**Appendix B**

Summarizing the results of Section 4.1, one obtains:

$$\psi\_{7,3}(a,\theta) = \frac{(\pi - 2)(5\pi - 14)Z^3}{340200\sqrt{5}\pi^{3/2}} \left\{ \frac{\vec{f}\_1 + \vec{f}\_2}{\sqrt{5}\pi^{3/2}} - 2Z \left[12(32\pi - 97)\vec{f}\_3 + (357\pi - 1112)\vec{f}\_4\right] \right\},\tag{A9}$$

where

$$\begin{split} \vec{f}\_{1} &= \left(\frac{41437\pi}{12} - \frac{74342}{7}\right) \vec{\xi}^{\top} + \left(36476 - \frac{35588\pi}{3}\right) \vec{\xi}^{5} + \\ &\quad + \frac{5}{2} (4931\pi - 15156) \vec{\xi}^{3} + 5 (2276 - 741\pi) \vec{\xi}. \end{split} \tag{A10}$$

$$\bar{f}\_{\mathfrak{I}} = -\frac{\rho(1+\rho)(29+\rho\{16+\rho[\rho(16+29\rho)-114]\})\cos\theta}{9\sqrt{5}\ \pi^{3/2}(\rho^2+1)^{7/2}},\tag{A11}$$

$$\bar{f}\_4 = -\frac{\sin^3 \alpha \sqrt{1 + \sin \alpha}}{2\sqrt{5} \,\, n^{3/2}} P\_3(\cos \theta),\tag{A12}$$

$$\bar{f}\_2 = \frac{1}{48} \sum\_{l=0}^{\infty} \frac{\bar{\zeta}\_l(\rho) P\_l(\cos \theta)}{(2l - 1)(2l + 3)}. \tag{A13}$$

The ¯ *ζ* function is defined as follows:

$$\mathcal{L}\_l(\rho) = \begin{cases} \bar{\chi}\_l(\rho)\_{\prime} & 0 \le \rho \le 1 \\ \bar{\chi}\_l(1/\rho)\_{\prime} & \rho \ge 1 \end{cases} \tag{A14}$$

where

$$\begin{split} \bar{\chi}\_{l}(\rho) &= -\frac{\rho^{l}}{(\rho^{2}+1)^{7/2}} \Big\{ \frac{(32l^{2}+26l-25)\rho^{6}}{2l+5} \Big[ \frac{(2l-1)\rho^{2}}{2l+9} + 4 \right] + \\ &+ \frac{1}{2l-3} \Big[ \frac{6(84l^{2}+84l-95)\rho^{4}}{2l+5} - (32l^{2}+38l-19) \Big( \frac{2l+3}{2l-7} + 4\rho^{2} \Big) \Big] .\end{split} \tag{A15}$$

Recall that variable *ξ* is defined by Equation (6), and special cases of the function ¯ *f*2 ≡ ¯ *f*2(*<sup>α</sup>*, *<sup>θ</sup>*), when they can be obtained in closed form, are represented by Equations (141)–(144). Summarizing the results of Section 4.2, one obtains:

$$\Psi\_{8,4} = \frac{Z^4(\pi - 2)(5\pi - 14)}{\pi^{5/2}} [b\_{80} Y\_{8,0}(a, \theta) + b\_{82} Y\_{8,2}(a, \theta) + b\_{84} Y\_{8,4}(a, \theta)],\tag{A16}$$

where

$$b\_{80} = \frac{\pi (150339\pi - 927292) + 1430792}{19289340000}, \quad b\_{82} = \frac{\pi (751965\pi - 4654046) + 7209976}{1928934000\sqrt{70}},$$

$$b\_{84} = \frac{\pi (3190317\pi - 19828996) + 30802176}{25719120000\sqrt{14}}.\tag{A17}$$
