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(May 16, 2021)

The state-targeted energy projection (STEP) method
^{
158
}
J. Chem. Theory Comput.

(2020),
16,
pp. 5067.
Link
supplies a robust
and cost-effective alternative to the maximum overlap
method (MOM) and the “initial” MOM (IMOM) procedures described in Section 7.6.
STEP applies a level shift via a simple modification of the Fock matrix,

$${\text{\mathbf{F}}}^{\prime}=\text{\mathbf{F}}+\eta \text{\mathbf{S}\mathbf{Q}\mathbf{S}},$$ | (7.38) |

where Q is the matrix representation of the projector onto the user-defined virtual space, and $\eta $ is a parameter. The level shift supplied by $\eta \text{\mathbf{S}\mathbf{Q}\mathbf{S}}$ elevates the energy of virtual orbital ${\psi}_{a}$ from ${\epsilon}_{a}$ to ${\epsilon}_{a}+\eta $ for each unoccupied orbital that is contained in Q. The parameter $\eta $ is chosen to provide the smallest level shift that retains the desired electron configuration and is defined as follows:

$$\eta =|{\epsilon}_{\text{HOMO}}-{\epsilon}_{\text{LUMO}}|+{\u03f5}^{\prime}$$ | (7.39) |

The HOMO/LUMO gap in Eq. (7.39) pertains to the HOMO and LUMO of the user-defined
configuration; meaning that like the MOM procedure, STEP requires a set of initial-guess molecular orbitals
(usually from a ground state calculation).
The small empirical parameter ${\u03f5}^{\prime}$ controls the magnitude of the gap between the occupied
and unoccupied orbitals and is settable by the *$rem* variable STEP_EPSILON.
Application of the STEP level shift constrains the solutions of the SCF equations to prevent variational
collapse by forcing an aufbau occupation of the desired occupied space at every SCF cycle.

The implementation of STEP in Q-Chem takes advantage of the fact that faster convergence is generally achieved by allowing a dynamic level shift parameter $\eta $ that changes each cycle depending on the new HOMO/LUMO gap, which differs from the static $\eta $ parameter reported in Ref. 158. In the most extreme of cases, if the desired aufbau configuration is trivially satisfied without application of a level shift projection, STEP will set $\eta =0$, which allows for unconstrained occupied/virtual rotations in optimizing the Fock matrix and thus for rapid convergence to the desired state. The parameter ${\u03f5}^{\prime}$ is nonetheless held constant as to allow control over the relative magnitude of the level shift in cases where one is necessary throughout the optimization.

STEP can be applied directly as a $\mathrm{\Delta}$SCF procedure, wherein spin contamination of the excited state is often introduced, or within a restricted open-shell framework (Section 7.8.1) in order to directly converge spin-pure excited states. tThe STEP algorithm is available for restricted, unrestricted, and restricted open-shell orbitals in Q-Chem.

Job control for $\mathrm{\Delta}$SCF (R- or U-STEP) and RO-STEP calculations:

STEP

Activates the STEP procedure.

TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

FALSE
Do not apply the STEP level-shift algorithm.
TRUE
Apply the STEP level-shift algorithm.

RECOMMENDATION:

None

STEP_EPSILON

Scales the size of the occupied/virtual gap imposed by the level-shift by $N$/100 Hartree.

TYPE:

INTEGER

DEFAULT:

10

OPTIONS:

$N$

RECOMMENDATION:

Use the default unless convergence issues arise, in which case a larger value
can be used until the desired state is found. Be aware that increasing the occupied/virtual
gap in level-shift algorithms slows convergence so it may be advisable to
increase SCF_MAX_CYCLES if large shifts are required.

STEP_PRINT

Controls the print level for STEP algorithm information.

TYPE:

INTEGER

DEFAULT:

1

OPTIONS:

0
Do not print any information about STEP between SCF cycles.
1
Print the level-shift applied at each SCF cycle (R- and U-STEP).
2
Print the level-shift for both mixed and triplet states at each SCF cycle (RO-STEP).

RECOMMENDATION:

Use the default. Level shifts of 0 indicate that an aufbau criterion
is sufficient to determine orbital occupation, and shifts $>0$ imply
non-aufbau selection of the occupied space.

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 Ground state calculation for reference orbitals $end $molecule 0 1 C 0.0000000000 -0.0000000177 -0.6043240964 C 0.0000000000 0.0000000000 0.6043240820 H 0.0000000000 0.0000000654 -1.6654864149 H 0.0000000000 0.0000000198 1.6654865011 $end $rem METHOD b3lyp BASIS def2-tzvpd SCF_CONVERGENCE 7 $end @@@ $comment Actual U-STEP calculation $end $molecule read $end $rem METHOD b3lyp BASIS def2-tzvpd SCF_GUESS read STEP true STEP_EPSILON 10 ! default is 0.1 Hartree UNRESTRICTED true SCF_CONVERGENCE 7 $end $occupied 1:7 1:6 8 $end

$comment Generates the ground-state reference orbitals $end $molecule 0 1 C 0.0000000000 -0.0000000177 -0.6043240964 C 0.0000000000 0.0000000000 0.6043240820 H 0.0000000000 0.0000000654 -1.6654864149 H 0.0000000000 0.0000000198 1.6654865011 $end $rem METHOD b3lyp BASIS def2-tzvpd SCF_CONVERGENCE 7 $end @@@ $comment Actual RO-STEP calculation $end $molecule read $end $rem METHOD b3lyp BASIS def2-tzvpd SCF_ALGORITHM gdm ! usually recommended with ROKS, but not necessary SCF_GUESS read ROKS true STEP true STEP_EPSILON 10 ! default is 0.1 Hartree SCF_CONVERGENCE 7 $end $reorder_mo 1 2 3 4 5 6 7 1 2 3 4 5 6 7 $end