\Delta

SCF Methods 7.8.3 Squared-Gradient Minimization 7.9 Correlated Excited State Methods: The CIS(D) Family

7.8.4 State-Targeted Energy Projection

(July 14, 2022)

The state-targeted energy projection (STEP) method ¹⁶⁶ Carter-Fenk K., Herbert J. M.
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{{F}}^{\prime}=\text{{F}}+\eta\text{{SQS}}\;,

(7.35)

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 $\eta\text{{SQS}}$ elevates the energy of virtual orbital $\psi_{a}$ from $\varepsilon_{a}$ to $\varepsilon_{a}+\eta$ 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=|\varepsilon_{\text{HOMO}}-\varepsilon_{\text{LUMO}}|+\epsilon^{\prime}

(7.36)

The HOMO/LUMO gap in Eq. (7.36) 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 $\epsilon^{\prime}$ 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. 166 Carter-Fenk K., Herbert J. M.
J. Chem. Theory Comput.
(2020), 16, pp. 5067. Link . 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 $\epsilon^{\prime}$ 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.

In more difficult cases, the dynamic level-shift approach (while more efficient) can lead to variational collapse. If the dynamic level-shift is insufficient, reverting to the implementation that was originally reported in Ref. 166 Carter-Fenk K., Herbert J. M.
J. Chem. Theory Comput.
(2020), 16, pp. 5067. Link can increase the robustness of STEP appreciably. In this approach, the level-shift parameter in Eq. (7.36) is always active at every SCF cycle such that $\eta$ is never zero.

STEP can be applied directly as a $\Delta$ SCF procedure, wherein spin contamination of the excited state is often introduced, or within a restricted open-shell framework (Section 7.8.2) in order to directly converge spin-pure excited states. The STEP algorithm is available for restricted, unrestricted, and restricted open-shell orbitals in Q-Chem.

Job control for $\Delta$ SCF (R- or U-STEP) and RO-STEP calculations: After STEP is activated in the $rem section, the remainder of the options for STEP are handled through the $step input section.

STEP

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

Epsilon
       Scales the size of the occupied/virtual gap imposed by the level-shift by $N$ /100 Hartree.
INPUT SECTION: $step
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.

Print
       Controls the print level for STEP algorithm information.
INPUT SECTION: $step
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.

Always_Active
       Toggles the original implementation of STEP where the level-shift is static (applied every cycle).
INPUT SECTION: $step
TYPE:
       STRING
DEFAULT:
       None
OPTIONS:
       alpha Apply a constant level-shift to the alpha spin orbitals. beta Apply a constant level-shift to the beta spin orbitals. both Apply a constant level-shift to both alpha and beta spin orbitals.
RECOMMENDATION:
       Use in cases where the dynamic level-shift does not achieve satisfactory results. In the case of U-STEP, the constraint need only be applied to orbitals that must maintain a non-Aufbau configuration (i.e.an alpha-electron promotion requires only the alpha constraint, but two-electron promotions will require both constraints). For RO-STEP this keyword is set to both by default and cannot be turned off. In R-STEP it is only sensible to use both if the desired doubly-excited configuration cannot be found with the dynamic level-shift parameter.

ROKS

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 7.34 Lowest energy $\pi\rightarrow\pi^{\ast}$ transition in acetylene using a $\Delta$ SCF approach with STEP.

$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
   UNRESTRICTED      true
   SCF_CONVERGENCE   7
$end

! default level-shift is 0.1 Hartree
! beta orbital promotion means only betas need constrained
$step
Epsilon 10
Always_Active   beta
$end

$occupied
   1:7
   1:6 8
$end

Example 7.35 A spin-pure lowest energy $\pi\rightarrow\pi^{\ast}$ transition in acetylene using ROKS with STEP.

$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
   SCF_CONVERGENCE   7
$end

! prints all level-shift information
$step
Epsilon 10
Print   2
$end

$reorder_mo
   1 2 3 4 5 6 7
   1 2 3 4 5 6 7
$end

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7.8.4 State-Targeted Energy Projection