Thomas-Fermi-Dirac statistics

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. 

Cowan RD, Ashkin J (1957) Extension of the Thomas-Fermi-Dirac statistical theory of the atom to finite temperatures. Phys Rev 105 144-157... 

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 Thomas-Fermi (TF) and related methods such as the Thomas-Fermi-Dirac (TFD) have played an important role in the study of complex fermionic systems due to their simplicity and statistical nature [1]. For atomic systems, they are able to provide some knowledge about general features such as the behaviour with the atomic number Z of different ground state properties [2,3]. 

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... 

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] ... 

The so-called Hartree-Fock-Slater method is much more widely utilized, and is a hybrid of the Hartree and Thomas-Fermi-Dirac methods. In this method the direct part of the potential is calculated using the Hartree-Fock approach, whereas the exchange part is approximated by some statistical expression of the model of free electrons. The Slater potential is given by ... 

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 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 equations of quantum statistical mechanics for a system of non-identical particles, for which all solutions of the wave equations are accepted, are closely analogous to the equations of classical statistical mechanics (Boltzmann statistics). The quantum statistics resulting from the acceptance of only antisymmetric wave functions is considerably different. This statistics, called Fermi-Dirac statistics, applies to many problems, such as the Pauli-Sommerfeld treatment of metallic electrons and the Thomas-Fermi treatment of many-electron atoms. The statistics corresponding to the acceptance of only the completely symmetric wave functions is called the Bose-Einstein statistics. These statistics will be briefly discussed in Section 49. 

The basic idea underlying the development of the various density functional theory (DFT) formulations is the hope of reducing complicated, many-body problems to effective one-body problems. The earlier, most popular approaches have indeed shown that a many-body system can be dealt with statistically as a one-body system by relating the local electron density p(r) to the total average potential, y(r), felt by the electron in the many-body situation. Such treatments, in fact, produced two well-known mean-field equations i.e. the Hartree-Fock-Slater (HFS) equation [14] and the Thomas-Fermi-Dirac (TFD) equation [15], It stemmed from such formulations that to base those equations on a density theory rather than on a wavefunction theory would avoid the full solution... 

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