Thomas-Fermi energy

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... 

The first term is the local Thomas-Fermi energy and the second gives the von Weisacker quantum correction [11]. A value j8 = 0.5 for the coefficient in this term has been found convenient for jellium clusters [85]. The results obtained for B are in good agreement with those from a full Kohn-Sham calculation. The ETF method is also useful to calculate the fission barrier F for very large clusters [86], where the importance of shell effects is expected to decrease and a full KS calculation becomes tedious. 

Similarly, for the Thomas-Fermi energy (see Townsend and Handler, 1962)... 

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 one of the earliest DFT models, the Thomas-Fermi theory, the kinetic energy of an atom or a molecule is approximated using the above type of treatment on a local level. That is, for each volume element in r space, one... 

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. 

Confusion is created by the often-quoted results of calculations by Latter that did predict some of the above ordering on the badis of the rather crude Thomas-Fermi method of approximation 20). More recent Hartree-Fock calculations on atoms show, for example, that the 3d level is definitely of lower energy than that of 4s (21). 

If this is combined with the classical expression for the nuclear-electron attractive potential and the electron-electron repulsive potential we have the famous Thomas-Fermi expression for the energy of an atom,... 

Of these, only J[p] is known, while the explicit forms of the other two contributions remain a mystery. The Thomas-Fermi and Thomas-Fermi-Dirac approximations that we briefly touched upon in Chapter 3 are actually realizations of this very concept. All terms present in these models, i. e., the kinetic energy, the potential due to the nuclei, the classical... 

In the Thomas-Fermi model,49 the kinetic energy density of the electron gas is written as... 

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... 

Bloch (1933a,b) first pointed out that in the Thomas-Fermi-Dirac statistical model the spectral distribution of atomic oscillator strength has the same shape for all atoms if the transition energy is scaled by Z. Therefore, in this model, I< Z Bloch estimated the constant of proportionality approximately as 10-15 eV. Another calculation using the Thomas-Fermi-Dirac model gives I tZ = a + bZ-2/3 with a = 9.2 and b = 4.5 as best adjusted values (Turner, 1964). This expression agrees rather well with experiments. Figure 2.3 shows the variation of IIZ vs. Z. 

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. 

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