New Algorithms to make Quantum Monte Carlo more Efficient Daniel R. Fisher William A. Goddard III Materials and Process Simulation Center (MSC) California Institute of Technology

 2003, D.R. Fisher Dirac’s Thoughts on Chemistry "The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the application of these laws leads to equations much too complicated to be soluble." - P. Dirac, Proc. Roy. Soc (London) 123:714 (1929)

 2003, D.R. Fisher Computational Quantum Mechanics Algorithms Adapted from: Morokuma, et al. IBM J. Res. & Dev. Vol. 45 No. 3/4 May July p

 2003, D.R. Fisher Hartree-Fock QM Calculations Average Position of the Other Electrons Replace explicit electron-electron interactions with average electron-electron interactions to get N,1-particle equations. This works BUT it does not deal correctly with electron-electron correlation

 2003, D.R. Fisher When Is Electron Correlation Important? Transition Metal Oxides –High Temprature Copper Oxide Superconductors Heavy Fermion Metals –Usually Rare Earths or Actinides Organic Charge Transfer Compounds One and Two Dimensional Electron Gas Systems –Nano-scale Wires Positron Chemistry –Defect Detection and Analysis Muon/Exotic Particle Physics –Catalyzed Fusion See 21 April 2000 Science for More Examples

 2003, D.R. Fisher QMC Basic Algorithm Molecule Change Jastrow to Optimize it VMC Equilibrate VMC Sampling DMC Equilibrate DMC Sampling Optimized? Enough Construct Initial Walker(s) Get  T (HF,DFT,MCSCF) Get Starting Jastrow Traditional QC package Very Accurate Result Task Conceived!!! -Accurate result -Small expense

 2003, D.R. Fisher Metropolis Algorithm and QMC Reformulate Energy Expectation Integral. Monte Carlo Integration for this 3N-dimensional integral. –Metropolis algorithm produces electron configurations with respect to  2. –System energy expectation value is average of the local energies.

 2003, D.R. Fisher VMC Wavefunction for a Molecule  HF-Type Wavefunction Function of how far apart the electrons are from each other Function of how far electrons are from nuclei What a typical  He might look like:

 2003, D.R. Fisher Diffusion QMC Change variables... Solution at  Ground State Excited States Key Point: No Matter What Wavefunction We Start With, It Decays to the True Ground State as We Take Steps in “time,”  !

 2003, D.R. Fisher Generic Jastrow Manager- Worker Parallelization New Algorithms and Areas Affected Molecule Change Jastrow to Optimize it VMC Equilibrate VMC Sampling DMC Equilibrate DMC Sampling Optimized? Enough Construct Initial Walker(s) Get  T (HF,DFT,MCSCF) Get Starting Jastrow Traditional QC package Very Accurate Result Task Conceived Decorrelation Algorithm

 2003, D.R. Fisher Problem With Metropolis Sampling Energies at these points are correlated! From intro statistics, the uncorrelated variance in energy is: This does not work in this case because the energies calculated at sequential points are serially correlated. Probability of a move being accepted is related to the transition probability. T(A  B) Rejected Move Accepted Move

 2003, D.R. Fisher Flyvbjerg-Petersen Decorrelation Algorithm x1x1 x2x2 x4x4 x3x3 x5x5 x6x6 x7x7 x9x9 x8x8 x n-3 x n-1 x n-2 xnxn Original Data x1x1 x2x2 x4x4 x3x3 x5x5 x6x6 x7x7 x9x9 x8x8 Blocked Data Average blocks of the original data into new data elements. If the data blocks are sufficiently large, the blocked data points are uncorrelated and the standard O(N) variance equation can be used.

 2003, D.R. Fisher VMC Particle-in-a-Box Standard Deviation Calculation Plateau True  Underestimate Constant True  Uncorrelated Data Correlated Data

 2003, D.R. Fisher New Statistical Analysis Algorithm Flyvbjerg-Petersen Algorithm One processor must do all of the work Must communicate O(N) data when used on a parallel computer Must store O(N) data Must be used at the end of the calculation (can’t check convergence on-the-fly). Dynamic Distributable Decorrelation Algorithm Feldmann M.T., D.R. Kent IV, R.P. Muller, W.A. Goddard III. “Efficient Algorithm for “On-the-fly” Error Analysis of Local or Distributed Serially-Correlated Data”, J. Chem. Phys. (submitted) Perfectly parallel even on inhomogeneous computers Must communicate O(log 2 N) data when used on a parallel computer Must store O(log 2 N) data Can provide on-the-fly results (convergence based termination) N = log 2 N = 23-40

 2003, D.R. Fisher New Manager Worker Parallelization Algorithm Current Algorithm - Pure Iterative All processors do an equal amount of work If one processor finishes first it must wait on all of the others Intended only for homogeneous machines (all processors the same) Convergence-based termination not possible New Algorithm - Manager Worker Feldmann M.T., D.R. Kent IV, R.P. Muller, W.A. Goddard III. “Manager-Worker Based Model for Massively Parallel and Perfectly Load-Balanced Quantum Monte Carlo”, J. Comp. Chem. (submitted) All processors do as much work as they can All processors complete at the same time Intended for either homogeneous or inhomogeneous machines (processors can be different) Convergence-based termination possible with DDDA

 2003, D.R. Fisher Parallel Algorithm Performance: Heterogeneous Computer 8 total processors. Mixture of Pentium II 200 MHz and Pentium III 866 MHz

 2003, D.R. Fisher Parallel Algorithm Performance: Heterogeneous Computer Not quite linear LLNL Blue Pacific Linear to 2048 Processors LANL Nirvana Performance depends on what is being measured…

 2003, D.R. Fisher Are Transferable Parameter Sets Possible?  HF-Type Wavefunction Function of how far apart the electrons are Function of how far electrons are from nuclei Can parameter sets be found that are transferable between different systems? Similar idea to contracted Gaussian basis sets (e.g. 6-31G**).

 2003, D.R. Fisher Correlation Energy Recovered by Generic Jastrow Function All these hydrocarbons have nearly the same optimal “b  ” parameter!

 2003, D.R. Fisher Generic Jastrow Performance DMC CH 4 DMC C 2 H 2

 2003, D.R. Fisher Why Isn’t QMC Really Linearly Scaling? Each processor needs to perform statistically independent calculations Each processor must begin calculating with independent, statistically significant points in configuration space A separate Metropolis calculation must be equilibrated for each independent point in configuration space Efficiency LANL Nirvana Number of Processors

 2003, D.R. Fisher Current Work- Initialization Solution Make algorithm that reduces the initialization time. –Better guess for placement of initial walkers in the Monte Carlo random walk. –Reduce the number of steps to equilibrate the walker. –Make algorithm for initialization which itself is parallelizable. There are two criteria by which we can judge the quality of a configuration. –The sum of one electron probability densities from the SCF calculation: –The distance of the electrons from each other.

 2003, D.R. Fisher Constructing Initial Configurations Orbitals are linear combinations of primitive gaussians: The square of the orbital is its probability density. We can change to spherical coordinates and separate the dependence in r, θ, and φ:

 2003, D.R. Fisher Probability Distribution Functions By integrating over one variable at a time, we can get the marginal probability distribution function in each direction. These are the probability distribution functions for the 2p x orbital of Ne: Now we can generate uniform random numbers between 0 and 1 to distribute electrons with respect to the probability density of this orbital.

 2003, D.R. Fisher Deciding which configurations to keep We can invert each orbital of the molecule and distribute electrons in them according to their occupancy. This will ensure that our initial configurations are in regions of high one electron probability density. This algorithm, however, will not prevent electrons from being placed near each other. We plan to develop a heuristic score function that will take a 3N-dimensional walker as input and return a real number, such that the larger the result, the fewer steps the walker will need to take to reach an equilibrium region of configuration space.

 2003, D.R. Fisher The Score function The score function will depend on the one electron density and the distance between pairs of electrons:, where Our goal is to determine a functional form of the score function G that is transferable between different molecular systems.

 2003, D.R. Fisher Using the score function Once a reliable score function is developed, it will be possible to construct high-quality walkers: - For an N-electron molecule, we could generate M>N electron positions and then find which combination gives the best walker. - Alternatively, a large set of N-electron configurations could be generated, and then the most favorable could be chosen. - The number of equilibration steps would be determined by the result of the score function ????? Could it be possible to use the score function to dynamically determine when a walker is equilibrated ????? If this is possible, we could terminate the initialization of each walker individually. This would mean we could start gathering data from a walker as soon as it is equilibrated, rather than waiting for the entire ensemble.

 2003, D.R. Fisher Conclusion For parallel calculations, the efficiency drops off drastically as the initialization time and number of processors increase. This new class of Metropolis initialization algorithms will greatly decrease the time required to initialize a QMC calculation. The efficient use of parallel processors will allow highly accurate quantum mechanical calculations of larger and more interesting chemical systems.

 2003, D.R. Fisher Acknowledgements General: Mike Feldmann Chip Kent Rick Muller William A. Goddard III Goddard Group CACR staff Funding: $$$ ASCI $$$ (Caltech-ASCI-MP)