# Mathematical Methods in Quantum Molecular Dynamics (13w5018)

Arriving in Banff, Alberta Sunday, April 28 and departing Friday May 3, 2013

## Objectives

We propose a workshop on new mathematical methods for studying the dynamics of molecules and reactions. Chemists, physicists, and mathematicians are working in this field, but communication between them is poor. The workshop will bring together people with different perspectives, experience and knowledge in different disciplines who are working on similar problems.

There is a crying need to develop new methods to study the dynamics of molecules. The problems are extremely challenging because they require solving partial differential equations in high dimensions. There is a very large group of numerical analysts and mathematicians working on ideas (e.g., finite differences, finite elements, adaptive grids, preconditioners, etc.) for solving problems in up to three dimensions. These ideas are widely applied by engineers who study real 3D problems (e.g., the design of airplane wings). In quantum mechanics, every particle may move 3 dimensions, and thus a molecule or reacting system with N atoms involves dynamics in 3N dimensions. If N is 5 (as is the case, for example, for the methane molecule, an important green house gas), one needs to solve a 15 dimensional problem! Methods that work well in 3D are completely useless in 15D. This difficulty is often referred to as the “curse of dimension.” If the motion of molecules were governed by the laws of classical mechanics the problem would be tractable, but molecules are intrinsically quantum mechanical. Because of the non-locality of quantum mechanics, the wavefunctions that describe the state of the system are functions of 3N independent coordinates. (For many problems the centre of mass motion and perhaps the angular momentum can be separated, but doing so reduces the dimension by at most 6.)

There has been considerable work in the development of semi-classical and adiabatic approximations for solving high-dimensional problems in quantum dynamics. Mathematicians, chemists, and physicists sometimes have developed similar ideas. It is crucial that scientists in these disciplines be brought together. Significant progress will be made possible by the resulting cross-fertilization. The understanding of mathematicians working in semi-classical analysis, the theory of adiabatic approximations and avoided crossings should enable chemists who apply these ideas to develop better computational methods. Gaussian basis methods for solving the time dependent Schrödinger equation are particularly promising.

To solve the Schrödinger equation without making approximations, one almost always expands the unknown wavefunctions as linear combinations of known basis functions. The expansion coefficients are most frequently obtained by applying the Hamiltonian operator to a linear combination of basis functions, multiplying on the left by a basis function and integrating. (Numerical analysts call this a Galerkin method.) For the purpose of calculating (ro-)vibrational spectra, cross sections, reaction rate constants, etc., it is common to use a direct product basis, each of whose functions is a product of functions of a single particle’s coordinates. Direct product bases are simple and systematically improvable. Typically one requires about 10 functions per coordinate so the size of the direct product basis required to solve an angular momentum zero Schrödinger equation in the centre of mass frame with N particles is about 103N-6. Two major problems impede progress. One is the size of the basis set, the other is the size of the quadrature grid required to compute integrals in 3N-6 dimensions. Sparse grid techniques developed by mathematicians can be used to overcome both problems. Some progress has been made on both fronts, but there is a real need to share information between mathematicians and chemists/physicists. More is known about reducing the basis size. Scientists are just starting to explore the use of sparse grid ideas for doing the quadratures necessary to solve the Schrödinger equation.

With this conference we propose to bring these disparate groups of researchers together to share information. Physicists and chemists will learn of recent progress by mathematicians and numerical analysts in solving these problems. Mathematicians will learn about the specific difficulties faced by physicists and chemists, as well as the new approach

There is a crying need to develop new methods to study the dynamics of molecules. The problems are extremely challenging because they require solving partial differential equations in high dimensions. There is a very large group of numerical analysts and mathematicians working on ideas (e.g., finite differences, finite elements, adaptive grids, preconditioners, etc.) for solving problems in up to three dimensions. These ideas are widely applied by engineers who study real 3D problems (e.g., the design of airplane wings). In quantum mechanics, every particle may move 3 dimensions, and thus a molecule or reacting system with N atoms involves dynamics in 3N dimensions. If N is 5 (as is the case, for example, for the methane molecule, an important green house gas), one needs to solve a 15 dimensional problem! Methods that work well in 3D are completely useless in 15D. This difficulty is often referred to as the “curse of dimension.” If the motion of molecules were governed by the laws of classical mechanics the problem would be tractable, but molecules are intrinsically quantum mechanical. Because of the non-locality of quantum mechanics, the wavefunctions that describe the state of the system are functions of 3N independent coordinates. (For many problems the centre of mass motion and perhaps the angular momentum can be separated, but doing so reduces the dimension by at most 6.)

There has been considerable work in the development of semi-classical and adiabatic approximations for solving high-dimensional problems in quantum dynamics. Mathematicians, chemists, and physicists sometimes have developed similar ideas. It is crucial that scientists in these disciplines be brought together. Significant progress will be made possible by the resulting cross-fertilization. The understanding of mathematicians working in semi-classical analysis, the theory of adiabatic approximations and avoided crossings should enable chemists who apply these ideas to develop better computational methods. Gaussian basis methods for solving the time dependent Schrödinger equation are particularly promising.

To solve the Schrödinger equation without making approximations, one almost always expands the unknown wavefunctions as linear combinations of known basis functions. The expansion coefficients are most frequently obtained by applying the Hamiltonian operator to a linear combination of basis functions, multiplying on the left by a basis function and integrating. (Numerical analysts call this a Galerkin method.) For the purpose of calculating (ro-)vibrational spectra, cross sections, reaction rate constants, etc., it is common to use a direct product basis, each of whose functions is a product of functions of a single particle’s coordinates. Direct product bases are simple and systematically improvable. Typically one requires about 10 functions per coordinate so the size of the direct product basis required to solve an angular momentum zero Schrödinger equation in the centre of mass frame with N particles is about 103N-6. Two major problems impede progress. One is the size of the basis set, the other is the size of the quadrature grid required to compute integrals in 3N-6 dimensions. Sparse grid techniques developed by mathematicians can be used to overcome both problems. Some progress has been made on both fronts, but there is a real need to share information between mathematicians and chemists/physicists. More is known about reducing the basis size. Scientists are just starting to explore the use of sparse grid ideas for doing the quadratures necessary to solve the Schrödinger equation.

With this conference we propose to bring these disparate groups of researchers together to share information. Physicists and chemists will learn of recent progress by mathematicians and numerical analysts in solving these problems. Mathematicians will learn about the specific difficulties faced by physicists and chemists, as well as the new approach