Pauli kinetic energy density

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... 

Fig. 1 Orbital localization measured by the orbital-contribution entropy 5orb for the Hartree-Fock calculation of the N2 molecule. From left to right. 1, canonical orbitals 2, natural localized orbitals 5, Pauli kinetic energy density fy 4, ELF...

Fig. 2 Data for the Ne atom using the wave function of Clement and Roetti. Solid line, ELF (the horizontal line indicates rj = 0.5) dashed line, Pauli kinetic energy density tp dotted line, tp h = Cp The circles, marked by capital letters, highlight the situation where tp = tpji. The arrows point to the corresponding positions with ij = 0.5...

The most important ingredient in ELF seems to be the curvature of the Fermi hole (respectively the value of Pauli kinetic energy density). However, the respective functions itself do not show the rich structure so typical for ELF. It is exclusively the calibration with respect to the unifonn electron gas that generates all the desired features. Thus, the ELF values depend on the function used for the calibration, which was arbitrarily chosen to be the kinetic energy density of the uniform electron gas. This arbitrariness of the choice was often criticized, e.g., by Bader [73]. Another calibration function was examined by Ayers [54]. He used nearly free electron gas, but found the results much less satisfactory. 

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 ... 

In a pericyclic reaction, the electron density is spread among the bonds involved in the rearrangement (the reason for aromatic TSs). On the other hand, pseudopericyclic reactions are characterized by electron accumulations and depletions on different atoms. Hence, the electron distributions in the TSs are not uniform for the bonds involved in the rearrangement. Recently some of us [121,122] showed that since the electron localization function (ELF), which measures the excess of kinetic energy density due to the Pauli repulsion, accounts for the electron distribution, we could expect connected (delocalized) pictures of bonds in pericyclic reactions, while pseudopericyclic reactions would give rise to disconnected (localized) pictures. Thus, ELF proves to be a valuable tool to differentiate between both reaction mechanisms. 

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 field Coulomb correlations since its quantum-mechanical source charge distribution is the pair-correlation density g(r,r ). On the other hand, the field Z,Jr) arises from the kinetic-energy-density tensor tj (r). It is the difference of the fields derived from the tensor for the interacting and Kohn-Sham noninteracting systems, and is thereby representative of the correlation-kinetic-energy. 

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) ... 

These functions provide a local measure of the effect of the Pauli repulsion on the kinetic energy density. In the region of space where the Pauli repulsion is weaker than in a uniform electron gas of identical density, we should say where the local parallel pairing is lower, i/ji r) and are... 

The ELF is defined in terms of the excess of local kinetic energy density due to the Pauli exclusion principle T(p(r)) and Thomas-Ferml kinetic energy density Th(p(r))... 

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 ... 

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