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Efficient Calculation of Electronic AbsorptionSpectra by Means of Intensity-Selected TD-DFTB

Robert Rüger,†,‡ Erik van Lenthe,† You Lu,†,‖ Johannes Frenzel,¶,⊥ ThomasHeine,§ and Lucas Visscher∗,‡

Scientific Computing & Modelling NV, De Boelelaan 1083, 1081 HV Amsterdam, TheNetherlands, Department of Theoretical Chemistry, VU University Amsterdam, De Boelelaan

1083, 1081 HV Amsterdam, The Netherlands, Department of Chemistry, University ofCalgary, 2500 University Drive, N.W., T2N 1N4 Calgary, Canada, and School of Engineering

and Science, Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany

E-mail: [email protected]

AbstractDuring the last two decades density functionalbased linear response approaches have becomethe de facto standard for the calculation ofoptical properties of small and medium-sizedmolecules. At the heart of these methods is thesolution of an eigenvalue equation in the space ofsingle-orbital transitions, whose quickly increas-ing number makes such calculations costly if notinfeasible for larger molecules. This is especiallytrue for time-dependent density functional tightbinding (TD-DFTB), where the evaluation ofthe matrix elements is inexpensive. For the rel-atively large systems that can be studied thesolution of the eigenvalue equation therefore de-termines the cost of the calculation. We proposeto do an oscillator strength based truncation ofthe single-orbital transition space to reduce thecomputational effort of TD-DFTB based absorp-tion spectra calculations. We show that even a∗To whom correspondence should be addressed†Scientific Computing & Modelling‡VU University Amsterdam¶University of Calgary§Jacobs University Bremen‖Present address: STFC Daresbury Laboratory,

Daresbury, Warrington WA4 4AD, United Kingdom⊥Present address: Lehrstuhl für Theoretische Chemie,

Ruhr-Universität Bochum, Universitätsstraße 150, 44780Bochum, Germany

sizeable truncation does not destroy the princi-pal features of the absorption spectrum, whilenaturally avoiding the unnecessary calculationof excitations with small oscillator strengths.We argue that the reduced computational costof intensity-selected TD-DFTB together withits ease of use compared to other methods low-ers the barrier of performing optical propertiescalculations of large molecules, and can serve tomake such calculations possible in a wider arrayof applications.

1 IntroductionDensity functional theory (DFT) based on theHohenberg-Kohn theorem1 and implemented interms of the Kohn-Sham equations2 is one ofthe most popular methods in both solid-statephysics and quantum chemistry. The reasonfor this popularity is that DFT is computation-ally relatively affordable and its accuracy formany systems not far behind more accurate butalso much more expensive wavefunction basedmethods. For systems which are too large tobe treated with DFT one can introduce furtherapproximations on top of the DFT framework,most notably density functional based tight bind-ing (DFTB).3,4 In DFTB, tight-binding approx-imations are made to the DFT total energy ex-

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pression, most notably an optimized minimumvalence orbital basis that reduces the linear alge-bra operations, and a two center-approximationthat allows to precalculate and store all integralsusing the Slater-Koster technique.5 The self-consistent charge (SCC) technique6 accountsfor density fluctuations and improves resultson polar bonds. Detailed information on theDFTB parameterization for all elements havebeen published recently.7As the underlying Hohenberg-Kohn theorem

is only a statement about the ground state, stan-dard DFT can not be applied to the broad classof problems involving excited states, most no-tably the study of optical properties of an elec-tronic system. The extension of DFT to excitedstates has been accomplished in the form of time-dependent density functional theory (TD-DFT)based on the Runge-Gross theorem,8 which isa time-dependent analogon to the Hohenberg-Kohn theorem. In quantum chemistry TD-DFTis in practice often used in the form of Casida’sformalism,9 where the electron density’s linearresponse to a perturbation in the external poten-tial is used to construct an eigenvalue equationin the space of single orbital transitions fromwhich the excitation energies and excited statescan be extracted. TD-DFT calculations of ex-cited states are much more expensive than theirground state counterpart, and therefore limitedin the size of the systems that can be treated. Atthe expense of accuracy the computational costof TD-DFT calculations can be reduced by mak-ing further approximations, most notably theTamm-Dancoff approximation10 (TDA) and re-lated techniques.11 It is interesting to note thatTDA results can even be better than unapproxi-mated TD-DFT results12 even though TDA vio-lates the Thomas-Reiche-Kuhn f -sum rule.13–15Another way to reduce the computational ef-fort is to translate Casida’s formalism to theDFTB framework. This was done by Niehauset al. and is known as time-dependent densityfunctional based tight binding (TD-DFTB).16Note that there is an alternative formulation ofTD-DFTB which has recently been developedby Trani et al..17At the heart of both TD-DFT and TD-DFTB

is the solution of Casida’s eigenvalue equation

in the space of single orbital transitions. As thenumber of transitions grows quadratically withthe size of the system, the resulting matrix canonly be diagonalized using iterative eigensolvers,and even then the huge size of the matrix quicklybecomes the limiting factor. This is especiallytrue for TD-DFTB where the calculation of thematrix elements is rather cheap, so that biggersystems with relatively larger matrices can beinvestigated.In this article we discuss practical methods

to deal with the increasing dimension of theeigenvalue problem encountered in TD-DFTBcalculations for large molecules. The remain-der of the article is organized as follows. Insection 2 we recapitulate the basic equations ofground state SCC-DFTB and review how adapt-ing Casida’s TD-DFT approach to the DFTBframework results in the TD-DFTB method.In section 3 we analyze the bottlenecks of themethod and show how the TD-DFTB equationscan be implemented efficiently. For the specificapplication of calculating electronic absorptionspectra we present ways to reduce the size of theeigenvalue problem through a physically moti-vated truncation of the single orbital transitionspace. In section 4 we use this truncation tocalculate the absorption spectra of a numberof example molecules ranging from small modelsystems to entire proteins in order to validatethe precision of the results as well as the compu-tational performance of the method. Section 5summarizes our results.

2 Review of the methods

2.1 DFTBLet us quickly recapitulate the most importantequations of SCC-DFTB.6 More comprehensivereviews can be found in reference 18 and 19.The total energy within the SCC-DFTB methodis given by

ESCC-DFTB = Eorb + ESCC + Erep (1)

Eorb =Nocc∑i

〈φi|H0|φi〉 (2)

2

ESCC = 12

Natom∑AB

∆qAγAB∆qB (3)

Erep = 12

Natom∑AB

UAB , (4)

where the individual terms are called the orbitalcontribution Eorb, the self-consistent charge cor-rection ESCC, and the repulsive energy Erep.DFTB uses a (typically minimal) basis of

atomic valence orbitals χµ(~r) to expand themolecular orbitals φi(~r) as

φi(~r) =Natom∑A

∑µ∈A

cµiχµ(~r) . (5)

In this basis the matrix elements of H0 are cal-culated as

〈χµ|H0|χν〉 =

εfree atomµ for µ = ν⟨χµ∣∣∣ T + V 0

AB

∣∣∣χν⟩for µ ∈ A, ν ∈ B, A 6= B

0 otherwise

,

(6)where εfree atom

µ is the energy of the correspond-ing atomic orbital of the free atom and V 0

AB isa strictly pairwise effective potential, usuallyimplemented in terms of the Kohn-Sham poten-tial and the atomic electron densities ρ0

A and ρ0B.

Note that the matrix elements of H0 only de-pend on the elements of atom A and B andthe distance RAB =

∣∣∣~RA − ~RB∣∣∣ between the two

nuclei. It is therefore possible to precalculatethem by running DFT calculations for all indi-vidual atoms as well as all possible dimers at asufficient number of internuclear distances RAB.Details on this parametrization can be found inthe literature.6,7The self-consistent charge contribution ESCC

accounts for the fact that the actual groundstate density

ρGS(~r) = ρ0(~r)+δρ(~r) with ρ0(~r) =∑Aρ0A(~r)

(7)differs from the sum of the atomic densities bya density fluctuation δρ(~r). Within SCC-DFTBthis density fluctuation is then decomposed into

atomic contributions δρA(~r) which are subjectedto a multipole expansion and a monopolar ap-proximation.

δρ(~r) =∑AδρA(~r) ≈

∑A∆qAξA(~r) (8)

Here ξA(~r) is a spherically symmetric func-tion centered on atom A and the transferredcharges ∆qA are calculated from the expan-sion coefficients and the overlap matrix Sµν =〈χµ|χν〉 through Mulliken population analysis.

∆qA = qA − qfree atomA with (9)

qA = 12

Nocc∑i

∑µ∈A

∑ν

(cµiSµνcνi + cνiSνµcµi

)

The elements of the matrix γ in equation (1) cannow be calculated with any exchange-correlationfunctional Exc[ρ] through

γAB =∫

d3~r∫

d3~r ′ ξA(~r) fHxc[ρ0](~r, ~r ′) ξB(~r ′)(10)

with

fHxc[ρ0](~r, ~r ′) = 1|~r − ~r ′|

+ δ2Exc

δρ(~r)δρ(~r ′)

∣∣∣∣ρ0.

(11)

Note that γAB only depends on the type ofatom A and B as well as the distance RAB be-tween their nuclei. Due to the locality of theexchange-correlation functional, the SCC con-tribution reduces in the limit of large RAB tojust the Coulomb interaction between two pointcharges at ~RA and ~RB. The on-site term γAAcan be approximated by the atom’s Hubbardparameter

UA ≈ 2ηA ≈ IA − AA , (12)

where IA is the atomic ionization potential, AAthe electron affinity, and ηA the chemical hard-ness which can be calculated by DFT as thesecond derivative of the energy with respectto the occupation number of the highest occu-pied atomic orbital. An interpolation formulais then used to calculate γAB for intermediatedistances RAB.6

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While the repulsive term Erep can also beparametrized from DFT calculations,6 it is forfixed nuclear positions only a global shift inenergy that does not influence the absorptionspectrum and is hence irrelevant for this article.Finally the molecular orbitals φi(~r) from equa-

tion (5) can be obtained by solving the Kohn-Sham equation of SCC-DFTB.∑

ν

Hµνcνi = εi∑ν

Sµνcνi (13)

Hµν = H0µν + 1

2Sµν∑C

(γAC + γBC)∆qC

with µ ∈ A, ν ∈ B(14)

Note that this has to be done self-consistentlyas the ∆qC depend on the expansion coefficientsvia equation (9).

2.2 TD-DFT(B)One of the most popular ways to apply time-dependent density functional theory (TD-DFT)in the field of quantum chemistry is withoutdoubt Casida’s formalism.9 Starting from theelectron density’s linear response to a small per-turbation in the external potential, Casida caststhe problem of calculating excitation energiesand excited states into an eigenvalue equationin the Ntrans = NoccNvirt dimensional space ofsingle orbital transitions c†aci |Ψ0〉, where |Ψ0〉 isthe Slater determinant of the occupied Kohn-Sham orbitals. The eigenvalue problem can bewritten as

Ω ~FI = ∆2I~FI , (15)

where ∆I is the excitation energy. The elementsof the matrix Ω are given by

Ωia,jb = δijδab∆2ia + 4

√∆ia∆jbKia,jb , (16)

where we have abbreviated ∆ia = εa − εi. Weadopt the usual convention of using the in-dices i, j for occupied and a, b for virtual orbitals.The form of the so-called coupling matrix Kdepends on the multiplicity of the excited state.Neglecting spin-orbit coupling, only the singletexcitations are relevant for the calculation ofthe absorption spectrum. We therefore restrictour discussion to the singlet case, for which the

coupling matrix is given by

Kia,jb =∫

d3~r∫

d3~r ′φi(~r)φa(~r) (17)

fHxc[ρGS](~r, ~r ′) φj(~r ′)φb(~r ′) .

Once the eigenvalue equation (15) has beensolved, information about the excited state canbe extracted from the eigenvectors ~Fi. Follow-ing Casida, we use the components of the eigen-vector ~F to expand the excited state |ΨI〉 insingle orbital excitations relative to the Kohn-Sham Slater determinant |Ψ0〉.

|ΨI〉 =∑ia

√2∆ia

∆I

Fia,I c†aci |Ψ0〉 (18)

While the resulting |ΨI〉 should only be viewedas approximation to the true excited state, thetransition dipole moment ~dI of the excitationcan be calculated as a linear combination of thetransition dipole moments ~dia of these singleorbital transitions.

~dI = 〈Ψ0|~r|ΨI〉 =∑ia

√2∆ia

∆I

Fia,I ~dia (19)

with ~dia = 〈φi|~r|φa〉 (20)

The oscillator strength fI of the excitation andthereby the absorption spectrum is then easilyobtained from

fI = 23∆I

∣∣∣~dI ∣∣∣2 . (21)

While a direct solution of equation (15) is inprinciple possible, the need to store the N2

transelements of Ω in practice limits the size ofthe treatable systems. In the common casethat only Nexcit Ntrans lowest excitations areneeded, this problem can be overcome by theuse of iterative eigensolvers, which only needto multiply Ω with a set of trial vectors, with-out ever storing Ω explicitly. Not storing theelements of Ω implies that they have to be re-calculated on-the-fly for every iteration of theeigensolver. The diagonal part of Ω is trivial,but the coupling matrix elements involve costlytwo-center integrals, and even though very effi-cient methods to calculate these are available,20

4

their evaluation still is the major bottleneck inCasida’s formulation of TD-DFT.Time-dependent density functional based

tight-binding is a method put forward byNiehaus et al. 16,21,22 that builds on SCC-DFTBto approximate the coupling matrix K to thepoint where the costly integrals can be parame-terized in advance. Let us quickly recapitulatethe most important steps of the derivation. Firstthe transition density pia(~r) = φi(~r)φa(~r) is de-composed into atomic contributions which arethen subjected to a multipole expansion andapproximated by their monopolar term.

pia(~r) =∑Apia,A(~r) ≈

∑Aqia,AξA(~r) (22)

Here the ξA(~r) are the same atom centered func-tions that are used in the SCC extension ofground state DFTB, and the atomic transitioncharges qia,A are calculated from the coefficientand overlap matrices through

qia,A = 12∑µ∈A

∑ν

(cµiSµνcνa+cνiSνµcµa

). (23)

Note that the definition of the atomic transitioncharges also makes it straightforward to calcu-late the transition dipole moments of the singleorbital transitions:

~dia =∑Aqia,A ~RA (24)

Inserting equation (22) into the expression forthe coupling matrix elements yields

Kia,jb =∑AB

qia,AγABqjb,B , (25)

where the atomic coupling matrix γ is given by

γAB =∫

d3~r∫

d3~r ′ ξA(~r) fHxc[ρGS](~r, ~r ′) ξB(~r ′).(26)

Comparison with equation (10) reveals that γand γ only differ in the density at which thederivative of the exchange-correlation energyfunctional Exc[ρ] is evaluated. At this pointNiehaus et al. argue that the second deriva-tive of the exchange-correlation energy is shortranged and therefore only contributes to the

on-site elements γAA, which are then in anal-ogy to ground state SCC-DFTB approximatedby the Hubbard parameters.6 Domínguez et al.furthermore show that neglecting the depen-dence of the Hubbard parameters on the atomiccharges is consistent within a linear responsetreatment based on ground state SCC-DFTB.22Using the Hubbard parameters of the neutralatoms reduces the atomic coupling matrix γ tothe γ matrix from ground state SCC-DFTB,which then leads to a simple equation for thematrix Ω.

Ωia,jb = δijδab∆2ia + 4

√∆ia∆jb

∑AB

qia,AγABqjb,B

(27)Note that the orbital energy differences ∆ia aswell as the coefficient matrix C and the over-lap matrix S can easily be extracted from anyDFTB ground state calculation, and that noTD-DFTB specific parameters are needed sincethe γ matrix already had to be parameterizedwithin the SCC-DFTB method. TD-DFTB cantherefore immediately be applied to any systemfor which ground state SCC-DFTB parametersare available. It would be beyond the scope ofthis article to validate the TD-DFTB methoditself. Such studies have of course been per-formed16,17,21,22 and while the approximationsmade in TD-DFTB seem drastic at first sight,the overall accuracy of the method has beenfound to be promising and TD-DFTB has sinceseen a wide variety of applications.23–32In summary, TD-DFTB is a computationally

rather simple approximation to TD-DFT wherethe computational bottleneck is the size of theresponse matrix Ω and the calculation of itseigenvectors. In the next section we will presentcomputational methods to solve the TD-DFTBequations efficiently.

3 Computational methodsAs both the number of occupied Nocc and thenumber of virtual orbitals Nvirt grow linearlywith the number of atoms Natom, the totalnumber of single orbital transitions Ntrans =NoccNvirt increases quadratically with the sys-tem size. This in practice limits the size of

5

the systems treatable with TD-DFT(B), whichuses the single orbital transitions as the basisof the space in which Casida’s eigenvalue equa-tion (15) has to be solved. An exact diagonal-ization of the full matrix Ω is only possible forthe smallest systems, as the memory requiredto store Ω scales as O(N2

trans), which equates toa prohibitive O(N4

atom) scaling. A lot of applica-tions only need a small part of the spectrum atits low energy end, so it is possible to use iter-ative eigensolvers that avoid storage of the fullmatrix Ω in favor of a series of matrix-vectormultiplications. Especially popular in the con-text of TD-DFT(B) is a class of methods basedon an idea by Davidson,33 in which the eigen-value problem is solved approximately in a smallsubspace, which is then iteratively extendedand refined to include the desired eigenvectorswithin a certain accuracy. There is a multitudeof different Davidson based diagonalization al-gorithms and reviewing them would be beyondthe scope of this article. As the eigensolver forTD-DFTB calculations we use a variant of theGD+k method developed by Stathopoulos andSaad34 and implemented in the PRIMME li-brary.35 While the eigensolver internally needsto store the subspace basis, this required mem-ory scales as O(Ntrans) and is often negligible incomparison to the (Ntrans×Nexcit) matrix of thedesired eigenvectors.

3.1 Efficient implementation ofthe matrix-vector multiplica-tion

Eigensolvers based on the Davidson method33

solve the eigenvalue problem approximately ina small subspace which is then iteratively ex-panded by adding new basis vectors until itcontains the desired eigenvectors. They onlyuse the matrix they diagonalize in terms ofa matrix-vector multiplication with the newlyadded basis vectors. In practice this is actuallya matrix-matrix multiplication as it is commonto add Nblock ≥ 1 basis vector per iteration.This is known as the block Davidson methodwhich was proposed by Liu36 as a method to in-crease computational efficiency and to improve

convergence for degenerate eigenvalues. In caseof the block Davidson method the only part ofthe algorithm that is referencing the originalmatrix Ω can be written as

R = ΩT , (28)

where T is an (Ntrans × Nblock) matrix whosecolumns are the newly added basis vectors.We want to discuss the implementation of thismatrix-vector multiplication in some more detailnow, as it is crucial to the performance of theentire TD-DFTB method.As storage of the full matrix Ω is certainly

impossible – hence the iterative solution in thefirst place – we need to recalculate its elementsduring every matrix-vector multiplication. Wecan, however, precalculate a set of smaller aux-iliary objects from which Ω can be obtainedmore quickly.Inserting equation (27) into (28) it is easy to

see that one can precalculate a scaled versionof the atomic transition charges qij,A in orderto turn the multiplication with the large cou-pling matrix K into a series of matrix-matrixmultiplications involving only smaller matrices.

Ria,I = ∆2iaTia,I + 4

∑A

√∆ia qia,A︸︷︷︸hia,A

(29)

∑BγAB

∑jb

√∆jb qjb,B︸︷︷︸hjb,B

Tjb,I

R = diag(∆2ia

)T + 4hγhTT (30)

Here h is of size (Ntrans × Natom) whereas γis (Natom ×Natom). In order to ensure the over-all cubic scaling of the matrix-matrix productswe need to evaluate the subexpressions via tem-porary objects.

XBI =∑jb

hjb,BTjb,I (31)

YAI =∑BγABXBI (32)

Ria,I = ∆2iaTia,I + 4

∑Ahia,AYAI (33)

Here the first and third step scale asO(NtransNatomNblock), whereas O(N2

atomNblock)operations are needed for the intermediate step,

6

which is negligible since Natom Ntrans. Notethat this is only the scaling of a single matrix-vector product, which is different from the totaltime spent in matrix-vector products: Con-sidering the entire calculation instead of thesingle product, the total number of trial vectorsrequired for convergence is roughly linear inthe number of requested excitations Nexcit, nomatter how the trial vectors are blocked duringthe multiplications. Ergo, it is more insight-ful to consider the scaling of the total timespent performing matrix-vector products, whichis O(NtransNatomNexcit). The scaling behavior ofthe different operations involved in TD-DFTBis summarized in table 1.Equation 30 provides an extremely fast way

to perform the matrix-vector product as onlybasic linear algebra operations are used whichcan be offloaded to highly optimized libraries.If for large systems the matrix h of the scaledatomic transition charges becomes too large tobe stored though, it is necessary to recalculateits elements during the matrix-vector multipli-cations. Looking again at equation (23) it iseasy to see that the sum over ν is just a reg-ular matrix-matrix multiplication between theoverlap matrix S and the coefficient matrix c.

hia,A = 12

√∆ia

∑µ∈A

∑ν

(cµiSµνcνa + cνiSνµcµa

)

= 12

√∆ia

∑µ∈A

(cµiΘµa + cµaΘµi

)(34)

The product matrix Θ = Sc can be calculatedin advance and stored instead of S without ad-ditional memory in a full matrix storage imple-mentation. The calculation of the scaled atomictransition charge hij,A then only contains a sumover the basis functions centered on atom A,which is usually a small number due to the min-imal basis set and the large frozen core typicallyused in DFTB calculations. Note that precal-culating Θ makes it possible to calculate theelements of h in a system-independent constanttime, so that evaluating them on-the-fly doesnot change the scaling of the matrix-vector mul-tiplication but only increases the prefactor.In case of precalculated atomic transition

charges one can rely on standard libraries to

perform the parallelization of the matrix-vectorproduct. This is no longer true for on-the-fly cal-culated transition charges, where one has to par-allelize equation (31) and (33) manually. Bothequations can easily be parallelized, but one hasto pay attention to distribute the work such thateach scaled atomic transition charge hij,A is intotal only calculated once per step: The ele-ment XBI in equation (31) depends both on theatom B as well as the trial vector index I, butthe element hjb,B only depends on the atom B.Therefore, the parallelization is chosen to bedone over the atoms B since parallelizing overthe index I would require every processor tocalculate hjb,B. The matrix-matrix product inequation (33) is chosen to be parallelized via thetransition index ia for the exact same reason.In summary, recalculating the atomic transi-tion charges on-the-fly during the matrix-vectormultiplications removes the need to store thematrix h of size (Ntrans ×Natom). The storagerequired for the coefficient matrix c and productmatrix Θ can usually be neglected compared tothe (Ntrans ×Nexcit) matrix of the desired eigen-vectors. The memory requirements for all thedifferent methods are summarized in table 2.At this point it is necessary to mention that

while its performance is certainly important, thematrix-vector multiplication is not always thebottleneck of the Davidson eigensolver. The rea-son for this is that in order to find the Nth eigen-vector it is necessary to orthonormalize it againstthe N − 1 already known eigenvectors. This hasan O(NtransN

2excit) scaling which for large Nexcit

dominates over the O(NtransNatomNexcit) scalingof the matrix-vector multiplication.

3.2 Basis size reduction by tran-sition selection

While iterative eigensolvers make TD-DFTB cal-culations of larger molecules possible in the firstplace, the huge dimension Ntrans of the singleorbital transition space still limits the size ofthe treatable systems. It is therefore worthwhileto investigate the possibility of working in asubspace of single orbital transitions in whichthe (approximately) same result can be obtainedusing fewer transitions.

7

The most obvious way to reduce the basis sizeis a truncation in energy: As the iterative so-lution of the eigenvalue problem only targetsa few of the lowest eigenvectors of a typicallydiagonally dominant matrix, the eigenvector canbe expected to have little overlap with basis vec-tors for which the diagonal element is large. Inphysical terms this just means that the transi-tions from the lowest most tightly bound molec-ular orbitals to the highest virtuals will usuallynot contribute to the lowest excitations, whichmostly consist of transitions close to the HOMO-LUMO gap.Our target application of TD-DFTB are

UV/Vis absorption spectra, for which the solu-tion of Casida’s eigenvalue equation (15) pro-duces the excitation energies ∆I , while the cor-responding oscillator strengths fI can be calcu-lated through equation (19). Together these canimmediately be used to plot a stick-like spec-trum, that using Dirac’s δ-distribution could bewritten as

Astick(E) =∑I

fI δ(E −∆I) . (35)

As these spectra are both hard to interpret andunrealistic, it is common practice to artificiallyintroduce line broadening through a convolutionwith a peaked function Γ (E).

Abroad.(E) =∫

dE ′ Γ (E ′ − E)Astick(E ′)

=∑I

fI Γ (E −∆I)(36)

Both Gaussian and Lorentzian functions arecommon choices for Γ (E). As the absorptionpeaks are scaled with the oscillator strength fIof the excitation, the absorption spectrum ismostly determined by the excitations whichhave a large oscillator strength. Looking atequation (19) for the transition dipole momentof the excitations, it is easy to see that sin-gle orbital transitions with a small transitiondipole moment ~dia contribute little to the tran-sition dipole moment of the excitation ~dI , andhence its oscillator strength fI . Consequentlyit appears to be a reasonable approximationto remove those single orbital transitions from

the basis for which the oscillator strength fiais small. Note that this is an approximation,as even leaving out a single orbital transitionwith fia = 0 might still influence the oscillatorstrength fI through an overall change in thecorresponding eigenvector ~FI . The benefit ofremoving single orbital transitions with smalloscillator strengths fia goes beyond the obvi-ous reduction in computational effort associatedwith the smaller dimension of the eigenvalueproblem: As one is essentially working in the os-cillator strength carrying subspace, many of theexcitations with small oscillator strength fI arealso removed from the final spectrum, makingit possible to calculate the absorption spectrumin a fixed energy window with fewer excitations.It is in fact an all too common problem that alarge number of excitations has to be calculatedin order to cover the energy window of interest,while only a few of them actually determine theshape of the absorption spectrum due to theirlarge oscillator strength fI .For a direct diagonalization of the Ω matrix

it is obvious that the relative reduction in basissize translates quadratically into memory sav-ings and cubically into reduced processor time,compare table 1 and 2. For the iterative solversthe situation is more complicated due to thefact that the number of excitations that have tobe calculated within a fixed energy interval isalso reduced: Depending on whether the matrix-vector multiplication or the orthonormalizationof the subspace basis is the bottleneck, the rela-tive reduction in basis size will translate eitherquadratically or cubically into reduced processortime.The idea to reduce the number of consid-

ered single orbital transitions is not entirelynew: A truncation of the single orbital tran-sition space based on orbital localization hassuccessfully been used by Besley for the spe-cial cases of molecules in solution and on sur-faces.37 The more generally applicable trunca-tion in energy or oscillator strength has recentlybeen also proposed and tested in the PhD thesisof Domínguez, but no in-depth evaluation of themethod was performed.38 In the next sectionwe will assess the validity of the approximationsintroduced by truncating the basis in energy or

8

oscillator strength, and we will show that thesetechniques can at negligible loss in accuracy leadto orders of magnitude reductions in computertime and required memory.

4 ExamplesThe accuracy loss due to the additional approx-imation introduced by the truncation of thesingle orbital transition basis certainly needs tobe investigated in order to judge whether theseapproximations can be used in practice. Further-more we need to determine to which extent theloss in accuracy is justified by the computationalbenefits of truncation. Detailed timings of thevarious example calculations can be found in ta-ble 3. Note that we can use arbitrary units as weare only comparing theoretical data in these ex-amples, for comparison with experimental dataone may insert the appropriate prefactors forthe desired unit system.

4.1 Fullerene C60

The fullerene C60 was used by Niehaus et al. inthe original TDDFTB article16 as a benchmarkto judge the quality of the approximations in-troduced by TD-DFTB in general. The authorsfound that the inclusion of coupling between thesingle orbital transitions is crucial in the descrip-tion of the optical properties of C60 and thatTD-DFTB qualitatively reproduces the mainfeatures of the experimental spectrum.39We have performed a series of calculations

with differently truncated single orbital transi-tion spaces. With 4 valence electrons per atom,the C60 molecule has 120 occupied and 120 vir-tual orbitals (assuming a minimal basis), whichresults in a total of 14400 single orbital tran-sitions. For this rather small number of tran-sitions it is still possible to perform an exactdiagonalization of the Ω matrix. We used thecarbon parameters included in the mio-1-1 pa-rameter set.6Figure 1 shows absorption spectra calculated

using a basis from which single orbital transi-tions with an oscillator strength fia smaller thana user defined threshold fmin

ia have been removed.

Absorption[arb.units]

2 3 4 5 6 7

∆I [eV]

all

fia > 0.001(25% Ntrans/3% twall)

all


all

fia > 0.01(15% Ntrans/0.8% twall)

all


Figure 1: TD-DFTB calculated absorption spectraof C60 fullerene with different intensity selectionthresholds. The percentage in the parentheses isthe size of the remaining basis and the requiredcomputational time relative to the full calculation.

As expected the quality of the approximationdecreases as the threshold fmin

ia is increased andmore and more of the single orbital transitionsare removed. Note that there is a slight blueshiftof the main peaks for larger fmin

ia . Looking atthe bottom plot in figure 1 one can see that alarge part of the basis does not seem to con-tribute to the absorption spectrum at all, asa threshold of fmin

ia = 0.001 already removesthree quarters of all single orbital transitionswhile leaving the obtained absorption spectrumpractically unchanged. The reason for this isthat for the highly symmetric fullerene C60 thereare a lot of single orbital transitions where thetransition dipole moment ~dia and hence the oscil-lator strength fia is zero purely due to symmetry.This is a great advantage for the use of intensityselection and leads to a wall time reduction bytwo orders of magnitude at a negligible loss in ac-

9


2 3 4 5 6 7

∆I [eV]

all

∆ia < 33eV(77% Ntrans/56% twall)

all


all


all


Figure 2: TD-DFTB calculated absorption spectraof C60 fullerene with different energy truncationthresholds. The percentage in the parentheses isthe size of the remaining basis and the requiredcomputational time relative to the full calculation.

curacy for a selection threshold of fminia = 0.002.

We will later look at less symmetric examplesthough, where this does not play a role.Figure 2 shows absorption spectra calculated

using a basis from which single orbital transi-tions with a large orbital energy difference ∆ia

have been removed. It is evident that truncationof the basis in energy has a relatively large effecton the absorption spectrum, at least comparedto the intensity selection. While the numberof peaks is preserved upon energy truncation,they are subject to a sizeable blueshift and theirrelative oscillator strength is not well preserved.Overall this results in a too strong absorptionband around 6eV that does not exist in thisform in calculations using the full basis. A pos-sible reason for the mediocre performance of theenergy truncation could be the fact that the or-bital energy difference directly enters into equa-

tion (19) for the transition dipole moment ~dI ofthe linear response excitations, giving high en-ergy transitions a disproportionately large effecton the low energy end of the absorption spec-trum, even though the associated eigenvectorelements Fia,I might be rather small. A majordisadvantage of the truncation in energy com-pared to the intensity selection is that it doesnot reduce the number of excitations per en-ergy interval, so that for the iterative solver therelative reduction in basis size translates onlylinearly into memory savings and reduced pro-cessor time. Our overall experience is that thetruncation in energy introduces non-negligibleerrors while offering only moderate computa-tional advantages. While it is easily possible tocombine truncation in energy with truncation inoscillator strength, we have found that even thisis consistently outperformed by pure intensityselection on which we will therefore focus in theremainder of this article.

4.2 Ir(ppy)3

The compound Tris(2-phenylpyridine)iridium,abbreviated as Ir(ppy)3, has recently been dis-cussed in the context of highly efficient organiclight emitting diodes.40 There are two geomet-rical isomers, facial (fac-Ir(ppy)3) and merid-ional (mer-Ir(ppy)3), where the former is lowerin energy. We will therefore only discuss thefac-Ir(ppy)3 isomer. While the triplet excita-tions of Ir(ppy)3 are technically more interestingdue to their role in the process called triplet-harvesting,41 theoretical as well as experimen-tally obtained absorption spectra can also befound in the literature.42,43 These show two ab-sorption bands around 3.5eV and 5eV. The for-mer band has been found to originate from metalto ligand charge transfer, while the latter moreintense band around 5eV has been attributed toπ − π∗ excitations in the ligand.We performed TD-DFTB calculations on fac-

Ir(ppy)3 using the parameters developed byWahiduzzaman et al. 7 , which include param-eters for the central Iridium atom. Ir(ppy)3 hasa total of 7830 single orbital transitions so thatthe Ω matrix can easily be diagonalized exactly.Figure 3 shows the TD-DFTB calculated ab-

10


2.5 3 3.5 4 4.5 5 5.5 6

∆I [eV]

all


all


all


all


Figure 3: TD-DFTB calculated absorption spec-tra of fac-Ir(ppy)3 with different intensity selectionthresholds. The percentage in the parentheses isthe size of the remaining basis and the requiredcomputational time relative to the full calculation.

sorption spectrum obtained with intensity se-lection at different oscillator strength thresh-olds. TD-DFTB reproduces the general shapeof the TD-DFT calculated absorption spectrapublished by Asada et al. 42, though the moreintense band at higher energies is blueshifted byabout 0.5eV. As was the case for the fullereneexample, the absorption spectrum is practicallyunchanged when imposing an intensity selec-tion threshold of fmin

ia = 0.001. In contrast tothe fullerene example though, the resulting re-duction of the basis size is far less drastic: Athreshold of fmin

ia = 0.001 removes 75% of thefullerene single orbital transitions, but only 28%of the transitions in Ir(ppy)3. This is due to thefact that the less symmetric fac-Ir(ppy)3 doesnot have any single orbital transitions whosetransition dipole moment vanishes purely dueto symmetry. Increasing the selection threshold

decreases the quality of the approximation asseen in figure 3, but it is not until fmin

ia = 0.03(which results in a 76% reduction) that the spec-trum starts to become qualitatively different.Overall, carefully used intensity selection in caseof fac-Ir(ppy)3 provides sizable computationaladvantages with wall time reductions up to oneorder of magnitude and little loss of accuracy.

4.3 UbiquitinUbiquitin44 is an extremely common small pro-tein that has various regulatory functions inalmost all eukaryotic cells.45–47 It has recentlybeen used as an example system for UV/VISspectroscopy of entire proteins in gas phase48so that both experimentally observed as wellas theoretically calculated absorption spectraare available.49 The low energy part of the ubiq-uitin absorption spectrum is dominated by ab-sorption in the single tyrosine amino acid, sothat Bellina et al. were able to calculate ubiq-uitin’s absorption spectrum using a QM/MMapproach,49 where the tyrosine chromophore isembedded into a classical environment (modeledwith the Amber force field50), while the chro-mophore itself is treated quantum mechanicallywith TD-DFT (B3LYP/aug-cc-pvdz).We performed TD-DFTB calculations using

the mio-1-1 parameter set6 based on the PBEfunctional.51 For such a large system the it-erative solution of the eigenvalue problem isessential, but with 1231 atoms and in total2 284 880 single orbital transitions the 22 giga-byte matrix of atomic transition charges canstill be precalculated and stored in memory, sothat the matrix-vector multiplication can be im-plemented as equation 30. If one attempts tocalculate the absorption spectrum up to 200nmwithout using intensity selection, one quicklyfinds that there are almost 16 000 single or-bital transitions within this window, so thatan equally large number of excitations wouldhave to be calculated to get the interesting partof the absorption spectrum. With 18 megabyteof memory per eigenvector, this would requirealmost 290 gigabyte to store the solution, whichis rather excessive. Analysis of the single orbitaltransitions reveals though, that many of them

11

1 2 3 4 5 6

∆ia [eV]

800 500 400 300 200

λ [nm]

1

10

100

1000

Number

ofsingleorbital

tran

sition

s

allfia > 0.001

fia > 0.02

fia > 0.1

Figure 4: Number of single orbital transitions perenergy interval for ubiquitin for different intensityselection thresholds. Note the large number of lowintensity transitions below 3.5eV that is removedby even a small threshold.

have a very small oscillator strength. This isvisualized in figure 4 where the number of singleorbital transitions per energy interval is plottedfor different oscillator strength thresholds. It isevident that almost all single orbital transitionsbelow 4eV have an oscillator strength fia < 0.001and would be removed if intensity selection wasapplied. Setting a threshold of fmin

ia = 0.001 intotal removes 29% of the single orbital transi-tions, but looking only at the relevant part of thespectrum up to 200nm it reduces the number oftransitions to about 1600, which is a reductionby one order of magnitude. This not only makesthe solution much faster, but also only requiresmemory for 1600 eigenvectors of 13 megabyteeach, which is 21 gigabyte in total and certainlymanageable.The absorption spectrum of ubiquitin cal-

culated using TD-DFTB with different inten-sity selection thresholds is shown in figure 5.Except for a slight redshift of the first absorp-tion band around 267nm, TD-DFTB overallvery well reproduces the spectrum obtained byBellina et al.. Concerning the intensity selec-tion, it is especially remarkable that imposingthe aforementioned oscillator strength thresholdof fmin

ia = 0.001 does not change the resultingabsorption spectrum at all, even though it re-duces the number of excitations in the shown

Absorption

[arb.units]

240 250 260 270 280 290 300

λ [nm]

all


all


all


all


Figure 5: TD-DFTB calculated absorption spectraof ubiquitin with different intensity selection thresh-olds. The percentage in the parentheses is the sizeof the remaining basis and the required computa-tional time relative to the full calculation. Notethat the intensity-selected calculations were run onfewer cluster nodes than the full calculation so thatthe shown wall times underestimate the speedup.Detailed timings can be found in table 3.

energy window by more than one order of mag-nitude. Increasing the threshold to fmin

ia = 0.02removes 70% of the basis while still producingan essentially perfect absorption spectrum at adrastically reduced computational cost: Whilethe calculation using the full basis took morethan 12 hours and had to be run on 8 clusternodes due to its substantial memory require-ments, the intensity-selected calculation witha fmin

ia = 0.02 threshold finished in less than15 minutes on only two cluster nodes. Furtherincreasing the threshold to fmin

ia = 0.05 the in-tensity selection’s influence on the spectrumbecomes more noticable: We observe a slightblueshift and an increase in intensity of theband around 267nm, and for large thresholds

12

we also see some excitations vanish, most no-tably two relatively intense excitations at 254nmand 277nm, whose disappearance further con-tributes to making the central absorption bandstand out.The reason why there are so many excita-

tions with practically zero oscillator strengthat low energies is that these are mostly charge-transfer excitations, where an electron is trans-ferred from one part of the molecule (the donor)to another part (the acceptor), possibly overa relatively long distance. It is widely knownthat Kohn-Sham DFT based calculations candrastically underestimate the excitation ener-gies of such charge-transfer excitations, due tothe fact that the LUMO energy of the acceptordoes not correspond to its electron affinity, aswould be correct in case of a charge-transferexcitation where the acceptor essentially gainsan additional electron.52 It is interesting to notethough that charge-transfer excited states typi-cally have a small overlap with the ground stateand thereby according to equation (19) also arather small transition dipole moment.53 Whileintensity selection by no means solves the un-derlying problem of too small charge-transferexcitation energies in Kohn-Sham DFT, it atleast helps to alleviate the worst of the asso-ciated computational problems for the specificapplication of calculating electronic absorptionspectra.As a last practical example we have tried to

reproduce the spectral shift associated with theinclusion of the tyrosine chromophore into theprotein environment. Figure 6 shows the TD-DFTB calculated absorption spectra of ubiq-uitin and the isolated tyrosine in comparison.TD-DFTB predicts a blueshift of about 4nmupon embedding of the chromophore into theprotein environment, which is in agreement withthe shift calculated by Bellina et al.. This goesto show that intensity-selected TD-DFTB is aviable alternative to QM/MM methods for thecalculation of electronic absorption spectra oflarge compounds. In addition to the more ac-curate treatment of the environment, a generaladvantage of TD-DFTB over QM/MM is that itis much easier to use, as the user does not haveto first identify the chromophore and does not

240 250 260 270 280 290 300

λ [nm]

Absorption

[arb.units]

Tyrosine

Ubiquitin

Figure 6: Comparison of the absorption spectraof tyrosine and ubiquitin. The embedding of thetyrosine into the protein environment produces aslight redshift of the absorption band at 264nm.

have to make decisions on which part to treatquantum mechanically and how to embed it intothe classically treated region.

4.4 Parallel scalingIn order to evaluate the performance of ourparallel implementation, we have conducted ascaling test for the example calculation of theubiquitin absorption spectrum with an intensityselection threshold of fmin

ia = 0.005. The scalingtest was performed on 1 to 8 cluster nodes withtwo octa-core Intel Xeon E5-2650 v2 processorseach and 64GB of memory per node. The par-ticular threshold of fmin

ia = 0.005 was chosen asit results in a calculation that (using precalcu-lated atomic transition charges) barely fits intothe memory of a single node. In this way weconducted the scaling test on the largest systemwe were able to solve in serial, allowing us toplot the scaling behavior for the entire rangefrom 1 to 128 cores. The result of the scalingtest for both precalculated and on-the-fly atomictransition charges is shown in figure 7.For precalculated transition charges we ob-

serve a good scaling both within a single nodeand across nodes. Within a single node (smallpanel in figure 7) it is interesting to note that weobserve a super linear speedup when going from1 to 2 or 4 cores, while for more than 8 coresthe additional speedup is rather small. We at-tribute this to the cores’ competition for sharedresources like cache and memory bandwidth,which become available as more subunits of the

13

0 1 2 3 4 5 6 7 8

Number of nodes

0

10

20

30

40

50

60

70

80

90

100

110

120

Excitationsper

minute

precalc. qia,I

on-the-fly qia,I

Number of cores1 4 8 12 16

0

5

10

15

20

Figure 7: Parallel scaling of our TD-DFTB imple-mentation for the test case of ubiquitin with anintensity selection threshold of fmin

ia = 0.005. Thecalculations were performed on 1 to 8 cluster nodeswith two octa-core Intel Xeon E5-2650 v2 processorseach and 64GB of memory per node.

machine (e.g. both sockets) are used, but areultimately exhausted when too many processorcores compete for them. The visible oscillationsin figure 7 are due to the use of ScaLAPACK54

to implement equation (30), which favors evenand especially power of two processor grid sizes.For on-the-fly calculated atomic transition

charges the overall performance is worse, but theparallel scaling both within a node and acrossnodes is better. Within a single node this isdue to the absence of the large matrix of atomictransition charges, which reduces reading frommain memory and thereby frees shared resources.The better scaling across nodes is simply due tothe fact that a relatively large amount of timeis spent on the trivially parallel task of recalcu-lating atomic transition charges, for which nocommunication is required.

5 ConclusionIn summary, we have shown that the compu-tational cost of absorption spectra calculationsusing time-dependent density functional tightbinding (TD-DFTB) can be significantly re-duced by not considering single orbital tran-sitions with small oscillator strengths. We havefound that small selection thresholds do not no-

ticeably affect the accuracy of the result, whilealready providing sizable computational bene-fits. This is especially true if the low energypart of the absorption spectrum contains a largenumber of spurious low-intensity charge-transferexcitations which can be removed through in-tensity selection, making otherwise infeasiblecalculations possible. As an example, we havecalculated the absorption spectrum of ubiqui-tin and the spectral shift upon embedding itstyrosine chromophore into the protein environ-ment, and have demonstrated that the accuracyof intensity-selected TD-DFTB is on a par withcompeting QM/MM methods, which tend torequire more work and expertise from the user.We believe that its ease of use together with

the moderate computational cost of intensity-selected TD-DFTB lower the barrier of per-forming optical properties calculations of largemolecules, and can serve to make such calcula-tions possible in a wider array of applications.Intensity-selected TD-DFTB has been imple-mented in the 2014 release of the ADF molecularmodeling suite.55

Acknowledgement The authors thank OnnoMeijers for his work on the eigensolver backend.The research leading to these results has receivedfunding from the European Union’s SeventhFramework Programme (FP7-PEOPLE-2012-ITN) under project PROPAGATE, Ref. 316897.Supporting Information Available: We

provide the molecular geometries for all exam-ple systems from section 4. This materialis available free of charge via the Internet athttp://pubs.acs.org/.

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Table 1: Computational complexity of operations within the TD-DFTB method.

Operation Computational complexity of operations per method:direct diag. Davidson (precalc) Davidson (on-the-fly)

direct diag. of full Ω N3trans – –

subspace basis orthon. – NtransN2excit

mat.-vec. multiplication – NtransNatomNexcitwith a small prefactor

NtransNatomNexcitwith a large prefactor

Table 2: Memory requirements of the TD-DFTB method. Note that this table only contains thelargest objects needed during the diagonalization itself.

Object Scaling of storage requirements per method:direct diag. Davidson (precalc) Davidson (on-the-fly)

full matrix Ω N2trans – –

eigenvectors ~FI N2trans NtransNexcit

subspace basis vectors – Ntrans with a large prefactorparameter matrix γ – N2

atomatomic transition charges h – NtransNatom –

coefficient matrix c – – N2atom

product matrix Θ = Sc – – N2atom

Table 3: Measured runtimes of the example TD-DFTB calculations using intensity selection. Thecalculations for C60 and Ir(ppy)3 were performed on a workstation with an Intel Core i7-4770processor and 16GB memory. The ubiquitin calculations were performed on 1 to 8 cluster nodeswith two octa-core Intel Xeon E5-2650 v2 processors each and 64GB of memory per node.

System Natom fminia Ntrans Nexcit #CPU twall tCPU

C60 60 – 14400 4 434s 1736sC60 60 0.001 3610 4 12s 49sC60 60 0.005 2581 4 5.5s 22sC60 60 0.01 2113 4 3.5s 14sC60 60 0.05 1032 4 0.8s 3.2s

Ir(ppy)3 61 – 7830 4 88s 352sIr(ppy)3 61 0.001 5656 4 37s 148sIr(ppy)3 61 0.01 3326 4 10s 40sIr(ppy)3 61 0.02 2426 4 4.9s 20sIr(ppy)3 61 0.03 1896 4 2.8s 11sUbiquitin 1231 – 2 284 880 15820 128 12.6h 67dUbiquitin 1231 0.001 1 628 370 1638 48 1.1h 2.1dUbiquitin 1231 0.02 689 208 552 32 635s 5.6hUbiquitin 1231 0.05 333 337 359 16 332s 1.5hUbiquitin 1231 0.1 156 488 232 16 137s 2192s

17

eﬃcientcalculationofelectronicabsorption ... · occ andthe number of virtual orbitals n virt grow...

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