Thomas-Fermi-Based Functionals

In Sections 1.3.2 and 1.3.3, we will briefly summarize the main families of kinetic energy functionals. Many fnnctionals are necessarily omitted the intention is to provide the flavor of mainstream approaches, rather than an encyclopedic list of fnnctionals. 

The first kinetic energy density functional was derived, independently, by Fermi and Thomas in 1928 and 1927, respectively. The Thomas-Fermi functional is the simplest local density approximation. 

In a uniform electron gas (UEG) of noninteracting fermions with density p, the local kinetic energy per fermion is a constant  

The local kinetic energy per unit volume is then the probability of observing a fermion at the point r times the local kinetic energy per fermion 

Integrating the overall volume gives the Thomas-Fermi functional 

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The nonnegativity constraints on the Pauli correction and its potential give stringent constraints on the types of functionals that can be considered. The most popular form for the kinetic energy has attempted to modify the enhancement factors from Thomas-Fermi-based kinetic energy functionals, defining ... 

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. 

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,... 

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

Various reasons have been advanced for the relative accuracy of spin-polarized Kohn-Sham calculations based on local (spin) density approximations for E c- However, two very favourable aspects of this procedure are clearly operative. First, the Kohn-Sham orbitals control the physical class of density functions which are allowed (in contrast, for example, to simpler Thomas-Fermi theories). Second, local density approximations for are mild-mannered,... 

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 )...

Methods of density functional theory (DFT) originate from the Xa method originally proposed by Slater [78] on the base of statistical description of atomic electron structure within the Thomas-Fermi theory [79]. From the point of view of Eq. (3), fundamental idea of the DFT based methods consist first of all in approximate treatment of the electron-electron interaction energy which is represented as ... 

Models for the electronic structure of polynuclear systems were also developed. Except for metals, where a free electron model of the valence electrons was used, all methods were based on a description of the electronic structure in terms of atomic orbitals. Direct numerical solutions of the Hartree-Fock equations were not feasible and the Thomas-Fermi density model gave ridiculous results. Instead, two different models were introduced. The valence bond formulation (5) followed closely the concepts of chemical bonds between atoms which predated quantum theory (and even the discovery of the electron). In this formulation certain reasonable "configurations" were constructed by drawing bonds between unpaired electrons on different atoms. A mathematical function formed from a sum of products of atomic orbitals was used to represent each configuration. The energy and electronic structure was then... 

The Hohenberg-Kohn theory of /V-clcctron ground states is based on consideration of the spin-indexed density function. Much earlier in the development of quantum mechanics, Thomas-Fermi theory [402, 108] (TFT) was formulated as exactly such a density-dependent formalism, justified as a semiclassical statistical theory [231, 232], Since Hohenberg-Kohn theory establishes the existence of an exact universal functional Fs [p] for ground states, it apparently implies the existence of an exact ground-state Thomas-Fermi theory. The variational theory that might support such a conclusion is considered here. 

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 Thomas-Fermi approximation (34) for T[n is not very good. A more accurate scheme for treating the kinetic-energy functional of interacting electrons, T[n], is based on decomposing it into one part that represents the kinetic energy of noninteracting particles of density n, i.e., the quantity called above Ts[n], and one that represents the remainder, denoted Tc[n (the sub-... 

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... 

For some computational techniques in quantum chemistry a simple zero-th order approximation of the electron density of any atom of the system can be useful as the starting point of an iterative procedure. A very simple description of the electron density and binding energy of any atom or ion allows a rapid evaluation of very complex stmctures. This is the spirit of the orbital-free, explicit density functional approaches, usually based on the Thomas-Fermi-Dirac model and its extensions [1]. 

Most importantly, these systems are amenable to the Electron Localization Function (ELF) method [21]. This is a local measure based on the reduced second-order density matrix, which as pioneered by Lennard-Jones [22] should retain the chemical significance and at the same time reduce the complexity of the information contained in the square of the wave function ELF is defined in terms of the excess of local kinetic energy density due to the Pauli exclusion principle, T r), and the Thomas-Fermi kinetic energy density, Th(r) ... 

Although the Thomas-Fermi method is an interesting theory representing the Hamiltonian operator as the functional only of the electron density, even qualitative discussions cannot be contemplated based on this method in actual electronic state calculations. Dirac considered that this problem may be attributed to the lack of exchange energy (see Sect. 2.4), which was proposed in the same year (Fock 1930), and proposed the first exchange functional of electron density p (Dirac 1930),... 

Gradually, the Thomas-Fermi method or its modem descendants, which are known as density functional theories, have become equally powerful compared to methods based on orbitals and wavefimctions and in many cases can outstrip the wavefunction approaches in terms of computational accuracy. The solution is expressed in terms of the variable Z, which represents atomic number, the crucial feature that distinguishes one kind of atom from any other element. One does not need to repeat the calculation separately for each atom, but this advantage applies only in principle, as discussed below. 

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