Multistate-Averaged Kohn–Sham ΔSCF Method for Excitation Energies, Transition Properties, and Absorption Spectra and Its Relation to Ensemble Density Functional Theory

The ΔSCF (Δ-Self-Consistent-Field) approach is an attractive alternative to time-dependent density-functional theory (TDDFT) for excited-state calculations, particularly for Rydberg, charge-transfer, and doubly excited states, for which conventional linear-response TDDFT performs poorly. However, conventional ΔSCF methods based on the unrestricted Kohn-Sham formalism suffer from spin- and spatial-symmetry breaking, optimize the ground- and excited-state determinants independently, leading to nonorthogonal Kohn-Sham determinants that complicate the calculation of transition properties, and require a separate SCF procedure for each excited state of interest. In order to address these limitations, we present a potential-averaged multistate-averaged Kohn-Sham (pa-MSTA-KS) method, which averages the Hartree-exchange-correlation potential of the ground state with state-averaged Hartree-exchange-correlation potentials of one or several excited states during a single SCF procedure. Unlike restricted open-shell Kohn-Sham (ROKS) approaches, pa-MSTA-KS yields canonical SCF equations and can be implemented in any standard KS code with minimal effort. The pa-MSTA-KS method avoids spin-symmetry breaking by construction, yields orbitals that describe ground and excited states simultaneously, and enables the direct evaluation of transition dipole moments and the simulation of full UV/vis absorption spectra based on a single SCF calculation. For excitation energies of the QUEST1 data set and for absorption spectra of water, ethylene, and ammonia, pa-MSTA-KS outperforms TDDFT in accuracy. By combining the generalized-adiabatic-connection Kohn-Sham formalism with ensemble density-functional theory, we provide a formal basis for the pa-MSTA-KS method and derive a new decomposition of the ensemble correlation energy into state- and density-driven contributions.