Linear-response time-dependent density functional theory (LR-TDDFT) is an efficient method for computing optical absorption spectra, often achieving errors of just 0.3 eV for valence excitations. However, size-extensive errors on the order of 10 eV for second-period elements and 60 eV for the third-period blunt the predictive powers of LR-TDDFT for core-excited states. A large contributor to this error is long-range self-interaction (LRSI), characterized by the inexact cancellation of the Coulomb interaction between the promoted electron in the virtual space and itself in the (previously occupied) core orbital. Lack of orbital relaxation also hinders LR-TDDFT, and although orbital-optimized DFT (OO-DFT) methods predict accurate energies by accounting for orbital relaxation, they are state-specific and inefficient for calculating dense excitation manifolds. An intermediate approach that optimizes an N-1-electron (core-ionized) reference, then applies an EA-CIS formalism to compute partially orbital-optimized excitation spectra is the static-exchange approximation (STEX). In order to eliminate the LRSI problem and incorporate orbital optimization into TDDFT, we have developed a generalization of STEX to DFT by means of an excited-state absorption formalism that applies two linear-responses directly to the N-1-electron reference state. We can show that this formalism reduces to the EA-CIS equations in the case of the Hartree-Fock functional and is otherwise free of LRSI error. Casting our response formalism in terms of the density matrix results in EA-TDDFT, an approach that is amenable to commonly used density functionals while reducing errors relative to 132 experimental K-edge transitions from 10 eV (LR-TDDFT) or 1 eV (STEX) to just 0.4 eV (EA-TDDFT) for second-period elements using rCAM-B3LYP under the Tamm-Dancoff approximation. The errors are consistent across the periodic table for a given functional, amplifying error reduction for the third period from 60 eV (LR-TDDFT) or 3.2 eV (STEX) to 0.4 eV (EA-TDDFT). Statistical errors of EA-TDDFT are competitive with those of OO-DFT, but EA-TDDFT offers distinct advantages of capturing the entire spectrum within a single calculation and (in some cases) significant improvements to transition properties. We anticipate that EA-TDDFT will be a powerful tool that promotes the application of TDDFT in a semi-quantitative, predictive capacity in a domain of spectroscopy where LR-TDDFT has remained qualitative at best.

Revitalizing DFT-based response theory for core-excited states with electron affinity TDDFT

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