Ensemble exchange energy, for multiplets

So the ensemble exchange energy for multiplets is linear in w. Using the Hartree-Fock values for the ground-state exchange energy E]. and the multi-plet separation energies AE and AE , the constants C2 and C3 can be calculated. Table IV presents these values for several atoms. 

The electron configuration is the same for the multiplets so ejy+i = c/y. Integrating Eq. (42) with respect to w we obtain for the ensemble exchange energy... 

The method of fractionally occupied states can be used to treat the multiplet problem, too. However, the exchange energy and potential are not known even for the ground state. They are unknown for the ensemble of excited states, too. 

---