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Molecules consist of atomic nuclei bound together by electrons. The energy that binds the nuclei together varies as a function of their relative position, and looks like a landscape, which is called a potential energy surface. A chemical reaction can usually be described by visualising atomic nuclei as tiny ball bearings rolling over the potential energy landscape. The path that the nuclei follow is called a reaction path: it connects together stable starting and end points on the landscape, which correspond to the reagent and product molecules. Sometimes, however, this simple picture breaks down, because there can exist two potential energy surfaces, one higher in energy than the other, which come together at a particular geometry of the nuclei, to form a funnel-shaped object called a 'conical intersection'. Typically, conical intersections occur in light-activated reactions, such as are found in photosynthesis and vision. By absorbing the light-energy, the molecules jump onto the upper surface; when they subsequently react, they may relax rapidly to the lower surface by following a reaction path which passes down the conical intersection funnel. This process is equivalent to exchanging energy between the electrons and the atomic nuclei, and is highly quantum mechanical. As a result, the motion of the nuclei can no longer be visualised as ball bearings rolling on a potential surface. Instead it must be visualised as a wave function, which extends over both the potential surfaces, and describes a complicated coupled motion in the vicinity of the conical intersection funnel. The aim of the proposed research is to understand more about what goes on inside such a wave function. In recent work, the applicant found that, when the wave function is confined to the lower of the two potential surfaces (and thus encircles the base of the funnel), it satisfies a topological theorem which allows it to be split into two pieces: one piece contains all the reaction paths that loop in a clockwise direction around the funnel, the other piece contains the anticlockwise paths. The recent application of this theorem solved a longstanding puzzle in experimental data measured for the hydrogen-exchange reaction, and this led to a general theory explaining the effect of the 'Berry phase' (an effect produced by looping around conical intersections) on chemical reaction mechanisms and rates [Science 309, 1227 (2005)]. We propose extending this theory to treat reactions which are not confined to the base of the funnel, but which can pass through it, from surface-to-surface. The new theory will decompose the wave functions of such reactions into all topologically distinct reaction paths; each path will be a combination of loops (around the funnel) and hops (between the two surfaces). The theory will show how the interaction between these types of reaction path facilitates or impedes passage through the funnel, and this will result in a general theory explaining the effect of topology on reaction mechanisms and rates at conical intersections. To develop the theory, we will incorporate it into accurate quantum simulations on the hydrogen-exchange reaction, which will permit direct comparison with new experimental data to be measured by the Zare group (Stanford). This will permit topological conjectures to be refined or eliminated, thus guiding us towards the final form of the theory. After developing the theory, we will use it to improve the efficiency and interpretational power of a variety of approximate methods which can be used to simulate complex light-harvesting reactions that take place in solution.
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