30.4 The projector $\hat Q_{12}$

$\hat Q_{12}$ is a projector that keeps the explicitly correlated terms orthogonal to the conventional MP2 wave function. Generally, we use ansatz 3 (cf. I), for which the projector has the form

$$\hat Q_{12} = (1-\hat o_1) (1 - \hat o_2) (1 - \hat v_1 \hat v_2),$$ (13)

where $\hat o_i$ is a one-electron projector for electron $i$ onto the occupied space, and $\hat v_i$ projects onto the virtual orbital space. In the case that domain approximations are used in local explicitly correlated wave functions, the operators $\hat v$ are replaced by operators $\hat d^{ij}$ that project just onto the domain for the orbital pair $ij$.