Constrained macromolecular chains at thermal equilibrium: A quantum-mechanical approach
1 Departamento de Física Teórica I, Facultad de Ciencias Físicas, Universidad Complutense, 28040 Madrid, Spain
2 Departamento de Matemáticas, ETS de Ingenieros de Caminos, Canales y Puertos and IMACI-Instituto de Matemática Aplicada a la Ciencia y la Ingeniería, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain
Revised: 10 October 2011
Published online: 13 December 2011
Many approaches to three-dimensional constrained macromolecular chains at thermal equilibrium, at about room temperatures, are based upon constrained Classical Hamiltonian Dynamics (cCHDa). Quantum-mechanical approaches (QMa) have also been treated by different researchers for decades. QMa address a fundamental issue (constraints versus the uncertainty principle) and are versatile: they also yield classical descriptions (which may not coincide with those from cCHDa, although they may agree for certain relevant quantities). Open issues include whether QMa have enough practical consequences which differ from and/or improve those from cCHDa. We shall treat cCHDa briefly and deal with QMa, by outlining old approaches and focusing on recent ones. In QMa, we start with Hamiltonians for N(≫ 1) non-relativistic quantum particles, interacting among themselves through potentials which include strong vibrational ones (constraining bond lengths and bond angles) and other weaker interactions. We get (by means of variational calculations) effective three-dimensional constrained quantum partition functions at equilibrium (ZQ) and Hamiltonians (HQ) for single-stranded (ss) macromolecules (freely-jointed, freely-rotating, open or closed) and for double-stranded (ds) open macromolecules. Due to crucial cancellations, we can neatly separate the constrained degrees of freedom (by getting the large constant vibrational zero-point energies associated to them) from the slow unconstrained angular variables (accounted for by ZQ and HQ). In the classical limit, we obtain classical partition functions Zc from ZQ. The Zc’s are respectively different from the classical partition functions found starting from cCHDa for similar chains. Thus, they differ in determinants of the sort referred to in a companion tutorial article : QMa determinants are simpler than cCHDa ones. For ss macromolecules, we compare several quantities (bond-bond correlations, squared end-to-end distances, etc) from QMa with the standard Gaussian model in Polymer Science: the comparisons display good consistencies (which are also met with cCHDa). For double-stranded DNA (dsDNA) macromolecules, the Zc’s from QMa have structures which enable one to study thermal denaturation.
© EDP Sciences, Springer-Verlag, 2011