Von Weizsacker kinetic energy

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)). 

The term [VrP F( )] /p vi ) is known as the von Weizsacker kinetic energy fw[P i (r)] [4]. Hence, in the other words, Corollary 1.1 tells that square-integrable. Usually, the von Weizsacker term is only a part of the total many-electron kinetic energy [4]. The exception is the Hartree-Fock 2-electron model systems for which fw[p>i (r)] is the exact kinetic energy. We further have... 

Chakraborty, D. Ayers, P. W. Derivation of generalized Von-Weizsacker kinetic energies from quasi-probabihty distribution functions. In Statistical Complexity. Springer Science, 2011. 

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 ... 

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... 

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),... 

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. 

The Fisher information, reminiscent of von Weizsacker s [70] inhomogeneity correction to electronic kinetic energy in the Thomas-Fermi theory, charactoizes the compactness of the probability density. For example, the Fisher information in normal distribution measures the inverse of its variance, called invariance, while the complementary Shannon entropy is proportional to the logarithm of variance, thus monotonically increasing with the spread of Gaussian distribution. Therefore, Shannon entropy and intrinsic accuracy describe complementary facets of the probability density the former reflects the distribution s ( spread ( disorder , a measure of uncertainty), while the latter measures its narrowness ( order ). 

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. 

Von Weizsacker found that the kinetic energy of the almost uniform gas is given by... 

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 ... 

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