A quantum mechanical Hamiltonian for unhindered macromolecular chains: consistency with Gaussian models

A model for an open unhindered three-dimensional macromolecular chain, based upon quantum mechanics and proposed in previous works, is studied in order to investigate its physical properties and consistency. The chain is formed by N particles interacting through harmonic-like vibrational potentials in the high-frequency limit (in which all successive bond lengths become fixed). This formulation leads to a specific Hamiltonian for the chain: it constitutes an improvement in comparison with standard Gaussian models, which do not. The classical partition function Z_c resulting from the quantum formulation is represented through an integral, which exhibits explicitly rotational invariance in the integrand and provides the basis for further approximations for large N. Approximate formulae are obtained for correlations between pairs of bond vectors, the distribution function for the end-to-end vector, distribution functions for individual bond vectors, “rubber eleasticity” (when stretching forces act) and the structure factor for small wave vector. In all cases, the results which have arisen from the quantum mechanical formulation coincide with those obtained for the standard Gaussian chain. This agreement appears to confirm the physical consistency of the quantum Hamiltonian characterizing the model.