On the Use of Classical Statistical Mechanics in the Treatment of Polymer Chain Conformation

Two different treatments of the degrees of freedom of bond stretching and bond angle bending in chain polymers by classical statistical mechanics lead to different and nonequivalent expressions of the partition functions. If we fix the bond lengths and bond angles at the outset and treat them as constraints (the classical rigid model), the partition function is given by an integral of (det G)-'12 exp(-@F(Q)) over the space of the dihedral angles Q in a poly- mer chain, where the elements of the matrix G are the coefficients in the quadratic expression (in terms of general- ized momenta conjugate to Q) for the kinetic energy of the polymer chain, and F(Q) is the conformational energy of the polymer chain. If we conceptually allow bond lengths and bond angles to vary under an infinitely strong potential (the classical flexible model) and perform the integration of the Boltzmann factor over the momenta conjugate to the Cartesian coordinates, we obtain the partition function in the form of an integral of exp(-@F(Q)) over the space of the dihedral angles Q. The origin of the difference in these two expressions lies in the different treatments of the vi- brational motions involving bond lengths and bond angles. In order to decide which of the two expressions is to be used as the basis of a statistical mechanical study of the polymer chain in equilibrium, an expression for the partition function that is quantum mechanically correct for these vibrational motions is derived, and the approximations in- volved to obtain each of the two non-equivalent classical expressions from the quantum mechanical expression are examined. The classical rigid model can be derived from the quantum mechanical one (a) by applying the ground state approximation for all vibrations associated with bond stretching and bond angle bending (i.e., by neglecting contributions to the partition function from excited vibrational states), and (b) by neglecting the conformational de- pendence of the zero-point energy of these vibrations. The classical flexible model can be derived by treating all these vibrations classically, which would appear to be unwarranted because many of these vibrations are of sufficiently high frequency to require a quantum mechanical treatment. However, a quantitative analysis of the approximations involved in each of the two models reveals that, of the two nonequivalent classical treatments, the classical flexible model is better than the classical rigid model.