Efficient formulation of the large generator matrices required for computation of the higher moments, and mixed moments, of conformation‐dependent properties of chain molecules with independent bonds

Abstract Dimensionless ratios of various moments of conformation‐dependent physical properties play an important role in the evaluation of the behavior of chain molecules. For example, the correlation coefficient, ρ xy , between two conformation‐dependent physical properties, denoted here as x and y , is determined by the three dimensionless ratios 〈 x 2 〉/〈 x 〉 2 , 〈 y 2 〉/〈 y 〉 2 , and 〈 xy 〉/〈 x 〉〈 y 〉. Angle brackets denote the statistical mechanical average of the enclosed property. In the rotational isomeric state approximation, generator matrices of modest size can often be used for calculation of 〈 x 〉 and 〈 y 〉. The dimensions of the matrices grow rapidly upon going to higher moments or to mixed moments, such as 〈 xy 〉. Formulation of these large matrices, while straight‐forward, has been extremely tedious for chains with independent bonds, arbitrary rotational potentials, and arbitrary bond angles (which need not be fixed). Here we describe an approach that quickly and accurately solves this problem for all cases in which the generator matrices required for 〈 x 〉 and 〈 y 〉 are known. The algorithm is validitated by successful use for the computation of ρ r 2 s 2 for r 2 and s 2 for freely jointed chains of various n , a case for which an exact analytical result in closed form is available in the literature. Here r 2 and s 2 denote the squared end‐to‐end distance and squared radius of gyration for a specified conformation, and n denotes the number of bonds.