1. Comp. Mat. Science School 20011 Linear Scaling ‘Order-N’ Methods in Electronic Structure Theory Richard M. Martin University of Illinois Acknowledgements: Pablo OrdejonDavid Drabold Matthew GrumbachUwe Stephan Daniel Sanchez-PortalSatoshi Itoh Thanks to: Jose Soler, Emilio Artacho, Giulia Galli,...

2. Comp. Mat. Science School 20012 Linear Scaling ‘Order-N’ Methods and Car-Parrinello Simulations Fundamental Issues of locality in quantum mechanics Paradigm for view of electronic properties Practical Algorithms Results

3. Comp. Mat. Science School 20013 Locality in Quantum Mechanics V. Heine (Sol. St. Phys. Vol. 35, 1980) “Throwing out k-space” Based on ideas of Friedel (1954),... Many properties of electrons in any region are independent of distant regions Walter Kohn “Nearsightness”

4. Comp. Mat. Science School 20014 Locality in Quantum Mechanics Which properties of electrons are independent of distant regions? Total integrated quantities Density, Forces on atoms,... Coulomb Forces are long range but they can be handled in O(N) fashion just as in classical systems

5. Comp. Mat. Science School 20015 Non-Locality in Quantum Mechanics Which properties of electrons are non-local? Individual Eigenstates in crystals Sharp features of the Fermi surface at low T Electrical Conductivity at T=0 Metals vs insulators: distinguished by delocalization of eigenstates at the Fermi energy (metals) vs localization of the entire many-electron system (insulators) Approach in the Order-N methods: Identify localized and delocalized aspects

6. Comp. Mat. Science School 20016 Density Matrix I Key property that describes the range of the non- locality is the density matrix  (r,r’) In an insulator  (r,r’) is exponentially localized In a metal  (r,r’) decays as a power law at T = 0, exponentially for T > 0. (Goedecker, Ismail-Beigi) For non-interacting Bosons or Fermions, Landau and Lifshitz show that the correlation function g(r,r’) is uniquely related to the square of  (r,r’) Thus correlation lengths and the density matrix generally become shorter range at high T

7. Comp. Mat. Science School 20017 Density Matrix II Key property that describes the range of the non- locality is the density matrix  (r,r’) Definition:  (r,r’) =  i  i * (r)  i (r’) Can be localized even if each  i * (r) is not! r fixed at r =0 r’ Atom positions

8. Comp. Mat. Science School 20018 Toward Working Algorithms I (My own personal view) Heine and Haydock laid the groundwork - but it was applied only to limited Hamiltonians, …. 1985 - Car-Parrinello Methods changed the picture Key quantity is the total energy E[{  i }] which does not require eigenstates - only traces over the occupied states - the {  i } can be linear combinations of eigenstates

9. Comp. Mat. Science School 20019 Toward Working Algorithms II How can we use the advantages of the Car-Parrinello and the local approaches? 1992 - Galli and Parrinello pointed out the key idea - to make a Car-Parrinello algorithm that takes advantage of the locality Require that the states in localized. Note this does not require a localized basis - it may be very convenient, but a localized basis is not essential to construct localized states (example: sum of plane waves can be localized)

10. Comp. Mat. Science School 200110 Toward Working Algorithms III What are localized combinations of the eigenfunctions? Wannier Functions (generalized)! Wannier Functions span the same space as the eigenstates - all traces are the same Wannier Functions One localized Wannier Ftn centered on each site Extended Bloch Eigenfunctions

11. Comp. Mat. Science School 200111 Toward Working Algorithms IV Can work with either localized Wannier functions w i (r) or localized density matrix  (r,r’) =  i  i * (r)  i (r’) =  i w i * (r) w i (r’) Functions of one variable But not unique Functions of two variables - more complex But unique

12. Comp. Mat. Science School 200112 Linear Scaling ‘Order-N’ Methods Computational complexity ~ N = number of atoms (Current methods scale as N 2 or N 3 ) Intrinsically Parallel “Divide and Conquer” Green’s Functions Fermi Operator Expansion Density matrix “purification” Generalized Wannier Functions Spectral “Telescoping” (Review by S. Goedecker in Rev Mod Phys)

13. Comp. Mat. Science School 200113 Divide and Conquer (Yang, 1991) Divide System into (Overlapping) Spatial Regions. Solve each region in terms only of its neighbors. (Terminate regions suitably) Use standard methods for each region Sum charge densities to get total density, Coulomb terms

14. Comp. Mat. Science School 200114 Expansion of the Fermi function Sankey, et al (1994); Goedecker, Colombo (1994); Wang et al (1995) Explicit T nonzero Projection into the occupied Subspace Multiply trial function  by “Fermi operator”: F  = [(H - E F )/K B T +1] -1  Localized  leads to localized projection since the Fermi operator (density matrix) is localized Accomplish by expanding F in power series in H operator -

15. Comp. Mat. Science School 200115 Density Matrix “Purification” Li, Nunes, Vanderbilt (1993); Daw (1993) Hernandez, Gillan (1995) Idea: A density matrix  at T=0 has eigenvalues = occupation = 0 or 1 Suppose we have an approximate  that does not have this property The relation  n+1 = 3 (  n ) 2 - 2 (  n ) 3 always produces a new matrix with eigenvalues closer to 0 or 1.

16. Comp. Mat. Science School 200116 Density Matrix “Purification” The relation  n+1 = 3 (  n ) 2 - 2 (  n ) 3 always produces a new matrix with eigenvalues closer to 0 or 1. x 3 x 2 - 2 x 3 1 1 instability

17. Comp. Mat. Science School 200117 Generalized Wannier Functions Divide System into (Overlapping) Spatial Regions. Require each Wannier function to be non-zero only in a given region Solve for the functions in each region requiring each to be orthogonal to the neighboring functions New functional invented to allow direct minimization without explicitly requiring orthogonalization Mauri, et al.; Ordejon, et al; 1993; Stechel, et al 1994; Kim et al 1995

18. Comp. Mat. Science School 200118 Generalized Wannier Functions Factorization of the density matrix  (r,r’) =  i w i * (r ) w i (r’) Can chose localized Wannier functions (really linear combinations of Wannier functions) Minimize functional: E = Tr [ (2 - S) H] Since this is a variational functional, the Car- Parrinello method can be used to use one calculation as the input to the next Mauri, et al.; Ordejon, et al; 1993; Stechel, et al 1994; Kim et al 1995 Overlap matrix

19. Comp. Mat. Science School 200119 Functional (2-S)(H - E F ) Minimization leads to orthonormal filled orbitals focres empty orbitals to have zero amplitude Each matrix element (S and H) contains two factors of the wavefunction - amplitude ~ x. For occupied states (eigenvalues below E F ) x - ( 2 x 2 - x 4 ) 1 1 Minimum for normalized wavefunction (x = 1) Minimum at zero for empty states above E F

20. Comp. Mat. Science School 200120 Example of Our work Prediction of Shapes of Giant Fullerenes S. Itoh, P. Ordejon, D. A. Drabold and R. M. Martin, Phys Rev B 53, 2132 (1996). See also C. Xu and G. Scuceria, Chem. Phys. Lett. 262, 219 (1996).

21. Comp. Mat. Science School 200121 Wannier Function in a-Si U. Stephan

22. Comp. Mat. Science School 200122 Combination of O(N) Methods

23. Comp. Mat. Science School 200123 Collision of C 60 Buckyballs on Diamond Galli and Mauri, PRL 73, 3471 (1994)

24. Comp. Mat. Science School 200124 Deposition of C 28 Buckyballs on Diamond Simulations with ~ 5000 atoms, TB Hamiltonian from Xu, et al. ( A. Canning, G.~Galli and J.Kim, Phys.Rev.Lett. 78, 4442 (1997).

25. Comp. Mat. Science School 200125 Example of DFT Simulation (not order N) Daniel Sanchez-Portal (Phys. Rev. Lett. 1999) Simulation of a gold nanowire pulled between two gold tips Full DFT simulation Explanation for very puzzling experiment! Thermal motion of the atoms makes some appear sharp, others weak in electron microscope

26. Comp. Mat. Science School 200126 Simulations of DNA with the SIESTA code Machado, Ordejon, Artacho, Sanchez-Portal, Soler (preprint) Self-Consistent Local Orbital O(N) Code Relaxation - ~15-60 min/step (~ 1 day with diagonalization) Iso-density surfaces

27. Comp. Mat. Science School 200127 HOMO and LUMO in DNA (SIESTA code) Eigenstates found by N 3 method after relaxation Could be O(N) for each state

28. Comp. Mat. Science School 200128 O(N) Simulation of Magnets at T > 0 Collaboration at ORNL, Ames, Brookhaven Snapshot of magnetic order in a finite temperature simulation of paramagnetic Fe. These calculations represent significant progress towards the goal of full implementation of a first principles theory of the finite temperature and non- equilibrium properties of magnetic materials. Record setting performance for large unit cell models (up to 1024-atoms) led to the award of the 1998 Gordon Bell prize. The calculations that were the basis for the award were performed using the locally self-consistent multiple scattering method, which is an O(N) Density Functional method Web Site: http://oldpc.ms.ornl.gov/~gms/MShome.html

29. Comp. Mat. Science School 200129 FUTURE! ---- Biological Systems Examples of oriented pigment molecules that today are being simulated by empirical potentials

30. Comp. Mat. Science School 200130 Conclusions It is possible to treat many thousands of atoms in a full simulation - on a workstation with approximate methods - intrinsically parallel for a supercomputer Why treat many thousands of atoms? Large scale structures in materials - defects, boundaries, …. Biological molecules The ideas are also relevant to understanding even small systems