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 in , where , is the spatial coordinate and is the spin coordinate, we have

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 1s^{2}2s^{1}, 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.