Physical boundary conditions for holes

Note that if a hole h were not spherically mmetric (Le. contained a spherically symmetric component plus some non-spherical components) with respect to the position of electron 1, the contribution of such a hole to the integral in (11.59) over d r2 would come only from its spherically mmetric component, because the operator l/ri2 is spherically symmetric. 

The hole functions aie of fuiidamental uiiporlaiice in the DFT, because thc have to fulfil some boundary requirements. The requirements are unable to fix the precise mathematical form of the hole functions (and therefore ol Exc), but the form is heavily restricted by the boundary conditions. 

What boundaries we are talking about I fiese boundaries are ditterent tor electrons with the same spin coordinates to those corresponding to the opposite spins. For a pair of electrons with the same spins, we have to have the following result of integration over 2 (for any 

Since for r = r2 = r wc have vu have (Pauli exclusion principle) i ) = 

If similar calculations were done for the electrons with opposite spins, we would obtain 

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