∆SCF Excited-State Approach: Theoretical Foundation, Linear Conditions for Fractional Charges, and Physical Meaning of Orbital Energies

The Kohn-Sham functionals have been employed for ∆SCF calculations of excited states, achieving accuracy comparable to ground state results, despite a lack of theoretical justification. Our recent functional development overcoming the systematic delocalization error leads to orbital energies accurately approximating the corresponding quasiparticle energies, which are the ionization energies and electron affinities to ground and excited state electron removal and addition states. Both observations indicate that the functional theoretically formulated and approximations constructed for grounds states contain excited state information. Indeed so. We establish the theoretical foundation for ∆SCF calculations of excited states, showing that it is necessary to go beyond electron density and use the first-order density matrix of the noninteracting reference system to define the energy functional. While the minimum of the same functional corresponds to the ground state energy, the stationary solutions yield excited-state energies and electron densities, establishing the foundation of ∆SCF calculations. We also extend the theory to fractional charges. Finally, we prove the general chemical potential theorem: the noninteracting one-electron orbital energies in DFT ground states and ∆SCF excited states are corresponding chemical potentials of electron addition or removal, from an N -particle ground or excited state to an (N ± 1)-particle ground or excited state.