\Delta

SCF Excited States 7.6 Maximum Overlap Method (MOM) for

\Delta

SCF Excited States 7.7 Non-Orthogonal CIS and Static Exchange (STEX)

7.6.1 Overview

(September 1, 2024)

The Maximum Overlap Method (MOM) is a useful alternative to CIS and TDDFT for obtaining low-cost excited states. ⁴¹¹ Gilbert A. T. B., Besley N. A., Gill P. M. W.
J. Phys. Chem. A
(2008), 112, pp. 13164. Link It works by modifying the orbital selection step in the SCF procedure. By choosing orbitals that most resemble those from the previous cycle, rather than those with the lowest eigenvalues, non-aufbau, excited-state SCF determinants can be determined, in what has sometimes been called excited-state Kohn-Sham theory. ⁴⁸¹ Hanson-Heine M. W. D., George M. W., Besley N. A.
J. Chem. Phys.
(2013), 138, pp. 064101. Link This represents a form of “ $\Delta$ SCF” approach to computing excitation energies, which has advantages over TDDFT is certain cases. For example, TDDFT exhibits systemic problems with the description of charge-transfer and Rydberg excitations, both of which can be modeled using the $\Delta$ SCF approach. The use of MOM also allows the user to easily target very high energy states, such as those involving excitation of core electrons, ¹¹¹ Besley N. A., Gilbert A. T. B., Gill P. M. W.
J. Chem. Phys.
(2009), 130, pp. 124308. Link which can be difficult to capture using other excited state methods. Other $\Delta$ SCF approaches are described in Section 7.8.

In order to calculate an excited state using MOM, the user must correctly identify the orbitals involved in the transition. For example, in a $\pi\rightarrow\pi^{\ast}$ transition, the $\pi$ and $\pi^{\ast}$ orbitals must be identified and this usually requires a preliminary calculation. The user then manipulates the orbital occupancies using the $occupied section, removing an electron from the $\pi$ and placing it in the $\pi^{\ast}$ . The MOM is then invoked to preserve this orbital occupancy. The success of the MOM relies on the quality of the initial guess for the calculation. If the virtual orbitals are of poor quality then the calculation may ‘fall down’ to a lower energy state of the same symmetry. Often the virtual orbitals of the corresponding cation are more appropriate for using as initial guess orbitals for the excited state.

Because the MOM states are single determinants, all of Q-Chem’s existing single determinant properties and derivatives are available. This allows, for example, analytic harmonic frequencies to be computed on excited states. The orbitals from a Hartree-Fock MOM calculation can also be used in an MP2 calculation. For all excited state calculations, it is important to add diffuse functions to the basis set. This is particularly true if Rydberg transitions are being sought. For DFT based methods, it is also advisable to increase the size of the quadrature grid so that the more diffuse densities are accurately integrated.

Due to its single-determinant description, the MOM algorithm for unrestricted $\Delta$ SCF calculations of open-shell singlets is often badly spin contaminated. Spin contamination can be alleviated by resorting to a restricted open-shell Kohn-Sham (ROKS) formalism (see Section 7.8.3), in which the objective function $E_{\text{S}}=2E_{\text{M}}-E_{\text{T}}$ is optimized, where $E_{\text{S}}$ is the spin-pure singlet energy, $E_{\text{M}}$ is the energy of the “mixed” (or spin-contaminated) open-shell singlet state, and $E_{\text{T}}$ is the triplet energy. MOM can be used in conjunction with ROKS to evaluate a spin-pure energy for arbitrary open-shell singlet electron configurations (formerly only $S_{1}$ was accessible within the ROKS formulation).

The MOM-based $\Delta$ SCF method can be combined with a PCM based solvation description for both equilibrium and non-equilibrium effects (for more details see Section 7.8.6).

The following $rem is used to invoke the MOM:

MOM_START

MOM_START
       Determines when MOM is switched on to preserve orbital occupancies.
TYPE:
       INTEGER
DEFAULT:
       0 (FALSE)
OPTIONS:
       0 (FALSE) MOM is not used $n$ MOM begins on cycle $n$ .
RECOMMENDATION:
       For calculations on excited states, an initial calculation without MOM is usually required to get satisfactory starting orbitals. These orbitals should be read in setting SCF_GUESS = TRUE and MOM_START = 1.

MOM_METHOD

MOM_METHOD
       Determines the target orbitals with which to maximize the overlap on each SCF cycle.
TYPE:
       INTEGER
DEFAULT:
       MOM
OPTIONS:
       MOM Maximize overlap with the orbitals from the previous SCF cycle. IMOM Maximize overlap with the initial guess orbitals.
RECOMMENDATION:
       If appropriate guess orbitals can be obtained, then IMOM can provide more reliable convergence to the desired solution. ⁶⁸ Barca G. M. J., Gilbert A. T. B., Gill P. M. W.
J. Chem. Theory Comput.
(2018), 14, pp. 1501. Link

Example 7.34 Calculation of the lowest singlet state of CO.

$comment
   CO spin-purified calculation
   Step 1: prepare MOs
$end

$molecule
   0 1
   C
   O  C  1.05
$end

$rem
   METHOD            B3LYP
   BASIS             6-31G*
$end

@@@

$comment
  Step 2: spin purification (OPSING=TRUE)
$end

$molecule
   read
$end

$rem
   METHOD            B3LYP
   BASIS             6-31G*
   SCF_GUESS         read
   MOM_START         1
   UNRESTRICTED      true
   OPSING            true
$end

$occupied
   1 2 3 4 5 6 7
   1 2 3 4 5 6 8
$end

Example 7.35 Input for obtaining the ${}^{2}$ A ${}^{\prime}$ excited state of formamide corresponding to the $\pi\rightarrow\pi^{\ast}$ transition. The ${}^{1}$ A ${}^{\prime}$ ground state is obtained if MOM is not used in the second calculation. Note the use of diffuse functions and a larger quadrature grid to accurately model the larger excited state.

$molecule
   1  2
   C
   H  1  1.091480
   O  1  1.214713  2  123.10
   N  1  1.359042  2  111.98  3  -180.00
   H  4  0.996369  1  121.06  2    -0.00
   H  4  0.998965  1  119.25  2  -180.00
$end

$rem
   METHOD         B3LYP
   BASIS          6-311(2+,2+)G(d,p)
   XC_GRID        000100000194
$end

@@@

$molecule
   0  1
   C
   H  1  1.091480
   O  1  1.214713  2  123.10
   N  1  1.359042  2  111.98  3  -180.00
   H  4  0.996369  1  121.06  2    -0.00
   H  4  0.998965  1  119.25  2  -180.00
$end

$rem
   METHOD         B3LYP
   BASIS          6-311(2+,2+)G(d,p)
   XC_GRID        000100000194
   MOM_START      1
   SCF_GUESS      read
   UNRESTRICTED   true
$end

$occupied
   1:12
   1:11 13
$end

Additionally, it is possible to perform a CIS/TDDFT calculation on top of the MOM excitation. This capability can be useful when modeling pump-probe spectra. In order to run MOM followed by CIS/TDDFT, the $rem variable CIS_N_ROOTS must be specified. Truncated subspaces may also be used, see Section 7.3.2.

Example 7.36 MOM valence excitation followed by core-state TDDFT using a restricted subspace

$molecule
   0 1
   O    0.0000    0.0000    0.1168
   H    0.0000    0.7629   -0.4672
   H    0.0000   -0.7629   -0.4672
$end

$rem
   METHOD       B3LYP
   BASIS        aug-cc-pvdz
   INTEGRAL_SYMMETRY    false
   POINT_GROUP_SYMMETRY false
$end

@@@

$molecule
   read
$end

$rem
   METHOD         B3LYP
   BASIS          aug-cc-pvdz
   SCF_GUESS      read
   MOM_START      1
   UNRESTRICTED   true
   CIS_N_ROOTS    5
   TRNSS          true  ! use truncated subspace for TDDFT
   TRTYPE         3     ! specify occupied orbitals
   CUTVIR         15    ! truncate high energy virtual orbitals
   N_SOL          1     ! number core orbitals, specified in $solute section
   INTEGRAL_SYMMETRY    false
   POINT_GROUP_SYMMETRY false
$end

$solute
   1
$end

$occupied
   1 2 3 4 5
   1 2 3 4 6
$end

If the MOM excitation corresponds to a core hole, a reduced subspace must be used to avoid de-excitations to the core hole. The $rem variable CORE_IONIZE allows only the hole to be specified so that not all occupied orbitals need to be entered in the $solute section.

CORE_IONIZE

CORE_IONIZE
       Indicates how orbitals are specified for reduced excitation spaces.
TYPE:
       INTEGER
DEFAULT:
       1
OPTIONS:
       1 all valence orbitals are listed in $solute section 2 only hole(s) are specified all other occupations same as ground state
RECOMMENDATION:
       For MOM + TDDFT this specifies the input form of the $solute section. If set to 1 all occupied orbitals must be specified, 2 only the empty orbitals to ignore must be specified.

Example 7.37 O(1s) core excited state using MOM followed by excitations among valence orbitals. Note that a reduced excitation subspace must be used to avoid “excitations” into the empty core hole

$molecule
   0 1
   O   0.0000   0.0000   0.1168
   H   0.0000   0.7629  -0.4672
   H   0.0000  -0.7629  -0.4672
$end

$rem
   METHOD       B3LYP
   BASIS        aug-cc-pvdz
   INTEGRAL_SYMMETRY    false
   POINT_GROUP_SYMMETRY false
$end

@@@

$molecule
   read
$end

$rem
   METHOD            B3LYP
   BASIS             aug-cc-pvdz
   SCF_GUESS         read
   MOM_START         1
   UNRESTRICTED      true
   CIS_N_ROOTS       5
   TRNSS             true  ! use truncated subspace for TDDFT
   TRTYPE            3     ! specify occupied orbitals
   N_SOL             1     ! number core holes, specified in $solute section
   CORE_IONIZE       2     ! hole orbital specified
   INTEGRAL_SYMMETRY false
   POINT_GROUP_SYMMETRY false
$end

$solute
   6
$end

$occupied
   1 2 3 4 5
   2 3 4 5 6
$end

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7.6.1 Overview