Dynamics of coarse-grained helical wormlike chains. II. Eigenvalue problems

Abstract

The eigenvalue problem for the matrix representation of the diffusion operator appearing in the diffusion equation previously derived for the coarse-grained helical wormlike (HW) chain model is considered. Following the procedure already established in the study of the original diffusion equation for the discrete HW chain, the problem is solved in the subspace and block-diagonal approximations in the subspace L(1), where L is the ‘‘total angular momentum quantum number’’ and (1) indicates the one-body excitation basis set. The 1(1) eigenvalues λ1,kj thus obtained form three branches of the eigenvalue spectrum specified by the index j as before, and it is shown that λ1,k0 in the j=0 (lowest) branch at small wave number k is approximately proportional to the corresponding Rouse–Zimm eigenvalue in the Hearst version. The behavior of the eigenvalue spectra is also numerically examined taking atactic polystyrene with the fraction of racemic diads fr=0.59 and atactic poly(methyl methacrylate) with fr=0.79 as typical examples of flexible polymers. It is found that the ratio λ1,k0/λ1,10 at small k depends somewhat (appreciably for large k) on chain stiffness and local chain conformation.