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Linear scaling solvers based on Wannier-like functions

Linear scaling solvers based on Wannier-like functions. P. Ordejón Institut de Ciència de Materials de Barcelona (CSIC). Linear scaling = Order(N). CPU load. 3. ~ N. ~ N. Early 90’s. ~ 100. N (# atoms). Order-N DFT. Find density and hamiltonian (80% of code)

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Linear scaling solvers based on Wannier-like functions

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  1. Linear scaling solvers based on Wannier-like functions P. Ordejón Institut de Ciència de Materials de Barcelona (CSIC)

  2. Linear scaling = Order(N) CPU load 3 ~ N ~ N Early 90’s ~ 100 N (# atoms)

  3. Order-N DFT • Find density and hamiltonian (80% of code) • Find “eigenvectors” and energy (20% of code) • Iterate SCF loop • Steps 1 and 3 spared in tight-binding schemes

  4. Key to O(N): locality Large system ``Divide and conquer’’W. Yang, Phys. Rev. Lett. 66, 1438 (1992) ``Nearsightedness’’W. Kohn, Phys. Rev. Lett. 76, 3168 (1996)

  5. 1 = 1/2 (Ψ1+Ψ2) Ψ2 Ψ1  2 = 1/2 (Ψ1-Ψ2) Locality of Wave Functions Wannier functions (crystals) Localized Molecular Orbitals (molecules)

  6. Locality of Wave Functions Energy: Unitary Transformation: We do NOT need eigenstates! We can compute energy with Loc. Wavefuncs.

  7. 610-21 7.6 Locality of Wave Functions Exponential localization (insulators): Wannier function in Carbon (diamond) Drabold et al.

  8. Locality of Wave Functions Insulators vs Metals:  Carbon (diamond)  Aluminium Goedecker & Teter, PRB 51, 9455 (1995)

  9. Linear Scaling Localization + Truncation • Sparse Matrices 5 4 • Truncation errors 2 1 3 In systems with a gap. Decay rate a depends on gap Eg

  10. Linear Scaling Approaches • (Localized) object which is computed: • - wave functions • - density matrix • Approach to obtain the solution: • - minimization • - projection • spectral • Reviews on O(N) Methods: Goedecker, RMP ’98 • Ordejón, Comp. Mat. Sci.’98

  11. Basis sets for linear-scaling DFT • LCAO: - Gaussian based + QC machinery • M. Challacombe, G. Scuseria, M. Head-Gordon ... • - Numerical atomic orbitals (NAO) • SIESTA • S. Kenny &. A Horsfield (PLATO) • OpenMX • Hybrid PW – Localized orbitals • - Gaussians J. Hutter, M. Parrinello • - “Localized PWs” • C. Skylaris, P, Haynes & M. Payne • B-splines in 3D grid • D. Bowler & M. Gillan • Finite-differences (nearly O(N)) • J. Bernholc

  12. x central b c b buffer buffer central buffer x’ Divide and conquer Weitao Yang (1992)

  13. 1 0 Emin EF Emax Fermi operator/projector Goedecker & Colombo (1994) f(E) = 1/(1+eE/kT)  n cn En F   cn Hn Etot = Tr[ F H ] Ntot = Tr[ F ] ^ ^ ^ ^ ^

  14.    = 3 2 - 2 3 Etot() =  H = min 1    0 -0.5 0 1 1.5 Density matrix functional Li, Nunes & Vanderbilt (1993)

  15. Sij = < i | j > | ’k > = j | j > Sjk-1/2 EKS = k< ’k | H | ’k > = ijk Ski-1/2< i | H | j > Sjk-1/2 = Trocc[ S-1 H ] Kohn-Sham EOM = Trocc[ (2I-S) H ] Order-N = Trocc[ H] + Trocc[(I-S) H ] ^ ^ Wannier O(N) functional • Mauri, Galli & Car, PRB 47, 9973 (1993) • Ordejón et al, PRB 48, 14646 (1993)

  16. O(N) Non-orthogonality penalty KS Sij = ij  EOM = EKS Order-N vs KS functionals

  17. Chemical potential Kim, Mauri & Galli, PRB 52, 1640 (1995) • (r) = 2ij i(r) (2ij-Sij) j(r) • EOM = Trocc[ (2I-S) H ]# states = # electron pairs •  Local minima • EKMG = Trocc+[ (2I-S) (H-S) ]# states > # electron pairs  = chemical potential (Fermi energy) Ei >   |i|  0 Ei <   |i|  1 Difficulties Solutions • Stability of N() Initial diagonalization / Estimate of  • First minimization of EKMG Reuse previous solutions

  18. i Rc rc Orbital localization  i(r) =  ci(r)

  19. Convergence with localisation radius Si supercell, 512 atoms Relative Error (%) Rc (Ang)

  20. x i x 0 2 5.37 5.37 3 1.85 8.29 1.85 = 15.34 3.14  0 = 0 1.15  0 = 0 ------- Sum 15.34 1.85 7 2.12 0 0 i y 0 2.12 3 8.29 0 4 3.14 8 1.15 Sparse vectors and matrices Restore to zero xi 0 only

  21. Actual linear scaling c-Si supercells, single- Single Pentium III 800 MHz. 1 Gb RAM 132.000 atoms in 64 nodes

  22. Linear scaling solver: practicalities in SIESTA P. Ordejón Institut de Ciència de Materials de Barcelona (CSIC)

  23. Order-N in SIESTA (1) Calculate Hamiltonian Minimize EKS with respect to WFs (GC minimization) Build new charge density from WFs SCF

  24. Energy Functional Minimization • Start from initial LWFs (from scratch or from previous step) • Minimize Energy Functional w.r.t. ci • EOM = Trocc[ (2I-S) H ] or • EKMG = Trocc+[ (2I-S) (H-S) ] • Obtain new density • (r) = 2ij i(r) (2ij-Sij) j(r) ci(r) =  ci(r)

  25. i Rc rc Orbital localization  i(r) =  ci(r)

  26. Order-N in SIESTA (2) • Practical problems: • Minimization of E versus WFs: • First minimization is hard!!! (~1000 CG iterations) • Next minimizations are much faster (next SCF and MD steps) • ALWAYS save SystemName.LWF and SystemName.DM files!!!! • The Chemical Potential (in Kim’s functional): • Data on input (ON.Eta). Problem: can change during SCF and dynamics. • Possibility to estimate the chemical potential in O(N) operations • If chosen ON.Eta is inside a band (conduction or valence), the minimization often becomes unstable and diverges • Solution I: use chemical potential estimated on the run • Solution II: do a previous diagonalization

  27. Example of instability related to a wrong chemical potential

  28. Order-N in SIESTA (3) • SolutionMethod OrderN • ON.Functional Ordejon-Mauri or Kim (def) • ON.MaxNumIter Max. iterations in CG minim. (WFs) • ON.Etol Tolerance in the energy minimization 2(En-En-1)/(En+En-1) < ON.Etol • ON.RcLWF Localisation radius of WFs

  29. Order-N in SIESTA (4) • ON.Eta (energy units) Chemical Potential (Kim) Shift of Hamiltonian (Ordejon-Mauri) • ON.ChemicalPotential • ON.ChemicalPotentialUse • ON.ChemicalPotentialRc • ON.ChemicalPotentialTemperature • ON.ChemicalPotentialOrder

  30. 1 0 Emin EF Emax Fermi operator/projector Goedecker & Colombo (1994) f(E) = 1/(1+eE/kT)  n cn En F   cn Hn Etot = Tr[ F H ] Ntot = Tr[ F ] ^ ^ ^ ^ ^

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