Sublinear scaling for time-dependent stochastic density functional theory

Sublinear scaling for time-dependent stochastic density functional theory

Yi Gao and Daniel NeuhauserDepartment of Chemistry and Biochemistry, University of California, Los Angeles, CA-90095 USA

Roi BaerFritz Haber Center for Molecular Dynamics, Institute of Chemistry,

The Hebrew University of Jerusalem, Jerusalem 91904, Israel

Eran RabaniDepartment of Chemistry, University of California and LawrenceBerkeley National Laboratory, Berkeley, California 94720, USA

A stochastic approach to time-dependent density functional theory (TDDFT) is developed forcomputing the absorption cross section and the random phase approximation (RPA) correlationenergy. The core idea of the approach involves time-propagation of a small set of stochastic orbitalswhich are first projected on the occupied space and then propagated in time according to the time-dependent Kohn-Sham equations. The evolving electron density is exactly represented when thenumber of random orbitals is infinite, but even a small number (≈ 16) of such orbitals is enoughto obtain meaningful results for absorption spectrum and the RPA correlation energy per electron.We implement the approach for silicon nanocrystals (NCs) using real-space grids and find that theoverall scaling of the algorithm is sublinear with computational time and memory.

I. INTRODUCTION

Time-dependent density functional theory (TDDFT)1allows for practical calculations of the time evolution ofelectronic densities under time-dependent perturbations.In principle TDDFT is an exact theory,1 but in ap-plications, several assumptions and approximations aretypically made.2–6 For example, for the most commonusage of TDDFT, namely the absorption spectrum7 ofmolecules and materials8 one uses the adiabatic approx-imation local/semi-local exchange-correlation potentials(time-dependent adiabatic local density approximation9(TDALDA) or time-dependent adiabatic generalized gra-dient approximation (TDAGGA)). TDDFT can alsobe used to compute the ground-state DFT correla-tion energy within the adiabatic-connection fluctuation-dissipation (ACFD) approach,10 or for studying strong-field nonpetrurbative dynamics.11–16

There are two types of challenges facing the applicationof TDDFT for large systems. One is the construction ofappropriate functionals, as the simplest, local and semi-local adiabatic functionals (TDALDA and TDAGGA) of-ten fail for large systems.17–22 The second issue is thedevelopment of a linear-scaling approach that overcomesnot only the quartic (O

(N4)) scaling in the frequency-

domain formulation23 but also the quadratic (O(N2))

limit achieved when real-time propagation according tothe time-dependent Kohn-Sham (TDKS) equations isused.24–26 This latter scaling is commonly considered thelowest theoretical scaling limit as it does not require fullresolution of the TDKS excitation energies. This is im-portant for large systems where the density of excitedstates is very large and there is no point in resolving ofall single-excited states as in small systems.

The present paper addresses the second challenge de-

scribed above and presents a stochastic formulation ofTDDFT (TDsDFT) formally equivalent to the TDKSmethod but without the Kohn-Sham (KS) orbitals. Thenew method is based on representing the time-dependentdensity as an average over densities produced by evolv-ing projected stochastic orbitals.27,28 We consider twodemonstrations of the TDsDFT within the linear re-sponse limit: The first concerns the calculation of thedipole absorption cross section and the second is basedon the ACFD approach to calculate the random phaseapproximation DFT correlation energy .

The paper is organized as follows: Section II first re-views the relation between linear response TDDFT andthe generalized susceptibility operator χλ (t). Next, weshow how TDsDFT can be used to perform the timeconsuming computational step in linear response appli-cations, i.e. the action of χλ (t) on a given potential. InSection III we show how the absorption spectrum and theACFD-RPA correlation energy can be calculated usingTDsDFT. We present results for a series of silicon NCsof varying sizes. We also analyze the scaling, accuracyand stability of the proposed TDsDFT. In Section IV weconclude.

II. THEORY

A. The Generalized Susceptibility Function andTime-Dependent Density Functional Theory

Consider a system of Ne electrons interacting via adamped Coulomb potential (λvC(|r−r′|) where 0 ≤ λ ≤1 and vC (r) = e2/4πε0r in their ground state |0λ〉and having a density n0 (r) = 〈0λ |n (r)| 0λ〉 . The lineardensity response of the system at time t (δnλ (r, t)) to

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a small external time-dependent potential perturbation(v (r′, t′)) is described by the following integral:29

δnλ (r, t) =

ˆ t

0

dt′ˆdr′χλ (r, r′, t− t′) δv (r′, t′) , (1)

where χλ (r, r′, t) is the generalized susceptibilityfunction,30 which is also given by retarded density-density correlation function of the system:

χλ (r, r′, t) = (i~)−1θ (t)

⟨0λ∣∣[nλ (r, t) , nλ (r′, 0)

]∣∣ 0λ⟩ ,(2)

where nλ (r, t) is the density operator at position rand time t. Eq. (2) is also known as the fluctuation-dissipation relation.31 χλ (r, r′, t) is used, for example,to compute the linear polarizability and energy absorp-tion of the system under external fields, the dielectricresponse, the conductivity and the correlation energies.

Rather than computing χλ (r, r′, t) directly (which inpractice requires a huge effort for large systems), a moreefficient approach is to obtain δnλ (r, t) by applying animpulsive perturbation, i.e., δv (r′, t′) = γv (r′) δ (t′) (γis a small constant with units of time):

δnλ (r, t) = γ

ˆdr′χλ (r, r′, t) v (r′) . (3)

Here, δnλ (r, t) can be computed by applying a perturba-tion e−iγ ˆv/~ and propagating the perturbed ground state:

δnλγ (r, t) =⟨

0λ

∣∣∣eiγv/~nλ (r, t) e−iγv/~∣∣∣ 0λ⟩− n0 (r) ,

(4)where v =

´n (r′) v (r′) dr′. To see this, expand the right

hand side of Eq. (4) to first order in γ: δnλγ (r, t) =

i~−1γ⟨0λ∣∣[v, nλ (r, t)

]∣∣ 0λ⟩ which, when combined with(2) gives Eq. (1).

To obtain the density response δnλγ (r, t) one needsto solve the many-electron time-dependent Schrödingerequation, which is prohibitive in general. A practicalalternative is to use TDDFT. Starting from the KS sys-tem of non-interacting electrons having the ground-statedensity n0 (r) = 2

∑j∈occ |φj (r)|2, one perturbs the KS

eigenstates φj (r) at t = 0:

ϕj (r, t = 0) = e−iγv(r)/~φj (r) , (5)

and then propagates in time according to the TDKSequations

i~∂ϕj (r, t)

∂t= hλ (t)ϕj (r, t) , (6)

where the TDKS Hamiltonian hλ (t) depends on thescreening parameter λ and the propagated density,nλγ (r, t) = 2

∑j∈occ |ϕj (r, t)|2. The density response of

Eq. (4) is then obtained from:

ˆdr′χλ (r, r′, t) v (r′) =

1

γ

(nλγ (r, t)− nλγ=0 (r, t)

)≡∆nλ (r,t) . (7)

Eq. (7) simply states that the integral of the susceptibil-ity and a potential v (r) can be computed from the differ-ence between the perturbed and unperturbed densities.This relation holds also for the half Fourier transformquantities (f (ω) =

´∞0dt eiωtf (t)):

ˆdr′χλ (r, r′, ω) v (r′) =

1

γ

(nλγ (r, ω)− nλγ=0 (r, ω)

)≡∆nλ (r, ω) . (8)

B. Time-Dependent Stochastic Density FunctionalTheory

The stochastic formulation of the density responseis identical to the deterministic version outlined abovebut instead of representing the time-dependent densitynλγ (r, t) as a sum over all occupied orbital densities(|ϕj (r, t) |2) we represent it as an average over the den-sities of stochastic orbitals ξj (r, t).27 Each stochastic or-bital is first projected onto the occupied space and thenpropagated in time. The advantage of the proposed ap-proach is immediately clear: If the number of stochas-tic orbitals needed to converge the results does not in-crease with the system size N , the scaling of the ap-proach is linear with N (rather than quadratic for thedeterministic version). Perhaps, in certain cases, due toself-averaging,27 the scaling will even be better than lin-ear, since the number of stochastic orbitals required toconverge the results to a predefined tolerance may de-crease with the system size.

The stochastic TDDFT (TDsDFT) procedure is out-lined as follows (for simplicity we use a real-space gridrepresentation, but the approach can be generalized toplane-waves or other basis sets):

1. Generate Nζ stochastic orbitals ζj (r) =

eiθj(r)/√δV , where θj (r) is a uniform ran-

dom variable in the range [0, 2π], δV is the volumeelement of the grid, and j = 1 , . . . , Nζ . Here,Nζ is typically much smaller than the number oftotal occupied orbitals (more details below). Thestochastic orbitals obey the relation 1 = 〈|ζ 〉〈 ζ|〉ζwhere 〈· · · 〉ζ denotes a statistical average over ζ.

2. Project each stochastic orbital ζj (r) onto the oc-

cupied space: |ξj〉 ≡√θβ |ζj〉, where θβ (x) =

12erfc (β (µ− x)) is a smooth representation of theHeaviside step function27 and µ is the chemical po-

tential. The action of√θβ is performed using a

3

suitable expansion in terms of Chebyshev polyno-mials32 in the static Hamiltonian with coefficientsthat depend on µ and β.

3. As in the deterministic case, apply a perturbationat t = 0: ξj (r, t = 0) = e−iγv(r)/~ξj (r) and propa-gate the orbitals according to the adiabatic stochas-tic TDKS equations:

i~∂ξj (r, t)

∂t= hλ (t) ξj (r, t) , (9)

with hλ (t) = hKS + vλHXC[nλγ (t)

](r) −

vλHXC[nλγ (0)

](r) and

vλHXC [n] (r) = λ

ˆdr′

n (r′)

|r− r′|+ vλXC (n (r)) , (10)

where vλXC (n (r)) is the local density (or semi-local) approximation for the exchange correlationpotential. For convergence reasons hKS is ob-tained with a rather large number of stochasticorbitals using the sDFT27 (or its more efficientversion, embedded fragment sDFT)28 and is fixedfor the entire propagation. The difference termvλHXC

[nλγ (t)

](r) − vλHXC

[nλγ (0)

](r) is generated

with a relatively small number of stochastic orbitalsNζ and the density

nλγ (r, t) = 2⟨|ξ (r, t)|2

⟩ζ≈ 2

Nζ

Nζ∑j=1

|ξj (r, t)|2 (11)

is obtained as an average over the stochastic orbitaldensities.

4. Generate ∆nλ (r, t) = 1γ

(nλγ (r, t)− nλγ=0 (r, t)

),

where γ is a small parameter, typically 10−3 −10−5~E−1h . We note in passing that for nλγ=0 (r, t)one has to carry out the full propagation since theunperturbed projected stochastic orbitals (|ξj〉) arenot eigenstates of the ground-state Hamiltonian.This propagation is not necessary for the determin-istic case.

III. RESULTS

A. TDsDFT Calculation of the Absorption CrossSection

The absorption cross section (ω ≥ 0) is given by theimaginary part of

σ (ω) =e2

3ε0cω

ˆdrdr′ r · χ (r, r′, ω) · r′. (12)

where c is the speed of light. For simplicity, we as-sume that the perturbing potential is in the z-direction

0 5 10 15ω (eV)

-300

-200

-100

0

100

200

300

400

σ(ω

) (Å

2 )

0 5 10 15 20ω (eV)

0

100

200

300

400

500

600

0 2 4 6 8 10time (fs)

d(t)

Nζ=16

Nζ=32

Nζ=64

Figure 1. Upper panels: The real (left) and imaginary (right)parts of σ (ω) calculated for Si705H300 using Nζ = 16 (redline), 32 (green line), and 64 (blue line). Lower panel showsthe corresponding dipole correlation d (t) as a function oftime.

(v (r) = z) and obtain σ (ω) in Eq. (12) from the Fouriertransform of the dipole-dipole correlation function:

dzz (t) =

ˆz∆nλ=1

z (r, t) d3r, (13)

where ∆nλ=1z (r, t) is obtained from Eq. (7) and σ (ω) =

e2

ε0cω´∞0dt eiωtdzz (t).

The real and imaginary parts of σ (ω) for Si705H300 areplotted in the upper panels of Fig. 1. These and all otherresults shown in this subsection were generated using thealgorithm above within the TDALDA approximation anda grid representation with grid spacing of δx = 0.6a0 em-ploying norm-conserving pseudopotentials33 and imagescreening methods.34 We used β = 0.01E−1h to repre-sent the smoothed step-function θβ , and a Chebyshevexpansion length of 3770 terms. The time-dependentdipole correlation was calculated using a time step ofδt = 0.0012fs up to tmax = 7.5fs. This signal was multi-plied by a Gaussian window function of width 2.5fs andthen Fourier transformed to give the absorption cross sec-tion.

The right upper panel of Fig. 1 shows the absorptioncross-section with a characteristic plasmon frequency of∼ 10eV.21,35–37 This feature is already captured withNζ = 16 stochastic orbitals compared to 1560 occupiedorbitals required in the full deterministic TDDFT. It isseen that further increase of Nζ reduces the statisticalfluctuations and provides a handle on the accuracy ofthe calculation. The convergence of the real part of σ (ω)shown in the upper left panel is similar to its imaginarycounterpart.

The calculated dipole correlation dzz (t) is shown in thelower panel of Fig. 1. For these large but finite systemswe expect dzz (t) to oscillate and decay to zero at interme-diate times followed by recurrences that appear at very

4

long times (much longer than the timescales shown here).Indeed, the stochastic approximation to dzz (t) oscillatesand decays to zero up to a time τC , but this is followedby a gradual increase which eventually leads to diver-gence. This is caused by an instability of the non-linearTDsDFT equations due to the stochastic representationof the time-dependent density. As Nζ increases and thestatistical fluctuations in the density decrease, the diver-gence onset time τC is increased.

0 2 4 6 8 10 12 14

time (fs)

106

107

108

109

1010

S(t

) (a

rb.

u.)

Si35

H36

Si87

H76

Si353

H196

Si705

H300

Figure 2. Divergence of the stochastic TDDFT calculation.Shown, the integrated dipole signal S(t) =

´ t0dzz (t

′)2dt′,

where dzz (t) is the calculated dipole correlation as a func-tion of time t, for Nζ = 16 (solid line) and Nζ = 64 (dashedline) stochastic orbitals for the different Si NCs. Because ofthe decay of the dipole correlation the signals reach a plateauafter which they diverge sharply due to a nonlinear instability.

In Fig. 2 we plot the integrated dipole signal S(t) =´ t0dzz (t′)

2dt′ on a semi-log scale. S(t) provides a clearer

measure of τC , which is determined as the onset of expo-nential divergence from the plateau (in practice we takethe value of τC to be at the middle of the plateau). Twoimportant observations on the onset of the divergencecan be noted:

1. τC increases for a fixed Nζ as the system size grows.For Nζ = 16, τC increases from ≈ 1.1fs for Si35H36

to ≈ 2.3fs for Si705H300. This is a rather moderate,but notable effect, that is a consequence of the socalled “self-averaging”.27

2. τC increases with Nζ for a fixed system size. Wefind that τC roughly scales as N1/2

ζ , namely, anincrease of τC by 2 requires an increase of Nζ by 4.

These findings indicate that the number of stochastic or-bitals not only determines the level of statistical noise(which scales as 1/

√Nζ) but also determines the spec-

tral resolution, given by τ−1C . To achieve converged re-sults for a fixed cutoff time of τC = 10fs, we find that Nζdecreases from ≈ 1300 for Si35H36 to ≈ 230 for Si705H300.

0 5 10 15 20ω (eV)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Im[σ

(ω)]

/ N

Si (

Å2 )

Si35

H36

Si87

H76

Si353

H196

Si705

H300

0 1000 2000 3000N

e

0

10

20

30

GP

U ti

me

(hrs

)

Figure 3. Upper panel: The absorption cross section =σ (ω)scaled for size as calculated for several silicon nanocrystalswith Nζ = 64. Lower panel: Extrapolated GPU computa-tional time (TGPU ) scaled for a cutoff time of τC = 10fs. Theline is a power law TGPU ∝ N0.5

e .

In the upper panel of Fig. 3 we show the absorptioncross section for the series of silicon NCs and a fixed num-ber of stochastic orbitals, Nζ = 64. As the NC size in-creases the plasmon frequency (peak near 10eV) slightlyshifts to lower energies and the width of the plasmon reso-nance slightly decreases. This is consistent with classicalMaxwell equations for which the plasmon frequency de-pends strongly on the shape but very mildly on the sizeof the NCs.38 The statistical fluctuations in the absorp-tion cross section decrease with the system size for a fixedNζ , as clearly evident in the figure (most notably at thelower energy range).

The lower panel of Fig. 3 shows the GPU computa-tional time of the approach for a predefined spectral res-olution (namely, for converged results up to a fixed cutofftime τC = 10fs). Each GPU performs roughly as 3 Intel3.5GHZ i7 third generation quad-core CPUs. Since thenumber of stochastic orbitals required to converge theresults for a fixed time decreases with the system size,the overall scaling of the TDsDFT is better than O (Ne)for the range of sizes studied here, significantly improvingthe O

(N2e

)scaling of the full deterministic TDDFT. The

overall computational effort does depend on the spectralresolution and thus, for small systems or for very highresolution the computational effort of the stochastic ap-proach may exceed that of the full deterministic calcu-lation with all occupied states. But this is certainly notthe case for the larger set of NCs studied here, where thewide plasmon resonance dominates the absorption crosssection, and thus the spectral features are converged forτC < 7.5fs.

5

0 10 20 30 40 50 60 70N

ζ

-1.6

-1.5

-1.4

-1.3

EC

RP

A /

Ne (

eV

)

Si35

H36

Si87

H76

Si353

H196

Si705

H300

0 1000 2000 3000N

e

0

10

20

30

GP

U t

ime (

hrs

)

Figure 4. Upper panel: The RPA correlation energy per elec-tron for the silicon NCs as a function of the number of stochas-tic orbitals Nζ used to represent the time-dependent density.The error bars are the standard deviation evaluated from 6statistically independent runs of the algorithm. Lower panel:GPU computational time (TGPU ) scaled for a statistical errorof 10meV in the total energy per electron. The line is a powerlaw TGPU ∝ N0.47

e .

B. Stochastic Approach to the Random PhaseApproximation Correlation Energy in DFT

The second application of the stochastic TDDFT isfor the RPA correlation energy, which is related toχλ (r, r′, ω) by the adiabatic-connection formula:39,40

ERPAC = − ~2π=ˆ 1

0

dλ

ˆ ∞0

dω

ˆdrdr′×(

χλ (r, r′, ω)− χ0 (r, r′, ω))vC (|r− r′|) , (14)

where the integral over λ adiabatically connects the non-interacting density response χ0 (r, r′, ω) to the interact-ing one χλ (r, r′, ω). To proceed, we rewrite Eq. (14) asan average over an additional set of stochastic orbitalsη (r) = eiθ(r)/

√δV

ERPAC = − ~2π=ˆ 1

0

dλ

ˆ ∞0

dω

ˆdrdr′dr′′×⟨

η∗ (r)(χλ (r, r′, ω)− χ0 (r, r′, ω)

)vC (r′′, r′) η (r′′)

⟩η.

(15)

This is done in order to rewrite the perturbationpotential as a single-variable potential: v (r′) =´dr′vC (r′, r) η (r′), which perturbs the stochastic or-

bitals at t = 0: ξj (r, t = 0) = e−iγv(r)/~ξj (r) and fromwhich the density nλγ (r, t) is computed using Eq. (11).For the propagation of ξj (r, t) according to Eq. (9) we setvλXC (n (r)) to zero, i.e. use the time-dependent Hartreeapproximation. Using this density and step 4 of the pro-cedure outlined above we compute the density response

∆nλ (r, t) from which our the RPA correlation energy iscalculated:

ERPAC = − ~2π=ˆ 1

0

dλ

ˆ ∞0

dω

ˆdr×⟨

η∗ (r)(∆nλ (r, ω)−∆nλ=0 (r, ω)

)⟩η. (16)

The stochastic formulation for Eq. (16) follows the algo-rithm described above in Sec. II.

We apply the stochastic RPA formulation to the vari-ous silicon NCs studied above. The integration over λ inEq. (16) was carried out using Gaussian quadrature with20 sampling points. For each value of λ we used a differ-ent set of ζ (for the TDsDFT) and η (for the applicationof v (r)) stochastic orbitals. The TDsDFT total prop-agation time was 1.5fs with a time step δt = 0.0012fs,sufficient to converge the RPA correlation energy.

In the upper panel of Fig. 4 we show the calculatedthe RPA correlation energy per particle for the varioussilicon NCs as a function of increasing Nζ , showing con-vergence asNζ increases. The correlation energy per elec-tron grows (in absolute value) with system size, in accor-dance with our findings in previous studies41,42 based ona semi-empirical Hamiltonian43. The standard deviation(indicated by error bars) evaluated over 6 different runsgenerally decreases as Nζ grows for a given system sizeand also decreases as system size grows for a given valueof Nζ . The magnitude of the error, however, is rathernoise due to the small number of independent runs usedto estimate it.

The lower panel of Fig. 4 shows the GPU computa-tional time of the approach for a fixed statistical error(estimated as the standard deviation based on the esti-mate of 20 independent runs) of 10meV. Our previousstochastic formulation of the RPA correlation energy re-lied on storing all occupied states (memory wise scaled asO(N2e

)) and the computational effort of the RPA stage

scaled as O (Nαe ) with 1 < α < 2,42 better than quadratic

scaling due to self-averaging. Comparing the current ap-proach with our previous work42, we find that the presentapproach shows significant improvements with respect tothe computational time and memory requirements. Thecomputational time scales as O

(N0.47e

)for the range of

NCs studied, better than linear scaling for the total RPAcorrelation energy per electron.

IV. SUMMARY

We have developed a stochastic approach to TDDFTfor computing the absorption cross section (via the time-dependent dipole correlation function) and the RPA cor-relation energy. The core idea of the approach involvestime propagation of a set of Nζ stochastic projected or-bitals ξj (r, t) according to the time-dependent Kohn-Sham equations. The evolving electron density is ex-actly represented when Nζ → ∞ but the strength of

6

the method appears when a small number of orbitalsNζ � Ne, where Ne is the number of electrons, is used.Such a truncation produces a statistical fluctuation dueto finite sampling. The magnitude of this error is pro-portional to 1/

√Nζ .

The finite sampling error coupled with a nonlinearinstability of the time-dependent Kohn-Sham equationsproduces a catastrophic exponential divergence that be-comes noticeable only after a certain propagation timeτC , which determines the spectral resolution of the ap-proach. The onset of divergence can be controlled byincreasing Nζ and empirically we determined that τC ∝√Nζ , consistent with the statistical nature of the error.The TDsDFT was applied to study the absorption

cross section and RPA correlation energy for a seriesof silicon NCs with sizes as large as Ne ≈ 3000. Forthis range of NC sizes, the computational time scalessub-linearly, roughly as O

(N

1/2e

)for both the absorp-

tion cross section and for the RPA correlation energy perelectron. For the former, the scaling holds for a givenspectral resolution τC . Since the computational time isalso proportional to NζNe, one can work backwards to

show that τC ∝√NζN

1/2e . For the RPA application,

the scaling holds for a given statistical error in the RPAcorrelation energy per electron.

The developed stochastic TDDFT approach adds an-other dimension to the arsenal of stochastic electronicstructure methods, such as the sDFT27 (and its moreaccurate fragmented version)28 and the sGW.44 Futurework will extend the approach to include exact andscreened exchange potentials in order to account forcharge-transfer excited states and multiple excitations.

ACKNOWLEDGMENTS

RB and ER are supported by The Israel Science Foun-dation – FIRST Program (grant No. 1700/14). Y. G.and D. N. are part of the Molecularly Engineered En-ergy Materials (MEEM), an Energy Frontier ResearchCenter funded by the DOE, Office of Science, Office ofBasic Energy Sciences under Award No. de-sc0001342.D. N. also acknowledges support by the National ScienceFoundation (NSF), Grant .CHE-1112500.

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