Knowledge Bank

Effective rotational Hamiltonians, written as power series in the body-fixed angular momentum components $P\alpha (\alpha =a,b,c)$, have been used extensively in the analysis of high-resolution pure rotation and vibration-rotation spectra. When written in terms of cylindrical tensor components, two alternative forms are in $use1$. In one form, the operators are constructed as Hermitian combinations of products of integral powers of $p2,PC2$ and $(Pa2-Pb2)$ where $p2=P2+P2+PC2$, and for the other form one seus integral powers of $P2$, and $PC2$ along with $(P+2n+P-2n),n=1,2,3,\dots$ where $P\pm =Pa\pm iPb$. Since $(Pa2-Pb2)n\ne (P+2n+P-2n)/2n$ except for $n=1$, the sets of empirical constants determined with the two formulations will not, in general be identical and the differences become the more extensive the larger the value of n. Since coefficients of octic, dectic and even higher powers of angular momentum are now regularly reported, it becomes important that the implications of the above inequality be examined in detail so that results from different laboratories can be compared more readily and so that empirical constants can be correctly related to theoretical expressions and reduction-invariant combinations which are specified through a model Hamiltonian written in terms of Hermitian combinations of products of integral powers of $Pa2,Pb2$ and $PC2$. A study of the above inequality will be described and its consequences for rotational Hamiltonians in general will be illustrated with a detailed consideration of quartic Hamiltonian.