Thomas-Fermi-Dirac model

Density functional theory-based methods ultimately derive from quantum mechanics research from the 1920 s, especially the Thomas-Fermi-Dirac model, and from Slater s fundamental work in quantum chemistry in the 1950 s. The DFT approach is based upon a strategy of modeling electron correlation via general functionals of the electron density. 

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 Xa multiple scattering method generates approximate singledeterminant wavefunctions, in which the non-local exchange interaction of the Hartree-Fock method has been replaced by a local term, as in the Thomas-Fermi-Dirac model. The orbitals are solutions of the one-electron differential equation (in atomic units)... 

In the simplest form, the Thomas-Fermi-Dirac model, the functionals are those which are valid for an electronic gas with slow spatial variations (the nearly free electron gas ). In this approximation, the kinetic energy T is given by... 

P. A. M. 1930). The combination of this expression with Eqs. (8.3)-(8.5) defines the Thomas-Fermi-Dirac model, although it too remains sufficiently inaccurate that it fails to see any modem use. 

There exists a whole number of approximate expressions for Vl(r) (see, for example [139]). The simplest, called the Thomas-Fermi potential, follows from the statistical model of an atom. Unfortunately, it leads to results of very low accuracy. More accurate is the Thomas-Fermi-Dirac model, in which an attempt is made to account for the exchange part of the potential energy of an electron in the framework of the free electron gas approach. Various forms of the parametric potential method are fairly widely utilized, particularly for multiply charged ions. Such potentials may look as follows [16] ... 

The idea of calculating atomic and molecular properties from electron density appears to have arisen from calculations made independently by Enrico Fermi and P.A.M. Dirac in the 1920s on an ideal electron gas, work now well-known as the Fermi-Dirac statistics [19]. In independent work by Fermi [20] and Thomas [21], atoms were modelled as systems with a positive potential (the nucleus) located in a uniform (homogeneous) electron gas. This obviously unrealistic idealization, the Thomas-Fermi model [22], or with embellishments by Dirac the Thomas-Fermi-Dirac model [22], gave surprisingly good results for atoms, but failed completely for molecules it predicted all molecules to be unstable toward dissociation into their atoms (indeed, this is a theorem in Thomas-Fermi theory). 

At the present time, by far the most useful non-empirical alternatives to Cl are the methods based on density functional theory (DFT) . The development of DFT can be traced from its pre-quantum-mechanical roots in Drude s treatment of the electron gas" in metals and Sommerfeld s quantum-statistical version of this, through the Thomas-Fermi-Dirac model of the atom. Slater s Xa method, the laying of the formal foundations by... 

The first generation is the local density approximation (LDA). This estimation involves the Dirac functional for exchange, which is nothing else than the functional proposed by Dirac [15] in 1927 for the so-called Thomas-Fermi-Dirac model of the atoms. For the correlation energy, some parameterizations have been proposed, and the formula can be considered as the limit of what can be obtained at this level of approximation [16-18], The Xa approximation falls into this category, since a known proportion of the exchange energy approximates the correlation. 

Thomas-Fermi model [18], or with embellishments by Dirac, the Thomas-Fermi-Dirac model [18], gave surprisingly good results for atoms, but failed completely for molecules it predicted all molecules to be unstable toward dissociation into their atoms (indeed, this is a theorem in Thomas-Fermi theory). 

For some computational techniques in quantum chemistry a simple zero-th order approximation of the electron density of any atom of the system can be useful as the starting point of an iterative procedure. A very simple description of the electron density and binding energy of any atom or ion allows a rapid evaluation of very complex stmctures. This is the spirit of the orbital-free, explicit density functional approaches, usually based on the Thomas-Fermi-Dirac model and its extensions [1]. 

However, this formula is found by including the Scott correction [3] to the energy of the Thomas-Fermi-Dirac model, and this approach does not provide a correspondingly corrected density. 

The minimization of the energy functional with respect to the density for r > ro leads to the integral equation of the Thomas-Fermi-Dirac model restricted to this region. This was solved numerically with some constraints that must be imposed because of a wrong asymptotic behavior of the exact solution when r 00,... 

In this work a simple analytical atomic density model is obtained from the expression of a modified Thomas-Fermi-Dirac model with quantum corrections near the nucleus as the minimization of a semiexplicit density functional. The use of a simple exponential analytical form for the density outside the near-nucleus region and the resolution of a single-particle Schrodinger equation with an effective potential near the origin allows us to solve easily the problem and obtain an asymptotic expression for the energy of an atom or ion in terms of the nuclear charge Z and the number of electrons N. 

Slater proposes an effective quantum number n = 3.7, the atomic factor can only be presented in the form of a sum with an infinite number of components. The series may be terminated if the effective quantum number for the N shell is taken as 3.5, 4.0, or 4.5. We calculated values of the atomic factor for the neutral Br atom with different values of n. The most satisfactory agreement with the theoretical form factors, calculated according to the Thomas—Fermi—Dirac model, was obtained at n — 4.5 screening coefficients proposed in [11] were used in the calculations. The equation of the atomic scattering function for the N shell in the case of a spherically symmetrical electron density distribution and n — 4.5 has the following form ... 

The energy functional, i.e., the energy expressed in terms of the density (r) in the TFD (Thomas-Fermi-Dirac) model is expressed as [206]... 

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