10.1: Hartree-Fock theory

Last updated
Save as PDF

Page ID: 28669

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

The idea is to solve the Schrödinger equation for an electron moving in the potential of the nucleus and all the other electrons. We start with a guess for the trial electron charge density, solve Z/2 one-particle Schrödinger equations (initially identical) to obtain Z electron wavefunctions. Then we construct the potential for each wavefunction from that of the nucleus and that of all the other electrons, symmetrise it, and solve the Z/2 Schrödinger equations again.

Fock improved on Hartree’s method by using the properly antisymmetrised wavefunction (Slater determinant) instead of simple one-electron wavefunctions. Without this, the exchange interaction is missing. This method is ideal for a computer, because it is easily written as an algorithm.

Although we are concerned here with atoms, the same methodology is used for molecules or even solids (with appropriate potential symmetries and boundary conditions). This is a variational method, so wherever we refer to wavefunctions, we assume that they are expanded in some appropriate basis set.

The full set of equations are

\[\epsilon_{i} \psi_{i}(\mathbf{r})=\left(-\frac{1}{2} \nabla^{2}+V_{i o n}(\mathbf{r})\right) \psi_{i}(\mathbf{r})+\sum_{j} \int d \mathbf{r}^{\prime} \frac{\left|\psi_{j}\left(\mathbf{r}^{\prime}\right)\right|^{2}}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} \psi_{i}(\mathbf{r})-\sum_{j} \delta_{\sigma_{i} \sigma_{j}} \int d \mathbf{r}^{\prime} \frac{\psi_{j}^{*}\left(\mathbf{r}^{\prime}\right) \psi_{i}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} \psi_{j}(\mathbf{r}) \]

The first term is the kinetic energy and electron-ion potential. The second “Hartree” term, is the electrostatic potential from the charge distribution of \(N\) electrons, including an unphysical self-interaction of electrons when \(j = i\). The third, “exchange” term, acts only on electrons with the same spin and comes from the Slater determinant form of the wavefunction.

Physically, the effect of exchange is for like-spin electrons to avoid each other. Each electron is surrounded by an “exchange hole”: there is one fewer like-spin electrons nearby than the mean-field would imply. The term \(i = j\) neatly cancels out the self interaction of the electron.