(May 16, 2021)

The squared-gradient minimization (SGM) algorithm sidesteps the challenge of optimizing a saddle point in the space of orbital rotation variables $\vec{\theta}$, 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 $\vec{\theta}$ and therefore the gradient $\hat{\bm{\nabla}}_{\vec{\theta}}E$ must be zero at convergence. It is therefore possible to obtain the same result by minimizing $\Delta(\vec{\theta})=\|\hat{\bm{\nabla}}_{\vec{\theta}}E\|^{2}$ to zero. However, all stationary points of $E$ are minima of $\Delta(\vec{\theta})$, 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 $\Delta(\vec{\theta})$. 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: $\Delta$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). ,, 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 $\Delta$SCF and ROKS, permitting geometry optimizations and ab initio molecular dynamics. Analytic frequencies are available for $\Delta$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 $\Delta$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 $\Delta$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 $\Delta(\vec{\theta})$ is computed analytically (except in the case of functionals that contain VV10 nonlocal correlation), for R-, U- and RO-$\Delta$SCF, at a cost equal to a single Fock build. However, the gradient $\Delta(\vec{\theta})$ 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 $\Delta(\vec{\theta})$ 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-$\Delta$SCF or ROKS calculations only. SGM_LS is recommended for R- or U-$\Delta$SCF, though it can also be used for RO-$\Delta$SCF or ROKS. SGM_QLS is a slower but more robust option for R- and U-$\Delta$SCF calculations. DELTA_GRADIENT_SCALE Scales the gradient of $\Delta$ 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. Example 7.28 Restricted $\Delta$SCF double excitation from $2s$ to $2p$ of Be atom, using SGM_QLS and the ground state orbitals as an initial guess. $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
$end$rem
METHOD            scan
BASIS             aug-cc-pVTZ
THRESH            14
SCF_ALGORITHM     sgm_qls
SYMMETRY          false
SYM_IGNORE        true
XC_GRID           000099000590
$end$occupied
1 3
1 3
$end  View output Example 7.29 ROKS single excitation from HOMO to LUMO for an NH${}_{3}$$\cdots$F${}_{2}$ model complex, which describes electron transfer from NH${}_{3}$ to F${}_{2}$. Ground state orbitals are used as an initial guess. $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
$end$rem
METHOD            pbe0
BASIS             cc-pVDZ
SYMMETRY          false
SYM_IGNORE        true
SCF_ALGORITHM     sgm
ROKS              true
$end$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  View output Example 7.30 Unrestricted $\Delta$SCF single excitation from HOMO$-1$ to LUMO$+1$ for HCHO using SGM_LS SCF convergence algorithm and the ground state orbitals as an initial guess. $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
$end$rem
METHOD            scan
BASIS             aug-cc-pVTZ
THRESH            14
SCF_ALGORITHM     sgm_ls
SYMMETRY          false
SYM_IGNORE        true
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
$end  View output Example 7.31 Combined RO-$\Delta$SCF core-ionization and $1s\rightarrow\text{LUMO}+1$ ROKS core-excited state. $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
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
$end$reorder_mo