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

Calculations of excited electronic states are carried out by finding saddle points on the surface describing the variation of the energy of the system as a function of the electronic degrees of freedom. This approach has several advantages over commonly used methods especially in the context of density functional calculations, as collapse to the ground state is avoided, and yet, the orbitals are variationally optimized for the excited state. Such a state-specific optimization makes it possible to describe excitations with large charge transfer, where calculations based on ground state orbitals, such as linear response time-dependent density functional theory, can be problematic. A generalized mode following method is presented where an nth-order saddle point is found by inverting the components of the gradient in the direction of the eigenvectors of the n lowest eigenvalues of the electronic Hessian matrix. This approach has the distinct advantage of following a chosen excited state by its saddle point order through molecular configurations where the symmetry of the single determinant wave function is broken, thereby making it possible to calculate potential energy curves even at avoided crossings, as demonstrated here in calculations of the ethylene and dihydrogen molecules. Results of calculations are, furthermore, presented for charge transfer excitations in nitrobenzene and N-phenylpyrrole, corresponding to fourth- and sixth-order saddle points, respectively, where an approximate initial estimate of the saddle point order could be found by energy minimization with excited electron and hole orbitals frozen. Finally, calculations of a diplatinum-silver complex are presented, illustrating the applicability of the method to larger molecules.