Hartree-Fock Theory

The simplest way to approximate electron-electron interactions is through the Hartree approximation, where the true $N$-electron wavefunction $\Psi$ is replaced by a product of single-particle orbitals, $\psi_i({\bf r}_i s_i)$,

$$\Psi({\bf r}_1 s_1,{\bf r}_2 s_2, \, \ldots \, , {\bf r}_N s_N) =... ...{\bf r}_1 s_1) \, \psi_2({\bf r}_2 s_2) \, \ldots \, \psi_N({\bf r}_N s_N) \, ,$$ (1.18)

where $\psi_i({\bf r}_i s_i)$ is composed of a spatial function $\phi_i({\bf r}_i)$, and an electron spin function $\sigma(s_i)$ such that,

$$\psi_i({\bf r}_i) = \phi_i({\bf r}_i) \, \sigma(s_i) \, ,$$ (1.19)

and $\sigma = \alpha,\beta$ represent up-spin and down-spin electrons respectively. However, as mentioned previously, the Hartree approximation does not account for exchange interactions since (1.19) does not satisfy,

$$\Psi({\bf r}_1 s_1, \ldots ,{\bf r}_i s_i, \ldots ,{\bf r}_j s_j,... ...,{\bf r}_j s_j, \ldots ,{\bf r}_i s_i, \ldots {\bf r}_N s_N) \, , \hspace{-1cm}$$

under the interchange of particle coordinates, which is required by the exclusion principle.

This problem was rectified by the Hartree-Fock approximation [5] which accounts for electron exchange interactions by writing the wavefunction as an antisymmetrised product of orbitals. The Hartree-Fock wavefunction $\Psi_{\hbox{{HF}}}$ amounts to a linear combination of the terms in (1.19), which includes all permutations of the electron coordinates with the corresponding weights $\pm 1$, i.e.

$$\Psi_{\hbox{{HF}}} = \frac{1}{\sqrt{N !}} \, \Big[ \psi_1({\bf r}_1 s_1) \, \psi_2({\bf r}_2 s_2) \, \ldots \, \psi_N({\bf r}_N s_N)$$

$$\hspace{1.00cm} - \psi_1({\bf r}_2 s_2) \, \psi_2({\bf r}_1 s_1) \, \ldots \, \psi_N({\bf r}_N s_N) + \dots \Big] \, ,$$ (1.20)

and so fulfils (1.21). In $1951$ Slater [6] realised that the Hartree-Fock wavefunction can be efficiently represented as an $N \times N$ determinant, now known as a Slater determinant:

$$\Psi_{\hbox{{HF}}} = \frac{1}{\sqrt{N !} \,} \, \left\vert \begin... ..._N({\bf r}_2 s_2) & \ldots & \psi_N({\bf r}_N s_N) \end{array} \right\vert \, ,$$ (1.21)

where the orbitals are subject to the orthonormal constraint,

$$\int \psi_i^*({\bf r}) \psi_j({\bf r}) \, d{\bf r} = \langle \psi_i \vert \psi_j \rangle = \delta_{ij} \, .$$ (1.22)

The Slater determinant can also be written in shorthand notation as,

$$\Psi_{\hbox{{HF}}} = \frac{1}{\sqrt{N!}} \, {\hbox{det}} [\psi_1({\bf r}_1 s_1) \psi_2({\bf r}_2 s_2) \dots \psi_N({\bf r}_N s_N)] \, .$$ (1.23)

The Hartree-Fock energy can be evaluated by taking the expectation value of the Hamiltonian (1.6) with the above Slater determinant. This yields,

$$E_{\hbox{{HF}}} = \langle \Psi_{\hbox{{HF}}} \vert \hat{H} \vert \Psi_{\hbox{{HF}}} \rangle$$

$$= \sum_i^N \int \psi_i^*({\bf r}) \left(- \frac{1}{2} \nabla^2 + v_{\hbox{{ext}}}({\bf r}) \right) \psi_i({\bf r}) \, d{\bf r}$$

$$+ \frac{1}{2} \sum_i^N \sum_{j}^{N} \int \int \frac{\vert\psi_i({... ...rt\psi_j({\bf r'})\vert^2}{\vert{\bf r}-{\bf r'}\vert} \, d{\bf r} \, d{\bf r'}$$

$$- \frac{1}{2} \sum_i^N \sum_{j}^{N} \int \int \frac{\psi_i^*({\bf... ...}{\vert{\bf r}-{\bf r'}\vert} \, \delta_{s_i s_j} \, d{\bf r} \, d{\bf r'} \, .$$ (1.24)

The last term is of significant interest since it arises from the antisymmetric nature of the Hartree-Fock wavefunction - it vanishes when $s_i \ne s_j$, which is an artefact of the Pauli principle. Consequently this term is called the exchange energy $E_{\hbox{{X}}}$. It should also be noted that in practice an extra term due to the repulsion energy between the ions must be added to (1.26) in order to obtain the total energy of the system.