BigDFT tutorial. BigDFT tutorial FIRST YARMOUK SCHOOL. Thierry Deutsch. November, L_Sim - CEA Grenoble. Introduction DFT.

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1 Big tutorial FIRST YARMOUK SCHOOL Big tutorial Thierry Deutsch L_Sim - CEA Grenoble November,

2 Outline 1 2 Density Functional Theory (quick view) 3 4

3 A basis for nanosciences: the Big project STREP European project: Big( ) Four partners, 15 contributors: CEA-INAC Grenoble, U. Basel, U. Louvain-la-Neuve, U. Kiel Aim: To develop an ab-initio code based on Daubechies Wavelets, to be integrated in ABINIT, distributed freely (GNU-GPL license) L. Genovese, A. Neelov, S. Goedecker, T. Deutsch, et al., Daubechies wavelets as a basis set for density functional pseudopotential calculations, J. Chem. Phys. 129, (2008)

4 Big version 1.3: capabilities Free, surface and periodic boundary conditions Local geometry optimizations (with constraints) Global geometry optimization Saddle point searches Vibrations External electric fields Born-Oppenheimer MD Unoccupied KS orbitals Collinear and Non-collinear magnetism All XC functionals of the ABINIT package Empirical van der Waals interactions Also available from within the ABINIT package

5 Big version 1.4 (June 2010) Better performance for periodic systems (convolutions) Exact exchange Hybrid functionals (libxc, G2-1 tests) Molecular Dynamics from ABINIT K points (periodic) Symmetries from ABINIT Preliminary version for XANES version with S_ Big/ABINIT training workshop: June 30 Juny 1

6 Spirit of Big Simple for developers: no parser, use input files with mandatory parameters Few input files Optional input files as input.kpt (for periodic systems) or input.perf Few input parameters ex. not parameters for parallel calculations

7 Big: Executables bigdft Single point calculation, geometry optimisation, molecular dynamics Execute simply: bigdft memguess number_of_cores Check input files and determine the memory amount frequencies Vibration properties abscalc XANES calculations bart Big + ART (global minima search) NEB Determination of saddle point

8 Big: input files posinp.xyz Atomic positions + boundary conditions input.dft Everything about calculations psppar.n file with the name of atom (ABINIT format) Optional files input.geopt Geometry optimisation or molecular dynamics parameters input.freq Frequencies calculations input.perf Options for performances input.kpt K-points mesh for periodic systems

9 Atomic positions: posinp.xyz or posinp.ascii units reduced, angstroem, bohr or atomic Boundary conditions periodic, surface 8 reduced periodic Si Si Si Si Si Si Si Si

10 Atomic positions: posinp.xyz or posinp.ascii units reduced, angstroem, bohr or atomic Boundary conditions periodic, surface 3 angstroem surface O H H

11 Atomic positions: posinp.xyz or posinp.ascii units reduced, angstroem, bohr or atomic Boundary conditions periodic, surface 5 atomic Si H H H H

12 Kohn-Sham formalism H-K theorem: E is an unknown functional of the density E = E[ρ] Density Functional Theory Kohn-Sham approach Mapping of an interacting many-electron system into a system with independent particles moving into an effective potential. Find a set of orthonormal orbitals Ψ i (r) that minimizes: E = 1 2 N/2 i=1 Ψ i (r) 2 Ψ i (r)dr + 1 ρ(r)v H (r)dr 2 + E xc [ρ(r)] + V ext (r)ρ(r)dr N/2 ρ(r) = 2 i=1 Ψ i (r)ψ i (r) 2 V H (r) = 4πρ(r)

13 Performing a calculation A self-consistent equation ρ(r) = Ψ i (r)ψ i (r), where ψ i satisfies i ( 12 ) 2 + V H [ρ] + V xc [ρ] + V ext + V pseudo ψ i = E i ψ i, (Kohn-Sham) Ingredients An XC potential, functional of the density several approximations exists (LDA,GGA,... ) A choice of the pseudopotential (if not all-electrons) (norm conserving, ultrasoft, PAW,... ) A basis set for expressing the ψ i An (iterative) algorithm for finding the wavefunctions ψ i A (good) computer...

14 Exchange-Correlation Energy (input.dft) E xc [ρ] = K exact [ρ] K KS [ρ] + V e e exact [ρ] V Hartree [ρ] K KS [ρ]: Kinetic energy for a non-interaction electron gas Kohn-Sham formalism is exact (mapping of an interacting electron system into a non-interacting system) Many types from ABINIT: Use same codes (ixc: LDA=1, PBE=11, see manual) libxc: code for exchange code for correlation ( see manual) Hybrid functionals from libxc

15 Performing a calculation A self-consistent equation ρ(r) = Ψ i (r)ψ i (r), where ψ i satisfies i ( 12 ) 2 + V H [ρ] + V xc [ρ] + V ext + V pseudo ψ i = E i ψ i, (Kohn-Sham) Ingredients An XC potential, functional of the density several approximations exists (LDA,GGA,... ) A choice of the pseudopotential (if not all-electrons) (norm conserving, ultrasoft, PAW,... ) A basis set for expressing the ψ i An (iterative) algorithm for finding the wavefunctions ψ i A (good) computer...

16 Gaussian type separable s (HGH) Local part V loc (r) = Z [ ] [ ion r erf + exp 1 ( ) ] 2 r r 2rloc 2 r loc { ( ) 2 ( ) 4 ( r r r C 1 + C 2 + C 3 + C 4 r loc r loc Nonlocal (separable) part H( r, r ) H sep ( r, r ) = 2 i=1 r loc Y s,m (ˆr) p s i (r) h s i p s i (r ) Ys,m(ˆr ) m + Y p,m (ˆr) p p 1 (r) hp 1 pp 1 (r ) Yp,m(ˆr ) m ) 6 } p l 1(r) = 2 r l e 1 2 ( r ) 2 r l p Γ(l l r l+2 e 1 2 ( r ) 2 r l 2(r) = ) Γ(l + 7) 2 r l+ 3 2 l r l+ 7 2 l

17 The nonlocal part of the pseudopotential H sep Ψ = P >< P Ψ > If the expansion coefficients c J of P in a Daubechies basis are known then ( )( ) < P Ψ >= c I φ(x I) Ψ J φ(x J) dx = c J Ψ J I J J Because a Gaussian is separable and a 3-dim scaling basis is a product of 1-dim scaling functions/wavelets, c i,j,k can easily be calculated with machine precision. c i,j,k = φ k i (x)φ k j (y)φ k k(z)exp( (x 2 + y 2 + z 2 )) dx dy dz = φ k i (x)exp( x 2 ) dx φ k i (y)exp( y 2 ) dy φ k i (z)exp( z 2 ) dz

18 file: psppar.n In the same directory as the calculation. Large library: Accurate and transferable Goedecker pseudopotential for N zatom,zion,pspdat pspcod,pspxc,lmax,lloc,mmax,r2well rloc nloc c1 c2 2 nnonloc rs ns hs rp np hp kp11 Save memory: projectors calculated on-the-fly (cost < 1%, experimental option)

19 Performing a calculation A self-consistent equation ρ(r) = Ψ i (r)ψ i (r), where ψ i satisfies i ( 12 ) 2 + V H [ρ] + V xc [ρ] + V ext + V pseudo ψ i = E i ψ i, (Kohn-Sham) Ingredients An XC potential, functional of the density several approximations exists (LDA,GGA,... ) A choice of the pseudopotential (if not all-electrons) (norm conserving, ultrasoft, PAW,... ) A basis set for expressing the ψ i An (iterative) algorithm for finding the wavefunctions ψ i A (good) computer...

20 s in real space: Wavelet basis sets Properties of wavelet basis sets: localized both in real and in Fourier space allow for adaptivity (for core electrons) are a systematic basis set φ(x) ψ(x) x A basis set both adaptive and systematic, real space based

21 A brief description of wavelet theory Two kind of basis functions A Multi-Resolution real space basis The functions can be classified following the resolution level they span. Scaling Functions The functions of low resolution level are a linear combination of high-resolution functions = + m φ(x) = h j φ(2x j) j= m Centered on a resolution-dependent grid: φ j = φ 0 (x j).

22 A brief description of wavelet theory Wavelets They contain the DoF needed to complete the information which is lacking due to the coarseness of the resolution. = φ(2x) = m j= m h j φ(x j) + m j= m g j ψ(x j) Increase the resolution without modifying grid space SF + W = same DoF of SF of higher resolution ψ(x) = m j= m g j φ(2x j) All functions have compact support, centered on grid points.

23 Adaptivity of the mesh Adaptivity Resolution can be refined following the grid point. The grid is divided in: Data can thus be stored in compressed form. Low resolution pts (SF, 1 DoF) High resolution pts (SF + W, 8 DoF) Points of different resolution belong to the same grid. Localization property Empty regions must not be filled with basis functions. Optimal for big inhomogeneous systems, O(N) approach.

24 Adaptivity of the mesh Atomic positions (H 2 O), hgrid

25 Adaptivity of the mesh Fine grid (high resolution, frmult)

26 Adaptivity of the mesh Coarse grid (low resolution, crmult)

27 input.dft: hgrid, crmult, frmult hgrid Step grid: twice as bigger than the lowest gaussian coefficient of pseudopotential Roughly 0.45 (only orthorhombic mesh) crmult Extension of the coarse resolution (factor 6.): depends on the extension of the HOMO for the atom frmult Extension of the fine resolution (factor 8.): depends on the pseudopotential coefficient In practice, we use only hgrid from 0.3 to 0.55.

28 No integration error Orthogonality, scaling relation dx φ k (x)φ j (x) = δ kj φ(x) = 1 2 m j= m h j φ(2x j) The hamiltonian-related quantities can be calculated up to machine precision in the given basis. The accuracy is only limited by the basis set (O ( h 14 grid) ) Exact evaluation of kinetic energy Obtained by convolution with filters: f(x) = c l φ l (x), 2 f(x) = c l φ l (x), l l c l = c j a l j, a l φ 0 (x) 2 xφ l (x), j

29 Separability in 3D The 3-dim scaling basis is a tensor product decomposition of 1-dim Scaling Functions/ Wavelets. φ e x,e y,e z j x,j y,j z (x,y,z) = φ e x j x (x)φ e y j y (y)φ e z j z (z) With (j x,j y,j z ) the node coordinates, φ (0) j and φ (1) j the SF and the W respectively. Gaussians and wavelets The separability of the basis allows us to save computational time when performing scalar products with separable functions (e.g. gaussians): Initial wavefunctions (input guess) Poisson solver Non-local pseudopotentials

30 Wavelet families used in Big code Daubechies f(x) = l c l φ l (x) Orthogonal c l = dx φ l (x)f(x) LEAST ASYMMETRIC DAUBECHIES-16 scaling function wavelet Interpolating f(x) = j f j ϕ j (x) Dual to dirac deltas f j = f(j) 1 INTERPOLATING (DESLAURIERS-DUBUC)-8 scaling function wavelet Used for wavefunctions, scalar products Used for charge density, function products Magic Filter method (A.Neelov, S. Goedecker) The passage between the two basis sets can be performed without losing accuracy

31 Performing a calculation A self-consistent equation ρ(r) = Ψ i (r)ψ i (r), where ψ i satisfies i ( 12 ) 2 + V H [ρ] + V xc [ρ] + V ext + V pseudo ψ i = E i ψ i, (Kohn-Sham) Ingredients An XC potential, functional of the density several approximations exists (LDA,GGA,... ) A choice of the pseudopotential (if not all-electrons) (norm conserving, ultrasoft, PAW,... ) A basis set for expressing the ψ i An (iterative) algorithm for finding the wavefunctions ψ i A (good) computer...

32 Kohn-Sham Equations: Operators Apply different operators Having a one-electron hamiltonian in a mean field. [ 1 ] V eff (r)(r) ψ i = ε i ψ i V eff (r) = V ext (r) + V ρ(r ) r r dr + µ xc (r) E[ρ] can be expressed by the orthonormalized states of one particule: ψ i (r) with the fractional occupancy number f i (0 f i 1) : ρ(r) = 2 f i ψ i (r) 2 i

33 Kohn-Sham Equations: Computing Energies Calculate different integrals E[ρ] = K [ρ] + U[ρ] U[ρ] = K [ρ] = m e i V dr ψ i 2 ψ i dr V ext(r)ρ(r)+ 1 ρ(r)ρ(r ) dr dr V 2 V r r }{{} Hartree We minimise with the variables ψ i (r) and f i with the constraint dr ρ(r) = N el. V + E xc [ρ] }{{} exchange correlation

34 KS Equations: Self-Consistent Field Set of self-consistent equations: { 1 } V eff ψ i = ε i ψ i 2 m e with an effective potential: V eff (r) = V ext (r)+ ) dr ρ(r V r r }{{} Hartree δe xc + δρ(r) }{{} exchange correlation and: ρ(r) = 2 i f i ψ i (r) 2

35 KS Equations: Self-Consistent Field Set of self-consistent equations: { 1 } V eff ψ i = ε i ψ i 2 m e with an effective potential: V eff (r) = V ext (r)+ ) dr ρ(r V r r }{{} Hartree δe xc + δρ(r) }{{} exchange correlation and: ρ(r) = 2 i f i ψ i (r) 2 Poisson Equation: V Hartree = ρ (Laplacian: = 2 x y z 2 )

36 KS Equations: Self-Consistent Field Set of self-consistent equations: { 1 } V eff ψ i = ε i ψ i 2 m e with an effective potential: V eff (r) = V ext (r)+ ) dr ρ(r V r r }{{} Hartree δe xc + δρ(r) }{{} exchange correlation and: ρ(r) = 2 i f i ψ i (r) 2 Poisson Equation: V Hartree = ρ (Laplacian: = 2 x y z 2 )

37 KS Equations: Self-Consistent Field Set of self-consistent equations: { 1 } V eff ψ i = ε i ψ i 2 m e with an effective potential: V eff (r) = V ext (r)+ ) dr ρ(r V r r }{{} Hartree δe xc + δρ(r) }{{} exchange correlation and: ρ(r) = 2 i f i ψ i (r) 2 Poisson Equation: V Hartree = ρ (Laplacian: = 2 x y z 2 )

38 KS Equations: Self-Consistent Field Set of self-consistent equations: { 1 } V eff ψ i = ε i ψ i 2 m e with an effective potential: V eff (r) = V ext (r)+ ) dr ρ(r V r r }{{} Hartree δe xc + δρ(r) }{{} exchange correlation and: ρ(r) = 2 i f i ψ i (r) 2 Poisson Equation: V Hartree = ρ (Laplacian: = 2 x y z 2 )

39 KS Equations: Self-Consistent Field Set of self-consistent equations: { 1 } V eff ψ i = ε i ψ i 2 m e with an effective potential: V eff (r) = V ext (r)+ ) dr ρ(r V r r }{{} Hartree δe xc + δρ(r) }{{} exchange correlation and: ρ(r) = 2 i f i ψ i (r) 2 Poisson Equation: V Hartree = ρ (Laplacian: = 2 x y z 2 )

40 Direct Minimisation: Flowchart { ψ i = a c i aφ a } Basis: adaptive mesh (0,...,N φ ) Orthonormalized

41 Direct Minimisation: Flowchart { ψ i = a c i aφ a } inv FWT + Magic Filter Basis: adaptive mesh (0,...,N φ ) Orthonormalized occ ρ(r) = i ψ i (r) 2 Fine non-adaptive mesh: < 8N φ

42 Direct Minimisation: Flowchart { ψ i = a c i aφ a } inv FWT + Magic Filter Basis: adaptive mesh (0,...,N φ ) Orthonormalized occ ρ(r) = i ψ i (r) 2 Fine non-adaptive mesh: < 8N φ Poisson solver V H (r) = V xc [ρ(r)] G(.)ρ V NL ({ψ i }) V effective

43 Direct Minimisation: Flowchart { ψ i = a c i aφ a } inv FWT + Magic Filter Basis: adaptive mesh (0,...,N φ ) Orthonormalized occ ρ(r) = i ψ i (r) 2 Fine non-adaptive mesh: < 8N φ Poisson solver V H (r) = V xc [ρ(r)] G(.)ρ V NL ({ψ i }) V effective Kinetic Term

44 Direct Minimisation: Flowchart { ψ i = a c i aφ a } inv FWT + Magic Filter Basis: adaptive mesh (0,...,N φ ) Orthonormalized FWT occ ρ(r) = i ψ i (r) 2 Fine non-adaptive mesh: < 8N φ Poisson solver V H (r) = V xc [ρ(r)] G(.)ρ V NL ({ψ i }) FWT δca i = E total c i (a) + Λ ij ca j j V effective Kinetic Term Λ ij =< ψ i H ψ j >

45 Direct Minimisation: Flowchart { ψ i = a c i aφ a } inv FWT + Magic Filter Basis: adaptive mesh (0,...,N φ ) Orthonormalized FWT occ ρ(r) = i ψ i (r) 2 Fine non-adaptive mesh: < 8N φ Poisson solver V H (r) = V xc [ρ(r)] G(.)ρ V NL ({ψ i }) δca i = E total c i (a) + Λ ij ca j j c new,i a FWT preconditioning = c i a + h step δc i a V effective Kinetic Term Λ ij =< ψ i H ψ j > Steepest Descent, DIIS

46 Direct Minimisation: Flowchart { ψ i = a c i aφ a } inv FWT + Magic Filter Basis: adaptive mesh (0,...,N φ ) Orthonormalized FWT occ ρ(r) = i ψ i (r) 2 Fine non-adaptive mesh: < 8N φ Poisson solver V H (r) = V xc [ρ(r)] G(.)ρ V NL ({ψ i }) Stop when δc i a small δca i = E total c i (a) + Λ ij ca j j c new,i a FWT preconditioning = c i a + h step δc i a V effective Kinetic Term Λ ij =< ψ i H ψ j > Steepest Descent, DIIS

47 input.dft: idsx DIIS history Use DIIS algorithm Direct Invesion of Iterative Subspace Use idsx=6 previous iterations Memory-consuming (decrease if too big=) Steepest descent if idsx=0 itermax=50 Specify maximum number of iterations Typically 15 iterations to converge gnrm_cv=1.e-4 convergence criterion Residue δψ 2 < gnrm_cv

48 Description of the file input.dft hx, hy, hz crmult, frmult, coarse and fine radius 1 ixc: XC parameter (LDA=1,PBE=11) ncharge: charge of the system, Electric field 1 0 nspin=1 non-spin polarization, total magnetic moment 1.E-04 gnrm_cv: convergence criterion gradient itermax, nrepmax 7 6 # CG (preconditioning), length of diis history 0 dispersion correction functional (0, 1, 2, 3) write "CUDA" to use (.config file needed) 0 F 0 InputPsiId, output_wf, output_grid length of the tail, # tail CG iterations davidson treatment, # virtual orbitals, # plotted orbitals 2 verbosity of the output 0=low, 2=high T disable the symmetry detection

49 Description of the file input.dft hx, hy, hz crmult, frmult, coarse and fine radius 1 ixc: XC parameter (LDA=1,PBE=11) ncharge: charge of the system, Electric field 1 0 nspin=1 non-spin polarization, total magnetic moment 1.E-04 gnrm_cv: convergence criterion gradient itermax, nrepmax 7 6 # CG (preconditioning), length of diis history 0 dispersion correction functional (0, 1, 2, 3) write "CUDA" to use (.config file needed) 0 F 0 InputPsiId, output_wf, output_grid length of the tail, # tail CG iterations davidson treatment, # virtual orbitals, # plotted orbitals 2 verbosity of the output 0=low, 2=high T disable the symmetry detection

50 Description of the file input.dft hx, hy, hz crmult, frmult, coarse and fine radius 1 ixc: XC parameter (LDA=1,PBE=11) ncharge: charge of the system, Electric field 1 0 nspin=1 non-spin polarization, total magnetic moment 1.E-04 gnrm_cv: convergence criterion gradient itermax, nrepmax 7 6 # CG (preconditioning), length of diis history 0 dispersion correction functional (0, 1, 2, 3) write "CUDA" to use (.config file needed) 0 F 0 InputPsiId, output_wf, output_grid length of the tail, # tail CG iterations davidson treatment, # virtual orbitals, # plotted orbitals 2 verbosity of the output 0=low, 2=high T disable the symmetry detection

51 Description of the file input.dft hx, hy, hz crmult, frmult, coarse and fine radius 1 ixc: XC parameter (LDA=1,PBE=11) ncharge: charge of the system, Electric field 1 0 nspin=1 non-spin polarization, total magnetic moment 1.E-04 gnrm_cv: convergence criterion gradient itermax, nrepmax 7 6 # CG (preconditioning), length of diis history 0 dispersion correction functional (0, 1, 2, 3) write "CUDA" to use (.config file needed) 0 F 0 InputPsiId, output_wf, output_grid length of the tail, # tail CG iterations davidson treatment, # virtual orbitals, # plotted orbitals 2 verbosity of the output 0=low, 2=high T disable the symmetry detection

52 Description of the file input.dft hx, hy, hz crmult, frmult, coarse and fine radius 1 ixc: XC parameter (LDA=1,PBE=11) ncharge: charge of the system, Electric field 1 0 nspin=1 non-spin polarization, total magnetic moment 1.E-04 gnrm_cv: convergence criterion gradient itermax, nrepmax 7 6 # CG (preconditioning), length of diis history 0 dispersion correction functional (0, 1, 2, 3) write "CUDA" to use (.config file needed) 0 F 0 InputPsiId, output_wf, output_grid length of the tail, # tail CG iterations davidson treatment, # virtual orbitals, # plotted orbitals 2 verbosity of the output 0=low, 2=high T disable the symmetry detection

53 Description of the file input.dft hx, hy, hz crmult, frmult, coarse and fine radius 1 ixc: XC parameter (LDA=1,PBE=11) ncharge: charge of the system, Electric field 1 0 nspin=1 non-spin polarization, total magnetic moment 1.E-04 gnrm_cv: convergence criterion gradient itermax, nrepmax 7 6 # CG (preconditioning), length of diis history 0 dispersion correction functional (0, 1, 2, 3) write "CUDA" to use (.config file needed) 0 F 0 InputPsiId, output_wf, output_grid length of the tail, # tail CG iterations davidson treatment, # virtual orbitals, # plotted orbitals 2 verbosity of the output 0=low, 2=high T disable the symmetry detection

54 Description of the file input.dft hx, hy, hz crmult, frmult, coarse and fine radius 1 ixc: XC parameter (LDA=1,PBE=11) ncharge: charge of the system, Electric field 1 0 nspin=1 non-spin polarization, total magnetic moment 1.E-04 gnrm_cv: convergence criterion gradient itermax, nrepmax 7 6 # CG (preconditioning), length of diis history 0 dispersion correction functional (0, 1, 2, 3) write "CUDA" to use (.config file needed) 0 F 0 InputPsiId, output_wf, output_grid length of the tail, # tail CG iterations davidson treatment, # virtual orbitals, # plotted orbitals 2 verbosity of the output 0=low, 2=high T disable the symmetry detection

55 Description of the file input.dft hx, hy, hz crmult, frmult, coarse and fine radius 1 ixc: XC parameter (LDA=1,PBE=11) ncharge: charge of the system, Electric field 1 0 nspin=1 non-spin polarization, total magnetic moment 1.E-04 gnrm_cv: convergence criterion gradient itermax, nrepmax 7 6 # CG (preconditioning), length of diis history 0 dispersion correction functional (0, 1, 2, 3) write "CUDA" to use (.config file needed) 0 F 0 InputPsiId, output_wf, output_grid length of the tail, # tail CG iterations davidson treatment, # virtual orbitals, # plotted orbitals 2 verbosity of the output 0=low, 2=high T disable the symmetry detection

56 Description of the file input.dft hx, hy, hz crmult, frmult, coarse and fine radius 1 ixc: XC parameter (LDA=1,PBE=11) ncharge: charge of the system, Electric field 1 0 nspin=1 non-spin polarization, total magnetic moment 1.E-04 gnrm_cv: convergence criterion gradient itermax, nrepmax 7 6 # CG (preconditioning), length of diis history 0 dispersion correction functional (0, 1, 2, 3) write "CUDA" to use (.config file needed) 0 F 0 InputPsiId, output_wf, output_grid length of the tail, # tail CG iterations davidson treatment, # virtual orbitals, # plotted orbitals 2 verbosity of the output 0=low, 2=high T disable the symmetry detection

57 Description of the file input.dft hx, hy, hz crmult, frmult, coarse and fine radius 1 ixc: XC parameter (LDA=1,PBE=11) ncharge: charge of the system, Electric field 1 0 nspin=1 non-spin polarization, total magnetic moment 1.E-04 gnrm_cv: convergence criterion gradient itermax, nrepmax 7 6 # CG (preconditioning), length of diis history 0 dispersion correction functional (0, 1, 2, 3) write "CUDA" to use (.config file needed) 0 F 0 InputPsiId, output_wf, output_grid length of the tail, # tail CG iterations davidson treatment, # virtual orbitals, # plotted orbitals 2 verbosity of the output 0=low, 2=high T disable the symmetry detection

58 Description of the file input.dft hx, hy, hz crmult, frmult, coarse and fine radius 1 ixc: XC parameter (LDA=1,PBE=11) ncharge: charge of the system, Electric field 1 0 nspin=1 non-spin polarization, total magnetic moment 1.E-04 gnrm_cv: convergence criterion gradient itermax, nrepmax 7 6 # CG (preconditioning), length of diis history 0 dispersion correction functional (0, 1, 2, 3) write "CUDA" to use (.config file needed) 0 F 0 InputPsiId, output_wf, output_grid length of the tail, # tail CG iterations davidson treatment, # virtual orbitals, # plotted orbitals 2 verbosity of the output 0=low, 2=high T disable the symmetry detection

59 Description of the file input.dft hx, hy, hz crmult, frmult, coarse and fine radius 1 ixc: XC parameter (LDA=1,PBE=11) ncharge: charge of the system, Electric field 1 0 nspin=1 non-spin polarization, total magnetic moment 1.E-04 gnrm_cv: convergence criterion gradient itermax, nrepmax 7 6 # CG (preconditioning), length of diis history 0 dispersion correction functional (0, 1, 2, 3) write "CUDA" to use (.config file needed) 0 F 0 InputPsiId, output_wf, output_grid length of the tail, # tail CG iterations davidson treatment, # virtual orbitals, # plotted orbitals 2 verbosity of the output 0=low, 2=high T disable the symmetry detection

60 Description of the file input.dft hx, hy, hz crmult, frmult, coarse and fine radius 1 ixc: XC parameter (LDA=1,PBE=11) ncharge: charge of the system, Electric field 1 0 nspin=1 non-spin polarization, total magnetic moment 1.E-04 gnrm_cv: convergence criterion gradient itermax, nrepmax 7 6 # CG (preconditioning), length of diis history 0 dispersion correction functional (0, 1, 2, 3) write "CUDA" to use (.config file needed) 0 F 0 InputPsiId, output_wf, output_grid length of the tail, # tail CG iterations davidson treatment, # virtual orbitals, # plotted orbitals 2 verbosity of the output 0=low, 2=high T disable the symmetry detection

61 Description of the file input.dft hx, hy, hz crmult, frmult, coarse and fine radius 1 ixc: XC parameter (LDA=1,PBE=11) ncharge: charge of the system, Electric field 1 0 nspin=1 non-spin polarization, total magnetic moment 1.E-04 gnrm_cv: convergence criterion gradient itermax, nrepmax 7 6 # CG (preconditioning), length of diis history 0 dispersion correction functional (0, 1, 2, 3) write "CUDA" to use (.config file needed) 0 F 0 InputPsiId, output_wf, output_grid length of the tail, # tail CG iterations davidson treatment, # virtual orbitals, # plotted orbitals 2 verbosity of the output 0=low, 2=high T disable the symmetry detection

62 s: wavelets self-consistent () Harris-Foulkes functional relativistic pseudopotential norm-conserving all electrons PAW Hybrid functionals beyond LDA GW non-relativistic [ N 3 scaling O(N) methods periodic non-periodic non-collinear collinear +V(r) non-spin polarized spin polarized LDA,GGA ] +µ xc (r) ψ k i = ε k i ψ k i atomic orbitals plane waves augmented real space LDA+U Gaussians Slater numerical finite difference Wavelet

63 Optimal for isolated systems 10 1 Test case: cinchonidine molecule (44 atoms) Plane waves Absolute energy precision (Ha) E c = 40 Ha Wavelets E c = 90 Ha h = 0.4bohr E c = 125 Ha h = 0.3bohr Number of degrees of freedom Allows a systematic approach for molecules Considerably faster than Plane Waves codes. 10 (5) times faster than ABINIT (CPMD) Charged systems can be treated explicitly with the same time

64 Systematic basis set Two parameters for tuning the basis The grid spacing hgrid The extension of the low resolution points crmult

65 Poisson Solver using interpolating scaling function The Hartree potential is calculated in the interpolating scaling function basis. Poisson solver with interpolating scaling functions From the density ρ( j) on an uniform grid it calculates: V H ( j) = d x ρ( x) x j Very fast and accurate, optimal parallelization Can be used independently from the code Integrated quantities (energies) are easy to extract Non-adaptive, needs data uncompression Explicitly free boundary conditions No need to subtract supercell interactions charged systems.

66 Poisson Solver for surface boundary conditions A Poisson solver for surface problems Based on a mixed reciprocal-direct space representation V px,p y (z) = 4π dz G( p ;z z )ρ px,p y (z), Can be applied both in real or reciprocal space codes No supercell or screening functions More precise than other existing approaches Example of the plane capacitor: Periodic Hockney Our approach V V V y y y x x x

67 input.geopt: Relaxation or molecular dynamics DIIS 500 ncount_cluster: max steps during geometry relaxation 1.d0 frac_fluct: geometry optimization stops if force norm less than frac_fluct of noise 0.0d0 randdis: random amplitude for atoms 4.d0 5 betax: steepest descent step size, diis history Different algorithms: DIIS: Direct Inversion in the iterative subspace VSSD: Variable Size Steepest Descent LBFGS: Limited BFGD AB6MD: Use molecular dynamics routines from ABINIT 6

68 input.geopt: Relaxation or molecular dynamics DIIS 500 ncount_cluster: max steps during geometry relaxation 1.d0 frac_fluct: geometry optimization stops if force norm less than frac_fluct of noise 0.0d0 randdis: random amplitude for atoms 4.d0 5 betax: steepest descent step size, diis history Different algorithms: DIIS: Direct Inversion in the iterative subspace VSSD: Variable Size Steepest Descent LBFGS: Limited BFGD AB6MD: Use molecular dynamics routines from ABINIT 6

69 input.geopt: Relaxation or molecular dynamics DIIS 500 ncount_cluster: max steps during geometry relaxation 1.d0 frac_fluct: geometry optimization stops if force norm less than frac_fluct of noise 0.0d0 randdis: random amplitude for atoms 4.d0 5 betax: steepest descent step size, diis history Different algorithms: DIIS: Direct Inversion in the iterative subspace VSSD: Variable Size Steepest Descent LBFGS: Limited BFGD AB6MD: Use molecular dynamics routines from ABINIT 6

70 input.geopt: Relaxation or molecular dynamics DIIS 500 ncount_cluster: max steps during geometry relaxation 1.d0 frac_fluct: geometry optimization stops if force norm less than frac_fluct of noise 0.0d0 randdis: random amplitude for atoms 4.d0 5 betax: steepest descent step size, diis history Different algorithms: DIIS: Direct Inversion in the iterative subspace VSSD: Variable Size Steepest Descent LBFGS: Limited BFGD AB6MD: Use molecular dynamics routines from ABINIT 6

71 input.geopt: Relaxation or molecular dynamics DIIS 500 ncount_cluster: max steps during geometry relaxation 1.d0 frac_fluct: geometry optimization stops if force norm less than frac_fluct of noise 0.0d0 randdis: random amplitude for atoms 4.d0 5 betax: steepest descent step size, diis history Different algorithms: DIIS: Direct Inversion in the iterative subspace VSSD: Variable Size Steepest Descent LBFGS: Limited BFGD AB6MD: Use molecular dynamics routines from ABINIT 6

72 Operations performed Different numerical operations performed: For each wavefunction (Hamiltonian application) Between wavefunctions (Linear algebra) Comput. operations Convolutions with short filters BLAS routines FFT (Poisson Solver) Interpolating Daubechies

73 Massively parallel Two kinds of parallelisation By orbitals (Hamiltonian application, preconditioning) By components (overlap matrices, orthogonalisation) A few (but big) packets of data More demanding in bandwidth than in latency No need of fast network Optimal speedup (eff. 85%), also for big systems Cubic scaling code For systems bigger than 500 atomes (1500 orbitals) : orthonormalisation operation is predominant (N 3 )

74 High Performance Computing Localisation & Orthogonality Data locality Principal code operations can be intensively optimised Optimal for application on supercomputers Little communication, big packets of data No need of fast network Optimal speedup Efficiency of the order of 90%, up to thousands of processors Efficiency (%) at 65 at 173 at 257 at 1025 at (with orbitals/proc) Number of processors Data repartition optimal for material accelerators () Graphic Processing Units can be used to speed up the computation

75 Performance in parallel (Big) Distribution per orbitals (wavelets) Nothing to specify in input files

76 -ported operations in Big (double precision) 20 Convolutions (CUDA rewritten) speedups between and 20 can be 6 obtained for different 4 sections 2 70 speedup (Double prec.) locden locham precond Wavefunction size (MB) speedup (Double prec.) DGEMM DSYRK Number of orbitals Linear algebra (CUBLAS library) The interfacing with CUBLAS is immediate, with considerable speedups

77 Data repartition on a node The S_ approach S_ library manages resource within the node S_ MPI 0 MPI 1 MPI 2 MPI 3 DATA DATA DATA DATA

78 Big code on Hybrid architectures Big code can run on hybrid CPU/ supercomputers In multi- environments, double precision calculations No Hot-spot operations Different code sections can be ported on up to 20x speedup for some operations, 7x for the full parallel code (under improvements) Percent Seconds (log. scale) CPU code Hybrid code (rel.) Number of cores 1 Time (sec) Comm Other PureCPU precond locham locden LinAlg

79 CUDA + S_ Put "CUDA" in input.dft Use a.config file (affinity between CPU and ) USE_SHARED=0 Set if the repartition is shared (1) or static (0) MPI_TASKS_PER_NODE=8 Define the number of mpi tasks per node NUM_=2 Define the number of device per node _CPUS_AFF_0=0,1,2,3 Define the affinity of the processes among the s. _CPUS_AFF_1=4,5,6,7 USE BLAS=1 Use CUDABLAS

80 OpenCL (without S_) Put "OCL" in input.dft No.config needed presently (should be removed also for CUDA) Fermi is better for concurrent access

81 Functionality: Mixed between physics and chemistry approach Order N (real space basis set as wavelets) QM/MM (Quantum Mechanics/Molecular Modeling) Multi-scale approach (more than 1000 atoms feasible for a better parametrization) Numerical experience: One-day simulation Better exploration of atomic configurations Molecular Dynamics (4s per step for 32 water molecules)

82 THE END

83 Future developments, Discussions

84 O(N) approach Versatile boundary conditions (surface, periodic, isolated) Localization regions Wannier functions Easily embedded T CPU Crossover point N

85 Embedding a system (QM/MM) QM/MM Charge defects in bulk materials Semi-infinite system as surface

86 Developments Embedding atoms, O(N) method, region of localisation Many-body: TD-, GW, Bethe-Salpeter PAW pseudopotentials (better accuracy, larger step grid) Mixed basis sets