Coupled Quantum – Atomistic and Quantum – Continuum Mechanics Methods in Materials Research

The interface of quantum mechanics methods with classical atomistic simulation techniques, such as molecular dynamics and Monte Carlo, continues to be an area of considerable promise and interest. Such coupled quantum–atomistic approaches have been developed and employed, for example, to gain a comprehensive understanding of the energetics, kinetics, and dynamics of chemical processes involving surfaces and interfaces of hard materials. More recently, it has become possible to directly couple first-principles electronic structure techniques to continuum solid mechanics, either on the fly with feedback between length scales or by information passing between length scales. We discuss, with tutorial examples, the merging of quantum mechanics with molecular dynamics and Monte Carlo simulations, as well as quantum–continuum coupled techniques. We illustrate the opportunities offered by incorporation of information from quantum mechanics (reducing assumptions in higher length-scale models) and outline the challenges associated with achieving full predictive capability for the behavior of materials. uum mechanics methods—conceptually, the molecular or continuum mechanics simulation region provides correct boundary conditions for the quantum mechanics region. While this coupling is motivated in part by necessity, the realization that it is unnecessary to use quantum mechanics to treat atoms that behave similarly provides further impetus for developing coupled methods. This viewpoint presented may be thought of as a bottom-up approach to coupled models. One may also adopt the top-down point of view wherein the breakdown of continuum models at small length scales necessitates the inclusion of additional physics from these smaller scales—this could be done by introducing microscopic parameters in the continuum model or, in cases where the coupling across scales is strong, by concurrent modeling at both scales. Whichever the point of view, coupled methods provide a powerful approach for accurate modeling of realistic situations with finite resources. The building blocks of coupled schemes exist in well-honed numerical techniques at each length scale (Figure 1). At the quantum mechanics level, quantum chemistry approaches (e.g., configuration interaction or quantum Monte Carlo) provide the most accurate description of electronic structure. Unfortunately, their computational cost is still too high to be viable for quantum mechanics/molecular mechanics (QM/MM) coupling in the materials context. A more practical approach is density functional theory (DFT),1,2 which reduces the task of determining the many-electron wave function to a less-demanding problem of optimizing the electronic density. DFT is the method of choice for large systems, since it usually provides sufficient accuracy at a lower computational cost per atom. Traditionally, empirical potentials are used for molecular mechanics modeling. These potentials are typically fit to equi librium properties and hence are less reliable far from equilibrium. Ab initio data are used increasingly to improve overall reliability, as will be discussed later. Nevertheless, processes such as bond formation, bond breaking, and charge transfer are best handled with quantum mechanics, leaving molecular mechanics to handle nearequilibrium situations. Finally, at the continuum level, the finite element method (FEM)3 continues to be the technique of choice, mostly due to the ease of modeling arbitrary geometries with wide-ranging boundary conditions. Defect kinetics (e.g., dislocation dynamics4–6) also can be included phenomenologically within FEM to produce mesoscopic-scale models. This list of methods is by no means exhaustive. Computational materials science is an intersection of many disciplines, each of which has developed specialized tools at differing scales of interest. The challenge for coupling schemes lies in integrating these tools while making controllable approximations and without introducing spurious physics through ad hoc assumptions. Coupled methods are broadly classifiable as either multiscale models seeking to couple two or more spatial and/or Introduction Over the past few decades, theory/ computation has firmly established itself as a partner to experiment in unraveling fundamental principles behind materials behavior. The ever-improving performance of computers and the development of accurate and efficient algorithms progressively bring predictive quantum mechanics models of materials within reach. Nevertheless, full quantum mechanics treatments remain all but intractable at present for more than a few hundred atoms. Several schemes have been devised to couple quantum mechanics with lessexpensive molecular mechanics or contin-