Polynomial coefficients. Application to spin-spin splitting by N equivalent nuclei of spin I > 1/2

Abstract

The NMR intensity pattern of a nucleus split by N identical nuclei of spin 1/2 is given by the familiar binomial coefficients. These are conveniently obtained from Pascal's Triangle, equivalent to the chemist's branching diagram. Much less well known is the pattern from splitting by N identical nuclei of spin I > 1/2. This had been presented originally in terms of multinomial coefficients, but polynomial coefficients are more convenient. These describe the number of ways that N objects can be distributed to 2I+1 numbered boxes. They arise in the polynomial expansion, which is well known to mathematicians but not to chemists and which is conveniently obtained from generalizations of Pascal's Triangle. Examples include the 13C NMR of methanol-d3 and the EPR of tetracyanoethylene radical anion. Predictions are made for the 1H NMR of HC(CD3)3 and the 31P{1H} NMR of (CD3CH2)4P+.