Ensemble density variational methods with self- and ghost-interaction-corrected functionals

Pastorczak, Ewa; Pernal, Katarzyna

doi:10.1063/1.4866998

Cited by 44 publications

(47 citation statements)

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“…Note that the u-dependence of e 0 and e 1 has been dropped in Eq. (27) for clarity. In order to expand the ensemble Legendre-Fenchel transform F w (n) around the symmetric n = 1 case, which is equivalent to expanding f w (δ) around δ = 0,…”

Section: Expansion Around the Symmetric Casementioning

confidence: 99%

Exploring weight-dependent density-functional approximations for ensembles in the Hubbard dimer

et al. 2018

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Gross-Oliveira-Kohn density-functional theory (GOK-DFT) is an extension of DFT to excited states where the basic variable is the ensemble density, i.e. the weighted sum of ground-and excitedstate densities. The ensemble energy (i.e. the weighted sum of ground-and excited-state energies) can be obtained variationally as a functional of the ensemble density. Like in DFT, the key ingredient to model in GOK-DFT is the exchange-correlation functional. Developing density-functional approximations (DFAs) for ensembles is a complicated task as both density and weight dependencies should in principle be reproduced. In a recent paper [Phys. Rev. B 95, 035120 (2017)], the authors applied exact GOK-DFT to the simple but nontrivial Hubbard dimer in order to investigate (numerically) the importance of weight dependence in the calculation of excitation energies. In this work, we derive analytical DFAs for various density and correlation regimes by means of a Legendre-Fenchel transform formalism. Both functional and density driven errors are evaluated for each DFA. Interestingly, when the ensemble exact-exchange-only functional is used, these errors can be large, in particular if the dimer is symmetric, but they cancel each other so that the excitation energies obtained by linear interpolation are always accurate, even in the strongly correlated regime.PACS. PACS-key discribing text of that key -PACS-key discribing text of that key

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Section: Expansion Around the Symmetric Casementioning

confidence: 99%

Exploring weight-dependent density-functional approximations for ensembles in the Hubbard dimer

et al. 2018

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“…EDFT remains an active research area because, being variational, it should not suffer from some of the limitations of standard TDDFT. Recent strides by Pernal and coworkers [20,21], Fromager and coworkers [22,23], and others attempt to create a useful practical alternative to TDDFT, but the difficulty remains in finding accurate low-cost approximations. EDFT usually requires running several different self-consistent ensemble calculations to extract several low-lying excitations.…”

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confidence: 99%

Direct Extraction of Excitation Energies from Ensemble Density-Functional Theory

et al. 2017

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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...

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“…On the other hand, also the elimination of the “ghost interaction” error has been shown to be best achieved by employing orbital‐dependent functionals of the form

E_{ens}^{μ ‐ eDFT} = \sum_{I = 1}^{m} \sum_{i}^{M} ω_{I} n_{i}^{I} h_{i i} + \frac{1}{2} \sum_{I = 1}^{m} \sum_{i, j}^{M} ω_{I} n_{i}^{I} n_{j}^{I} {〈 i j | | i j 〉}^{LR} + \sum_{I = 1}^{m} ω_{I} E_{Hxc}^{SR} true[ρ_{I} true]

and taking into account the spin polarization:

E_{ens}^{μ ‐ speDFT} = \sum_{I = 1}^{m} \sum_{i}^{M} ω_{I} n_{i}^{I} h_{i i} + \frac{1}{2} \sum_{I = 1}^{m} \sum_{i, j}^{M} ω_{I} n_{i}^{I} n_{j}^{I} {〈 i j | | i j 〉}^{LR} + \sum_{I = 1}^{m} ω_{I} E_{Hxc}^{SR} true[ρ_{I}^{α}, ρ_{I}^{β} true] .

…”

Section: Theorymentioning

confidence: 99%

“…On the other hand, also the elimination of the "ghost interaction" error [19,30,31] has been shown to be best achieved by employing orbital-dependent functionals of the form…”

Section: Theorymentioning

confidence: 99%

A road to a multiconfigurational ensemble density functional theory without ghost interactions

Pastorczak

Pernal

2016

Int J of Quantum Chemistry

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Ensemble density functional theory (DFT) is a theory potentially able to describe electronic states inaccessible to traditional time‐dependent DFT approaches, e.g. Rydberg and double excitations. When combined with ensemble wavefunction approaches through a range‐separation scheme of Stoll and Savin (Density Functional Methods in Physics, 1985, 177‐207), a resulting multiconfiguration ensemble DFT is able to address also such challenging phenomena as bond breaking in the electronically excited molecules. Ensemble DFT is, however, crippled by the so‐called “ghost interaction” error, analogous to the self‐interaction error in the ground‐state DFT. We are exploring ways to alleviate this effect. We also study the importance of spin polarization in the density functional, the self‐consistency effects and the impact of tunable parameters on the quality of shapes of potential energy surfaces. © 2016 Wiley Periodicals, Inc.

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Ensemble density variational methods with self- and ghost-interaction-corrected functionals

Cited by 44 publications

References 43 publications

Exploring weight-dependent density-functional approximations for ensembles in the Hubbard dimer

Exploring weight-dependent density-functional approximations for ensembles in the Hubbard dimer

Direct Extraction of Excitation Energies from Ensemble Density-Functional Theory

A road to a multiconfigurational ensemble density functional theory without ghost interactions

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