Quantifying local exciton, charge resonance, and multiexciton character incorrelated wave functions of multichromophoric systemsDavid Casanova and Anna I. Krylov Citation: The Journal of Chemical Physics 144, 014102 (2016); doi: 10.1063/1.4939222 View online: http://dx.doi.org/10.1063/1.4939222 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/144/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Quantifying charge resonance and multiexciton character in coupled chromophores by charge and spincumulant analysis J. Chem. Phys. 142, 224104 (2015); 10.1063/1.4921635 Automatic determination of important mode–mode correlations in many-mode vibrational wave functions J. Chem. Phys. 142, 144115 (2015); 10.1063/1.4916518 Density functional theory embedding for correlated wavefunctions: Improved methods for open-shell systemsand transition metal complexes J. Chem. Phys. 137, 224113 (2012); 10.1063/1.4770226 Self-consistent embedding theory for locally correlated configuration interaction wave functions in condensedmatter J. Chem. Phys. 125, 084102 (2006); 10.1063/1.2336428 Correlated geminal wave function for molecules: An efficient resonating valence bond approach J. Chem. Phys. 121, 7110 (2004); 10.1063/1.1794632

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THE JOURNAL OF CHEMICAL PHYSICS 144, 014102 (2016)

Quantifying local exciton, charge resonance, and multiexciton characterin correlated wave functions of multichromophoric systems

David Casanova1,2,a) and Anna I. Krylov31Kimika Fakultatea, Euskal Herriko Unibersitatea (UPV/EHU) and Donostia International Physics Center(DIPC), P.K. 1072, 20018 Donostia, Spain2IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Spain3Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482, USA

(Received 28 October 2015; accepted 18 December 2015; published online 5 January 2016)

A new method for quantifying the contributions of local excitation, charge resonance, and multiex-citon configurations in correlated wave functions of multichromophoric systems is presented. Theapproach relies on fragment-localized orbitals and employs spin correlators. Its utility is illustratedby calculations on model clusters of hydrogen, ethylene, and tetracene molecules using adiabaticrestricted-active-space configuration interaction wave functions. In addition to the wave functionanalysis, this approach provides a basis for a simple state-specific energy correction accountingfor insufficient description of electron correlation. The decomposition scheme also allows one tocompute energies of the diabatic states of the local excitonic, charge-resonance, and multi-excitoniccharacter. The new method provides insight into electronic structure of multichromophoric sys-tems and delivers valuable reference data for validating excitonic models. C 2016 AIP PublishingLLC. [http://dx.doi.org/10.1063/1.4939222]

I. INTRODUCTION

The assemblies of chromophores appear in natural andartificial light-harvesting systems as well as in molecularphotovoltaic materials.1–4 The energy flow5–8 and the nature ofthe excited states in multichromophoric systems are governedby the electronic structure of the individual chromophoresand their interactions. These interactions, which depend onthe relative orientations of the chromophores, perturb thestates of the individual moieties and also couple them, oftenleading to delocalized excited states. The resulting adiabaticwave functions can be described as linear combinations of thelocal excitations (so-called excitonic configurations), charge-resonance configurations (akin to ionic configurations inmolecular wave functions), and multiexcitons (configurationsin which more than one chromophore is excited). Thewave function composition determines the energy levels,band structure, and non-adiabatic couplings between thestates.

Analysis of complex excited-state wave functions interms of such physically distinct contributions provides aninsight into the nature of the states and their properties. Forexample, in the context of singlet fission,9,10 the key electronicstates are an initially excited bright state of an excitoniccharacter and a dark multiexciton state. Both states includecontributions from charge-resonance configurations, whichgovern the couplings.9–13 There are several wave functionanalysis schemes that allow one to quantify the degree oflocal excitonic and charge-resonance character from the wavefunction amplitudes and/or density matrices.14,15 Recently,we introduced a method for identifying and characterizing

a)david.casanova@ehu.eus

multi-exciton contributions in the wave functions of moleculardimers.16 Here we extend this scheme for multichromophoricsystems.

In our approach, we compute correlated adiabatic wavefunctions of an aggregate and then decompose them interms of the diabatic-like configurations of the excitonic,charge-resonance, and multiexcitonic character. We employthe RASCI (restricted-active-space configuration interaction)method,17–19 which describes singly (both excitonic andcharge-resonance) and multiply excited states on an equalfooting. Importantly, in the spin-flip (SF) formulation themethod is size-intensive,18,20–22 so that the calculations forthe monomers, dimers, trimers, etc., can be meaningfullycompared. Owing to these properties, RASCI-SF has beensuccessfully employed to model various aspects of singletfission.11,23–28

In addition to interpretation purposes, the analysisof multichromophoric wave functions in terms of simplediabatic-like configurations provides valuable reference datafor devising various excitonic models for computing excitedstates in extended systems.29–36 In these approaches, whichare similar in spirit to the valence-bond method, electronicstates of an extended system are described using productsof the wave functions of the individual chromophores. Thecoefficients are obtained by solving a small CI-like problemfor a model Hamiltonian in this physically motivated diabaticbasis. As in the original valence-bond method, the resultsof such calculations depend crucially on which states of theindividual chromophores are used to construct the diabaticbasis (e.g., how many singly excited states are included,whether charge-resonance configurations are included, etc.).Thus, the analysis of accurate correlated adiabatic wavefunctions in multi-chromophoric systems will help to validate

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the choice of diabatic basis in such excitonic models and aidin their development.

The structure of the paper is as follows. Section IIdescribes the electronic states in multichromophoric systems.It explains the use of spin correlators for the excited-state analysis and the quantification of charge-resonancecharacter. This section also introduces corrections to theexcitation energies by means of exciton decomposition andthe computation of diabatic states. Section III illustrates theapplication of this methodology to the singlet excited states inclusters of hydrogen molecules, ethylene aggregates, and intetracene trimers.

II. THEORY

A. Electronic states of multichromophores

Our analysis relies on fragment-localized molecularorbitals. A general adiabatic wave function of a multichro-mophore system can be written as a combination of neutralexcitons (NEs), local excitations conserving the number ofelectrons in the individual fragments, and charge-resonance(CR) configurations in which two or more chromophores arecharged (but the total charge of the assembly is conserved),

|Ψ⟩ = |ΨNE⟩ + |ΨCR⟩. (1)

The NE contributions can be further classified by excitationlevels, i.e., as local single excitations (LEs, the states in whichonly one chromophore is excited) and simultaneous multipleexcitations (MEs, the states in which multiple fragments areexcited). This decomposition can be written as

|ΨNE⟩ =Mk=1

|Ψ(k)⟩, (2)

where k is the number of simultaneously excited chromo-phores. In notations of Eq. (2), the LE terms correspond toΨ(k = 1), whereas the ME states consist of the Ψ(k > 1)configurations. Formally, the sum over the excitation levelk in Eq. (2) also includes the k = 0 case correspondingto the ground-state configuration in which none of thechromophores is excited. Each Ψ(k) comprises multiplepossible configurations,

|Ψ(k)⟩ =L(k)l=1

cl(k)|φl(k)⟩, (3)

where φl(k) corresponds to a unique configuration with kexcited chromophores and L(k) is the dimension of each k-space, which depends on the total number of the chromophoricmoieties in the system M as

L(k) ≡ L(k; M) = M!k!(M − k)! . (4)

Note that each configuration φl(k), indicating which chro-mophores are excited, implicitly contains the contributionsfrom all electronic excitations within each moiety, i.e., allexcited states of the individual chromophores. The weightof each k-level excitation can be evaluated from the set ofamplitudes of the adiabatic wave functions obtained from a

given electronic structure model,

ω(k) =l

c†l(k)cl(k). (5)

In this work, we go beyond a simple assignment of singleor multiexcitonic character of the excited state and analyzethe excited-state character in more detail. Excited states canbe described in terms of excitations of the monomers, pairs ofmonomers, and so on, which allows one to determine whichchromophores are involved in the given electronic transitionand, therefore, to describe the extent of delocalization of theexcited state. In addition, it is also desirable to characterizeexcited states in terms of coupled monomer excitations withwell defined (local) spins.37 This is especially relevant inprocesses such as singlet fission9,10 in which the key electronicstate is of the ME character that can be described as two tripletstates localized on two adjacent chromophores and coupledinto an overall singlet state, 1TT. This state is also involved inthe reverse photophysical process, triplet-triplet annihilation.38

Another example of a ME state is the precursor of a recentlyproposed photophysical mechanism called bright fission;39

this ME state is derived from two coupled singlet states, SS.

B. Spin correlators

In our previous work,16 we employed spin correlators40

to distinguish between the singlet TT and SS contributions tothe ME states in molecular dimers. Here we extend the use ofspin cumulants to the general case of multiple chromophores.The spin correlator matrix of M coupled molecules is definedas

ZI J = ⟨Ψ|SIz SJ

z |Ψ⟩, (6)

where SIz and SJ

z are local spin operators acting onchromophores I and J, respectively,

SIz =

12

p∈I

(p†↑p↑ − p†↓p↓

)=

12

(NαI − N β

I

). (7)

In the case of singlet states, the spin correlator matrix elementsobey the sum rule,41

MJ=1

ZI J = 0;∀I = 1,M. (8)

Also, in order for ZI J , 0, I and J monomers cannot be ina local singlet spin state. As a simple example, consider thewave function of the 1TT state of a dimer, which can beexpressed as

1|TT⟩ = 1√

3[|T1T−1⟩ + |T−1T1⟩ + |T0T0⟩] , (9)

where the subindices denote the MS values of the localtriplets. The spin-cumulant matrix elements for this statecan be directly evaluated as ZI I =

23 and ZI J = − 2

3 , whichcomply with the sum rule in Eq. (8). A pure TT state of amultichromophore can be expressed as a linear combinationof the TT states localized on different pairs,

|TT⟩ =K<L

cTTKL |TKTL⟩, (10)

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014102-3 D. Casanova and A. I. Krylov J. Chem. Phys. 144, 014102 (2016)

where TKTL denotes the two triplet states localized on K andL monomers. Then the trace of the spin-correlator matrix ofthe TT state is

MI=1

⟨TT |(SIz )2|TT⟩ = 4

3

K<L

|cTTKL |2 =

43, (11)

where ⟨TKTL |(SIz )2|TKTL⟩ = 2

3 (δK I + δLI). The off-diagonalelements are directly related to the contributions from eachpair of chromophores,

⟨TT |SIz SJ

z |TT⟩ =K<L

ωTTKL⟨TKTL |SI

z SJz |TKTL⟩ = −2

3ωTT

I J,

(12)

where ωTTI J = |cTT

I J |2 and ⟨TKTL |SIz SJ

z |TKTL⟩ = − 23 (δK IδLJ

+ δLIδKJ). For mixed states, these relations hold true ifinstead of the full spin-correlator matrix we only considerΨ(k = 2) contributions and disregard the quintet-quintet (QQ)and higher-spin coupled pairs in the wave function, whichwe expect to be a reasonable approximation in the case oflow-lying states. Now, the overall weight of the TT states,ωTT =

K<L |cTT

KL |2 ≤ 1, can be computed as

ωTT =34

Tr[Z(k = 2)] (13)

and the SS contributions can be approximated as ωSS

≈ ω(k = 2) − ωTT. Moreover, off-diagonal elements of theZ(k = 2) matrix allow the decomposition of the contributionsfrom each individual chromophore pair,

ωTTI J = −

32

ZI J(k = 2), (14)

ωSSI J ≈ ωI J(k = 2) − ωTT

I J . (15)

C. Charge resonances

The CR contributions in the wave function can beexpressed as linear combinations of configurations withcharged fragments conserving the total charge of the sys-tem,

|ΨCR⟩ =q

′|ΨCR(q)⟩, (16)

where the q runs over charge resonances with {nI} set ofmonomer charges and the prime indicates that the sum isrestricted to the charged configurations fulfilling the totalcharge conservation condition,

MI=1

nI = C (17)

where C is the total charge of the system.

D. Exciton energy corrections

Often, the accuracy in excited-state calculations deterio-rates relative to the ground-state description. In particular, inthe case of large systems, where highly accurate excited-state methods are not affordable, drastic approximationsare invoked introducing noticeable errors in the excitation

energies. Importantly, the errors strongly depend on thenature of the electronic states introducing the imbalancein relative state energies. This is particularly important inquantum chemistry models such as RASCI with low-ordertruncation of the excitation operators (such as in hole-particle approximation which is roughly comparable to theconfiguration interaction singles method).17–19 The RASCImethod in its SF variant provides a good description of theelectronic structure of the ground and low-lying excited statesin strongly correlated systems11,23–27,42,43 but misses mostof the dynamical correlation. Consequently, the computedexcitation energies need to be corrected, which is sometimesdone in an empirical fashion.11,25 The magnitude of theexcitation energy correction is strongly state-dependent, e.g., itis well known that the states of a diradical type (e.g., triplets)are less affected by dynamical correlation, in contrast tosinglet excited states with significant CR contributions (see,for example, Refs. 44–48). Thus, the decomposition of thewave functions in terms of LE, ME, and CR can be used tocorrect the computed transition energies based on the weightsof the different contributions and an estimate of the errorthat each type introduces to the final computed energy. Thisapproach was recently used in the computation of transitionenergies in anthracene dimers.39 Here we generalize thisscheme to the case of multiple chromophores. The correctedenergies can be expressed as

E [Ψ] = E0 [Ψ] +X

ωXEC

�Ψ

X�, (18)

where X corresponds to a specific type of excitationcontribution (i.e., LE, ME, CR) and EC

�ΨX

�are correction

energies associated with each X contribution. The set of{EC

�ΨX

�} energies can be related to the excitation energiesof individual chromophores (this is the case of the LE andME contributions) or dimers for the lowest CR configurations.The correction is defined as the difference between a referencevalue and the computed energy,

EC

�Ψ

X�= Eref

�Ψ

X�− E0

�Ψ

X�. (19)

The reference value can be obtained either from experimentor from accurate theoretical calculations. The validity of theenergy correction from Equations (18) and (19) depends on theaccuracy of the reference values and on the correspondencebetween the reference and diabatic states.

E. Construction of diabatic states

Decomposition of the adiabatic wave functions in termsof specific configurations allows us to compute diabatic statesby restricting or allowing electronic configurations of differentcharacters in the computation of eigenstates. In this context,it is natural to compute the CR and NE diabatic states bynot allowing them to mix in the diagonalization of theHamiltonian. The comparison of the so-computed diabatswith adiabatic states allows us to better understand the shapesof the adiabatic potential energy surfaces. Furthermore, wecan decompose the neutral excitations in the LE and ME statesto quantify the participation of individual chromophores inthe overall adiabatic excited state.

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014102-4 D. Casanova and A. I. Krylov J. Chem. Phys. 144, 014102 (2016)

III. RESULTS AND DISCUSSION

We apply the methodology introduced in Sec. II tocompute and analyze singlet excited states in model molecularaggregates. We begin by investigating the character of theelectronic states in clusters of hydrogen molecules. Wethen analyze the lowest ME state in the coplanar ethylenetrimer and tetramer. Finally, we describe the ME statesin tetracene clusters representing the crystal structure anddiscuss their relevance in the context of singlet fission. Inall cases, we quantify the LE, ME, and CR weights in theexciton wave function from the wave function amplitudescomputed using fragment-localized orbitals. Then we separatethe ME contributions into SS and TT by analyzing thespin-correlator matrix from the Ψ(k = 2) part. We employfragment-localized orbitals obtained using the Mulliken-charges criterion described in Ref. 16. All calculations wereperformed with the RASCI method17–19 implemented in adevelopment version of the Q-Chem program.49,50

A. Clusters of H2 molecules

Here we compute and characterize singlet excited statesof model (H2)M clusters with 2 ≤ M ≤ 5. In all clusters,rHH = 0.711 44 Å and the molecules are aligned parallelto each other forming a D2h structure with 3.0 Å separationbetween the neighbors. Ground and excited singlet states werecomputed at the RASCI/cc-pVDZ level with 2M electrons in2M orbitals as the RAS2 space.

Figure 1 shows the excitation energies and the LE, ME,and CR weights in the respective wave functions for (H2)Mwith M = 2-5. The lowest M excited singlet states correspondto the LE states composed of linear combinations of theH2 molecular S1 state (1Σu), with small contributions ofthe CR configurations (ωCR ∼ 0.03-0.11), and even smalleradmixture of the ME states (ωME ∼ 0.01-0.03). The energysplittings between these states, which provide direct measuresof the exciton coupling, are of the order of a few tenths of anelectron-volt. The energy distribution of the M lowest excitedsinglets can be rationalized in terms of exciton mixing and theCR contributions (Figure 2). For the studied cases, the excitonmixing determines the state ordering, while interaction withCR configurations systematically stabilizes their excitationenergies.

At higher energies, we obtain a set of M(M − 1)CR states mainly corresponding to linear combinations of(+,−) configurations, i.e., with one H2 molecule positivelycharged and one with an extra electron. The gap betweenthe highest LE and the lowest CR states changes from3.2 eV (M = 2) to 2.6 eV (M = 5). These states includesmall contributions of the LE terms. The lowest ME stateappears at twice the energy of the lowest H2 triplet (3Σu). TheTT character of these M(M − 1)/2 states is confirmed by thewave function decomposition, which yields ωTT ≥ 0.95 and asmall admixture of the CR configurations. The SS ME statesappear at twice the energy of the molecular S1 state. Thesestates feature much larger weights of the CR configurationsthan the ME states derived from the coupled moleculartriplets.

FIG. 1. Excitation energies (empty circles) and the composition of the singletexcited states in terms of the LE (blue bars), ME (green bars), and CR (orangebars) configurations in equidistant eclipsed (H2)M clusters with M = 2 ((a),left), M = 3 ((a), right), M = 4 (b), and M = 5 (c) computed with RASCI/cc-pVDZ. Dashed lines show the excitation energies of a single H2 molecule.Arrows in the M = 3 case mark the states analyzed in Fig. 3.

We now take a closer look at the exciton composition interms of the monomers involved in the electronic transitionusing the (H2)3 cluster as an example (Figure 3). The threelowest excited singlets are mainly of the LE character andshow energy splittings of the order of ∼0.4 eV (Figure 1(a)).The energy ordering of these three states is determined

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014102-5 D. Casanova and A. I. Krylov J. Chem. Phys. 144, 014102 (2016)

FIG. 2. Excitation energies of the lowest singlet in H2 monomer, i.e., noexciton interaction (EI) or CR (empty blue circles), the three lowest singletstates with no CR (full blue circles), and the three lowest singlet adiabaticstates (grey circles) in (H2)3.

by exciton interaction (Figure 2) and agrees with the stateordering obtained by the dipole-dipole interaction betweenthe molecular transition moments for the 1Σu state (alignedalong the molecular axis) in each of the H2 fragments. Thelowest-energy state is mostly located on the central moleculewith smaller contributions from the edge molecules (Figure 3,left). Exciton interaction stabilizes this state with respectto the monomer due to the antiparallel alignment of thetransition dipoles. The third excited singlet corresponds tothe parallel alignment of the transition dipoles with similarcontributions from each of the H2 molecules. As a result, thisstate is destabilized with respect to the monomer excitationand the electronic transition from the ground state has arather large oscillator strength. The excited singlet state lyingbetween these two LE states contains equal contributionsof S1 excitations from the two edge monomers. The smallexciton coupling in this state results in a transition energyrather close to the excitation energy of the H2 monomer.Based on their energies, the six lowest states with large CRweights are divided into two sets (Figure 1(a)). The fourlowest CR excitons correspond to linear combinations of the(+,−) configurations with the central molecule positivelyor negatively charged, whereas the other two CR statescorrespond to the ionic configurations of the edge molecules.The three lowest singlets with large ME character correspond

to a mixture of the TT states. Their energy splittings arerather small, with transition energies very close to twicethe excitation energy of 3Σu in H2. Two of these stateshave an equal mixture of the center-edge and edge-edgeTT configurations. The remaining state has no contributionfrom the terms with two triplets located on the two edgemolecules. The decomposition of the ME part for the stateswith significant SS character follows a pattern similar to thatof the TT states.

B. Ethylene clusters

Now we turn our attention to the analysis of singletexcited states in ethylene coplanar eclipsed oligomers (D2hsymmetry). We also compute the lowest LE, CR, andME diabatic states, following the algorithm described inSection II E. Ethylene’s molecular geometry was taken fromthe ground-state structure optimized at the B3LYP/6-31G(d)level (rCC = 1.331 Å, rCH = 1.088 Å α(CCH) = 121.868◦).The molecular orbitals are defined with the molecular planesparallel to the x y-plane and with the double bonds parallel tothe x-axis. The low-lying excited singlet states were computedat the RASCI/6-31G(d) level with 2M electrons in 2M orbitalsas the RAS2 space, where M is the number of ethylenemolecules. Excitation energies for the ethylene dimer havebeen corrected following Eqs. (18) and (19). Reference valuesemployed for the lowest valence singlet (11B3u) and triplet(13B3u) states are 8.00 eV51 and 4.36 eV,52,53 respectively. Thereference CR energy has been computed with constrained DFT(CDFT)30,31 with the BLYP,54,55 B3LYP,56 and CAMB3LYP57

density functionals and with the 6-31G(d) basis. Correctionenergies (Equation (19)) obtained with the three functionalswere within 0.20 eV from each other. Here we only presentthe CAMB3LYP results.

1. Ethylene dimer

We first analyze the two lowest excited singlet statesof the ethylene dimer (11B2g and 21Ag) and the dependenceof their character on the intermolecular distance. Figure 4shows the energy profiles for the 11B2g and 21Ag adiabaticstates and for the lowest two LE, one TT, and two CR diabatic

FIG. 3. Weights (in %) of the fragmentcontributions in the singlet LE (S1-S3),CR (S4-S9), TT (S10-S12), and SS (S20,S21 and S24) states in (H2)3. The statesanalyzed here are marked in Figure 1(a)by small arrows.

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014102-6 D. Casanova and A. I. Krylov J. Chem. Phys. 144, 014102 (2016)

FIG. 4. Energies of the two lowest adiabatic states and the LE, TT, and CRdiabats in the eclipsed ethylene dimer as a function of intermolecular distance.

states. Asymptotically, 11B2g correlates with the 11B3u valencestate of the ethylene molecule, whereas 21Ag converges totwice the energy of 13B3u, consistently with the excitondecomposition shown in Figure 5. The CR character of 11B2gincreases from 1% at 4.6 Å up to 50% at 2.2 Å indicatingexcimer formation. The 21Ag state has a pure TT characterat dissociation limit and the weight of the CR contributionsalso increases when the two molecules become closer, butthe ionic configurations appear at much shorter distances thanin the 11B2g state. It is worth mentioning that the energyprofiles of the LE and TT diabats show no bonding character,suggesting that the CR contributions at short distances areresponsible for the bonding interactions in the 11B2g and 21Ag

states.

FIG. 5. Exciton decomposition of the two lowest singlet states in terms ofthe LE, TT, and CR contributions in the eclipsed ethylene dimer as a functionof intermolecular distance.

FIG. 6. Energies of the lowest adiabatic polyexcimer singlet state and theLE and CR diabats in the eclipsed ethylene trimer along the symmetricdissociation.

2. Lowest polyexcimer in ethylene trimer

Here we consider the lowest excited state of three ethylenemolecules in equidistant eclipsed structure (D2h) along itssymmetric dissociation coordinate. Figure 6 shows the energyprofile of the lowest excited singlet of ethylene trimer (the11B2u state) and the lowest LE and CR diabats for different(symmetric) distances between the central and edge molecules.

The potential energy curve for the 11B2u state is similarto that of the excimer state in the ethylene dimer, suggestingpolyexcimer character. The three neutral diabats involved in

FIG. 7. Exciton decomposition of the lowest singlet state in terms of the LEand CR contributions in the eclipsed ethylene trimer along the symmetricdissociation. LE(c) and LE(e) denote the LE configurations located on thecentral and edge molecules, whereas CR(c-e) and CR(e-e) correspond to thecenter-edge and edge-edge charge resonances, respectively.

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014102-7 D. Casanova and A. I. Krylov J. Chem. Phys. 144, 014102 (2016)

the polyexcimer state of the ethylene trimer mainly correspondto the mixing between the LE states (Figure 7). At largeintermolecular distances, the three diabats converge to theenergy of the lowest excited singlet of the ethylene molecule.As in the ethylene dimer, the lowest LE diabats do notshow bonding character. Hence, the polyexcimer nature of11B2u results from the LE-CR mixing. The four lowest CRdiabats correspond to the mixture of the configurations witha single electron transfer within pairs of ethylenes. At largeseparation, these four diabats correspond to the four stateswith ±1 and ∓1 charges in the central and one of the edgeethylene molecules, respectively, with degenerate potentialenergy surfaces.

3. Lowest polyexcimer in ethylene tetramer

The lowest excited singlet of the ethylene tetramer (the11B2g state) also shows polyexcimeric behavior (Figure 8).Exciton decomposition in terms of the LE and CRcontributions shows qualitatively the same trend as in theethylene dimer and trimer, but now the CR contributionsappear at larger intermolecular distance. For example, theweight of the CR configurations reaches 0.5 at ∼3.1 Å, whilein the trimer and dimer cases this value is obtained at muchshorter separations, that is, 2.6 Å and 2.2 Å for the ethylenetrimer and dimer, respectively.

Analysis of the LE and CR configurations in terms of thecontributions from individual ethylene molecules (Figure 9)indicates that LE in the 11B2g state is mainly located onthe two central monomers, whereas CR is observed betweencenter-center (c-c) and center-edge (c-e) pairs.

C. Multiexciton states in tetracene trimers

Crystalline tetracene is of a paramount importance in thecontext of singlet fission.11,27,58–61 Singlet fission proceeds viathe ME state of the 1TT character, which can be described

FIG. 8. Energies of the lowest adiabatic polyexcimer singlet state and theweights of the LE, ME, and CR configurations in the eclipsed ethylenetetramer along its symmetric dissociation.

FIG. 9. Exciton decomposition lowest singlet state in terms of the LE andCR contributions in the eclipsed ethylene tetramer along its symmetric disso-ciation. The LE configurations are further decomposed in terms of the centerand edge excitations and the CR contributions in center to center (c-c), centerto adjacent edge (c-e) and to secondary edge (c-e′).

as two triplet states coupled into an overall singlet. Thefeasibility of singlet fission depends on the relative energyand electronic characteristics of this state. Here we analyzethe 1TT character of the low-lying ME states in modeltetracene trimers and compare them to tetracene dimers.The geometries of the dimers and trimers are taken fromthe crystal structure.62 Figure 10 shows the structure andthe labels denoting the individual molecules and selectedtrimers. The calculations for tetracene dimer (n = 2) andtrimer (n = 3) models were performed at the RASCI-nSF/6-31G(d) level with the lowest ROHF (2n + 1)-tuplet as thehigh-spin reference state. The excitations from the carbon 1sorbitals or to the n × 36 highest energy virtual orbitals wereexcluded from the calculations.

Excitation energies and wave function analysis for thethree lowest singlet and quintet ME states of the four tetracenetrimers are shown in Table I. The distribution of the ME statesin trimers 2, 3, and 12 is very similar. In these three cases,the two lowest 1TT states are degenerate or nearly degenerate,with excitation energies of ∼3.7 eV, slightly lower thanthe excitation energy obtained for the 1ME state in dimer2; they correspond to the TT excitations localized on thetwo non-coplanar neighboring tetracene molecules. The twolowest TT states of trimer 3 show a small delocalization overthe third tetracene molecule. This delocalization is smaller intrimer 12 and nearly non-existent in trimer 1. These statescontain 2%-3% of the CR configurations. In these states, themultiexciton binding energy11 (the energy difference betweenthe singlet and quintet ME states, Eb = E(5ME) − E(1ME))is within the 13-24 meV range (comparable to the bindingenergy in dimer 2), with trimer 3 and trimer 2 having thelargest and the smallest Eb values, respectively. These resultsare consistent with the extent of the delocalization of these

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014102-8 D. Casanova and A. I. Krylov J. Chem. Phys. 144, 014102 (2016)

FIG. 10. Herringbone structure in tetracene crystal withthe labels denoting monomers and trimers.

TABLE I. Analysis of the lowest singlet and quintet ME states in tetracene trimers, and couplings (∥γ∥2) to thelowest LE state computed at the RAS-nSF/6-31G(d) level. The subindices 0, X, and Y refer to the tetracenemonomers in Figure 10, with X= 1, 2 for dimers 1 and 2, and (X,Y)= (1,1′), (2,2′), (3,3′), and (1,2) for trimers 1,2, 3 and 12, respectively.

Structure State Eex (eV) ωTT0X ωTT

0Y ωTTXY ωCR Eb (meV) ∥γ∥2

Dimer 111ME 3.67 1.00 . . . . . . 0.00 1 0.0015ME 3.67 1.00 . . . . . . 0.00

Dimer 211ME 3.76 0.97 . . . . . . 0.03 19 0.1715ME 3.78 1.00 . . . . . . 0.00

Trimer 111ME 3.61 0.04 0.00 0.87 0.00 0 0.0021ME 3.61 0.00 0.91 0.00 0.00 2 0.0131ME 3.62 0.87 0.00 0.04 0.00 1 0.0015ME 3.61 0.03 0.00 0.89 0.0025ME 3.62 0.89 0.00 0.03 0.0035ME 3.62 0.00 0.91 0.00 0.00

Trimer 211ME 3.70 0.89 0.00 0.00 0.03 19 0.2421ME 3.70 0.00 0.90 0.00 0.02 13 0.1231ME 3.83 0.00 0.00 0.91 0.00 0 0.0015ME 3.71 0.00 0.91 0.00 0.0025ME 3.71 0.91 0.00 0.00 0.0035ME 3.83 0.00 0.00 0.91 0.00

Trimer 311ME 3.69 0.86 0.03 0.00 0.03 23 0.1621ME 3.69 0.03 0.85 0.00 0.03 24 0.1531ME 3.83 0.00 0.00 0.91 0.01 0 0.0015ME 3.71 0.90 0.01 0.00 0.0025ME 3.71 0.01 0.90 0.00 0.0035ME 3.83 0.00 0.00 0.91 0.00

Trimer 1211ME 3.69 0.00 0.89 0.01 0.02 21 0.0621ME 3.70 0.00 0.01 0.89 0.03 18 0.1131ME 3.84 0.91 0.00 0.00 0.00 0 0.0115ME 3.71 0.00 0.91 0.00 0.0025ME 3.71 0.00 0.00 0.91 0.0035ME 3.84 0.91 0.00 0.00 0.00

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014102-9 D. Casanova and A. I. Krylov J. Chem. Phys. 144, 014102 (2016)

states and indicate that the strength of the inter-chromophoriccoupling follows the following order: trimer 3 > trimer 12> trimer 1. In these three trimers, the third 1ME state hasno CR contributions and no stabilization relative to theuncoupled triplets (Eb = 0 meV). The strength of the non-adiabatic coupling of the 1TT states to the optically brightstate depends on the weight of the CR configurations inthe wave functions.11 Thus, the lack of ionic configurationsin the highest 1ME state would prevent its population fromthe LE states. This is confirmed by the evaluation of 1ME-LEcoupling by the squared norm of the transition density matrix11

(Equations (20) and (21)) between 1ME states and the lowestLE (Table I). In Equation (20), γpq corresponds to the matrixelement of the transition density between the initial and finalstates, Ψi and Ψf ,

γpq ≡ ⟨Ψi |p†q|Ψf ⟩, (20)

∥γ∥2 ≡ Tr[γγ†]. (21)

Trimer 1 corresponds to a coplanar arrangement of threetetracenes slipped along the molecular short axis with nosuperposition of the planes, directly related to dimer 1.Consequently, trimer 1 exhibits much weaker intermolecularinteractions compared to the other three. The computed threelowest ME states are degenerate, with no CR contributions inany of them, with very small ME binding energies and verylow couplings, that is, ∥γ∥2 ≤ 0.01 in all cases.

Note that the sum of all the TT configurations in the statesof tetracene trimers listed in Table I does not equal 1. Theanalysis of the triply excited fragment cumulant spin matrix(Z(k = 3)) indicates that the missing contributions (∼8% inall cases) correspond to the triply excited configurations ofthe STT character, where the TT fragment is located on thesame dimer with the largest ωTT

I J value.

IV. CONCLUSIONS

We presented a new analysis scheme designed formultichromophoric assemblies. The analysis is based on thefragment-localized orbitals and employs spin correlators todistinguish between different types of the ME states. Theutility of the scheme is illustrated by the analysis of theexcited states in clusters of H2, ethylene, and tetracene. Theadiabatic wave functions in these molecular aggregates werecomputed by RASCI. Owing to its size-intensivity, the spin-flip flavor of the RASCI type of wave function allows formeaningful comparison between systems of varying size.The new method provides insight into electronic structure ofmultichromophoric systems and delivers valuable referencedata for validating excitonic models. The decomposition ofthe wave functions in terms of the LE, CR, and varioustypes of ME configurations can be used for computing simpleenergy correction, to account for deficiencies of underlyingexcited-state methods.

The analysis of the ME states in model tetracene trimersreveals that the lowest TT states are localized on tetracenedimers, which justifies using dimer-based models for thedescription of these ME states, and in rate calculations if singleexcitons are also mainly localized on two chromophores.23,24

The multiexciton stabilization energy in these states is relatedto CR contributions, which, in turn, controls the strength ofnon-adiabatic couplings between the LE and ME states.

ACKNOWLEDGMENTS

The authors are grateful to Professor Anatoliy V. Luzanovfor motivating this study and for valuable insights. D.C.gratefully acknowledges the Basque Government (ProjectNo. IT588-13) and IKERBASQUE, Basque Foundation forScience, for financial support, and to SGIker for allocationof computational resources. A.I.K. was supported by theScientific Discovery through Advanced Computing (SciDAC)program funded by U.S. Department of Energy, Officeof Science, Advanced Scientific Computing Research andBasic Energy Sciences and through Grant No. DE-FG02-05ER15685 (DOE).

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