Analytical multiconfiguration treatment to one-center many-electron He-isoelectronic ions and Period-II elements with H-like bound-states

Abstract

Employing H-like spin-orbitals (SOs) in electronic structure theory is a long-awaited quantum problem as the analytical integral of Coulomb interaction is very difficult to solve for one-center many-electron (1c-ne) system. He-isoelectronic ions become a benchmark. Complexity grows fast for Period-II s- and p-block elements with increasing number of electrons. Moreover, Hartree-Fock Self-Consistent Field (SCF) and post Hartree-Fock SCF theories generally make use of closed-shell, restricted and unrestricted open-shell single configurations (SCs) but actual electronic bound states urge for multiconfigurations (MCs). After Born-Oppenheimer (BO) approximation, utilization of associated Laguerre polynomial/Whittaker-M function basis sets of H-like SOs for the Coulomb Green$'$s function among electrons furnishes analytical, terminable, simple and finitely summed integrals in terms of Lauricella functions. MCs complying with so-called ground, singly and multiply excited states incurring s- and p-SOs are constructed to capture monopole and dipole factors only. However, we believe that quadrupole and higher order poles can be achieved as a product of angular integrals using Wigner 3-j symbols and closed forms of radial integrals. We have observed good agreement among literature and exact ground state energies (GSEs) of He-isoelectronic ions and Period-II elements with both their so-called ground electronic configurations as well as MCs. For certain elements, we have found satisfactory results for ionization energies (IEs).

Appendices

Appendix A Standard equations and integrals

Associated Laguerre Polynomial [17,18,19,20, 41]

$$\begin{aligned} L^{\mu }_{k}(x)=\sum _{m=0}^{k}(-1)^m\frac{(\mu +k)!}{(k-m)!(\mu +m)!m!}x^m \hspace{0.5cm}\text {where}\hspace{0.5cm}\mu >-1 \end{aligned}$$

(A.1)

Lower and Upper Incomplete Gamma function [17,18,19,20, 41]

$$\begin{aligned} &\gamma (a,x)=\int _{0}^{x}e^{-t}t^{a-1}dt=(a-1)!\bigg (1-e^{-x}\sum _{s=0}^{a-1}\frac{x^s}{s!}\bigg ) \nonumber \\ &\Gamma (a,x)=\int _{x}^{\infty }e^{-t}t^{a-1}dt=(a-1)!\bigg (e^{-x}\sum _{s=0}^{a-1}\frac{x^s}{s!}\bigg )\\&\text {where}\hspace{0.5cm} Re(a)\ge 0 \end{aligned}$$

(A.2)

Standard Integral-I (Erd$\acute{e}$lyi’s Integral) [17,18,19,20, 55]

$$\begin{aligned} \begin{aligned}&\int _0^{\infty } x^{\left( \varrho -1\right) } e^{-cx}M_{\kappa _1,\gamma _1-\frac{1}{2}}(a_1x)M_{\kappa _2, \gamma _2-\frac{1}{2}}(a_2x) dx\\&=a_1^{\gamma _1}a_2^{\gamma _2}(c+A)^{-\varrho -M}\Gamma (\varrho +M)\\&\quad \times F_2\left( \varrho +M;\gamma _1-\kappa _1,\gamma _2-\kappa _2;2\gamma _1,2\gamma _2;\frac{a_1}{c+A},\frac{a_2}{c+A}\right) \\&\text {where } Re(\varrho +M)>0, Re\left( c\pm \frac{1}{2}a_1\pm \frac{1}{2}a_2\right) >0 \text { and } M\\&=\gamma _1+\gamma _2 \end{aligned} \end{aligned}$$

(A.3)

Recurrence Relations [19, 41]

$$\begin{aligned} a) &\int Y_l^{m*}Y_0^0Y_l^md\Omega=\sqrt{\frac{1}{4\pi }}\nonumber \\ b) &\int Y_{l+1}^{m*}Y_1^0Y_l^md\Omega=\sqrt{\frac{3}{4\pi }}\sqrt{\frac{(l-m+1)(l+m+1)}{(2l+1)(2l+3)}}\nonumber \\ c) &\int Y_{l-1}^{m*}Y_1^0Y_l^md\Omega=\sqrt{\frac{3}{4\pi }}\sqrt{\frac{(l-m)(l+m)}{(2l+1)(2l-1)}}\nonumber \\ d) &\int Y_{l+1}^{m+1*}Y_1^1Y_l^md\Omega=\sqrt{\frac{3}{8\pi }}\sqrt{\frac{(l+m+1)(l+m+2)}{(2l+1)(2l+3)}}\nonumber \\ e) &\int Y_{l-1}^{m+1*}Y_1^1Y_l^md\Omega=-\sqrt{\frac{3}{8\pi }}\sqrt{\frac{(l-m)(l-m-1)}{(2l+1)(2l-1)}}\nonumber \\ f) &\int Y_{l+1}^{m-1*}Y_1^{-1}Y_l^md\Omega=\sqrt{\frac{3}{8\pi }}\sqrt{\frac{(l-m+1)(l-m+2)}{(2l+1)(2l+3)}}\nonumber \\ g) &\int Y_{l-1}^{m-1*}Y_1^{-1}Y_l^md\Omega=-\sqrt{\frac{3}{8\pi }}\sqrt{\frac{(l+m)(l+m-1)}{(2l+1)(2l-1)}} \end{aligned}$$

(A.4)

Appendix B Degeneracies of lowest bound states

Table 4 Lowest bound states obtained after diagonalization for He-isoelectronic ions

Table 5 Lowest bound states obtained after diagonalization for atoms B, C, O and F