Upscaling DFT Capabilitiesfor the Many Thousand

Atoms Regime

Laura E. Ratcliff

30.10.2017

DFT for

1000s of

Atoms

Laura

Ratcliff

O (N )DFT

Fragment

Approach

Simulating

OLEDs

Outline

Linear-Scaling DFT

QM of large systems

0

10

20

30

40

50

5000 10000 15000 20000 25000

Wallt

ime (m

in.)

WalltimeLinear extrap. from 10725 atoms

Number of atoms

BigDFT

www.bigdft.org

ONETEP

www.onetep.org

Fragment Approach

exploiting repetition

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Simulating OLEDs

excitations in an environment

DFT for

1000s of

Atoms

Laura

Ratcliff

O (N )DFT

Fragment

Approach

Simulating

OLEDs

Outline

1 O (N ) DFT

2 Fragment Approach

3 Simulating OLEDs

DFT for

1000s of

Atoms

Laura

Ratcliff

O (N )DFT

Fragment

Approach

Simulating

OLEDs

Beyond the Cubic-Scaling Limit

Size Limits

thanks to supercomputers wecan treat up ∼ 1000 atoms withDFT, but O(N 3) scaling limitssystem sizes 0

10

20

30

40

50

60

0 1000 2000

Tim

e / i

tera

tion

(s)

Number of atoms

Nearsightedness

• the behaviour of large systems is short-ranged, or“nearsighted”

• the density matrix decays exponentially in systems with a gap

⇒ how can we exploit nearsightedness to treat large systems?

DFT for

1000s of

Atoms

Laura

Ratcliff

O (N )DFT

Fragment

Approach

Simulating

OLEDs

Key Quantities

Support Functions (SFs)

write KS orbitals as linearcombinations of SFs φα(r):

Ψi(r) =∑

α

cαi φα(r)

• localized (user-defined radius)

• atom-centred

• expanded in systematic basis

Density Kernel

define the density matrixρ(r, r′) and kernel Kαβ:

ρ(r, r′) =∑

i

fi∣

∣Ψi(r)〉〈Ψi(r′)∣

∣

=∑

α,β

∣

∣

∣

φα(r)〉Kαβ〈φβ(r

′)∣

∣

∣

Total Energy

Hαβ = 〈φα|H|φβ〉; Sαβ = 〈φα|φβ〉

E = Tr (KH) ; N = Tr (KS)

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DFT for

1000s of

Atoms

Laura

Ratcliff

O (N )DFT

Fragment

Approach

Simulating

OLEDs

The Algorithm

optimize SFs

atomicorbitals

optimize K

energy& forces

Accurate Minimal Basis

• energy is minimized with respect to bothSFs and kernel

• SFs adapt to the environment

• different options for kernel optimization:Fermi Operator Expansion (FOE), penaltyfunctional, purification, LNV. . .

⇒ minimal, localized basis with the samehigh accuracy as underlying systematic basis

DFT for

1000s of

Atoms

Laura

Ratcliff

O (N )DFT

Fragment

Approach

Simulating

OLEDs

Basis Sets – Psincs and WaveletsCommon Features

• localized • orthogonal • systematic • use of PSPs

Wavelet Features

• flexible boundary conditions

• analytic operators

• multiresolution grid: 1 grid, 2resolution levels

-1.5

-1

-0.5

0

0.5

1

1.5

-6 -4 -2 0 2 4 6

LEAST ASYMMETRIC DAUBECHIES-16

scaling functionwavelet

Psinc Features

• periodic boundary conditions

• equivalence with plane waves

• regular grid: 1 psinc functioncentred at each grid point

0.0

0.5

1.0

-1.0 -0.5 0.0 0.5 1.0

DFT for

1000s of

Atoms

Laura

Ratcliff

O (N )DFT

Fragment

Approach

Simulating

OLEDs

Support Functions

Common Features

• strict localization (∼ 6 – 8 bohr)

• minimal number: e.g. 1 SF per H, 4 per C/N/O etc.

BigDFT

• quasi-orthogonal SFs

• use of a confining potential

ONETEP

• Non-orthogonal GeneralizedWannier Functions (NGWFs)

DFT for

1000s of

Atoms

Laura

Ratcliff

O (N )DFT

Fragment

Approach

Simulating

OLEDs

From Sparsity to Linear Scaling

Sparse Matrices

• strict localization leads to sparse matrices

• sparsity depends on size anddimensionality

• make use of sparse matrix algebra

• sparsity impacts crossover point

0

200

400

600

800

1000

0 200 400 600 800 1000

Hamiltonian, alkane with 512 atoms

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

0

100

200

300

400

500

600

700

800

0 100 200 300 400 500 600 700 800

Hamiltonian, Si cluster with 291 atoms

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

0

10

20

30

40

50

5000 10000 15000 20000 25000

Wa

lltim

e (

min

.)Number of atoms

O (N ) DFT

• can treat 1000s of atoms using DFT at a high accuracy

• many functionalities and features available

DFT for

1000s of

Atoms

Laura

Ratcliff

O (N )DFT

Fragment

Approach

Simulating

OLEDs

Outline

1 O (N ) DFT

2 Fragment Approach

3 Simulating OLEDs

DFT for

1000s of

Atoms

Laura

Ratcliff

O (N )DFT

Fragment

Approach

Simulating

OLEDs

Exploiting Repetition

Calculation Bottleneck

• SF optimization takes the majority of compute time

• what happens in similar chemical environments?

Water Droplet

• internal molecularenvironment dominates

• differences between watermolecules are small

• can we use the same SFsfor each molecule?

DFT for

1000s of

Atoms

Laura

Ratcliff

O (N )DFT

Fragment

Approach

Simulating

OLEDs

ReformattingRototranslations

how do we account for varying orientations and positions?• scheme to detect rototranslations with respect to reference(“template”) coordinates

• accurate and efficient wavelet interpolation scheme

0.001

0.01

0.1

1

10

180 0 180 0 180 0 180 0 180 0 180 0 1800

E−

Ere

f(m

eV

/ato

m)

θ (°), u

(0,0,1) (0,1,0) (0,1,1) (1,0,0) (1,0,1) (1,1,0) (1,1,1)

cubic eggboxlinear eggbox

template

DFT for

1000s of

Atoms

Laura

Ratcliff

O (N )DFT

Fragment

Approach

Simulating

OLEDs

Fragment Approach

Calculation Steps

• template calculation: optimize SFs for isolated fragment

• reformatting: replicate and rototranslate template SFs foreach fragment instance

• full calculation: use fragment SFs as a fixed basis, optimizingdensity kernel only

DFT for

1000s of

Atoms

Laura

Ratcliff

O (N )DFT

Fragment

Approach

Simulating

OLEDs

Fragments in Action

R (Å)

−0.4

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

0.5

1 1.5 2 2.5 3 3.5 4 4.5 5

E −

Ere

f(e

V)

R (Å)

cubiclinear

fragment 1,4fragment 2,5fragment 5,8

1.75

1.80

1.85

1.90

1.95

2.00

Req

(Å)

1e−04

1e−03

1e−02

1e−01

1e+00

E −

Ecu

bic

(eV

)

H2O Dimer

• energies affected by basis set superposition error

• but equilibrium bond length less affected

• can increase basis to improve accuracy

• approach is suited to weakly interacting fragments

DFT for

1000s of

Atoms

Laura

Ratcliff

O (N )DFT

Fragment

Approach

Simulating

OLEDs

Constrained DFT I

Constrained DFT (CDFT)

• wavelet basis is ideal for adding a (constrained) charge

• in CDFT we find the lowest energy state satisfying a given(charge) constraint on the density

• we want to associate a given charge with a particular fragment

• can be used to reduce the self-interaction problem and includeenvironmental effects on site energies

CDFT with SFs

W [n, Vc] = EKS [n] + Vc (2Tr [Kwc]−Nc)

The SF basis lends itself to a Lowdin like approach for theweight function: wc = S

12PS

12

DFT for

1000s of

Atoms

Laura

Ratcliff

O (N )DFT

Fragment

Approach

Simulating

OLEDs

Constrained DFT II

-0.1

0.0

0.1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Vc

Nc

0.0

1.0

2.0

3.0

4.0

∆E

(eV

)

ZnBC-BC complex

• varying charge separation between two molecules

• can find charge transfer states

DFT for

1000s of

Atoms

Laura

Ratcliff

O (N )DFT

Fragment

Approach

Simulating

OLEDs

Outline

1 O (N ) DFT

2 Fragment Approach

3 Simulating OLEDs

DFT for

1000s of

Atoms

Laura

Ratcliff

O (N )DFT

Fragment

Approach

Simulating

OLEDs

OLED Transport Parameters (a)

Jholeij = 〈ψHOMO

i(mol.)

∣

∣

∣H∣

∣

∣ψHOMO

j(mol.)〉

(b)

Eholeon-site = E+1

tot − E0tot

Fragments in Disordered Host-Guest Material

• extract cluster of nearest neighbours for each molecule

• template calculations for isolated host and guest molecules

• use fragment basis to calculate transfer integrals in clusters (a)

• use CDFT to add charge to central molecule in each cluster

• calculate on site energies using CDFT results (b)

DFT for

1000s of

Atoms

Laura

Ratcliff

O (N )DFT

Fragment

Approach

Simulating

OLEDs

OLED Transport Parameters

6.2 6.4 6.6 6.8 7.0 7.2 7.4Energy (eV)

guest

pure host

host

Statistics

• disorder ⇒ dispersionof values for Eon-site

and Jij

Environmental Effects

• shift in average Eon-site (- - -) vs.isolated molecules (—)

• differences between pure host andhost guest materials

Future Challenge for Simulating OLEDs

improve the description of excitations in realistic morphologies

DFT for

1000s of

Atoms

Laura

Ratcliff

O (N )DFT

Fragment

Approach

Simulating

OLEDs

Summary

Linear-Scaling DFT

QM of large systems

0

10

20

30

40

50

5000 10000 15000 20000 25000

Wallt

ime (m

in.)

WalltimeLinear extrap. from 10725 atoms

Number of atoms

BigDFT

www.bigdft.org

ONETEP

www.onetep.org

Fragment Approach

exploiting repetition

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Simulating OLEDs

excitations in an environment

PhD and postdoc position available

Thank you for your attention!