Fermi hole kinetic energy

As noted above, the curl of the expression on the right-hand side of Equation 7.47 vanishes. However, it does not mean that the Coulombic and non-Coulombic components—the former is the electric field produced by the Fermi-Coulomb hole and the latter arises from the kinetic energy tensor—of this field also have vanishing curl. Thus the potential Wxc of Equation 7.38 may sometimes be path dependent [21]. 

The Fermi-Dirac distributions of carriers within the subbands means that even when the subband separation is less than the energy of an LO phonon, an electron (or hole) with sufficient in-plane kinetic energy in the upper subband can still emit an LO phonon. Although this effect decreases with temperature and subband separation, it is nonetheless present and competing with radiative transitions and as... 

Soon after the appearance of the ELF, some interesting interpretations and remarks were given by Dobson [41]. He stated that the kernel of the ELF formula (cf. Eqs. (10) and (11)) is valid for states with zero Schrodinger current density, which explicitly means that time dependency would change the formula. Additionally, Dobson connected the Fermi hole curvature with the kinetic energy of the relative motion of same-spin electron pairs. 

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. 

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