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10.1: Hartree-Fock theory

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    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.


    Figure \(\PageIndex{1}\) : Algorithm for Self-consistent field theory.

    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.

    This page titled 10.1: Hartree-Fock theory is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Graeme Ackland via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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