Simple Atomic Model and its Associated Wave Function

A simple atomic model, with features characteristic of the screening-theory approach, is constructed and provides simple analytic trial wave functions ${\psi }_t$ which describe the motion of the inner electrons particularly well. Basing our model on the energy-extremum principle, in contrast to a semiempirical or phenomenological approach, we can judge improvements in the model in a consistent manner. The single-particle orbitals from which ${\psi }_t$ is constructed are taken to be orthogonal hydrogenlike functions with effective nuclear charges ${\tilde{Z}}_l$ which depend on the quantum number $l$; the ${\tilde{Z}}_l$ serve as variational parameters. By a succession of reasonably good upper bounds the evaluation of the expectation value of the $\frac{1}{r_{\mathrm{ij}}}$ interelectronic terms is reduced to trivial algebraic expressions involving only the occupation number of electrons for each of the ($n,l$) subshells; no numerical work is necessary to determine the energy-optimized set of ${\tilde{Z}}_l$ With the resulting ${\psi }_t$, which is most appropriate for closed-shell atoms, a variety of atomic entities can be obtained directly in analytic form. This model is considered physically more realistic than the Thomas-Fermi model, yielding results of greater accuracy in general. (For large $Z$, the results obtained yield the usual Thomas-Fermi scaling laws.) Much more importantly, these wave functions will be shown elsewhere to be very useful in exploratory analytic studies to judge the effectiveness of variational techniques in the evaluation of the expectation values of various atomic operators. Also, a straightforward adaptation of the present approach will be shown elsewhere to provide a much better understanding of atomic structure in the presence of the intense magnetic fields characteristic of pulsars, a regime in which studies of the heavier atoms have been limited to a statistical Thomas-Fermi-like approach.