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Variational Monte Carlo Studies of Atoms View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by NORA - Norwegian Open Research Archives Variational Monte Carlo studies of Atoms by HÅVARD SANDSDALEN THESIS for the degree of MASTER OF SCIENCE (Master in Computational Physics) Faculty of Mathematics and Natural Sciences Department of Physics University of Oslo June 2010 Det matematisk-naturvitenskapelige fakultet Universitetet i Oslo Contents 1 Introduction 7 I Theory 9 2 Quantum Physics 11 2.1 Quantum Mechanics in one dimension . 11 2.1.1 Probability and statistics . 11 2.1.2 The time-independent Schrödinger Equation . 12 2.2 Bra-Ketnotation ............................... 15 2.3 Quantum Mechanics in three dimensions . 15 2.3.1 Separation of variables - quantum numbers l and m ....... 16 2.3.2 The Angular equation . 16 2.3.3 The Radial equation and solution for the hydrogen atom ..... 18 3 Many-particle Systems 21 3.1 Atoms..................................... 21 3.1.1 Two-particle systems . 21 3.1.2 Wave functions for N-particle atoms . 23 3.1.3 Electron configuration . 24 3.1.4 Hamiltonian and scaling . 25 3.2 Hartree-FockTheory ............................. 27 3.2.1 Hartree-Fock equations . 29 4 Quantum Monte Carlo 35 4.1 Markovchains................................. 36 4.2 Randomnumbers............................... 37 4.3 The Variational Principle . 38 4.4 Variational Monte Carlo . 39 4.4.1 VMC and the simple Metropolis algorithm . 39 4.4.2 Metropolis-Hastings algorithm and importance sampling . 41 5 Wave Functions 47 5.1 Cusp conditions and the Jastrow factor . 47 5.1.1 Single particle cusp conditions . 47 5.1.2 Correlation cusp conditions . 49 5.2 Rewriting the Slater determinant . 49 5.3 Variational Monte Carlo wave function . 50 5.4 OrbitalsforVMC............................... 51 5.4.1 S-orbitals ............................... 51 5.4.2 P-orbitals ............................... 52 5.5 Roothaan-Hartree-Fock with Slater-type orbitals . ......... 55 5.5.1 Derivatives of Slater type orbitals . 57 II Implementation and results 59 6 Implementation of the Variational Monte Carlo Method 61 6.1 Optimizing the calculations . 62 new old 6.1.1 Optimizing the ratio - ΨT (r )/ΨT (r ) ............. 63 6.1.2 Derivative ratios . 67 6.2 Implementation of Metropolis-Hastings algorithm . ......... 73 6.3 Blocking.................................... 74 6.4 Time step extrapolation . 76 6.5 Parallel computing . 76 7 Hartree-Fock implementation 77 7.1 Interaction matrix elements . 77 7.2 AlgorithmandHFresults .......................... 78 8 VMC Results 81 8.1 Validationruns ................................ 81 8.1.1 Hydrogen-like orbitals . 81 8.1.2 Slater-type orbitals - Hartree Fock results . ..... 82 8.2 Variationalplots ............................... 83 8.2.1 Hydrogen-like orbitals . 83 8.2.2 Slater-type orbitals - Hartree Fock results . ..... 87 8.3 Optimal parameters with DFP . 88 8.3.1 Hydrogen-like orbitals . 90 8.3.2 Slater-type orbitals - Hartree Fock results . ..... 91 8.4 Time step analysis - extrapolated results . ...... 91 8.4.1 Hydrogenic wave functions . 91 8.4.2 Slater-type orbitals . 91 8.5 Discussions .................................. 96 9 Conclusion 99 A Roothaan-Hartree-Fock results 101 B Statistics 105 C DFP and energy minimization 111 Bibliography 114 Acknowledgements I would like to thank my advisor Morten Hjorth-Jensen for always being available and full of helpful adivce. Also, I would like to thank my trusted fellow students Magnus Pedersen Lohne, Sigurd Wenner, Lars Eivind Lervåg and Dwarkanath Pramanik for being positive and inspirational through the rough patches. I would not have able to complete this work without the support of my family and friends. Thank you! Håvard Sandsdalen Chapter 1 Introduction The aim of this thesis is to study the binding energy of atoms. Atoms of interest in this work are helium, beryllium, neon, magnesium and silicon. The smallest of all atoms, the hydrogen atom, consists of a nucleus of charge e and one electron with charge e. The − size of this atom is approximately the famous Bohr radius, a 5 10−11 (see [1]). To be 0 ≈ · able to describe such small systems, we must utilize the concepts of quantum physics. Many-particle systems such as atoms, cannot be solved exactly using analytic methods, and must be solved numerically. The Variational Monte Carlo (VMC) method is a technique for simulating physical systems numerically, and is used to perform ab initio calculations on the system. The term “ab initio” means that the method is based on first principle calculations with strictly controlled approximations being made (e.g. the Born-Oppenheimer approximation, see section 3.1.4). For our case this means solving the Schrödinger equation (see e.g. [2]). Our main goal will be to use VMC to solve the time-independent Schrödinger equation in order to calculate the energy of an atomic system. We will study the ground state of the atom, which is the state corresponding to the lowest energy. Variational Monte Carlo calculations have been performed on atoms on several occasions (see ref. [3]), but for the atoms magnesium and silicon this is not so common. In this work we will perform VMC calculations on both well-explored systems such as helium, beryllium and neon in addition to the less examined magnesium and silicon atoms. The helium, beryllium and neon calculations from [3] will serve as benchmark calculations for our VMC machinery, while for silicon and magnesium we have compared with results from [4]. Chapter 2 in this thesis will cover the quantum mechanics we need in order to implement a VMC calculation on atoms numerically. We start off by introducing the basic concepts of single-particle quantum mechanical systems, and move on to describe the quantum mechanics of many-particle systems in chapter 3. Furthermore, we will introduce the Variational Monte Carlo method in chapter 4, and a detailed discussion of how to approximate the ground state of the system by using Slater determinants is included in chapter 5. A large part of this thesis will also be devoted to describing the implementation of the VMC machinery and the implementation of a simple Hartree-Fock method. This will be discussed in chapters 6 and 7. We will develop a code in the C++ programming language (see ref. [5]) that is flexible with respect to the size of the system and several other parameters introduced throughout the discussions. Chapter 8 and 9 will contain the results produced by the VMC program we have Chapter 1. Introduction developed, as well as a discussion and analysis of the results we have obtained. 8 Part I Theory Chapter 2 Quantum Physics In the 1920’s European physicists developed quantum mechanics in order to describe the physical phenomena they had been discovering for some years. Such phenomena as the photoelectric effect, compton scattering, x-rays, blackbody radiation and the diffraction patterns (see ref. [1]) from the double-slit experiment indicated that physicists needed a new set of tools when handling systems on a very small scale, e.g. the behavior of single particles and isolated atoms. This chapter will give an introduction to the relevant topics in quantum physics needed to describe the atomic systems in this project. Some parts will closely follow the discussions in the books [2] and [6], while other parts only contain elements from the sources listed in the bibliography. 2.1 Quantum Mechanics in one dimension The general idea and goal of quantum mechanics is to solve the complex, time-dependent Schrödinger-equation (S.E.) for a specific physical system which cannot be described by classical mechanics. Once solved, the S.E. will give you the quantum mechanical wave function, Ψ, a mathematical function which contains all information needed about such a non-classical system. It introduces probabilities and statistical concepts which contradict with the deterministic properties of systems described by classical mechanics. The full one-dimensional S.E. for an arbitrary potential, V , reads: ∂Ψ ~2 ∂2Ψ i~ = + V Ψ. (2.1) ∂t −2m ∂x2 The Schrödinger equation is the classical analogy to Newton’s second law in classical physics, and describes the dynamics of virtually any physical system, but is only useable in small scale systems, quantum systems. 2.1.1 Probability and statistics The wave function, Ψ, is now a function of both position, x, and time, t. Quantum mechanics uses the concept of probability and statistics via Ψ, and these solutions of the S.E. may be complex as the equation Eq. (2.1) is complex itself. To comply with the statistical interpretation we must have a function that is both real and non-negative. As described in [2], Born’s statistical interpretation takes care of this problem by introducing the complex conjugate of the wave function. The product Ψ∗Ψ= Ψ 2 is interpreted as | | Chapter 2. Quantum Physics the probability density for the system state. If the system consists of only one particle, the integral b Ψ(x,t) 2 dx, (2.2) | | Za is then interpreted as the probability of finding the particle between positions a and b at an instance t. To further have a correct correspondence with probability, we need the total probability of finding the particle anywhere in the universe to be one. That is ∞ Ψ(x,t) 2 dx = 1. (2.3) | | Z−∞ In quantum mechanics, operators represent the observables we wish to find, given a wave function, Ψ. The operator representing the position variable, x, is just x itself, while the momentum operator is p = i~(∂/∂x). All classical dynamical variables are − expressed in terms of just momentum and position (see [2]). Another importantb operator is the Hamilton operator. The Hamiltonb operator gives the time evolution of the system and is the sum of the kinetic energy operator T and the potential energy operator V, H = T + V.
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