Born-Oppenheimer approximations magnetic dipole moments

The Born-Oppenheimer approximation [96], created to simpUfy the electronic calculus for frozen nuclei approximation, breaks down when computing, for instance, the magnetic dipole moment and its derivative with respect to the nuclear velocities or momenta for assessing the molecular properties of surfaces [97]. 

Whereas the dipole moment gradient (the atomic polar tensor) is well defined and it is the same quantity that determines conventional infrared absorption spectra (see the chapter on vibrational spectroscopy), the gradient of the magnetic dipole moment is zero within the Born-Oppenheimer approximation. This is due to the fact that the magnetic dipole moment for a closed-shell molecule is quenched (since it corresponds to an expectation value of an imaginary operator), making the rotational strength in Eq. 2.150 zero. 

Deutsche and Moscowitz, a number erf semiempirical models have been proposed to explain features in experimental spectra. Although none has enjoyed any but the most modest successes, experience is gradually accumulating on the limits of applicability. Some of these are reviewed below. It is readily shown that within the Born-Oppenheimer (BO) approximation, the electronic contribution to the magnetic dipole transition moment associated with a vibrational transition of a molecule in its ground electronic state vanishes. A non-BO theory of VCD intensities was independently developed by several groups.2° 2" ... 

For vibrational transitions it is not appropriate to use the Born-Oppenheimer (BO) approximation because the ground and excited states in the context of vibrational transitions have the same electronic wavefunction and differ only in the nuclear wavefunctions, a consequence of which is that the electronic contribution to the magnetic dipole transition matrix element vanishes in the BO approximation. In order to include the important electronic contribution to magnetic dipole transition moments, one has to choose either to make further approximations to the magnetic dipole operator yielding effective non-vanishing magnetic transition moments, or to go beyond the BO approximation. Various approximate models and exact a priori methods have resulted in the last 25 years. 

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