Exact conditions on the exchange-correlation hole

We may now list some of the simple physical conditions that the exact exchange-correlation hole satisfies. A common decomposition of the hole is into its separate exchange and correlation contributions. The exchange (or Fermi) hole is the hole due to the Pauli exclusion principle, and obeys the exact conditions  

for a spin-unpolarized two-electron system like the Hooke s atom, the exact parallel hole is made up entirely of exchange. This can be seen most easily in the spin-decomposed second-order density matrix. Since the ground state wavefunction is a spin singlet, it contains no contribution in which both electrons have the same spin. Therefore 

Another more subtle condition is the electron-electron cusp condition. As two electrons approach each other, their Coulomb interaction dominates, and this leads to a cusp in the exchange-correlation hole at zero separation[15]. It is most simply expressed in terms of the pair distribution function. We define its spherically-averaged derivative at zero separation as 

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