Prof Robert MacKay

Bifurcations of transition states and incorporation of electronic degrees of freedom

In each energy level for an interval above any index-1 saddle of an n-DoF Hamiltonian system
is a normally hyperbolic (2n-3)-sphere. It is a transition state. But as the energy is increased
further it may lose normal hyperbolicity and the question arose what happens to it. We show
that for a substantial class of systems it just develops a singularity at a critical energy and then
re-emerges as a normally hyperbolic submanifold of a different topological type.

Secondly, the interacting dynamics of classical nuclei and quantum electrons can be
formulated as a Hamiltonian system using Fubini-Study symplectic form on the projectivised
Hilbert space or a Lie-Poisson structure on the space of Hermitian operators. For practical
purposes, however, the dimension of the space of electronic degrees of freedom is too large
and a way to restrict attention to relevant electronic degrees of freedom is required. I propose
one based on analysis of persistence of spectral projections for operators on large tensor
products.

References:
[1] RS MacKay, DC Strub, Bifurcations of transition states: Morse bifurcations, Nonlinearity 27
(2014) 859–95

[2] RS MacKay, DC Strub, Morse bifurcations of transition states in bimolecular reactions,
Nonlinearity 28 (2015) 4303–29