Asymptotic Solution of the Thomas-Fermi Equation for Large Atom Radius

The general form of the solution $\phi$ of the Thomas-Fermi equation in the limit of large atom radius $x_b$ is determined as an asymptotic series valid for points $x$ near the atom boundary, by a perturbation on the Sommerfeld inverse-cube form. This generalization of the Coulson-March solution yields the complete form of that part of the asymptotic solution which depends on disposable constants of integration (two in number). The general terms of the series are determined explicitly, and coefficients of leading terms are given numerically as ratios in terms of the disposable constants. The two constants of integration are fixed simultaneously, in general, by the requirement of continuity in value and slope of $\phi$ with the result of a series solution for small $x$, and by the boundary condition. The general method of obtaining the two constants of integration for a compressed atom as power series in an inverse power ${x_b}^{-{\lambda }_2}$ of $x_b$ is outlined, where ${\lambda }_2=\frac{1}{2}({73}^{\frac{1}{2}}-7)$; coefficients of leading terms are calculated by making use of numerical results of Kobayashi et al. The products ${x_b}^3{\phi }_b$, where ${\phi }_b$ is the boundary value of $\phi$, and ${x_b}^7({{\phi }_i}^{\prime }-{{\phi }_{i,\infty }}^{\prime })$, where ${{\phi }_i}^{\prime }-{{\phi }_{i,\infty }}^{\prime }$ is the difference of the initial slope of $\phi$ from its value for an isolated atom, are each represented as an asymptotic power series in ${x_b}^{-{\lambda }_2}$ for $x_b$ large. The two leading coefficients in the former series and the leading coefficient in the latter are determined numerically to relatively high accuracy and compared with values obtained by Gilvarry from the approximate solution of Sauvenier. The results are used to construct a fitted function for ${\phi }_b$ in which only one of five terms appearing is evaluated empirically, and which reproduces accurate numerical values of ${\phi }_b$ from solutions of the Thomas-Fermi equation within 0.1%, in general. This accuracy exceeds by a factor of about ten that obtained previously by March and by Gilvarry with fitted functions containing three and two theoretical terms, respectively. The connection between the general asymptotic solution and previous approximate results is discussed.