### Bifurcations of transition states and incorporation of electronic degrees of freedom

Fluids and Materials Seminar

21st March 2019, 2:00 pm – 3:00 pm

Main Maths Building, SM2

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

*Organiser*: Rachel Bennett

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