Thomas-Fermi approach

On the other hand the Thomas-Fermi method, which treats the electrons around the nucleus as a perfectly homogeneous electron gas, yields a mathematical solution that is universal, meaning that it can be solved once and for all. This feature already represents an improvement over the method which seeks to solve Schrodinger equation for every atom separately. This was one of the features that made people go back to the Thomas-Fermi approach in the hope of... 

The Thomas-Fermi approach is based on the minimization of an energy functional of... 

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 ... 

The primary particle involved in the screening process is the mobile electron. One has then the problem of a self-consistent calculation of the charge distribution in the neighborhood of a test charge. The Thomas-Fermi approach to this problem is the analog of the Debye-Huckel calculation wherein allowance has been made for the Pauli exclusion principle. From any standard text one can obtain the Poisson equation (19)... 

Domps, A., Reinhard, P.-G. and Suraud, E. (1998). Time-dependent Thomas-Fermi approach to electron dynamics in metal clusters, Phys. Rev. Lett. 80, 5520-5523. 

Third, the fact that the energy functional Ee[p] is unknown manifests itself in Eq. (21) as the fact that Exc is unknown. However, Exc is only a smaller part of Ee, and, compared with the Thomas-Fermi approach, the final results are significantly less sensitive to approximations in Exc. Such approximations are often introduced by first writing Exc as... 

However, the application of semiclassical models derived from the Thomas-Fermi approach has some drawbacks, including an overestimation of the electron density near the nucleus, which affects the dependence of low-order radial expectation values with Z, and the impossibility of describing atomic anions [1]. 

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. 

In this section we will approach the question which is at the very heart of density functional theory can we possibly replace the complicated N-electron wave function with its dependence on 3N spatial plus N spin variables by a simpler quantity, such as the electron density After using plausibility arguments to demonstrate that this seems to be a sensible thing to do, we introduce two early realizations of this idea, the Thomas-Fermi model and Slater s approximation of Hartree-Fock exchange defining the X(/ method. The discussion in this chapter will prepare us for the next steps, where we will encounter physically sound reasons why the density is really all we need. 

Actually, the first attempts to use the electron density rather than the wave function for obtaining information about atomic and molecular systems are almost as old as is quantum mechanics itself and date back to the early work of Thomas, 1927 and Fermi, 1927. In the present context, their approach is of only historical interest. We therefore refrain from an in-depth discussion of the Thomas-Fermi model and restrict ourselves to a brief summary of the conclusions important to the general discussion of DFT. The reader interested in learning more about this approach is encouraged to consult the rich review literature on this subject, for example by March, 1975, 1992 or by Parr and Yang, 1989. 

The calculations were subsequently extended to moderate surface charges and electrolyte concentrations.8 The compact-layer capacitance, in this approach, clearly depends on the nature of the solvent, the nature of the metal electrode, and the interaction between solvent and metal. The work8,109 describing the electrodesolvent system with the use of nonlocal dielectric functions e(x, x ) is reviewed and discussed by Vorotyntsev, Kornyshev, and coworkers.6,77 With several assumptions for e(x,x ), related to the Thomas-Fermi model, an explicit expression6 for the compact-layer capacitance could be derived ... 

The electronic wave function of an n-electron molecule is defined in 3n-dimensional configuration space, consistent with any conceivable molecular geometry. If the only aim is to characterize a molecule of fixed Born-Oppenheimer geometry the amount of information contained in the molecular wave function is therefore quite excessive. It turns out that the three-dimensional electron density function contains adequate information to uniquely determine the ground-state electronic properties of the molecule, as first demonstrated by Hohenberg and Kohn [104]. The approach is equivalent to the Thomas-Fermi model of an atom applied to molecules. 

Since the early days of quantum mechanics, the wave function theory has proven to be very successful in describing many different quantum processes and phenomena. However, in many problems of quantum chemistry and solid-state physics, where the dimensionality of the systems studied is relatively high, ab initio calculations of the structure of atoms, molecules, clusters, and crystals, and their interactions are very often prohibitive. Hence, alternative formulations based on the direct use of the probability density, gathered under what is generally known as the density matrix theory [1], were also developed since the very beginning of the new mechanics. The independent electron approximation or Thomas-Fermi model, and the Hartree and Hartree-Fock approaches are former statistical models developed in that direction [2]. These models can be considered direct predecessors of the more recent density functional theory (DFT) [3], whose principles were established by Hohenberg,... 

After reviewing the different approaches in the literature for the formulation of a relativistic Thomas-Fermi procedure for the study of complex electron systems, we will make contact between quantum mechanics with first-order relativistic corrections and the weak relativistic limit of quantum electrodynamics for finding explicit energy functionals that will be studied. In addition to this the possibility of using alternative near-nuclear corrections instead of gradient ones is discussed. 

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...

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... 

Expectation values for ground-state atoms from a modified Thomas-Fermi- 215 Dirac approach... 

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