Non-Adiabatic Rotational and Vibrational Reduced Masses

The two alternative expressions for the spin rotation tensor in equations (6.53) and (6.56) are equivalent, if we are dealing with the exact eigenstates of the unperturbed Hamiltonian. However, in approximate calculations this does not always hold and different values might be obtained from Eqs. (6.53) and (6.56). 

This relation is of great importance for NMR spectroscopy because it allows us to determine the absolute shielding tensor by a combination of the measured spin rotation tensor with its nuclear contribution, which can easily be calculated from the nuclear coordinates, and a calculated diamagnetic contribution 

This is the only possibility to determine experimental, or rather semi-experimental, absolute shielding constants, as one can only obtain differences in the shielding constants, i.e. chemical shifts, from NMR spectra as discussed in Section 5.7. However, one has to be careful in applying this relation similar to the relation between the rotational g tensor and the magnetizability, Eq. (6.30). First, one has to take care of the vibrational corrections in the measured spin rotation tensors and secondly NMR spectra are normally measured in the liquid phase so that solvent effects would have to be considered as well. Nevertheless, it has been used to establish absolute shielding scales for several light nuclei (Flygare, 1964 Hindermann and Cornwell, 1968 Jameson et al, 1980 Vaara et al, 1998 Puzzarini et al, 2009). 

In the previous sections we have studied Born-Oppenheimer-breakdown corrections to two molecular properties, the rotational g tensor and the nuclear spin-rotation constant, i.e. the effect of the coupling between nuclear and electronic motion on the electronic energies. In this and the following sections we will now turn our attention to the effect of this coupling on the motion of the nuclei and will discuss Born-Oppenheimer-breakdown corrections to the rotational and vibrational energies. For the sake of a simpler presentation we will illustrate it for a diatomic molecule AB, where there is only one vibrational mode that involves changes in the internuclear 

Born-Oppenheimer expressions for vibration-rotation energies of diatomic molecules have been derived several times (Herman and Asgharian, 1966 Watson, 1973 Bunker and Moss, 1977 Watson, 1980 Herman and Ogilvie, 1998). Here we will follow mainly the derivation by Brniker and Moss (Bunker and Moss, 1977). After separation of the translation of the whole molecule and transformation to nuclear centre of mass coordinates one can write the field-free Hamiltonian for the electronic ground state of symmetry of a diatomic molecule as 

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The corrections to the vibrational and rotational reduced nuclear masses, on the other hand, can be related to molecular electromagnetic properties. They also involve sum over excited states and are therefore non-adiabatic corrections... 

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