Hartree-Fock method - Aetherwisp

The Hartree-Fock method is a variational method that uses a Slater determinant of single-particle wavefunctions as a trial function. It is used to solve many fermion systems, such as the many-electron atom.

Description#

The method starts by claiming that the state of the system is described by a Slater determinant

\Psi(q_{1},q_{2},\ldots,q_{N})=\frac{1}{\sqrt{ N! }}\begin{vmatrix} u_{\alpha}(q_{1}) & u_{\beta}(q_{1}) & \ldots & u_{\nu}(q_{1}) \\ u_{\alpha}(q_{2}) & u_{\beta}(q_{2}) & \ldots & u_{\nu}(q_{2}) \\ \vdots \\ u_{\alpha}(q_{N}) & u_{\beta}(q_{N}) & \ldots & u_{\nu}(q_{N}) \end{vmatrix}

where $u(q)$ are the single-particle wavefunctions, which are assumed to be orthonormal to each other, $\braket{ u_{\lambda} | u_{\mu} }=\delta_{\lambda \mu}$ . As per the variational method, the energy is this state is the minimum of the Functional

E[\Psi]=\braket{ \Psi | \hat{H}| \Psi }

A given Hamiltonian $\hat{H}$ can be split in an independent term $\hat{H}_{1}$ and an interaction term $\hat{H}_{2}$ in the form $\hat{H}=\hat{H}_{1}+\hat{H}_{2}$ . $\hat{H}_{1}$ is given by a sum of single-particle Hamiltonians $\hat{h}_{i}$ , whereas $\hat{H}_{2}$ is a sum of interaction terms dependent on the distance between particles. It consists of doing the typical variational method workflow using the Slater determinant as a trial function.

The energy in $\hat{H}_{1}$ is

\braket{ \Psi |\hat{H}_{1}|\Psi }=\sum_{\lambda}\braket{ u_{\lambda}(q_{i}) | \hat{h}_{i} | u_{\lambda}(q_{i}) } =\sum_{\lambda}\mathcal{I}_{\lambda}

Meanwhile, the energy in $\hat{H}_{2}$ is

\begin{align} \braket{ \Psi | \hat{H}_{2} | \Psi } &=\frac{1}{2}\sum_{\lambda}\sum_{\mu}\left[ \left\langle u_{\lambda}(q_{i})u_{\mu}(q_{j})| \frac{1}{r_{ij}} | u_{\lambda}(q_{i})u_{\mu}(q_{j}) \right\rangle \right. \quad&(\text{direct term }\mathcal{F}_{\lambda \mu})\\ &\left.- \left\langle u_{\lambda}(q_{i})u_{\mu}(q_{j})| \frac{1}{r_{ij}} | u_{\mu}(q_{i})u_{\lambda}(q_{j}) \right\rangle \right]&(\text{exchange term } \mathcal{K}_{\lambda \mu}) \end{align}

The total energy to minimize then is

\boxed{E[\Psi]=\sum_{\lambda}\mathcal{I}_{\lambda}+ \frac{1}{2}\sum_{\lambda}\sum_{\mu}[\mathcal{F}_{\lambda \mu}-\mathcal{K}_{\lambda \mu}]}

If we impose stationariety on this functional we can obtain a set of single-particle eigenvalue equations

\boxed{\left[ \hat{h}_{i}+\sum_{\mu}V_{\text{direct}}^{\mu}(q_{i})-\sum_{\mu}V_{\text{exchange}}^{\mu}(q_{i}) \right]u_{\lambda}(\mathbf{r}_{i})=E_{\lambda}u_{\lambda}(q_{i})}

where

\begin{align} V_\text{direct}^{\mu}(\mathbf{r}_{i})&=\int u_{\mu}^{*}(\mathbf{r}_{j}) \frac{1}{r_{ij}} u_{\mu}(\mathbf{r}_{j})dq_{j} \\ V_\text{exchange}^{\mu}(q_{i})u_{\lambda }(q_{i})&=\left[ \int u_{\mu}^{*}(q_{j}) \frac{1}{r_{ij}}u_{\lambda}(q_{j})dq_{j}\right]u_{\mu}(q_{i}) \end{align}

These are known as the Hartree-Fock equations. They can be solved computationally through an iterative algorithm.