The Quasi-Diabatic Hamiltonian Approach to Accurate and Efficient Non-adiabatic Dynamics with

266 Pages·2014·8.04 MB·English

by  Xiaolei Zhu

THE QUASI-DIABATIC HAMILTONIAN APPROACH
TO ACCURATE AND EFFICIENT NON-ADIABATIC DYNAMICS
WITH CORRECT TREATMENT OF CONICAL INTERSECTION SEAMS

By
Xiaolei Zhu

A dissertation submitted to Johns Hopkins University in conformity with the
requirements for the degree of Doctor of Philosophy

Baltimore, Maryland
January 22, 2014

© 2014 Xiaolei Zhu
All Rights Reserved
Abstract
A method to simulate photoelectron spectra using quadratic local quasi-diabatic
Hamiltonians (Hd) is generalized and augmented to enable high accuracy dynamics
simulations of nonadiabatic processes that involve large amplitude motions, including
dissociation. The improvement is achieved by using a flexible symmetry adapted
analytical expansion to approximate the representation of electronic Hamiltonian operator
in a quasi-diabatic basis, the diabaticity of which is achieved by minimization of residual
coupling between quasi-diabatic states.
Although previous theoretical treatments have been used to treat adiabatic
dissociation and rearrangement processes with success, difficulties have been
encountered in systems complicated by seams of conical intersections. Existing methods
are either too expensive to be applied, or could not provide sufficient accuracy. Even for
nonadiabatic reactions of very small systems, such as photodissociation of NH , all
3
previous theoretical treatments have been unable to accurately reproduce experimental
measurements.
In this work, inspired by the success of bound-state Hd approach, a rigorous and
flexible framework is established to create a more robust method for accurate and
efficient nonadiabatic dynamics simulations, through the construction of quasi-diabatic
Hamiltonians(Hd) that correctly describes reactions. This new method requires no
assumption on the properties of individual systems. The application of local intersection
adapted representations and partially diagonalized representations enabled entire seams
of conical intersections as well as the nearby regions to be accurately described. No ad
hoc approximation is made in the diabatization procedure, and the residual coupling of
ii
the underlying quasi-diabatic representation is minimized in a least squares sense and can
be exactly quantified. Polynomials of arbitrary functions of internal coordinates are used
to construct an extremely flexible basis for Hd, and generic symmetry treatment allows
incorporation of arbitrary point group or Complete Nuclear Permutation Inversion
(CNPI) group symmetry1.
With the Hd ← photodissociation
process of NH was simulated. New results, obtained using Hd constructed with the
3
method described in this work, accurately reproduce experimental measurements,
illustrating its promising potential.
The method is then further enhanced to allow application to much larger systems,
with the coupled potential energy surfaces of the 1,2,31A states for the photodissociation
of phenol used as an example. A partially diagonalized representation approach is
developed to accurately treat near degenerate points, and a null-space analysis procedure
is added to guide the selection of monomial basis and to remove linear dependencies in
the fitting procedure. Coupled potential energy surfaces that fully incorporate all 33
degrees of freedom, many different large amplitude motions, and multiple seams of
conical intersections, are successfully constructed from ab initio data.
Thesis Advisor:
Professor David R. Yarkony, Johns Hopkins University
Additional Readers:
Professor Paul J. Dagdigian, Johns Hopkins University
Professor Harris J Silverstone, Johns Hopkins University

1 P. R. Bunker, Molecular Symmetry and Spectroscopy. (Academic Press, New York, 1979)
iii
This dissertation is dedicated to my dear wife Jin Yang.

iv
Acknowledgement
My deepest gratitude is to my advisor, Dr. David R. Yarkony for granting me the
opportunity to work in his research group. His guidance made the journey to Ph.D.
rewarding and stimulating. His experience and expertise in the area is unparalleled and
was indespensable during the course of every research project. Most of all, I sincerely
appreciate him for being extraordinarily kind and patient to always support me to pursue
my ideas, even after repeated mistakes and failures and periods of frustration.
I would like to thank Dr. Michael Schuurman and Dr. Joseph Dillion for their help
and friendship. Not only did they teach me all they know without reservation about
research in the group, they also helped me to adapt to life in graduate school, and in a
foreign country. A great many ideas incorporated in this dissertation are results of
discussions with them.
I would like to thank current and former members of the group, Christopher
Malbon, Sara Marquez and Nathan Kopf, along with Joseph and Michael, for countless
inspiring conversations both during group meetings and in private. Speical thanks to
Chris for sharing a cheerful office with me.
I would like to express my gratitude to our collaborator, Dr. Hua Guo, Dr. Jianyi
Ma and Dr. Changjian Xie, for performing full quantum dynamics simulations on our
constructed Hamiltonians. It was an incredible learning experience to work with Dr
G ’ g . I i i -of-the art quantum dynamics simulations that
served as the verification of the quality of our constructed coupled potential energy
surfaces. As the first users of our code outside our group, they also served as the beta
v
tester of our program and their feedbacks resulted in a number of improvement and bug
fixes. Thank you for bearing with us through the bugs and problems!
I would like to say thank you to all my friends and colleagues at Hopkins for their
friendship and company, especially Xin Tang, Ting Zhang, Yao Li, Yuqi Li, Dr. Yang Li,
Man Li, Jing Chen, Dr. Xiang Li, Xuan Li, Xinxing Zhang, Jing Li, Ying Zhang and
Yuchong Shao. I would like to thank Jin Yang, whom I met and married during the years
of my Ph.D. studies. She is the source of my passion and confidence, and is the main
reason I had a life outside of research lab.

Copyright and Permission Notices:
This dissertation contains the following original papers that have been previously
published in peer-reviewed journals:
Chapter Reprint with Permission from Copyright
X. Zhu, D.R. Yarkony, The Journal of
Chapter 2 ©2010, AIP Publishing LLC.
Chemical Physics 132, 104101 (2010)
X. Zhu, D.R. Yarkony, The Journal of
Chapter 3 ©2012, AIP Publishing LLC.
Chemical Physics 137, 22A511 (2012)
X. Zhu, J. Ma, D.R. Yarknony, H. Guo,
The Journal of Chemical Physics 136,
234301 (2012)
Chapter 4 ©2012, AIP Publishing LLC.
J. Ma, X. Zhu, H. Guo, D.R. Yarkony,
The Journal of Chemical Physics 137,
22A541 (2012)
X. Zhu, D.R. Yarknony, The Journal of
Chapter 5 ©2014, AIP Publishing LLC.
Chemical Physics 140, 024112 (2014)

vi
The work presented in Chapter 2 was supported by NSF grant CHE-0513952 to
DRY. The work presented in Chapter 3 was supported by NSF grant CHE-1010644 to
D.R.Y. The portion of work presented in Chapter 4 that involve construction of Hd was
supported by NSF grant CHE-1010644 to DRY. JYM and HG, who performed the 6-
dimensional quantum mechanical calculations, acknowledge the support of the NSF grant
CHE-0910828 to HG. The work presented in Chapter 5 was supported by NSF grant
CHE-1010644 to D.R.Y.

vii
Table of Contents
ABSTRACT ...................................................................................................................... II
ACKNOWLEDGEMENT ............................................................................................... V
LIST OF TABLES ....................................................................................................... XIV
LIST OF FIGURES ................................................................................................... XVII
CHAPTER 1 INTRODUCTION ..................................................................................... 1
1.1 CHEMISTRY BEYOND THE BORN-OPPENHEIMER APPROXIMATION ............................. 1
1.2 ELIMINATING THE COMPUTATIONAL BOTTLENECK WITH ANALYTICAL
APPROXIMATION .............................................................................................................. 3
1.3 INADEQUACIES OF EXISTING METHODS ..................................................................... 4
1.4 THE NON-LOCAL QUASI-DIABATIC HAMILTONIAN (HD) APPROACH .......................... 5
REFERENCES .................................................................................................................... 8
CHAPTER 2 TOWARD HIGHLY EFFICIENT NONADIABATIC DYNAMICS
ON THE FLY: AN ALGORITHM TO FIT NON-LOCAL QUASI-DIABATIC,
COUPLED ELECTRONIC STATE HAMILTONIANS BASED ON AB INITIO
ELECTRONIC STRUCTURE DATA .......................................................................... 10
2.1 ABSTRACT ................................................................................................................ 10
2.2 INTRODUCTION......................................................................................................... 11
2.3 THE QUASI-DIABATIC HAMILTONIAN AND ITS DETERMINATION .............................. 14
2.3.1 The Quasi-Diabatic Hamiltonian .................................................................... 14
2.3.2 Equations defining Hd ...................................................................................... 14
2.3.3 Constrained Pseudo Normal Equations........................................................... 16
viii
2.3.4 Advantages of the Pseudo Constrained Normal Equations Approach ............ 18
2.3.5 Nuclear Coordinates. ....................................................................................... 19
2.3.6 Global Symmetry of Hd .................................................................................... 21
2.4 HD FOR THE 1,21A ELECTRONIC STATES OF NH3 ...................................................... 22
2.4.1 Electronic Structure description of NH . ......................................................... 23
3
2.4.2 Symmetry-adapted internal coordinates .......................................................... 23
2.4.3 Construction of Hd ........................................................................................... 25
2.4.4 Accuracy of Hd ................................................................................................. 26
2.5 SUMMARY AND CONCLUSIONS ................................................................................. 44
2.6 APPENDICES ............................................................................................................. 45
2.6.1 Complete Nuclear Permutation Inversion Group of NH ................................ 45
3
2.6.2 Transformation Properties of Dot-Cross-Product Function ........................... 46
2.6.3 CNPI Irreducible Representations of Electronic States .................................. 47
2.6.4 Ab Initio and Hd Predicted Energies of All Data Points ................................. 48
2.6.5 Geometries of All Data Points ......................................................................... 55
REFERENCES .................................................................................................................. 63
CHAPTER 3 QUASI-DIABATIC REPRESENTATIONS OF ADIABATIC
POTENTIAL ENERGY SURFACES COUPLED BY CONICAL
INTERSECTIONS INCLUDING BOND BREAKING: A MORE GENERAL
CONSTRUCTION PROCEDURE AND AN ANALYSIS OF THE DIABATIC
REPRESENTATION ..................................................................................................... 68
3.1 ABSTRACT ................................................................................................................ 68
3.2 INTRODUCTION......................................................................................................... 69
ix
3.3 THE ALGORITHM ...................................................................................................... 73
3.3.1 General Definitions .......................................................................................... 73
3.3.2 Defining equations ........................................................................................... 74
3.3.3 Original Algorithm........................................................................................... 78
3.3.4 Newton Raphson equations .............................................................................. 78
3.4 COMPUTATIONAL RESULTS ...................................................................................... 79
3.4.1 Derivative Couplings ....................................................................................... 81
3.4.2 Newton-Raphson equations ............................................................................. 83
3.4.3 The Diabatic Character of the Representation ................................................ 88
3.5 SUMMARY AND CONCLUSIONS ................................................................................. 94
3.6 APPENDICES ............................................................................................................. 94
3.6.1 Form of Singule Coordinate Functions Used to Construct Hd ........................ 94
3.6.2 Second derivatives of Unknown Coefficients ................................................... 96
dI(R )
3.6.3 Evaluation of n .................................................................................... 98
z
k
3.6.4 Formulation and solution of Eq. ( 3-23 ) ....................................................... 100
REFERENCES ................................................................................................................ 104
CHAPTER 4 COMPUTATIONAL DETERMINATION OF
ABSORPTION SPECTRA AND PHOTODISSOCIATION PRODUCT
BRANCHING RATIOS OF NH AND OF ND USING HD AND FULL SIX
3 3
DIMENSIONAL QUANTUM DYNAMICS .............................................................. 112
4.1 ABSTRACT .............................................................................................................. 112
4.2 INTRODUCTION....................................................................................................... 112
x