Mathematical Methods in Quantum Molecular Dynamics

The workshop brought together chemists developing mathematical and computational tools for studying the motion of atoms in polyatomic molecules and mathematicians interested in numerical methods for highdimensional problems and semiclassical mechanics. Quantum and semiclassical methods applicable to diatomic molecules are well known and widely used, but the outstanding problem in this field is devising new mathematical and computation tools for studying larger molecules. This is difficult due to the dimensionality of the problem. In principle, molecular dynamics can be understood by solving the time-dependent Schrödinger equation. However, because 3N coordinates are required to specify the configuration of the nuclei in a molecular or reacting system with N atoms, quantum molecular dynamics calculations must deal with very high dimension. This is typically referred to as “the curse of dimensionality.” Effective computational approaches exist for solving differential equations in up to three dimensions, but for a molecule with 6 atoms one must deal with 18 dimensions! Three of the 18 coordinates can be chosen to specify the position of the centre of mass of the system, and are therefore easy to separate. It is common to select coordinates so that three others describe the rotational orientation of the system, and if this is done there are 3N − 6 coordinates describing the shape of the molecule or reacting system. Rotation, however, does not separate because of Coriolis and centrifugal coupling. Although one can easily write down molecular Schrödinger equations, one cannot solve them. So, one resorts to various approximations, primarily to deal with the very high dimensionality of the problem. At the workshop, mathematicians learned what theoretical chemists are doing and what difficulties they must overcome. Chemists learned about rigorous mathematical results obtained recently by mathematicians. This primarily involved theoretical work, but also included ways to deal with high dimensionality when using computers for approximations. In prior conferences and workshops in this subject, there have been significant difficulties getting chemists and mathematicians to talk to one another in a meaningful way. There are differences in nomenclature, and people in the two disciplines often have different aims and priorities. A main goal of this workshop was to facilitate as much interaction between the two groups of individuals as possible, and in this regard, the workshop was very successful.