Microscopic theory of singlet exciton fission. III. Crystalline pentaceneTimothy C. Berkelbach,1, a) Mark S. Hybertsen,2, b) and David R. Reichman1, c)

1)Department of Chemistry, Columbia University, 3000 Broadway, New York, New York 10027,USA2)Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, New York 11973-5000,USA

We extend our previous work on singlet exciton fission in isolated dimers to the case of crystalline materials, focusingon pentacene as a canonical and concrete example. We discussthe proper interpretation of the character of low-lyingexcited states of relevance to singlet fission. In particular, we consider a variety of metrics for measuring charge-transfercharacter, conclusively demonstrating significant charge-transfer character in the low-lying excited states. The impact ofthis electronic structure on the subsequent singlet fissiondynamics is assessed by performing real-time master-equationcalculations involving hundreds of quantum states. We makedirect comparisons with experimental absorption spectraand singlet fission rates, finding good quantitative agreement in both cases, and we discuss the mechanistic distinctionsthat exist between small isolated aggregates and bulk systems.

I. INTRODUCTION

Singlet exciton fission is a spin-conserving energy trans-fer event whereby a photogenerated singlet exciton splits intotwo lower-energy triplet excitons.1,2 When combined witha secondary absorbing material, this type of multiple ex-citon generation3 can boost the maximal power conversionefficiency of a solar cell to almost 50%, well above theShockley-Queisser limit of 31%.4 Despite a significant num-ber of experimental5–18 and theoretical19–31 studies on a widerange of materials, the microscopic mechanism by which sin-glet fission takes place is still a matter of debate. The status ofthe field has been expertly summarized by Smith and Michl inRef. 1 and more recently in Ref. 2.

Theoretical and computational contributions to a mech-anistic understanding of singlet fission have almost en-tirely focused on pairs (or dimers) of chromophoremolecules.19,20,22,26,29–31Although this two-molecule pictureis indeed a minimal model for singlet fission, it must be ques-tioned whether such a model is quantitatively – or even qual-itatively – relevant for experimental investigations in crys-talline materials. It must be kept in mind that essentially allreported experimental findings of rapid and efficient singletfission occur in the crystal phase9,10,12–18whereas singlet fis-sion in covalent dimers in solution has been reported with van-ishingly small yield.32,33 Enormous variation in basic experi-mental signatures of electronic energy levels, including ion-ization energies and absorption spectroscopies, further hint atthe potential differences between single molecules and bulkcrystals.

This undeniable gap between theory and experiment can belargely attributed to the theoretical and computational chal-lenges associated with large system sizes; a unified treatmentof the static and dynamic properties of large systems of inter-acting electrons and phonons is an ongoing challenge in con-densed phase chemical dynamics. For example, even the static

a)Electronic mail: tcb2112@columbia.edub)Electronic mail: mhyberts@bnl.govc)Electronic mail: drr2103@columbia.edu

ab initio energy ordering of the low-energy states of isolatedacene dimers is still under debate.20,23,29–31The current paperbridges this critical gap by extending our previous work onsinglet fission in pentacene dimers25,26 to the problem of sin-glet fission incrystallinepentacene,treated at the same levelof theory. Our ability to treat such large system sizes origi-nates from a more pragmatic computational approach, whicheschewsab initio predilection, favoring a careful combina-tion of quantum chemical and semi-empirical methods guidedby experiment. In this way, we are able to consistently com-pare our mechanistic predictions between dimers and crystals,and more importantly between theory and experiment. Fur-thermore, our theoretical framework continues to be the onlyone to microscopically treat the exciton dynamics in the pres-ence of coupling to thermal vibrations. This latter featureismandatory for any meaningful prediction ofrates, which arethe most robust experimental observables for singlet fission.

The layout of this paper is as follows. We begin in Sec. IIwith a summary of the major results of our past and presentwork. This is followed in Sec. III by a review of the method-ology relevant for a treatment of larger clusters and crystals.In Sec. IV, we present our results, focusing on the linear ab-sorption, charge-transfer characterization, and singletfissiondynamics of crystalline pentacene. We conclude in Sec. V.An appendix is devoted to a discussion of the validity of thediabatic approach to singlet fission used in this work.

II. SUMMARY OF MAJOR RESULTS

In the first paper of this series25 (hereafter referred to asI), we formulated a generic, microscopic theory of singletfission, which unified static electronic structure and subse-quent quantum dynamics in the presence of a vibrational bath.In the second paper26 (hereafter II), we applied this formal-ism to the study of singlet fission dynamics inpairs of pen-tacene molecules. One aspect of II concerned the role, orlack thereof, of so-called charge-transfer (CT) states in theearly stages of singlet fission as a means of generating a cor-related triplet pair state (TT) from the initially photoexcitedFrenkel excitation (FE). We found that for pentacene the “di-rect,” two-electron coupling is quite weak, capable of produc-

2

ing fission dynamics on a 10 ps timescale at best. We foundthat a higher-order superexchange-like singlet fission mecha-nism mediated by CT states, with an effective (second order)matrix elementVeff = VFE,CTVCT,TT/∆E(CT), dominated thedirect two-electron transfer process. This higher-order mech-anism for singlet fission has since been adopted in a numberof works.2,28,30However, in II it was demonstrated that a purenon-adiabatic rate theory based on the superexchange matrixelement is notquantitativelyaccurate due to fact that the ma-trix elements that control the mixing with CT states is of thesame order as the energy difference between the FE and CTstates themselves. In this large coupling limit,Veff > 10 meV,our dynamics calculations showed the onset of a plateau inthe singlet fission rate (see Fig. 7(b) of II). Remarkably thisbehavior, including even the value of the plateau onset, hasbeen very recently observed in a large-scale joint experimen-tal and theoretical study by Yostet al.34 These authors ratio-nalized their findings in a manner consistent with the theorypresented in II.

In this paper we apply the methodology developed in I andII to treat fission in bulk crystals, focusing on the prototypicalcase of pentacene. Hundreds of electronic states coupled tolattice (phonon) degrees of freedom are treated within a fullymicroscopic framework for the quantum dynamics followingthe initial photo-excitation of the bright, low-lying S1 singletstate. On a superficial level, we find that the time-dependentpopulations of theS1 and TT state in the pentacene crystal ap-pears to proceed in a manner similar to that of an isolated pen-tacene dimer whose geometry is taken from the crystal (videinfra, compare Fig. 5 of this work and Fig. 5 of II). In partic-ular, in both cases we find that the temporal growth of the TTstate is ultrafast (occurring on the order of 250 fs) with nearlyexponential kinetics. This fact would seem to support the no-tion that in studying fission kinetics, a dimer, perhaps embed-ded in a continuum dielectric environment, is a good proxyfor the bulk. However, a closer analysis reveals that there arequalitative distinctions between these extreme situations.

For a pentacene dimer, as described above, the couplingbetween the FE and TT states is enhanced by mixing withhigh-lying, virtual CT states. In the bulk case investigatedhere, we will clearly demonstrate that the FE and CT statesare strongly mixed into theadiabatic S1 state, a mixing facil-itated by the lowering of CT energy levels engendered by thepolarizability of the surrounding molecules. In this sense, thelow-lying photoexcited singlet already has a large CT com-ponent. Hence, fission in pentacene essentially occurs viaa direct one-electron, as opposed to a two-electron, trans-fer process. It should be noted however, that the degree ofCT mixing is material dependent. In systems with an inter-molecular spacing larger than that of pentacene, it is expectedthat the two-electron coupling will eventually dominate overthe terms responsible for CT mixing. In such systems fissionmay occur via a direct two-electron transfer mediated by theCoulomb operator. In II, we investigated such a scenario andpredicted that in this case fission would proceed on time scalesof tens of picoseconds or slower. This prediction also appearsconsistent with the experimental findings of Yostet al.34

With respect to the treatment of electronic structure, our

theory – developed and applied in I, II, and the present work– employs a real-space, tight-binding (diabatic) formulation.Here, adjustment of parameters is carried out to optimizeagreement with thea-polarized absorption spectrum, similarto the approach taken in earlier studies that targeted the Davy-dov shift.27,35 General aspects pertaining to the validity of thediabatic approach to fission are discussed in an appendix. Thevalues of electronic coupling parameters that we find for pen-tacene are in semi-quantitative agreement with those foundindependently by block-localized DFT,36 constrained DFT,34

and both restricted30 and complete31 active space methods. Incomparison with calculations for bulk pentacene at theGWandGW+Bethe-Salpeter equation levels of electronic struc-ture theory, we find good agreement both with respect to theband structure (Fig. 1) and the qualitative real-space charac-teristics of the electron-hole distributions (Fig. 4).

We further provide a detailed analysis of the real-spaceelectronic character of theS1 state in Sec. IV B. Althoughthe deficiency of assessing CT character via examinationof static dipole moments has been discussed in several re-cent studies,29,37,38we make this point explicit by calculatingdipole moments associated with low-lying singlet states forclusters derived from the the crystal structure of pentacene.For clusters that break the natural inversion symmetry of thecrystal (such as those employed in Ref. 21), a non-zero staticdipole moment may be detected in low-lying singlet states,while clusters that obey the symmetry of the periodic crys-tal exhibit a vanishing dipole moment for all excited singletstates. This analysis makes clear that such dipole momentsare not directly associated with CT character, but instead arisefrom surface-localized charge density in systems with brokensymmetry. We will also explicitly show that for the assign-ment of CT character, as well as for quantification of the ex-citon size, the examination of the exciton correlation functionprovides more useful information than a natural transitionor-bital analysis in periodic crystals with inversion symmetry.

III. METHODOLOGY

Because our adopted theoretical methodology has been de-scribed and utilized in I and II,25,26 we will only briefly de-scribe the formalism, with a focus on the differences betweenthe present crystal-phase calculation and our previous studyof acene dimers. We employ a system-bath excitonic Hamil-tonian describing the coupling of the electron and phonon de-grees of freedom,39,40 Htot = Hel + Hel−ph + Hph, with

Hel =∑

i

|i〉Ei〈i| +∑

i, j

|i〉Vi j 〈 j|, (1)

Hel−ph =∑

i

|i〉〈i|∑

k

ck,i qk +∑

i, j

|i〉〈 j|∑

k

ck,i j qk, (2)

and

Hph =∑

k

p2k

2+

12ω2

kq2k

. (3)

3

We emphasize that the Hamiltonian above is constructed ina basis of many-body electronic states. The parameters ofthis Hamiltonian are determined via a variety ofab initio andsemi-empirical methods, as discussed in Sec. III B.

A. Reduced density matrix quantum dynamics

The reduced dynamics generated by the above Hamiltonianare calculated with a Redfield-type, weak-coupling quantummaster equation.25,39,41Specifically, in the basis of electroniceigenstatesHel|α〉 = ~ωα|α〉, the reduced density matrix (i.e.averaged over the phonon degrees of freedom) obeys the equa-tion of motion

dραβ(t)

dt= −i(ωα − ωβ)ραβ(t) +

∑

γ,δ

Rαβγδργδ(t). (4)

Explicit expressions for the Redfield tensor,Rαβγδ, which in-troduces population relaxation and coherence dephasing, canbe found in Appendix C of I. Such expressions depend onlyon the spectral density of the phonons, which we take to beidentical for all electronic states, and to be of the Ohmic formwith a Debye cutoff, i.e.

J(ω) =π

2

∑

k

c2k

ωkδ(ω − ωk) = 2λΩω

1ω2 + Ω2

. (5)

We takeλ = 50 meV and~Ω = 180 meV. Consistent with ourfindings in II, we neglect the off-diagonal Peierls coupling.As an aside, we would like to point out that the “true” spec-tral density as defined in the first equality of Eq. (5) has amaximum phonon frequency above whichJ(ω) = 0 (roughlyaround 3000 cm−1 for C-H stretching modes in acenes). How-ever, the addition of a high-frequency component to the spec-tral density, when treated to lowest order (as in Redfield the-ory), can serve as a proxy for multi-phonon relaxation pro-cesses arising from lower-energy phonons. In other words, theelectron-phonon interaction is written generically asHel−ph =∑

i |i〉〈i|Φi(qk) and the spectral density is then understood tobe related to the Fourier transform of the phonon autocorrela-tion function〈Φi(qk(t))Φi(qk(0))〉ph. This observation ex-tends the applicability of our approach to systems with largeelectronic energy gaps, as is found for example in hexacene.42

With large quantum dynamics calculations in mind, let usbriefly comment on aspects of computational tractability. Firstand foremost, a reduced density matrix calculation is likelythe only microscopic quantum dynamics formalism with thepotential to scale to large clusters approaching crystal behav-ior, primarily because such an approach has already averagedover the continuous vibrational degrees of freedom. Withoutinvoking the secular approximation (which decouples popula-tion and coherence evolution in the electronic eigenstate ba-sis), the solution of the Redfield equation requires the appli-cation of theN2 × N2 Redfield tensor onto theN × N densitymatrix (whereN is the number of electronic states). Such aneffective matrix-vector multiplication scales asO(N4). Pol-lard and Friesner43 showed that by exploiting the structure

of most system-bath interactions, the application of the Red-field tensor can be reduced toB matrix-matrix multiplications,scaling asO(BN3), whereB is the number of collective bathvariables to which the system is coupled (and can be as fewas B = 1). In the case where each electronic state is cou-pled to its own set of bath variables, thenB = N, and thereis nooperationalsavings in the Pollard and Friesner scheme;even worse, in the presence of Peierls-like off-diagonal cou-pling to B = N2 independent sets of bath variables, it scalesasO(N5), which isworsethan the naive approach. However,this scheme never requires the storage of the Redfield tensorand so there can indeed be savings in memory storage require-ments, depending on implementation. In our calculations, wehave exploited the techniques of Pollard and Friesner, whichwere sufficient to treat the desired system sizes. Furthermore,the results presented have invoked the secular approximation,although the non-secular Redfield dynamics exhibit the samequalitative behavior.25

B. Geometry and electronic structure

Following I, we parametrize our electronic Hamiltonian ina tight-binding basis of localized, diabatic states with excita-tions spanning one or two molecules per basis state. The basisstates are characterized as being in one of three spin singletclasses: (1) intramolecular Frenkel excitations (FE); (2)inter-molecular charge-transfer excitations (CT); and (3) correlated,triplet pair excitations (TT). Note that in I and II, we followedthe convention of the field and referred to localized FE statesasS1 states. We avoid this potentially confusing nomenclaturehere because the lowest-lying (adiabatic) singlet excitation incrystals is no longer dominated by FE states,vide infra.

For a cluster ofM molecules, there are in principleM FEstates,M(M − 1) CT states, andM(M − 1)/2 TT states, suchthat the number of states grows quadratically with the num-ber of molecules, i.e. N ∼ M2. However, in molecularcrystals like pentacene, the molecules largely retain their iso-lated properties and intermolecular interactions are reasonablyweak, suggesting a semi-empirical truncation of the excita-tions to nearest-neighbors only (allowing for weakly Wannier-like excitons). With this approximation, the number of elec-tronic states grows linearly withM. Our Hamiltonian is thusgiven byHel = HFE + HCT + HTT + HFE,CT + HCT,TT, where

HFE =∑

m

|FEm〉EFE〈FEm| (6)

HCT =∑

〈mn〉|CmAn〉ECT(m,n)〈CmAn| (7)

HTT =∑

〈mn〉|TmTn〉ETT〈TmTn| (8)

4

HFE,CT =∑

〈mn〉

[

|FEm〉tLL(m,n)〈CmAn| (9)

− |FEm〉tHH(m,n)〈CnAm| + H.c.]

(10)

HCT,TT =√

3/2∑

〈mn〉

[

|TmTn〉tHL(m,n)〈CmAn|

+ |TmTn〉tLH(m,n)〈CnAm| + H.c.]

, (11)

where〈mn〉 denotes thatm andn are nearest-neighbors. Wehave neglected the so-called “direct” coupling, two-electronmatrix element〈FEm|Hel|TmTn〉 due to its relative smallness,as quantified statically and dynamically in II. It is straightfor-ward to include this contribution if necessary.

The molecules in a pentacene crystal adopt a familiar her-ringbone packing arrangement. As discussed in II, there areapproximately three symmetry-unique nearest-neighbor pairsof molecules with different electronic interactions. Follow-ing the Appendix of I, we employ notation such thatεg isthe single-molecule HOMO-LUMO gapin the crystal envi-ronmentand tAB(m,n) are one-electron coupling matrix el-ements. These properties follow from an appropriate one-electron theory, for which we have used Hartree-Fock the-ory, supplemented with renormalizations appropriate for thecrystal. Specifically,εg is increased, as in theGW correc-tion to DFT band structures, and the one-electron couplingstAB are rescaled to incorporate screening effects. In the ab-sence of electron-hole interaction effects, these parametersyield the single-particle band structure shown in Fig. 1, whichis in good agreement with previously publishedab initio GWcalculations.44 Although the LUMO band in particular showsslightly less dispersion than that obtained byGW, our ap-proach yields an absorption spectrum in better agreement thanthat fromGW plus the Bethe-Salpeter equation (BSE). Fur-thermore the quasiparticle gap ofEQP = 2.0 eV, located at theC-point in the Brillouin zone, is in good agreement with theexperimental value of 2.2 eV,45,46especially when accountingfor surface effects. The reduction as compared to the gas-phase single molecule gap of about 5 eV47 can be ascribed toelectronic polarization of the surrounding crystal medium.

Electron-hole interaction effects are incorporated with aconfiguration interaction approach, from which we write thediagonal energies as

EFE = εg + 2K − J (12)

ECT(i, j) = εg − J′(i, j) (13)

ETT ≈ 2E(T) = 2(εg − J). (14)

In the above,

J =∫

d3r1

∫

d3r2H2A(r1)r−1

12L2A(r2) (15)

is a direct, intramolecular Coulomb integral;

K =∫

d3r1

∫

d3r2HA(r1)LA(r2)r−112LA(r1)HA(r2) (16)

εg = 2.35 eV EQP = 2.0 eV

Γ X C Y Γ

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Ene

rgy

(eV

)

Present theory GW

FIG. 1. Band structure of crystalline pentacene calculatedwithin the presenttheory compared to the many-bodyGW results of Ref. 44. Also shown are thebare single-molecule gapεg in the absence of dispersion, and the minimumquasiparticle gapEQP when dispersion is included.

is an intramolecular exchange integral; and

J′1 =∫

d3r1

∫

d3r2H2A(r1)r−1

12L2B(r2) (17)

J′2 =∫

d3r1

∫

d3r2H2B(r1)r−1

12L2A(r2) (18)

are direct, intermolecular Coulomb integrals. For simplic-ity, we assumeJ′1 = J′2 ≡ J′ (although we point out thatvery recently Zenget al.31 have shown that this is not accu-rate for certain molecular geometries). We have neglected theintermolecular exchange integral due to the smallness of thedifferential overlap. We adopt the Bethe-Salpeter viewpointto configuration-interaction excitations, such thatK is an un-screened exchange integral, whereasJ andJ′ are screened di-rect integrals.48 With this convention,K approximately satis-fies the experimental, single-molecule singlet-triplet exchangesplitting, 2K ≈ E(S1) − E(T1) which we take to be 1.10eV.49,50 The unscreened direct interaction for a gas-phasemolecule would beJ ≈ Eg − E(T1) ≈ 4 eV, however it isreduced in the crystal by approximately a factor of 3 due toscreening, and we useJ = 1.45 eV. The value ofJ′, which setsthe CT state energies, depends on the specific dimer-pair, andis accordingly larger for nearer neighbors (stronger electron-hole interaction). Our employed CT state energies are in goodagreement with those obtained experimentally by Sebastienetal.51 and theoretically by Yamagataet al.35 The values for allparameters used throughout this work are collected in Tab. I.

The ultimate validation of these parameters lies in the theo-retically predicted absorption spectrum which includes all theeffects of state-mixing and spatial delocalization. We turn tothis topic next. In all results presented henceforth, we haveperformed calculations on finite clusters ofN = 10,27, and52 molecules (roughly 3× 3, 5× 5, and 7× 7 respectively)until converged. Without the Hilbert space truncation de-scribed above (i.e. nearest-neighbor CT and TT pairs only),these clusters would have 145, 1080, and 4030 electronicstates. With such truncation, these clusters have 67, 213, and439 electronic states, which still provides significant compu-tational demands in light of the scaling of density matrix dy-namics described above.

5

εg J E(T1) 2K E(FE)

Monomer 2.35 1.45 0.90 1.10 2.0

[a b] J′ E(CT) tHH tLL tHL tLH

[1 0] 0.10 2.25 26 -24 -26 26[1/2 1/2] 0.30 2.05 -44 46 38 -43[−1/2 1/2] 0.35 2.00 68 -44 38 -47

TABLE I. Electronic structure values (all in eV, exceptti j in meV) for the pen-tacene crystal for monomers (top) and for the three nearest-neighbor dimertypes (bottom). All values should be understood to be in the crystal environ-ment, not the gas phase.

IV. RESULTS

A. Linear absorption spectroscopy

A crucially important metric in assessing the accuracy ofthe employed electronic structure model is the calculated lin-ear absorption signal. Furthermore, because of the anisotropyof the pentacene herringbone lattice, polarized absorptionspectroscopy permits access to more detailed structural infor-mation. Specifically, we calculate the imaginary part of thedielectric function,

ε2(ω) ∝ 1ω2

∑

α

|e · µα|2D(Eα − ~ω) (19)

whereD(E) is a lineshape function (taken to be Gaussian witha broadening of 0.04 eV),e is the optical polarization vector,and

µα ≈ 〈0|µ|α〉 =∑

n

CαFEn

µFE (20)

is the transition dipole moment to the adiabatic excited stateα.Note that we assume that the localized Frenkel exciton statescarry all the oscillator strength, and thus the optical bright-ness of a given adiabatic excited state is dependent on thedegree of Frenkel exciton character. The absorption (or ex-tinction) coefficientα(ω) is related to the dielectric functionvia α(ω) ∝ ωε2(ω), where the index of refraction is assumedconstant over the energy range of interest.

To assess our electronic structure model, we seek tocompare to experimental measurements whenever possible.Unfortunately, experimental polarization-resolved absorptionspectra for pentacene are in fact quite varied. This mayin part be due to variations in methodology necessitated bycrystal anisotropy, which include reflectivity,52 generalizedellipsometry,53 and electron energy-loss spectroscopy.52,54,55

Spectra for thea-polarized absorption are more uniform, andin Fig. 2(a) we compare to the ellipsometry spectrum fromRef. 53. The total absorption spectrum, i.e. the average ofthe a + b components, can be obtained from polycrystallinesamples. In Fig. 2(b), we compare to low temperature (4 K)absorption spectra of such a polycrystalline film presentedinRef. 16. The agreement for all observed peaks can be seento be quite good, giving credence to our employed electronic

(a)

1.6 1.8 2.0 2.2 2.4

Energy (eV)

0.0

0.2

0.4

0.6

0.8

1.0

ε2(ω

)(a

.u.)

Exp’t || a

Theory|| a

Theory|| b

(b)

1.6 1.8 2.0 2.2 2.4

Energy (eV)

0.0

0.2

0.4

0.6

0.8

1.0Exp’t a+ b

Theorya+ b

FIG. 2. Comparison between theoretically predicted and experimental ab-sorption spectra of crystalline pentacene. Panel (a) showsthe theoretical spec-trum for botha andb polarization, compared to the experimentala-polarizedspectrum from Ref. 53. Panel (b) compares the theoretical averaged spectrumwith that of a polycrystalline sample given in Ref. 16, where we have con-verted absorption data viaε2(ω) ∝ α(ω)/ω. All spectra are normalized to thepeak value of the lowest-lying transition. The Davydov shift is given by theenergy separation of the first two peaks in panel (b), as emphasized by thevertical dashed lines.

structure model. In Ref. 55, it was remarked that the struc-ture in pentacene’sa-polarized spectrum does not appear tooriginate from a vibronic progression, in contrast to that oftetracene and smaller acenes. This observation is consistentwith our calculation, which neglects explicit phonon couplingin the absorption spectra. Such behavior can be understoodin terms of the reduced exciton-phonon coupling in largermolecules, combined with an increased CT-mixing that ac-counts for the additional peaks (discussed further below).

Importantly, such polarization-resolved spectra offer accessto the Davydov shift, i.e. the signed difference between thefirst peaks of theb- anda-polarized spectra. Our calculationsyield a Davydov shift of 0.16 eV, in good agreement with var-ious experimental values (0.13 eV,16,56,570.14 eV,58 and 0.15eV53) and theoretical values (0.13 eV27,35,44and 0.20 eV59).Yamagataet al.have identified an important trend in their the-oretical study of acene spectroscopy: an increasing (positive)Davydov shift is correlated with an increasing degree of CTcharacterphysically mixedinto the low-lying brightS1 exci-ton. Indeed, in our calculations on sufficiently large clusters ofpentacene molecules, we find that the lowest-lying, bright sin-glet state comprises roughly 50% FE and 50% CT character.These observations are consistent with the CT exciton char-acteristics inferred from solid-stateGW+BSE calculations onperiodic pentacene crystals,37,44,59but inconsistent with recentquantum chemical TD-DFT calculations on finite pentaceneclusters embedded in a polarizable environment.21 Given theimportant prospective role of CT states in facilitating rapidsinglet fission, we now turn to an analysis of CT character,seeking to unify the theoretical results put forth to date.

B. Characterizing charge-transfer character

Because of the localized basis employed in our electronicstructure model, we can definitively quantify the degree of CTcharacter (∼50%) in our adiabatic excited states (and likewise

6

(b)(a)

FIG. 3. Two finite molecular clusters considered for an evaluation of the staticdipole moment as a measurement of CT character. The cluster shownin (a)was the one used in Ref. 21, however only the cluster in (b) preserves theinversion symmetry of the crystal. Calculatedab initio dipoles moments ofthe excited states are given in Tab. II.

in the work of Yamagataet al.35 and Beljonneet al.27). How-ever, inab initio approaches that work directly in the adiabaticbasis, such as TD-DFT orGW+BSE, no such decompositionis possible. Thus, to put our work in perspective, we will con-sider more generic quantities which are in principle availableto all levels of theory. Specifically, we will consider threepos-sible metrics for characterizing CT character: the static dipolemoment of an excited state, natural transition orbitals (NTOs),and exciton correlation functions.

Let us first briefly consider the static dipole moment, aswas used in Ref. 21 to argue against CT character in low-lying excited states of tetracene and pentacene. Specificallythe authors of that work found a small dipole moment forlow-lying excited states but a larger dipole moment for high-lying excited states, and from this concluded that low-lyingstates were of Frenkel type and only high-lying states hadsignificant CT character. Although such an analysis is use-ful in asymmetric small molecules (such as metal-to-ligandcharge transfer60), for symmetric systems (like periodic crys-tals) there can be nonet charge-transfer in any direction,as this would break symmetry. We believe the non-zerodipole moments of Ref. 21 are only due to the use of a fi-nite cluster which breaks the inversion symmetry inherentin the crystal. This concern has been raised in a numberof other recent works.37,38 To demonstrate this assertion, weperformedab initio HF+CIS calculations with a modest 6-31G(d) basis set on two different molecular clusters of pen-tacene (calculations were carried out at the experimental ge-ometry with the GAMESS(US) software package61). Thefirst is the four-molecule cluster considered in Ref. 21 andthe second is a seven-molecule cluster which restores inver-sion symmetry, as shown in Fig. 3. The dipole momentsof the first five excited states for each cluster are given inTab. II. Clearly, the four-molecule cluster exhibits largeandincreasing dipole moments which are completely absent inthe inversion-symmetric seven-molecule cluster. We thereforeconclude that a dipole-moment based analysis fails to reporton CT character in solid-state materials, and turn now to theother two common metrics: NTO analysis and exciton corre-lation functions.

The NTO analysis62 is common in quantum chemistry andwas employed in Ref. 21 as a second means of characterizingCT character for singlet fission materials. Exciton (electron-hole) correlation functions are more common in the solid-state

TABLE II. Total static dipole moments (in Debye) of the ground state andfirst five excited states for the 4- and 7-molecule clusters shown in Fig. 3.

S0 S1 S2 S3 S4 S5

4-molecule cluster 0.0 0.5 3.6 16.5 23.0 15.77-molecule cluster 0.0 0.0 0.0 0.0 0.0 0.0

GW+BSE community for such characterization.48 Here weunify these two viewpoints in terms of the transition densitymatrix, relevant for any single-excitation theory (HF+CIS,TD-DFT, GW+BSE).

Consider the lowest lying bright excited state,S1. Within asingle-excitation theory, and employing a minimal, localizedbasis of HOMO (H) and LUMO (L) orbitals (M each), we canwrite

|S1〉 =∑

m,n

TmnEnm|GS〉 (21)

where the singlet excitation operator is given byEnm =

(c†L,n,αcH,m,α + c†L,n,βcH,m,β)/√

2. The transition density matrix

Tmn = 〈GS|Emn|S1〉 is M×M and reflects the weight of excita-tion from the HOMO of moleculem to the LUMO of moleculen. In other words, the diagonal of the matrix correspondsto Frenkel excitations and the off-diagonal to CT excitations.Such an interpretation is not so straightforward in the basis ofdelocalized ground-state (HF-like) orbitals.

Calculating NTOs first requires rotating the transition den-sity matrix from the localized basis to the HF-like basis:T′ = CHTC†L, where

[

φHFL

]

i=∑

j [CL] i j

[

φlocL

]

jand likewise

for the HOMOs. The NTOs then result from a singular-valuedecompositionUT′V† = Λ. The matrixU defines a transfor-mation of the occupied orbitals into a set of NTOs represent-ing the hole, and likewiseV defines a transformation of the un-occupied orbitals into a set of NTOs representing the excitedelectron. The pair of electron and hole NTOs correspondingto the largest element of the diagonal matrixΛ gives, in somesense, the best electron-hole picture of the excited state,i.e.

|S1〉 =Λ00√

2

(

a†L,0,αaH,0,α + a†L,0,βaH,0,β

)

|GS〉 +O(Λ11) (22)

wherea†L,n,σ =∑

v Vnvc†v,σ andaH,n,σ =

∑

o Unoco,σ.Turning now to the use of exciton correlation functions,

we consider the wavefunction of an electron-hole excitationin real space,

χS1(re, rh) =∑

m,n

TmnLn(re)Hm(rh) =∑

v,c

T′vcφc(re)φv(rh),

(23)wherev andc denote valence and conduction bands; the lat-ter equality is to emphasize that (unlike NTOs), the exci-ton wavefunction is independent of the choice of reference.This two-particle wavefunction is six-dimensional but canbe visualized in three dimensions by fixing (for example)the hole near moleculeA and plotting the electron density,

7

FIG. 4. Panels (a) and (b) depict, for the lowest-lying bright excited singlet state, the dominant natural transition orbitals of the electron and hole, respectively,projected onto a central cluster for clarity (these orbitals in reality extend over the entire system). Despite the apparent spatial overlap, the excitation still hassignificant CT character. Panels (c) and (d) depict, for the same state, the electron and hole density for a fixed hole and electron position, respectively. Thehole and electron, depicted by the blue and red spheres, are fixed near the central molecule. Density on neighboring molecules unambiguously demonstratessignificant CT character.

|χS1(re; rh = RA)|2, where

χS1(re; rh = RA) ≈ HA(RA)∑

n

TAnLn(re), (24)

and we have utilized the locality of the MOs. In this way,onecorrelatesthe locations of the particle pair. Importantly,if the exciton is completely of Frenkel type, then the transitiondensity matrix is completely diagonal (in the localized basis),TAn = TAAδAn, and

χS1(re; rh = RA) ≈ TAAHA(RA)LA(re) ∝ LA(re). (25)

Thus the electron density,for a hole fixed near one molecule,will be simply proportional to the (squared) LUMO orbitalonthe same molecule. In contrast, if the exciton is completelyof the CT type, then the transition density matrix is entirelyoff-diagonal,TAn = TAn(1− δAn), and

χS1(re; rh = RA) ≈ HA(RA)∑

n,A

TAnLn(re). (26)

In this case, the electron density will be proportional to thesum of the squared LUMO orbitals onneighboringmolecules.Again, although this interpretation is made most apparent inthe localized basis, the exciton wavefunction and thus theanalysis, is completely independent of basis.

In Fig. 4, for the same excited state, we compare the domi-nant electron and hole NTOs (panels (a) and (b)) with the elec-tron and hole correlation functions, for fixed hole and electronrespectively (panels (c) and (d)). We find that the NTOs are

extended over the entire system, and so for clarity we onlyplot the components on a central seven-molecule cluster. Forthe exciton correlation functions, these seven molecules aresufficient because in our model, CT excitations are restrictedto nearest-neighbors only. As a reminder, the excited stateunder consideration has a 50%/50% mix of FE and CT com-ponents. Looking first at the NTOs, we see that they are bothcompletely delocalized over the entire cluster, and show noelectron-hole correlation. However, in contrast to the claimsof Ref. 21, this has no implication for CT character. Such aresult is especially apparent here because we know the exci-ton wavefunction being analyzed has significant CT character,due to our use of a localized basis. We are forced to con-clude again that NTO analysis, although useful for asymmet-ric molecules and excitations with a net transfer of charge,62

is not useful for inversion symmetric or periodic systems, justas we found for the static dipole moment above.

On the contrary, if we look at the exciton correlation func-tions in Fig. 4(c) and (d), we clearly see evidence of both FEand CT character. As explicated above, the electron densityonthe same molecule as the (fixed) hole reports on FE compo-nents, whereas the electron density on neighboring moleculessignals significant CT mixing (and vice versa for the holedensity for fixed electron). In particular, we find significantCT excitations along the [−1/2 1/2] direction, which can betraced back to the low energy of these states and the strongcoupling in this direction (see Tab. I). These latter pointsare also reflected in the anisotropic mobility tensor of acenecrystals.63

8

This behavior is qualitatively similar, although quantita-tively different, than that observed in analogous plots obtainedvia earlyGW+BSE44 calculations, where essentially all elec-tron density is on neighboring molecules, indicating a nearlypure CT exciton. However, our results appear to be similarto more recent calculations at the same level of theory.59,64

To the extent that the transition dipole moment is purely in-tramolecular, i.e.〈Hm|r|Ln〉 = δmnµFE, the brightness of a sin-glet excited state is proportional to its FE character (as men-tioned in Sec. III). Thus a pure CT exciton would be nearlydark spectroscopically, and so differences in the spatial exci-ton wavefunction may originate from different choices of theanalyzed excited state, i.e. perhapsGW+BSE predicts low-lying, dark CT excitons, while the bright excitons necessarilyhave more FE character. For these reasons, we believe ouranalysis shown in Fig. 4 is generic for the lowest-lyingbrightexciton, independent of methodology.

As an aside, we would like to address some apparent confu-sion in the literature regarding the “size” of an exciton. Intheabsence of exciton-phonon coupling and crystal defects, therein no mechanism to break symmetry in a crystal, and thus theexciton is delocalized over the entire system. This behavior isreflected in the NTO analysis, as the electron and hole NTOsare completely delocalized (as observed in Ref. 21) and there-fore give no insight into the size of the exciton. A meaning-ful quantity is the exciton radius, or theaverageelectron-holeseparation. This is the variable used in Wannier-like mod-els of excitons, wherein the electron (or hole) is fixed at theorigin giving rise to hydrogenic wavefunctions and energy re-lations. This symmetry-breaking imposed by fixing one of thetwo particles is exactly the same as that employed in the cor-relation function analysis presented in Fig. 4(c) and (d). Shar-ifzadehet al.have developed a variety of other useful metricsfor assessing CT character, and similarly conclude that low-lying excitons in pentacene exhibit significant CT character,with average electron-hole separations greater than 6 Å.37

Let us conclude this section by emphasizing that, in light ofthe provisos explained above, the electronic structure calcula-tions of Ref. 21 are in no way inconsistent with significant CTcharacter in the low-lying excited states of tetracene and pen-tacene. We therefore anticipate that when analyzed carefullyand consistently, most electronic structure calculationsin theliterature (semiempirical, TDDFT, orGW+BSE) would be inqualitative agreement concerning the role of CT mixing intolow-lying, bright singlet states of larger oligoacenes. Withthis apparent discrepancy resolved, and having found strongevidence for significant CT character in these excited states,let us turn to the dynamical implications for the subsequentsinglet fission.

C. Singlet fission dynamics

As an idealized initial condition, we first consider an ex-citation to the single lowest-lying, bright adiabatic eigenstate.Although an electronic eigenstate, this initial conditionis non-stationary with respect to the nuclei (alternatively, it isa su-perposition in the localized, diabatic basis) and thus the re-

(a)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

State

pop

ulation

SinglesTT

FE

CT

(b)

0.0 0.2 0.4 0.6 0.8 1.0

Time (ps)

0.0

0.2

0.4

0.6

0.8

1.0

State

pop

ulation

FIG. 5. Singlet fission population dynamics for all single excitations (thickblack line), TT excitations (thinner red line), FE excitations (dashed blueline), and CT excitations (dot-dashed green line). Note that the singles pop-ulation is simply the sum of the FE and CT populations. Panel (a) is foran initial condition which populates only a single, low-lying bright adiabaticstate (near 1.8 eV) and panel (b) is for an impulsive (wide band) spectroscopicpreparation where all adiabatic states are populated proportional to their os-cillator strength.

laxation dynamics will proceed towards thermalization. Theresults of our Redfield master equation dynamics are shownin Fig. 5(a); specifically we plot the summed population of allsingle excitations (FE+CT) and the summed population of allTT excitations. Furthermore, we decompose the single exci-tons into their FE and CT components; as described above,the initial condition is roughly 50%/50% in its FE/CT com-position. As the dynamics proceed, the adiabaticS1 state isdepleted and the TT state is populated with a time constant of270 fs, in qualitatively good agreement with the experimen-tal value of 80–110 fs,9,10,16 especially given the complexityof the process. Minor changes to the electronic structure orphonon spectral density can easily account for this difference,though we prefer not to explore such endeavors here.

With regards to the underlying mechanism, we emphasizethat because the initial condition already contains a signif-icant degree of CT character, the low-energy TT states canbe rapidly accessed via localized one-electron transfer eventsthroughout the crystal, as was also suggested in Ref. 27. Weemphasize again that in spite of the similarity in timescales,the mechanisms for CT-mediated singlet fission in dimers(presented in II) and crystals (presented here) are qualitativelydifferent. Nonetheless, we again find dynamics which appear“direct” in the one-step, kinetic sense. One must simply becareful in recognizing that the initial condition under photoex-citation isnot of the pure FE type and so concerns regardingtwo-electron transfer processes are unnecessary (at leastforcrystalline materials with sufficient CT character in their low-lying excited states).

9

We now consider a more realistic spectroscopicpreparation40 in the presence of a radiation field withHamiltonian

Hrad = −∑

α

|α〉µα · E(t)〈0| (27)

whereα is an eigenstate of the electronic Hamiltonian (i.e. anadiabatic excited state at the ground-state geometry),µα =

〈0|µ|α〉 is the transition dipole moment (vector) from theground state to the excited stateα, and E(t) is the polar-ized time-dependent electric field. Henceforth we considera-polarized fields, i.e.E(t) = E(t)a. To lowest non-trivialorder in the electric field, the initial excited state populationsfollowing the pulse are given by

ραα(0) ≈ 1~2

∫ 0

t0

dt1

∫ t1

t0

dt2|µα,a|2E(t1)E(t2)

×[

Trph

U†0(t1 − t2)Uα(t1 − t2)ρph

+ H.c.]

(28)

where Ui(t) corresponds to adiabatic evolution in the statei (we neglect non-adiabatic vibrational effects during thepulse). In the impulsive limitE(t) = Aδ(t), the above ex-pression reduces to

ραα(0) ≈ 1~2

∣

∣

∣µα,aA∣

∣

∣

2. (29)

In this scenario, the exciting pump probe is infinitely shortand thus has an infinite spectral bandwidth, populating all ex-cited states in proportion to their oscillator strength. For sim-plicity, we chooseA to give the initial RDM unity trace, i.e.A−1 =

∑

α |µα,a|2/~2. The results of such impulsive prepara-tion are shown in Fig. 5(b). Although the short time, tran-sient dynamics of the individual FE and CT populations areslightly different (in fact, the initial condition hasmore CTcharacter), it is clear that the overall dynamics are qualita-tively unchanged. Quantitatively, this initial conditiondoesexhibit slightly faster singlet fission, with a time constant of250 fs.

The results shown here demonstrate that for reasonable ini-tial conditions, the CT character of such low-lying states fa-cilitates ultrafast fission dynamics. The presented study alsosuggests an interesting endeavor towards engineering initialconditions (either structurally or spectroscopically) tomaxi-mize singlet fission rates. Another topic of concern relatedto spectroscopic preparation involves the vibrational degreesof freedom. In particular, if vibrational relaxation in theex-cited state manifold out-competes singlet fission, then thenew“initial condition” is equilibrated in theS1 state. Such relax-ation can set up a reorganization energy barrier for subsequentfission, thereby inhibiting the process. Such behavior likelycannot be accounted for using weak-coupling Redfield-likeperturbation theory, however the modified Redfield theory65,66

for adiabatic state populations could be readily applied inthisregime.

V. CONCLUSIONS

In conclusion, we have extended the treatment of excitonfission presented in I and II to the crystalline case focusingonpentacene as a concrete example. Our theory combines ele-ments of electronic structure theory with a full treatment of thequantum dynamics that result from the coupling of the elec-tronic degrees of freedom to lattice vibrations. A completediscussion of the role of the CT character in the excited statesthat mediate the fission process has been presented. Whilesome adjustment of parameters constrained by experimentaldata has been employed (as is standard in excited state elec-tronic structure calculations of large systems21), various non-trivial predictions and features emerge from our theoreticalapproach with no further alterations that lend strong supportto the correctness of the framework presented in I, II, andthe present paper. These successful predictions include thenon-adiabatic to adiabatic crossover predicted in II and dis-cussed above, the magnitude and direction of the Davydovshift, the full polarization-dependence of the absorptionspec-trum, and the absolute rate of fission in crystalline pentacene.Finally, using the framework presented here, we have also cal-culated the absorption line shape properties and fission rate inthe more challenging case of hexacene, and have confirmedthat the calculated properties are in striking agreement withexperiment in this case as well.42

The ability to treat fission in systems of this size allowsus to distinguish the mechanistic features that arise in dimersor in small aggregates from those uniquely associated withbulk systems. Combined with our dimer study in II, we arethus able to make rather general qualitative and quantitativestatements about the earliest steps of singlet fission in bothfinite and periodic systems. While we have used pentaceneas a paradigmatic example, the generality of our approach hasramification for the understanding of singlet fission in a widerange of organic materials. Given these successes, we are nowin position to use the theoretical apparatus presented in I,II,and this work to explore the possibilities of more exotic fissionphenomena such as the potential conversion of the S1 stateinto more than two triplet excitons, as well as realistic “inversedesign” and optimization of new fissionable materials.

ACKNOWLEDGMENTS

We thank Joe Subotnik, Garnet Chan, and Eran Rabani forhelpful conversations. This work was supported in part bythe Center for Re-Defining Photovoltaic Efficiency throughMolecule Scale Control, an Energy Frontier Research Cen-ter funded by the US Department of Energy, Office of Sci-ence, Office of Basic Energy Sciences under Award Num-ber DE-SC0001085. This work was carried out in part atthe Center for Functional Nanomaterials, Brookhaven Na-tional Laboratory, which is supported by the U.S. Departmentof Energy, Office of Basic Energy Sciences under ContractNo. DE-AC02-98CH10886 (M.S.H). T.C.B. was supported inpart by the Department of Energy Office of Science GraduateFellowship Program (DOE SCGF), made possible in part by

10

the American Recovery and Reinvestment Act of 2009, ad-ministered by ORISE-ORAU under Contract No. DE-AC05-06OR23100.

Appendix A: On the use of diabatic states in singlet fission

The theoretical framework employed in I, II and in thispaper makes use of a physically motivated, real-space tight-binding basis to define singlet, triplet, charge-transfer,andmulti-exciton configurations, as well as the coupling betweenthese configurations and the vibrational modes of the lattice.The true eigenstates of the electronic Hamiltonian, necessaryfor interpretation of dynamics following photo-excitation aswell as the computation of absorption line shapes, are foundvia diagonalization of the electronic Hamiltonian in this basis.While not defined via a formal diabatization procedure start-ing from the adiabatic eigenstates as found, e.g. from a quan-tum chemistry program, this basis may be considered an ap-proximate diabatic basis. Such an approach lies at the heartofnearly all successful theories of dynamical processes in con-densed phases,67–69 including the description of charge,40,70,71

energy,40,72,73 and spin transport.74 In addition, the diabaticapproach has formed the core of essentially all successful pre-dictions made in the field of singlet fission, as described inSec. II and V of the present paper. Given these facts, theclaims of the recent paper by Fenget al. in Ref. 29, that the di-abatic picture cannot not be employed to study singlet fissionare rather surprising and deserve attention.

The first aspect of the critique of the use of the diabatic ap-proach is based on the analysis of the configurations that con-tribute to the states relevant to singlet fission. For the case thatmost overlaps with the work in II and in this paper, namelythe “X-ray” tetracene and pentacene dimers, the authors ofRef. 29 conclude that initial and final states in the fissionprocess are mixed with states of charge-transfer character(or“charge-resonance” character as referred to in Ref. 29). Thisis in agreementwith previous work, as well as that presentedin II and in this paper. More specifically, Fenget al. find thatthe adiabaticS1 state is composed of configurations largely ofFE and CT character (∼ 80% of total) while the singlet TTstate (called1ME in Ref. 29) is overwhelmingly composed ofconfigurations of TT and CT character (> 90%). However,Fenget al. consider this a failure of the diabatic picture be-cause these states have a small mixing component with con-figurations other than those of CT character. While it wouldbe quite simple to include additional states into the approachdescribed in this paper (which is clearly capable of accommo-dating hundreds of states), this is not necessary. Considerthe“correct” adiabatic S1 state,|S1〉 = α|S(0)

1 〉+ (1−α)|X〉, where|S(0)

1 〉 is an approximate adiabatic state, and|X〉 schematicallydescribes all additional states that are not captured by a FE/CTdiabatic framework. In terms of the rate, this missing weightwill enter as (at least) the square, such that the importanceofthe neglected configurations is roughly (1− α)2/α2 ∼ 0.01for α = 0.9 (α2 ≈ 0.8), i.e. two orders of magnitude smaller.Thus, in this quantitative sense, such configurations may beignored.

The second argument against the diabatic approach pre-sented by Fenget al. involves the ability of the diabatic pictureto describe the correct dynamics at particular high-symmetrydimer configurations. As a concrete example, Fenget al. con-sider the case of the cofacial (D2h) geometry for acene dimers.While this configuration hasno relevance for fission in thecrystalline acenes such as pentacene,7,75 it provides an instruc-tive example of the subtleties that arise when considering theconsistency of interpretation of state transitions in the diabaticand adiabatic bases. To be concrete, consider a simple 2× 2Hamiltonian in a diabatic basis,

H = T(Pz,PR) +

E1 V(z,R)V(z,R) E2

, (A1)

whereT is the total kinetic energy operator,Ei are the dia-batic energies of the two states (assumed for simplicity to beindependent of nuclear configuration), andV(z,R) is the di-abatic coupling between the states. We take thez-axis to beperpendicular to the plane of the dimers in theD2h cofacial ge-ometry and the magnitude ofz to denote the distance betweenthe molecules, whileRschematically denotes coordinates per-pendicular to thez axis. Thusz is the “D2h-scan” coordinateof Fenget al. Due to theD2h symmetry, bothV(z,R= 0) and∂V(z,R= 0)/∂zare identically zero.

We first calculate the non-adiabatic coupling (NAC) alongthez-axis (R= 0),

Ax+,−(z,R= 0) =

⟨

ψ+

∣

∣

∣

∣

∣

∂

∂x

∣

∣

∣

∣

∣

ψ−

⟩

(z,R=0)

(A2)

where x is a nuclear coordinate which can be taken sepa-rately as eitherz or R and |ψ±〉 are the adiabatic eigenstatesof the Hamiltonian, Eq. (A1). Because of the symmetry de-scribed above, it is clear thatAz

+,− = 0, whereas it is straight-forward to show thatAR

+,− = φ(z)/(E1 − E2), whereφ(z) =∂V(z,R)/∂R|R=0. While V(z,R= 0) is zero, clearlyφ(z) is not,howeverφ(z) will rapidly decay with increasingz.

Physically, the above analysis shows that transitions in theadiabatic picture cannot occur if nuclear motion is constrainedto occur along the “D2h-scan” (z) direction, but is facilitatedby nuclear motiontransverseto z along theRdirections. Fur-ther, becauseφ(z) is a decreasing function ofz, the probabil-ity of transition decreased for increasing molecular separation.We now show that the very same physical picture emerges inthe diabatic basis. At first this may seem surprising, becausealong thez axis (R = 0) the mixing between diabatic statesV(z,R) is identically zero, hence states of potentially distinctsymmetry cannot be connected. HoweverV(z,R) is not just afunction ofz, but varies as a function ofR as well. In the di-abatic picture the facilitator of transitions is the vibronic cou-pling, found by expansion ofV(z,R). Here, to lowest orderV(z,R) ≈ φ(z)Rand the Hamiltonian in the diabatic basis maybe written

H ≈ T(Pz,PR) +

E1 φ(z)Rφ(z)R E2

, (A3)

which is closely related to the linear vibronic couplingmodel frequently employed in the simulation of non-adiabatic

11

processes.76,77 If we assume for simplicity thatz in a non-dynamical parameter (i.e.z is fixed), then the golden-ruletransition rate between diabatic states may be written as78

k =φ(z)2

~2

∫ ∞

−∞dtei(E1−E2)t/~〈R(t)R(0)〉 (A4)

where〈R(t)R(0)〉 is the time correlation function of transversenuclear fluctuations. Despite the difference in precise mean-ing of electronic transitions in the two bases, the same grossfeatures emerge: a rate for transitions that is zero if nuclearmotion is constrained to proceed only alongz, a finite ratefacilitated by transverse nuclear motion, and a rate that de-creases asz increases. The error in the reasoning of Fengetal. lies in not recognizing thatV(z,R= 0) = 0 does not implyφ(z) = ∂V(z,R)/∂R|R=0 = 0. It is this term which in the adi-abatic picture provides a non-zero NAC, and in the diabaticpicture defines the strength of the vibronic coupling.

Lastly, as an alternative to computing the full NAC, Fenget al. note that the NAC is a “one-electron” object and pro-pose to bound the NAC by the norm of the (dimensionless)transition density matrix. It should be noted that the NACcan only be treated as a one-electron objectif the exact adia-batic wave functionsare known and employed. In practice, forDFT as well as wave-function based approaches, both the oneand two-particle density operators are needed to calculatetheNAC.79–81 In this sense, the proposed “bound” is in generalnot valid.

In conclusion, we believe the claims of Fenget al. regard-ing the utility of the diabatic basis are faulty. Indeed, severalstudies have very recently appeared, using an electronic struc-ture methodology identical to that employed by Fenget al.,that have constructed a diabatic basis starting from numeri-cally determined adiabatic wave functions and have success-fully computed fission-related properties.30,31 The physically-motivated diabatic approach described in I, II and in this workis in semi-quantitative agreement with these studies with re-spect to the description of the electronic structure.

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