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This article, considers in detail P. Gombás's idea of grouping electrons into n- and nl-shells in the Thomas–Fermi theory of free atoms briefly, the TFG n- and TFG nl-models respectively). Using these models, exact analytical expressions for the total energy E and the atomic form factor F(κ) are obtained. All integrals of the TFG nl-model are computed by means of the hypergeometric functions ₂F₁(x), ₃F₂(x), F₂(x,y) and F_A(x₁,…,x₆) for the first time. In particular, Weizsäcker's gradient correction to the kinetic energy of the nl-th shell generates a new numerical triangle with coefficients bw=n+2l(n-l-1).

We summarize recent progress in the formulation of a theory for white dwarfs and neutron stars based on the general relativistic Thomas-Fermi equations of equilibrium.

For independent electrons moving in a one-body potential V(r) of arbitrarily low symmetry, a partial differential equation is set up for the Slater sum Z (r, β), where β = (k_BT)⁻¹. As an example, the case when V(r) is the result of an applied time-independent and spatially constant electric field of arbitrary strength is briefly considered.