Linear-scaling time-dependentdensity-functional theory (LS-TDDFT):

Implementation and applications

T. J. Zuehlsdorff1 N. D. M. Hine2 M. C. Payne1 P. D.Haynes3

1Cavendish LaboratoryUniversity of Cambridge, UK

2University of Warwick, UK

3Imperial College London, UK

ONETEP Masterclass, September 2015

Outline

Motivation

Linear-scaling DFT in ONETEP

Linear-scaling TDDFT

Converging targeted localised excitations

Applications

Optical properties of large systems

Bacteriochlorophyll in FMO

I Effects due to proteinenvironment

I Optical properties differentfrom isolated molecule

figure taken from http://www.ks.uiuc.edu/Research/fmo/

Semiconductor nanoparticle

I Surface effects andquantum confinement

I Optical properties differentfrom bulk crystal

Linear Response TDDFT

Solve (A(ω) B(ω)−B(ω) −A(ω)

)(XY

)= ω

(XY

)where

Acv ,c′v ′(ω) = δcc′δvv ′(εKSc′ − εKS

v ′ ) + Kcv ,c′v ′(ω)

Bcv ,c′v ′(ω) = Kcv ,v ′c′(ω)

and

Kcv ,c′v ′(ω) =

⟨ψKS

c ψKSv

∣∣∣∣ 1|r− r′|

+ fxc(r, r′, ω)∣∣∣∣ψKS

c′ ψKSv ′

⟩Tamm-Dancoff approximation: Y = 0; B = 0; AX = ωXUse iterative eigensolvers: Only the action q = Ax is required.

DFT vs. Linear-scaling DFT

DFT

I KS-orbitals are delocalisedover the entire system

I Keeping O(N) orbitalsorthogonal to each other→O(N3) scaling

LS-DFT

I Use atom-centerednonorthogonal orbitals thatare very localised

I Move from a KS stateformalism to a densitymatrix formalism

Linear-scaling DFT in ONETEP

In linear-scaling DFT, the valence density matrix is expanded interms of atom-centered localised functions {φα}

ρ(r, r′) =occ∑v

ψKSv (r)ψKS∗

v (r′) =∑αβ

φα(r)P{v}αβφ∗β(r′)

I {φα} are optimised in situ during a ground state calculation→ minimal number is needed to span valence space

EDFT = EDFT

[P{v}, {φα}

]I Linear scaling is achieved by truncating the density matrix

with distance→ P{v} is sparseI No reference to individual Kohn-Sham eigenstates and

energies

Conduction optimisation of phthalocyanine

30

20

10

10

20

30

0

-8 -6 -4 -2 0 2 4 6 8

DO

S

Energy / eV

PWPPONETEP

ONETEP+cond

I Optimisation of a second set of localised functions {χα}and effective density matrix P{c} to represent low energypart of conduction space 1

1L. E. Ratcliff, N. D. M. Hine, and P. D. Haynes, Phys. Rev. B 84,165131(2011)

Optimisation of the Rayleigh-Ritz value in Kohn-Shamspace

ω = minx

x†Axx†x

Differentiating the Rayleigh-Ritz value yields an energygradient:

∂ω

∂x= 2Ax−

[x†Ax

]x

Write q = Ax in terms of an effective transition density ρ{1}(r)and the Kohn-Sham eigenvalue differences

qcv = (εKSc − εKS

v )xcv +(

V {1}SCF

[ρ{1}

])cv

where ρ{1}(r) =∑

cv ψKSc (r)xcvψ

KSv (r)

Transition density matrix in ONETEP

ρ{1}(r) =∑cv

ψKSc (r)xcvψ

KSv (r)

=∑αβ

χα(r)P{1}αβφβ(r)

I Express the transitiondensity in terms of densitymatrix P{1}

I Hole well described by{φα}

I Electron well described by{χα}

I ρ{1}(r) well described by{φα} and {χβ}

The Tamm-Dancoff TDDFT gradient in {φ} and {χ}representation

The TDDFT gradient q = Ax in mixed {φ} and {χ}representation:

qχφ = P{c}HχKSP{1} − P{1}Hφ

KSP{v} + P{c}V{1}χφSCF P{v}

I Fully O(N) if all involved density matrices are truncatedI Can be used to generate a gradient for conjugate gradient

algorithm

I Multiple excitations: Optimise {P{1}i } silmutaneously→O(N2

ω) due to orthonormalisations

Full TDDFT: An effective variational principle

In full TDDFT, define an effective variational principle for thepositive eigenvalues:

ωmin = min{P{p},P{q}}

Tr[P{p}†Sχqχφ{p}S

φ]

2∣∣∣Tr[P{p}†SχP{q}Sφ

]∣∣∣+

Tr[P{q}†Sχqχφ{q}S

φ]

2∣∣∣Tr[P{p}†SχP{q}Sφ

]∣∣∣

where

qχφ{p} = P{c}HχKSP{1} − P{1}Hφ

KSP{v}

qχφ{q} = P{c}HχKSP{1} − P{1}Hφ

KSP{v} + 2P{c}V{1}χφSCF P{v}

Full TDDFT vs. Tamm-Dancoff: Bacteriochlorophyll

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

Ab

so

rptio

n s

tre

ng

th (

arb

. u

nits)

Energy (eV)

Full TDDFTTDA

Experiment

(10,0) Carbon nanotube: Linear scaling test

0

200

400

600

800

1000

1200

0 500 1000 1500 2000

Tim

e(s

)

Number of atoms

No truncationTruncation: 60 Bohr

I Time taken for a single conjugate gradient iteration vs.system size

Constraining the TDDFT transition density matrix

Fully dense P{1}

I P{1} filling: 100.0 %I ω = 5.1950 eVI f = 0.177 × 10−6

I 10th excitation converged

P{1} only on one Benzene

I P{1} filling: 25.0 %I ω = 5.1953 eVI f = 0.111 × 10−6

I Lowest excitation converged

Interactions between subsystems: Exciton coupling

Reintroduce subsystem coupling perturbatively:P{1}tot =

∑i αiP

{1}Ai

+∑

j βjP{1}Bj

4.39

4.395

4.4

4.405

4.41

4.415

4.42

4.425

4.43

4.435

4.5 5 5.5 6 6.5 7 7.5 8

Ene

rgy

(eV

)

Separation (Å)

full systemCoupled subsystems

αi , βj obtained in a single subspacediagonalisation as a post-processing step

Solvatochromic shift of alizarin (C14H8O4) in water

How much of the environment has to be treated explicitly inorder to converge localised excitations of alizarin?

I Create a model system from a single classical MDsnapshot

I All water within 12 Å of alizarin is included in thecalculation ≈ 1800 atoms

Solvatochromic shift of alizarin (C14H8O4) in water

2.3

2.35

2.4

2.45

2.5

2.55

2.6

2.65

0 Å 4 Å 6 Å ∞

Ene

rgy

(eV

)

P{1} truncation constraint

explicit watervacuum

implicit solvent

I Strong solvatochromic shift of 0.12 eV due tocharge-transfer delocalisation in P{1} onto the water

I Delocalisation confined to within 6 Å from alizarin ( ≈ 300atoms)

The Fenna-Matthews-Olson (FMO) complex

I 7 Bacteriochlorophyll pigments per monomerI Including protein environment ≈ 10.000 atoms per

monomerI Exciton dynamics studied with model Hamiltonians→

accurate pigment site energies needed

figure taken from http://www.ks.uiuc.edu/Research/fmo/

Site energies from linear-scaling TDDFT: Case studyof BChl site 1

Two model systems of the protein environment:

562 atoms10 Å radius

1646 atoms15 Å radius

Absorption spectrum, full TDDFT

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

1.2 1.4 1.6 1.8 2 2.2

Abs

orpt

ion

(arb

.un

its)

Energy (eV)

10 Å system15 Å system

Fully dense P{1}: large number of spurious charge-transferstates with local fxc (34 states needed for 15 Å spectrum)

Quantifying exciton delocalisation into the protein

1.65

1.7

1.75

1.8

1.85

0 Å 8 Å 10 Å 12 Å 14 Å

Ene

rgy

(eV

)

P{1} truncation constraint

Qy transition Bchl1Qx transition Bchl1

I 12 Å radius of explicit protein environment necessary toconverge localised site energies ≈ 1000 atom systems

I Constraining P{1}: Localised site energies become thelowest excited states of the system

Conclusion

I ONETEP provides an efficient framework for calculatinglow energy excited states in systems containing thousandsof atoms (available features: (semi)-local functionals,PAW; Future work: Hybrid functionals, TDDFT forces→excited states dynamics)

I Fully linear-scaling for sufficiently large systemsI Truncation of P{1} can be used to converge targeted,

localised excitations of larger systemsI Warning: QM/MM methods can require very large QM

regions to converge environmental effects on localisedexcitations.

Acknowledgements

I This research was funded by the EPSRC grantsEP/G036888/1, EPSRC Grant EP/J017639/1 and theARCHER eCSE programme

I All calculations were performed using the ONETEP code:J. Chem. Phys. 122, 084119 (2005), www.onetep.org

I Further information and benchmark tests regardinglinear-scaling TDDFT in ONETEP: J. Chem. Phys. 139,064104(2013)