In most U.S. English style guides, the decision about whether to double- or single-hyphenate a phrase such as "spherical Gaussian based approximations" rests on whether the first word in the string attaches primarily to the noun or primarily to the modifier closer to the noun. In other words, if you are talking about Gaussian-based approximations that are spherical, you would normally use a single hyphen:
We propose spherical Gaussian-based approximations to calculate this analytically.
But if you are talking about the kind of Gaussian basis (namely, spherical) that underlies the approximations, you would normally use two hyphens:
We propose spherical-Gaussian-based approximations to calculate this analytically.
I don't know anything about this area of physics (or mathematics), but it seems to me likely that the first of these associations is the one that the writer intends here, which would make the single-hyphen approach suitable.