bigdft les ondelettes, une base ﬂexible permettant un ... 2014/deutsch.pdfbigdft introduction...

BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Journees Horizon-Maths

Les mathematiques se devoilent aux industriels

IFPEN, RUEIL-MALMAISON

Les ondelettes, une base flexible permettantun controle fin de la precision et la mise aupoint des methode ordre N pour le calcul de

la structure electronique via BigDFT

Thierry Deutsch

CEA Grenoble, INAC

December 15, 2014Laboratoire de Simulation Atomistique http://inac.cea.fr/L Sim Thierry Deutsch

BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

A basis for nanosciences: the BigDFT project

STREP European project: BigDFT(2005-2008)Four partners, 15 contributors:CEA-INAC Grenoble, U. Basel, U. Louvain-la-Neuve, U. Kiel

Aim: To develop an ab-initio DFT code basedon Daubechies Wavelets, to be integrated inABINIT, distributed freely (GNU-GPL license)

“Daubechies wavelets as a basis set for density functionalpseudopotential calculations”,L. Genovese, A. Neelov, S. Goedecker, T. Deutsch, et al., J. Chem. Phys. 129, 014109 (2008)

“Daubechies wavelets for linear scaling density functional theory”,S. Mohr, L. Genovese, , T. Deutsch, S. Goedecker, et al., J. Chem. Phys. 140, 204110 (2014)

Laboratoire de Simulation Atomistique http://inac.cea.fr/L Sim Thierry Deutsch

BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Goal

Waveletsan ideal basis for electronicstructure calculations – flexible,systematic etc.

(Linear-scaling) DFTallows us to access very largesystem sizes via the use of alocalized minimal basis set

we want to combine the two...


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Goal

Massively parallel architectures... and run calculations on large, realistic systems, using massivelyparallel machines


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Configuration space of the cage-like boron clusters

Stabilize the buckyball configuration of B80 systemsPRB 83, 081403(R) (2011)

-2

-1.5

-1

-0.5

0

0.5

-2

-1.5

-1

-0.5

0

0.5

B @B12

B 80

14:6

68


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Configuration space of the cage-like boron clusters

Stabilize the buckyball configuration of B80 systemsPRB 83, 081403(R) (2011)

-2

-1.5

-1

-0.5

0

0.5

-2

-1.5

-1

-0.5

0

0.5

-12.76 eV

-13.55 eV on B @B

12 68

@ B 80

8:12

Sc 3N


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

http://bigdft.org version 1.7.x

• Isolated, surfaces and 3D-periodic boundary conditions(k-points, symmetries)

• All XC functionals of the ABINIT package (libXC library)

• Hybrid functionals, Fock exchange operator

• Direct Minimisation and Mixing routines (metals)

• Local geometry optimizations (with constraints)

• External electric fields (surfaces BC)

• Born-Oppenheimer MD

• Vibrations

• Unoccupied states

• Empirical van der Waals interactions

• Saddle point searches (NEB, Granot & Bear)

• All these functionalities are GPU-compatible


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Kohn-Sham formalism in DFT

H-K theorem: E is an unknown functional of the densityE = E[ρ]→ Density Functional Theory

Kohn-Sham approachMapping of an interacting many-electron system into a system withN independent particles moving into an effective potential.

Find a set of orthonormal orbitals ψi(r) that minimizes:

E =−12

N

∑i=1

∫ψ∗i (r)∇

2ψi (r)dr +

12

∫ρ(r)VH(r)dr

+ Exc[ρ(r)] +∫

Vext (r)ρ(r)dr

ρ(r) =N

∑i=1

ψ∗i (r)ψi (r) ∇

2VH(r) =−4πρ(r)


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

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O(N )

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Fragment

Perspectives

Kohn-Sham Equations: Computing Energies

Calculate different integrals

E[ρ] = K [ρ] + U[ρ]

K [ρ] =−12~2

me

N

∑i

∫V

drψ∗i ∇

2ψi

U[ρ] =∫

VdrVext (r)ρ(r) +

12

∫V

drdr′ρ(r)ρ(r ′)|r− r′|︸︷︷︸

Hartree

+ Exc[ρ]︸︷︷︸exchange−correlation

We minimise the total energy along the orbitals ψi (r) with the

constraint∫

Vdrρ(r) = Nel and 〈ψi |ψj〉= δij


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Kohn-Sham Equations in Density Functional Theory

Apply different operators

Having a one-electron hamiltonian in an effective potentiel Veff [ρ]

[−1

2∇

2 + Veff [ρ] (r)

]ψi (r) = εi ψi (r)

Veff [ρ] (r) = Vext (r) +∫

V

ρ(r ′)|r − r ′|

dr ′+ µxc [ρ] (r)

The electronic density ρ(r) can be expressed by the N occupiedorthonormalized orbitals ψi (r) i.e. 〈ψi |ψj〉= δij .

ρ(r) =N

∑i|ψi (r)|2


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Kohn-Sham Equations: Self-Consistent Field

Set of self-consistent equations:{−1

2~2

me∇

2 + Veff

}ψi = εi ψi

with an effective potential:

Veff [ρ] (r) = Vext (r)+∫

Vdr ′

ρ(r ′)|r − r ′|︸︷︷︸

Hartree

+ µxc [ρ]︸︷︷︸exchange−correlation

and: ρ(r) = ∑Ni |ψi (r)|2

Poisson Equation: ∆VHartree = ρ (Laplacian: ∆ = ∂2

∂x2 + ∂2

∂y2 + ∂2

∂z2 )

Real Mesh (1003 = 106): 106×106 = 1012 evaluations !


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Basis sets for electronic structure calculation

How can we express the Kohn-Sham wavefunctions?

Plane Waves4 Localization in Fourier space, efficient preconditioning

4 Systematic convergence properties

8 No localization in real space. Empty regions must be “filled”with PW. Non adaptive

Gaussians, Slater type Orbitals4 Real space localized, well suited for molecules and other open

structures

4 Small number of basis functions for moderate accuracy

8 Many different recipes for generating basis sets

8 Over-completeness before convergence.Non systematic basis set.


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Outline

1 IntroductionKohn-Sham formalism

2 The machinery of BigDFT (cubic scaling version)Mathematics of the waveletsMagic FilterSolver for the Poisson equation

3 O(N ) (linear scaling) BigDFT approachStrategyPerformance and AccuracyFragment approach

4 Perspectives


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Why do we use wavelets in BigDFT?

AdaptivityOne grid, two resolution levels in BigDFT:

• 1 scaling function (“coarse region”)

• 1 scaling function and 7 wavelets(“fine region”)

Ideal for big inhomogeneous systemsEfficient Poisson solver, capable ofhandling different boundary conditions –free, wire, surface, periodicExplicit treatment of charged systemsEstablished code with many capabilites

-1.5

-1

-0.5

0

0.5

1

1.5

-6 -4 -2 0 2 4 6 8

x

φ(x)

ψ(x)


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

A brief description of wavelet theory

Two kind of basis functions

A Multi-Resolution real space basisThe functions can be classified following the resolution level theyspan.

Scaling FunctionsThe functions of low resolution level are a linear combination ofhigh-resolution functions

= +

φ(x) =m

∑j=−m

hjφ(2x− j)

Centered on a resolution-dependent grid: φj = φ0(x− j).


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

A brief description of wavelet theory

WaveletsThey contain the DoF needed to complete the information which islacking due to the coarseness of the resolution.

= 12 + 1

2

φ(2x) =m

∑j=−m

hjφ(x− j) +m

∑j=−m

gjψ(x− j)

Increase the resolution without modifying grid spaceSF + W = same DoF of SF of higher resolution

ψ(x) =m

∑j=−m

gjφ(2x− j)

All functions have compact support, centered on grid points.


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Adaptivity of the mesh

Atomic positions (H2O)


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives


Fine grid (high resolution)


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives


Coarse grid (low resolution)


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Separability in 3D

The 3-dim scaling basis is a tensor product decomposition of 1-dimScaling Functions/ Wavelets.

φex ,ey ,ezjx ,jy ,jz (x ,y ,z) = φ

exjx (x)φ

eyjy (y)φ

ezjz (z)

With (jx , jy , jz) the node coordinates,

φ(0)j and φ

(1)j the SF and the W respectively.

Gaussians and waveletsThe separability of the basis allows us to save computational timewhen performing scalar products with separable functions (e.g.gaussians):

Initial wavefunctions (input guess)

Poisson solver

Non-local pseudopotentials


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Basis set features

Tensor product decomposition of the basisThe 3D basis is separable in 1D SF/ W.

φex ,ey ,ezjx ,jy ,jz (x ,y ,z) = φ

exjx (x)φ

eyjy (y)φ

ezjz (z)

(jx , jy , jz) are the grid points, φ(0)j and φ

(1)j the SF and the W.

Orthogonality, scaling relationDaubechies wavelets are orthogonal and multi-resolution∫

dxφk (x)φ`(x) = δk` φ(x) =1√2

m

∑j=−m

hjφ(2x− j)

Hamiltonian-related quantities can be calculated analytically

The accuracy is only limited by the basis set (O(h14grid)))


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

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Perspectives

No integration error

Orthogonality, scaling relation∫dx φk (x)φj (x) = δkj φ(x) =

1√2

m

∑j=−m

hjφ(2x− j)

The hamiltonian-related quantities can be calculated up to machineprecision in the given basis.

The accuracy is only limited by the basis set (O(

h14grid

))

Exact evaluation of kinetic energyObtained by short convolution with filters:

f (x) = ∑`

c`φ`(x) , ∇2f (x) = ∑

`

c` φ`(x) ,

c` = ∑j

cj a`−j , a` ≡∫

φ0(x)∂2x φ`(x) ,


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Systematic basis set

Two parameters for tuning the basisThe grid spacing hgrid

The extension of the low resolution points crmult


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Wavelet families used in BigDFT code

Daubechies f (x) = ∑` c`φ`(x)

Orthogonal c` =∫

dx φ`(x)f (x)

-1.5

-1

-0.5

0

0.5

1

1.5

-6 -4 -2 0 2 4 6

LEAST ASYMMETRIC DAUBECHIES-16

waveletscaling function

Used for wavefunctions,scalar products

Interpolating f (x) = ∑j fj ϕj (x)

Dual to dirac deltas fj = f (j)

-1

-0.5

0

0.5

1

-4 -2 0 2 4

scaling functionwavelet

INTERPOLATING (DESLAURIERS-DUBUC)-8

Used for charge density, functionproducts

Magic Filter method (A. Neelov, S. Goedecker)The passage between the two basis sets can be performed withoutlosing accuracy


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Magic Filter for the density and the potential

Wavefunctions are expressed using Daubechies basis set.

Wavefunctions

Ψ(x) = ∑i

Ψiφi (x)

Examples of operations that must be performed:

Express the point values of the density

ρ(x) = ∑i|Ψi (x))|2

Calculate the potential energy matrix elements

Uij =∫

dxφ(x− i)V (x)φ(x− j)

with a fast and precise method.

These issues can be addressed with the Magic Filter method.


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Problem: Calculating the potential energy

Daubechies are not smooth enough for conventional integrationschemes.

The calculation of the potential energy matrix elements Uij :

Uij =∫

dxφ(x− i)V (x)φ(x− j)

can in principle be done by the following methods

Triple product method (Beylkin)

The collocation method

The “magic filter” method (A. Neelov, S. Goedecker)


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

The triple product method

We assume that the grid spacing h=1. If

V (x) = ∑i

ViφIP(x− i)

where for interpolating scaling functions

Vi = V (i)

andΨ(x) = ∑

iΨiφ(x− i)

with Daubechies scaling functions φ then

Epot =∫

Ψ(x)V (x)Ψ(x)dx = ∑i,j,k

ΨiVj Ψk Ii−j,k−j

where Ii,k is a short array

Ii,k =∫

φ(x− i)φIP(x)φ(x− k)dx


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

The triple product ∝ L4

ProblemThe amount of computation per grid point scales like L4 in 3DL is the dimension of the matrix Iij .In particular, for 2m = 8, L = 20 and L4 = 1.6∗105

On the other hand, the triple product method is very precise: Theonly source of error is the approximation of the potential.


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

The collocation method

U =∫

Ψ(x)V (x)Ψ(x)dx ≈∑i

Ψ(i)Vi Ψ(i)

where

Ψ(i) = ∑j

Ψjφ(i− j)

Computation per grid point

in 3D: ∝ 6m.For 2m = 8,this gives only 24.The scaling functions arenot very smooth, so thecollocation method is notsufficiently precise.

1e-14

1e-12

1e-10

1e-08

1e-06

1e-04

0.01

1

1 10 100

EN

ER

GY

ER

RO

R

1/h

|Evar −E0 ||Ec −E0 ||Ec −Evar |


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

The “magic filter” method

A. Neelov, S. Goedecker, J. Comp. Phys. (2006)

We do not calculate the values of the Ψ on grid points, but wecalculate values that represent best Ψ in a neighborhood aroundthe grid point

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-6 -4 -2 0 2 4 6 8

φ(y)ωl


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

The “magic filter” method

We calculate the magic real space values Ψi by

Ψi = ∑j

Ψjωi−j , Epot = ∑i

ΨiVi Ψi

Ψi is the value at i of the polynomial P(x) of degree 2m−1∫dxP(x)φ(x− j) = Ψj

The magic filter ’restores’ the original smooth wavefunction Ψ thatgives rise to a certain expansion in a Daubechies basis set. If Ψ(x)is a polynomial of degree m−1 or lower, then Ψ(i) = Ψi .

Optimal accuracyIf the error in the wavefunction is O(hm), the error in the energy isO(h2m−2).

Used also for calculating the point-values of the density (themultipole moments are reproduced with error O(h2m))


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Kohn-Sham Equations: Self-Consistent Field

Set of self-consistent equations:{−1

2~2

me∇

2 + Veff

}ψi = εi ψi

with an effective potential:

Veff [ρ] (r) = Vext (r)+∫

Vdr ′

ρ(r ′)|r − r ′|︸︷︷︸

Hartree

+ µxc [ρ]︸︷︷︸exchange−correlation

and: ρ(r) = ∑Ni |ψi (r)|2

Poisson Equation: ∆VHartree = ρ (Laplacian: ∆ = ∂2

∂x2 + ∂2

∂y2 + ∂2

∂z2 )

Real Mesh (1003 = 106): 106×106 = 1012 evaluations !


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Methods for the solution of Poisson’s equation

Poisson’s equation establishes the relation between a chargedensity ρ and its resulting potential V

∇2V (r) =−4πρ(r)

For non-periodic systems such as atoms and molecules, free

boundary conditions where the potential vanishes at infinity are theappropriate ones. Formally the solution can then be written as

V (r) =∫

ρ(r′)|r− r′|

dr′

The numerical solution of Poisons equation is frequently based on

the differential form rather than the integral form.


BigDFT

Introduction

Kohn-Sham

BigDFT

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O(N )

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Perspectives

Plane wave techniques

ρ(r′) on an equally spaced real space grid.N 3-dim grid points, the cost of FFT: N log2(N).Fourier space representation of ρ

ρ(r) = ∑G

cG exp(iG · r)

the Fourier space representation of the potential is

V (r) = ∑G

4πcG

G2 exp(iG · r)

Under periodic boundary conditions it is necessary that the systemhas no net charge, i.e. that c0 = 0.

The real space values of the potential on the grid are obtained byusing a backward Fourier transformation.


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Poisson Solver based on plane waves

Isolated systems, surfaces, wires

Use FFT (zero padding technics)

Uniform mesh

Correct multipole interactions better as possible

R. Hockney (1970)

G. Makov, M. Payne (1995)

G. Martyna, M. Tuckerman (1999)

L. Fusti-Molnar, P. Pulay (2002)

Use large box: only the center has the correct behavior.


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Poisson using Green function

What is still missing is a method that can solve Poisson’s equationwith free boundary conditions for arbitrary charge densities.

V (r) =∫

ρ(r′)|r− r′|

dr′

Problem: Evaluation of the Green function in each point

Real Mesh (1003): 106 integral evaluations


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Poisson Solver using interpolating scaling function

The Hartree potential is calculated in the interpolating scalingfunction basis.

Poisson solver with interpolating scaling functions

From the density ρ(~j) on an uniform grid it calculates:

VH(~j) =∫

d~xρ(~x)

|~x−~j|

4 Very fast and accurate, optimal parallelization

4 Can be used independently from the DFT code

4 Integrated quantities (energies) are easy to extract

8 Non-adaptive, needs data uncompression

Explicitly free boundary conditionsNo need to subtract supercell interactions→ charged systems.


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Tensor decomposition of 1/r

Feasible if the 1/r kernel is made separable.Represent it as a sum of Gaussians. The representation is bestbased on the identity

1r

=2√π

∫∞

−∞

e−r2 exp(2s)+sds

Discretizing this integral we obtain

1r

= ∑k

ωk e−pk r2

With 89 well optimized values for ωk and pk it turns out that 1/rcan be represented in the interval from 10−9 to 1 with an relativeerror of 10−8.


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Tensor decomposition of 1/r

-11

-10.5

-10

-9.5

-9

-8.5

-8

-7.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Erreur sur la decomposition de 1/r


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

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Fragment

Perspectives

Used of separability

The 3-dimensional integral becomes then a sum of 89 products of1-dimensional integrals.∫

dx∫

dy∫

dzφ(x− i1) φ(y− i2) φ(z− i3)√

(x− j1)2 + (y− j2)2 + (z− j3)2=

89

∑k

ωk

∫dx

∫dy

∫dz

φ(x− i1) φ(y− i2) φ(z− i3) e−pk ((x−j1)2+(y−j2)2+(z−j3)2) =

89

∑k

ωk

(∫dx φ(x− i1)e−pk (x−j1)2

)×(∫

dy φ(y− i2)e−pk (y−j2)2)(∫

dz φ(z− i3)e−pk (z−j3)2)

G. Beylkin et L. Monzon, On approximation of functions by exponential

sums, Applied and Computational Harmonic Analysis 19, p. 17 (2002)


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Poisson Solver for surface boundary conditions

This technology can be generalized for different BC.

A Poisson solver for surface problemsBased on a mixed reciprocal-direct space representation

Vpx ,py (z) =−4π

∫dz ′G(|~p|;z− z ′)ρpx ,py (z) ,

4 Can be applied both in real or reciprocal space codes

4 No supercell or screening functions

4 More precise than other existing approaches

8 Non-adaptive, needs data uncompression


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Poisson Solver for surface boundary conditions

A Poisson solver for surface problemsBased on a mixed reciprocal-direct space representation

Vpx ,py (z) =−4π

∫dz ′G(|~p|;z− z ′)ρpx ,py (z) ,

4 Can be applied both in real or reciprocal space codes

4 No supercell or screening functions

4 More precise than other existing approaches

Example of the plane capacitor:

Periodic

x

y

V

x

Hockney

x

y

V

x

Our approach

x

y

V

x


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Flexibility example: yet another Poisson Solver

(Screened) Poisson Equation for any BCWire-like periodicity, non-orthorhombic cells (surface BC):

(∇2−µ20)V (x ,y ,z) =−4πρ(x ,y ,z)

Very good accuracy J. Chem. Phys. 137, 13 (2012)

Toy examples for µ20 = {0,1,10,100} bohr−2

Future developmentsRange-separated Coulomb operator

1r

[erf r

r0+ erfc r

r0

]Laboratoire de Simulation Atomistique http://inac.cea.fr/L Sim Thierry Deutsch

BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Performances of the Poisson Solvers

Elapsed Time on a Cray XT3, 1283 grid

#proc 1 2 4 8 16 32 64Free sec .92 .55 .27 .16 .11 .08 .09

Surface sec .43 .26 .16 .10 .07 .05 .04

More precise than other existing Poisson Solvers:

Free BC

1e-11

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0.001

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

Max

Err

or

Grid step

HockneyTuckerman

8th14th20th30th40th50th60th

100th

L.G. et al., J. Chem. Phys.125, 74105 (06)

Surfaces BC

1e-16

1e-14

1e-12

1e-10

1e-08

1e-06

1e-04

0.01

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Abs

olut

e re

lativ

e er

ror

grid spacing

Mortensen8th

16th24th40th

h8 curve

L.G. et al., J. Chem. Phys.127, 54704 (07)


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

A versatile formalism

Flexible Boundary ConditionsIsolated (free) BC

Surfaces BC

Periodic (3D) BC

Wires BC (soon. . . )

Systematic approachOnly relevant degrees of freedom are taken into accountBoundary conditions can be implemented explicitly

-15

-10

-5

0

-6 -4 -2 0 2 4 6

avg.

V(z

) (e

V)

layer distance (Bohr)

E.g.: Surfaces BC2D Periodic + 1D isolatedOptimal to treat dipolar systemswithout corrections


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Order N method

Minimal basis set


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Outline




4 Perspectives


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Direct Minimisation: Flowchart

{ψi = ∑

ac i

aφa

}Adaptive mesh (0, . . . ,Nφ) (N2)

Orthonormalized

ρ(r) =occ

∑i|ψi (r)|2 Fine non-adaptive mesh: < 8Nφ

inv FWT + Magic Filter (N2)

VH (r) =∫

G(.)ρ Veffective (N2)Vxc [ρ(r)] VNL({ψi})

Poisson solver

− 12 ∇2 Kinetic Term (N2)

δc ia =− ∂Etotal

∂c∗i (a)+∑

jΛij c

ja

Λij = 〈ψi |H|ψj〉 (N3)

FWT

FWT

cnew ,ia = c i

a + hstepδc ia

Steepest Descent,DIIS (N2)

preconditioningStop when δc ia small


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives


{ψi = ∑

ac i

aφa


Orthonormalized

ρ(r) =occ



VH (r) =∫


Poisson solver



∂c∗i (a)+∑

jΛij c

ja


FWT

FWT

cnew ,ia = c i

a + hstepδc ia


preconditioning

Stop when δc ia small


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives


{ψi = ∑

ac i

aφa


Orthonormalized

ρ(r) =occ



VH (r) =∫


Poisson solver



∂c∗i (a)+∑

jΛij c

ja


FWT

FWT

cnew ,ia = c i

a + hstepδc ia


preconditioningStop when δc ia small


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Cubic-scaling operations

Application of the Hamiltonian: H ψi(r)

ψi(r) overall system (N) for N orbitals: N2

Overlap matrix: Λij =∫

ψi(r)ψ∗j (r)dr

N2 terms over the whole system (N): N3

Orthogonalization:

ψi(r) = ψi(r)−∑j<i

Λijψj(r)

ScalingN(N−1)

2∗N


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Cubic-scaling behavior

0

20

40

60

80

100

1 5 8 17 32 65 128 257 512 1025 0.1

1

10

100

1000

Per

cent

Sec

onds

(lo

g. s

cale

)

Number of atoms

LinAlgsumrhoPSolverHamAppPrecondOtherComm (%)Time (sec)


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Defining localized functions: Localization region


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Revisit cubic-scaling operations

Application of the Hamiltonian: H ψi(r)ψi (r) in a region of localization (κ) for N orbitals: κN

Overlap matrix: Λij =∫

ψi(r)ψ∗j (r)drOverlap matrix between neighbor orbitals (M): κMN

Orthogonalization (Cholesky)

ψi(r) = ψi(r)−∑j<i

Λijψj(r)

Orthogonalization with neighbor orbital(M): κMN


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Localization of the basis functions

The grid for a small molecule. We see the global box (coarse andfine) and the localization region (coarse and fine).


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Creation of the basis functions

Take the standard DFT Hamiltonian and add a confining potentialto it:

H i = H DFT + ci (r−Ri )4

where Ri is the position of the center of the localisation region i .

This confining potential ensures that the basis functions are welllocalized, but close to their center they feel the “correct”Hamiltonian and should therefore be of excellent quality.

DFT potentialconfining potentialeffective potential


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Creation of the basis functions

Since the Hamiltonian is different for each localisation region, wecan not search for eigenfunctions to determine the basis functionsφj .

Instead we have to minimize the “trace” of the Hamiltonian, i.e. ourbasis functions are given by the condition

minφi

∑i〈φi |Hi |φi〉 with 〈φi |φj〉= δij


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

The expansion coefficients

Once we have determined the basis functions φj , we can calculatethe Hamiltonian matrix in this basis, i.e.

HDFTjk = 〈φj |H DFT |φk〉

Diagonalizing this matrix to get the eigenvectors

HDFT ci = εici

provides us with the expansion coefficients for the physical

orbitals, i.e.Ψi = ∑cijφj


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Λij Overlap matrix (before N3)

To calculate the overlap matrix, we need to communicate betweenthe processes handling two overlapping basis functions. Only thatoverlap will be communicated.

In this way each process calculates a small part of the matrix.Using a collective communication call brings the entire matrix to allprocesses.


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Calculation of charge density (before N2)

To calculate the charge density each process will do the job for oneslice. No additional communication is required later, since thePoisson solver uses the same slices distribution.

Therefore each process has to gather all orbitals extending into itsslice. Only the part in the range of that slice will be communicated.


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Minimal basis and density kernel

Write the KS orbitals as linearcombinations of minimal basisset φα(r):

Ψi (r) = ∑α

cαi φα(r)

• localized

• atom-centred

• expanded in wavelets

Define the density matrix ρ(r, r′)and kernel K αβ:

ρ(r, r′) = ∑i

∣∣Ψi (r)〉〈Ψi (r′)∣∣

= ∑α,β

∣∣∣φα(r)〉K αβ〈φβ(r′)∣∣∣


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

The algorithm

Minimal basis set• Initialized to atomic orbitals, optimized

using a quartic confining potential subjectto an orthogonality constraint

• Minimize the ‘trace’, band structureenergy or a combination at fixed potential

• SD or DIIS with k.e. preconditioning

Density kernelThree methods for self-consistent optimization:

• Diagonalization

• Direct minimization

• Fermi operator expansion (linear-scaling)


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Scaling (alcanes)

0

10

20

30

40

50

60

0 1000 2000 3000 4000 5000 6000 7000 8000

Tim

e/itera

tion

(s)

Number of atoms

ax3

+ bx2

+ cx:1.0e+00; 3.0e+03; 1.5e+06

4.6e-02; 2.2e+02; 1.4e+06

7.6e-03; 4.3e-03; 1.4e+06

CubicDminFOE

Improved time and memory scaling• 301 MPI, 8 OpenMP for above results

• ∼ O (N ) for FOE

• need sparse matrix algebra for directminimisation→ O (N )


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Standard case: DNA

Use 3 000 cores (MPI+OpenMP)

20 min for 18 000 atoms (single point calculation)

0

200

400

600

800

1000

1200

1400

1600

1800

5000 10000 15000 20000 25000

time

(sec

onds

)

number of atoms

total runtimelinear extrapolation, reference 5720 atoms

0

1

2

3

4

5

6

7

8

9

10

11

5000 10000 15000 20000 25000m

emor

y pe

ak (

GB

)

number of atoms

memory peak MPI masterlinear extrapolation, reference 5720 atoms

Need sparse linear algebra (10-15%)


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Performance for many systems

cubic linear dif erence

energy force norm energy force norm energy force normhartree hartree/ bohr hartree hartree/ bohr meV/ atom hartree/ bohr

Vitamin B12 −926.78 2.13 · 10− 3 −926.71 1.93 · 10− 2 11.43 1.72 · 10− 2

Chlorophyll −476.70 3.05 · 10− 3 −476.64 1.24 · 10− 2 12.33 9.38 · 10− 3

C60 −341.06 2.69 · 10− 4 −341.02 7.36 · 10− 3 17.23 7.09 · 10− 3

Si87H76 −386.79 7.80 · 10− 4 −386.69 1.01 · 10− 2 17.20 9.27 · 10− 3

(H2O)100 −1722.99 5.23 · 10− 1 −1722.87 5.26 · 10− 1 10.89 2.80 · 10− 3

DNA −4483.12 5.60 · 10− 1 −4482.84 5.63 · 10− 1 10.69 2.40 · 10− 3


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Fragment approach• By dividing a system into fragments,

we can avoid optimizing the supportfunctions entirely for large systems

• Substantially reduces the cost(need an efficient reformatting)

• Many useful applications, including theexplicit treatment of solvents

• A necessary first step towards atight-binding like approach, with eachatom as a fragment

Reformatting the minimal basis set in the same grid


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Applications: Transfer integrals for OLEDs (I)

We want to consider environment effects in realistic ‘host-guest’morphologies: 6192 atoms, 100 molecules

Host molecule

Guest molecule


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Transfer integrals and site energies

0.00

0.20

0.40

0.60

0.80

1.00

0 30 60 90

Ene

rgy

(eV

)

Rotation angle

DZ J12 LUMOTZP J12 LUMOTMB J12 LUMO

0.00

0.20

0.40

0.60

0.80

1.00

0 30 60 90

Ene

rgy

(eV

)

Rotation angle

DZ J12 HOMOTZP J12 HOMOTMB J12 HOMO

−4.00

−3.50

−3.00

−2.50

−2.00

−1.50

0 30 60 90

Ene

rgy

(eV

)

Rotation angle

DZ e1,2 LUMOTZP e1,2 LUMOTMB e1,2 LUMO

−6.00

−5.50

−5.00

−4.50

−4.00

0 30 60 90E

nerg

y (e

V)

Rotation angle

DZ e1,2 HOMOTZP e1,2 HOMOTMB e1,2 HOMO

BigDFT compared to ADF fragment approach• support functions from molecules reused in dimers

• Application to OLED (Organic Light-Emitting Diode)


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Transfer integrals for OLEDs (II)

Transfer integrals (left) and site energies (right) calculated with(bottom) and without (top) constrained DFT

-0.06 -0.04 -0.02 0 0.02 0.04 0.06Energy (eV)

all

host-host

guest-guest

host-guest-5.5 -5.4 -5.3 -5.2 -5.1 -5 -4.9 -4.8 -4.7 -4.6 -4.5

-7.3 -7.2 -7.1 -7 -6.9 -6.8 -6.7 -6.6 -6.5 -6.4 -6.3

Energy (eV)

hostguest

Generate good statistics.Then use of Markus model to calculate the efficiency of OLEDS.


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Outline




4 Perspectives


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Perspectives of linear scaling

Combination of wavelets and localized basis functions• Highly accurate (energies and forces) and efficient

• Linear scaling – Thousands of atoms

• Ideal for massively parallel calculations

• Flexible e.g. reuse of support functions to accelerategeometry optimizations and for charged systems

Fragment approach• Accurate reformatting scheme – can accelerate calculations

on large systems e.g. explicit treatment of solvents

• Wide range of applications: constrained DFT, transfer integrals

• Crucial for future developments e.g. tight-binding, embeddedsystems (QM/QM)


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

QM/QM embedding in solids


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Conclusions

Wavelets: Powerful formalism• Linear scaling – Thousands of atoms

• Accurate reformatting scheme – can accelerate calculationson large systems e.g. explicit treatment of solvents

• Wide range of applications: constrained DFT, transfer integrals

• Towards multi-scale approach (embedded systems)

• Numerical stable scheme: multipole preserving (large gridstep)

Beyond DFT• Resonant states are the way to describe fully a system

• Give a compact description (few states) of the systems

• Linear-response, TD-DFT, ...

• Many-body perturbation theory (GW, ...)


BigDFT

Introduction

Kohn-Sham

BigDFT

Wavelets

Magic Filter

Poisson

O(N )

Strategy

Performance

Fragment

Perspectives

Acknowledgments

Main maintainer Luigi Genovese

Group of Stefan Goedecker A. Ghazemi, A. Willand, S. De,A. Sadeghi, N. Dugan (Basel University)

Link with ABINIT and bindings Damien Caliste (CEA-Grenoble)

Order N methods Stefan Mohr, Laura Ratcliff, Paul Boulanger

Exploring the PES N. Mousseau (Montreal University)

Resonant states A. Cerioni, A. Mirone (ESRF, Grenoble),I. Duchemin (CEA), M. Moriniere (CEA)

Optimized convolutions Brice Videau, Jean-Francois Mehaut (LIG,computer scientist, Grenoble)

Applications P. Pochet, D. Timerkaevai, E. Zvereva(CEA-Grenoble)


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