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.^{Dreuw:2005}
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. Hait:2020b, 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. Hait:2020b.

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. Barca:2018 and Hait:2020b).
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 7.6) 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 7.7 and 7.7 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 7.7 and 7.7 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.4.43), 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.

SCF_ALGORITHM

Algorithm used for converging the SCF.

TYPE:

STRING

DEFAULT:

None

OPTIONS:

SGM
SGM_LS
SGM_QLS

RECOMMENDATION:

SGM should be used for RO-$\mathrm{\Delta}$SCF or ROKS calculations only. SGM_LS is reccomended for R- or U- $\mathrm{\Delta}$SCF, though it can also be used for RO-$\mathrm{\Delta}$SCF or ROKS. SGM_QLS is a slower, but more robust option for R- and U- $\mathrm{\Delta}$SCF calculations.

DELTA_GRADIENT_SCALE

Scales the gradient of $\mathrm{\Delta}$ by N/100, which can be useful for cases with troublesome convergence by reducing step size.

TYPE:

INTEGER

DEFAULT:

100

OPTIONS:

N

RECOMMENDATION:

Use default. For problematic cases 50, 25, 10 or even 1 could be useful.

ROKS

Controls whether ROKS calculation will be performed.

TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

FALSE
ROKS is not performed.
TRUE
ROKS will be performed.

RECOMMENDATION:

Set to TRUE if ROKS calculation is desired.
UNRESTRICTED = FALSE should also be ensured.

$comment Calculates Delta-SCF excitation energy for the 2s^2 -> 2p^2 excitation of Be using SCAN and SGM_QLS scf converger $end $molecule 0 1 Be $end $rem method scan basis aug-cc-pVTZ thresh 14 scf_convergence 8 scf_algorithm diis symmetry false sym_ignore true xc_grid 000099000590 gen_scfman true $end @@@@@@@@@ $molecule read $end $rem method scan basis aug-cc-pVTZ thresh 14 scf_algorithm sgm_qls symmetry false sym_ignore true scf_guess read xc_grid 000099000590 gen_scfman true $end $occupied 1 3 1 3 $end

$molecule 0 1 N 0.0000 0.0000 0.0000 H 0.0000 -0.9377 -0.3816 H 0.8121 0.4689 -0.3816 H -0.8121 0.4689 -0.3816 F 0.0000 0.0000 6.0000 F 0.0000 0.0000 7.4120 $end $rem method pbe0 basis cc-pVDZ symmetry false sym_ignore true scf_convergence 8 $end @@@@@@@@@@@ $comment The reorder section is superfluous here since the excitation is HOMO to LUMO and thus the unpaired electron orbitals are already at the frontier. $end $molecule read $end $rem method pbe0 basis cc-pVDZ symmetry false sym_ignore true scf_algorithm sgm roks true scf_guess read $end $reorder_mo 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 $end

$comment Calculates Delta-SCF excitation energy for the HOMO-1 -> LUMO+1 excitation of HCHO using SCAN and SGM_LS scf converger $end $molecule 0 1 O1 0.0000 0.0000 1.2050 C2 0.0000 0.0000 0.0000 H3 0.0000 0.9429 -0.5876 H4 0.0000 -0.9429 -0.5876 $end $rem method scan basis aug-cc-pVTZ thresh 14 scf_convergence 8 scf_algorithm diis symmetry false sym_ignore true xc_grid 000099000590 gen_scfman true $end @@@@@@@@@ $molecule read $end $rem method scan basis aug-cc-pVTZ thresh 14 scf_algorithm sgm_ls symmetry false sym_ignore true scf_guess read xc_grid 000099000590 gen_scfman true unrestricted true scf_convergence 7 max_scf_cycles 500 $end $occupied 1 2 3 4 5 6 8 10 1 2 3 4 5 6 7 8 $end

$molecule 0 1 F 0.0000 0.0000 0.0000 H 0.0000 0.0000 0.9168 $end $rem method scan basis aug-cc-pCVTZ symmetry false $end @@@@@@@@@@@@ $comment Calculates the RO-DeltaSCF core ionized state. The reorder section pushes the F1s orbital (first orbital by energy) to the frontier, instead of the HOMO (5th orbital). $end $molecule 1 2 F 0.0000 0.0000 0.0000 H 0.0000 0.0000 0.9168 $end $rem method scan basis aug-cc-pCVTZ unrestricted false scf_guess read symmetry false scf_algorithm sgm $end $reorder_mo 2 3 4 5 1 2 3 4 5 1 $end @@@@@@@@@@@@ $comment Calculates the ROKS core excited state. The O1s orbital is in the place of the HOMO from the previous reordering. The present reorder section pushes the LUMO+1 orbital (the 7th orbital) to the frontier, instead of the LUMO (6th orbital). $end $molecule 0 1 F 0.0000 0.0000 0.0000 H 0.0000 0.0000 0.9168 $end $rem method scan roks true basis aug-cc-pCVTZ scf_guess read symmetry false scf_algorithm sgm max_scf_cycles 200 $end $reorder_mo 1 2 3 4 5 7 6 1 2 3 4 5 7 6 $end