Relativistic Independent-Particle Model for Atoms, Based on Analytical Potentials

Abstract

A two-parameter analytic potential for electrons in atoms, introduced by Green, Sellin, and Zachor, is used in conjunction with Dirac's equation to determine independent-particlemodel electronic energy levels for neutral atoms in their ground states. The parameters aredetermined for each atom so that the predicted energy levels approximate observed electronic separation energies in accordance with a weighted least-squares sum, The agreement with experimental data is as good as for relativistic Hartree-Fock results. Electronic energies are given for a random selection of elements with Z =10, 20, . . . , 90.In a recent study, Green, Sellin, and Zachor' used a simple analytical potential to treat bound states of electrons in an atom in the nonrelativistic independent-particle-model approximation.For an atom containing N electrons and Z nuclear protrons, their potential is of the form V(r) = 2[(N -1)T(r) -Z]/r, 'P (r) = 1 -1/[(e' " -1) H + 1] .(1) (2) It was shown that the parameters d and H can be suitably adjusted for each atom to reproduce approximately observed or calculated electronic energies for neutral atoms in their ground states. Such analytical potentials are clearly much more convenient for many applications, such as the calculation of scattering cross sections or transition probabilities, than numerically generated ones such as those obtained from Hartree-Fock-Slater calculations. This is particularly the case when computations for very many atoms are being undertaken. Furthermore, as we shall demonstrate, the electronic energies obtained from the potential {1)are certainly not in worse agreement with experimentally determined values (insofar as the latter can be identified) than results of Hartree-Fock and related calculations.The Green-Sellin-Zachor work has the drawback that the electrons are treated nonrelativistically. This, as is known, leads to poor results for innercore electrons, particularly for heavy elements. Though the relativistic correction, and spin-orbit splitting, can be accounted for by adding suitable terms to the nonrelativistic Hamiltonian, one might as well use the Dirac equation in place of the Schrodinger equation. This has been done in the present work, thus permitting direct comparison with experimental data or relativistic Hartree-Fock calculations. The Dirac equation for an electron moving in a spherically symmetrical potential V(r) is [ca~p+ pmc + V(r)] (=Eg, {3) where all the symbols have their usual meaning. As is well known, the equation can be reduced to radial form in this case: dF/dr= -(K/r)F+ &a [e -V(r)+ 4/a ]G, dG/dr (K/r) G =--, 'a[e -V(r)]F, (4) (5) d u/dr + Qu=O, where 1 d V 3 dV K dV 4B dr 2 16B~d r 2Br dr (6)K(K+ 1) + -'a (e -V)B-where F = F""(r) and G = G. ""(r) are the large and small radial components, respectively. Here, e = Emc, a = e /Ac, K= (lj) {2j+1) while n, l, and j = t + -, ' are the usual principal orbital-angularmomentum and total-angular-momentum quantum numbers. We use rydberg units, 2m = 8= &e~= 1, so that lengths are measured in units of B...

Numerical electronic structure calculations for atoms. II. Generalized variable transformation and relativistic calculations

“…The inclusion of these effects in calculations for the general case of a many‐electron atom in an arbitrary electronic state is not yet solved satisfactorily. Calculations for a many‐electron atom with an effective single‐particle potential: This type of calculations includes the application of the independent particle model potential proposed by Darewych et al 31, which is of essentially the same analytical form as the nonrelativistic analog described in 32. The parameters H and d are functions of the nuclear charge number Z and the number of electrons N .…”

Relativistic effects in the variable-screening model

“…Using the former procedure, values of the relativistic screening parameters have been calculated [23] only for a few atomic systems, and at present no relativistic ab initio values for these parameters are available.…”