Alkane isomer combinatorics: Stereostructure enumeration and graph-invariant and molecular-property distributions
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Combinatoric manipulative methodology to characterize different isomer-class features is extended and illustrated in application to alkane isomers. Stereoisomeric enumerations and various molecular - property characteristics for structural-isomer classes are made. For a range of carbon-atom counts N we enumerate the following: the mean number of diastereomers per N-carbon structural isomer; the mean number of stereoisomers per N-carbon diastereomer; and the mean number of "conformers" per N-carbon stereoisomer. Also we consider distributions of several graph-theoretic properties, characterizing the distributions in terms of averages, variances, covariances, and extremal values. The results for these graph-theoretic invariants are used to predict (via substructural cluster expansion) some molecular properties (heats of formation and magnetic susceptibilities) averaged over these classes. The associated proper standard deviations as well as the extremal values for these cluster-expansion approximants are also obtained, and correspondent extremal structures are identified. Thence combinatoric techniques to deal neatly with astronomical numbers (even if greater than a mole) of different molecular structures and some associated physical and chemical properties are developed.
