Direct orbital optimization methods for variational density functional calculations of excited electronic states

Excited states can be calculated as stationary states of the energy expressed as a density functional. This variational approach provides atomic forces, can describe doubly excited states and, thanks to state-specific orbital relaxation, gives better approximations of charge-transfer, Rydberg and core-level excitations compared to the widely used linear-response time-dependent density functional theory (TDDFT) formalism. However, widespread application in excited-state geometry optimization and molecular dynamics is hindered by limitations of standard self-consistent field (SCF) algorithms. Such algorithms perform well in calculations of non-degenerate ground states (minima of the energy functionals) but commonly fail to converge excited-state solutions because the latter are typically saddle points and are often nearly degenerate to each other or the ground state. To overcome this impediment we develop robust direct orbital optimization methods for variational excited-state DFT calculations [1-3]. It will be shown how a novel approach using quasi-Newton algorithms that can develop negative Hessian eigenvalues in combination with the maximum overlap method can converge excited states in molecules where conventional SCF approaches usually fail, such as charge-transfer excitations in nitrobenzene [1-3]. The new method can be used to calculate potential energy surfaces (PESs) of challenging electronically degenerate systems, as it will be shown for the double bond breaking and conical intesection in ethylene. There, unrestricted broken-symmetry solutions can provide PESs in agreement with multireference calculations. Challenges for application in excited-state structural optimization or dynamics arise  due  to  the  presence  of  lower symmetry  solutions  with unphysical PESs. It will be illustrated how these issues can be overcome using strategies based on following the lowest eigenmodes of the electronic Hessian to converge on the target excited-state solutions [4].