Electronic kinetic energy densities

The electronic interaction in the H2 production process in Section 2.2.3 can be expressed in terms of the quantum energy densities108-113 based on the regional DFT.108-114 The electronic kinetic energy density nT(r) is defined as... 

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. 

Meta-GGA functionals incorporate the Laplacian of the electron density and generally depend on the electron kinetic energy density. These features allow a systematic improvement of results for many quantum chemical calculations. We have used the exchange functional BR89 (Becke-Roussel 1989 represented in analytic form) [87,90] and the correlation functional B94 (Becke 1994) [90, 91]. Minnesota functionals tested are M05 [92], M05-2X [93], M06-L [94], M06-HF [95], M06 [96], M06-2X [96], and Ml 1 [97]. Another global hybrid studied is BMK [98]. A hybrid extension of the nonempirical exchange-correlation TPSS (Tao, Perdew, Staroverov, and Scuseria) [99] and functional TPSSh [100] is also examined. 

Mata et al. [49] have recently proposed a relationship between H-bonding energy and electronic kinetic energy density at the BCP based on gas phase studies of of FH... FR (R = H, Li, A1,C1, CCH) complexes (in a.u.) ... 

These two linear relationships between and and G appear to be quite simple (BSSE-free) and useful approximations, which enable the evaluation of the H-bonding energy in solid state using theoretical [66] or experimental (derived from X-ray or synchrotron diffraction experiment [67, 68]) energy densities. In the latter case the electronic kinetic energy density distribution, G, is derived from the following accurate approximation for closed-shell interactions (like H-bond is) in terms of experimental electron density, p, and its Laplacian V p at the BCP (in a.u.) [69] ... 

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. 

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... 

Note that, for a non-interacting system of electrons, the kinetic energy is just the sum of the individual electronic kinetic energies. Within an orbital expression for the density, Eq. (8.14) may then be rewritten as... 

LT2A Local square kinetic energy density exchange functional depending only on Ihe kinetic energy density (i.e.. not at all on the electron density). Maximoff, S. N., Enzerhof, M., Scuseria, G. E. 2002. 7. Chem. Phys., 117, 3074. 

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