Commutation relations anomalous

Because of these anomalous commutation relations the transformation to space-fixed angular momentum components takes a special form. An equation similar to Eq. (2.80) yields... 

In computing the rotation Hamiltonian matrix in eqn (14.25), we should note that Hj is the projection of the angular momentum operator H along the molecular axis. Thus the angular momentum operator H,- satisfies the anomalous commutation relation ... 

We present a detailed description of angular momentum theory in chapter 5, and the reader may wish to examine the results given there at this stage. It emerges that the angular momentum operator / commutes with L and S in this axis system and its molecule-fixed components obey the usual commutation relations for angular momentum operators provided that the anomalous sign of i is used,... 

These components are now independent of the orbital and spin variables and so commute with L and S. They also obey the commutation relations for a general angular momentum, provided only that the anomalous sign of i is used. 

A major difficulty for molecular as opposed to atomic systems arises from the fact that two different reference axis systems are important, the molecule-fixed and the space-fixed system. Many perturbation related quantities require calculation of matrix elements of molecule-fixed components of angular momentum operators. Particular care is required with molecule-fixed matrix elements of operators that include an angular momentum operator associated with rotation of the molecule-fixed axis system relative to the space-fixed system. The molecule-fixed components of such operators have a physical meaning that is not intuitively obvious, as reflected by anomalous angular momentum commutation rules. 

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