Rotational g Tensor and Electric Quadrupole Moment

Recalling the definition of the electric dipole moment, Eq. (4.25), introducing the total charge q of the molecule 

This relation allows us to determine experimentally the electric dipole moment by simply measuring the rotational g tensor of two isotopologues (Rosenblum et al., 1958). However, one should keep in mind that the expressions for the dipole moment as well as for the rotational g tensor, which were used in the derivation, are for a particular nuclear geometry. The effects of vibrational motion of the nuclei, as discussed in Chapter 8, are thus not included, while experimentally measured rotational g tensors are always for a particular vibrational state. Equation (6.38) can thus only be applied to measured rotational g tensors, if the changes due to the vibrational motion of the nuclei are negligible or explicitly corrected for prior to application of Eq. (6.38). 

The rotational g tensor can also be related to the electric quadrupole moment tensor . We will in the following derive this relation for the xx diagonal component, which according to Eq. (4.27) is given as 

Recalling the relation between the magnetizability and the rotational g tensor, Eq. (6.30), we can replace the diamagnetic and nuclear contributions by the total magnetizability and rotational g tensor and obtain finally 

Corresponding relations for the other diagonal components can be obtained by cyclic permutations of the coordinate triple xyz to yzx or zxy. Since the quadrupole moment of a polar molecule depends on the origin of the coordinate system it is important to remember that the centre of nuclear masses is automatically chosen as the origin in Eq. (6.41). Similar to the previously derived relations between the rotational g tensor and other molecular properties, Eqs. (6.30) and (6.38), one has to take care of vibrational corrections to the properties in Eq. (6.41), when it is applied to measured rotational g tensors and magnetizabilities. 

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