# 7.8.3 State-Targeted Energy Projection

(June 30, 2021)

The state-targeted energy projection (STEP) method 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,

 $\text{{F}}^{\prime}=\text{{F}}+\eta\text{{SQS}}\;,$ (7.38)

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 $\eta\text{{SQS}}$ 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:

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

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 $\epsilon^{\prime}$ 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. 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 $\epsilon^{\prime}$ 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. 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 7.8.1) 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 $\Delta$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. Example 7.32 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
$end$rem
METHOD            b3lyp
BASIS             def2-tzvpd
STEP              true
STEP_EPSILON      10 ! default is 0.1 Hartree
UNRESTRICTED      true
SCF_CONVERGENCE   7
$end$occupied
1:7
1:6 8
$end  View output Example 7.33 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
$end$rem
METHOD            b3lyp
BASIS             def2-tzvpd
SCF_ALGORITHM     gdm ! usually recommended with ROKS, but not necessary
$end$reorder_mo