Fermi-Thomas-Dirac atomic model

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 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). 

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approximate singledeterminant wavefunctions, in which the non-local exchange interaction of the Hartree-Fock method has been replaced by a local term, as in the Thomas-Fermi-Dirac model.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)... 

It is not absolutely necessary to have accurate interatomic potentials to perform reasonably good calculations because the many collisions involved tend to obscure the details of the interaction. This, together with the fact that accurate potentials are only known for a few systems makes the Thomas-Fermi approach quite attractive. The Thomas-Fermi statistical model assumes that the atomic potential V(r) varies slowly enough within an electron wavelength so that many electrons can be localized within a volume over which the potential changes by a fraction of itself. The electrons can then be treated by statistical mechanics and obey Fermi-Dirac statistics. In this approximation, the potential in the atom is given by ... 

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] ... 

Nevertheless, it seems likely that the model treatment of atomic ions in hot, non-degenerate plasmas presented in this work, is well worth further study, the intermediate Fermi-Dirac degeneracy being of obvious importance. Under these conditions, an appropriate starting point to introduce the potential would be the elevated temperature Thomas-Fermi theory [46]. 

While Eq. (12) represents the correct prediction of the non-relativistic Schro-dinger equation as Z —> oo, in the range of the Periodic Table corrections are needed. One is due to the fact that Eq. (11) near the point atomic nucleus assumed, shows that pit) diverges as r 3/2 and this is due to neglecting density gradients in the Fermi gas model employed. This, as was shown by Scott [12], corrects [13,14] Eq. (12) with a term (1/2)Z2. Earlier Dirac had introduced the exchange energy A into the Thomas-Fermi atom, with the result... 

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 statistical models of these interactions are the well-known Thomas-Fermi (TF) and Thomas-Fermi-Dirac (TFD) theories based on the idea of approximating the behavior of electrons by that of the uniform negatively charged gas. Some authors (Sheldon, 1955 Teller, 1962 Balazs, 1967 Firsov, 1953,1957 Townsend and Handler, 1962 Townsend and Keller, 1963 Goodisman, 1971) proved that these theories provide an adequate description of purely repulsive diatomic interactions. Abraham-son (1963, 1964) and Konowalow et al. (Konowalow, 1969 Konowalow and Zakheim, 1972) extended this region to intermediate internuclear distances, but Gombas (1949) and March (1957) showed that the Abraham-son approach is incorrect, and so the question of how adequately the TFD theory provides diatomic interactions for closed-shell atoms is still open. Here we need to note that until recently, there has existed only work by Sheldon (1955), as far as we know, in which the TFD interaction potential is actually calculated by solving the TFD equation for a series of internuclear distances (see also, Kaplan, 1982). 

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]. 

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 ... 

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