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Explore the scalability of quantum chemistry through innovative linear scaling methodologies for faster computations. Learn about the current status and potential advancements in computational efficiency.
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Linear Scaling Quantum Chemistry Richard P. Muller1, Bob Ward2, and William A. Goddard, III1 1Materials and Process Simulation Center California Institute of Technology and 2 Department of Computer Science University of Tennessee, Knoxville
Guess r Form H Diagonalize r Did r Change? Yes No Done Why QM Calculations Take So Long O(N4) PS/Jaguar O(N2) O(N3) Difficult to reduce: Krylof space, Conjugate gradient Currently only important if N > 2000
Psuedospectral Technology (with Columbia U.) • Multigrids • Dealiasing functions • Replace N4 4-center Integrals with N3 potentials • Use Potentials to Form Euler-Lagrange Operator: • CURRENT STATUS: • Single processor speed 9 times faster than best alternate methodology • Scales a factor of N2 better than best alternate methodology QM Methodology (Jaguar) Gaussian Log CPU Time Jaguar Log (number basis functions) Collaboration with Columbia U. and Schrödinger Inc.
QM Scalability: Comments • Algorithm ill-suited to massive parallelizability • Seriel diagonalization • Local data • Two steps in Quantum Chemistry • Hamiltonian H formation • H diagonalization to produce density r • Because H is a function of r, this is a nonlinear problem • Linearization and parallelization in Quantum Chemistry requires techniques to localize the density. • Modified Divide-and-Conquer technique • Solves the H-formation and H-diagonalization problems • Generalize to metallic systems
nbf Divide and Conquer H Hamiltonian: Divided into fragments and buffer zones
Divide and Conquer Shortcomings • GOOD: • Solves H-formation, H-diagonalization, and parallelization simultaneously! • BAD if: Correlation lengths > fragment size! • Metals, surfaces, conjugated systems • Must hierarchically correct error in fragments • Pairwise recombination of fragments to yield larger fragments • Hierarchically combine larger fragments to yield still-larger fragments • Continue until converged • At each level, include additional H elements: • Few, since fall off as 1/r3 (dipole potential)
Testing Divide and Conquer • Linear Alkanes • 14-98 atoms • 170-817 basis functions • Use standard Jaguar B3LYP/6-31G** techniques • Simple integration for testing
Beyond Simple Divide and Conquer • Buffer zones • Only way to correct for errors in D&C • Require large buffer zones (7x size of fragment); we only use small ones here. • Impractical for large systems/long correlation lengths -- ultimately start scaling as N3 • Renormalization-type approach • Combine pairs of lowest level of blocks to make larger blocks • …pairs of larger blocks to make still larger blocks • …etc. Continue until converged
Divide, Conquer, and Recombine A B • Eigenvalue Solving Going Up • Already have eigs of HA and HB. • Make good guess at eigs of H(A+B) • Can use fast (linear) diagonalization: • Krylov-space • Conjugate gradient • Don’t have to do O(N3) diagonalization