1. Linear Scaling Quantum Chemistry Richard P. Muller 1, Bob Ward 2, and William A. Goddard, III 1 1 Materials and Process Simulation Center California Institute of Technology and 2 Department of Computer Science University of Tennessee, Knoxville

2. Why QM Calculations Take So Long Form H Diagonalize  Did  Change? Guess  No Done Yes O(N 4 ) PS/Jaguar  O(N 2 ) O(N 3 ) Difficult to reduce: Krylof space, Conjugate gradient Currently only important if N > 2000

3. QM Methodology (Jaguar) Psuedospectral Technology (with Columbia U.) Multigrids Dealiasing functions Replace N 4 4-center Integrals with N 3 potentials Use Potentials to Form Euler-Lagrange Operator: CURRENT STATUS: Single processor speed 9 times faster than best alternate methodology Scales a factor of N 2 better than best alternate methodology Log (number basis functions) Jaguar Gaussian Log CPU Time Collaboration with Columbia U. and Schrödinger Inc.

4. QM Scalability: IBM SP2

5. 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  –Because H is a function of , 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

6. Divide and Conquer H Hamiltonian: Divided into fragments and buffer zones nbf

7. 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/r 3 (dipole potential)

8. Testing Divide and Conquer Linear Alkanes –14-98 atoms –170-817 basis functions Use standard Jaguar B3LYP/6-31G** techniques –Simple integration for testing

9. Simple Divide and Conquer Results

10. Error in Simple Divide and Conquer

11. Buffered Divide and Conquer Results

12. Errors in Buffered Divide and Conquer

13. 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 N 3 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

14. Divide, Conquer, and Recombine A B Eigenvalue Solving Going Up Already have eigs of H A and H B.  Make good guess at eigs of H (A+B) Can use fast (linear) diagonalization: Krylov-space Conjugate gradient Don’t have to do O(N 3 ) diagonalization