Finite-basis approximations. Closed-shell systems

2 FINITE-BASIS APPROXIMATIONS. CLOSED-SHELL SYSTEMS 

First we develop the finite-basis approach in a purely formal way, assuming that the n occupied spin-orbitals are expressed in terms 

This matrix is a matrix representative of the 1-electron density operator p, as mentioned in Section 5.3 in the basis the operator is 

The energy expression (6.1.3) may also be expressed in terms of basis-function integrals thus 

In other words, G is the difference of coulomb and exchange matrices  

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To go a step further we need to explicitly define the solvent terms. To do that, we prefer to introduce the common finite-basis approximation and a closed shell system. In this approximation we can eliminate the spin dependence (the occupied spin-orbitals occur in pairs) and expand the molecular orbitals (MOs) as a linear combination of atomic orbitals (LCAO). On performing the spin integration in the equations used so far, we find, for the electronic free energy ... 

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