Fermi-Thomas theory

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. 

The orbital theory of Kohn and Sham [205] differs from Thomas-Fermi theory in that the density function p = J4iniPi = / , is postulated to have an 

In Kohn-Sham theory, densities are postulated to be sums of orbital densities, for functions (pi in the orbital Hilbert space. This generates a Banach space [102] of density functions. Thomas-Fermi theory can be derived if an energy functional E[p] = I p + F [ p is postulated to exist, defined for all normalized ground-state 

If Fs is defined for unrestricted variations of p in any infinitesimal function neighborhood of a solution, this implies the Thomas-Fermi (TF) equation 

The Lagrange multiplier p. determined by normalization, is the chemical potential [232], such that pt = dE/dN when the indicated derivative is defined. This derivation requires the locality hypothesis, that a Frechet derivative of Fs p exists as a local function (r). 

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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... 

Rice5 used this Thomas-Fermi theory for the metal in the interface. This equation is multiplied by 2 d(n)213/dx9 and then integrated from x = 0 to x = —oo. Since dn(x)/dx - 0 when x -> — oo one gets... 

Since this capacitance is supposed to be in series with that of the solution and since capacitances of mercury-solution interfaces are much larger than 2 F/cm2, this number is too low. The Thomas-Fermi theory as well as the neglect of interactions between metal electrons and the electrolyte are at fault. To reduce the metal s contribution to the inverse capacitance, a model must include6 penetration of the electron tail of the metal into the solvent region, where the dielectric constant is higher, as the models discussed below do. 

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]... 

Review of Relativistic Extensions of Thomas-Fermi Theory... 

In nonrelativistic Thomas-Fermi theory the functional is given by... 

The relativistic formulation of Thomas-Fermi theory started at the same time as the original non-relativistic one, the first work being of Vallarta and Rosen [9] in 1932. The result they arrived at can be found by replacing the kinetic energy fimctional by the result of the integration of the relativistic kinetic energy in terms of the momentum p times the number of electrons with a given momentum p from /i = 0 to the Fermi momentum p = Pp. ... 

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,... 

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... 

Hohenberg and Kohn have proved generally that the total ground state energy E of a collection of electrons in the presence of an externally applied potential (e.g. the valence electrons in the presence of the periodic potential due to the cores in a lattice), when no net magnetic moment is present, depends only on the average density of electrons n(R). By this proof, n(R) becomes the fundamental variable of the system (as it is in the Thomas-Fermi theory ). Variational minimization of the most general form of E, with respect to n lends to the Hartree-Fock equations formalism. 

For the non-local term E c, defined in (22), Kohn and Sham ° suggested afunctional similar to that of the Thomas-Fermi theory. In this way, Eq. (20) is completely reduced to functionals of n(R) and then to a local form. 

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 ... 

The failure is not limited to metal-ammonia solutions nor to the linear Thomas-Fermi theory (19). The metals physicist has known for 30 years that the theory of electron interactions is unsatisfactory. E. Wigner showed in 1934 that a dilute electron gas (in the presence of a uniform positive charge density) would condense into an electron crystal wherein the electrons occupy the fixed positions of a lattice. Weaker correlations doubtless exist in the present case and have not been properly treated as yet. Studies on metal-ammonia solutions may help resolve this problem. But one or another form of this problem—the inadequate understanding of electron correlations—precludes any conclusive theoretical treatment of the conductivity in terms of, say, effective mass at present. The effective mass may be introduced to account for errors in the density of states—not in the electron correlations. 

The idea of calculating atomic and molecular properties from electron density appears to have arisen from calculations made independently by Enrico Fermi and P.A.M. Dirac in the 1920s on an ideal electron gas, work now well-known as the Fermi-Dirac statistics [19]. In independent work by Fermi [20] and Thomas [21], atoms were modelled as systems with a positive potential (the nucleus) located in a uniform (homogeneous) electron gas. This obviously unrealistic idealization, the Thomas-Fermi model [22], or with embellishments by Dirac the Thomas-Fermi-Dirac model [22], gave surprisingly good results for atoms, but failed completely for molecules it predicted all molecules to be unstable toward dissociation into their atoms (indeed, this is a theorem in Thomas-Fermi theory). 

Since J2ini = N anexclusion principle for any compact system with more than two electrons. The failure of this sum rule implies that in general the assumed Frechet derivative of l s [p] cannot exist for more than two electrons, and there can be no exact Thomas-Fermi theory. 

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. 

Thomas-Fermi theory) requires a Frechet derivative for the kinetic energy, and cannot exist for more than two electrons [288],... 

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