Thomas-Fermi functional

The above form includes also the Thomas-Fermi functional (LDA) for which F(s) = const = 1. Fig. 5 shows the considered enhancement factors GEA2 -... 

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. 

Integrating the overall volume gives the Thomas-Fermi functional... 

Since the Thomas-Fermi functional is exact for the uniform electron gas, its failings must arise because the electron densities of chemical substances are far from uniform. This suggests that we construct the gradient expansion about the uniform electron gas limit such functionals will be exact for nearly uniform electron gases. An alternative perspective is to recall that the Thomas-Fermi theory is exact in the classical high-quantum number limit. The gradient expansion can be derived as a Maclaurin series in powers of ti it adds additional quantum effects to the Thomas-Fermi model. 

The Weizsacker functional is a lower bound to the true kinetic energy, but it is a very weak lower bound. " " It is even less accurate than the Thomas-Fermi functional. 

Corrected Weizsacker-Based Functionals 1.3.4.1 Weizsacker Plus Thomas-Fermi Functionals... 

The simplest approach to correcting the Weizsacker functional is to add a fraction of Thomas-Fermi functional, forming a W + XTF functional. 

The conventional WDA for the kinetic energy results when the effective Fermi vector is determined by substituting the asymmetric one-matrix (Equation 1.110) into the diagonal idempotency condition (Equation 1.115). This functional is also exact for the uniform electron gas, but the kinetic energies of atoms and molecules are still predicted to be far too high. Indeed, this functional is only slightly more accurate than the Thomas-Fermi functional. This is surprising, since the WDA and the TF functional were derived from the same formula for the one-matrix, but the WDA adds an additional exact constraint. 

Garcia-Aldea, D. AlvareUos, J. E. Kinetic-energy density functionals with nonlocal terms with the structure of the Thomas-Fermi functional. Phys. Rev. A 2007, 76,052504. 

It is very easy to write the noninteracting kinetic energy for the split fc-space. In the zeroth order (akin to the Thomas-Fermi functional), it is given by the formula [32]... 

These results show that while the traditional Thomas-Fermi functional underestimates the exact kinetic energy, the modified Thomas-Fermi functional gives numbers closer to the exact values. This indicates the correcmess of splitting A -space to obtain functionals for excited states. 

For the ground-state theory, the Thomas-Fermi functional can be made more accurate by adding the gradient correction to it. For ground states, the gradient correction up to the second order is given as [3-5]... 

Lieb and Thirring [44] have conjectured that Ts[n] is boimded from below by the Thomas-Fermi functional... 

Unfortunately, the Thomas-Fermi energy functional does not produce results that are of sufficiently high accuracy to be of great use in chemistry. What is missing in this... 

Thomas-Fermi total energy Eg.j.p [p] gives the so-called Thomas-Fermi-Dirac (TFD) energy functional. 

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. 

The foundation for the use of DFT methods in computational chemistry was the introduction of orbitals by Kohn and Sham. 5 The main problem in Thomas-Fermi models is that the kinetic energy is represented poorly. The basic idea in the Kohn and Sham (KS) formalism is splitting the kinetic energy functional into two parts, one of which can be calculated exactly, and a small correction term. 

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. 

Actually, the first attempts to use the electron density rather than the wave function for obtaining information about atomic and molecular systems are almost as old as is quantum mechanics itself and date back to the early work of Thomas, 1927 and Fermi, 1927. In the present context, their approach is of only historical interest. We therefore refrain from an in-depth discussion of the Thomas-Fermi model and restrict ourselves to a brief summary of the conclusions important to the general discussion of DFT. The reader interested in learning more about this approach is encouraged to consult the rich review literature on this subject, for example by March, 1975, 1992 or by Parr and Yang, 1989. 

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... 

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 universal function x(x) obtained by numerical integration and valid for all neutral atoms decreases monotonically. The electron density is similar for all atoms, except for a different length scale, which is determined by the quantity b and proportional to Z. The density is poorly determined at both small and large values of r. However, since most electrons in complex atoms are at intermediate distances from the nucleus the Thomas-Fermi model is useful for calculating quantities that depend on the average electron density, such as the total energy. The Thomas-Fermi model therefore cannot account for the periodic properties of atoms, but provides a good estimate of initial fields used in more elaborate calculations like those to be discussed in the next section. 

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. 

The electronic wave function of an n-electron molecule is defined in 3n-dimensional configuration space, consistent with any conceivable molecular geometry. If the only aim is to characterize a molecule of fixed Born-Oppenheimer geometry the amount of information contained in the molecular wave function is therefore quite excessive. It turns out that the three-dimensional electron density function contains adequate information to uniquely determine the ground-state electronic properties of the molecule, as first demonstrated by Hohenberg and Kohn [104]. The approach is equivalent to the Thomas-Fermi model of an atom applied to molecules. 

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