Determining the number of isomers in X-, XY-, XYZ-, and XYZU-substituted D2d allenes: Additive schemes for calculating enthalpies of vaporization

Abstract

Using Polya’s theorem, cycle indices are derived that identify the chiral and achiral substitution isomers of an allene (1,2-propadiene). Equations of symmetry that allow us to determine the number of isomers in a series and arrange the isomers according to families, depending on the number of substitution sites, are obtained. Eight- and nine-constant additive schemes based on the similarity between subgraphs in the molecular graphs (MGs) of a series of 120 molecules of X-, XY-, XYZ-, and XYZU-substituted allenes and the expansion of polygonal numbers (triangular, tetrahedral, and others) of a Pascal triangle are devised. Enthalpies of vaporization LNBP are calculated for 21 XY- and 120 XYZU-substituted allenes, respectively, not yet studied experimentally. It is shown that each coefficient of the scheme (the number of ways for superpositioning subgraphs with lengths i1, i2, ⋯ on molecular graphs of allene) is a result of partitioning triangular, tetrahedral, or four-dimensional tetrahedral numbers of a Pascal triangle.