  Variational calculations of excited electronic states are carried out by
finding saddle points on the surface that describes how the energy of the
system varies 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. This
optimization makes it possible to describe excitations with large charge
transfer where calculations based on ground state orbitals are problematic, as
in linear response time-dependent density functional theory. A generalized mode
following method is presented where an $n^{\text{th}}$-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
through atomic configurations where the symmetry of the single determinant wave
function is broken, as demonstrated in calculations of potential energy curves
for nuclear motion in the ethylene and dihydrogen molecules. The method is
implemented using a generalized Davidson algorithm and an exponential
transformation for updating the orbitals within a generalized gradient
approximation of the energy functional. Convergence is found to be more robust
than for a direct optimization approach previously shown to outperform standard
self-consistent field approaches, as illustrated here for charge transfer
excitations 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 of
large molecules.
