Potential-averaged ∆SCF methods: underlying formalism and evaluation of accuracy of excitation energies

Several new ∆SCF (∆-Self-Consistent-Field) methods to calculate energies of excited electronic states within a density-functional framework are presented. Like ∆SCF methods in general, the new methods have a firm formal basis in the fundamental density theorem and the associated generalizedadiabatic-connection Kohn-Sham (GAC-KS) formalism derived in Phys. Rev. A 59, 3359 (1999). The GAC-KS formalism is elucidated, its meaning as a formal justification of ∆SCF methods is explained, and its relation to the optimized effective potential (OEP) method is discussed. Four ∆SCF 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 ∆SCF calculation, 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, and all yield more accurate singlet and triplet excitation energies than the standard UKS method. 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.