Density Functional Theory for Electronic Excited States

Abstract

This chapter provides a basic introduction to excited-state extensions of density functional theory (DFT), including time-dependent (TD-)DFT in both its linear-response and its explicitly time-dependent formulations. As applied to the Kohn-Sham DFT ground state, linear-response theory affords an eigenvaluetype problem for the excitation energies in a basis of singly-excited Slater determinants, and is widely known simply as "TDDFT" despite its frequency-domain formulation. This form of TDDFT is the mostly widely-used quantum-chemical method for excited states, due to a favorable combination of low cost and reasonable accuracy. The chapter surveys the accuracy of linear-response TDDFT, which is generally more sensitive to the details of the exchange-correlation functional as compared to ground-state DFT, and also describes some known systemic problems exhibited by this approach. Some of those problems can be corrected on a case-by-case basis using orbital-optimized, excited-state self-consistent field (SCF) calculations, in what is known as excited-state Kohn-Sham theory or a "∆SCF" procedure, a class of methods that includes restricted open-shell Kohn-Sham theory. Recent successes of these approaches are highlighted, including double excitations and core-level excitations. Finally, explicitly time-dependent (or "real-time") TDDFT involves propagation of the molecular orbitals in time following an external perturbation, according to the Kohn-Sham analogue of the time-dependent Schrödinger equation. The time-dependent approach has been used to model strong-field electron dynamics, and in the weak-field limit it provides a route to broadband spectra based on the time evolution of the dipole moment function. This is useful for describing high-energy excitations (as in x-ray spectroscopy) and in systems where the density of states is high, as demonstrated by a few examples.

Computational approaches for XANES, VtC-XES, and RIXS using linear-response time-dependent density functional theory based methods

“…However, TDDFT is not a panacea and many open questions remain and new theoretical developments are needed. 184,185 As we have emphasized earlier, care must be taken especially in terms of the exchange-correlation functional and basis set choices, which have to be carefully assessed when performing excited-state computations in general including core-level spectra. Other areas of development include simulations of non-linear X-ray spectroscopic signals, 186 transient spectroscopies 187 including dynamics, real-time domain simulations, and TDDFT coupled to strong fields.…”

Slater transition methods for core-level electron binding energies

“…76 Calculations reported below were performed using a locally-modified version of Q-Chem 5.4, 77 which contains several different algorithms that can be used to optimize a non-aufbau determinant that contains a core hole or fractional core hole. 17 The simplest of these is the maximum overlap method (MOM), 61 and for the present calculations we have found it sufficient to use the "initial MOM" (IMOM) algorithm. 62 This differs from the original MOM procedure only in that the reference orbitals used for computing overlaps are not updated during the SCF iterations, but are taken from an initial closed-shell, integer-occupancy SCF calculation that is used to obtain E initial 0 (N ) in Eq.…”

“…Slater considered this energy to be a continuous function of the MO occupation numbers n i . 17 We follow a slightly different formulation, 50 taking E(q) to be a function of a single continuous variable q, equal to the fraction of an electron that is removed from whichever MO is to be ionized. [The index of this MO, corresponding to i in Eq.…”

“…[6][7][8][9][10][11][12][13][14] There are several approaches to compute CEBEs using selfconsistent field (SCF) methods such as density functional theory (DFT). [15][16][17] The most widely used procedure, at least for molecules, is the "∆SCF" method, 17 in which one explicitly computes a final-state determinant containing a core hole. This approach has been benchmarked and thoroughly studied for both molecules and solids, [18][19][20][21][22][23] yet is not without problems.…”

“…24,[48][49][50][51][52][53][54][55] The latter approaches are the ones considered here. Fractional-occupancy SCF calculations have their basis in Slater's transition method 17,48,49 (STM) and generalizations thereof, 24,50-52 but have received less attention as compared to ∆SCF methods. However, they may hold some advantages over ∆SCF for modeling complex systems or experiments, such as transient spectroscopy at x-ray or extreme ultraviolet wavelengths, where the state that is probed is not the ground state.…”