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Motivations for Bigdft Formalism; Overview of Daubechies Wavelets in DFT

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Motivations for Bigdft Formalism; Overview of Daubechies Wavelets in DFT BigDFT Basics Max Webinar Thierry Deutsch BigDFT code Wavelets BigDFT Precision O(N) BigDFT Motivations for BigDFT formalism; overview of Daubechies wavelets in DFT Thierry Deutsch Laboratoire de Simulation Atomistique - L_Sim November 12, 2020 Virtual Room Outline BigDFT Basics Thierry Deutsch BigDFT code Wavelets Precision O(N) BigDFT BigDFT code Properties of wavelets Precision of the description O(N) BigDFT Basis sets for DFT calculations BigDFT Basics Thierry Deutsch BigDFT code Wavelets Plane Waves Gaussians, Slater Orbitals Precision O(N) BigDFT ABINIT, CPMD, VASP, . CP2K,Gaussian,AIMPRO, . Systematic convergence Real space localized 4 Accuracy increases 4 Small number of basis with the number of functions for moderate basis elements accuracy 4 Non-localised, 4 Well suited for optimal for periodic molecules and other systems open structures 8 Non adaptive 8 Non systematic FFT Analytic functions Robust, easy to Kinetic and overlap matrices are parallelise calculated analytically Origin of the BigDFT project BigDFT Basics Thierry Deutsch BigDFT code STREP European project: BigDFT(2005-2008) Wavelets Precision (N) BigDFT In the beginning: Four partners, 15 people O Now: around 10 active developers, Grenoble, Basel, Barcelona, London, Uppsala, Kobe Used in production since twelve years. Aim: To develop an ab-initio DFT code based on Daubechies Wavelets, to be integrated in ABINIT. BigDFT 1.0 −! January 2008 Why have we done this? • Test the potential advantages of a new formalism * A lot of outcomes and interesting results • Future opportunities and ideas A DFT code based on Daubechies wavelets BigDFT Basics Thierry Deutsch BigDFT code Wavelets Wavelets Precision O(N) BigDFT A basis with optimal properties for expanding localised information • Localised in real space • Smooth (localised in Fourier space) • Orthogonal basis • Daubechies Wavelets Multi-resolution basis 1.5 φ(x) • Adaptive 1 ψ(x) • Systematic 0.5 0 From early 80’s -0.5 -1 Applied in several domains -1.5 -6 -4 -2 0 2 4 6 8 Interesting properties for DFT x A brief description of wavelet theory BigDFT Basics Thierry Deutsch BigDFT code Wavelets A Multi-Resolution real space basis Precision O(N) BigDFT All functions w/ compact support, centered on grid points. In the wavelet theory we have two kind of basis functions. Wavelets (W) Scaling Functions (SF) Contain the DoF needed to The functions of low resolution complete the information lacking level are a linear combination due to the coarseness of the of high-resolution functions. resolution. = + 1 1 = 2 + 2 Increase the resolution without modifying grid space SF + W = Degrees of Freedom of SF of higher resolution Wavelet properties: adaptivity BigDFT Basics Thierry Deutsch BigDFT code Wavelets Adaptivity Precision O(N) BigDFT Resolution can be refined following the grid point. The grid is divided in Low (1 DoF) and High (8 DoF) resolution points. Points of different resolution belong to the same grid. Empty regions must not be “filled” with basis functions. Localization property,real space description Optimal for big & inhomogeneous systems, highly flexible Basis set features BigDFT Basics Thierry Deutsch BigDFT code Tensor product decomposition of the basis Wavelets Precision O(N) BigDFT The 3D basis is separable in 1D SF/ W. e ;e ;e e f x y z (x;y;z) = fex (x)f y (y)fez (z) jx ;jy ;jz jx jy jz (0) (1) (jx ;jy ;jz ) are the grid points, fj and fj the SF and the W. Orthogonality, scaling relation Daubechies wavelets are orthogonal and multi-resolution Z 1 m dxfk (x)f`(x) = dk` f(x) = p ∑ hj f(2x − j) 2 j=−m Hamiltonian-related quantities are calculated analytically 14 The accuracy is only limited by the basis set (O(hgrid))) Operator expressions, pseudopotentials BigDFT Basics Thierry Deutsch BigDFT code Wavelets Exact evaluation of kinetic energy Precision O(N) BigDFT Expressed analytically by a convolution: 2 f (x) = ∑c`f`(x); ∇ f (x) = ∑~c` f`(x) ; ` ` Z 2 ~c` = ∑cj a`−j ; a` ≡ f0(x)¶x f`(x) ; j From N3 to 3N calculations for separable objects We save computational time when performing scalar products with separable functions (e.g. gaussians). Z 2 Z 2 − 1 ( r ) − 1 ( x ) dry(r)e 2 ra = ∑cijk di dj dk ; di = fi (x)e 2 ra ijk di coefficients can be calculated at machine precision. GTH-HGH: analytic and separable pseudopotentials Systematic basis set BigDFT Basics Thierry Deutsch BigDFT code The absolute accuracy of the calculation is directly proportional Wavelets Precision to the number of the basis functions O(N) BigDFT Two parameters for tuning the basis • The grid spacing hgrid • The extension of the Low resolution points crmult Optimal for inhomogeneous systems BigDFT Basics Thierry Deutsch Test case: cinchonidine molecule (44 atoms) BigDFT code Wavelets 1 10 Precision Plane waves O(N) BigDFT 100 Wavelets Ec = 40 Ha 10-1 Ec = 90 Ha 10-2 h = 0.4bohr Ec = 125 Ha 10-3 Absolute energy precision (Ha) h = 0.3bohr 10-4 105 106 Number of degrees of freedom Enables a systematic approach for molecules Considerably faster than Plane Waves codes. the above run :10 (5) times faster than ABINIT (CPMD) Charged systems can be treated explicitly with the same time All-electron precision with HGH PSP BigDFT Basics Thierry Deutsch Non-linear core correction J. Chem. Phys. 138, 104109 (2013) BigDFT code Wavelets Precision • Simple analytic form (a single gaussian as rc) O(N) BigDFT • Same hardness as HGH * a systematic localized basis is fundamental Precision considerably improved G2-1 test set (Atomization energies) kcal/mol MAD RMSD MSD maxAD minAD Old HGH 6.85 9.13 -6.76 23.94 0.10 NLCC-HGH 0.51 0.63 0.16 1.50 0.03 PAW Paier 0.46 0.56 -0.43 1.13 0.01 ∆ AE (geopt) 0.29 0.70 -0.29 4.21 0.00 AE precision for quantities in different environments Bond lengths, Pressure (Bulk systems), Dispersion-corrected interaction energies, . Local orbitals and linear scaling BigDFT Basics Thierry Deutsch BigDFT code KS orbitals Density Matrix Wavelets Precision (N) BigDFT Linear combinations of support Defined via the kernel K ab in O functions fa(r): the fa(r) basis: Y (r) = caf (r) i ∑ i a 0 0 a r(r;r ) = ∑fi Yi (r)Yi (r ) • localized around atoms i ab 0 • expanded in wavelets = ∑fa(r)K fb(r ) a;b • optimized in-situ Localization ! sparse matrices ! O (N) Overall scheme BigDFT Basics Thierry Deutsch BigDFT code Initial Guess: Wavelets Atomic Orbitals Precision Two-step optimization O(N) BigDFT scheme Basis optimization: • The fa minimize the “trace” of a confining KS or hamiltionian • Coefficients minimize KS energy Kernel Optimization: Diagonalize Hamiltonian Update potential High quality results • Good precision • No need of Pulay forces Forces: due to basis completeness! Derivative basis functions A flexible formalism BigDFT Basics Thierry Deutsch BigDFT code Flexible Boundary Conditions Wavelets Precision O(N) BigDFT • Isolated (free) BC • Surfaces BC • Periodic (3D) BC • Wires BC Systematic approach Only relevant degrees of freedom are taken into account Boundary conditions can be implemented explicitly 0 E.g.: Surfaces BC -5 2D Periodic + 1D isolated -10 avg. V(z) (eV) Optimal to treat dipolar systems -15 without corrections -6 -4 -2 0 2 4 6 layer distance (Bohr).
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