Many-body perturbation theory

The intermediate states given by \( \hat{Q} \) can at most be of a \( 2p-2h \) nature if we have a two-body Hamiltonian. This means that second order in the perturbation theory can have \( 1p-1h \) and \( 2p-2h \) at most as intermediate states. When we diagonalize, these contributions are included to infinite order. This means that higher-orders in perturbation theory bring in more complicated correlations.

If we limit the attention to a Hartree-Fock basis, then we have that \( \langle\Phi_0\vert \hat{H}_I \vert 2p-2h\rangle \) is the only contribution and the contribution to the energy reduces to

$$ \Delta E^{(2)}=\frac{1}{4}\sum_{abij}\langle ij\vert \hat{v}\vert ab\rangle \frac{\langle ab\vert \hat{v}\vert ij\rangle}{\epsilon_i+\epsilon_j-\epsilon_a-\epsilon_b}. $$