  We present a general approach to converge excited state solutions to any
quantum chemistry orbital optimization process, without the risk of variational
collapse. The resulting Square Gradient Minimization (SGM) approach only
requires analytic energy/Lagrangian orbital gradients and merely costs 3 times
as much as ground state orbital optimization (per iteration), when implemented
via a finite difference approach. SGM is applied to both single determinant
$\Delta$SCF and spin-purified Restricted Open-Shell Kohn-Sham (ROKS) approaches
to study the accuracy of orbital optimized DFT excited states. It is found that
SGM can converge challenging states where the Maximum Overlap Method (MOM) or
analogues either collapse to the ground state or fail to converge. We also
report that $\Delta$SCF/ROKS predict highly accurate excitation energies for
doubly excited states (which are inaccessible via TDDFT). Singly excited states
obtained via ROKS are also found to be quite accurate, especially for Rydberg
states that frustrate (semi)local TDDFT. Our results suggest that orbital
optimized excited state DFT methods can be used to push past the limitations of
TDDFT to doubly excited, charge-transfer or Rydberg states, making them a
useful tool for the practical quantum chemist's toolbox for studying excited
states in large systems.
