Semi-stochastic Diagonalization Approach to Quantum Chemistry

Cornell University, Ithaca NY

Investigators

Abstract

Cyrus Umrigar of Cornell University is supported by an award from the Chemical Theory, Models and Computational Methods Program in extending a high-level algorithm for electronic structure including configuration interaction. Strongly correlated systems, for example, molecules with stretched bonds, pose a particularly severe challenge for high-level quantum chemistry methods, and this is an area where the new method may have considerable impact. The Alavi group has already performed full configuration interaction (FCI) calculations in a space of 10^16 (ten to the sixteenth) determinants, well beyond the 10^10 determinants feasible with traditional iterative diagonalization techniques. The diffculty posed by a problem depends not only on the size of the space but also upon distribution of wave function weights and the severity of the sign problem. The work centers around improvements to enhance the range of applicability of the method. A hybrid deterministic-cum-stochastic approach is being developed that preserves the desirable features of both approaches (reduced statistical error from having a deterministic component as well as the ability to explore very large spaces made possible by a stochastic approach). Preliminary work has demonstrated that improved trial wavefunctions can yield a large reduction in the statistical errors of the computed expectation values. Since the walk is done in determinant space, the Fermion sign problem, which is severe in competing methods due to the exponential growth of the Bosonic wavefunction relative to the Fermionic wavefunction, is absent in the present method. The method does have a less severe form of the sign problem and practical ways are demonstrated for keeping that under control. A procedure for improving the convergence with respect to basis size is also being developed. These improvements are to make possible highly accurate calculations on molecular systems that were hitherto not feasible with existing quantum chemistry methods, such as the dissociation curves for transition metal dimers. The general mathematical problem being addressed extends well beyond the realm of quantum chemistry, namely that of finding the dominant eigenvalue of a matrix that is so large that even a single column of the matrix cannot be stored in computer memory. Although the basic equation of quantum mechanics, the Schroedinger equation, has been known for almost a century, efficient methods for finding accurate solutions to this equation continue to be an active area of research. Such methods are essential for making useful predictions about a variety of properties of molecules and materials of technological interest. It is the purpose of this work to improve upon a method recently developed by the Alavi group at Cambridge University that uses probabilistic techniques to effectively employ orders of magnitude more basis functions (to get less error) than was hitherto possible.