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 n
th-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.