Excited-state electronic structure of molecules using many-body Green’sfunctions: quasiparticles and electron-hole excitations with VOTCA-XTP

G. Tirimbo,1, 2 V. Sundaram,1, 2 O. Caylak,1, 2, 3 W. Scharpach,1, 2 J. Sijen,1, 2 C. Junghans,4 J. Brown,5, 6 F.Zapata Ruiz,7 N. Renaud,7 J. Wehner,1, 2 and B. Baumeier1, 2, a)1)Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513,5600 MB Eindhoven, The Netherlands2)Institute for Complex Molecular Systems, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven,The Netherlands3)Institute for Pure and Applied Mathematics, University of California Los Angeles, 460 Portola Plaza, Los Angeles,CA 90095, USA4)Computer, Computational, and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos,NM 87545, USA5)Dept. of Electrical Computer and Energy Engineering, University of Colorado Boulder, 425 UCB, Boulder,CO 80309, US6)Renewable and Sustainable Energy Institute, University of Colorado Boulder, 4001 Discovery Dr, Boulder,CO 80303, USA7)Netherlands eScience Center, Science Park 140, 1098 XG Amsterdam, The Netherlands

(Dated: 30 December 2019)

We present the open-source VOTCA-XTP software for the calculation of the excited-state electronic structureof molecules using many-body Green’s functions theory in the GW approximation with the Bethe–SalpeterEquation (BSE). This work provides a summary of the underlying theory and discusses details of its imple-mentation based on Gaussian orbitals, including, i.a., resolution-of-identity techniques, different approachesto the frequency integration of the self-energy or acceleration by offloading compute-intensive matrix opera-tions using GPUs in a hybrid OpenMP/Cuda scheme. A distinctive feature of VOTCA-XTP is the capabilityto couple the calculation of electronic excitations to a classical polarizable environment on atomistic levelin a coupled quantum- and molecular-mechanics (QM/MM) scheme, where a complex morphology can beimported from Molecular Dynamics simulations. The capabilities and limitations of the GW -BSE implemen-tation are illustrated with two examples. First, we study the dependence of optically active electron-holeexcitations in a series of diketopyrrolopyrrole-based oligomers on molecular-architecture modifications andthe number of repeat units. Second, we use the GW -BSE/MM setup to investigate the effect of polarizationon localized and intermolecular charge-transfer excited states in morphologies of low-donor content rubrene-fullerene mixtures. These showcases demonstrate that our implementation currently allows to treat systemswith up to 2500 basis functions on regular shared-memory workstations, providing accurate descriptions ofquasiparticle and coupled electron-hole excited states of various character on an equal footing.

Keywords: excited states, Green’s functions, electronic structure, GW -BSE, QM/MM

I. INTRODUCTION

The functional role molecules play in either fabricateddevices, such as organic light-emitting diodes,1,2 photo-voltaic cells3,4 and photodetectors,5 or in naturally oc-curring mechanisms,6,7 is determined by characteristicsof their electronically excited states. In many cases thefocus lies on dynamical processes, like the diffusion ofa photoexcited state, its conversion into free charges(or vice versa), or drift-diffusion of carriers. While anadequate description of such dynamics depends on thespecifics of material composition and/or operating condi-tions in a device, basic understanding and control of theprocesses are often sought via analysis of the excited-state electronic structure in terms of energy levels orstate diagrams.8 In this context, gaining insight from

a)https://www.baumeiergroup.com

first-principles calculations requires methods that reli-ably predict properties for a variety of excited states,ranging from single-particle, ionizing excitations to cou-pled electron-hole pairs on one or more molecules. Meth-ods also need to be practical, in the sense that they canbe applied to molecular complexes of the size relevant forthe aforementioned applications at manageable compu-tational cost. Finding the appropriate balance betweencomputational cost and accuracy plays a crucial role inanalyzing large complex systems.

Post-Hartree–Fock wave function-based approaches,such as coupled cluster or configuration interaction, cangenerally be used to determine excited-state propertieswith good accuracy.9–12 However, their large baselinecomputational cost and rapid scaling with system sizemake studying large molecular systems practically impos-sible in many situations. In contrast, density functionaltheory13 (DFT) is commonly used to calculate ground-state properties, while the properties of non-ionizing ex-citations are formally accessible by its time-dependent

2

(TD-DFT) extension.14,15 While the computational ef-fort of both techniques is moderate compared to thewave-function approaches, they rely on the choice of anapproximate exchange-correlation functional. Separatecalculations are necessary to determine reliable ioniza-tion energies, and the quality of predictions of differentexcited states, in particular of charge-transfer (CT) char-acter, is sensitive to the functional choice.16–18

Recently, the use of many-body Green’s functions the-ory,15,19–21 typically more rooted in the solid-state com-munity, has received increasing attention in quantum-chemical applications.22–33 It consists of a two-step pro-cedure, in which at first the addition and removal of anelectron is calculated within the GW approximation asquasiparticle excitations to an N -electron (DFT) groundstate. In the second step, neutral excitations are con-structed in the same framework as coupled electron-holepairs by solving the so-called Bethe–Salpeter Equation(BSE).15 Many-body effects, or electron correlation, aretaken into account via a non-local, energy-dependentelectron self-energy operator.34 This GW -BSE methodhas been shown to yield excitation energies with goodaccuracy for different types of excitations, e.g., localized(Frenkel) and bimolecular CT excitons,25,26 on an equalfooting. The computational cost of GW -BSE is com-parable to TD-DFT, which makes it an attractive tech-nique for the calculation of the excited-state electronicstructure of complex molecular materials — provided ad-equate software tools are available. Most common imple-mentations stem from the solid-state community and usea plane-wave basis with periodic boundary conditions.While it is possible to study molecules by choosing alarge unit cell in such calculations, this is computation-ally inefficient.

In this paper, we present an open-source imple-mentation of GW -BSE in the VOTCA-XTP package35

that expresses the electronic states using atom-centeredGaussian-type orbitals (GTOs). VOTCA-XTP is partof the VOTCA software suite,36–38 written in C++,and freely available on GitHub. It contains an in-ternal module for calculating the DFT ground statemostly for development purposes, and additionally pro-vides extensible interfaces to standard packages (Gaus-sian,39 NWChem,40 Orca41). One distinct feature ofthe GW -BSE implementation in VOTCA-XTP is thatit comes in two modes. The standalone mode operateslike any common quantum-chemistry software, and re-quires only the atom coordinates of the molecule andcalculation parameters as input. In the second mode, acomplex molecular morphology of, e.g., a solute-solventmixture or a donor-acceptor blend, is first simulated byMolecular Dynamics (MD) or similar techniques, andthen translated as a whole into an internal data struc-ture. This morphology mode facilitates the evaluationof the excited-state electronic structure in a complex en-vironment38,42–45 in a hybrid quantum- and molecular-mechanics GW -BSE/MM scheme,38 and is linked to amultiscale framework for the determination of dynam-

ical electronic properties. Additional features of thisframework include, i.a., the calculation of electronic46

and excitonic47 intermolecular coupling elements (trans-fer integrals), the prediction of ultraviolet photoelectronspectra including carrier-vibration coupling,45 the sim-ulation of optical absorption and emission spectra fromcoupled solute-solvent relaxation,48 and the determina-tion of charge-carrier mobilities37 and exciton diffusionlengths49 via kinetic Monte Carlo.

Here, we give a concise overview of the underlying the-ory and details of the implementation, including, e.g.,the use of resolution-of-identity techniques, different ap-proaches to the frequency integration of the self-energy,or acceleration by offloading compute-intensive matrixoperations using GPUs in a hybrid OpenMP/Cudascheme. We highlight VOTCA-XTP’s capabilities us-ing two showcase applications related to organic photo-voltaics. The first one uses the standalone mode andcomprises a study of the lowest optically active excita-tions in a series of diketopyrrolopyrrole-derived (DPP)oligomers depending on molecular-architecture modifica-tions and the number of repeat units. DPP polymers areused as donor materials in bulk heterojunctions with afullerene acceptor.50–52 The polymeric nature of the ma-terial allows a systematic demonstration of the accessi-ble system sizes and the computational cost involved. Atthe same time, this study yields insight into the effects ofdifferent approximations on the excitation energies andthe localization behavior of the photoexcited electron-hole pair. The second case study uses the morphologymode to illustrate the importance of accounting for en-vironment effects on excited states.26 We consider a low-donor content amorphous mixture of rubrene and C60

as a realistic model system for a small-molecule donor-acceptor blend.53 Particular focus lies here on evaluatingthe effects of atomistic polarization on single-molecule lo-calized Frenkel and bimolecular CT excitons in differentvariants of the GW -BSE/MM setup.

This paper is organized as follows: In Sec. II, we brieflysummarize the essential theoretical aspects underlyingGW -BSE, while technical details of the implementationas well as software development strategies are discussedin Sec. III. In Sec. IV, we showcase the capabilities ofVOTCA-XTP in two examples: the size dependence ofelectronic excitations in the DPP family of polymers,and the stabilization of CT excitations in rubrene-C60

mixtures due to environment polarization. Finally, weoutline current limitations and future development di-rections in Sec. V.

II. THEORY

For the sake of compactness, we restrict the followingdiscussion of the main concepts underlying GW -BSE toa spin-singlet, closed-shell system of N electrons. Us-ing DFT, the ground state |N, 0〉 is determined from the

3

solutions of the Kohn–Sham (KS) equations15

HKS|φKSi 〉 = [H0 + Vxc]|φKS

i 〉 = εKSi |φKS

i 〉. (1)

Here, H0 = T0 + Vext + VH, with T0 being the kinetic en-

ergy, Vext an external potential, VH the Hartree potential,

and Vxc the exchange-correlation potential.Excitations that add (N → N + 1) or remove (N →

N − 1) an electron from the system, referred to asquasiparticles (QPs), are determined by the one-particleGreen’s function19,54

G1(r1, t1, r2, t2) = −i〈N, 0|T (ψ(r1, t1)ψ†(r2, t2))|N, 0〉,(2)

where T is the time-ordering operator, and ψ and ψ† arethe annihilation and the creation electron field operator,respectively. This Green’s function obeys a Dyson-typeequation of motion, which reads in spectral representa-tion as

[H0 + Σ(E)]G1(E) = EG1(E) , (3)

where the electron self-energy operator Σ(E) containsthe exchange-correlation effects. This equation is partof a closed set of coupled equations, known as Hedin’sequations.55,56 An approximate solution to this systemis provided by the GW approximation, in which the self-energy takes the form

Σ(r, r′, ω) =i

2π

∫dω′G1 (r, r′, ω + ω′)W (r, r′, ω) ,

(4)i.e., it is a convolution of G1 with the screened Coulombinteraction W = ε−1vc, where vc(r, r

′) = |r − r′|−1 isthe bare Coulomb interaction and ε−1(r, r′, ω) is the in-verse dielectric function calculated in the Random-PhaseApproximation (RPA).20

Using this approximation, Eq. 3 is converted into aDyson-type equation of motion for the quasiparticles(i.e., the QP electron and hole states):23,57[

H0 + Σ(εQPi )

]|φQPi 〉 = εQP

i |φQPi 〉 , (5)

where εQPi are the one-particle excitation energies of the

system, and |φQPi 〉 are the QP wave functions.

In practice, the QP wave functions are expressed in

a basis of KS states, i.e., |φQPi 〉 =

∑j a

ij |φKS

j 〉. With

H0 = HKS − Vxc, diagonalizing the energy-dependentQP Hamiltonian in this basis as

HQPij (E) = εKS

i δij + 〈φKSi |Σ(E)− Vxc|φKS

j 〉 (6)

yields the QP states and energies.If the off-diagonal elements of Eq. 6 are small, i.e.

|φQPi 〉 ≈ |φKS

i 〉, the quasiparticle energies can be eval-uated perturbatively according to

εQPi = εKS

i +∆εGWi = εKSi +〈φKS

i |Σ(εQPi )−Vxc|φKS

i 〉. (7)

Computing εQPi requires determination of ∆εGWi , which

consequently leads to self-consistently solving Eq. 7.This involves either a linearization on a grid or fixed-point methods. Conventionally this is referred to asthe G0W0 approximation. To improve upon this one-shot approach, the evGW procedure can be used in-stead: QP energies are updated in the calculation of thenon-local, energy-dependent microscopic dielectric func-tion calculated within the RPA until eigenvalue (ev) self-consistency is reached.

Neutral excitations with a conserved number of elec-trons and a change in their configuration S (|N, 0〉 →|N,S〉) rely instead on the two-particle Green’s func-tion.19 This can be obtained from another Dyson-likeequation of motion, known as the Bethe–Salpeter Equa-tion (BSE).56 It determines the four-point density re-sponse function of the interacting system from the non-interacting system, see Refs. 15, 21, and 38 and referencestherein for a detailed derivation. In the specific case ofoptical excitations, one can employ a product basis of QPwave functions for coupled electron-hole amplitudes, i.e.,

χS(re, rh) =

occ∑v

unocc∑c

ASvcφc(re)φ∗v(rh) +BSvcφv(re)φ

∗c(rh),

(8)

where re (rh) is for the electron (hole) coordinate, and wedrop the label QP for clarity. With this choice of basis,the BSE can be transformed into an effective two-particleHamiltonian problem of the form(

Hres K−K −Hres

)(AS

BS

)= ΩS

(AS

BS

). (9)

Explicitly, the matrix elements of Hres and K are calcu-lated as

Hresvc,v′c′ = Dvc,v′c′ + κKx

vc,v′c′ +Kdvc,v′c′ (10)

Kcv,v′c′ = κKxcv,v′c′ +Kd

cv,v′c′ , (11)

where κ = 2 (0) for spin singlet (triplet) excitations, and

Dvc,v′c′ = (εv − εc)δvv′δcc′ , (12)

Kxvc,v′c′ =

∫d3re d3rh φ

∗c(re)φv(re)vc(re, rh)

× φc′(rh)φ∗v′(rh) (13)

Kdvc,v′c′ =

∫d3re d3rh φ

∗c(re)φc′(re)W (re, rh, ω = 0)

× φv(rh)φ∗v′(rh) . (14)

The exchange interaction Kx originates from the bareinteraction vc, and is responsible for the singlet-tripletsplitting. The direct interaction Kd contains the attrac-tive, but screened, interaction W between electron andhole. This interaction is responsible for the binding ofthe electron-hole pair. Furthermore, it is assumed herethat the dynamic properties of W (ω) are negligible, and

4

the computationally less demanding static approxima-tion ω = 0 is sufficient.

In systems for which the elements of the off-diagonalblocks K in Eq. 9 are negligible, it is legitimate to usethe Tamm–Dancoff Approximation (TDA),58 in whichthe electron-hole amplitude is expressed as

χTDAS (re, rh) =

occ∑v

unocc∑c

ASTDA,vcφc(re)φ∗v(rh), (15)

i.e., by resonant transitions from occupied v to emptyc states only. The effective Hamiltonian reduces to theupper diagonal block of Eq. 9:

HresASTDA = ΩTDA

S ASTDA. (16)

The TDA is known to reduce triplet instabilities.59 Onthe other hand, the coupling between resonant and anti-resonant parts is significant, and its neglect can causedeviations of several 0.1 eV from results obtained withthe full approach,22 in particular for small molecules.

III. TECHNICAL DETAILS AND IMPLEMENTATION

A. GW-BSE with Gaussian orbitals

VOTCA-XTP uses Gaussian-type orbitals (GTOs)ϕα(r) = ϕ`,m`,γ,I(r) = N`γY`m`

|r − RI |l exp(−γ|r −RI |2) with decay constant γ centered around atom posi-tion RI , where Y`m`

are spherical harmonics with angu-lar momentum number ` and magnetic quantum numbersm`, and N`γ is a normalization constant, to expand theone- and two-point quantities involved in both DFT andGW -BSE steps. Specifically, the KS states in Eq. 1 areexpressed using these basis functions in real space as

φKSi (r) =

∑α

ciαϕα(r) (17)

turning Eqs. 1, 6, and 9 into (generalized) eigenvalueproblems in matrix form. See also the discussion aboutefficient solvers for the BSE in Sec. III E. VOTCA-XTPevaluates the integrals of the respective operators overthe Gaussian basis functions using the modified recur-sive algorithms by Obara and Saika60,61 for contractedGaussian basis functions with l ≤ 4.

Of particular importance is the computation of 4-center repulsion integrals over the GTOs

(αβ|α′β′) =

∫∫d3r d3r′

ϕα(r)ϕβ(r)ϕα′(r′)ϕβ′(r′)

|r − r′| ,

(18)which scales with N4

b (with Nb the number of basis func-

tions) and occurs in the KS Hamiltonian term VH and inthe self-energy. The set of N2

b unique product functionsϕα(r)ϕβ(r) can be approximated by a smaller auxiliarybasis containing only 3Nb to 5Nb functions ξµ. This

reduces the scaling from N4b to N3

b by rewriting the 4-center integrals as a combination of 3-center and 2-centerrepulsion integrals:62

(αβ|α′β′) ≈∑µ,ν

(αβ|µ)(µ|ν)−1(ν|α′β′), (19)

where (µ|ν)−1 is an element of the inverse of the 2-centerrepulsion matrix

(µ|ν) =

∫∫d3r d3r′ ξµ(r)

1

|r − r′|ξν(r′) (20)

and (αβ|µ) is an element of the 3-center repulsion tensor

(αβ|µ) =

∫∫d3r d3r′ ϕα(r)ϕβ(r)

1

|r − r′|ξµ(r′). (21)

The expression in Eq. 19 appears formally as the insertionof a resolution-of-identity (RI) with metric (ν|µ)−1.

Within the RI approximation, the elements of the QPHamiltonian in the basis of KS states contain Σmn(E) =

〈φKSm |Σ(E)|φKS

n 〉 (Eqs. 6 and 7), which are determined as

Σmn(E) =∑µ,ν

∑l

Imlµ Inlνi

2π

∫dω

eiωθε−1µν (ω)

E + ω − εl ± iη,

(22)where the factor with θ → 0+ ensures convergence ofthe integral, and the imaginary perturbations ±η avoidsingularities on the real axis, where the plus (minus) istaken when l is occupied (empty). Further,

Imlµ =∑ν

(µ|ν)−1/2∑α,β

cmα clβ(αβ|ν) =

∑ν

(µ|ν)−1/2Mmlν

(23)and

εµν(ω) = δµν−2

occ∑m

unocc∑l

Imlµ Imlν

[1

ω − (εm − εl) + 2iη

− 1

ω + (εm − εl)− 2iη

].

(24)

is called the dielectric matrix. In the G0W0 approach,we take the KS energies εi = εKS

i , whereas in the evGW

approach, we take the QP energies εi = εQPi . Currently,

VOTCA-XTP pre-calculates all integrals at the start ofthe calculation and keeps Imlµ in memory.

B. Frequency dependence of the self-energy

The frequency integration, in Eq. 4, is one of the majordifficulties in a GW calculation. Although it is possibleto perform a numerical integration, this is likely unsta-ble, since the integrand needs to be evaluated in regionsin which it is ill-behaved. VOTCA-XTP offers differ-ent alternatives for an approximate or exact integration,

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summarized in the following. We present three methods:one that is exact yet takes too much computational ef-fort, one that is exact and is reduced in scaling, and onethat is approximate yet very well-scaled.

In the GW approach, it is customary to separate theself-energy Σ = iGW into its bare exchange part Σx =

iGvc and its correlation part Σc = iGW , where W =W−vc. In the following, several approaches to determinethe correlation part of Eq. 22 are presented.

1. Fully Analytical Approach (FAA)

The integral in Eq. 22 can be evaluated analytically,yielding an exact expression of the correlation part of theself-energy. Its evaluation requires the calculation of the

reducible polarizability P . We can express it in termsof an eigenvalue decomposition of the RPA Hamiltonian

HRPA

P (ω) = [HRPA − ω]−1 =∑S

|χS〉〈χS |ΩS − ω

, (25)

where the RPA Hamiltonian obeys a BSE as in Eq. 9with κ = −1 and Kd = 0 in Eqs. 10 and 11.

We can apply analytic continuation to the complexplane and contour deformation techniques to the con-volution Eq. 4.30,63 The resulting matrix entries of thecorrelation part of the self-energy are given by

Σc,mn(E) = 2∑l,S

RSmlRSnl

E − εl ± (ΩS − iη), (26)

where ± denotes + (−) for l occupied (unoccupied), andthe factor 2 accounts for spin. The residues RSmn arecalculated as

RSmn =∑µ,ν

occ∑v

unocc∑c

Imnµ Ivcν (ASvc +BSvc). (27)

While this approach is analytically exact, it is not fea-

sible for large systems as the diagonalization of HRPA

scales as N6 in computational effort and N4 in memoryrequired.21

2. Contour Deformation Approach (CDA)

To avoid the computational bottleneck due to the scal-ing of the FAA, an alternative approach, also involvinganalytic continuation of the integral over the real axisfrom Eq. 22 to the complex plane and contour deforma-tion techniques, can be employed. It yields a differentrewriting, containing an integral over the imaginary axisand residual contributions. Since the new integral is verypeaked around the origin when E ≈ ε` for some `, we adda Gaussian function inside the integral and subtract itsintegral value on the outside. We then can calculate the

matrix entries Σc,mn = Σintc,mn(E) + Σres

c,mn(E), with theintegral term

Σintc,mn(E) =

1

π

∑µ,ν,l

Imlµ Inlν

∫dω

E − εlω2 + (E − εl)2

×[κµν(0)e−α

2ω2 − κµν(iω)

],

(28)

which can be numerically evaluated using a Gauss–Laguerre quadrature method, and the residual term

Σresc,mn(E) =

∑µ,ν,l

Imlµ Inlν

[κµν(E − εl ± iη)Θl(E)

− 1

2κµν(0)eα

2(E−εl)2sgn(E − εl)erfc(α|E − εl|)].

(29)

Here, κ = ε−1−I, and α is a Gaussian scaling parameteradaptive to the numerical model to be used.64,65 Further-more, for l occupied Θl(E) is −1 if E < εl and −1/2 ifE = εl; for l unoccupied it is 1 if E > εl and 1/2 if E = εl.By design, the FAA is in principle the most exact one,since it is parameter-free except for the dependence onthe basis set. However, the same results can already beachieved with the CDA using a moderately-sized numer-ical integration grid for the Gauss–Laguerre quadrature,which reduces the scaling. This asset, however, may de-pend on the implementation, which is currently work inprogress for the CDA.

3. Plasmon-Pole Model (PPM)

Instead of a formally exact treatment of the frequencydependence, it can be approximated within a generalizedplasmon-pole model (PPM). The dielectric matrix can beexpressed in terms of its eigenvalues λµ′ and eigenvectorsΦµ′ as

εµν(ω) =∑µ′

Φµµ′(ω)λµ′(ω) Φνµ′(ω), (30)

In the PPM, eigenvectors are assumed to be frequency-independent, so only the eigenvalues λµ′ depend on ω.In particular, this approximate dependence reads

λ−1µ′ (ω) ≈ 1 +zµ′ωµ′

2

[1

ω − (ωµ′ − iη)− 1

ω + (ωµ′ − iη)

].

(31)Here, zµ′ denotes the plasmon-pole weight and ωµ′ de-notes the plasmon-pole frequency. These two model pa-rameters are found by fitting the plasmon-pole model tothe exact dielectric function, as shown in Eq. 24, for thefrequencies ω = 0 and ω = iE0, with E0 an additionalmodel parameter, typically E0 = 0.5 Ha. The correla-tion part of the self-energy results from the second termof Eq. 31, and its matrix entries are obtained as

Σc,mn(E) = 2∑l,µ′

1

4

zµ′ωµ′Imlµ′ Inlµ′

E − εl ± ωµ′, (32)

6

−4

−3

−2

−1

0

1

2

3

−30 −20 −10 0 10

ethene

−30 −20 −10 0 10

adenine

εQP−

εKS

(eV

)

εKS (eV)

FAAPPM

εKS (eV)

FIG. 1. Comparison of the QP corrections εQP−εKS vs the KSenergies for ethene (left) and adenine (right), obtained withthe frequency integration of Eq. 4 using the FAA (triangles)and the PPM (circles), respectively.

where ± denotes + (−) for l occupied (unoccupied), andthe factor 2 accounts for spin.

4. Comparison of FAA and PPM for QP excitations ofethene and adenine

Figure 1 illustrates the influence of the choice of tech-nique for the frequency integration on the obtainedQP corrections to the KS state energies for two smallmolecules: ethene and adenine. In both cases, the cal-culations have been performed using the aug-cc-pVTZbasis66 and the PBE067 functional for the DFT ground-state calculation, with the whole range of states (121 forethene, 399 of adenine) included in the RPA and QPsteps. The corrections for all occupied levels and thesame number of unoccupied levels is shown. Clearly,the QP corrections obtained with the PPM are slightlymore positive (about 0.3 eV) than with the exact methodfor the valence and lowest empty levels in both cases,whereas there is hardly any deviation for empty levels athigher energy. It should be noted that, due to the na-ture of the deviations, energy differences near the gap arevery similar for both methods. In particular, the QP gapbetween HOMO and LUMO levels as predicted by thePPM and the exact method differs by only 0.05 eV forthe two molecules considered here. However, this closeagreement is paralleled by a significant difference in com-putational cost:68 while the use of Eq. 26 increases thecomputation time from 4 min to 12 min compared to thePPM in ethene, we find that for adenine the same causesan increase from 174 min to 2566 min.

Considering the above, we find that the PPM appearsto be a suitable “low-cost” approximation to the exactfrequency dependence of the self-energy when the maininterest is in near-gap excitations such as HOMO andLUMO energies or HOMO-LUMO transitions in larger

Time (s)Molecule Nb Naux CPU GPUBenzene 174 774 21 9Naphthalene 274 1228 68 28Phenanthrene 374 1682 216 44Coronene 600 2724 1241 266

TABLE I. Comparison of calculation times for the convolu-tion of the 3-center repulsion tensor with the molecular or-bital coefficients on CPU (single thread of Intel(R) Xeon(R)Gold 5120) and GPU (Nvidia Titan Xp), respectively, for fourpolycyclic aromatic hydrocarbons with different numbers offunctions in the basis (Nb) and auxiliary basis (Naux).

molecular systems. Therefore, we use the PPM for theshowcase applications in Sec. IV.

C. Software Development Strategies

VOTCA-XTP is written in C++ and mostly adheresto the C++14 standard.69 It can be obtained on github.com/votca/xtp. We use the Git feature branch workflowcombined with code review and continuous integration,which executes code formatting, static analyzers, debugand release builds and the test-suite. We use CMakeas the build system, which also resolves the inclusionof external dependencies. The linear algebra is handledby Eigen,70 which can be accelerated by internally call-ing the Intel Math Kernel Library.71 For serialization,the HDF5 format is used via the canonical libraries.72

Exchange-correlation functionals are provided by theLIBXC package.73 Various boost packages74 are used forfile system operations, file parsing and string operations.Doxygen is used to document the APIs of VOTCA-XTPand automatically deploys to http://doc.votca.org.

VOTCA-XTP is designed as a library, which is linkedinto very thin executables, which can execute a varietyof calculators by adding keywords on the command line.Virtual interfaces and factory patterns make the additionof new calculators simple. The same architecture is usedfor external DFT and MD codes, making VOTCA-XTPeasily extensible. Lower-level data structures make use oftemplate metaprogramming to support a variety of datatypes.

D. GPU acceleration

The computation of theMmlν =

∑α,β c

mα c

lβ(αβ|ν) term

described in Eq. 23 requires the convolution of the 3-center repulsion tensor from Eq. 21 with the molecularorbital coefficients. These convolutions involve a largenumber of matrix-matrix multiplications that take a sig-nificant fraction of the computation time. In VOTCA-XTP, we have accelerated these operations by offloadingthem to a GPU, using the highly optimized CuBLAS li-

7

brary.75 Moreover, due to the typically limited memoryof the GPU and the latency required to copy the matri-ces back and forth from the device, there are limitationsto the theoretical speedup for a given system size.

Table I illustrates the time trend for the 3-center inte-grals calculation on GPU (Nvidia Titan Xp) and singleCPU thread (Intel(R) Xeon(R) Gold 5120) as the molec-ular size increases. For systems smaller than benzenethe acceleration achieved by the GPU does not compen-sate for much of the communication latency, and conse-quently the reduction of calculation time is limited, albeiton an already overall low level. For benzene and naph-thalene, we observe a GPU speedup of about 60 % com-pared to the respective single CPU thread values, whilefor the larger systems (phenanthrene and coronene) itincreases to 80 %. For even larger systems, the speedupincreases until it eventually plateaus due to limits in theGPU memory and hardware bandwidth. Even thoughthe observed quantitative speedups depend on the givenGPU/CPU hardware combination, a qualitatively similarbehavior is expected in a general setting. Note that over-all, best performance is obtained by combining all CPUthreads and the GPU in a mixed hybrid OpenMP/Cudamode, which is also implemented in VOTCA-XTP.

E. Iterative matrix-free eigensolvers for the BSE

The size of the BSE matrix in Eq. 9 increases rapidlywith the number of occupied (Nocc) and unoccupied(Nunocc) states included in the product basis, with itsdimension being 2NBSE × 2NBSE, where NBSE = Nocc ×Nunocc. Even if the TDA (Eq. 16) is used, the dimen-sion of Hres is still NBSE × NBSE. This leads to acomputational as well as a memory bottleneck for GW -BSE calculations. The Davidson algorithms76,77 form afamily of subspace-iterative diagonalization schemes thatare extensively used in large-scale quantum chemistryapplications.78,79 These methods allow rapid computa-tion of a selected numbers of eigenvalues of large matri-ces, while reducing memory requirements when comparedto other methods.

For an eigenproblem HX = ΩX, the Davidsonmethod starts from a set of N guess eigenvectors V =v1,v2, . . . ,vN, where each vi is a column vector. Thesevectors are used to obtain a small eigenvalue problem:

(V THV )x = ωx. (33)

The Ritz eigenpairs of this problem (ωi,yi = V xi)are approximate solutions of the large eigenvalue prob-lem. The residues of the Ritz eigenpairs, ri = H yi −ωiyi, are then used to construct additional basis vec-tors, ti, that are appended to the projector: V =v1,v2, . . . ,vN , t1, t2, . . . tn. This new projector is thenorthogonalized, using either a Gram-Schmidt or a QRapproach, and used to obtain a better approximation ofthe large eigenpairs. This is repeated until the residuesof all Ritz eigenpairs respect the condition ||ri|| ≤ ε,

where ε is a fixed threshold parameter. When the size ofthe projector V becomes too large, it is reset to N Ritzeigenvectors.

In addition to considerably accelerating the diagonal-ization of the BSE matrix, these methods do not requirethe matrix to be stored in memory and only the actionof this matrix on vectors is required. This matrix-freeapproach naturally decreases the memory requirement ofthe calculation. Different methods based on the generalidea behind the Davidson algorithm have been developed.These methods differ in the way the correction vectorsti are calculated and on which part of the spectrum istargeted. We briefly present the solutions we have im-plemented in VOTCA-XTP to solve the BSE equationusing the TDA or the full matrix.

If the TDA is used, the Davidson method allows rapidcomputation of the lowestN eigenvalues and eigenvectorsof the Hermitian matrix Hres. The initial projector V isthen set to select the N transitions with the lowest energydifference. We have implemented different methods tocompute the correction vectors. Following Davidson’soriginal idea,76 the correction vectors can be obtainedvia

ti = −(Dres − ωiI)−1ri, (34)

where Dres is the diagonal of the matrix Hres. Note that(Dres − ωiI) is a diagonal matrix and that therefore itis not necessary to explicitly diagonalize it. It is impor-tant to mention that the method outlined above requiresonly one evaluation ofHresV per iteration. This productcan be calculated without having to form the completeHres matrix, decreasing the memory requirement at theexpense of a slight increase in computational time.

If the TDA is not used, the lowest transitions corre-spond to the interior eigenvalues of the non-Hermitianmatrix in Eq. 9, as all its eigenvalues come in pairs(−ΩS ,ΩS). The procedure outlined above is optimal forexterior eigenvalues but often leads to spurious eigen-values when applied to the calculation of interior eigen-values. Following Morgan,80 the original Rayleigh–Ritzapproach is modified to map the interior eigenvalues tothe exterior of the spectrum of an inverted matrix. Tothis end, the small eigenvalue problem Eq. 33 is replacedby the generalized eigenvalue problem

(V THV )x = ω(V TH2V )x. (35)

Solving Eq. 35 gives the harmonic Ritz eigenpairs(ωi,yi = V xi) with ωi = xTi V

THV xi. As before, theresidues of these eigenpairs are used to construct correc-tion vectors using Eq. 34, which are then appended tothe projector matrix V . It is worth noting that the gen-eralized eigenvalue problem Eq. 35 requires two matrix-vector products per iteration to evaluateHV andH2V ,which significantly increases the computational cost ofthe method.

To illustrate the performance of our implementationof the Davidson algorithm, we show in Tab. II the com-putation time and the memory requirement of the BSE

8

Time (s) Memory (GB)Molecule NBSE DSYVEX DPR DPR-MF DSYVEX DPR DPR-MFUracil 7743 83.44 1.33 32.63 1.08 1.20 0.71Cytosine 8149 98.68 1.52 33.16 1.21 1.31 0.79Thymine 10593 217.85 2.03 61.99 1.88 2.02 1.16Adenine 11725 293.49 2.39 75.48 2.18 2.38 1.28

TABLE II. Performance benchmark (computation time and memory requirements) of different diagonalization schemes imple-mented in VOTCA-XTP during the calculation of the 25 lowest singlet excitations of the NBSE × NBSE BSE matrices usingthe TDA on a single thread.

calculations of 4 nucleobases. Here, the Davidson method(Diagonal-Preconditioned-Residue, DPR) using Eq. 34 toobtain the correction vectors, as well as its matrix-freeimplementation (DPR-MF), are compared to the highly-optimized Lapack routine DSYVEX, which also allowsthe calculation of the lowest part of the spectrum. Asseen in Tab. II, the DPR method is up to 2 orders ofmagnitude faster than DSYVEX, while having similarmemory requirements. The matrix-free approach is fasterthan DSYVEX, but significantly reduces the memoryrequirement, hence enabling BSE calculations on muchlarger systems.

F. QM/MM embedding schemes for QP andelectron-hole excitations

As mentioned in Sec. I, VOTCA-XTP provides a cou-pled QM/MM framework in which the quantum (excited)state is linked to a polarizable atomistic model for theenvironment. Our scheme makes use of a distributedatomic multipole representation for molecules in the MMregion, which allows treatment of both the effects of staticelectric fields and the polarization response as a self-consistent reaction field. Specifically, static atomic mul-tipole moments81 Qmt are employed, where t indicatesthe multipole rank and m the associated atom in themolecule M . The tensor Tmm

′

tu describes the interactions

between the multipoles moments Qmt and Qm′

u . To modelthe polarization effects, each atom can additionally be as-signed a polarizability αmm

′

tu for the creation of inducedmoments ∆Qmt due to the field generated by moments uon a different atom m′. If we split a classical MM systemS in state s into regions R and R′ with S = R∪R′, itstotal energy is given by

E(s)class(S) = E(s)(R) + E(s)(R′) + E(s)(R,R′), (36)

where

E(s)(R) =1

2

∑M∈R

∑M ′∈RM ′ 6=M

E(s)MM ′ +

1

2

∑M∈R

E(s)M (37)

E(s)(R,R′) =∑M∈R

∑M ′∈R′

E(s)MM ′ (38)

with

E(s)MM ′ =

∑m∈M

∑m′∈M ′

∑tu

(Qm(s)t + ∆Q

m(s)t )

× Tmm′

tu (Qm′(s)

u + ∆Qm′(s)

u ) (39)

and

E(s)M =

∑m∈M

∑m′∈Mm′ 6=m

∑tu

∆Qm(s)t (α−1)mm

′

tu(s)∆Qm′(s)u . (40)

Eq. 36 follows a variational principle with respect to theinduced moments, and a preconditioned conjugate gra-dient method is used to find the ∆Qmt , which give theminimum energy. Induced interactions are modified us-ing Thole’s damping functions82,83 to avoid overpolar-ization. Then, the MM multipoles can interact with theQM region as an additional external potential to Eq. 1,while the explicit electrostatic field from the QM densityis used to polarize the MM region.

More specifically, VOTCA-XTP can partition a sys-tem into multiple active QM regions (possibly treatedat different levels of theory) and multiple classical MMregions, in which static and polarizable multipoles of dif-ferent orders can be defined, generalizing Eq. 36. In or-der to evaluate excitation energies within this QM/MMscheme, a self-consistent procedure is required if polar-ization is included in the MM region. Within a singleiteration step p, a QM level calculation (DFT for theground state s = n, DFT+GW -BSE for excited s = xstates) is performed in the electric field generated by thetotal moments in the MM region. The resulting QM en-ergy then reads

E(s),pQM = E

(s),pDFT + δsx ΩpS . (41)

The associated total electron density is then evaluated ona grid. In VOTCA-XTP, the default quadrature on a gridused for the numerical integration is an Euler–MacLaurinscheme for the radial components and a Lebedev schemefor the angular components, and the discretized densityis then used to self-consistently determine new induceddipoles in the MM region. The minimized classical energy

E(s),pclass is used to update the total energy of the coupled

QM/MM system

E(s),pQM/MM = E

(s),pQM + E

(s),pclass . (42)

9

HN

NH

O

O

N

N

S

H

Hm

FIG. 2. Chemical structure of a monomer (n = 1) buildingblock of the n-DPP2PymT oligomers with m = 1, 2, 3.

The whole procedure is repeated until the change of to-tal energy is less then a preselected accuracy, typically10−5 Ha.

To obtain the excitation energy Ω(s) of a complex in thepolarizable environment, total energies of the combinedQM/MM system are obtained self-consistently for boththe ground and excited state, and their difference defines

Ω(s) = E(s)QM/MM − E

(n)QM/MM. (43)

IV. SHOWCASE APPLICATIONS

In this section, we demonstrate the application ofVOTCA-XTP in two showcases related to materials usedin organic photovoltaics. First, we investigate the opti-cal excitation energies in a series of diketopyrrolopyrrole-derived oligomers. Second, we study the effects of donor-acceptor complex conformations and environment polar-ization on the CT excitations in a low-donor contentblend of rubrene and fullerene.

A. Optical excitations of diketopyrrolopyrrole-derivedoligomers

As a first showcase application, we investigate theexcited-state properties of a series of diketopyrrolopyr-role (DPP) oligomers as model systems for the respec-tive polymers. The chemical structure of the monomerbuilding blocks is shown in Fig. 2. We consider structuralvariants containing m = 1, 2, 3 thiophene rings (in transorientation) in the monomer unit, as well as oligomersconsisting of up to n = 4 repeat units, further referredto as n-DPP2PymT. This choice allows not only for astudy of the physical system but simultaneously enablesus to highlight the systematic increase in size of the com-putational system and the associated powers and limita-tions of the current implementation. In its ground state,the DPP core is highly electron-withdrawing. Hence, nu-cleophilic aromatic substituents are added to the DPPcore.50 In the compounds studied, pyridine is used asan aromatic substituent which lowers the highest occu-pied molecular orbital (HOMO) and the lowest occu-pied molecular orbital (LUMO) levels due to the pres-ence of sp2-hybridized nitrogen.51 Further addition of

thiophene (a stronger nucleophile) increases the numberof aromatic substituents in conjugation with DPP. Thedonor strength increases consequently as evident fromthe HOMO energy levels determined by cyclic voltam-metry measurements.52 This addition of thiophene alsominimally influences the LUMO and the optical excita-tion energy is lowered from 2.22 eV for m = 1 to 2.05 eVfor m = 3.52

Geometry optimizations for the n-DPP2PymToligomers have been carried out within DFT based onthe cc-pVTZ basis set84 with the PBE0 hybrid func-tional67 using the ORCA package.41 Initial structuresfor all oligomers have been prepared as suggested byexperiments with the thiophenes in trans orientation toeach other and the final obtained geometries are nearlyplanar.

For the subsequent GW -BSE calculations, we chosethe evGW approach with the PPM. An optimized aux-iliary basis set for cc-pVTZ66 taken from the Basis SetExchange85 has been used in the resolution-of-identitysteps. Aiming at a consistent benchmark, we use in eachcase the full spectrum of single-particle states in the RPA(NRPA), and determine QP corrections to all occupied(Nocc) and Nunocc = 2Nocc unoccupied states. The lat-ter are also taken into account in the product basis forthe BSE, i.e., NBSE = Nocc ·Nunocc. The explicit valuesas listed in Tab. III for all oligomers. Due to computa-tional limitations, only systems with n = 1, 2, 3 could betreated for n-DPP2Py3T. The HOMO and LUMO ener-gies as obtained in the KS and QP approaches, as wellas the optical excitation energies resulting from the TDAand the full BSE are also given in Tab. III.

The quasi-particle energies for HOMO and LUMO fol-low the expected trend with increasing number of thio-phene units m, as discussed above. We observe that forall oligomers increasing the number of repeat units n ini-tially, i.e., from monomer to dimer, affects the HOMOenergies by about 0.2 eV. Upon further additions of re-peat units the values tend to saturate, indicating thatthe electronic excitations remain localized.

A similar trend is apparent for the energy of the op-tically active singlet excitations as shown in Fig. 3 forcalculations based on the full BSE (solid lines) and us-ing the TDA (dashed lines), respectively. The values forΩfull

opt obtained for the largest systems are close to those

obtained in solution52 (2.22 eV for DPP2Py1T, 2.16 eV inDPP2Py2T, 2.05 eV in DPP2Py3T). It is also noteworthythat even for the biggest oligomers the difference betweenthe TDA and the full BSE is about 0.2 eV. This is a fur-ther indication of the fact that the electronic states andexcitations do not delocalize completely over the wholeoligomer but remain localized on a few repeat units.26

This notion is corroborated by the visualizations ofthe electron density difference upon excitation shown asinsets in Fig. 3. Although the DPP bicyclic core is elec-tron withdrawing, upon exciton generation the electrondensity is reduced there. In the case of 1-DPP2Py1Tthe change in electronic density is visible over the entire

10

NRPA Nocc Nunocc NBSE εKSHOMO εKS

LUMO εQPHOMO εQP

LUMO ΩTDAopt Ωfull

opt

1-DPP2Py1T 982 96 191 18336 -5.73 -2.93 -6.76 -1.31 2.79 2.442-DPP2Py1T 1935 191 381 72771 -5.66 -3.19 -6.57 -1.73 2.45 2.223-DPP2Py1Ta 2890 286 571 163306 -5.65 -3.30 -6.52 -1.90 2.31 2.114-DPP2Py1Ta 3844 381 761 289941 -5.64 -3.36 -6.50 -1.97 2.25 2.06

1-DPP2Py2T 1164 117 233 27261 -5.64 -2.95 -6.65 -1.39 2.68 2.382-DPP2Py2T 2300 233 465 108345 -5.55 -3.15 -6.42 -1.73 2.35 2.153-DPP2Py2Ta 3436 349 697 243253 -5.53 -3.23 -6.38 -1.84 2.25 2.074-DPP2Py2Tb 4572 465 929 431985 -5.53 -3.26 -6.36 -1.90 2.19 2.03

1-DPP2Py3T 1346 138 275 37950 -5.55 -2.96 -6.53 -1.45 2.60 2.342-DPP2Py3T 2664 275 549 150975 -5.46 -3.11 -6.32 -1.72 2.29 2.113-DPP2Py3Ta 3982 412 823 339076 -5.45 -3.17 -6.28 -1.81 2.20 2.05

a needs more than 256 GB RAMb needs more than 512 GB RAM

TABLE III. Number of RPA levels (NRPA), number of occupied (Nocc) and unoccupied levels (Nunocc = 2Nocc), the product basis

set size for BSE calculations (NBSE), Kohn-Sham HOMO (εKSHOMO) and LUMO (εKS

LUMO) energies, quasi-particle HOMO (εQPHOMO)

and LUMO (εQPLUMO) energies and optical excitation energies for TDA (ΩTDA

opt ) and full BSE (Ωfullopt) for the n-DPP2PymT

oligomers. All energies are given in eV.

1.8

2

2.2

2.4

2.6

2.8

3

1 2 3 4

Ω(e

V)

# of repeat units

DPP2Py1T DPP2Py2T DPP2Py3T

FIG. 3. Excitation energy Ω (in eV) for the DPP2PymToligomers as a function of the number of repeat units. Thesolid lines represent the energies obtained using the fullBSE, while the respective dashed ones result from the TDA.The blue and red colour around the molecule shows the in-creased electron density and increased hole density (Isovalue±2 · 10−4 e/a3

B.) respectively after exciton formation. Themolecules shown are 4-DPP2PyT (top) and 1-DPP2PyT (bot-tom).

small molecule. In contrast, for 4-DPP2Py1T we noticethat the termini of the oligomer are nearly free of eitherhole or electron density, with the excitation appearing tobe localized on about three repeat units.

Turning now towards the computational effort requiredto obtain these results, we first observe that, as also notedin Tab. III, systems with up to roughly Nb = 2500 ba-sis functions (the same as NRPA) can be treated on ma-chines with less than 256 GB of memory. Due to thescaling of the RPA step more than 512 GB are needed for

FIG. 4. View of the two rubrene-C60 configurations takenfrom the MD simulation. The QM complexes are surroundedby the MM atomic sites, here pictured as small spheres. In(a) the C60 is closer to one of the phenyl ring of rubrene, andthis complex is referred to as CTP. In the CTA configuration(b) the fullerene is close to the anthracene core.

the biggest oligomer (4-DPP2Py2T). These requirementscan be considered typical for most organic compoundswith similar composition and arise from the aforemen-tioned storage of the three-center integrals Imlµ (Eq. 23)in memory, see Sec. III A. It should be emphasized thatthis apparent limitation of the system sizes accessible onregular workstations can be overcome by choosing a lowerNRPA or a smaller basis set than the large cc-pVTZ setused here for demonstration purposes.38 We also stressthat the memory is required in the RPA of the GW step,not for obtaining the solution of the BSE as we are mak-ing use of the matrix-free Davidson (DPR-MF) methodas introduced in Sec. III E.

11

B. Molecular and charge-transfer excitations inrubrene-fullerene mixtures

The second application example showcases VOTCA-XTP’s capability to determine the different excited statesof molecular complexes within the GW -BSE/MM em-bedding schemes. Here we focus on representativestructures of a donor-acceptor heterojunction as used,e.g., in organic photovoltaics. In particular, we studyan amorphous morphology with low-donor content (<10 mol%), composed of fullerene (C60) and 5,6,11,12-tetraphenyltetracene (rubrene). Because of the low-donor content, a C60 cluster will surround the donormolecule, making the interaction between the singledonor molecule with a close shell of neighbouring C60 ac-ceptors representative of the properties of the system asa whole. These complexes are therefore meaningful can-didates for a computational analysis of the influence ofdonor-acceptor conformations and environment polariza-tion effects in the GW -BSE/MM framework introducedin Sec. III F.

To obtain representative structures, mixed morpholo-gies have been simulated with ab-initio MD based onDensity Functional Tight Binding theory using linearscaling self consistent field calculations within the CP2Kcode.86 Initial configurations have been prepared usingpackmol,87 targeting experimental values53 for densitiesand mole percentages. This structure is first equilibratedat 700 K in NpT (with velocity rescaling themostat88 atatmospheric pressure89) for 7 ps (timestep 1 fs), then an-nealed to 300 K within 10 ps. A final NpT equilibrationfollowed for 5 ps.

In the final morphology, donor-acceptor complexeswith many different orientations between fullerene andrubrene molecules are present and we focus here onthe two cases shown in Fig. 4. The first configura-tion, labeled CTP, displays a center-of-mass distance of∆COM = 11.3 A between the two molecules and theirclosest contact distance ∆CC = 3.3 A is found between acarbon atom in C60 and an hydrogen atom of one phenylgroup in rubrene. On the other hand in the CTA config-uration the two molecules are closer (∆COM = 8.9 A) and∆CC = 1.8 A is found with respect to an atom in the an-thracene core of rubrene. Such variations of the mutualorientations of rubrene and C60 in the two complexes arealso expected to give rise to distinctly different charac-teristics of electronic excitations, intermolecular charge-transfer excitations above all.

For both configurations, we apply the GW -BSE/MMmethod described in Sec. III F. To keep the computa-tional costs tractable, the calculations make use of effec-tive core potentials and an associated basis set90 that hasbeen augmented by a single shell of polarization functionstaken from the 6-311G** basis.91 An associated auxil-iary basis was constructed using the technique employedin the SAPT code.92 The PBE067 functional has beenused for the underlying DFT ground-state calculations.In the GW -BSE steps, a total of NRPA = 2366 levels have

been included in the RPA, and QP corrections have beendetermined for Nocc = 218 occupied and Nunocc = 435unoccupied states. The corresponding product basis forthe BSE in TDA is of size NBSE = 94830. In the fol-lowing we determine the electron-hole excitations of thecomplexes in three embedding variants: in the gas-phase(vacuum, GW -BSE), in a static environment in which nopolarization effects are included in the MM region withina cutoff of 2.2 nm (GW -BSE/MMs), and in a polarizableenvironment (GW -BSE/MMp), respectively.

The resulting excitation energies are shown in Fig. 5.In all cases, local excitations on C60 (rubrene) are markedin green (blue), and excitations with CT character aregiven by varying shades of red depending on the amountof charge transfer ∆Q. Note in case of GW -BSE/MMand GW -BSE/MMs, one obtains the full spectrum of ex-citations from a single calculation. Due to the state-dependent response of the system in the polarizable case— requiring the self-consistent procedure as describedin Sec. III F — only the lowest energy excitations withthe respective characters are shown for GW -BSE/MMp.

For both configurations, the spectra are hardly affectedby the embedding in a static environment. This is due tothe fact that most of the environment of the complexesis formed by C60 molecules, which only exhibit a faintelectrostatic potential.

For the CTP configuration (Fig. 5(a)), the lowest 15excitations, as obtained by GW -BSE/MMs, are localizedon C60, with the optically active rubrene excitation foundright above them at about 2.4 eV. Charge-transfer exci-tations (with ∆Q = 1, see also isosurface plot in theinset) result another 0.2 eV higher in energy. Polariza-tion as taken into account in the GW -BSE/MMp calcula-tions has very little effect on the two localized excitationssince both C60 and rubrene exhibit only a small changeof molecular dipole moment upon excitation. In con-trast, the creation of a CT exciton causes a large changein dipole moment to which the environment strongly re-sponds. The electron-hole pair is also less strongly boundin the CT exciton and therefore itself more susceptibleto changes in the environment. In total, this results in astrong energetic stabilization around 0.46 eV so that theCT state energy is found 0.24 eV below the rubrene ex-citation. This is qualitatively different to the situationobserved without the inclusion of polarization effects, in-dicating that the formation of the CT state could beexothermic rather then endothermic.

The spectra of the CTA are significantly different. Ina vacuum or static environment, the three lowest exci-tations between 1.6–1.8 eV show partial charge transfer(∆Q = 0.38) due to a delocalization of the single-particleHOMO level over both molecules. Between 2.0–2.4 eVthe spectra are generally dense with excitations of differ-ent character being energetically close. The localized ex-citations result at very similar energies as in CTP in theGW -BSE/MMp calculation. Similarly, also the energyof the CT excitation is only lowered by 0.03 eV, whichwe attribute to a combination of smaller dipole moment

12

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

GW -BSE GW -BSE/MMs GW -BSE/MMp

∆Q = 1

∆Q = 1

(a)

C60

CTP

rubrene

GW -BSE GW -BSE/MMs GW -BSE/MMp

∆Q = 0.38∆Q = 0.68

(b)

C60

CTA

rubreneΩ(e

V)

FIG. 5. Excitation energies Ω (in eV) as obtained for the CTP (a) and CTA (b) complexes the from GW -BSE calculationsin vacuum and embedded in a static (GW -BSE/MMs) and polarizable (GW -BSE/MMp) environment, respectively. Localexcitations on C60 (rubrene) are marked in green (blue), and excitations with CT character are given by varying shades of reddepending on the amount of charge transfer ∆Q. Isosurfaces of the electron-hole densities with isovalue ±10−4 e/a3

B are shownas insets.

and lower polarizability of the excited state as comparedto the one in CTP. It is noteworthy, however, that theamount of charge transferred is increased to ∆Q = 0.68 .The non-integer type of CT can also be seen from theisosurface plot in the inset of Fig. 5(b), in which part ofthe hole density (red) is present on C60. Conversion toCT excitations appear energetically favorable from bothtypes of localized excitations in CTA.

V. DISCUSSION

The two examples discussed in the previous sectionemphatically showcased the capabilities of VOTCA-XTPin calculating the excited-state electronic structure ofmolecular materials based on GW -BSE both in isolatedmolecular systems and in complex molecular aggregates.Our implementation is particularly suited for the study ofnear-gap excitations, and performs well on regular work-stations. There are, however, as briefly already discussed,some limitations, e.g., in terms of accessible system sizes,or ultimate accuracy depending on the method used forthe frequency integration of the self energy.

All in all, the design of VOTCA-XTP aims at strik-ing a balance between performance, accuracy and im-plementation complexity. For instance, the RI approxi-mation simplifies the implementation and speeds up thecalculations. In particular, the N3

b scaling of the mem-ory requirements of the RPA makes keeping the three-center integrals in memory feasible for moderate-sizedmolecules described by up to approximately 1500 basisfunctions, thus allowing GW -BSE calculations even ondesktop hardware. As VOTCA-XTP is built with sharedmemory parallelization, large memory hardware is re-quired for the treatment of larger systems. This could, in

principle, be overcome either by calculating the necessaryterms on the fly or by efficiently storing them on disk.Both would come with a noticeable drop in performance,and are currently not actively pursued. Instead, we con-sider as an alternative the inclusion of a framework forDensity Functional Perturbation Theory93 (DFPT) intoVOTCA-XTP. It has been shown before94 that the ex-plicit summation over unoccupied states in the RPA canbe omitted, if the whole GW is rephrased in terms ofa Sternheimer equation. In the same spirit, DFPT canalso yield beneficial improvements in terms of speed andmemory consumption for the BSE.95

Besides the above limitations, the GW -BSE implemen-tation in VOTCA-XTP currently supports only closed-shell calculations and thus systems with explicit spin,e.g., for the optical spectra of cations or anions cannotbe treated. This restriction is planned to be lifted in fu-ture versions. Furthermore, VOTCA-XTP can only de-termine excited-state response properties such as atomicforces, excited-state vibrational modes, or polarizationtensors via numerical derivatives due to the lack of ana-lytic gradient expressions.96 It is also known that whilethe GTO-based implementation is adequate for the repre-sentation of bound electronic states, it is less effective indescribing delocalized electronic states close to or abovethe vacuum level. As a consequence, Rydberg excitationsare not as reliably accounted for. Such states typically re-quire the addition of very diffuse atomic basis functions,causing significant linear dependencies in the molecularbasis set which require careful treatment.97

In the GW -BSE/MM framework, classical polariza-tion effects of the environment are currently modeledvia atomic induced dipoles as described in Sec. III F.This model cannot be expected to yield an accurate de-scription of the response, if the environment is strongly

13

polarizable to the extent that charge flow effects arepresent. Furthermore, our GW -BSE/MM framework re-lies on the assumption that the environment responseis single-reference in character, allowing for an unam-biguous identification of the excited states in the self-consistent procedure required to evaluate Eq. 41. Thiscan be achieved by analyzing excited-state characteris-tics, such as oscillator strengths for optical transitions orthe amount of transferred charge between two molecules,or directly the density matrix of the excitation. Any ofthese state-tracking methods are not straightforwardlyapplicable, e.g., at or close to intersections where at leasttwo states mix strongly. The inclusion of automatic solu-tions to this problem 98,99 is left for future developments.

ACKNOWLEDGEMENTS

B.B. acknowledges support by the Innovational Re-search Incentives Scheme Vidi of the Netherlands Organ-isation for Scientific Research (NWO) with project num-ber 723.016.002. Funding is also provided by NWO andthe Netherlands eScience Center through project number027.017.G15, within the Joint CSER and eScience pro-gram for Energy Research (JCER 2017). We gratefullyacknowledge the support of NVIDIA Corporation withthe donation of the Titan X Pascal GPU used for thisresearch.

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