Quantum Monte Carlo algorithms for electronic structure at the petascale; the endstation project

Over the past two decades, continuum quantum Monte Carlo (QMC) has proved to be an invaluable tool for predicting of the properties of matter from fundamental principles. By solving the Schrödinger equation through a stochastic projection, it achieves the greatest accuracy and reliability of methods available for physical systems containing more than a few quantum particles. QMC enjoys scaling favorable to quantum chemical methods, with a computational effort which grows with the second or third power of system size. This accuracy and scalability has enabled scientific discovery across a broad spectrum of disciplines. The current methods perform very efficiently at the terascale. The quantumMonte Carlo Endstation project is a collaborative effort among researchers in the field to develop a new generation of algorithms, and their efficient implementations, which will take advantage of the upcoming petaflop architectures. Some aspects of these developments are discussed here. These tools will expand the accuracy, efficiency and range of QMC applicability and enable us to tackle challenges which are currently out of reach. The methods will be applied to several important problems including electronic and structural properties of water, transition metal oxides, nanosystems and ultracold atoms. SciDAC 2008 IOP Publishing Journal of Physics: Conference Series 125 (2008) 012057 doi:10.1088/1742-6596/125/1/012057 c © 2008 IOP Publishing Ltd 1 1. Computational quantum mechanics and quantum Monte Carlo “The underlying physical laws necessary for a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact applications of these laws lead to equations much too complicated to be soluble.” – Paul Dirac, 1929 This quotation, colloquially dubbed Dirac’s Challenge, reflects the understanding that, at least in principle, the exact solution of the Dirac equation (or its non-relativistic counterpart, the Schrödinger equation) would yield all the information necessary to predict the behavior of matter at normal energy scales, but finding the solution by traditional pencil and paper methods is very difficult. For a system containingN electrons, the Schrödinger equation is a partial differential equation in 3N dimensions. Because of the high dimensionality of the problem, exact solutions to these equations have thus far been found only simple cases such as one dimensional systems, so that Dirac’s Challenge is still unmet. Nonetheless, much of theoretical physics and chemistry has been devoted to finding ever more accurate approximate solutions to these equations governing the behavior of matter at the quantum level. Through the decades, the coupling of more advanced methods with the exponential increase in computing power has enabled us to progress from qualitatively correct models based on empirically determined parameters to truly ab initio calculations, i.e. those starting from the beginning. These calculations, taking the Challenge head on, have as their input only the atomic numbers of system’s constituent atoms. Of these ab initio methods, the mostly widely known are based on density functional theory (DFT). The basis of these methods is an approximate mapping of the 3N -dimensional equation onto a coupled set of N three-dimensional equations. In this mapping, the details of the correlated motion of individual electrons are approximated in an average “mean-field” sense. DFT has enjoyed a tremendous degree of success, and continues to do so, with new insights being gained every day. Nevertheless, there are many outstanding problems for which the level of accuracy provided by DFT has proved insufficient. As an example, DFT predicts a number of known insulators, including important transition metal oxides, to be metallic. In these cases, methods which treat the many-body electron more directly are required. The most accurate method available, known as full configuration interaction, aims at a near-exact solution to the Schrödinger equation. Unfortunately, the required computation scales exponentially with N , the number of electrons in the system, limiting its application to atoms and small molecules. Other methods from quantum chemistry address larger systems by truncating the many-particle basis intelligently using chemical insight. These provide highly accurate approximate solutions, but are still very limited in system size since the cost scales with a high power of N . Quantum Monte Carlo (QMC) methods achieve very high accuracy by treating the Schrödinger equation in the original 3N -dimensional space using a stochastic sampling of the many-body wave function. Since the cost of Monte Carlo sampling is relatively insensitive to dimensionality, large systems can be simulated at reasonable computational expense. In current implementations, the cost scales as N2 to N3, depending on the quantities of interest, although order-N methods are currently under development. Since electron correlation is treated explicitly, QMC can achieve accuracy similar to the best quantum chemistry methods, but can be applied to much larger systems, including nanoclusters, macromolecules, and crystals. For these reasons, it is generally accorded the distinction of being the most accurate method available for systems of more than a few atoms. Furthermore, the methods are extremely general and have been successfully applied to molecules, fluids, perfect crystals, and material defects. Finally, the methods afford several avenues for parallelization which, when combined, will allow very efficient operation at the petascale. SciDAC 2008 IOP Publishing Journal of Physics: Conference Series 125 (2008) 012057 doi:10.1088/1742-6596/125/1/012057