Wavelets for density functional calculations: Four families and three. applications

Transcription

1 Wavelets for density functional calculations: Four families and three Haar wavelets Daubechies wavelets: BigDFT code applications Stefan Goedecker Interpolating wavelets: Poisson solver and direct exchange library Multi-wavelets: High accuracy atomization energies

2 Wavelets: A family of relatively new mathematical basis sets with astonishing properties All families share the properties: of being localized both in real and in Fourier space. This allows for the so-called multi-resolution analysis that gives information about the localization properties both in real and in Fourier space of a function (signal)

3 of being a systematic basis set, i.e the error is guaranteed to approach zero as the basis set tends to infinity an arbitrary high degree of adaptivity can be obtained.

4 Wavelet basis functions Each family is characterized by two functions. The mother scaling function ψ and the mother wavelet φ. A basis set is generated by translations and dilatation-s of these two functions j: Translation, Localization in real space k: Dilatation, Localization in Fourier space ψ k j(x) ψ(2 k x j) φ k j(x) φ(2 k x j) Both the wavelet and the scaling function at a certain resolution level can be written as a linear combination of scaling functions at a higher resolution level (refinement relations) φ(x)= m h j φ(2x j) j= m

5 Haar wavelet ψ and scaling function φ φ ψ Scaling function representation φ 4 0 x 1 f(x)= s 4 j φ 4 j(x) j

6 Haar wavelet basis set ψ φ ψ ψ ψ Wavelet representation: f(x)=s 0 1φ 0 1(x)+d 0 1ψ 0 1(x)+ 2 di 1 ψ 1 i(x)+ i=1 4 di 2 ψ 2 i(x)+ i=1 8 di 3 ψ 3 i(x) (1) i=1

7 Operators in a wavelet basis: The standard form c S 0 0 D D 1 A b S 0 0 D D 1 D 2 = * D 2 D 3 D 3 Coupling between different resolution levels Works fine for a limited number (two) of resolution levels. Complicated and numerically inefficient structure for a large number of resolution levels

8 Operators in a wavelet basis: The non-standard operator form S D S 0 D S D S D b * = S D S 0 D S D S D c A No coupling between different resolution levels For a small number of resolution levels larger prefactor than standard form Easy and numerically efficient structure if the transition region between the different resolution levels is large compared to the support length of the wavelet

9 Daubechies wavelets Ingrid Daubechies, 1988: ORTHONORMAL BASES OF COMPACTLY SUPPORTED WAVELETS Daubechies wavelet basis set combines advantages of Plane waves: Systematic, orthogonal basis set Localization in Fourier space allows for efficient preconditioning techniques. Gaussians: Localized in real space: well suited for molecules and other open structures, direct way to linear scaling Adaptivity

10 Daubechies wavelet of order DAUBECHIES-4 SCALING FUNCTION AND WAVELET scaling function wavelet Even though the function is not very smooth polynomials up to degree 4 can be represented exactly

11 High order Daubechies wavelets are used to represent wave functions in BigDFT LEAST ASYMMETRIC DAUBECHIES-16 scaling function wavelet Polynomial up to degree 16 can be represented exactly: convergence rate h 14

12 BigDFT solves the many-electron Schrödinger equation in the Kohn-Sham density functional approximation using a Daubechies wavelet basis All boundary conditions All XC functionals of the LibXC library (LDA, GGA, hybrid functionals) Local geometry optimizations (with constraints) Saddle point searches Global geometry optimization: Minima Hopping Implicit solvation models (dielectric, electrolyte) linear scaling version Vibrations Born Oppenheimer MD collinear and non-collinear magnetism Empirical van der Waals terms High degree of parallelization and GPU acceleration: fast time to solution Website: inac.cea.fr/l Sim/BigDFT/

13 Wavelet basis sets in three dimensions 1 scaling function 7 wavelets all are products of 1-dim scaling functions and wavelets φ i, j,k (x,y,z) = φ(x i)φ(y j)φ(z k) ψ 1 i, j,k (x,y,z) = φ(x i)φ(y j)ψ(z k) ψ 2 i, j,k (x,y,z) = φ(x i)ψ(y j)φ(z k) ψ 3 i, j,k (x,y,z) = φ(x i)ψ(y j)ψ(z k) ψ 4 i, j,k (x,y,z) = ψ(x i)φ(y j)φ(z k) ψ 5 i, j,k (x,y,z) = ψ(x i)φ(y j)ψ(z k) ψ 6 i, j,k (x,y,z) = ψ(x i)ψ(y j)φ(z k) ψ 7 i, j,k (x,y,z) = ψ(x i)ψ(y j)ψ(z k)

14 Pseudopotentials: two resolution levels are used in BigDFT

15 Kinetic energy matrix elements between scaling functions a i = Can be calculated exactly (Beylkin): a i = φ(x) 2 x 2 φ(x i)dx = 2h ν h µ ν,µ = 2h ν h µ ν,µ = h ν h µ 2 2 ν,µ = h ν h µ 2 2 a 2i ν+µ ν,µ φ(x) 2 x 2 φ(x i)dx φ(2x ν) 2 x 2 φ(2x 2i µ)dx φ(y ν) 2 y 2 φ(y 2i µ)dy φ(y) 2 y l φ(y 2i µ+ν)dy We thus have to find the eigenvector a associated with the eigenvalue of 2 2, ( ) 1 2 A i, j a j = a i j 2

16 where the matrix A i, j is given by A i, j = h ν h µ δ j,2i ν+µ ν,µ Operator approach: Applying the kinetic energy operator results in a convolution with the filter a (KΨ) i = Ψ j a i j j In the 3-dim case applying the kinetic energy operator requires a 3-dim convolution with a filter that is a product of 3 1-dim filters. The operation count is 3 N 1 N 2 N 3 L where L is the length of the filter.

17 Interpolating scaling functions: PSolver library of BigDFT for solution of Poisson s equation Compact support Many continuous derivatives for a given support length Trivial transformation from a real space data set to a scaling function expansion Not orthogonal

18 Construction of interpolating scaling functions Recursive interpolation from Kronecker data set: Example linear interpolation scf

19 High order interpolating scaling functions represent charge densities and potentials 1 Interpolating scf 14 Interpolating scf

20 Solution of Poisson s equation for free boundary conditions L. Genovese, T. Deutsch, A. Neelov, S. Goedecker, G. Beylkin, J. Chem. Phys. 125 (2006) Given the values of the charge density on a regular grid, ρ i, j,k, the continuous charge distribution is represented in terms of interpolating scaling functions ρ(r)= ρ i, j,k φ(x i)φ(y j)φ(z k) i, j,k The moments of the discrete and continuous charge distributions ρ i, j,k and ρ(r) are identical i l 1 j l 2 k l 3 ρ i, j,k = i, j,k dr x l 1 y l 2 z l 3 ρ(r) (2) if l 1,l 2,l 3 < m, where m is the order of the scaling functions. The potential at a grid point i 1,i 2,i 3 is given by V i1,i 2,i 3 = φ(x j 1 )φ(y j 2 )φ(z j 3 ) ρ j1, j 2, j 3 j 1, j 2, j 3 r j1, j 2, j 3 r dr = ρ j1, j 2, j 3 K i1 j 1,i 2 j 2,i 3 j 3 j 1, j 2, j 3 The above convolution can be calculated rapidly with Fourier methods

21 Boundary conditions The Poisson equation is solved exactly for all boundary conditions free wire surface periodic Highly accurate treatment of charged clusters dipolar surfaces clusters, surfaces in electric fields

22 Efficient and accurate hybrid functional calculations with the PSolver package of BigDFT The exact exchange energy E x is given by E X = N i=1 N j=i+1 dr dr ψ i (r) ψ j(r) ψ i (r ) ψ j (r ) r r (3) For a system of N electrons it requires the solution of N(N 1)/2 Poisson equations for all pairwise charge densities ψ i (r ) ψ j (r At greatly reduced cost hybrid functional calculations should become much more widespread in systematic basis sets since: High accuracy can nearly be reached in atomization energies Materials that are problematic with other functionals such as transition metal oxides can be well treated Gaps in solids are reasonably accurate

23 A direct hybrid functional calculation is about two orders of magnitude more expensive than a GGA calculation with all the state of the art plane wave codes BigDFT gives extremely high speed in the expensive exact exchange energy part: Since only a few basic operations have to be performed in our scaling function basis, all the computations are implemented in CUDA and are executed on the GPU Communication is overlapped with computation Direct GPU communication

24 Hybrid functional calculations possible up to 1000 atoms: Cray (Piz Daint) at CSCS

25 Exact ionic forces In contrast to other approaches that gain speed by evaluating the exact exchange based on localized orbitals, no cutoffs or other approximations are necessary and the forces are the exact derivative of the energy. This leads for instance to a perfect energy conservation in MD.

26 PBE0 pseudopotentials are available and should be used in PBE0 calculations XC used in calculation PBE PBE0 PBE0 XC of pseudopotential used PBE PBE PBE0 C 2 H s CH 3 Cl s CH 3 OH s CH s CO s H 2 CO s H 2 O s HOCl s OH d Atomization energy of molecules in kcal/mol for consistent (PBE/PBE and PBE0/PBE0) and inconsistent (PBE/PBE0) use of the exchange correlation functionals in the molecular calculation and for the generation of the pseudopotential. Dual space Gaussian pseudopotentials were used in the BigDFT code with free boundary conditions.

27 Overview: Poisson Solver package of BigDFT Solves the Poisson equation both for a constant and a spatially varying dielectric constant as well as the Poisson Boltzmann equation to describe electrolytes all standard boundary conditions Input and output are given on equally spaced Cartesian grids. In the periodic case both orthorombic and non-orthorombic cells can be treated The hybrid functional package is based on the Poisson solver package and the LibXC library

28 compact support Multi-wavelets: All electron calculations several basis functions per support interval representable as polynomials orthogonal symmetric Continuous derivatives within interval, possibly discontinuities among neighboring intervals (discontinuous Galerkin) Only integral equations can be solved

29 Solving Schrödingers equation in integral form Ψ i (r)+v(r)ψ i (r)=ε i Ψ i (r) ( 2 + 2ε i ) Ψi (r)=4π 1 2π V(r)Ψ i(r) Helmholtz equation: The inverse operator of 2 + 2ε i is exp( 2εi r r ) r r dr. Hence we obtain the following iteration scheme for the Kohn-Sham orbitals Ψ new i (r)= exp( 2εi r r ) r r Ψ old i (r )dr

30 MRChem Frediani et al.: Real-space numerical grid methods in quantum chemistry, Phys. Chem. Chem. Phys. 2015, 17, Developed in the group of Luca Frediani in Norway by Stig Jensen et al Performs non-relativistic all-electron density functional calculations for LDA, GGA and hybrid functionals by using multi-wavelets Allows for an arbitrary number of resolution levels and uses the non-standard operator form Any preset accuracy for the wave functions/energies can be obtained (if enough memory/cores are available) Same underlying method as in MADNESS, highly stable implementation Program under development to include more features

31 Accuracy of DFT calculations Nearly all codes use approximations that go beyond the XC functional: basis sets, pseudopotentials Kurt Lejaeghere et al.: Reproducibility in density functional theory calculations of solids, Science 351, 6280 (2016) A large number of electronic structure codes give more or less identical results for the energy versus volume curve For more difficult quantities significant disagreement between different codes can still be found Atomization energies with µha accuracy obtained with MRChem for a test set of nearly 300 molecules: S. Jensen et al.: The Elephant in the Room of Density Functional Theory Calculations. Phys. Chem. Lett., 2017, 8 (7), pp For codes with a systematic basis set the accuracy is only limited by the pseudopotential or PAW scheme.

32 Basis set errors

33 Pseudopotential errors

34 People involved in this work Basel: Bastian Schaefer, Augustin Degomme, Giuseppe Fisicaro, Santanu Saha, Huan Tran, Alireza Ghasemi Grenoble: Luigi Genovese, Damien Caliste, Thierry Deutsch Arctic University of Norway: Stig Jensen, Luca Frediani

35 Selected Applications of BigDFT Geometric ground state of metal decorated boron clusters Santanu Saha et al., accepted by Nature Scientific reports

36 Selected Applications of BigDFT Disconnectivity graph for a 26 atom gold cluster Au + 26 Bastian Schaefer et al. ACS Nano, (2014)