Thomas-Fermi-Dirac

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

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. 

TF) theory, including the Ko[p exchange part (first derived by Block but commonly associated with the name of Dirac" (constitutes the Thomas-Fermi-Dirac (TFD) model. 

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... 

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. 

In summary, the original Thomas-Fermi-Dirac DFT was unable to give binding in molecules. This was corrected by Kohn-Sham, [11] who chose to use an orbital rather than density evaluation of the kinetic energy. By the virial theorem, = —E, so this was a necessity to obtain realistic results for energies. Next, it was shown that the exact exchange requires an orbital-dependent form, too. [47,48] The future seems to demand an orbital-dependent form for the correlation. 

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)... 

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]. 

Gradient corrections to the energy density for improving the Thomas-Fermi-(Dirac)... 

The minimization of this functional, which includes second order gradient corrections leads to the relativistic analogous of the Thomas-Fermi-Dirac-Weizsacker model and constitutes the state of the art in relativistic semiclassical approaches for many-electron systems. 

Table 2 Values of relativistic energies (E) and differences among relativistic and non-relativistic energies (AE) for neutral atoms in atomic units with the present approach using thefunctional given by Eq. (46) not including (1) or including (2) the term, compared to the results of Engel and Dreizler (ED) [23] using the relativistic Thomas-Fermi-Dirac- Weirsacker approach described in Section 2.6, and to Dirac-Fock values...

Two types of corrections to the Thomas-Fermi-Dirac non-relativistic energy density appear. The first is the correction to the kinetic energy given by the mass-variation term ... 

Here we will present a semi-relativistic Thomas-Fermi-Dirac approach for the evaluation of ground state atomic properties for not too large Z specifically designed for being used with near-nuclear corrections as those mentioned in a previous section, which becomes an alternative approach of gradient corrections procedures. 

A. Moya and I. Porras Expectation values for ground state atoms from a modified Thomas-Fermi-Dirac approach, in A. Hemandez-Laguna et al. (eds.), Quantum Systems in Chemistry and Physics, Vol. I, Kluwer Academic Publishers (2000). 

Expectation Values for Ground-State Atoms from a Modified Thomas-Fermi-Dirac Approach... 

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