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

A method is developed for calculating even moments ${\mathrm{〈r}}^{\text{2k}}〉$ 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 ${\mathrm{〈r}}^{\text{2k}}〉$ is ${}_5{H}_k{\text{s\hspace{0.17em}=\hspace{0.17em}(k\hspace{0.17em}+\hspace{0.17em}4)!(4!k!)}}^{\text{−1}}s$ , 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 ${\mathrm{〈r}}^{12}〉$ for $n$ through 400, where $n$ is the number of C–C bonds. Approximate values of the distribution function $W_n(\mathrm{r})$ of the end‐to‐end vector r, $W_n\text{(0)}$ , and ${\mathrm{〈r}}^{\text{−\hspace{0.17em}1}}〉$ , are calculated from these even moments by a method previously developed, and mathematical implications of them are discussed.