Kantorovich dual solution for strictly correlated electrons in atoms and molecules

Mendl, Christian B.; Lin, Lin

doi:10.1103/physrevb.87.125106

Cited by 43 publications

(46 citation statements)

References 26 publications

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“…(39) (38) and Eq. (39). As r → ∞, one step function inside g NLR XC (r,r ) will vanish-the term with R(r )-since R(r ) is rather small near the molecular center.…”

Section: A Nonlocal Model Of the XC Hole For Strong Correlationmentioning

confidence: 97%

“…There are still challenges to meet before KS-SCE is ready for practical applications, however. In the three dimensional (3D) case, algorithms to compute the SCE functional are currently applicable only to spherically symmetric systems, although progress is being made by several groups exploiting the formal similarity between the SCE problem and optimal transport (or mass transportation) theory [36][37][38][39][40][41], a field of mathematics and economics [42]. Another crucial point is that SCE requires suitable corrections [34,43] (e.g., local or semilocal) to make it useful for chemistry and solid-state physics, where both the strong and the weak correlation regimes need to be treated accurately by the same methodology.…”

Section: Introductionmentioning

confidence: 99%

“…There is no shortcut; the co-motion functions come out of W SCE [n] being minimized subject to constraints (29) and (30). There is an alternative approach to evaluate W SCE [n], the Kantorovich dual formulation [36,39], which bypasses the co-motion functions, and proves feasible for nonspherical systems.…”

Section: Introductionmentioning

confidence: 99%

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Electron avoidance: A nonlocal radius for strong correlation

Wagner

Gori‐Giorgi

2014

Phys. Rev. A

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We present here a model of the exchange-correlation hole for strongly correlated systems using a simple nonlocal generalization of the Wigner-Seitz radius. The model behaves similarly to the strictly correlated electron approach, which gives the infinitely correlated limit of density functional theory. Unlike the strictly correlated method, however, the energies and potentials of this model can be presently calculated for arbitrary geometries in three dimensions. We discuss how to evaluate the energies and potentials of the nonlocal model, and provide results for many systems where it is also possible to compare to the strictly correlated electron treatment.

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“…(39) (38) and Eq. (39). As r → ∞, one step function inside g NLR XC (r,r ) will vanish-the term with R(r )-since R(r ) is rather small near the molecular center.…”

Section: A Nonlocal Model Of the XC Hole For Strong Correlationmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Electron avoidance: A nonlocal radius for strong correlation

Wagner

Gori‐Giorgi

2014

Phys. Rev. A

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“…The dual variable w can be seen as an approximation to the Kantorovich potential v in (28). As pointed out in the literatures of DFT [11,12,5], the Kantorovich potential allows the functional derivative of V SCE ee (·) to be taken. From (32), we make the following identification:…”

Section: 2mentioning

confidence: 99%

Convex Relaxation Approaches for Strictly Correlated Density Functional Theory

Khoo¹,

Ying²

2019

SIAM J. Sci. Comput.

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In this paper, we introduce methods from convex optimization to solve the multimarginal transport type problems arise in the context of density functional theory. Convex relaxations are used to provide outer approximation to the set of N -representable 2-marginals and 3-marginals, which in turn provide lower bounds to the energy. We further propose rounding schemes to obtain upper bound to the energy. Numerical experiments demonstrate a gap of the order of 10 −3 to 10 −2 between the upper and lower bounds. The Kantorovich potential of the multi-marginal transport problem is also approximated with a similar accuracy.2010 Mathematics Subject Classification. 49M20, 90C22, 90C25.

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“…Even more interesting is the case of general dipolar anisotropic interactions: the SCP functional combined with the KS kinetic energy (fermionic or bosonic) should be able to capture many of the interesting phenomena observed in the strongcorrelation regime [29,53]. In this case, the SCP solution can be constructed from the dual Kantorovich formulation [41], for which few results have started to appear recently [54,55]. Applications to low-dimensional dipolar ultracold gasesWe consider N ultracold bosonic or fermionic particles with dipole moment d in quasi-one-(Q1D) and quasitwo-dimensional (Q2D) geometries.…”

mentioning

confidence: 99%

Density-Functional Theory for Strongly Correlated Bosonic and Fermionic Ultracold Dipolar and Ionic Gases

et al. 2015

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We introduce a density functional formalism to study the ground-state properties of stronglycorrelated dipolar and ionic ultracold bosonic and fermionic gases, based on the self-consistent combination of the weak and the strong coupling limits. Contrary to conventional density functional approaches, our formalism does not require a previous calculation of the interacting homogeneous gas, and it is thus very suitable to treat systems with tunable long-range interactions. Due to its asymptotic exactness in the regime of strong correlation, the formalism works for systems in which standard mean-field theories fail.Introduction -In contrast with its widespread use and success in areas as diverse as quantum chemistry [1], materials science [2] or semiconductor nanostructures [3], Density Functional Theory (DFT) has received relatively little attention in the very active field of ultracold atomic gases. It is well known that the Hohenberg-Kohn theorems, originally formulated in terms of the electron gas [4,5], hold for both fermionic and bosonic systems, as well as for interactions different than the Coulomb one. However, the lack of adequate density functionals has hindered the role of DFT in the study of ultracold atomic gases in favour of other well-established approaches, such as the widely used Gross-Pitaevskii (GP) method in the case of Bose gases. The latter is a mean-field approach and does not allow treating the effect of correlations, which play a crucial role in many different phenomena occurring in ultracold quantum gases [6]. One then often turns to configuration-interaction (CI), quantum Monte Carlo (QMC) or Green's-function methods (for recent reviews, see, e.g., Refs. 6-8).

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Kantorovich dual solution for strictly correlated electrons in atoms and molecules

Cited by 43 publications

References 26 publications

Electron avoidance: A nonlocal radius for strong correlation

Electron avoidance: A nonlocal radius for strong correlation

Convex Relaxation Approaches for Strictly Correlated Density Functional Theory

Density-Functional Theory for Strongly Correlated Bosonic and Fermionic Ultracold Dipolar and Ionic Gases

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