Thomas-Fermi energy functional

Unfortunately, the Thomas-Fermi energy functional does not produce results that are of sufficiently high accuracy to be of great use in chemistry. What is missing in this... 

Furthermore, the Thomas-Fermi energy density functional cannot be inserted in the density functional philosophy presented by the mappings (13) and (14) for all p(r) e T>n since the ground-state energies of many Thomas-Fermi atoms and ions" lie below the exact ones. ... 

In 1930, Dirac [9] proposed that a density functional for exchange be added to the Thomas-Fermi energy expression (Eq. 1). Dirac s exchange functional... 

In this equation, n is the conduction electron density, Ep the Fermi energy and kp the radius of the Fermi sphere. e(Q) given by (4.159) is also known as the Lindhard dielectric constant [4.69]. For Q 0, the quantity in the square brackets is equal to 1 and e(Q) then reduces to the Thomas-Fermi dielectric function [4.69],... 

Thomas-Fermi total energy Eg.j.p [p] gives the so-called Thomas-Fermi-Dirac (TFD) energy functional. 

The foundation for the use of DFT methods in computational chemistry was the introduction of orbitals by Kohn and Sham. 5 The main problem in Thomas-Fermi models is that the kinetic energy is represented poorly. The basic idea in the Kohn and Sham (KS) formalism is splitting the kinetic energy functional into two parts, one of which can be calculated exactly, and a small correction term. 

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 universal function x(x) obtained by numerical integration and valid for all neutral atoms decreases monotonically. The electron density is similar for all atoms, except for a different length scale, which is determined by the quantity b and proportional to Z. The density is poorly determined at both small and large values of r. However, since most electrons in complex atoms are at intermediate distances from the nucleus the Thomas-Fermi model is useful for calculating quantities that depend on the average electron density, such as the total energy. The Thomas-Fermi model therefore cannot account for the periodic properties of atoms, but provides a good estimate of initial fields used in more elaborate calculations like those to be discussed in the next section. 

The Self-Consistent-Field (SCF) procedure can be initiated with hydrogenic wave functions and Thomas-Fermi potentials. It leads to a set of solutions w(fj), each with k nodes between 0 and oo, with zero nodes for the lowest energy and increasing by one for each higher energy level. The quantum number n can now be defined asn = / + l + A to give rise to Is, 2s, 2p, etc. orbitals. 

The hardness kernels in Equation 24.110 depend on the kinetic energy functional as well as on the electron-electron interactions. Thomas-Fermi models can be used to evaluate the kinetic part of these hardness kernels and can be combined with a band structure calculation of the linear response X -... 

After reviewing the different approaches in the literature for the formulation of a relativistic Thomas-Fermi procedure for the study of complex electron systems, we will make contact between quantum mechanics with first-order relativistic corrections and the weak relativistic limit of quantum electrodynamics for finding explicit energy functionals that will be studied. In addition to this the possibility of using alternative near-nuclear corrections instead of gradient ones is discussed. 

The Thomas-Fermi approach is based on the minimization of an energy functional of... 

The minimization of the energy functional using this kinetic energy term leads to a relativistic Thomas-Fermi differential equation... 

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