The Rouse Normal Modes of Motion

Besides the diffusion motion, the dynamics of a polymer chain are also characterized by its internal motions, which can be expressed in terms of the normal modes. 

Here the normal procedure is to do the transformation from the bond vectors b. to the normal coordinates qt which will diagonalize the Rouse matrix (Agt)- As shown in Appendix 3.A, the Rouse matrix has the eigenvalues  

Following the procedure shown in Appendix 3.B for solving the Langevin equation of a particle in a simple harmonic potential, we obtain from Eq. (3.50) in combination with Eqs. (3.52) and (3.53) 

The normal coordinate qj, with p being a positive integer represents the pth internal or intramolecular mode of motion of the polymer chain. Equation (3.54) indicates that ifp q, the pth mode of motion is not correlated with the gth mode. Equation (3.55) can be expressed as 

The above results of the normal modes qp(f) allow the numerous internal motions of the polymer chain to be classified into independent modes. Consider for example the end-to-end vector 

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