Throughout, Einstein notation is employed, meaning that indices appearing twice on only one side of an equation are summed over, e.g.:

\[c = \braket{i}{i} \iff c \equiv \sum_i \braket{i}{i},\]

whereas

\[c_{ka} = \braket{a}{k}\]

implies no summation.

# Spin-orbitals

\[\begin{equation} \chi_a(\tau_i) \defd \psi_a(\vec{r}_i)\sigma_a(s_i), \end{equation}\]

where $\sigma(s)$ is either $\alpha$ (spin up) or $\beta$ (spin down).

The overlap between two spin-orbitals is given by

\[\begin{equation} \braket{i}{j} \defd \int\diff{\tau} \conj{\chi_i}(\tau) \chi_j(\tau), \end{equation}\]

which, the case of orthogonal bases (this is a matter of choice), simplifies to

\[\braket{i}{j} = \delta_{ij}.\]

# One-body matrix elements

\[\begin{equation} I(a,b) \equiv \onebody{a}{b} \defd \matrixel{a}{\hamiltonian}{b}, \end{equation}\]

where

\[\begin{equation} \hamiltonian \defd T + V \end{equation}\]

is the (possibly time-dependent) one-body Hamiltonian.

# Two-body matrix elements

\[\begin{equation} \twobodydx{ab}{cd} \defd \twobody{ab}{cd} - \twobody{ab}{dc}, \end{equation}\]

where

\[\begin{equation} \begin{aligned} \twobody{ab}{cd} &\defd \int\diff{\tau_1}\diff{\tau_2} \conj{\chi_a}(\tau_1) \conj{\chi_b}(\tau_2) \frac{1}{r_{12}} \chi_c(\tau_1) \chi_d(\tau_2) \\ &= \delta(\sigma_a,\sigma_c) \delta(\sigma_b,\sigma_d) \int\diff{\vec{r}_1}\diff{\vec{r}_2} \conj{\chi_a}(\vec{r}_1) \conj{\chi_b}(\vec{r}_2) \frac{1}{r_{12}} \chi_c(\vec{r}_1) \chi_d(\vec{r}_2). \end{aligned} \end{equation}\]

The special case

\[\begin{equation} F(a,b) \defd \twobody{ab}{ab} \end{equation}\]

is called the *direct interaction* (gives rise to the screening potential), and the other special case

\[\begin{equation} G(a,b) \defd \twobody{ab}{ba} \end{equation}\]

is called the *exchange interaction* (gives rise to the non-local potential).

Since

\[\begin{equation} \twobodydx{ii}{ii} = 0, \end{equation}\]

we have

\[\begin{equation} \frac{1}{2}\twobodydx{ij}{ij} \equiv \sum_{i,j<i}\twobodydx{ij}{ij}, \end{equation}\]

i.e. we sum over $i,j$, divide by two to avoid double-counting and avoid automatically the case $i=j$.

# Repulsion potential

We also have the repulsion potential formed from two orbitals $a,b$:

\[\begin{equation} \twobody{a}{b} \defd \int\diff{\tau_1}\conj{\chi_a}(\tau_1)\frac{1}{r_{12}}\chi_b(\tau_1)= \delta(\sigma_a,\sigma_b)\int\diff{\vec{r}_1}\conj{\chi_a}(\vec{r}_1)\frac{1}{r_{12}}\chi_b(\vec{r}_1). \end{equation}\]