Wavelets for density functional calculations: Four ... · • All XC functionals of the LibXC...
Transcript of Wavelets for density functional calculations: Four ... · • All XC functionals of the LibXC...
Wavelets for density functional
calculations: Four families and three
applications
Stefan Goedecker
http://comphys.unibas.ch/
• Haar wavelets
• Daubechies wavelets: BigDFT code
• Interpolating wavelets: Poisson solver and direct exchange library
• Multi-wavelets: High accuracy atomization energies
Wavelets: A family of relatively new mathematical basis sets withastonishing 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)
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• 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.
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
ψkj(x) ∝ ψ(2kx− j)
φkj(x) ∝ φ(2kx− 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
∑j=−m
h j φ(2x− j)
Haar wavelet ψ and scaling function φ
φ ψ
1 0
1
0
Scaling function representation
0 1x
φ4
f (x) = ∑j
s4j φ4
j(x)
Haar wavelet basis set
φ
ψ
ψ
ψ
ψ
0
1
2
3
4
Wavelet representation:
f (x) = s01φ0
1(x)+d01ψ0
1(x)+2
∑i=1
d1i ψ1
i (x)+4
∑i=1
d2i ψ2
i (x)+8
∑i=1
d3i ψ3
i (x) (1)
Operators in a wavelet basis: The standard form
3
2
1
0
D
D
D
DS 0
3
2
1
0
D
D
D
DS 0
b
*=
c A
• 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
Operators in a wavelet basis: The non-standard operator form
SDS
0
D
S
D
S
D
0
1
1
2
3
3
2
b
*=
SDS
0
D
S
D
S
D
0
1
1
2
3
3
2
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
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
Daubechies wavelet of order 4
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DAUBECHIES-4 SCALING FUNCTION AND WAVELET
scaling functionwavelet
Even though the function is not very smooth polynomials up to degree 4 can be repre-
sented exactly
High order Daubechies wavelets are used to represent wave functionsin BigDFT
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LEAST ASYMMETRIC DAUBECHIES-16
scaling functionwavelet
Polynomial up to degree 16 can be represented exactly: convergence rate h14
BigDFT solves the many-electron Schrodinger 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/
Wavelet basis sets in three dimensions
1 scaling function
7 wavelets
all are products of 1-dim scaling functions and wavelets
φi, j,k(x,y,z) = φ(x− i)φ(y− j)φ(z− k)
ψ1i, j,k(x,y,z) = φ(x− i)φ(y− j)ψ(z− k)
ψ2i, j,k(x,y,z) = φ(x− i)ψ(y− j)φ(z− k)
ψ3i, j,k(x,y,z) = φ(x− i)ψ(y− j)ψ(z− k)
ψ4i, j,k(x,y,z) = ψ(x− i)φ(y− j)φ(z− k)
ψ5i, j,k(x,y,z) = ψ(x− i)φ(y− j)ψ(z− k)
ψ6i, j,k(x,y,z) = ψ(x− i)ψ(y− j)φ(z− k)
ψ7i, j,k(x,y,z) = ψ(x− i)ψ(y− j)ψ(z− k)
Pseudopotentials: two resolution levels are used in BigDFT
Kinetic energy matrix elements between scaling functions
ai =
∫φ(x)
∂2
∂x2φ(x− i)dx
Can be calculated exactly (Beylkin):
ai =∫
φ(x)∂2
∂x2φ(x− i)dx
= ∑ν,µ
2hνhµ
∫φ(2x−ν)
∂2
∂x2φ(2x−2i−µ)dx
= ∑ν,µ
2hνhµ22−1∫
φ(y−ν)∂2
∂y2φ(y−2i−µ)dy
= ∑ν,µ
hνhµ22∫
φ(y)∂2
∂ylφ(y−2i−µ+ν)dy
= ∑ν,µ
hνhµ 22 a2i−ν+µ
We thus have to find the eigenvector a associated with the eigenvalue of 2−2,
∑j
Ai, j a j =
(
1
2
)2
ai
where the matrix Ai, j is given by
Ai, j = ∑ν,µ
hνhµ δ j,2i−ν+µ
Operator approach: Applying the kinetic energy operator results in a convolution with the
filter a
(KΨ)i = ∑j
Ψ jai− 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 N1N2N3L where L is
the length of the filter.
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
Construction of interpolating scaling functions
Recursive interpolation from Kronecker data set: Example linear interpolation
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’scf2’
High order interpolating scaling functions represent charge densities
and potentials
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Interpolating scf 14Interpolating scf 100
Solution of Poisson’s equation for free boundary conditionsL. 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, ρi, j,k, the continuous charge
distribution is represented in terms of interpolating scaling functions
ρ(r) = ∑i, j,k
ρi, j,kφ(x− i)φ(y− j)φ(z− k)
The moments of the discrete and continuous charge distributions ρi, j,k and ρ(r) are iden-
tical
∑i, j,k
il1 jl2 kl3 ρi, j,k =∫
dr xl1 yl2 zl3 ρ(r) (2)
if l1, l2, l3 < m, where m is the order of the scaling functions. The potential at a grid point
i1, i2, i3 is given by
Vi1,i2,i3 = ∑j1, j2, j3
ρ j1, j2, j3
∫φ(x′− j1)φ(y
′− j2)φ(z′− j3)
|r j1, j2, j3 − r′| dr′
= ∑j1, j2, j3
ρ j1, j2, j3 Ki1− j1,i2− j2,i3− j3
The above convolution can be calculated rapidly with Fourier methods
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
Efficient and accurate hybrid functional calculations with the PSolver
package of BigDFT
The exact exchange energy Ex is given by
EX =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
A direct hybrid functional calculation is about two orders of magnitude more expen-
sive 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
Hybrid functional calculations possible up to 1000 atoms: Cray (Piz Daint) at CSCS
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.
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
C2Hs2 416.35 399.38 405.79
CH3Cls 400.13 393.13 395.53
CH3OHs 518.58 505.43 508.56
CHs4 420.50 415.18 418.13
COs2 413.48 384.89 389.52
H2COs 385.02 368.52 371.66
H2Os 232.98 225.17 226.07
HOCls 174.07 161.60 162.71
OHd 109.20 104.74 105.14
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 molec-
ular calculation and for the generation of the pseudopotential. Dual space Gaussian
pseudopotentials were used in the BigDFT code with free boundary conditions.
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
Multi-wavelets: All electron calculations
• compact support
• 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
Solving Schrodingers equation in integral form
−1
2∇2Ψi(r)+V (r)Ψi(r) = εiΨi(r)
(
∇2 +2εi
)
Ψi(r) = 4π1
2πV (r)Ψi(r)
Helmholtz equation: The inverse operator of
∇2 +2εi
is ∫exp(−
√−2εi|r− r′|)
|r− r′| dr′
. Hence we obtain the following iteration scheme for the Kohn-Sham orbitals
Ψnewi (r) =
∫exp(−
√−2εi|r− r′|)
|r− r′| Ψoldi (r′)dr′
MRChem
Frediani et al.: Real-space numerical grid methods in quantum chemistry, Phys. Chem.
Chem. Phys. 2015, 17, 31357
• 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 oper-
ator 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
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 fora 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 14491457
• For codes with a systematic basis set the accuracy is only limited by the pseudopo-
tential or PAW scheme.
Basis set errors
Pseudopotential errors
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
Selected Applications of BigDFTGeometric ground state of metal decorated boron clusters
Santanu Saha et al., accepted by Nature Scientific reports
Selected Applications of BigDFTDisconnectivity graph for a 26 atom gold cluster
Au+26 Bastian Schaefer et al. ACS Nano, 8 7413 (2014)