Potentials in the spherical HO basis

In the spherical basis, the secondary densities to be used in Eq. (46) have the form given in Eq. (102). Therefore, the potentials in Eq. (60) acquire the form

$$\tilde{U}^{J''M''}_{n'L'v'J'M'}(\vec{r})\!\!\!\! = \sum_{TM_T} \tilde{U}_{n'L'v'J'}^{TJ''}(br) e^{-(br)^2} C^{TM_T}_{J'M'J''M''} Y_{TM_T}(\theta,\phi) ,$$

where the radial form factors read:

$$\tilde{U}_{n'L'v'J'}^{TJ''}(br) = \sum_{mI,nLvJ} \left\{C^{n'L'v'J'}_{mI,nLvJ}+(-1)^{J-J'}C^{nLvJ}_{mI,n'L'v'J'}\right\} \tilde{\rho}_{mI,nLvJ,J'}^{TJ''}(br).$$ (100)

Similarly, potentials ${U}^{J''M''}_{n'L'v'J'M'}(\vec{r})$ in Eq. (61) read

$${U}^{J''M''}_{n'L'v'J'M'}(\vec{r})\!\!\!\! = \sum_{TM_T} {U}_{n'L'v'J'}^{TJ''}(br) e^{-(br)^2} C^{TM_T}_{J'M'J''M''} Y_{TM_T}(\theta,\phi) ,$$ (101)

and the radial from factors ${U}_{n'L'v'J'}^{TJ''}(br)$ can be calculated in the complete analogy to the results outlined in Section 2, see Eq. (8), namely,

$${U}_{n'L'v'J'}^{TJ''}(br) = \sum_{a\alpha\beta;mI,nLvJ}C_{a,\alpha}^{\beta}\chi_{a,\alpha;n'L'v'J'}^{\beta;mI,nLvJ} \tilde{\rho}_{mI,nLvJ,J'}^{TJ''}(br).$$ (102)