where annihilation and creation operators without bar operate only on alpha spin-orbitals, while the ones with bar operate only on beta spin-orbitals. The density matrix coefficients can then easily be rotated to the AO product basis via the MO coefficients. ρ F,αα and ρ F,ββ are actually alpha and beta densities, and are (in principle) non-zero for all the state densities and for transition densities between the states with the same m s values, while they are zero for transition densities between the states with different m s values. On the other hand, the ρ F,αβ and ρ F,βα are non-zero only for transition densities between the site states with ∆m s = ±1 respectively. Finally, after integrating over ω 1 and ω 2 and expressing the partial densities in the basis of AO-products, the K integral simplifies to