An improved algorithm for the evaluation of intra- and inter-molecular contributions to the NMR second moment is presented. The final expressions are given in terms of quadratic forms of the second moment tensor which depends on the equilibrium crystal structure as well as on parameters describing average atomic and molecular motion, assuming the latter to be expressible by the rigid body motion formalism. These parameters are readily available once the crystal structure of the compound considered has been determined by X-ray or neutron diffraction methods, and may serve for the evaluation of second moments of single-crystal as well as polycrystalline specimens. The use of the second moment tensor formulation considerably reduces the computational effort which is required by such calculations when the Van Vleck formula is being directly evaluated. An explicit introduction of crystal symmetry into the second moment tensor formalism, presented in this paper, further enhances the efficiency and ease of application of the proposed method.