APPROXIMATIONS FOR THE INERTIAL DEFECTS OF PLANAR MOLECULES

The observed zero-point inertial defect DELTA0 = I0cc - I0aa - I0bb is often used as a test of the planarity of a molecule, but this test is difficult to make quantitative when the information on the vibrations and vibration-rotation interactions is limited. The general formula for the inertial defect to order kappa2I(e) is separated into harmonic and Coriolis contributions DELTA0harm and DELTA0Cor, which can be expressed in various ways by use of sum rules. It is shown here that the zero-point harmonic term is given approximately by DELTA0harm almost-equal-to DELTA0tau = 3K(1/omega(A) + 1/omega(B) + 1/omega(AB) - 1/omega(C)), where K is h2BAR/2hc and the effective vibrational frequencies are given in terms of the centrifugal tensor tau(alphabetagammadelta) by tau(aaaa) = -16A3/omega(A)2, tau(bbbb) = -16B3/omega(B)2, tau(cccc) = -16C3/omega(C)2, and tau(abab) = -16A BC/omega(AB)2, so that DELTA0harm can be estimated from centrifugal distortion constants. A fit of empirical data gives DELTA0 almost-equal-to DELTA0tau[1.0292-0.0911(N-3)], where N is the number of atoms and the correction factor makes approximate allowance for DELTA0Cor. Thus an observed DELTA0 that is significantly negative relative to this approximation may indicate that the molecule is nonplanar or has unusually low out-of-plane frequencies.