Extended method for adiabatic mode reordering.

The task of vibrational mode reordering is very important for reaction valley studies and for the determination of small curvature tunneling effects. An extended algorithm for adiabatic mode reordering is presented. It is based on the method introduced by Konkoli et al. [J Comput Chem 1997, 18, 1282], which is shown to suffer from numerical problems in the region of frequency-crossings and avoided crossings. One improvement is the use of cubic splines for the interpolation of the projected matrix of force constants, which allows larger step sizes between the discrete points along the reaction path, where vibrational analysis is performed. The main improvement is the use of perturbation theory to resolve crossings and avoided crossings. Within this theoretical framework it becomes clear why the method of the maximal overlap between the normal modes cannot work properly, as eigenvectors associated with nearby eigenvalues tend to become "wobbly". Thus a perturbative procedure is designed that is used for all cases where two harmonic frequencies approach each other and the overlap of the associated normal modes is of no practical use. Advantages of the new procedure are the use of larger step sizes along the minimum energy path and the much more reliable resolution of mode-crossings and avoided crossings independent of the systems symmetry. In addition to that it is shown that one should be very cautious in all computational situations when working with eigenvectors associated with nearby eigenvalues.