Even Moments of the End-to-End Distance of Polymeric Chains

A method is developed for calculating even moments 〈r2k〉 of the end-to-end distance r of polymeric chains, on the basis of the rotational-isomeric-state approximation for rotations about skeletal bonds. Expressions are obtained in a form which is applicable in principle to arbitrary k, but practical applications are limited by a tremendous increase in orders of matrices to be treated, with increasing k. The order of a matrix derived for 〈r2k〉 is Hk5s = (k + 4)!(4!k!)−1s, where s is the number of states in which each bond can exist. An application is made to the polyethylene chain by using the familiar, three-state (trans, gauche, and gauche prime) model. With a computer of relatively low ability, we have obtained even moments through 〈r12〉 for n through 400, where n is the number of C–C bonds. Approximate values of the distribution function Wn(r) of the end-to-end vector r, Wn(0), and 〈r− 1〉, are calculated from these even moments by a method previously developed, and mathematical implications of them are discussed.