Energy functional Thomas-Fermi theory

In Kohn-Sham theory, densities are postulated to be sums of orbital densities, for functions (pi in the orbital Hilbert space. This generates a Banach space [102] of density functions. Thomas-Fermi theory can be derived if an energy functional E[p] = I p + F [ p is postulated to exist, defined for all normalized ground-state... 

Methods of density functional theory (DFT) originate from the Xa method originally proposed by Slater [78] on the base of statistical description of atomic electron structure within the Thomas-Fermi theory [79]. From the point of view of Eq. (3), fundamental idea of the DFT based methods consist first of all in approximate treatment of the electron-electron interaction energy which is represented as ... 

This argument shows that the locality hypothesis fails for more than two electrons because the assumed Frechet derivative must be generalized to a Gateaux derivative, equivalent in the context of OEL equations to a linear operator that acts on orbital wave functions. The conclusion is that the use by Kohn and Sham of Schrodinger s operator t is variationally correct, but no equivalent Thomas-Fermi theory exists for more than two electrons. Empirical evidence (atomic shell structure, chemical binding) supports the Kohn-Sham choice of the nonlocal kinetic energy operator, in comparison with Thomas-Fermi theory [288]. A further implication is that if an explicit approximate local density functional Exc is postulated, as in the local-density approximation (LDA) [205], the resulting Kohn-Sham theory is variation-ally correct. Typically, for Exc = f exc(p)p d3r, the density functional derivative is a Frechet derivative, the local potential function vxc = exc + p dexc/dp. 

Thomas-Fermi theory — This is an early version of density functional theory, which accounts for the kinetic energy of the electrons and for Coulomb interactions, but neglects exchange and correlations. It... 

This began with a landmark paper by Slater in which he proposed that the effects of exchange in the wave function could be replaced by an exchange potential, proportional to p where p is the electron density function. This followed earlier work by Dirac, in which he showed how to add the exchange energy to the Thomas-Fermi theory of the atom. The exchange potential is also dependent on a factor, cv, which is allowed to vary somewhat from its value in a uniform electron gas. The method is called the Xa method and involves solving a series of one-electron wave equations in a self-consistent manner. 

Lee, H., Lee, C.,and Parr, R. G. (1991) Conjoint gradient correction to the Hartree-Fock kinetic-and exchange-energy density functionals. Phys. Rev., A44, 768-771. Yang, W. (1986) Gradient correcttion in Thomas-Fermi theory. Phys. Rev., A34, 4575-4585. 

For liquid metals, one has to set up density functionals for the electrons and for the particles making up the positive background (ion cores). Since the electrons are to be treated quantum mechanically, their density functional will not be the same as that used for the ions. The simplest quantum statistical theories of electrons, such as the Thomas-Fermi and Thomas-Fermi-Dirac theories, write the electronic energy as the integral of an energy density e(n), a function of the local density n. Then, the actual density is found by minimizing e(n) + vn, where v is the potential energy. Such... 

The origins of density functional theory (DFT) are to be found in the statistical theory of atoms proposed independently by Thomas in 1926 [1] and Fermi in 1928 [2]. The inclusion of exchange in this theory was proposed by Dirac in 1930 [3]. In his paper, Dirac introduced the idempotent first-order density matrix which now carries his name and is the result of a total wave function which is approximated by a single Slater determinant. The total energy underlying the Thomas-Fermi-Dirac (TFD) theory can be written (see, e.g. March [4], [5]) as... 

Note that the various T and V terms defined in Eqs. (8.3)—(8.5) are functions of the density, while the density itself is a function of three-dimensional spatial coordinates. A function whose argument is also a function is called a functional , and thus the T and V terms are density functionals . The Thomas-Fermi equations, together with an assumed variational principle, represented the first effort to define a density functional theory (DFT) the energy is computed with no reference to a wave function. However, while these equations are of significant historical interest, the underlying assumptions are sufficiently inaccurate that they find no use in modem chemistry (in Thomas-Fermi DFT, all molecules are unstable relative to dissociation into their constituent atoms...)... 

Models for the electronic structure of polynuclear systems were also developed. Except for metals, where a free electron model of the valence electrons was used, all methods were based on a description of the electronic structure in terms of atomic orbitals. Direct numerical solutions of the Hartree-Fock equations were not feasible and the Thomas-Fermi density model gave ridiculous results. Instead, two different models were introduced. The valence bond formulation (5) followed closely the concepts of chemical bonds between atoms which predated quantum theory (and even the discovery of the electron). In this formulation certain reasonable "configurations" were constructed by drawing bonds between unpaired electrons on different atoms. A mathematical function formed from a sum of products of atomic orbitals was used to represent each configuration. The energy and electronic structure was then... 

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