Section 3

ATOMIC AND MOLECULAR SPECTRA

Topics

Project objectives

• The ``good quantum numbers'' used to describe bands, branches, multiplets, polyads, etc in quantum energy spectra will be related to the geometry and approximate dynamical symmetries of the corresponding classical problem. Bifurcations and re-arrangments of these structures that occur as excitation increases or external parameters are varied will be classified. A `quantum-classical dictionary' for these phenomena will be compiled.
• Global bifurcation diagrams for REs and RPOs will be computed for (a) small, mainly triatomic, molecules and (b) Rydberg atoms and molecules. Normal forms for the dynamics near REs and RPOs will be constructed, analysed and the quantum-classical correspondence principle and results of objective 3.1 applied to describe the energies and symmetries of associated localized quantum states.
• Existing semiclassical methods based on invariant tori and periodic orbits, and used to approximate quantum spectra, will be extended to EBK-type quantisation schemes for singular torus-structures (pinched tori, heteroclinic cycles, separatrices) and for near-integrable dynamics including Arnold diffusion. The methods will be applied to atoms and small molecules.

Scientific Originality

3.1      Qualitative analysis of classial limits of quantum structures began with the late 1970's work on rotational energy surfaces and has since been extended to vibrational polyads and electronic states. These techniques will be developed further in this project, using recent results on singular reduction, and extended to mixed quantum-classical rotational energy surfaces and models which combine rotational energy surfaces with vibrational polyads. The links between re-arrangements of these structures and quantum monodromy will also be explored. Quantum monodromy has been shown to be present in physical systems, such as water molecules and hydrogen atoms in electric and magnetic fields, and current indications are that it will become as widely accepted as a universal phenomenon as the quantum Hall effect.

3.2     The realisation that classical REs are important organizing centres for molecular spectra has been growing over the past few years culminating in the computation of global RE bifurcation diagrams for a number of small molecules and quantum predictions based on these. Periodic orbits have similarly been used as organizing centres for highly excited spectra with angular momentum J = 0 and global bifurcation diagrams computed. However there has been virtually no analogous work on RPOs for J > 0, one of the aims of this project.

3.3      Semiclassical methods for obtaining quantum spectra and wavefunctions from classical inputs have attracted renewed interest in recent years. They are especially useful for unveiling `hidden' structures in quantum spectra due to stable periodic orbits, approximate tori and short unstable periodic orbits. Quantitative results can be obtained by direct semiclassical quantisation or scaled Fourier transformation of quantum spectra. However relatively little work has been done on classical systems with more than 2 degrees of freedom. Semiclassical studies of the integrable and near-integrable dynamics (including Arnold diffusion) of atomic and molecular systems such as helium, hydrogen in electric and/or magnetic fields and water will be undertaken in this part of the project.

Research Method

3.1      For the qualitative analysis of quantum atomic and molecular problems the quantum-classical correspondence principle will be used to construct classical symbols and the topological and exact or approximate symmetry properties of these analysed using singular reduction and invariant theories. Evolution of the classical dynamical system under the variation of control parameters (such as external fields, exact or approximate integrals of motion, particle numbers, masses and charges) is of particular importance and will be studied using singularity theory.

3.2      Relative equilibria will be computed as equilibrium points of reduced Hamiltonians. The computation of global bifurcation diagrams for small atoms and molecules from the reduced Hamiltonians is a straightforward exercise and can be achieved within a package such as MATHEMATICA or MAPLE. The computation of RPO bifurcation diagrams can similarly be reduced to the computation of periodic orbits of reduced Hamiltonians. This will be treated as a boundary value problem and numerical geometric path-following techniques applied. The Floquet exponents will also be computed. Normal forms for the classical dynamics near RE/RPOs will be obtained using the methods developed under objective 1.1 and analysed from classical and quantum viewpoints using the methods of 3.1.

3.3     Semiclassical quantisation schemes for near-integrable dynamics with more than two degrees of freedom will be developed using approximate EBK-quantisation and periodic orbit summation. The effects of Arnold diffusion on quantum propagation will be studied with the help of numerical integrator methods developed in 2.2. Short periodic orbits which play an important role in semiclassical quantisation can be identified either by analysing quantum spectra with the help of Fourier transformation or by a systematic phase space search by means of suitable numerical techniques used in 2.2. The influence of classical topological properties like monodromy will be studied in a semiclassical context for integrable and near-integrable dynamics. Classical, semiclassical and quantum calculations will focus on atomic systems such as helium, hydrogen in external fields, and small molecules.