Quantum Monte Carlo for Transition Metal Systems: Method Developments and Applications

(1)

WAGNER, LUCAS K. Quantum Monte Carlo for Transition Metal Systems: Method Developments and Applications. (Under the direction of Professor Lubos Mitas).

Quantum Monte Carlo (QMC) is a powerful computational tool to study correlated systems of electrons, allowing us to explicitly treat many-body interactions with favorable scaling in the number of particles. It has been regarded as a benchmark tool for condensed matter systems containing elements from the first and second row of the periodic table. It holds particular promise for the more complicated transition metals, because QMC treats the correlations between electrons explicitly, and has a computational cost that scales well with the system size.

(2)

by

Lucas K. Wagner

A dissertation submitted to the Graduate Faculty of North Carolina State University

in partial fulfillment of the requirements for the Degree of

Doctor of Philosophy

Physics

Raleigh 2006

Approved By:

Dr. M. Buongiorno-Nardelli Dr. D. Lee

Dr. L. Mitas Dr. K. Ito

(3)

(4)

Biography

I was born September 21, 1980 in the city of Danville, in the southwestern part of Virginia to James and Deborah Wagner. I grew up near the town of Chatham, about 10 miles north of Danville with a population of around 1,300 people. I spent kindergarten and first grade at Chatham Elementary School, moved to Climax Elem for second through fifth, Central Middle for sixth and seventh, and finished primary schooling in Chatham High School in June of 1998.

I entered North Carolina State University as a major in physics. I worked one semester for the WebAssign project, and immediately sought out a place to work in a ‘real’ lab. I approached Dr. Jacqueline Krim for a place as an undergraduate researcher and she agreed. I started in my second semester at NCSU. The first day, I leveled all the desks in her office and painted her filing cabinet. I spent some time helping her group move in and set up the lab, since she had just moved from Northeastern University. After a while, I did perform some research in her lab, none of which was probably worthy of publishing, but I very much enjoyed it. I worked on various projects, finding a quartz crystal that would still oscillate at 500 Celsius, building a ultra-high vacuum vapor deposition chamber, and ripping postdoc Brian Mason’s carefully constructed superconductivity-dependent friction experiment to pieces.

(5)

(6)

Acknowledgements

First, I would like to thank Dr. Lubos Mitas for bringing me through the six long years from the middle of my undergraduate education to the end of the PhD program. I have learned a lot from him in that time. Also, thanks for the trips to Italy and France, which altered the course of my life tremendously.

Thanks to Jeff Grossman for working with me throughout my graduate career, as well as my other colleagues who I’ve worked with within the Mitas group: Michal Bajdich, Prasenjit Sen, Jindrich Kolerenc, Gabriel Drobny, Ji-Woo Lee, and David Sulock. I’d also like to thank Mark Losego for stimulating conversations about BiFeO3.

Thanks to Marta Cerruti who provided me with love, support, and excellent food! Thanks to my parents who have provided much of the same, although for a longer time and a bit differently.

(7)

List of Figures

3.1 DMC simulation for a one dimensional harmonic oscillator . . . 18 3.2 Finite size scaling of cubic BaTiO3using a Slater determinant of LDA orbitals

in VMC without a Jastrow factor. The averaged and weighted k-points are over the four real k-points in the cubic cell. . . 25 4.1 Calculation structure for the VMC method on a molecule using a

Slater-Jastrow wave function. . . 28 4.2 Simple VMC code . . . 30 4.3 Flow of a QMC calculation . . . 38 4.4 Example input file for VMC evaluation of properties. This corresponds to

the fourth box in Fig 4.3. . . 38 4.5 Example input file for optimization of variational parameters. This is the

fifth box in Fig 4.3. . . 39 4.6 Example input file for evaluation of properties of the correlated wave function,

plus a DMC calculation. This corresponds to the sixth and seventh block in Fig 4.3. . . 40 4.7 Scaling of QWalk code over processors in Monte Carlo steps per second. The

system is a 2x2x2 cell of BaTiO3 with 320 electrons and one walker per node. This is VMC; DMC is very much the same, because of the constant walker algorithm. This is close to a worst-case scenario for QMC, since the run was only approximately 40 seconds per processor. . . 42 5.1 Conservation of energy for different methods of calculating the force on H2.

Both the F-U and P-C approximations are equally good in this case. . . 48 5.2 Calculated energy difference as a function of the error in a Jastrow parameter.

An accurate 3-body Jastrow factor was computed and one parameter was varied by multiplying it by a, so ata= 1, the accurate Jastrow is used. . . 49 5.3 Test of conservation of energy on N2 using a two-body Jastrow factor. The

(11)

5.4 Energy difference as a function of the error in the Jastrow factor for the TiO molecule. This was done in the same way as in Fig 5.2. It is not possible to extrapolate to the correct answer since the error in the Jastrow factor is not a function of only one variable. . . 52 6.1 Convergence of the minimum as we add data points starting from one side . 57 6.2 Convergence of minimum as we add data points starting from both sides and

filling in the middle. . . 58 6.3 Calculated minimum energy lattice constant of BaTiO3 by LDA for different

fitting functions . . . 60 7.1 The d-p hybridization orbital (doubly occupied) for TiO in Hartree-Fock

(top) and B3LYP (bottom). B3LYP enhances the hybridization significantly, which leads to lower energy in QMC. . . 66 7.2 The d-p hybridization orbital (doubly occupied) for MnO in Hartree-Fock

(top) and B3LYP (bottom). B3LYP enhances the hybridization significantly, which leads to lower energy in QMC. . . 67 7.3 The energy gain in DMC from using B3LYP orbitals as a function of the

metal monoxide. The line is a guide to the eye. . . 68 7.4 The number of determinants versus the energy and dipole moment for TiO.

The dipole moments are shifted downwards by 0.1 Debye to correct for the pseudopotential error. . . 71 8.1 BaTiO3 structure. The titanium is the central atom, with barium at the

corners and oxygen on the faces. . . 74 8.2 BaTiO3 one particle density: Hartree-Fock versus LDA . . . 76 8.3 BaTiO3 one particle density isosurface in the tetragonal phase. . . 77 8.4 Parameters for the fitting functionE(a) =b(1−exp(−c(a−a0)))2+E0 from

a 2x2x2 supercell calculation of BaTiO3, along with the mean and standard deviation of the gaussian fits. . . 79 8.5 Zero point energy correction for BaTiO3 . . . 81 8.6 Dynamic shift in the lattice constant for BaTiO3 along with the energy per

cell, as a function of the soft-mode distortion. This is calculated in LDA using Crystal . . . 82 8.7 Formula unit cell of BiFeO3 in the ferroelectric phase, with Fe in the center

and Bi at the corners. The oxygen cage is rotated and the cell is slightly distorted from the cubic phase. The antiferromagnetic ordering is in the 111 direction. . . 82 8.8 Experimental measurements of thin films for BiFeO3. The references are

(12)

List of Tables

1.1 Significant accomplishments reported in this dissertation . . . 7 3.1 List of approximations in the QMC method. They are decomposed into

sys-tematic approximations, which can be resolved by changing parameters in the simulation, and methodological approximations, which are inherent in the method. In principle, it is known how to remove each of these approx-imations, although it may be prohibitively expensive to do so for a large system. . . 26 4.1 The central objects of the code and their physical correspondents . . . 28 4.2 Optimization objective functions implemented . . . 31 5.1 Summary of what methods work on which materials with which Jastrow factor. 51 7.1 Binding energies of the first five transition metal monoxides by different

theo-retical methods, along with RMS deviations from the experiment(all in eV). Statistical uncertainties in units of 10−2 _{eV are shown in parentheses for} Monte Carlo and experimental results. Zero point energy corrections are estimated to be much less than the size of the uncertainty in experiment. . 65 7.2 Bond lengths in ˚A. The statistical uncertainties for ScO,TiO,VO,CrO, and

MnO are respectively 0.002,0.003,0.003,0.004, and 0.004. . . 69 7.3 Dipole moments in Debye. The fixed-node RMC results have been obtained

with a single determinant of B3LYP orbitals. See text for an analysis of the errors involved for the case of TiO. . . 69 8.1 Cohesive energy of BaTiO3 by various first principles methods. All

theoret-ical calculations are shifted by a 0.13 eV zero point energy correction. . . . 77 8.2 DMC data for the energy of BaTiO3 as a function of lattice constant.(2x2x2

cell, averaging over 4 real k-points) . . . 78 8.3 Structural parameters for BiFeO3 in our calculations, given in terms of the

(13)

B.1 The Barium pseudopotential given as a sum of gaussians: V =P

kckrnke−αkr

2 .

(14)

List of Symbols and Abbreviations

˚

A 10−10 meters

CC Coupled Cluster technique

CCSD(T) Coupled Cluster with singles, doubles, and perturbative triples

DFT Density Functional Theory

DMC Diffusion Monte Carlo

En Energy eigenvalue of the Hamiltonian

GGA Generalized Gradient approximation of DFT

H or ˆH Hamiltonian operator

LDA Local Density approximation of DFT

Ne Number of electrons

QMC Quantum Monte Carlo

r Position of a single electron

R Point in many-body space

RMC Reptation Monte Carlo

VMC Variational Monte Carlo

Z Nuclear charge

Φn Eigenfunction of the Hamiltonian

ΨT Trial wave function

(15)

Chapter 1

Introduction

1.1 Ab Initio Calculations

With the Schr¨odinger equation, the rules of non-relativistic quantum mechanics, we can describe almost all of condensed matter. We are confident that these rules, if solved appropriately, can predict the behavior of materials, from what configurations atoms will be found to the hardness of a material to what color and the optical behavior of some object. We have a universal equation for all these things–the only problem is that it is extremely difficult to solve for many interacting particles: the well-known many-body problem. This is worse than the many-body problem for classical systems, like planetary motion, because there does not exist a numerically exact, polynomially scaling method that solves the Schr¨odinger equation. So we are left performing some kind of approximations, which necessarily cause a loss of universality.

(16)

power-ful, and allows us to understand by analogy. It reduces the dimensionality of the problem significantly, allowing us to have a good understanding of a material by specifying a few parameters for the model system. If we want to calculate something about a specific system, however, we either have to estimate or calculate those parameters for that system. To use an admittedly old (but still in use) example, the electrons in a metal can be modeled by free electrons with an effective mass that is different from the real electronic mass and includes the effects of many-body interactions. One can easily calculate the behavior of the metal as a function of the effective mass, but to study a particular system, the effective mass has to be gotten from somewhere, usually from an experiment.

To go beyond the ‘capture the physics’ model, we wish to use the Schr¨odinger equation to obtain a predictive theory without any adjustable parameters. Given the types of nuclei in a system, we want to be able to know what kind of material they will form, what their lowest energy geometry will be, and the properties of the material. In principle, this is possible, but as the number of atoms increases, the Schr¨odinger equation becomes infea-sible to solve exactly. We first make a small general approximation: the Born-Oppenheimer approximation, which assumes that the wave function of nuclei and electrons are separable; that is, Ψ(Rnuclei,Relectron) = Ψn(Rnuclei)Ψe(Relectron). This is usually a good

approxima-tion because the nuclei weigh at least one thousand times more than the electrons, which means that their motions can to good approximation be decoupled. In this thesis, we will focus on solving for the electronic wave function Ψe. Ψncan be solved for using many of the

same methods, especially Quantum Monte Carlo. Often, heavy nuclei can be treated clas-sically with very little approximation. If we were able to solve for Ψe, much of condensed

matter physics would be described to extremely high accuracy, barring only high-energy physics and extremely relativistic particles.

The Schr¨odinger equation for Ψe, which we will refer to as Ψ for the rest of the

dissertation is as follows (in atomic units):

−idΨ(R, t)

dt = ˆHΨ(R, t). (1.1)

where ˆH is the electronic Hamiltonian operator ˆ

H= 1 2∇

2₋X

iI

ZI

riI

+X

i6=j

1

rij

(1.2)

and large/small indices stand for nuclei/electrons. ZI is the nuclear charge. Since the

(17)

Φn such that ˆHΦn = EnΦn. While at finite temperatures, the electrons will occupy a

combination of states according to quantum statistical mechanics, at low temperatures, they will stay in the ground state to a very good approximation, only visiting the excited states on some kind of perturbation to the system, such as an incident photon. How low the temperature needs to be depends on the system, but for insulators and molecules, room temperature is well within this envelope. So we can further restrict ourselves to finding the ground state Φ0 and the first few excited states above that, while still covering a large portion of materials of interest. To current knowledge, this is still an exponentially hard problem.

Given the high desirability of Φ0, there have been many attempts to calculate it over the past 60 or so years. Many theories and approximations have been developed, the most common of which are Hartree-Fock and post Hartree-Fock methods, and Den-sity Functional Theory (DFT). These methods and many others function by a reduction technique, where the many-body problem is reduced onto a set of one-body problems. In Hartree-Fock and post Hartree-Fock, the functional form of the many-body wave function is limited, which in turn limits the physics that can be described. In Density Functional Theory, the problem is reduced from working with the many-body wave function to working with the one-body density, a three-dimensional object. This reduction is in principle exact, but extracting the energy of the system requires an unknown functional. While compu-tationally quite efficient, these reductions are also limiting. With Quantum Monte Carlo (QMC), we treat the full many-body problem at the cost of computer time.

By this point, QMC has established itself as a generator of excellent energetic properties, usually producing the most accurate cohesive energies available in a reasonably scaling method in materials ranging from organic molecules to transition metal systems[6, 7, 8, 9]. Band gaps and excitation energies have been calculated as well[10, 8, 11] with excellent agreement with experiment. There has been much work on forces[12, 13, 14, 15, 16], but there has not been a definitive method for calculating the energy derivative in all systems. There has also been some work on non-energetic quantities, such as the dipole moment[17], but most of these studies used the so-called pure Diffusion Monte Carlo method[18, 19], which does not work for larger systems. Recently, the Reptation Monte Carlo[20] method has been developed, which shows promise for larger systems, but has not been applied to chemical problems.

(18)

Quan-tum Monte Carlo calculations. Discounting pure research codes, to my knowledge there are only three that can perform calculations with periodic boundary conditions: CHAMP of Umrigar and Filippi, CASINO from Cambridge, and QWalk, developed in this dissertation. Although CHAMP is currently (at the time of this writing) only available to collabora-tors with their respective groups, while CASINO is available under a somewhat restrictive license. On the other hand, we plan to release QWalk under the GNU Public License, which will make it the first general purpose (meaning that it can calculate both solids and molecules) QMC program available under that license.

1.2 Transition Metal Oxides

Transition metal chemistry is an exciting area of research that has implications in fields from biological physics to astrophysics. Transition metals can form many types of bonds and transition metal solids exhibit useful properties like ferroelectric and ferromag-netic ordering. This interesting physics comes from thed-shell states which are very close in energy to the outers−states, along with strongly correlated electrons that make accurate first principles calculations particularly challenging.

(19)

Bayesian analysis techniques in Chapter 6.

1.3 Outline

The dissertation is laid out as follows:

• In Chapter 2, I go over several methods for solving the Schr¨odinger equation, includ-ing the wave function based Hartree-Fock and post Hartree-Fock, as well as Density Functional Theory.

• In Chapter 3, I present the theoretical basis for the Quantum Monte Carlo methods used in this thesis, along with a few small enhancements for this work.

• In Chapter 4, I describe a new program for electronic structure Quantum Monte Carlo called QWalk, developed primarily by myself. QWalk includes state of the art algorithms and is capable of treating systems with over a thousand electrons.

• In Chapter 5, I develop a unified framework for two separate methods of obtaining the forces within QMC. This allows me to compare them directly and find which is more robust to a poor wave function. I find that one formulation is superior, perhaps to some surprise, but neither is sufficiently robust to allow application to transition metal oxides.

• In Chapter 6, I develop a new application of Bayesian inference to fitting the reliable total energy of Quantum Monte Carlo data. The method not only generates well-defined and correct uncertainties on arbitrary quantities such as the minimum energy point, it also provides a means to estimate the optimal sets of data points to minimize the uncertainty.

(20)

• In Chapter 8, I extend the study to extended materials with ABO3 perovskite struc-ture, focusing on the two ferroelectric materials BaTiO3 and BiFeO3. For the classic ferroelectric BaTiO3, I obtain an estimate of the minimum energy cubic lattice con-stant in excellent agreement with experiment, which has been a long-standing problem in electronic structure calculations. In the BiFeO3 work, done in collaboration with undergraduate researcher David Sulock, we investigate the cohesive energy and the ferroelectric well depth. We find that the cohesive energy is smaller than typical per-ovskites and the well depth is large, which suggests that the material is difficult to grow without defects, which will then hamper measurement of the ferroelectric effect.

1.4 Accomplishments

(21)

Table 1.1: Significant accomplishments reported in this dissertation

Problem Resolution

Fast and general QMC code needed I wrote the general purpose QMC pack-age QWalk, which is being used by several groups

There are many suggested methods for calculating forces within QMC, which are generally of uncertain accuracy and effi-ciency

I generalized the implementation of the force methods using Reptation Monte Carlo, tested several methods consistently, and found none of them accurate enough on transition metal systems. For finite dif-ference methods, I narrowed the problem down to the Green’s function approxima-tion and suggested a way to improve it. Quantum Monte Carlo data has stochastic

uncertainties, making minimization diffi-cult

I used Bayesian inference to create a con-sistent framework to handle the stochastic uncertainties and to predict the most effi-cient distribution of data points

The accuracy of fixed-node Diffusion Monte Carlo/Reptation Monte Carlo is not well studied on transition metal ox-ides

I studied the bond length, binding en-ergy, and dipole moment of several TMO molecules and compared them to experi-ment. I found that for bond lengths and binding energies, the method is extremely accurate and that the dipole moment is very sensitive to the treatment of elec-tronic correlation.

Perovskite lattice constants are often poorly predicted by traditional electronic structure techniques.

I used Quantum Monte Carlo to calcu-late the cubic lattice constant of BaTiO3, which showed that 1) it is possible to per-form such a calculation on a challenging transition metal oxide solid while control-ling the errors and 2) QMC is quite accu-rate, at least on this material, and quite likely also on others.

The perovskite BiFeO3 has a large disper-sion in experimental measurements of the polarization

(22)

Chapter 2

Electronic Structure Methods

Other than Quantum Monte Carlo

2.1 Hartree-Fock and Post Hartree-Fock

Hartree-Fock and post-Hartree-Fock are based on the variational theorem, which states that the energy expectation value of a given wave functionhEi=hΨ|Hˆ|Ψi is always greater than or equal to the ground state energy. Within a given functional form, an approximation to the true ground state wave function is thus the one that minimizes hEi. An excellent treatment of these methods is found in Szabo and Ostlund[22].

Hartree-Fock (HF) takes as the variational wave function a determinant of one-particle orbitals:

ΨHF(R) =|M|, (2.1)

where

Mij =φi(rj). (2.2)

(23)

Typically the orbitals are expanded in some sort of basis:

φi(r) =

X

k

c(_ki)bk(r). (2.3)

The basis functions are usually chosen so that the integrals can be evaluated analytically. This usually boils down to an expansion in plane waves or Gaussian functions. This wave function satisfies fermionic antisymmetry, but does not include any other electronic correla-tion. The difference between the energy of the Hartree-Fock wave function and the ground state is called the correlation energy, which is typically approximately 5% of the total en-ergy in chemical systems. While this may seem small, most of chemistry occurs through the correlation of electrons, so it is quite important. For this reason, Hartree-Fock is not a very accurate method for chemistry, but it provides a useful starting point for other calculations. Post HF methods expand the wave function in terms of determinants, for a wave function looking like

ΨpHF(R) =

X

i

ci|Mi|. (2.4)

EachMi is a determinant of a different set of orbitals, and one can optimize either only the

coefficients (Configuration Interaction or CI) or the orbitals (MCSCF). While for a complete basis and summing over all possible orbitals, post HF is and exact solution, the number of determinants increases combinatorially with the basis set and the solution converges very slowly with respect to the basis set. In practical calculations, only some determinants and a small basis are treated, which limits the accuracy. The convergence with respect to the basis set is so slow because post HF gets to the electronic correlation in a very roundabout way, since it expands it in a one-particle basis.

Coupled Cluster[23] is similar in idea to Configuration Interaction, but done in such a way as to retain size consistency at all orders. Size consistency ensures that individual atoms and the molecule will be treated on the same footing, improving binding energies over CI. The wave function is given as

eTˆ|ΨHFi, (2.5)

where ˆT is the excitation operator. For example, the single excitation operator written in second quantization is ˆT1 = P_iP_ataiˆaiˆa†a, which describes a sum of excitations from

(24)

and two unoccupied, ˆT3 is over six indices, etc. The Coupled Cluster wave function is found by solving for the unknown coefficients ta_i for each of the different orders of excitation.

Whether Coupled Cluster or Configuration Interaction is used, they both suffer from poor scaling with respect to the number of electrons in the system (greater thanO(N_e4)) and slow convergence of the energy with respect to the basis set. These disadvantages generally limit their application to relatively small molecular systems.

2.2 Density Functional Theory

Density Functional Theory (DFT) takes a completely different approach. It works in terms of the one-particle density ρ(r). One can prove[24] that there exists a one-to-one mapping between the ground state density and ground state wave function, and therefore between the density and the ground state energy. Therefore, if we know the mapping E=

F[ρ(r)], we can minimize the energy with respect to the three-dimensional density function instead of the many-body wave function. As it turns out, the mapping is a functional, which can depend on integrals of, derivatives of, etc of the density. There is also no prescription for finding the density functional, so it must be either constructed from general principles or fitted. The breakthrough in DFT came when Kohn and Sham[25] wrote an ansatz for the density as a sum of one-particle orbitals:

ρ(r) =X

i

φ2_i(r). (2.6)

This allows the kinetic energy to be evaluated as if the orbitals were from a Slater de-terminant, which, along with the classical electrostatic energy (also known as the Hartree potential), covers most of the total energy of a system. The energy of a system is then written as

E[n(r)] =−1

2 X

i

Z

φ∗_i(r)∇2φi(r)dr+

1 2

Z

ρ(r)ρ(r0)

|r−r0_| drdr 0

+ Z

ρ(r)Vext(r)dr+Exc[ρ] (2.7)

The missing quantum correlations are folded into the so called exchange-correlation energy,

Exc[ρ]. Any improvement in the accuracy of DFT goes into the improvement of Exc. The

Local Density Approximation (LDA) approximates the exchange correlation energy as a local function:

E_xcLDA[ρ] = Z

(25)

whereexc(ρ) is a simple function. The canonical version of this is a fit to Quantum Monte

Carlo data[26] for the homogeneous electron gas. Further refinements such as the General-ized Gradient Approximation (GGA) have built on the LDA, by adding a dependence on the gradient:

E_xcGGA[ρ] = Z

exc(ρ(r),∇ρ(r))ρ(r)dr. (2.9)

This allows for further accuracy, although there is some room for creativity in the for-mulation of the GGA. In chemistry, the BLYP[27, 28] functional is quite popular, and in solid-state, the PBE[29] functional is commonly used. Note that there is not a strict ladder of accuracy, however. The GGAs can be less accurate than the LDA depending on the formulation and the material.

Both LDA and GGA suffer from the self-interaction problem, where each elec-tron interacts with itself. This is seen in the second term in Eqn 2.7, where the elecelec-tron density interacts with itself. This is valid within classical electrostatics, but not in quan-tum mechanics. There have been methods to fix this, including LDA+U[30] and the self-interaction correction[31], although there is some arbitrariness in their application. A large collaboration[32] recently tested various corrections to LDA on the MnO solid, and found that they show rather large variation. It is thus difficult to know which of the corrections, if any, are appropriate. Another method to improve the DFT functional is to add some per-centage of the Hartree-Fock energy to the functional. There is no theoretical way to know the correct percentage, however, so it is typically fitted to experiment, as in B3LYP, which uses the LYP functional in the 3-parameter fit by Becke[33]. While theoretically not a first principles method, B3LYP is heavily used in the chemistry community as a very accurate semi-empirical model. Unfortunately, for solids, B3LYP often performs much worse than PBE. There are also the new meta-GGAs, which add a dependence on the second derivative of the density. Of these, TPSS[34, 35] and its semi-empirical hybrid[36] are claimed to be quite accurate, and they will be compared to QMC in Chapter 7.

(26)

(27)

Chapter 3

Quantum Monte Carlo

Here, we go over the QMC methods used in the dissertation in detail. Diffusion Monte Carlo and Variational Monte Carlo are covered in more depth in a review article[38].

3.1 Variational Monte Carlo

The expectation value for an arbitrary operatorOand a given real trial variational wave function ΨT is given by

hOi= hΨT|O|ΨTi

hΨT|ΨTi

= R

Ψ2_T(R)[OΨT(R)/ΨT(R)]dR

R

Ψ2_T(R)dR

whereR= (r1,r2, ...,rNe) denotes a set of Ne electron coordinates in 3D space. Typically, such integrals are evaluated by reducing the multi-dimensional integral into a set of low-dimensional integrals. Unfortunately, this either restricts the functional form of ΨT(R) or

makes the calculations undoable for more than a few electrons. One of the key motivations for employing stochastic approaches is to eliminate this restriction and to gain qualitatively new variational freedom for describing many-body effects.

In order to evaluate the expectation value integral stochastically we first generate a set {Rm} of statistically independent sampling points distributed according to Ψ2T(R)

(28)

quantity called local energy (Eloc)

EV M C =

1

M

X

m=1

HΨT(Rm)

ΨT(Rm)

+ε = 1

M

X

m=1

Eloc(Rm) +ε (3.1)

with the statistical errorεproportional to 1/√M.

It is straightforward to apply the variational theorem in this framework. One starts with a variational wave function ΨT(R, P), where R is the set of all the electron

positions and P is the set of variational parameters in the wave function

E(P) = R

ΨT(R, P)HΨT(R, P)dR

R Ψ2

T(R, P)dR

(3.2)

A (hopefully) good approximation to the ground state is then the wave function with the set of parameters P that minimizes E(P). The stochastic method of integration allows us to use explicitly correlated trial wave functions such as, for example, Slater-Jastrow and other functional forms as explained later. In fact, as long as the trial function and its derivatives can be evaluated quickly, any functional form can be used.

This procedure is broken down into two parts: sampling Ψ2_T while evaluating energy and other properties, and optimizing the wave function. The first part, sampling Ψ2

T, is carried out using the Metropolis-Hastings[39, 40] algorithm. We start with a point

R in 3Ne dimensional space and generate a second point R0 according to the transition

probabilityT(R0 ←R). T is a completely arbitrary function so long asT(R0 ←R)6= 0⇔

T(R←R0)6= 0; that is, all moves are reversible. We then accept the move with probability

a= min

1,Ψ

2

T(R0)T(R←R0)

Ψ2

T(R)T(R0 ←R)

. (3.3)

After a few steps, the distribution converges to Ψ2_T, and we continue making the moves until the statistical uncertainties are small enough. For atoms with pseudopotentials, we use the moves as outlined in Ref [38], modified with a delayed rejection step similar to Ref [41], although developed independently, and for full-core calculations, we use the accelerated Metropolis method from Ref [42]. The total energy and its components are evaluated, as well as other properties.

(29)

quantities other than energy that, upon being minimized, will provide an approximation to the ground state wave function. One important one is the variance of the local energy; that is

σ2 = Z

dRΨ2_T(R)(Eloc− hEloci)2/

Z

dRΨ2_T(R). (3.4) Since Eloc is a constant when | ΨTi =| Φ0i, the variance will go to zero for an exact eigenstate. There are several other possible functions, listed in Sec 4.2.2, but variance and energy are generally the only ones optimized in this dissertation.

3.2 Projector Monte Carlo

The variational wave functions are only as accurate as the guessed functional form, whose error is difficult to control in a systematic manner across variety of systems. In order to minimize this bias we employ the projector method, which projects out the ground state of a given symmetry from any trial wave function. To do this, we follow the time propagation of the so-called imaginary-time Schr¨odinger equation

−d|Ψ(τ)i

dτ = (H−E0)|Ψ(τ)i. (3.5)

The time evolution operator ise−(H−E0)τ_{, and} _E

0 will be determined self-consistently. For any state |Ψ(0)i with a non-zero overlap with the ground state |Φ0i, we can expand in a spectral decomposition in the eigenkets:

e−(H−E0)τ_|_Ψ(0)_i₌_c

0|Φ0i+

∞

X

i=1

cie−(Ei−E0)τ|Φii. (3.6)

Since the ground state has the lowest energy, all states except |Φ0i are filtered out in the limit asτ → ∞.

For small τ, we can write the projection operator inR-representation as

hR0|e−(H−E0)τ_|_R_i₌_G₍_R0_,_R_{, τ}₎_'_exp(₋₍_R0₋_R₎2_/₂_τ_{) exp(}₋1

2(V(R) +V(R

0₎₋₂_E

0)) (3.7) (we will write symmetric Green’s functions with a comma, and non-symmetric Green’s functions with an arrow indicating what direction it is acting). This can be interpreted as a dynamic diffusion kernel GD(R0,R, τ) = exp(−(R0 −R)2/2τ) plus a branching kernel

(30)

to sample a path G(RN,RN−1, τ)...G(R2,R1, τ)ΨT(R1), that is, to work in the extended space with dimension 3NeN. ForN large enough, the distribution ofRN will approach Φ0. However, to interpret this as a stochastic process, the path distribution must be positive; that is, the product of allG’s with ΨT must be positive. This gives rise to the fixed node

approximation, where the nodes(the places where the wave function equals zero) of the ground state wave function are approximated as the nodes of the trial wave function. One can avoid this restriction by performing a released-node calculation[26], although the price is a change from polynomial to exponential scaling with system size.

In actual calculations, we perform an importance-sampling transformation, where

G(R0,R, τ) is replaced by the importance sampled Green’s function ˜

G(R0 ←R, τ) = ΨT(R0)G(R0,R, τ)/ΨT(R) (3.8)

The dynamic part of the Green’s function then becomes ˜

GD(R0 ←R, τ) = exp(−(R0−R−τ∇lnΨT(R))2/2τ), (3.9)

a drifting Gaussian, and the branching part becomes ˜

GB(R0,R, τ) = exp(−

τ

2(EL(R) +EL(R

0₎₋₂_E

0)), (3.10) both of which are much better-behaved stochastically, since the ’force’ ∇lnΨT(R) biases

the walk to where the wave function is large, and the local energyEL(R) is much smoother

than the potential energy. This expansion is easiest to see by multiplying both sides of Eqn 3.5 by ΨT and rewriting the differential equation in terms off(R) = ΨT(R)Ψ(R, τ). The

application of the importance sampled Green’s function is

f(R0, τ) = Z

˜

G(R,R0, τ)Ψ2_T(R)dR =

Z

ΨT(R0)G(R0,R, τ)/ΨT(R)Ψ2T(R)dR

= ΨT(R0)

Z

G(R0,R, τ)ΨT(R).

= ΨT(R0)Ψ(R0, τ).

Therefore, if we generate the path ˜G(RN,RN−1, τ)...G˜(R2,R1, τ)Ψ2_T(R1), for large enough

N, the distribution of RN is ΨT(RN)Φ0(RN), which is called the mixed distribution. The

ground state energy is obtainable by evaluating the integral R

ΨTΦ0HΨT/ΨTdR

R

ΨTΦ0dR

= R

Φ0HΨTdR

R

ΨTΦ0dR

(31)

since Φ0 is an eigenstate of H within the nodal boundaries. We use two versions of the projector method: Diffusion Monte Carlo, which has the advantage that the large N limit is easily obtained, and Reptation Monte Carlo, which makes the ‘pure’ distribution Φ2₀ available. Note that in Monte Carlo, the extra time dimension do not cause an extra cost in efficiency over VMC, since only the variance of quantities matters for efficiency, not the number of dimensions. Projector Monte Carlo is typically approximately ten times slower than VMC, though, because the short time Green’s function approximation requires us to move in space much more slowly than in VMC.

3.2.1 Diffusion Monte Carlo

Diffusion Monte Carlo has been discussed by many authors (for a review, see Ref [38, 42]); it attains the f(R, τ) by starting with a distribution of Ψ2_T and interpreting the action of the Green’s function as a stochastic process with killing and branching, eventually ending up with the mixed distribution ΨTΦ0. It has the advantage that the τ → ∞ limit

is easy to achieve, but the disadvantage of not having access to the pure distribution Φ2₀. To elucidate the Diffusion Monte Carlo algorithm, we start with a number of walkers distributed according to Ψ2₍_R_,_{0) = Ψ}2

T(R). We then take one Monte Carlo step

using the importance sampling Green’s function, which is separated into the dynamic and branching parts, as above. The dynamic Green’s function is the same as the one for Brow-nian motion with a drift term, so each walker moves according to the ‘quantum force’

τ∇lnΨT(R), then diffuses. The branching part of the Green’s function is not a

(32)

No importance sampling Importance sampling

Figure 3.1: 100 walker DMC simulation for a one dimensional harmonic oscillator. The horizontal axis is the position coordinate and the vertical axis is the time coordinate moving upwards. The non-importance-sampling method samples the probability distribution Φ0= exp(−1₂x2), while the importance-sampling method samples ΨTΦ0= exp(−2₅x2) exp(−1₂x2)

(33)

correct way by choosing two walkers, one with a large weightw1 and the other with a small weight w2. Walker 1 is branched and walker 2 is killed with probability _w₁w₊1_w₂ and each copy takes on the weight w1+w2

2 . Otherwise, walker 2 is branched and walker 1 is killed, with each copy having the same weight. This control of the weights creates a very small bias that goes to zero as the number of walkers goes to infinity. Therefore, we choose a sizable number of walkers, approximately 1000, to keep this error to a minimum. The extra walkers do not affect efficiency, since they serve as averaging points to reduce the stochastic uncertainty. After∼100 steps the distribution is converged, and the energy can be obtained by averaging over many generations of walkers.

Another large efficiency improvement can be made in the dynamical part of the Green’s function. The exact non-importance-sampled Green’s function must be symmetric, soG(R0,R, τ) =G(R,R0, τ). Since the importance-sampled Green’s function satisfies

˜

G(R0 ←R, τ) = ΨT(R0)G(R0,R, τ)/ΨT(R), (3.12)

it must then satisfy ˜

G(R0←R, τ)ΨT(R) ΨT(R0)

= ˜G(R←R0)ΨT(R

0₎

ΨT(R)

(3.13)

which can be rewritten as ˜

G(R0 ←R)Ψ2_T(R) = ˜G(R←R0)Ψ2_T(R0). (3.14) This is simply the detailed balance condition, which can be enforced the same way as we did in VMC by introducing an acceptance step with the same probability as Eqn 3.3, using only the dynamic Green’s function, since the branching part is already symmetric. This modification has been shown[42] to have very small time step errors, and has the pleasing result of having the DMC algorithm be essentially VMC dynamics at a small time step with reweighting according to the branching Green’s function.

(34)

3.2.2 Reptation Monte Carlo

For local operators that do not commute with the Hamiltonian, we use Reptation Monte Carlo[20] with the bounce algorithm[43]. We sample the path distribution

Π(s) = ΨT(R0)G(R0, R1, τ). . . G(Rn−1,Rn, τ)ΨT(Rn) (3.15)

where s= [R0,R1, . . . ,Rn−1,Rn] is a projection path. This can be interpreted in several

different ways. If we examine the distribution at R0, we can view the samples of Green’s functions as acting on ΨT(Rn), and therefore the probability distribution ofR0isPR0(R0) = ΨT(R0)Φ0(R0). This is the same distribution as we obtain in DMC as the path length goes

to infinity. Alternatively, since G is symmetric on exchange of the two R coordinates, the probability distribution of Rn is the same. Finally, we can split the path in two, one

projecting on ΨT(R0), and the other projecting on ΨT(Rn). We then have

PRn/2(Rn/2) = (G(Rn/2,Rn/2−1). . . G(R1,R2)ΨT(R0))

×(G(Rn/2,Rn/2+1). . . G(Rn−1,Rn)ΨT(Rn))

= Φ2₀(Rn/2)

for n→ ∞, which allows us to obtain correct expectation values of operators that do not commute with the Hamiltonian. We thus evaluate the energy as ERM C = h[EL(R0) +

EL(Rn)]/2i and operators non-commuting with H as ORM C = hO(Rn/2)i. Reptation Monte Carlo does not include branching; it instead uses an acceptance/rejection step. This is a tradeoff, allowing us to project only for a finite τ, since otherwise the probability distribution function is not normalizable, but allowing access to the pure distribution and imaginary time correlations.

The algorithm is realized as follows: we first start with a single walker R0 dis-tributed according to Ψ2

T. We then apply the importance-sampled dynamic Green’s

func-tion to the walker n times, while keeping the path in memory. This results in a path

s= [R0,R1, . . . ,Rn−1,Rn] distributed close to the distribution we wish to generate,

miss-ing only the branchmiss-ing part of the Green’s function. Next, we choose a direction to go, forwards or backwards. Suppose we choose to go forwards. We generate a new position in 3Ne dimensional space X by applying the importance-sampled dynamic Green’s function

starting from the endRn to form a new paths0 = [R1,R2, . . . ,Rn,X]. The idea is to move

(35)

rejected according to the Metropolis algorithm:

a=min

1,Π(s

0₎_T₍_s_←_s0₎

Π(s)T(s0 _←_s₎

. (3.16)

The acceptance reduces to a very simple term. First, we look at the ratio of the desired distribution Π(s0)/Π(s). By writing down Eqn 3.15 for each path and canceling like terms, we obtain

Π(s0) Π(s) =

ΨT(R1)G(Rn,X, τ)ΨT(X)

ΨT(R0)G(R0,R1, τ)ΨT(Rn)

. (3.17)

The transfer probability T(s0 ←s) is the probability to generate the new point ˜GD(X ←

Rn, τ), since we used the importance-sampled dynamics. The reverse move is similarly

˜

GD(R0 ←R1, τ), since we could have gone backwards. Since ˜

GD(R0←R, τ) ˜GB(R0 ←R, τ)'G˜(R0 ←R, τ) = ΨT(R0)G(R,R0, τ)/ΨT(R), (3.18)

after substitution for ˜GD and cancellation of common terms, we end up with

a=min

1, exp(−

τ

2(EL(Rn) +EL(X)) exp(−τ₂(EL(R0) +EL(R1))

, (3.19)

the ratio of the branching parts of the importance-sampled Green’s function. In retrospect, this might have been expected, since we generated the new point with the dynamic part; all that is left is the branching part. Unlike DMC, RMC is a pure Metropolis algorithm–the walk is just in the path space. This is a big advantage in terms of parallel scalability and simplicity of implementation. The biggest disadvantage is that since the path can move forward or backward each step, it takes more steps to generate a new path the larger n

is. This can be remedied to a large part by the bounce[43] modification, but for very long paths it still becomes slower.

The bounce algorithm makes one minor modification to the algorithm as above. Rather than choosing the direction randomly at each step, the bounce algorithm instead continues in the same direction until a step is rejected. This does not satisfy detailed balance, but the transfer matrix does have the correct distribution as an eigenfunction, which is sufficient to show that the bounce algorithm generates Π(s). This increases the efficiency of RMC tremendously, because instead of the path moving one step right, then one step left while getting nowhere, it moves perhaps 100 steps to the right, replacing the entire path, then 100 to the left, generating a whole new path.

(36)

energy, while also offering an unbiased estimation of quantities that do not commute with the Hamiltonian. The only cost is that the projection length must be chosen long enough that all quantities are converged, but short enough that the path does not slow down. This is not too much of a restriction, because the cost of the calculation only rises when the path is not completely replaced before a bounce.

We mentioned that RMC only obtains unbiased estimates for local operators. This is only partially true. Nonlocal operators are possible, but difficult. To evaluate a general operatorO at the center of the path in RMC, we are calculating the integral

hOi= Z

dR0. . . dRnΨT(R0)G(R0,R1). . .

G(R_n/₂−1,Rn/2)OG(Rn/2,Rn/2+1)

. . . G(Rn−1,Rn)ΨT(Rn),

where we have suppressed the τ dependency and the normalization over paths. If the operator is local, then it is simply evaluated by the procedure explained above. If it is not, then one must evaluate OG(R_n/₂,R_n/₂₊₁) in one way or another. This has little relevance for the dissertation, but is discussed at length for finite temperature path integral calculations[44].

3.3 Pseudopotentials

Heavy atoms cause a problem with QMC because of the high energy core region. There, the large potential energy causes the core electrons to have large kinetic energy and the local energy to fluctuate wildly. The core electrons contribute very little to the chemical properties of a material, but make the calculation much more difficult. This problem has been solved in Hartree-Fock, DFT, etc. by removing the core electrons and introducing a pseudopotential to replace them. The end result is a pseudo-atom which has only valence electrons that mimic the behavior of the valence electrons of the real atom. This is done by adding a local term in the Hamiltonian, which does not cause problems, and a term that is dependent on the angular momentuml:

Vpseudo =

X

l

(37)

The projection operator|lihl|mimics the action of the core electrons, sincescore electrons interact differently with other s electrons than with other angular momenta, etc. While the pseudopotential has the desired effect of removing the core electrons, the projection operator introduces nonlocality in the potential energy, which in turn makes the stochastic interpretation of the imaginary time Schr¨odinger equation difficult. This has recently been overcome[45] within DMC and RMC, although the work in this thesis uses a simpler method from Mitas et al.[46]. Instead of using the nonlocal pseudopotential Hamiltonian, we use the Hamiltonian Vloc(R) =

VpseudoΨT(R)

ΨT(R) , which, similar to the local energy, is a local function of R. The DMC (or RMC) calculation is then performed normally. This approximation is called the locality approximation, and is exact when ΨT is the same as the ground state

wave function. It converges as the square of the error in ΨT, so for a good Jastrow factor,

the error is generally fairly small, on the order of the fixed node approximation or less.

3.4 Periodic Boundary Conditions

For molecules, the implementation of QMC methods are fairly straightforward, but for infinite(periodic) systems, there are several techniques to obtain the approach to the infinite limit. The complicating factor for solids is that the real system that we are trying to simulate is a molecule containing at least 1023particles, distributed in some regular way. If we enforce periodic boundary conditions, where an electron escaping a simulation cell reappears on the other side, it is a good approximation to the large molecule that we call a solid.

3.4.1 Finite Size Error

(38)

be added at the cost of complex arithmetic by using the fixed-phase Diffusion Monte Carlo algorithm[50]. We didn’t find that necessary, since the 2x2x2 cell with all k-points was sufficient for the calculations in this work.

For a cubic cell with lattice constant a, there are four real independent k-points: Γ(0,0,0), X(π/a,0,0), R(π/a,π/a,0), and L(π/a,π/a,π/a). By counting the number of symmetry-related k-points, we can assign respective weights of 1₈,3₈,3₈, and 1₈. It is somewhat more convenient to weight them all the same in DMC, since then the optimal errors are obtained by performing all of the calculations for the same length of time. It turns out that setting the weights to one does not affect the calculated energy significantly, at least for the perovskite materials studied here(see Fig 3.2). For cases in which the weights do matter, the error bar is given as

= P1

iwi

s X

i

w2

i2i, (3.21)

where the sum is over all k-points treated. The error bar is proportional to 1/√Ti, where

Ti is the computer time spent on that k-point. Therefore, we must minimize the function

X

i

w_i2 Ti

(3.22)

subject to the constraint thatP

iTi =T, the total amount of computer time spent. For the

cubic case, this reduces to finding the minimum of 1/x+ 9/(T−x) (settingT1 =T4=xand

T2=T3 =T−x), which is approximatelyx= 0.2T, although the minimum is very shallow. Therefore, optimally, one should spend four times longer on the k-points with weight equal to three than with one. Spending equal time (x= 0.5T) results in an efficiency penalty of about 20%.

The coulomb finite size effect only appears in many-body simulations, and is the result of the interaction of an electron with its periodic image. With PBCs, instead of an infinite solid with all the electrons moving more or less independently, we simulate lattices of electrons interacting with each other. For large enough cells, the lattices are spaced far enough apart that it is a good approximation, but the coulomb finite size error precludes simulation at the primitive cell size for most materials. There have been various attempts to correct this size effect in a consistent way[47, 51], but we use a heuristic which seems to work just about as well. For the homogeneous electron gas, the potential energy scales as 1/rs, where rs is the average separation between electrons. Therefore, the importance

(39)

1

2

3 Supercell size (nxnxn)

-130.6

-130.4

-130.2

-130

Energy/cell (Hartrees)

averaged k-points, coulomb correction weighted k-points, coulomb correction L point, coulomb correction

L point, no coulomb correction LDA(2x2x2 kmesh, shifted)

Figure 3.2: Finite size scaling of cubic BaTiO3 using a Slater determinant of LDA orbitals in VMC without a Jastrow factor. The averaged and weighted k-points are over the four real k-points in the cubic cell.

By doing finite size extrapolations for several materials, Mitas(unpublished) came to a correction equal to 0_r.36

s Hartrees for most solids, which appears to work quite well. Note that this correction is the same for all cell sizes, so the relative importance decreases as the inverse of the system size, as it is generally expected.

Fig 3.2 shows the performance of this scheme, where we see that our scheme performs quite well, converging to less than 10 mH per cell with the 2x2x2 cell. Corrections using the LDA, which have been used in the past, are actually in the wrong direction.

3.5 Summary

(40)

Table 3.1: List of approximations in the QMC method. They are decomposed into system-atic approximations, which can be resolved by changing parameters in the simulation, and methodological approximations, which are inherent in the method. In principle, it is known how to remove each of these approximations, although it may be prohibitively expensive to do so for a large system.

Approximation Resolution

Systematic

Small time step expansion for the Green’s func-tion

extrapolate to zero time step Population control in DMC add more walkers

Finite projection length in RMC extrapolate to infinite path length Finite size simulation cell in periodic boundary

conditions

Sample k-points and extrapolate to infinite size

Methodological

Fixed-node approximation Vary nodal structure to minimize fixed-node energy. This is still an approximation in this case. Only completely resolvable by exact pro-jection, which is exponentially scal-ing

Pseudopotential transferability error Check full-core calculations in DFT or Hartree-Fock. Mostly cancels out in energy differences

Localization approximation for pseudopoten-tials

Use Casula’s algorithm[45] or use a very good trial wave function; the Jastrow factor helps. Typically quite small error

the approximations made and their resolutions in Table 3.5

(41)

Chapter 4

QWalk: An Implementation of

Quantum Monte Carlo

We have developed a new program QWalk for QMC calculations written in C++ with modern programming techniques and incorporating state of the art algorithms in a fast and scalable code. QWalk has already been used in several publications[7, 8, 52, 53, 54] not including the work in this dissertation.

4.1 Organization and Implementation

(42)

Method (VMC)

Wavefunction_type (Multiply) System

(Molecule)

Dynamics generator (Delayed Rejection)

Wavefunction_data (Slater Determinant)

Wavefunction_data (Jastrow factor)

Molecular orbital evaluator(Linear)

One-body Jastrow Two-body Jastrow

Basis_function (Gaussian)

Basis_function (cutoff Pade)

Most Abstraction

Least abstraction

Figure 4.1: Calculation structure for the VMC method on a molecule using a Slater-Jastrow wave function.

Table 4.1: The central objects of the code and their physical correspondents Module name Mathematical object

System parameters and form of the Hamiltonian Sample point R, the integration variables

Wave function type Wave function ansatz Wave function ΨT(R), ∇ΨT(R), ∇2ΨT(R)

(43)

The modules form a tree of successive abstractions (Fig 4.1). At the top of the tree is the QMC method, VMC in this case. It works only in terms of the objects directly below it, which are the concepts of System, Wave function data, etc. (see Table 4.1). These in turn may have further abstractions below them, as we’ve shown for the wave function object. The highest wave function object is of type ‘Multiply’, which uses two wave function types to create a combined wave function. In this case, it multiplies a Slater determinant with a Jastrow correlation factor to form a Slater-Jastrow function. Since the wave functions are pluggable, the Slater determinant can be replaced with any antisymmetric function, as well as the Jastrow factor. The type is listed along with the specific instant of that type in parenthesis. At each level, the part in parenthesis could be replaced with another module of the same type.

We present an implementation of the VMC algorithm as an example of how the code is organized(Fig 4.2). For reasons of space, we do not write the function line-by-line, which includes monitoring variables, etc., but instead give a sketch of the algorithm. The VMC method works at the highest level of abstraction, only in terms of the wave function, system, and random dynamics. It does not care what kind of system, wave function, etc. are plugged in, only that they conform to the correct interfaces. In an App C, we give an example of how to create a new module.

We will now provide a listing of the available modules for the major types, along with some details of their implementation.

4.2 Methods

4.2.1 Variational Monte Carlo

(44)

Vmc_method::run(vector <string> & vmc_section, vector <string> & system_section,

vector <string> & wavefunction_section) {

//Allocate the objects we will be working with System * sys=NULL;

allocate(sys, system_section);

Wavefunction_data * wfdata=NULL;

allocate(wfdata, sys, wavefunction_section);

Sample_point * sample=NULL; sys->generateSample(sample); Wavefunction * wf=NULL;

wfdata->generateWavefunction(wf);

//the Sample_point will tell the Wavefunction when we move an //electron

sample->attachWavefunction(wf); sample->randomGuess();

//This is the entire VMC algorithm for(int s=0; s< nsteps; s++) {

for(int e=0; e < nelectrons; e++) {

dynamics_generator->sample(e,timestep,wf,sample); } //end electron loop

//gather averages } //end step loop

//report final averages

(45)

Table 4.2: Optimization objective functions implemented

Function Minimized quantity

Variance h(EL(R)−Eref)2i

Absolute value h|EL(R)−Eref|i

Lorentz hln(1 + (EL(R)−Eref)2/2)i

Energy hEL(R)i

Mixed aEnergy+ (1−a)V ariance,0< a <1

4.2.2 Optimization of Wave Functions

We have implemented three different methods for optimization. All methods are capable of optimizing several objective functions(See Table 4.2). Any of these objective functions will obtain the correct ground state with an infinitely flexible function, but may obtain different minima for incomplete wave functions and some are easier to optimize than others. The first (OPTIMIZE) is based on Umrigar et al.’s[55] variance optimization. The method minimizes the objective function on a set of fixed configurations from VMC using a conjugate gradient technique, usually not reweighting the averages as the wave function changes. Optimizing the energy using OPTIMIZE is quite expensive, because it requires many configurations to evaluate an unbiased estimate of the energy derivative.

The next two are based on Umrigar and Filippi’s Newton optimization[56] method. OPTIMIZE2 also uses a fixed set of configurations, but instead of evaluating only the first derivatives of the objective function, as conjugate gradients do, it uses a low-variance esti-mator for the Hessian matrix and Newton’s method to find the zeros of the first derivatives. OPTIMIZE2 is able to produce better wave functions with lower energies than OPTIMIZE by directly optimizing the energy even for very large systems (we have applied it for up to 320 electrons) while costing slightly more. This method was used throughout this work to optimize a linear combination of 95% energy and 5% variance.

(46)

4.2.3 Diffusion Monte Carlo

DMC is implemented almost identically to VMC, except that the time step is typically much smaller and each walker accumulates a weight equal to exp(−τef f

2 (EL(R

0_{) +} EL(R)−2Eref)). Since we use an acceptance/rejection step, τef f is chosen somewhat

smaller thanτ asτef f =pτ, wherep is the acceptance ratio. To control the fluctuations in

the weights, we employ a constant-walker branching algorithm, which improves the parallel load balancing properties of DMC. Every few steps, on each node, we choose a set of walkers that have large weights (w1) for branching. Each one of these walkers is matched with a smaller weight walker (w2) which is due for killing. The large weight walker is branched and the small weight walker is killed with probability w1

w1+w2, with each copy gaining a weight of w1+w2

2 . Otherwise, the small weight walker is branched and the large weight walker is killed, with the copies having the same weight as before. Occasionally, walkers are exchanged between nodes to keep the weight per processor approximately constant. This conserves the total weight and keeps the number of walkers constant. QWalk keeps track of two numbers: Eref and E0. Eref is first set to the VMC average energy, and then to

the energy of the last block. The energy that goes into the weights, E0, is then calculated every few steps as

E0=Eref−log

P

wi

Nconf

, (4.1)

whereNconf is the number of sample points (configurations) in the simulation.

During the DMC calculation, the local energy will very occasionally fluctuate down drastically, causing the weight to increase too much. This can be fixed by cutting off the weights. For fluctuations beyond ten standard deviations of the energy, we smoothly bring the effective time step to zero for the weights, which avoids the efficiency problem without introducing a noticeable error.

4.2.4 Reptation Monte Carlo

(47)

the Green’s function as explained in Sec 3.2.2. We use the bounce algorithm[43], which improves the efficiency by allowing the path to explore the many-body phase space much more quickly.

4.2.5 Correlated Sampling

Correlated sampling is a technique where one samples two very similar systems with the same sets of samples. The variance in the difference will decrease asV ar(X−Y) =

V ar(X) +V ar(Y)−2Cov(X, Y), so for perfectly correlated sampling, the variance will be zero for the difference. In QWalk, this is handled by performing a primary walk that samples some probability distributionP1(X). Averages are obtained as usual by calculating the integral hO1i=

R

P1(X)O1dX. Suppose we wish to findhO2−O1i. It can be written as

Z

P2(X)O2−P1(X)O1dX= Z

P1(X)

P2

P1

O2−O1

dX. (4.2)

Since we are sampling P1(X), in the Monte Carlo averaging, this integral is evaluated by averaging the weighted difference over sample points:

N

X

i

"

wi(Xi)O2(Xi)

P

jwi(Xi)

−O1(Xi)

N

#

(4.3)

The difference in the methods is only in how they determine the weights.

VMC, DMC and RMC all support correlated sampling between arbitrary systems. In VMC, the weights are w(X) = Ψ22(X)

Ψ2 1(X)

, which is an exact relationship. DMC and RMC both require some approximation to the Green’s function to weight the secondary averages properly. In both, we use the approximation of Filippi and Umrigar[12], who discuss the problem in detail. See also Chapter 5 for a discussion on correlated sampling and its limitations.

4.3 Systems

4.3.1 Boundary Conditions

(48)

with arbitrary atoms, charge, spin state, and with finite electric field are supported. In 3D periodic systems, the calculation can be done at any real k-point, allowing k-point integrations, and there is also a finite size correction as discussed in Sec 3.4.1. The code has been used on systems with up to 135 atoms and 1080 electrons; the limiting factor is the amount of computer time needed to reduce the stochastic uncertainties.

4.3.2 Pseudopotentials

QWalk accepts pseudopotentials as an expansion of nonlocal angular momentum operators:

ˆ

VECP =Vlocal(R) + lmax

X

l=0

Vl(R)|lihl| (4.4)

for arbitrary maximum angular moment. Vl is a basis function object that is typically a

spline interpolation of a grid or a sum of Gaussian functions. While any pseudopotential of this form can be used, we use soft potentials in which the Z_r divergence has been removed from the nuclei-electron interaction. These potentials have been created specifically for QMC and are available in the literature[57, 58, 59], although more traditional Hartree-Fock or DFT pseudopotentials in the Troullier-Martins form work as well.

4.4 Forms of Wave function

For chemical problems, the first-order trial function is usually written as a Slater determinant taken from Hartree-Fock or Density Functional Theory multiplied by a cor-relation factor(known as a Jastrow factor) which is optimized in Variational Monte Carlo. Between 90% and 95% of the correlation energy is typically obtained with this trial wave function in Diffusion Monte Carlo.

(49)

then fills in a Slater determinant object and a Jastrow object. For a Pfaffian-Jastrow wave function, the user replaces the Slater determinant input with the Pfaffian input. Obviously, it is up to the user to make sure that the total wave function is properly antisymmetric and represents the problem correctly.

4.4.1 Slater Determinant(s)

This is the standard sum of Slater determinants, written as ΨT =

P

ciD

↑

iD

↓

i,

where D↑_i(↓) is a determinant of the spin up(down) one-particle orbitals. The coefficients are optionally optimizable within VMC.

4.4.2 Jastrow Factor

The Jastrow factor is written as eU, where

U =X

iIk

cei_kak(riI) +

X

ijk

cee_k bk(rij) +

X

ijIklm

ceei_klm[ak(riI)al(rjI) +ak(rjI)al(riI)]bm(rij), (4.5)

i, j are electron indices, and I is a nuclear index, and sums over these indices are implied. Both the coefficients and parameters within the basis functions can be optimized. For the basis functions, we satisfy the exact electron-electron cusp conditions with the function

b(r) = cp(r/rcut)/(1 +γp(r/rcut)), where p(z) =z−z2+z3/3, γ is the curvature, which is optimized, and c is the cusp(0.25 for like spins and 0.50 for unlike spins). Further correlation is added by including functions of the formbk(r) =ak(r) = ₁₊1−_βzppzpp(₍r/rcut_r/rcut)₎ where

(50)

4.4.3 Pfaffian Pairing Wave Function

Pairing wave functions with a Jastrow factor for molecules were first investigated by Casula and coworkers[60], who studied the constant number of particles projection of the BCS wave function. We write the wave function as ΨT =eUdetΦ, whereeU is the Jastrow

factor of above and the matrix Φij =χ(ri, rj) is the pairing function between opposite-spin

electrons(the function is easily extended forNup6=Ndown). This function contains the Slater

determinant as a special case whenχis written as the sum over the occupied single-particle orbitals: χ(ri, rj) =PN_keφk(ri)φk(rj). We have implemented the Pfaffian[52] pairing wave

function, which allows not only unlike-spin pairing, as the canonical projection of the BCS wave function does, but also allows like-spin pairing. The wave function is written as the Pfaffian of the matrix P, appears as the following:

P = pf 

  

ξ↑↑ Φ↑↓ ϕ↑

−Φ↑↓T ξ↓↓ ϕ↓ −ϕ↑T _−ϕ↓T ₀



  

. (4.6)

The Φ matrices are the same as in the BCS wave function, and the ϕ matrices are made up of the one-particle orbitals for a spin-polarized system. Theξ are antisymmetric triplet pairing matrices. The operation of the Pfaffian ensures that the entire wave function is antisymmetric. The Pfaffian wave function contains the BCS wave function as a special case without triplet pairing, and thus contains the Slater determinant wave function as well. The general expansion for Φ is

Φ(r1,r2) = X

kl

cklφk(r1)φl(r2) (4.7)

under the constraint thatckl=clk. ξ is written in a very similar way:

ξ(r1,r2) =

X

kl

dklφk(r1)φl(r2) (4.8) under the constraint that dkl =−dlk. The sum extends over the virtual space of orbitals.

Quantum Monte Carlo for Transition Metal Systems: Method Developments and Applications

Contents

List of Figures

List of Tables

List of Symbols and Abbreviations

Chapter 1

Introduction

1.1

Ab Initio Calculations

1.2

Transition Metal Oxides

1.3

Outline

1.4

Accomplishments

Chapter 2

Electronic Structure Methods

Other than Quantum Monte Carlo

2.1

Hartree-Fock and Post Hartree-Fock

2.2

Density Functional Theory

Chapter 3

Quantum Monte Carlo

3.1

Variational Monte Carlo

3.2

Projector Monte Carlo

3.3

Pseudopotentials

3.4

Periodic Boundary Conditions

1

2

3

Supercell size (nxnxn)

-130.6

-130.4

-130.2

-130

Energy/cell (Hartrees)

3.5

Summary

Chapter 4

QWalk: An Implementation of

Quantum Monte Carlo

4.1

Organization and Implementation

4.2

Methods

4.3

Systems

4.4

Forms of Wave function