The Born-Oppenheimer approximation (BOA) is one of the most widely used methods used to study the quantum dynamics of molecules. Intuitively, it is motivated by the fact that the electrons are much lighter, and therefore much faster, than the nuclei, and hence rapidly adjust their positions with respect to those of the nuclei, allowing the dynamics to be decoupled. This simplifies numerical studies of such problems by requiring ‘only’ the solution of a one-level PDE.
However, there are interesting situations in which the BOA breaks down; in many photochemical processes, two electronic energy surfaces become close, or even cross. In such cases, parts of the nuclear wavepacket are transmitted between different electronic energy levels. In the former case, known as an avoided crossing, the BOA is still valid to leading order (in a small parameter, the square root of the ratio of the electronic and nuclear masses), but the remaining corrections are of fundamental interest and, in fact, determine the associated chemistry. In the latter case the BOA breaks down completely. Such systems, at least naively, require the solution of multi-level PDEs.
The paradigmatic associated mathematical problem is the description of transitions in a two-level partial differential equation, with one effective spatial degree of freedom – the internuclear separation. Given a wavepacket that travels on the upper electronic level, the aim is to determine the size and shape of the part of the wavepacket transmitted to the lower level at large times. The challenge lies in the multi-scale, highly oscillatory nature of the problem, as well as the exponentially small transmitted wavepacket, which prohibits the use of standard numerical methods.
In this talk, we will briefly review some state-of-the-art approaches from mathematics and chemistry, before describing a novel method which, crucially, requires only one-level evolution of wavepackets and retains the full phase information, allowing the study of interference effects arising from multiple transitions. We will demonstrate its effectiveness for both model systems and a real-world example, and explain how standard surface-hopping methodologies and the famous Landau-Zener formula can be recovered after appropriate approximations. Finally, if time allows, we will outline the extension of the method to higher dimensions.
Joint work with Volker Betz (TU Darmstadt), Tim Hurst (Edinburgh), Uwe Manthe (Bielefeld) and Stefan Teufel (Tuebingen)