7.6.1 Overview

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 “$\mathrm{\Delta }$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 $\mathrm{\Delta }$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 $\mathrm{\Delta }$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 $\pi \to {\pi }^{\ast }$ 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 $\mathrm{\Delta }$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 $\mathrm{\Delta }$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:

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.

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.