4.5.14 Square Gradient Minimization (SGM)

The GDM method (Section 4.5.7) is an extremely effective energy minimizer but it cannot reliably be applied to optimize excited-state orbitals, as such states are typically unstable stationary points in orbital-rotation space. Energy minimization based approaches therefore tend to ‘slip’ from these saddle points to some local minima (often the ground state, a phenomenon often described as ‘variational collapse’).

Diptarka Hait and Martin Head-Gordon have proposed an alternative way to optimize excited state orbitals, by minimizing the square of the energy gradient against orbital degrees of freedom. ⁴⁷³ Hait D., Head-Gordon M.
J. Chem. Theory Comput.
(2020), 16, pp. 1699. Link This energy gradient should be zero for all stationary points in energy, and thus all such stationary points are global minima of the squared energy gradient $\mathrm{\Delta }$. Quasi-Newton methods therefore can reliably converge to the stationary point closest to the initial guess orbitals by minimizing $\mathrm{\Delta }$, without the risk of variational collapse. The resulting SGM approach is thus essentially an extension of GDM that converges to the closest state (i.e., stationary point in orbital space) to the initial guess, as opposed to the closest energy minimum. SGM consequently can be used for reliable excited state optimization within a direct minimization framework, similar to how the MOM algorithm of Section 4.5.13 can be used in conjunction with iterative diagonalization methods like DIIS. Further details about SGM applying for excited-state orbital optimization can be found in Section 7.8.4. Full details of the SGM algorithm are provided in Ref.  473 Hait D., Head-Gordon M.
J. Chem. Theory Comput.
(2020), 16, pp. 1699. Link .

The use of SGM is controlled by the SCF_ALGORITHM variable in the $rem section: