adapted ab initio theory: a simplified kohn-sham...
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ADAPTED AB INITIO THEORY: A SIMPLIFIED KOHN-SHAM DENSITYFUNCTIONAL THEORY
By
JOSHUA J. McCLELLAN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2008
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c©2008 Joshua J. McClellan
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To my parents.
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ACKNOWLEDGMENTS
I thank the chair and members of my supervisory committee for their mentoring. I
would also like to give special thanks to Mark Ponton, Victor Lotrich, and Dr. Deumens
for their extensive help in the last few months. I owe a debt of gratitude to my family,
whose support has instilled a deep yearning to pursue my interests. Several people made
many other contributions to my life, both personally and academically. Larry urged me
to appreciate the finer things life has to offer and kept me sane. I thank the two real
chemists, Travis and Lani, for their unending quest to answer the basic question man asks
of life: What is fire? The people of the Quantum Theory Project offered a constant source
of inspiration, and often a source of morning headaches. I offer many thanks to Andrew,
Tom, Dan, Christina, Georgios, Seonah, Martin, Joey, Lena, Julio, and of course Merve.
I also acknowledge Prof. Merz for the use of the DIVCON program and Dr. Martin
Peters for helping to implement and test some initial ideas for cubic spline interpolation
(Chapter 4). Special thanks go to Andrew Taube and Tom Hughes for the numerous
extensive discussions that were invaluable in my research. Also, I acknowledge Tom
Hughes’ work in implementing the transfer Hamiltonian parameters for nitromethane
clusters (Chapter 2). I would like to recognize the help from Dr. Ajith Perera for help
throughout my graduate studies.
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TABLE OF CONTENTS
page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
CHAPTER
1 INTRODUCTION AND BACKGROUND . . . . . . . . . . . . . . . . . . . . . 10
1.1 Who Ordered a New Semiempirical Theory? . . . . . . . . . . . . . . . . . 101.2 Quantum Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3 Hartree-Fock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.4 Coupled-Cluster Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.5 Kohn-Sham Density Functional Theory . . . . . . . . . . . . . . . . . . . . 221.6 Huckel Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.7 Neglect of Diatomic Differential Overlap . . . . . . . . . . . . . . . . . . . 251.8 Tight-Binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.9 Density-Functional Tight-Binding . . . . . . . . . . . . . . . . . . . . . . . 271.10 Self-Consistent-Charge Density-Functional Tight-Binding . . . . . . . . . . 29
2 THE TRANSFER HAMILTONIAN . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1 Theoretical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.2 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.3 Nitrogen-Containing Energetic Materials . . . . . . . . . . . . . . . . . . . 37
2.3.1 Nitromethane Monomer . . . . . . . . . . . . . . . . . . . . . . . . . 402.3.2 Nitromethane Dimer . . . . . . . . . . . . . . . . . . . . . . . . . . 412.3.3 Nitromethane Trimer . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4 Silica and Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.4.1 Silicon Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.4.2 Silica Polymorphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3 IMPETUS FOR A NEW SEMIEMPIRICAL THEORY . . . . . . . . . . . . . . 58
3.1 Neglect of Diatomic Differential Overlap Based Methods . . . . . . . . . . 583.1.1 Does NDDO Approximate HF? . . . . . . . . . . . . . . . . . . . . 583.1.2 Artificial Repulsive Bump . . . . . . . . . . . . . . . . . . . . . . . 603.1.3 Separation of Electronic and Nuclear Energy Contributions . . . . . 613.1.4 Total Energy Expression . . . . . . . . . . . . . . . . . . . . . . . . 61
5
3.2 Self-Consistent Charge Density-Functional Tight-Binding . . . . . . . . . . 623.2.1 Self-Consistent Charge Density-Functional Tight-Binding in an AO
Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.2.2 Mulliken Approximation Connection to SCC-DFTB . . . . . . . . . 653.2.3 SCC-DFTB Repulsive Energy . . . . . . . . . . . . . . . . . . . . . 66
3.3 Systematic Comparison of HF, NDDO, DFTB, SCC-DFTB, KS-DFT in aSingle Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.3.1 General Energy Expression . . . . . . . . . . . . . . . . . . . . . . . 703.3.2 Density Independent Fock-Like Matrix Contribution . . . . . . . . . 713.3.3 Density Dependent Fock-Like Matrix Contribution . . . . . . . . . . 723.3.4 Residual Electronic Energy . . . . . . . . . . . . . . . . . . . . . . . 733.3.5 General Gradient Expression . . . . . . . . . . . . . . . . . . . . . . 743.3.6 Requirements for an Accurate Simplified Method . . . . . . . . . . . 75
4 ADAPTED AB INITIO THEORY . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.1 Two-Electron Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.2 Kohn-Sham Exchange and Correlation . . . . . . . . . . . . . . . . . . . . 894.3 Adapted ab initio Theory Model Zero . . . . . . . . . . . . . . . . . . . . . 924.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.5 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
APPENDIX PARALLEL IMPLEMENTATION IN THE ACES III ENVIRONMENT 117
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
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LIST OF TABLES
Table page
1-1 The NDDO one-center two-electron integrals . . . . . . . . . . . . . . . . . . . . 32
1-2 The NDDO two-center two-electron integrals . . . . . . . . . . . . . . . . . . . . 33
1-3 Multipole distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2-1 Equilibrium geometry for NMT, in A and degrees . . . . . . . . . . . . . . . . . 49
4-1 Adapted integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4-2 Average percentage of Fock matrix contribution by adapted approximation . . . 104
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LIST OF FIGURES
Figure page
2-1 The NMT force curve for C-N rupture . . . . . . . . . . . . . . . . . . . . . . . 50
2-2 The NMT energy curve for C-N rupture . . . . . . . . . . . . . . . . . . . . . . 51
2-3 The NMT dimer energies relative minimum in the intermolecular hydrogen bondingdistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2-4 Snapshots of the geometry optimizations for NMT dimer and trimer performedwith the TH-CCSD and AM1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2-5 Average deviation of Si-O stretch . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2-6 Average deviation of Si-Si stretch . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2-7 Average deviation of O-Si-O angle . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2-8 Density deviation from PW91 DFT . . . . . . . . . . . . . . . . . . . . . . . . . 57
3-1 Multi-center contribution to the C2sN2s matrix element of NMT. . . . . . . . . 76
3-2 The AM1 artificial repulsive bump for NMT direct bond fission. . . . . . . . . 77
3-3 The SCC-DFTB total energy breakdown. . . . . . . . . . . . . . . . . . . . . . 78
3-4 Energies from Rudenberg and Mulliken approximations. . . . . . . . . . . . . . 79
4-1 Scatter plots of agreement between approximate and exact two-electron integrals. 105
4-2 Dissociation curve of C-N with 4-center Rudenberg approximation. . . . . . . . 106
4-3 Orbital products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4-4 Dissociation curve of C-N with GZERO(B3LYP) approximation. . . . . . . . . . 108
4-5 Dissociation curve of C-N with AATM0 approximation. . . . . . . . . . . . . . . 109
4-6 Timing of ab initio versus Rudenberg four-center integrals . . . . . . . . . . . . 110
4-7 Percentage of multi-center terms versus system size . . . . . . . . . . . . . . . . 111
4-8 Timing of Fock build using traditional NDDO and NDDO with cubic splines asa function of system size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4-9 Error introduced per atom from cubic splines as a function of system size . . . . 113
4-10 Pseudo-reaction-path splitting C20 . . . . . . . . . . . . . . . . . . . . . . . . . 114
4-11 Force RMSD from MP2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4-12 Force RMSD from B3LYP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
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Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy
ADAPTED AB INITIO THEORY: A SIMPLIFIED KOHN-SHAM DENSITYFUNCTIONAL THEORY
By
Joshua J. McClellan
May 2008
Chair: Rodney J. BartlettMajor: Chemistry
We present work toward a one-electron Hamiltonian whose solution provides
electronic energies, forces, and properties for more than 1000 atoms fast enough to drive
large scale molecular dynamics. Ideally, such a method would be as predictive as accurate
ab initio quantum chemistry for such systems. To design the Hamiltonian requires that
we investigate rigorous one-particle theories including density functional theory and
the analogous wavefunction theory construction. These two complementary approaches
help identify the essential features required by an exact one-particle theory of electronic
structure. The intent is to incorporate these into a simple approximation that can provide
the accuracy required but at a speed orders of magnitude faster than today’s DFT.
We call the framework developed adapted ab initio theory. It retains many of the
computationally attractive features of the widely-used neglect of diatomic differential
overlap and self-consistent-charge density-functional tight-binding semiempirical methods,
but is instead, a simplified method as it allows for precise connections to high-level ab
initio methods. Working within this novel formal structure we explore computational
aspects that exploit modern computer architectures while maintaining a desired level of
accuracy.
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CHAPTER 1INTRODUCTION AND BACKGROUND
The underlying physical laws for the mathematical theory of a large partof physics and the whole of chemistry are thus completely known, and thedifficulty is only that the exact application of these laws leads to equationsmuch too complicated to be soluble. — P.A.M. Dirac
1.1 Who Ordered a New Semiempirical Theory?
Available quantum chemical methods are incapable of achieving so-called chemical
accuracy (1 to 2 kcal/mol for bond energies) for a system composed of thousands of
atoms within a computationally efficient framework (approximately one day on a typical
desktop machine). A computational method with this capability would revolutionize the
way that matter is studied in biology, chemistry and physics. In the field of quantum
chemistry, there are many unexploited connections among wavefunction, density functional
and semiempirical theories. Use of such connections will aid in the development of
quantum chemical methods with the desired accuracy and computational efficiency. The
ultimate goal then is to provide scientists with a computational tool that makes materials
simulations predictive, chemically specific, and applicable to a wide variety of mechanical
and optical properties of realistic, complex systems.
Though high-level quantum methods approach the desired accuracy, the computational
scaling of such methods precludes their use for systems larger than a tens of atoms (e.g.,
coupled-cluster with singles and doubles (CCSD) scales O(N6), where N is the number
of basis functions). In contrast, the computational scaling of semiempirical (SE) methods
is acceptable (typically scales O(N3)), though they can be made to scale linearly with N .
For instance, if the size of a system is increased by an order of magnitude then a CCSD
calculation would incur a factor of 106 more computational time. For the same increase
in system size, the computational time of a normal SE method would only increase by
a factor of a thousand, but with linear scaling only by ten. In the regime of very large
systems (thousands of atoms) it is apparent that a calculation that could be performed in
one day using SE methods might take several years for CCSD. Unfortunately, SE methods
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or, better, simplified methods (SM) are not yet sufficiently reliable, although they do
provide an excellent opportunity for improvement. There are two general approaches to
improve SE methods: first, existing reference data can be expanded to include additional
systems and properties or improved fitting procedures can be used to help locate new
minima in the model Hamiltonian’s parameter space; second, more rigorous model
Hamiltonians can be derived to satisfy more demanding desiderata, such as uniform
accuracy of the energy and forces across the entire potential energy surfaces (PES) and
improved electronic densities, ionization potentials, and associated first-order properties.
The first route is a mainstay of SE method development, though we will show that the
second SM strategy has the potential to be more valuable in the long run. To this end,
a new SM is discussed that incorporates rigorous formal connections to high-level ab
initio methods while retaining computational features of commonly used SE methods. An
important computational aspect of existing SE methods is that their speed is dictated by
the solution of a set of one-particle eigenvalue equations (typically matrix diagonalization)
and not by the construction of those one-particle equations or any ensuing step. This
computational restriction, in addition to a modicum of accuracy, guides the development
of the method under discussion, adapted ab initio theory (AAT). AAT not only provides
a general theoretical SM framework into which existing SE methods can fit, but also has
the advantage that there are well-defined connections between it and high-level ab initio
theories. This novel framework not only allows for improvements to existing SE methods
that enhance transferability and extendibility, but also has the flexibility needed for more
systematic improvement in future versions of SE methodology.
Before we begin the detailed description of our quantum mechanical method the
need for a fast and accurate method to describe the properties of large molecules will be
demonstrated. Phenomena that involve bond-breaking or bond-forming often necessitate
a detailed knowledge of the electronic structure and the PES. Such quantum chemical
descriptions are important in many complex chemical processes; for example:
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1. Gas phase studies.• Plasmon absorption of organometallic nanoparticles.• Degradation pathways of chemical warfare agents.• Environmentally safe green explosives.
2. Condensed matter studies.• Defects in semiconductors.• Crack-tip propagation and hydrolytic weakening of silica.• Bulk tensile properties of silica polymorphs.
3. Biological studies.• Bioinformatics: fragmentation patterns of peptides for proteomics.• Biocatalysis: enzymatic reactions on oligosaccharides for biofuel generation.• Photoreceptors such as rhodopsin.• Drug design.
Quantum chemical methods are used to some extent in the analysis of these types
of problems. The ideal quantum chemical method could be applied to any size system
and achieve any desired accuracy, while using minimal computational resources, but, of
course, this cannot be achieved. Hence, there are often practical limitations dictated by
the desired accuracy and the available computational resources. Our goal is to create a
method with chemical accuracy that can be applied to systems with thousands of atoms.
A method with these features would be able to readily provide the quantum mechanical
contribution to the solution of the systems mentioned.
The two main approaches in quantum chemistry are wavefunction theory (WFT) and
density functional theory (DFT). Likewise, there are two commonly used SE techniques:
neglect of diatomic differential overlap (NDDO) methods, which originated in WFT, and
density-functional tight-binding (DFTB), including its self-consistent-charge extension
(SCC-DFTB), which originated in DFT. Both NDDO and SCC-DFTB can be considered
semiempirical since they both approximate the formal structure of the original methods
and are parameterized to reproduce either experimental or theoretical reference data.
Just as the formal connections between WFT and DFT have recently been uncovered
and exploited, via ab initio DFT [1], there are similar formal connections between their
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semiempirical counterparts: NDDO and SCC-DFTB. These connections form the basis of
AAT, which is a SM rather than a SE method.
Existing SE methods are widely used and, when applied judiciously, perform quite
well. Regrettably, they are often used in situations for which the results are not reliable,
such as locating transition states. To expand the applicability of SE methods, we combine
their primary features with those from rigorous WFT or DFT, and define the new AAT
framework. Hence, AAT encompasses both NDDO and SCC-DFTB.
Despite their formal deficiencies, both NDDO and SCC-DFTB have been quite
successful in a variety of applications [2]. Nonetheless, there remain several aspects
of these SE methods that fall short of expectation that could be greatly improved. In
addition to obvious formal limitations, existing SE methods were designed to reproduce
the energetics of systems at equilibrium and often yield unphysical results near transition
states and along dissociation pathways. Uniform accuracy of the PES is critical for
molecular dynamics (MD) simulations, for determining the lowest energy (global
minimum) structure, and for calculating rate constants. Associated with these failings
are unrealistic electronic densities that, in turn, make combining these methods into
multi-scale modeling schemes difficult. Furthermore, errors in first-order properties such
as the dipole and higher-order moments would be remedied by a method that delivers
adequate electronic densities. Attempts to improve such properties have met with modest
success, typically at the expense of other aspects of the model.
By far, the most common route to incorporate such additional features is to
reparameterize the model. Unfortunately, building in accuracy for non-equilibrium
structures, excited states, and other properties, by simple reparameterization of existing
NDDO-based SE methods [3] is not readily achievable, at least not in a way that
maintains transferability (similar accuracy regardless of bonding characteristics or system
size). This difficulty highlights an important distinction between the current AAT route of
designing a new simplified model Hamiltonian based on rigorous ab initio principles versus
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attempting to extend existing model Hamiltonians to include features that the model was
not originally intended to capture. To avoid such complications the AAT SM is primarily
designed to yield accurate electronic structure (electronic density) and, secondarily, the
total energy and forces. Excited and ionized states are of interest. Clearly, the electronic
structure and the energy of a molecule are closely connected. Consequently, the distinction
between the two is subtle, nevertheless, their separation is crucial in order to ensure that
the AAT achieves uniform accuracy for energies, forces and first-order properties over an
entire PES. The general computational approach is as follows:
• Approximate four-center terms with a sum of two-center terms using the Rudenbergapproximation [4, 5].
• Determine exchange-correlation potentials to guarantee the exact result at theseparated-atom limit, starting from a sum of atom-based densities.
• Use cubic splines to interpolate these two-center potential terms (a lookup table ofsplines need only be generated once and is very fast).
• Solve a set of one-particle equations with explicit inclusion of orbital overlap.
• Include explicit exact (HF-like/ non-local) exchange.
• At convergence of self-consistency, make a final energy correction by inserting thedensity into a DFT functional (B3LYP) and performing a single numeric integrationfor the exchange-correlation energy.
An important computational consideration is the inclusion of the overlap between
orbitals, which is a feature of AAT that distinguishes it from standard NDDO methods.
The zero-differential overlap approximation, which is the fundamental approximation of
NDDO, implies that there is no overlap between orbitals. Though this approximation
enhances the speed of the method, it neglects an important physical feature of electrons
and the basis sets used to describe them. There have been attempts to reintroduce such
overlap effects in NDDO methods, most notably Thiel’s orthogonalization correction
methods (OM1 and OM2) [6]. Still, Elstner’s development of SCC-DFTB [7] demonstrates
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that one can include overlap explicitly and still retain a computationally efficient
framework. Thus, AAT explicitly includes the overlap between atomic orbitals.
The predominant computational aspects of the method have been highlighted; now,
consider the formal development of the AAT approach. There is an exact reference to
the exchange-correlation potential with which we can directly compare. The electronic
density can be corrected iteratively by an inverse procedure, such as that of Zhao,
Morrison, and Parr [8] or that of van Leeuwen and Baerends [9]. These procedures
take a high-quality reference density and generate the numerical exchange-correlation
potential that best reproduces it. In the limit that the reference density is exact, such
procedures produce an exact local exchange-correlation potential. This exact limit
defines a set of one-particle equations, which in turn, define the first-order electronic
energy. When added to the nuclear-nuclear repulsive energy, the electronic energy gives
the total energy correct through first-order in the trial wavefunction. In existing SE
methods, the remaining higher-order energy terms are combined with the nuclear-nuclear
repulsion to give a parameterized atom-pair based total energy correction that would be
predominantly nuclear repulsion. Instead, we choose to retain a separate nuclear-nuclear
repulsion contribution to ensure that every aspect of our model has a direct one-to-one
correspondence to ab initio theory. The connections to the exact limits for both density
and total energy components of the overall parameterization provide the necessary
structure so that every approximation of the model can be rigorously verified.
1.2 Quantum Chemistry
Electronic structure theory describes the distribution of electrons around nuclei in
atoms, molecules, and solids. Many observable features of a molecule are given by its
electronic structure, though computational efficiency requires approximations that can
limit the accuracy with which we can describe some chemical phenomena. There is a series
of standard approximations that permeate quantum chemistry that restrict its universal
applicability, but that are valid for many cases of interest:
15
• Time-independent Schrodinger equation.
• Born-Oppenheimer approximation.
• Nonrelativistic Hamiltonian.
• Linear combination of atomic orbitals.
The time-independent Schrodinger equation provides the solutions needed for the
time-dependent Schrodinger equation, and though some problems require a time-dependent
solution, the vast majority of problems in quantum chemistry can be treated adequately
with the time-independent Schrodinger equation. The Born-Oppenheimer approximation
is based on the simple observation that electrons move much faster than nuclei. This
difference in speed effectively implies nuclear motion cannot affect the electronic
structure significantly and we can solve the Schrodinger equation for a set of fixed nuclear
coordinates. For example in the hydrogen atom the electron is traveling at about two
million meters per second, which is orders of magnitude faster than the motion of the
nucleus. Also, this speed is less than 1% of the speed of light, which also implies that we
need not concern ourselves with relativstic effects that would only become important for
atoms with heavier nuclei (typically those with atomic number larger than 25). For such
heavy atoms the speed of an electron near the nucleus will approach the speed of light and
would require a relativistic Hamiltonian for an accurate description. In such cases effective
core potentials that approximate the passive (Darwin and mass-velocity) relativistic effects
can be used, which still allows one to keep the nonrelativistic Hamiltonian framework.
Finally, the linear combination of atomic orbitals (LCAO) approximation introduces basis
sets into the problem and gives a general framework for systematically expanding the
reference space of the orbitals that the electrons may occupy.
The time-independent Schrodinger equation within the Born-Oppenheimer (clamped-
nucleus) approximation is
HΨ = EΨ (1–1)
16
where E is the energy of the system, Ψ is the wavefunction of the electrons, and the
nonrelativistic Hamiltonian in Hartree atomic units is
H =− 1
2
∑i
∇2i −
∑A,i
ZA
rAi
+∑i>j
1
rij
+∑A>B
ZAZB
RAB
=∑
i
h(i) +∑i>j
1
rij
+ VNN
(1–2)
with i and j referring to electrons and A and B to nuclei. The terms in Equation 1–2
are the operators that correspond to the kinetic energy of the electrons, electron-nuclear
attraction, electron-electron repulsion, and nuclear-nuclear repulsion. The wavefunction
Ψ describes the state of our quantum system. We choose to approximate Ψ within a basis
set of atomic orbitals (AOs). Enforcing antisymmetry and satisfying the Pauli exclusion
principle can be achieved with an antisymmetric product of orthonormalized AOs within a
Slater determinant, the LCAO approximation
|ψ〉 = |χ1(r1)χ2(r2) · · ·χn(rn)〉 (1–3)
where n is the number of electrons and the spin orbital χ(r1) = φ(r1)α, β and α, β denote
spin and φ(r) is a spatial orbital.
1.3 Hartree-Fock
Focusing on the ground electronic state, the most widely known approximate solution
of the electronic Schrodinger equation is made by a mean-field approximation: HF theory.
That is, we insist that a zeroth-order approximation to the n-electron wavefunction (Φ0)
be written as an antisymmetrized product of n one-electron molecular spin-orbitals,
Φ0 = A[φ1(1)φ2(2)...φn(n)]. Here A is the antisymmetrization operator. Then using the
Raleigh-Ritz variational principle, EHF ≥ Eexact, we derive the one-particle HF equations
which yield the energetically best possible single determinant wavefunction. Each of the
orbitals in its canonical form is an eigen-function of the effective one-particle operator, f .
It is fitting to start with the most well-known approximation in electronic structure
theory. HF is an excellent touchstone since correlation energy is defined by it (Ecorrelation =
17
Eexact − EHF ) and because many quantum chemists have developed an intuitive
understanding of it. Also, HF provides a convenient way to introduce notation that will be
used throughout the current work. The one-electron operator of the full Hamiltonian is
h(1)|χi〉 = (−1
2∇2
1 −∑
A
ZA
rA1
)|χi〉 (1–4)
and the two-electron operators are Coulomb J and exchange K,
J(1)j|χi〉 =
∫ |χj(r2)|2r12
dr2|χi〉 (1–5)
K(1)j|χi〉 =∑
j 6=i
∫χ∗j(r2)χi(r2)
r12
dr2|χj〉 (1–6)
and
f(1)|χi〉 = [h(r1) + (J(1)− K(1))]|χi〉 = εi|χi〉 (1–7)
denoting the molecular spin orbital χi as a linear combination of atomic spin orbitals χi
χi(r1) =∑
µ
Cµiχµ(r1) (1–8)
Using Lagrange’s method of undetermined multipliers and enforcing orthonormality of the
orbitals gives a set of one-particle matrix equations, the Hartree-Fock-Roothaan equations:
FC = SCε (1–9)
where
Sµν =
∫χµ(r1)χν(r1)dr1 (1–10)
and
Fµν =
∫χµ(r1)f(r1)χν(r1)dr1 (1–11)
The construction of a Fock-like (or effective one-particle) matrix is a central theme of this
work. A Fock-like matrix is an orbital representation of an effective one-particle operator,
18
such as found in HF or DFT. It is also possible to define others, as in the correlated
orbital potential (COP) method [10, 11].
HF illustrates many of the necessary features of a one-particle theory. One distinct
feature, which sets HF apart from most other one-particle theories, is that HF enforces the
Pauli exclusion principle exactly via the exchange operator. Such exact exchange terms
are missing in Kohn-Sham density functional theory [12] which leads to the so-called self-
interaction error. Another benefit of HF is that there is Koopmans’ theorem that gives
meaning to the eigenvalues of the Fock matrix by relating them to ionization potentials
and electron affinities. However, the exact DFT structure can provide a formally correct
one-particle theory that includes electron correlation, which HF cannot do because by
definition it does not include electron correlation. COP is another exact alternative that
can be defined from WFT.
1.4 Coupled-Cluster Theory
The introduction of electron correlation in wavefunction theory can be achieved by
generalizing the mean-field wavefunction to allow electrons to be excited into the orbitals
that are unoccupied (virtual, i.e., do not appear in the Hartree-Fock determinant) orbitals,
a,b,c, . . . . Those additional orbitals allow the electrons to avoid each other in a more
detailed way (hence lower the energy) than in the mean-field wavefunction. That is, for
single excitations (replacements), double excitations, triple excitations, etc., we define
excitation operators Tj, j = 1, 2, 3, . . .
T1Φ0 =∑i,a
tai Φai (1–12)
T2Φ0 =∑
i<j,a<b
tabij Φab
ij (1–13)
T3Φ0 =∑
i<j<k,a<b<c
tabcijkΦ
abcijk (1–14)
...
19
with the excited determinants, Φabij , meaning the antisymmetrized product,
Φabij = A[φ1(1)φ2(2)..φa(i)..φb(j)...φn(n)], (1–15)
where unoccupied orbitals a and b replace the occupied ones, i and j. The coefficient
(amplitude) in front of the excitation introduces its proper weight into the wavefunction.
The ultimate version of such a sequence would include all possible excitations up to n-fold
for n electrons and is called the full configuration interaction or full CI. Full CI is:
• The best possible solution in a given basis set.
• Invariant to all unitary transformations (rotations) of the orbitals.
• Size-extensive [13] (meaning that it scales properly with the number of electrons.)
• An upper bound to the exact energy for a molecular electronic state.
The full CI can be written conveniently in the coupled-cluster (CC) form,
Ψ = exp(T1 + T2 + T3 + ... + Tn)Φ0. (1–16)
[13–16]. Approximations to CC theory are made by restricting the Tp operators to singles
and doubles, as in CCSD, or to other approximations that decouple higher-excitations
from lower ones, e.g., CCSD(T). Here the triple excitations are approximated from T1 and
T2, which can be obtained from CCSD, e.g., without allowing the new T3 estimate then to
contribute back to T1 and T2. The details of these methods, which can be found elsewhere
[17], are not important to the present discussion. What is important is that all truncated
CC methods are size-extensive, but do not necessarily give upper bounds to the exact
energy.
The full CC equations provide the exact electronic energy,
ECC = 〈Φ0|H|Φ0〉 = EHF +∑
i<j,a<b
〈ij||ab〉(tabij + tai t
bj − taj t
ai ) +
∑i,a
〈i|f |a〉tai (1–17)
20
where the two-electron and one-electron integrals are, respectively,
〈pq||rs〉 =
∫φ∗p(1)φ∗q(2)(1− P12)r
−112 φr(1)φs(1)d1d2 (1–18)
〈i|f |a〉 =
∫φ∗i (1)f(1)φa(1)d1 (1–19)
and P12 is the operator that permutes electrons 1 and 2. The amplitudes, {tai , tabij , · · · }
come from projection of exp(−T )H exp(T ) = H onto single, double, etc. excitations,
〈Φai |H|Φ0〉 = 0 (1–20)
〈Φabij |H|Φ0〉 = 0 (1–21)
. . .
The CC equations are a set of complicated, coupled non-linear equations. Unlike the
truncated CI equations, even when the CC equations are restricted to some subset of
excitations, like singles and doubles (CCSD), they still guarantee a size-extensive result.
They converge much more rapidly to the full CI reference solution than does the sequence
of partial sums of the CI expansion itself. For these reasons coupled-cluster has become
the method of choice in high-accuracy quantum chemical applications for small molecules
[17–19].
CC theory provides a route to exact reference data. It is well-known that the
hierarchy of CC methods are systematically improvable, and lead to the full CI solution
for a particular basis set. Hence, for small molecules we have ready access to reference
energies, forces, densities, and electronic properties to which we can compare directly.
The wealth of information that is afforded by a high-level ab initio calculation is superior
to the typical amount of information available from experiment and provides unique
insight into how well an approximation is working. For instance, besides getting energetic
information about a molecule we simultaneously have access to the electronic density.
This increased level of control, when compared to experimental reference data, allows for
systematically verifiable approximations of our effective one-particle Hamiltonian. The
21
ability to verify every approximation systematically puts the current work into sharp
relief from current semiempirical methods that depend intrinsically upon uncontrolled
approximations.
1.5 Kohn-Sham Density Functional Theory
The Kohn-Sham [12] equations may be written as
fKS|χi〉 = εi|χi〉 (1–22)
or,
[−1
2∇2 + νext(r) + νJ(r) + νxc(r)]|χi〉 = εi|χi〉 (1–23)
where
νext(r) = −∑
A
ZA
|RA − r| , (1–24)
νJ(r1) =
∫ρ(r2)
r12
dr2, (1–25)
ρ(r) =∑
i
|φi(r)|2, (1–26)
and νxc(r) is the exchange-correlation potential. Constrained-search minimization of the
noninteracting kinetic energy alone determines orbitals that yield the ground state density
[20]. The other terms do not affect the minimization because they are explicit functionals
of the density.
The Hohenberg-Kohn-Sham theorems guarantee that the density obtained from the
exact ground state wavefunction will be returned by Equation 1–26 when using the exact
exchange-correlation potential. In a pragmatic sense, these arguments give a formal proof
of existence, though they do not guide the construction of the exact one-particle operator,
or, more specifically, the exchange-correlation potential. However, in combination with
so-called inverse procedures, such as that of Zhao, Morrison and Parr (ZMP) [21–23],
which have been explored by Tozer et al. [24], or the procedure by Peirs et al. [25], the
potential may be refined iteratively such that the solutions to Equation 1–23 will generate
a reference density, presumably a high-quality ab initio reference density.
22
This is only part of the overall problem. In addition to the purely electronic
properties, we also want total energies and forces. The total energy in KS-DFT can
be written,
EKSelec =
occ∑i
εi − 1
2
∫ ∫ρ(r)ρ(r′)|r − r′| drdr′ −
∫vxc(r)ρ(r)dr + Exc[ρ] (1–27)
where the exchange-correlation potential is the functional derivative of the exchange-
correlation energy functional
vxc(r) =δExc[ρ]
δρ(r). (1–28)
Unfortunately, currently there is no procedure that allows us to take a numerically
derived exchange-correlation potential, one that is determined such that it reproduces a
high-quality reference density, and generate the exchange-correlation functional, which is
an explicit functional of the density. This inability to determine the exchange-correlation
energy functional from a reference density alone is the first indication of a fundamental
problem in semiempirical methods, which will be discussed in further detail later: how
does one design a model Hamiltonian that will simultaneously achieve the correct result
for both electronic properties and energetics? However, if we are willing to assume the
form of a particular exchange-correlation energy functional then the potential is well
defined, as is the effective one-particle Hamiltonian.
1.6 Huckel Method
The Huckel π-electron method is perhaps the simplest quantum chemical approximation.
It is intended to highlight the basic underlying effects of symmetry involved with π
electrons for alternant hydrocarbons, though it can be made more general. In relation
to HF, the same type of one-particle matrix equations are solved, though with the
overlap matrix set to the identity matrix and the Fock-like matrix is composed of only a
one-electron (density independent) contribution, F = H
HC = Cε (1–29)
23
The total energy is
E =∑
i
niεi (1–30)
where ni is the occupation number for the ith MO. Note there is no explicit consideration
of the nuclear repulsion in Equation 1–30. The Huckel approximations are
Hij =
α if i = j
β if i are nearest neighbors j
0 otherwise.
(1–31)
The extended Huckel method [26], as its name implies, is merely an extension of the
standard Huckel method with the same total energy formula. The full matrix equation
HC = SCε is solved, with overlap explicitly included. And the Hamiltonian is subject to a
slightly more sophisticated approximation:
Hij =
α if i = j
KSij(Hii + Hjj) if i and j are nearest neighbors
0 otherwise
(1–32)
where Hii is based on the ith AO of an isolated atom and K is a dimensionless adjustable
parameter for an atom-pair. Hoffmann suggested a value for K based on its effect on the
energy of ethane (C2H6). He argued that, since the energy dependence on K is nearly
linear beyond K = 1.5, the MO energies and, in turn, the total energy, will not be too
sensitive to K for values greater than that. He then considered the energy difference
between the staggered and eclipsed conformations of ethane and optimized the value of
K to minimize the error between calculated and experimental values. The final value he
arrived at was K = 1.75, a value is still in common use.
The important aspect to note about the Huckel method, as it pertains to the
development of improved semiempirical methods, is that a very simple model can
reproduce a few features qualitatively well. The spatial symmetry is correct due to the
inclusion of the overlap matrix. However, despite the ability of such basic approximations
24
to capture a large fraction of the total energy, more sophisticated descriptions are needed
to guarantee chemical accuracy.
1.7 Neglect of Diatomic Differential Overlap
Neglect of diatomic differential overlap (NDDO) is the model that most people
associate with semiempirical theory in quantum chemistry today. The NDDO approach
is meant to approximate the HF solution, not the exact solution, though its parameters
can be varied to minimize error to reference data [27–32]. It is a minimal basis set,
valence-only approach. NDDO is based on the zero-differential overlap (ZDO) approximation:
products of orbitals with respect the same electron coordinate that are on different centers
are defined to be zero,
χAµ (1)χB
ν (1) = 0. (1–33)
The orbitals are considered to be Lowdin symmetrically orthogonalized AOs, so that
the overlap matrix is the identity matrix, and the set of one-particle matrix equations
solved is FNDDOC = Cε. The Fock-like matrix in NDDO contains one-electron (H) and
two-electron (G) components. The one-electron part is approximated as
Hµν =
Uµµ −∑
B 6=A ZB(µµ|BB) if µ = ν
12Sµν(βµ + βν) if µ 6= ν
(1–34)
where µ is on atom A, ν is on atom B, and Uµµ and βµ are adjustable parameters,
(µµ|BB) models the repulsion between a product of orbitals (µ) and a point charge
at center B, (〈µ(1)| 1r1B|µ(1)〉). Note that the two-center one-electron terms include
overlap and are therefore not consistent with the ZDO approximation. The one-center
two-electron integrals are given in Table 1-1. The parameters Gss, Gpp, Hsp, Gpp, and Gp2
are adjustable.
Within a local framework there are twenty-two unique two-center NDDO-type
two-electron integrals for each atom pair (Table 1-2). The NDDO two-center two-electron
integrals are approximated by a multipole-multipole expansion [33]. Each product of
25
orbitals is assigned an approximate charge distribution (Table 1-3). Adjustable parameters
arise from the distances between the point charges used in the multipole expansion and
by enforcing the correct limit as R → 0 where the equivalent one-center term should be
reproduced.
As mentioned above, originally the NDDO Fock-like matrix was designed to model
the HF one-particle Hamiltonian. The HF solution neglects all electron correlation
effects and on the scale of chemical accuracy one should not a priori expect the NDDO
approximation to reproduce experimental results. However, because some parameters are
adjustable and fit to experimental data the typical argument is that electron correlation
is implicitly included. In fact, it is clear that electron correlation is no more implicit in
the NDDO model than are relativistic effects, time-dependence, non-Born-Oppenheimer
effects, basis set incompleteness, and experimental errors in the reference data. Despite
such formal gaps, the NDDO method has enjoyed a tremendous amount of success and
reproduces heats of formation and other properties with surprising accuracy. The difficulty
arises when better accuracy is required in situations for which the model was not originally
intended to describe, such as transition states or other nonequilibrium structures. In such
cases one quickly approaches a point of diminishing returns for parameterizing to new
reference data. The precise origin of these limitations is explored later.
1.8 Tight-Binding
The basic equation for the total energy in tight-binding (TB) theory is assumed to be
of the form,
ETB =n∑
i=1
εi +1
2
∑
A6=B
U(|RA −RB|). (1–35)
where the εi’s are the eigenvalues of
heff |i〉 = (−1
2∇2 + vext)|i〉 = εi|i〉, (1–36)
vext is the nuclear-electron Coulomb attraction term and U(|RA − RB|) is considered to
represent all the remaining terms. In total, the latter constitute a repulsive function of the
26
set {RAB} between atoms A and B. For the one-particle Hamiltonian, the usual LCAO
approximation is made and the φi’s are expanded in a set of χµ’s giving the equation,
HeffC = SCε (1–37)
where
(Heff )µν = 〈µ|heff |ν〉 (1–38)
and
Sµν = 〈µ|ν〉. (1–39)
In practice, these two-center matrix elements are evaluated using the Slater-Koster
parameterization [34]. The TB approximation has been applied successfully to many solid
state problems and works especially well when the density of the chemical system is well
approximated as a sum of spherically symmetric atomic densities. In the case of molecules,
a more advanced approach that “treats the delicate balance of charge” [7] is needed. If we
compare TB to HF, we see that U(|RA −RB|) has to account for J − K and VNN . In DFT
it would account for J , VNN , and would combine exchange with correlation by adding vxc.
1.9 Density-Functional Tight-Binding
To vest the TB energy expression with more rigor, Foulkes and Haydock [35] derived
it from first-principles DFT considerations. The energy expression in KS-DFT, Equation
1–27, may be written as
EDFT =occ∑i
〈φi| − ∇2
2+ vext +
1
2
∫ ′ ρ(r′)|r − r′| |φi〉+ Exc[ρ(r)] +
1
2
N∑A,B
ZAZB
|RA −RB| . (1–40)
Equation 1–40 is exact if ρ =∑occ
i |φ2i |, where ρ is the exact density. However, it should
be noted that φi are eigensolutions of hs = (−∇2
2+ vext +
∫ ′ ρ′0|r−r′| + vxc[ρ0]) and not
(−∇2
2+ vext + 1
2
∫ ′ ρ(r′)|r−r′|). Following Foulkes and Haydock, the density ρ is replaced by a
reference density ρ0 and a small deviation δρ, such that ρ = ρ0 + δρ. Then we can rewrite
27
the energy expression to be
EDFT =occ∑i
〈φi| − ∇2
2+ vext +
∫ ′ ρ′0|r − r′| + vxc[ρ0]|φi〉
−1
2
∫ ∫ ′ ρ′0(ρ0 + δρ)
|r − r′| −∫
Vxc[ρ0](ρ0 + δρ) (1–41)
+1
2
∫ ∫ ′ δρ′(ρ0 + δρ)
|r − r′| + Exc[ρ0 + δρ] +1
2
N∑A,B
ZAZB
|RA −RB|
to enable us to introduce a form dependent upon the one-particle Hamiltonian, hs, plus its
corrections. The first new term accounts for the double-counting of the Coulomb potential,
the next accounts for the addition of vxc in the first term, and the third new term is the
remaining part of the Coulomb potential. Next, the exchange-correlation contribution
Exc is expanded about the reference density, ρ0, with a Volterra (Taylor) expansion for
functionals,
Exc[ρ0 + δρ] = Exc[ρ0] +
∫νxc[ρ0]δρ +
1
2
∫ ∫ ′ δ2Exc
δρδρ′
∣∣∣∣ρ0
δρδρ′
+1
6
∫ ∫ ′ ∫ ′′ δ3Exc
δρδρ′δρ′′
∣∣∣∣ρ0
δρδρ′δρ′′ + · · · (1–42)
Substitution of this expression for Exc into the energy equation yields an expression for the
energy that is formally exact, yet only includes terms that are second-order and higher in
δρ.
EDFT =occ∑i
〈φi| − ∇2
2+ vext +
∫ ′ ρ′0|r − r′| + vxc[ρ0]|φi〉
− 1
2
∫ ∫ ′ ρ′0ρ0
|r − r′| + Exc[ρ0] +1
2
N∑A,B
ZAZB
|RA −RB| −∫
vxc[ρ0](ρ0)
+1
2
∫ ∫ ′ δρ′δρ|r − r′| +
1
2
∫ ∫ ′ δ2Exc
δρδρ′
∣∣∣∣ρ0
δρδρ′ (1–43)
+1
6
∫ ∫ ′ ∫ ′′ δ3Exc
δρδρ′δρ′′
∣∣∣∣ρ0
δρδρ′δρ′′ + · · ·
28
All of the terms in this expression that are independent of δρ comprise the standard
TB equation. Hence,
U(|RA −RB|) = −1
2
∫ ∫ ′ ρ′0ρ0
|r − r′| −∫
vxc[ρ0](ρ0) + Exc[ρ0] +1
2
N∑A,B
ZAZB
|RA −RB| (1–44)
which obviously is correct to second order in ρ. Note the inclusion of nuclear-nuclear
repulsion in this term.
One of the great advantages of DFT approaches is that they offer a reasonable
zeroth-order approximation to the energy simply from knowing some approximation to the
reference density, ρ0. However, since Exc[ρ] is not known, getting the exchange-correlation
potential, vxc(r) = δExc/δρ(r), and its kernel, fxc(r, r′) = δ2Exc/δρ(r)δρ(r′), from first
principles, which are required to set up a rigorous DFT, is difficult [1, 36]. However, were
we to accept one of the plethora of approximations to Exc (LDA, LYP, PBE, B3LYP, B88,
etc.) the equations can be solved for a few hundred atoms, but not on a time-scale that
can be tied to MD for thousands of atoms.
Furthermore, to provide a correct description of bond breaking the electronic charge
has to be free to relocate. A common example is LiF, in which at R = Req we have
virtually an exact Li+F− form, but at separation in a vacuum, we must have Li0F0.
Such self-consistency usually is introduced by solving the AO-based DFT equations via
diagonalization, but this approach imposes an iterative step that scales as N3 to the
procedure. Since the model is meant to be approximate anyway, the alternative charge
self-consistency approach, familiar from iterative extended Huckel theory [37, 38], can be
used more efficiently. This extension leads to the self-consistent-charge density-functional
tight-binding (SCC-DFTB) approach of Elstner et al. [7].
1.10 Self-Consistent-Charge Density-Functional Tight-Binding
To introduce charge self-consistency, the terms that are second-order in δρ are
approximated as a short-range Coulombic interaction based upon Mulliken atomic charge
29
definitions,
E2nd =1
2
∫ ∫ ′(
δρ′δρ|r − r′| +
δ2Exc
δρδρ′
∣∣∣∣ρ0
)δρδρ′ ≈ 1
2
N∑A,B
∆qA∆qBγAB. (1–45)
The ∆qA are the differences between some reference charge, q0A, and the charge on atom A,
qA =occ∑i
∑µ∈A
∑ν
cµicνiSµν (1–46)
γAB is a pair-potential, whose common functional forms are taken from traditional
semiempirical quantum chemistry schemes such as those due to Klopman [39], Ohno [40],
and Mataga-Nishimoto [41]. For example, the latter form is
γAB = 1/[2(γA + γB)−1 + RAB] (1–47)
where the single index γA is meant to be a purely atomic two-electron repulsion term.
These are typically chosen as fixed atomic parameters in the calculation. For an
assessment of how the SCC-DFTB model performs compared to standard semiempirical
quantum chemical methods, see Morokuma, et al. [42]. For example, unlike SCC-DFTB,
standard AM1 does not give the most and least stable isomers of C28 when compared
to DFT calculations with the B3LYP Exc and a 6-31G(d) basis set. In contrast, the
relative energies of the isomers of C28 with B3LYP/6-31G(d)//SCC-DFTB are in very
good agreement with B3LYP/6-31G(d), having a linear regression R2 coefficient of 0.9925.
Though the geometries generated by SCC-DFTB are excellent, the SCC-DFTB relative
energies are poor, with a linear regression R2 of 0.7571.
SCC-DFTB is a fairly rigorous semiempirical method. The framework provided by
Foulkes and Haydock exposed how higher-order terms could be included. The practical
implementation of a well-behaved approximation of higher-order was subsequently
provided by Elstner. Later, we will develop the SCC-DFTB formalism from an MO
viewpoint to facilitate the comparison between it and other methods.
30
We have discussed the basic features of HF, CC, KS-DFT, Huckel, NDDO, TB,
DFTB, and SCC-DFTB, and have alluded to the role each plays in the development
of an improved semiempirical method. We take two approaches to develop this target
method. First, we demonstrate the transfer Hamiltonian, which provides an exact
one-particle framework in WFT. Second, we develop adapted ab initio theory which
provides rigorously verifiable approximations to reproduce ab initio reference data.
31
Table 1-1. The NDDO one-center two-electron integrals
Parameter Two-electron integralGss (ss|ss)Gsp (ss|pp)Hsp (sp|sp)Gpp (pp|pp)Gp2 (pp|p′p′)
32
Table 1-2. The NDDO two-center two-electron integrals
Term Term1 (ss|ss) 12 (spσ|pπpπ)2 (ss|pπpπ) 13 (spσ|pσpσ)3 (ss|pσpσ) 14 (ss|spσ)4 (pπpπ|ss) 15 (pπpπ|spσ)5 (pσpσ|ss) 16 (pσpσ|spσ)6 (pπpπ|pπpπ) 17 (spπ|spπ)7 (pπpπ|pπ′pπ′) 18 (spσ|spσ)8 (pπpπ|pσpσ) 19 (spπ|pπpσ)9 (pσpσ|pπpπ) 20 (pπpσ|spπ)10 (pσpσ|pσpσ) 21 (pπpσ|pπpσ)11 (spσ|ss) 22 (pπpπ′|pπpπ′)
33
Table 1-3. Multipole distributions
Term Charge distribution Point charges(ss| Monopole 1(sp| Dipole 2(pp| Linear Quadrapole 4(pp′| Square Quadrapole 4
34
CHAPTER 2THE TRANSFER HAMILTONIAN
2.1 Theoretical Method
Historically SE method development has been guided by experiment, the goal being
to reproduce a set of experimental quantities. While any QM method should in some
limit reproduce experimentally observed phenomena, the exact WFT imposes more
demanding and detailed requirements on a model than experiment alone. For instance,
the wavefunction is not an observable yet contains all the information about the system.
In this sense a model based in WFT aspires to reproduce the exact wavefunction and the
effective Hamiltonian that generates it. The expectation value of this exact wavefunction
for a particular Hermitian operator will yield a particular experimental observable.
High-quality ab initio reference data is readily available from CC theory and provides an
unambiguous point of reference for our model. Furthermore, by reproducing the features of
an exact WFT we are guaranteed to reproduce all experimental phenomena.
However, practical limitations restrict the functional form of the wavefunction and
the Hamiltonian. The challenge is to incorporate the many-body effects of exact WFT
into a simple (low-rank) model Hamiltonian. One aspect of the approach taken here is to
develop a formally exact one-particle theory, the transfer Hamiltonian (Th), which includes
all many-particle effects. Another aspect is to connect a simple approximate model
Hamiltonian to the exact Th. Our initial work used the well-known NDDO-type model
Hamiltonian. Because these models have limited applicability, it has become evident that
more complete models are needed that accurately incorporate the many-particle effects
of the Th. We develop the AAT simplified approach to include the features of the Th in a
systematic way, which is discussed in detail later.
Here we introduce the Th [43], a one-particle operator with a correlation contribution,
as an extension of the similarity transformed Hamiltonian from CC theory [44]. This
approach provides a rigorous formal framework for an exact one-particle WFT. Also,
unlike other methods, such as the specific reaction coordinate approach of Truhlar [45],
35
which attempt to fit a known energy surface to a SE form via reparameterization, the Th
fits to the Hamiltonian itself, to provide a form that is simple enough that it can be solved
on a time scale consistent with large scale MD.
We start from the exponential ansatz in which the exact ground state wavefunction is
given by
|Ψg〉 = eT |Φ0〉 (2–1)
where Φ0 is a reference function, which can be taken as a single determinant and T is the
excitation operator of CC theory. When defined this way, we may write the Schrodinger
equation in terms of the similarity transformed Hamiltonian, H.
H|Φ0〉 = e−T HeT |Φ0〉 = E|Φ0〉. (2–2)
The solution of this equation has the same eigenvalues as the bare Hamiltonian. The total
energy, E, is exact, thus the forces, ∂E∂RA
, are exact as well. In second-quantized form H
may be written as,
H =∑p,q
f pqp†q +
1
2!
∑p,q,r,s
〈pq||rs〉p†q†sr
+1
3!
∑p,q,r,s,t,u
〈pqr||stu〉p†q†r†uts + · · · (2–3)
where we show the inclusion of three- and higher-body interaction terms. To include
the higher-order terms we renormalize with respect to the one- and two-body terms and
obtain a generalized, correlated Fock-like operator, called the transfer Hamiltonian [10, 43]
Th =∑µ,ν
[hµν +∑
λ,σ
Pλσ˜〈pq||rs〉]µ†ν (2–4)
where Pλσ is the single particle density matrix and both hµν and ˜〈pq||rs〉 can be
approximated with a suitable functional of atom-based parameters. We have used the
NDDO form in the current work, but the formalism developed thus far is not limited by
this choice of Fock-like operator.
36
2.2 Computational Details
Parameters were optimized by a quasi-Newton-Raphson optimization algorithm in
tandem with a genetic algorithm to fit Austin Model 1 (AM1) NDDO parameters [27] to
reproduce the forces determined with CC theory. The parameterization was performed
by a combination of the PIKAIA [46] implementation of a genetic algorithm and the
linearized quasi-Newton-Raphson minimization routine of L-BFGS [47].
The reference forces used in this parameterization were obtained with the ACES
II [48] program system. We have used CCSD in a triple zeta basis, with a unrestricted
reference and Hartree-Fock stability following (to ensure that we arrive at the proper
unrestricted spin solution.) All semiempirical AM1 and OM1 calculations were performed
using a modified version of MNDO97 Version 5.0 [30–32, 49].
DFT calculations for the nitromethane (NMT) monomer were performed at the
B3LYP/6-31G(d) level with the HONDOPLUS program [50]. The initial NMT dimer
and trimer geometries were taken from a DFT study using B3LYP/6-31++G** [51]. Due
to the large number of correlated reference calculations needed in modeling our clusters,
we used the Turbomole Version 5 electronic structure package [52, 53] to do all-electron
calculations at the resolution-of-the-identity second order Møller-Plesset (RI-MP2) level
in a TZVP basis. We chose an all-electron model in a large auxiliary basis to ensure an
accurate treatment of correlation energies and geometries that were demonstrated on a
series of compounds representative of atoms in various oxidation states [54].
2.3 Nitrogen-Containing Energetic Materials
The motivation of this study is to develop a Th that can accurately predict the
quantum mechanical behavior of nitrogen-containing energetic materials. We present a
new parameterization for a semiempirical NDDO Hamiltonian which improves upon the
standard AM1 parameter set. This new parameterization reproduces the decomposition
of NMT to the CH3 and NO2 radicals as predicted with CCSD/TZP. We demonstrate
transferability to small NMT clusters. For NMT dimers and trimers this model Th predicts
37
rearrangements similar to previously calculated unimolecular mechanisms. The NMT
trimer undergoes a concerted rearrangement to the methylnitrite trimer when one of the
NMT molecules becomes geometrically strained.
NMT has been the subject of many theoretical and experimental studies due to the
significant role that nitrogen-containing compounds play in the chemistry of explosives,
propellants, and atmospheric pollution [55]. The description of the decomposition of NMT
and its products is crucial to the understanding of the kinetics and dynamics of larger
energetic materials for which NMT is a model. NMT is small enough to allow detailed
investigation of its complex decomposition [56, 57]. Specifically, even for a system of
deceiving simplicity, its decomposition to radicals via simple C-N bond rupture versus the
competing NMT-methylnitrite (MNT) rearrangement has been the subject of much debate
[56–59].
The use of classical potentials and standard semiempirical NDDO methods often
yield incorrect behavior for energies, forces, and geometries, especially in non-equilibrium
cases. Therefore, the use of such techniques to do MD will often lead to incorrect results
when bond breaking and forming are studied. Alternatively, ab initio quantum chemistry
is known to be predictive, in the sense that it can reproduce most quantities adequately
compared to the experimental values in those regions without knowing the result prior
to calculation. However, these methods are too expensive and, therefore, currently
impractical for MD because of the time and disk space requirements.
Previous NDDO parameterizations, such as AM1, can reproduce near-equilibrium
structures for a large number of molecules, yet they often fail for non-equilibrium
geometries. Thus, for the purposes of doing MD simulations, we are willing to sacrifice the
ability to accurately treat a large number of molecules near equilibrium, for the ability to
treat a small class of molecules accurately at all geometries. When modeling combustion
or detonation it is particularly important to have a method that works equally well for
equilibrium and non-equilibrium geometries if the chemistry of bond breaking and forming
38
is to be described. However, this sacrifice means that the new Th parameter set must be
applied judiciously. If used for a class of molecules for which it was not trained, then the
accuracy of calculated properties could be diminished. If the appropriate set of reference
data and a sufficiently complete model are chosen, then the Th will provide an equally
appropriate and complete description of the system.
Among the first to study NMT theoretically were Dewar et al. [56] who used
MINDO/3 and found that the NMT-MNT rearrangement occurred with an energy
barrier of 47.0 kcal/mol and that MNT dissociated to H2CO+HNO with an energy of
32.4 kcal/mol. Hence they proposed that NMT decomposition occurs via rearrangement.
McKee [57] reported the NMT-MNT rearrangement barrier to be 73.5 kcal/mol calculated
at the MP2 level of theory with a 6-31G(d) basis and that MNT dissociated to H2CO+HNO
with an energy of 44.1 kcal/mol. The high barrier height of the rearrangement suggests
that the decomposition pathway is simple bond rupture to NO2 and the CH3 radical.
Hu et al. [59] performed calculations with the G2MP2 approximation and found that
the NMT dissociation occurs via direct C-N bond rupture with an energy of 61.9
kcal/mol. The NMT dissociation is lower than both the NMT-MNT rearrangement
and the NMT-aci-nitromethane rearrangement by 2.7 and 2.1 kcal/mol, respectively.
Nguyen et al. [58] reported that the NMT-MNT rearrangement barrier is 69 kcal/mol
and NMT direct dissociation to CH3 and NO2 radicals is 63 kcal/mol calculated at the
CCSD(T)/cc-pVTZ level using CCSD(T)/cc-pVDZ geometries. Most recently Taube and
Bartlett [60] report the NMT-MNT rearrangement at 70 kcal/mol using ΛCCSD(T).
Hu et al. [59] also performed a detailed study of unimolecular NMT decomposition
and isomerization channels using DFT with G2MP2//B3LYP level of theory in a
6-311++G(2d,2p) basis. Single points were basis set extrapolated along the minimum
energy paths to correct for incompleteness using the G2MP2 method. Their article shows
that from a particular intermediate state (IS2b) one possible reaction path is the rupture
of the N-O bond to give the methoxy radical and nitrous oxide. Hu et al. indicate that on
39
this reaction path no transition state could be located. Results from photodissociation and
thermolysis experiments suggest that one of the primary steps of NMT decomposition is
the elimination of oxygen [58, 61, 62].
However, the reliability of both the B3LYP and, to a lesser extent, CCSD(T) at
non-equilibrium geometries [59] is limited and the same accuracy should not be expected
in regions other than equilibrium.
The work of Li et al. [51] suggests that the three-body interaction energy in bulk
NMT is a critical component in accurately describing the potential energy surface. We
investigated the use of our Th in predicting how an accurate treatment of three-body
effects dictates chemical reactivity.
2.3.1 Nitromethane Monomer
We have developed a set of NDDO parameters to reproduce the force curve along
the intrinsic reaction coordinate for C-N bond rupture and to reproduce the CCSD/TZP
equilibrium geometry. As seen in Table 2-1 the agreement between CCSD and TH-CCSD
is to within a degree for the angles and less than one hundredth of an A for bond lengths.
TH-CCSD is shown in Figure 2-1 to reproduce the original force curve for C-N
bond rupture to within an accuracy of 0.005 Hartree/Bohr. The AM1 parameters were
designed to reproduce equilibrium structures, and thus they have the correct force at
the equilibrium bond length. However, at 0.1 A away from equilibrium the forces are in
considerable error. A recurring feature for AM1 force curves, which is problematic for
MD, is the unphysical repulsive behavior at large internuclear distances. The transfer
Hamiltonian removes this artificial repulsion and gives the qualitatively and quantitatively
correct dissociation.
One way of determining energy differences in a single reference theory, other than
Hartree-Fock, is by integration of the force curve. For AM1 the error in the forces is
reflected in the energy as shown in Figure 2-2. Also shown in Figure 2-2 the DFT solution
40
yields a potential energy surface and force curve which are incorrect at non-equilibrium
geometries.
2.3.2 Nitromethane Dimer
In analogy with how Bukowski et al. [63] have studied the interaction energy of
dimers as a function of rigid monomer separation, we calculate the relative energies of
the NMT dimer as a function of hydrogen bonding distance. The minimum chosen in our
NMT dimer calculations from [51] is stable due to the formation of two hydrogen bonds of
equal length and antiparallel arrangement of the molecular dipoles. In Figure 2-3, we plot
the energies relative to the respective methods minima in ROH .
Our data indicate that in the case of frozen monomer geometries, AM1 overestimates
the repulsive wall at small intermolecular distances. It appears as if the H, C, N, and
O parameters in the core-core repulsion function for AM1 were unable to cancel the
effect from which MNDO has traditionally suffered, namely overestimation of energies
in crowded molecular systems [31, 64]. The TH-CCSD reproduces the interaction forces
quite well for intermolecular distances up to ∼3 A , while AM1 is shown to slightly
overbind. Thereafter the TH-CCSD underbinds slightly in disagreement with RI-MP2 but
in agreement with the OM1 Hamiltonian. The OM1 method has credence as a method
which models the dipole and thus hydrogen bonding interaction properly by correcting the
overlap matrix [32, 65, 66].
Adiabatic C-N bond dissociation for small NMT clusters was investigated starting
from published local minima [51] using the TH-CCSD and AM1. In studying the dimer
interaction, for which one of the monomers is undergoing unrestricted C-N bond rupture,
the TH-CCSD forces predict that a bimolecular process takes place when the stretched
bond is at a distance of 2.42 A (1.6Req). The reaction products are nitrosomethane,
methoxy radical, and nitrogen dioxide radical. Figure 2-4 shows snapshots of this
particular dimer decomposition channel.
41
The TH-CCSD geometry optimization shows the monomer-monomer distance
decreasing from 3.50 A to 2.90 A as the methyl group on monomer one rotates to face an
oxygen on monomer two. As this occurs, the C-N bond on monomer two increases from
1.5 A to 1.6 A and the length of the N-O bond participating in the reaction increases
from 1.3 A to 1.7 A . The AM1 Hamiltonian does not predict any chemistry as a result of
the C-N bond breaking because the forces governing proper dissociation are incorrect, as
shown in the case of the NMT monomer (Figure 2-1). The geometry optimization using
AM1 reveals that the monomers actually separate by 0.45 A and that no methyl rotation
takes place.
The bimolecular dimer reaction predicted by the TH-CCSD is supported by the
unimolecular rearrangement of intermediate IS2b [59] for which no transition state
was located. The similarity is that the reaction products both involve the formation of
methoxy radical via the cleavage of an N-O bond. Unlike the study in [59], our process
is bimolecular, where methoxy radical formed is by NMT undergoing oxygen elimination
or N-O rupture. Contrary to [59], where the methyl group picks up an oxygen from its
own monomer, we observe that the oxygen is picked up by a methyl group on a different
monomer. Our results for oxygen elimination are in agreement with previous studies
[61, 62], which suggest this is the primary process in detonation.
2.3.3 Nitromethane Trimer
The NMT trimer local minimum taken from [51] has a ring structure involving three
hydrogen bonds. Performing the same study done for the dimer on the reference structure
trimer from [51] shows that at a C-N bond distance of 1.94 A or 1.3Req a concerted
rearrangement of NMT to MNT occurs. The decomposition of NMT in trimer thus occurs
at a C-N distance that is 0.5 A less than that in the dimer. This indicates a cooperative
reactivity in clusters of NMT initiated by the vibrational excitation of a C-N bond. The
monomer rearrangement to MNT is widely supported [58, 59] and is believed to be a
key step in the detonation of NMT. The initial intermolecular N-N distances are 4.5 A ,
42
4.6 A , and 4.8 A and angles are 62◦, 58◦, and 60◦ (Figure 2-4). During the geometry
optimization, because one of the monomers has a stretched C-N bond, the system is
unable to reach its ring-like hydrogen-bonded local minimum, leading to ring strain. The
TH-CCSD predicts that, as a result, the one monomer cleaves at a C-N bond distance of
1.8 A and, subsequently, a second monomer dissociates at the same C-N bond distance.
At this point, all the intermolecular N-N bond distances have decreased to 4.3 A , 4.5
A , and 4.6 A ; the angles, however, remain unchanged. Geometry optimization with
AM1 shows that the stretched monomer shifts slightly from the remaining two monomers
allowing one of them to rotate to a lower energy structure where it forms a bifurcated
hydrogen bond each of which is 2.44 A.
We have demonstrated that the TH-CCSD parameter set trained for NMT monomer
transfers to the dimer and trimer calculations. The criteria for transferability used
here regards the extent to which these forces yield products that are in accord with
theoretically supported decomposition mechanisms of NMT [58, 59]. Note that the
training of our Hamiltonian was not tailored to favor any particular rearrangement
mechanism, evident by the fact that the dimer decomposition differs markedly from the
trimer mechanism. Along the adiabatic surface AM1 does not predict any bond breaking
to occur. Therefore, the accuracy of modeling bond breaking and forming processes in
bulk simulations using AM1 is questionable, at best.
We have shown a parameterization of the NDDO Hamiltonian that reproduces proper
dissociation of NMT into radicals, both for the energy and forces. This parameter set
reproduces the reference CCSD/TZP geometries and forces for unrestricted C-N bond
breaking. This NDDO parameterization would appear to potentially offer improved
accuracy in MD simulations for clusters of high-energy materials. We also show that
concerted reactions for both the dimer (bimolecular rearrangement) and trimer (trimolecular
rearrangement) are not generated with an AM1 parameterization, yet are required for
the proper description of NMT clusters. This new transfer Hamiltonian will allow for
43
rapid and accurate MD studies due to the inherent speed of NDDO and the fitting of
the parameters to high-level CC reference forces. In this sense we are only limited by
the reference data and the quality of the model Hamiltonian. We have demonstrated
transferability of monomer parameters to nitromethane dimer and trimer. These forces
predict chemical reactivity in clusters that is not observed in AM1 simulations and, in
the case of the trimer, we see a threefold rearrangement of NMT to MNT giving further
insight into the detonation process.
2.4 Silica and Silicon
The TH depends upon the choice of the form of the Hamiltonian. So far we have
used the NDDO form [67, 68] because of its wide use and prior successes. However, such
methods using standard parameters like AM1 [27], MNDO [30–32] and PM3 [28, 29]
are also known to have significant failings compared to ab initio methods. Our TH, on
the other hand, benefits from the fact that we only require specific parameters [45] for
the systems of interest, for example, SiO2 clusters and H2O in our hydrolytic weakening
studies [69]. These parameters permit all appropriate clustering and bond breaking for
all possible paths for all the components, to allow MD to go where the physics leads.
Furthermore, unlike traditional semiempirical methods, we do not use a minimal basis set,
preferring to use polarization functions on all atoms, and in some cases, diffuse functions.
The form of the Hamiltonian emphasizes two-center interactions (but is not limited to
nearest neighbors as in TB), which allows our parameters to saturate for relatively small
clusters. We check this by doing ab initio calculations on larger clusters involving the
same units to document that the TH is effectively unchanged with respect to system size.
At that point, we have a TH that we can expect to describe extended systems composed
of SiO2, and its solutions properly, including all multi-center effects on the energies and
forces. Only the Hamiltonian is short-range. We also check our TH by applying it to
related systems. We will show that our TH determined from forces for bond rupture in
disiloxane, pyrosilicic acid, and disilane also provide correct structures for several small
44
cluster models for silica and silicon. Moreover, multiple bonds and high coordination Si,
though not typical in silica polymorphs, are described well. Finally, we will show that our
TH performs well using periodic boundary conditions for silica polymorphs.
2.4.1 Silicon Clusters
A method based on rigorous quantum mechanics should be transferable. We define
two types of transferability: first, an appropriate method should apply equally well
to systems of similar size, but with different bonding characteristics, e.g., single- and
double-bonds; and second, a method should perform equally well for small and large
systems, and even infinite systems as modeled by periodic boundary conditions.
Now we will show that the TH-CCSD parameters for an NDDO-type Hamiltonian
trained on a small set of reference molecules are transferable in the first sense. We chose
our training set to be reasonable representatives of the local behavior of extended systems:
disiloxane (H3SiOSiH3), pyrosilicic acid ((OH)3SiOSi(OH)3) and disilane (H3SiSiH3).
The initial parameters used were from MNDO/d [70] because of availability, overall
performance on the training set, and inclusion of d-orbitals on Si.
The fit was based on forces from the CCSD approximation using DZP basis sets.
Primarily the TH-CCSD fit was to the forces along the intrinsic reaction coordinate for
Si-Si bond breaking in disilane and Si-O bond breaking in both disiloxane and pyrosilicic
acid. Further refinement was achieved by fitting to the forces for the Si-O-Si bends
of disiloxane and pyrosilicic acid. The most reliable error function to use in the type
of fitting seems to be the square of the difference between the reference CCSD and
TH-CCSD forces. The fitting procedure itself, though nonlinear, involved iteratively
performing a linear minimization of each parameter, starting with those that most reduced
the error.
To verify the transferability of the parameters one should evaluate properties for
molecules outside the reference set. Verification for bond breaking tests were performed
on Si(OH)4, Si(SiH3)(OH)3, Si(SiH3)2(OH)2, Si(SiH3)3(OH), Si(SiH3)4, H2Si=SiH2,
45
(SiH3)Si≡Si(SiH3), (SiH3)2Si=Si(SiH3)2, and (SiH3)3Si-Si(SiH3)3. Verification testing for
the O-Si-O bend involved calculations on Si(OH)4, Si(SiH3)(OH)3, and Si(SiH3)2(OH)2.
Figures 2-5 and 2-6 show the results relative to DFT (with the B3LYP approximate
exchange-correlation functional) values.
It is apparent that the TH-CCSD parameters outperform many of the commonly used
NDDO methods for a variety of different bonding environments. Furthermore, although
the parameters were designed to reproduce Si-O-Si, Si-O, and Si-Si forces, they transfer
well to equilibrium O-Si-O (Figure 2-7).
2.4.2 Silica Polymorphs
Now we demonstrate transferability of the TH-CCSD parameters to periodic systems.
We choose to do so for crystalline forms of SiO2 first. Note that we need a model that
describes amorphous and crystalline phases equally well; if a method is comparatively
more accurate (or less accurate) for crystalline or amorphous materials then non-physical
restrictions would be imposed. For example, crack tip propagation in silica is attracted to
regions of low potential. Crystalline SiO2 is a good first step because crystalline phases
have been observed at the interface in the otherwise amorphous silica gate oxide layer in
metal-oxide semiconductors (MOS) [71]. Understanding silica and silicon brings us closer
to understanding defects in MOS. In them, phenomena such as current leakage, electrical
breakdown over time, and the thinning trends of the dielectric layer in gate oxides are
strongly influenced by defects [72, 73].
It also should be noted that we focus on training models that perform well primarily
for localized units (i.e., clusters) and secondarily for bulk systems with periodic boundary
conditions. This is because it is much harder to build local phenomena into a model
used primarily for extended systems than it is to construct a model based on local
chemistry that is applicable for small molecules, clusters, and bulk. For example, periodic
calculations with plane waves require exceedingly large numbers of plane waves to describe
oxygen defects in silica [74].
46
Furthermore, there is evidence that any model that is only dependent upon
pair-potentials, (i.e., nearest-neighbor distances) such as standard orthogonal TB schemes,
lacks the ability to describe some fundamental properties of silica. It is difficult to
prove such a claim, as we are able to imagine an expansion of the energy about some
parameter, such as nearest-neighbor distances. However, Stixrude et al. [75] have
shown experimentally, based on the two-fold compression of liquid SiO2, that there is
essentially no change in nearest-neighbor distances though there are significant changes
in medium-range structure. This medium-range structure is best described by ring- and
cluster-statistics and cluster geometry [76], which would be not be incorporated in any
two-body potential.
To avoid the limitations of empirical two-body potentials, one may consider NDDO
or non-orthogonal TB methods discussed above, which include some many-body effects
due to inclusion of the overlap matrix [77]. Especially promising is the SCC-DFTB
approach [7] which includes the largest contributions to long-range Coulomb interactions
while handling charge in a self-consistent way. These methods offer a practical balance of
chemical accuracy and computational efficiency, as evidenced from their recent increase in
popularity.
Figure 2-8 shows the deviations of the TH-CCSD parameterization, AM1, and
MNDO/d NDDO calculated unit cell densities from reference DFT densities. The
reference DFT calculations for all silica polymorphs are from the VASP code using
the gradient-corrected PW91 GGA exchange-correlation functional with 3D periodicity, an
ultrasoft pseudopotential, and a plane wave basis set (395 eV cutoff). A 2x2x2 k-point
grid is employed, which converges total energies to within ±0.05 eV/cell, and the
convergence of self-consistent steps is precise to 10−4 eV/cell. The optimization of cell
parameters and atomic positions is precise to < 10−3 eV A−1.
It is clear that the TH-CCSD parameters trained on small silica clusters are
transferable to several silica polymorphs. It is also interesting to note that not only is
47
quantitative improvement achieved but the error from the PW91 DFT calculations is
systematic and parallels similar improvement found when this new parameterization is
applied to molecular systems. Further refinement of the TH-CCSD parameters must be
made to accurately describe the silicon-silica interface, since the current parameterization
fails to converge to the proper topology.
The application of the transfer Hamiltonian methodology has been presented for
nitrogen containing energetic materials and silicon containing compounds. In both cases
we have demonstrated significant improvement over the underlying semiempirical method
by reparameterizing with a training set of high-quality CC forces for representative
molecules. It is apparent that for focused MD studies that use semiempirical model
Hamiltonians there is an advantage to creating a special set of parameters that is tailored
to the chemical system being studied. However, the transferability of the model is affected
by which molecules are selected for the training set and the PES reference points used.
These models will often break down when applied to systems to which they have not been
parameterized. One route to achieve this level of transferability is to design a simplified
method that does not require parameterization, only controlled approximations, which
leads to the AAT approach to defining a superior Th.
48
Table 2-1. Equilibrium geometry for NMT, in A and degrees
Method RCN RCH RNO AHCN AONC
CCSD/TZP 1.498 1.087 1.219 107 117TH-CCSD 1.504 1.084 1.225 108 118
AM1 1.500 1.119 1.201 107 119
49
-0.1
-0.05
0
0.05
0.1
1 1.5 2 2.5 3 3.5
Forc
e (H
artr
ee/B
ohr)
C-N reaction coordinate (Angstrom)
CCSD/TZP UHFAM1 UHF
TH(CCSD) UHFB3LYP/6-31G* UHF
Figure 2-1. The NMT force curve for C-N rupture
50
0
20
40
60
80
1 1.5 2 2.5 3 3.5
Ene
rgy
(kca
l/mol
)
C-N reaction coordinate (Angstrom)
CCSD/TZP UHFAM1 UHF
TH(CCSD) UHFB3LYP/6-31G* UHF
Figure 2-2. The NMT energy curve for C-N rupture
51
-5
0
5
10
15
1 1.5 2 2.5 3 3.5 4 4.5E(R
OH
)-E
(RO
H-e
q) (
kcal
/mol
)
O-H reaction coordinate (Angstrom)
RIMP2/TZVP RHFAM1 RHFOM1 RHF
TH(CCSD) RHF
Figure 2-3. The NMT dimer energies relative minimum in the intermolecular hydrogenbonding distance
52
A
B
C
D
Figure 2-4. Snapshots of the geometry optimizations for NMT dimer and trimerperformed with the TH-CCSD and AM1. A) NMT dimer treated withTH-CCSD. B) NMT dimer treated with AM1. C) NMT trimer treated withTH-CCSD. D) NMT trimer treated with AM1
53
0
0.02
0.04
0.06
0.08
0.1
TH(CCSD)MNDO/dPM3AM1
Ang
stro
m
Method
Figure 2-5. Average deviation of Si-O stretch
54
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
TH(CCSD)MNDO/dPM3AM1
Ang
stro
m
Method
Figure 2-6. Average deviation of Si-Si stretch
55
0
1
2
3
4
5
6
7
8
TH(CCSD)MNDO/dPM3AM1
Ang
le
Method
Figure 2-7. Average deviation of O-Si-O angle
56
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9∆
dens
ity (
g/cm
3 )
α-Cris
tobali
te
α-Qua
rtz
β-Cris
tobali
te
β-Qua
rtz
β-Trid
ymite
Coesit
e
TH(CCSD)AM1
MNDO/d
Figure 2-8. Density deviation from PW91 DFT
57
CHAPTER 3IMPETUS FOR A NEW SEMIEMPIRICAL THEORY
Semiempirical method development has a long history dating back to the 1930’s
starting with Huckel theory for describing even-alternant hydrocarbons. In the current
study we have been guided by the developments over the last seventy years in regard to
both what should and should not be approximated in our model. We focus here on the
NDDO and SCC-DFTB models. Both models have been widely-used with reasonable
success. The relative numerical performance of the most recent versions of each model has
been reported recently [2], and extensive comparisons of the formal similarities of each
model have been made [78, 79]. It is important to understand the context in which these
models were designed to fully appreciate the difference in the design philosophy of the
current AAT model. NDDO-based methods were originally constructed to reproduce the
experimental heats of formation for small organic molecules. Since the heat of formation
is defined only for molecules in their equilibrium structures, it is not surprising that
NDDO models are parameterized specifically to reproduce equilibrium properties. For
nonequilibrium structures the accuracy of NDDO methods varies greatly, as we will show.
Similarly, SCC-DFTB has been parameterized to reproduce equilibrium structures.
The goal of the AAT approach is to achieve uniform accuracy of electronic properties,
structures, and energetics over the entire PES to the extent possible for a minimal ba-
sis set effective one-particle theory. Uniform accuracy is required to achieve meaningful
results when performing MD simulations, in which transition states and other nonequilibrium
structures play a critical role.
3.1 Neglect of Diatomic Differential Overlap Based Methods
3.1.1 Does NDDO Approximate HF?
A common misconception about NDDO-based methods is that the NDDO model
Hamiltonian approximates the HF Hamiltonian. This assumption is not the case and it
can be shown that the density dependent contribution to the Fock-like matrix in NDDO
has little to do with the density dependent Fock matrix of HF theory, though they are
58
deceptively similar:
Gσλµν
HF= (µν|λσ)− 1
2(µσ|λν) (3–1)
Gσλµν
NDDO= (µν|λσ)δABδCD − 1
2(µσ|λν)δADδBC (3–2)
where µ, ν, λ, and σ are on centers A, B, C, and D respectively and the terms (µν|λσ)
are subject to the multipole-multipole expansion approximation. It is true that a typical
NDDO-type two-electron integral encountered is often larger in magnitude than a
corresponding non-NDDO-type two-center, three-center, or four-center two-electron
integral. For molecules with more that three heavy atoms in a valence-only minimal basis
set representation, there are significantly more terms involving three- and four-center
two-electron integrals. The combined effect of this large number of multi-center terms
has a significant effect on the density dependent Fock matrix contributions. We consider
a simple example. Shown in Figure 3-1 are the individual contributions to the NMT
density dependent Fock matrix element of the carbon 2s-type orbital and nitrogen 2s-type
orbital in a minimal basis set for the combined three- and four-center, two-center, ab initio
NDDO-type, and full multi-center contributions. Though only a single matrix element
is plotted over this reaction coordinate, the trend is the same for other matrix elements
and other molecules. The three- and four-center contributions are nearly equal and oppo-
site to the two-center contributions. This cancellation is a universal trend and holds at
all energy scales. And, while it could be said that the NDDO-type class of two-electron
integrals approximates the full set of two-center two-electron integrals, it is clear that
the NDDO-type integrals alone bear little resemblance to the full two-electron integral
contribution. Instead, it is the delicate balance among all multi-center two-electron
integrals that is critical to obtaining the proper Fock matrix contribution.
It has been known since studies were first performed on the NDDO approximation
that a parameterized set of NDDO integrals perform better that the ab initio NDDO
integrals. However, what has been less clear is the lack of correspondence between the
NDDO approximation and the quantity it is supposedly approximating. Instead it seems
59
that the adjustable parameters involved in the multipole-multipole expansion are being fit
to the full density dependent part of the Fock matrix, and not the integrals themselves.
Furthermore, this fit is non-linear because of the two-electron exchange integrals, which
means that improving the NDDO approximation by modifying the integrals or trying to
add three- or four-center contributions becomes increasingly convoluted. For the AAT
model, as we will show, all integral approximations can be systematically improved and
do not involve any adjustable parameters. By avoiding the need to parameterize integral
contributions we can retain uniform accuracy of the PES because no structural bias has
been introduced.
3.1.2 Artificial Repulsive Bump
There are many chemically interesting cases in which existing semiempirical methods
seem to work quite well. For example Jorgensen et al. [80] show that their PDDG/PM3
reparameterization of PM3 yields heats of formation with a mean absolute error of 3.2
kcal/mol for a test set of 622 molecules. However, there is little evidence to suggest that
the same performance should be expected for chemical structures that are significantly
distorted from equilibrium structures. Even for simple activation barriers, the error is
typically 10-15 kcal/mol [80]. For the determination of rate constants and in molecular
dynamics studies the PES should be described within 1-2 kcal/mol. For example, since at
300K an error of 1 kcal/mol in the activation barrier will result in an order of magnitude
change in the rate constant, clearly a 10-15 kcal/mol error will not yield meaningful
rate constants. The situation is especially bad in cases in which direct bond fission
is treated with NDDO-based model Hamiltonians. In such cases there is an artificial
repulsive region near 1.2Req to 1.6Req, as is shown in Figure 3-2 for NMT UHF direct
bond fission. Generally, in cases where it is known that there is barrierless dissociation,
one would recognize such features as an error in the method and subsequent results would
be questioned. For more complicated chemical processes, for example transition states or
60
concerted reaction mechanisms, such dramatic errors are much more difficult to recognize,
though it is precisely those regions that are often of primary interest.
3.1.3 Separation of Electronic and Nuclear Energy Contributions
In ab initio theory (without consideration of non-BO effects) inclusion of the
nuclear-nuclear repulsion term (VNN) is a trivial matter when evaluating total energies
and associated forces. While VNN has no part in the determination of purely electronic
properties (such as the electronic density, electronic spectra, ionization potentials,
or electron affinities) it is critical in the determination of molecular structure, the
identification of transition states, and their associated vibration spectra. In simplified
theories the role of VNN is obfuscated because the electronic and nuclear repulsion effects
are not separated.
The core-core repulsion term in NDDO Hamiltonians involves parameters that are
designed for energetics at equilibrium. For such a specific set of properties the difference
of two large numbers, the attractive (negative) electronic energy and the repulsive
(positive) nuclear-nuclear repulsion energy, is easier to model than each separately. In
our opinion, combining such unrelated terms is the origin of the long-standing contention
that electronic properties and total energy properties cannot be described within the same
set of parameters. Also, once the core-core terms are parameterized, the correspondence
between rigorous theory and semiempirical theory is lost. To solve this problem, we need
to search for a better form for our model Hamiltonian that does not require that VNN be
combined with terms that are purely electronic.
3.1.4 Total Energy Expression
Here we illustrate the origin of the combined electronic and nuclear energy contributions
of the total energy expression, the core-core term in NDDO methods, so that we may
avoid this pitfall in the development of AAT. Consider the case in which the exact
KS-DFT exchange-correlation energy functional and corresponding potential are known,
and further are represented in a complete basis set. Then our effective one-particle
61
Hamiltonian will yield a set of orbitals which yield the exact density. The exact energy is
given by the KS-DFT total energy expression, Equation 1–27.
Now, if we expect that the NDDO model Hamiltonian will generate the correct
electronic density, which is a requirement if we want any of the first-order electronic
properties of our system, then we can relate the NDDO and KS-DFT energy expressions.
Making the temporary assumption that fNDDO ≈ fKS, and consequently ρNDDO ≈ ρKS,
then the total energy difference is
ENDDO − EKS =∑
A,B>A
ENDDOAB (RAB)− Exc[ρ] +
1
2
∫vxc(r)ρ(r)dr −
∑A,B>A
ZAZB
RAB
(3–3)
Now it is clear that if the NDDO one-particle Hamiltonian were to yield the correct
electronic density then the two-center core-core term (ENDDOAB ) must account for not only
the simple nuclear-nuclear repulsion term (ZAZB
RAB), but also some pair-based contribution,
dependent only upon RAB, of the the complicated multi-center exchange-correlation
contribution (Exc[ρ] − 12
∫vxc(r)ρ(r)). The motivation for approximating this last term
is obvious: it drastically simplifies and avoids a complicated and expensive step in the
determination of the total energy. However, there is no reason to expect a priori that this
approximation would be accurate. For our improved model Hamiltonian we make a special
effort to ensure that all approximations are controllable and systematically verifiable.
3.2 Self-Consistent Charge Density-Functional Tight-Binding
3.2.1 Self-Consistent Charge Density-Functional Tight-Binding in an AOFramework
Equations 3–4 through 3–8 are the working equations for SCC-DFTB which come
directly from Elstner’s original paper [7]:
∆qA = qA − q0A (3–4)
qA =1
2
occ∑i
ni
∑µ∈A
atoms∑ν
(c∗µicνiSµν + c∗νicµiSνµ) (3–5)
62
M∑ν
cνi(HSCCµν − εiSµν) = 0 (3–6)
HSCCµν = 〈χµ|H0|χν〉+
1
2Sµν
atoms∑
ξ
(γAξ + γBξ)∆qξ
= H0µν + H1
µν (3–7)
ETB2 =
occ∑i
〈φi|H0|φi〉+1
2
atoms∑A,B
γAB∆qA∆qB +∑
A6=B
Erep(RAB) (3–8)
and the corresponding gradients are
~Fα = −occ∑i
ni
∑µν
cµicνi[∂H0
µν
∂α− (εi −
H1µν
Sµν
)∂Sµν
∂α]−∆qα
N∑
ξ
∂γαξ
∂α∆qξ −
∂∑
A 6=B Erep(RAB)
∂α
(3–9)
we have used µ ∈ A and ν ∈ B.
To facilitate comparison of the SCC-DFTB model Hamiltonian to other methods
we force it into a structure that is comprised of a density independent part H and a
density dependent part G. Using Mulliken population analysis to determine the net charge
associated with atom A yields
∆qA = −q0A +
∑µ∈A,ν
PµνSνµ (3–10)
where the occupation numbers ni = 1 and the MO coefficients are real.
F SCC−DFTBµν = H0
µν + H1µν
= H0µν −
1
2Sµν
atoms∑
ξ
(γAξ + γBξ)q0ξ +
1
2Sµν
atoms∑
ξ
∑
λ∈ξ,σ
(γAξ + γBξ)PλσSσλ
= (HSCC)µν +∑
λσ
Pλσ(GSCC)σλµν (3–11)
we have introduced the notation that
(HSCC)µν = H0µν −
1
2Sµν
atoms∑
ξ
(γAξ + γBξ)q0ξ (3–12)
63
(GSCC)σλµν =
1
2SµνSσλ(γAξ + γBξ) (3–13)
Note that unlike HF or NDDO, in SCC-DFTB it appears that it is not generally true that
Gσλµν = Gµν
σλ, as λ ∈ ξ and σ is unrestricted. If SCC-DFTB were to have this property, that
would be useful both for the calculation of forces and for comparison among all effective
one-particle methods we consider. To enforce this similarity we make the following
observation:
∑
λσ
Pλσ[1
2SµνSσλ(γAC + γBC)] =
∑
λσ
Pλσ[1
4SµνSσλ(γAC + γBC + γAD + γBD)] (3–14)
where µ, ν, λ, and σ are on A, B, C, and D respectively, and our working equation for the
density dependent contribution in SCC-DFTB is
(GSCC)σλµν =
1
4SµνSσλ(γAC + γBC + γAD + γBD) (3–15)
Equation 3–15 bears a similarity to the Mulliken approximation to Coulomb repulsion, we
will address this in further detail later.
Now we revisit the total energy expression. Using our new definition of the density
dependent and independent terms in the SCC-DFTB Fock-like matrix.
ESCC =1
2
∑µν
Pνµ[2HSCCµν +
∑
λσ
Pλσ(GSCC)σλµν ] +
∑
A6=B
(1
2γABq0
Aq0B + ESCC
rep (RAB)) (3–16)
This energy expression closely resembles other one-particle Hamiltonians, yet is equal to
Equation 3–8. Similarly, after significant manipulation the gradient expression can be
shown to be
~Fα =1
2
∑µν
Pνµ[2∂HSCC
µν
∂α+
∑
λσ
Pλσ
∂(GSCC)σλµν
∂α]
−∑µν
∂Sµν
∂α
N∑i
cµicνiεi +∂
∂α
∑
A6=B
(1
2γABq0
Aq0B + ESCC
rep (RAB)) (3–17)
which again is equal to the original expression, Equation 3–9, but allows a more
systematic comparison with other methods.
64
3.2.2 Mulliken Approximation Connection to SCC-DFTB
To facilitate understanding of the density-dependent term in SCC-DFTB in relation
to concepts in wavefunction theory we must recognize that the term ∆qA is the charge
fluctuation on atom A from some reference charge for that atom-type (q0A), with the
charge on atom A being defined by the Mulliken population analysis. In the expanded
representation in terms of the AO density matrix (P) the additional term to the density
dependent part of the Fock-like matrix is given by Equation 3–13, which has been shown
to be equivalent to Equation 3–15, though the latter has a form that some may recognize
as the Mulliken expansion for two-electron integrals,
(µν|λσ) ≈ 1
4SµνSσλ(γAC + γBC + γAD + γBD) (3–18)
The origin of the Mulliken expansion and the further extension of the basic approximation
to the Rudenberg approximation will be explored later. For now it is sufficient that
we discuss the numerical properties of this approximation and what advantages and
limitations it entails.
It is perhaps easiest to consider the nature of the Mulliken approximation by way of
a simple example. For an NDDO-type two-center two-electron integral with normalized
orbitals it follows directly from Equation 3–18
γAB ≈ (µAµA|λBλB) (3–19)
There have been many studies on the various forms for γ, most notably by Klopman [39],
Ohno [40], and Nishimoto-Mataga [41]. The particular form within SCC-DFTB model
is the Coulomb repulsion between two 1s Slater type orbitals with an added condition
that as the separation between the two centers approaches zero γAB → UA where UA is
the Hubbard parameter, which is related to chemical hardness, which in turn is related
to the difference between the first IP and EA of that atom. The particular form of the
γAB is only as significant as the underlying approximation in which it is used. In this
65
case the Mulliken approximation ought to be accurate. The Mulliken approximation in
SCC-DFTB essentially converts three- and four-center two-electron integral contributions
to the Coulomb repulsion energy term into overlap-weighted two-center repulsive terms.
If we apply the Mulliken approximation in standard HF and use the ab initio (µµ|λλ)
two-center integrals, the behavior is quite unphysical. Shown in Figure 3-4 are errors
from the full HF when the Mulliken and Rudenberg approximations are applied to the
two classes of three-center terms, (AA|BC) and (AB|AC), and the four-center terms in
NMT over the carbon-nitrogen reaction coordinate. The error introduced by the Mulliken
approximation is drastic, particularly for the (AA|BC) three-center term, for which
the error is over 1000 kcal/mol. Given that the only explicit density dependence of the
Fock-like matrix in SCC-DFTB is introduced by the Mulliken term, and yet the relative
energetics of SCC-DFTB in practice do not introduce such errors, then some density
independent part of the model must be accounting for the difference. We will avoid such
uncontrolled approximations in the AAT approach.
3.2.3 SCC-DFTB Repulsive Energy
The SCC-DFTB total energy expression relies on the original TB form for the total
energy, in which a sum of repulsive atom-pair potentials is used. An error is introduced
because of this reliance on atom-pair potentials to describe quantities that are inherently
density dependent. Under an ideal parameterization, using the same argument as
used earlier for NDDO, where we assume the effective one-particle model SCC-DFTB
Hamiltonian returns the correct density, the error in the SCC-DFTB energy to the exact
form is
ESCC − EKS =∑
A,B>A
ESCCAB (RAB)− Exc[ρ] +
1
2
∫vxc(r)ρ(r)dr −
∑A,B>A
ZAZB
RAB
(3–20)
The numerical behavior of this error is shown in Figure 3-3. Clearly, this repulsive
potential is carrying the burden of a portion of the electronic energy. This is not by
accident; the value of a function (in this case the total energy) that results from the
66
cancellation of two unrelated large functions (the electronic and nuclear repulsion energies)
is easier to parameterize against than the values of the larger component functions.
Alternatively, if the parameterization is against the larger functions there is a risk that
any error introduced in the parameterization will be magnified when they are subsequently
combined. If only the total energies or heats of formation (as in NDDO), and hence
only some combined function, are of interest then such an average approach would be
suitable. However, for our purposes we are interested in two separate and distinct sets of
information, the electronic properties and the total energy. Because of this added demand
on our semiempirical method we must generate the individual components of the total
energy expression with equal accuracy. We recognize three distinct components: first-order
electronic energy, residual electronic energy, and nuclear-nuclear repulsion. The first-order
electronic component has been the focus of most semiempirical method development, and
yields, in addition to the first-order electronic energy, the first-order electronic properties:
IPs, EAs, dipole, etc. The residual energy term presents a unique challenge, though
we will demonstrate how it can be included in a computationally efficient way. The
nuclear-nuclear repulsion is trivial to calculate and the only complication arises when
screened charges must be used if we use a valence-only Hamiltonian.
3.3 Systematic Comparison of HF, NDDO, DFTB, SCC-DFTB, KS-DFT in aSingle Framework
One of the goals of this research is to bring many seemingly disparate approximate
methods under a single common framework. For now, we will not deal explicitly with
so-called post-HF methods that include correlation based upon a reference single
determinant (because of the required computational effort). Instead, we are trying to push
the limits of what can be done within a one-particle theory. The problem we are solving is
the set of Hartree-Fock-Roothaan matrix equations, FC = SCε, for a general one-particle
operator f . We use the notation that F = H + G, where H is the density independent
contribution and G is the density dependent contribution to F. The spin-free one-particle
67
density matrix P is defined by a set of coefficients C for the real spin-unrestricted case,
Pµν =N∑i
(CαµiC
ανi + Cβ
µiCβνi) (3–21)
This dependence allows us to use a self-consistent field (SCF) procedure: determine the
Fock-like matrix from a guess density and determine a new set of coefficients and the
corresponding density, repeat until the density output is within some threshold of the
input density. Alternatively, FPS = PSF and 〈a|heff |i〉εi−εa
= 0 would be good measures of
convergence.
Upon convergence of the SCF equations we have a set of AO coefficients that define
the set of MOs
φi =∑
µ
Cµiχµ, (3–22)
a reference single determinant wavefunction is then generated as an antisymmetrized
product of the MOs,
Φ0 = A{φ1(1)φ2(2) . . . φn(n)} (3–23)
It is useful at this point to define the first-order electronic energy given a zeroth-order
wavefunction, |Φ0〉. The exact electronic Hamiltonian is
H =∑
i
h(i) +∑i,j>i
1
rij
, (3–24)
where the core Hamiltonian is
h(1) = −1
2∇2
1 −∑
A
ZA
r1A
(3–25)
and in its matrix representation the core Hamiltonian is
Hcoreµν =
∫χµ(r1)h(r1)χν(r1)dr1 (3–26)
68
The electronic energy correct through first order is
E = 〈Φ0|H|Φ0〉
= Tr[PHcore] +1
2
∑i,j
〈ij||ij〉 (3–27)
Note that the first-order electronic energy is exactly the same as the HF energy expression
and furthermore has no explicit dependence on the particular form of the one-particle
operator f .
However, the goal is not the first-order electronic energy but the exact electronic
energy. We can consider the CC route, where we know that for a single determinant the
exact electronic energy is (with any choice of orbitals)
ECCelec = E +
∑
i<j,a<b
〈ij||ab〉(tabij + tai t
bj − taj t
ai ) +
∑i,a
〈i|f |a〉tai (3–28)
The CC energy expression provides great insight, but the explicit calculation of tabij and tai
amplitudes requires much more computational effort than we are able to expend for a fast
semiempirical model. A more natural environment for a semiempirical methods is given
by KS-DFT. Assuming that we have the exact exchange-correlation energy functional, the
exact KS-DFT energy is (for KS-DFT orbitals)
EKSelec = E +
1
2
∑i,j
〈ij|ji〉+ Exc[ρ]
= Tr[PHcore] +1
2
∑i,j
〈ij|ij〉+ Exc[ρ] (3–29)
There is another form of the KS-DFT energy expression that is more useful for the
current development, though it requires that we use a slightly more specific form for
our one-particle operator. If the target is a one-particle Hamiltonian that returns the
correct electronic density, and hence the correct first-order electronic properties, then the
ideal one-particle operator is indeed the KS-DFT one-particle operator. And, if only the
total operator and not its individual components, i.e., −12∇2, νext(r), νJ(r), and νxc(r), is
69
pertinent then the first-order energy expression given by Equation 3–29 would be suitable.
By separating terms that are dependent on the density (G) we can write
EKSelec =
1
2Tr[P(2HKS + GKS)]− 1
2Tr[PGKS
xc ] + Exc[ρ] (3–30)
where
(GKSxc )µν = 〈µ|vxc(r)|ν〉 (3–31)
and F = H+G. Equation 3–30 is a central theme of our work, as it gives the unambiguous
definition of the exact electronic energy. Tied to it are the one-particle KS equations that
define the orbitals, the density and associated first-order properties. In this way, the
problem for the total energy and gradients is completely specified by
• F that generates the ground state one-particle density matrix P.
• F that can be decomposed into density independent and (H) and dependent (G)components.
• The exchange-correlation energy functional.
3.3.1 General Energy Expression
We now choose a general electronic energy expression. Including the nuclear-nuclear
repulsion we have the total energy expression,
E =1
2Tr[P(2H + G)] + Eresidual[P] +
∑A,B>A
ZAZB
RAB
(3–32)
We will bring HF, NDDO, DFTB, SCC-DFTB and KS-DFT into this framework, to make
it clear what the three distinct components are:
• Density independent H .
• Density dependent G.
• Residual electronic energy Eresidual[P].
If a method is to give the correct electronic density, and we can know the exact
KS-DFT HKS and GKS, then the model components for H and G absolutely must
70
correspond to them. There is no flexibility with this interpretation. In addition, if we
have the F that yields the correct density then to get the exact total energy, Eresidual[P]
must be EKSresidual[P]. Clearly, we are dealing with approximate models, though instead of
defining the target or reference as being a single number, such as an experimental heat
of formation, we instead have three separate and distinct functional forms, for which the
exact forms are, in principle, known.
3.3.2 Density Independent Fock-Like Matrix Contribution
This component generally accounts for nuclear-electron attraction and the electron
kinetic contribution. The following definitions are based on µ ∈ A and ν ∈ B.
• HF:
HHFµν = 〈µ| − 1
2∇2
1 −∑
A
ZA
r1A
|ν〉 (3–33)
• NDDO:
Hµν =
{Uµµ −
∑B 6=A ZB(µµ|BB) if µ = ν
12Sµν(βµ + βν) if µ 6= ν
(3–34)
• DFTB:
HDFTBµν =
1
2H0
µν (3–35)
• SCC-DFTB:
HSCC−DFTBµν =
1
2H0
µν −1
2Sµν
N∑
ξ
(γAξ + γBξ)q0ξ (3–36)
• KS-DFT:
HKSµν = 〈µ| − 1
2∇2 −
∑A
ZA
r1A
|ν〉 (3–37)
The only density independent contribution to the ideal Fock-like matrix (the Fock-like
matrix which returns the correct one-particle ground state density) is the exact core
Hamiltonian, Hcore. The biggest bottleneck for computing contributions for Hcore is the
three-center one-electron integrals, which scale as n2orbitalsNatoms. Still, they represent
a small computational effort for large systems when compared to the n3orbitals terms
71
for the three-center two-electron integrals. Though there is some cancellation, the
possibility of complete cancellation among the three-center one-electron attraction and
two-electron repulsion integrals is limited. All three-center one-electron terms appear only
in off-diagonal blocks of the Fock-like matrix and are small compared to the dominant
three-center two-electron terms which appear primarily in the diagonal blocks.
3.3.3 Density Dependent Fock-Like Matrix Contribution
This component generally accounts for electron-electron contributions.
• HF:
GHFµν =
∑
λσ
Pλσ((µν|λσ)− 1
2(µλ|νσ)) (3–38)
• NDDO:
GNDDOµν =
∑
λσ
Pλσ((µν|λσ)δABδCD − 1
2(µλ|νσ)δADδBC) (3–39)
• DFTB:GDFTB
µν = 0 (3–40)
• SCC-DFTB:
GSCC−DFTBµν =
∑
λσ
Pλσ[1
4SµνSσλ(γAC + γBC + γAD + γBD)] (3–41)
• KS-DFT:GKS
µν = 〈µ|vJ [P ](r) + vxc[P ](r)|ν〉 (3–42)
The density dependent term of the ideal Fock-like matrix is GKS. This choice
follows directly from the chosen form for the density independent component and the
requirement that we reproduce the exact electronic density. The density dependence of
the Fock-like matrix accounts for the complex interactions among electrons. This term
is dominated by Coulombic repulsion. The second most important term is generally
called exchange. The third component is correlation, and can be considered to be the
highly complex way in which electrons interact with one another, i.e., their motion is
72
correlated. For large systems, the density dependent part of a Fock-like matrix is by
far the most computationally costly. Unlike the density independent part, this term
must be recalculated for every iteration of the SCF procedure. In addition, the terms
involved here are more computationally intensive, e.g., the four-center integrals and the
numeric integration required for the full treatment of vxc. Generally, reduction in the
computational cost of the density dependent term translates into an equivalent reduction
in cost for the entire calculation. Therefore, the bulk of our attention is on systematic
approximations that can be applied that simultaneously reduce the computational cost but
maintain a desired accuracy.
3.3.4 Residual Electronic Energy
This term places no restriction on the Fock-like matrix and hence should have no
direct effect on the electronic density. However, its accurate determination is critical for
high-quality electronic energies and consequently total energies and structures.
• HF:EHF
residual = 0 (3–43)
• NDDO:
ENDDOresidual =
∑A,B>A
(Ecore−coreAB (RAB)− ZAZB
RAB
) (3–44)
• DFTB:
EDFTBresidual =
∑A,B>A
(EDFTBAB (RAB)− ZAZB
RAB
) (3–45)
• SCC-DFTB:
ESCC−DFTBresidual =
∑A,B>A
(γABq0Aq0
B + ESCCAB (RAB)− ZAZB
RAB
) (3–46)
• KS-DFT:
EKSresidual = −1
2Tr[PGKS
xc ] + Exc[ρ] (3–47)
73
Because we insist that an approximate Fock-like matrix reproduce the KS Fock-like
matrix, and thereby define the first-order electronic energy, the residual electronic energy
must be EKSresidual. The residual electronic energy and its combination with the unrelated
nuclear-nuclear repulsion term in SE methods is notorious. We have attempted to include
this term in a simple and computationally efficient way in our method. If our Fock-like
operator included exact (non-local) exchange then this term would introduce only the
residual correlation energy contribution. Although the correlation energy is a small
component (< 1%) of the total energy, its accurate inclusion is required to obtain chemical
accuracy.
3.3.5 General Gradient Expression
One of the overarching objectives of our work is to create a model Hamiltonian
to drive large-scale MD. Achieving this goal requires an efficient method for obtaining
gradients. Given the general energy expression provided by Equation 3–30, the gradient is
[81]
dE
dα= −2
∂Sµν
∂α
∑i
CµiCνiεa + Pνµ[∂Hµν
∂α+
1
2
∑
λσ
Pσλ∂Gµν
λσ
∂α] +
∂Eresidual
∂α(3–48)
which is based on∑
i Cνi(Fµν − εiSµν) = 0 and orthonormality. Note that there is no
explicit dependence on ∂P∂α
. Equation 3–48 only holds if terms in G are linear in the
density
Gµν =∑
λσ
PσλGµνλσ (3–49)
The derivatives of Eresidual, which are not present in HF, are simple for those
methods that only depend on a sum of atom-pair terms, such as NDDO, DFTB, and
SCC-DFTB. In KS-DFT additional numerical integration is required for the perturbed
density. Potential approaches for reducing the computational cost associated with the
derivatives in KS-DFT will be discussed in further detail later.
74
3.3.6 Requirements for an Accurate Simplified Method
We have discussed the features of a general simplified method aimed at large systems
that can be used to reliably obtain electronic properties as well as energetics. It is clear
that the framework is essentially that of KS-DFT, though we have structured the basic
equations in a form amenable to further approximation. The two essential elements for the
determination of electronic structure are the terms in the Fock-like matrix that are either
completely independent (H) or dependent (G) on the electronic density. The final element,
which is needed to acquire accurate energetics and structures, is the residual electronic
energy term. Furthermore, we have well-defined exact limits for each of these terms. When
we discuss the AAT approach we have unambiguous connections to DFT and, from ab
initio DFT, an analogous connection to WFT and the transfer Hamiltonian.
75
-6
-4
-2
0
2
4
6
1 1.5 2 2.5 3
Ene
rgy
(Har
tree
)
C-N reaction coordinate (Angstrom)
G3,4-C
GFull
G2-C
GNDDO
Figure 3-1. Multi-center contribution to the C2sN2s matrix element of NMT.
76
-10
0
10
20
30
40
50
1 1.5 2 2.5 3
Ene
rgy
(kca
l/mol
)
C-N reaction coordinate (Angstrom)
AM1 UHF
Figure 3-2. The AM1 artificial repulsive bump for NMT direct bond fission.
77
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
1 1.5 2 2.5 3
Ene
rgy
(Har
tree
)
C-C reaction coordinate (Angstrom)
EtotErep
Eelec
Figure 3-3. The SCC-DFTB total energy breakdown.
78
A
0
200
400
600
800
1000
1200
1 1.5 2 2.5 3 3.5
Err
or f
rom
HF
(kca
l/mol
)
C-N reaction coordinate (Angstrom)
RudenbergMulliken
B
10
20
30
40
50
60
70
80
90
100
1 1.5 2 2.5 3 3.5
Err
or f
rom
HF
(kca
l/mol
)
C-N reaction coordinate (Angstrom)
RudenbergMulliken
C
-2 0 2 4 6 8
10 12 14 16 18 20
1 1.5 2 2.5 3 3.5
Err
or f
rom
HF
(kca
l/mol
)
C-N reaction coordinate (Angstrom)
RudenbergMulliken
Figure 3-4. Energies from Rudenberg and Mulliken approximations. A) Three-center(AABC). B) Three-center (ABAC). C) Four-center (ABCD).
79
CHAPTER 4ADAPTED AB INITIO THEORY
We have specified the target framework of our simplified quantum mechanical
method. The task now is to demonstrate simple well-controlled approximations that
significantly reduce computational effort while maintaining a desired level of accuracy. To
this end, a primary focus is on the density dependent component of the Fock-like matrix,
as it is the most computationally intensive part of the SCF procedure. Furthermore, the
residual electronic energy contribution is examined because we have shown that its proper
treatment is essential for a SM to achieve uniform accuracy of electronic structure and
energetics.
Our strategy is to start with a reference theoretical method that is reasonably
well-behaved, in this case KS-DFT. Though there is much debate about the best DFT
energy functional, we choose a hybrid DFT functional, specifically the widely-used
B3LYP, because it includes a portion of exact exchange which adds a substantial degree
of flexibility to our simplified method. To simplify the computational procedure we
then explore various approximations and verify their accuracy. It may be too much to
demand a simplified method that is systematically improvable (in the CC wavefunction
theory sense), however, every approximation should be systematically verifiable. We
will show that by making a couple of well-controlled approximations we are able to
reproduce the topographical features of a hybrid DFT PES with a substantial reduction in
computational effort.
4.1 Two-Electron Integrals
Explicit calculation of two-electron integrals is a computational bottleneck in DFT
because the number of integrals scales as n4orbitals. For the Coulomb integrals this can
be reduced to n3orbitals by fitting the density prior to the evaluation of the integrals [82],
but this cannot be done for the exact exchange, leaving n4orbitals dependence. Though
most quantum chemistry programs employ screening techniques to reduce the number
of two-electron integrals calculated, they remain a significant cost, especially when exact
80
exchange is included. There are two ways to lower the computational cost associated with
two-electron integrals: first, the number of integrals can be lowered by applying screening
methods; second, the cost associated with the evaluation of each individual integral can
be reduced. Several techniques for reducing the number of integrals have been successfully
applied. They generally involve a procedure that recognizes a threshold for significant
contributions to the Fock matrix. These include fast matrix multipole (FMM) [83–85], the
linear exact exchange [86], and quantum chemical tree code (QCTC) [87]. These methods
still rely on the explicit calculation of integrals using standard techniques for Cartesian
Gaussian-type functions. For large systems the most numerous class of integrals are the
four-center integrals. Furthermore, four-center integrals are not only the largest class of
integrals, but also the most computational involved integrals. According to Gill et al. [88]
a four-center integral is typically an order of magnitude more computationally demanding
than a two-center integral. Our focus here is on systematic approximations that can be
made to these ab initio multi-center integrals that will reduce the computational effort
associated with calculating these terms while retaining acceptable accuracy.
The cornerstone of our approach is based on ideas originally presented by Rudenberg
in 1951 [5]. The underlying approximation is to expand an orbital on center A in terms of
a set of auxiliary orbitals located on center B:
φµA(1) =
∞∑
k=1
SkB , µAφkB
(1) (4–1)
where
SkB , µA= 〈kB|µA〉 (4–2)
If the auxiliary orbitals on center B are complete and orthogonal then the expansion is
exact. By repeated application of Equation 4–1 the number of centers that are involved in
three- and four-center two-electron integrals can be reduced. The possible combinations
of this type of expansion has been explored and enumerated by Koch et al. [89], though
without comparing the numerical accuracy of such approximations. We will show the
81
numerical precision of the Rudenberg approximation in an effort to avoid the explicit
calculation of the numerous four-center integrals. By reducing the cost of each four-center
integral we improve the most computationally expensive step of building the Fock matrix
for large systems.
First, consider the effect of the orbital expansion on a two-center product of two
orbitals with respect to the same electron coordinate,
φµA(1)φνB
(1) =1
2
∞∑
k=1
[SµAkBφkB
(1)φνB(1) + SkAνB
φµA(1)φkA
(1)] (4–3)
Since a general multi-center two-electron integral is given as
〈µAλC |νBσD〉 = (µAνB|λCσD) =
∫ ∫φµA
(1)φνB(1)φλC
(2)φσD(2)
r12
d1d2 (4–4)
substituting Equation 4–3 into Equation 4–4 for both the AB and CD orbital products
yields the following expression, which is the Rudenberg two-electron integral form,
(µAνB|λCσD) =1
4
∞∑
k=1
∞∑j=1
[SµAkBSλCjD
(kBνB|jDσD) + SµAkBSjCσD
(kBνB|λCjC)
+SkAνBSλCjD
(µAkA|jDσD) + SkAνBSjCσD
(µAkA|λCjC)]
(4–5)
In the limit that the sums over auxiliary orbitals k and j are complete this expansion
is exact. In practice, the expansion is approximated by limiting the orbitals over which
we are expanding to those included in the AO basis functions, though using a separate
auxiliary set of orbitals may also be suitable.
The Rudenberg two-electron integral expression (Equation 4–5), is more general than
the well-known Mulliken two-electron integral approximation. The Mulliken approximation
is equivalent to restricting the sum in Equation 4–1 to be only over orbitals of the same
type,
φµA(1)φνB
(1) =1
2SµAνB
[φµA(1)φµA
(1) + φνB(1)φνB
(1)] (4–6)
82
which captures a large portion of the Rudenberg expansion and when applied to
two-electron integrals yields,
(µAνB|λCσD) =1
4SµAνB
SλCσD[(µAµA|λCλC) + (µAµA|σDσD)
+(νBνB|λCλC) + (νBνB|σDσD)].
(4–7)
Note the similarity of this expression to that which appears in SCC-DFTB, Equation
3–18, in which
γAB ≈ (µAµA|νBνB) (4–8)
though in SCC-DFTB the orbitals µ and ν are restricted to be s-type functions.
We consider here all eight different approximations that result from the application
of Equation 4–1 to three- and four-center two-electron integrals. The two classes of
three-center terms are of the form (µAνA|λBσC) and (µAνB|λAσC), which will be referred
to as types A and B respectively. To facilitate our discussion we introduce the numbering
convention in Table 4-1.
The Rudenberg and Mulliken formulas can be applied to either class of three-center or
the four-center integrals (approximations I-VI) using Equations 4–5 and 4–7, respectively.
For the partial approximations to four-center terms (VII and VIII), partial means that the
corresponding approximation has been applied only once. The expansion for these partial
approximations is then in terms of three-center two-electron integrals (instead of only
two-center terms). For the Mulliken partial (VII) we have
(µAνB|λCσD) =1
4{SµAνB
[(µAµA|λCσD) + (νBνB|λCσD)]
+SλCσD[(µAνB|λCλC) + (µAνB|σDσD)]}.
(4–9)
and the Rudenberg partial (VIII) is
(µAνB|λCσD) =1
4
∞∑
k=1
[SµAkB(kBνB|λCσD) + SkAνB
(µAkA|λCσD)
+SλCkD(µAνB|kDσD) + SkCσD
(µAνB|λCkC)]
(4–10)
83
With all eight of the two-electron integral approximations now defined, the accuracy
of each will be considered. Since one requirement of our approach is that it reproduce the
PES that would arise from a full ab initio treatment, we require that our approximations
be uniformly accurate over the entire PES. This is in contrast with many commonly
used semiempirical methods, in which the focus is typically on accuracy only about an
equilibrium structure. However, uniform accuracy is critical to getting accurate forces
to drive large-scale MD and to aid in locating transition states for rate constant studies.
Unfortunately there is no simple measure for uniform accuracy, due to the complexity of
the 3N − 6 dimensional PES, so our analysis will be statistical.
Note that the total electronic energy is the sum of all the one-, two-, three-, and
four-center components. And, each of these components individually bear little similarity
to the total electronic energy, again consider each contribution to any particular matrix
element shown in Figure 3-1, instead the total energy is the result of the delicate
cancellation among these terms. In addition, since only the non-parallelity errors are
of importance in reconstructing the PES, the energetic contribution arising from each
individual class of terms may be shifted by a constant value independently of one another
without affecting the features of a PES. These subtleties, compounded by the 3N − 6
degrees of freedom of the PES, make ascribing significance to any single class of terms
difficult. Because we want to assess the accuracy of a particular approximation as it
applies to a single class of terms we will instead focus on the Fock matrix elements, which
contain all the information needed for the zeroth-order electronic energy, to overcome some
of these difficulties.
Given any set of points on the PES (for example, a dissociation curve) and the Fock
matrix for each point, each class of multi-center two-electron integrals will contribute a
particular percentage to the full density dependent Fock matrix. The best approximation
for a particular class of integrals is then given by how well it reproduces the average
percentage of the Fock matrix for the corresponding ab initio set of integrals. Consider
84
the scatter plots of the full density-dependent contribution of the Fock matrix versus the
individual contributions arising from the multi-center terms. Performing a least-squares
linear regression yields a line whose slope is a measure for the overall contribution from
each term. Summing the one-, two-, three-, and four-center slopes yields unity. An
individual slope is, in a sense, the average percentage of the density-dependent part of the
Fock matrix elements (entries). However, it should not be interpreted as the percentage
of the energetic contribution. Because we are interested in not only the energy but also
the density (which is given by the Fock matrix) then this average percentage serves
as an acceptable measure of the importance of each term. Furthermore, in assessing
approximations to individual classes of terms then we have an average way of measuring
its significance. This is useful because the contribution from each multi-center component
can vary greatly from molecule to molecule and also depending on internuclear separation.
Take as an example NMT, and the dissociation of the C-N bond. Plotted in Figure 4-1
are the scatter plots for each multi-center contribution to the Coulomb and exchange
contributions of the Fock matrix from HF theory in a minimal basis. The C-N reaction
coordinate ranges from 1.0 A to 5.0 A in steps of 0.2 A . The slopes of the corresponding
best-fit lines of the one-, two-, three- and four-center contributions are 0.0063, 0.5199,
0.4522, and 0.0216 respectively, note that these slopes represent the fraction of the full
contribution, and that their sum is one.
Table 4-2 shows the statistical averages of the density-dependent contributions to the
Fock matrix for several molecules along the specified reaction coordinate (Rxn) at the HF
level of theory in a minimal basis set. The dissociation in all cases is from 1.0 A to 5.0 A
in steps of 0.2 A . On average, the two-center two-electron integral component accounts
for roughly half of the full matrix elements and the three-center type A integrals recover
another third. Again, the corresponding energetic contributions will not exhibit the
same ratios. The purpose here is to validate individual approximations made to various
multi-center two-electron integrals. To this end we consider the average contribution: the
85
better the approximation to a particular multi-center class of integrals the closer their
average contributions will be.
The a priori best approximation to the two types of three-center contributions is that
of Rudenberg. Shown in Figure 3-4 are the errors introduced by making approximations
I-VI, for NMT direct bond fission along the carbon-nitrogen reaction coordinate. For type
A and B three-center integrals (3-C A and 3-C B) the average matrix contributions are
35.67% and 4.98% respectively. The Rudenberg approximations II and IV differ by about
half a percent at 35.12% and 5.50%. Though the percent error for both approximations
II and IV are roughly the same, the corresponding error introduced energetically is two
orders of magnitude larger for the (AA|BC) three-center terms than for the (AB|AC)
three-center terms. In contrast, the Rudenberg approximation applied to four-center
terms (VI) is 2.47% which differs from the four-center ab initio (4-C) value of 2.39%
by only 0.08%. The error introduced by the full Rudenberg four-center approximation
(VI) is surprisingly small. One would expect that the partial Rudenberg approximation
would have the best agreement with the exact value, instead, at least empirically, the
full Rudenberg approximation performs best, and has the smallest error relative to the
average contribution to the Fock-like matrix. Indeed, the effect this 0.08% difference has
on the energetics of a dissociation curve is within an acceptable range and reproduces the
energetics well for a variety of different dissociation curves. We will focus on the molecules
in Table 4-2 that involve the dissociation of a C-N bond. Plotted in Figure 4-2 are the
dissociation curves for NMT, nitroethane (NET), COHNO2, and CH3NH2 at the B3LYP
level of theory in a minimal basis set. There is a clear trend for the four-center Rudenberg
approximation to overestimate the average contribution to the Fock-like matrix. The
energetics, on the other hand, do not show this clear-cut trend since the energy is lowered
for NET, raised for CH3NH2, and essentially unchanged for NMT. However, even though
the PES are not precisely reproduced they still exhibit all the basic features of the ab
86
initio surface, surprisingly without the explicit calculation of four-center two-electron
integrals.
We have demonstrated that the Mulliken and Rudenberg approximations do not
approximate the three-center one- or two-electron integrals well. These classes of integrals
pose a challenge as they are the only remaining terms that require explicit integration
since all one-center terms can be tabulated and two-center terms can be interpolated by
cubic splines, as will be demonstrated later. However, based on the Gaussian product rule
these terms can, without approximation, be reduced to two-center terms. In principle then
cubic spline interpolation could be used to store the intermediates and completely avoid
the need to do any integrals explicitly. The Gaussian product rule is
e−αµ(r−RA)2e−αν(r−RB)2 = e− αµαν
αµ+αν(RA−RB)2
e−(αµ+αν)(r−αµRA+ανRB
αµ+αν)2
(4–11)
We can generate an auxiliary set of orbitals for each Gaussian-type orbital-pair that have
the orbital exponent αµ +αν . Also, because products of l = 1 (p-type) orbitals are involved
this auxiliary set must also now span l = 2 (d -type). The prefactor is not included in the
stored orbitals, instead it is computed on-the-fly to retain the two-center character of the
terms in which it will be used.
There is a significant difficulty with using the Gaussian product in a simple
way. Because we work in contracted Gaussian-type orbitals the number of orbital
pairs increases as the number of contracted functions squared. The centers of these
resulting product Gaussians are not necessarily the same, αµRA+ανRB
αµ+αν. To use cubic spline
interpolation an auxiliary orbital must have a constant exponent, e.g., αµ + αν in the case
of the simple product of uncontracted Gaussians. We envision an approximation to the
exact contracted Gaussian product, which is a sum of several Gaussians with different
centers and different exponents, that is a single contracted Gaussian located at a single
center. The approximate auxiliary Gaussian could be determined by fitting directly to the
87
exact product. However, this fit will generally yield different exponents and coefficients for
two contracted Gaussian functions as there separation changes.
If we consider the basis set STO-nG, a sum of contracted Gaussians approximate a
Slater-type orbital. There is no Slater product rule that reduces a product of two Slaters
to a single Slater. Instead we have
e−αµ|r−RA|e−αν |r−RB | = e−αµ|r−RA|−αν |r−RB | (4–12)
Consider a simple example of the product of two Gaussian functions versus two Slater
functions separated by one Bohr, which is represented by the blue curve in Figure 4-3.
The challenge is now clearer, the underlying basis functions we use are fit to Slater
functions, and the product of two Slater functions does not resemble a simple Gaussian,
or even a sum of contracted Gaussians. The Slater product is defined exactly by the
intersection of two functions:
e−αµ|r−RA|e−αν |r−RB | = e−αγ |r−RC | ∩ e−αδ|r−RD| (4–13)
where, for αµ < αν and RA < RB
γ = αµ + αν (4–14)
δ = αν − αµ (4–15)
RC =αµRA + ανRB
αµ + αν
(4–16)
RD =αµRA − ανRB
αµ − αν
(4–17)
And for this particular case RA < RC < RB < RD. For the Slater orbital product
given in Figure 4-3 the exponents used were αµ = αν so αγ = αµ
2, αδ = 0, RC = 0, and
RD → ∞. The purple curve is e−αγ |r−RC | which is truncated by a horizontal line that
results from e−αδ|r−RD|. For αµ 6= αν the situation is somewhat more complicated. However
we now have a more direct path to two-center intermediate terms that have a constant
exponent for each atom-pair, and, are therefore, amenable to cubic spline interpolation
88
with universally transferable parameters. The difficulty still remains on how best to
incorporate the intersection of these two functions in a practical way.
We have developed the strategy and framework for eventually including the
three-center terms. For now we are satisfied with reducing the computational cost through
only the four-center Rudenberg approximation because the four-center terms account for
most of the computational effort associated with construction of the Fock matrix for large
systems.
4.2 Kohn-Sham Exchange and Correlation
If we envision an approximation to KS-DFT then a primary focus ought to be
avoiding the numerical integration that must be performed for each SCF iteration. The
formal scaling of the numerical integration step is n3orbitals, as it is performed in real space.
In practice, the scaling is lower because the exchange-correlation potential is local and
the density decays rapidly. Still, we want to avoid any extra computational effort that
contributes little to the overall energetics. To approximate the exchange-correlation
potential contribution consider the expansion of the potential in terms of atom-centered
densities, ρ =∑
A ρA. Then we have the following telescoping series
vxc[∑
A
ρA](r) =∑
A
vxc[ρA](r) +∑A>B
{vxc[ρA + ρB](r)− vxc[ρA](r)− vxc[ρB](r)}
+∑
A>B>C
{vxc[ρA + ρB + ρC ](r)
− vxc[ρA + ρB](r)− vxc[ρA + ρC ](r)− vxc[ρB + ρC ](r)
+ vxc[ρA](r) + vxc[ρB](r) + vxc[ρC ](r)}+ · · ·
=v1−centerxc + v2−center
xc + v3−centerxc + · · · (4–18)
Equation 4–18 is exact. It only requires an arbitrary partition of the density into
atomic parts, though convergence of the series is affected by the choice of partition.
Our target is to include contributions from the exchange-correlation potential to the
KS Fock-like matrix in a way that only requires the storage of a limited set of two-center
89
terms. Such two-center terms can be interpolated using cubic splines once, then the
cubic spline coefficients can be stored in RAM to be efficiently retrieved and computed
on-the-fly. With such a target in mind only terms in Equation 4–18 that can be retained
are v1−centerxc and v2−center
xc . Furthermore, such contributions ought to be independent of
chemical environment and thus completely transferable. Therefore, we insist on densities
that are transferable. The natural choice is spherically symmetric atom-centered densities
of neutral atoms, ρA ≈ ρ0A, recognizing we can have α and β spin components. Based on
these considerations we have an approximate exchange-correlation potential,
vxc[∑
A
ρA](r) ≡∑
A
vxc[ρ0A](r) +
∑A>B
[vxc[ρ0A + ρ0
B](r)− vxc[ρ0A](r)− vxc[ρ
0B](r)] (4–19)
We do not want to map and store the real-space exchange-correlation potential,
because of the obvious high degree of complexity. Instead, all that is ever needed are
the contributions from the potential projected in the basis set. The Fock-like matrix
contribution from this approximate exchange-correlation potential is
(Gxc)µAνB[ρ] = 〈µA|vxc[ρ]|νB〉 (4–20)
Restricting this to contributions that only depend on two centers yields, for the diagonal
block (A = B)
(Gxc)µAνA=〈µA|vxc[
∑A
ρ0A]|νA〉
=〈µA|vxc[ρ0A]|νA〉
+1
2
∑
B 6=A
〈µA|vxc[ρ0A + ρ0
B]− vxc[ρ0A]|νA〉
(4–21)
and for the off-diagonal block (A 6= B)
(Gxc)µAνB=〈µA|vxc[
∑A
ρ0A]|νB〉
=〈µA|vxc[ρ0A + ρ0
B]|νB〉(4–22)
90
The off-diagonal block terms can be further improved by applying the Rudenberg
approximation. Note that the diagonal block terms do not benefit from this approximation
because the orbitals only depend upon one center. The class of neglected terms 〈µA|vxc[ρC ]|νB〉becomes
〈µA|vxc[ρ0C ]|νB〉 =
1
2
∑
C 6=A,B
∞∑
k
[SµAkB〈kB|vxc[ρ
0C ]|νB〉+ SkAνB
〈µA|vxc[ρ0C ]|kA〉] (4–23)
The leading order terms are
(G0xc)µAνB
=
〈µA|vxc[ρ
0A]|νA〉 if A = B
〈µA|vxc[ρ0A + ρ0
B]|νB〉 if A 6= B(4–24)
and a slight improvement could be achieved with the addition of
(G1xc)µAνB
=
12
∑B 6=A〈µA|vxc[ρ
0A + ρ0
B]− vxc[ρ0A]|νA〉 if A = B
12
∑C 6=A,B
∑∞k [SµAkB
〈kB|vxc[ρ0C ]|νB〉+ SkAνB
〈µA|vxc[ρ0C ]|kA〉] if A 6= B
(4–25)
Shown in Figure 4-4 are results for the GZERO(B3LYP) approximation: G0xc is used with
the B3LYP energy functional interpolated over all the atom pairs CNOH. The errors
introduced are relatively small and GZERO(B3LYP) appears to be generally transferable.
In terms of computational feasibility for GZERO we have introduced a single
one-center matrix per atom-type and a single two-center matrix per atom-pair. For C,
N, O and H atom types the number of one-center matrices is trivial and does not present
a challenge, as they are the dimension of the number of orbitals per atom squared. The
two-center matrices are stored in the local coordinate framework of the atom pair and are
roughly the dimension of the number of orbitals on atom A times the number of orbitals
on atom B. The actual dimension can be reduced because most of the matrix elements are
zero due to orthogonality. To interpolate the values of these two-center matrix elements
cubic splines are used. Cubic splines require five parameters per number interpolated. We
used adaptive grids (non-uniformly spaced) that are defined for every pair term from 0.5
91
A to infinity (effectively). To a good level of numerical precision this fit can be performed
with roughly 600 reference points, but more can be added should higher levels of precision
be required. For CNOH in a minimal basis set (there are ten unique atom pairs) splines
require roughly half a million double precision numbers. Going to a double zeta type basis
increases this storage requirement by about a factor of nine. Even for the double zeta case
that cost does not present a challenge.
4.3 Adapted ab initio Theory Model Zero
To briefly summarize, we have applied two novel approximations to speed up the
construction of the Fock-like matrix in a KS-DFT framework. We have shown that the
Rudenberg approximation applied to four-center two-electron integrals retains the features
of the ab initio integrals, thus avoiding the most costly class of Cartesian Gaussian
integrals. Furthermore, this approximation is readily applied to hybrid KS-DFT methods,
which include some fraction of exact exchange.
In addition to approximations for integrals we also demonstrated approximations that
can be applied to the exchange-correlation potential contribution of the Fock-like matrix,
and thereby avoid the need to perform numerical integration for every SCF iteration.
Extensions to include three-center contributions from the telescoping series approximation
to the potential were also presented.
The last component we will consider involves the residual total energy. Though it
is worthwhile to avoid numerical integration to generate the potential and subsequent
matrix contribution, a final one-time evaluation of Exc[ρ] upon SCF convergence should
not be a bottleneck. Furthermore, as demonstrated by Bartlett and Oliphant [90] the
exchange-correlation energy functional is relatively insensitive to the input density. Indeed,
as we have seen already in Figures 4-2 and 4-4 the B3LYP energy evaluation was used and
introduced no significant error.
92
The current implementation of AAT (AAT Model Zero) is given by
HAATµν = 〈µ| − 1
2∇2
1 +∑
A
ZA
r1A
|ν〉 (4–26)
all one-electron terms are calculated exactly as they represent a small portion of the
total computational cost for large systems. Given that we are using a hybrid KS-DFT
functional we have a fraction of exact exchange in the density dependent Fock-like matrix,
for B3LYP, α = 0.2.
(GJ−αK1,2,3−center)µν =
∑
λσ
Pλσ((µν|λσ)− α(µλ|νσ)) (4–27)
(GJ−αK4−center)µν =
∑
λσ
Pλσ( ˜(µν|λσ)− α ˜(µλ|νσ)) (4–28)
where
˜(µν|λσ) =1
4
∑
k=1
∑j=1
[SµAkBSλCjD
(kBνB|jDσD) + SµAkBSjCσD
(kBνB|λCjC)
+SkAνBSλCjD
(µAkA|jDσD) + SkAνBSjCσD
(µAkA|λCjC)]
(4–29)
(G0xc)µν =
〈µA|vxc[ρ
0A]|νA〉 if A = B
〈µA|vxc[ρ0A + ρ0
B]|νB〉 if A 6= B(4–30)
GAATµν = (GJ−αK
1,2,3−center)µν + (GJ−αK4−center)µν + (G0
xc)µν (4–31)
and the residual electronic energy is
EAATresidual = −1
2Tr[PG0
xc] + Exc[P] (4–32)
Finally, shown in Figure 4-5 are UHF dissociation curves for AAT Model Zero (AATM0),
where both the Rudenberg and GZERO(B3LYP) approximations have been applied
simultaneously. This model reproduces the underlying B3LYP dissociation curves well
especially when compared to AM1.
The corresponding gradient expression for AATM0 is given by Equation 4–34. The
only difference between standard KS-type derivatives arise from the modified four-center
93
term dependence on overlap. The evaluation of terms ∂ ˜(µν|λσ)∂α
is straight-forward and only
dependent upon derivatives of the overlap, ∂Sµν
∂α, and two-center NDDO-type integrals,
∂(µAνA|λBσB)∂α
.
∂ ˜(µν|λσ)
∂α=
1
4
∑
k=1
∑j=1
[∂SµAkB
∂α{SλCjD
(kBνB|jDσD) + SjCσD(kBνB|λCjC)}
+∂SkAνB
∂α{SλCjD
(µAkA|jDσD) + SjCσD(µAkA|λCjC)}
+∂SλCjD
∂α{SµAkB
(kBνB|jDσD) + SkAνB(µAkA|jDσD)}
+∂SjCσD
∂α{SµAkB
(kBνB|λCjC) + SkAνB(µAkA|λCjC)}
+SµAkBSλCjD
∂(kBνB|jDσD)
∂α+ SµAkB
SjCσD
∂(kBνB|λCjC)
∂α
+SkAνBSλCjD
∂(µAkA|jDσD)
∂α+ SkAνB
SjCσD
∂(µAkA|λCjC)
∂α]
(4–33)
dE
dα= −2
∂Sµν
∂α
∑i
CµiCνiεi + Pνµ[∂{HAAT
µν + (G0xc)µν}
∂α+
1
2
∑
λσ
Pσλ∂{(Gµν
λσ)1,2,3−center + (Gµνλσ)4−center]}
∂α] +
∂Eresidual
∂α
(4–34)
4.4 Implementation
The implementation of the approximations outlined above must be efficient because
the construction of the Fock-like matrix can be a rate limiting step for large systems.
The Rudenberg approximation can be implemented in a stored integral or direct SCF
procedure. For the Rudenberg approximation to be competitive requires that the
contraction of the overlap with the appropriate two-center two-electron integrals be
significantly faster than the four-center two-electron integrals. Shown in Figure 4-6 are
the relative timings for the evaluation of the ab initio four-center integrals versus the
Rudenberg approximation. These timings are preliminary and more work could be done
to optimize both methods for evaluating this four-center contributions. However, it is
clear that there is a significant advantage to using the Rudenberg approximation and
94
the possibility of improving the speed over traditional DFT methods by two-orders of
magnitude is well within range. It is sufficient to focus only on timings for the four-center
integrals as they quickly become the dominate contribution for large systems, as shown in
Figure 4-7.
Screening of small two-electron integrals can lead to linear scaling for large systems.
Such screening methods can be applied to the Rudenberg approximation very efficiently
because of the explicit dependence on overlap.
In an effort to minimize computational effort with building the Fock-like matrix we
have made extensive use of storing transferable one-center and two-center terms. In our
current implementation all integrals up through three-centers are calculated exactly. There
are two reasons for explicitly evaluating these integrals: first, they are a small contribution
when compared to the explicit calculation of the four-center terms; second, the application
of the Rudenberg or Mulliken approximations to three-center terms (decomposing them
into two-center terms) introduces significant errors. If an accurate, simple and transferable
approximation for the three-center terms were available then all terms in the model could
be two-center. It is a straight forward matter to interpolate functions of two centers using
cubic splines. We have implemented such techniques for the evaluation of the matrix
elements of the exchange-correlation: (G0xc)µAνB
(RAB).
Cubic splines are applicable for numerical 2-center functions, where the exact
functional form is not known, and for complicated analytic 2-center functions. In the
first case, the benefit of splines is to provide an interpolation scheme for a set of empirical
reference points. In the second case splines provide a computationally efficient framework
for the evaluation of otherwise prohibitively expensive analytic functions. We are primarily
interested in the latter.
The ability to store a large number of parameters in computer memory (RAM) is
a requirement for the efficient implementation of cubic splines. The memory required
is directly determined by the number of cubic spline parameters. A pertinent example
95
is the evaluation of two-center two-electron integrals terms in MNDO, specifically, the
multipole-multipole expansion of the NDDO-type two-electron integrals. There are
twenty-two unique NDDO-type integrals for every atom-pair involving two heavy atoms
in standard NDDO methods such as AM1, PM3, etc. This means that for CNOH atom
types, for which there are ten unique atom-pairs (CC, CN, CO, CH, etc.), there are fewer
than 220 analytic functions. The evaluation of a cubic spline requires fewer operations
than even the simplest of the twenty-two unique NDDO-type analytic expressions. As a
simple demonstration of the speed of cubic splines we have implemented them to replace
the explicit multipole-multipole evaluation for each of the 22 unique NDDO-type integrals.
The multipole-multipole approximation is already significantly faster than the ab initio
integral evaluation. As shown in Figure 4-8 a modest speed-up is achieved over the already
fast NDDO Fock-like matrix construction. These timings are for 16 protein systems up
to 20000 orbitals. Using cubic splines is up to 15% faster than the multipole-multipole
expansion. There is an error introduced, though it is small and controllable. For example
the average error per atom is given in Figure 4-9. Furthermore, while going to higher
multipoles in the expansion of integrals is increasingly expensive, cubic splines cost the
same regardless of the underlying function being interpolated. However, the number of
parameters involved in a typical NDDO multipole-multipole expansion is relatively small
versus the number of parameters needed for cubic splines. The premium placed on RAM
(or core) memory when NDDO methods were first developed thirty years ago would have
made cubic splines impractical. Now, the amount of memory typically available on most
desktop machines is more than sufficient to make cubic splines the method of choice when
evaluating nearly any two-center integral in quantum chemistry.
A general cubic spline has the form
f(x) = Ak + Bk(x− xk) + Ck(x− xk)2 + Dk(x− xk)
3 (4–35)
96
where xk is a reference point (node), and f(xk) = Ak. The determination of the
parameters Bk, Ck, and Dk is straight-forward and unambiguous. We have used the freely
available Duris routine for discrete cubic spline interpolation and smoothing [91]. The only
degrees of flexibility when using cubic spline interpolation are the number of nodes used
and if they are evenly spaced (discrete) or based on an adaptive grid. In our case, for both
integrals and matrix elements, the functions in all cases go to zero in the separated limit
and at smaller length scales the function requires a dense grid to completely capture the
subtle changes that occur in that region. A dense node spacing over the entire length scale
is unnecessary because the functions change very little at long length scales.
To determine a suitable node mapping function we tried several different functional
forms and concluded that a simple exponential function provides a good balance between
having a dense grid in the more critical bonding region (from around 0.5 A to 5 A) and a
sparse grid at long bond lengths (hundreds of A). The actual node mapping function used
is
node(RAB) = Int(1000 ∗ e−6/(RAB+1))− 1. (4–36)
All of the functions we have currently implemented, those involving two-center exchange-
correlation approximations, are zero beyond 10 A. Using the node mapping function then
requires approximately 600 reference points. Since there are 5 variables per reference
point we need 3000 parameters that would form a look-up table for a single function. If
double precision numbers are used to store the spline parameters the storage requirement
is about 0.023 MB per function. Further details of the current implementation of the AAT
procedure in the parallel ACES III program system can be found in Appendix A.
4.5 Verification
We have shown that AATM0 reproduces well the basic features of a PES for a small
test set of molecules involving the direct fission of a C-N bond. Now we turn our attention
to a more thorough comparison to test the behavior of AATM0 for a variety of different
properties. We use the diverse set of molecules in the so-called G2 test of molecules
97
[92]. The G2 set is made up of 148 neutral molecules with singlet, doublet, and triplet
spin multiplicities, though we focus on a subset of these that contain only C, N, O, and
H because the current implementation of AATM0 includes only the spline parameters
generated for those atom atom-pairs. There are 71 molecules in this truncated set ranging
from 2 to 14 atoms.
The separation of electronic and nuclear components in the energy expression
of our AAT approach sets it apart from commonly used SE methods. So it follows
that we expect more consistent results for purely electronic properties. To this end we
have considered the vertical ionization potentials of the relevant molecules in the G2
set. The vertical ionization potential is purely electronic as it does not depend on the
nuclear-nuclear repulsion. We have used the simple definition EIP = E(N) − E(N − 1).
This requires the total energy of the molecule and, since all molecules in the G2 set
are neutral, the corresponding cation with the properly modified multiplicity. We have
compared AATM0 against the underlying B3LYP minimal basis set results and find
a average RMSD from B3LYP for our entire test set of 0.057 Hartree (1.6 eV). The
analogous comparison to AM1 yields an RMSD from B3LYP of 0.331 Hartree (9.0 eV).
This clearly demonstrates the ability of AATM0 to reproduce the electronic structure of
the underlying B3LYP. The error of AM1 from B3LYP is substantial, though a more valid
comparison for AM1 would be against experimental IPs, the important issue here is the
ability our AATM0 to reproduce the physics of the method it is approximating.
We now consider the dipole moment. The dipole RMSD deviation of AATM0 and
AM1 from minimal basis set B3LYP is 0.219 a.u. (0.56 Debye) and 0.595 a.u. (1.51
Debye) respectively. There were no sign errors observed for the orientation of the
dipole moment for either AATM0 or AM1. Though the error introduced by AATM0
is significantly smaller than that of AM1, it is larger than would be expected given the
performance of other properties generated by the AATM0 procedure.
98
Another test of a purely electronic property is simply the electronic density. In this
case a direct comparison to existing SE methods is not possible in a straight forward
way because NDDO-based and DFTB-based methods drop the core orbitals since they
are using a valence-only approach. Instead, we have used B3LYP as the reference and
compared AATM0, minimal basis HF with the Rudenberg approximation applied to
four-center two-electron integrals, and full ab initio minimal basis HF. Considering the
one-particle density matrix (AO density matrix P) from B3LYP the average RMSDs
are 0.061, 0.136 and 0.029 for AATM0, HF (with Rudenberg), and full HF, respectively.
These results demonstrate some surprising behavior. On average the full HF has the best
agreement to B3LYP, at least in the case of a minimal basis set. The AATM0 agreement
is somewhat worse and HF with the Rudenberg approximation introduces a significant
error. Since AATM0 includes the Rudenberg approximation, we would expect its deviation
from B3LYP to be similar to the deviation of HF with the Rudenberg approximation from
the full ab initio HF results, especially since several systems in the test have only two or
three atoms and the four-center approximation does not affect those cases. We see two
probable origins of this unexpected discrepancy; first, there is a cooperative cancellation of
errors between the Rudenberg approximation and the exchange-correlation approximation
(G0xc); second, because the hybrid B3LYP functional includes only 20% nonlocal exchange
the corresponding error introduced by the Rudenberg approximation is not as significant
when compared to the HF treatment in which full nonlocal exchange is used.
In addition to the average RMSD of the density of AATM0 and full HF we consider
the standard deviation of the RMSD of the density for each molecule in the test set.
The standard deviations are 0.042 and 0.057 for AATM0 and full HF respectively. This
suggests that while the overall average RMSD is somewhat higher from AATM0 the error
is slightly more systematic.
Turning our attention to energetic properties we consider the atomization energies of
the relevant portion of the G2 set. We take as the atomic reference energies the neutral
99
atoms for each method. Since the AATM0 is designed to reproduce the exact B3LYP
result in the separated atom limit, the atomic energies are identical. So, the error between
AATM0 and B3LYP is equivalent to the total energy differences. Since we also want to
compare B3LYP to AM1 we must also have atomization energy for B3LYP. Similarly, for
AM1 we perform the neutral atom calculations and subtract the sum of atomic energies
from the total energy. The AATM0 error from B3LYP is 0.057 Hartree. For AM1 the
corresponding error is 0.331 Hartree. This indicates that the AATM0 total energies are in
better agreement with the underlying B3LYP total energies.
Since one goal of our approach is to reproduce a complex PES of large systems we
have applied the AATM0 approximation to describe a somewhat unphysical reaction
mechanism for C20 in order to subject the approximation to an extreme case of bond
breaking. The pseudo-reaction-path corresponds to splitting the C20 molecule in half,
simultaneously breaking ten C-C bonds. Shown in Figure 4-10 are the total energies of
B3LYP, AATM0, AM1, and SCC-DFTB. The reaction coordinate is the distance between
the end-caps and the equilibrium value is 3.223 A, the range is from 2.8 A through 6
A. As expected we see that AM1 exhibits unphysical behavior, and some convergence
problems beyond 4.5 A. The SCC-DFTB actually performs quite well for C20, this result
may not be too surprising since the repulsive energy term in SCC-DFTB is parameterized
from B3LYP total energies for C-C bond breaking and the energy scale is very similar to
the sum of C-C bond energies. The AATM0 overbinds this system significantly and may
in fact be converging to a different electronic state, as the eigenvalue spectrum deviates
significantly from the B3LYP result. Further work is needed to determine the precise
origin of this discrepancy and to correct it.
The final comparison we make is to the RMSD of the gradients for the G2 set. We
will consider two sets of data, one for the MP2 equilibrium structures and the other
for the B3LYP minimal basis set equilibrium structures. Shown in Figure 4-11 are the
average RMSD of the Cartesian gradients from the MP2 reference structures calculated
100
with AATM0, B3LYP, AM1, and SCC-DFTB for the G2 set. The standard deviation of
the gradients is also shown. AM1 appears to have the best agreement to the MP2 result
with an RMSD of about 0.02 Hartree/Bohr, the corresponding standard deviation is also
about 0.02 Hartree/Bohr. SCC-DFTB does not perform well for the G2 set, however
if we eliminate those molecules that contain two or three atoms from the set then the
SCC-DFTB performance improves significantly and the RMSD is 0.007 Hartree/Bohr,
which lower than the AM1 RMSD of 0.017 Hartree/Bohr for this truncated set. In both
cases the AATM0 and B3LYP RMSD forces are nearly the same at 0.05 Hartree/Bohr,
with a standard deviation of 0.01 Hartree/Bohr.
A better comparison for how well the AATM0 approximates the underlying
B3LYP structure is shown in Figure 4-12. At the B3LYP geometries the RMSD is 0.05
Hartree/Bohr and 0.09 Hartree/Bohr for AM1 and SCC-DFTB, respectively. We see the
AATM0 has an average RMSD of 0.006 Hartree/Bohr with a standard deviation of only
0.003 Hartree/Bohr. This clearly shows that AATM0 is reproducing the B3LYP gradients
with excellent accuracy.
4.6 Conclusion
Current SE methods are not uniformly accurate over an entire PES, owing to severe
approximations that are only valid near equilibrium. AAT has been shown to be able to
describe bond breaking in its spin-polarized form at least as well as B3LYP, and much
better than traditional SE methods, like AM1. The AAT SM framework enables the
systematic verification of approximations via comparisons to CC and B3LYP references
for prototypical molecules. Furthermore, all approximations can be further refined. Unlike
existing SE model Hamiltonians, there are no adjustable parameters.
Fast and accurate approximations to four-center two-electron integrals via Rudenberg
(for Coulomb or exact exchange) enable substantial savings for very large systems. Since
the Rudenberg approximation merely reduces the cost associated with calculating each
four-center two-electron integral, methods for screening two-electron integrals to achieve
101
linear scaling are equally applicable here. The nonlocal exchange that is not considered in
SCC-DFTB is allowed in AAT, so that B3LYP and other hybrid DFT methods can be a
target of the approach. We have also developed a route toward a fully two-center model
based on considerations of the three-center two-electron repulsion integrals.
The Fock build requires no numerical integration, but the AAT approach does
explicitly include exchange and correlation. In the current approach we have used
the minimal basis set B3LYP exchange-correlation potentials as a reference, further
improvement is possible by evaluating large basis set ab initio DFT exchange-correlation
potentials and subsequently projecting them in a minimal basis set representation.
Extensions of the AATM0 that include a more complete expansion of the telescoping series
for the exchange-correlation potential have also been described and their implementation
would be a further improvement on the results presented.
102
Table 4-1. Adapted integrals
(µAνA|λBσC) I Mulliken3-center Type A II Rudenberg(µAνB|λAσC) III Mulliken3-center Type B IV Rudenberg(µAνB|λCσD) V Mulliken4-center VI Rudenberg
VII Mulliken PartialVIII Rudenberg Partial
103
Tab
le4-
2.A
vera
geper
centa
geof
Fock
mat
rix
contr
ibuti
onby
adap
ted
appro
xim
atio
n
Rxn
1-C
2-C
3-C
AI
II3-
CB
III
IV4-
CV
VI
VII
VII
IC
OH
NO
2C
N0.
6155
.11
40.5
941
.85
40.7
22.
282.
522.
361.
401.
411.
401.
381.
39N
MT
CN
0.63
51.9
941
.07
42.3
040
.85
4.15
4.87
4.47
2.16
2.25
2.20
2.17
2.16
CH
2O
CO
1.49
62.5
331
.27
31.7
730
.61
4.84
6.18
5.44
-0.1
3-0
.14
-0.1
4-0
.13
-0.1
3C
H3N
H2
CN
0.52
55.5
633
.13
33.1
632
.19
8.07
10.0
48.
962.
712.
982.
832.
752.
73H
OO
HO
O2.
2176
.34
19.8
520
.15
19.2
20.
891.
191.
100.
710.
850.
790.
760.
74N
ET
CN
0.35
41.8
148
.28
48.9
947
.84
5.15
5.98
5.57
4.40
4.65
4.52
4.43
4.41
C2H
6C
C0.
1949
.40
35.4
735
.18
34.3
99.
4911
.95
10.6
05.
455.
935.
685.
555.
50A
vera
ge0.
8656
.11
35.6
736
.20
35.1
24.
986.
105.
502.
392.
562.
472.
412.
40
104
A
-0.2-0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-10 -5 0 5 101-ce
nter
2-e
- inte
gral
(a.
u.)
Full 2-e- integral (a.u.)
y = 0.0063xR2 = -0.0065
B
-4-3-2-1 0 1 2 3 4
-10 -5 0 5 102-ce
nter
2-e
- inte
gral
(a.
u.)
Full 2-e- integral (a.u.)
y = 0.5199xR2 = 0.8988
C
-4-3-2-1 0 1 2 3 4
-10 -5 0 5 103-ce
nter
2-e
- inte
gral
(a.
u.)
Full 2-e- integral (a.u.)
y = 0.4522xR2 = 0.8699
D
-0.2-0.15-0.1
-0.05 0
0.05 0.1
0.15 0.2
-10 -5 0 5 104-ce
nter
2-e
- inte
gral
(a.
u.)
Full 2-e- integral (a.u.)
y = 0.0216x
R2 = 0.6841
Figure 4-1. Scatter plots of agreement between approximate and exact two-electronintegrals. A) One-center. B) Two-center. C) Three-center. D) Four-center
105
A
-60
-40
-20
0
20
40
60
1 1.5 2 2.5 3
Ene
rgy
(kca
l/mol
)
C-N reaction coordinate (Angstrom)
Rudenberg (4C)B3LYP
B
-80
-60
-40
-20
0
20
40
60
1 1.2 1.4 1.6 1.8 2 2.2 2.4
Ene
rgy
(kca
l/mol
)
C-N reaction coordinate (Angstrom)
Rudenberg (4C)B3LYP
C
-20
-10
0
10
20
30
40
50
60
1 1.2 1.4 1.6 1.8 2 2.2 2.4
Ene
rgy
(kca
l/mol
)
C-N reaction coordinate (Angstrom)
Rudenberg (4C)B3LYP
D
-100
-50
0
50
1 1.5 2 2.5 3 3.5 4
Ene
rgy
(kca
l/mol
)
C-N reaction coordinate (Angstrom)
Rudenberg (4C)B3LYP
Figure 4-2. Dissociation curve of C-N with 4-center Rudenberg approximation. A) NMT.B) NET. C) COHNO2. D) CH3NH2
106
A
0.4
0.2
0-3 -2 -1 0 1 2 3
Separation (Bohr)
B
0.4
0.2
0-4 -2 0 2 4
Separation (Bohr)
Figure 4-3. Orbital products. A) Gaussian orbital product. B) Slater orbital product
107
A
-60
-40
-20
0
20
40
60
1 1.5 2 2.5 3
Ene
rgy
(kca
l/mol
)
C-N reaction coordinate (Angstrom)
GZERO(B3LYP)B3LYP
B
-80
-60
-40
-20
0
20
40
60
1 1.2 1.4 1.6 1.8 2 2.2 2.4
Ene
rgy
(kca
l/mol
)
C-N reaction coordinate (Angstrom)
GZERO(B3LYP)B3LYP
C
-20
-10
0
10
20
30
40
50
60
1 1.2 1.4 1.6 1.8 2 2.2 2.4
Ene
rgy
(kca
l/mol
)
C-N reaction coordinate (Angstrom)
GZERO(B3LYP)B3LYP
D
-100
-50
0
50
1 1.5 2 2.5 3 3.5 4
Ene
rgy
(kca
l/mol
)
C-N reaction coordinate (Angstrom)
GZERO(B3LYP)B3LYP
Figure 4-4. Dissociation curve of C-N with AAT approximation. A) NMT. B) NET. C)COHNO2. D) CH3NH2
108
A
-60
-40
-20
0
20
40
60
1 1.5 2 2.5 3
Ene
rgy
(kca
l/mol
)
C-N reaction coordinate (Angstrom)
AATM0B3LYP
AM1
B
-80
-60
-40
-20
0
20
40
60
1 1.2 1.4 1.6 1.8 2 2.2 2.4
Ene
rgy
(kca
l/mol
)
C-N reaction coordinate (Angstrom)
AATM0B3LYP
AM1
C
-20
-10
0
10
20
30
40
50
60
1 1.2 1.4 1.6 1.8 2 2.2 2.4
Ene
rgy
(kca
l/mol
)
C-N reaction coordinate (Angstrom)
AATM0B3LYP
AM1
D
-100
-50
0
50
1 1.5 2 2.5 3 3.5 4
Ene
rgy
(kca
l/mol
)
C-N reaction coordinate (Angstrom)
AATM0B3LYP
AM1
Figure 4-5. Dissociation curve of C-N with AATM0 approximation. A) NMT. B) NET. C)COHNO2. D) CH3NH2
109
0
5000
10000
15000
20000
25000
30000
35000
40000
400 600 800 1000 1200 1400 1600
Tim
e (S
econ
ds)
Orbitals
ab initioRudenberg
Figure 4-6. Timing of ab initio versus Rudenberg four-center integrals
110
-10 0
10 20 30 40 50 60 70 80 90
100
0 100 200 300 400 500 600 700 800 900 1000
Perc
ent o
f to
tal
Number of atoms (5 orbitals per atom)
1-center2-center3-center4-center
Figure 4-7. Percentage of multi-center terms versus system size
111
0 10 20 30 40 50 60 70 80 90
100
0 5000 10000 15000 20000 25000
Seco
nds
Orbitals
StandardSplines
Figure 4-8. Timing of Fock build using traditional NDDO and NDDO with cubic splinesas a function of system size
112
-1.92e-05-1.9e-05
-1.88e-05-1.86e-05-1.84e-05-1.82e-05-1.8e-05
-1.78e-05-1.76e-05-1.74e-05-1.72e-05
0 5000 10000 15000 20000 25000
Err
or (
kcal
/mol
)
Orbitals
Error
Figure 4-9. Error introduced per atom from cubic splines as a function of system size
113
-4-3.5
-3-2.5
-2-1.5
-1-0.5
0 0.5
2.5 3 3.5 4 4.5 5 5.5 6
Ene
rgy
(Har
tree
)
R (Angstrom)
B3LYPAATM0
AM1SCC-DFTB
Figure 4-10. Pseudo-reaction-path splitting C20
114
0
0.05
0.1
0.15
0.2
AATM0
B3LYP
AM1
SCC-DFTB
RMSDStandard Deviation
Figure 4-11. Force RMSD from MP2
115
0
0.05
0.1
0.15
0.2
AATM0
AM1
SCC-DFTB
RMSDStandard Deviation
Figure 4-12. Force RMSD from B3LYP
116
APPENDIX Parallel Implementation in the ACES III Environment
The ACES III program system environment has been designed for the efficient parallel
implementation of wavefunction theory methods in computational chemistry. The ACES
III program is suited to treating large systems, typically 500-1000 basis functions and up
to 300 electrons with post-HF ab initio methods. The super instruction processor (SIP)
manages the communication and processing of blocks of data, such blocks are intended to
be somewhat large (on the order of 500,000 floating point numbers). To provide a more
direct route for implementing new methods the SIP reads the code written in the high
level symbolic language, the super instruction assembly language (SIAL) (pronounced
“sail”). The SIP hides many of the more complicated aspects of parallel programing
allowing the SIAL programmer to focus on algorithm and method development. Within
SIAL each index of an array is divided into segments, the segments map elements of
the array into blocks. An algorithm then involves operations for which the segment is
the fundamental unit of indexing. Operations that involve the contraction of two arrays
can then be broken into several smaller contractions over two blocks. Each block-pair
contraction can then be distributed over many processors, allowing for the parallel
implementation of the contraction of two large arrays.
A general issue in parallel codes is ensuring that latency does not become a rate
limiting step. This can be done by making sure that the amount of computation done is
on a par with the latency of any operation. Such balancing is machine dependent, so it
requires some amount of testing to determine an optimal solution for a particular machine.
For the implementation of the AAT methodology this balancing is critical and requires a
different treatment than is usually used for post-HF methods.
The structure of the AAT equations are quite different than post-HF equations.
In the former the atomic indices are used as the primary variable in the loops used to
construct the Fock-like matrix. For post-HF methods the orbital indices are the important
variable. Moreover, the number of orbitals per atom in AAT is small relative to the
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number of atoms, the reverse of many post-HF calculations, which are generally used to
treat systems with fewer atoms but with much larger basis sets.
The speed of the AAT approach is achieved by avoiding the explicit calculation of the
numerous and costly four-center integrals. Instead, these terms are introduced by a sum of
contracted two-center terms. Initially, we used small segment block sizes. The block size
ultimately determines the computational efficiency by affecting how the IO, or fetching
of data, is done. In these initial studies each block spanned only one atom. In this case,
when the two-electron integrals were evaluated it was straight forward to simply neglect
the integrals that involved four atoms (four-centers.) This procedure worked for systems
with few atoms (<20), but, as the system size increased, the number of blocks rapidly
increased, and latency became an problem.
An improved approach that was implemented, which is more consistent with the
original ACES III design philosophy, involved removing the atom-spanning block
size restriction. This was achieved by rewriting the original routines that returned all
two-electron integrals spanning four-block units to explicitly exclude integrals involving
four-centers. In addition, the Rudenberg approximation, which relies on contractions
between the overlap matrix and NDDO-type two-center two-electron integrals ((AA|BB)),
required modification to avoid over-counting some terms. Similar to the routine that
excludes four-center integrals, the modified routine to generate the integrals used by the
Rudenberg approximation excluded all integrals that were not of the type (AA|BB).
These two modified routines involve some overhead that could be reduced with further
refinement. These modified routines are specific to the AAT approach and are not
included in the standard release version of ACES III.
Since two-electron integrals possess an eight-fold spatial symmetry the overall number
of integrals that need to be calculated can be reduced by roughly a factor of eight. To
do so requires recognizing the symmetry of every integral type. In ACES III the problem
is slightly different. Instead of having this eight-fold symmetry manifest itself by AO
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orbital index, it instead deals with block indices. In the pseudo code for the integral direct
implementation given below, µ, ν, λ, and σ are block indices and each index generally
spans several atoms. The following code for generating the two-electron integrals evaluates
every block of integrals only once, implying that the eight-fold degeneracy is optimally
incorporated.
Do µDo ν
If ν = µ(µµ|µµ)
End If (ν = µ)If ν 6= µ
(µµ|µν) → (µµ|νµ), (µν|µµ), (νµ|µµ)Do λ
If λ 6= µ and λ > ν(µµ|νλ) → (µµ|λν), (νλ|µµ), (λν|µµ)(µν|µλ) → (µν|λµ), (νµ|µλ), (νµ|λµ), (µλ|µν), (λµ|µν), (µλ|νµ), (λµ|νµ)End If (λ 6= µ and λ > ν)
End Do (λ)End If (ν 6= µ)If ν > µ
(µµ|νν) → (νν|µµ)[Apply Rudenberg approximation to the terms (µµ|νν) and Sj,k]
(µν|µν) → (νµ|µν), (νµ|νµ), (µν|νµ)Do λ
If λ > µ and λ 6= νDo σ
If σ > λ and σ 6= µ and σ 6= ν(µν|λσ) → (µν|σλ), (νµ|λσ), (νµ|σλ), (λσ|µν), (σλ|µν), (λσ|νµ), (σλ|νµ)
End If (σ > λ and σ 6= µ and σ 6= ν)End Do (σ)
End If (λ > µ and λ 6= ν)End Do (λ)
End If (ν > µ)End Do (ν)End Do (µ)
In addition to the Rudenberg approximation, we have also implemented an interface
to the stored cubic spline parameters needed to construct the approximation to the
exchange-correlation contribution to the Fock-like matrix. This component is much
faster than the two-electron integral component, therefore it is not a rate limiting step.
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Still, there are many aspects that need to be implemented efficiently to have a viable,
stable code. We have previously discussed the use of cubic splines are an efficient way of
balancing the numerical accuracy of the two-center functions and the number of reference
points stored.
Another aspect that needs to be considered is the rotation of the two-center matrix
elements in a local atom-pair coordinate system to the global coordinate system of the
molecule. This rotation is unambiguous once the global coordinate system is specified,
and although orbitals of the molecule can be rotated simultaneously without affecting
the energetics, the corresponding matrix elements will change on rotation. This degree
of flexibility must be incorporated to ensure that the Fock-like matrix contribution that
arises from every atom-pair corresponds to the orbital orientation of the global coordinate
system of the molecule. To achieve this we use a simple procedure that generates the
matrix that rotates a pair of atomic coordinates so that they are aligned to the axis of
the the stored cubic spline, and then perform the reverse rotation on that sub-block of the
Fock-like matrix.
Finally, the third component implemented was the numerical integration that is
performed once the SCF procedure has converged. The ACES III program system
generates a job archive file (JOBARC) that can be read by ACES II executables, assuming
it has been generated on the same computer architecture. To test the efficacy of the AAT
approximations, the quickest route was to modify the ACES II xintgrt executable.
The xintgrt module performs numerical integration on a density to determine the
exchange-correlation energy that is needed for the residual electronic energy contribution
of AAT. Effectively, xintgrt pulls the AO eigenvector matrix from the JOBARC, as well
as other information about the system that would typically be generated by the xvscf ks
executable in ACES II when running a KS-DFT calculation (e.g. occupation numbers).
Since ACES III does not create all of the necessary records in the JOBARC, some small
adjustments to the xintgrt routine led to a new modified executable that performed the
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needed numerical integration for the density generated by the AAT SCF procedure.
The largest modification in this regard was the proper inclusion of the exchange terms.
In the original xintgrt routine the Coulomb and exchange energy contributions are
recalculated from the integral file (IIII). This integral file is not created by ACES III
(because it works in an integral direct environment), removing the dependence on the IIII
file and eliminating the revaluation of the Coulomb and exchange terms. Instead those
intermediates are stored in the JOBARC by ACES III when they are evaluated in the
normal course of evaluating the Fock-like matrix. With these modifications all energy
contributions needed for the AAT approximations becomes available.
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BIOGRAPHICAL SKETCH
I was born in Michigan but have done time in Ohio, Washington, Oregon, and now
Florida. My tumultuous career in science began in sixth grade when my science teacher
called me a dreamer (I know I am not the only one) for proposing the construction of an
elevator to the moon to avoid the costs associated with space flight. In hindsight, I should
not have specified my original design as a rigid structure. That same year I was awarded
a certificate as the classroom’s strangest kid. After making the transition to high school,
I doubled-up on math and science and finished all available classes in those areas by my
Junior year, resulting in a misspent Senior year.
Somehow I managed to get into the only college to which I applied. I arrived a
Freshman filled with youthful optimism and persuaded my academic advisor to enroll me
in a chemistry course 2 years beyond my level. I continued my Sophomore year without
the youthful optimism. I participated in two Research Experience for Undergraduates
programs sponsored by the National Science Foundation at Cornell and the University
of Chicago. In Chicago I worked for Prof. Karl Freed, essentially spending the summer
building overly complicated internal coordinate input files. I must have enjoyed it because
that same summer I tattooed myself with the mark of a quantum chemist, literally.
Eventually, I took an interdisciplinary BA in chemistry and physics from Reed College in
beautiful Portland, Oregon, in 2001.
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