7.8.1 Introduction

Standard TDDFT is prone to catastrophic failure in regimes where there is a substantial difference in density between ground and excited states, such as charge-transfer, Rydberg, or core excitations.²⁹⁰ This can be greatly ameliorated via inclusion of orbital relaxation beyond linear response, via explicit optimization of excited-state orbitals in a manner analogous to a ground-state SCF calculation, and several methods for doing so are described in this section. Several of these methods recognize that a single-determinant description of any singlet excited state cannot be spin pure, and the minimal description of an open-shell singlet state requires two determinants. This can be handled in a computationally tractable manner using restricted open-shell Kohn-Sham (ROKS) calculation, as described in Section 7.8.2.

Excited-state orbital optimization is generally more challenging as compared to the ground-state SCF problem because excited-state solutions of the SCF equations are generally not local minima in the orbital rotation space, but are instead typically saddle points. Traditional orbital optimizers like DIIS or GDM often fail to locate these excited-state, non-aufbau solutions to the SCF equations, and instead collapse to lower-energy solutions (usually the ground state). This problem of “variational collapse” restricts the utility of excited-state orbital-optimization methods. The maximum overlap method (MOM, Section 7.6) can prevent this in many cases, but is constrained by the convergence issues stemming from the underlying SCF algorithm. Q-Chem includes two alternative procedures that tend to be more robust as compared to MOM and can be used to find excited-state solutions to the SCF equations: squared-gradient minimization (Section 7.8.3),⁴¹⁰ and state-targeted energy projection (Section 7.8.4).¹⁶⁰ Both methods can be used on their own to produce a single-determinant “$\mathrm{\Delta }$SCF” estimate of an excitation energy, using ground-state machinery, or else combined with the ROKS procedure in order to avoid the spin contamination problems associated with the $\mathrm{\Delta }$SCF approach.