The geometry of reaction dynamics

Abstract

The geometrical structures which regulate transformations in dynamicalsystems with three or more degrees of freedom (DOFs) form the subject of thispaper. Our treatment focuses on the (2n-3)-dimensional normallyhyperbolic invariant manifold (NHIM) in nDOF systemsassociated with a centre×···×centre×saddle in the phase space flow in the(2n-1)-dimensional energy surface. The NHIM bounds a (2n-2)-dimensionalsurface, called a `transition state' (TS) in chemical reaction dynamics,which partitions the energy surface into volumes characterized as`before' and `after' the transformation. This surface is thelong-sought momentum-dependent TS beyond two DOFs. The (2n-2)-dimensional stable and unstable manifolds associatedwith the (2n-3)-dimensional NHIM are impenetrable barriers with thetopology of multidimensional spherical cylinders. The phase flowinterior to these spherical cylinders passes through the TS as the system undergoes its transformation. The phase flowexterior to these spherical cylinders is directed away from theTS and, consequently, will never undergo the transition.The explicit forms of these phase space barriers can be evaluated usingnormal form theory. Our treatment has the advantage of supplying apractical algorithm, and we demonstrate its use on a strongly couplednonlinear Hamiltonian, the hydrogen atom in crossed electric andmagnetic fields.