Fermi equation

Coulson, C., Compt. rend. 239, 868, Sur une relation d Odiot et Daudel entre la density lectronique et le potentiel lectrique d un atome." The relation is derived from the Thomas-Fermi equation. 

The Thomas-Fermi equation (21) is independent of Z or any other physical constants and is valid for all atoms. The normalization condition in dimensionless form simply becomes... 

Note that the various T and V terms defined in Eqs. (8.3)—(8.5) are functions of the density, while the density itself is a function of three-dimensional spatial coordinates. A function whose argument is also a function is called a functional , and thus the T and V terms are density functionals . The Thomas-Fermi equations, together with an assumed variational principle, represented the first effort to define a density functional theory (DFT) the energy is computed with no reference to a wave function. However, while these equations are of significant historical interest, the underlying assumptions are sufficiently inaccurate that they find no use in modem chemistry (in Thomas-Fermi DFT, all molecules are unstable relative to dissociation into their constituent atoms...)... 

Here, q is the inverse of a screening length related to the valence electron density which contributes to the screening and /u. is a Lagrange multiplier controlling the total number of particles. The boundary conditions to be used with Equation (23) are that V(r) must match Vc r) at Rs and that rV(r) -> -1 as r -> 0. Once we have solved the Thomas-Fermi equation, we have calculated the screened function, defined as the bare impurity potential divided to the screened one, namely Vb/V. 

If the locality hypothesis is valid, then = vT(r). and the Thomas-Fermi equation... 

Figure 1 Types of solution of the dimensionless Thomas-Fermi equation (10). Function 4> expresses the potential distribution in the atomic ion as a function of distance from the nucleus.

D. A. Kirzhnits, Quantum corrections to the Thomas-Fermi equation, Sov. Phys. JETP 5, 64-72 (1957). 

From the form of the Fermi function, it is seen that when it is used for an electrode process where Ep is modulated according to Ep = E°p Ve, the Ve term does not simply factorize from the Fermi equation giving a normal Tafel relation even with an empirically included j8 factor (cf. Gurney ), since... 

For fixed normalization the Lagrange multiplier terms in 8Ts vanish. If these constants are undetermined, it might appear that they could be replaced by a single global constant pt. If so, this would result in the formula [22] 8Ts = J d3r p, — v(r) 8p(r). Then the density functional derivative would be a local function vr(v) such that STj/Sp = Vj-(r) = ix — v(r). This is the Thomas-Fermi equation, so that the locality hypothesis for vT implies an exact Thomas-Fermi theory for noninteracting electrons. 

It has since been discovered that a more accurate equation, which takes account of radial correlations the g Thomas-Fermi equation, does allow d-orbital collapse to occur. Thus Mayer was actually not far from her objective... 

E2.2 The Fermi equation for the hyperfine coupling a = iM>glJ-BgNl NP(0) contains the factor p(0) = lt(f(0)p, the spin density at the nucleus. Its value for the hydrogen atom is obtained with the wave-function i r(r) =, ... 

The two cases from above can be compressed in the formal Thomas-Fermi equation, as follows ... 

An equation in the distributions sense, equivalent with the Thomas-Fermi equation, can be introduced by taking into account the expression of the Coulomb-Poisson potential, resulting the Thomas-Fermi differential equation with 0(x) instead of ( ), thus ... 

The first relation is obtained from the Thomas-Fermi equation, multiplied with p and then integrated. Alternatively, one can proof the assertion (a) by noting... 

The discussion of the two group equations (10), (10 a) leads only to rather obvious results in the case of a simple, bare pile and their usefulness becomes apparent only in less simple cases. The situation is rather the opposite for the modifled Fermi equations (7), (7 a) which can be solved easily only for a simple bare pile. Even finding the adjoint operator to... 

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