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Guaranteed Precision and Speed – Multiresolution Quantum Chemistry

Robert J. Harrison and George I. Fann, Oak Ridge National Laboratory
in collaboration with Gregory Beylkin, University of Colorado

Summary

The objective of this project is the complete elimination of basis set error while maintaining correct scaling of the computational cost with system size in calculations of molecular electronic structure. Additional expected benefits to chemistry include a computational framework that is substantially simpler than conventional atomic orbital methods, that has robust guarantees of both speed and precision, and that is applicable to large systems. In contrast, current conventional methods are severely limited in both the attainable precision and size of system that may be studied. For effective one-electron theories, we have largely achieved our objective and we are now studying many-body theories. The electronic structure methods should be valuable in many disciplines, and the underlying numerical methods are broadly applicable.

Conventional electronic structure methods build molecular wave functions from combinations of atom-centered functions. This provides a compact and physically insightful representation, but long-range interactions are present, for instance, between the fine structure near one atom, and distant parts of the molecule. As a result, conventional methods are very inefficient for either large systems or high precision.

In contrast, multiresolution analysis generates efficient representations of many functions and operators by separating the behavior at different length scales. Many important long-range interactions decay much faster in a multiresolution representation. For instance, the electrostatic interaction between charges that decays as r-1 decays as r-k-1 in a multiwavelet basis of order k. Furthermore, the so-called non-standard form eliminates interactions between length scales. This form is particularly advantageous since it is easy to compute, efficient to apply on modern cache-based computers, and consumes only a small amount of memory . Multiresolution analysis also provides a simple truncation criterion, which enables adaptive local refinement while rigorously enforcing a global error bound.

Figure 1. Molecular orbital of benzene showing the adaptively refined grid and an isosurface.

Besides using a multiresolution approach, a critical step in attaining the objective has been the use of new separable representations for kernels of Green functions. In particular, we construct and use separable representations of Green functions for the Poisson and bound-state Helmholtz equations. These constructions, combined with multiresolution representations make our approach practical in three and higher dimensions.

We have achieved our objective for effective one electron theories with the implementation and validation of a parallel, prototype computer code for density functional theory (DFT) and Hartree-Fock (HF) energies, analytic derivatives w.r.t. atomic positions, and linear response theory for excited states. This code has been applied to atoms as heavy as barium and many molecules comprising first, second and third row atoms.

Demonstration of a practical approach for solution of one-electron methods is an essential precursor to direct numerical solution of two- and many-electron problems. This will require solution of equations in six dimensions, which is a challenge since most numerical approaches are formulated for up to three dimensions. However, it is for many-electron methods that we an-ticipate the greatest benefit from our approach.

The prototype code is written in Python, with C and Fortran for computationally intensive kernels. The use of Python, a high-level, object-oriented scripting language, has greatly accelerated the development. Our object-oriented design enables applications to be written at a very high level, composed in terms of operations performed upon functions. This permits the application developer to focus upon the physics/chemistry instead of computational details.

The overall computational framework is named MADNESS (multiresolution adaptive numerical scientific simulation). Presently it includes classes that provide functions and operators in 1, 3 and 6 (limited capability to date) dimensions, with free-space boundary conditions and separated forms for critical kernels. We will be adding other boundary conditions as operators as need arises. The chemistry application code is an independent module and other application areas will be prototyping within the same framework.

Access to massive computer resources at ORNL has enabled rapid exploration of the parameter space that control speed and precision. The current parallel algorithm is limited to processors within a single shared memory computer, so the 32-processor IBM Power-4 nodes at ORNL have been particularly useful. The next version of the code will use distributed memory and scale to many more processors. As we start to implement a more production-oriented computer program we will draw more heavily upon the resources of the ISICs for scalable iterative solvers, visualization, CCA, and performance analysis. This will require some close collaboration since the multiresolution formulation differs significantly from many other numerical approaches. Furthermore, in the near future, we will be solving in six spatial dimensions.

This is the first significant application in three or higher dimensions of multiwavelet bases for partial differential equations, and the separated representations are new. The close collaboration with the mathematicians, George Fann (funded by SciDAC MICS) and Prof. Gregory Beylkin, has greatly benefited both the application (chemistry) and the development of the new applied mathematics and numerical methods.
For more information and for downloads please visit http://www.ornl.gov/~rj3.

¹The non-standard form of operators with translation invariance is a Toeplitz matrix.

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