Extended method for adiabatic mode reordering

Abstract

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.

Analysis and dynamics of unusual double proton transfer reactions based on the reaction path Hamiltonian

“…K denotes the Cartesian Hessian matrix and P the projector. 24 A correct and physically meaningful ordering of the vibrations along the MEP was performed by applying the approach of Konkoli et al 45 Resolution of all forbidden crossings could be achieved by spline interpolation 46 applied to the eigenvectors L that depend on s. The dynamic coupling terms B m (s), which are defined as…”

“…This indicates that the plateau is not an artifact of the computational method. 45,46 In order to resolve all forbidden crossings we used the approach of Konkoli et al in combination with a cubic spline interpolation scheme for the projected eigenvectors L m which was found to be very reliable with respect to the chosen step size within the MEP tracing. For comparison, frequencies were also calculated for the same reaction after deuteration of the two nitrogen atoms involved in the DPTR.…”

“…Fig. 3 depicts the frequencies of the orthogonal vibrational modes along the MEP for reaction P. As pointed out by Konkoli et al and Tautermann et al, the adiabatic re-ordering of the vibrational modes along the reaction path might become a difficult task for reactions in C 1 symmetry 45,46. In order to resolve all forbidden crossings we used the approach of Konkoli et al in combination with a cubic spline interpolation scheme for the projected eigenvectors L m which was found to…”