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You Need to Know About Bigdft Basis-Set Solving KS Equations

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You Need to Know About Bigdft Basis-Set Solving KS Equations BigDFT hands-on 2012 OAK RIDGE,TENNESSEE, USA Introduction to BigDFT Wavelets All you need to know about BigDFT Basis-set Solving KS equations Damien Caliste, Luigi Genovese & Thierry Deutsch Manual input.xxx Compile L_Sim - CEA Grenoble Run 8 February 2012 Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim Damien Caliste, Luigi Genovese & Thierry Deutsch Outline 1 Introduction to the wavelet calculations Definition of the basis-set Structure of the calculation Introduction to BigDFT The electronic minimisation loop The input guess Wavelets Basis-set Possible add-on calculations after fix point is reached Solving KS equations 2 A quick look from the user point of view Manual The input parameters and atomic data input.xxx Compile How to compile BigDFT Run How to run BigDFT Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim Damien Caliste, Luigi Genovese & Thierry Deutsch A DFT code based on Daubechies wavelets BigDFT: a PSP Kohn-Sham code A Daubechies wavelets basis has unique properties for DFT usage Systematic, Orthogonal Introduction to BigDFT Localised, Adaptive Wavelets Basis-set Kohn-Sham operators are analytic Solving KS equations Daubechies Wavelets Short, Separable convolutions 1.5 φ(x) ~c` = ∑ aj c`−j 1 Manual j ψ(x) input.xxx 0.5 Compile Peculiar numerical properties Run 0 Real space based, highly flexible -0.5 -1 Big & inhomogeneous systems -1.5 -6 -4 -2 0 2 4 6 8 x Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim Damien Caliste, Luigi Genovese & Thierry Deutsch Wavelet properties: multi-resolution Example of two resolution levels: step function (Haar Wavelet) Scaling Functions: Multi-Resolution basis Low and High resolution functions related each other Introduction to BigDFT = + Wavelets Basis-set Solving KS equations Wavelets: complete the low resolution description Defined on the same grid as the low resolution functions Manual input.xxx Compile Run 1 1 = 2 + 2 Scaling Function + Wavelet = High resolution We increase the resolution without modifying the grid spacing Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim Damien Caliste, Luigi Genovese & Thierry Deutsch Going to a 3D basis-set On a discretised mesh [i;j;k] for one reslution: Nx Ny Nz f (i;j;k) = ∑ cl;m;nΦl (i)Φm (j)Φn (k) l;m;n Introduction Schematic view for a multi-resolution grid in 2D to BigDFT Wavelets Basis-set Solving KS equations Manual input.xxx Compile Run Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim Damien Caliste, Luigi Genovese & Thierry Deutsch Wavelet properties: adaptivity Adaptivity Resolution can be refined following the grid point. The grid is divided in Introduction Low (1 DoF) and High (8 to BigDFT DoF) resolution points. Wavelets Basis-set Points of different resolution Solving KS equations belong to the same grid. Empty regions must not be Manual “filled” with basis functions. input.xxx Compile Run Localization property,real space description Optimal for big & inhomogeneous systems, highly flexible Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim Damien Caliste, Luigi Genovese & Thierry Deutsch Overview of wavelet families used in BigDFT code Daubechies f (x) = ∑` c`f`(x) Interpolating f (x) = ∑j fj jj (x) Orthogonal set Dual to dirac deltas R c` = dx f`(x)f (x) fj = f (j) LEAST ASYMMETRIC DAUBECHIES-16 INTERPOLATING (DESLAURIERS-DUBUC)-8 1.5 1 Introduction scaling function scaling function wavelet wavelet 1 to BigDFT 0.5 0.5 0 0 Wavelets -0.5 Basis-set -0.5 -1 Solving KS -1.5 -1 equations -6 -4 -2 0 2 4 6 -4 -2 0 2 4 No need to calculate overlap The expansion coefficients are Manual matrix of basis functions the point values on a grid input.xxx Compile Used for wavefunctions, scalar Used for charge density, Run products function products Magic Filter method (A.Neelov, S. Goedecker) The passage between the two basis sets can be performed without losing accuracy Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim Damien Caliste, Luigi Genovese & Thierry Deutsch Basis set features BigDFT features in a nutshell 4 Arbitrary absolute precision can be achieved Good convergence ratio for real-space approach (O(h14)) 4 Optimal usage of the degrees of freedom (adaptivity) Introduction Optimal speed for a systematic approach (less memory) to BigDFT 4 Hartree potential accurate for various boundary Wavelets Basis-set conditions Solving KS equations Free and Surfaces BC Poisson Solver (present also in CP2K, ABINIT, OCTOPUS) Manual * Data repartition is suitable for optimal scalability input.xxx Compile Simple communications paradigm, multi-level parallelisation Run possible (and implemented) Improve and develop know-how Optimal for advanced DFT functionalities in HPC framework Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim Damien Caliste, Luigi Genovese & Thierry Deutsch The DFT framework The ground state is obtained by solving: Schrödinger equation HΨ = EΨ Introduction Expressed only for the electrons, using the to BigDFT Born-Oppenheimer approximation ; Wavelets Basis-set Solved in the framework of Kohn-Sham, using one Solving KS equations particule wavefunctions ; Use of a self-consistency loop since the Hamiltonian H Manual depends on Ψ. input.xxx Compile Run What about wavelets? Wavelets are the basis set for the representation of the wavefunctions. Thus things like XC approximation or Hellmann-Feynman theorem are still usable. Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim Damien Caliste, Luigi Genovese & Thierry Deutsch Elements of the Hamiltonian H All classical parts present in plane-waves, can also be treated with wavelets. The kinetic operator ∇2Ψ Computed from the wavefunctions Ψ, applying a local filter. Introduction to BigDFT Length for Daubechies is 28, see isf_to_daub_kinetic(). Wavelets Basis-set Finite differences in 1D: Solving KS f (xn−1)−2f (x)+f (xn+1) equations ∆f (xn) = h2 is a filter of (1;−2;1). Manual 14 input.xxx Precision is O(h ) for arbitrary function. Even exact for Compile Run linear combination of Daubechies functions: 2 f (x) = ∑c`f`(x); ∇ f (x) = ∑~c` f`(x) ; ` ` Z 2 ~c` = ∑cj a`−j ; a` ≡ f0(x)¶x f`(x) ; j Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim Damien Caliste, Luigi Genovese & Thierry Deutsch Elements of the Hamiltonian H The non-local part of pseudo-potentials Computed from the scalar product of wavefunctions Ψ and the projectors, in routine applyprojectorsonthefly(). Introduction to BigDFT ` (`) `;m `;m Wavelets Vnonloc = ∑∑ ∑ hij jpi ihpj j Basis-set ij Solving KS ` m=−` equations 1 r `;m (`) − 2 r` hrjpi i = Y`;m(q;j)fi (r`)e Manual input.xxx Compile This is possible for GTH and HGH pseudo-potentials because Run of the spatial separability in their analytic expression. Ongoing developments are done to include PAW pseudo-potentials. Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim Damien Caliste, Luigi Genovese & Thierry Deutsch Elements of the Hamiltonian H The local potentials, VH Vxc Vpsploc & Vionic + Hartree and ionic potentials are computed with a Poisson solver from the density r(~j). Introduction R r(~x) to BigDFT V (~j) = d~x , V (~j) with LibXC routines and H j~x−~jj xc Z Wavelets ~ R k Vionic(j) = ∑k d~x Basis-set j~x−~jj Solving KS equations This Poisson solver is: Manual 4 very fast and accurate, with optimal parallelisation ; input.xxx Compile 4 can be used independently from the DFT code ; Run 4 integrated quantities (energies) are easy to extract ; 4 correct treatment of isolated boundary conditions and of charge offset ; 8 non-adaptive, needs data uncompression. Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim Damien Caliste, Luigi Genovese & Thierry Deutsch Finding the fix point in Schrödinger equation Two possible methods: the diagonalisation and the direct minimisation. The latter is fast and simple but restricted to systems with a non-zero HOMO-LUMO gap. Direct minisation (systems with gap) Introduction to BigDFT Classical gradient algorithms are applied for electronic Wavelets minimisation: steepest descent or DIIS. Basis-set Solving KS equations Diagonalisation (metallic systems) Done within two loops, the inner one, non self consistent to Manual input.xxx find wavefunctions of a given hamiltonian, the outer one Compile Run mixing either potential or density. To improve convergency, preconditioning is done on the gradients HΨ using the routine preconditionall(). 1 2 ~ ~ ~ ( 2 ∇ + C)p = g p is computed in an iterative way (CG). Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim Damien Caliste, Luigi Genovese & Thierry Deutsch The input guess, take a good start! On the contrary to plane waves, wavelet representation requires a good input start for the minimisation loop. Using atomic orbitals Introduction to BigDFT The input guess is based on atomic orbitals, using the following scheme: Wavelets Basis-set Solving KS project gaussian atomic orbitals in a Daubechies equations wavelet basis set (see the inputguess_gaussian_orbitals() routine) ; Manual input.xxx compute the associated hamiltonian ; Compile Run diagonalise it in this basis set ; use the lowest states as input for the calculation. This is done in input_wf_diag() routine. Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim Damien Caliste, Luigi Genovese & Thierry Deutsch Additional computation after minimisation Getting the Kohn-Sham wavefunctions A diagonalisation is done on output of the minimisation loop by the last_orthon() routine. Compute the forces Introduction to BigDFT The forces are sum of three parts: Wavelets Basis-set
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