Variational principle of classical density functional theory via Levy’s constrained search method

Dwandaru, Wipsar Sunu Brams; Schmidt, Matthias

doi:10.1103/physreve.83.061133

Cited by 18 publications

(31 citation statements)

References 27 publications

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“…Dwandaru and Schmidt 37 have recently considered this double maximization process when relating the DFT and MT levels of description. In the entropy language, their Eqs.…”

Section: The Bridge Theoremmentioning

confidence: 99%

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Functional thermo-dynamics: A generalization of dynamic density functional theory to non-isothermal situations

Anero

Español

Tarazona

2013

The Journal of Chemical Physics

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We present a generalization of Density Functional Theory (DFT) to non-equilibrium non-isothermal situations. By using the original approach set forth by Gibbs in his consideration of Macroscopic Thermodynamics (MT), we consider a Functional Thermo-Dynamics (FTD) description based on the density field and the energy density field. A crucial ingredient of the theory is an entropy functional, which is a concave functional. Therefore, there is a one to one connection between the density and energy fields with the conjugate thermodynamic fields. The connection between the three levels of description (MT, DFT, FTD) is clarified through a bridge theorem that relates the entropy of different levels of description and that constitutes a generalization of Mermin's theorem to arbitrary levels of description whose relevant variables are connected linearly. Although the FTD level of description does not provide any new information about averages and correlations at equilibrium, it is a crucial ingredient for the dynamics in non-equilibrium states. We obtain with the technique of projection operators the set of dynamic equations that describe the evolution of the density and energy density fields from an initial non-equilibrium state towards equilibrium. These equations generalize time dependent density functional theory to non-isothermal situations. We also present an explicit model for the entropy functional for hard spheres. © 2013 AIP Publishing LLC.

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“…Dwandaru and Schmidt 37 have recently considered this double maximization process when relating the DFT and MT levels of description. In the entropy language, their Eqs.…”

Section: The Bridge Theoremmentioning

confidence: 99%

“…Microscopically these variables are defined in Eq. (37). The averages of these phase functions with respect to the solution ρ t (z) of the Liouville equation are denoted by…”

Section: Dynamic Transport Equations For Ftdmentioning

confidence: 99%

Functional thermo-dynamics: A generalization of dynamic density functional theory to non-isothermal situations

Anero

Español

Tarazona

2013

The Journal of Chemical Physics

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“…If it is considered by many primitive unit cells, the potential is shown on the Figure 3. Next, NaCl crystal potential (8) is substituted to the umbrella function (5). Indeed, we also have to remember that the potential has to be taken lower than or equal to 4/15 so that the minimizer has the real value.…”

Section: Estimation Of the Minimizer Of The Nacl Crystal Latticementioning

confidence: 99%

“…They found that systematic variations of valence electron density are depending on the atomic number of anion and cation. The existence of a minimizer is an interesting theme to be investigated [5]. The existence of the minimizer for TFW model was proved by Benguria, Brezis and Lieb in 1981 [6].…”

Section: Introductionmentioning

confidence: 99%

Estimation of the Minimizer of the Thomas-Fermi-Dirac-von Weizsäcker Functional of NaCl Crystal Lattice

Wahyuni

Dwandaru

Rosyid

2016

J. Phys.: Conf. Ser.

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Abstract. Estimation of the minimizer of the Thomas-Fermi-Dirac-von Weizsäcker functional of NaCl crystal lattice is calculated by making use of a direct method we have developed recently. The method is referred to as direct method for the reason that in the course of the calculation of the estimation, we do not derive the Euler-Lagrange equation at all. By using the graph of the umbrella functions, the estimation of the minimizer function is represented. The minimizer is bounded from above by the umbrella functions. IntroductionThe Thomas-Fermi (TF) theory was proposed by Thomas and Fermi independently in 1927. Dirac introduced the exchange correction to the theory in 1930, so that the corrected model is called the Thomas-Fermi-Dirac (TFD). Then the gradient correction of the kinetic energy was added by von Weizsäcker in 1935, so that it is called the Thomas-Fermi-von Weizsäcker (TFW) model. Those theories were placed on a firm mathematical footing by Lieb [1]. Although the Thomas-Fermi-Dirac-von Weizsäcker (TFDW) theory has not been studied as extensively as other theories, some authors studied it, such as Engel and Dreizler [2], who presented an accurate numerical solution of the variational equation resulting from the standard TFDW energy functional plus fourth-order gradient terms of the kinetic energy.Chan et.al.[3] perform a numerical study of the TFDW theory in finite systems to gain an understanding of the variational behavior of kinetic energy functionals. Their findings indicate that the TFDW theory can give an approximate description of matter, with atomic energies, binding energies, and bond lengths of the correct order of magnitude, though not to the accuracy required of a qualitative chemical theory.Zhuravlev et.al.[4] calculate valence electron densities in series of crystal substances MA (M = Li, Na, K, Rb, Ag, Mg, Ca; A = F, Cl, Br, O, S) with a NaCl lattice using the ab initio local DFT method combined with the ab initio pseudopotential technique. They found that systematic variations of valence electron density are depending on the atomic number of anion and cation.

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“…The method was subsequently applied to a system of one-dimensional rods [12]. The formal extension of the DFT formalism to the canonical ensemble can be based on the Mermin-Evans argument [13] or on Levy's constrained search method [14].…”

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confidence: 99%

Full Canonical Information from Grand-Potential Density-Functional Theory

Heras

Schmidt

2014

Phys. Rev. Lett.

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We present a general and formally exact method to obtain the canonical one-body density distribution and the canonical free energy from direct decomposition of classical density functional results in the grand ensemble. We test the method for confined one-dimensional hard-core particles for which the exact grand potential density functional is explicitly known. The results agree to within high accuracy with those from exact methods and our Monte Carlo many-body simulations. The method is relevant for treating finite systems and for dynamical density functional theory.

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Variational principle of classical density functional theory via Levy’s constrained search method

Cited by 18 publications

References 27 publications

Functional thermo-dynamics: A generalization of dynamic density functional theory to non-isothermal situations

Functional thermo-dynamics: A generalization of dynamic density functional theory to non-isothermal situations

Estimation of the Minimizer of the Thomas-Fermi-Dirac-von Weizsäcker Functional of NaCl Crystal Lattice

Full Canonical Information from Grand-Potential Density-Functional Theory

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