I think there might be a typo in the equation for the Cartesian electric field gradient integrals for
, and that the prefactor of (1/3) should not be present.
Ignoring symmetry and their prefactors, each component of the EFG integrals should be the same as for the spin-dipole integrals. That is, when I divide the matrix representation of
by the matrix representation of
, for example, each entry should be 1 after the prefactors are divided out as well. However, this is only true when (g_e/2) is divided out of the SD integrals, and not (1/3) from the EFG integrals.
I did not check the spherical EFG integrals.