Kinetic electron energy density

The characteristics of BCP inform on the nature of atom-atom interaction since attractors are attributed to the positions of atoms these are electron density at BCP (pbcp), laplacian of this electron density (V pbcp), the total electron energy density at BCP (Hbcp) and the components of the latter value the kinetic electron energy density (Gbcp - always positive value) and the potential electron energy density (Vbcp - always negative value). There are the following relations between the mentioned above values Eqs. 17.1 and 17.2. 

Gbcp and Vbcp are the components of the mentioned above Hbcp energy density and represent the kinetic electron energy density and the potential electron energy density, respectively. Gbcp is always a positive value while Vbcp is always negative. 

The electronic potential energy density (r), the virial of the forces exerted on the electrons (eqn (6.30)), and the electronic kinetic energy density, G(r) (eqn (5.49)), define the electronic energy density. 

To compare the kinetic and potential energy densities on an equal standing, instead of the 2 1 virial ratio, Cremer and Kraka [110] evaluate the total electronic energy density at the BCP ... 

Similarly to Eq. (2.6), fCis a proportionality constant containing fixed operating conditions, for example incident electron current density, transmission of the analyzer at the kinetic energy Ea, efficiency of the detector at the kinetic energy Ea, and the probability of the Auger transition XYZ. 

DFT methods compute electron correlation via general functionals of the electron density (see Appendix A for details). DFT functionals partition the electronic energy into several components which are computed separately the kinetic energy, the electron-nuclear interaction, the Coulomb repulsion, and an exchange-correlation term accounting for the remainder of the electron-electron interaction (which is itself... 

Here, ej f are the vibration-rotation energies of the initial (anion) and final (neutral) states, and E denotes the kinetic energy carried away by the ejected electron (e.g., the initial state corresponds to an anion and the final state to a neutral molecule plus an ejected electron). The density of translational energy states of the ejected electron is p(E) = 4 nneL (2meE) /h. We have used the short-hand notation involving P P/p to symbolize the multidimensional derivative operators that arise in the non BO couplings as discussed above ... 

In the Thomas-Fermi model,49 the kinetic energy density of the electron gas is written as... 

One knows, however, that the simple density-functional theories cannot produce an oscillatory density profile. The energy obtained by Schmickler and Henderson55 is, of course, lower than that of Smith54 because of the extra parameters, but the oscillations in the profile found are smaller than the true Friedel oscillations. Further, the density-functional theories often give seriously inexact results. The problem is in the incorrect treatment of the electronic kinetic energy, which is, of course, a major contributor to the total electronic energy. The electronic kinetic energy is not a simple functional of the electron density like e(n) + c Vn 2/n, but a... 

Modern theories of electronic structure at a metal surface, which have proved their accuracy for bare metal surfaces, have now been applied to the calculation of electron density profiles in the presence of adsorbed species or other external sources of potential. The spillover of the negative (electronic) charge density from the positive (ionic) background and the overlap of the former with the electrolyte are the crucial effects. Self-consistent calculations, in which the electronic kinetic energy is correctly taken into account, may have to replace the simpler density-functional treatments which have been used most often. The situation for liquid metals, for which the density profile for the positive (ionic) charge density is required, is not as satisfactory as for solid metals, for which the crystal structure is known. 

Glossman, M. D., L. C. Baibas, A. Rubio, and I. A. Alonso. 1994. Nonlocal Exchange and Kinetic Energy Density Functionals with Correct Asymptotic Behavior for Electronic Systems. Int. I. Q. Chem. 49, 171. 

In Eq. (4.23d), N is the number density of molecules, cross section for production of any primary species x at electron energy E having an energetic threshold Ix for its production, and N/T) is the total number of such species formed by the complete absorption of an electron of kinetic energy T. In this sense, y can be thought of as a distribution function. Specifically, if x refers to ionization, Ix is simply the ionization potential I, Ox is the ionization cross section a, and N = n., the total number of ionizations. 

Glossman, M. D., Baibas, L. C., Rubio, A. and Alonso, J. A. Nonlocal exchange and kinetic energy density functionals with correct asymptotic behavior for electronic systems, Int.J. Quantum ( hem., 49 (1994), 171-184... 

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 energy of the electron gas is composed of two terms, one Hartree-Fock term (T)hp) and one correlation term (Hartree-Fock term comprises the zero-point kinetic energy density and the exchange contribution (first and second terms on the right in equation 1.148, respectively) ... 

With the purpose of evaluate not only the energy but also the electron density itself, Ashby and Holzman [15] performed calculations in which the relativistic TF density was replaced at short distancies from the nucleus from the one obtained for the 1 s Dirac orbital for an hydrogenic atom, matched continuously to the semiclassical density at a switching radius rg where the kinetic energy density of both descriptions also match. 

A differential virial theorem represents an exact, local (at space point r) relation involving the external potential u(r), the (ee) interaction potential u r,r ), the diagonal elements of the 1st and 2nd order DMs, n(r) and n2(r,r ), and the 1st order DM p(ri r2) close to diagonal , for a particular system. As it will be shown, it is a very useful tool for establishing various exact relations for a many electron systems. The mentioned dependence on p may be written in terms of the kinetic energy density tensor, defined as... 

Compared to computational approaches based on the g-electron density, approaches based on the g-electron reduced density matrix have the advantage that the kinetic energy functional can be written in an explicit form ... 

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