Positive kinetic energy density

The positive kinetic energy density of the occupied Kohn-Sham orbitals is... 

K y) appears to be the sum of two contributions the first one Tfr) = yields the expectation value of the kinetie energy when integrated over all space. This contribution is always positive, it is called the definite positive kinetic energy density. The integral over space of the remaining contribution vanishes. For either a real wavefunetion or a stationary state the quantum mechanical current is zero and therefore this remaining term... 

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... 

The definite positive kinetic energy density has received a considerable attention in order to build approximate kinetic energy functionals to be used in a density functional theory not based on orbitals (for a review of such functionals see Lacks and Gordon[33]). Among the most promising routes to this goal, we can mention the approximation proposed by Lee, Lee and Parr[34] ... 

Equation (18) is again simply the kinetic energy density of a uniform Fermi gas, but now applied locally to the electron cloud at position r. The total kinetic energy T is given by... 

The principle underlying the whole of the density theory of atoms and molecules has essentially been exposed by the argument leading to equation (24). We shall see below, however, that there is a need to refine the elementary approximation (18) for the kinetic energy density, in order to transcend TF statistical theory. Nevertheless, even without refinement, certain useful relations follow from this simplest form of density theory and we will discuss these now for positive atomic ions. 

The interaction between bonded atoms is characterized by the values of p(r), V-p(r), G r) and V(r) at the bond critical point. G(r) is the positive definite kinetic energy density and V(r) is the potential energy density. At a bond critical point, the kinetic and potential energy densities are related to the Laplacian by the local form of the virial relation ... 

The local value of the total energy density at a point r, H(r), is another useful topological descriptor that provides supplementary information about the nature of the interaction at r. The total energy density H(r) is the sum of the kinetic energy density G(r), a positive quantity, and the potential energy density F(r), a negative quantity, both densities related with the Laplacian of p(r) through the local expression for the virial theorem [45, 46] ... 

G(r) is the positive definite form of the kinetic energy density [11] and the virial field v(r) may be expressed as... 

As a proof-of-principle for our fitting, we show in Appendix 1 that we can fit the exact positive noninteracting kinetic energy density of the Airy gas to a known second-order gradient expansion. 

Appendix 1 Proof-of-Principle for the Fitting The Positive Noninteracting Kinetic Energy Density... 

Fig. 7 Exact and fitted (using Eq. (A2)) deviation of the positive noninteracting kinetic energy density from its LDA or TF approximation, as a function of position z, for the Airy gas model with force F = 0.10 (atomic units)...

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]. 

Since is that wavefunction which yields density n and minimizes (T), (1.68) shows that c[ ] is the sum of a positive kinetic energy piece and a negative potential energy piece. These pieces of Ec contribute respectively to the first and second terms of the virial theorem, (1.45). Clearly for any one-electron system [9] ... 

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 attempt to determine ELF directly fi om the experimental electron densities was realized in 2002 by Tsirelson and Stash [52]. Again, the ELF formulation of Savin was used with t replaced by second-order expansion of the kinetic energy density (Kirzhnits approximation [53]). The proposed modification of ELF is dependent only on the electron density and its derivatives. This modified ELF reveals the atomic shell structure. However, due to a deformation of the atomic shell shape toward the nucleus, a saddle point emerges at the position between the bonded atoms, where the original ELF displays a maximum. Several different... 

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