Transition from order to chaos in molecular wave functions and spectra

Abstract

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. © 1996 American Institute of Physics. ͓S0021-9606͑96͒03316-6͔In 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 minimum 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 t...

Avoided crossings, scars, and transition to chaos

“…In the present paper, which is an extension of a previous Communication to this journal, 23 the correlation diagram of the LiNC/LiCN system as a function of ប, taken as a parameter, has been calculated and studied in detail. Two series of ACs of very different nature have been described, and the corresponding interaction matrix elements between the involved adiabatic states ͗ m ‫͉‪ប‬ץ/ץ͉‬ n ͘ analytically calculated using the Hellman-Feynman theorem.…”

“…22 In a previous Communication to this journal, 23 hereafter called paper I, we presented a preliminary study of the correlation diagram for the vibrational states of the LiNC/LiCN isomerizing system as a function of ប ͑or equivalently the atomic masses͒ taken as a parameter. This molecule has been extensively studied in the past [23][24][25][26][27] in connection with the quantum manifestations of classical chaos. The Li-NC/ Li-CN bending motion is very floppy and chaos sets in at relatively low energies.…”

Saddle-node bifurcations in the LiNC/LiCN molecular system: Classical aspects and quantum manifestations

“…As was discussed in Ref. 16, and due to the well known bobsled effect, this is a region where important interaction or mixing between the R and modes takes place.…”

“…[14][15][16] In the first two references it was shown how to project eigenstates on the region around a given PO of the system, making the argument of Bogomolny more quantitative. Also it has been demonstrated, 16 by considering variations with ប, taken as a parameter, how the scars emerge as the result of the interaction between regular wave functions. 17 This localization is also, and often more clearly, visible in quantum phase space.…”

Bifurcations of the normal modes of the Ne⋯Br2 complex

“…However, not much attention has been paid to the vibrational dynamics of the rare gas around the dihalogen dimer. With this kind of study, widely applied in the LiCN and HCP molecules [12], it is possible to determine the structure of the phase space. As it is well known, periodic orbits are the backbone of the phase space.…”