Restricted Closed Shell Hartree-Fock

In the restricted formalism each spatial orbital contains two electrons, one spin up, the other spin down. That is,
$$\begin{align}\begin{split} \chi_{2 i - 1}({\bf x}) = \psi_{i}({\bf r}) \alpha(s) \\ \chi_{2 i}({\bf x}) = \psi_{i}({\bf r}) \beta(s). \end{split}\end{align}$$
Using these orbitals the HF energy is, in terms of the spatial components of molecular orbitals (remembering that the wavefunction is normalized and that the orbitals are orthonormal),
 $$\begin{align} \begin{split} E = & \langle \Psi \vert \hat{H} \vert \Psi \rangle ... ...i_{j}\psi_{j}) - (\psi_{i}\psi_{j}\vert\psi_{i}\psi_{j})] \end{split}\end{align}$$
where $\hat{h}$ represents the one-electron operators and $2\sum(\psi_{i}\psi_{i}\vert\psi_{j}\psi_{j})$ is the Coulombic repulsion between all electrons. Note that this term includes a spurious self-repulsion. The final term of equation (1.30), $-\sum (\psi_{i}\psi_{j}\vert\psi_{i}\psi_{j})$, has arisen from making the wavefunction antisymmetric. It is termed the exchange energy and has no classical analogue. Most importantly, the exchange term contains elements which exactly cancel the spurious self-interaction of the Coulomb energy. Another important effect of the exchange term is that, while an electron feels only the average field of all other electrons, it does feel an instantaneous effect of all electrons of the same spin. That is, the probability of finding two electrons at the same point at the same time is non-zero, but the probability of finding two electrons at the same point (at the same time) of the same spin is zero.

To find the orbitals which minimize the energy we make the energy stationary with respect to variations of the MO coefficients, $C_{\mu i}$. If there are m basis functions and n occupied orbitals, $\psi_{i}$, then solving the Schrödinger equation will produce (m-n) unoccupied (or virtual) orbitals, $\psi_{a}$, which obey $(\psi_{a}\vert\psi_{i}) = 0$ (with the standard notation of using i,j to denote occupied orbitals; a,b for virtual; and p,q to denote any MOs).

At the minimum the energy is stationary with respect to the variation
 $$\begin{align} \psi_{i} \rightarrow \psi_{i} + \lambda \psi_{a} \qquad (i=1,\ldots,n;a=n+1,\ldots,m). \end{align}$$
This variation preserves orbital orthonormality through first order in $\lambda$. Substituting equation (1.31) into equation (1.30), picking out the coefficient of $\lambda$ and setting it to zero yields the stationary condition;
$$\begin{align}(\psi_{a}\vert\hat{h}\vert\psi_{i}) + \sum_{j} [2(ai\vert jj) - (aj\vert ij)] = 0 \end{align}$$
It is easier to use these equations when expressed in operator form. We define the Fock operator such that
$$\begin{align}\begin{split} F_{ai} = & (\psi_{a}\vert\hat{F}\vert\psi_{i}) \\ =... ...rt\psi_{i}) + \sum_{j} [2(ai\vert jj) - (aj\vert ij)] = 0 \end{split}\end{align}$$
or, in operator form
 $$\begin{align} \hat{F}({\bf r}_{1}) = \hat{h}({\bf r}_{1}) + 2 \hat{J}({\bf r}_{1}) - \hat{K}({\bf r}_{1}), \end{align}$$
where the Coulomb operator, $\hat{J}$, is
$$\begin{align}\hat{J}({\bf r}_{1}) = \sum_{j} \int \frac{\psi_{j}^{2}({\bf r}_{2})}{r_{12}} \, d {\bf r}_{2} \end{align}$$
and the exchange operator, $\hat{K}$, is
$$\begin{align}\hat{K}({\bf r}_{1}) = \sum_{j} \int \frac{\psi_{j}({\bf r}_{2})\psi_{j}({\bf r}_{1})}{r_{12}}\hat{P}_{12} \, d {\bf r}_{2}, \end{align}$$
where $\hat{P}_{12}$ is the permutation operator, that is
 $$\begin{align} \hat{P}_{12} \psi_{p}({\bf r}_{1}) = \psi_{p}({\bf r}_{2}). \end{align}$$
The Fock operator is therefore an effective one-electron Hamiltonian.

Orbitals that satisfy the condition F_ai=0 are obtained by solving the Roothaan-Hall equations [20,21]
$$\begin{align}\sum_{\nu}(\phi_{\mu}\vert\hat{F} - \varepsilon_{i}\vert\phi_{\nu}) C_{\nu i} = 0. \end{align}$$
The resulting orbitals will not only satisfy F_ai=0, but also
$$\begin{align}F_{ij} = \varepsilon_{i} \delta_{ij}. \end{align}$$
This, however, does not matter as the SCF energy is invariant to a mixing of the occupied orbitals. When the Fock matrix is completely diagonal the orbitals are termed canonical.