7.8 Square Gradient Minimization for Calculation of $\mathrm{\Delta }$SCF and ROKS Excited States

Standard TDDFT is prone to catrastrophic failure in regimes where there is a substantial difference in density between ground and excited states, such as charge-transfer, Rydberg, or core excitations.²³³ This can be greatly ameliorated via inclusion of orbital relaxation beyond linear response, via explict optimization of excited state orbitals in a manner analogous to ground state SCF. However, this is a challenging task as excited states are typically not minimums of the energy in orbital space, but are usually saddle points. Traditional orbital optimizers like DIIS or GDM therefore tend to fail in converging to excited state orbitals, and instead collapse to lower energy configurations (usually the ground state). This variational collapse problem consequently restricts the utility of excited state orbital optimized methods. The MOM approach (see Sec 7.5) can prevent this in many cases, but is constrained by the convergence issues stemming from the underlying DIIS method.

Diptarka Hait and Martin Head-Gordon presented an alternative approach to excited state orbital optimization in Ref. 334, which avoids the challenges of energy extremization by minimizing the square of the energy gradient instead. Ground state HF and DFT approaches seek to minimize the energy $E$ against orbital degrees of freedom $\overrightarrow{\theta }$ and therefore the gradient of $E$ against $\overrightarrow{\theta }$ (${\nabla }_{\overrightarrow{\theta }}E$) must be zero at convergence. It is therefore possible to obtain the same result by minimizing $\mathrm{\Delta }={|{\nabla }_{\overrightarrow{\theta }}E|}^2$ to zero. However, all stationary points of $E$ are minimums of $\mathrm{\Delta }$, 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 $\mathrm{\Delta }$. This avoids all the pitfalls of attempting to optimize unstable stationary points in $E$ and thus averts variational collapse. Further details about this Square Gradient Minimization method are provided in Ref. 334.

SGM in Q-Chem can be used to optimize orbitals for two different excited state approaches: $\mathrm{\Delta }$SCF and ROKS. $\mathrm{\Delta }$SCF simply attempts to minimize the energy of a single Slater determinant, which is often sufficient for many challenging excitations (including many double excitations as shown in Refs. 40 and 334). 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 filled determinant post SCF, but this requires at least two separate orbital optimizations. ROKS (described in detail in Sec 7.7) offers an alternative solution to this problem by directly optimizing 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 contain VV10 nonlocal correlation), while finite-difference frequency calculations (with IDERIV =1) could be run for ROKS (or $\mathrm{\Delta }$SCF with non VV10 functionals).

There are some slight differences between use of SGM for different orbital classes due to ease of implementation. $\mathrm{\Delta }$SCF 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 7.5 and the examples 7.8 and 7.8 below). $\mathrm{\Delta }$SCF with restricted open-shell (RO) orbitals or ROKS calculations can be performed via SCF_ALGORITHM = SGM or SCF_ALGORITHM = SGM_LS, and a reordering of orbitals to ensure that the unpaired ones lie at the frontier (as described in examples 7.8 and 7.8 below). The $\mathrm{\Delta }$ gradient (vs $\overrightarrow{\theta }$) employed by SGM is computed analytically (except for VV10 containing functionals) for R-,U- and RO- $\mathrm{\Delta }$SCF, at a cost equal to a single Fock build. However, the ROKS $\mathrm{\Delta }$ gradient (as well as the $\mathrm{\Delta }$SCF cases with VV10 containing functionals) is found via a finite difference approach (see Eq.4.44), which leads to the cost being equal to that of two Fock builds. Cumulatively, a single SGM iteration costs twice as much as a single GDM iteration when analytic $\mathrm{\Delta }$ gradients are available, and three times as much if the finite difference approximation has to be used. This does not affect the overall asymptotic scaling of the calculation, however. It is worth noting that excited state convergence could take more iterations than corresponding ground state jobs, and so MAX_SCF_CYCLES should be set to a large value (like 200) instead of the default 50. SCF_CONVERGENCE = 4 (instead of the default value of 5) 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 (variation greater than 0.001 hartrees or 0.03 eV would be quite problematic). Lowering SCF_CONVERGENCE further however would likely significantly compromise properties like dipole moments or nuclear gradients, and is therefore not reccomended.