Mathematical Relationships of Individual Stereocenter er Values to dr Values

A mathematical relationship is derived for relating the enantiomeric ratios (er values) of two individual stereocenters within a single chiral molecule to the diastereomeric ratio (dr). Whereas the er (or enantiomeric excess, ee) of chiral molecules is readily determined by chiral chromatography and dr values can be determined by chromatography or NMR, modern methods for the optical determination of er values at individual functional groups do not normally determine the er and dr of the entire molecule. We find there is only a special circumstance when knowledge of the er of two individual stereocenters can be used to predict the er of the enantiomers in each diastereomeric set, along with the dr of the stereoisomers. Under circumstances where this relationship fails, one will require a dr assay in addition to two individual er assays to fully characterize the stereochemical parameters of a reaction. Thus, with these circumstances in mind, we give mathematical relationships for determining complete stereoisomer speciation having the knowledge of individual stereocenter er values and a dr value.