{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "# Hartree-Fock theory\n", "\n", " **Morten Hjorth-Jensen**, [National Superconducting Cyclotron Laboratory](http://www.nscl.msu.edu/) and [Department of Physics and Astronomy](https://www.pa.msu.edu/), [Michigan State University](http://www.msu.edu/), East Lansing, MI 48824, USA\n", "\n", "Date: **May 16-20 2016**\n", "\n", "## Why Hartree-Fock?\n", "Hartree-Fock (HF) theory is an algorithm for finding an approximative expression for the ground state of a given Hamiltonian. The basic ingredients are\n", " * Define a single-particle basis $\\{\\psi_{\\alpha}\\}$ so that" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\hat{h}^{\\mathrm{HF}}\\psi_{\\alpha} = \\varepsilon_{\\alpha}\\psi_{\\alpha}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "with the Hartree-Fock Hamiltonian defined as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\hat{h}^{\\mathrm{HF}}=\\hat{t}+\\hat{u}_{\\mathrm{ext}}+\\hat{u}^{\\mathrm{HF}}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "* The term $\\hat{u}^{\\mathrm{HF}}$ is a single-particle potential to be determined by the HF algorithm.\n", "\n", " * The HF algorithm means to choose $\\hat{u}^{\\mathrm{HF}}$ in order to have" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\langle \\hat{H} \\rangle = E^{\\mathrm{HF}}= \\langle \\Phi_0 | \\hat{H}|\\Phi_0 \\rangle\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "that is to find a local minimum with a Slater determinant $\\Phi_0$ being the ansatz for the ground state. \n", " * The variational principle ensures that $E^{\\mathrm{HF}} \\ge E_0$, with $E_0$ the exact ground state energy.\n", "\n", "\n", "\n", "\n", "## Why Hartree-Fock?\n", "We will show that the Hartree-Fock Hamiltonian $\\hat{h}^{\\mathrm{HF}}$ equals our definition of the operator $\\hat{f}$ discussed in connection with the new definition of the normal-ordered Hamiltonian (see later lectures), that is we have, for a specific matrix element" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\langle p |\\hat{h}^{\\mathrm{HF}}| q \\rangle =\\langle p |\\hat{f}| q \\rangle=\\langle p|\\hat{t}+\\hat{u}_{\\mathrm{ext}}|q \\rangle +\\sum_{i\\le F} \\langle pi | \\hat{V} | qi\\rangle_{AS},\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "meaning that" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\langle p|\\hat{u}^{\\mathrm{HF}}|q\\rangle = \\sum_{i\\le F} \\langle pi | \\hat{V} | qi\\rangle_{AS}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The so-called Hartree-Fock potential $\\hat{u}^{\\mathrm{HF}}$ brings an explicit medium dependence due to the summation over all single-particle states below the Fermi level $F$. It brings also in an explicit dependence on the two-body interaction (in nuclear physics we can also have complicated three- or higher-body forces). The two-body interaction, with its contribution from the other bystanding fermions, creates an effective mean field in which a given fermion moves, in addition to the external potential $\\hat{u}_{\\mathrm{ext}}$ which confines the motion of the fermion. For systems like nuclei, there is no external confining potential. Nuclei are examples of self-bound systems, where the binding arises due to the intrinsic nature of the strong force. For nuclear systems thus, there would be no external one-body potential in the Hartree-Fock Hamiltonian.\n", "\n", "\n", "\n", "\n", "\n", "\n", "## Definitions and notations\n", "Before we proceed we need some definitions.\n", "We will assume that the interacting part of the Hamiltonian\n", "can be approximated by a two-body interaction.\n", "This means that our Hamiltonian is written as the sum of some onebody part and a twobody part" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "
\n", "\n", "$$\n", "\\begin{equation}\n", " \\hat{H} = \\hat{H}_0 + \\hat{H}_I \n", " = \\sum_{i=1}^A \\hat{h}_0(x_i) + \\sum_{i < j}^A \\hat{v}(r_{ij}),\n", "\\label{Hnuclei} \\tag{1}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "with" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation}\n", " H_0=\\sum_{i=1}^A \\hat{h}_0(x_i).\n", "\\label{hinuclei} \\tag{2}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The onebody part $u_{\\mathrm{ext}}(x_i)$ is normally approximated by a harmonic oscillator or Woods-Saxon potential or for electronic systems the Coulomb interaction an electron feels from the nucleus. However, other potentials are fully possible, such as \n", "one derived from the self-consistent solution of the Hartree-Fock equations to be discussed here.\n", "\n", "\n", "\n", "\n", "## Definitions and notations\n", "Our Hamiltonian is invariant under the permutation (interchange) of two particles.\n", "Since we deal with fermions however, the total wave function is antisymmetric.\n", "Let $\\hat{P}$ be an operator which interchanges two particles.\n", "Due to the symmetries we have ascribed to our Hamiltonian, this operator commutes with the total Hamiltonian," ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "[\\hat{H},\\hat{P}] = 0,\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "meaning that $\\Psi_{\\lambda}(x_1, x_2, \\dots , x_A)$ is an eigenfunction of \n", "$\\hat{P}$ as well, that is" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\hat{P}_{ij}\\Psi_{\\lambda}(x_1, x_2, \\dots,x_i,\\dots,x_j,\\dots,x_A)=\n", "\\beta\\Psi_{\\lambda}(x_1, x_2, \\dots,x_i,\\dots,x_j,\\dots,x_A),\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where $\\beta$ is the eigenvalue of $\\hat{P}$. We have introduced the suffix $ij$ in order to indicate that we permute particles $i$ and $j$.\n", "The Pauli principle tells us that the total wave function for a system of fermions\n", "has to be antisymmetric, resulting in the eigenvalue $\\beta = -1$.\n", "\n", "\n", "\n", "## Definitions and notations\n", "In our case we assume that we can approximate the exact eigenfunction with a Slater determinant" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation}\n", " \\Phi(x_1, x_2,\\dots ,x_A,\\alpha,\\beta,\\dots, \\sigma)=\\frac{1}{\\sqrt{A!}}\n", "\\left| \\begin{array}{ccccc} \\psi_{\\alpha}(x_1)& \\psi_{\\alpha}(x_2)& \\dots & \\dots & \\psi_{\\alpha}(x_A)\\\\\n", " \\psi_{\\beta}(x_1)&\\psi_{\\beta}(x_2)& \\dots & \\dots & \\psi_{\\beta}(x_A)\\\\ \n", " \\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", " \\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", " \\psi_{\\sigma}(x_1)&\\psi_{\\sigma}(x_2)& \\dots & \\dots & \\psi_{\\sigma}(x_A)\\end{array} \\right|, \\label{eq:HartreeFockDet} \\tag{3}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where $x_i$ stand for the coordinates and spin values of a particle $i$ and $\\alpha,\\beta,\\dots, \\gamma$ \n", "are quantum numbers needed to describe remaining quantum numbers.\n", "\n", "\n", "\n", "## Definitions and notations\n", "The single-particle function $\\psi_{\\alpha}(x_i)$ are eigenfunctions of the onebody\n", "Hamiltonian $h_i$, that is" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\hat{h}_0(x_i)=\\hat{t}(x_i) + \\hat{u}_{\\mathrm{ext}}(x_i),\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "with eigenvalues" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\hat{h}_0(x_i) \\psi_{\\alpha}(x_i)=\\left(\\hat{t}(x_i) + \\hat{u}_{\\mathrm{ext}}(x_i)\\right)\\psi_{\\alpha}(x_i)=\\varepsilon_{\\alpha}\\psi_{\\alpha}(x_i).\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The energies $\\varepsilon_{\\alpha}$ are the so-called non-interacting single-particle energies, or unperturbed energies. \n", "The total energy is in this case the sum over all single-particle energies, if no two-body or more complicated\n", "many-body interactions are present.\n", "\n", "\n", "\n", "## Definitions and notations\n", "Let us denote the ground state energy by $E_0$. According to the\n", "variational principle we have" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "E_0 \\le E[\\Phi] = \\int \\Phi^*\\hat{H}\\Phi d\\mathbf{\\tau}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where $\\Phi$ is a trial function which we assume to be normalized" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\int \\Phi^*\\Phi d\\mathbf{\\tau} = 1,\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "where we have used the shorthand $d\\mathbf{\\tau}=dx_1dr_2\\dots dr_A$.\n", "\n", "\n", "\n", "## Brief reminder on some linear algebra properties\n", "Before we proceed with a more compact representation of a Slater determinant, we would like to repeat some linear algebra properties which will be useful for our derivations of the energy as function of a Slater determinant, Hartree-Fock theory and later variational Monte Carlo.\n", "\n", "The inverse of a matrix is defined by" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\mathbf{A}^{-1} \\cdot \\mathbf{A} = I\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A unitary matrix $\\mathbf{A}$ is one whose inverse is its adjoint" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\mathbf{A}^{-1}=\\mathbf{A}^{\\dagger}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A real unitary matrix is called orthogonal and its inverse is equal to its transpose.\n", "A hermitian matrix is its own self-adjoint, that is" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\mathbf{A}=\\mathbf{A}^{\\dagger}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Basic Matrix Features\n", "\n", "**Matrix Properties Reminder.**\n", "\n", "\n", "Relations | Name | matrix elements |
---|---|---|
$A = A^{T}$ | symmetric | $a_{ij} = a_{ji}$ |
$A = \\left (A^{T} \\right )^{-1}$ | real orthogonal | $\\sum_k a_{ik} a_{jk} = \\sum_k a_{ki} a_{kj} = \\delta_{ij}$ |
$A = A^{ * }$ | real matrix | $a_{ij} = a_{ij}^{ * }$ |
$A = A^{\\dagger}$ | hermitian | $a_{ij} = a_{ji}^{ * }$ |
$A = \\left (A^{\\dagger} \\right )^{-1}$ | unitary | $\\sum_k a_{ik} a_{jk}^{ * } = \\sum_k a_{ki}^{ * } a_{kj} = \\delta_{ij}$ |