Electron correlation effects in electron-hole recombination in organic ...

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likely processes,and hence there is no reason for to be. 0.25. Experim entally, hasbeen found to rangefrom. 0.25. { 0.668,9,10,11 in di erent m aterials. In O LED ...
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arXiv:cond-mat/0208529v1 [cond-mat.mtrl-sci] 27 Aug 2002

E lectron correlation e ects in electron-hole recom bination in organic light-em itting diodes K unj Tandony,1 S. R am asesha,1 and S. M azum dar2 1

2

Solid State and Structural C hem istry U nit, Indian Institute of Science, B angalore 560 012, India D epartm ent of Physics, and T he O pticalSciences C enter, U niversity of A rizona, Tucson, A Z 08721 (D ated: M arch 18,2013) W e develop a generaltheory ofelectron{hole recom bination in organic light em itting diodes that leads to form ation ofem issive singlet excitons and nonem issive triplet excitons. W e brie y review other existing theories and show how our approach is substantively di erent from these theories. U sing an exact tim e-dependent approach to the interchain/interm olecular charge{transfer w ithin a long{range interacting m odelw e nd that,(i) the relative yield of the singlet exciton in polym ers is considerably larger than the 25% predicted from statisticalconsiderations,(ii) the singlet exciton yield increases w ith chain length in oligom ers, and, (iii) in sm all m olecules containing nitrogen heteroatom s,the relative yield ofthe singlet exciton is considerably sm aller and m ay be even close to 25% . T he above results are independent of w hether or not the bond-charge repulsion, X ? , is included in the interchain part ofthe H am iltonian for the tw o-chain system . T he larger (sm aller) yield ofthe singlet (triplet) exciton in carbon-based long-chain polym ers is a consequence ofboth its ionic (covalent) nature and sm aller (larger) binding energy. In nitrogen containing m onom ers, w avefunctions are closer to the noninteracting lim it,and this decreases (increases) the relative yield of the singlet (triplet) exciton. O ur results are in qualitative agreem ent w ith electrolum inescence experim ents involving both m olecular and polym eric light em itters. T he tim e-dependent approach developed here fordescribing interm olecularcharge-transferprocessesiscom pletely generaland m ay be applied to m any other such processes. PA C S num bers: 78.60.Fi,73.50.P z,72.80.Le,71.35.-y,31.15.D v

I. IN T R O D U C T IO N

C harge recom bination and photoinduced chargetransfer lie at the heart of current attem pts to construct viable optoelectronic devices using organic sem iconducting m aterials consisting of -conjugated polym ers or m olecules. C harge recom bination is the fundam entalprocessofinterestin organiclightem itting diodes (O LED S). Electrolum inescence (EL) in O LED S results from , (a) the injection of electrons and holes into thin lm s containing the em issive m aterial,(b) m igration of these charges, w hich can involve both coherent m otion on a singlechain and interchain orinterm olecularchargetransferbetween neutraland charged species,(c) recom bination ofelectronsand holeson thesam epolym erchain orm olecule1,2,3. Ifthe recom bination leadsto the singlet opticalexciton,lightem ission can occur.If,on the other hand,the nalproduct ofthe recom bination is a triplet exciton, only nonradiative relaxation can occur in the absence ofstrong spin-orbit coupling. EL in O LED S is of strong current interest, both because of applications in display devices4,5 and the potentialfor obtaining organic solid state lasers6. T he fundam entalprocess that occurs in photoinduced charge-transfer is the exact reverse ofthat in EL:opticalexcitation to the singlet exciton in a donor m olecule is followed by charge separation and m igration of charge to a neighboring acceptor m olecule.T he latterprocessisofinterestin photovoltaic applications7.

T he fundam entalelectronic processofcharge recom bination or separation is therefore ofstrong current interest. Especially in the context ofEL in O LED S,charge recom bination has received both experim entaland theoreticalattention (see below ). T he overallquantum e ciency ofthe EL depends on,(i) the fraction ofthe total num ber of injected carriers that end up as excitons on the sam e polym eric chain or m olecule, (ii) the fraction ofthese excitonsthatare spin singlets,since only singlet excitonsareem issive,and (iii)thefraction ofsingletexcitonsthatactually undergo radiativedecay.In thepresent paper we focus on (ii),w hich determ ines the m axim um possible EL e ciency. Form ally, the charge recom bination process can be w ritten as, P+ + P

! G + S=T

(1)

w here P are charged polaronic states of the em issive m olecule,G is the ground state ofthe neutralm olecule, and S and T are singlet and triplet excited states ofthe neutralm olecule. Eq.1 indicates that both singlet and triplet excitons are likely products ofthe charge recom bination process. W e shalldenote the fraction ofsinglet excitons generated in O LED S by the above recom bination process as . Early discussions of were based on statisticalargum ents alone.Since electronsand holes are injected independently from thetwo electrodes,and sincetwo spin-1/2 particles can give three independent spin 1 states (w ith

2 M S = {1,0 and + 1) but only one spin 0 state (M S = 0),it follow s that is 0.25. N ote,however,that this argum entisstrictly valid only fornoninteracting electrons, such that single-con guration m olecular orbitaldescriptions of all eigenstates are valid. In such a case, the highest occupied and lowest unoccupied m olecular orbitals (H O M O and LU M O ) are identicalfor the singlet and tripletexcited states. C harge recom bination (Eq.1) then involves m erely the m igration of an electron from the doubly (singly) occupied H O M O (LU M O ) ofP to the singly occupied (unoccupied)H O M O (LU M O )ofthe P + , for both singlet and triplet channels. T he singlet channeland allthree triplet channels of the charge recom bination process are equally likely w ithin the M O schem e.Ifelectronsare interacting,however,thissim ple single-con guration description breaks dow n, as allthe states included in Eq.1 are now superpositionsofm ultiple con gurations. T here is no longer any fundam ental reason for the singlet and triplet channels to be equally likely processes,and hence there is no reason for to be 0.25. Experim entally, hasbeen found to rangefrom 0.25 { 0.668,9,10,11 in di erent m aterials. In O LED S w ith the m olecular species A lum inum tris (8-hydroxyquinoline) (A lq3 )as the em issive m aterialB aldo et. al. have determ ined 0.22 0.03,in agreem entw ith thatexpected from statisticalargum ents8. O n the other hand,considerably larger 0.45 has been found in derivatives of poly(para-phenylenevinylene) (PPV ) by C ao et al. and H o et.al.9,10.W ohlgenanntet. al.,using spin-dependent recom bination spectroscopy,have determ ined the form ation cross-sectionsofsingletand tripletexcitons, S and T ,respectively,fora large num berofpolym eric m aterials (including nonem issive polym ers in w hich the lowest two-photon state 2A g occursbelow the optical1B u exciton),and found that S = T is strongly m aterialdependent and in allcases considerably largerthan 1 (thereby im plying that is m aterialdependent and m uch larger than 0.25)11. M ore recently,W ilson etal.12 and W ohlgenannt et. al.13 have show n that can depend strongly on the e ective conjugation length,w ith values ranging from 0.25 for sm allm onom ers to considerably larger than 0.25 for long chain oligom ers. T heoretically, has been investigated by a num ber of groups11,14,15,16,17 including ourselves. T here is general agreem ent that can be substantially greater than 0.25 in -conjugated polym ers and that this is an electron correlation e ect. T here exist,however,substantialdifferences between the assum ptions and form alism s that go into these theories.T he goalofthe presentwork is to develop a form alism thatgivesa clearphysicalpicture of the electron-hole recom bination and explainsw hy substantially larger than 0.25 is to be expected in organic polym eric system s. Ideally, since photoinduced chargetransfer is the exact reverse process ofelectron-hole recom bination, it should also be possible to extend our approach to photoinduced charge-transfer in the future. B riefpresentation ofour work has been m ade earlier11,

w here, however,the em phasis was m ore on the experim entaltechniqueused by ourexperim entalcollaborators. H ere we present the fulltheoreticaldetails ofour earlier work,provide a critique ofthe earlier theories and also report on the new and interesting results of our investigation of externalelectric eld e ects on , albeit for arti cially large elds, and also on the role of nitrogen heteratom s in electron{hole recom bination. Speci cally, our theoreticalapproach involves a tim e-dependent form alism ,w ithin w hich the initialstate com posed oftwo oppositely charged polarons is allowed to propagate in tim e under the in uence of the com plete H am iltonian that includes both on-chain and interchain interactions. Forthe sake ofcom pleteness,we also discussotherexisting theoreticalapproaches14,15,16,17,and theirapplicability to realsystem s.In particular,thereexistsa super cial sim ilarity between the approach used in references16,17 and ours. For a physicalunderstanding ofthe electronholerecom bination processitisessential(seebelow )that thedi erencebetween ourapproach and thatused by the authors ofreference 16,17 is precisely understood. T he plan ofthe paper is as follow s. In section II,we present our theoreticalm odels for intrachain and interchain interactions, and also discuss the m odel system s that are studied. In section III we present a brief critique ofthe existing theories. A m ore extended discussion ofthe approach used by Shuaiet. al.16,17 is given in A ppendix 1. In section IV we present the m ethod of propagation of the initial state, w hile in section V we presentournum ericalresults.In thissection we also discuss an alternate approach to the tim e propagation for the sim plest case oftwo ethylenes that con rm s the validity of the m ore generalapproach used in section IV , and that also gives a physicalpicture ofthe recom bination process.W hile > 0.25 is found in ourcalculations w ith interacting electrons,theabsoluteyieldsofboth singletand tripletexcitonsare found to be extrem ely sm all w ith standard electron correlation param eters.W etherefore investigate the e ectsofthe externalelectric eld on these yieldsw ithin a highly sim pli ed m odel. Itis found that for su ciently large elds the yields w ith interacting electrons are as large as those w ith noninteracting electrons in the eld-free case,and that in the relatively sm all eld region continuesto be greaterthan 0.25.In the very high eld regim e it is found that can even be sm aller than 0.25. W hile the bare electric elds required to see the reversalofthe singlet-triplet ratio are rather large and therefore only of academ ic interest, if internal eld e ects are taken into account, it is possible to envisage situations w here the e ective electric eld is large enough to bring about such a reversal in the singlet-tripletratio. Follow ing the discussion ofelectric eld e ects,we discuss how the chain length dependence of , as observed experim entally12,13, can be understood w ithin ourtheory.W e then considerthe role of heteroatom s,especially in the contextofm olecularem itters. W e show that in sm allsystem s w ith heteroatom s can approach the statisticallim it,thus explaining quali-

3 tatively the m onom erresultsofW ilson et. al.12,and the resultsofB aldo et.al.forA lq38.T heem phasisin allour calculations is on understanding the qualitative aspects ofchargerecom bination and noton detailed quantitative aspects. Finally in section V Iwe discuss the conclusions and scope offuture work.

II. T H E O R E T IC A L M O D E L

T he goal of the present work is to provide benchm ark resultsforthe chargerecom bination reaction w hich are valid for the strong C oulom b interactions that characterize -conjugated system s. A ccurate treatm ents of electron-electron interactions are not possible for long chain system s,and in thisinitialstudy we have therefore chosen pairs ofshort polyene chains,w ith 2 { 6 carbon atom sin each chain asourm odelsystem s.Since polyene eigenstates possess m irror-plane and inversion sym m etries, we shall henceforth refer to the ground state G (see Eq.1)as 11A g ,and S and T as 11B u and 13B u ,respectively.T he m odelsystem containing two hexatrienes (12 carbon atom s overall) is the largest system that can be treated exactly at present w ithin correlated electron m odels. O urapproach su ersfrom two apparentdisadvantages. First, polyenes and polyacetylenes are weakly em issive because the 21A g state in these occur below the optical11B u state. T his presents no problem as far as the analysis of the EL in em issive m aterials is concerned, as the spectroscopic technique ofW ohlgenannt et. al.11 nd a strong deviation of S = T from 1 even in system s w ith energy ordering sim ilar to that in polyenes11 (see results for PT V in this paper,for instance), and as we show in the follow ing,this is a direct consequence ofthe large energy di erence between the singlet 11B u exciton and the triplet 13B u exciton,as wellas the fundam ental di erence in their electronic structures. B oth, in turn, are consequencesofstrong electron-electron interactions, w hich also characterize system s like PPV and polyparaphenylene (PPP),asevidenced from the large di erence in energies between the singlet and triplet excitons in these system s,determ ined experim entally18,19,20,21,as wellas theoretically22,23. A second apparent disadvantage ofour procedure is related to the lim itation ofour calculations to short system s. T his prevents direct evaluation ofthe chain length dependence of . W e believe that this problem can be circum vented once the m echanism of the physical process that leads to the di erence between singlet and triplet generation is precisely understood,and for this purpose it is essentialthat the electron correlation e ects are investigated thoroughly using exactly solvable m odels. A swe show later,ourapproach gives a precise though qualitative explanation of the chain length dependence. O ur m odel system consists of two polyene chains of equallengths that lie directly on top ofeach other,separated by 4 A . W e consider the charge recom bination

processofEq.1,and there are two possible initialstates: (i) a speci c chain (say chain 1) is positively charged, w ith the other (chain 2) having negative charge,a conguration that hereafter we denote as P 1+ P 2 ,w here the subscripts 1 and 2 are chain indices, or (ii) the superposition P 1+ P 2 P 2+ P 1 ,in the sam e notation. In our calculationswe have chosen the rstasthe properinitial state, since experim entally in the O LED S the sym m etry between the chains is broken by the externalelectric eld (we em phasize thatthe consequence ofchoosing the sym m etric or antisym m etric superposition can be easily predicted from our all our num erical calculations that follow ). Even w ith initial state (i), the nal state can consist of both (11A g )1 (11B u )2 and (11A g )2 (11B u )1 in the singletchannel.T he sam e istrue in the tripletchannel, i.e., either of the two chains can be in the ground (excited) state. H ereafter we w illw rite the initialstates asjiS iand jiT i,w herethesubscriptsS and T correspond to spin states S = 0 and 1. W e consider only the M S = 0 triplet state. T he initialstates are sim ply the product states w ith appropriate spin com binations,

jiS i= 2

1=2

+ (jP 1; " ijP 2;# i

+ jP 1; # ijP 2;" i)

(2)

jiT i= 2

1=2

+ + (jP 1; " ijP 2;# i+ jP 1;# ijP 2;" i)

(3)

T here exist of course two other initial triplet states w ith M S = 1. T he overallH am iltonian for our com posite two-chain system consists ofan intrachain term s H in tra and interchain interactionsH in ter . A dditionalinteractions m ust be explicitly included to discuss externalin uences like the electric eld etc. H in tra describing individual chains is the Pariser-Parr-Pople (PPP) H am iltonian24,25 for -electron system s,w ritten as, X H in tra = X

tij(ayi; aj; + H :C :)+

< ij> ;

i

ini + i

X Vij(ni

U ini;" ni;# +

X

zi)(nj

zj) (4)

i> j

w here ayi; createsa -electron ofspin on carbon atom i, ni; = ayi; ai; is the num ber of electrons on atom i P ni; isthe totalnum berofelecw ith spin and ni = tronson atom i, i is the site energy and zi are the local chem icalpotentials. T he hopping m atrix elem ent tij in the above are restricted to nearestneighborsand in principle can contain electron-phonon interactions,although a rigid bond approxim ation is used here. U i and Vij are theon-siteand intrachain intersiteC oulom b interactions. W e use standard param eterizations for H in tra . T he hopping integrals for single and double bonds are taken to be 2.232 eV and 2.568 eV ,respectively and allthe site energiesofcarbon atom sin a polym erw ith allequivalent sites are setto zero.W e choose the H ubbard interaction param eterU C for carbon to be 11.26 eV ,and for the Vij

4 we choose the O hno param eterization26, #

" 28:794 Ui + Uj

Vij = 14:397

2

+

1 2

ri2j

(5)

w here the distance rij is in A , Vij is in eV and the local chem icalpotentialzC forsp2 carbon isone. Itshould be noted then w hen hetero atom s like nitrogen are present, the on-site correlation energy, the site energy and the localchem icalpotentialcould be di erentfrom those for carbon. For H in ter ,we choose the follow ing form , X (ayi a0i; + H :C :)+ H in ter = t? X +X? i;

i;

(ni + n0i)(ayi a0i; + H :C :)+ X Vi;j(ni

zi)(n0j

zj0)

(6)

i;j

In theabove,prim ed and unprim ed operatorscorrespond to siteson di erentchains.N ote thatthe interchain hopping t? is restricted to corresponding sites on the two chains,w hich are nearest interchain neighbors. T he interchain C oulom b interaction Vi;j,however,includes interaction between any site on one chain w ith any other site on the other chain. In addition to the usual oneelectron hopping that occurs w ithin the zero di erential overlap approxim ation24,25 wehavealso included a m anyelectron sitecharge-bond chargerepulsion X ? (operating between nearestinterchain neighbors only) that consists ofm ulticenter C oulom b integrals. T his term should also occurw ithin H in tra ,butisusually ignored there because ofitssm allm agnitude,relative to allotherterm s24,25,27. In contrast, the t? in H in ter is expected to be m uch sm aller, and X ? cannot be ignored in interchain processes, especially at large interchain separations28. W e have done calculations for both X ? = 0 and X ? 6 = 0. III. B R IE F C R IT IQ U E O F E X IST IN G T H E O R IE S

To put our work in the proper context we present a discussion of the existing theories of charge recom bination14,15,16,17 in this section. T he natures of H in tra w ithin all these m odels are sim ilar in the sense that they all incorporate intrachain C oulom b interactions,w ithout w hich ofcourse there cannot be any difference between singlet and triplet generation. Follow ing this, there is a fundam ental di erence between the m odels ofreferences 14,15 on the one hand,and those of references 16,17 and ours on the other. W ithin the theory ofreferences 14,15,there is no di erence in singlet or triplet generation in the rst stage ofthe chargerecom bination process,w hich involvesinterchain chargetransfer.W ithin thesem odels,interchain charge-transfer yields high energy singlet and triplet excited states of

long chains that occur in the continuum , and the low est singlet and triplet excitons result from relaxations of these high energy states. D i erences in the relative yields ofthe lowest singlet and triplet excitons are consequences ofdi erences in the intrachain relaxation processesin thesingletand tripletchannels,thatoccurin the second stage ofthe overallprocess. In contrast,w ithin our theory11 and the theory ofreferences 16 and 17,the lowest singlet and triplet excitons are generated directly from two oppositely charged polarons,and their di erentyieldsare consequencesofthe di erentcross-sections ofthe interchain charge-transfer reactions in the singlet and triplet channels. W ithin the m odel of H ong and M eng14, the continuum singlet state decays to the lowest singlet exciton, w hile the continuum triplet state decays to a high energy tripletstate T2 consisting ofa loosely bound triplet exciton,w hich then relaxes nonradiatively to the lowest tightly bound triplet exciton T1. T he energy gap between T2 and T1 is large, and according to H ong and M eng, this nonradiative relaxation has to be a m ultiphonon cascade process. T he large energy gap and the m ultiphonon nature ofthe relaxation creates a \bottleneck" in the T2 ! T1 nonradiative transition,and spinorbit coupling leads to intersystem crossing from T2 to the singlet exciton,thereby increasing the relative yield ofsinglets14. W e believe that the key problem w ith this approach is thatthe m odelisin disagreem entw ith w hat isknow n aboutthespectrum oftripletstatesfrom triplet absorptions in -conjugated polym ers20 and theoretical solutions to the PPP m odel23. Experim entally,in PPV , forinstance,the lowesttripletoccursatabout1.55 eV 20, w hile in M EH -PPV this state occursat 1.3 eV 18 . T he triplet absorption energy in these system s is about 1.4 eV .T heoretically,the nalstate in tripletabsorption occurs slightly below the continuum band23, and this is therefore the T2 state (also referred to as the m 3 A g 20 ). T he energy region between T2 and T1 (m 3A g and 13B u ) in the tripletsubspace isnotatallsparse,asassum ed by H ong and M eng,but rather,w ithin the correlated PPP H am iltonian H in tra in Eq. 4,thisenergy region contains num erous other triplet states29,30. T hus any nonradiative relaxation from T2 to T1 in the realistic system s should involve a num ber of interm ediate triplet states w ith sm allenergy gaps between them ,and therefore the phonon bottleneck sim ply w illnot occur. A n additional problem w ith them odelofH ong and M eng isthateven in the singletchannel,generation ofthe lowestexciton from a continuum singletstate cannotbe directbutcan occur only through the m 1A g loosely bound singlet exciton23. In principle,thiscan lead to a bottleneck even in the singlet channel. To sum m arize,we believe that the m odel of H ong and M eng is in disagreem ent w ith the know n singletand tripletenergy spectra w ithin the PPP m odel. W ithin the m odel of K obrak and B ittner15 also polaron pairs are form ed on the single chain rst. T hese authors take into account the electron-phonon interactions explicitly, and the two-particle states on a single

5 chain are allowed to evolve by interacting w ith a onedim ensionalclassicalvibrationallattice. D i erent crosssections for singlet and triplet excitons are found w ithin theauthors’m odel,and thedi erenceoriginatesfrom the di erence in the m ixing between the polaron and exciton states w ith di erent spin. T he theory includes only the C oulom b interactions between the polaron charges and not the C oulom b interactions between allthe electrons that appear in the PPP H am iltonian. T he theory also assum es large quantum e ciency for the generation of the high energy states w ith the two polaron charges on the sam e chain,starting from a state w ith the chargeson di erentchains.A recentcalculation by Ye et.al.17 indicates very weak cross-sectionsfor the generation ofhigh energy 1 B u and 3B u statesstarting from the initialstate containing the charges on di erent chains (see Fig.8 in reference 17). T his is supported also by our exactcalculations (see below ). H owever,the calculations by Ye et al.17 as wellas ours are for relatively short chains,and further work is needed to test the validity ofthe m odel of K obrak and B ittner. A s we show in section V , > 0.25 is predicted from considerations ofthe initialstage ofinterchain charge-transfer alone. W hether additional contributionscan com e from di erencesin the intrachain relaxation processes needs to be studied further. W e now com e to the work by Shuai et. al.16,17, w ho,like us, have determ ined > 0.25 in oligom ers of PPV from considerations of interchain charge-transfer. Precisely because of the apparent sim ilarity of our approaches,it is essentialthat we discuss the approach of Shuaiet. al. in detail,since our ultim ate goalis to arrive at a physicalexplanation ofthe greater yield ofthe singlet exciton than w hat is predicted from statistical considerations,and as we show later,the physicalm echanism s w ithin references 16,17 and w ithin our work are quite di erent. T he quantity that is calculated in references 16, 17 is S = T , viz., the ratio of the form ation cross-sections ofthe 11B u singlet and 13B u triplet exciton. For fast spin-lattice interaction,the expression for in term s of S and T can be w ritten as11,13, =

S =( S

+3

T)

(7)

and thus,for S = T > 1, > 0.25. Shuai et. al. consider the sam e H in tra as us, and H in ter that is sim ilar (see below ). T he authors then use the Ferm i\G olden R ule" approach to calculate S and T . A ccording to the authors,the cross-section ratio is given by, S= T

= jhiS jH in ter jfS ij2 =jhiT jH in ter jfT ij2

(8)

w herejiS iand jiT iarethesingletand tripletinitialstates (see Eqs. 2 and 3), and jfS i and jfT i are the corresponding nalstates,respectively. Since the interchain C oulom b interaction isdiagonalin the spaceofthe states considered in Eq.8,the authors ignore Vi;j in Eq.6 but retain the other term s. Shuaiet. al. nd that for X ? = 0 in Eq.6,w hen the interchain charge-transfer is due

to the hopping t? only,the righthand side ofEq.8 is 1,a result we agree w ith (see A ppendix 1). T he authors then claim thatfornonzero positive X ? ,and forpositive t? (note negative sign in front ofthe one-electron term in Eq.6),the right hand side ofEq.8 can be substantially larger than 1. T he authors calculated the m atrix elem ents in Eq.8 for pairs ofPPV oligom ers in parallel con guration using approxim atem ethods(singlescon guration interaction16 and coupled-clusterm ethod17 ),and have found the right hand side of Eq.8 to show divergent behavior over a broad range ofX ? =t? (see Fig. 1 in reference 16 and Figs. 3,4,6 and 7 in reference 17). B ased on these calculations the authors conclude that a m oderate to large X ? is essentialfor the experim entally observed large S = T 9,10,11,12,13. T his result is surprising,in view of the fact that the sitecharge-bond chargerepulsion isspin-independent,exactly as the one-electron interchain hopping in Eq. 6. Since this question is intim ately linked w ith the m echanism ofcharge recom bination that we are after we have re-exam ined this issue by perform ing exact calculations for pairs of polyene chains w ith lengths N = 2, 4 and 6. T he conclusions from these exact calculations are described below . A s discussed above, even w ith P 1+ P 2 as the initial state (w ith, of course, appropriate spin functions) the nal state contains two term s, w ith one of the two chains in the ground state and the other in the excited state. Instead of working w ith di erent superpositions of the nal states we consider S to be proportional to jhiS jH in ter j(11A g )1(11B u )2ij2 + jhiS jH in ter j(11A g )2(11B u )1ij2 . Sim ilarly, T is taken to be proportional to jhiT jH in ter j(11A g )1(13B u )2ij2 + jhiT jH in ter j(11A g )2 (13B u )1ij2 . A s show n explicitly in the A ppendix, the m agnitudes of the m atrix elem ents ofthe initialsinglet[triplet]P 1+ P 2 w ith (11A g )1(11B u )2 [(11A g )1(13B u )2] and (11A g )2 (11B u )1 [(11A g )2(13B u )1] aredi erentforX ? 6 = 0,and hencethe nalstatescannot be 1:1 superpositions ofthese con gurations. N ote that by taking the sum softhe squaresweexhaustallpossibilities autom atically. For the conclusions ofreferences 16, 17 to be valid the calculated S = T w ithin Eq.8 should now show strong dependence on X ? =t? (as m entioned above divergent S = T is im plied in references 16,17). O urexactresultsforthe three di erentchain lengthsare show n in Fig. 1 below ,w here we see thatonly forX ? =t? very close to 0.5 is S = T ,as calculated w ithin Eq.8,is substantially di erent from 1. A t allother X ? =t? the R H S ofEq.8 is very close to 1. Furtherm ore,except for X ? =t? = 0.5 the chain length dependence of S = T is weak. If we now recallthat allchain length dependent quantities(forexam ple,opticaland otherenergy gapsin polyenes29) exhibit strongest length dependence at the shortest lengths,the conclusion that em erges is that except for the unique point X ? =t? = 0.5, S = T rem ains 1 w ithin the G olden R ule approach even in the long chain lim it. In orderto understand this di erence from the results

6 ofShuaiet. al.16,17 in further detailwe present analytic resultsforthe case oftwo ethylenes(N = 2)in A ppendix 1. T hese results are im portant in so far as they begin to give a physicalpicture for the charge recom bination reaction,even asthey indicate that the site charge-bond chargerepulsion isnotthe origin oflarge .T he analytic calculationsalso m ake the origin ofthe uniquenessofthe pointX ? =t? = 0.5 absolutely clear.Indeed itisseen that precisely atthis point both S and T ,asde ned in Eq. 8, approach zero. M ore im portantly, the chain lengthindependence,as suggested in Fig. 1 can be understood very clearly from theanalyticcalculations.Finally,itcan also be seen from these calculations that had we taken the initialstate to be the superposition P 1+ P 2 P 2+ P 1 , instead of only one of these, the S = T , as calculated from Eq. 8 would be exactly 1 for allX ? =t? . O ur basic conclusion then is that the Ferm i G olden R ule approach isnotvalid forcalculationsof S = T or . T hisisto be expected also from a di erentconsideration, viz.,the Ferm iG olden R ule approach isvalid forcalculations ofstates that lie w ithin a narrow band,w hereas in the presentcase the energy di erence between the initial and nalstates,and thatbetween the singletand triplet excitons are both m uch larger than t? and X ? . T he origin of the di erence between our exact calculations ofm atrix elem ents and the approxim ate calculations of Shuaiet. al. is harder to ascertain. O ne possibility is that the polaron wavefunctions are open shell, and approxim ating these w ithin m ean eld or lim ited C I could lead to w rong conclusions. In the follow ing sections we therefore go beyond the Ferm iG olden R ule approach to understand the origin of large .

IV . T IM E E V O L U T IO N O F T H E P O L A R O N P A IR ST A T E

A straightforward num erical solution of H in tra + H in ter w illm erely give the electronic structure of the com posite two-chain system . Such a calculation does not contain any inform ation about the relative yields of speci c nal states starting from the initial two-polaron states. O ur approach therefore consists of propagating the initial state in tim e under the in uence of the com plete H am iltonian, and m onitoring the tim e-evolved state to obtain inform ation about the nalproducts. In principle, given a H am iltonian, propagation of any initial state is easily achieved by solving the tim edependent Schrodinger equation. O ne could use the interaction picture to separate the nontrivialevolution of the initialstate from the trivialcom ponentw hich occurs as a result ofthe evolution ofthe product ofthe eigenstates of the H am iltonian of the subsystem s31. In the contextofthe m any-body PPP H am iltonian such an approach is di cult to im plem ent num erically. T his is because the totalnum ber ofeigenstates for the two-chain

system is very large: the num ber ofsuch states for two chains of six carbon atom s each is 853,776 in the M s = 0 subspace. O btaining allthe eigenstates ofthe twocom ponentsystem and expressing the m atrix elem entsof H in ter in the basis ofthese eigenstates is therefore very intensive com putationally. It is sim pler to calculate the tim e evolution in the Schrodinger representation,determ ine the tim e-evolved states,and projectthem on to the desired nal eigenstates (for instance, j11A g i1j11B u i2 ). T his is the approach we take. W e rst obtain the eigenstates jP 1+ i,jP 2 i as wellas the product states exactly in the valence bond (V B ) basis29 (in w hich the total spin S is a good quantum num ber)in orderto avoid spin contam ination.Follow ing the tim e-evolution,however,we need to calculate overlaps ofthe tim e-evolved states w ith various nalstates (see below ),w hich is cum bersom e w ithin the nonorthogonalV B basis. A fter calculating the exact spin singlet and tripletinitialstates,we therefore expand these in an orthonorm albasis that has only well de ned total M S value. H enceforth we refer to the initialstates jiS i and jiT i collectively as (0) and the tim e-evolved states as (t). In principle,the tim e evolution can be done by operating on (0) w ith the tim e evolution operator, U (0;t)= exp( iH t)

(9)

w here H is the totalH am iltonian H in tra + H in ter . T his approach would,however,requireobtaining a m atrix representation of the exponential tim e evolution operator, w hich in turn requires the determ ination of the prohibitively large num ber of eigenstates of the com posite two-chain system . W e can avoid this problem by using sm alldiscrete tim e intervalsand expanding the exponentialoperatorin a Taylorseries,and stopping atthe linear term . Such an approach,however,has the undesirable e ect ofspoiling unitarity,and for long tim e evolutions would lead to loss ofnorm alization ofthe evolved state. T he way around this dilem m a has been proposed and used by others32,33 in di erent contexts and involves using the follow ing truncated tim e-evolution schem e, (1 + iH

t ) (t+ t)= (1 2

iH

t ) (t) 2

(10)

In the above equation,on the left hand side,we evolve the state at tim e (t+ t) backwards by t=2 w hile on the righthand side,we evolve the state attim e tforward by t=2. B y forcing these two to be equal, we ensure unitarity in the tim e evolution of the state. It can be seen easily that this tim e evolution w hich is accurate to t2 2 is unitary. For a given m any-body H am iltonian and initialstate,the right hand side ofEq.10 is a vector in the H ilbertspace ofthe two-chain H am iltonian. T he left hand side corresponds to the action of a m atrix on an as yet unknow n vector,that is obtained by solving the above set of linear algebraic equations. Further details ofthe num ericalprocedure can be found in A ppendix 2.

7 A ftereach evolution step,theevolved stateisprojected onto the space ofneutralproducteigenstatesofthe twochain system . T he relative yield Im n (t)fora given product state jm ;ni = jm i1 jni2 is then obtained from , Im n (t)= jh (t)jm ;nij2

(11)

In ourcase the statesjm ;nican be any ofthe nalstates ofinterest,viz.,j(11A g )1 (11B u )2 i,j(11A g )1 (13B u )2 i,etc. Itisfore cientcalculationsofthe overlaps(w hile atthe sam e tim e m aintaining spin purity) in Eq. 11 that we expand our exact eigenstates of the neutral system in the V B basis to the totalM S basis. W e em phasize that Im n (t) is a m easure of the yield of the state jm ;ni at tim e tand is not a cross-section.

V . N U M E R IC A L R E SU LT S

In thissection we reportthe resultsofourcalculations ofrecom bination dynam icsforforpairsofethylenes,butadienesand hexatrienes,both w ithin the noninteracting H uckel m odel (U i = Vij = X ? = 0) and the interacting PPP m odel. Follow ing this,we show the results of our investigation ofelectric eld e ects on the sam e system s,discuss the chain length dependence of ,and nally present the num erical results for a m odel system containing nitrogen heteroatom s. T he calculations for the noninteracting case providesa check ofournum erical procedure,and the com parison between the noninteracting and the interacting m odelallow susto determ ine the e ect ofelectron-electron interactions.

A . D ynam ics in the H uckel M odel

H am iltonian. T he frequency of oscillation is higher for larger interm olecular transfer integral t? , as expected. T he frequency of the oscillation also depends upon the size ofthe m olecule and islowerforlargerm olecules(see below for an explanation ofthis). T he equalities in the yields ofthe singlet and triplet excited states found num erically conform sto thesim plefreespin statisticsw hich predicts that in the M S = 0 state form ed from electronhole recom bination,the probability ofsingletand triplet form ation are equal. Since the M S = 1 cases always yield triplets, the spin statistics corresponding to 25% singlets and 75% triplets is vindicated in this case. A lthough the H uckelcalculationsdo notyield any new inform ation,it is usefulto pursue them further in order to arrive at a physicalm echanism ofthe charge recom bination process. To this end we have developed an alternate procedure forcalculating the above dynam icsfor the sm allestm odelsystem ,viz.,a pairofethylenes.T his alternate approach consistsofexpanding the initialstate (0)asa superposition ofthe eigenstates i ofthe com posite two-chain system w ith eigenvalues E i, X cij i(0)i (12) j (0)i= i

T he evolution ofthe state (0) is now sim ply given by X cij i(0)iexp( iE it=h) (13) j (t)i= i

T heyield Im n (t)in a given channelw ith nalstatejm ;ni is then obtained from , Im n (t)= jhm ;nj (t)ij2 X jcihm ;nj i(0)iexp( iE it=h)j2 = i

X W hile there is no di erence in energy between singlets and triplets in the H uckelM odel,it is nevertheless possible to have spin singlet and triplet initial states jiS i and jiT i, as well as singlet and triplet nal states. In Fig. 2 we show the yield for the electron-hole recom bination in the singlet channel, for pairs of ethylenes, butadienes and hexatrienes. T he yields for the triplet channelsare notshow n separately in thiscase,{ we have ascertained that these are identicalto those in the singletchannelin thiscase,asexpected. T hese calculations are for t? = 0.1 eV w ithin Eq. 6. W e note that the yields Im n (t) oscillate w ith tim e. T his is to be expected w ithin our purely electronic H am iltonian,w ithin w hich an electron or hole jum ps back and forth between the two m olecularspecies. T hese oscillationsare the analogs ofthe R abioscillations34,35 that occur upon the stim ulation of a system w ith light,w here absorption of light can occuronly w ith nonzero dam ping.W ithin ourpurely electronic H am iltonian,com plete transition to the nal states can only occur in the presence of dam ping (for exam ple,radiative and nonradiativerelaxationsofthe nalstates),that has not been explicitly included in our

jcihm ;nj i(0)ij2 +

= X

i

X 2 R e fcicjhm ;nj i(0)ih j(0)jm ;nig

i j> i

cos((E i

E j)t=h)

(14)

T he quantitieshm ;nj i(0)iarereadily obtained from the eigenstates ofthe neutralone-chain subsystem s and the com posite eigenstatesofthe two-chain system .In Tables I and II we list the nonzero values ofthe coe cients c i and the hm ;nj i(0)ivaluesforthe caseoftwo ethylenes. It is seen that sets of degenerate states of the com posite system together contribute equally to the singletand triplet channels,although individualm em bers ofthe set m ay contribute unequally. W e have determ ined that the tim e evolution obtained from thisapproach isexactly the sam easthatobtained from thegeneralm ethod described in the previous section. T he contribution arising from the right hand side of Eq. 14 has been separated into tim e-independent and tim e-dependent parts. T he latter com es about w henever the two eigenstates in question are nondegenerate.

8 TA B LE I:Signi cant ci = < (0)j i > and the < m ;nj i > values and their product in the H uckel m odel for a pair ethylenes in singlet channel. T he index i corresponds to the index of ‘signi cant’ eigenstates of the total system and E i the corresponding energy eigenvalue. i 2 3 4 5 6 7 8 9

E i (eV ) ci < m ;nj i > < m ;nj i > ci -4.3360 0.3691 0.1362 0.0503 -4.3360 -0.3373 0.1138 -0.0384 -4.1360 -0.0171 0.0120 -0.0002 -4.1360 -0.5000 0.0058 -0.0029 -4.1360 0.4989 0.0120 0.0057 -4.1360 -0.0285 0.0059 -0.0002 -3.9360 -0.3558 0.1266 -0.0450 -3.9360 0.3513 0.1234 0.0433

TA B LE II: Signi cant ci = < (0)j i > and < m ;nj for the triplet channel,for a pair ofethylenes. i 2 3 4 5 6 7 8 9

E i (eV ) < -4.3360 -4.3360 -4.1360 -4.1360 -4.1360 -4.1360 -3.9360 -3.9360

TA B LE III:Signi cantci = < (0)j i > and the < m ;nj i > values and their product for PPP m odel in the absence of electric eld,fora pairofethylenesin thesingletchannel. T he index icorresponds to the index of‘signi cant’eigenstates of the totalsystem and E i the corresponding energy eigenvalue. 4 5 11 13 29 30 32 34

i

>,

(0)j i > < m ;nj i > < m ;nj i > ci 0.3373 0.1138 0.0384 0.3691 0.1362 0.0503 -0.0179 0.0037 -0.0001 0.4985 0.0180 0.0090 0.5005 0.0047 0.0024 0.0251 0.0170 0.0004 0.3513 0.1234 0.0433 0.3558 0.1266 0.0450

Furtherm ore,att= 0 the contribution from the tim e independentpartexactly cancelsthe contribution from the tim e dependent part. W hen the sign ofthe tim e dependentpartbecom espositive the two contributionsadd up to givethe m axim um yield of0.25 in both the singletand the tripletchannelsobserved in the discrete calculations. T he periodicity ofthe oscillation corresponds to the energy di erence between the two pairs ofthe degenerate states. T his analysis could in principle be extended to the case ofthe largersystem sbutwould be quite tedious in view ofthe largerH ilbertspace dim ensions.N ote that the decrease ofthe oscillation frequency ofIm n (t) w ith increasing chain length (Fig. 2) is explained w ithin the abovealternateprocedure.T helength dependence ofthe oscillation frequency originatesfrom thesm aller(E i E j) in longer chains.

B . D ynam ics in the P P P m odel

W e now present our results for interacting electrons in H in tra and H in ter . In allcases for the interchain Vi;j we have chosen the O hno param eters,and the interchain hopping t? = 0.1 eV .For X ? ,we present the results of calculationsw ith both X ? = 0 and 0.1 eV .In Figs.3 (a)

Ei 0.5295 0.7328 3.7748 3.7844 11.2503 11.6379 14.0483 14.0611

ci < m ;nj i > ci < m ;nj i > -0.0249 -0.6992 .0174 -0.0458 -0.6953 .0318 0.7066 -0.0258 -.0182 -0.7056 0.0446 -.0315 0.0082 0.1020 .0008 -0.0025 0.1206 -.0003 0.0050 0.0028 .00001 -0.0054 0.0081 -.00004

and 3 (b) we show the plots ofIm n (t) in the singlet and triplet channels for pairs of ethylenes, butadienes and hexatrienes, respectively, for the case of X ? = 0. T he sam e results are show n in Figs 3 (c) and 3(d) for X ? = 0.1 eV . T he m ost obvious di erence from the H uckelm odelis thattheyieldsIm n (t)in both thesingletand tripletchannels are considerably reduced in the present cases. T wo otherpointsare to be noted.First,there isnow substantialdi erence between the singlet and triplet channels, w ith the singlet yield higher in all cases. Second, the strong di erences in singlet and triplet yields are true for both X ? = 0 and X ? 6 = 0. T his is in contradiction to the G olden R ule approach16,17 ,w hich ignoresthe energy di erence between the 11B u and the 13B u . T he only consequence of nonzero X ? is the asym m etry between the yields of(11A g )1(11B u )2 and (11A g )2(11B u )1 in the singlet channels,and a sim ilar asym m etry in the triplet channels. Further discussion of this asym m etry can be found in A ppendix 1. T he overallconclusion that em erges from the results of Figs. 3 (a) - (d) is that nonzero electron-electron interactions substantially enhances . In orderto understand the above resultsin furtherdetailwe have also carried outthe dynam icscalculation for pairs ofethylenes according to Eq.14. A s in the H uckel case these calculationsyield the sam e resultsasthe m ore generalm ethod. O urresultsforthe wavefunctionsofthe com positetwo-chain system and theoverlapsoftheproduct eigenstates ofthe nalneutralm olecules w ith these are show n in TablesIIIand IV .T he degeneracies in the eigenstates of the com posite system that characterized the H uckelm odelare now lifted,w hich is a know n electron correlation e ect. W hat is m ore signi cant in the present case is that the com posite state wavefunctions thathave large overlapsw ith (0)are now notthe sam e onesthathavelargeoverlapsw ith the productwavefunctionsofthe nalstates.T hisisw hatreducestheyieldsof the charge-transferprocesses in the PPP m odel,relative to the H uckelm odel.

9 TA B LE IV :Signi cantci = < (0)j i > and the < m ;nj i > values and their product for PPP m odel in the absence of electric eld,for a pair ofethylenes in the triplet channel. 2 3 10 12 19 20 31 33

Ei -2.7283 -2.7238 3.7697 3.7775 8.0804 8.0875 14.0475 14.0515

ci < m n;nj i > ci < m ;nj i > -0.0215 -0.6980 .0150 -0.0091 0.6982 -.0064 0.7068 0.0060 .0042 -0.7067 0.0203 -.0143 -0.0056 0.1115 -.0006 -0.0190 -0.1114 .0021 0.0051 -0.0056 -.00003 -0.0052 -0.0023 .00001

Tables III and IV give a clear physicalpicture ofthe charge recom bination process. For a large yield w hat appears to be essentialis that the com posite two-chain system m ust have at least som e eigenstates which have sim ultaneously large overlaps with both the directproduct of the initial polaronic states and the direct product of the pair of eigenstates of the neutralsubsystem s in the the chosen channel. T his can be interpreted as a \transition state theory" for the charge recom bination reaction ofEq.1. Large overlaps w ith the initialpolaronic pair states occur for the states 11 and 13 in the singlet channel(see Table III),and for the states 10 and 12 in the triplet channel (see Table IV ). T his is in contrast to the H uckel case, w here the large overlaps w ith the polaron pair wavefunctions were w ith the sam e com posite two-chain eigenstates. T he overlaps ofthese speci c two chain eigenstates are larger for products of singlet nal states j11A g i1 j11B u i2 than for triplet nal states j11A g i1j13B u i2 ,and this is w hat gives a larger yield for the singlet exciton.

C . E ects of external electric eld

O ur results in the previous subsection already indicate that can be substantially larger than 0.25 for the correlated electron H am iltonian ofEq.4. From com parison of Fig. 2 and Figs. 3 (a) - (d), we see however, that the relative yields Im n (t) are lower by orders of m agnitude for interacting electrons. T his is easily understandable w ithin tim e-independent second order perturbation theory,w ithin w hich the extent to w hich the initialpolaron-pair state is m odi ed is directly proportionalto the m atrix elem entofH in ter between the initial and nalstates,and inversely proportionalto the zeroth order energy di erence. Since the energy di erences between the polaron-pairstatesand the nalneutralstates are substantialw ithin the PPP H am iltonian,the yields are low . T here are two possible interpretations ofthese results. First, the actual yields of excitons in O LED S is indeed low ,com pared to the theoreticalm axim um for noninteracting electrons (recall that no direct com par-

ison of the experim entallight em ission intensities w ith the theoreticalm axim um ispossible).Second,the experim entally observed yields are in uenced substantially by externalfactors ignored so far. W e consider this second possibility here,and calculatew ithin ourtim e-dependent form alism the yields Im n (t) in the presence ofan externalelectric eld (\external" in the follow ing includesthe e ects ofboth the actualbias voltage as wellas allinternal eld e ects). W hat follow s m ay be thought ofas overly sim ple,but nevertheless,we believe that it gives the correctphysicalpicture.W e rstpresentourform alism and num ericalcalculations,and only then we discuss the interpretation ofthese results. A sbefore,we considerpairsofm oleculesthatare parallelto each other,w ith the m olecularchain-axesaligned parallelto the x-axis. T he electric eld is chosen along the y-axis,such that the totalH am iltonian now has an additionalcontribution, X ((ni 1):yi + (n0i 1):yi0) (15) H field = E i

In the above E isthe strength ofthe electric eld,and yi (yi0)givesthe y-com ponentofthe location ofthe ith (i0th carbon atom in m olecule 1 (2). W e now perform our dynam ical calculations w ith the com plete H am iltonian including H field . In Figs.4 and 5 weshow the e ectofthe externalelectric eld on the yield in the singlet and triplet channels fora pairofethylenes. W e see thatin allthe casesthere is a strong nonlinear dependence ofthe yield on the external eld. In both the singlet and the triplet channels, we see sharp increases in the yields over a range of eld strengths. T he eld strengths at w hich the increases in the yieldsoccurareabouttwo ordersofm agnitudelarger than the experim ental eldsin the O LED S,and we com m ent upon this below . H ere we only observe that the eld strength E over w hich the singlet yield is larger is sm aller than eld strength over w hich the triplet yield dom inates. W e have perform ed sim ilar calculations for the longer chain system s,and in allcases the e ects are the sam e, viz.,there exists a range of eld strength w here a sudden increase in the singlet yield occurs, w hile at still larger elds there occurs a sim ilar jum p in the triplet yield. In Figs 6 (a) and 6 (b) we have show n the singlet and triplet yields for eld strengths of0.3 V /A and 1.0 V /A ,respectively,forhexatriene.In general,fora given spin channelthe threshold eld strength decreases w ith the chain length (the threshold eld forthe singletchanneldecreases from 0.7 V /A to 0.3 V /A on going from ethylene to hexatriene,w hile the threshold eld for the tripletchanneldecreasesfrom 1.6 V /A to 1.0 V A ).T he m ost im portant conclusions that em erge from these calculations are that,(a) m acroscopically observable yields, com parable to the zero- eld yields w ithin the noninteracting H uckelm odel,are found for large elds,and (b) w hile the calculated are greater than 0.25 for sm aller elds, this is reversed w ith further increase in the eld

10 TA B LE V : In case of a pair of ethylenes the states w ith signi cant ci = < (0)j i > and < m ;nj i > , for the PPP m odelw ith electric eld of0.7 V /A in the singlet channel. i 4 5 11 21 27 30

E i (eV ) ci < m ;nj i > ci < m ;nj i > 0.5153 0.2086 0.7629 0.1591 0.6828 0.3824 0.4933 0.1886 1.0479 -0.9001 0.3868 -0.3482 6.5753 0.0005 -0.0055 0.0000 11.2881 -0.0016 -0.0577 0.0001 11.6946 0.0040 -0.1175 -0.0005

strength. In orderto understand theorigin oftheincreased yields overrangesoftheelectric eld,wehaveanalyzed thecase ofa pair ofethylenes extensively,w ithin Eq.14. Firstly it is worth noting that the geom etry in w hich the eld is introduced,the productstates ofthe neutralH am iltonian are una ected by the electric eld. W e also notice thattheeigenvaluesofthetotalH am iltonian arenotvery sensitive to the external eld. A s in the eld-free cases, we haveobtained the projectionsofthe eigenstatesofthe two-chain system on the initialstate aswellasthe productofthe nalstates,asa function ofthe applied electric eld in both the singlet and the triplet channels. In Fig. 7, we plot the coe cients h ijm ;ni as function of the electric eld for the singlet and the triplet channels. W e see that there are severalstates that show strong variation in both cases as a function ofthe eld. H owever, w hen a productofthese coe cients w ith < (0)j i > is analyzed,the num ber ofthe states that sim ultaneously have a large value ofthese coe cients at the sam e electric eld is sm aller. In Fig. 8, we plot the dom inant coe cients ofthese projections,as a function ofthe applied eld. W e note that only a few states have both projectionssim ultaneously large.W e also note thatboth the projectionspeak atthe sam e eld strength.Itisthis thatleadsto an abruptincrease in the yield atthat eld strength. T he eigenstatesofthe fullH am iltonian thathave large projections sim ultaneously to both the initial and the nal states can in fact be expressed alm ost com pletely as a linear com bination of the initial polaron product state and the nalproduct state of the neutralsystem eigenstates. In Tables V and V I, we show the projections of the eigenstates of the full H am iltonian at the resonant electric eld on to (i) the initialstate and (ii) to the product of eigenstates of the neutral system for w hich resonance is observed. W e note that there are a few eigenstates ofthe fullH am iltonian w hich have large coe cients for both projections. T his seem s to be independent of the energy of the eigenstate of the total system . T he energetics decide the period ofoscillations and not the am plitude ofthe oscillations. W e now com e to our interpretations ofthe above num erical calculations. In all cases the applied elds in

TA B LE V I: In case of a pair of ethylenes the states w ith signi cant ci = < (0)j i > and < m ;nj i > , for the PPP m odelw ith electric eld of1.6 V /A in the triplet channel. i 2 3 5 20 23 24 26

E i (eV ) ci < m ;nj i > ci < m ;nj i > -2.8347 0.5927 0.3031 0.1796 -2.7237 0.0052 0.9142 0.0048 -2.5174 0.8053 -0.2179 -0.1755 7.5928 0.0089 -0.0180 -0.0002 8.0821 0.0054 0.1518 0.0008 8.1387 -0.0031 0.0364 -0.0001 10.1769 0.0001 0.0163 0.0000

ourcalculationsare substantially largerthan w hatis expected from the externally applied voltage in O LED S. N ote,however,that our m olecules are rather sm all,and the calculated threshold elds at w hich the e ect becom es observable decrease w ith the m olecular size. In this context, it is worth recalling a previous exact calculation ofelectroabsorption for short nite polyenes36. T here the electric eld was parallelto the chain axis (as opposed to perpendicular, as in the present case), and it was found that the calculated electroabsorption can sim ulate the experim entally observed behavior in long chain polym ers37,38, provided the electric eld used in the short chain calculation was larger by two orders of m agnitude than the experim ental eld. T his is because ofthe large energy gapsin shortchains.W e believe that a sim ilar argum ent applies in the present calculations of interchain charge-transfer:the energy di erence between the initialpolaron-pairstate and the nalstatesism uch larger in the sm allm olecule-pair system than in the experim entalsystem s,even w hen oligom eric. T he analogy to electroabsorption would then im ply thatthe enhanced m acroscopic yields would occur in the real system s at m uch sm aller,perhaps even realistic elds. O ne nalpointconcernsthe geom etry used in ourcalculations. In real O LED S the relative orientations of the m olecules ofa given pair,as wellas the orientation of the electric eld w ith respect to individualm em bers of the pair w illboth be di erent from that assum ed in our sim ple calculations above. Electric elds that are nonorthogonalto the chain axis ofa m olecule w illhave even stronger e ects than found in our calculations36, w hile the random arrangem ents of the m olecule pairs w ith respectto the eld in the experim entalsystem sim pliesthatthe range of eld overw hich a given spin channeldom inatesw illbesubstantially largerthan thatfound in our calculations. W e therefore believe that a proper interpretation of our calculations is that in the experim entalsystem s,there occurm acroscopically large yields of both singlet and triplet excitons over a broad range of electric eld. For sm allto m oderate eld strengths, the singlet channel dom inates over the triplet channel. H owever,atstilllarger elds it is possible that this situation reverses.W hetherornotthishigherregim e of eld

11 strength is experim entally accessible is a topic offuture theoreticaland experim entalresearch.

D . C hain length dependence

W e now discuss the chain length dependence of as has recently been determ ined experim entally12,13. From carefulm easurem ents using di erent techniques,W ilson et. al.12 and W ohlgenannt et. al.13 have established that increases w ith conjugation length. W ilson et. al. have show n that w hile is close to the statistically expected 0.25 in the m onom er12,it is substantially larger in the polym er. W ohlgenannt et. al. have show n that T = S increaseslinearly w ith the inverse ofthe conjugation length. W ithin ournum ericalprocedure,itisdi cultto determ ine the chain length dependence of directly. T his is because ofm ultiple reasons,w hich include,(i) the lim itation to rathersm allsizes,(ii)the necessity to integrate Im n (t)overa com plete period in each case in Figs. 3 (a) -(d) to obtain the totalyield over that period,and (iii) the di erence in the periods for singlet and triplet channels,aswellasthe di erencesam ong di erentsingletand triplet channels. W e therefore present our discussion of chain length dependence w ithin a sim pli ed form alism that is consistent w ith our tim e-dependent procedure. C onsider a transition (w hich could be of the chargetransfer type) between the states jki and jsi of a twostate system , such that at tim e t = 0 the system is in state jsi (cs(0) = 1, ck (0) = 0). W e are interested in the yield jck (t)j2 at a later tim e t due to a perturbation Vks. T his is a standard textbook problem 39, and the tim e-dependent Schrodinger equation in this case is, ih

@ ck = Vksexp(i!kst) @t

(16)

For tim e-independent Vks, as is true here (Vks in the presentcaseissim ply H in ter ofEq.6)the aboveequation is easily integrated to give, jck (t)j2 = 2jhkjV jsij2

1

cos!kst (0)

(E k

(0)

(17)

E s )2

In our case jsi = jP 1+ P 2 i, w ith the appropriate spin com binations, and jki = j(11A g )1 (11B u )2i for the singlet channel,and jki = j(11A g )1(13B u )2i for the triplet channel (as usual, jki can also have the chain indices reversed). W e have already dem onstrated (see section III and A ppendix 1) that the m atrix elem ent hkjV jsi is nearly the sam e for the singlet and triplet channels,except near the unique point X ? =t? = 0.5. Ignoring the oscillation involving !ks wenote thatthe relativeyield of the singletexciton isinversely proportionalto the square of E S = [E (P + )+ E (P ) E (11A g ) E (11B u )],w hile that of the triplet exciton is inversely proportional to the square of E T , in w hich E (11 B u ) in the above is replaced w ith E (13 B u ) (note, however,that w ithin the

two-state approxim ation we have assum ed that the singlet and triplet states that are ofinterest are the lowest singletand tripletstates;thiscan only be justi ed by the com plete m any-state calculationsofthe previoussubsections). W e see im m ediately that this sim ple two-state form alism predicts higher singlet yield, since E (11B u ) is considerably higher than E (13B u ). Im portantly, the chain-length dependence of isalso understood from the above. B oth E S and E T decrease w ith increasing chain length. H owever, the ratio E S = E T also decreases, because of the covalent character of the 13B u and the ionic character ofthe 11B u . T his is seen m ost easily in the lim it ofthe sim ple H ubbard m odelfor the individualchains(zero intersite C oulom b interaction and zero bond alternation in Eq.4),w here E S approaches 0 and E T approachesU in the long chain lim it. W ehavecalculated E S and E T exactly forallchain lengths N = 4 - 10 w ithin the PPP-O hno potential. W hile the ratio E T = E S show s the correct qualitative trend (viz., increasing E T = E S w ith increasing N ) necessary for increasing w ith increasing N ,the actualvariation is sm all. T his is to be expected,since our chain length variation is sm all,and the O hno potential decays very slow ly. W ith our lim itation on N ,it is necessary that the C oulom b potentialis short range, such that we have the sam e H am iltonian at allchain lengths, as is approxim ately true for the experim ental system s investigated12,13. W e have therefore done exact calculationsof E S and E T forthe extended H ubbard H am iltonian (Vij in Eq.4 lim ited to nearest neighbor interaction V )w ith param eterstij = 1.08t0 and 0.92t0 for double and single bonds,V=t0 = 2 and U =t0 = 5 and 6. In Fig. 9 we show our calculated results for E T = E S for the two cases,for di erent N .In both cases,increasing E T = E S w ith increasing N indicateslarger forlonger chain lengths.Energy convergencesarefasterw ith larger U ,w hich explainsthe steeperbehaviorof E T = E S for largerU ,and gives additionalsupport to our argum ent.

E . R ole of heteroatom s

T he experim ents by B aldo et. al.8 and W ilson et. al.12 both indicate that in sm all m olecular system s can be close to 0.25. T his is in contrast to our results for ethylene (see Fig. 3). for w hich is calculated to be substantially larger. O ne reason for this m ight be that the C oulom b correlation e ects in thin lm sam ples are sm aller than w ithin the PPP H am iltonian due to interm olecularinteractions.T hedom inante ect,how ever,is due to the heteroatom s in the m olecules investigated by these authors,as we show below . Speci cally, the site-energy (electronegativity)di erence between the heteroatom and carbon atom sm akesthesesystem scloser to the H uckellim it and this is w hat decreases . In orderto com parew ith them odelpolyenesystem swe consider pairs of(C H = N )2 in the follow ing calculations. T he single chain H am iltonian (Eq.4)isthen m odi ed as

12 follow s40. T he H ubbard U forthe nitrogen atom s,U N = 12.34. eV .T he localchem icalpotentialzN for nitrogen w ith lone pair involved in conjugation is 2. Finally, nitrogen hassite energy = { 2.96 eV relative to thatof the carbon atom s. T here are two possible arrangem ents for the two chains in a parallelcon guration,(i) carbon (nitrogen) on one chain lying directly above carbon (nitrogen) atom on the other,and (ii) carbon atom on one chain lying abovenitrogen atom on the second chain.W e have chosen arrangem ent (i),{ there is no fundam ental reason for arrangem ent(ii) to have a very di erent . In Figs. 10 (a) and (b) we have plotted the Im n (t) for the singlet and triplet channels,respectively,for the case of X ? = 0. Figs. 10 (c) and (d) show the sam e for X ? = 0.1 eV .T he m ost im portant conclusion that em ergesfrom these calculationsisthatthe relative yields oftriplets are substantially larger in the present case,so m uch so that can be even close to the statisticallim it of 0.25 (note that there are three triplet channels and the gures show the results for only one of these). W e believe that these results give a qualitative explanation ofthe observation ofB aldo et. al.8. Taken togetherw ith the chain length dependence of , as found in the previous subsection,these results also explain qualitatively theobservationsby W ilson et.al.12,sincethesam echain length dependence found in the case ofsim ple polyenes should be true here also,although it is conceivable that rate ofincrease of w ith N here m ay be slower.

V I. D ISC U SSIO N S A N D C O N C L U SIO N S

W ith a parallel arrangem ent of two polyene chains, we have show n that several experim entally observed qualitative features of the singlet-to-triplet yield ratios in -conjugated system s can be understood w ithin a well-de ned totalH am iltonian for the two-chain system . W hile our m odel system s are rather sim ple, our theoreticaltreatm ent ofthe charge-transfer process between the two chains is exact. W e have given a full tim edependent approach to the interchain charge-transfer process,and have show n thatin system scontaining only carbon atom s,the overallyield ofthe singlet exciton is considerably largerthan thatoftriplet excitons and > 0.25. T his is a direct consequence ofm oderate electronelectron C oulom b interactions w hich has strong e ects on both the energiesand the wavefunctionsofthe singlet and tripletexcitons.T hem echanism oftheexciton yields that em erges from our calculations is as follow s. For large yields,it is essentialthat there exist excited states ofthe com posite two-chain system w hose wavefunctions have sim ultaneously large overlaps w ith the wavefunction ofthe initialstate consisting ofpolaron pairs,and the nalstate consisting ofthe two chainsin the neutral states. O verlaps of the excited states of the two-chain system w ith nalstates in the singlet channelare considerably larger than for nalstates in the triplet channel, and this is w hat gives a large yield for the singlet

exciton. T his is a consequence of the di erent natures of the singlet and triplet excitons, w hich are ionic and covalent, respectively, in the V B notation. O ur result hereisconsistentw ith experim entson long oligom ersand polym ers9,10,11,12,13. A lthough ourexactcalculationsare lim ited to short chains, w ithin a two-state approxim ation thatisconsistentw ith thefullm ulti-levelcalculation we have show n that increases w ith the chain length, in agreem ent w ith experim entalobservations12,13. T he two-state approxim ation gives an alternate explanation ofthe higher yield ofthe singlet exciton that is related to the singletand tripletexciton binding energies,w hich are substantially di erent in -conjugated polym ers. Finally, we have exam ined the role of heteroatom s, and have show n that in sm allm olecular system s w ith nitrogen as the heteroatom , is substantially sm aller, and m ay be even close to the statistically expected value of 0.25. T he wavefunctions in this case,due to the strong electronegativity di erence between the heteroatom and carbon atom s,are closer to the H uckellim it,and this is w hat increases the relative yield of the triplet exciton. O ur results here successfully explain the di erence between A lq38 and heteroatom containing m onom ers12 on the one hand,and polym eric system s on the other,and thereby provide additional strong support to our theoreticalapproach. T he tim e-dependent approach to the charge-transfer process developed here is com pletely generaland can be applied to m any other sim ilar processes, for exam ple, photoinduced charge-transfer,triplet-triplet collisions in O LED S,etc. T hese and other applications are currently being investigated. Sim ilarly, for a m ore com plete understanding ofthe chain length dependence of ,we w ill investigatecharge-transferprocessw ithin thedensity m atrix renorm alization group technique. W hile this m anuscript was under preparation we received a m anuscript(S.K arabunarliev and E.R .B ittner, cond-m at/0206015) from E. B ittner that discusses the relative yields ofsinglet and triplet excitons w ithin the context of intrachain processes (see section III) as opposed to the interchain processdiscussed here.A lthough the approach ofthese authorsisdi erentfrom ours,they also nd thatthe relative yieldsofsingletand tripletexcitons are determ ined by their binding energies (sm aller binding energies giving larger yields). It is not clear w hether the approach used by these authors applies to m olecule-based O LED S.T hese authorshave also investigated the e ectofbroken electron-hole sym m etry,w hich isrelated to ourcalculationson chainsof(C H = N )2 . O ur results here are di erent. W hile K arabunarliev and B ittner nd even higher relative yield of singlet excitons (com pared to electron-hole sym m etric case)we nd that here is sm aller (see above). W hile a com plete analysis ofthe electron-hole recom bination m ust include both interchain and intrachain processes (and is a subject of future work in this area),we believe thatthis lastresult, w hen com pared to experim ents,justify ourbasicassum ption thatspin-dependenceofthe yieldsofexcitonscan be

13 understood largely w ithin the context ofinterm olecular and interchain charge-transfer.

V II. A C K N O W L E D G M E N T S

W ork in B angalore was supported by the C SIR ,India and D ST ,India,through /IN T /U S (N SF-R P078)/2001. W ork in A rizona was supported by N SF-D M R -0101659 and N SF-IN T -0138051. W e are grateful to our experim ental colleagues Z.V . Vardeny and M . W ohlgenannt fornum erousstim ulating discussions.S.M .acknow ledges the hospitality ofthe Indian Institute ofScience,B angalore,w here this work was com pleted.

V III. A P P E N D IX 1

W e present here detailed analytic calculations of the m atrix elem ents ofH in ter for the case oftwo ethylenes. W e believe thatthese calculationsgive clearunderstandings ofthe chain-length independence of the calculated S = T w ithin the Ferm iG olden R ule approach (Eq.8) that was presented in section III (see Fig. 1). W e also believe that even as these calculations show the inadequacy ofthe G olden R ule approach they provide an indirectunderstanding ofthe actualm echanism behind large in long chain polym ers. A s in the rest of the paper we consider parallel arrangem entsofthe ethylene m olecules,w ith sites 1 and 2 (3 and 4) corresponding to the lower (upper) m olecule. Subscripts 1 and 2 that are assigned to wavefunctions describe the lowerand upperm olecule,respectively.T he relevantsingle-m oleculeeigenstates,corresponding to the lower m olecule then can be w ritten as p j11A g i1 = (c1 = 2)(ay1;" ay1;# + ay2;" ay2;# )j0i + p (16a) (c2 = 2)(ay1;" ay2;# ay1;# ay2;" )j0i p y y y y j11B u i1 = (1= 2)(a1;" a1;# a2;" a2;# )j0i (16b) p y y y y 3 j1 B u i1 = (1= 2)(a1;" a2;# + a1;# a2;" )j0i (16c) p y y (16d) jP + i1 = (1= 2)(a1; + a2; )j0i p y y y y y y jP i1 = (1= 2)(a1;" a1;# a2; a2;" a2;# a1; )j0i (16e) In the above j0i is the vacuum for chain 1 and c1 and c2 are the coe cients of the ionic and covalent con gurations in the 1A g ground state that are to be determ ined by solving for the 2 2 A g subspace ofet p hylene w ithin the PPP H am iltonian (c1 = c2 = 1/ 2 in the H uckel H am iltonian and them atrix elem entsin Eq. 8 in thesinglet and triplet channels are exactly equalin this case). W e have chosen the M S = 0 wavefunction for the 13B u , but w hat follow s is equally true for the M S = 1 wavefunctions. W e have not assigned de nite spin states to the charged polaronic wavefunctions, since the charged m oleculescan haveeitherspin,and since di erentcom binationsofthese spin statesgive the initialspin singletor

triplet product eigenstates for jP + P i. N ote,however, the relative m inus signs between the two con gurations in jP i,asopposed to the relativeplussignsbetween the two con gurationsin jP + i. T hisisw hatensuresthatthe productwavefunctionsofthe type jP 1+ P 2 i,w ith positive charge on m olecule 1 and negative charge on m olecule 2 has odd parity w ith respect to the center ofinversion on a single chain, and charge recom bination can therefore only generate neutral states that have odd parity (for exam ple,j11A g i1j11B u i2 but not j11A g i1 j21A g i2 ). W e considerthe initialstatesjiS iand jiT i rst,w hich are constructed from taking products of the polaronic wavefunctions given above. Since these product functions contain four term s each, and also since one of our goals is to arrive at a visual representation of the charge recom bination process in con guration space,we have chosen not to w rite their explicit form but have given in Fig. 11 the wavefunctions in the V B notation, w here a singlet bond between sites i and j is de ned as 2 1=2(ayi;" ayj;# ayi;# ayj;" )j0i, a triplet bond (w ith an arrow pointing from site i to site j) is de ned as 2 1=2(ayi;" ayj;# + ayi;# ayj;" )j0i,and crosses correspond to doubly occupied sites ayi;" ayj;# j0i. G iven the initialand nalstates,itisnow easily seen thatVi;j in H in ter (Eq.6) playsno rolew ithin the G olden R ule approach16,17,since this term causes no transition between the initialand nalstates (note,however,Vi;j can play a signi cant role in the fulldynam ics calculation ofsection IV ). T he m atrix elem ents ofthe rem aining term s in H in ter are now readily evaluated and these are given below , h(11A g )1 (11B u )2 jH in ter jP 1+ P 2 iS p p = (c1= 2)( t? + 2X ? ) (c2= 2)( t? + X ? ) (17a) h(11A g )2 (11B u )1 jH in ter jP 1+ P 2 iS p p = (c1 = 2)( t? + 2X ? )+ (c2= 2)( t? + 3X ? ) (17b) h(11A g )1 (13B u )2jH in ter jP 1+ P 2 iT p p = (c1= 2)( t? + X ? ) (c2= 2)( t? + 2X ? ) (17c) h(11A g )1 (13B u )2jH in ter jP 1+ P 2 iT p p = (c1= 2)( t? + 3X ? )+ (c2 = 2)( t? + 2X ? ) (17d) Severalpointsareto benoted now .First,forX ? = 0,the squares ofallthe m atrix elem ents are equal,and hence there is no di erence between singlet and triplet generation w ithin the G olden R ule approach in this lim it,and we agree on this w ith Shuaiet. al.16 Second, however, de ning overall S as the sum ofthe squares ofthe m atrix elem ents in Eqs. 17(a) and and (b),and T as the sum ofthe squares ofthe m atrix elem ents in Eqs. 17(c) and (d), respectively, we see that S = T depends very weakly on X ? =t? at allX ? =t? except for X ? =t? very close to 0.5,w here t? + 2X ? = 0 and t? + X ? and t? + 3X ? have opposite signs. T his is particularly so

14 for the calculated c1 and c2 for PPP-O hno param eters (c1 = 0.5786,c2 = 0.8156). W e now exam ine the di erent term s in Eqs. 17(a) - (d) in detail. From Fig. 11 we note that there are three classes of interchain electron transfers: (i) charge-transferbetween sites that are both singly occupied,leading to a doubly occupied site or an em pty site (denoted by 1 + 1 ! 2 + 0, w here the num bers denote site occupancies) { or the exact reverse process,(ii)charge-transfersofthe type 1 + 0 ! 0 + 1,using the sam e notation,and (iii) charge-transfers of the type 2 + 1 ! 1 + 2, again w ith the sam e notation. T hese three processes have di erent m atrix elem ents ( t? + 2X ? ),( t? + X ? ) and ( t? + 3X ? ),respectively.T he role ofX ? now becom esabsolutely clear. N onzero X ? creates an asym m etry between the upper and lower m olecule,leading to a di erence between the yieldsofj11A g i1j11B u i2 and j11A g i2j11B u i1,butitdoes not create a signi cant di erence between S and T . A texactly X ? =t? = 0.5 term scontaining ( t? + 2X ? ) in the m atrix elem ents vanish,w hile the other term s are also sm all and of opposite signs. T he singlet channel m atrix elem ents now involve only c2 , w hile the triplet channel m atrix elem ents involve only c1. Since for repulsive C oulom b interactionsc2 > c1,the sum ofthe the squaresofthe m atrix elem entshere arelargerforthe singletchannelthan for the triplet channel. T his is w hat is re ected in our plot ofFig.8. N ote,however,that the calculated yields approach zero in both cases here. It is also clear from Eqs.17 that this di erence between the singlet and triplet channels persist over a very narrow region about X ? =t? = 0.5. W e therefore do not believe that this is ofany relevance for realistic system s. O ur nalpointconcernsthechain-length independence ofour results in Fig.8 (except near X ? =t? = 0.5). For arbitrary chain lengths there can occur only the three classes ofinterchain charge-transfers discussed above (1 + 1 ! 2 + 0,1 + 0 ! 0 + 1 and 2 + 1 ! 1 + 2). T he detailed wavefunctionsoflongerchainsare di erent,but the expectation values hni;" ni;# i for the di erent wavefunctions are nearly the sam e for xed intrachain correlation param eters. T hus although in long chains there can in principle occur m any m ore interchain hops that are ofthe type 1 + 1 ! 2 + 0,such charge-transferslead to additional double occupancies (relative to the overallinitialstates) that are energetically costly because of electron correlation e ects. Such charge-transfers therefore m ake weak contributions to the overall interchain charge-transfer. T he net consequence is the weak chainlength dependence found in Fig. 1 at all points other than X ? =t? = 0.5. W e believe that the above detailed calculation,aside from indicating the inapplicability of the G olden R ule, also indicatesthatthe propertheoreticaltreatm entm ust include the di erences in the energies and wavefunctions of the nalstates, as indeed is done in our tim edependent calculations.

IX . A P P E N D IX 2

D etails ofthe num ericalprocedure that were not discussed in section V aregiven below .T he charged aswell as neutraleigenstates ofH in tra for individualchains are obtained in the V B representation by using a diagram m atic V B approach29 w ith bitrepresentation ofthe basis states.T he eigenstatesofa given spin S are obtained for M S = S.T heV B eigenstatesarethen transform ed to the basis ofSlater determ inants w ith M S = S by expanding the term s in each singlet pair and assigning an up-spin at each unpaired site w ith single occupancy. T hus, a triplet V B basis consisting oftwo singlet pairs and corresponding to a function w ith M S = 1 is expanded into four Slater determ inants each w ith M S = 1. To obtain eigenstate corresponding to other M S values w ith Slater determ inantal basis, we apply the S^ operator on the state,as m any tim es as is necessary. W e use the eigenstates ofjP + i and jP i to form the initialstate ofchosen spin in the form , " 1 1 1 1 1 j ;+ i1 ? j ; i2 (18) 1;0 (0)= p 2 2 2 2 2 # 1 1 1 1 j ; i1 ? j ;+ i2 2 2 2 2 w here the subscripts 1 and 0 refer to the total spin of the initial state. T he direct product of the states are expressed in the Slater determ inantalbasis ofthe com posite system w ith the coe cientc k ofthe basisstate jki in the com posite system being given by X ck = dldm < kjl? m > (19) l;m

w here dl and dm are the coe cients ofthe basis states jliand jm iin the ground states ofthe subsystem s 1 and 2 respectively. T he direct product itself is e ected by shifting the 2n1 bits ofthe integerrepresenting the basis stateofsystem 1 w ith n1 sitesto theim m ediateleftofthe 2n2 bits in the integer that represents the basis state of system 2 w ith n2 sites. T he resulting largerinteger w ith 2(n1 + n2) bits correspond to an integer that represents oneofthebasisstatesofthecom positesystem of(n1 + n2) sites. T he evolution ofthe initialstate involves solving the linear algebraic equations,A x(t+ t) = b, w here,the m atrix elem ents ofthe m atrix A and the com ponents of b are given by, A ij = ( ij + ihH ij X ( ij

bi = j

t ) 2

ihH ij

t )xj(t) 2

(20) (21)

T he m atrix A in the largest system we have studied is nearly of order one m illion and for reasonable convergence ofthe solution ofthe system ofequations we need

15 a toftheorderof0:05eV=h w hich istypically 0.033fem toseconds,and thisguaranteesdiagonaldom inanceofthe m atrix A . T hus,ifone w ishesto follow the dynam icsfor even aslong assay 60 fem tosecondsoneneedsto solvethe linearsystem about2000 tim es.T hisisrendered possible by the sparseness ofthe m atrix A . For e cient convergence,we use a sm allm atrix algorithm 41 w hich is very sim ilarto the D avidson’salgorithm form atrix eigenvalue problem . In the case ofthe largest system size,it takes abouttwelve hourson a D EC A lpha 333 M H z system to evolve the state by 60fs for a given channel. A t the end of each iteration in the evolution of the state, we obtain the intensity or the yield in a pair of

1

2 3 4 5 6

7

8 9 10 11

12 13

14

15

16

17

18 19 20

21 22

23

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28

29

30

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states in the neutral subsystem . T he num ber of such pairs is enorm ous w hen we dealw ith say two system s of six siteseach.T he num berofpairsin the singletchannel is 30,625 w hile that in the triplet channelis 33,075. W e can reduce the pairs onto w hich we project the evolved state by restricting ourselves to a few low -energy states of the neutral subsystem . H owever, even in this case, the num ber ofpairs could be rather large. To overcom e this problem , we select only those pairs w hich have a m inim um yield ofsay 10 4 at alltim es. T his restriction w hen im plem ented judiciously leaves us w ith only a few pairs w ith signi cantyields.

31

32

33

34 35

36 37 38 39

40

41 42

A .M essiah,in Q uantum M echanics (N orth-H olland (A m sterdam ),1972). J.C rank and P.N icholson,Proc.C am bridge Phil.Soc.43, 50 (1947). R .Varga,in M atrix Iterative A nalysis (Prentice-H allInc., N ew Jersey,1962). I.I.R abi,Phys.R ev.51,652 (1937). L.A llen and J.H .Eberly,in O pticalResonance and T wo LevelA tom s (D over,1987). D .G uo et al.,Phys.R ev.B 48,1433 (1993). G .W eiser,Phys.R ev.B 45,14076 (1992). M .Liess et al.,Phys.R ev.B 56,15712 (1997). E. M erzbacher, in Q uantum M echanics (John W iley & Sons (N ew York),1970). S.R am asesha,I.D .L.A lbert,and P.K .D as,Phys.R ev. B 43,7013 (1991). S.R ettrup,J.C om put.Phys.45,100 (1982). y Present address: G E John F.W elch Technology C enter, Sy 152, Export Prom otion Industrial Park, Ph 2, H oodi V illage,W hite Field R oad,B angalore 560 066,India

16 F igure C aptions F igure 1 : T he ratio ofthe squaresofthe singletand tripletm atrix elem entsofH in ter ( S = T according to Eq. 8),asa function ofX ? =t? forpairsofethylenes(circles), butadienes (squares)and hexatrienes (diam onds). F igure 2 : Y ield in the singlet channelas a function of tim e, for pairs of ethylenes (top panel), butadienes (m iddle panel) and hexatrienes (bottom panel),w ithin the sim ple H uckel m odel (U = Vij = X ? = 0). Signi cant yield in all cases occur only for nal states j(11A +g )1 (11B u )2 > and j(11B u )1 (11A +g )2 > , between w hich the yields are identical. Y ields in the triplet channelj(11 A +g )1 (13B u+ )2 > and j(13B u+ )1 (11A +g )2 > are identicalto those in the singlet channel. F igure 3 : Y ields in the singlet and triplet channels w ithin the PPP H am iltonian. In allcases the top panelcorresponds to pair ofethylenes,the m iddle panel to pairs ofbutadienes,and the bottom panelto pairs of hexatrienes. (a) Singlet channel,t? = 0:1eV ; X ? = 0; (b) Triplet channel, t? = 0:1eV ; X ? = 0; (c) Singlet channel,t? = 0:1eV ; X ? = 0:1eV ; (d) Triplet channel t? = 0:1eV ; X ? = 0:1eV . Evolution in case of hexatrienes is tracked for 20 fs w hile in other cases,the evolution is tracked for 60fs. Signi cant yields in singlet channeloccurs only for nalstates j(11A +g )1 (11B u )2 > and j(11B u )1(11A +g )2 > , between w hich the yields are identical in (a) and (b) but di erent in (c) and (d). Sim ilarly, yields in triplet channel are to the states j(11A +g )1 (13B u+ )2 > and j(13B u+ )1 (11A +g )2 > , between w hich the yieldsare identicalin (a)and (b)butdi erent in (c) and (d). F igure 4 : Y ields in the singlet channels (a) j(11A g )1 (1Bu )2 i,(b) j(11A g )2(1Bu )1 i,as a function ofthe electric eld (V /A )and tim e (fs). H ere t? = 0.1 eV and X ? = 0.1 eV . F igure 5 : Y ields in the triplet channels (a) j(11A g )1 (3Bu )2 i,(b) j(11A g )2(3Bu )1 i,as a function ofthe electric eld (V /A ) and tim e (fs). Param eters are sam e

as in Fig. 4. F igure 6 : Y ields in the singlet channel for pairs ofhexatriene m olecules,as a function oftim e (fs) w ith t? = 0:1eV and X ? = 0:1eV in an externalelectric eld. (a) Singlet channel at 0.3 V /A , (b) singlet channel at 0.42 V /A and triplet channelat 1.0 V /A . F igure 7 : Evolution of signi cant < ijm ;n > as a function ofelectric eld (V /A ),in case ofthe explicit tim e evolution ofeigenvectors the PPP H am iltonian for a pairofethylenesin (a)singletand (b)tripletchannels. F igure 8 : < m nj i > < (0)j i > plotted as a function of electric eld (V /A ), for signi cant states ’i’ for (a) the singlet-singlet channeland (b) the singlet-triplet channelfor a pair ofethylenes. T he singlet-singletchannel in (a) corresponds to jm S 1 > and jnS 2 > and the singlet-triplet channelin (b) corresponds to jm S 1 > and jnT > . F igure 9 : E T = E S vs 1=N for the case oflinear chains w ith "U -V " m odel H am iltonian for the case of (i) U = 5eV and V = 2eV (squares) and (ii) U = 6eV and V = 2eV (circles). F igure 10 : Y ieldsin thePPP m odelforthe(C H = N )2 system .(a)singletand (b)tripletchannelsw ith X ? = 0; (c)and (d)singletand tripletchannelsw ith X ? = 0:1eV . T hestateto w hich theyield issigni cantin (a)isjS 0S1 > w hile in (b)itis to the state jS0T > . T he yield to states jS1S0 > in singlet channeland jT S0 > in triplet channel are identicalto those for jS0S1 > and jS0T > in (a) and (b)respectively.In (c)the yieldsto jS1S0 > and jS0S1 > are not the sam e and are show n separately.Sim ilarly,in (d) yields to jT S0 > and jS0T > are show n separately. F igure 11 : T he initial (a) singlet and (b) triplet states jP 1+ P 2 i for the case of two ethylenes, and the result ofoperating w ith H in ter . T he upper (lower) two sites correspond to m olecule 1 (m olecule 2). T he result (c) is a linear relationship between covalent triplet V B diagram s.

Figure 1

4

’’σS/σT’’

3

2

1

0

0.5

1

X/t

1

1.5

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Figure 2

0.4 0.2 0

Yield

0.4 0.2 0 0.4 0.2 0 0

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20

30 Time (fs)

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Figure 3 (a)

0.015

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Figure 3 (b)

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Figure 3 ( )

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Figure 3 (d)

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Figure 4

1

Yield

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(a)

0.6 0.4 0.2 0 0

60 0.5

40

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Elec

tric

20

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2

s)

e (f

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(b)

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60 0.5

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tric

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V/A 0

1

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s)

e (f

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Figure 5

1

Yield

0.8

(a)

0.6 0.4 0.2 0 0

60 0.5

40

1

Elec

tric

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s)

e (f

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60 0.5

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tric

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s)

e (f

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Figure 6

0.6 0.3

(a)

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Yield

0.6 0.3

(b)

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(c)

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Figure 7 (a)

1



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El

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ec

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ld

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A0

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ex of H tot

e ind Eigenstat

Figure 7 (b)

1



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ld

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A0

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e ind Eigenstat

Figure 8 (a)

i=3

0.4 0.2 0 0

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ci

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i=4

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2 o

Electric field V/A

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Electric field V/A

Figure 8 (b)

i=2

ci

0.4 0.2 0 0

i=5

0.4 0.2

1

2 o

Electric field V/A

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Figure 9

12

T

∆ E /∆ E

S

10

8

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0.15

0.2

1/N

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Figure 10 (a)

Singlet channel yield

0.015

0.01

0.005

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40 Time (fs)

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Figure 10 (b)

Triplet channel yield

0.015

0.01

0.005

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Figure 10 ( )

0.04

Singlet Channel Yield

0.03

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Figure 10 (d)

Triplet channel yield

0.06

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Figure 11

.

|iS > =

x

.



x

.

+

x

. .

(a)

x

(−t + 3 X )



x

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Hinter

|iT > =

+



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x

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x

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+

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(−t + X )

x

+

+

+

.

x

x

1



2 (−t + 2 X )

(c) =

x

x

(−t + 3 X )



+

. . .

(b)

x



.

2 (−t + 2 X )

.

. . . . . .

Hinter |iS > = (−t + X )

x



+