Direct calculation of coupling constant

In calculating the energy differences (transition energies) the perturbation theory becomes very effective calculating the transition energy is much faster than calculating the correlated total molecular energies [5]. In the framework 

Then the second-order correction to the exchange coupling constant is 

The configuration functions of the lowest singlet and triplet states are constructed with the help of the OMOs hence 

Here (... ii) represents a subspace of the doubly occupied MOs. Using these Slater determinants, the expression for J(2) can be rewritten in a simpler form 

Only those configuration functions PK contribute to the singlet-triplet separation which simultaneously interact with the reference Slater determinants a i and S2 involving OMOs. 

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