Thomas-Fermi potential

The Self-Consistent-Field (SCF) procedure can be initiated with hydrogenic wave functions and Thomas-Fermi potentials. It leads to a set of solutions w(fj), each with k nodes between 0 and oo, with zero nodes for the lowest energy and increasing by one for each higher energy level. The quantum number n can now be defined asn = / + l + A to give rise to Is, 2s, 2p, etc. orbitals. 

Equation 5 is often used to decribe the interaction between the incoming ion and the target atoms. The interaction between two target atoms generally occurs at low energy where the Thomas-Fermi potential overestimates the interaction. Under this situation a Born-Mayer potential is more appropriate , i.e. ... 

An especially useful approximation for the Thomas-Fermi potential has been developed by Lindhard and co-workers where the screening function is assumed to have the form ... 

Fig. 21. Reduced nuclear stopping power, s (e), as a function of e, bottom scale and of E for Ar —Cu, top scale. Based on Thomas-Fermi potential (see Equation 5). (Based on data presented in ref )...

R is the range and E the incident ion energy. See Sect. 2.2.2 for definitions of symbols. This curve is calculated assuming a Thomas-Fermi potential and with neglect of electronic stopping... 

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] ... 

Hoffman, G. G. and Pratt, L. R., Optimized Thomas-Fermi potential for discrete propagator electron density functional calculations. In J. D. Doll and J. E. Gubernatis (eds.). Proceedings of the International Workshop on Quantum Simulation of Condensed Matter Phenomena, TeaneckNJ, pp. 105-115. Singapore World Scientific Publishing (1990). 

Lindhard et al. used a Thomas-Fermi potential and calculated the differential scattering cross section for multiple collisions as... 

Another model potential can be obtained utilizing the statistical theory for the electron distribution in an atom due to the studies done by Thomas and Fermi [266, p. 145-156]. The Thomas-Fermi potential can be seen as the simplest potential possible within the framework of density functional theory in section 8.8. This close connection to DFT shows that exchange and correlation functionals can easily be introduced into the program code despite numerical... 

Returning to the Thomas-Fermi potential, it satisfies, as we have seen, the classical Poisson equation... 

But alas most of what has been described so far concerning density theory applies in theory rather than in practice. The fact that the Thomas-Fermi method is capable of yielding a universal solution for all atoms in the periodic table is a potentially attractive feature but is generally not realized in practice. The attempts to implement the ideas originally due to Thomas and Fermi have not quite materialized. This has meant a return to the need to solve a number of equations separately for each individual atom as one does in the Hartree-Fock method and other ab initio methods using atomic orbitals. 

If this is combined with the classical expression for the nuclear-electron attractive potential and the electron-electron repulsive potential we have the famous Thomas-Fermi expression for the energy of an atom,... 

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... 

Consider two different metals in contact and assume that both are well described by the Thomas-Fermi model (see Problem 3.3) with a decay length of Ltf 0.5 A. (a) Calculate the dipole potential drop at the contact if both metals carry equal and opposite charges of 0.1 C m 2. (b) If the work functions of the two metals differ by 0.5 eV, how large is the surface-charge density on each meted ... 

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 shape function had a role in theoretical chemistry and physics long before it was named by Parr and Bartolotti. For example, in x-ray measurements of the electron density, what one actually measures is the shape function—the relative abundance of electrons at different locations in the molecule. Determining the actual electron density requires calibration to a standard with known electron density. On the theoretical side, the shape function appears early in the history of Thomas-Fermi theory. For example, the Majorana-Fermi-Amaldi approximation to the exchange potential is just [3,4]... 

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