect of changing thedetails of the isomer potential
energy surface, initial conditions and the solvent structure have been explored. Based on these
changes, we believe our results to be accurate to within a factor of 2. We have determined that
the two most important features that dictate timescales arethe normal mode frequencies and the
initial partitioning of energy among the normal modes.
32
The two decay rate constants computed experimentally were interpreted as the fast forma-
tion and initial relaxation of the isomer and the slower decay of the energy to the surroundings.4
We have identified that this slower decay is due to VER of mode 2. Our calculated decay rate
constant of mode 1 matches the experimental rate constant offast formation and initial relax-
ation of the isomer, but because of the adiabatic approximation made in this work, further study
is needed to establish this early time correspondence. The third lowest frequency mode is the
C-Cl stretch (frequency 880 cm−1). This mode is shown to relax much slower than the observed
timescales.
LT theory predictions, which are shown to be in qualitative agreement with NEMD results,
clarify the influence of vibrational frequencies on decay rate constants as well provide additional
insights into the energy flow mechanisms. It shows that mode 1decays faster, not only because
it has lower frequency, but also due to its stronger interaction with the bath modes.
An interesting direction for future research is to examine the role of different solvents on
the energy flow. Experimental decay rate constants of 0.002 ps−1 and 0.02 ps−1 are observed
for nitrogen and methane matrices, respectively.4 Further study is required to shed light on the
reasons for this order of magnitude difference in decay rateconstants.
33
Chapter 3
Rates of symmetric proton tunneling -
surface hopping and ring polymer
molecular dynamics
Contents
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.1 Exact quantum dynamics . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.2 Surface hopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.3 Ring polymer molecular dynamics . . . . . . . . . . . . . . . . . . 49
3.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4.1 Exact quantum dynamics . . . . . . . . . . . . . . . . . . . . . . . 51
3.4.2 Surface hopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4.3 Ring polymer molecular dynamics . . . . . . . . . . . . . . . . . . 56
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Figures
3.1 Equipotential contour plots of the potential energy surface modelling the
formic acid dimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
34
3.2 Slice of the adiabatic surface alongq4 at q6 = 0.6 A . . . . . . . . . . . . . . 44
3.3 The full adiabatic surface as a function ofq4 andq6 for gas phase FAD . . . . 44
3.4 Comparison of this diabatic surface with the adiabatic surface withq4 = 0 . . 46
3.5 The first 8 tunneling splittings as a function of the coupling strength for the
formic acid dimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.6 The decay ofPR computed using exact quantum dynamics . . . . . . . . . . 53
3.7 Comparison of decay ofPR computed using LZ formalism and the fewest
switching criterion of surface hopping . . . . . . . . . . . . . . . . . . . . . 54
3.8 The decay ofPR computed using NVE and NVT calcultions of surface hopping 54
3.9 The decay ofPR computed using a hybrid calculation of surface hopping . . 55
3.10 The decay ofPR computed with the restriction on the intrinsic bath coordinate
applied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.11 Comparison of the decay ofPR computed exactly and assuming first order
kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.12 The ratio of the rates at 300 K to 200 K computed using RPMD and exact
calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Tables
3.1 Parameters for the potential energy surface of FAD . . . . . . . . . . . .. . 39
3.2 The parameters for the DVR basis for gas-phase FAD . . . . . . . . . . .. . 41
3.3 The parameters for the DVR basis for gas-phase FAD coupled to a harmonic
bath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 Tunneling splittings for gas-phase FAD . . . . . . . . . . . . . . . . . . . . 51
3.1 Introduction
Proton transfer plays a key role in many chemical and biological systems. Quantitative compu-
tation of its rates has been the focus of many studies, and still remains a challenging problem.
Its sensitivity to the details of the potential energy surface and temperature makes it an ideal
process for testing approximate methods.
For systems with only a few degrees of freedom, one can solve the time independent
Schrodinger equation (TISE) and obtain exact dynamics. For large systems, several semiclas-
sical methods have been developed over the last two decades that incorporate tunneling effects
35
into the dynamics. Topaler and Makri68 have applied the quasi-adiabatic path integral (QUAPI)
approach to obtain the numerically exact thermal rate constants for a system-bath model which
comprises a one dimensional double well linearly coupled toa harmonic bath. This model has
since served as a benchmark for several methods, including numerically exact multilayer multi-
configuration time-dependent Hartree (ML-MCTDH)69,70 and hierarchical equation of motion
(HEOM),71–73 as well as semiclassical and mixed quantum-classical methods such as surface
hopping (SH),74 ring polymer molecular dynamics (RPMD),8 and centroid molecular dynamics
(CMD).75,76 All of these methods have their advantages and disadvantages. The exact meth-
ods are of course accurate, but are applicable to either onlysmall systems or specific functional
form of coupling. The semiclassical methods are applicableto large systems; their disadvantage
being inaccurate treatment of coherence and entanglement effects.
Despite the success of this model to simulate the effects of the condensed phase environment
on proton tunneling, it may not adequately represent certain aspects of tunneling in systems
where the tunneling mode is a high frequency hydrogen stretch. Another important feature that
is absent from the system-bath model is the acceptor-donor distance mode that plays a central
role in the Azzouz-Borgis model.77 The effects of this mode have been incorporated in several
studies by symmetrically coupling the double well potential to a low frequency mode.72,78,79
In this work, we investigate a model Hamiltonian that simulates the tunneling splitting patterns
of the formic acid dimer (FAD), where the tunneling mode in this system is represented by a
symmetric hydrogen stretch, which is strongly coupled to two low frequency modes - the sym-
metric dimer rock and the dimer stretch. The couplings of these modes represent the symmetric
and the anti-symmetric coupling, the two important forms ofsymmetry that influence tunnel-
ing.72,80–82 The system-bath model can be understood as an adiabatic limit of this system in
the classical Hynes picture,83 where the dimer rock plays the role of the solvent coordinateand
the dimer stretch modulates the tunneling probabilities. In order to simulate the effects of the
condensed phase, linear coupling of the dimer rock with a harmonic bath is investigated. We in
this chapter obtain the exact dynamics by solving the TISE, and benchmark SH and RPMD to
test their validity to our model system.
RPMD, a method based on the path integral formulation of quantum mechanics, can ac-
curately compute correlation functions from classical trajectories in an extended phase space.
Manolopoulos and coworkers developed this formalism to compute the thermal rate constants
36
using flux-side correlation function, getting excellent agreement with the results of Topaler and
Makri.8 Subsequently, RPMD has been employed extensively to computerate constants of
tunneling.84–87 There are, however, limitations of this method. These limitations include poor
performance for the correlation functions of nonlinear operators, and for rate constant of elec-
tron transfer in the Marcus inverted regime.84,88,89 Surprisingly, Althorpe and coworkers have
shown that RPMD may still predict the rate constants in the deep tunneling regime due to its
connection with the semiclassical instanton theory.9
The SH algorithm, originally developed by Tully,10,11 is a widely used mixed quantum-
classical method that can predict the timescales of tunneling for a wide range of systems.12–14
In this method, the classical degrees of freedom evolve on a single potential energy surface.
The nonadiabatic effects are partially incorporated by theinstantaneous hops between the sur-
faces dictated by evolving the quantum mechanical state amplitudes. References 90–92 provide
excellent reviews on this theory. Hammes-Schiffer and coworkers extended the SH method to
compute the thermal rate constant within the reactive flux formalism.74 Quantitative agreement
with the exact Topaler and Makri results are obtained in the strong coupling regime, although
in the weak coupling regime much larger rates are obtained.
It is interesting to note how these methods will perform in the deep tunneling regime where
the quantum coherences play a central role. In the weak coupling regime, the details of energy
flow into and out of the system are important. In addition, thechange in path length of the
tunneling path can have important effects. The interplay ofboth these factors governs the exact
dynamics. We in this chapter show that SH and RPMD can incorporate the effects of energy
diffusion, but struggle with the corresponding effects of the changes in path length. This is
expected because of the inaccurate treatment of coherence in both these methods.
This chapter is organized as follows: In Sec. 3.2, the Hamiltonian that models the tunneling
splittings of the FAD is presented. In Sec. 3.3 we describe various methods employed to
compute the timescales of tunneling. The details of the exact method to solve the TISE, SH
and RPMD are summarized in this section. The results are provided and discussed in Sec. 3.4;
Section 3.5 concludes the chapter.
37
3.2 Model Hamiltonian
Barnes and Sibert have performed extensive studies on the tunneling splittings of FAD.82,93–95
One of their conclusions was that the qualitative features of the splittings can be retained by
considering a three dimensional model. The three modes are defined corresponding to the
transition state (T.S.) geometry, and represent the symmetric proton stretch (q1), the symmetric
dimer rock (q4), and the dimer stretch (q6). We maintain the same notation used by Barnes
and Sibert. These three modes will be collectively referredto as the gas-phase FAD, and its
Hamiltonian is given by
HG =∑
i=1,4,6
p2i
2m+ VG(q1, q4, q6), (3.1)
wherem = 1 a.m.u. is the mass, andpi is the momentum.
We fit the potential of Barnes and Sibert assuming a simple functional form that retains
the qualitative features of the tunneling splittings. Thisform is obtained by first constructing
the diabatic potentialsV dL (V d
R) that describe the region about the left (right) minimum. These
potentials are related by the symmetry relationV dR(q1, q4, q6) = V d
L (−q1,−q4, q6). The full
adiabatic potential is obtained by solving for the lowest eigenvalue of the 2× 2 matrix96
VG =
(
V dL V12
V12 V dR
)
, (3.2)
whereV12 is an interaction matrix element introduced to couple the diabatic surfaces.V dL is
taken to be of the form
V dL (q1, q4, q6) = a1(1 − ea2∆q1)2 + a3∆q2
4 + a4(1 − e−a5∆q6)2 + a6∆q1∆q4+ (3.3)
(a7∆q6∆q1 + a8∆q4∆q6)S(α(qc6 − ∆q6)),
with ∆qi = qi − q0i , where{q0
1, q04, q
06} = {0.582,−0.897, 1.26}A are the equilibrium val-
ues of the coordinates. Coupling to mode 6 is multiplied by thesigmoid functionS(x) =
[1 + exp(−x)]−1 to ensure correct asymptotic properties. The parameters are given in Table
3.1.
38
Figure 3.1: Equipotential contour plots of the potential energy surfaceVG of Eq. (3.1). Contours
are separated by 1000 cm−1. The constant coordinate is fixed at the transition state value (a)
q4 = 0 and (b)q6 = 0.
V12 is computed based on the method described in Ref. 96. The potential is expanded near
the transition state
VG = V0 +1
2qTK0q, (3.4)
whereq ≡ {q1, q4, q6}, V0 is the barrier height, andK0 is the diagonal force constant matrix.
These values are also listed in Table 3.1
The contour plots of two surfaces are shown in Fig. 3.1. Theseplots show the different
effects of the coupling. Mode 4 is coupled anti-symmetrically to the tunneling modeq1, and
acts as a solvent coordinate changing its sign as the reaction proceeds from the left minimum to
the right minimum. Mode 6, on the other hand, is coupled symmetrically to q1, and modulates
the width and height of the barrier, hence changing the tunneling probability.83 The symmetry
of these two forms of coupling play a key role in determining the effects of the vibrational
excitation on the tunneling splittings.72,80–82 These effects will be discussed in detail in the
results section.
The effect of a bath is simulated by the linear coupling ofq4 to a bath of harmonic oscilla-
tors.68 The Hamiltonian for this model is
H(p,q) = HG + Hbath, (3.5)
39
Parameter Value
a1 2291994.6 cm−1
a2 0.18A−1
a3 811.6 cm−1A−2
a4 331322.4 cm−1
a5 0.094A−1
a6 9612.7 cm−1A−2
a7 -25454.9 cm−1A−2
a8 -1589.2 cm−1A−2
α 2 A−1
qc6 4 A
V0 2909 cm−1
K1,10 -48284.6 cm−1A−2
K2,20 1504.6 cm−1A−2
K3,30 7869.4 cm−1A−2
Table 3.1: Parameters for the potential energy surface.
with
Hbath =
f∑
i=1
p2xi
2m+
1
2mω2
i
(
xi −ciq4
mω2i
)2
, (3.6)
wherem = 1 a.m.u.,xi andpxirepresents the coordinates and the corresponding momenta,
respectively of thef bath modes. The discretization scheme used for the continuous Ohmic
bath is8
ωi = −ωc ln
(
i − 0.5
f
)
, (3.7)
40
and
ci = ωi
[
2ηmωc
fπ
]1/2
, (3.8)
with the cutoff frequencyωc = 500cm−1, andη represent the coupling strength of the bath with
FAD.
3.3 Methods
We will consider several methods for obtaining the timescales of tunneling. These methods are
outlined here. We first describe the details of solving the TISE, that gives numerically exact
results. A brief summary and the simulation details of SH andRPMD methods are discussed
next.
3.3.1 Exact quantum dynamics
We obtain numerically exact dynamics by solving for the eigenvalues of the full Hamiltonian in
the diabatic basis, closely following the details given in Ref. 82. The diabatic basis functions are
the eigenfunctions (φdL,R) of the Hamiltonian with the potentialV d
L andV dR , respectively. This
diabatic Hamiltonian is constructed in the DVR basis set. Wetransform this Hamiltonian in a
two-step contraction scheme. In the first contraction step,the DVR basis set with respect toq1
is contracted to the eigenfunctions of the diabatic Hamiltonian with the potentialV dL (q1; q4, q6)
computed at fixed values of mode 4 and 6 at their correspondingDVR grid points. This is
similar to making an adiabatic approximation to the high frequency mode 1, while treating the
low frequency modes 4 and 6 classically. Since multiple adiabatic surfaces are computed, and
the off-diagonal elements are retained, the transformation is numerically exact. In the second
contraction step, the DVR basis functions forq4 andq6 are used as a basis calculate the eigen-
functions of the adiabatic surfaces obtained after the firstcontraction step. Table 3.2 shows the
details of the DVR basis and the contraction steps. In the first contraction step, 30 DVR func-
tions are transformed to 6 basis functions (for each of the 20x21 DVR points corresponding to
mode 4 and 6). In the second step, 420 DVR basis functions for mode 4 and 6 are contracted
to 25 basis functions (for each of the 6 basis functions for mode 1). The final diabatic Hamil-
41
i Ni qmin(A) qmax(A) Ncont
1 30 -1.20 1.20 6
4 21 -3.14 3.1425
6 20 -0.60 3.40
Table 3.2: The parameters for the equally spaced Ni DVR points. Ncont is the number of basis
functions used in the contraction steps discussed in the text.
tonian obtained after these steps is a 150x150 matrix. Diagonalizing this Hamiltonian gives the
diabatic basis setφdL in the DVR representation. The basis functionsφd
R are computed using
the symmetry relationφdR(q1, q4, q6) = φd
L(−q1,−q4, q6). Finally an orthonormal basis set, ob-
tained via Lowdin orthogonalization,97 is used to calculate the eigenvalues of the full gas-phase
FAD HamiltonianHG [Eq. (3.1)]. All diagonalization of matrices have been performed using
the package LAPACK.98
To compute the eigenvalues for FAD coupled to a bath, we use a direct product basis. For
the FAD degrees of freedom, we use the gas-phase eigenfunctions. For the bath modes, eigen-
functions of the corresponding simple harmonic oscillatorare used. Five bath modes are con-
sidered, and the tunneling splittings are computed in the weak coupling regime (η/mωb ≤ 0.1).
The eigenvalues can be readily evaluated using the Lanczos algorithm implemented in the pack-
age ARPACK.99 The DVR grid for the bath modes, and the number of basis functions whose
direct-product generate the full basis functions are givenin Table 3.3. The first 60 tunneling
splittings are converged to within 5% accuracy with respectto increasing any basis set size by
a factor of 1.5.
For a comparison to the thermal rate constants obtained using RPMD, we develop a kinetic
model for tunneling at temperatureT . The exact probability of tunneling in a two-state model
is given by
PR(t) =1
2
[
1 +∑
i
c2i cos(∆it/h)
]
, (3.9)
42
i ωi Ni xmin(A) xmax(A) Ncont
1 1151 200 -3 3 5
2 601 200 -3 3 5
3 346 200 -3 3 5
4 178 200 -4 4 10
5 52 200 -4 4 15
FAD - - - - 45
Table 3.3: The parameters for the equally spaced Ni DVR points. Ncont is the number of basis
functions used for the particular mode. The frequencyωi is given by Eq. (3.7). The last row
shows the number of basis functions used for FAD.
where the sum is truncated for the first 60 tunneling splittings,∆i is the splitting for thei’th
excited state, andci represent the coefficients of the wavefunction at time 0 in the left diabatic
basisφdL
c2i =
e−βEi
∑
i e−βEi
, (3.10)
with β = 1/kBT . Ei is the average energy of the tunneling doublet. This two state model is
tested to be accurate to numerical accuracy for gas-phase FAD.
Since the decay rate att = 0 obtained from Eq. 3.9 is 0, we develop a first order kinetic
model as
PR(t) =1
2
[
1 +∑
i
c2i e
−∆it/h
]
. (3.11)
The thermal rate constant can now be defined as
k = −(
dPR
dt
)
t=0
=1
2
∑
i
c2i ∆i
h. (3.12)
43
This will be useful in comparing the thermal rate constants obtained using RPMD.
3.3.2 Surface hopping
SH methods90,92 separate the Hamiltonian into classical and quantum degrees of freedom.
We treatq1 quantum mechanically, and all the other degrees of freedom classically: R ≡{q4, q6,xi}. Figure 3.2 shows the plot of the adiabatic potential forq6 = 0.6A, and serves
to illustrate a fundamental point. The introduction of the separation of quantum and classical
degrees of freedom converts a 2900 cm−1 barrier crossing problem into a relatively low energy
surface hopping problem. The full adiabatic surface for gas-phase FAD is shown in Fig. 3.3.
The time-dependent wavefunction for theq1 degree of freedomψ(q1;R, t) is expanded in an
orthonormal basis setφn(q1;R(t))
ψ(q1;R, t) =∑
n
cn(t)φn(q1;R(t)). (3.13)
Substituting this into the time-dependent Schrodinger equation and using the fact that
φn(q1;R(t)) depends on timet only throughR(t) gives
ihcj =∑
n
cn(Vjn − djn.R), (3.14)
whereVjn = 〈φj| H| φn〉, and the nonadiabatic coupling vectordjn(R) = 〈φj| ∇Rφn〉. The
brackets denote integration over only the quantum degrees of the freedom.
The forces for the classical equation of motion are determined by a single basis functionφi
mR = −〈φi| ∇RH| φi〉 , (3.15)
wherem is the mass. The system can ‘hop’ fromφi to φj at timet according to the rate of the
quantum probabilities|ci(t)|2. The fewest switching algorithm gives the probability of this hop
as a function of the variables propagated via Eqs. (3.14-3.15).
There are several choices present in the literature for the basis functionsφi.100–102 The
representation that leads to fewest hops is generally considered better. In this adiabatic rep-
resentation, the surfaces come close to each other as shown in Fig. 3.2, leading to large but
44
Figure 3.2: The adiabatic surface alongq4 at q6 = 0.6A. The inset shows a closeup of the
avoided crossing.
Figure 3.3: The full adiabatic surface as a function ofq4 andq6 for gas phase FAD.
highly localized nonadiabatic couplingdij values nearq4 = 0. Hence the probability of the
hop is nearly one whenever the classical trajectory samplesthese regions of largedij, and in
order to sample these regions one needs to take very small time steps. The small time steps
45
required for the evolution of the quantum amplitudes can be mitigated to some extent using the
recent formulation of Wang and Prezhdo, where the quantum amplitudes are evolved in a more
convenient diabatic basis with the transition probabilityto the state with largedij computed
through the difference between the probability of hopping out of the current state and the sum
of probabilities of hopping to all other states.103 As an alternative, trajectories can be run in
the diabatic representation where there are fewer hops. Theprobability of these hops is mostly
governed byVij. This coupling, however, is de-localized and presents the problem that most of
the hops are frustrated. For these reasons we choose to evolve the classical trajectories on the
diabatic surface, and use an appropriate nonadiabatic transition probability formula whenever
the trajectory crosses the dividing surfaceq4 = 0, similar to the Tully and Preston method.10 As
the diabatic surfaces meet at this point, the issue of frustrated hops does not arise. The diabatic
surface is defined as
EdL/R =
⟨
φdL/R
∣
∣ H∣
∣ φdL/R
⟩
, (3.16)
whereφdL/R is the diabatic eigenfunction, as was defined in the previoussection, andH is the full
Hamiltonian. Using the full Hamiltonian in the definition ofthe diabatic surface includes some
of the effects of the anharmonicity near the transition state. The comparison of this diabatic
surface with the adiabatic surface forq4 = 0 is shown in Fig. 3.4.
The nonadiabatic transition probabilityP12 is often computed through the well known
Landau-Zener (LZ) formula104
P12 = 1 − e−p, (3.17)
with
p =2πH2
12
h|v4F12|, (3.18)
whereH12 is the coupling between the diabatic representation of LZ formalism,v4 is the com-
ponent of velocity along mode 4, and|F12| is the magnitude of the difference in the slopes of the
diabats. All of the above quantities are computed at the dividing surface. This equation can be
46
Figure 3.4: Comparison of this diabatic surface with the adiabatic surface withq4 = 0.
rearranged in terms of quantities that are easy to compute inthe adiabatic representation as105
p =2π∆0
8h|v4d012|
, (3.19)
where∆0 is the energy difference between the adiabats, shown in the inset of Fig. 3.2, andd012
is the component of the nonadiabatic coupling vector along mode 4.
The multidimensional nature of tunneling can be challenging to capture in the one dimen-
sional LZ formalism.106–108 Nakamura and coworkers have made significant improvements to
the LZ formalism by casting it as a scattering matrix problem, and have derived transition proba-
bilities across all regimes. The formalisms by Lasser and coworkers and Lebedev and coworkers
were compared recently,109 and were shown to produce similar results. Borrowing ideas from
the above formalisms, we develop a theory of our own to incorporate the multi-dimensional
effects. For a two dimensional potential, assuming that∆ varies exponentially with changes
alongq6 (which is to be expected from WKB theory)
∆(q6) = ∆0e−α(q6−q0
6), (3.20)
whereq06 is the point where the trajectory crosses the dividing surface, it can be shown within
47
the LZ formalism that
p =π
4h2
∆20
|k1|exp
[
−k22Re(k1)
2|k1|2]
, (3.21)
where
k1 = −1
2αa6 + ih−1v4d
012∆0, (3.22)
a6 is the acceleration along mode 6, andk2 = αv6. The detailed proof is given in Appendix B.
Numerical simulations designed to test this theory show that the qualitative trends for changes
in transition probability with changes inv6 are correctly captured with very little extra compu-
tational cost.
We have applied the standard 1D LZ formalism, the above 2D formalism, and the one devel-
oped by Nakamura and coworkers106 for a standard case, and the results are in close agreement.
This is because of relatively small multidimensional effects in the high energy regime. All re-
sults shown later are computed using the classical 1D LZ theory. To further test the validity of
LZ theory, we also performed calculations on the adiabatic surface with the hops governed by
evolutions of quantum amplitudes.11 The details of these calculations are provided in Appendix
C. The results are generally in agreement with those obtainedusing LZ formalism.
Choosing appropriate initial conditions poses another challenge. At 300 K, many trajecto-
ries do not have enough energy to reach the dividing surfaceq4 = 0. This causesPR not to
equilibrate at 0.5 as should be expected.
Even in the presence of the bath we will see that the energy flowbetween the FAD states is
limited. It is interesting therefore to explore the role of energy diffusion. For this purpose, we
have implemented two different approaches. In the first, we simulate a canonical ensemble by
running trajectories with a thermostat which ensures that all trajectories can reach the dividing
surface. We have applied the Anderson as well as the Nose-Hoover thermostat, and no notice-
able difference is observed between them. We note that even though these trajectories do not
have dynamical information in them, it would be expected that they sample the dividing surface
with the correct thermal distribution. Hence the LZ rate obtained should still be accurate.73 This
is similar to the reactive flux approach to compute rates,74 and should represent the thermal rate
constant.
48
This approach has a drawback that it reports only the averagerate. The multiple timescales
that characterize tunneling are lost in this approach. To extract these timescales, we also run the
trajectories at constant energy with initial conditions sampled from a Boltzmann distribution
and averaging over only those trajectories that tunnel on the timescale of the problem. Any
trajectory that does not tunnel in 1 ns is discarded. We find that if the initial energy is not
sufficient to reach (q4,q6)=(0,0.7)A, the trajectories do not tunnel, and hence we impose this
condition on the initial conditions for faster computations.
For the calculations performed using LZ method, an average is taken over 1200 trajectories
for NVE simulations, and 300 trajectories for NVT simulations using the velocity-Verlet inte-
gration scheme with a time step of 2 fs. The trajectories are evolved for a duration of 4 ns. The
variables∆0 andd012 in Eq. (3.19) are computed by solving the lowest two eigenvalues of the
1D potential whenever the trajectory crosses the dividing surface. The trajectory is back inte-
grated with the appropriate time step to reach theq4 = 0 dividing surface. An equally spaced
30 points DVR grid ranging from -1.2A to 1.2A is used for diagonalizing the Hamiltonian.
The Nose-Hover thermostat is implemented by coupling the system with a Nose-Hover
chain of length two. The fictitious masses (Q1 andQ2 in Eqs. (6.1.29-6.1.34) of Ref. 110) are
computed byQ1 = NclkBTτ 2 andQ2 = kBTτ 2, whereNcl are the number of the classical
degrees of freedom, andτ is the characteristic time scale of the system.111 We chooseτ = 10
fs, and confirm that the results are independent of variations to this choice.
For both the constant energy and constant temperature ensembles, the trajectories are ini-
tiated in the left well by using the action-angle variables for the normal modes around the
minimum. The angle variable is chosen randomly from the set [0,2π], and the action vari-
able which is quantized is chosen from a Boltzmann distribution of the corresponding energies.
The inclusion of the zero-point energies in the bath do not affect the results appreciably since
the higher frequency bath modes are less important to the dynamics. Calculations were also
performed with initial conditions chosen from a Boltzmann distribution, while discarding the
trajectories that do not tunnel, and similar results are obtained. For constant temperature calcu-
lations, the tunneling times are much longer than the thermal equilibration time, and hence an
initial equilibration time is not necessarily required.
49
3.3.3 Ring polymer molecular dynamics
The RPMD formulation8,84,112is based on the path integral formulation of quantum mechanics,
where each degree of freedom is mapped onto a fictitious ring polymer withk beads connected
through springs. We use the notationR = {q1, q4, q6,x} for the full position vector, andn = 8
as the total number of coordinates. The ring-polymer HamiltonianHk is given by
Hk =n
∑
i=1
k∑
j=1
p2i,j
2m+ Vk(Rk), (3.23)
where
Vk(Rk) =n
∑
i=1
k∑
j=1
1
2mω2
k(Ri,j − Ri,j−1)2 +
k∑
j=1
V (R1,j, R2,j, ..., Rn,j), (3.24)
with ωk = k/βh. The variablesRk,pk represent the extended phase space of the system with
Ri,0 = Ri,k. HereV (R) is the potential of the Hamiltonian defined in Eq. (3.5).
The dividing surfaces(R) is usually chosen as a coordinate of the system for simplicity:
s(R) = Rs − R‡s. We chooseRs = q1, with its optimal transition valueR‡
s = 0. The thermal
rate constant is given by
k(T ) = limt→∞
κ(t)kQTST , (3.25)
wherekQTST is the quantum transition state approximation for the rate constant given by the
Bennett-Chandler method85
kQTST =p(R0
s)√2πβm
exp
[
−β
∫ R‡s
R0s
dR′s
dW (R′s)
dR′s
]
, (3.26)
whereR0s is a reference point in the reactant region. The probabilitydistributionp and the free
50
energy functionW are given by
p(R′s) =
⟨
δ(R′s − Rs)
⟩
⟨
h(R‡s − Rs)
⟩ , (3.27)
W (R′s) = − 1
βln p(R′
s), (3.28)
whereh is a Heaviside step function,Rs = 1k
∑kj=1 q1,j is the centroid position of the reaction
coordinateq1, and〈...〉 denotes the equilibrium ensemble average.
A time dependent transmission coefficient
κ(t) =
⟨
δ[x‡s − xs]vs(pk,xk)h[xs(t) − x‡
s]⟩
⟨
δ[x‡s − xs]vs(pk,xk)h[vs]
⟩ , (3.29)
that accounts for the recrossing of the trajectories. Herexs(t) is computed by evolving the
classical equations of motion in the extended phase space using the ring polymer Hamiltonian
given by Eq. (3.23).
The calculation details follow the algorithm of Ref. 85. We computekQTST with an inte-
gration time step of 0.5 fs, in an algorithm with alternatingmomentum and free harmonic ring
polymer updates. An Anderson thermostat with a time constant of 10 fs is applied to sample the
Boltzmann distribution. The histogram forp(x0s) is calculated using one RPMD trajectory with
a total simulation time of 1 ns. This time is long enough for the system to tunnel. Whenever this
occurs, the trajectory is restarted in a random configuration in the left minimum region. The
constrained simulations for computing the mean free forceW are performed using the RATTLE
algorithm.113 The transmission coefficient is computed by averaging Eq. (3.29) over 200000
trajectories without any thermostatic control. A plateau time of 500 fs is used in finding the
infinite limit in Eq. (3.25).
3.4 Results and discussion
We first present the results for the exact dynamics. The results of SH and RPMD are discussed
next.
51
(n4,n6) Diabatic (cm−1) Adiabatic (cm−1) Splitting (cm−1)
(0,0) 0.0 0.0 0.0017
(1,0) 180.2 179.9 0.019
(0,1) 226.0 223.2 0.0080
(2,0) 360.3 359.6 0.080
(1,1) 406.6 403.2 0.090
(0,2) 453.2 446.8 0.023
(5,0) 899.5 897.8 -0.074
Table 3.4: Diabatic and adiabatic energies and tunneling splittings for the gas-phase FAD
Hamiltonian. The energies are given with respect to the ground state, and (n4,n6) refers to
the corresponding excitation inq4 andq6 mode, respectively.
3.4.1 Exact quantum dynamics
The tunneling splittings for the gas-phase FAD are shown in Table 3.4 along with the energies
for the diabatic and adiabatic potentials of Eq. (2.1). The tunneling splittings show mode se-
lectivity upon vibrational excitations. Low vibrational excitations in mode 4 are more effective
at increasing tunneling splittings than excitations in mode 6, even though mode 6 has a higher
frequency. This selectivity has been observed in many systems, and is generally understood as a
consequence of the change in path length and barrier height upon coupling to the corresponding
mode. Inclusion of the modes with anti-symmetric coupling leads to an increase of the path
length, and hence decrease in the tunneling splittings.
The multi-dimensional nature of this selectivity is important. This can be observed in the
strong dependence of the tunneling splittings on the coupling between mode 4 and 6. Changing
the coupling parametera8 from -1589 cm−1A−2 to -1389 cm−1A−2 changes the ratio of the
first excited mode 6 tunneling splitting to the ground state tunneling splitting by a factor of
4, while keeping the ratio in tunneling splittings for mode 4relatively unchanged. This effect
can be understood in the context of overlap of wavefunctionsacross the symmetric dividing
52
Figure 3.5: The first 8 tunneling splittings as a function of the coupling strength.
surfaceq1 = 0. Changes in the coupling affects the orientation of the wavefunction, resulting
in exponential changes in the overlap.
Couplings to the bath have complex effects on the tunneling splittings. Figure 3.5 shows
the first 8 tunneling splittings as a function of the dimensionless coupling parameterη/mωb.
The non-monotonic behavior and the negative tunneling splittings are a signature of the Franck-
Condon-like factors.93,114 The negative tunneling splittings at zero coupling strength corre-
spond to odd number of quanta in the bath modes. Due to the symmetry of the potential
V (q1, q4, q6,x) = V (−q1,−q4, q6,−x), negative tunneling splitting occurs when the lower
energy state obeysψ(q1, q4, q6,x) = −ψ(−q1,−q4, q6,−x), which is true for odd quanta of
excitation in the bath modes. In contrast we will see that thenegative tunneling splitting for the
state (5,0) (Table 3.4) for gas-phase FAD, is caused by the negative contributions of the overlap
of diabatic wavefunctions.
The decay ofPR for different temperatures and the coupling strength is shown in Fig. 3.6.
The timescale of tunneling is roughly 1 ns in the gas-phase, and roughly 5 ns for the weakly
coupled case. These timescales neither show strong dependence on the temperature for the
gas-phase nor the coupled case. The decay slows down with theincrease in the coupling.
This is a consequence of the strong quantum effects, and the non-monotonic behaviour that
53
Figure 3.6: The decay ofPR with time computed using exact quantum dynamics. Plots for 200
K and 300 K correspond to the gas-phase decay.
the tunneling splittings demonstrate upon the vibrationalexcitations. The transition state based
theories though would predict an exponential dependence onthe temperature, and an increase
in the rate in the weak coupling regime, which is energy diffusion controlled.
3.4.2 Surface hopping
We compare the results obtained using the LZ formalism and the fewest switching criterion in
Fig. 3.7 for gas-phase FAD at 300 K. We obtain faster results with the fewest switching criterion
compared to the LZ method when using the thermostat. One possibility is the lower barrier
height of the adiabatic surface compared to the diabatic surface (see Fig. 3.4). The results are
in close agreement for the NVE case, where the possible effects due to the barrier height would
be mitigated to some extent as the average is performed over only those trajectories that tunnel
on the 1 ns timescale.
The results forPR for NVE and Nose-Hoover thermostat calculations using the LZ formal-
ism are shown in Fig. 3.8 for different temperatures and coupling strength, and compared to the
exact results. SH cannot capture the long time quantum coherences, and hence the computed
PR decays exponentially to 0.5.
54
Figure 3.7: Comparison of decay ofPR computed using LZ formalism (black) and the fewest
switching criterion (red) for (a) Nose-Hoover thermostat,and (b) NVE calculations.
Figure 3.8: The same as in Fig. 3.6, exceptPR is computed using the SH method with (a)
Nose-Hoover thermostat, and (b) NVE calculations.
The results obtained with a thermostat show a single exponential decay as all trajectories
sample the available phase space on a timescale that is shortcompared to that for successful
hops, producing an average rate. Removing the thermostat, wefind in Fig. 3.8 (b) that there
is a fast initial decay followed by a slower decay that is on a similar timescale to that of the
quantum results. The fast decay is due to the high energy trajectories that are hopping on times
that are faster than any timescale observed in the quantum results. The remaining trajectories
hop on a timescale of 1 ns, a time that is similar to the quantumresults. However, in contrast
to the quantum results, coupling to the bath enhances the rate. This increase is due to the
energy diffusion in the FAD modes. Even with the bath modes this diffusion is limited and
55
Figure 3.9: Comparison of the results obtained using a hybridcalculation, where only the bath
modes are coupled to Anderson thermostat.
some trajectories do not gain enough energy (on 1 ns timescale) to ever sample that part of
phase-space where hopping occurs; these trajectories are not included in the results.
As we have just seen, the bath for the NVE calculations provides limited energy diffusion.
When we enhanced the energy diffusion by coupling the system to a thermostat the role of
the bath modes is diminished, this leading to similar decay curves at 300 K in Fig. 3.8 (a).
To observe the effects of the bath and the energy diffusion more clearly we perform a hybrid
calculation, where we apply the Anderson thermostat only tothe bath modes. The weakest
coupling for which almost all trajectories tunnel on 1 ns timescale isη/mωb = 0.01, and the
decay ofPR for this value of the coupling strength is compared toη/mωb = 0.1 in Fig. 3.9.
The increase in the rates due to the coupling with bath modes is more apparent. This increase
is due to the increased energy diffusion amongst the FAD modes.
To summarize, coupling to the bath provides limited energy diffusion . This diffusion is
sufficiently limited that the dominant effect of the increasing coupling to the bath in the quantum
decay is the corresponding increase in the tunneling path length with the result that decay times
decrease. This effect as we have seen is not observed in the SHresults.
We believe that the source of this qualitative discrepancy is the separation of the classical
56
and quantum degrees of freedom. The quantum degree of freedom q1 is directly coupled to
q4 andq6. But as the bath modes are coupled toq4 only, their coordinates do not affect the
probability of the hop; they only change the probability of reaching the dividing surface. Since
these bath modes have the same symmetry as mode 4, one should expect that tunneling should
have high probability only when some “intrinsic” bath coordinate has close to zero value. In the
system studied here, the coupling takes the formVC = −q4
∑
cixi, and hencexb =∑
cixi is a
reasonable choice for the intrinsic bath coordinate. This idea is similar in spirit to the classical
Hynes model,83 where a bath coordinate drives the reaction. To identify what the cutoffxcb
should be, we note that the non-adiabatic vector1/dij at the dividing surface is a measure of the
distance over which the quantum amplitudes change, or the hopping can take place. Sincexb is
coupled toq4, we need to treat the mode 4 quantum-mechanically while computing the value of
dij. Every time the dividing surface (q4 = 0) is reached, we compute the value of1/dij along
xb while treating bothq1 andq4 quantum mechanically. This quantum treatment of mode 4 is
limited only for the computation ofdij. Since we started with quantized action for the initial
condition of the phase space, one would expect from the normal mode picture that the action
should remain relatively constant. This we found to be usually true. Hence we find the value
of 1/dij for the current action value corresponding to mode 4. In summary, a hop is allowed to
happen if
|xb| < 1/dij(xb) (3.30)
holds. We computedij, while treating bothq1 andq4 quantum mechanically, atxb =∑
cixi =
0, i is the nearest integer to the instantaneous action of mode 4,andj = i + 1.
The results obtained with this restriction are compared with the exact results as shown in
Fig. 3.10. The results are encouraging, and we hope that thisapproach can pave the way to
produce generalized expressions of the above restriction for more complicated systems.
3.4.3 Ring polymer molecular dynamics
We now present the results obtained using the RPMD method which computes a thermal rate
constant. These thermal rate constants stay relatively constant with coupling bath strength, and
are approximately 1.1 ns−1 at 300 K and 0.05 ns−1 at 200 K. Similar rate constants are obtained
by fitting the decay ofPR computed using SH with the nose-hover thermostat [Fig. 3.8 (b)] to
57
Figure 3.10: The decay ofPR with time for η/mωb = 0.1. Restriction on the intrinsic bath
coordinate given by Eq. (3.30) has been applied.
an exponential. We compare these rate constants with the quantum rate constants given by Eq.
(3.12). Unfortunately converging these numbers requires far more tunneling splittings, and is
beyond the scope of this work. But the ratio of the rates for different temperatures and coupling
strength are relatively converged; they change by roughly 5% upon changing the number of
converged tunneling splittings from 60 to 120. This non-convergence is only for the initial
rate; the long time dynamics shown in Fig. 3.6 is converged computed using the 60 tunneling
splittings. Figure 3.11 compares the decay ofPR computed using Eq. (3.9) and the first order
kinetics given by Eq. (3.11) for the coupled case (η/mωb = 0.1) computed using 60 and 120
tunneling splittings. It can be seen that the long time decayis similar for 60 and 120 tunneling
splittings, though the initial decay rate has changed.
The ratio of the exact rate constants for the two different temperatures is nearly 1.3, in con-
trast to more than an order of magnitude difference for the RPMD rate constants as shown in
Fig. 3.12. It is known that RPMD can breakdown for systems withstrong quantum coherences.
The symmetric double well presents an extreme example wherethe effects of quantum coher-
ences are crucial, and hence it is not surprising that the thermal rate constants computed using
RPMD vary by an order of magnitude at different temperatures.The exact results are dictated
58
Figure 3.11: Comparison of the decay ofPR computed exactly [see Eq. (3.9)] and assuming first
order kinetics [see Eq. (3.11)]. The solid and dashed black lines correspond to exact quantum
dynamics computed using 60 and 120 tunneling splittings, respectively. The solid and dashed
red lines show the corresponding decay for first order kinetics.
by the tunneling splittings, which are a function of the overlap of the diabatic wavefunctions
at the dividing surface. In a simple one dimensional picture, this overlap will indeed change
exponentially with energy, and consequently with temperature. But a closer analysis shows
the importance of the multi-dimensional nature of the problem, which causes non-monotonic
change of the tunneling splittings with the energy.
These results also indicate that the rate constant given by the flux-flux correlation function
may not adequately represent the timescales of tunneling for our model. The thermal rate con-
stant for both RPMD and exact results are almost constant withchange in the coupling strength
in the weak coupling regime. But the long time dynamics shown in Fig. 3.6 slows down sub-
stantially with the coupling strength parameterη. This indicates that the additional coherence
effects of the bath only change the longer timescales, whichare dictated by the ground state tun-
neling splittings. The importance of the multiple timescales comes out clearly here; the faster
timescales represented by the thermal rate constant are relatively independent of the coupling
and are qualitatively described by RPMD. But the exact quantumcoherences are necessary in
59
0
5
10
15
20
25
0 0.02 0.04 0.06 0.08 0.1
k30
0/k
200
η/mωb
RPMD
Exact
Figure 3.12: The ratio of the rates at 300 K to 200 K computed using RPMD and exact calcula-
tions as a function of the coupling strengthη/mωb.
predicting the slower timescales. The SH method, which struggles with the effects of the strong
quantum coherences, also did not perform well in describingthe effects of coupling to the bath.
We note that this breakdown is only for the symmetric double well coupled to only a small
number of bath modes. Recently, corrections to RPMD have been proposed to obtain the cor-
rect rate constants in the Marcus inverted regime for electron transfer. Richardson and Thoss115
show the limitations of the fluctuating flux correlation functions, which is true in the present
case, to accurately predict the rate constants. A modified correlation function based on the lin-
ear response theory was proposed and encouraging predictions for the weak coupling regime
of the Marcus theory was obtained. Additionally, Miller andcoworkers116 have developed a
kinetically controlled RPMD method that incorporates the effects of fluctuations in the elec-
tronic state variables in the kink pair formalism, leading to excellent agreement in the Marcus
turnover regime. These developments are very encouraging,and similar analysis can perhaps
be performed for proton transfer.
60
3.5 Conclusions
The timescales for proton tunneling for a three dimensionalmodel of the formic acid dimer
(FAD) weakly coupled to a bath have been calculated using surface hopping (SH) and ring
polymer molecular dynamics (RPMD). The results have been compared to exact quantum cal-
culations.
In the absence of bath coupling we found that SH is in reasonable agreement with the quan-
tum results, although the higher energy trajectories tunneled too rapidly and low energy tra-
jectories never reached regions of configuration space where tunneling was possible on a 1 ns
timescale. However, when coupling to the bath was included the quantum rates decreased and
the surface hopping rates increased. Two effects are at playin causing the opposite trends.
Increasing the bath coupling increases the tunneling path lengths and therefore increases the
tunneling times. Increasing the coupling also increases the energy diffusion, and this decreases
the tunneling times. This energy diffusion is limited, the main effect being the increase in hop-
ping probabilities due to an increase in the fluctuations in the dimer rockq4. Even with the bath,
direct comparison to the quantum results is problematic dueto that fact that here too many low
energy FAD trajectories never gain enough energy to attemptto hop.
It is not too surprising that whenever coherence effects areimportant in the deep tunneling
regime, increased coupling may lead to increased tunnelingpath lengths. How one includes
these effects in SH models remains an open question. We have shown that the role of the bath
modes can be partially treated by including an additional constraint [see Eq. (3.30)] on the
intrinsic bath coordinate. This constraint leads to the correct qualitative trends of rates with
coupling in the weak coupling limit. Further work is needed to test its validity to more general
systems.
The role of energy diffusion in the surface hopping results was explored further by coupling
the bath, consisting of 5 harmonic oscillators, to a thermostat as well as coupling both FAD and
the bath to a thermostat. In both these cases, there is a single decay rate; the phase space of the
FAD degrees of freedom is explored on a timescale that is faster than that of the tunneling. In the
latter case, when weaker FAD-bath couplings are considered, one finds that the rates decrease
as the system-bath coupling decreases. This is consistent with the decreasing rate of energy
diffusion among the FAD degrees of freedom. The rates are slower than the SH results without
the thermostat, because here all all trajectories eventually tunnel, and hence all are included.
61
The thermal rate constants obtained through RPMD are similarto the SH results with NVT
calculations; they show strong dependence on the temperature and little sensitivity to the cou-
pling to the bath in the weak coupling regime. When we extrapolate to the very short time
dynamics of the quantum model, in order to obtain rate constants, we observe that these re-
sults are also insensitive to the coupling to the bath. This presents the picture that the effect
of the quantum coherences of the bath changes the slower dynamics more than the initial fast
dynamics.
62
Chapter 4
Tunneling splittings - the Makri-Miller
method
Contents
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Makri-Miller method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Figures
4.1 An illustration of the straight line tunneling path for the Makri-Miller method 65
4.2 Comparison of the instanton trajectory, the minimum energy path and the
tunneling path chosen for Makri-Miller method . . . . . . . . . . . . . . . . 67
4.3 Comparison of exact tunneling splittings with the tunneling splittings ob-
tained through the Makri-Miller method . . . . . . . . . . . . . . . . . . . . 68
4.4 Comparison of decay ofPR computed using exact dynamics and the Makri-
Miller method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
63
4.1 Introduction
In Chapter 3 we presented numerically exact results of the timescales of proton tunneling in a
model system that qualitatively represents formic acid dimer (FAD) coupled to a harmonic bath.
The FAD modes considered are the symmetric proton stretch (q1), the symmetric dimer rock
(q4), and the dimer stretch (q6). The bath is represented by 5 simple harmonic oscillators coupled
linearly toq4. The numerically exact results were compared with the timescales obtained using
two methods, namely surface hopping (SH) and ring polymer molecular dynamics (RPMD)
Perhaps not surprisingly, in the deep tunneling regime investigated in this work, we find that
these dynamical methods do not accurately describe the timescales of tunneling. The underlying
assumption in RPMD of representing the timescales of tunneling by an average thermal rate
constant may not hold in this regime due to the inherent importance of the coherence effects.
The effects of entanglement between the bath modes and the tunneling mode, another important
feature that dictates the timescales, are also hard to incorporate in SH and RPMD. Certain
modifications in SH were suggested work to partially includethese entanglement effects.
An alternate approach to obtain the timescales of tunnelingis to compute the tunneling
splittings, which dictate the exact dynamics in the deep tunneling regime. Computation of
tunneling splittings remain an outstanding problem. For example, values ranging from 0.001
cm−1 to 0.3 cm−1 have been computed for FAD.117 Wenzel, Kramers, and Brillouin (WKB)15
theory developed in 1926 reproduces the exact quantum results within ∼ 10% accuracy for a
single dimensional problem. Several attempts have been made to generalize WKB theory to
multidimensional tunneling by identifying a tunneling path. The instanton model, originally
proposed by Miller and Coleman,118,119 identifies the tunneling path as the classical trajectory
that connects the two identical minima on the upside-down potential. Later, this model was
applied to compute tunneling splittings under the semiclassical approximation, with integration
of the Herring’s formula performed using the method of the steepest descent along this instan-
ton path.80,120–122 Mil’nikov and Nakamura81,123,124developed a practical implementation of
this theory, the invariant instanton theory, to obtain tunneling splittings for high dimensional
systems.
There are several related methods that have been developed based on the tunneling path
approach. Fernandez-Ramos and coworkers have developed therainbow instanton method, a
new scheme that avoids computation of the instanton trajectory.125,126 A recent formulation
64
by Althorpe and coworkers127 applies the ring polymer approach to the semiclassical instan-
ton model, where the tunneling splitting is computed in terms of the partition functions of
the system with and without inclusion of tunneling, and obtained very encouraging results.
Bowman and coworkers have developed aQim-path approach, where the tunneling splittings
are computed for the one-dimensional, fully relaxed potential along the imaginary frequency
mode.128,129 This method has shown excellent experimental agreement forvarious systems.
Another method developed by Makri and Miller16 identifies the tunneling path as the straight
line joining the caustics of classical trajectories. Tunneling splittings are computed by averaging
over the classical trajectories turning points along the tunneling path under the WKB formalism.
The multiple spawning approach to semiclassical dynamics has also been extended to compute
the tunneling splittings by Ben-Nun and Martinez.130
The majority of these methods have been developed to computeground state or funda-
mentally excited tunneling splittings. The Makri-Miller formalism readily extends to compute
excited state tunneling splittings, and has shown great promise to compute these excited state
splittings.17–19 We in this chapter implement this method, and show that this method retains
all the qualitative features. We describe the details of theMakri-Miller formalism in Sec. 4.2.
The results obtained using this formalism are presented in Sec. 4.3, and the conclusions are
provided in Sec. 4.4.
4.2 Makri-Miller method
The Makri-Miller formalism16 incorporates tunneling in classical simulations under theWKB
formalism.131,132 A tunneling direction and the tunneling path are identified,and the tunneling
amplitudeexp(−θ) is computed at each turning point (x0) of the trajectory along the tunneling
direction. The model Hamiltonian under consideration is described in Sec. 3.2. Figure 4.1
shows the picture of a trajectory on theq4 = 0 surface of the gas-phase FAD [Fig. 2.2 (a)], along
with a straight line tunneling path.θ corresponds to the classical action along the tunneling path
θ = h−1
∫ zmax
0
dz√
2m[V (z) − V (0)], (4.1)
wherez parametrizes the tunneling path withz = 0 corresponding tox0, andzmax is the value
of z at whichV (zmax) = V (0). In Fig. 4.1,z = 0 and z = zmax correspond to the end
65
Figure 4.1: A sample trajectory (red) on theq4 = 0 surface, with the straight line tunneling path
shown in blue for a particular turning point.
points of the tunneling path. The tunneling splitting for the corresponding action variables of
the trajectory is given by
∆ = 2hd
dt〈S(t)〉, (4.2)
where S is the accumulated tunneling amplitude
S(t) =∑
n
h(t − tn) exp(−θn). (4.3)
Heretn are the times at which the trajectory experiences classicalturning points, andh(t − tn)
is the Heaviside step function.
The central issue of the method is the choice of the tunnelingpath. The instanton path gives
a good approximation, but is expensive to compute for each turning point. Instead Makri and
Miller chose the tunneling path to be a straight line that connects the caustics of the classical
trajectories in the shortest way, as shown by the blue line inFig. 4.1.16 This is based on the idea
that along this line the semiclassical wavefunctions will have the maximum overlap.
Further study on this by Thompson and coworkers18,19 suggests two extreme regimes: adi-
66
abatic (small curvature) and sudden (large curvature). In the adiabatic regime, the orthogonal
degrees of freedom are fast compared to the motion along the reaction path, such that the trans-
verse coordinates adjust adiabatically as the system movesalong the reaction path. In this
regime, the tunneling path coincides with the minimum energy path. In the sudden regime the
reaction path is fast compared to the orthogonal coordinates. In this regime, the transversal co-
ordinates are held constant while the reaction path coordinate changes from the reactant region
to the product region.
Our preliminary calculations with the sudden tunneling path resulted in incorrect qualitative
trends of the tunneling splittings with respect to the coupling strengthη. A possible issue with
the sudden tunneling path is that it ignores the curvature ofthe slow coordinates. This curvature
is important in the weak coupling regime, and hence we favor the adiabatic approach. For gas-
phase FAD, our choice of the tunneling path is comprised of two straight lines; the first segment
joins the classical turning point (q4,q6) to the transition state (0,0), and the second segment joins
the transition state to the symmetrically opposite coordinate (-q4,q6). The energy along this
path is computed while treatingq1 adiabatically, similar to the SH method. Additionally, this
incorporates the zero-point effects. In the presence of weak coupling with the bath, the tunneling
path along the bath coordinates move along the direction of the straight line connecting the two
symmetric bath minima. In this case, the path is followed until the conditionV (zmax) = V (0)
is satisfied.
This tunneling path is motivated by the instanton path for gas-phase FAD. The details of the
computations of the instanton path are described in Appendix D and its corresponding results
for tunneling splittings will be discussed in Chapter 5. Figure 4.2 shows the comparison of the
instanton path, the minimum energy path and the path we have chosen for a particular turning
point. This chosen path has a similar curvature compared to the instanton path. We chose the
tunneling path to cross the dividing surface through the transition state valueq6 = 0A, rather
thanq6 = 0.6A (where the instanton trajectory crosses the dividing surface) since the tunneling
path starts at the turning point rather than the global minima. Choosingq6 = 0.6A as the
crossing point would lead to much lower curvature and path length.
The choice of the tunneling path can occasionally lead to complex action. The distinc-
tion between the complex and pure-imaginary action was recognized by Takada and Naka-
mura,114,133 and this was incorporated later into the Makri-Miller formalism by propagating
67
Figure 4.2: Comparison of the instanton trajectory (black),the minimum energy path (red) and
the tunneling path chosen for Makri-Miller method (green).
generalized classical trajectories.17,134,135 For our model Hamiltonian, we consider the contri-
butions from the pure-imaginary action only, since the contribution from the complex action is
small.
Trajectories are initiated in the left minimum region usingthe action-angle transformation.
The action variable is quantized asIi = (ni + 1/2) h whereni is the number of quanta in
modei. The trajectories are evolved on the diabatic surface defined by Eq. (3.16). This choice
prevents the trajectories of highly excited states from sampling the other minimum. An average
over 100 trajectories is performed to compute the average tunneling amplitude〈S(t)〉 using
a time step of 2 fs. Trajectories are evolved for a total timeτ of 4 ps to compute the slopeddt〈S(t)〉 = 〈S(τ)〉/τ . To evaluate the actionS, the straight line joining the turning point with
the transition state is divided into 100 equal steps. The adiabatic energy at each step, required for
the evaluation of the integrand in Eq. (4.1), is computed by solving for the lowest eigenvalue
of the 1D Hamiltonian alongq1 (with constantq4 andq6). Thirty equally spaced DVR basis
functions ranging from -1.2A to 1.2A are used.
4.3 Results and discussion
The tunneling splittings for the excited states of mode 4 and6 computed using the Makri-
Miller method are compared with the exact results in Fig. 4.3. The comparison of the first
8 tunneling splittings when FAD is weakly coupled to the bathis also shown. The details of
the computation of the exact results are given in Sec. 3.3.1.Since this method cannot capture
68
Figure 4.3: Comparison of exact tunneling splittings (black) with the tunneling splittings ob-
tained through the Makri-Miller method (red). (a) Gas-phase FAD with n4 = 0, (b) gas-phase
FAD with n6 = 0, (c) first four tunneling splittings in presence of the bath,and (d) tunneling
splittings for states 5 to 8 in presence of the bath.
negative tunneling splittings, only the magnitude of tunneling splittings are compared. The
results are in qualitative agreement, even for the highly excited states. One can see in Fig. 4.3
(a) that the tunneling splittings increase too rapidly withexcitations in mode 6. This is due
to the decrease in path length that results upon excitationsin mode 6. This phenomenon has
been recognized by Takada and Nakamura previously.133 The reason for the slowdown of the
increase in tunneling splitting upon mode 4 excitation shown in Fig. 4.3 (b) is the crossover of
the turning point across the dividing surfaceq4 = 0, leading to an increase in the path length.
Once we have the tunneling splittings, we can easily computethe dynamics given by Eq.
(3.9). This comparison is shown in Fig. 4.4. The results are in qualitative agreement with
the exact results for changing temperature and coupling strength. It is very encouraging that
this method can get the qualitative trends of the tunneling splittings in the presence of the
bath as shown in Fig. 4.3 (c) and (d). The decrease of the ground state and several excited
state tunneling splittings correctly predicts slower timescales with the increase in the coupling
strength parameterη, a feature that both SH and RPMD fail to obtain. Although, it overestimates
this slowdown of the decay rates as seen in Fig. 4.4 (c). Thereare two competing factors that
69
Figure 4.4: Comparison of decay ofPR computed using exact dynamics (black) and the Makri-
Miller method (red). (a) Gas-phase with T=300 K, (b) gas-phase with T=200 K, and (c) coupled
to bath (η/mωb = 0.1) with T=300 K.
decide these timescales in this weak coupling regime. Firstis the increase in the path length
that decreases tunneling splittings, and the second is the larger fluctuations of the tunneling
coordinate in this energy diffusion regime which leads to faster timescales. The Makri-Miller
method correctly identifies that the effects of the former factor dominates. The balance between
these two factors is only qualitatively captured though, asis demonstrated by the overestimation
of the slowdown. It should also be noted that one of the bath modes is in resonance with mode
4. Hence some of the tunneling splittings show a sharp jump, which is well described using the
classical trajectories.
The qualitative feature that this method cannot capture is the non-monotonic behaviour
shown in Fig. 4.3 (c). These tunneling splittings correspond to excitations in the bath mode
with 52 cm−1 frequency. Excitation in this mode introduces Franck-Condon like factors, which
are not incorporated in this formalism. We will show that these can be captured in the adiabatic
Herring estimate in Chapter 5. Nonetheless, the results are in good agreement and present a sim-
ple approach to estimate the tunneling splittings in both the gas-phase and the weak coupling
regime.
70
4.4 Conclusions
We have employed the Makri-Miller method to compute the tunneling splittings for a three di-
mensional model representing formic acid dimer, and its further coupling to a bath. It captures
the trends upon mode 4 and mode 6 excitations, as well as resonance effects of bath modes.
Further, in contrast to the semiclassical and quantum-classical methods, this method correctly
predicts the slowdown of the tunneling timescales with coupling to the bath. In the weak cou-
pling regime, as the coupling to the bath is increased there is greater energy diffusion. This
competes with the increase in the path length, which leads tosmaller tunneling splittings. The
net effect of these two factors is what governs the dynamics.This method demonstrates that the
effect of the increased path length dominates the effects ofenergy diffusion.
Despite the qualitative agreement, there is quantitative discrepancy present. The slowdown
with increased friction with the bath is overestimated. Further, excitations in the mode 6 lead
to a much faster increase in the tunneling splittings compared to the exact results. This can be
understood in terms of decrease of path length upon mode 6 excitations. Franck-Condon factors
that lead to non-monotonic trends are also not well capturedin this method as is expected.
Further work is needed to quantitatively capture the above mentioned effects into this method.
71
Chapter 5
Tunneling splittings - the adiabatic
Herring method
Contents
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2 Adiabatic Herring estimate . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4 Additional methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Figures
5.1 Comparison of exact tunneling splittings with the tunneling splittings ob-
tained through the adiabatic Herring method . . . . . . . . . . . . . . . . . . 78
5.2 Comparison of decay ofPR computed using exact dynamics and adiabatic
Herring estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3 Comparison of tunneling splittings obtained through the various levels of ap-
proximation made in the adiabatic Herring method . . . . . . . . . . . . . . 79
5.4 The diabatic wavefunctions and the contributions to the ground state tunneling
splitting from various regions of the coordinate space given by the adiabatic
Herring theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
72
5.5 The diabatic wavefunctions and the contributions to an excited state tunneling
splitting from various regions of the coordinate space given by the adiabatic
Herring theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.6 The instanton trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.1 Introduction
Tunneling splittings of energy levels are a fundamental feature in symmetric double well po-
tentials, and are an important issue in spectroscopy of gas-phase molecules. These tunneling
splittings dictate the timescales of symmetric proton tunneling. We computed the numerically
exact tunneling splittings for a model Hamiltonian that qualitatively describes symmetric proton
tunneling in the formic acid dimer (FAD) and its coupling to abath in Chapter 3. The model
Hamiltonian comprises three FAD modes – the symmetric proton stretch (q1), the symmetric
dimer rock (q4), and the dimer stretch (q6), and 5 bath modes linearly coupled toq4. The cou-
pling between these modes represent the symmetric and the anti-symmetric coupling, the two
important forms of symmetry that influence tunneling.72,80–82The timescales of proton transfer
obtained using these tunneling splittings were shown to slowdown with increasing friction to
the bath, in contrast to the results obtained using surface hopping and ring polymer molecular
dynamics.
Computation of the tunneling splittings is a challenging task for multidimensional systems.
In Chapter 4 we benchmarked the Makri-Miller approach, whichis capable in computing ex-
cited state tunneling splittings. All the qualitative features of the numerically exact results are
reproduced by this method, including the slowdown of decay rates with increasing friction to the
bath modes. This slowdown is explained by the increase of thepath length with increased cou-
pling to the bath modes, which dominates the effects of energy diffusion. It further demonstrates
that coherence effect is the missing ingredient in surface hopping and ring polymer molecular
dynamics.
The Makri-Miller approach, however, has a few limitations.The Franck-Condon factors that
leads to non-monotonic behavior93,114 cannot be captured under the ambit of this theory. Fur-
ther, since this method is based on tunneling path approach,quantitative computation of highly
excited tunneling splittings is challenging. Takada and Nakamura133 suggested dependence of
73
the tunneling path on the caustics of the classical trajectory. Further work is needed to include
these effects into the Makri-Miller method.
There are several methods that do not rely on the tunneling path. Stuchebrukhov and
coworkers developed the tunneling current approach, wherethe tunneling matrix element is
given by sum over contributions of the tunneling current through various paths.136 The path
integral approach137–139and the diffusion Monte-Carlo formalism140–142 to compute tunneling
splittings also present alternate approaches to compute the tunneling splittings and have been
applied to many systems.
We develop a theory of our own in this chapter to compute excited state tunneling splittings
that does not rely on the tunneling path approach by making anadiabatic approximation to
the Herring estimate.20 Our theory also incorporates the effects of Franck-Condon factors.
Excellent agreement with the exact results are obtained forthe model Hamiltonian. This theory
is applicable to systems with a high frequency tunneling mode coupled strongly to a few other
low frequency modes. These modes can then be further coupledto bath like modes. The
shortcomings of this theory include dependence of the results on the diabatic potential, and will
be further discussed.
We have also implemented the instanton theory developed by Nakamura and coworkers, one
of the most general instanton formalisms, to compute groundstate tunneling splitting within 10
% accuracy of the exact result, further shedding light on thenature of the tunneling path. This
is further compared to the results obtained by theQim-path approach, an easily implemented
formalism, to illustrate the importance of the curvature inthe current system.
We describe the adiabatic Herring formalism in Sec. 5.2. Theresults obtained using this
formalism are presented in Sec. 5.3. These results are discussed in the context of the instanton
method and theQim-path approach in Sec. 5.4. The conclusions are provided in Sec. 5.5
5.2 Adiabatic Herring estimate
Although the Makri-Miller method can compute excited statetunneling splittings qualitatively,
it misses the effects of Franck-Condon factors that can significantly affect the tunneling split-
tings. Here we develop a model for computing tunneling splittings for higher excitations with
the Franck-Condon factors included. The model is developed for the case of the adiabatic sep-
74
aration of the vibrational degrees of freedom. We shall showthat this model captures all the
trends of the tunneling splittings, including negative tunneling splittings, correctly.
Without loss of generality, the modes of any symmetric tunneling system can be divided into
three categories -q, xa, andxs such that the potential obeysV (q,xa,xs) = V (−q,−xa,xs).
Hereq refers to the tunneling mode, andxa andxa represent the modes with anti-symmetric
and symmetric coupling respectively. The Schrodinger equation for the symmetric (ψs) and the
anti-symmetric (ψa) pair of states is
[
T + V]
ψs = Esψs (5.1)[
T + V]
ψa = Eaψa, (5.2)
whereT = −h2∇2/2m is the kinetic energy operator, andEs andEa are the corresponding
eigen values. Multiplying Eq. 5.2 byψs and Eq. 5.1 byψa and taking the difference gives the
tunneling splitting∆ = Ea − Es as
∆ψaψs = (Tψa)ψs − (Tψs)ψa. (5.3)
DefiningψL ≡ 1√2(ψs + ψa) andψR ≡ 1√
2(ψs − ψa), substituting in Eq. 5.3 and integrating
over the left coordinate space defined byq = −∞ to q = 0 gives
∆
∫
dxa
∫
dxs
∫ 0
−∞dq
[
ψ2L − ψ2
R
]
= − h2
m
∫
dxa
∫
dxs
∫ 0
−∞dq
[
(∇2ψL)ψR − (∇2ψR)ψL
]
(5.4)
= − h2
m
∫
dxa
∫
dxs
∫ 0
−∞dq∇ [(∇ψL)ψR − (∇ψR)ψL] .
(5.5)
Using the divergence theorem, and the symmetry relationψL(q,xa,xs) = ψR(−q,−xa,xs),
one obtains the well-known Herring’s formula20
∆ = −2h2
m
∫
dxa
∫
dxs
[
dψL
dqψR
]
q=0∫
dxa
∫
dxs
∫ 0
−∞ dq(ψ2L − ψ2
R), (5.6)
75
Note that the dividing surface chosen hereq = 0 is different from the one used in the derivation
of the instanton theory whereψL = ψR defines the dividing surface. AssumingψL (ψR) is
normalized and localized in the left (right) well leads to the denominator being equal to one.
ψL can be understood approximately as the eigen function of theleft diabatic potential.
We make an adiabatic approximation on this diabatic potential treatingq as the high frequency
mode
ψL(q,xa,xs) = χL(xa,xs)φL(q;xa,xs), (5.7)
whereφL(q;xa,xs) is the eigenfunction of the left diabatic potential at a given value of{xa,xs}.
As the frequencies of the diabatic potential do not change dramatically with the coordinates, this
approximation should hold well for all regions. It leads to
∆ = −2h2
m
∫
dxs
∫
dxaχLχR
(
dφL
dqφR
)
q=0
. (5.8)
Further progress can be made by approximating
−2h2
m
(
dφL
dqφR
)
q=0
≈ δ(xs), (5.9)
whereδ(xs) is the tunneling splitting alongq as a function ofxs only. This can be understood in
a context of a two state model wherexs modulates the tunneling splittings, whilexa dictates the
energy difference between the two minima. This gives our expression for the tunneling splitting
as
∆ =
∫
dxs
∫
dxaχLχRδ(xs). (5.10)
We assumeχL andχR given by Eq. (5.7) to be eigenfunctions of a diabatic potential. We note
that this formulation depends upon the choice of the diabat,and will be further discussed in the
results section.
We now consider several levels of approximation for computing the above expression for
∆. These simplifications depend on the form of the couplings. If only a few modes are strongly
coupled toq, then the integration can be simplified. All the weakly coupledxa would approxi-
76
mately integrate out as 1, while the weakly coupledxs modes give Franck-Condon factors. For
FAD weakly coupled to a bath [see Sec. 3.2]q corresponds toq1, xs corresponds to mode 6,
and all the other modes are represented byxa. In the simplest approximation, we consider full
separability ofχL = χq6(q6)χq4
L (q4)∏5
i=1 χiL(xi). Substituting this into the Eq. (5.10) gives
∆ =
[∫
dq6δ(q6)χ2(q6)
]
IFCq4
5∏
i=1
IFCi (5.11)
where
IFCi =
∫
dxiχiL(xi)χ
iR(xi), (5.12)
and
IFCq4
=
∫
dq4χq4
L (q4)χq4
R (q4) (5.13)
are the corresponding Franck-Condon factor associated withthe bath modei and mode 4, re-
spectively. This simple approximation performs poorly as it ignores the strong coupling of mode
1 and 4, as well as the resonance of one of the bath modes, whichwe will refer to asx1, with
q4. We include these couplings intoχL in successive forms asχL = χL(q6, q4)∏5
i=1 χiL(xi) and
χL = χL(q6, q4, x1)∏5
i=2 χiL(xi). Inserting these forms into Eq. (5.10) yield
∆ =
∫
dq6δ(q6)
∫
dq4χL(q6, q4)χR(q6, q4)5
∏
i=1
IFCi , (5.14)
and
∆ =
∫
dq6δ(q6)
∫
dq4
∫
dx1χL(q6, q4, x1)χR(q6, q4, x1)5
∏
i=2
IFCi , (5.15)
respectively. The difference in the results obtained usingEq. (5.14) and Eq. (5.15) demonstrates
the importance of the effects of resonance with the bath modex1. These formulae highlight the
role of the different symmetries, and this will be discussedlater. The diabatic wavefunction
χL(q6, q4, x1) is computed under the normal mode approximation made at the left minimum
77
while treatingq1 adiabatically. The parameters for the DVR grid are the same as shown in Table
3.2 and Table 3.3.
5.3 Results and discussion
The results for the tunneling splittings for the model Hamiltonian described in Sec. 3.2 given
by the adiabatic Herring estimate using Eq. (5.15) are shownin Fig. 5.1. Most of the tunneling
splittings are within a factor of 2 of the exact results. To illustrate the importance of the coupling
of mode 4 and the resonance of the bath mode, we compare the results obtained from Eq. (5.11)
and Eq. (5.14) in Fig. 5.3. Only the tunneling splittings forstates 5 to 8 for FAD coupled
to the bath are shown for clarity. It can be seen that including the effects of the coupling to
mode 4 is essential, as should be expected. Ignoring the resonance effects of the bath mode
does not capture the sharp jumps observed in the exact results. It performs reasonably well
otherwise. We compute the decay ofPR given by Eq. (3.9) as was performed in Chapter 4.
This comparison is shown in Fig. 5.2. The comparison is very good, which is expected from
the good agreement of the tunneling splittings.
The tunneling splittings in the ambit of this theory are to beunderstood in terms of the
contributions from the different regions of the surfaceq1 = 0. To illustrate the point, Fig.
5.4 (a) showsχL for the ground state plotted as a function ofq4 andq6. Also shown are its
contours along with the contours ofχR. The contributions to the tunneling splittings from
the various regions of (q4,q6) is given by the integrand of Eq. (5.14) shown in Fig. 5.4 (b),
which has a maximum nearq4 = 0, q6 = 0.5 A. Figure 5.5 shows the corresponding plots
for n4 = 0, n6 = 5. In this caseχL andχR overlaps at much larger values ofq6, but large
contributions to tunneling splittings comes from smaller values ofq6 sinceδ(q6) is much larger
there as shown in Fig. 5.5 (b). The negative and positive contributions in this excited state
partially cancel each other, leading to the decrease of the tunneling splittings. This cancellation
has close resemblance to the work by Stuchebrukhov and coworkers, where the tunneling rates
are computed using the tunneling current theory. The constructive and destructive interference
of these currents give the transfer matrix element.136
The role of the different symmetries of the coupling on the tunneling splittings is well recog-
nized in literature,72,80–82and is also well highlighted in the adiabatic Herring theory. In a zeroth
78
Figure 5.1: Comparison of exact tunneling splittings (black) with the tunneling splittings ob-
tained through the adiabatic Herring method (red). (a) Gas-phase FAD withn4 = 0, (b) gas-
phase FAD withn6 = 0, (c) first four tunneling splittings in presence of the bath,and (d)
tunneling splittings for states 5 to 8 in presence of the bath.
Figure 5.2: Comparison of decay ofPR computed using exact dynamics (black) and adiabatic
Herring estimate (red). (a) Gas-phase with T=300 K, (b) gas-phase with T=200 K, and (c)
coupled to bath (η/mωb = 0.1) with T=300 K.
order picture, the modes with anti-symmetric coupling contribute through the Franck-Condon
factors, for example the weakly coupled bath modes in the present case. The contribution from
the modes with symmetric coupling is given by the average over the tunneling splitting (δ(q6))
79
Figure 5.3: Comparison of panel (d) of Fig. 5.1 with the red line corresponding to tunneling
splittings computed using (a) Eq. (5.11), (b) Eq. (5.14)
Figure 5.4: (a) Plots ofχL andχR, and (b) the contributions to the tunneling splittings from
various regions ofq4 andq6 (integrand of Eq. (5.14)). These plots are for the computation of
ground state tunneling splitting.
weighted by the corresponding wavefunction amplitudeχ2(q6) [see Eq. (5.11)].114 The cou-
pling between the symmetric and the anti-symmetric coupledmodes change the orientation of
the wavefunctionχL(q6, q4, x1) [Eq. (5.15)], and hence cause an exponential difference in the
tunneling splittings. This explains the sensitivity of thetunneling splitting to the coupling be-
tween mode 4 and 6 as was mentioned in the exact results. It further explains the origin of
the turnover in the tunneling splittings with vibrational excitation in the modes with symmetric
80
Figure 5.5: (a) Plots ofχL andχR, and (b) the contributions to the tunneling splittings from
various regions ofq4 andq6 (integrand of Eq. (5.14)). These plots are for the computation of
tunneling splitting for the staten4 = 0, n6 = 5.
coupling as the destructive interference of the contributions of the tunneling splittings.
A possible shortcoming of our approach is computation ofχL in Eq. (5.7) using the normal
mode approximation of the diabatic potential. We test this approximation by studying the effect
of different parameters, particularly the barrier heightV0 and coupling parametera8 (Table
3.1). Since the information of the barrier height is not present in the normal modes, this is a
stringent test of the method. The trends of the tunneling splittings for excitations in mode 4
are qualitatively captured, though the deviation is large.The method does not perform well for
excitations in mode 6. The anharmonicities in mode 6, as wellas the diabatic wavefunction
approximation are the root causes of this problem.
5.4 Additional methods
The tunneling path plays a central role in the determinationof the tunneling splittings, and nu-
merous approaches have been developed that are based on thispath. We apply the invariant
instanton method to compute tunneling splittings, this being one of the most accurate tunnel-
ing path approaches.81,123,124 The details of its computation are provided in the Appendix D.
The instanton path is shown in Fig. 5.6. It is oriented along mode 4 at the minimum, and
81
Figure 5.6: Instanton trajectory plotted as a function ofq1.
passes through roughlyq6 = 0.6A at the dividing surface. The ground state tunneling splitting
is computed to be 0.00156 cm−1, and the tunneling splitting corresponding to mode 4 funda-
mental vibrational excitation is 0.0195 cm−1, in excellent agreement with the exact results of
0.00170 cm−1 and 0.0186 cm−1, respectively. The computation of the tunneling splittings for
the transversal mode excitation leads to numerical instability in evolution of the matrixU [Eq.
(2.43) of Ref. 81], and was not performed.
It is interesting to connect these approaches to the findingsof the adiabatic Herring treatment
developed in this work. This treatment does not rely on a tunneling path, but integrates over
contributions from all regions of the coordinate space. Nonetheless, we can use this approach
to see what portions of configuration space are important to tunneling. The region important to
the computation of the tunneling splittings is given by the integrand of Eq. (5.14). The ground
state tunneling splitting is primarily given by theq4 = 0 surface, and maximum contribution
comes from(q4, q6) = (0, 0.5)A as is shown in Fig. 5.4. This is close to where the instanton
trajectory reaches the dividing surface. The maximum contributions to the excited state tunnel-
ing splittings comes from significantly lower values ofq6, as can be seen from the contour plots
of Fig. 5.5, indicating that the tunneling path depends on the state under consideration. Takada
and Nakamura133 in an effort to include such effects have suggested that the tunneling path is
82
dependent on the caustics of the classical trajectory. Further work is needed in choosing the
appropriate tunneling path for excited states.
Bowman and coworkers128,129 have developed a one dimensional zero curvatureQim for-
malism which neglects the path length of the tunneling path.Excellent agreement with the
experimental results for tunneling splittings and the kinetic isotope effects is obtained for var-
ious systems. This method provides an opportunity to test the importance of the curvature of
the tunneling path. Applying this approach to the current Hamiltonian of gas-phase FAD gives
the ground state tunneling splitting of 0.55 cm−1, roughly two orders of magnitude larger than
the exact result. Including the zero point energies of mode 4and 6 gives tunneling splitting of
0.48 cm−1; this fairly small change is in agreement with the results ofBowman and coworkers
on the tunneling splittings of malonaldehyde.128 This lack of agreement demonstrates the im-
portance of the effects of the curvature to the current system. This importance can be further
demonstrated by applying this formalism to study the effectof the coupling of the bath on the
ground state tunneling splitting. For the system-bath coupling of the form of Eq. (3.6),V (Qim)
is independent of the coupling strength and hence it would predict no change in the ground state
tunneling splitting.
5.5 Conclusions
We have presented a new formalism in this chapter — the adiabatic Herring estimate — to com-
pute tunneling splittings. This formalism is developed forsystems with separation of timescales
for the vibrational degrees of freedom. It develops over theHerring estimate of tunneling split-
ting, which is known to be highly accurate. The key approximations made in this theory are (a)
the one dimensional tunneling splitting is a function of only the symmetrically coupled modes
[see Eq. 5.9] and (b) the computation ofχL using the normal mode approximation made at the
left minimum while treatingq1 adiabatically.
Tunneling splittings computed using this method are in excellent agreement with the exact
results, including negative tunneling splittings. The turnover of the tunneling splittings, and the
negative tunneling splittings, can be understood in the context of the positive and the negative
contributions from different regions of the dividing surface. It further illustrates the effects of
the different coupling symmetries on the tunneling splittings. While the anti-symmetric modes
83
contribute through Franck-Condon terms, the symmetric modes contribution is an average of 1D
tunneling splittings along the symmetric coordinate weighted by the amplitude of the diabatic
wavefunction. The coupling between symmetrically and anti-symmetrically modes change the
orientation of the diabatic wavefunction, resulting in an exponential change in the tunneling
splittings.
The dependence of the method on the diabatic wavefunction isa limitation of the method.
Computation of tunneling splittings for different barrier heights has large deviations from the
exact results, though all the qualitative trends are captured. The anharmonicities of the potential
can be important and further work is needed to address this issue.
84
Chapter 6
Conclusions
Contents
6.1 Vibrational energy relaxation . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.2 Symmetric proton tunneling . . . . . . . . . . . . . . . . . . . . . . . . . 85
We have investigated two of the important aspects of chemical reactions. In the first part we
studied in detail the timescales of vibrational energy relaxation (VER) and the various factors
that influence it. In the second half, we investigated several methods to obtain the timescales of
proton tunneling, an important step in various chemical andbiological reactions.
6.1 Vibrational energy relaxation
Non-equilibrium molecular dynamics simulations have beenperformed for two and three di-
mensional models to investigate the pathways of VER in iso-chloroiodomethane (CH2Cl-I).
There is a loss of roughly 15000 cm−1 of energy in the first 100 fs due to the impulsive col-
lisions with argon atoms. The energy decay of the molecule after 1 ps can be described in a
normal mode picture. The two normal modes that play the most important role correspond to
C-Cl-I bend (frequency∼ 167 cm−1) and Cl-I stretch (frequency∼ 247 cm−1). The long time
energy decay rate of roughly 0.05 ns−1 corresponds to the decay of energy of Cl-I stretch mode,
and is in agreement with the experimentally computed decay rate of 0.074 ns−1.
The effect of variation of various parameters on the decay rate have been explored. Two
85
most important features that affect the thermal decay rate constants are the normal mode fre-
quencies and the initial partitioning of energy among the normal modes. The effects of the
initial partitioning of energy is not as well highlighted inthe literature as are the effects of the
normal mode frequencies. This work demonstrates that afterimpulsive collisions, the over-
all decay rates can be substantially lowered if the higher frequency normal mode gains more
energy.
Landau-Teller theory predictions are in qualitative agreement with the non-equilibrium
molecular dynamics results. This theory clarifies the role of the normal mode frequencies,
and further sheds light on the different interactions of thenormal modes with the bath modes.
We also show that the anharmonicities in the form of couplingbetween the normal modes can
have adverse effects on the predictions of the Landau-Teller theory.
A possible direction for future research is to examine the effects of different solvents on the
timescales of VER. The experimental decay rates of 0.002 ps−1 and 0.02 ps−1 are computed for
nitrogen and methane matrices, respectively.4 The origins of this order of magnitude difference
will be interesting to note.
6.2 Symmetric proton tunneling
The second half of the thesis focuses on a model Hamiltonian that qualitatively represents sym-
metric proton tunneling in formic acid dimer and its coupling to a bath. The timescales of proton
tunneling are obtained using numerically exact quantum mechanics. One of the key results of
the computed exact dynamics is that the timescales of tunneling increase with increasing bath
friction in the weak coupling regime. This is in contrast to the classical picture of energy dif-
fusion which predicts a decrease in the timescales. We demonstrate that for the model system
investigated, which is in the deep tunneling regime, the coherence effects cannot be ignored.
These coherence effects manifest themselves as an increasein path length with increasing fric-
tion which dominates the effects of the energy diffusion.
We use this system as a severe test for two popular approximate methods - surface hopping
and ring polymer molecular dynamics. Not surprisingly, both these methods do not capture the
effects of the bath on the timescales, as the effects of the path length is not accurately treated in
these methods. We suggest modifications in surface hopping method to partially include these
86
coherences.
In order to predict the correct trends, we take an alternate route by computing the tunneling
splittings, which dictate the dynamics. Makri-Miller method, which can compute excited state
tunneling splittings, is benchmarked against the exact results. The results of this formalism
are in qualitative agreement with the exact results, including the increase of timescales with
increasing bath friction. This method correctly identifiesthe change in the path length as the
reason for this increase of timescales. There are quantitative discrepancies though. The tunnel-
ing splittings corresponding to the highly excited vibrational modes are hard to compute in this
method. Franck-Condon factors, that cause non-monotonic changes in the tunneling splittings,
also are not captured in this method.
We develop a new formalism to compute tunneling splittings –adiabatic Herring estimate.
The results obtained with this formalism are in quantitative agreement with the exact results for
the model Hamiltonian, including prediction of negative tunneling splittings. The integration
over contributions from different regions of the slow vibrational degrees of freedom gives the
tunneling splittings. These contributions can be positiveor negative, and the destructive inter-
ference is the reason for the non-monotonic trends as well asthe negative tunneling splittings.
This approach further highlights the effects of the different symmetries on the tunneling split-
tings. In a zeroth order picture, while the anti-symmetrically coupled modes contribute through
Franck-Condon factors, the symmetrically coupled modes contribute through a weighted av-
erage of 1D tunneling splittings over the symmetric mode. The dependence on the diabatic
wavefunction is a limitation of the method.
There are multiple avenues for future research. One of the burning question is what happens
at stronger coupling strengths and inclusion of more bath modes. With increasing computation
resources, the answer to this question can be revealed in thefuture. The comparison of these
exact results to the vast amount of literature present on theKramer’s turnover will be interesting
to note in various regimes. Another question that emerges out is what is the best way to obtain
a thermal rate constant in exact dynamics, which is dictatedby oscillatory functions, as well as
exhibit multiple timescales.
The results obtained through surface hopping in this work demonstrate an important short-
coming of this method. The modes that do not have a coupling term in the Hamiltonian with
the quantum degree of freedom do not affect the quantum amplitudes. This neglect usually
87
does not have significant effects, but in systems where coherences are important, this may play
a crucial role. We partially included these effects by including an additional constraint. More
work is needed to generalize this constraint, as well as testit for a broader class of systems.
These coherence effects are also missing in the RPMD method. The works of Richardson and
Thoss115 and Miller and coworkers116 for including non-adiabatic effects in electron transfer are
very encouraging, and a similar line of thought can perhaps be pursued for proton transfer.
The simplicity of the Makri-Miller method is remarkable, and the results obtained from this
method are very encouraging. An important issue though is the choice of the tunneling path.
More work is needed to obtain general guidelines for this choice. Further, it is desirable to
include the effects of the vibrational excitations on the tunneling path. The works of Takada
and Nakamura133 can perhaps serve as a good starting point.
The adiabatic-Herring method that we have developed has room for significant improve-
ments and generalizations. Its dependence on the choice of the diabatic surface is an important
issue. There have been significant advances in the choice of the diabatic surfaces, which can be
helpful in improving this theory. The effects of anharmonicities is also not well captured, and
needs significant improvement. This theory further needs tobe benchmarked against a larger
class of systems. The predictions of this theory to full dimensional formic acid dimer and mal-
onaldehyde compared with the experimental results will be valuable in the development of this
method.
88
Appendices
89
Appendix A
Introduction for broader audience
I welcome the initiative by the Wisconsin Initiative of for Science Literacy to include a
chapter in my thesis to convey the work I have done to non-specialists. I have adopted the style
of a dialogue between the reader and myself for this chapter to overcome the monotony of long
paragraphs. I, like most scientists, love talking about my research and this chapter is partly
based on conversations I had over the years about my research with several friends.
Q. What branch of chemistry do you study?
I am a physical chemist. This branch uses our knowledge of physics to better understand
chemical reactions. Specifically, I am a theoretician, who attempts to simulate chemical reac-
tions on computers as well as develop theories that have experimentally verifiable predictions.
Q. Can you summarize your thesis as if you are describing it your grandmother?
Wow, that’s exactly the question asked to me in my visa interview. My thesis comprises
two projects. First, I study a phenomenon found in nature called tunneling. It is so bizarre that
we do not have any direct physical experience of it. Imagine hitting a baseball with a bat, and
imagine a full head-on collision. Well the ball will bounce back from the bat, right? Almost
always! It turns out that if you follow the laws of quantum mechanics, a theory developed in
the last century, there exists a very, very small probability that the ball will pass right through
the bat. These probabilities are much larger at atomic scales.
90
Q. Why is tunneling relevant?
Implications of tunneling can be seen all around us. Considerthe sun, where energy is
created by fusing together hydrogen atoms. Even the sun doesnot have enough energy to
surmount the huge repulsion that the two positively chargednuclei feel, and bring them close
enough to fuse. The hydrogen atom tunnels through the repulsion barrier, leading to fusion.
Radioactivity is also a consequence of tunneling. There are several examples in biological
systems as well where tunneling is crucial. DNA, the basic ingredient of life, has two strands
that are connected through ‘hydrogen bonds’. Sometimes hydrogen tunnels from one strand to
another, which is believed to be the cause of certain diseases as well as aging.
Q. Can you be specific about what tunneling is?
Tunneling occurs when a particle passes through a ‘barrier’without ever being at the barrier.
This barrier is whatever makes two things interact or feel each other. In the case of baseball,
it is the collision of the atoms of the baseball with the atomsof the bat. In the sun, it is the
electrostatic interaction between the protons. For DNA, itis the forces responsible for the
hydrogen bond.
Q. So a particle can go from one place to another, without everpassing through regions
in between. Is that possible?
That is a very interesting question, and has puzzled many, including me. The short answer is
that quantum mechanics predict it, and people have observedthis phenomenon by constructing
very careful experiments. Hence it must be. Sir Arthur Canon Doyle in one of his novels said
‘When you have eliminated the impossible, whatever remains,however improbable, must be
the truth’.
In quantum mechanics, matter can act as both particle and wave. That is strange, but a
thoroughly verified fact. When you think of a wave, it is not really ‘present’ at any one point,
but spread out. It is possible to imagine a situation where this wave can have a large amplitude
on one side of the barrier initially, and a large amplitude onthe other side at a later time.
Q. What exactly do you study about tunneling?
91
Tunneling, as I mentioned earlier, can be important in many biological systems. Yet we do
not know how to include its effects in conventional large scale simulations, which have become
an indispensable tool in chemistry. I have investigated a mathematical model, involving only
a few variables, that describes proton tunneling. As the proton tunnels, the surrounding atoms
and bonds reorganize which can have profound effects on tunneling. I employ the model with
the aim to develop new methods, as well as improve existing ones, to include the reorganization
effects on tunneling in the large scale simulations. The small number of variables simplify
equations that describe tunneling, greatly helping in development of new methods, as well as
provide a deeper understanding of tunneling.
Q. What are the key results of this project?
A very interesting feature I discovered is that under certain conditions, the reorganization of
the surrounding atoms can increase the time it takes for the proton to tunnel. I tested two of the
most promising methods available for large scale simulations, and observed that they miss this
feature. I suggested improvements in one of them. It remainsto be seen if this improvement will
work for larger, and more complex, systems. I further testedanother method, that is somewhat
specifically designed for this particular mathematical model. This method works reasonably
well, and includes all the features to some degree.
I also developed a method of my own. This new method works extremely well for the
model I have studied. Although more work will be required to include this method in large
scale simulations.
Q. Please briefly describe the other project you have worked on.
In my second project, I investigate the flow of energy in molecules. Imagine a cup of hot tea
left out. After a while, it will cool down to the room temperature. Something similar happens
in molecules. Any chemical reaction requires energy. When the products of the reaction form,
this energy dissipates away by a process known as vibrational energy relaxation. I have studied
in detail the time it takes for the molecule to cool down.
Q. Why do you care about how much time it takes for energy to dissipate?
When the product of the reaction forms, it always has an optionof returning back to the
92
original reactants. If it takes too long for the products to cool down, then there is a good
chance that the product will revert back to its reactants, leading to no reaction. That is why
understanding what factors affect this time is important.
Q. What methods are available to study this, and which do you work with?
One method is to heat up a molecule very precisely using lasers. Then we wait, and see how
much energy the molecule has after a certain period of time. The molecules move really fast,
on the timescale of femtoseconds which is a millionth of a billionth of a second. Ahmed Zewail
developed a technique of ‘pumping’ energy into, and ‘probing’ the molecules on femtosecond
timescales in 1980-1990, and received a Nobel prize for thiswork! Another way, the approach
I use, is to simulate this process on a computer. The moleculeis modelled in some form – a
simple way is to think about billiard balls connected through springs. The balls represent atoms
and the springs represent chemical bonds. We can tune how stiff these springs are to model
different kinds of bonds. We can even see how these springs will act in time based on methods
proposed by Newton. Voila, now we can put energy in by stretching the bond, and see how it
dissipates in time. Computers can be very efficient in simulating this.
Q. How do you know this computer modelling works? It seems a stretch of imagination
to compare springs with bonds.
This is where the kind of experiments designed by Ahmed Zewail greatly helps. Fleming
Crim, a professor at UW Madison, and his group performed a similar experiment on a molecule
called chloroiodomethane. This molecule is important for environmental reasons. If our com-
puter model agrees with the experiments, it gives us a great deal of confidence. This kind of
computer modelling was started in 1965 by four scientists – Fermi, Pasta, Ulam and Tsingou,
and has made great progress since then. In fact, last year’s Nobel prize was given to Karplus,
Levitt and Warshel for the developments they have done in these computer simulations. These
simulations have been proven to simulate chemical reactions with some accuracy.
Q. But why perform these simulations if you can watch the real molecule itself?
Simulations contain far more information. You know the motion of every atom, and the
details of the energy flow. Not only that, one can also change the stiffness of the bonds, or how
93
much energy was initially put in and observe the change in theresults. This gives us the ability
to generalize and predict the time of energy flow in new molecules. These changes are very
hard to do in the experiments.
Q. Briefly, what are the key results of this work?
The time of cooling I obtained roughly matches the time seen in the experiment. What I
further obtained is the speed and pathways of energy flow – very rapid at the beginning, slowing
down later. The slower time depends on how stiff the bonds are. The stiffer the bond, the slower
the energy loss. Another crucial factor is the partitioningof the energy among the bonds after
the rapid energy flow at the start. If the stiffer bonds retainmore energy, then the overall energy
flow will be slower.
Q. How does your research affect society?
All applications originate from a better understanding of how nature works. Galileo’s thor-
ough study of celestial motion became the precursor to the Newtonian mechanics, which has a
vast range of applications. My work adds to our understanding of tunneling and energy flow in
molecules, both of which are relevant to society. For example, based on the methods presented
in this thesis on tunneling, more simulations and better experiments can be designed to develop
medicines to fight tumors. Similarly, with a better understanding of energy flow, we can control
the flow of energy. In doing so, we can manipulate reactions toour advantage – for example we
can use this research to improve solar cells by increasing the fraction of the sunlight energy that
converts into chemical and electrical energy.
94
Appendix B
2D Landau-Zener formalism
Figures
B.1 Plot of∆0 and∆0d012 with respect toq6 . . . . . . . . . . . . . . . . . . . . 96
B.2 Comparison of transition probabilities obtained through the 2D LZ formalism
with the exact transition probabilities . . . . . . . . . . . . . . . . . . . . . . 98
We present a formalism to partially include multi-dimensional effects in the 1D LZ formula.
Assuming the basis is comprised of 2 wavefunctions∣
∣φbi
⟩
, which do not explicitly depend on
time. The total wavefunction at any time t is given by:
|ψ〉 =2
∑
i=1
ci(t)∣
∣φbi
⟩
e−iR
t
0dτHii/h, (B.1)
whereHij =⟨
φbi
∣
∣ H∣
∣ φbj
⟩
. Substituting this into the Schrodinger equation leads to:
ihc1 =c2
[
H12 − ih⟨
φb1
∣
∣
∣φb
2
⟩]
exp
[
−i
∫ t
0
dτ(H22 − H11)/h
]
. (B.2)
In contrast to the original LZ formulation, we treat∣
∣φbi
⟩
to be a function of mode 6 as
∣
∣φb1/2
⟩
=1√2
[∣
∣ψb1
⟩
±∣
∣ψb2
⟩]
, (B.3)
where∣
∣ψbi (q6)
⟩
is the eigenfunction of the symmetric double well potentialVb(q4 = 0, q6). Let
Eib be the corresponding eigenvalues,V represent the true potential at (q4,q6), andδV = V −Vb
95
be the difference of potentials.
In this representation,
⟨
φb1
∣
∣
∣
∣
dφb2
dt
⟩
=
⟨
φb1
∣
∣
∣
∣
dφb2
dq6
⟩
dq6
dt(B.4)
= −⟨
ψb1
∣
∣
∣
∣
dψb2
dq6
⟩
dq6
dt(B.5)
= −
⟨
ψb1
∣
∣
∣
dVb
dq6
∣
∣
∣ψb
2
⟩
E2b − E1
b
dq6
dt(B.6)
= 0. (B.7)
The last equality is obtained by noting thatψb1 and dVb
dq6
are even functions ofq1, while ψb2 is an
odd function ofq1.
Solving forH12, we find
H12 =⟨
φb1
∣
∣ H∣
∣ φb2
⟩
(B.8)
=⟨
φb1
∣
∣ T + Vb + δV∣
∣ φb2
⟩
(B.9)
= −(E2b − E1
b )/2 +⟨
φb1
∣
∣ δV∣
∣ φb2
⟩
. (B.10)
Near the dividing surfaceq4 = 0,⟨
φb1
∣
∣ δV∣
∣ φb2
⟩
is small, and is ignored in the current formalism.
Hence
H12 ≃ −∆
2, (B.11)
where∆ = E2b − E1
b is the tunneling splitting corresponding to the potentialVb.
Finally, solving forH22 − H11,
H22 − H11 =⟨
φb2
∣
∣ H∣
∣ φb2
⟩
−⟨
φb1
∣
∣ H∣
∣ φb1
⟩
(B.12)
= −2⟨
ψb1
∣
∣ δV∣
∣ ψb2
⟩
(B.13)
≃−2
⟨
ψb1
∣
∣
∣
∂V∂q4
∣
∣
∣ψb
2
⟩
E2b − E1
b
(E2b − E1
b )(δq4) (B.14)
≃ −2d12∆(vq4t), (B.15)
96
Figure B.1: Plot of (a)∆0 and (b)∆0d012 with respect toq6. An exponential fit to∆0 is also
shown.
wheret = 0 is defined whenq4 = 0, vq4≃ δq4
tis the velocity along mode 4 atq4 = 0, andd12
is the nonadiabatic coupling vector along mode 4.
To simplify the equations,∆ is assumed to be an exponential with respect toq6:
∆(q6) = ∆0e−α(q6−q0
6), (B.16)
whereq06 is the point where the trajectory crosses the dividing surface, and
∆d12 =
⟨
ψb1
∣
∣
∣
∣
∂V
∂q4
∣
∣
∣
∣
ψb2
⟩
(B.17)
≃ ∆0d012, (B.18)
is assumed to be constant. We show the plot of∆0 as well as∆0d012 in Fig. B.1. It can be seen
that∆0 fits well to an exponential. Although∆0d012 is not a constant, it varies only by a factor
of 2 over 1A variation inq6.
Substituting forH12 andH22 − H11 in equation B.2 gives
ihc1 = c2(−∆0/2) exp(
−α(q6 − q06
)
) exp
[
−i
∫ t
0
dτ(−2d012∆0vq4
t)/h
]
(B.19)
≃ c2(−∆0/2) exp(
−α(vq6t + aq6
t2/2))
exp[
id012∆0vq4
t2/h]
, (B.20)
whereq6 is expanded in a Taylor expansion asq6(t) ≃ q06 + vq6
t+ aq6t2/2. We solve this differ-
97
ential equation, with the boundary conditionc1(−∞) = 0, by approximatingc2(t) = 1. This
leads to an integral over a Gaussian with the result being (without any further approximation)
c1(∞) =i∆0
2h
√
π
k1
e−k2
2/4k1 , (B.21)
k1 = −αaq6/2 + id0
12vq4∆0/h, (B.22)
k2 = αvq6. (B.23)
To have the correct asymptotic properties, the transition probability is defined as:
P12 = 1 − e−|c1(∞)|2 , (B.24)
with
|c1(∞)|2 =π
4h2
∆20
|k1|exp
[
−k22Re(k1)
2|k1|2]
. (B.25)
This is the final result. Ifaq6= 0, thenRe(k1) = 0, and the above equation will give
the result for 1D LZ theory. Numerical simulations are performed to test the accuracy of this
method by initiating the trajectories at(q4, q6) = (0, q06) with different velocities alongq4 andq6.
Trajectories are then propagated back in time by 10.5 fs, to obtain the classical initial condition.
These trajectories are then evolved together with the quantum amplitudes (ci) using a time step
of 0.01 fs. Figure B.2 shows the comparison of the transition probability obtained through the
2-D LZ formalism developed above and the exact quantum results as a function ofvq6(velocity
of mode 6 atq4 = 0) for q06 = 0.5 A andvq4
= 2000 m/s. The probability given by the classical
1-D LZ formalism is also shown. The good agreement between the exact results and the 2-D
LZ formalism is very encouraging.
98
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0 500 1000 1500 2000
P12
vq6(m/s)
Quantum2D-LZ1D-LZ
Figure B.2: Comparison of transition probabilities obtainedthrough the 2D LZ formalism with
the exact transition probabilities as a function ofvq6. The trajectory crosses the dividing surface
at q06 = 0.5 A with vq4
= 2000 m/s.
99
Appendix C
Fewest switches surface hopping
We provide the computational details for implementing the fewest switches surface hopping.11
For the largest values ofq6 accessed by the classical trajectories, the nonadiabatic coupling
vector along mode 4d12 has very large values (∼ 1010 A−1), and this would require very small
time steps. Since the hopping probability for these regionsis expected to be nearly one, we
introduce a cutoff value forq6 above which the trajectory hops with unit probability whenever
it crosses theq4 = 0 surface. Below this cutoff, the hopping is given by the fewestswitching
criteria of Tully. The cutoff forq6 is chosen to be 0.8A which is slightly larger than 0.7A, above
which trajectories did not to tunnel on a 1 ns timescale whileperforming the LZ calculations.
The results are insensitive to variation to this value.
The velocity-Verlet integration method is used to evolve the classical equations of motion
with a time step of 0.5 fs, except near the dividing surfaceq4 = 0 where a smaller time step is
used. Thus when the trajectory either crosses the dividing surface, or the condition|q4| < 0.001
A holds, using a time step of 0.5 fs, the trajectory is returned to the previous classical step. A
new time step is estimated asdt = 1/(105d12), whered12 is the non-adiabatic coupling vector
along mode 4 atq4 = 0 in m−1. If the time step is smaller than 0.005 fs, we setdt = 0.005 fs.
Similarly if the time step is larger than 0.5 fs, it is set to 0.5 fs. This estimate is based on an
average velocity of 1000 m/s. Using a uniform time step of 0.005 fs produces similar results.
The quantum amplitudes are evolved using the fourth-order Runge-Kutta integration method
with a uniform time step of 0.005 fs. Decoherence is introduced by setting the coefficient of
the current state to be one whenever|q4| > 0.2A, which is chosen based on the negligible
values of the non-adiabatic coupling vectord12 outside this region. In case frustrated hops are
100
encountered, the time uncertainty algorithm proposed by Truhlar and coworkers143 is used. The
trajectory can hop at some timeth if |t0 − th| < h/2∆E holds, wheret0 is the time of the
attempt and∆E is the amount of extra energy that was required to make the hoppossible att0.
Variations to both the classical and the quantum time steps do not change the results appreciably.
An average over 300 trajectories is performed to compare thedecay ofPR to the ones
obtained using the LZ formalism. Both NVE and NVT calculations are performed, and the
same computational details are used for these two methods asfor the LZ method. The energy
of the system shows some long time fluctuations as well as a small drift, which are perhaps a
consequence of the chaos introduced by the multiple hops.144 The results obtained using this
method are generally in agreement with the results obtainedusing LZ formalism, discussed in
Chapter 3.
101
Appendix D
Invariant instanton theory
Here we summarize the invariant instanton method to computeground state and low vibra-
tionally excited state tunneling splittings. We closely follow the details described in Ref. 81.
The instanton trajectory (q) is taken of the form
q(z) = q0(z) +∑
n
cnφn(z), (D.1)
wherez ∈ [−1 : 1] parameterizes the trajectory,q0(z) is some approximate instanton path,
φn(z) are a set of basis functions, andcn are corresponding coefficients. Newton’s method is
used to optimize action in the space of the coefficientscn. We use 20000 grid points forz and
50 basis functionsφn(z) = (1 − z2)Pn(z), wherePn(z) are Legendre polynomials.
We compute the ground state and fundamental vibrationally excited tunneling splitting un-
der the instanton formalism developed by Nakamura and coworkers.81,123,124The ground state
tunneling splitting is given by
∆0 =1
m
√
4hdetAl
π
pΣ.pΣ√
detAΣ(pTA−1p)Σ
exp[−S0 − S1], (D.2)
whereS0 is the classical action along the instanton, andS1 is given by
S1 =1
m
∫ 0
−∞dtTr[A(t) − Al]. (D.3)
Heret is to be understood as a function ofq with t = −∞ representingz = −1, andt = 0 is
102
when the trajectory crosses the dividing surfaceΣ or whenz = 0; see Eq. (27) of Ref. 124.
The matrixA satisfies
dA
dt+
A2
m= H, (D.4)
whereH is the matrix of the second derivatives of the potential. Theinitial condition ofA(t =
−∞) = Al is given by
A2l = mH(t = −∞), (D.5)
The tunneling splitting for the excitation in the longitudinal mode, this being the normal
mode oriented along the instanton at the minimum, is given by
∆(n=1) = ∆04V (q)Σ
hω||exp
[
2
∫ 0
−∞dt
(
ω|| −1
2V
dV
dt
)]
, (D.6)
whereω|| is the frequency of the longitudinal mode.
To avoid the problem of stiffness atz = −1, a linear expansion is implemented as
A(z) = Al + aǫ,
H(z) = H(−1) + hǫ, (D.7)
z(z) = αǫ,
with ǫ = (1 + z), andα = ω||. Computation ofh is performed by a linear numerical fit.
After some simple manipulations, one obtainsa = Ua′U†, whereU is a unitary matrix that
diagonalizesAl/m with eigenvaluesd. The elements of the matrixa′ are
a′ij =
h′ij
α + di + dj
, (D.8)
with h′ = U†hU. We chooseǫ = 0.0001 as the grid size in z, and use the fourth order Runge-
Kutta method for numerically integrating Eq. (D.4). Performing a higher order fit atz = −1 to
the matrixA gives the same answer to within numerical accuracy.
The computed instanton trajectory for gas-phase FAD is shown as a function ofq1 in Fig.
103
5.6, and has action of 15.82h. The instanton is parallel to mode 4 at the minimum. This is con-
firmed by the time dependence of the parameterz at the minimumz(−1 + ǫ)/ǫ = 182.7cm−1,
which is close to the mode 4 frequency of 180.3 cm−1. The angle between the vector pointing
along this mode and the velocity of the instanton trajectoryis cos−1(0.989), this being close
to the expected value of 0. The ground state tunneling splitting is computed to be 0.00156
cm−1, and the tunneling splitting corresponding to mode 4 fundamental vibrational excitation
is 0.0195 cm−1. This is in excellent agreement to the exact results of 0.00170 cm−1 and 0.0186
cm−1 for the ground state and the excited state tunneling splittings.
104
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