Analytical two-center one-electron overlap and exchange integrals for $^{1}\Sigma$ states: Lah number guided Coulomb Green function of H-like s-orbitals

Abstract

Theoretical studies of two-center one-electron (2c-1e) small microcluster are associated with hurdles in Schr$\ddot{o}$dinger equation (SE) born out of divergence of Coulomb interactions and nuclear separation (R). The SE deals with morphologically altered H-like AOs, Slater type orbitals (STO), Gaussian type orbitals (GTO), B-spline, Sturmian function and etc in both VBT and MOT calculations. Few elegant computational and analytical methods are available for STO, GTO and other square integrable trial wavefunction under Born-Oppenheimer approximation. Even so, analytical treatment for H-like AOs has become very necessary. Utilizing Sheffer identity in associated Laguerre polynomial/Whittaker-M H-like AOs and adopting elliptic coordinates provide exact, analytical and simple 2c-1e Coulomb exchange interactions (Ks) and overlap integrals as functions of R with different scaling factors associated with electrons. The energetics of diatomic molecule is evident to be the function of R with extrema as Lah number moderated $L_n^{-1}$ for nuclear coordinates.

Graphical abstract

Analytical and simple 2c-1e Coulomb exchange interactions (Ks) and overlap integrals (Ss) as functions of R are presented utilizing Sheffer identity in associated Laguerre polynomial/ Whittaker-M H-like AOs and adopting elliptic coordinates mediated through Lah number. The energetics of the diatomic molecule is evident to be the function of R.

Appendix A Standard Equations and Integrals

Associated Laguerre Polynomial³⁹

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

(A1)

Lower and Upper Incomplete Gamma function³⁹

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

(A2)

Sheffer identity³³

$$\begin{aligned} L_n^\alpha (x+y)=\sum _{k=0}^{n}\left( {\begin{array}{c}n\\ k\end{array}}\right) L_k^\alpha (x)L_{n-k}^{-1}(y) \end{aligned}$$

(A3)

$\underline{{\textbf {Relation between}}\ L_n^{-1}\ {\textbf {and Lah number,}}\ L(n,k)}$^34,35

$$\begin{aligned} L_n^{-1}(x)=\frac{1}{n!}\sum _{k=0}^{n}L(n,k)(-x)^k=\sum _{k=0}^{n}\left( {\begin{array}{c}n-1\\ k-1\end{array}}\right) \frac{(-x)^k}{k!} \end{aligned}$$

(A4)

Standard integral⁴²

$$\begin{aligned}&\int x^ne^{bx}dx=e^{bx}\sum _{k=0}^{n}\frac{(-1)^k\,{^nP_k}x^{n-k}}{b^{k+1}}\nonumber \\ {}&\int x^ne^{-bx}dx=-e^{-bx}\sum _{k=0}^{n}\frac{{^nP_k}\,x^{n-k}}{b^{k+1}} \end{aligned}$$

(A5)

Recurrence Relations³⁹

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

(A6)