Open three‐dimensional unhindered molecular chains: Partition function

Abstract

AbstractAn open molecular chain, formed by N classical particles moving in three dimensions and having N − 1 bonds of constant lengths between successive neighbours, is considered. Hamiltonian methods with manifest rotational invariance allow to characterize all constraints. The classical partition function, Z0, for a large open chain in thermal equilibrium is analyzed in general. Simpler integral representations for Z0 are obtained, with the following advantages: (i) explicit rotational invariance in the integrands, (ii) only tridiagonal matrices appear and, moreover, approximations for their determinants can be obtained. For a two‐dimensional open chain, an approximate factorized formula for Z0 is presented, and its essentials are generalized to the three‐dimensional case. In both cases, the features of the partition functions bear certain similarities to that for a classical ideal gas. The approximate partition functions lead to approximate analytical computations of the correlations between pairs of different bond vectors, of the squared end‐to‐end distance, of the probability distribution for the end‐to‐end vector and of the structure factor, which display some novel features. Some comparisons with the corresponding results for the Gaussian chain are made.