Thomas-Fermi equations

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). 

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... 

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... 

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