**Appendix A**

Summarizing the results of Section 2, one obtains.

$$\psi\_{5,2}(\mathbf{a},\theta) = -\frac{Z^2(\pi - 2)(5\pi - 14)}{270\sqrt{\pi}} \left[ 3\pi^{-3/2}(2f\_1 + f\_2) - 2Z(f\_3 + \sqrt{2}f\_4) \right],\tag{A1}$$

where

$$f\_1 = -\frac{1}{60}\sqrt{1 - \sin a \cos \theta} [\sin a \cos \theta (4 + 13 \sin a \cos \theta) - 2],\tag{A2}$$

$$f\_3(a) = -\frac{1}{60\pi^{3/2}} [11\sin a + 21\cos(2a) + 2] \sqrt{1 + \sin a} \tag{A3}$$

$$f\_4(a,\theta) = -\frac{\sqrt{2}}{6\pi^{3/2}}(\sin a)^2 \sqrt{1+\sin a} \,\, P\_2(\cos \theta),\tag{A4}$$

$$f\_2(\mathfrak{a}, \theta) = \frac{1}{6} \sum\_{l=0}^{\infty} \frac{\zeta\_l(\rho) P\_l(\cos \theta)}{(2l - 1)(2l + 3)}.\tag{A5}$$

The *ζ* function is defined as follows:

$$\mathcal{J}\_l(\rho) = \begin{cases} \begin{array}{c} \chi\_l(\rho), \\ \chi\_l(1/\rho), \end{array} & 0 \le \rho \le 1 \end{cases} \tag{A6}$$

where

$$\chi\_l(\rho) = \frac{\rho^l}{(\rho^2 + 1)^{5/2}} \left[ \frac{(l-3)(2l-1)\rho^6}{2l+7} + 9l\rho^4 - 9(l+1)\rho^2 - \frac{(l+4)(2l+3)}{2l-5} \right]. \tag{A7}$$

Recall that *ρ* = tan(*α*/2), 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 (93)–(96).

Summarizing the results of Section 3, one obtains:

$$\Psi\_{6,3}(\mathbf{a},\theta) = \frac{Z^3(\pi - 2)(5\pi - 14)}{56700\pi^{3/2}\sqrt{5}} \left[ (97 - 32\pi)\mathcal{Y}\_{6,1}(\mathbf{a},\theta) + \frac{(1112 - 357\pi)}{12}\mathcal{Y}\_{6,3}(\mathbf{a},\theta) \right],\tag{A8}$$

where *Yn*,*<sup>l</sup>*(*<sup>α</sup>*, *θ*) are the normalized hyperspherical harmonics.
