Transition from order to chaos in molecular wave functions and spectra
F. J. Arranz, F. Borondo, R. M. Benito
1996
Journal of Chemical Physics
In this Communication we describe how the transition from regularity to classical chaos in molecular Hamiltonian systems shows up at the quantum level in the structure of the corresponding wave functions and spectra. By changing the value of ប we show how the scars result from combinations of regular wave functions. In the early stages of vibrational spectroscopy typical studies were only concerned with low lying vibrational states in which the nuclei move in a localized region around the
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... m of the Born-Oppenheimer potential energy surface ͑PES͒. Anharmonic terms, responsible for the overtone and combination frequencies, were considered only as weak perturbations. 1 In this regime, the intramolecular dynamics are completely regular and the spectra consist ͑at least in the ideal case͒ of a progression of bands, corresponding to the different excitations of each normal mode, that can easily be assigned, due to the lack of irregularities. The corresponding wave functions exhibit a very regular nodal pattern 2 and quantum numbers can also, in principle, be assigned easily. Special care has to be exerted in the presence of classical resonances since they have a profound influence on the nodal structure of wavefunctions, as was demonstrated in the work of DeLeon, Davis, and Heller. 3 At higher levels of excitation the dynamics of molecular systems change very much, and the interactions between normal modes 4 cause the structure of the spectra to be more complicated. The KAM theorem dictates that more and more regular tori are destroyed as energy increases, rendering a multitude of resonant chains of islands, overlapping bands of stochasticity and embedded cantori. 5 Nonlinear interactions among normal modes lead to irreversible intramolecular vibrational energy flow, which is controlled by all those classical structures. Also, the rate of many intramolecular processes, such as isomerization, unimolecular decomposition, etc., is determined by this intramolecular vibrational relaxation ͑IVR͒. 6 Modern spectroscopy has broadened this horizon by the introduction of new techniques, such as IR overtone excitation, multiphonon excitation, stimulated emission pumping or electron photodetachment, 7 in which extensive regions of the PES, sometimes very far from the equilibrium geometries, are probed. On the theoretical side, efficient methods have been developed to calculate accurately the eigenvalues and eigenfunctions of the vibrationally excited states involved. A special mention is due to those based on the discrete variable representation ͑DVR͒ method. 8 In this energy regime, the interpretation of corresponding wave functions in simple terms is much more difficult. For moderate excitation energies choosing an adequate ͑curvilinear͒ coordinate system 9 can help. On the other hand, for very high vibrational energies some wave functions appear localized on periodic orbits ͑POs͒ of the system. This effect, known as "scarring", has received much attention in the literature 10 although its origin is not yet fully understood. The POs of chaotic systems also determine the corresponding spectra, both at high 11 and low resolution. 12 The chaotic dynamics are very complex and only fully resolved when long times are considered; 13 conversely if only low resolution is required merely the details of short trajectories or pieces of trajectories are needed. 12, 14 Time dependent methods have also witnessed a tremendous development in recent years. 15 Wave packet propagation has also contributed greatly to our understanding of the relationship between spectra and the underlying dynamics. 16 In this Communication, we discuss how the transition from order to classical chaos is reflected in the structure of the wave functions as energy increases. To illustrate this effect we use a 2D model for the molecular vibrations of the LiNC/LiCN system 17 that has been extensively studied in the literature. In it, the C-N distance is held constant at its equilibrium value, r e , and the two remaining vibrational coordinates, R and , are defined as the distance from the Li atom to the CN center of mass and the angle between these two vectors, respectively. Motion in the bending angle is very floppy, sampling extensive regions of the PES. This is the origin of chaos in this system. The classical dynamics of this model can best be monitored by constructing Poincaré surfaces of sections ͑PSOS͒ using the minimum energy path ͑MEP͒, which connects the two linear isomers: LiNC ͑ ϭ180°͒ and LiCN ͑ϭ0͒, as sectioning plane. 18 In Fig. 1 a composite PSOS computed at an energy intermediate between those of the 24th and 25th eigenvalues is shown. At this energy the LiCN well is barely accessible ͑the first state located in this well is state number 31͒, and two very distinct regions on the LiNC can be distinguished: the inner one ͑around the elliptic fixed point marked with a star and corresponding to the stable isomer LiNC͒ which is regular, and the outer one that corresponds to chaotic motion. In this chaotic region we have also marked ͑with squares and triangles͒ two POs, stable and unstable, respectively, that 6401
doi:10.1063/1.471301
fatcat:4anquan77bbojjaszrx4fgz4cy