Hulthén Approximations to 1s and 2p Orbitals of Atoms

It is shown that a substantial energy improvement over the Slater-orbital description is obtained by the variational use of 0s and 1p functions in the 1s and 1p shells of atoms, where 0s and 1p represent differences of two Slater 0s and Slater 1p functions, respectively. Reduction of the number of parameters by enforcing nuclear cusp conditions does not affect the variational energy significantly. The reduction yields one-parameter 0s and 1p orbitals giving a better energy than both one-parameter Slater functions and two-parameter variable-principal-quantum-number functions. It is also shown that in correlated functions of the form ψ = φ (r1) φ (r2) χ (r12), the orbital 0s when used as a one-term approximation to the best possible φ gives an excellent result for the energy of heliumlike systems and the correlation function 1 + c exp(− ηr12) gives a very good approximation to the best χ(r12). Reduction of the number of parameters by requiring both nuclear and correlation cusp conditions yields a two-parameter trial function which for helium has the energy − 2.90002 a.u., vs the Roothaan–Weiss best energy, − 2.90039 a.u., for a function of this form.