High-dimensional Quantum Dynamics: Challenges and Opportunities
- Hans-Dieter Meyer (University of Heidelberg, Germany)
- Graham A. Worth (University College London, United Kingdom)
Please see http://www.stchem.bham.ac.uk/~worthgrp/hdqd2012/index.html for more details.
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The study and understanding of the motion of atoms in a molecule, i.e. rotation, vibrations, dissociation, reaction, etc., is a central problem in the molecular sciences: not only in chemistry but also in astro- and atmospheric-physics and in biology. Interpretation of experiments probing the fine details of molecular behaviour often requires computer simulations. Typical examples are found in scattering experiments and the expanding field of femtochemistry which uses ultra short laser pluses to "watch" the atoms move.
The computation of the motion, known as molecular dynamics simulations, links observables such as cross sections, reaction rates, spectra, and information on internal energy transfer to the underlying molecular properties. As experiments probe materials at the molecular level with ever finer detail, the importance of simulations is increasing to aid the interpretation. The correct theory to describe the dynamics is quantum dynamics. Due to its complexity, however, a full quantum-mechanical treatment is possible only for rather small molecules. For larger molecules, classical and semi-classical mechanics can be used, although important features such as interferences and tunnelling are ignored, or only partially taken into account.
Much effort has been invested over the last decades to improve the situation and make quantum dynamical simulations possible for larger molecules. This requires not only bigger computers, but, more, importantly, better algorithms. A method that exemplifies present progress in the field is the multi-configuration time-dependent Hartree (MCTDH) method. We propose to bring together scientists who are exploring the use of a range of high-dimensional dynamics methods, including MCTDH, to discuss challenging new applications and to identify new ways of improving and extending thel toolbox available.
The full numerical solution of the time-dependent Schroedinger equation (TDSE) has a fundamental bottleneck in that the computer resources required grow exponentially with the number of degrees of freedom treated. Until recently this restricted accurate simulations of molecular dynamics to three atoms . While such simulations can give detailed insight into fundamental reactivity, they are clearly unsuited to describing the time evolution of the polyatomic molecules of chemistry. At present, however, the combination of ever increasing computer power and improving algorithms means progress is being made towards the accurate treatment of these systems.
For example, the multi-configuration time-dependent Hartree (MCTDH) method galvanised the field by enabling accurate quantum dynamics simulations of up to 10 atoms to be made [2,3]. In fact the most efficient version of MCTDH can treat hundreds of degrees of freedom in model problems quantum mechanically . This opens up the treatment of system-bath problems explicitly and allows comparison with approximate methods.
In parallel to the development in full quantum dynamics, a number of semiclassical methods that use trajectories to approximately solve the TDSE have also been worked out. Examples include the Initial Value Representation  Bohmian trajectories  and Gaussian wavepacket methods [7,8]. These have a lower computational cost, but care needs to be taken as the error is sometimes difficult to know. Semiclassical methods have had particular success in non-adiabatic problems, where coupled potential surfaces are involved, for example in the forms of trajectory surface hopping  and multiple spawning .
A second bottleneck in molecular dynamics simulations is the need for accurate potential energy surfaces. Obtaining these requires fitting a function to points calculated over a range of geometries, and again grows exponentially with system size. A present strand of research to overcome this problem is direct dynamics in which the potential surface is calculated on-the-fly as and when required, saving the need to describe regions never visited by the system . In addition to the potential energy surface problem, deriving a kinetic energy operator for polyatomic systems is also non-trivial .
Efficient coding and the use of modern parallel computer architectures is required to get the most out of any scientific program. Quantum dynamics is no exception, and examples are to be found in the DDPHP parallel implementation of wavepacket dynamics  and the use of parallel coding in MCTDH . These bring huge advances in performance, enabling yet larger systems to be treated and the use of careful programming will be crucial for the next phase of quantum dynamics development.
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