Chebyshev modes

Equation (1.16) is a Helmholtz equation with the unknown sum j + c This is first transformed in the spectral domain where, due to the separability of the Helmholtz equation, the equations for each pair of Fourier modes (r, k) are fully decoupled with the only coupling appearing among the Chebyshev modes. Thus, by solving the Helmholtz equation with a fast solver [86], the updated solution for the conformation tensor is obtained, from which, with the aid of Eq. (1.6), the extra stress tensor n+i i calculated. 

Table XVII shows similar comparisons of the deviations from the first rule of the mean for an 0-disproportionation between carbon dioxide species. Here, the normal modes have been divided into 2 linear modes and 2 non-linear modes. The approximations for the non-linear modes are much better than those for the linear modes, simply because v max for the former is only a quarter of v max for the latter. Calculations (not shown in the Table) made with the Chebyshev (L = 5) yielded In K s of 2.550 X 10 and 2.086 X 10 for n = 2 and 3, respectively, compared with the exact value of 3.546 X 10 at 300°K. Rule of the Mean tests are especially useful for examining the suitability of a polynomial to a particular molecule, because they permit an examination of the correspondence between individual terms of the expansion and motions of groups of atoms in the molecule.

Figure 7 Time evolution of the initial 6v,) state, computed using the Chebyshev propagation scheme (a) survival probability (lower curve) and overall chromophore population (upper curve, defined in Eq. (123). (b), (c), and (d) Vibrational mode populations, defined in Eq. (124).

The above linear system of differential equations can be solved by various propagation methods [2], as the time split [28, 29], Chebyshev [30], or Lanczos [31, 32] method, provided an efficient and accurate method for computing the matrix elements Hij is given (see, e.g., Appendix B of ref. 3). The basic limitation of the method stays in its bad scaling properties, since the number of equations goes as rf, where n is the dimension of the single-mode basis set (supposed here to be identical for all the modes). 

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