Approximate Analytical Solutions of the Thomasndash;Fermindash;Dirac and Thomasndash;Fermindash;Diracndash;Gombaacute;s Equations

摘要

Approximate analytical solutions of the Thomasndash;Fermindash;Dirac (TFD) and Thomasndash;Fermindash;Diracndash;Gombaacute;s (TFDG) differential equations are obtained for neutral and positively charged atoms. The approximate solutions are of the formpsgr;thinsp;equals;thinsp;phgr;0thinsp;plus;thinsp;eegr;andPSgr;thinsp;equals;thinsp;phgr;0thinsp;plus;thinsp;eegr;tilde;, respectively, wherephgr;0is a universal approximate analytical solution of the Thomasndash;Fermi (TF) differential equation for neutral atoms andeegr;andeegr;tilde;are correction functions. The functionphgr;0has been obtained previously, by making use of a variational principle (the equivalent of the TF equation), and the functionseegr;andeegr;tilde;are determined by solving secondhyphen;order inhomogeneous linear differential equations with the boundary and subsidiary (normalization) conditions that are required in the TFD and TFDG models, respectively. The correction functionseegr;andeegr;tilde;depend on the atomic or ionic radius, which, in turn, depends on the atomic numberZand electron numberNof the system considered. The approximate solutionspsgr;andPSgr;are tested by calculating the first ionization energies of Ar, Kr, and Xe. The values obtained are about the same as those calculated from the exact numerical solution of the TFD and TFDG equations. In three Appendices approaches for improved approximate analytical solutions of the TFD and TFDG equations are outlined. Appendix A proposes an iterative procedure for obtaining the correction functionseegr;andeegr;tilde;in successively more refined approximations, which is expected to be of importance for multiply charged ions or for neutral atoms of low atomic number. Appendix B establishes a variational principle (which to the author's knowledge has not been formulated before) for obtaining approximate analytical solutions of the TFD and TFDG equations, or for achieving further refinements in the approximate solutionspsgr;andPSgr;that are derived in this paper. Appendix C describes Jensen's modification of the TFD and TFDG models, which, with the technique developed in this paper, enables one to obtain approximate analytical solutions for negative ions.