Optical activity of electronically delocalized molecular aggregates: Nonlocal response formulation

Transcription

1 Optial ativity of eletronially deloalized moleular aggregates: Nonloal response formulation Thomas Wagersreiter and Shaul Mukamel Department of Chemistry, University of Rohester, Rohester, New York Reeived 14 June 1996; aepted 26 July 1996 A unified desription of irular dihroism and optial rotation in small optially ative moleules, larger onjugated moleules, and moleular aggregates is developed using spatially nonloal eletri and magneti optial response tensors (r,r,). Making use of the time dependent Hartree Fok equations, we express these tensors in terms of deloalized eletroni osillators. We avoid the ommonly-used long wavelength dipole approximation k r1 and inlude the full multipolar form of the moleule field interation. The response of moleular aggregates is expressed in terms of monomer response funtions. Intermoleular Coulomb interations are rigorously taken into aount thus eliminating the neessity to resort to the loal field approximation or to a perturbative alulation of the aggregate wave funtions. Appliations to naphthalene dimers and trimers show signifiant orretions to the standard interating point dipoles treatment Amerian Institute of Physis. S I. INTRODUCTION Cirular dihroism CD, i.e., the differene in absorption of right and left irularly polarized light has long been used in the investigation of strutural properties of extended hiral moleules or aggregates. This tehnique exploits the spatial variation of the eletromagneti field aross the moleule and yields valuable strutural information, whih is missed by the ordinary linear absorption LA. It has been used in the investigation of a variety of systems inluding polythiophenes, 1 polynuleotides, 2,3 helial polymers, 4 6 moleular rystals, 7,8 and antenna systems and the reation enter of photosynthesis Also ommonly used are other related signatures of optial ativity OA, i.e., optial rotation OR and elliptiity. A moleular theory for optial ativity was first formulated by Born, 13 and almost simultaneously by Oseen. 14 Born attributed the origin of optial ativity to the spatial variation of the eletromagneti field over the system under investigation. Rosenfeld reformulated these ideas in terms of the quantum-mehanial theory of dispersion. 15 These results were later derived as sattering problems in quantum field theory. 16 The next step was the development of polarizability theories, whih express the optial rotatory power ORP of a moleule in terms of the onfiguration and polarizabilities of its onstituent groups In transparent spetral regions, Kirkwood ahieved this goal by applying Borns theory to a moleule whih he separated into smaller units, and perturbatively alulating the wavefuntions modified by the interation of these subunits. Moffitt and Mosowitz also studied absorptive spetral regions. By using omplex polarizabilities they worked out a unified desription of dispersive refration, optial rotation and absorptive linear absorption, irular dihroism properties. 21 They established Kramers Kronig relations between OR and CD and further disussed the effets of vibrational degrees of freedom on these observables. De Voe approahed this many-body problem by applying the loal field approximation LFA whih aounts for intermoleular eletrostati interations via a loal field whih at a point like moleule in an aggregate is the superposition of the external field and the field indued by the surrounding moleules modeled as point dipoles. He thus expressed LA and OR of unaggregated monomers, aggregates in solution, and moleular rystals in terms of the aggregate geometry and the omplex polarizabilities of the onstituent monomers. Using this method, Applequist alulated the polarizabilities of halomethanes, 22 and presented a normal mode analysis of polarizabilities and OR. 23,24 When dealing with solvated moleules or aggregates, the loal field should also inlude the ontribution from the solvent. 25,26 In all models disussed sine then, the spatial variation of the field with wavevetor k was aounted for by the expansion e ikr 1 ikr. Usually the series is trunated after the seond dipole or third quadrupole term. The moleule itself is modeled by an eletri and a magneti transition dipole moment. Larger moleules or aggregates are regarded as an assembly of point-like units, eah having an eletri and a magneti transition dipole moment. Charge transfer between units is negleted and their mutual oupling is given by the dipole dipole interation. De Voe also improved the desription of the interation between units, replaing the dipole dipole interation term T ij by the intermoleular interation of all point-harges point monopole approximation ontained in units i and j. 25 This orretion is essential when the dimensions of the units are omparable to their separation, as is the ase in typial aggregates of polynuleotides. 2,3 The purpose of this paper is to develop a omputationally feasible proedure for the investigation of OA whih avoids the ommonly used multipolar expansion and replaes the LFA by a rigorous treatment of intermoleular eletrostati interations. To set the stage, we shall briefly review the onventional theory of OA using the notation of Moffitt and Mosowitz. 21 It is based on the observation, that J. Chem. Phys. 105 (18), 8 November /96/105(18)/7995/16/$ Amerian Institute of Physis 7995

2 7996 T. Wagersreiter and S. Mukamel: Optial ativity of moleular aggregates the indued eletri dipole moment is not only proportional to the eletri field, but also to the magneti field and its time derivative. Analogously, the indued magneti dipole moment is proportional to the magneti field, as well as to the eletri field and its time derivative E H H, 1 t m H E E, 2 t where, are the polarizability and magnetizability of the moleule and E,H are monohromati fields with frequeny. The oeffiient has no effet on OA, and whih is responsible for OA, an be separated into real and imaginary parts aording to 1 i 2, whih are related to eah other via the Kramers Krönig relations Eq. 52 in Ref. 21. For transparent regions one obtains R i i 2 i0 2, 4 where the entral quantities desribing optial ativity are the rotational strengths R i0 for transitions from the ground state 0 to an exited state i with transition frequeny i0. For systems muh smaller than the optial wavelength they are given by R 0i Im0ˆ i imˆ 0, where ˆ, mˆ denote the eletri, and magneti dipole moment operators of the entire system. In an optially ative medium, left and right irularly polarized light propagate with different speeds and are absorbed differently. In transparent regions the polarization vetor of linearly polarized light is rotated by an angle whih is proportional to the propagation distane. The optial rotation in radians per unit length is given by 21 N n , 6 where N is the number density of moleules, and a orretion fator for the surrounding medium with refration index n 0 has been added. In absorptive regions linearly polarized light turns into elliptially polarized light after propagating through the medium, and denotes the angle between the long axis of the ellipse and the diretion of polarization of the inident linearly polarized wave. The elliptiity per unit length is given by 21 N n It is diretly proportional to the differene in absorption of right and left irularly polarized light, i.e., the CD signal An early yet very instrutive review on theories of optial rotatory power was written by Condon. 27 A lear disussion of CD at a minimum level of omplexity time dependent perturbation theory with simple eletri and magneti dipole interation an be found in Ref. 28. Finally there are numerous textbooks overing theory and experiment exhaustively In order to develop a unified mirosopi approah to OA whih applies to a broad lass of moleular systems we shall employ nonloal response tensors, whih relate the indued linear polarization P (r,) and magnetization M (r,) to the driving eletri and magneti fields E ext (r,),h ext (r,) via the linearized relationships P r, dr r,r; E ext r, r,r; H ext r,, M r, dr r,r; E ext r, r,r; H ext r, In this nonloal response formulation NLRF, the tensors (r,r;),,,, are intrinsi moleular properties that desribe all linear optial phenomena. Strutural and hemial details of different systems enter through their effets on these tensors. By using nonloal response tensors we ompletely avoid the ommonly employed multipolar expansion, whih uses the indued eletri and magneti dipole, quadrupole, otopole, et. moments. If so desired, the latter an be alulated by integrating Eqs. 9,10 over r and expanding the driving eletri and magneti fields around a referene point of the moleule. By formulating the problem this way we an defer the introdution of details to the final stage of the alulation. This enables us to unify many of the treatments in the literature that are restrited to speifi models e.g., aggregates of point dipoles, et.. We already employed the nonloal polarizability tensor (r,r,) for investigating the anisotropy of the linear absorption, and for gaining insight into the role of intramoleular oherenes When magneti interations ontribute to the linear response, the magnetizability (r,r,) and the ross response tensors (r,r,), (r,r,) enter the piture as well. We will show that these tensors fully determine the CD signal, so that the level of rigor is determined by the approximations applied for their alulation. By using the multipolar Hamiltonian and the time dependent Hartree Fok TDHF proedure we derive general formulas for these response tensors whih ontain the full multipolar expansion, thus avoiding the dipole approximation (kr1). When investigating moleular aggregates we will go beyond the ommonly used loal field approximation LFA, see Eqs. 65, 67 below and rigorously express the nonloal response tensors in terms of those of the monomers. Using this method, intermoleular interations are fully inorporated, and the need for a perturbative expansion of the aggregate

3 T. Wagersreiter and S. Mukamel: Optial ativity of moleular aggregates 7997 wavefuntions is eliminated, providing a lear advantage over the ommonly used point dipole or point monopole approximations. In Se. II we derive an expression for the CD signal of a spatially extended moleular system using the eletri and magneti nonloal response tensors. We next alulate the latter in Se. III using the TDHF equations. In Se. IV we express the aggregate response in terms of the response funtions of its onstituent monomers. Applying the point dipole and loal field approximations we retrieve Tinoos formulas for the rotational strengths in Se. V. We then apply these results to naphthalene dimers and trimers in Se. VI, and summarize in Se. VII. II. NONLOCAL RESPONSE FORMULATION OF CIRCULAR DICHROISM The multipolar Hamiltonian desribing a moleular system interating with a lassial eletromagneti field is given by 32,37 Ĥ s Ĥ m Vˆ interĥ drpˆ r E r Mˆ r H r. Here H m denotes the moleular Hamiltonian, and Vˆ inter st dr drpˆ strrpˆ tmˆ strrmˆ t, desribes the interation between the eletri and magneti dipole densities of the eletrons, defined as Pˆ srerˆsr 0 1 d rrrˆsr, Mˆ srerˆsrpˆs 0 1 drrrˆsr, where R is an arbitrary referene point, e is the eletron harge, rˆs denote the position operators of the harges labeled by s, and pˆ s is the orresponding anonial momentum. The polarization P(r,t)(t)Pˆ (r)(t) is given by the expetation value of the mirosopi polarization operator 32 Pˆ r s Pˆ sr, 13 where (t) is the eletroni wavefuntion. The magnetization M(r,t)(t)Mˆ (r)(t) is similarly defined as the expetation value of the mirosopi magnetization operator 32 F r dr rr Fr, with the transverse -funtion r 2 3 1r 1 4r 3 13rˆrˆ. Here we use a dyadi notation for the tensor produt rˆrˆ. Both polarization and magnetization depend on the referene point R, but their transverse parts entering Maxwells equations for the observable fields see below, do not. This follows from the gauge invariane of the observable transverse fields. The third term in the Hamiltonian H s dr m s r 2 2 dr m 3 s r m t r, s,t is quadrati in the magnetization densities mˆ s, and we have omitted terms quadrati in the magneti field, as they do not ontribute to the linear response. Finally, the last term in the Hamiltonian represents the oupling of the system to the external transverse eletri and magneti fields. We start our analysis with the mirosopi Maxwell equations for the eletri and magneti fields E(r,t) and H(r,t) Ref. 32 E r,t 1 t Hr,t4Mr,t, 15 Hr,t 1 t Er,t4Pr,t. 16 Here is the vauum speed of light. The rhs of Eqs. 15, 16 are transverse, as D B0 with DE4P and BH4M. However, for the sake of larity we add the symbol expliitly in the derivation below. We shall swith to the frequeny domain by the Fourier transform f dt e it ft, ft 2 1 d e t f. The indued eletri and magneti dipole moment densities an be expanded in a power series in the driving fields. For linear absorption, the first terms Eq. 9, 10 in these expansions are needed. (r,r;) denotes the nonloal eletri polarizability tensor, (r,r;) the nonloal magnetizability tensor, and the remaining ross response tensors desribe the effets of the magneti eletri field on the polarization magnetization. They have the symmetries Mˆ r s Mˆ sr, 14 and F denotes the transverse part of a vetor field F, defined by ij r,r; ji r,r;, ij r,r; ji r,r;, ij r,r; ji r,r;,

4 7998 T. Wagersreiter and S. Mukamel: Optial ativity of moleular aggregates where the indies i, j denote Cartesian omponents. Note that we have defined these tensors with respet to the external fields E ext, H ext Eqs. 9, 10. The absorption ross setion A of an arbitrary system an be alulated by integrating the Poynting vetor of the eletromagneti field over a losed surfae enompassing it. For a monohromati field with frequeny 2/T we find see Appendix A 1 1 T in T dt 0 dre r,t Pr,t V t H r,t Mr,t t, 20 where in (1/T)dt(/4)E ext H ext is the timeaveraged inident energy flux. The solution of the oupled Maxwell equations Eqs. 15, 16 an be written in the form see Appendix E E ext E, H H ext H, where E, H are the solutions of the inhomogeneous part of the wave equations and an be expressed in terms of the retarded Greens funtion. Consequently the absorption ross setion an be written as see Appendix A A 0 A 1. The first term 0 A 1 1 T in T dt 0 dre ext r,t Pr,t V t 21 H ext r,t Mr,t t 22 is obtained from Eq. 20 by replaing the transverse Maxwell field by the external field. A (1) whih is given in Eq. A15 ontains the effets of multiple sattering in the sample. It beomes important when sattering phenomena of large partiles suh as proteins or DNA strands have to be aounted for. Hereafter we shall only onsider A (0).To establish the onnetion with marosopi observables we assume a sample with N noninterating aggregates per unit volume. The hange of intensity I with the penetration depth l is given by dii N A dl, resulting in IlI 0 e N A l. Substituting a linearly polarized inident plane wave propagating in the z diretion E ext r,t E 0 2 e xe ik rt e ik rt, H ext r,t H 0 2 e ye ik rt e ik rt, with H 0 ne 0, and e y e z e x into Eq. 22, yields the absorption ross setion for a moleule embedded in a medium with refrative index n 0 A 4 dr ik rr n Im dr e xx r,r, n n yy r,r, yx r,r, xy r,r,. 23 Here ij e i e j, et. denote the tensor omponents. In the long wavelength approximation e ik (rr) 1 the ross terms, anel eah other beause of the symmetry Eq. 19 and we obtain the familiar result 38,37 0 A 4 Im tot tot n, n lim k 0 with tot ()drdre 1 (r,r,) e 1, tot () drdre 2 (r,r,) e 2. By speializing to irularly polarized plane waves we an express the CD signal in terms of the nonloal eletri and magneti response funtions. A right () or left () irularly polarized plane wave propagating in z diretion k (n/)e z, k (2/) an be represented in the form E ext r,te 0 e x osk rte y sink rt, 24 H ext r,th 0 e x sink rte y osk rt. 25 It is ustomary to use the separation with A A A, A A A, 2 A A A Here A denotes the average absorption, and A is the CD signal. When substituting these fields into Eq. 22 we find see Appendix A0 4 dr Re dre ik rr xx r,r, yy r,r, n xx r,r, yy r,r,n yx r,r, xy r,r, yx r,r, xy r,r,. 27

5 T. Wagersreiter and S. Mukamel: Optial ativity of moleular aggregates A 4 dr Re dre ik rr yx r,r, xy r,r, n yx r,r, xy r,r,n xx r,r, yy r,r, xx r,r, yy r,r,. 28 This exat expression whih relates the CD signal of an arbitrary moleular system to its nonloal eletri and magneti response tensors defined by Eqs. 9, 10 forms the basis for our subsequent analysis. Equation 28 is not restrited to a speifi order of the widely used multipolar expansion. The latter is ompletely avoided by using nonloal response tensors, whih impliitly inlude all multipoles. The problem of the unphysial origin dependene onneted with the eletri quadrupole and higher moments, disussed e.g., in Refs. 39,40, is naturally resolved with our approah. As an be seen from Eq. 22, our result does not depend on the origin. The salar produt with the external fields E ext,h ext whih are intrinsially transverse, ensures, that only the transverse part of the polarization P and the magnetization M, respetively enter the absorption ross setion, and as indiated earlier they do not depend on R. No speifi model of the system has been assumed yet for example, these expressions hold whether the moleular states are loalized or deloalized, and no assumptions were made on the system-size in omparison with the optial wavelength. All approximations will enter through the different levels of rigor used in modeling the nonloal response tensors. In the long wavelength limit k(rr)0 the CD signal vanishes, whih follows from Eqs and interhanging the integration variables r r. This is to be expeted sine a finite spatial extent of the system ompared with the optial wavelength is neessary in order to observe hirality. The leading term in a multipolar expansion is thus obtained assuming e ik(rr) 1ik(rr) whih will lead to the elebrated Tinoo formulae. 29,41 III. CALCULATION OF THE NONLOCAL RESPONSE TENSORS USING THE TIME DEPENDENT HARTREE FOCK EQUATIONS Consider an N-eletron system a moleule or an aggregate whih we desribe using single eletron basis funtions m,m1,...,n. We define ĉ m (ĉ m ) as the reation annihilation operators of an eletron at site m. We further introdue the redued single eletron density matrix with matrix elements mn ĉ m ĉ n. Sine the polarization and magnetization are single-eletron operators, their expetation values an be expanded using the density matrix Pr, mn P mn r, 29 mn where P mn r m Pˆ n, 30 are the matrix elements of the polarization operator Eq. 13. Similarly, the expetation value of the magnetization is given by Mr, mn mn M mn r, 31 where the matrix elements of the magnetization operator are defined by M mn r m Mˆ n. 32 To alulate the linear optial response we expand the density matrix to first order in the driving fields setting mn () mn () (1) mn (). Here mn is the redued one eletron ground state density matrix obtained by iterative diagonalization. 42 The dynamis of the linear response (1) mn () in Liouville spae will be alulated using the time dependent Hartree Fok TDHF proedure, resulting in the equation of motion 35,36,42,43 1 mn kl F mn. The soure F mn is given by with F mn,v mn, A mn,kl 1 kl i kl mn,kl 1 kl V mn drp mn r Er,M mn r Hr,, 35 and the matries A, are defined in Ref. 43. Terms quadrati in the magneti field have been negleted in Eq. 33, aswe are interested in linear absorption. Equation 33 generalizes the results of Refs. 35, 36, 43 to inlude the magneti interation M H as well as off-diagonal matrix elements P mn in F mn. The TDHF proedure deals with physially relevant quantities harges nn and eletroni oherenes mn ), and a dominant mode piture an be developed whih greatly redues omputational ost. 44 By using position-dependent eletri and magneti dipole densities whih an be alulated from the eletroni basis set we avoid the long wavelength approximation and the following expressions for the polarizabilities hold for arbitrary system size. The solution of this TDHF equation an be written in the form 1 mn mn,kl V kl, 36 kl with the nonloal response matrix S mn, S mn,kl 1,tl tk S 1,kt lt. 37 t i Here S mn, diagonalizes the matrix A mn,kl aording to

6 8000 T. Wagersreiter and S. Mukamel: Optial ativity of moleular aggregates S,mn mnkl 1 A mn,kl S kl,,, 38 and are radiative damping onstants. The expetation values of Eqs. 29, 31 then satisfy Eqs. 9, 10, with r,r, mn,kl mn,kl P mn rp kl r, r,r, mn,kl mn,kl P mn rm kl r, r,r, mn,kl mn,kl M mn rp kl r, r,r, mn,kl mn,kl M mn rm kl r All response tensors fatorize into the disrete nonloal response matrix Eq. 37 and position-dependent tensor produts of eletri and magneti dipole densities suh as P mn (r)p kl (r). In pratie these expressions are limited to small systems, as the required memory sales with the fourth power of the basis set size. This problem an be remedied by the oupled eletroni osillator CEO piture. 35,36,42 44 The matrix S kl, an be regarded as a transformation matrix from the site representation to a nonloal mode representation. Sine A is not symmetri, S is in general a omplex matrix. However, one an show that all eigenvalues are real and ome in pairs,. 43 Moreover, in appliations to onjugated polymers the S matrix is real. 35,36 The transformations of the eletri and magneti transition dipole moment densities to this representation are given by r mn P mn rs mn,, 43 m r mn M mn rs mn,. 44 Without loss of generality we an hoose the basis funtions n to be real. The polarization densities P mn (r) are then purely real, and the magnetization densities M mn (r) are zero for mn, and purely imaginary for m n. Consequently the eletri transition dipole moments (r) are real and the magneti transition dipole moments m (r) are imaginary. One an therefore follow similar steps as in Ref. 43, whih finally yields r,r; 2 r,r; 2 r,r; 2 r *r i 2 2, 45 i rm *r i 2 2, 46 i m r *r i 2 2, 47 r,r; 2 m rm*r i Here the summation runs over partile-hole osillators with frequeny 0. The advantage of this piture is that typially very few osillators ontribute to the optial response, 42,44 leading to a drasti redution in the number of terms ompared with the (mn) summation in the site representation. Note that this form immediately yields the symmetry Eq. 19. IV. NONLOCAL POLARIZABILITIES OF SPATIALLY EXTENDED AGGREGATES We onsider moleular aggregates with eletrostati intermoleular fores no intermoleular harge exhange. In this ase it is possible to rigorously express the global nonloal response tensors in terms of nonloal response tensors of the onstituent monomers. These expressions greatly redue omputational ost and simplify the theoretial analysis. Usually moleular aggregates are treated within the LFA Refs. 20, 25, 26, 22 whih assume the individual moleules as point dipoles. The nonloal expressions derived here using the TDHF equations are exat and provide a onrete means for modeling aggregates when moleular sizes are omparable to their separation so that the point dipole approximation is not expeted to hold. Consider an aggregate made out of M moleules labeled a,b,,..., eah desribed by a basis set of N a, N b,... orbitals indiated by subsripts a i,b j,.... We have shown earlier, 36 that all intermoleular oherenes vanish ( ai b j (1) ai b 0) when harge exhange is negligible. Eqs. 35, j 29, 31, 36 lead to r,r, a,b r,r, a,b r,r, a,b i, ai a j,k,l j,b k b l P ai a j rp bk b l r, 49 i, ai a j,k,l j,b k b l P ai a j rm bk b l r, 50 i, ai a j,k,l j,b k b l M ai a j rp bk b l r, 51 r,r, a,b i, ai a j,k,l j,b k b l M ai a j rm bk b l r, 52 where intermoleular omponents P ai b j,m ai b j were negleted. Following similar steps as used in Ref. 36 we first derive a losed EOM in the spae of moleule a 1 ai a j kl l 1 Ā ai a j,a k a l ak a l ai a l V lo al a V lo j ai a l al a j, where the generalized loal potential is defined as 53

7 T. Wagersreiter and S. Mukamel: Optial ativity of moleular aggregates 8001 V lo ai b V j ai b j 2 ai b j a,l U ai l l l. 54 The differene from Ref. 36 is that we now inlude nondiagonal matrix elements V ai b j. The EOM is modified by the ground state harge distribution of the surrounding moleules aording to Ā ai a j,a k a l A ai a j,a k a l ik jl a,l U ai l U aj l 2 l l 1, 55 r,r, a,b a,b m a r b r,, r,r, a,b a,b m a rm b r. 62, The CEO representation greatly redues omputational effort, as the number of dominant osillators is generally muh smaller than the square of the basis set dimensionality. Moreover, the matrix a,b provides information regarding the oupling between eletroni osillators of different moleules. where U ai l denotes the Coulomb interation between orbitals a i and l. The solution of Eq. 53 an then be expressed in terms of the moleular response matries ai a j,a r a s 1 ai a j rs lo ai a j,a r a s V ar a. 56 s Here is defined in analogy to Eq. 37, with S replaed by the matrix S whih diagonalizes Ā aording to 1 S a ijkl,a i a Ā j ai a j,a k a l S a k a l,a a,. 57 Inserting the definition of the loal field Eq. 54, equating to Eq. 36, and assuming further, that V ai b j is negligible for a b yields an equation for, whose solution is ai a j,b k b l nm S ai a j,b n b m bn b m,b k b l, 58 with 1 S ai a j,b k b l ai b k a j b l 21 ab m ai a j,a m a m U am b k kl. 59 This exat result provided intermoleular harge exhange as well as V ai b j are negligible, together with Eq. 49 allows the alulation of aggregate response tensors using the monomer response tensors. Sine off diagonal intramoleular matrix elements P am a n,m am a n are inluded, the number of indies in and V doubled. Swithing to the CEO representation leads to a,b 1 S a,b 1 S a,a i a j S a,b b,b, 1 S ai a j,b k b l S b k b l,b, 60 where a denotes the -th osillator of moleule a note that a i denotes the i-th site in moleule a). The nonloal response tensors finally assume the form r,r, a,b r,r, a,b a,b a r b r,, a,b a rm b r,, 61 V. OPTICAL ACTIVITY IN THE POINT DIPOLE APPROXIMATION So far we have formulated optial ativity in terms of nonloal response funtions, ompletely avoiding the multipolar expansion. We also paved the way for omputational appliations by rigorously expressing aggregate response funtions in terms of monomer response funtions, provided that exhange interations between the monomers are negligible. In order to establish the onnetion to ommonly employed approximations in the investigation of optial ativity we now speialize to the lowest order in the multipolar expansion of the polarization and magnetization densities P, M. This amounts to keeping only the lowest order term in the funtions appearing in Eqs. 13, 14, resulting in P mn r mn r, M mn rm mn r, with mn e dr m *rr n r, m mn ie dr m *rr n r, where we have set R0, and the dipole densities in the CEO representation Eqs. 43, 44 assume the form r r, m rm r, with mn mn S mn,, m mn m mn S mn,. Under these assumptions the nonloal polarizability Eq. 45 beomes r,r; rr, 63 where the ontribution of the -th osillator to the polarizability tensor labeled by Greek subsripts is

8 8002 T. Wagersreiter and S. Mukamel: Optial ativity of moleular aggregates 2 i Analogous expressions hold for the other response tensors,, and. We now onsider an aggregate made of N point-like moleules loated at r a,r b,... whih are haraterized by their linear response tensors a, b,... with,,,. The label in Eq. 64 is then replaed by the double index a denoting the ontribution from the -th osillator of moleule a. Intermoleular harge transfer is negleted, so that dipole-dipole oupling is the only interation. The polarization and magnetization an then be written as Pr a Mr a P a rr a, M a rr a. The indued eletri and magneti dipole moments at the a-th moleule an be alulated from the moleular response funtions by employing the LFA P a a E a lo a H a lo, M a a H a lo a E a lo, where the loal eletri and magneti fields are given by the sum of the external fields and the fields indued by the surrounding moleules E a lo E a b T ab P b, 67 ad be Y ae S eb bd d b T b S d d, ad be Y ae S eb bd d b T b S d d The various tensors appearing in these equations are defined as follows: S 1 ab 1 ab a T ab,,, X 1 ae 1 ae S ab b T b S d d T de bd Y 1 ae 1 ae S ab b T b S d d T de, bd where R ab r a r b, e ab R ab /R ab. By omparing Eqs. 69, 70 with Eqs. 9, 10 we obtain the nonloal response tensors for this model r.r; ab rr a rr b ab,,,,, so that the differene in absorption ross setion for right and left irularly polarized light Eq. 28 beomes 0 A 4 Re e ik R ab ab ab yx xy ab n yx ab xy ab n xx ab yy ab H a lo H a b T ab M b. 68 xx ab yy ab. 75 Here T ab is given by T ab 1 ab 3e abe ab 1 R ab 3. Substitution of Eqs. 67, 68 into Eqs. 65, 66 yields a linear system of equations for P a and M a, whose solution is The response tensors ab,,,, an be omputed by numerial inversion of 3N3N matries, where N is the basis set dimensionality. However, this expression is simplified by expanding the matries S,X,Y to first order in T and negleting terms of the order m 2, whih results in ab ab a a T ab b, 76 P a d ad E d ad H d, 69 ab ab a a T ab b, 77 M a ad E d ad H d, 70 d with the nonloal response matries ad be X ae S eb bd d b T b S d d, ad be X ae S eb bd d b T b S d d, ab ab a a T ab b, ab Setting e ik r 1ik r, performing orientational averaging (A kˆ)(b kˆ) kˆ 1 3 A B, and using ij ji ab ba gives A n a R a a 2 a 2 a a 2 2, a 80 where we have defined the rotational strength R a for the -th transition of moleule a

9 T. Wagersreiter and S. Mukamel: Optial ativity of moleular aggregates 8003 R a a with 2 ba Im a m a 2 ba ImV ab V ab a T ab b. a a m b b b m a 2 2 a b a b V ab a b R ab a 2 2 b, The CD-signal vanishes as the damping rates go to zero, and it sales as 1 for large frequenies. The first term in the rotational strengths, desribing the intrinsi CD of the individual moleules stems from the diagonal terms in Eqs. 77, 78. The seond term in Eq. 81 is due to the mixed terms in Eqs. 77, 78, and the third term omes from the seond term in Eq. 76. Terms inluding moleular magneti transition dipole moments are finite even when setting e ikr 1, whereas the last term in Eq. 81 requires e ikr 1ikr. This last term an also be identified with Kirkwoods polarizability approximation 20 A 4 n Im ab,ij a yi T ij ab b jx a xi T ij ab b jy k R ab, 8 n Im ab,ij a yi T ij ab b jx k R ab. The elebrated Tinoo formulas 29,41 for the rotation of the polarization plane of linearly polarized inident light per unit length are readily obtained using the Kramers Krönig relation 22 P A d 0 2 2, FIG. 1. Orientation of the monomer in the x z plane and labeling of the atom, used in Fig. 3. where P denotes the prinipal part of the integral. Negleting damping ( 0) results in the familiar expression for OR in transparent regions of the aggregate 162 R 3n a a 2 2. a The remaining two terms in Tinoos result see e.g., p. 70 in Ref. 29 representing interations with the ground state harge distribution are inluded impliitly in Eq. 81, aswe use moleular response tensors a,,,, modified by the ground state harge distribution of the neighboring moleules. 36 They an be retrieved by a perturbative expansion of a in the interation Hamiltonian. VI. CD SPECTROSCOPY OF NAPHTHALENE CLUSTERS We have applied the present nonloal formalism to alulate the CD spetrum of naphthalene dimers and trimers. Comparison with the standard loal formulation shows signifiant differenes, whih illustrates the need for a nonloal formulation. Small naphthalene lusters have been investigated by studying isotopially mixed rystals 45 and by supersoni moleular beam tehniques 46 revealing insight into their geometry. Although CD spetra of these systems are not available, they were hosen for their simpliity. They also serve as models for more omplex onjugated systems that appear in biologial aggregates. 4 6,9 12 We assume irularly polarized light propagating in z-diretion and investigate the trimer geometry suggested in Fig. 7 of Ref. 46. Using the SPARTAN pakage for a HF ab initio alulation with geometry optimization in an STG6-11* basis set yields the monomer geometry. The dimer is onstruted from a monomer in the xz-plane shown in Fig. 1 by a 60 rotation around the z-axis and a translation of 3.7 Å in the z-diretion. The trimer is obtained by adding one more moleule rotated by 120 around the z-axis and translated by 7.4 Å in the z-diretion relative to the first moleule. Instead of the absorption ross setion, we swith to the absorption oeffiient via the relation 37 A 4, so that Eq. 26 beomes. 83 In the NLRF alulation we have onsidered only the polarization term in Eq. 28 and negleted the magneti terms,,. Negleting further nondiagonal matrix elements P ai a j, setting n1, and substituting Eq. 49 into Eq. 28 leads to 0 x x Im ai a ab,ij i,b j b j P ai a i kp bj b j k y P ai a i y kp bj b j k,

10 8004 T. Wagersreiter and S. Mukamel: Optial ativity of moleular aggregates 0 Re ab,ij ai a i,b j b j P ai a i k P bj b j k e k, where we have defined the Fourier transform of the polarization densities P ai a j k dre ik r P ai a j r. Assuming point harges 2 ai (r)(rr ai ) results in k r ai sin k r 2 ai P ai a i ker ai e i 2, k r ai 2 and the matrix is alulated using Eq. 58. This pointmonopole approximation is widely used for the alulation of intermoleular interations. 3,4,25,26 However, here we also keep the full matter-field interation to all orders in the multipolar expansion by using nonloal response funtions, whih have a non-trivial r or k dependene. Sine Tinoos formulae Eqs whih are derived using the approximative matrix inversion (1S) 1 1S) are not appliable for the small intermoleular distanes of the investigated aggregates, we resort to the more aurate preursor, Eq. 75 for an oriented sample as a omparison. Negleting again magneti ontributions (m a 0) results in 0, and X ae 1 ae, so that with LFA Im ab LFA Re ab ad S ad d, S 1 ad 1 ad a T ad. e ik R ab xx ab yy ab, e ik R ab xy ab yx ab, FIG. 2. Absorption spetrum () of a naphthalene monomer top panel, the dimer middle panel and the trimer bottom panel. The solid lines show the NLRF alulations, and dashed lines depit the LFA results for the dimer and trimer ev, and for the other parameters see the text. In Fig. 2 we display the average absorption oeffiient () see Eq. 26 for the naphthalene monomer top panel, the dimer middle panel, and the trimer bottom panel alulated via the NLRF approah solid lines. For the dimer and trimer we also show the LFA results for omparison dashed lines. We only show the region of the lowest absorption band. Other bands are smaller by a fator of 20. We further hoose a small damping ev. The monomer has one absorption peak at 5.58 ev. In ase of the dimer the exat alulation yields four peaks. Approximating the monomers by point dipoles leads to two absorption peaks. Their separation exeeds the absorption range as obtained by the NLRF. For the trimer the NLRF alulation also yields four peaks, and the LFA results in three. The LFA results an be understood with a simple two or three dimensional model Hamiltonian. For the dimer the monomer peak splits into two peaks whih are symmetrially positioned around the monomer peak. Assuming M a b H t E M a E M a b a E for the trimer yields the eigenvalues 1 E M b, 84 2,3 E M b8a2 b 2, 85 2 whih explains the position of the peaks observed in Fig. 2. The additional peaks in the NLRF reflet the fat that this alulation fully aounts for the intermoleular interations, whereas the LFA result retains only the leading dipole-dipole interation. In order to study inter and intramoleular oherenes, we depit the nonloal response matrix at the frequenies of the four peaks of the NLRF in Fig. 3. The real part left panel is naturally onsiderable smaller than the imaginary part right panel near resonanes. Resorting to Fig. 1 for the labeling of the atoms we find that the seond and third transitions predominantly orrelate the losest sites of eah monomer, whereas the first and fourth transitions orrelate the most distant ones.

11 T. Wagersreiter and S. Mukamel: Optial ativity of moleular aggregates 8005 FIG. 3. Real left and imaginary right part of the nonloal response matrix () of a naphthalene dimer for the four peaks shown in Fig. 2 from top to bottom (5.37,5.50,5.61,5.72 ev). Atoms 1 10, 11 20, orrespond to the three monomers see Fig. 1. For aggregate geometry see the text. Figure 4 shows CD of the dimer left panel and the trimer right panel alulated in the NLRF solid lines and the LFA dashed lines. Eah absorption peak leads to a distint peak in the CD signal, whose amplitude reflets the rotational strength of the optial transition. Apart from the magnitude of these peaks, their sign arries important information, refleting the handedness of the moleule. In Fig. 5 we used a larger damping 0.08 ev whih yielded reasonable absorption line shapes for PPV oligomers. 35,36 The top panels show the absorption oeffiient of the dimer left and the trimer right, alulated using the NLRF solid and the LFA dashed. The bottom panels display the CD signal (). The absorption oeffiients for both systems are now haraterized by a single peak with internal struture. In order to show the transitions involved we also plot the results for small damping. In ase of the dimer the zero of the CD

12 8006 T. Wagersreiter and S. Mukamel: Optial ativity of moleular aggregates FIG. 4. CD signal () alulated with the NLRF of the naphthalene dimer left and trimer right for a damping rate of ev. signal oinides with the monomer frequeny M. In the LFA the two rotational strengths are equal in magnitude and opposite in sign, so that the signal for finite damping is symmetri with respet to M. For the trimer suh a relation does not hold, beause of the form of the transition frequenies Eqs. 84, 85. Due to the small intermoleular separation smaller than the monomer size the LFA gives markedly different results ompared to the LFA. However, as shown in Fig. 6 the LFA onvergenes towards the NLRF results with inreasing intermoleular separation. This series learly demonstrates the breakdown of the LFA, as the intermoleular separation approahes moleular dimensions. The NLRF must be employed for distanes smaller than 10 Å. Sine we used Ohnos formula FIG. 5. Comparison of linear absorption () top panels and CD () bottom panels alulated with the NLRF solid line and the LFA dashed line of a naphthalene dimer left panels and trimer right panels for a damping rate 0.08 ev. The vertial lines show the small damping results.

13 T. Wagersreiter and S. Mukamel: Optial ativity of moleular aggregates 8007 U U mn 86 1r mn /a 0 2 with UI ev, a Å for the eletrostati interations in the PPP Hamiltonian, 35 the long distane behaviour is k/r with k ev Å. Considering the approximation (1B) 1 1B whih applies at large separations and is neessary to obtain Tinoos formula, the CD signal is determined to first order by the seond term in Eq. 59, so that k enters the CD signal as an overall saling fator. In order to obtain agreement with the LFA results whih assume 1/r mn for the Coulomb interation for large intermoleular separation we divided the NLRFs result by k. Naphthalene moleules do not have a permanent dipole moment and the ground state harges are idential for all sites ( nn 0.5. Therefore the modifiations of the moleular response funtions by the ground state harge distribution of the surrounding moleules anel see Eq. 55, and the response matrix oinides with that of isolated moleules. This is not expeted to be the ase for donor aeptor substituted moleules, in whih the ground state harge distribution is nonuniform. VII. SUMMARY The onventional approah to the desription of optial ativity is based on the multipolar expansion. Cirular dihroism or optial rotation are alulated term by term for systems muh smaller than the optial wavelength. The present theory is based on the onept of nonloal response tensors and ompletely avoids these approximations. We first alulated irular dihroism CD defined as the differene in absorption ross setion for right and left irularly polarized light. Eq. 20 ontains the polarization P r, the magnetization M r and the eletri and magneti fields E r, H r. Utilizing the linearized relationships P E,H, M E,H Eqs. 9, 10 lead to our final expression Eq. 28 whih holds for arbitrary wavelengths and system sizes. With this result the theory of optial ativity OA is fully aptured through four nonloal response tensors (r,r,), (r,r,), (r,r,), (r,r,) whih relate polarization and magnetization to the driving external fields. It is formally exat; approximations neessary for pratial omputations enter in the alulation of the response tensors. We then foused on eletronially deloalized onjugated systems desribed by a Pariser Parr Pople Hamiltonian. Using the time dependent Hartree Fok equations and the multipolar form of the interation Hamiltonian we derived the expressions Eqs for the response tensors. They fatorize into a entral, nonloal response matrix and dyadis P (r)p (r), P (r)m(r), et. One a onvenient eletroni basis set has been hosen, the polarization and magnetization densities P r,m(r) an be alulated without any further approximations. In partiular, our final expression Eq. 28 impliitly inludes all multipolar moments, suh as eletri and magneti dipole, quadrupole, otopole, et. Moreover we ould show easily the independene of the absorption ross setion, and hene of the CD-signal of the hoie of origin. A onsiderable redution of omputational ost in the investigation of spatially extended moleular aggregates is possible if intermoleular harge exhange is negligible. In this ase, intermoleular eletroni oherenes, i.e., offdiagonal elements of the single eletron density matrix between basis funtions loated at different monomers, vanish identially. This makes it possible to rigorously express the aggregate response in terms of monomer response tensors. The exat relationship Eq. 58 derived in this artile eliminates the need for the ommonly used loal field approximation whih ompletely neglets the moleular internal struture. The alulation was arried out by deriving moleular equations of motion for the redued one eletron density matrix in Liouville spae, whih fully aount for intermoleular Coulomb interations. This way we ompletely avoided a perturbative alulation of aggregate wavefuntions. To aount for exhange interations and harge transfer, intermoleular eletroni oherenes need to be inluded in the alulation of the nonloal response tensors. Note that the nonloal response tensors (r,r,), where r and r are at different monomers imply the existene of exitoni oherenes, whereby eletron hole pairs an hop among monomers retaining their phase. Eletroni oherenes on the other hand imply oherent motion of eletrons between monomers and reflets a phase between an eletron at one FIG. 6. Convergene of the LFA dashed towards the NLRF solid with inreasing intermoleular distane (3.7,5,10 Å from top to bottom for a naphthalene dimer. Shown are the absorption oeffiient left panels and the CD signal right panels.

14 8008 T. Wagersreiter and S. Mukamel: Optial ativity of moleular aggregates monomer and a hole at another. Eletroni oherene is not required to attain exitoni oherene whih an arise from purely eletrostati interations. However, the presene of intermoleular harge exhange does affet the exitoni oherene and the optial response. 35,36 We have further formulated the response funtions using the oupled eletroni osillator representation see Eqs , As demonstrated in Refs. 42, 44 only very few osillators ontribute signifiantly to the optial response whih leads to a drasti redution in omputational ost, as an be inferred from the dimension of the matries ai a j,b k b l and a,b. Moreover, the latter quantities arry information about the oupling between osillators of different moleules and thus provide most valuable insight into intermoleular interations. The well known Tinoo formula Eq. 81 for the rotational strengths was reovered by introduing the loal field approximation, the long wavelength approximation, and an approximative matrix inversion (1T) 1 1T to our general result for A (0) (). Finally we ompared our results with those obtained from Tinoos formulae by appliation to naphthalene dimers and trimers. The aggregate geometries were taken from Ref. 46, and we first hose a damping rate small enough to reveal the involved eletroni transitions. The resulting spetra then reflet the struture of the osillator and rotational strengths. For the monomer, the average frequeny dependent absorption oeffiient of irularly polarized light is haraterized by one absorption peak at 5.58 ev see Fig. 2. This peak splits up into two three peaks for the dimer trimer when alulated with the LFA, whih an be understood with a simple two three dimensional exiton Hamiltonian. The exat NLRF yields four absorption peaks for the dimer and the trimer, whih are not as muh separated as the LFA indiates. Eah absorption peak has a orresponding peak in the CD spetrum Fig. 4. This figure also ontains the sign information of the rotational strengths, whih is essential for differentiating between right or left handed geometries. The inreased number of transitions reflets the orretions due to nonloal interations between spatially extended moleules. We finally demonstrated the breakdown of the LFA for intermoleular separations omparable to moleular dimensions in Fig. 6. For intermoleular distanes smaller than 10 Å the NLRF has to be employed. ACKNOWLEDGMENTS One of us Th. W. holds an E. Shrödinger Stipendium Grant No. J PHY from the Fonds zur Förderung der Wissenshaftlihen Forshung, Austria, whih is gratefully aknowledged. We gratefully aknowledge the support of the Air Fore Offie of Sientifi Researh, the National Siene Foundation and pu time on a C90 granted by Air Fore major shared resoure enters MSRC. APPENDIX: THE ABSORPTION CROSS SECTION OF AN AGGREGATE The energy flux density of an eletromagneti wave in a dieletri is given by the pointing vetor see Se. 61 in Ref. 38 Sr, 4 Er,Hr,. A1 Integrating S (/4)(H EE H) over a volume V with surfae O and making use of Maxwells equations 15, 16 leads to df S 1 d drh O 8 dt 2 E 2 V dre P H M V t t. A2 In this energy balane equation the left-hand side is the derease of energy per unit time, due to the flux through the surfae O, and the right hand side has been separated into the time derivative of the energy of the fields E, H, and a dissipation term due to the indued eletri and magneti dipoles. We now assume V to be the entire spae, and apply this result to an isolated aggregate. The absorbed energy per unit time is then given by the time average of Eq. A2. Dividing by the time averaged inident energy flux in 1 T 0 T dt 4 E exth ext A3 yields the effetive absorption ross setion A. For monohromati fields with frequeny 2/T the time average of the first term on the rhs of Eq. A2 vanishes as the fields have the same values at t0 and tt), and using dr f r g r dr f r g r we arrive at A 1 1 T in T dt 0 dre P V t H M t. A4 The transverse eletri and magneti Maxwell fields E and H are solutions of the wave equations E t 2 E 4 2 t t P M, A5 H t 2 H 4 2 tp t M, A6 whih follow from the Maxwell Eqs. 15, 16. These linear equations an be solved using Greens funtions. The Maxwell fields are given to lowest order by the external fields E ext (r,t),h ext (r,t), and orretions are due to multiple sattering of the inoming wave by the harge distribution: E r,te ext r,ter,t, H r,th ext r,thr,t. A7 A8 The primed fields an be obtained from Eqs. A5, A6 after transforming to k, spae via

15 T. Wagersreiter and S. Mukamel: Optial ativity of moleular aggregates 8009 f k, dr dt fr,te ikrt, resulting in Ek, 42 2 k 2 2 P k,kˆmk,, Hk, 42 2 k 2 2 kˆpk,m k,, where kˆk/k. Equations 9, 10 beome P k, dkk,k; E ext k, A9 A10 A11 k,k; H ext k,, A12 M k, dkk,k; E ext k, k,k; H ext k,, A13 and the transverse vetor omponents in k spae are obtained by multipliation with the projetor (1kˆkˆ). Substituting Eqs. A7, A8 into Eq. A4 yields to the separation Eq. 21 A A 0 A 1, where A (0) is given by Eq. 22, and A 1 1 in 1 T 0 T dt dkek,t Pk,t t A14 Hk,t Mk,t t. A15 A linearly polarized plane wave with wavevetor k (n/ ) e k and polarization diretion e 1 perpendiular to e k an be represented by E ext r,t E 0 2 e 1e ikrt e ikrt, A16 H ext r,t H 0 2 e 2e ikrt e ikrt, A17 with H 0 ne 0, and e 2 e k e 1. Upon substitution into Eq. 20 we find in (E 0 H 0 /4), whih results in Eq. 23 Choosing e 1 e x and kˆe z, this assumes the form 0 A 4 dr n Im dr e ikrr 1 n xx r,r,n yy r,r, yx r,r, xy r,r,. Cirularly polarized fields an be written as E ext r,t E 0 2 e e ikrt e e ikrt, H ext r,ti H 0 2 e e ikrt e e ikrt, where the omplex unit vetors are defined as 32 A18 A19 A20 A21 e e xie y, A22 2 e e xie y. A23 2 We find again in (/4)E 0 H 0 and the differene in absorption ross setions for right and left irularly polarized light Eq. 22 beomes 0 A 0 0 A A A24 8 dr n Im dre * r,r, e e * r,r, e e ik rr 8 dr Re dr e * r,r, e e * r,r, e e ik rr 8 dr Re dre * r,r, e e * r,r, e e ik rr 8n Im dr dre * r,r, e e * r,r, e e ik rr. A25 Swithing bak to the real basis vetors e x,e y we reover Eq M. M. Bouman et al., Mol. Cryst. Liquid Cryst. 256, ; M.M. Bouman and E. W. Meijer, Adv. Mater. 7, W. C. Johnson, Jr. and I. Tinoo, Jr., Biopolymers 7, C. L. Ceh, W. Hug, and I. Tinoo, Jr., Biopolymers 15, R. W. Woody and I. Tinoo Jr., J. Chem. Phys. 46, Craig W. Deutshe, J. Chem. Phys. 52, F. M. Loxsom, J. Chem. Phys. 51, V. Devarajan and A. M. Glazer, Ata Crystallogr. A 42, A. M. Glazer and K. Stadnika, J. Appl. Cryst. 19, K. Sauer and L. A. Austin, Biohem. 17, ; R. J. Cogdell and H. Sheer, Photohem. Photobiol. 42, ; K. Sauer et al. unpublished.