Thomas-Fermi kinetic functional

Following the success of the von Weizsacker approach in improving the Thomas-Fermi kinetic functional. Sham showed in 1971 that an analogous correction to the Dirac exchange functional can be derived, Kleinman later demonstrated that the Sham derivation was flawed and that his correction was too small by exactly 10/7, It is now agreed that the correct second-order alpha exchange functional is... 

For a homogeneous electron density in a given volume V, the kinetic energy can be derived form the electron gas theory. For a pair of homogeneous electron densities Pa t) = phA and Pb t) = phg) the analytic form of T ad pA, Pb] obtained using Thomas-Fermi kinetic energy functional applied to Ts[pA + pb], Ts[pa], and Ts[pb] reads ... 

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) ... 

This functional is found to be the exact LDA exchange functional. Furthermore, von Weizsacker proposed a correction term using the gradient of electron density for the Thomas-Fermi kinetic energy functional (von Weizsacker 1935),... 

Apart from these indexes, the electron localization function (ELF) is another approach to measure aromaticity [34]. It is based on the properties of the electron density. Introduced by Becke and Edgecombe [34(a)], ELE is defined in terms of excess local kinetic energy density due to Pauli exclusion principle, T[p(f)], and Thomas-Fermi kinetic energy density, Th[p(r)], as follows ... 

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. 

The hardness kernels in Equation 24.110 depend on the kinetic energy functional as well as on the electron-electron interactions. Thomas-Fermi models can be used to evaluate the kinetic part of these hardness kernels and can be combined with a band structure calculation of the linear response X -... 

The minimization of the energy functional using this kinetic energy term leads to a relativistic Thomas-Fermi differential equation... 

In the simplest form, the Thomas-Fermi-Dirac model, the functionals are those which are valid for an electronic gas with slow spatial variations (the nearly free electron gas ). In this approximation, the kinetic energy T is given by... 

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. 

The kinetic and exchange energy functionals given by Eqs. (8) and (12), respectively, contain universal terms that just depend upon the one-particle density. In the case of the former, such term is p6/3, the Thomas-Fermi term [22,23] and for the latter, the set p(ri)(4+fc 3, where the first term p4 3 (for k = 0) is the Dirac exchange expression [24]. But in addition, in Eq. (8) we observe the presence of a factor, which we call Fis([p]jr) defined as ... 

Thomas-Fermi theory — This is an early version of density functional theory, which accounts for the kinetic energy of the electrons and for Coulomb interactions, but neglects exchange and correlations. It... 

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