Energy of microstates

The electronic energy, Er, of any microstate Ψr is the sum   on the one and two-electron energies:

begin{displaymath}E_r = sum_i^pH_{ii}+sum_i^qH_{ii}+frac{1}{2}sum_{ij}^p(J_......+frac{1}{2}sum_{ij}^q(J_{ij}-K_{ij}) +sum_i^psum_j^qJ_{ij}end{displaymath}

where Hii is the one-electron energy of M.O. ψi, Jii is the two-electron Coulomb integral <ψiψi|ψjψj>, and Kii is the two-electron exchange integral <ψiψj|ψiψj>. In this section it is more convenient to express it in terms of molecular orbital occupancies:

begin{displaymath}E_r = sum_{i=B}^AH_{ii}(O_i^{alpha r}+ O_i^{beta r})+su......_i^{beta r}O_j^{beta r})+J_{ij}O_i^{alpha r}O_j^{beta r})end{displaymath}

Similarly, the orbital energies can be written  

begin{displaymath}epsilon_{ii}^{alpha r} = H_{ii}+sum_j^p(J_{ij}-K_{ij})+sum_j^qJ_{ij}end{displaymath}

or, in terms of orbital occupancies

begin{displaymath}epsilon_{ii}^{alpha r} = H_{ii}+sum_{j=B}^A(J_{ij}-K_{ij})O_j^{alpha r}+sum_{j=B}^AJ_{ij}O_j^{beta r}.end{displaymath}