Eigenvalue equation local modes

For a molecule of N atoms with its structure at a local energy minimum, the normal modes can be calculated from a 3A x 3A1 mass-weighted second derivative matrix H, the Hessian matrix, defined in a molecular force field such as CHARMM [32-34] or AMBER [35-38]. For each mode, the eigenvalue X and the 3A x 1 eigenvector r satisfy the eigenvalue equation. Hr = Ar. 

The modal propagation constant /3(z) satisfies the local eigenvalue equation for each value of z. This equation has the plane-wave form of Eqs. (36-12) and (36-13) for high-order modes on step- and graded-profile fibers. Using the relationships between mode and ray parameters in Table 36-1, page 695, we find that the local eigenvalue equation has the form of the adiabatic invariant of Eq. (5-41). 

In general, the local-mode propagation constants are contained implicitly within the local eigenvalue equation, and their precise values must be obtained numerically. However, analytical expressions can be derived for the coupling coefficients, as we show in the example below. 

---