A Novel Quantum/Classical Hybrid Simulation Technique Mike C. Payne, [c] Gµbor Csµnyi, [c] Tristan Albaret, [d] and Alessandro De Vita* [a, b] Natural phenomena occur on a variety of length scales. These length scales not only tend to define scientific disciplines (physics and chemistry for the very small, biology for inter- mediate, geology for very long times and large length scales, and cosmology for the largest) but can also delineate subfields within a given discipline due to the very different experimental methods and theoretical models that are applicable at each scale. In some cases, these models form a multiscale hierarchy in which the parameters used in the larger scale models can be measured or calculated using modelling carried out on a smaller scale. [1] However, in a large class of problems the length scales cannot be separated in this way because the coupling between them is strong, and “bidirectional”. This occurs, for instance, when microscopic phenomena are not only driven by macroscopic forces but also change these mac- roscopic forces. In this case, there is a feedback loop between the microscopic and macroscopic scales. Stress-induced defect processes in solids are a good example, and brittle fracture is the prototypical problem. If we wish to simulate such process- es, we have to accurately simulate both length scales simulta- neously. Crucially, one cannot use the smallest-scale model to simulate the entire system, because it would be too computa- tionally demanding and it would also be hugely wasteful of re- sources. For instance, a typical quantum mechanical molecular dynamics simulation can deal with hundreds of atoms but the minimum system size that can capture the larger scale aspects of the problem, such as the long-range stress fields, could reach into hundreds of thousands of atoms or more. In the past decade there has been a growing effort to devise so- called hybrid simulation techniques, that seamlessly integrate a wide variety of different models, ranging from first-principles methods to finite elements techniques, into a concurrent simu- lation. [2, 3] A pioneering work in this field was the quasicontin- uum method, [2] which successfully linked classical atomistic and continuum elasticity models. Here we concentrate on a lower level linkage: that of quantum mechanical to classical model- ling. After mentioning general issues related to hybrid atomis- tic modelling, we discuss our scheme which was proposed in ref. [4]. Finally, we present several validation tests and an ex- ample application to brittle fracture in silicon. Hybrid Schemes The objective of hybrid atomistic modelling schemes is the fol- lowing: given a large atomic system, some (perhaps small) re- gions of it need to be simulated with model A (which we take to be quantum mechanical), and the rest with model B (which we take to be classical atomistic). The major issues that have to be addressed are: * Handshaking: The two models need to interact at the border that separates the two regions. If, as is natural, the system is partitioned by atoms into the regions A and B, how do each of the models react to having an artificial sur- face where we switch to using the other model? * Selection: How do we select which regions are to be treated by which model? In some cases, this might be straightfor- ward, for example, if we are investigating an impurity or a stationary defect. But in general, the region of interest where inherently quantum mechanical processes take place can appear, disappear and move, and we need to be able to track that. * Validation: Given any particular hybrid scheme, it has to be validated against fully accurate quantum mechanical calcu- lations. A successful scheme will be insensitive to the pre- cise location of the border between the two models, and all observables should be able to be converged by increasing the size of the quantum region. A New Scheme We have recently proposed a novel approach to the handshak- ing problem. [4] While some of the analysis put forward will not be repeated here, we note that the main philosophical differ- ence with the previous approaches is that we do not try to make a combined Hamiltonian from the separate model Hamil- tonians H classical and H quantum , rather, we focus on a purely local quantity, the force on each atom. Then, to make a seamless transition between the two models, one can compute the forces in the transition region using both models, and smooth- ly cross over from using one set of forces on one side of the seam to using the other set on the other side. This can be done in a number of ways, for example, by averaging the forces, or by making the transition region large enough so that at the outer edge the two models yield essentially identical forces. In practice however, we choose a third way. To see why, let us consider how the resulting forces are used in the simula- tion. Having given up the notion of a combined Hamiltonian, we do not have a strict conservation of the total energy any- more. What we do have are forces on each atom in any given configuration, and thus we can perform molecular dynamics [a] Prof. A. De Vita Physics Department, King’s College London, Strand London WC2R 2LS (UK) E-mail : alessandro.de_vita@kcl.ac.uk [b] Prof. A. De Vita INFM-DEMOCRITOS National Simulation Center and Center of Excellence for Nanostructured Materials (CENMAT) University of Trieste (Italy) Fax: (+ 39) 040-572-044 [c] Prof. M. C. Payne, Dr. G. Csµnyi Cavendish Laboratory, Madingley Road, Cambridge, CB3 0HE (UK) [d] Dr. T. Albaret Laboratoire de Physique de la Matiere Condensee et Nanostructures Bat. Leon Brillouin, Campus de la Doua Universite Claude Bernard Lyon 1 43 Bld du 11 novembre 1918, 69622 Villeurbanne Cedex (France) ChemPhysChem 2005, 6, 1731 –1734 DOI: 10.1002/cphc.200400585 # 2005 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 1731