Origin of the Singlet Excited Electronic Energy Shifts in ΔSCF with Fractional Occupation Numbers and Hybrid Density Functionals

Singlet excited electronic states can be directly constructed with the ΔSCF method using a single electronic density by applying fractional occupation numbers. While the advantages and disadvantages of such a DFT-based ΔSCF method were demonstrated in several studies, the spurious energy shift of the singlet electronic states when constructed using hybrid DFT functionals remains unexplained. Here, we explain in detail the origin of the effect of losing the idempotence properties of density matrices on the DFT energy terms with hybrid-based DFT functionals, as well as a procedure to eliminate the artificial shift from ΔSCF singlet excited state energies when obtained using the hybrid-based DFT functionals.