Density functions for 1-determinant wavefunctions

We recall that a single antisymmetrized product, or determinant, of 

1- electron functions may sometimes provide a useful approximate wave-function. This is the basis of MO theory, and it is therefore useful to obtain the corresponding orbital forms of the density matrices. 

We consider the single spin-orbital configuration in which spin-orbitals i A i B i x are occupied by the N electrons, each spin-orbital having an orbital factor that may be referred to in the present discussion as an MO, though its actual form is at present immaterial and we write the corresponding 1-determinant wavefunction (3.3.3) as 

Instead of calculating the density matrices directly, from (5.2.6) and (5.2.7), it is sufficient to consider the energy expression already obtained in (3.3.7a)  

Expression (5.3.2) affords a simple method of writing down the density functions, for the two sums arise as the expectation values of the 1- and 

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