Potential-Averaged ΔSCF Methods: Underlying Formalism and Evaluation of Accuracy of Excitation Energies

Several new ΔSCF (Δ-Self-Consistent-Field) methods to calculate energies of excited electronic states within a density-functional framework are presented. Like ΔSCF methods in general, the new methods have a firm formal basis in the fundamental density theorem and the associated generalized-adiabatic-connection Kohn-Sham (GAC-KS) formalism derived in [Görling, A.Phys. Rev. A1999, 59, 3359].. The GAC-KS formalism is elucidated, its meaning as a formal justification of ΔSCF methods is explained, and its relation to the optimized effective potential (OEP) method is discussed. Four ΔSCF approaches employing the OEP method are presented that differ by the treatment of open shells and states that cannot be represented by a single Slater determinant. From these four parent KS-OEP methods, four simplified methods are derived. One is the well-known unrestricted Kohn-Sham (UKS) method commonly used in ΔSCF calculations, the other three new methods replace the OEP step by a simple averaging of exchange-correlation potentials and are named potential-averaged Kohn-Sham (pa-KS) methods. The pa-KS methods, as well as their parent KS-OEP methods, are free of spin contamination, all yield more accurate singlet and triplet excitation energies than the standard UKS method, and outperform methods based on time-dependent density-functional theory. Interestingly, the pa-KS methods are slightly more accurate than their parent KS-OEP methods. Because pa-KS methods require minimal implementation effort in existing KS codes while outperforming time-dependent density-functional methods, they are highly promising for practical applications.