Other open-shell situations. Ensemble averaging

6 OTHER OPEN-SHELL SITUATIONS. ENSEMBLE AVERAGING 

Roothaan (1960) first noted that certain states of atoms and linear molecules, even those requiring many-determinant wavefunctions, yield energy expressions that can be put in the form 

Values of V2, a and b for various states may be obtained by setting up the appropriate combinations of determinants and writing down the expressions for the energy, and have been listed for many states by Roothaan (1960). The SCF equations for all functions that give energy expressions of the form (6.6.1) can be obtained by the procedure developed in Section 6.5, with only trivial modifications. In fact, with the revised definition of V2, (6.5.3) and the analysis that follows may be adopted in its entirety, provided that G in (6.5.4) is replaced by 

The iterative introduction of self-consistency, using the Hamiltonian (6.5.18), is unaffected by these minor dianges. 

The theoretical basis of this approach is not immediately obvious what, for example, is the meaning of the fractional occupation numbers To establish its status, let us take a simple example, the boron atom with electron configuration ls 2s 2p and a ground state. If we use a 1-determinant wavefunction with A = %, B = 2, W = 2p then the energy expression will be 

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