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(May 16, 2021)

The squared-gradient minimization (SGM) algorithm
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J. Chem. Theory Comput.

(2020),
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pp. 1699.
Link
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).
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J. Chem. Theory Comput.

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J. Chem. Theory Comput.

(2020),
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pp. 1699.
Link
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J. Chem. Theory Comput.

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pp. 5067.
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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 7.8.1, 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 7.8.2
and 7.8.2 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 7.8.2 and
7.8.2 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. (4.44)]. 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.

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 recommended 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, $N=$50, 25, 10 or even $N=1$ 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 convergence $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 $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 $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 convergence $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 F1s (first 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