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Variational calculations of excited electronic states are carried out byfinding saddle points on the surface that describes how the energy of thesystem varies as a function of the electronic degrees of freedom. This approachhas several advantages over commonly used methods especially in the context ofdensity functional calculations, as collapse to the ground state is avoided andyet, the orbitals are variationally optimized for the excited state. Thisoptimization makes it possible to describe excitations with large chargetransfer where calculations based on ground state orbitals are problematic, asin linear response time-dependent density functional theory. A generalized modefollowing method is presented where an $n^{\text{th}}$-order saddle point isfound by inverting the components of the gradient in the direction of theeigenvectors of the $n$ lowest eigenvalues of the electronic Hessian matrix.This approach has the distinct advantage of following a chosen excited statethrough atomic configurations where the symmetry of the single determinant wavefunction is broken, as demonstrated in calculations of potential energy curvesfor nuclear motion in the ethylene and dihydrogen molecules. The method isimplemented using a generalized Davidson algorithm and an exponentialtransformation for updating the orbitals within a generalized gradientapproximation of the energy functional. Convergence is found to be more robustthan for a direct optimization approach previously shown to outperform standardself-consistent field approaches, as illustrated here for charge transferexcitations in nitrobenzene and N-phenylpyrrole, involving calculations of$4^{\text{th}}$- and $6^{\text{th}}$-order saddle points, respectively.Finally, calculations of a diplatinum and silver complex are presented,illustrating the applicability of the method to excited state energy curves oflarge molecules.

Calculations of Excited Electronic States by Converging on Saddle Points Using Generalized Mode Following