Multicenter molecular integrals over the spherical Laguerre Gaussian-type functions (LGTFs), L-n(l+1/2)(ar(2)) r(l)Y(lm) ((r) over cap)e(-ar2), are evaluated analytically by using the generalized spherical gradient operator method. Addition theorem to expand the generalized spherical gradient operator, Y-nlm (del), is developed. Integrals are evaluated by transforming the product of the gradient operators instead of the product of LGTFs. The transformation G coefficients for the gradient operators are explicitly given in terms of the vector-coupling coefficients, and they are much simpler to evaluate than the Talmi coefficients which transform the product of LGTFs. The integral formulas obtained are compact and general for LGTFs of unrestricted quantum numbers n, l, and m. They are four-center (as well as three-center and two-center) integrals of two-electron irregular solid harmonic operator, Y-lm((r) over cap(12))/r(12)(l+1), where l=0, l=1, or l=2 corresponds to electron repulsion, spin-other-orbit, or spin-spin interaction, respectively. The two-center exchange-type and Coulomb-type integrals are also evaluated, In the case of two-center and three-center, integral formulas are also obtained for the one-electron irregular solid harmonic operator, Y-lm ((r) over cap)/r(l+1), were l=0, l=1, or l=2 corresponds to nuclear attraction, spin-orbit interaction or electron-spin nuclear-spin interaction, respectively, Integrals of multicenter overlap as well as transition multipole moment operator, r(l)Y(lm) ((r) over cap), have also been derived. All of the integral formulas are explicitly in terms of the vector-coupling coefficients and LGTFs of the internuclear coordinates, where the analytical derivatives of these integrals with respect to the geometrical variables can be easily obtained.