GENERAL SCHEME FOR SEPARATING ELECTRONIC VARIABLES

The GF form of the trial wave function is, of course, an approximation. In general, the electron transfers between the groups do take place and in general destroy the variable separation built in the structure of the GF approximation. It would be desirable to take the effect of these transfers into account without destroying the attractive features of the GF wave function the separation of the electronic variables describing different groups. This is done using the Lowdin partition applied to derive the GF approximate form for the wave function. 

We start from a most general form [53-56] of the wave function. It differs from the GF approximation eq. (1.181) in that respect that the number of electrons in each group is not fixed, so that the generalized group function (GGF) expansion is a linear combination of functions which are antisymmetrized products of multipliers with a different number of electrons in the groups [53-56]  

24Subsequent material is based on A.L. Tchougreeff, Group Functions, Loewdin Partition, and Hybrid QC/MM Methods for Large Molecular Systems. Phys. Chem. Chem. Phys. 1, 1051, 1999. Reproduced by permission of the PCCP Owner Societies. 

In other words, the entire space originally spanned by the orbitals ipm is represented as the direct sum of orthogonal subspaces serving as the carrier subspaces to different groups, but the numbers of residing electrons are not fixed for each carrier subspace and all their possible distributions enter the expansion. In the expansion eq. (1.214) each distribution a of electrons among the groups satisfies the condition  

Also (ua) is the i -th -electron function where only the one-electron states of the A-th group may be occupied. Expansion coefficients (, f f nA ) are thought to be determined on the basis of the variational principle. 

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