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. 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.