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

$$$$$\chi_a(\tau_i) \defd \psi_a(\vec{r}_i)\sigma_a(s_i),$$$$$

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

The overlap between two spin-orbitals is given by

$$$$$\braket{i}{j} \defd \int\diff{\tau} \conj{\chi_i}(\tau) \chi_j(\tau),$$$$$

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

$$$\braket{i}{j} = \delta_{ij}.$$$

One-body matrix elements

$$$$$I(a,b) \equiv \onebody{a}{b} \defd \matrixel{a}{\hamiltonian}{b},$$$$$

where

$$$$$\hamiltonian \defd T + V$$$$$

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

Two-body matrix elements

$$$$$\twobodydx{ab}{cd} \defd \twobody{ab}{cd} - \twobody{ab}{dc},$$$$$

where

\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}

The special case

$$$$$F(a,b) \defd \twobody{ab}{ab}$$$$$

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

$$$$$G(a,b) \defd \twobody{ab}{ba}$$$$$

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

Note

Since

$$$$$\twobodydx{ii}{ii} = 0,$$$$$

we have