I am now having a problem regarding the integral over two spherical Bessel function. If anyone can give any help, it would be so nice of you. Thank you so much for any help.
Specifically, I intend to solve the problem $\langle n^\prime l^\prime m^\prime|z|n l m\rangle$, where $|nlm\rangle$ is the eigenvector for spherical Kohn-Sham equation (in DFT method) and it can be written as: $$|nlm\rangle=R_{nl}(r)Y_l^m(\theta,\phi)$$ where $R_{nl}(r)$ is the radial function, and it can be expressed as: $$R_{nl}(r)\propto j_l(\alpha_{nl}\ r)$$ in which $j_l$ is the $l$-th order of spherical Bessel function, $\alpha_{nl}$ is the $n$-th zeros of $j_l(x)$, i.e. $j_l(\alpha_{nl})=0$. Since $z=r\cos \theta$, in the calculation of the equation $\langle n^\prime l^\prime m^\prime|z|n l m\rangle$, the integral over $\theta, \phi$ can be easily achieved, but I didn't find the solution for the integral over spherical Bessel function. This integral is:
$$\langle n^\prime l^\prime m^\prime|z|n l m\rangle\propto \int_0^\infty j_{n^\prime l^\prime}(\alpha_{n^\prime l^\prime}\ x)j_l(\alpha_{nl}\ x)x^3\ dx$$
where $\alpha_{nl}$ is the $n$-th zeros of $j_l(x)$, i.e. $j_l(\alpha_{nl})=0$.Noticeable, in my problem the radial eigenvector $R_{nl}(r)$ is the solution of an infinity well depth, so the integral region can also be changed to $(0,1)$, i.e.,
$$\langle n^\prime l^\prime m^\prime|z|n l m\rangle\propto \int_0^{\color{blue}{1}} j_{n^\prime l^\prime}(\alpha_{n^\prime l^\prime}\ x)j_l(\alpha_{nl}\ x) x^3\ dx$$
My problem is, how to calculate the above integral? Can anyone give any hints? Thank you so much for any help.