Asymptotically Correct Thomas-Fermi Model Density

Abstract

The Thomas-Fermi (TF) model provides a fairly useful description of the average properties of atoms and ions. Its importance lies in its great simplicity. It can be regarded as a statistical model with the local density deduced from phase space considerations. Alternatively, the local density can be obtained from the WKB radial wave functions in the classically allowed region. This approach makes it clear that the model is not accurate for atoms and ions, near the nucleus where the potential is expected to vary rapidly, and in the classically forbidden region far away from the nucleus. However, the Thomas-Fermi density does provide a good description of the electron density at intermediate distances from the nucleus. On the other hand, in (2.15) we do have a rigorous relation for the asymptotic behaviour of the wave function in terms of the separation energy of the last electron. This relation can be used to obtain the correct form of the asymptotic density. Here we consider a density which has the TF form in the intermediate distance r region, and continues smoothly into the correct asymptotic form for large values of r. This density allows us to deduce several useful properties of atoms and ions, such as expectation values 〈r²ⁿ〉, polarizabilities, etc.. Implicit in this approach is the suggestion that while the TF density is not correct for very large values of r, and the asymptotic expression for the density is not valid for small values of r, there is some domain of intermediate values of rwhere both the densities are reliable. We use this region to continue from TF density in the intermediate region into the asymptotic form of the density.