The exchange integral 
(or exchange force) is the contribution to the expectation value of a multi-electron system, as a result of the interaction of electrons with the same spin.
As shown in the article ‘Slater-Condon rule for a two-electron operator’, the integral arises from the expectation value of the two-electron operator 
, where 
and 
is the Slater determinant given by eq66.
To illustrate the properties of the exchange integral, we refer to the ground state of He. If we substitute =\phi_i(\boldsymbol{\mathit{ r}}_i)\sigma(\omega_i))
in , where \phi_j^*(x_j)\frac{\phi_j(x_i)\phi_i(x_j)}{r_{ij}}dx_idx_j)
, 
is the spatial coordinate and 
is the spin coordinate, we have
\phi_2^*(\boldsymbol{\mathit{ r}}_2)\frac{\phi_1(\boldsymbol{\mathit{ r}}_2)\phi_2(\boldsymbol{\mathit{ r}}_1)}{r_{12}}d\boldsymbol{\mathit{ r}}_1d\boldsymbol{\mathit{ r}}_2\int\alpha(\omega_1)^*\beta(\omega_1)d\omega_1\int\beta(\omega_2)^*\alpha(\omega_2)d\omega_2)
Due to spin orthogonality, 
. In other words, the exchange energy between two electrons with antiparallel spins is zero. This implies that for the ground state of lithium with electronic configuration 1s22s1, there is an exchange integral term of 
, i.e. between the sole 2s electron and one of the two 1s electrons that has the same spin as the 2s electron.
