Paper

Excited electronic states of molecules and solids play a fundamental role in fields such as catalysis and electronics. In electronic structure calculations, excited states typically correspond to saddle points on the surface described by the variation of the energy as a function of the electronic degrees of freedom. A direct optimization algorithm based on generalized mode following is presented for density functional calculations of excited states. While conventional direct optimization methods based on quasi-Newton algorithms usually converge to the stationary point closest to the initial guess, even minima, the generalized mode following approach systematically targets a saddle point of a specific order $l$ by following the $l$ lowest eigenvectors of the electronic Hessian up in energy. This approach thereby recasts the challenging saddle point search as a minimization, enabling the use of efficient and robust minimization algorithms. The initial guess orbitals and the saddle point order of the target excited state solution are evaluated by performing an initial step of constrained optimization freezing the electronic degrees of freedom involved in the excitation. In the context of Kohn-Sham density functional calculations, typical approximations to the exchange-and-correlation functional suffer from a self-interaction error. The Perdew and Zunger self-interaction correction can alleviate this problem, but makes the energy variant to unitary transformations in the occupied orbital space, introducing a large amount of unphysical solutions that do not fully minimize the self-interaction error. An extension of the generalized mode following method is proposed that ensures convergence to the solution minimizing the self-interaction error.