Quantum statistical mechanics of closed-ring molecular chains

Abstract

A model for a closed-ring unhindered three-dimensional macromolecular chain, based on Quantum Mechanics, is presented. Upon starting from an exact non-relativistic Hamiltonian operator, we integrate out all electronic degrees of freedom, in the Born-Oppenheimer framework, giving rise to an effective vibro-rotational Hamiltonian for the chain. Then, assuming a harmonic oscillator-like vibrational potential between nearest-neighbour atoms, the integration of the atomic radial degrees of freedom is carried in the limit of high frequencies. Thus, all bond lengths become fixed, including the one which makes the chain to become fixed, including the one which makes the chain to become a closed ring. This formulation leads to a specific Hamiltonian for the remaining angular variables of the closed-ring chain, and constitutes an alternative in comparison with standard Gaussian models, which do not. Use is made of a variational inequality by Peierls to find an approximate quantum partition function for the angular variables of the system. We then proceed to obtain approximately another representation for the angular partition function in the classical limit. Several features of the classical partition function are disscussed.