Electric field gradients, generalized Sternheimer shielding constants, and electric-field-gradient polarizabilities are calculated for H-2, N-2, F-2,HF HCl, CO, HCN, HNC, H2O, and NH3. The calculations are performed at both the Hartree-Fock and second order Moller-Plesset levels of approximation using large basis sets. For most of these molecules this is the first time that the shielding constants and electric field gradient polarizabilities have been determined. Electron correlation is generally found to be a significant factor.