Open three-dimensional unhindered molecular chains: Partition function

An 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.