The rotational partition function for linear molecules

Abstract

An important function in the statistical treatment of a gas of linear molecules is $S(\alpha ) = \sum\limits_{J = 0}^\infty {(2J + 1)e^{ - \alpha J(J + 1)} }$ This sum is convenient to use mainly when α is large and alternate expressions, generally asymptotic expansions, are often used when α is small. In this paper, the sum is evaluated to yield a single expression that is valid for large and small values of α. The expression is composed of three terms, each of which involves the theta functions of Jacobi. One term is in the form of an integral, but is small relative to the other two and easily evaluated by numerical means. The expression is readily differentiated and can be used for the general evaluation of the rotational partition function for gases of linear molecules at all temperatures.