Analytical Expression of Molecular Integrals over Slater-Type Functions for Generating Their Polynomial Expressions

Abstract

A method to generate analytical expressions with respect to atomic distance and orbital exponent of STF for all types of one-center and two-center molecular integrals over Slater-type functions is presented. The analytical integral obtained in this study is intended to generate approximate polynomial expressions of molecular integrals by which the algebraic molecular orbital equation is defined. As an example, formulation of two-center overlap integral and Coulomb-type electron repulsion integral is demonstrated in detail. Diatomic molecule is a minimum quantum system in which electrons and nuclei coexist with nonadiabatic interactions. To solve the dynamical coupling of nuclei and electrons as a multivariable problem is a fundamental problem in chemistry. Analytical molecular integral over STFs for diatomic molecules is indispensable to a revolutionary step for the construction of algebraic quantum chemistry. Symbolic calculation system is competent to formulate the molecular integrals in this study.

Appendices

Appendices

1.1 Appendix A: Real Spherical Harmonic Function

This appendix contains many relations with respect to normalized real spherical harmonics used in this article. Some duplicates in the body are shown to help following formulations.

$$ {Y}_l^m\left({\theta}_A,\;{\phi}_A\right)={p}_l^m\left( \cos\;{\theta}_A\right)\;{f}_m\left(\phi \right) $$

(3.A1)

1.1.1 Normalized Real Function with Respect to ϕ

$$ {f}_m\left(\phi \right)={N}_{\phi }(m)\left\{\begin{array}{c}\hfill \cos m\phi \hfill \\ {}\hfill \sin \left|m\right|\phi \hfill \end{array}\right\}\begin{array}{c}\hfill for\hfill \\ {}\hfill for\hfill \end{array}\begin{array}{c}\hfill m\ge 0\hfill \\ {}\hfill m<0\hfill \end{array} $$

(3.A2)

$$ {N}_{\phi }(m)={\left(-1\right)}^m{\left[\pi \left(1+{\delta}_{m,0}\right)\right]}^{-\frac{1}{2}} $$

(3.A3)

$$ {\displaystyle {\int}_0^{2\pi }d\phi }{f}_{m_a}\left(\phi \right){f}_{m_b}\left(\phi \right)={\delta}_{m_a,{m}_b} $$

(3.A4)

$$ {\displaystyle {\int}_0^{2\pi }d\phi }{f}_m\left(\phi \right)=\sqrt{2\pi }{\delta}_{m,0} $$

(3.A5)

$$ \Phi \left({m}_a,{m}_b,{m}_c\right)\equiv {\displaystyle {\int}_0^{2\pi }d\phi }{f}_{m_a}\left(\phi \right){f}_{m_b}\left(\phi \right){f}_{m_c}\left(\phi \right) $$

(3.A6)

$$ \begin{array}{l}I\left({m}_a,{m}_b,{m}_c\right)\\ {}\kern1em ={\displaystyle {\int}_0^{2\pi }d\phi}\left\{\begin{array}{c}\hfill \cos {m}_a\phi \hfill \\ {}\hfill \sin \left|{m}_a\right|\phi \hfill \end{array}\right\}\left\{\begin{array}{c}\hfill \cos {m}_b\phi \hfill \\ {}\hfill \sin \left|{m}_b\right|\phi \hfill \end{array}\right\}\left\{\begin{array}{c}\hfill \cos {m}_c\phi \hfill \\ {}\hfill \sin \left|{m}_c\right|\phi \hfill \end{array}\right\}\end{array} $$

(3.A7)

$$ \Phi \left({m}_a,{m}_b,{m}_c\right)={N}_{\phi}\left({m}_a\right){N}_{\phi}\left({m}_b\right){N}_{\phi}\left({m}_c\right)I\left({m}_a,{m}_b,{m}_c\right) $$

(3.A8)

1.1.2 Normalized Associated Legendre Function

$$ {p}_l^m(x)={N}_x\left(l,m\right){P}_l^m(x) $$

(3.A9)

$$ {N}_x\left(l,m\right)={\left[\frac{2l+1}{2}\frac{\left(l-\left|m\right|\right)!}{\left(l+\left|m\right|\right)!}\right]}^{1/2} $$

(3.A10)

$$ {P}_l^m(x)={\left(1-{x}^2\right)}^{\frac{\left|m\right|}{2}}{\displaystyle \sum_{\nu =0}^{\left[\frac{l-\left|m\right|}{2}\right]}{\omega}_{\nu}^{l,m}}{x}^{l-\left|m\right|-2\nu } $$

(3.A11)

$$ {\omega}_{\nu}^{l,m}=\frac{{\left(-1\right)}^{\nu}\left(2l-2\nu -1\right)!}{\nu !{2}^{\nu}\left(l-\left|m\right|-2\nu \right)!} $$

(3.A12)

$$ {\displaystyle {\int}_{-1}^1dx}{p}_{l_a}^{m_a}(x){p}_{l_b}^{m_b}(x)={\delta}_{m_a,{m}_b}{\delta}_{l_a,{l}_b} $$

(3.A13)

$$ {\displaystyle {\int}_{-1}^1dx}{p}_l^0(x)={\delta}_{l, even}2{N}_x\left(l,0\right){\displaystyle \sum_{\nu =0}^{\left[\frac{l}{2}\right]}\frac{\omega_{\nu}^{l,0}}{l-2\nu +1}} $$

(3.A14)

$$ \Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_a\\ {}{l}_a\end{array}\hfill & \hfill \begin{array}{l}{m}_b\\ {}{l}_b\end{array}\hfill & \hfill \begin{array}{l}{m}_c\\ {}{l}_c\end{array}\hfill \end{array}\right)\equiv {\displaystyle {\int}_{-1}^1{p}_{l_a}^{m_a}(x){p}_{l_b}^{m_b}(x){p}_{l_c}^{m_c}(x)dx} $$

(3.A15)

$$ \left|{m}_a\right|+\left|{m}_b\right|+\left|{m}_c\right|=2m $$

(3.A16)

$$ \begin{array}{l}{\displaystyle {\int}_{-1}^1{x}^{l_a+{l}_b+{l}_c-2m-2\left({\nu}_a+{\nu}_b+{\nu}_c\right)+2i}}dx\\ {}\begin{array}{cc}\hfill \kern1em =\frac{2}{l_a+{l}_b+{l}_c-2m-2\left({\nu}_a+{\nu}_b+{\nu}_c\right)+2i+1}\hfill & \hfill for\; \mod \left({l}_a+{l}_b+{l}_c,2\right)=0\hfill \end{array}\end{array} $$

(3.A17)

$$ {\displaystyle {\int}_{-1}^1{x}^{l_a+{l}_b+{l}_c-2m-2\left({\nu}_a+{\nu}_b+{\nu}_c\right)+2i}}\kern0.5em dx=0\begin{array}{cc}\hfill \hfill & \hfill for\hfill \end{array} \mod \left({l}_a+{l}_b+{l}_c,2\right)\ne 0 $$

(3.A18)

$$ \begin{array}{l}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_a\\ {}{l}_a\end{array}\hfill & \hfill \begin{array}{l}{m}_b\\ {}{l}_b\end{array}\hfill & \hfill \begin{array}{l}{m}_c\\ {}{l}_c\end{array}\hfill \end{array}\right)={N}_x\left({l}_a,{m}_a\right){N}_x\left({l}_b,{m}_b\right){N}_x\left({l}_c,{m}_c\right)\\ {}\begin{array}{cc}\hfill \hfill & \hfill \kern1em \times {\displaystyle \sum_{\nu_a=0}^{\left[\frac{l_a-\left|{m}_a\right|}{2}\right]}{\omega}_{\nu_a}^{l_a,{m}_a}}{\displaystyle \sum_{\nu_b=0}^{\left[\frac{l_b-\left|{m}_b\right|}{2}\right]}{\omega}_{\nu_b}^{l_b,{m}_b}}{\displaystyle \sum_{\nu_c=0}^{\left[\frac{l_c-\left|{m}_c\right|}{2}\right]}{\omega}_{\nu_c}^{l_c,{m}_c}}{\displaystyle \sum_i^m\left(\begin{array}{c}\hfill m\hfill \\ {}\hfill i\hfill \end{array}\right)}{\left(-1\right)}^i\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill \kern1em \times {\displaystyle {\int}_{-1}^1{x}^{l_a+{l}_b+{l}_c-2m-2\left({\nu}_a+{\nu}_b+{\nu}_c\right)+2i}}dx\hfill \end{array}\end{array} $$

(3.A19)

1.1.3 Normalized Real Spherical Harmonic Function

$$ {\displaystyle \int d\Omega}{Y}_l^m\left(\theta, \phi \right)=\sqrt{2\pi }{\delta}_{m,0}{\displaystyle {\int}_{-1}^1dx}{p}_l^0(x) $$

(3.A20)

$$ {\displaystyle \int d\Omega}{Y}_{l_a}^{m_a}\left(\theta, \phi \right){Y}_{l_a}^{m_a}\left(\theta, \phi \right)={\delta}_{l_a,{l}_b}{\delta}_{m_a,{m}_b} $$

(3.A21)

$$ {Y}_{l_a}^{m_a}\left(\theta, \phi \right){Y}_{l_b}^{m_{{}_b}}\left(\theta, \phi \right)={\displaystyle \sum_{m_{ab}}\Phi \left({m}_a,{m}_b,{m}_{ab}\right)}{\displaystyle \sum_{l_{ab}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_a\\ {}{l}_a\end{array}\hfill & \hfill \begin{array}{l}{m}_b\\ {}{l}_b\end{array}\hfill & \hfill \begin{array}{l}{m}_{ab}\\ {}{l}_{ab}\end{array}\hfill \end{array}\right)}{Y}_{l_{ab}}^{m_{ab}}\left(\theta, \phi \right) $$

(3.A22)

where

$$ \max \left(\left|m\right|,\left|{l}_a-{l}_b\right|\right)+n\le {l}_{ab}\le {l}_a+{l}_b $$

(3.A23)

$$ n=\left\{\begin{array}{c}\hfill 0,\hfill \\ {}\hfill 1,\hfill \end{array}\begin{array}{c}\hfill \left(m\equiv \left|{l}_a-{l}_b\right|\left( \mod 2\right)\right)\hfill \\ {}\hfill (otherwise)\hfill \end{array}\right. $$

(3.A24)

$$ \begin{array}{l}G\left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_a\\ {}{l}_a\end{array}\hfill & \hfill \begin{array}{l}{m}_b\\ {}{l}_b\end{array}\hfill & \hfill \begin{array}{l}{m}_{ab}\\ {}{l}_{ab}\end{array}\hfill \end{array}\right)\\ {}\begin{array}{cc}\hfill \hfill & \hfill \equiv {\displaystyle \int d\Omega}{Y}_{l_a}^{m_a}\left(\theta, \phi \right){Y}_{l_b}^{m_{{}_b}}\left(\theta, \phi \right){Y}_{l_c}^{m_c}\left(\theta, \phi \right)\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill ={\displaystyle \sum_{m_{ab}}\Phi \left({m}_a,{m}_b,{m}_{ab}\right)}{\displaystyle \sum_{l_{ab}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_a\\ {}{l}_a\end{array}\hfill & \hfill \begin{array}{l}{m}_b\\ {}{l}_b\end{array}\hfill & \hfill \begin{array}{l}{m}_{ab}\\ {}{l}_{ab}\end{array}\hfill \end{array}\right){\displaystyle \int d\Omega {Y}_{l_{ab}}^{m_{ab}}\left(\theta, \phi \right){Y}_{l_c}^{m_c}\left(\theta, \phi \right)}}\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill ={\displaystyle \sum_{m_{ab}}\Phi \left({m}_a,{m}_b,{m}_{ab}\right)}{\displaystyle \sum_{l_{ab}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_a\\ {}{l}_a\end{array}\hfill & \hfill \begin{array}{l}{m}_b\\ {}{l}_b\end{array}\hfill & \hfill \begin{array}{l}{m}_{ab}\\ {}{l}_{ab}\end{array}\hfill \end{array}\right){\delta}_{m_{ab},{m}_c}{\delta}_{l_{ab},{l}_c}}\hfill \end{array}\end{array} $$

(3.A25)

1.1.4 Triple Product of Normalized Real Spherical Harmonic Function (1)

$$ \begin{array}{l}{Y}_{l_a}^{m_a}\left(\theta, \phi \right){Y}_{l_b}^{m_{{}_b}}\left(\theta, \phi \right){Y}_{l_c}^{m_c}\left(\theta, \phi \right)\\ {}\kern1em ={\displaystyle \sum_{m_{ab}}\Phi \left({m}_a,{m}_b,{m}_{ab}\right)}{\displaystyle \sum_{l_{ab}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_a\\ {}{l}_a\end{array}\hfill & \hfill \begin{array}{l}{m}_b\\ {}{l}_b\end{array}\hfill & \hfill \begin{array}{l}{m}_{ab}\\ {}{l}_{ab}\end{array}\hfill \end{array}\right)}\\ {}\kern2.35em {\displaystyle \sum_{m_{abc}}\Phi \left({m}_a,{m}_b,{m}_{abc}\right)}{\displaystyle \sum_{l_{ab}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_a\\ {}{l}_a\end{array}\hfill & \hfill \begin{array}{l}{m}_b\\ {}{l}_b\end{array}\hfill & \hfill \begin{array}{l}{m}_{abc}\\ {}{l}_{abc}\end{array}\hfill \end{array}\right){Y}_{l_{abc}}^{m_{abc}}\left(\theta, \phi \right)}\end{array} $$

(3.A26)

$$ \begin{array}{l}{\displaystyle \int d\Omega}{Y}_{l_a}^{m_a}\left(\theta, \phi \right){Y}_{l_b}^{m_{{}_b}}\left(\theta, \phi \right){Y}_{l_c}^{m_c}\left(\theta, \phi \right)\\ {}\kern1em ={\displaystyle \sum_{m_{ab}}\Phi \left({m}_a,{m}_b,{m}_{ab}\right)}{\displaystyle \sum_{l_{ab}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_a\\ {}{l}_a\end{array}\hfill & \hfill \begin{array}{l}{m}_b\\ {}{l}_b\end{array}\hfill & \hfill \begin{array}{l}{m}_{ab}\\ {}{l}_{ab}\end{array}\hfill \end{array}\right)}\\ {}\kern2.35em {\displaystyle \sum_{m_{abc}}\Phi \left({m}_a,{m}_b,{m}_{abc}\right)}{\displaystyle \sum_{l_{ab}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_a\\ {}{l}_a\end{array}\hfill & \hfill \begin{array}{l}{m}_b\\ {}{l}_b\end{array}\hfill & \hfill \begin{array}{l}{m}_{abc}\\ {}{l}_{abc}\end{array}\hfill \end{array}\right)}\\ {}\kern2.35em {\displaystyle \int d\Omega}{Y}_{l_{abc}}^{m_{abc}}\left(\theta, \phi \right)\end{array} $$

(3.A27)

$$ \begin{array}{l}\begin{array}{cc}\hfill \hfill & \hfill ={\displaystyle \sum_{m_{ab}}\Phi \left({m}_a,{m}_b,{m}_{ab}\right)}{\displaystyle \sum_{l_{ab}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_a\\ {}{l}_a\end{array}\hfill & \hfill \begin{array}{l}{m}_b\\ {}{l}_b\end{array}\hfill & \hfill \begin{array}{l}{m}_{ab}\\ {}{l}_{ab}\end{array}\hfill \end{array}\right)}\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill \kern0.78em \times {\displaystyle \sum_{m_{abc}}\Phi \left({m}_a,{m}_b,{m}_{abc}\right)}{\displaystyle \sum_{l_{ab}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_a\\ {}{l}_a\end{array}\hfill & \hfill \begin{array}{l}{m}_b\\ {}{l}_b\end{array}\hfill & \hfill \begin{array}{l}{m}_{abc}\\ {}{l}_{abc}\end{array}\hfill \end{array}\right)}\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill \kern-3.9em \times {\delta}_{l_{abc}, even}2{N}_x\left({l}_{abc},0\right){\displaystyle \sum_{\nu =0}^{\left[\frac{l_{abc}}{2}\right]}\frac{\omega_{\nu}^{l_{abc},0}}{l_{abc}-2\nu +1}}\hfill \end{array}\end{array} $$

(3.A28)

1.1.5 Transfer of Origin of Spherical Harmonics from B to A

$$ {Y}_l^m\left({\theta}_B,\;\phi \right)={\displaystyle \sum_{k=\left|m\right|}^l{T}_k^{lm}\;{R}^{l-k}\;{r}_A^k{r}_B^{-l}{Y}_k^m}\left({\theta}_A,\;\phi \right) $$

(3.A29)

$$ {T}_k^{lm}={\left(-1\right)}^{k+m}\frac{N_x\left(l,m\right)}{N_x\left(k,m\right)}\frac{\left(l+\left|m\right|\right)!}{\left(l-k\right)!\left(k+\left|m\right|\right)!} $$

(3.A30)

1.1.6 Triple Product of Normalized Real Spherical Harmonic Function (2)

$$ \begin{array}{l}{Y}_{l_a}^{m_a}\left({\theta}_A,\phi \right){Y}_{l_b}^{m_{{}_b}}\left({\theta}_B,\phi \right)\\ {}\begin{array}{cc}\hfill \hfill & \hfill ={\displaystyle \sum_{k_b=\left|{m}_b\right|}^{l_b}{T}_{k_b}^{l_b{m}_b}}{\displaystyle \sum_{m_{ab}}\Phi \left({m}_a,{m}_b,{m}_{ab}\right)}{\displaystyle \sum_{l_{ab}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_a\\ {}{l}_a\end{array}\hfill & \hfill \begin{array}{l}{m}_b\\ {}{k}_b\end{array}\hfill & \hfill \begin{array}{l}{m}_{ab}\\ {}{l}_{ab}\end{array}\hfill \end{array}\right)}\hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \times {R}^{l_b-{k}_b}{r}_A^{k_b}{r}_B^{-{l}_b}{Y}_{l_{ab}}^{m_{ab}}\left({\theta}_A,\phi \right)\hfill \end{array}\end{array} $$

(3.A31)

$$ \begin{array}{l}{Y}_{l_a}^{m_a}\left({\theta}_A,\phi \right){Y}_{l_b}^{m_{{}_b}}\left({\theta}_B,\phi \right){Y}_{l_c}^{m_c}\left({\theta}_A,\phi \right)\\ {}\begin{array}{cc}\hfill \hfill & \hfill ={\displaystyle \sum_{k_b=\left|{m}_b\right|}^{l_b}{T}_{k_b}^{l_b{m}_b}}{R}^{l_b-{k}_b}{r}_A^{k_b}{r}_B^{-{l}_b}{Y}_{l_a}^{m_a}\left({\theta}_A,\phi \right){Y}_{k_b}^{m_{{}_b}}\left({\theta}_A,\phi \right){Y}_{l_c}^{m_c}\left({\theta}_A,\phi \right)\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill ={\displaystyle \sum_{k_b=\left|{m}_b\right|}^{l_b}{T}_{k_b}^{l_b{m}_b}}{R}^{l_b-{k}_b}{r}_A^{k_b}{r}_B^{-{l}_b}{\displaystyle \sum_{m_{ab}}\Phi \left({m}_a,{m}_b,{m}_{ab}\right)}{\displaystyle \sum_{l_{ab}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_a\\ {}{l}_a\end{array}\hfill & \hfill \begin{array}{l}{m}_b\\ {}{k}_b\end{array}\hfill & \hfill \begin{array}{l}{m}_{ab}\\ {}{l}_{ab}\end{array}\hfill \end{array}\right)}\hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \times {\displaystyle \sum_{m_{abc}}\Phi \left(\begin{array}{ccc}\hfill {m}_{ab}\hfill & \hfill {m}_c\hfill & \hfill {m}_{abc}\hfill \end{array}\right)}{\displaystyle \sum_{l_{abc}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_{ab}\\ {}{l}_{ab}\end{array}\hfill & \hfill \begin{array}{l}{m}_c\\ {}{l}_c\end{array}\hfill & \hfill \begin{array}{l}{m}_{abc}\\ {}{l}_{abc}\end{array}\hfill \end{array}\right)}{Y}_{l_{abc}}^{m_{abc}}\left({\theta}_A,\phi \right)\hfill \end{array}\end{array} $$

(3.A32)

$$ \begin{array}{l}{\displaystyle {\int}_0^{2\pi }d\phi }{Y}_{l_a}^{m_a}\left({\theta}_A,\phi \right){Y}_{l_b}^{m_{{}_b}}\left({\theta}_B,\phi \right){Y}_{l_c}^{m_c}\left({\theta}_A,\phi \right)\\ {}\begin{array}{cc}\hfill \hfill & \hfill ={\displaystyle \sum_{k_b=\left|{m}_b\right|}^{l_b}{T}_{k_b}^{l_b{m}_b}}{R}^{l_b-{k}_b}{r}_A^{k_b}{r}_B^{-{l}_b}{\displaystyle {\int}_0^{2\pi }d\phi }{Y}_{l_a}^{m_a}\left({\theta}_A,\phi \right){Y}_{k_b}^{m_{{}_b}}\left({\theta}_A,\phi \right){Y}_{l_c}^{m_c}\left({\theta}_A,\phi \right)\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill =2\pi {N}_{\phi }(0){\displaystyle \sum_{k_b=\left|{m}_b\right|}^{l_b}{T}_{k_b}^{l_b{m}_b}}{R}^{l_b-{k}_b}{r}_A^{k_b}{r}_B^{-{l}_b}\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\kern0.55em \times {\displaystyle \sum_{m_{ab}}\Phi \left({m}_a,{m}_b,{m}_{ab}\right)}{\displaystyle \sum_{l_{ab}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_a\\ {}{l}_a\end{array}\hfill & \hfill \begin{array}{l}{m}_b\\ {}{k}_b\end{array}\hfill & \hfill \begin{array}{l}{m}_{ab}\\ {}{l}_{ab}\end{array}\hfill \end{array}\right)}\\ {}\begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\kern0.55em \times {\displaystyle \sum_{m_{abc}}\Phi \left(\begin{array}{ccc}\hfill {m}_{ab}\hfill & \hfill {m}_c\hfill & \hfill {m}_{abc}\hfill \end{array}\right)}{\displaystyle \sum_{l_{abc}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_{ab}\\ {}{l}_{ab}\end{array}\hfill & \hfill \begin{array}{l}{m}_c\\ {}{l}_c\end{array}\hfill & \hfill \begin{array}{l}{m}_{abc}\\ {}{l}_{abc}\end{array}\hfill \end{array}\right)}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}\kern-0.45em \times {p}_{l_{abc}}^0\left( \cos {\theta}_A\right)\end{array} $$

(3.A33)

1.1.7 Triple Product of Normalized Real Spherical Harmonic Function (3)

$$ \begin{array}{l}{Y}_{l_a}^{m_a}\left({\theta}_B,\phi \right){Y}_{l_b}^{m_b}\left({\theta}_B,\phi \right)\\ {}\begin{array}{cc}\hfill \hfill & \hfill ={\displaystyle \sum_{m_{ad}}\Phi \left({m}_a,{m}_b,{m}_{ab}\right)}{\displaystyle \sum_{k_{cd}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_a\\ {}{l}_a\end{array}\hfill & \hfill \begin{array}{l}{m}_b\\ {}{l}_b\end{array}\hfill & \hfill \begin{array}{l}{m}_{ab}\\ {}{l}_{ab}\end{array}\hfill \end{array}\right)}{Y}_{l_{ab}}^{m_{ab}}\left({\theta}_B,\phi \right)\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill ={\displaystyle \sum_{m_{ad}}\Phi \left({m}_a,{m}_b,{m}_{ab}\right)}{\displaystyle \sum_{k_{cd}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_a\\ {}{l}_a\end{array}\hfill & \hfill \begin{array}{l}{m}_b\\ {}{l}_b\end{array}\hfill & \hfill \begin{array}{l}{m}_{ab}\\ {}{l}_{ab}\end{array}\hfill \end{array}\right)}\hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \kern-0.25em \times {\displaystyle \sum_{k_{ab}=\left|{m}_{ab}\right|}^{l_{ab}}{T}_{k_c}^{l_{ab}{m}_{ab}}}{R}^{l_{ab}-{k}_{ab}}{r}_A^{k_{ab}}{r}_B^{-{l}_{ab}}{Y}_{k_{ab}}^{m_{ab}}\left({\theta}_A,\phi \right)\hfill \end{array}\end{array} $$

(3.A34)

$$ \begin{array}{l}{Y}_{l_a}^{m_a}\left({\theta}_B,\phi \right){Y}_{l_b}^{m_b}\left({\theta}_B,\phi \right){Y}_{l_c}^{m_c}\left({\theta}_A,\phi \right)\\ {}\begin{array}{cc}\hfill \hfill & \hfill ={\displaystyle \sum_{m_{ab}}\Phi \left({m}_a,{m}_b,{m}_{ab}\right)}{\displaystyle \sum_{l_{ab}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_a\\ {}{l}_a\end{array}\hfill & \hfill \begin{array}{l}{m}_b\\ {}{l}_b\end{array}\hfill & \hfill \begin{array}{l}{m}_{ab}\\ {}{l}_{ab}\end{array}\hfill \end{array}\right)}{Y}_{l_{ab}}^{m_{ab}}\left({\theta}_B,\phi \right){Y}_{l_c}^{m_c}\left({\theta}_A,\phi \right)\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill ={\displaystyle \sum_{m_{ab}}\Phi \left({m}_a,{m}_b,{m}_{ab}\right)}{\displaystyle \sum_{l_{ab}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_a\\ {}{l}_a\end{array}\hfill & \hfill \begin{array}{l}{m}_b\\ {}{l}_b\end{array}\hfill & \hfill \begin{array}{l}{m}_{ab}\\ {}{l}_{ab}\end{array}\hfill \end{array}\right)}\hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \kern-0.25em \times {\displaystyle \sum_{k_{ab}=\left|{m}_{ab}\right|}^{l_{ab}}{T}_{k_c}^{l_{ab}{m}_{ab}}}{R}^{l_{ab}-{k}_{ab}}{r}_A^{k_{ab}}{r}_B^{-{l}_{ab}}{Y}_{k_{ab}}^{m_{ab}}\left({\theta}_A,\phi \right){Y}_{l_c}^{m_c}\left({\theta}_A,\phi \right)\hfill \end{array}\end{array} $$

(3.A35)

$$ \begin{array}{l}\begin{array}{cc}\hfill \hfill & \hfill ={\displaystyle \sum_{m_{ab}}\Phi \left({m}_a,{m}_b,{m}_{ab}\right)}{\displaystyle \sum_{l_{ab}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_a\\ {}{l}_a\end{array}\hfill & \hfill \begin{array}{l}{m}_b\\ {}{l}_b\end{array}\hfill & \hfill \begin{array}{l}{m}_{ab}\\ {}{l}_{ab}\end{array}\hfill \end{array}\right)}\hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \begin{array}{l}\kern-0.43em \times {\displaystyle \sum_{k_{ab}=\left|{m}_{ab}\right|}^{l_{ab}}{T}_{k_c}^{l_{ab}{m}_{ab}}}{R}^{l_{ab}-{k}_{ab}}{r}_A^{k_{ab}}{r}_B^{-{l}_{ab}}\\ {}\kern-0.43em \times {\displaystyle \sum_{m_{abc}}\Phi \left({m}_{ab},{m}_c,{m}_{abc}\right)}{\displaystyle \sum_{l_{abc}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_{ab}\\ {}{k}_{ab}\end{array}\hfill & \hfill \begin{array}{l}{m}_c\\ {}{l}_c\end{array}\hfill & \hfill \begin{array}{l}{m}_{abc}\\ {}{l}_{abc}\end{array}\hfill \end{array}\right)}\\ {}\kern-0.43em \times {Y}_{l_{abc}}^{m_{abc}}\left({\theta}_A,\phi \right)\end{array}\hfill \end{array}\\ {}\end{array} $$

(3.A36)

$$ \begin{array}{l}{\displaystyle \int d\Omega}{Y}_{l_a}^{m_a}\left({\theta}_B,\phi \right){Y}_{l_b}^{m_b}\left({\theta}_B,\phi \right){Y}_{l_c}^{m_c}\left({\theta}_A,\phi \right)\\ {}\begin{array}{cc}\hfill \hfill & \hfill =2\pi {N}_{\phi }(0){\displaystyle \sum_{m_{ad}}\Phi \left({m}_a,{m}_b,{m}_{ab}\right)}{\displaystyle \sum_{k_{cd}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_a\\ {}{l}_a\end{array}\hfill & \hfill \begin{array}{l}{m}_b\\ {}{l}_b\end{array}\hfill & \hfill \begin{array}{l}{m}_{ab}\\ {}{l}_{ab}\end{array}\hfill \end{array}\right)}\hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \kern-0.3em \times {\displaystyle \sum_{k_{ab}=\left|{m}_{ab}\right|}^{l_{ab}}{T}_{k_c}^{l_{ab}{m}_{ab}}}{R}^{l_{ab}-{k}_{ab}}{r}_A^{k_{ab}}{r}_B^{-{l}_{ab}}\hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \kern-0.3em \times {\displaystyle \sum_{m_{abc}}\Phi \left({m}_{ab},{m}_c,{m}_{abc}\right)}{\displaystyle \sum_{l_{abc}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_{ab}\\ {}{k}_{ab}\end{array}\hfill & \hfill \begin{array}{l}{m}_c\\ {}{l}_c\end{array}\hfill & \hfill \begin{array}{l}{m}_{abc}\\ {}{l}_{abc}\end{array}\hfill \end{array}\right)}\hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \kern-0.3em \times {\displaystyle {\int}_0^{\pi } \sin {\theta}_Ad{\theta}_A}\hfill \end{array}{p}_{l_{abc}}^0\left( \cos {\theta}_A\right)\end{array} $$

(3.A37)

$$ \begin{array}{l}\begin{array}{cc}\hfill \hfill & \hfill =2\pi {N}_{\phi }(0){\displaystyle \sum_{m_{ad}}\Phi \left({m}_a,{m}_b,{m}_{ab}\right)}{\displaystyle \sum_{k_{cd}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_a\\ {}{l}_a\end{array}\hfill & \hfill \begin{array}{l}{m}_b\\ {}{l}_b\end{array}\hfill & \hfill \begin{array}{l}{m}_{ab}\\ {}{l}_{ab}\end{array}\hfill \end{array}\right)}\hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \kern-0.3em \times {\displaystyle \sum_{k_{ab}=\left|{m}_{ab}\right|}^{l_{ab}}{T}_{k_c}^{l_{ab}{m}_{ab}}}{R}^{l_{ab}-{k}_{ab}}{r}_A^{k_{ab}}{r}_B^{-{l}_{ab}}\hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \kern-0.3em \times {\displaystyle \sum_{m_{ab c}}\Phi \left({m}_{ab},{m}_c,{m}_{ab c}\right)}{\displaystyle \sum_{l_{ab c}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_{ab}\\ {}{k}_{ab}\end{array}\hfill & \hfill \begin{array}{l}{m}_c\\ {}{l}_c\end{array}\hfill & \hfill \begin{array}{l}{m}_{ab c}\\ {}{l}_{ab c}\end{array}\hfill \end{array}\right)}\hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \kern-0.3em \times {\displaystyle {\int}_0^{\pi } \sin {\theta}_Ad{\theta}_A}\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\kern0.55em \times {\displaystyle \sum_{\nu =0}^{\left[\frac{l_{ab c}}{2}\right]}{\displaystyle \sum_{i=0}^{l_{ab c}-2\nu}\kern0.20em {\displaystyle \sum_{j=0}^{l_{ab c}-2\nu -i}\mathrm{P}\left({l}_{ab c},\nu, i,j\right)}}}{R}^{-{l}_{ab c}+2\left(\nu +i\right)}{r}_A^{l_{ab c}-2\left(\nu +i+j\right)}{r}_B^{2i}\end{array} $$

(3.A38)

$$ \begin{array}{l}{\displaystyle \int d\Omega {Y}_{l_a}^{m_a}\left({\theta}_B,\;\phi \right)}\;{Y}_{l_b}^{m_b}\left({\theta}_B,\;\phi \right)\;{Y}_{l_c}^{m_c}\left({\theta}_A,\;\phi \right)\\ {}\begin{array}{cc}\hfill \hfill & \hfill =2\pi {N}_{\phi }(0){\displaystyle \sum_{m_{ad}}\Phi \left({m}_a,{m}_b,{m}_{ab}\right)}{\displaystyle \sum_{k_{cd}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_a\\ {}{l}_a\end{array}\hfill & \hfill \begin{array}{l}{m}_b\\ {}{l}_b\end{array}\hfill & \hfill \begin{array}{l}{m}_{ab}\\ {}{l}_{ab}\end{array}\hfill \end{array}\right)}\hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \kern0.7em \times {\displaystyle \sum_{k_{ab}=\left|{m}_{ab}\right|}^{l_{ab}}{T}_{k_c}^{l_{ab}{m}_{ab}}}{\displaystyle \sum_{m_{ab c}}\Phi \left({m}_{ab},{m}_c,{m}_{ab c}\right)}{\displaystyle \sum_{l_{ab c}}\Theta \left(\begin{array}{ccc}\hfill \begin{array}{l}{m}_{ab}\\ {}{k}_{ab}\end{array}\hfill & \hfill \begin{array}{l}{m}_c\\ {}{l}_c\end{array}\hfill & \hfill \begin{array}{l}{m}_{ab c}\\ {}{l}_{ab c}\end{array}\hfill \end{array}\right)}\hfill & \hfill \hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \kern0.7em \times {\displaystyle {\int}_0^{\pi } \sin {\theta}_Ad{\theta}_A}\hfill & \hfill \hfill \end{array}\\ {}\kern1.86em \times {\displaystyle \sum_{\nu =0}^{\left[\frac{l_{ab c}}{2}\right]}{\displaystyle \sum_{i=0}^{l_{ab c}-2\nu}\kern0.20em {\displaystyle \sum_{j=0}^{l_{ab c}-2\nu -i}\mathrm{P}\mathrm{P}\left({l}_{ab c},\nu, i,j\right)}}}{R}^{l_{ab}-{k}_{ab}-{l}_{ab c}+2\left(\nu +i\right)}{r}_A^{k_{ab}+{l}_{ab c}-2\left(\nu +i+j\right)}{r}_B^{2i-{l}_{ab}}\end{array} $$

(3.A39)

1.2 Appendix B: Half Definite Integral, H and Hh

We define a definite integral, H and Hh, in this appendix.

$$ H\left[X,n,\alpha \right]\equiv {\displaystyle {\int}^X{x}^n} \exp \left(-\alpha x\right)dx $$

(3.B1)

We begin formulation for H in case a ≠ 0.

$$ H\left[X,n,\alpha \Big|n\ge 0,\alpha \ne 0\right]=-{\displaystyle \sum_{k=0}^nA\left(n,k\right)}{\alpha}^{-\left(k+1\right)}{X}^{n-k} \exp \left(-\alpha X\right) $$

(3.B2)

$$ A\left(n,k\right)=\frac{n!}{\left(n-k\right)!} $$

(3.B3)

$$ \left\langle H\left[0,n,\alpha \Big|n\ge 0,\alpha \ne 0\right]\right\rangle =-n!{\alpha}^{-\left(n+1\right)} $$

(3.B4)

$$ \left\langle H\left[\infty, n,\alpha \Big|n\ge 0,\alpha \ne 0\right]\right\rangle =0 $$

(3.B5)

$$ \left\langle H\left[0,n,\alpha \Big|n\ge 0,\alpha \ne 0\right]\right\rangle =-n!{\alpha}^{-\left(n+1\right)} $$

(3.B6)

$$ \left\langle H\left[1,n,\alpha \Big|n\ge 0,\alpha \ne 0\right]\right\rangle =-{\displaystyle \sum_{k=0}^nA\left(n,k\right)}{\alpha}^{-\left(k+1\right)} \exp \left(-\alpha \right) $$

(3.B7)

$$ \left\langle H\left[R,n,\alpha \Big|n\ge 0,\alpha \ne 0\right]\right\rangle =-{\displaystyle \sum_{k=0}^nA\left(n,k\right)}{\alpha}^{-\left(k+1\right)}{R}^{n-k} \exp \left(-\alpha R\right) $$

(3.B8)

For n = − 1

$$ H\left[X,n,\alpha \Big|n=-1,\alpha \ne 0\right]=Ei\left(-\alpha X\right) $$

(3.B9)

$$ Ei\left(-\alpha X\right)={\displaystyle {\int}^X\frac{1}{x}} \exp \left(-\alpha x\right)dx $$

(3.B10)

For n ≤ − 2,

$$ \begin{array}{l}H\left[X,n,\alpha \Big|n\le -2,\alpha \ne 0\right]\\ {}\kern0.5em =\frac{{\left(-\alpha \right)}^{-\left(n+1\right)}}{\left(-\left(n+1\right)\right)!}\kern0.5em Ei\left(-\alpha X\right)\\ {}\kern1.6em -{\displaystyle \sum_{k=0}^{-\left(n+1\right)}{\left(-1\right)}^{k-1}\frac{\left(-\left(n+1+k\right)\right)}{\left(-\left(n+1\right)\right)!}}\kern0.5em {\alpha}^{k-1}{X}^{n+k} \exp \left(-\alpha X\right)\end{array} $$

(3.B11)

In case α = 0 and n ≠ − 1,

$$ H\left[X,n,\alpha \Big|n\ne -1,\alpha =0\right]={\displaystyle {\int}^X{x}^n}dx=\frac{1}{n+1}{X}^{n+1} $$

(3.B12)

In case α = 0 and n = − 1,

$$ H\left[X,n,\alpha \Big|n=-1,\alpha =0\right]={\displaystyle {\int}^X{x}^{-1}}dx= \log \kern0.20em X $$

(3.B13)

The half definite integral Hh is defined using H as follows,

$$ \begin{array}{l}Hh\left[X,{n}_1,{\alpha}_1,{n}_2,{\alpha}_2\right]\\ {}\begin{array}{ccc}\hfill \hfill & \hfill ={\displaystyle {\int}^Xd{r}_1{r}_1^{n_1} \exp \left(-{\alpha}_1{r}_1\right)}\hfill & \hfill {\displaystyle {\int}^{r_1}d{r}_2{r}_2^{n_2} \exp \left(-{\alpha}_2{r}_2\right)}\hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill ={\displaystyle {\int}^Xd{r}_1{r}_1^{n_1} \exp \left(-{\alpha}_1{r}_1\right)}\hfill & \hfill H\left[{r}_1,{n}_2,{\alpha}_2\right]\hfill \end{array}\end{array} $$

(3.B14)

$$ \begin{array}{l}Hh\left[X,{n}_1,{\alpha}_1,{n}_2,{\alpha}_2\Big|{n}_2\ge 0,{\alpha}_2\ne 0\right]\\ {}\begin{array}{ccc}\hfill \hfill & \hfill ={\displaystyle {\int}^Xd{r}_1{r}_1^{n_1} \exp \left(-{\alpha}_1{r}_1\right)H\left[{r}_1,{n}_2,{\alpha}_2\Big|{n}_2\ge 0,{\alpha}_2\ne 0\right]}\hfill & \hfill \hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill =-{\displaystyle \sum_{k_2=0}^{n_2}A\left({n}_2,{k}_2\right)}{\alpha_2}^{-\left({k}_2+1\right)}{\displaystyle {\int}^Xd{r}_1{r}_1^{n_1+{n}_2-{k}_2} \exp \left(-\left({\alpha}_1+{\alpha}_2\right){r}_1\right)}\hfill & \hfill \hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill =-{\displaystyle \sum_{k_2=0}^{n_2}A\left({n}_2,{k}_2\right)}{\alpha_2}^{-\left({k}_2+1\right)}H\left[X,{n}_1+{n}_2-{k}_2,{\alpha}_1+{\alpha}_2\right]\hfill & \hfill \hfill \end{array}\end{array} $$

(3.B15)

$$ \begin{array}{l}Hh\left[X,{n}_1,{\alpha}_1,{n}_2,{\alpha}_2\Big|{n}_1+{n}_2\ge 0,{n}_2\ge 0,{\alpha}_2\ne 0\right]\\ {}\begin{array}{ccc}\hfill \hfill & \hfill =-{\displaystyle \sum_{k_2=0}^{n_2}A\left({n}_2,{k}_2\right)}{\alpha_2}^{-\left({k}_2+1\right)}H\left[X,{n}_1+{n}_2-{k}_2,{\alpha}_1+{\alpha}_2\right]\hfill & \hfill \hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill ={\displaystyle \sum_{k_2=0}^{n_2}A\left({n}_2,{k}_2\right)}{\alpha_2}^{-\left({k}_2+1\right)}{\displaystyle \sum_{k_1=0}^{n_1+{n}_2-{k}_2}A\left({n}_1+{n}_2-{k}_2,{k}_1\right)}{\left({\alpha}_1+{\alpha}_2\right)}^{-\left({k}_1+1\right)}\hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \kern-0.2em \times {X}^{n_1+{n}_2-\left({k}_1+{k}_2\right)} \exp \left(-\left({\alpha}_1+{\alpha}_2\right)X\right)\hfill \end{array}\end{array} $$

(3.B16)

$$ \begin{array}{l}Hh\left[X,{n}_1,{\alpha}_1,{n}_2,{\alpha}_2\Big|{n}_2=-1,{\alpha}_2\ne 0\right]\\ {}\begin{array}{ccc}\hfill \hfill & \hfill ={\displaystyle {\int}^Xd{r}_1{r}_1^{n_1} \exp \left(-{\alpha}_1{r}_1\right)H\left[{r}_1,-1,{\alpha}_2\Big|{\alpha}_2\ne 0\right]}\hfill & \hfill \hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill ={\displaystyle {\int}^Xd{r}_1{r}_1^{n_1} \exp \left(-{\alpha}_1{r}_1\right)Ei\left(-{\alpha}_2{r}_1\right)}\hfill & \hfill \hfill \end{array}\end{array} $$

(3.B17)

$$ \begin{array}{l}Hh\left[X,{n}_1,{\alpha}_1,{n}_2,{\alpha}_2\Big|{n}_2\le 2,{\alpha}_2\ne 0\right]\\ {}\begin{array}{ccc}\hfill \hfill & \hfill ={\displaystyle {\int}^Xd{r}_1{r}_1^{n_1} \exp \left(-{\alpha}_1{r}_1\right)H\left[{r}_1,{n}_2,{\alpha}_2\Big|{n}_2\le 2,{\alpha}_2\ne 0\right]}\hfill & \hfill \hfill \end{array}\end{array} $$

(3.B18)

$$ \begin{array}{l}Hh\left[X,{n}_1,{\alpha}_1,{n}_2,{\alpha}_2\Big|{n}_2\ne -1,{\alpha}_2=0\right]\\ {}\begin{array}{ccc}\hfill \hfill & \hfill ={\displaystyle {\int}^Xd{r}_1{r}_1^{n_1} \exp \left(-{\alpha}_1{r}_1\right)H\left[{r}_1,{n}_2,{\alpha}_2\Big|{n}_2\ne -1,{\alpha}_2=0\right]}\hfill & \hfill \hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill =\frac{1}{n_2+1}{\displaystyle {\int}^Xd{r}_1{r}_1^{n_1+{n}_2+1} \exp \left(-{\alpha}_1{r}_1\right)}\hfill & \hfill \hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill =\frac{1}{n_2+1}H\left[X,{n}_1+{n}_2+1,{\alpha}_1\right]\hfill & \hfill \hfill \end{array}\end{array} $$

(3.B19)

$$ \begin{array}{l}Hh\left[X,{n}_1,{\alpha}_1,{n}_2,{\alpha}_2\Big|{n}_2=-1,{\alpha}_2=0\right]\\ {}\begin{array}{ccc}\hfill \hfill & \hfill ={\displaystyle {\int}^Xd{r}_1{r}_1^{n_1} \exp \left(-{\alpha}_1{r}_1\right)H\left[{r}_1,{n}_2,{\alpha}_2\Big|{n}_2=-1,{\alpha}_2=0\right]}\hfill & \hfill \hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill ={\displaystyle {\int}^Xd{r}_1{r}_1^{n_1} \exp \left(-{\alpha}_1{r}_1\right) \log {r}_1}\hfill & \hfill \hfill \end{array}\end{array} $$

(3.B20)

1.3 Appendix C: B Function

We define B function in this appendix. B function is an integral in which STF centered on B center is looked from A center.

$$ \begin{array}{l}B\left(n,\zeta, R,{r}_A\right)\equiv {\displaystyle {\int}_0^{\pi } \sin {\theta}_Ad{\theta}_A}{r}_B^n \exp \left(-\zeta {r}_B\right)\\ {}\begin{array}{cccc}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}\kern1.87em ={\displaystyle {\int}_{-1}^1J\left({r}_A,R\right)dx}{r}_B^n \exp \left(-\zeta {r}_B\right)\end{array} $$

(3.C1)

If r _A  ≤ R, r _B  = R + r _A x. We obtain B function in Eq. (3.C1) as follows

$$ \begin{array}{l}{r}_B^n \exp \left(-\zeta {r}_B\right)={r}_B^n \exp \left(-\zeta {r}_B\right)\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}\kern2.95em = \exp \left(-\zeta R\right){\displaystyle \sum_{s=0}^n\left(\begin{array}{c}\hfill n\hfill \\ {}\hfill s\hfill \end{array}\right)}{R}^s{r}_A^{n-s}\left\{{x}^{n-s} \exp \left(-\zeta {r}_Ax\right)\right\}\end{array} $$

(3.C2)

$$ \begin{array}{l}J\left({r}_A,R\Big|{r}_A\le R\right){r}_B^n \exp \left(-\zeta {r}_B\right)\\ {}\begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}={R}^{-1}{r}_B{r}_B^n \exp \left(-\zeta {r}_B\right)\\ {}\begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}= \exp \left(-\zeta R\right){\displaystyle \sum_{s=0}^{n+1}\left(\begin{array}{c}\hfill n+1\hfill \\ {}\hfill s\hfill \end{array}\right)}{R}^{s-1}{r}_A^{n+1-s}\left\{{x}^{n+1-s} \exp \left(-\zeta {r}_Ax\right)\right\}\end{array} $$

(3.C3)

$$ \begin{array}{l}B\left(n,\zeta, R,{r}_A\Big|{r}_A\le R\right)\\ {}\begin{array}{cc}\hfill \hfill & \hfill = \exp \left(-\zeta R\right){\displaystyle \sum_{s=0}^{n+1}\left(\begin{array}{c}\hfill n+1\hfill \\ {}\hfill s\hfill \end{array}\right)}{R}^{s-1}{r}_A^{n+1-s}\left\{{\displaystyle {\int}_{-1}^1dx}{x}^{n+1-s} \exp \left(-\zeta {r}_Ax\right)\right\}\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill ={\displaystyle \sum_{s=0}^{n+1}\left(\begin{array}{c}\hfill n+1\hfill \\ {}\hfill s\hfill \end{array}\right)}{R}^{s-1} \exp \left(-\zeta R\right){r}_A^{n+1-s}\hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \left\{H\left[1,n+1-s,\zeta {r}_A\right]-H\left[-1,n+1-s,\zeta {r}_A\right]\right\}\hfill \end{array}\end{array} $$

(3.C4)

$$ H\left[1,n+1-s,\zeta {r}_A\right]=-{\displaystyle \sum_{k=0}^{n+1-s}A\left(n+1-s,k\right)}{\zeta}^{-\left(k+1\right)}{r}_A^{-\left(k+1\right)} \exp \left(-\zeta {r}_A\right) $$

(3.C5)

$$ \begin{array}{l}H\left[-1,n+1-s,\zeta {r}_A\right]=-{\displaystyle \sum_{k=0}^{n+1-s}{\left(-1\right)}^{n+1-\left(s+k\right)}A\left(n+1-s,k\right)}{\zeta}^{-\left(k+1\right)}\\ {}\begin{array}{cccc}\hfill \begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}\times {r_A}^{-\left(k+1\right)} \exp \left(\zeta {r}_A\right)\end{array} $$

(3.C6)

If r _A  ≥ R, we obtain B function in Eq. (3.C1) as follows

$$ {r}_B^n \exp \left(-\zeta {r}_B\right)= \exp \left(-\zeta {r}_A\right){\displaystyle \sum_{s=0}^n\left(\begin{array}{c}\hfill n\hfill \\ {}\hfill s\hfill \end{array}\right)}{R}^{n-s}{r}_A^s\left\{{x}^{n-s} \exp \left(-\zeta Rx\right)\right\} $$

(3.C7)

$$ \begin{array}{l}J\left({r}_A,R\Big|{r}_A\ge R\right){r}_B^n \exp \left(-\zeta {r}_B\right)\\ {}\begin{array}{cc}\hfill \hfill & \hfill ={r}_A^{-1}{r}_B{r}_B^n \exp \left(-\zeta {r}_B\right)\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill = \exp \left(-\zeta {r}_A\right){\displaystyle \sum_{s=0}^{n+1}\left(\begin{array}{c}\hfill n+1\hfill \\ {}\hfill s\hfill \end{array}\right)}{R}^{n+1-s}{r}_A^{s-1}\left\{{x}^{n+1-s} \exp \left(-\zeta Rx\right)\right\}\hfill \end{array}\end{array} $$

(3.C8)

$$ \begin{array}{l}B\left(n,\zeta, R,{r}_A\Big|{r}_A\ge R\right)\\ {}\kern1em = \exp \left(-\zeta {r}_A\right){\displaystyle \sum_{s=0}^{n+1}\left(\begin{array}{c}\hfill n+1\hfill \\ {}\hfill s\hfill \end{array}\right)}{R}^{n+1-s}{r}_A^{s-1}\left\{{\displaystyle {\int}_{-1}^1dx}{x}^{n+1-s} \exp \left(-\zeta Rx\right)\right\}\\ {}\kern1em ={\displaystyle \sum_{s=0}^{n+1}\left(\begin{array}{c}\hfill n+1\hfill \\ {}\hfill s\hfill \end{array}\right)}{R}^{n+1-s}{r}_A^s \exp \left(-\zeta {r}_A\right)\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}\kern-0.35em \times \left\{\left\langle H\left[1,n+1-s,\zeta R\right]\right\rangle -\left\langle H\left[-1,n+1-s,\zeta R\right]\right\rangle \right\}\end{array} $$

(3.C9)

$$ \left\langle H\left[1,n+1-s,\zeta R\right]\right\rangle =-{\displaystyle \sum_{k=0}^{n+1-s}A\left(n+1-s,k\right)}{\zeta}^{-\left(k+1\right)}{R}^{-\left(k+1\right)} \exp \left(-\zeta R\right) $$

(3.C10)

$$ \begin{array}{l}\left\langle H\left[-1,n+1-s,\zeta R\right]\right\rangle \\ {}\begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}=-{\displaystyle \sum_{k=0}^{n+1-s}{\left(-1\right)}^{n+1-\left(s+k\right)}A\left(n+1-s,k\right)}{\zeta}^{-\left(k+1\right)}{R}^{-\left(k+1\right)} \exp \left(\zeta R\right)\end{array} $$

(3.C11)

where <……. > means a definite integral. We obtain B function in Eq. (3.C1) as

$$ \begin{array}{l}B\left(n,\zeta, R,{r}_A\Big|{r}_A\le R\right)\\ {}=-{\displaystyle \sum_{s=0}^{n+1}\left(\begin{array}{c}\hfill n+1\hfill \\ {}\hfill s\hfill \end{array}\right)}{R}^{s-1} \exp \left(-\zeta R\right){\displaystyle \sum_{k=0}^{n+1-s}A\left(n+1-s,k\right)}{\zeta}^{-\left(k+1\right)}{r}_A^{n-\left(s+k\right)} \exp \left(-\zeta {r}_A\right)\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \kern-1.32em +{\displaystyle \sum_{s=0}^{n+1}\left(\begin{array}{c}\hfill n+1\hfill \\ {}\hfill s\hfill \end{array}\right)}{R}^{s-1} \exp \left(-\zeta R\right){\displaystyle \sum_{k=0}^{n+1-s}{\left(-1\right)}^{n+1-\left(s+k\right)}A\left(n+1-s,k\right)}{\zeta}^{-\left(k+1\right)}\hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \kern-0.4em \times {r}_A^{n-\left(s+k\right)} \exp \left(\zeta {r}_A\right)\hfill \end{array}\end{array} $$

(3.C12)

$$ \begin{array}{l}B\left(n,\zeta, R,{r}_A\Big|{r}_A\ge R\right)\\ {}\begin{array}{cc}\hfill \hfill & \hfill =-{\displaystyle \sum_{s=0}^{n+1}\left(\begin{array}{c}\hfill n+1\hfill \\ {}\hfill s\hfill \end{array}\right)}{\displaystyle \sum_{k=0}^{n+1-s}A\left(n+1-s,k\right)}{\zeta}^{-\left(k+1\right)}{R}^{n-\left(s+k\right)} \exp \left(-\zeta R\right)\hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \kern-0.25em \times {r}_A^s \exp \left(-\zeta {r}_A\right)\hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \kern-0.15em +{\displaystyle \sum_{s=0}^{n+1}\left(\begin{array}{c}\hfill n+1\hfill \\ {}\hfill s\hfill \end{array}\right)}{\displaystyle \sum_{k=0}^{n+1-s}{\left(-1\right)}^{n+1-\left(s+k\right)}A\left(n+1-s,k\right)}{\zeta}^{-\left(k+1\right)}{R}^{n-\left(s+k\right)} \exp \left(\zeta R\right)\hfill \end{array}\end{array} $$

(3.C13)

1.4 Appendix D: Half Definite Integral, Hb, Hbhb and Hhb

We define three half definite integrals, Hb, Hbhb and Hhb in this appendix.

$$ \begin{array}{l}Hb\left[X,{n}_1,{\zeta}_a,{n}_2,{\zeta}_b\right]\\ {}\begin{array}{cc}\hfill \hfill & \hfill \equiv {\displaystyle {\int}^Xd{r}_A}{r}_A^{n_1} \exp \left(-{\zeta}_a{r}_A\right){\displaystyle {\int}_0^{\pi } \sin {\theta}_Ad{\theta}_A}{r}_B^{n_2} \exp \left(-{\zeta}_b{r}_B\right)\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill ={\displaystyle {\int}^Xd{r}_A}{r}_A^{n_1} \exp \left(-{\zeta}_a{r}_A\right){\displaystyle {\int}_{-1}^1J\left({r}_A,R\right)dx}{r}_B^{n_2} \exp \left(-{\zeta}_b{r}_B\right)\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill ={\displaystyle {\int}^Xd{r}_A}{r}_A^{n_1} \exp \left(-{\zeta}_a{r}_A\right)B\left({n}_2,{\zeta}_b,R,{r}_A\right)\hfill \end{array}\end{array} $$

(3.D1)

If X ≤ R,

$$ \begin{array}{l}Hb\left[X,{n}_1,{\zeta}_a,{n}_2,{\zeta}_b\Big|{\zeta}_a\ne {\zeta}_b,X\le R\right]\\ {}\begin{array}{cc}\hfill \hfill & \hfill ={\displaystyle \sum_{s=0}^{n_2+1}\left(\begin{array}{c}\hfill {n}_2+1\hfill \\ {}\hfill s\hfill \end{array}\right)}{R}^{s-1} \exp \left(-{\zeta}_bR\right)\left[{\displaystyle \sum_{k=0}^{n_2+1-s}A\left({n}_2+1-s,k\right)}{\zeta}_b^{-\left(k+1\right)}\right.\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill \kern0.7em \times {\displaystyle \sum_{t=0}^{n_1+{n}_2-\left(s+k\right)}A\left({n}_1+{n}_2-\left(s+k\right),t\right)}{\left({\zeta}_a+{\zeta}_b\right)}^{-\left(t+1\right)}\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill \kern0.7em \times \left.{X}^{n_1+{n}_2-\left(s+k\right)-t} \exp \left(-\left({\zeta}_a+{\zeta}_b\right)X\right)\right]\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill \kern0.75em -{\displaystyle \sum_{s=0}^{n_2+1}\left(\begin{array}{c}\hfill {n}_2+1\hfill \\ {}\hfill s\hfill \end{array}\right)}{R}^{s-1} \exp \left(-{\zeta}_bR\right){\displaystyle \sum_{k=0}^{n_2+1-s}{\left(-1\right)}^{n_2+1-\left(s+k\right)}A\left({n}_2+1-s,k\right)}\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill \kern0.7em \times {\displaystyle \sum_{t=0}^{n_1+{n}_2-\left(s+k\right)}A\left({n}_1+{n}_2-\left(s+k\right),t\right)}{\left({\zeta}_a+{\zeta}_b\right)}^{-\left(t+1\right)}\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill \kern0.7em \times {X}^{n_1+{n}_2-\left(s+k\right)-t} \exp \left(-\left({\zeta}_a+{\zeta}_b\right)X\right)\hfill \end{array}\end{array} $$

(3.D2)

$$ \begin{array}{l}Hb\left[R,{n}_1,{\zeta}_a,{n}_2,{\zeta}_b\Big|{\zeta}_a\ne {\zeta}_b,X\le R\right]\\ {}\begin{array}{cc}\hfill \hfill & \hfill = \exp \left(-\left({\zeta}_a+2{\zeta}_b\right)R\right)\left[{\displaystyle \sum_{s=0}^{n_2+1}\left(\begin{array}{c}\hfill {n}_2+1\hfill \\ {}\hfill s\hfill \end{array}\right){\displaystyle \sum_{k=0}^{n_2+1-s}A\left({n}_2+1-s,k\right)}{\zeta}_b^{-\left(k+1\right)}}\right.\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill \kern0.7em \times \left.{\displaystyle \sum_{t=0}^{n_1+{n}_2-\left(s+k\right)}A\left({n}_1+{n}_2-\left(s+k\right),t\right)}{\left({\zeta}_a+{\zeta}_b\right)}^{-\left(t+1\right)}{R}^{n_1+{n}_2-\left(k+t+1\right)}\right]\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill \kern0.75em - \exp \left(-\left({\zeta}_a+2{\zeta}_b\right)R\right)\left[{\displaystyle \sum_{s=0}^{n_2+1}\left(\begin{array}{c}\hfill {n}_2+1\hfill \\ {}\hfill s\hfill \end{array}\right){\displaystyle \sum_{k=0}^{n_2+1-s}{\left(-1\right)}^{n_2+1-\left(s+k\right)}A\left({n}_2+1-s,k\right)}}\right.\hfill \end{array}\\ {}\left.\begin{array}{cc}\hfill \hfill & \hfill \kern0.7em \times {\displaystyle \sum_{t=0}^{n_1+{n}_2-\left(s+k\right)}A\left({n}_1+{n}_2-\left(s+k\right),t\right)}{\left({\zeta}_a+{\zeta}_b\right)}^{-\left(t+1\right)}{R}^{n_1+{n}_2-\left(k+t+1\right)}\hfill \end{array}\right]\end{array} $$

(3.D3)

$$ \begin{array}{l} Hb\left[0,{n}_1,{\zeta}_a,{n}_2,{\zeta}_b\Big|{\zeta}_a\ne {\zeta}_b,X\le R\right]\\ {}\begin{array}{cc}\hfill \hfill & \hfill = \exp \left(-{\zeta}_bR\right)\left[{\displaystyle \sum_{s=0}^{n_2+1}\left(\begin{array}{c}\hfill {n}_2+1\hfill \\ {}\hfill s\hfill \end{array}\right)}{R}^{s-1}{\displaystyle \sum_{k=0}^{n_2+1-s}A\left({n}_2+1-s,k\right)}{\zeta}_b^{-\left(k+1\right)}\right.\hfill \end{array}\\ {}\begin{array}{ccc}\hfill \begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}\hfill & \hfill \hfill & \hfill \kern-2.65em \times \left({n}_1+{n}_2-\left(s+k\right)\right)!{\left({\zeta}_a+{\zeta}_b\right)}^{n_1+{n}_2+1-\left(s+k\right)}]\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill \kern0.75em - \exp \left(-{\zeta}_bR\right)\left[{\displaystyle \sum_{s=0}^{n_2+1}\left(\begin{array}{c}\hfill {n}_2+1\hfill \\ {}\hfill s\hfill \end{array}\right)}{R}^{s-1}{\displaystyle \sum_{k=0}^{n_2+1-s}{\left(-1\right)}^{n_2+1-\left(s+k\right)}A\left({n}_2+1-s,k\right)}{\zeta}_b^{-\left(k+1\right)}\right.\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill \kern0.7em \times \left({n}_1+{n}_2-\left(s+k\right)\right)!{\left({\zeta}_a-{\zeta}_b\right)}^{n_1+{n}_2+1-\left(s+k\right)}]\hfill \end{array}\end{array} $$

(3.D4)

$$ \begin{array}{l}Hb\left[X,{n}_1,{\zeta}_a,{n}_2,{\zeta}_b\Big|{\zeta}_a={\zeta}_b,X\le R\right]\\ {}\begin{array}{cc}\hfill \hfill & \hfill ={\displaystyle \sum_{s=0}^{n_2+1}\left(\begin{array}{c}\hfill {n}_2+1\hfill \\ {}\hfill s\hfill \end{array}\right)}{R}^{s-1} \exp \left(-{\zeta}_bR\right)\left[{\displaystyle \sum_{k=0}^{n_2+1-s}A\left({n}_2+1-s,k\right)}{\zeta}_b^{-\left(k+1\right)}\right.\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill \kern0.7em \times {\displaystyle \sum_{t=0}^{n_1+{n}_2+2-\left(s+k\right)}A\left({n}_1+{n}_2-\left(s+k\right),t\right)}{\left({\zeta}_a+{\zeta}_b\right)}^{-\left(t+1\right)}\hfill \end{array}{X}^{n_1+{n}_2-\left(s+k+t\right)}\\ {}\left.\begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\kern0.5em \times \exp \left(-\left({\zeta}_a+{\zeta}_b\right)X\right)\right]+{\displaystyle \sum_{s=0}^{n_2+1}\left(\begin{array}{c}\hfill {n}_2+1\hfill \\ {}\hfill s\hfill \end{array}\right)}{R}^{s-1} \exp \left(-{\zeta}_bR\right)\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \kern-0.32em \times {\displaystyle \sum_{k=0}^{n_2+1-s}{\left(-1\right)}^{n_2+1-\left(s+k\right)}A\left({n}_2+1-s,k\right)}{\zeta}_b^{-\left(k+1\right)}\frac{1}{n_1+{n}_2+1-\left(s+k\right)}X\hfill \end{array}\end{array} $$

(3.D5)

$$ \begin{array}{l} Hb\left[R,{n}_1,{\zeta}_a,{n}_2,{\zeta}_b\Big|{\zeta}_a={\zeta}_b,X\le R\right]\\ {}\begin{array}{cc}\hfill \hfill & \hfill = \exp \left(-\left({\zeta}_a+2{\zeta}_b\right)R\right)\left[{\displaystyle \sum_{s=0}^{n_2+1}\left(\begin{array}{c}\hfill {n}_2+1\hfill \\ {}\hfill s\hfill \end{array}\right)}{\displaystyle \sum_{k=0}^{n_2+1-s}A\left({n}_2+1-s,k\right)}{\zeta}_b^{-\left(k+1\right)}\right.\hfill \end{array}\\ {}\begin{array}{cc}\hfill \begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\hfill & \hfill \kern-0.6em \times \left[{\displaystyle \sum_{s=0}^{n_2+1}\left(\begin{array}{c}\hfill {n}_2+1\hfill \\ {}\hfill s\hfill \end{array}\right)}{\displaystyle \sum_{k=0}^{n_2+1-s}A\left({n}_2+1-s,k\right)}{\zeta}_b^{-\left(k+1\right)}\right.\hfill \end{array}\\ {}\begin{array}{ccc}\hfill \begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\hfill & \hfill \kern-0.6em \times \left.{\displaystyle \sum_{t=0}^{n_1+{n}_2+2-\left(s+k\right)}A\left({n}_1+{n}_2-\left(s+k\right),t\right)}{\left({\zeta}_a+{\zeta}_b\right)}^{-\left(t+1\right)}{R}^{n_1+{n}_2-\left(k+t+1\right)}\right]\hfill & \hfill \hfill \end{array}\\ {}\begin{array}{cc}\hfill \begin{array}{cc}\hfill \hfill & \hfill \kern0.65em + \exp \left(-{\zeta}_bR\right){\displaystyle \sum_{s=0}^{n_2+1}\left(\begin{array}{c}\hfill {n}_2+1\hfill \\ {}\hfill s\hfill \end{array}\right)}\hfill \end{array}\hfill & \hfill \hfill \end{array}\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \kern-0.26em \times {\displaystyle \sum_{k=0}^{n_2+1-s}{\left(-1\right)}^{n_2+1-\left(s+k\right)}A\left({n}_2+1-s,k\right)}{\zeta}_b^{-\left(k+1\right)}\frac{1}{n_1+{n}_2+1-\left(s+k\right)}{R}^{n_1+{n}_2+1-\left(k+1\right)}\hfill \end{array}\end{array} $$

(3.D6)

$$ Hb\left[0,{n}_1,{\zeta}_a,{n}_2,{\zeta}_b\Big|{\zeta}_a={\zeta}_b,X\le R\right]=0 $$

(3.D7)

$$ \begin{array}{l}Hb\left[X,{n}_1,{\zeta}_a,{n}_2,{\zeta}_b\Big|X\ge R\right]\\ {}\begin{array}{cc}\hfill \hfill & \hfill \equiv {\displaystyle {\int}^Xd{r}_A}{r}_A^{n_1} \exp \left(-{\zeta}_a{r}_A\right){\displaystyle {\int}_0^{\pi } \sin {\theta}_Ad{\theta}_A}{r}_B^n\kern0.20em \exp \left(-{\zeta}_b{r}_B\right)\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill ={\displaystyle {\int}^Xd{r}_A}{r}_A^{n_1} \exp \left(-{\zeta}_a{r}_A\right){\displaystyle {\int}_{-1}^1J\left({r}_A,R\Big|{r}_A\ge R\right)dx}{r}_B^n\kern0.20em \exp \left(-{\zeta}_b{r}_B\right)\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill ={\displaystyle {\int}^Xd{r}_A}{r}_A^{n_1} \exp \left(-{\zeta}_a{r}_A\right)B\left({n}_2,{\zeta}_b,R,{r}_A\Big|{r}_A\ge R\right)\hfill \end{array}\end{array} $$

(3.D8)

$$ \begin{array}{l} Hb\left[X,{n}_1,{\zeta}_a,{n}_2,{\zeta}_b\Big|X\ge R\right]\\ {}\begin{array}{cc}\hfill \hfill & \hfill ={\displaystyle \sum_{s=0}^{n_2+1}\left(\begin{array}{c}\hfill {n}_2+1\hfill \\ {}\hfill s\hfill \end{array}\right)}\left\{{\displaystyle \sum_{k=0}^{n_2+1-s}A\left({n}_2+1-s,k\right)}{\zeta}_b^{-\left(k+1\right)}{R}^{n_2-\left(s+k\right)} \exp \left(-{\zeta}_bR\right)\right\}\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill \kern0.66em \times {\displaystyle \sum_{t=0}^{n_1+s}A\left({n}_1+s,t\right)}{\left({\zeta}_a+{\zeta}_b\right)}^{-\left(t+1\right)}{X}^{n_1+s-t} \exp \left(-\left({\zeta}_a+{\zeta}_b\right)X\right)\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill \kern0.7em -{\displaystyle \sum_{s=0}^{n_2+1}\left(\begin{array}{c}\hfill {n}_2+1\hfill \\ {}\hfill s\hfill \end{array}\right)}\left\{{\displaystyle \sum_{k=0}^{n_2+1-s}{\left(-1\right)}^{n_2+1-\left(s+k\right)}A\left({n}_2+1-s,k\right)}{\zeta}_b^{-\left(k+1\right)}{R}^{n_2-\left(s+k\right)} \exp \left({\zeta}_bR\right)\right\}\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill \kern0.65em \times {\displaystyle \sum_{t=0}^{n_1+s}A\left({n}_1+s,t\right)}{\left({\zeta}_a+{\zeta}_b\right)}^{-\left(t+1\right)}{X}^{n_1+s-t} \exp \left(-\left({\zeta}_a+{\zeta}_b\right)X\right)\hfill \end{array}\end{array} $$

(3.D9)

$$ \begin{array}{l} Hb\left[R,{n}_1,{\zeta}_a,{n}_2,{\zeta}_b\Big|X\ge R\right]\\ {}\begin{array}{cc}\hfill \hfill & \hfill = \exp \left(-\left({\zeta}_a+2{\zeta}_b\right)R\right){\displaystyle \sum_{s=0}^{n_2+1}\left(\begin{array}{c}\hfill {n}_2+1\hfill \\ {}\hfill s\hfill \end{array}\right)}\hfill \end{array}\left\{{\displaystyle \sum_{k=0}^{n_2+1-s}A\left({n}_2+1-s,k\right)}{\zeta}_b^{-\left(k+1\right)}{R}^{n_2-\left(s+k\right)}\right\}\\ {}\begin{array}{ccc}\hfill \begin{array}{cc}\hfill \hfill & \hfill \kern0.68em \left\{{\displaystyle \sum_{t=0}^{n_1+s}A\left({n}_1+s,t\right)}{\left({\zeta}_a+{\zeta}_b\right)}^{-\left(t+1\right)}{R}^{n_1+s-t}\right\}\hfill \end{array}\hfill & \hfill \hfill & \hfill \hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill \kern0.75em - \exp \left(-{\zeta}_aR\right){\displaystyle \sum_{s=0}^{n_2+1}\left(\begin{array}{c}\hfill {n}_2+1\hfill \\ {}\hfill s\hfill \end{array}\right)}\hfill \end{array}\left\{{\displaystyle \sum_{k=0}^{n_2+1-s}{\left(-1\right)}^{n_2+1-\left(s+k\right)}A\left({n}_2+1-s,k\right)}{\zeta}_b^{-\left(k+1\right)}{R}^{n_2-\left(s+k\right)}\right\}\\ {}\begin{array}{ccc}\hfill \begin{array}{cc}\hfill \hfill & \hfill \kern0.7em \left\{{\displaystyle \sum_{t=0}^{n_1+s}A\left({n}_1+s,t\right)}{\left({\zeta}_a+{\zeta}_b\right)}^{-\left(t+1\right)}{R}^{n_1+s-t}\right\}\hfill \end{array}\hfill & \hfill \hfill & \hfill \hfill \end{array}\end{array} $$

(3.D10)

$$ \begin{array}{l}Hb\left[X,{n}_1,{\zeta}_a,{n}_2,{\zeta}_b\Big|X\ge R\right]\\ {}\begin{array}{cc}\hfill \hfill & \hfill \equiv {\displaystyle {\int}^Xd{r}_A}{r}_A^{n_1} \exp \left(-{\zeta}_a{r}_A\right){\displaystyle {\int}_0^{\pi } \sin {\theta}_Ad{\theta}_A}{r}_B^n\kern0.20em \exp \left(-{\zeta}_b{r}_B\right)\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill ={\displaystyle {\int}^Xd{r}_A}{r}_A^{n_1} \exp \left(-{\zeta}_a{r}_A\right){\displaystyle {\int}_{-1}^1J\left({r}_A,R\Big|{r}_A\ge R\right)dx}{r}_B^n\kern0.20em \exp \left(-{\zeta}_b{r}_B\right)\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill ={\displaystyle {\int}^Xd{r}_A}{r}_A^{n_1} \exp \left(-{\zeta}_a{r}_A\right)B\left({n}_2,{\zeta}_b,R,{r}_A\Big|{r}_A\ge R\right)\hfill \end{array}\end{array} $$

(3.D11)

$$ \begin{array}{l}Hhb\left[X,{n}_1,{\zeta}_a,{n}_2,{\zeta}_b,{n}_3,{\zeta}_c\right]\\ {}\begin{array}{cc}\hfill \hfill & \hfill \begin{array}{l}\equiv {\displaystyle {\int}^Xd{r}_{A_1}}{r}_{A_1}^{n_1} \exp \left(-{\zeta}_a{r}_{A_1}\right){\displaystyle {\int}^{r_{A_1}}d{r}_{A_2}}{r}_{A_2}^{n_2}\kern0.20em \exp \left(-{\zeta}_a{r}_{A_2}\right)\\ {}\begin{array}{cc}\hfill \hfill & \hfill \hfill \end{array}\kern-0.6em \times {\displaystyle {\int}_0^{\pi } \sin {\theta}_{A_2}d{\theta}_{A_2}}{r}_{B_2}^{n_3} \exp \left(-{\zeta}_c{r}_{B_2}\right)\end{array}\hfill \end{array}\\ {}\begin{array}{cc}\hfill \hfill & \hfill \kern-0.15em ={\displaystyle {\int}^Xd{r}_A}{r}_A^n \exp \left(-{\zeta}_a{r}_A\right)Hb\left[{r}_A,{n}_2,{\zeta}_b,{n}_3,{\zeta}_c\right]\hfill \end{array}\end{array} $$

(3.D12)

$$ \begin{array}{l} Hbhb\left[X,{n}_1,{\zeta}_a,{n}_2,{\zeta}_b,{n}_3,{\zeta}_c,{n}_4,{\zeta}_d\right]\\ {}\begin{array}{cc}\hfill \hfill & \hfill \equiv {\displaystyle {\int}^Xd{r}_{A_1}}{r}_{A_1}^{n_1} \exp \left(-{\zeta}_a{r}_{A_1}\right)B\left({n}_2,{\zeta}_b,R,{r_A}_{1_1}\right)Hb\left[{r}_{A_1},{n}_3,{\zeta}_c,{n}_4,{\zeta}_d\right]\hfill \end{array}\end{array} $$

(3.D13)

In some cases of multiple integral, two types of integral appear.

$$ {\displaystyle {\int}^X{x}^n} \exp \left(- ax\right) \log\;x $$

(3.D14)

and

$$ {\displaystyle {\int}^X{x}^n} \exp \left(- ax\right)Ei\left(-bx\right) $$

(3.D15)

These integrals are expressed by the exponential integral function, the Euler gamma function, the generalized hypergeometric function, and log x.