Conformational Analysis of Macromolecules. II. The Rotational Isomeric States of the Normal Hydrocarbons

Abstract

The conformational potential energy of the normal hydrocarbon molecules is considered to be a function of the dihedral angles of internal rotation about the carbon—carbon single bonds, the bond lengths and valence angles being held fixed. The energy equation contains terms of two kinds, a sum over torsional or internal‐rotation terms, one for each carbon—carbon bond, which are attributed to exchange interactions of electrons in bonds adjacent to the bond about which internal rotation occurs, and a sum over all the pairwise nonbonded interactions between all of the atoms in the molecule. The torsional terms are dependent on the dihedral angles of internal rotation and on the choice of fixed valence angles, but each torsional term reduces to the form ½U 0(1+cos3ω i ) when all the fixed valence angles are selected to be tetrahedral. The nonbonded‐interaction terms are represented by Lennard‐Jones ``6–12'' potential functions, Uij =(dij/rij 12) − (eij/rij 6), the parameters eij being calculated from the Slater—Kirkwood equation. The energy equation contains three adjustable parameters: U 0, the barrier to internal rotation about a carbon—carbon single bond when the fixed valence angles are all tetrahedral, d HH, and d CC, d CH being taken as the geometric mean of d HH and d CC. These three parameters were selected to obtain agreement between the calculated and experimental values for the barriers to rotating a methyl group in ethane and propane, the difference in energy between the trans and the two gauche forms of butane, and the location of the gauche minima in butane. These parameters were then used, along with the known bond lengths and valence angles, to calculate the locations and energies of the various rotational isomeric states (potential‐energy minima) for pentane, hexane, and heptane. A method of energy minimization was used which allowed all the dihedral angles of internal rotation to vary independently. These calculations demonstrate that the traditional procedure of assuming that each bond in a hydrocarbon molecule or a polyethylene molecule can exist in three rotational isomeric states, therefore T (180°), G + (60°), and G − (300°), is an oversimplification and that more minima occur than is predicted by the traditional procedure. The implications of these results for formulating future statistical‐mechanical theories of the normal hydrocarbons and of polyethylene are discussed.