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Applied Hartree-Fock Methods

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Applied Hartree-Fock Methods Applied Hartree-Fock methods Atomic and diatomic energy computations Anders Larsen Rolf Sommer Poulsen Supervisor : Thomas Garm Pedersen Masters project Department of Physics Aalborg University Denmark Spring, 2015 ii Department of Physics and Nanotechnology Aalborg University http://www.aau.dk Title: Abstract: Applied Hartree-Fock methods The aim of this project was to Theme: calculate the electron and poten- Atomic and diatomic energy calcula- tial energy functions of diatomic tions molecules using the Hartree-Fock method. In this report the theory be- Project Period: hind the Hartree-Fock method is de- Spring semester 2015 rived, explained and the implementa- tion through Hartree-Fock-Roothaan Project Group: equations has been presented. anro-F15 To demonstrate the Hartree-Fock method of central field problems it Participants: was used to calculate the electron en- Anders Larsen ergy of neutral atoms from Z = 1 Rolf Sommer Poulsen through Z = 103, as well as for an- ions and cations of the 52 first ele- Supervisor: ments with good accuracy. Thomas Garm Pedersen Calculations on static atomic polar- izabilities has been attempted. Re- Copies: 4 sults for the Hydrogen atom has been presented, but further research is Page Numbers: 68 needed for the rest of the atoms. Applying a Cartesian Gaussian ba- Date of Completion: sis the energy functions and potential June 2, 2015 energy functions of the molecules H2, HeH, He2, LiH and Li2 were calcu- lated. The calculations yielded am- biguous results and need further re- search. The content of this report is freely available, but publication (with reference) may only be pursued due to agreement with the author. iv Institut for Fysik og Nanoteknologi Aalborg Universitet http://www.aau.dk Titel: Abstract: Applied Hartree-Fock methods M˚aletmed dette projekt var at be- Tema: regne elektron energier og potentialer Atomic and diatomic energy calcula- for diatomare molekyler ved anven- tions delse af Hartree-Fock metoder. I den- ne rapport udledes og forklares teo- Projektperiode: rien bag Hartree-Fock metoden, og Spring semester 2015 implementeringen gennem Hartree- Fock-Roothaan ligningerne er blevet Projektgruppe: præsenteret. anro-F15 For at demonstrere Hartree-Fock me- toden anvendt p˚aet centralpotential- Deltagere: problem er elektronenergierne for Anders Larsen atomerne Z = 1 til og med Z = Rolf Sommer Poulsen 103 blevet beregnet, ligesom energi- erne for anioner og kationer for de 52 Vejleder: første atomer er præsenteret med ac- Thomas Garm Pedersen ceptabel nøjagtighed. Beregninger for atomare polarisabili- Oplag: 4 teter er blevet forsøgt. Resultater for Hydrogen er præsenteret, men yderli- Sidetal: 68 gere undersøgelser er nødvendige for de resterende atomer. Afleveringsdato: Ved at anvende en kartesisk Gaus- 2. juni 2015 sisk basis er elektronenergier og po- tentialer for molekylerne H2, HeH, He2, LiH and Li2 blevet beregnet. Da beregningerne viser tvetydige resul- tater er der brug for nærmere un- dersøgelser. Rapportens indhold er frit tilgængeligt, men offentliggørelse (med kildeangivelse) m˚akun ske efter aftale med forfatterne. vi Preface This report is written by two FYS10 students during the spring of 2015 at Aalborg University. The aim of the report is to model multi-particle quantum mechanical systems, namely electron energies of atoms and diatomic molecules. Furthermore the project sought to calculate the polarizability of the elements using the Hartree-Fock method. The target group is students who are familiar with basic quantum mechanics, differential equations, calculus and linear algebra. Figures and equations will be denoted and referred to with two numbers like (x:y) where x denotes the chapter and y denotes the number of the given figure or equation. Citations are referred to using the Vancouver method where a reference is denoted by a number like [3]. In the report the following notations have been used: A vector named r is denoted r. A unit vector in the z-direction ∗ is denoted in ez The complex conjugate of a is denoted a , matrices are denoted like M and operators are denoted like H^ . The report is typeset in LATEX. Illustrations are made in TikZ and all cal- culations and figures have been produced in Python 3.0 using matplotlib, the numpy and scipy packages. To some of the analytical calculations the CAS-tool Wolfram Mathematica 10 has been used. The used scripts are presented on a DVD attachment in appendix D Anders Larsen Rolf Sommer Poulsen viii Contents 1 Preliminaries 1 1.1 The many-electron problem . .1 1.2 The Aufbau principle . .2 1.3 The Adiabatic approximation . .2 1.4 Slater Determinants . .4 1.5 Matrix operations on Slater Determinants . .4 1.6 The Hartree-Fock equations . .6 1.7 The Hartree-Fock-Roothaan equations . .9 2 Implementation for atoms 11 2.1 Gaussian basis functions . 11 2.2 The matrix elements . 12 2.3 Self Consistent Field procedure . 13 2.4 Convergence schemes . 13 2.5 Results . 15 2.6 Discussion . 17 2.7 Ions . 18 2.8 Conclusion . 21 3 Polarizability calculations 23 3.1 Polarizability . 23 3.2 The Hamiltonian with an external electric field . 23 3.3 Implementation . 24 3.4 Results . 24 4 Implementation for molecules 27 4.1 Calculation of molecular matrix elements . 27 4.2 Procedure . 34 4.3 Results . 35 4.4 Discussion . 37 4.5 Vibrational states . 40 4.6 Conclusion . 44 5 Conclusion 45 A Matrix elements for atomic calculations 47 A.1 The coulomb elements . 47 A.2 The exchange elements . 48 ix x CONTENTS B Hermite Polynomials 51 C Python code 53 C.1 Hartree Fock.py............................ 53 C.2 Polarizability.py . 56 C.3 Molecules.py . 60 C.4 Vibrational Diatomic.py . 63 D DVD with code 65 Chapter 1 Preliminaries The statements of this chapter can be found in any typical text book on quantum mechanics such as [3] or [1]. 1.1 The many-electron problem As it is commonly known the single electron Schr¨odingerequation in atomic units for a single-nucleus atom is 1 Z 2 = E : (1.1) −2r − r Let the Hamiltonian for this problem here forward be denoted h^i such that ^ 1 2 Z hi = i : (1.2) −2r − ri Adding N electrons to the atom gives rise to an electron repulsion term so that the Hamiltonian becomes N N ! X X 1 h^i + Ψ = EΨ: (1.3) r i=1 i<j ij The collected wave functions, Ψ, contains information about both the spacial coordinates and spin of all electrons. Moreover the term rij = ri rj (1.4) j − j simply describes the distance between two electrons. If the many-electron Hamiltonian is introduced as N N X X 1 H^ = h^i + ; (1.5) r i=1 i<j ij the collected Schr¨odingerequation for the many-electron atom becomes H^ Ψ = EΨ: (1.6) 1 2 CHAPTER 1. PRELIMINARIES 1.2 The Aufbau principle In order to understand and investigate the electronic energies of the atoms it is important to know the electronic states around the nucleus. More specifically the electron configuration of the atom is desired. Using numerous chemical and spectroscopic evidence a few guidelines for placing electrons around the nucleus of the atom have been derived. These rules are known as the Aufbau principle. The idea behind the Aufbau principle is that the formation of an atom can be considered as a successive capturing of electrons and maintaining them in their most stable conditions. In this way the hydrogen atom is created when a proton capture a single electron and holding it in its most stable state called the 1s orbital. The chemical and spectroscopic evidence suggests that a given spacial state only can hold two electrons resulting in the fact that the electrons of the Lithium atom are in two different spacial states. Since the atom tend to be in the most stable condition the two different spacial states of the Lithium atom must be the ones with the lowest possible energies being the 1s and 2s orbitals. The electron configuration of Lithium is then stated as 1s22s1. Using this procedure the Aufbau principle can be stated as the following rules: The electron configuration of an element may be determined by filling the orbitals after increasing values of n + l. For a fixed value of n + l orbitals with lower n-values are filled first. Here n is the principal quantum number whereas l is the azimuthal quantum number. Due to repulsion between electrons and the nucleus not all electron configu- rations obey the Aufbau principle but its a good guideline for the construction of atoms. 1.3 The Adiabatic approximation In all the quantum mechanical calculations of this project the adiabatic approxi- mation also known as the Born-Oppenheimer approximation is used to simplify the problems. The approximation assumes that because the electrons are so much lighter than the nuclei they rearrange instantaneously according to the positions of the nuclei. In this picture the nuclei can be considered as stationary and independent of the electron motion such that the molecular wave function may be written Ψmol (x; ρ) = Ψ (x; ρ) κ (ρ) ; (1.7) where x denotes the spin and radial coordinates of the electrons and ρ denotes the coordinates of the nuclei. Hereby the molecular wave function is written as a product of an electron wave function Ψ (x; ρ) of electrons at x assuming that the nuclei are at position ρ and a nuclear wave function κ (ρ). This approximation yields a easy way to implement the quantum mechanical calculations used in this project. Consider a system consisting of N electrons and M nuclei the Hamiltonian neglecting spin in atomic units yields M M N M N N ^ X 1 2 X ZkZl X 1 2 X X Zk X 1 Hmol = k + i + ; (1.8) − 2Mk r rkl − 2r − rik rij k=1 k<l i=1 k=1 i=1 i<j 1.3.
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