Large-Scale Density Functional Calculations

1. Large-Scale Density Functional Calculations James E. Raynolds, College of Nanoscale Science and Engineering Lenore R. Mullin, College of Computing and Information

2. Overview • Using computers to carry out “numerical experiments” in Materials Science, Chemistry and Physics • Quantum Mechanical equations solved for a system of atoms in a representative unit cell • Measurable properties obtained from “first-principles” • mechanical, thermodynamic, electronic • optical, magnetic, transport

3. + V Phenolate/Benzenediazonium Benzene Example: Transport in molecular wire

4. Peierls Distortion dimerized pair Pi stacked pair mechanical relaxation insulator metal

5. Frontier Problems • Non-equilibrium spin-transport in metals and semiconductors (Spintronics) • Transport and coupled mechanical / electronic interactions in molecules (metal - insulator transition due to mechanical relaxation) • Industrial applications: Modeling Chemical Vapor Deposition (CVD) processes atom by atom • Challenges: correlated motion of electrons • Coupled electron-phonon interactions (electron - vibration coupling)

6. Density Functional Theory • Density Functional Theory (DFT) is a “mean-field” solution to the many-electron problem. • Each electron interacts with an effective average field produced by all of the other electrons • Non-linear set of coupled differential equations

7. Density Functional Equations Looks linear but... depends on the charge density through: Example: Local density approximation

8. DFT solution approach • Expand the wave-functions in a basis set: • Matrix eigenvalue-eigenvector problem: • Orthogonality: • Iterative solution to “self-consistency” (i.e. output V(r) coincides with input)

9. Popular implementations • Plane wave basis functions (Fourier Series): • Drawback: • Benefit: easy to code, sophisticated non-linear response calculations possible • Localized “atomic-like” basis functions scaling • - exponential distance decay for • insulators • power law distance decay for • metals

10. Contrasting Implementations • Abinit: www.abinit.org • Very sophisticated array of calculated properties • Calculations become prohibitive for more than a few dozen atoms • VASP (Vienna Ab-Initio Simulation Package) • Less sophisticated by much faster • few hundred atoms possible • Siesta: (Spanish Initiative for Electronic Simulations with Thousands of Atoms) • O(N) scaling: fast but less sophisticated • few thousand atoms possible

11. Public Access • Many codes are freely available: go to http://psi-k.dl.ac.uk/data/codes.html for a list of more than 20 • Most codes still not user-friendly and take months to years to master

12. The Brick Wall!! • All of these methods run out of steam very quickly in terms of run time and memory • Calculations with scaling take days or weeks to run!! • Even calculations with scaling run into memory bottlenecks • Materials Science simulations require thousands of atoms for thousands of time steps

13. Key Algorithms • For plane wave based codes: the Fast Fourier Transform • We have gained factor’s of 4 improvement in speed and storage using Conformal Computing • A number of new developments are being implemented for further increases • Matrix diagonalization routines for very large matrices

14. Conformal Computing • Density Functional Calculations are an ideal setting for Conformal Computing! • In fact: any array (matrix) based computational setting is ripe for Conformal Computing • Why? Conformal Computing eliminates temporary arrays and un-necessary loops!

15. Opportunities • Current electronic band structures fairly fast (on the order of one hour):

16. Contrasting: electron-phonon • Electron-phonon calculations: on the order of 1 day for small systems • Superconductivity in • “conventional” materials • determined by the electron - • phonon interaction • Aluminum (1 atom) takes roughly 1 day of computing • Imagine several dozen atoms with scaling

17. Electron-Phonon improvements • Many quantities currently written to files then later combined • The size and number of these files is becoming prohibitively expensive • Opportunities for parallelization of integrals • Opportunities to eliminate temporaries through the use of direct indexing

18. Grid Computing • Even with highly optimized code (which is still a way off) there is always a need for more and more resources • For example: electron-phonon calculations involve dozens of separate calculations that could be run on independent machines • Grid computing allows many independent calculations to be run in parallel

19. Grid Computing: First Steps • QMolDyn GAT: a template for submitting Density Functional Calculations over the grid • Vision: QMolDyn will eventually have a variety of codes (modules) • Presently: Siesta ( ) running on the grid, 8, 16, 32, 64, 128, 256, 512- atom systems

20. Summary / Conclusions • There is a great demand for large-scale array (matrix) based calculations in materials science • Quantum calculations are increasingly important for Materials Science, Chemistry and Physics • Grid computing combined with Conformal Computing techniques is very promising