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

Calculations of excited electronic states are carriedout by findingsaddle points on the surface describing the variation of the energyof the system as a function of the electronic degrees of freedom.This approach has several advantages over commonly used methods especiallyin the context of density functional calculations, as collapse tothe ground state is avoided, and yet, the orbitals are variationallyoptimized for the excited state. Such a state-specific optimizationmakes it possible to describe excitations with large charge transfer,where calculations based on ground state orbitals, such as linearresponse time-dependent density functional theory, can be problematic.A generalized mode following method is presented where an n (th)-order saddle point is found by invertingthe components of the gradient in the direction of the eigenvectorsof the n lowest eigenvalues of the electronic Hessianmatrix. This approach has the distinct advantage of following a chosenexcited state by its saddle point order through molecular configurationswhere the symmetry of the single determinant wave function is broken,thereby making it possible to calculate potential energy curves evenat avoided crossings, as demonstrated here in calculations of theethylene and dihydrogen molecules. Results of calculations are, furthermore,presented for charge transfer excitations in nitrobenzene and N-phenylpyrrole, corresponding to fourth- and sixth-ordersaddle points, respectively, where an approximate initial estimateof the saddle point order could be found by energy minimization withexcited electron and hole orbitals frozen. Finally, calculations ofa diplatinum-silver complex are presented, illustrating theapplicability of the method to larger molecules.