Multiresolution Quantum Chemistry – Guaranteed Precision and Speed

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. Apparently 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.

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 represen tations 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. Applications include benchmark structures and stacking energies of benzene and DNA base pairs (the figure displays the cytosine dimer), and electronic excitation energies of molecules with asymptotically corrected potentials.

The program demonstrates the expected scaling w.r.t. system size, which varies between O(N) (e.g., water cluster) and O(N 2 ) (e.g., conjugated systems) depending on the extent of the canonical orbitals. Conventional methods exhibit much worse scaling for high-precision calculations especially on small systems.

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 requires 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 anticipate the greatest benefit from our approach.

Effective computation in six dimensions has been a major research thrust. Our approach readily and efficiently applies integral operators in arbitrarily high dimensions, so the major concern is the representation of other functions and operators. In particular, we wish to represent the one-particle density matrix, the one-particle Green function, and the two-particle wave function. With these capabilities, we would be able to address major basis set and computational problems in molecular chemistry, quantum transport, and areas of solid state physics. Many representations for high-dimension functions have been explored, ranging from globally separated forms, to adaptive forms with local separation, and using standard and non-standard compressed forms. The major bottleneck so far has been efficient algorithms to maintain a separated representation and this is a focus of ongoing mathematical investigation.

Access to massive computer resources at ORNL has enabled rapid exploration of the parameter space that control speed and precision. The current prototype is limited to processors within a single shared-memory computer, so the 32-processor IBM Power-4 nodes and 256 processor SGI Altix at ORNL are particularly useful. We anticipate drawing 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 including solving in six spatial dimensions.

Essential to the rapid and significant progress of this project and much of the novel numerical development has been the matching funding from MICS for the work of George Fann.

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

For more information and for downloads please visit http://www.ornl.gov/~rj3

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