The nuclear shielding constants and their change in the presence of an external uniform electric field are investigated for H2, N2, HF, and CO. The relevant properties are calculated analytically at the self-consistent field (SCF) level of approximation. Particular attention is paid to the choice of basis sets and gauge variance, both of which are measured by the satisfaction of certain sum rules. These calculations of electric field dependence are more accurate than those presented hitherto and, in the case of N2, are the first to be published.