Thomas-Fermi theory approximations

In one of the earliest DFT models, the Thomas-Fermi theory, the kinetic energy of an atom or a molecule is approximated using the above type of treatment on a local level. That is, for each volume element in r space, one... 

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]... 

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,... 

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 ... 

This argument shows that the locality hypothesis fails for more than two electrons because the assumed Frechet derivative must be generalized to a Gateaux derivative, equivalent in the context of OEL equations to a linear operator that acts on orbital wave functions. The conclusion is that the use by Kohn and Sham of Schrodinger s operator t is variationally correct, but no equivalent Thomas-Fermi theory exists for more than two electrons. Empirical evidence (atomic shell structure, chemical binding) supports the Kohn-Sham choice of the nonlocal kinetic energy operator, in comparison with Thomas-Fermi theory [288]. A further implication is that if an explicit approximate local density functional Exc is postulated, as in the local-density approximation (LDA) [205], the resulting Kohn-Sham theory is variation-ally correct. Typically, for Exc = f exc(p)p d3r, the density functional derivative is a Frechet derivative, the local potential function vxc = exc + p dexc/dp. 

Physical properties of atoms and ions in intense magnetic fields are hence obtained in the statistical limit of Thomas-Fermi theory. This discussion is then supplemented by the hyperstrong limit, considered especially by Lieb and co-workers. Chemistry in intense magnetic fields is thereby compared and contrasted with terrestrial chemistry. Some emphasis is then placed on a model of confined atoms in intense electric fields the statistical Thomas-Fermi approximation again being the central tool employed. 

While a great deal of progress has proved possible for the case of the hydrogen atom by direct solution of the Schrodinger wave equation, some of which will be summarized below, at the time of writing the treatment of many-electron atoms necessitates a simpler approach. This is afforded by the semi-classical Thomas-Fermi theory [4-6], the first explicit form of what today is termed density functional theory [7,8]. We shall summarize below the work of Hill et al. [9], who solved the Thomas-Fermi (TF) equation for heavy positive ions in the limit of extremely strong magnetic fields. This will lead naturally into the formulation of relativistic Thomas-Fermi (TF) theory [10] and to a discussion of the role of the virial in this approximate theory [11]. 

It should be appreciated that in contrast to the simple free electron models used in much of our discussion of metals and semiconductors, a treatment of screening necessarily involves taking into account, on some level, the interaction between charge carriers. In the Thomas-Fermi theory this is done by combining a semiclassical approximation for the response of the electron density to an external potential with a mean field approximation on the Hartree level—assuming that each electron is moving in the mean electrostatic potential of the other electrons. 

There is a parallel between the wave packet-based semi-classical treatment of nuclear motion developed here and the Thomas-Fermi theory of an electronic system in a slowly varying vector potential. In the semi-classical electronic theory as well as here, one naturally arrives at a locally linear approximation to the scalar-potential-derived forces and a locally uniform approximation to the magnetic force derived from the vector potential. See, R. A. Harris and J. A. Cina, J. Chem. Phys. 79, 1381 (1983) C. J. Grayce and R. A. Harris, Molec. Phys. 71, 1 (1990). 

The principal conclusion of the present analysis is that an exact DFT is not possible if restricted to local potential functions. This excludes an exact Thomas-Fermi theory for more than two electrons, but it is shown here that DFT in the local density approximation (LDA) and the optimimized effective potential (OEP) model are sound variational theories. An exact OFT exists, but must be implemented using nonlocal potentials. 

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. 

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,... 

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