Approximate analytical self-consistent field wavefunctions were calculated for the ground state and for certain excited states of the laser-active ion Nd3 +. The following states of the 4f3 configuration of Nd3 + were covered by our calculations: S4,D4,F4,G4,, and I4. For the purpose of this work we combined the minimum and the saturated basis set approximations. All the orbital exponents were accurately optimized. The d orbitals, and particularly the 4f orbital, were calculated unusually accurately. Except the optically unaffected orbitals 1s, 2s, and 2p, all the orbitals of the s, p, d, and f symmetries exhibited zero deviation (to five decimals) from the tail nodal extinction. The virial theorem was always satisfied to at least five significant figures.

screened potential were calculated and compared with the result of exact phase-shift calculations, JR CX . Table III lists these results in detail and permits the recognition of the following:(1) The Moliere approximation renders R by Eqs.(15) and (16) with an error of the order £ [Eq. (5) and Table I].(2) The classical approximation renders R c \^s by Eqs. (22) and (23) with an error which is comparable to that of R for scattering angles larger than 10°.(3) The large-angle approximation renders RL.A. by Eqs. (25) and (26) with errors generally larger than °f ^class* (4) The first-order Born approximation gives values, i?Bom, the errors of which exceed those of all the other approximations. Even for the light element Z=29 and for small angles the errors are larger than 10%. ACKNOWLEDGMENTSWe are indebted to J. W. Motz for numerous discussions and for pointing out to us the need for rather simple and yet sufficiently accurate calculations of the elastic electron-scattering cross section.We would also like to acknowledge the valuable assistance of R. L. Scott in preparing the program and performing the computations.The analytical self-consistent field (SCF) theory, based on the relativistic Breit equation generalized for many particles, was developed for closed-shell systems. The relativistic SCF equations, both of the absolute and of the expansion method type, were derived in the four-component spinor representation. The Breit operator was considered in the first-order perturbation theory. The formulas for the relativistic atomic integrals were derived in terms of simple functions. * The work was originated at

Self-consistent field calculations by the expansion method were carried out for the ground states and several excited states of atoms and ions of Al and Cu. The results for the total energies of the states computed represent very accurately the absolute Hartree-Fock solutions. The wave functions were calculated with the requirement to satisfy identically the cusp condition so that they can be considered to be particularly accurate in the immediate vicinity of the nucleus. Comparison with experiment is carried out in particular for the calculated energy levels.

Approximate analytical wavefunctions were calculated for the atoms Xe and Rn. The minimum basis set expansion used makes the wavefunctions sufficiently simple for orientational calculations. The Hartree—Fock total energies are represented to three significant figures. The virial theorem and the SCF thresholds are quite satisfactory.

The accurate analytical self-consistent-field wavefunctions were calculated for the Fe3+ ground state, and for the Fe3+ excited configurations with two and three open shells of different symmetries. All the expansion basis sets were selected so as to satisfy precisely the cusp condition. The deviations from the tail nodal extinction for the radial functions Piλ(r) never exceeded 0.00010. Our wavefunctions for Fe3+ are the most accurate ones available. For the two- and three-open-shell states of Fe3+ the calculated wavefunctions and total energies are presented for the first time. The experimental energy levels were available for comparison with one of the excited states calculated.