Challenging electronic excitations calculated by converging on saddle points of the electronic energy surface*

Excited electronic states can be calculated as solutions to the Kohn-Sham equations higher in energy than the ground state. As the orbitals are variationally optimized, this time-independent approach is well-suited for describing challenging excitations involving significant rearrangement of the electron density, such as charge transfer and Rydberg excited states, where methods based on linear response time dependent density functional theory (TDDFT) tend to fail. While conceptually simple, the time-independent approach faces the challenge that the excited state solutions typically correspond to saddle points on the surface describing the variation of the energy as a function of the electronic degrees of freedom, making the calculations prone to collapsing to the ground state. Variational collapse can be avoided in a generalized mode following approach by inverting the gradient along the modes of the electronic Hessian corresponding to the lowest n eigenvalues, thereby converging on a saddle point of order n. Several applications of time-independent density functional calculations of excited states both in condensed phase and molecular systems will be presented. The approach has been used to compute electronic excitations in the charged nitrogen-vacancy center in diamond. Contrary to previous contradictory reports, a correct ordering of the excited states is obtained, with meta-GGA functionals giving values of excitation energy in close agreement with high-level many-body calculations. It will also be shown how excited state molecular dynamics simulations with explicit solvation helped elucidate the dynamics following photoexcitation of a Cu complex photosensitizer to a metal-to-ligand charge transfer state. For molecules, remarkably good results are obtained even at the generalized gradient approximation level for Rydberg excitations. The results are improved when including Perdew-Zunger self-interaction correction, achieving close agreement with experimental values of excitation energy. Time-independent density functional calculations are also found to improve the description of intramolecular charge transfer excited states, where TDDFT with GGA functionals can underestimate the excitation energy by as much as 2 eV.