Improved Lieb–Oxford bound on the indirect and exchange energies

Lewin, Mathieu; Lieb, Élliott H.; Seiringer, Robert

doi:10.1007/s11005-022-01584-5

Cited by 13 publications

(6 citation statements)

References 57 publications

(46 reference statements)

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“…In summary, we have reported an ab initio attempt to construct universal GGA and MGGA exchange enhancement factors based on a family of jellium-slab exact-exchange selfconsistent calculations. We found that both F x (s) and F x (s, α) lie within very well defined bands that stay, for all s and α, below the local form of the Lieb-Oxford bound B = 1.3423 reported very recently for a situation where the wave function can be written as a single Slater determinant [32]. In the case of the GGA, our first-principles calculations allow us to devise a parametrization of F x (s) that yields very accurate exchange energies per particle.…”

Section: Discussionsupporting

confidence: 47%

“…with B = 1.804. Recent work has found a tighter lower bond, B = 1.3423, when the wave function can be written as a single Slater determinant [32], while in the case of a spinunpolarized two-electron ground state B = 1.174 has been found [21]. Similar lower bounds have been derived for the xc energy of a many-electron system in reduced dimensions [33].…”

Section: Towards a Universal First-principles Exchange Enhancement Fa...mentioning

confidence: 53%

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Towards a universal exchange enhancement factor in density functional theory

2023

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Section: Discussionsupporting

confidence: 47%

Section: Towards a Universal First-principles Exchange Enhancement Fa...mentioning

confidence: 53%

Towards a universal exchange enhancement factor in density functional theory

2023

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“…For example, the SCE limit is directly connected to the Lieb–Oxford (LO) inequality, 92,93 a key exact property used in the construction of XC approximations 8,94 . The LO inequality limits the value of the XC energy by bounding from below the AC integrand of Equation ():

W_{λ} [ρ] \geq - C_{LO} \int ρ^{4 / 3} (boldr) d r,

where the optimal

C_{LO}

is rigorously known to be between 1.4442 and 1.5765 66,95,96 . More generally,

- C_{LO} \int ρ^{4 / 3} (boldr) d r

bounds from below the indirect energy (electron–electron repulsion minus the Hartree energy) of any correctly normalized and antisymmetric Ψ[ ρ ].…”

Section: From Sce To Practical Methodsmentioning

confidence: 99%

“…where the optimal C LO is rigorously known to be between 1.4442 and 1.5765. 66,95,96 More generally, ÀC LO R ρ 4=3 r ð Þdr bounds from below the indirect energy (electron-electron repulsion minus the Hartree energy) of any correctly normalized and antisymmetric Ψ[ρ]. Letting Ψ[ρ] be Ψ λ [ρ], we obtain Equation (29).…”

Section: Other Applications Of Sce: Lower Bounds To XC Energies and C...mentioning

confidence: 99%

Density functionals based on the mathematical structure of the strong‐interaction limit of DFT

Vuckovic

Gerolin

Daas

et al. 2022

WIREs Comput Mol Sci

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While in principle exact, Kohn-Sham density functional theory-the workhorse of computational chemistry-must rely on approximations for the exchange-correlation functional. Despite staggering successes, present-day approximations still struggle when the effects of electron-electron correlation play a prominent role. The limit in which the electronic Coulomb repulsion completely dominates the exchange-correlation functional offers a well-defined mathematical framework that provides insight for new approximations able to deal with strong correlation. In particular, the mathematical structure of this limit, which is now well-established thanks to its reformulation as an optimal transport problem, points to the use of very different ingredients (or features) with respect to the traditional ones used in present approximations. We focus on strategies to use these new ingredients to build approximations for computational chemistry and highlight future promising directions.

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“…We then turn to another important exact constraint for the XC functional: the Lieb-Oxford (LO) inequality, 30,[297][298][299] which has been turned into a useful tool for constraining approximations, again, by John Perdew. 82,299 In previous works, 179,279,300 the SIL functional has been used to establish lower bounds for the optimal constant appearing in the LO inequality at given electrons number N .…”

Section: Introductionmentioning

confidence: 99%

The Strong Interaction Limit of the Møller-Plesset Adiabatic Connection

Daas

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All the information needed to describe chemical reactions can be obtained by solving the time dependent Schrödinger equation. However, this is only viable for the smallest of systems and for any chemically relevant system approximations are needed. The most used method for quantum chemical simulations is Density Functional Theory (DFT), which is formally exact and uses instead of the multidimensional wavefunction the three-dimensional electron density. However, for DFT calculations to be feasible and computationally inexpensive, approximations need to be made to the exchange correlation energy, which is the only part that cannot be calculated exactly. A systematic way to make approximations to the exchange correlation energy is the adiabatic connection, which links the non-interacting Kohn-Sham system, which can be solved exactly, to the physical system. Most DFT functionals can now be understood as linear extrapolations from the Kohn-Sham system to the physical system. However, getting accurate results from extrapolating information is generally a hard problem, which even the most sophisticated machine learning algorithms struggle with, compared to interpolating them. Therefore, the mathematical structure of the strong interaction limit of DFT was derived, which is an exactly solvable system that can be used as the endpoint for future interpolations, which in turn lead to the creation of a new class of DFT functionals. Interestingly, we can also link a different non-interacting system, the Hartree-Fock system, to the physical system, which lead to the creation of the Møller-Plesset Adiabatic Connection (MPAC), which has as small coupling limit the Møller-Plesset Perturbation Theory (MPPT) series. This adiabatic connection allows us to obtain the correlation energy via interpolations functionals, although the strong coupling limit needs to be derived first. In this work, the strong coupling limit of the MPAC was studied for the hydrogen atom and the results of the first three leading order terms were generalized to many-electron systems with different spins. After that accurate gradient expansion approximations were derived for the first two leading order terms to make these functionals more computationally viable. For this a compact representation, based on the large Z limit, is introduced to find the best coefficient for the gradient expansion. This inspired a subsequent investigation into the DFT adiabatic connection, which showed a surprising symmetry between the magnitude of the gradient expansion coefficient of the weak interaction limit (exchange energy) and strong interaction limit. Lastly a new class of interpolation functions was introduced for the MPAC to tackle non-covalent interactions (NCIs). These were shown to be competitive with dispersion corrected hybrid and double hybrid functionals at the computational cost of double hybrids. However, cost saving techniques as well as regularization were used to not only make the calculations computationally cheaper but also to further improve the accuracy of the method. This resulted in new functionals that can tackle a variety of NCIs and outperform dispersion corrected (double) hybrid functionals.

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Improved Lieb–Oxford bound on the indirect and exchange energies

Cited by 13 publications

References 57 publications

Towards a universal exchange enhancement factor in density functional theory

Towards a universal exchange enhancement factor in density functional theory

Density functionals based on the mathematical structure of the strong‐interaction limit of DFT

The Strong Interaction Limit of the Møller-Plesset Adiabatic Connection

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