Semiclassical Assignment of the Vibrational Spectrum of N₂O

Abstract

The vibrational spectrum of N 2 O as given by an effective spectroscopic Hamiltonian based on the existence of a superpolyad number is analyzed and assigned in terms of classical motions. The effective Hamiltonian includes a large number of resonances of which only one is dominant for low and intermediate superpolyad numbers. In this energy range, the corresponding classical system is quasi-integrable and can be described in terms of a system with only one nontrivial degree of freedom. This integrable system can be analyzed by considering the so-called "quantizing trajectories" on a "polyad sphere". This method is no longer applicable when the superpolyad number is further increased and classical chaos comes into play. We then turn to a powerful universal method based on the graphical representation of semiclassical wave functions on a naturally appearing toroidal configuration space. These wave functions are obtained using the already known transformation matrix used in fitting the effective Hamiltonian. Experience with the interpretation of the resulting figures allows one to draw conclusions on the classical internal motions and therefore on the assignment of the quantum states without any further calculation. As such, the method is of particular interest to nontheorists and to nonspecialists in the fields of nonlinear dynamics and quantum calculation. For higher superpolyad numbers, the chaos remains mainly concentrated about the direct neighborhood of a separatrix of the former integrable system so that a great part of the vibrational spectrum can still be assigned in terms of the EBK quantum numbers of quantized tori.

Spectroscopic interpretation: The high vibrations of CDBrClF

“…In recent years we have developed methods to investigate algebraic models ( spectroscopic Hamiltonians ) for the vibrations of molecules [1,2,3,4,5]. The Hamiltonians reproduce and encode by their construction the experimental data [6,7,8,9,10,11,12].…”

Analysis of an algebraic model for the chromophore vibrations of CF3CHFI

“…In recent years we have developed methods to investigate algebraic models (spectroscopic Hamiltonians) for the vibrations of molecules. [1][2][3][4][5] The Hamiltonians reproduce and encode by their construction the experimental data. [6][7][8][9][10][11][12] In the mentioned examples the system is reducible to two degrees of freedom.…”

Evaluation of an algebraic model for the vibrations of water, effects of a discrete symmetry

“…Therefore, also in the classically chaotic case we may find surprisingly simple and clean structures in a large part of the quantum wave functions. This was surprising for us in our previous examples [1][2][3][4][5]. In such cases, it can be appropriate to imagine simple idealized classical guiding centres and interpret the quantum states as quantum excitations of these idealized structures.…”

“…Next a few comments on chaos: as in our previous examples [1][2][3][4][5] we investigate a non-integrable system where in parts of the classical phase space large scale chaos dominates. On the other hand, we could find specific structures in all quantum states which allowed one to give a complete set of quantum numbers for any state.…”

“…During recent years we have developed methods to evaluate algebraic models ('spectroscopic Hamiltonians') for the vibrations of molecules and could give a complete assignment for various examples [1][2][3][4][5]. The central idea always was to study the skeleton of the classical dynamics and to interpret the quantum states as excitations of the motion corresponding to an appropriate classical guiding centre.…”