(untitled) (Contemporary Mathematics, 2022)
The present work has a twofold purpose. a) It proposes a quantum-mechanical approach to constrained molecular chains and their small vibrations and rotations, by employing in a compact way vector variables and operators associated to the constituent units of the chain. The methods here differ from standard approaches based upon cartesian coordinates and normal modes and generalize previous quantum Hamiltonians describing only rotational degrees of freedom. Several models in D = 2, 3 spatial dimensions, with new Hermitean Hamiltonians, are formulated and analyzed. The chains studied successively display an increasing number of constraints: freely-jointed, freely-rotating and with constrained torsions. Conservation of total orbital angular momentum is analyzed. As a partial test, by using the present approach, the vibrational frequencies of certain triatomic molecules (water vapour, hydrogen sulfide, heavy water and sulfur dioxide) are computed and shown to be consistent with experimental data. b) A new (quantum-mechanical) analysis of polymerization, namely, the growth of a freely-jointed molecular chain (of the kind considered above) by binding an additional unit 1 to the chain, is presented. They move in a very dilute solution in a fluid at rest in thermal equilibrium about room temperature. The analysis is based upon a mixed (quantum-classical) distribution function in phase-space: a quantum Wigner-like one for unit 1 and a classical Liouville one for the chain. That leads to an approximate Schmolukowski equation for unit 1 alone and, through it, to compute the mean first passage time (MFPT) for unit 1 to become bound by the chain. The resulting MFPT displays a temperature dependence consistent with the Arrhenius formula for rate constants in chemical reactions.