Matrix Elements and Notation

Before presenting the HF equations it is useful to define some matrix elements common to quantum chemistry. The overlap matrix, S, represents the overlap of basis functions
$$\begin{align}S_{\mu \nu} = \int \phi_{\mu}({\bf r}) \phi_{\nu}({\bf r}) \, d{\bf r}. \end{align}$$

A familiar concept to chemists is the electron density, $\rho({\bf r})$, which can be obtained by integrating the square of the wavefunction over all N electrons but one, and all spin variables.
$$\begin{align}\rho({\bf r}) = N \int \Psi^{2} \, ds_{1} \, d{\bf x}_{2} \, d{\bf x}_{3} \, \ldots \, d{\bf x}_{N} \end{align}$$
which is simply the square of the sum over all the occupied orbitals:
$$\begin{align}\begin{split} \rho({\bf r}) = & \sum_{i}^{occ} \psi_{i}({\bf r}) \p... ..._{\nu i} \right) \phi_{\mu}({\bf r}) \phi_{\nu}({\bf r}). \end{split}\end{align}$$
The bracwww-theored term is frequently required in solving the HF equations, and hence is usually precomputed and stored as the density matrix, P
$$\begin{align}P_{\mu \nu} = \sum_{i}^{occ} C_{\mu i} C_{\nu i}. \end{align}$$

The one-electron operators are also used to form their own matrices, T and V.

$$T_{\mu \nu} = \int \phi_{\mu}({\bf r}) \left( - \frac{1}{2} \nabla^{2} \right) \phi_{\nu}({\bf r}) \, d{\bf r}$$ (5.1)

$$V_{\mu \nu} = \int \phi_{\mu}({\bf r}) \left( - \sum_{A} \frac{Z_{A}}{\vert{\bf R}_{A} - {\bf r}\vert} \right) \phi_{\nu}({\bf r}) \, d{\bf r}.$$ (5.2)

These two matrices are added together to form the core Hamiltonian matrix, H, where $H_{\mu \nu}$ represents the energy of an isolated electron (in the presence of the nuclei) in the distribution $\phi_{\mu}\phi_{\nu}$.

The repulsion between an electron in the MO distribution $\psi_{i}\psi_{j}$ and the MO distribution $\psi_{k}\psi_{l}$ has a shorthand notation:
 $$\begin{align} (ij\vert kl) = \iint \frac{\psi_{i}^{*}({\bf r}_{1})\psi_{j}({\bf ... ... r}_{2})\psi_{l}({\bf r}_{2})}{r_{12}} \, d{\bf r}_{1} d{\bf r}_{2}, \end{align}$$
where
$$\begin{align}r_{12} = \vert{\bf r}_{1} - {\bf r}_{2}\vert. \end{align}$$
This shorthand also exists for repulsions between basis function distributions:
$$\begin{align}(\mu\nu\vert\lambda\sigma) = \iint \frac{\phi_{\mu}^{*}({\bf r}_{1}... ...2})\phi_{\sigma}({\bf r}_{2})}{r_{12}} \, d{\bf r}_{1} d{\bf r}_{2}. \end{align}$$
The relationship between the two is, of course
$$\begin{align}(ij\vert kl) = \sum_{\mu\nu\lambda\sigma} C_{\mu i} C_{\nu j} C_{\lambda k} C_{\sigma l} (\mu\nu\vert\lambda\sigma). \end{align}$$