6.4 Maximum Overlap Method (MOM) for SCF Excited States

The Maximum Overlap Method (MOM) [244] is a useful alternative to CIS and TDDFT for obtaining low-cost excited states. 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, excited SCF determinants are able to be obtained. The MOM has several advantages over existing low-cost excited state methods. Current implementations of TDDFT usually struggle to accurately model charge-transfer and Rydberg transitions, both of which can be well-modeled using the MOM. The MOM also allows the user to target very high energy states, such as those involving excitation of core electrons [435], which are hard to capture using other excited state methods.

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.

Example 6.117 Calculation of the lowest singlet state of CO.

$comment
   CO spin-purified calculation
$end

$molecule
   0 1
   C
   O  C  1.05
$end

$rem
   METHOD            B3LYP
   BASIS             6-31G*
$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

The following $rem is used to invoke the MOM:

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 using SCF_GUESS = true and MOM_START = 1.

Example 6.118 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 6.3.2.

Example 6.119 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
   SYMMETRY     false
   SYM_IGNORE   true
$end

@@@

$molecule
read
$end

$rem
   METHOD         B3LYP
   BASIS          aug-cc-pvdz
   SCF_GUESS      read
   MOM_START      1
   UNRESTRICTED   true
   SYMMETRY       false
   SYM_IGNORE     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
$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

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 6.120 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
   SYMMETRY     false
   SYM_IGNORE   true
$end

@@@

$molecule
   read
$end

$rem
   METHOD            B3LYP
   BASIS             aug-cc-pvdz
   SCF_GUESS         read
   MOM_START         1
   UNRESTRICTED      true
   SYMMETRY          false
   SYMMETRY_IGNORE   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
$end

$solute
   6
$end

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

1	all valence orbitals are listed in $solute section
2	only hole(s) are specified all other occupations same as ground state