A Green Function Method for Large Scale Electronic Structure Calculations. Rudolf Zeller

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1 A Green Function Method for Large Scale Electronic Structure Calculations Rudolf Zeller Institute for Advanced Simulation, Forschungszentrum Jülich Electronic structure calculations (density functional theory) Large scale systems (up to atoms) Green function (the classical one): George Green Our methodology and experience with more than processors

2 Electronic structure calculations Born-Oppenheimer approximation (fixed nuclei) Ĥ = N 2 i + i ĤΨ = EΨ N i<j e 2 r i r j 1 + many electron Schrödinger equation N N v ext (r i ) i Density functional theory Hohenberg-Kohn theorem: all properties are determined by density n(r) v ext (r) Ψ n(r) v ext (r). ground-state energy E 0 = min n E[n(r)]

3 Auxiliary Kohn-Sham system no interaction e 2 = 0 identical density provided by another potential v eff (r) single-particle equations [ ] 2 r + v eff(r) Ψ i (r) = E i Ψ i (r) Publications / Year DFT Publications n(r) = i Ψ i (r) Year from

4 Large scale systems Ge growth on Si biological molecules Gillan et al., CPC 2007 (CONQUEST) Ozaki, PRB 2006 (OPENMX)

5 Large scale systems JUGENE Jaguar XT5

6 Solution methods [ 2 r + V eff(r) E i ]Ψ i (r) = 0 basis function methods Ψ i (r) = c ν ϕ ν (r) ν orthonormality drϕ ν (r)ϕ ν (r) = δ ν,ν algebraic eigenvalue problem [H ν,ν Eδ ν,ν ] c ν = 0 ν about 80 to 2000 basis functions ϕ ν (r) per atom for instance, for atoms: matrix dimension to

7 Instead of use Green function method ν [H ν,ν E] c ν = 0 ν [H ν,ν E] G ν,ν (E) = δ ν,ν no eigenvalue solver, but linear equations solver required n(r) = 2 EF π Im de ϕ ν (r)g ν,ν(e)ϕ ν (r) why Green function matrix? seems to be much more work immediate difficulty: structure of integrand ν Density of states (1/eV) Cu Energy (ev)

8 Complex energy integration iγ z = E + iγ E F E n(r) = 2 π Im i w i G ( r, r, E i ) Solid State Comm. 44, 993, (1982)

9 Answer: none is necessary Question: which choice of basis set? Instead of [ 2 r + v eff(r) E i ]Ψ i (r) = 0 use [ ] 2 r + v eff(r) E G(r, r, E) = δ(r r ) immediate difficulty: G(r, r, E) is function of seven variables very strong structure introduced by δ function

10 Concept of reference systems instead of differential equation use integral equation G(r, r, E) = G r (r, r, E) + G r (r, r, E) [v eff (r ) v r (r )] G(r, r, E)dr with known Green function G r of a suitably chosen reference system free space reference system with potential v r (r) = 0 Green function is known analytically repulsive muffin-tin reference system Green function can be calculated easily with high precision

11 Green function of a complicated system G = G r + G r [V V r ]G by successive calculation of Green functions of simpler systems 1. free space ideal crystal 2. ideal crystal crystal with surface 3. crystal with surface adsorbate atoms Advantages 1. the region, where the potential changes, is exactly embedded in a simpler system 2. integral equation is restricted in space to the region, where the potential changes 3. all interactions, e.g., between adsorbed clusters and the substrate are included

12 Example: vicinal surface with nanowires x 1. Green function of free space 2. periodic bulk system (three-dimensional periodicity) 3. two-dimensional perturbation (empty layers)

13 Further examples one-dimensional perturbation zero-dimensional perturbation

14 Multiple scattering theory based on free space Green function G 0 (r, r, E) = 1 and on Gegenbauer s addition theorem 1875 e i E r r 4π r r. Lord Rayleigh 1892 Korringa 1947 Kohn and Rostoker 1954 KKR Green function version (developed in Jülich since 1978 with Dederichs and many other)

15 KKR Green function method G(r + R n, r + R n ) = δ nn G n s (r, r ) + R n L (r)gnn LL R n L (r ) LL divide space into cells n solve single-cell problems G n s (r, r ) = G 0 (r, r ) + R n L (r) = J L(r) + use matrix equation G nn LL n n = G r,nn LL + t n LL = n G 0 (r, r )V (r )G n s (r, r )dr G 0 (r, r )V (r )R n L (r )dr G r,nn LL n L L t n L L G n n L L R r,n L (r) V (r)rn L (r)dr x single-cell problems can be solved in parallel with O(N) work matrix equation is independent of the radial resolution used angular variation leads to L, L indices, usually 16 or 25 enough

16 Accuracy in comparison with FLAPW results lattice constant [a.u.] Al Fe Ni Cu Rh Pd Ag FPKKR FLAPW bulk modulus [Mbar] Al Fe Ni Cu Rh Pd Ag FPKKR FLAPW from Asato et al., PRB 60, 5202 (1999)

17 Forces and relaxations Displacement of first NN (%) comparison with experiment EXAFS Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge displacement of nearest Cu neighbour atoms around impurities comparison with ab-initio plane-wave results Si 7.2 (7.5)% Si In 14.1 (13.4)% D In Sb Si Si 0.8 (1.1) % 8.3 (8.2)% 1.0 (1.3)% Si In 1.8 (0.6)% 2.6 (2.8)% Si In 4.8 (4.5)% 6.5 (7.0)% P As 6.1 (5.7)% Si 5.3 (5.1)% 6.7 (6.5)% indium-donor complexes in Si Si

18 Large scale density functional calculations conventional techniques are limited to about several hundred atoms bottleneck: computational effort scales cubically with system size x principles of linear scaling methods (some exist now) reduce effort by tolerating small loss of precision exploit locality of potential and quasi-locality of 2 r by using local basis sets sparse matrices use iterative methods to benefit from sparse matrices nearsightedness of electronic matter (Prodan and Kohn, PNAS 2005) negligible effects of potential changes far away

19 . Linear strategy in KKR Green function method. use repulsive reference system exponentially decaying matrix elements neglect small matrix elements aiming at 1 mev error for total energy ṣimple iterations G (i+1) = G r + G r tg (i) do not work but QMR iterations (Freund and Nachtigal) work very well important gain by iteration: O(N 2 ) computational effort, O(N) storage easy parallelization over the atoms and L components works for insulating, semiconducting, metallic materials

20 How many iterations are needed? depends on size and chosen temperature T 5000 Number of iterations Pd 400 K 800 K 1600 K Number of atoms. curves fitted to n it = n it n(r) = 1 π Im αe γn 1/3 f(e, T )G(r, r, E)dE f(e, T ) = Fermi-Dirac function

21 KKRnano x our newly developed code (with Alexander Thiess, GRS, FZJ) why KKRnano? nanosystems contain many atoms (8000 in a cube of 6 nm length) accuracy (in mev) scaling behaviour. present applications x MgO: 1-10 % N, C J ij T c GaN:Gd with 1-5 % interstitial N, O (LDA+U) T c

22 How to use more processors than atoms? KKRnano uses four levels of parallelization with MPI groups and communicators and point-to-point and collective messages. parallelization over atoms (is efficient) parallelization over two spin directions (is trivial and efficient) parallelization over energy points (2 or 3 panels dynamically load balanced) parallelization over L components (until now only in matrix equation) optionally MPI or OPENMP speedup test system: NiPd number of processors

23 How to achieve linear scaling n truncate: G nn LL = 0 for R n R n > r cut G decays like the density matrix truncation leads to O(1) storage truncation leads to O(N) computing time processors ( ACC. A RC A CR A RR x use X (i+1) C ) ( X (i) C 0 = A CC X (i) C ) = ( A CC X (i) C A CR X (i) R and X(i+1) R = 0 ) time per iteration (s) Ni x Pd 1-x x=3% number of atoms

24 Truncation error for total energy 6 Cu 2 Pd Total energy error (mev) K 800 K 1600 K Total energy error (mev) K 800 K 1600 K Number of atoms Number of atoms computational details: model systems supercell with identical atoms in fcc geometry energy without truncation error was obtained for a small cubic cell with 5984 k points this corresponds to one special point k = 2π a (1 4, 1 4, 1 ) in the large supercell 4

25 KKRnano for a realistic system 4e+06 error in total energy per atom [mev] Mg 500 O 475 N 25 energy error per atom number of atoms in truncation zone total number of iterations 3e+06 2e+06 1e+06 0 energy error per atom Mg 500 O 475 N number of atoms in truncation zone for large truncation region (local interaction zone) concepts in KKRnano are similar to LSMS code (Oak Ridge)

26 How to improve the efficiency? preconditioning instead of (1 G r t)g = G r solve P (1 G r t)g = P G r if P (1 G r t) 1 fewer iterations required good initialization use results of previous self-consistency step. problems traditional preconditioners like incomplete LU are difficult to parallelize if many iterations are needed, good initialization is difficult

27 Multilevel block-circulant preconditioner developed together with Alexander Thiess (Jülich) Matthias Bolten (Wuppertal) and Irad Yavneh (Haifa) nz = structure of matrix eigenvalues of P (1 G r t).

28 Initialization with previous self-consistency step

29 Summary. 1. Accuracy of the KKR-GF method in general and for O(N) calculations 2. Key concepts for linear-scaling DFT calculations: sparse Hamiltonians (matrices) iterative solutions nearsightedness of electronic matter 3. Implementation in KKR Green function method by: repulsive reference system complex energy integration and QMR method local truncation of the Green function 4. Our code KKRnano for arbitrarily shaped supercells with thousands of atoms runs efficiently with many thousand processors 5. All electrons are treated, linear scaling effort possible also for metals 6. Fully relativistic extension (Dirac equation) is planned