Screening of Many-Electron Atoms

The problem of the screening of many-electron atoms (ions) is considered. Using the scaled Thomas-Fermi (STF) method of Stewart and Rotenberg for approximating the core potential of an atom (ion) with nuclear charge $Z$ and net core charge $Z^{*}$, a screened scaled Thomas-Fermi potential (SSTF) is presented: $V\left(r\right)=-\left(\frac{Z{e}^2}{r}\right)U_{\mathrm{SSTF}}$, where $\{U_{\mathrm{SSTF}},=U(r)-Ar, 0\le r\le a\}\{=BU(r)\mathrm{}{e}^{\frac{\mathrm{}(a-r)\mathrm{}}{D}}, r\ge a.\mathrm{}\}$ Here $D$ is the screening radius; $A$ and $B$ are constants; and $U\left(r\right)\mathrm{}$ is given by $\left\{U\right(r)=U_{\mathrm{STF}}\mathrm{}(r)\mathrm{},=\varphi (\frac{r}{\alpha \mu })+\frac{\mathrm{qr}}{r_c}, r\le r_c\}\{=q, r\ge r_c\}$ where $\varphi \left(\frac{r}{\alpha \mu }\right)$ is the well-known Thomas-Fermi function, $r_c$ is the STF core radius with $\mu =\frac{1}{4}{\left(\frac{9{\pi }^2}{2Z}\right)}^{\frac{1}{3}}=0.885\mathrm{}34Z^{\frac{-1\mathrm{}}{3}}$, $\alpha$ is the adjustable scaling factor, and $q=\frac{Z^{*}}{Z}$. The constants $B$ and $A$ are given by $B=\frac{D\left[a{U}^{\prime }\mathrm{}\right(a)-U(a\left)\right]}{\left\{D\right[a{U}^{\prime }\mathrm{}\left(a\right)-U\left(a\right)]-aU(a\left)\right\}}$ and $A=(1-B)\mathrm{}\frac{U\left(a\right)\mathrm{}}{a}$, where $U^{\prime }\mathrm{}\left(a\right)=\frac{\partial U}{\partial r}$ at $r=a$. Eigenvalues of the Schrödinger equation with the SSTF potential are given for the $3d$, $4s$, $4p$, $4d$, $4f$, and $5s$ orbitals of FeI and FeVIII. Comparisons between the ions and corresponding hydrogenic orbitals show that the variations of the SSTF eigen-values and the limiting screening radii are generally very different, with different level crossings and different relative energies. It is concluded that in order to obtain a correlation between a limiting screening radius and the observed disappearance of lines from a many-electron atom (ion), SSTF solutions are needed for the ion of interest. It is also concluded that until an accurate external screening function is obtained and applied to the screening of Hartree-Fock isolated-atom (-ion) solutions, SSTF solutions will be useful for the very important astrophysical problems of calculating equations of state and opacities for high-$Z$ matter at stellar densities and temperatures.