Boris Zhilinskii

Qualitative analysis of vibration-rotation hamiltonians for spherical top molecules

A new approach to the qualitative analysis of the rotational structure of a group of degenerate or quasidegenerate vibrational states is proposed. The approach is based on a quantum description of resonant vibrational states and on using classical methods when describing the rotational structure of these states. The approach is used for studying the rotational structure of two- and three-fold degenerate vibrational states of the tetrahedral AB 4 molecule as well as for studying the v 1, v 3 diad of 28SiH4. The general scheme of the analysis is realized by computer.

Symmetry, invariants, topology. Basic tools

Abstract Elementary concepts of group actions: orbits and their stabilizers, orbit types and their strata are introduced and illustrated by simple examples. We give the unified description of these notions which are often used in the different domains of physics under different names. We also explain some basic facts about rings of invariant functions and their module structure. This leads to a geometrical study of the orbit space and of the level surfaces of invariant functions (e.g. energy levels of Hamiltonians). Combining these tools with Morse theory we study the extrema of invariant functions. Some physical applications (not studied in other chapters) are sketched.

Critical phenomena in rotational spectra

Abstract Nonlinear effects in rotational spectra of molecules and atomic nuclei caused by a centrifugal distortion for high values of an angular momentum quantum number J are investigated. The theoretical analysis is based on a new expansion form of the effective rotational Hamiltonian. It is shown that qualitative changes of rotational motion may occur in the rotational spectra for some value J c of the quantum number J . These phenomena which are in some sense analogous to the macroscopic phase transitions are called critical phenomena and correspond to bifurcations in classical mechanics. The classification of critical phenomena for a purely rotational motion is given. This classification is based on the concept of a local symmetry group G . There exist five types of critical phenomena in the rotational spectra. The local critical phenomena occuring in a bounded region of the rotational motion phase space are specially discussed. In the classical limit the local critical phenomena are characterized by a broken symmetry G and discontinuity of the second derivative of the rotational energy with respect to J at J c . A universal rotational Hamiltonian is shown to exist in the neighborhood of J c , which does not depend on the dynamical systems internal structure up to numerical values of its parameters. A phenomenological theory of the local critical phenomena is developed with the aid of universal Hamiltonians. The difference between the local critical phenomena and second-order phase transitions in macroscopic systems is shown.

Symmetry, invariants, and topology in molecular models

Abstract Description of intra-molecular dynamical behavior is usually made in terms of effective Hamiltonians for different degrees of freedom. In such a way, rotational, vibrational, rovibrational, etc., dynamical systems arise in a natural way in the classical limit as corresponding to effective quantum Hamiltonians. The main idea of this paper is to answer the following general question: What kind of features of the quantum energy spectra can be predicted on the basis of qualitative (symmetry+topology) analysis of corresponding classical systems.

Vibration-rotation problem for triatomic molecules with two large-amplitude coordinates: Spherical model☆

Abstract The vibration-rotation problem of a triatomic molecule composed of a rigid diatomic core and an atom which possesses almost free motion around this core is discussed. Such molecules as LiCN and KCN are appropriate examples. A model with two large-amplitude coordinates is used for calculations. Numerical results are presented using model potentials corresponding to the ab initio potential surface for the LiCN molecule.

Counting levels within vibrational polyads: Generating function approach

Simple analytical formulas for the number of energy levels in the vibrational polyads are given. These formulas account for the resonances between the vibrational modes, and for the symmetry of the problem, so that the number of states of a particular symmetry type can be computed. The formulas are used to estimate the differential and integral densities of states from the minimum initial information about the molecule. Examples of the vibrational structure of triatomic molecules A3, tetrahedral molecules AB4, and linear molecules AB2 are considered. The analytical formulas are compared to the ab initio results for H3+ [J. R. Henderson et al., J. Chem. Phys. 98, 7191 (1993)].

On the Dynamical Meaning of the Diabolic Points

Using a simple exactly soluble quantum model, it is shown that the diabolic points may be associated with the qualitative phenomenon of the redistribution of the energy levels between different branches in the energy spectra.

Monodromy of the quantum 1:1:2 resonant swing spring

We describe the qualitative features of the joint spectrum of the quantum 1:1:2 resonant swing spring. The monodromy of the classical analogue of this problem is studied in Dullin et al. [Physica D 190, 15–37 (2004)]. Using symmetry arguments and numerical calculations we compute its three-dimensional (3D) lattice of quantum states and show that it possesses a codimension 2 defect characterized by a nontrivial 3D-monodromy matrix. The form of the monodromy matrix is obtained from the lattice of quantum states and depends on the choice of an elementary cell of the lattice. We compute the quantum monodromy matrix, that is the inverse transpose of the classical monodromy matrix. Finally we show that the lattice of quantum states for the 1:1:2 quantum swing spring can be obtained—preserving the symmetries—from the regular 3D-cubic lattice by means of three “elementary monodromy cuts.”

Fractional monodromy of resonant classical and quantum oscillators

Abstract We introduce fractional monodromy for a class of integrable fibrations which naturally arise for classical nonlinear oscillator systems with resonance. We show that the same fractional monodromy characterizes the lattice of quantum states in the joint spectrum of the corresponding quantum systems. Results are presented on the example of a two-dimensional oscillator with resonance 1:(−1) and 1:(−2). To cite this article: N.N. Nekhoroshev et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 985–988.

Qualitative analysis of vibrational polyads: N-mode case

A general method for the qualitative analysis of vibrational polyads formed by N quasidegenerate modes is proposed. The construction of effective Hamiltonians for vibrational polyads and the general scheme of the qualitative analysis are outlined. The description of the limiting classic manifold is given and the analysis of the group action on it is performed. The three-mode problem is used as the simplest nontrivial example. The possible types of classical bifurcations and corresponding critical phenomena in the energy spectra of quantum systems appropriate for molecular problem with different symmetry groups are found. Brief discussion of different molecular problems tightly connected with the model considered is given along with some generalizations.