7.8 Electron-Affinity Time-Dependent Density Functional Theory (EA-TDDFT)

X-ray absorption spectra (XAS) pose a significant challenge to time-dependent density functional theory (TDDFT). The errors in peak positions for the K-edge XAS of small gas-phase molecules are routinely in excess of 10 eV, and in condensed phase the errors in peak positions and intensities cause severe distortions in K-edge spectra. These errors can be traced to two sources: (1) A lack of orbital relaxation effects that is poorly compensated by linear-response theory with only single excitations atop ground state reference orbitals, and (2) a form of excited-state self-interaction error in the TDDFT potential called particle-hole self-interaction error. Both can be rectified by choosing a more suitable set of reference orbitals for the core-excitation problem (e.g., a core-ionized reference determinant) as done with the static-exchange approximation in Section 7.7.2. The generalization of STEX to include density functional theory correlation is accomplished through two separate linear responses. First, from an initial set of core-ionized molecular orbitals, the missing core electron is reattached (response #1), then a second linear response is done to excite the core electron. The second linear response occurs atop a non-stationary initial density, which is problematic due to the adiabatic approximation. Fortunately, an exact first-order correction to errors associated with the adiabatic approximation can be derived and is applied automatically in all EA-TDDFT calculations. This correction happens to take a form reminiscent of excited-state self-interaction error, so the EA-TDDFT equations are naturally excited-state self-interaction free. Because of this, EA-TDDFT can achieve statistical errors of just 0.5 eV root-mean-squared deviation with standard density functionals. ¹⁶⁹ Carter-Fenk K. et al.
J. Phys. Chem. Lett.
(2022), 13, pp. 9664. Link Inclusion of orbital relaxation effects at the level of the reference orbitals in EA-TDDFT also offers critical improvements to the spectra of solvated molecules relative to TDDFT. ¹⁷⁰ Carter-Fenk K., Head-Gordon M.
Phys. Chem. Chem. Phys.
(2022), 24, pp. 26170. Link EA-TDDFT is only defined for restricted open-shell orbitals and produces spin-pure spectra.

The full EA-TDDFT equations take a familiar form,

where $E^{+}$ is the self-consistently optimized energy of the core-ionized determinant, $F_{ab}^{+}$ are elements of the virtual-virtual block of the core-ion Fock matrix, and all integrals are computed using the core-ion orbitals. While the full EA-TDDFT equations can be solved at low cost (by requesting EA_RPA in the $nocis input section), it is seldom necessary to solve the whole RPA-like expression for K-edge XAS. Instead, for K-edge transitions the Tamm-Dancoff approximation can be used by setting EA_TDA in $nocis without sacrificing any accuracy. ¹⁷⁰ Carter-Fenk K., Head-Gordon M.
Phys. Chem. Chem. Phys.
(2022), 24, pp. 26170. Link Note only one occupied orbital index is considered in the EA-TDDFT equations, meaning that the core-valence separation approximation is invoked implicitly. Additionally, because EA-TDDFT is based on linear-response theory, there is no construction of non-orthogonal matrix elements between the ground state and the singly-excited determinants as is done in STEX or 1C-NOCIS. This means that EA-CIS (EA-TDDFT with the Hartree-Fock functional) will produce slightly different results than STEX, but the differences generally do not exceed 0.1 eV. ¹⁶⁹ Carter-Fenk K. et al.
J. Phys. Chem. Lett.
(2022), 13, pp. 9664. Link

The presence of $E^{+}$ in the above equation means that core-ionized reference orbitals must be optimized for each core orbital from which excitations will be generated. In the NOCIS and STEX implementations, this has been done exclusively with the Maximum Overlap Method (MOM) from section 7.6. However, Q-Chem now has several excellent algorithms for converging non-Aufbau solutions to the SCF problem such as State-Targeted Energy Projection (STEP) ¹⁷¹ Carter-Fenk K., Herbert J. M.
J. Chem. Theory Comput.
(2020), 16, pp. 5067. Link and Square Gradient Minimization (SGM); ⁴⁵¹ Hait D., Head-Gordon M.
J. Chem. Theory Comput.
(2020), 16, pp. 1699. Link see Sections 7.9.5 and 7.9.4, respectively. All of these approaches are now available in the EA-TDDFT code by setting either DSCF_ALGORITHM or REF_SCF_ALGORITHM within the $nocis input section. A hybrid algorithm consisting of running STEP in the initial cycles of an optimization to guide the solution towards a set of well-conditioned guess orbitals for a subsequent MOM optimization has been used with great success when other algorithms fail to converge. This can be invoked by setting DSCF_ALGORITHM = STEP_MOM and setting the point at which STEP turns off and MOM turns on with the keyword STEP_MOM_START.

There are two points at which localization can be used to improve the results or make orbital selection more convenient within EA-TDDFT. First, when two or more reference orbitals are selected a Boys localization is automatically performed on the core orbitals of interest to improve convergence of the core-ion SCF to a proper minimum. This localization routine may be controlled manually (e.g., localizing several orbitals but only computing excitations out of one of them, or turning off localization altogether) with the keyword LOCALIZE_ORBITALS. In large systems with many core orbitals of similar energy (e.g., water clusters where the response out of a particular O(1s) orbital is desired), Subsystem Projected AO Decomposition (SPADE) orbitals ²²³ Claudino D., Mayhall N. J.
J. Chem. Theory Comput.
(2019), 15, pp. 1053. Link can be used to localize orbitals onto a particular subset of atoms before ORB_OFFSET is considered. For instance, in the water cluster example given above, if the first water molecule in the $molecule section is of interest then one would set “SUBSYSTEM_ATOMS 1 2 3” within the $nocis section and the variable ORB_OFFSET would be set to 0 for the O(1s) orbital on this subset of atoms.