Symmetry Numbers Continued, Comments, Polyatomics

Proceeding in the spirit above it seems reasonable to inquire why s is equal to the number of equivalent rotations, rather than to the total number of symmetry operations for the molecule of interest. Rotational partition functions of the diatomic molecule were discussed immediately above. It was pointed out that symmetry requirements mandate that homonuclear diatomics occupy rotational states with either even or odd values of the rotational quantum number J depending on the nuclear spin quantum number I. Heteronuclear diatomics populate both even and odd J states. Similar behaviors are expected for polyatomic molecules but the analysis of polyatomic rotational wave functions is far more complex than it is for diatomics. Moreover the spacing between polyatomic rotational energy levels is small compared to kT and classical analysis is appropriate. These factors appreciated there is little motivation to study the quantum rules applying to individual rotational states of polyatomic molecules. 

For practical purposes the rules for diatomic molecules concerning even and odd J reduce to the statement that for homonuclear diatomic molecules the molecular partition function must be divided by two (s = 2), while for heteronuclear diatomic molecules no division is necessary (s = 1). The idea of the symmetry number, s, 

How does one determine the symmetry number As illustrated in the section above it is equal to the number of rotations that take the molecule into itself. Another and very attractive method is based on the use of group theory. Students who have taken a course in inorganic chemistry have been introduced to group theory. If the reader is uncomfortable with this topic the next few paragraphs can be skipped, especially since this method of finding molecular symmetry numbers need not to be used for finding the ratios of symmetry numbers, Si/s2, required to understand isotopomer fractionation. 

4 Isotope Effects on Equilibrium Constants of Chemical Reactions 

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