Weizsacker kinetic energy density

Hartree-Fock or Kohn-Sham approach and (1 = 1,..., N) are the corresponding molecular orbitals. Here, tyf[p r)] = (Vp) /8p is the Weizsacker kinetic energy density determined by the one-electron density p(r) = V (r)p, and finally tTplpix)) =... 

The ELF rj(r) has a rather simple normalized Lorentzian-type form and thus its domain lies in the interval 0 < (r) < 1. The upper limit of t] v) = 1 corresponds to the electron system whose kinetic energy density becomes identical to the Weizsacker one. Bearing in mind that the latter was derived on the basis of the Pauli principle, r] r) = 1 implies that all electrons are paired if 2/N, and there is only one unpaired electron in the opposite case. Its value r](r) = j determining the FWHM (= full width at half maximum) describes a case when t = %[p(r)] trF[p(r)], where the lower sign is valid if tvv[p(r)] > f7r-[p(r)]-... 

The logical extension of GGA methods is to allow the exchange and correlation functionals to depend on higher order derivatives of the electron density, with the Laplacian (V ) being the second-order term. Alternatively, the functional can be taken to depend on the orbital kinetic energy density t, which for a single orbital is identical to the von Weizsacker kinetic energy Tw (eq. (6.3)). 

Moreover, this term is the difference of the kinetic energy density of the actual system and of that of a system of spin-free independent particles both with identical one-particle densities />(r).For real wavefunctions or for stationary states, it is simply the difference of the definite positive kinetic energies since the (unwanted) remaining contributions cancel one another. Another attractive property of the non-von Weizsacker contribution is that it appears to be the trace of the Fisher s Information matrix[28]. 

For practical applications, we will not consider T(r) itself but rather the definite positive kinetic energy density of independent particles r (r) which appears in the exact density functional theory[31j. Within this framework, the non-von Weizsacker term accounts only for the Fermi correlation and is usually referred to as Pauli kinetic energy density[32]. Another propery of r ff(r) is its relationship to the conditional probability rO for... 

Using this density matrix, the kinetic energy density, r, becomes the Weizsacker one, in Eq. (4.4) (Dreizler and Gross 1990), such as... 

Garcia-Aldea, D. Alvarellos, J. E. Approach to kinetic energy density functionals nonlocal terms with the structure of the von Weizsacker fimctionaL Phys. Rev. A 2008, 77,022502. 

Unfortunately, adding the Dirac exchange formula to the TF model does not improve the quality of the calculated electron density. The TFD density suffers from the same undesirable characteristics as does the TF density. A major enhancement of these two overly simplified models was made through addition of an inhomogeneity the electron density correction to the kinetic energy density functional. This was first investigated by von Weizsacker, who derived a correction that depends upon the gradient of the density, namely. 

A year later, in 1983, Deb and Ghosh investigated an expression for the kinetic energy density t consisting of the full von Weizsacker term together with the Thomas-Fermi term modified by a position-dependent correction term/(r) [39] ... 

The ELF kernel xs (cf- Eq. (14)) is based on the Pauli kinetic energy density which contains the von Weizsacker term (Vp) ISp. It is obvious that this term cannot be in general written as a sum of contributions in the form ... 

GEA was tested. But the case of a nAS-representable pair of electron densities is much more general. It is striking to note that, the second-order GEA contribution to T ad pA, Pb is non-positive for all pairs of electron densities (for uAB-representable and not uAB-representable pairs alike). This follows from its explicit analytic form given here in the second line of Eq. 71. We recall now that the second-order GEA term (T2 in Eq. 69) is closely related to the von Weizsacker functional T [p 75 (T [p = 9T2[p]), which is the exact kinetic energy functional for one- and two-spin-compensated electron systems. Using Tsw p] in approximating T ad pA, Pb]... 

For a quantum system with a given one particle density, FCr) is the only term which is sensitive to the nature (fermion or boson) of the particles. For a many-fermion system, T(r) czxi be formally expressed as the siun of two contributions, one of which accounting for the Pauli principle and the other not. However, another paatition scheme in which the total kinetic energy is written as the sum of the von Weizsacker term 7 pf (r)[26] and of a remaining non-von Weizsacker term T w r) term has been generally adopted[27, 28, 29, 30]. The von Weizsacker term ... 

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),... 

If one can model the noninteracting one-matrix yj(r,r ) as a functional of the electron density, then, using Equation 1.47 or Equation 1.48, one can compute the kinetic energy. This is the most straightforward approach to deriving kinetic energy functionals, and the Thomas-Eermi functional and the Weizsacker functional can both be derived in this way. (Indeed, all of the most popular functionals can be derived in several different ways.) The one-matrix can also be modeled based on weighted density approximation (WDA), which we will discuss subsequently. 

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