7.8.3 Squared-Gradient Minimization

The squared-gradient minimization (SGM) algorithm⁴¹⁰ sidesteps the challenge of optimizing a saddle point in the space of orbital rotation variables $\overrightarrow{\theta }$, by instead minimizing the square of the energy gradient with respect to those variables. Ground-state SCF methods seek to minimize the energy $E$ with respect to $\overrightarrow{\theta }$ and therefore the gradient ${\widehat{\mathbf{\nabla }}}_{\overrightarrow{\theta }}E$ must be zero at convergence. It is therefore possible to obtain the same result by minimizing $\mathrm{\Delta }(\overrightarrow{\theta })={\parallel {\widehat{\mathbf{\nabla }}}_{\overrightarrow{\theta }}E\parallel }^2$ to zero. However, all stationary points of $E$ are minima of $\mathrm{\Delta }(\overrightarrow{\theta })$, not just the ground state. It is therefore possible to optimize excited-state orbitals by starting from a reasonable guess (such as a non-aufbau configuration corresponding to the excitation) and minimizing $\mathrm{\Delta }(\overrightarrow{\theta })$. This avoids all the pitfalls of attempting to optimize unstable stationary points in $E$ and thus averts variational collapse.

The SGM algorithm in Q-Chem can be used to optimize orbitals for two different excited state approaches: $\mathrm{\Delta }$SCF and ROKS. The former simply attempts to minimize the energy of a single Slater determinant, which is often sufficient for many challenging excitations (including many double excitations).^{60, 410, 160} However, many excitations (including all single excitations from a closed shell ground state) break electron pairs, leading to states that cannot be described with a single determinant. It is possible to spin-purify the energy of a spin-contaminated, non-aufbau determinant a posteriori, but this requires at least two separate orbital optimizations. An alternative is ROKS (as described in Section 7.8.2, which requires optimization of only a single set of orbitals, for which the spin-purified energy is stationary. Analytic nuclear gradients are available for both $\mathrm{\Delta }$SCF and ROKS, permitting geometry optimizations and ab initio molecular dynamics. Analytic frequencies are available for $\mathrm{\Delta }$SCF, except with functionals that contain VV10 nonlocal correlation.

There are some slight differences between use of SGM for different orbital classes due to ease of implementation. The $\mathrm{\Delta }$SCF procedure with restricted closed-shell (R) and unrestricted (U) orbitals can be run with SCF_ALGORITHM = SGM_LS or SCF_ALGORITHM = SGM_QLS, with initial orbital occupation specified by the $occupied block (as described in Section 7.6 and in Examples 7.8.3 and 7.8.3 below). A $\mathrm{\Delta }$SCF calculation with restricted open-shell (RO) orbitals or an ROKS calculation can be performed via SCF_ALGORITHM = SGM or SCF_ALGORITHM = SGM_LS, and a re-ordering of orbitals to ensure that the unpaired ones lie at the frontier. (See Examples 7.8.3 and 7.8.3 below.) The gradient of $\mathrm{\Delta }(\overrightarrow{\theta })$ is computed analytically (except in the case of functionals that contain VV10 nonlocal correlation), for R-, U- and RO-$\mathrm{\Delta }$SCF, at a cost equal to a single Fock build. However, the gradient $\mathrm{\Delta }(\overrightarrow{\theta })$ in the ROKS case, and for functionals containing VV10, is computed with a finite-difference approach [see Eq. (4.46)]. In those cases, the cost is equal to that of two Fock builds. Cumulatively, a single SGM iteration costs twice as much as a single GDM iteration when the analytic $\mathrm{\Delta }(\overrightarrow{\theta })$ gradient is available, and three times as much if the finite difference construction must be used, although this does not affect the asymptotic scaling of the calculation with respect to system size.

Excited-state orbital optimization sometimes requires more iterations than what is typical for ground-state SCF calculations, so MAX_SCF_CYCLES should be set to a large value (perhaps 200), rather than the default value of 50. A loose convergence threshold of SCF_CONVERGENCE = 4 is also permissible if only energies are desired, as long as it is explicitly confirmed that the variation in energy over several iterations is much less than the desired accuracy after job completion. (A variation greater than ${10}^{-3}{E}_h$ or 0.03 eV would be quite problematic, for example.) Further reduction of SCF_CONVERGENCE likely compromises properties such as dipole moments or nuclear gradients, and is not recommended.