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Thomas-Fermi Model

The Thomas-Fermi model is a density variational approximation to a many-electron Coulomb Hamiltonian. For a nucleus of charge Z and two electron spin states in atomic units, its energy functional is

E_Z(rho)=Tint_(R^3)rho^(5/3)dx-Zint_(R^3)(rho(x))/(|x|)dx+1/2int_(R^3)int_(R^3)(rho(x)rho(y))/(|x-y|)dxdy,

where T=3(3pi^2)^(2/3)/10. Its N-electron energy is the infimum over nonnegative densities of integral N with finite displayed terms. The density replaces the antisymmetric many-electron vector, so this is a different variational problem from the full quantum Hamiltonian.

The radial Euler equation leads to the Thomas-Fermi differential equation. Its outer-density and ionization asymptotics specify the constants in the generalized ionization conjecture for full Coulomb atoms.

See also

Coulomb Hamiltonian, Ionization Conjecture, Thomas-Fermi Differential Equation

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References

Solovej, J. P. "The Ionization Conjecture in Hartree-Fock Theory." Ann. Math. 158, 509-576, 2003. https://doi.org/10.4007/annals.2003.158.509.Solovej, J. P. "A New Look at Thomas-Fermi Theory." Mol. Phys. 114, 1036-1040, 2016. https://doi.org/10.1080/00268976.2015.1130273.

Cite this as:

Weisstein, Eric W. "Thomas-Fermi Model." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Thomas-FermiModel.html

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