Direct Extraction of Excitation Energies from Ensemble Density-Functional Theory

Abstract

A very specific ensemble of ground and excited states is shown to yield an exact formula for any excitation energy as a simple correction to the energy difference between orbitals of the KohnSham ground state. This alternative scheme avoids either the need to calculate many unoccupied levels as in time-dependent density functional theory (TDDFT) or the need for many self-consistent ensemble calculations. The symmetry-eigenstate Hartree-exchange (SEHX) approximation yields results comparable to standard TDDFT for atoms. With this formalism, SEHX yields approximate double-excitations, which are missed by adiabatic TDDFT.The Hohenberg-Kohn (HK) theorem [1-4] of groundstate density-functional theory (DFT) [1,5] has several parts. The most-used in practice is the establishment of an exact density functional, F [n], whose minimum yields the exact ground-state density and energy of a given system. Almost all practical calculations use the KohnSham (KS) scheme [5] to minimize F with an approximation to the small exchange-correlation contribution, E XC [n]. In fact, many properties of interest in a modern chemical or materials calculation can be extracted from knowledge of the ground-state energy as a function of nuclear coordinates, or in response to a perturbing field.However, except under very special circumstances, most optical excitation frequencies cannot be deduced. Hence there has always been interest in extending ground-state DFT to include such excitations. Moreover, another part of the HK theorem guarantees that such frequencies (and all properties) are indeed functionals of the ground-state density. In recent years, linearresponse time-dependent DFT (TDDFT) [6][7][8][9][10] has become a popular route for extracting low-lying excitation energies of molecules, because of its unprecedented balance of accuracy with computational speed [11]. For significantly sized molecules, more CPU time will be expended on a geometry optimization than a single TDDFT calculation on the optimized geometry.However, while formally exact, TDDFT with standard approximations is far from perfect. If the unknown exchange-correlation (XC) kernel of TDDFT is approximated by its zero-frequency (and hence ground-state) limit, no multiple excitations survive [11]. While a useful work-around exists for cases where a double is close to a single excitation [12,13], there is as yet no simple and efficient general procedure for extracting double excitations within adiabatic TDDFT [14].Ensemble DFT (EDFT) [15,16] applies the principles of ground-state DFT to a convex ensemble of the lowest M levels of a system, for which a KS system can be defined [17]. EDFT is formally exact, but practical calculations require approximations, and initial attempts yielded disappointing results [18]. Accuracy is greatly improved when so-called "ghost interactions" between distinct states are removed from the approximations [19]. EDFT remains an active research area because, being variational, it should not suffer from some of the limitations of standard TDDFT. Rece...

Combining extrapolation with ghost interaction correction in range-separated ensemble density functional theory for excited states

“…Note also that LIM applies to any approximate ensemble energies (WIDFA or GIC, with or without range separation 44 ). Let us finally mention that other approaches can be used for extracting excitation energies from an ensemble, in particular by making another choice for the ensemble weights and by using derivatives of the ensemble energy for a direct extraction 30 . In the latter case, weight dependence of the xc functional must be introduced, which is not trivial.…”

“…The exchange-correlation ensemble-density-functional energy depends in principle on these weights. It still remains a challenge to model this weight-dependence which actually plays a crucial role in the calculation of excitation energies 13,30,32,33,38,39 . The recent resurgence of eDFT in the literature is partly arXiv:1708.03478v2 [physics.chem-ph] 16 Oct 2017 due to the fact that, when combined with wavefunction theory by means of range separation for example 31,32 , it leads to a rigorous state-averaged multiconfigurational DFT.…”

Capturing multireference excited states by constrained-density-functional theory

“…Conical intersections, charge transfer states and Rydberg states are among those cases where practical implementations of TDDFT struggle to provide a physical model. More recently, multiconfigurational DFT methods, such as ensemble DFT [1][2][3][4][5], constrained DFT [6][7][8], block-localized DFT [9,10], DFT/MRCI [11], and even flavors of ground state DFT [12] have been proposed as innovative protocols for extracting excitation energies in a computationally efficient way while still making use of density functionals in their formulation.…”

Charge transfer excitations from exact and approximate ensemble Kohn-Sham theory

“…EEXX calculations can yield good results in small atoms, [38][39][40] even for excitations that are very difficult for approximations to time-dependent Kohn-Sham theory. EEXX can be calculated in two ways: it can be obtained as a functional of the exact density, using the exact orbitals, which is the course we pursue in this work to avoid densitydriven errors 64 .…”

“…EDFT can yield energy differences directly, as discussed in detail below. Indeed, excited state EDFT has seen increasing interest of late [33][34][35][36][37][38][39][40] as a potential alternative to TDDFT for excitation energies. This recent resurgence of GOK EDFT mirrors a growing interest in more general forms of EDFT, which can deal, e.g., with degenerate ground states 30,[41][42][43][44] and "open" systems with a noninteger number of electrons.…”