Exchange Term

The exchange term,

$$\quad {\sum_i \int d{\bf r'} \psi_{\nu_{i}}^*({\bf r'}) U(\vert{r... ...t) \psi_{\nu_{j}}({\bf r'}) \psi_{\nu_{i}}({\bf r}) \delta_{\sigma_i \sigma_j}}$$ (3.31)

is more complicated and cannot be written as a simple re-normalisation of the one-particle Hamiltonian $\hat{H}_0$. Its spin-dependence indicates that it originates from the exchange interaction between indistinguishable Fermions.

What we have achieved, though, is a self-consistent description of the interacting $N$-Fermion systems in terms of a single Slater determinant built from the states $\psi_i({\bf r},\sigma)$. Actually, for spin-independent $\hat{H}_0$ and $U$ only the orbital parts $\psi_i({\bf r}$ enter the Hartree-Fock equations, although the spin-indices do play a role in the exchange term.