Direct Term

The direct term,

$$\quad \left[{\sum_i \int d{\bf r'} \vert\psi_{\nu_{i}}({\bf r'})\vert^2 U(\vert{r-r'}\vert)}\right]\psi_{\nu_{j}}({\bf r})$$ (3.30)

acts like a local one-particle potential on particle $j$: it depends on all the wave functions $\psi_i({\bf r'})$ that have still to be determined. The direct term has a simple physical interpretation: it is the potential at position ${\bf r}$ generated by the total density $\sum_i\vert\psi_i({\bf r'})\vert^2$ of all the individual electrons in their states $\vert\nu_i\rangle$ at position ${\bf r'}$. The direct term can be interpreted as a `direct' re-normalisation of the one-particle Hamiltonian $\hat{H}_0$.