Local Fermi wavelength

The Seitz radius rs is the radius of a sphere which on average contains one electron, so rs —> 0 in the high-density limit valence electrons typically have 1 bohr < r, < 6 bohr. The relative spin polarization ( vanishes for a spin-unpolarized system, and equals + 1 or -1 where all the electron spins are up or down, respectively. The inhomogeneity parameter s measures how fast the density varies on the scale of the local Fermi wavelength valence electrons typically have 0 < s < 3. Both rs and s diverge in the low-density tail of an atom or molecule. 

This leads to a local Fermi wave number, ky(r), given by / k. (r) /2m = L i.(r), and from Lq. (15-4), a local electron density N(r) =/<. (r) /(37c ). This is called the Fermi-Thomas approximation. The essential assumption that is required is that the potential does not vary greatly over the distance corresponding to the electron wavelength. 

Note that PW91 becomes more local as the density is reduced. The explanation is that, as n decreases from oo to 0, the hole density at the origin drops from -n/2 to -n, and, because it integrates to exactly -1, it becomes more localized, on the scale of the Fermi wavelength. 

By construction, LSD is exact for a uniform density, or more generally for a density that varies slowly over space [6]. More precisely, LSD should be valid when the length scale of the density variation is large in comparison with length scales set by the local density, such as the Fermi wavelength 27r/fcF or the screening length l/feg. This condition is rarely satisfied in real electronic systems, so we must look elsewhere to understand why LSD works. 

It is not absolutely necessary to have accurate interatomic potentials to perform reasonably good calculations because the many collisions involved tend to obscure the details of the interaction. This, together with the fact that accurate potentials are only known for a few systems makes the Thomas-Fermi approach quite attractive. The Thomas-Fermi statistical model assumes that the atomic potential V(r) varies slowly enough within an electron wavelength so that many electrons can be localized within a volume over which the potential changes by a fraction of itself. The electrons can then be treated by statistical mechanics and obey Fermi-Dirac statistics. In this approximation, the potential in the atom is given by ... 

As already stated for other experimental parameters, two factors may account for the nonlinear optical response dependence on excitation wavelength Local field factor, f, and intrinsic nonlinear properties of the particles, x2 - The interband contribution to x2 expected to vary only for photon energies at least equal to the IB transition threshold, provided the intraband contribution remains negligible. On the other hand, the hot electron contribution, which accounts for the Fermi smearing mechanism, presents spectral variations for photon energies close to the IB transition threshold, since the electron distribution is modified around the Fermi level by the temperature increase subsequent to light absorption (see 3.2.3). The wavelength dependence of x has been already discussed in Section 6. 

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