# Coulomb integrals of spherical Bessel functions

• A
Hi, I'm no expert in math so I'm struggling with solving these integrals, I believe there's an analytical solution (maybe in http://www.hfa1.physics.msstate.edu/046.pdf).
$$V_{1234}=\int_{x=0}^{\infty}\int_{y=0}^{\infty}d^3\pmb{x}d^3\pmb{y}\, j_{l_1}^*(a_1\pmb{x})j_{l_2}(a_2\pmb{x})|\pmb{x}-\pmb{y}|^{-1}j_{l_3}^*(a_3\pmb{y})j_{l_4}(a_4\pmb{y})$$
where $j_l(r)$ are spherical Bessel functions. Does anyone know how to solve these integrals analytically?

DrDu
I would try to express the spherical Bessel functions in terms of eq. 10.54.2 from http://dlmf.nist.gov/10.54
i don't know how this helps, could you elaborate on your next step?

what if the integral was... $$V_{1234}=\int_{x=0}^{R}\int_{y=0}^{R}dxdy\, j_{l_1}^*(z_{l_1}x/R)j_{l_2}(z_{l_2}x/R)|x-y|^{-1}j_{l_3}^*(z_{l_3}y/R)j_{l_4}(z_{l_4}y/R)$$
where z_l is the first root of the l-th order spherical Bessel function

DrDu
I think it would be helpful if you could state your original problem.

I think it would be helpful if you could state your original problem.
im lookin for an analytical way of solving these integrals, specifically the second one, for use as a basis set in a full CI calculation on idealized colloidal nanostructures, hence infinite spherical well solutions - the bessel functions :)

if i turn my integrand into a product of integrals themselve, does this lead to some simplifications? how does this work?

im lookin for an analytical way of solving these integrals, specifically the second one, for use as a basis set in a full CI calculation on idealized colloidal nanostructures, hence infinite spherical well solutions - the bessel functions :)

if i turn my integrand into a product of integrals themselve, does this lead to some simplifications? how does this work?
and to give you a clue of my level of math, I have no idea what a Wronskian is haha

I've tried solving this using mathematica but i cant figure out how to do it.

I think it would be helpful if you could state your original problem.
to further elaborate, I'm trying show that a CI code works, and so I chose to have single particle states in an inf sph potential thinking that my Coulomb integrals i need in my CI hamiltonian for my system would have some nice neat forms. I'm starting to doubt this haha.

I'm contemplating switching to a cubic potential but that impacts on demonstrating angular momenta in the way I want to from my CI code. Would you expect things to be easier for me with solutions to a cubic potential?(i.e. same integrals but the bessell functions are replaced with sines and cosines)

DrDu
i don't know how this helps, could you elaborate on your next step?

what if the integral was... $$V_{1234}=\int_{x=0}^{R}\int_{y=0}^{R}dxdy\, j_{l_1}^*(z_{l_1}x/R)j_{l_2}(z_{l_2}x/R)|x-y|^{-1}j_{l_3}^*(z_{l_3}y/R)j_{l_4}(z_{l_4}y/R)$$
where z_l is the first root of the l-th order spherical Bessel function
Don't forget the ##x^2## and ##y^2## from the volume elements!

DrDu
Ben, have a look at "Application of the Legendre polynomials in physics":
https://en.wikipedia.org/wiki/Legendre_polynomials
The formula stated there is used to convert the coulomb operator into a sum of two operators depending only on x and y and some Legendre polynomials.
The integral over the latter together with the angular dependence of your wavefunctions restricts the summation to a small number of terms.
The remaining integrals involve only products of two Bessel functions and powers of x or y. This is the kind of integrals covered by the article you cited.
I expect that this problem has been tackled before, probably in nuclear physics. So maybe you find a solution there more readily.
Edit: Have a look here:
http://k2.chem.uh.edu/library/Index/UnCatagorized/PRB35118.pdf

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Don't forget the ##x^2## and ##y^2## from the volume elements!
do you mean $$d^3\pmb{x} \to x^2 dx$$ or something else?

DrDu
do you mean $$d^3\pmb{x} \to x^2 dx$$ or something else?