Variation theory for subsystems

To simplify matters, we take first a two-group system AB, with wavefunction 

A-group orbitals alone, so that will remain automatically strong- 

This means that the whole wavefunction may be optimized by considering just one group at a time, replacing an N-electron calculation effectively by a succession of smaller calculations on systems of N, Mb. - - -electrons. To complete this reduction, we express the interaction terms in (14.2.2) as matrix elements of actual one-electron operators describing the potential due to the electrons outside group R. Let us introduce for any group S the following operators, defined by their effect on any spin-orbital x  

Clearly, J (l) just multiplies xi) by the value of the potential at point JTi due to electrons distributed according to the density function for group S in state s. On the other hand K (l) is an integral operator, of the kind used in Hartree-Fock theory (Sections 6.1 and 6.4). Tliese two operators are the coulomb and exchange operators for an electron in the effective field due to the electrons of group S. It is now possible to write the interaction terms in (14.2.2) in the form 

All groups other than R have been formally eliminated, their presence being absorbed into the effective one-electron Hamiltonian (14.2.5), and the optimization of the R-group wavefunction is essentially an A r-electron problem. 

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