Thomas-Fermi relation

Coulson, C., Compt. rend. 239, 868, Sur une relation d Odiot et Daudel entre la density lectronique et le potentiel lectrique d un atome." The relation is derived from the Thomas-Fermi equation. 

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 Thomas-Fermi (TF) model (1927) for a homogeneous electron gas provides the underpinnings of modern DFT. In the following discussion, it will be shown that the model generates several useful concepts, relates the electron density to the potential, and gives a universal differential equation for the direct calculation of electron density. The two main assumptions of the TF model are as follows ... 

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]. 

In the Thomas-Fermi theory (March 1957), the electrostatic potential at r is related to the electron density of a neutral atom by the density functional... 

In Thomas-Fermi theory, adoption of the simple central field model for neutral molecules at equihbrium leads to simple energy relations—well supported by SCF calculations [12]—such as = (Vne + 2Vnn)- Evidently, nothing of the like apphes to vaience> but wc may well inquire how things are with... 

Here, q is the inverse of a screening length related to the valence electron density which contributes to the screening and /u. is a Lagrange multiplier controlling the total number of particles. The boundary conditions to be used with Equation (23) are that V(r) must match Vc r) at Rs and that rV(r) -> -1 as r -> 0. Once we have solved the Thomas-Fermi equation, we have calculated the screened function, defined as the bare impurity potential divided to the screened one, namely Vb/V. 

Density-Potential Relation of Thomas-Fermi Statistical Theory... 

Sections 13-1.5 are then concerned with relating the above model to atomic ions in a hot, non-degenerate plasma in an external electric field. The first step is to add an atomic-like potential energy V(r) to the model. Strictly, V(r) should be calculated self-consistently as a function of p, F and the plasma density. While this has not been achieved numerically at the time of writing, a model potential F(r) is incorporated into the treatment of Sect. 7.3 by means of the semiclassical Thomas-Fermi approximation. The second step taken by Amovilli et al. [41] is to connect the strength of the harmonic potential with the plasma density (Sect. 7.4). 

The Thomas-Fermi method and the Xa scheme were at the time of their inceptions considered as useful models based on the notion that the energy of an electronic system can be expressed in terms of its density. A formal proof of this notion came in 1964 when it was shown by Hohenberg and Kohn [38] that there is a unique relation between density and energy. The year after Kohn and Sham put forward a practical variational DFT approach in which they replaced E of (2) with a combined exchange and correlation term... 

The Thomas-Fermi approximation for screening, discussed in many books on solid-state physics, is obtained by minimizing ETF[n] with respect to n and linearizing the resulting relation between v(r) and n(r). It thus involves one more approximation (the linearization) compared to what is called the Thomas-Fermi approximation in DFT [44]. In two dimensions no linearization is required and both become equivalent [44]. 

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... 

N- 00 or Z oo the bosonic limit due to the scaled equivalence T or T 7 when the system is thermally expanded, being it related with the Thomas-Fermi theory, in Chapter 5 of the present volume exposed. 

The first relation is obtained from the Thomas-Fermi equation, multiplied with p and then integrated. Alternatively, one can proof the assertion (a) by noting... 

Lieb, E. H. (1981). Thomas-fermi and related theories of atoms and molecules. Rev. Mod. Phys. 53, 603-641. 

The last formula is circumvented to the high-density total correlation density approaches rooting at their turn on the Thomas-Fermi atomic theory. Very interesting, the relation (4.477) may be seen as an atomic reflection of the (solid state) high-density regime (r < 1) given by Perdew et al. (Perdew, 1986 Wang Perdew, 1989 Seidl et al., 1999 Perdew et al., 1996) ... 

We briefly describe three methods that can be considered as related to DFT and that are sometimes used for cluster computations the Thomas-Fermi (TF) approximation, the so-called DFT-based tight-binding (TB), and the many-body potentials of the embedded-atom type. 

Some models, such as e.g. the Thomas-Fermi and related models,have involved the construction of explicit approximate forms for T and Fee in T hk this constitutes a direct and simple approach since the resulting... 

For this purpose, we reanalyze all the available static EOS data for Th, as shown in table 1, with a set of three different EOS forms, and compare the effect of the different EOS forms with the effects resulting from different data sets. As EOS forms we use the Birch equation (Birch 1978) in second order, BE2, and two recently proposed forms (Holzapfel 1990,1991) in second-order form, H02 and HI2, which are related to the Thomas-Fermi theory and are distinguished by the fact that H12 is bound to approach the Fermi-gas limit at infinite compression. A close inspection of table 1 shows very clearly that most of the data are fitted almost equally well by any of these forms, without any significant difference in the fitted parameters Kq and K g or in the minimized standard deviation of the pressure, Tp. In contrast to many other publications, table 1 presents the unrestricted standard deviations of Kq and K, which correspond to the extreme values of the error ellipsoids presented in fig. 11, and not just to the width of the error ellipsoids along and K at the center points, which are usually given as (restricted) statistical errors. Thus, it becomes obvious that... 

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