Accuracy benchmarking of DFT results, domain libraries for electrostatics, hybrid functional and solvation

Transcription

1 Accuracy benchmarking of DFT results, domain libraries for electrostatics, hybrid functional and solvation Stefan Goedecker Multi-wavelets: High accuracy atomization energies Benchmarking of basis sets and pseudopotentials Libraries: Poisson solver, direct exchange library, solvation library

2 General dilemma when performing electronic structure calculations You have to accept either on of the two evils: Use a pseudopotential or PAW scheme Use a nonsystematic basis set

3 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. 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 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

8 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

9 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 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

10 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 For codes with a systematic basis set the accuracy is only limited by the pseudopotential or PAW scheme. 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

11 Basis set errors

12 Pseudopotential errors

13 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

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

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

16 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

17 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

18 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 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

19 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 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

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

21 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.

22 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.

23 ENVIRON: The solvation package Experimental processes frequently take place in neutral and ionic solutions: ENVI- RON library solves the Poisson equation both for a constant and a spatially varying dielectric constant as well as the Poisson Boltzmann equation to describe electrolytes Generalized Poisson equation Poisson-Boltzmann equation ε(r) φ(r)= 4πρ(r) ε(r) φ(r)= 4π [ ρ(r)+ρ ions [φ](r) ] Highly efficient iterative solution of GPe based on the PSolver library Continuous with respect to movements of the atoms (correct forces, energy conservation in MD)

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25 Accuracy comparable to best PCM (Polarizable Continuum Model)

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27 Benchmark for surfaces: contact angle Correct surface angles for a few materials that were tested: CaF 2, Si) 2, diamond, graphite

28 People involved in this work Basel: Santanu Saha, Giuseppe Fisicaro, Augustin Degomme, Jose Flores-Livas Grenoble: Luigi Genovese, Damien Caliste, Thierry Deutsch Arctic University of Norway: Stig Jensen, Luca Frediani Duke University: William Huhn, Volker Blum