Energy Partitions

Suppose that the set of occupied spin orbitals $\chi_{i}$ are split into two non-overlapping subsets $\chi_{i}^{A}$ and $\chi_{i}^{B}$. Since the density $\gamma$ is the sum of the squares of the spin orbitals, it will consequently be partitioned into two parts $\gamma^{A}$ and $\gamma^{B}$ with
$$\begin{align}\gamma({\bf x}) = \gamma^{A}({\bf x}) + \gamma^{B}({\bf x}). \end{align}$$
Any energy functional $E[\gamma^{A},\gamma^{B}]$ can then be split into `pure A', `pure B' and `interacting AB' parts by the partition
$$\begin{align}E^{A} &= E[\gamma^{A},0] \\ E^{B} &= E[0,\gamma^{B}] \\ E^{AB} &= E[\gamma^{A},\gamma^{B}] - E^{A} - E^{B}. \end{align}$$
Such a partition has been proposed by Stoll et al. [152,153] for the spin components, but it can be applied elsewhere. A similar partition for the conventional energies can be achieved by treating the energies as functionals of the sets $\chi_{i}^{A}$ and $\chi_{i}^{B}$
 $$\begin{align} E^{A} &= E[\chi^{A}_{i},0] \\ E^{B} &= E[0,\chi^{B}_{i}] \\ E^{AB} &= E[\chi^{A}_{i},\chi^{B}_{i}] - E^{A} - E^{B} \end{align}$$
where $E[\chi^{A}_{i},0]$ denotes that all integrals involving B spin orbitals are zeroed.

The above partition will cleanly split the MP2 correlation energy and also the exchange energy of the Fock functional. At higher levels of correlation treatment the partitions via equation (3.18) become questionable. There are numerous complicated many-body interactions involved which would be somewhat arbitrarily assigned. However, this is not addressed here, as only MP2 computations are performed. It seems reasonable that the DFT and conventional partitions should be comparable at a coarse level, since MP2 usually accounts for the majority of correlation in simple pair terms.