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An explicit formula is derived for the zero point energy of a system of coupled harmonic oscillators. It is shown that for the. case of isotopic homologues this leads to an approximate linear relation in the number of equivalent isotopic substituents. It is shown that the zero point energy differences of successive members of an isotopic homologous series increases with further substitution of the heavy isotope. Through a study of the parabolic deviations from the linear rule, it is shown thatdoi:10.1515/zna-1963-0217 fatcat:gammewe72ncuxpq3vmgqvxkq5a