Rotational g Tensor and Electric Dipole Moment

On comparison of the electronic contribution with the expression for the paramagnetic contribution to the magnetizability in Eq. (5.44) we can see that they are 

Combined with a calculated diamagnetic contribution to the magnetizability one can thus obtain semi-experimental values of the magnetizability 

However, one should keep in mind that this relation only holds for a given fixed nuclear geometry, while the measured rotational g tensor is for a particular vibrational state. Direct application of it will therefore neglect the possibly large contributions from vibrational motion of the nuclei (see e.g. Lutnaes et al. (2009)), as discussed in Chapter 8. For an accurate determination of the magnetizability it is therefore necessary to correct explicitly for the vibrational corrections included in the measured rotational g tensor. 

In the previous section we have seen that the rotational g tensor is related to the paramagnetic contribution to the magnetizability, Eq. (6.29). Here, we will explore a relation between the rotational g tensor and the electric dipole moment. We will see that the latter is related to the difference between the rotational g tensor of two isotopologues of the same molecule, i.e. two molecules that differ only in the isotopes of one or more nuclei. We consider therefore a component of the rotational g tensor, d japy of one isotopologue with moment of inertia tensor I and centre of nuclear masses, R qm, which is shifted by a vector D = R q — Rcm from the centre of nuclear masses of the second isotopologue with moment of inertia tensor I 

Using the relation between the position vectors of the two centres of mass we can rewrite this as 

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