Optical Activity of Electronically Delocalized

Optical activity of electronically delocalized molecular aggregates: Nonlocal response formulation Thomas Wagersreiter and Shaul Mukamel Department of Chemistry, University of Rochester, Rochester, New York 14627 ͑Received 14 June 1996; accepted 26 July 1996͒ A unified description of circular dichroism and optical rotation in small optically active molecules, larger conjugated molecules, and molecular aggregates is developed using spatially nonlocal electric and magnetic optical response tensors ␹(r,rЈ,␻). Making use of the time dependent Hartree Fock equations, we express these tensors in terms of delocalized electronic oscillators. We avoid the commonly-used long wavelength ͑dipole͒ approximation k–rӶ1 and include the full multipolar form of the molecule–field interaction. The response of molecular aggregates is expressed in terms of monomer response functions. Intermolecular Coulomb interactions are rigorously taken into account thus eliminating the necessity to resort to the local field approximation or to a perturbative calculation of the aggregate wave functions. Applications to naphthalene dimers and trimers show significant corrections to the standard interacting point dipoles treatment. © 1996 American Institute of Physics. ͓S0021-9606͑96͒01541-3͔

I. INTRODUCTION ing the local field approximation ͑LFA͒ which accounts for intermolecular electrostatic interactions via a local field Circular dichroism ͑CD͒, i.e., the difference in absorp- which—at a point like molecule in an aggregate—is the su- tion of right and left circularly polarized light has long been perposition of the external field and the field induced by the used in the investigation of structural properties of extended surrounding molecules modeled as point dipoles. He thus chiral molecules or aggregates. This technique exploits the expressed LA and OR of unaggregated monomers, aggre- spatial variation of the electromagnetic field across the mol- gates in solution, and molecular crystals in terms of the ag- ecule and yields valuable structural information, which is gregate geometry and the complex polarizabilities of the missed by the ordinary linear absorption ͑LA͒. It has been constituent monomers. Using this method, Applequist calcu- used in the investigation of a variety of systems including lated the polarizabilities of halomethanes,22 and presented a polythiophenes,1 polynucleotides,2,3 helical polymers,4–6 mo- normal mode analysis of polarizabilities and OR.23,24 When lecular crystals,7,8 and antenna systems and the reaction cen- dealing with solvated molecules or aggregates, the local field ter of photosynthesis.9–12 Also commonly used are other re- should also include the contribution from the solvent.25,26 lated signatures of optical activity ͑OA͒, i.e., optical rotation In all models discussed since then, the spatial variation ͑OR͒ and ellipticity. of the field with wavevector k was accounted for by the A molecular theory for optical activity was first formu- ikr lated by Born,13 and almost simultaneously by Oseen.14 Born expansion e Ϸ 1 ϩ ikr ϩ ••• . Usually the series is trun- attributed the origin of optical activity to the spatial variation cated after the second ͑dipole͒ or third ͑quadrupole͒ term. of the electromagnetic field over the system under investiga- The molecule itself is modeled by an electric and a magnetic tion. Rosenfeld reformulated these ideas in terms of the transition dipole moment. Larger molecules or aggregates quantum-mechanical theory of dispersion.15 These results are regarded as an assembly of point-like units, each having were later derived as scattering problems in quantum field an electric and a magnetic transition dipole moment. Charge theory.16 The next step was the development of polarizability transfer between units is neglected and their mutual coupling theories, which express the optical rotatory power ͑ORP͒ of a is given by the dipole–dipole interaction. De Voe also im- molecule in terms of the configuration and polarizabilities of proved the description of the interaction between units, re- 17–20 its constituent groups. In transparent spectral regions, placing the dipole–dipole interaction term Tij by the inter- Kirkwood achieved this goal by applying Borns’ theory to a molecular interaction of all point-charges ͑point monopole 25 molecule which he separated into smaller units, and pertur- approximation͒ contained in units i and j. This correction batively calculating the wavefunctions modified by the inter- is essential when the dimensions of the units are comparable action of these subunits. Moffitt and Moscowitz also studied to their separation, as is the case in typical aggregates of absorptive spectral regions. By using complex polarizabil- polynucleotides.2,3 ities they worked out a unified description of dispersive ͑re- The purpose of this paper is to develop a computation- fraction, optical rotation͒ and absorptive ͑linear absorption, ally feasible procedure for the investigation of OA which circular dichroism͒ properties.21 They established Kramers– avoids the commonly used multipolar expansion and re- Kronig relations between OR and CD and further discussed places the LFA by a rigorous treatment of intermolecular the effects of vibrational degrees of freedom on these observ- electrostatic interactions. To set the stage, we shall briefly ables. review the conventional theory of OA using the notation of De Voe approached this many-body problem by apply- Moffitt and Moscowitz.21 It is based on the observation, that

Downloaded¬07¬Mar¬2001¬to¬128.151.176.185.¬Redistribution¬subject¬to¬AIP¬copyright,¬see¬http://ojps.aip.org/jcpo/jcpcpyrts.html 7996 T. Wagersreiter and S. Mukamel: Optical activity of molecular aggregates the induced electric dipole moment is not only proportional An early yet very instructive review on theories of opti- to the electric field, but also to the magnetic field and its time cal rotatory power was written by Condon.27 A clear discus- derivative. Analogously, the induced magnetic dipole mo- sion of CD at a minimum level of complexity ͑time depen- ment is proportional to the magnetic field, as well as to the dent perturbation theory with simple electric and magnetic electric field and its time derivative dipole interaction͒ can be found in Ref. 28. Finally there are numerous textbooks covering theory and experiment Hץ ␻ exhaustively.29–32 ␮␻ϭ␣͑␻͒E␻ϩ␥͑␻͒H␻Ϫ␤͑␻͒ , ͑1͒ t In order to develop a unified microscopic approach toץ OA which applies to a broad class of molecular systems we E␻ץ m ϭ␹͑␻͒H ϩ␥͑␻͒E Ϫ␤͑␻͒ , ͑2͒ shall employ nonlocal response tensors, which relate the in- tץ ␻ ␻ ␻ duced linear polarization P˜(r,␻) and magnetization where ␣,␬ are the polarizability and magnetizability of the M˜ (r,␻) to the driving electric and magnetic fields molecule and E␻ ,H␻ are monochromatic fields with fre- Eext(r,␻),Hext(r,␻) via the linearized relationships quency ␻. The coefficient ␥ has no effect on OA, and ␤ which is responsible for OA, can be separated into real and ˜ imaginary parts according to P͑r,␻͒ϭ ͵ drЈ ͓␣͑r,rЈ;␻͒•Eext͑rЈ,␻͒ ␤ϭ␤ ϩi␤ , ͑3͒ 1 2 ϩ␤͑r,rЈ;␻͒•Hext͑rЈ,␻͔͒, ͑9͒ which are related to each other via the Kramers Kro¨nig relations ͓Eq. ͑52͒ in Ref. 21͔. For transparent regions one ob- ˜ tains M͑r,␻͒ϭ ͵ drЈ ͓␥͑r,rЈ;␻͒•Eext͑rЈ,␻͒ 4␲c R i0 ϩ␬͑r,rЈ;␻͒•Hext͑rЈ,␻͔͒. ͑10͒ ␤1ϭ ͚ 2 2 , ͑4͒ 33ប i ␻i0Ϫ␻ In this nonlocal response formulation ͑NLRF͒, the tensors where the central quantities describing optical activity are ␹(r,rЈ;␻),␹ϭ␣,␤,␥,␬ are intrinsic molecular properties the rotational strengths R for transitions from the ground i0 that describe all linear optical phenomena. Structural and state 0 to an excited state i with transition frequency ͉ ͘ ͉ ͘ chemical details of different systems enter through their ef- ␻ . For systems much smaller than the optical wavelength i0 fects on these tensors. By using nonlocal response tensors they are given by we completely avoid the commonly employed multipolar ex- ˆ ˆ pansion, which uses the induced electric and magnetic di- R0iϭIm͑͗0͉␮͉i͘•͗i͉m͉0͒͘, ͑5͒ pole, quadrupole, octopole, etc. moments. If so desired, the ˆ ˆ where ␮, m denote the electric, and magnetic dipole mo- latter can be calculated by integrating Eqs. ͑9,10͒ over r and ment operators of the entire system. expanding the driving electric and magnetic fields around a In an optically active medium, left and right circularly reference point of the molecule. By formulating the problem polarized light propagate with different speeds and are ab- this way we can defer the introduction of details to the final sorbed differently. In transparent regions the polarization stage of the calculation. This enables us to unify many of the vector of linearly polarized light is rotated by an angle which treatments in the literature that are restricted to specific mod- is proportional to the propagation distance. The optical rota- 21 els ͑e.g., aggregates of point dipoles, etc.͒. tion in radians per unit length is given by We already employed the nonlocal polarizability tensor 2 2 ␣(r,rЈ,␻) for investigating the anisotropy of the linear ab- n0ϩ2 4␲␻ ␾ϭN ␤ , ͑6͒ sorption, and for gaining insight into the role of intramolecu- 3 c 1 lar coherences.33–36 When magnetic interactions contribute where N is the number density of molecules, and a correction to the linear response, the magnetizability ␬(r,rЈ,␻) and the factor for the surrounding medium with refraction index n0 cross response tensors ␤(r,rЈ,␻), ␥(r,rЈ,␻) enter the pic- has been added. In absorptive regions linearly polarized light ture as well. We will show that these tensors fully determine turns into elliptically polarized light after propagating the CD signal, so that the level of rigor is determined by the through the medium, and ␾ denotes the angle between the approximations applied for their calculation. By using the long axis of the ellipse and the direction of polarization of multipolar Hamiltonian and the time dependent Hartree Fock the incident linearly polarized wave. The ellipticity ⌰ per ͑TDHF͒ procedure we derive general formulas for these re- unit length is given by21 sponse tensors which contain the full multipolar expansion, thus avoiding the dipole approximation (krӶ1). When in- 2 2 n0ϩ2 4␲␻ ⌰ϭN ␤ . ͑7͒ vestigating molecular aggregates we will go beyond the 3 c 2 commonly used local field approximation ͓LFA, see Eqs. ͑65, 67͒ below͔ and rigorously express the nonlocal response It is directly proportional to the difference in absorption of tensors in terms of those of the monomers. Using this right and left circularly polarized light ⌬⑀, i.e., the CD signal method, intermolecular interactions are fully incorporated, ⌬⑀ϭ33⌰. ͑8͒ and the need for a perturbative expansion of the aggregate

J. Chem. Phys., Vol. 105, No. 18, 8 November 1996

Downloaded¬07¬Mar¬2001¬to¬128.151.176.185.¬Redistribution¬subject¬to¬AIP¬copyright,¬see¬http://ojps.aip.org/jcpo/jcpcpyrts.html T. Wagersreiter and S. Mukamel: Optical activity of molecular aggregates 7997 wavefunctions is eliminated, providing a clear advantage Ќ Ќ over the commonly used point dipole or point monopole ap- F ͑r͒ϭ ͵ drЈ␦ ͑rϪrЈ͒•F͑rЈ͒, proximations. In Sec. II we derive an expression for the CD signal of a with the transverse ␦-function spatially extended molecular system using the electric and 2 1 magnetic nonlocal response tensors. We next calculate the Ќ ␦ ͑r͒ϭ 1␦͑r͒Ϫ 3 ͑1Ϫ3rˆrˆ͒. latter in Sec. III using the TDHF equations. In Sec. IV we 3 4␲r express the aggregate response in terms of the response func- Here we use a dyadic notation for the tensor product rˆrˆ. tions of its constituent monomers. Applying the point dipole Both polarization and magnetization depend on the ref- and local field approximations we retrieve Tinoccos’ formu- erence point R, but their transverse parts entering Maxwells’ las for the rotational strengths in Sec. V. We then apply these equations for the observable fields ͑see below͒, do not. This results to naphthalene dimers and trimers in Sec. VI, and follows from the gauge invariance of the observable trans- summarize in Sec. VII. verse fields. The third term in the Hamiltonian 2 II. NONLOCAL RESPONSE FORMULATION OF Ќ 2 Ј HЈϭ͚ dr ͉ms ͑r͉͒ ϩ ͚ drms͑r͒•mt͑r͒, CIRCULAR DICHROISM s ͵ 3 s,t ͵

The multipolar Hamiltonian describing a molecular sys- is quadratic in the magnetization densities mˆ s , and we have tem interacting with a classical electromagnetic field is given omitted terms quadratic in the magnetic field, as they do not by32,37 contribute to the linear response. Finally, the last term in the Hamiltonian represents the coupling of the system to the ˆ sc ˆ ˆ ˆ H ϭHmϩVinterϩHЈ external transverse electric and magnetic fields. We start our analysis with the microscopic Maxwell ˆ Ќ ˆ Ќ Ϫ ͵ dr͓P„r…–E „r…؉M„r…–H „r…͔. equations for the electric and magnetic fields E(r,t) and H(r,t) ͑Ref. 32͒ Here H denotes the molecular Hamiltonian, and ץ m 1 ϫEЌ͑r,t͒ϭϪ ͓H͑r,t͒ϩ4␲M͑r,t͔͒Ќ, ͑15ٌ͒ tץ c ˆ ˆ ˆ ˆ ˆ ,Vinterϭ dr drЈ͓PsT͑r؊rЈ͒PtϩMsT͑r؊rЈ͒Mt͔ s͚Ͻt ͵ ͵ ץ 1 describes the interaction between the electric and magnetic ٌϫH͑r,t͒ϭ ͓E͑r,t͒ϩ4␲P͑r,t͔͒Ќ. ͑16͒ tץ c dipole densities of the electrons, defined as

1 Here c is the vacuum speed of light. The rhs of Eqs. ͑15, 16͒ ˆ Ps͑r͒ϭϪe͑rˆsϪR͒ d␭ ␦͑rϪRϪ␭͑rˆsϪR͒͒, ͑11͒ are transverse, as ٌ•Dϭٌ•Bϭ0 with DϵEϩ4␲P and ͵0 BϵHϩ4␲M. However, for the sake of clarity we add the 1 symbol Ќ explicitly in the derivation below. ˆ ˆ ˆ ˆ Ms͑r͒ϭϪe͑rsϪR͒ϫps ͵ d␭␭␦͑rϪRϪ␭͑rsϪR͒͒, We shall switch to the frequency domain by the Fourier 0 transform ͑12͒ ϱ where R is an arbitrary reference point, Ϫe is the electron ˜f ͑␻͒ϭ ͵ dt ei␻tf͑t͒, charge, rˆs denote the position operators of the charges la- Ϫϱ beled by s, and pˆ s is the corresponding canonical momen- ˆ 1 ϱ tum. The polarization P(r,t)ϭ͗␺(t)͉P(r)͉␺(t)͘ is given by f͑t͒ϭ d␻ eϪ␻t˜f͑␻͒. the expectation value of the microscopic polarization 2␲͵Ϫϱ operator32 The induced electric and magnetic dipole moment densities ˆ ˆ can be expanded in a power series in the driving fields. For P͑r͒ϭ Ps͑r͒, ͑13͒ ͚s linear absorption, the first terms Eq. ͑9, 10͒ in these expan- sions are needed. ␣(r,rЈ;␻) denotes the nonlocal electric where ͉␺(t)͘ is the electronic wavefunction. The magnetiza- polarizability tensor, ␬(r,rЈ;␻) the nonlocal magnetizability ˆ tion M(r,t)ϭ͗␺(t)͉M(r)͉␺(t)͘ is similarly defined as the tensor, and the remaining cross response tensors describe the expectation value of the microscopic magnetization effects of the magnetic ͑electric͒ field on the polarization 32 operator ͑magnetization͒. They have the symmetries

ˆ ˆ ␣ij͑r,rЈ;␻͒ϭ␣ji͑rЈ,r;␻͒, ͑17͒ M͑r͒ϭ Ms͑r͒, ͑14͒ ͚s ␬ij͑r,rЈ;␻͒ϭ␬ji͑rЈ,r;␻͒, ͑18͒ and FЌ denotes the transverse part of a vector ﬁeld F, deﬁned by ␤ij͑r,rЈ;␻͒ϭϪ␥ji͑rЈ,r;␻͒, ͑19͒

J. Chem. Phys., Vol. 105, No. 18, 8 November 1996

Downloaded¬07¬Mar¬2001¬to¬128.151.176.185.¬Redistribution¬subject¬to¬AIP¬copyright,¬see¬http://ojps.aip.org/jcpo/jcpcpyrts.html 7998 T. Wagersreiter and S. Mukamel: Optical activity of molecular aggregates where the indices i, j denote Cartesian components. Note with H0ϭnE0, and eyϭezϫex into Eq. ͑22͒, yields the ab- that we have defined these tensors with respect to the exter- sorption cross section for a molecule embedded in a medium nal fields Eext , Hext ͓Eqs. ͑9, 10͔͒. with refractive index n The absorption cross section ␴A of an arbitrary system xx can be calculated by integrating the Poynting vector of the 4␲␻ ␣ ͑r,rЈ,␻͒ ␴͑0͒͑␻͒ϭ Im dr drЈ eϪik•͑r؊rЈ͒ electromagnetic field over a closed surface encompassing it. A nc ͵ ͵ ͫ n For a monochromatic field with frequency ␻ϭ2␲/T we find ͑see Appendix͒ ϩn␬yy͑r,rЈ,␻͒ϩ␥yx͑r,rЈ,␻͒ϩ␤xy͑r,rЈ,␻͒ͬ. P͑r,t͒ץ T 1 1 Ќ ͑23͒ ␴Aϭ dt dr E ͑r,t͒• tץ ͫ ⌽ T ͵0 ͵V in ij Here ␣ ϵei•␣•ej , etc. denote the tensor components. In the M͑r,t͒ץ Ќ long wavelength approximation eϪik•(r؊rЈ)Ϸ1 the cross ϩH ͑r,t͒• , ͑20͒ .t ͬ terms ␤,␥ cancel each other because of the symmetry Eqץ ͑19͒ and we obtain the familiar result38,37 where ⌽inϭ(1/T)͐dt(c/4␲)͉EextϫHext͉ is the time- averaged incident energy flux. The solution of the coupled 4␲␻ ␣tot͑␻͒ Maxwell equations Eqs. ͑15, 16͒ can be written in the form lim␴͑0͒͑␻͒ϭ Im ϩ␬totn , A c ͫ n ͬ k 0 ͑see Appendix͒ → Ќ tot tot E ϭEextϩEЈ, with ␣ (␻)ϭ͐dr͐drЈe1•␣(r,rЈ,␻)•e1 , ␬ (␻) ϭ͐dr͐drЈe2 ␬(r,rЈ,␻) e2. Ќ • • H ϭHextϩHЈ, By specializing to circularly polarized plane waves we can express the CD signal in terms of the nonlocal electric where E , H are the solutions of the inhomogeneous part Ј Ј and magnetic response functions. A right (ϩ) or left (Ϫ) of the wave equations and can be expressed in terms of the circularly polarized plane wave propagating in z direction retarded Greens’ function. Consequently the absorption cross ͓k ϭ (n␻/c)e , k ϭ (2␲/␭)͔ can be represented in the form section can be written as ͑see Appendix͒ z Ϯ ͑0͒ ͑1͒ Eext͑r,t͒ϭE0͓excos͑k•rϪ␻t͒Ϯeysin͑k•rϪ␻t͔͒, ͑24͒ ␴Aϭ␴A ϩ␴A . ͑21͒ Ϯ The first term Hext͑r,t͒ϭH0͓ϯexsin͑k•rϪ␻t͒ϩeycos͑k•rϪ␻t͔͒. ͑25͒ T P͑r,t͒ץ 1 1 0͒͑ ␴A ϭ dt dr Eext͑r,t͒• It is customary to use the separation tץ ͫ ⌽ T ͵0 ͵V in ¯ M͑r,t͒ ␴AϮϭ␴AϮ⌬␴A , ͑26͒ץ ϩHext͑r,t͒• ͑22͒ t ͬ withץ is obtained from Eq. ͑20͒ by replacing the transverse Max- ␴Aϩϩ␴AϪ well field by the external field. ␴(1) which is given in Eq. ¯␴ ϭ , A A 2 ͑A15͒ contains the effects of multiple scattering in the sample. It becomes important when scattering phenomena of ␴AϩϪ␴AϪ large particles such as proteins or DNA strands have to be ⌬␴ ϭ . (0) A 2 accounted for. Hereafter we shall only consider ␴A .To establish the connection with macroscopic observables we Here ¯␴ denotes the average absorption, and ⌬␴ is the CD assume a sample with ␳ noninteracting aggregates per unit A A N signal. When substituting these fields into Eq. ͑22͒ we find volume. The change of intensity I with the penetration depth ͑see Appendix͒ l is given by dIϭϪI␳N␴Adl, resulting in

Ϫ␳N␴Al 4␲␻ …I͑l͒ϭI0e . ¯␴͑0͒͑␻͒ϭ Re dr drЈeϪik–„r؊rЈ A c ͵ ͵ Substituting a linearly polarized incident plane wave propa- ␣xx͑r,rЈ,␻͒ϩ␣yy͑r,rЈ,␻͒ gating in the z direction ϫ ͭ n E0 i͑k•rϪ␻t͒ Ϫi͑k•rϪ␻t͒ xx yy yx Eext͑r,t͒ϭ ex͑e ϩe ͒, ϩ͓␬ ͑r,rЈ,␻͒Ϫ␬ ͑r,rЈ,␻͔͒nϪ͓␤ ͑r,rЈ,␻͒ ͱ2 Ϫ␤xy͑r,rЈ,␻͒Ϫ␥yx͑r,rЈ,␻͒ϩ␥xy͑r,rЈ,␻͔͒ . H0 ͮ i͑k•rϪ␻t͒ Ϫi͑k•rϪ␻t͒ Hext͑r,t͒ϭ ey͑e ϩe ͒, ͱ2 ͑27͒

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4␲␻ P ͑r͒ϭ͗␺ ͉Pˆ͉␺ ͘, ͑30͒ ␴͑0͒͑␻͒ϭ Re dr drЈeϪik–„r؊rЈ… mn m n⌬ A c ͵ ͵ are the matrix elements of the polarization operator Eq. ͑13͒. ␣yx͑r,rЈ,␻͒Ϫ␣xy͑r,rЈ,␻͒ Similarly, the expectation value of the magnetization is given ϫ ͭ n by ϩ͓␬yx͑r,rЈ,␻͒Ϫ␬xy͑r,rЈ,␻͔͒nϪ͓␤xx͑r,rЈ,␻͒ M͑r,␻͒ϭ ␳mn͑␻͒Mmn͑r͒, ͑31͒ ͚mn yy xx yy ϩ␤ ͑r,rЈ,␻͒Ϫ␥ ͑r,rЈ,␻͒Ϫ␥ ͑r,rЈ,␻͔͒ͮ. where the matrix elements of the magnetization operator are defined by ͑28͒ ˆ This exact expression which relates the CD signal of an ar- Mmn͑r͒ϭ͗␺m͉M͉␺n͘. ͑32͒ bitrary molecular system to its nonlocal electric and mag- To calculate the linear optical response we expand the den- netic response tensors ͓defined by Eqs. ͑9, 10͔͒ forms the sity matrix to first order in the driving fields setting basis for our subsequent analysis. Equation ͑28͒ is not re- (1) ␳mn(␻)ϭ¯␳mn(␻)ϩ␦␳mn(␻). Here ¯␳mn is the reduced one stricted to a specific order of the widely used multipolar ex- electron ground state density matrix obtained by iterative pansion. The latter is completely avoided by using nonlocal diagonalization.42 The dynamics of the linear response response tensors, which implicitly include all multipoles. (1) ␦␳mn(␻) in Liouville space will be calculated using the time The problem of the unphysical origin dependence connected dependent Hartree Fock ͑TDHF͒ procedure, resulting in the with the electric quadrupole and higher moments, discussed equation of motion35,36,42,43 e.g., in Refs. 39,40, is naturally resolved with our approach.

As can be seen from Eq. ͑22͒, our result does not depend on ͑1͒ ͑1͒ ͑1͒ ប␻␦␳mn͑␻͒Ϫ Amn,kl␦␳kl ͑␻͒ϩi ⌫mn,kl␦␳kl ͑␻͒ the origin. The scalar product with the external fields ͚kl ͚kl Eext ,Hext which are intrinsically transverse, ensures, that only the transverse part of the polarization P and the magne- ϭFmn͑␻͒. ͑33͒ tization M, respectively enter the absorption cross section, The source F is given by and—as indicated earlier—they do not depend on R. No spe- mn cific model of the system has been assumed yet ͑for example, Fmn͑␻͒ϭ͓¯␳,V͑␻͔͒mn , ͑34͒ these expressions hold whether the molecular states are lo- calized or delocalized͒, and no assumptions were made on with the system-size in comparison with the optical wavelength. All approximations will enter through the different levels of Vmn͑␻͒ϭ͵ dr͓P mn͑r͒•E͑r,␻͒ϩMmn͑r͒•H͑r,␻͔͒, rigor used in modeling the nonlocal response tensors. ͑35͒ In the long wavelength limit k(r؊rЈ)ϭ0 the CD signal vanishes, which follows from Eqs. ͑17–19͒ and interchang- and the matrices A,⌫ are defined in Ref. 43. Terms quadratic ing the integration variables r rЈ. This is to be expected in the magnetic field have been neglected in Eq. ͑33͒,aswe since a finite spatial extent of the↔ system ͑compared with the are interested in linear absorption. Equation ͑33͒ generalizes optical wavelength͒ is necessary in order to observe chirality. the results of Refs. 35, 36, 43 to include the magnetic inter- The leading term in a multipolar expansion is thus obtained action M–H as well as off-diagonal matrix elements P mn in assuming eϪik(r؊rЈ)Ϸ1Ϫik(r؊rЈ) which will lead to the Fmn . The TDHF procedure deals with physically relevant celebrated Tinoco formulae.29,41 quantities ͑charges ␳nn and electronic coherences ␳mn), and a dominant mode picture can be developed which greatly reduces computational cost.44 By using position-dependent III. CALCULATION OF THE NONLOCAL RESPONSE electric and magnetic dipole densities which can be calcu- TENSORS USING THE TIME DEPENDENT HARTREE FOCK EQUATIONS lated from the electronic basis set we avoid the long wavelength approximation and the following expressions for the Consider an N-electron system ͑a molecule or an aggre- polarizabilities hold for arbitrary system size. gate͒ which we describe using single electron basis functions The solution of this TDHF equation can be written in the † ␺m ,mϭ1,...,N. We define cˆ m (cˆ m) as the creation ͑anni- form hilation͒ operators of an electron at site m. We further intro- duce the reduced single electron density matrix ␳ with matrix ͑1͒ ˜ † ␦␳mn͑␻͒ϭ͚ ␣mn,kl͑␻͒Vkl͑␻͒, ͑36͒ elements ␳mnϵ͗cˆmcˆn͘. Since the polarization and magneti- kl zation are single-electron operators, their expectation values can be expanded using the density matrix with the nonlocal response matrix S ͓SϪ1¯␳ ϪSϪ1 ¯␳ ͔ ˜ mn,␯ ␯,tl tk ␯,kt lt P͑r,␻͒ϭ ␳mn͑␻͒P mn͑r͒, ͑29͒ ␣mn,kl͑␻͒ϭ͚ ͚ . ͑37͒ ͚mn ␯ t ␻Ϫ␻␯ϩi␥␯ where Here Smn,␯ diagonalizes the matrix Amn,kl according to

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2 ␻ m ͑r͒m*͑rЈ͒ SϪ1 A S ϭប␻ ␦ , ͑38͒ ␯ ␯ ␯ ͚ ␯,mn mn,kl kl,␯Ј ␯ ␯,␯Ј ␬͑r,rЈ;␻͒ϭ ͚ 2 2 . ͑48͒ mnkl ប ␯ ͑␻ϩi␥␯͒ Ϫ␻␯ and ␥␯ are radiative damping constants. The expectation val- Here the ␯ summation runs over particle-hole oscillators ues of Eqs. ͑29, 31͒ then satisfy Eqs. ͑9, 10͒, with with frequency ␻␯Ͼ0. The advantage of this picture is that typically very few oscillators contribute to the optical response,42,44 leading to a drastic reduction in the number of ␣͑r,rЈ,␻͒ϭ ˜␣mn,kl͑␻͒P mn͑r͒P kl͑rЈ͒, ͑39͒ mn͚,kl terms compared with the (mn) summation in the site representation. Note that this form immediately yields the symmetry Eq. ͑19͒. ␤͑r,rЈ,␻͒ϭ ˜␣mn,kl͑␻͒P mn͑r͒Mkl͑rЈ͒, ͑40͒ mn͚,kl IV. NONLOCAL POLARIZABILITIES OF SPATIALLY EXTENDED AGGREGATES ˜ ␥͑r,rЈ,␻͒ϭ ͚ ␣mn,kl͑␻͒Mmn͑r͒P kl͑rЈ͒, ͑41͒ mn,kl We consider molecular aggregates with electrostatic intermolecular forces ͑no intermolecular charge exchange͒.In ˜ this case it is possible to rigorously express the global non- ␬͑r,rЈ,␻͒ϭ ͚ ␣mn,kl͑␻͒Mmn͑r͒Mkl͑rЈ͒. ͑42͒ mn,kl local response tensors in terms of nonlocal response tensors of the constituent monomers. These expressions greatly re- All response tensors factorize into the discrete nonlocal re- duce computational cost and simplify the theoretical analy- sponse matrix Eq. ͑37͒ and position-dependent tensor prod- sis. Usually molecular aggregates are treated within the LFA ucts of electric and magnetic dipole densities such as ͑Refs. 20, 25, 26, 22͒ which assume the individual molecules P (r)P (r ). In practice these expressions are limited to mn kl Ј as point dipoles. The nonlocal expressions derived here using small systems, as the required memory scales with the fourth the TDHF equations are exact and provide a concrete means power of the basis set size. This problem can be remedied by for modeling aggregates when molecular sizes are compa- the coupled electronic oscillator ͑CEO͒ picture.35,36,42–44 The rable to their separation so that the point dipole approxima- matrix S can be regarded as a transformation matrix from kl,␯ tion is not expected to hold. the site representation to a nonlocal mode representation. Consider an aggregate made out of M molecules labeled Since A is not symmetric, S is in general a complex matrix. a,b,c,... , each described by a basis set of N , N ,... However, one can show that all eigenvalues ␻ are real and a b ␯ orbitals indicated by subscripts a ,b ,... . We have shown come in pairs ␻ , ␻ ϭϪ␻ .43 Moreover, in applications to i j ␯ ¯␯ ␯ earlier,36 that all intermolecular coherences vanish (¯␳ conjugated polymers the S matrix is real.35,36 The transfor- aib j ϭ ␦␳(1) ϭ0) when charge exchange is negligible. Eqs. ͑35, mations of the electric and magnetic transition dipole mo- aib j ment densities to this representation are given by 29, 31, 36͒ lead to

␣͑r,rЈ,␻͒ϭ ˜␣a a ,b b ͑␻͒P a a ͑r͒P b b ͑rЈ͒, ␮␯͑r͒ϭ P mn͑r͒Smn,␯ , ͑43͒ ͚a,b i,͚j,k,l i j k l i j k l ͚mn ͑49͒

m ͑r͒ϭ M ͑r͒S . ͑44͒ ␯ ͚ mn mn,␯ ␤͑r,rЈ,␻͒ϭ ˜␣a a ,b b ͑␻͒P a a ͑r͒Mb b ͑rЈ͒, mn ͚a,b i,͚j,k,l i j k l i j k l ͑50͒ Without loss of generality we can choose the basis functions ␺n to be real. The polarization densities P mn(r) are then ␥͑r,rЈ,␻͒ϭ ˜␣a a ,b b ͑␻͒Ma a ͑r͒P b b ͑rЈ͒, purely real, and the magnetization densities Mmn(r) are ͚a,b i,͚j,k,l i j k l i j k l zero for mϭn, and purely imaginary for m Þ n. Conse- ͑51͒ quently the electric transition dipole moments ␮␯(r) are real and the magnetic transition dipole moments m␯(r) are ␬͑r,rЈ,␻͒ϭ ˜␣ ͑␻͒M ͑r͒M ͑rЈ͒, ͚ ͚ aia j ,bkbl aia j bkbl imaginary. a,b i, j,k,l One can therefore follow similar steps as in Ref. 43, ͑52͒ which finally yields where intermolecular components P ,M were ne- aib j aib j glected. Following similar steps as used in Ref. 36 we first 2 ␻␯␮␯͑r͒␮␯*͑rЈ͒ ␣͑r,rЈ;␻͒ϭ ͚ 2 2 , ͑45͒ derive a closed EOM in the space of molecule a ប ␯ ͑␻ϩi␥␯͒ Ϫ␻␯ ͑1͒ ¯ ͑1͒ ប␻␦␳a a Ϫ Aa a ,a a ␦␳a a 2 ͑␻ϩi␥␯͒␮␯͑r͒m␯*͑rЈ͒ i j ͚kl i j k l k l ␤͑r,rЈ;␻͒ϭ ͚ 2 2 , ͑46͒ ប ␯ ͑␻ϩi␥␯͒ Ϫ␻␯ loc loc ϭ ¯␳a a Va a ϪVa a ¯␳a a , ͑53͒ ͚l i l l j i l l j 2 ͑␻ϩi␥␯͒m␯͑r͒␮␯*͑rЈ͒ ␥͑r,rЈ;␻͒ϭ ͚ 2 2 , ͑47͒ ប ␯ ͑␻ϩi␥␯͒ Ϫ␻␯ where the generalized local potential is defined as

J. Chem. Phys., Vol. 105, No. 18, 8 November 1996

loc V ϭVa b Ϫ2␦a b Ua c ␦␳c c . ͑54͒ ␥͑r,rЈ,␻͒ϭ ˜␣a ,b ͑␻͒ma ͑r͒␮b ͑rЈ͒, aib j i j i j ͚ i l l l ͚ ͚ ␯ ␯Ј ␯ ␯Ј cÞa,l a,b ␯,␯Ј The difference from Ref. 36 is that we now include nondi- ␬͑r,rЈ,␻͒ϭ ˜␣ ͑␻͒m ͑r͒m ͑rЈ͒. ͑62͒ agonal matrix elements Va b . The EOM is modified by the ͚ ͚ a␯ ,b␯Ј a␯ b␯Ј i j a,b ␯,␯Ј ground state charge distribution of the surrounding molecules according to The CEO representation greatly reduces computational ef- fort, as the number of dominant oscillators is generally much ¯A ϭA smaller than the square of the basis set dimensionality. aia j ,akal aia j ,akal Moreover, the matrix ˜␣ provides information regarding a␯ ,b␯Ј ϩ␦ ␦ ͑U ϪU ͒͑2¯␳ Ϫ1͒, ͑55͒ the coupling between electronic oscillators of different mol- ik jl ͚ aicl ajcl clcl cÞa,l ecules. where U denotes the Coulomb interaction between orbit- aicl als ai and cl . The solution of Eq. ͑53͒ can then be expressed V. OPTICAL ACTIVITY IN THE POINT DIPOLE in terms of the molecular response matrices ¯␣ APPROXIMATION aia j ,aras So far we have formulated optical activity in terms of ␦␳͑1͒ ͑␻͒ϭ ¯␣ Vloc . ͑56͒ nonlocal response functions, completely avoiding the multi- aia j ͚ aiaj ,aras aras rs polar expansion. We also paved the way for computational Here ¯␣ is defined in analogy to Eq. ͑37͒, with S replaced by applications by rigorously expressing aggregate response functions in terms of monomer response functions, provided the matrix ¯S which diagonalizes ¯A according to that exchange interactions between the monomers are negli- ¯Ϫ1 ¯ ¯ gible. In order to establish the connection to commonly em- Sa ,a a Aa a ,a a Sa a ,a ϭប␻a ␦␯,␯ . ͑57͒ ͚ijkl ␯ i j i j k l k l ␯Ј ␯ Ј ployed approximations in the investigation of optical activity we now specialize to the lowest order in the multipolar ex- Inserting the definition of the local field Eq. ͑54͒, equating to pansion of the polarization and magnetization densities P , Eq. ͑36͒, and assuming further, that V is negligible for M. This amounts to keeping only the lowest order term in aib j a Þ b yields an equation for ˜␣, whose solution is the␦ functions appearing in Eqs. ͑13, 14͒, resulting in

P mn͑r͒Ϸ␮mn␦͑r͒, ˜␣a a ,b b ͑␻͒ϭ S a a ,b b ͑␻͒¯␣b b ,b b ͑␻͒, ͑58͒ i j k l ͚nm i j n m n m k l Mmn͑r͒Ϸmmn␦͑r͒, with with S Ϫ1 ͑␻͒ϭ␦ ␦ ϩ2͑1Ϫ␦ ͒ aia j ,bkbl aibk a jbl ab * ␮mnϭϪe͵ drЈ␺m͑rЈ͒rЈ␺n͑rЈ͒, ϫ ¯␣a a ,a a Ua b ␦kl . ͑59͒ ͚m i j m m m k * ,mmnϭieប͵ drЈ␺m͑rЈ͒rЈϫٌЈ␺n͑rЈ͒ This exact result ͑provided intermolecular charge exchange and the dipole densities in the CEO ,0؍as well as V are negligible͒, together with Eq. ͑49͒ allows where we have set R aib j the calculation of aggregate response tensors using the representation Eqs. ͑43, 44͒ assume the form monomer response tensors. Since off diagonal ͑intramolecu- ␮␯͑r͒ϭ␮␯␦͑r͒, lar͒ matrix elements P ,M are included, the num- aman aman ber of indices in ␣ and V doubled. m␯͑r͒ϭm␯␦͑r͒, Switching to the CEO representation leads to with

˜␣ ͑␻͒ϭ S ͑␻͒¯␣ ͑␻͒, a␯ ,b␯Ј ͚ a␯ ,b␯Љ b␯Љ,b␯Ј ␮␯ϭ͚ ␮mnSmn,␯ , ␯Љ mn Ϫ1 ¯Ϫ1 Ϫ1 ¯ S ͑␻͒ϭS S ͑␻͒Sb b ,b , ͑60͒ a␯ ,b␯Ј a␯ ,aia j aia j ,bkbl k l ␯ m␯ϭ mmnSmn,␯ . ͚mn where a␯ denotes the ␯-th oscillator of molecule a ͑note that Under these assumptions the nonlocal polarizability Eq. ͑45͒ ai denotes the i-th site in molecule a). The nonlocal response tensors ﬁnally assume the form becomes

␣͑r,rЈ,␻͒ϭ ˜␣a ,b ͑␻͒␮a ͑r͒␮b ͑rЈ͒, ␣͑r,rЈ;␻͒ϭ ␣␯͑␻͒␦͑r͒␦͑rЈ͒, ͑63͒ ͚ ͚ ␯ ␯Ј ␯ ␯Ј ͚␯ a,b ␯,␯Ј where the contribution of the ␯-th oscillator to the polariz- ␤͑r,rЈ,␻͒ϭ ˜␣ ͑␻͒␮ ͑r͒m ͑rЈ͒, ͑61͒ ability tensor ͑labeled by Greek subscripts͒ is ͚ ͚ a␯ ,b␯Ј a␯ b␯Ј a,b ␯,␯Ј J. Chem. Phys., Vol. 105, No. 18, 8 November 1996

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2␻ ␮ ␮ ␯ ␯ ␯ ¯ ¯ ¯ ¯ ¯ ␣␯͑␻͒ϭ 2 2 . ͑64͒ ␥adϭ͚ Yae•Seb͑␬͒• ␦bd␥dϩ͚ ␥b•Tbc•Scd͑␣͒•␣d , ប ͑␻ϩi␥␯͒ Ϫ␻ be ͫ c ͬ ␯ ͑73͒ Analogous expressions hold for the other response tensors ␤,␥, and ␬. ¯ ¯ ¯ ¯ ¯ ␬adϭ͚ Yae•Seb͑␬͒• ␦bd␬dϩ͚ ␥b•Tbc•Scd͑␣͒•␤d . We now consider an aggregate made of N point-like be ͫ c ͬ molecules located at ra ,rb ,... which are characterized by ͑74͒ their linear response tensors ␹¯a ,␹¯b ,... with ␹ϭ␣,␤,␥,␬. The label ␯ in Eq. ͑64͒ is then replaced by the double index The various tensors appearing in these equations are defined a␯ denoting the contribution from the ␯-th oscillator of mol- as follows: ecule a. Intermolecular charge transfer is neglected, so that Ϫ1 S ͑¯␹͒ϭ1␦ Ϫ␹¯ T , ␹ϭ␣,␬, dipole-dipole coupling is the only interaction. The polariza- ab ab a• ab tion and magnetization can then be written as Ϫ1 ¯ Xae ϭ1␦aeϪ Sab͑¯␣͒ ␤b Tbc Scd͑¯␬͒ ␥¯d Tde ͚bcd • • • • • P͑r͒ϭ Pa␦͑rϪra͒, ͚a Ϫ1 ¯ Yae ϭ1␦aeϪ Sab͑¯␬͒ ␥¯b Tbc Scd͑¯␣͒ ␤d Tde , ͚bcd • • • • • M͑r͒ϭ Ma␦͑rϪra͒. ͚a where RabϭraϪrb , eabϭRab /͉Rab͉. By comparing Eqs. The induced electric and magnetic dipole moments at the ͑69, 70͒ with Eqs. ͑9, 10͒ we obtain the nonlocal response a-th molecule can be calculated from the molecular response tensors for this model functions by employing the LFA ,␹͑r.rЈ;␻͒ϭ͚ ␦͑r؊ra͒␦͑rЈϪrb͒␹ab͑␻͒ ¯ loc ¯ loc ab Paϭ␣a•Ea ϩ␤a•Ha , ͑65͒ ¯ loc ¯ loc ␹ϭ␣,␤,␥,␬, Maϭ␬a•Ha ϩ␥a•Ea , ͑66͒ where the local electric and magnetic fields are given by the so that the difference in absorption cross section for right and sum of the external fields and the fields induced by the sur- left circularly polarized light Eq. ͑28͒ becomes rounding molecules yx xy 4␲␻ ␣ab͑␻͒Ϫ␣ab͑␻͒ ͑0͒ Ϫik–Rab ⌬␴A ͑␻͒ϭ Re͚ e loc c ab ͭ n Ea ϭEaϩ͚ Tab•Pb , ͑67͒ b yx xy xx yy ϩ͓␬ab͑␻͒Ϫ␬ab͑␻͔͒nϪ͓␤ab͑␻͒ϩ␤ab͑␻͒

loc Ha ϭHaϩ͚ Tab•Mb . ͑68͒ xx yy b Ϫ␥ab͑␻͒Ϫ␥ab͑␻͔͒ͮ. ͑75͒ Here Tab is given by The response tensors ␹ab ,␹ϭ␣,␤,␥,␬ can be computed by numerical inversion of 3Nϫ3N matrices, where N is the 3eabeabϪ1 T ϭ͑1Ϫ␦ ͒ . basis set dimensionality. However, this expression is simpli- ab ab ͉R ͉3 ab ﬁed by expanding the matrices S,X,Y to ﬁrst order in T and Substitution of Eqs. ͑67, 68͒ into Eqs. ͑65, 66͒ yields a linear neglecting terms of the order ͉m͉2, which results in system of equations for P and M , whose solution is a a ¯ ¯ ¯ ␣abϭ␦ab␣aϩ␣a•Tab•␣b , ͑76͒

Paϭ ͑␣ad Edϩ␤ad Hd͒, ͑69͒ ␤ ϭ␦ ␤¯ ϩ␣¯ T ␤¯ , ͑77͒ ͚d • • ab ab a a• ab• b ¯ ¯ ¯ ␥abϭ␦ab␥aϩ␥a•Tab•␣b , ͑78͒ Maϭ ͑␥ad Edϩ␬ad Hd͒, ͑70͒ ͚d • • ␬abϭ0. ͑79͒

with the nonlocal response matrices Ϫik–r Setting e Ϸ1Ϫik–r, performing orientational averaging ˆ ˆ 1 ij ji ͗(A–k)(B–k)͘kˆϭ 3 A–B, and using ␤abϭϪ␥ba gives ¯ ¯ ¯ ¯ ¯ ␣adϭ͚ Xae•Seb͑␣͒• ␦bd␣dϩ͚ ␤b•Tbc•Scd͑␬͒•␥d , 3 be ͫ c ͬ 32␻ R ␥ ͑0͒ a␯ a␯ ͑71͒ ⌬␴A ϭ ͚ ͚ 2 2 2 2 2 2 , 3nc a ␯ ͓␻ Ϫ␻a␯Ϫ␥a␯͔ ϩ4␥a␯␻a␯ ͑80͒ ¯ ¯ ¯ ¯ ¯ ␤adϭ͚ Xae•Seb͑␣͒• ␦bd␤dϩ͚ ␤b•Tbc•Scd͑␬͒•␬d , be ͫ c ͬ where we have deﬁned the rotational strength Ra␯ for the ͑72͒ ␯-th transition of molecule a

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where P denotes the principal part of the integral. Neglect- Ra␯ϭ Im͑␮a␯ ma␯͒ ing damping (␥ 0) results in the familiar expression for ͚a • ␯ OR in transparent→ regions of the aggregate ␯␯Ј Im͓Vab ͑␻a␮a␯ mb␯ ϩ␻b␮b␯ ma␯͔͒ 2 • Ј Ј• 16␻ Ra␯ Ϫ2 ͚ ͚ 2 2 ϭ . bÞa ␻ Ϫ␻ ␾͑␻͒ ͚ ͚ 2 2 ␯Þ␯Ј a␯ b␯Ј 3nc a ␯ ␻ Ϫ␻a␯ ␯␯Ј 2␲ ␻a␯␻b␯ЈVab ␮a␯ϫ␮b␯Ј•Rab The remaining two terms in Tinoccos’ result ͑see e.g., p. 70 Ϫ ͚ ͚ 2 2 , ͑81͒ in Ref. 29͒ representing interactions with the ground state c bÞa ͓␻ Ϫ␻ ͔ ␯ЈÞ␯ a␯ b␯Ј charge distribution are included implicitly in Eq. ͑81͒,aswe with use molecular response tensors ␹a ,␹ϭ␣,␤,␥,␬ modified by the ground state charge distribution of the neighboring ␯␯Ј molecules.36 They can be retrieved by a perturbative expan- Vab ϭ␮a␯•Tab•␮b␯Ј. ͑82͒ sion of ␹a in the interaction Hamiltonian. The CD-signal vanishes as the damping rates go to zero, and it scales as ␻Ϫ1 for large frequencies. The first term in the rotational strengths, describing the intrinsic CD of the indi- VI. CD SPECTROSCOPY OF NAPHTHALENE vidual molecules stems from the diagonal terms in Eqs. 77, ͑ CLUSTERS 78͒. The second term in Eq. ͑81͒ is due to the mixed terms in Eqs. ͑77, 78͒, and the third term comes from the second term We have applied the present nonlocal formalism to cal- in Eq. ͑76͒. Terms including molecular magnetic transition culate the CD spectrum of naphthalene dimers and trimers. dipole moments are finite even when setting eϪikrϷ1, Comparison with the standard ͑local͒ formulation shows sig- whereas the last term in Eq. ͑81͒ requires eϪikrϷ1Ϫikr. nificant differences, which illustrates the need for a nonlocal This last term can also be identified with Kirkwoods polar- formulation. Small naphthalene clusters have been investi- izability approximation20 gated by studying isotopically mixed crystals45 and by super- sonic molecular beam techniques46 revealing insight into 4␲␻ ¯yi ij ¯jx ¯xi ij ¯jy their geometry. Although CD spectra of these systems are ⌬␴AϷ Im ͚ ͓␣a Tab␣b Ϫ␣a Tab␣b ͔k•Rab , nc ab,ij not available, they were chosen for their simplicity. They also serve as models for more complex conjugated systems 8␲␻ 4–6,9–12 ¯yi ij ¯jx that appear in biological aggregates. ϭ Im ͚ ␣a Tab␣b k•Rab . nc ab,ij We assume circularly polarized light propagating in z-direction and investigate the trimer geometry suggested in The celebrated Tinocco formulas29,41 for the rotation of Fig. 7 of Ref. 46. Using the SPARTAN package for a HF ab the polarization plane of linearly polarized incident light per initio calculation with geometry optimization in an STG6- ¨ unit length ␾ are readily obtained using the Kramers Kronig 11* basis set yields the monomer geometry. The dimer is relation constructed from a monomer in the xz-plane shown in Fig. 1

2 ϱ by a 60° rotation around the z-axis and a translation of 3.7 Å 2␻ ␴A͑␻Ј͒ ␾͑␻͒ϭ P d␻Ј , in the z-direction. The trimer is obtained by adding one more ͵ 2Ϫ 2 ␲ 0 ␻Ј͑␻Ј ␻ ͒ molecule rotated by 120° around the z-axis and translated by 7.4 Å in the z-direction relative to the ﬁrst molecule. Instead of the absorption cross section, we switch to the absorption coefﬁcient via the relation37 4␲␻ ␴ ϭ ⑀, A c so that Eq. ͑26͒ becomes

⑀Ϯϭ¯⑀Ϯ⌬⑀. ͑83͒ In the NLRF calculation we have considered only the polarization term ␣ in Eq. ͑28͒ and neglected the magnetic terms ␤,␥,␬. Neglecting further nondiagonal matrix elements P , setting nϭ1, and substituting Eq. ͑49͒ into Eq. aia j ͑28͒ leads to

͑0͒ x x ¯⑀ ͑␻͒ϭIm ˜␣a a ,b b ͑␻͓͒P a a ͑k͒P b b ͑Ϫk͒ ab͚,ij i i j j i i j j

FIG. 1. Orientation of the monomer in the x – z plane and labeling of the ϩP y ͑k͒P y ͑Ϫk͔͒, atom, used in Fig. 3. aiai bjbj

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͑0͒ ⌬⑀ ͑␻͒ϭϪRe ˜␣a a ,b b ͑␻͒ P a a ͑k͒ ab͚,ij i i j j i i

ϫP ͑Ϫk͒ e , bjbj • k where we have deﬁned the Fourier transform of the polarization densities

P ͑k͒ϭ drЈeϪik–rP ͑r͒. aia j ͵ aia j Assuming point charges ␺2 (r)ϭ␦(rϪr ) results in ai ai k r – ai sin k r 2 – ai P ͑k͒ϭϪer eϪi 2 , aiai ai k r – ai 2 and the matrix ˜␣ is calculated using Eq. ͑58͒. This point- monopole approximation is widely used for the calculation of intermolecular interactions.3,4,25,26 However, here we also keep the full matter-ﬁeld interaction to all orders in the multipolar expansion by using nonlocal response functions, which have a non-trivial r or k dependence. Since Tinoccos’ formulae Eqs. ͑80–82͒͑which are derived using the approximative matrix inversion (1Ϫ␣S)Ϫ1 ¯ Ϸ 1ϩ␣S) are not applicable for the small intermolecular dis- FIG. 2. Absorption spectrum ⑀(␻) of a naphthalene monomer ͑top panel͒, the dimer ͑middle panel͒ and the trimer ͑bottom panel͒. The solid lines show tances of the investigated aggregates, we resort to the more the NLRF calculations, and dashed lines depict the LFA results for the accurate precursor, Eq. ͑75͒ for an oriented sample as a com- dimer and trimer. ␥ϭ0.001 eV, and for the other parameters see the text. parison. Neglecting again magnetic contributions (ma␯ϭ0) results in ␤ϭ␥ϭ␬ϭ0, and Xaeϭ1␦ae , so that sional model Hamiltonian. For the dimer the monomer peak splits into two peaks which are symmetrically positioned LFA Ϫik–Rab xx yy ¯⑀ ͑␻͒ϷIm e ͓␣ab͑␻͒ϩ␣ab͑␻͔͒, ͚ab around the monomer peak. Assuming

EM ab LFA Ϫik–Rab xy yx ⌬⑀ ͑␻͒ϷϪRe e ͓␣ab͑␻͒Ϫ␣ab͑␻͔͒, Htϭ aEM a ͚ab ͩ baEͪ M with for the trimer yields the eigenvalues ␣ ϭS ͑␣͒␣ , ad ad d ␭1ϭEMϩb, ͑84͒ Ϫ1 2 2 Sad ͑␣͒ϭ1␦adϪ␣a•Tad . ϪbϮͱ8a ϩb ␭ ϭE ϩ , ͑85͒ 2,3 M 2 In Fig. 2 we display the average absorption coefficient ¯⑀(␻) ͓see Eq. ͑26͔͒ for the naphthalene monomer ͑top which explains the position of the peaks observed in Fig. 2. panel͒, the dimer ͑middle panel͒, and the trimer ͑bottom The additional peaks in the NLRF reflect the fact that this panel͒ calculated via the NLRF approach ͑solid lines͒. For calculation fully accounts for the intermolecular interactions, the dimer and trimer we also show the LFA results for com- whereas the LFA result retains only the leading dipole-dipole parison ͑dashed lines͒. We only show the region of the low- interaction. est absorption band. Other bands are smaller by a factor of In order to study inter and intramolecular coherences, we Ͼ 20. We further choose a small damping ␥ ϭ 0.001 eV. The depict the nonlocal response matrix ˜␣ at the frequencies of monomer has one absorption peak at 5.58 eV. In case of the the four peaks of the NLRF in Fig. 3. The real part ͑left dimer the exact calculation yields four peaks. Approximating panel͒ is naturally considerable smaller than the imaginary the monomers by point dipoles leads to two absorption part ͑right panel͒ near resonances. Resorting to Fig. 1 for the peaks. Their separation exceeds the absorption range as ob- labeling of the atoms we find that the second and third tran- tained by the NLRF. For the trimer the NLRF calculation sitions predominantly correlate the closest sites of each also yields four peaks, and the LFA results in three. The LFA monomer, whereas the first and fourth transitions correlate results can be understood with a simple two or three dimen- the most distant ones.

J. Chem. Phys., Vol. 105, No. 18, 8 November 1996

FIG. 3. Real ͑left͒ and imaginary ͑right͒ part of the nonlocal response matrix ˜␣(␻) of a naphthalene dimer for the four peaks shown in Fig. 2 from top to bottom (␻ϭ5.37,5.50,5.61,5.72 eV). Atoms 1–10, 11–20, 21–30 correspond to the three monomers ͑see Fig. 1͒. For aggregate geometry see the text.

Figure 4 shows CD of the dimer ͑left panel͒ and the able absorption line shapes for PPV oligomers.35,36 The top trimer ͑right panel͒ calculated in the NLRF ͑solid lines͒ and panels show the absorption coefficient ¯⑀ of the dimer ͑left͒ the LFA ͑dashed lines͒. Each absorption peak leads to a dis- and the trimer ͑right͒, calculated using the NLRF ͑solid͒ and tinct peak in the CD signal, whose amplitude reflects the the LFA ͑dashed͒. The bottom panels display the CD signal rotational strength of the optical transition. Apart from the ⌬⑀(␻). The absorption coefficients for both systems are now magnitude of these peaks, their sign carries important infor- characterized by a single peak with internal structure. In or- mation, reflecting the handedness of the molecule. In Fig. 5 der to show the transitions involved we also plot the results we used a larger damping ␥ϭ0.08 eV which yielded reason- for small damping. In case of the dimer the zero of the CD

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signal coincides with the monomer frequency ␻M . In the LFA the two rotational strengths are equal in magnitude and opposite in sign, so that the signal for ﬁnite damping is sym-

metric with respect to ␻M . For the trimer such a relation does not hold, because of the form of the transition frequencies Eqs. ͑84, 85͒. Due to the small intermolecular separation ͑smaller than the monomer size͒ the LFA gives markedly different results compared to the LFA. However, as shown in Fig. 6 the LFA convergences towards the NLRF results with increasing intermolecular separation. This series clearly demonstrates the breakdown of the LFA, as the intermolecular separation ap- proaches molecular dimensions. The NLRF must be employed for distances smaller than 10 Å. FIG. 4. CD signal ⌬⑀(␻) calculated with the NLRF of the naphthalene dimer ͑left͒ and trimer ͑right͒ for a damping rate of ␥ϭ0.001 eV. Since we used Ohnos’ formula

FIG. 5. Comparison of linear absorption ¯⑀(␻) ͑top panels͒ and CD ⌬⑀(␻) ͑bottom panels͒ calculated with the NLRF ͑solid line͒ and the LFA ͑dashed line͒ of a naphthalene dimer ͑left panels͒ and trimer ͑right panels͒ for a damping rate ␥ϭ0.08 eV. The vertical lines show the small damping results.

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UЈ etc. Moreover we could show easily the independence of the U ϭ ͑86͒ absorption cross section, and hence of the CD-signal of the mn 2 ͱ1ϩ͑rmn /a0͒ choice of origin. with UIЈ ϭ 7.4200 eV, a0 ϭ 1.2935 Å for the electrostatic A considerable reduction of computational cost in the interactions in the PPP Hamiltonian,35 the long distance be- investigation of spatially extended molecular aggregates is haviour is k/r with kϭ9.5977 eV Å. Considering the ap- possible if intermolecular charge exchange is negligible. In -proximation (1؉B)Ϫ1Ϸ1؊B ͑which applies at large separa- this case, intermolecular electronic coherences, i.e., off tions and is necessary to obtain Tinoccos’ formula͒, the CD diagonal elements of the single electron density matrix be- signal is determined to first order by the second term in Eq. tween basis functions located at different monomers, vanish ͑59͒, so that k enters the CD signal as an overall scaling identically. This makes it possible to rigorously express the factor. In order to obtain agreement with the LFA results aggregate response in terms of monomer response tensors. ͑which assume 1/rmn for the Coulomb interaction͒ for large The exact relationship Eq. ͑58͒ derived in this article elimi- intermolecular separation we divided the NLRFs result nates the need for the commonly used local field approxima- by k. tion which completely neglects the molecular internal struc- Naphthalene molecules do not have a permanent dipole ture. The calculation was carried out by deriving molecular moment and the ground state charges are identical for all equations of motion for the reduced one electron density sites (¯␳nnϭ0.5͒. Therefore the modifications of the molecu- matrix in Liouville space, which fully account for intermo- lar response functions by the ground state charge distribution lecular Coulomb interactions. This way we completely of the surrounding molecules cancel ͓see Eq. ͑55͔͒, and the avoided a perturbative calculation of aggregate wavefunc- response matrix ¯␣ coincides with that of isolated molecules. tions. To account for exchange interactions and charge trans- This is not expected to be the case for donor acceptor sub- fer, intermolecular electronic coherences need to be included stituted molecules, in which the ground state charge distribu- in the calculation of the nonlocal response tensors. Note that tion is nonuniform. the nonlocal response tensors ␹(r,rЈ,␻), where r and rЈ are at different monomers imply the existence of excitonic coherences, whereby electron–hole pairs can hop among VII. SUMMARY monomers retaining their phase. Electronic coherences on The conventional approach to the description of optical the other hand imply coherent motion of electrons between activity is based on the multipolar expansion. Circular di- monomers and reflects a phase between an electron at one chroism or optical rotation are calculated term by term for systems much smaller than the optical wavelength. The present theory is based on the concept of nonlocal response tensors and completely avoids these approximations. We first calculated circular dichroism ͑CD͒ defined as the difference in absorption cross section for right and left circularly polarized light. Eq. ͑20͒ contains the polarization P„r…, the magnetization M„r… and the electric and magnetic fields E„r…, H„r…. Utilizing the linearized relationships P„E,H…, M„E,H… Eqs. ͑9, 10͒ lead to our final expression Eq. ͑28͒ which holds for arbitrary wavelengths and system sizes. With this result the theory of optical activity ͑OA͒ is fully captured through four nonlocal response tensors ␣(r,rЈ,␻), ␤(r,rЈ,␻), ␥(r,rЈ,␻), ␬(r,rЈ,␻) which relate polarization and magnetization to the driving external fields. It is formally exact; approximations necessary for practical computations enter in the calculation of the response tensors ␹. We then focused on electronically delocalized conjugated systems described by a Pariser–Parr–Pople Hamil- tonian. Using the time dependent Hartree Fock equations and the multipolar form of the interaction Hamiltonian we derived the expressions Eqs. ͑39–42͒ for the response tensors. They factorize into a central, nonlocal response matrix ˜␣ and dyadics P (r)P (r), P (r)M(r), etc. Once a convenient electronic basis set has been chosen, the polarization and magnetization densities P „r…,M(r) can be calculated without any further approximations. In particular, our final ex- FIG. 6. Convergence of the LFA ͑dashed͒ towards the NLRF ͑solid͒ with increasing intermolecular distance (3.7,5,10 Å from top to bottom͒ for a pression Eq. ͑28͒ implicitly includes all multipolar moments, naphthalene dimer. Shown are the absorption coefficient ¯⑀ ͑left panels͒ and such as electric and magnetic dipole, quadrupole, octopole, the CD signal ⌬⑀ ͑right panels͒.

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Downloaded¬07¬Mar¬2001¬to¬128.151.176.185.¬Redistribution¬subject¬to¬AIP¬copyright,¬see¬http://ojps.aip.org/jcpo/jcpcpyrts.html 8008 T. Wagersreiter and S. Mukamel: Optical activity of molecular aggregates monomer and a hole at another. Electronic coherence is not APPENDIX: THE ABSORPTION CROSS SECTION OF required to attain excitonic coherence which can arise from AN AGGREGATE purely electrostatic interactions. However, the presence of The energy flux density of an electromagnetic wave in a intermolecular charge exchange does affect the excitonic co- dielectric is given by the pointing vector ͑see Sec. 61 in Ref. herence and the optical response.35,36 38͒ We have further formulated the response functions using the coupled electronic oscillator representation ͓see Eqs. c S͑r,␻͒ϭ E͑r,␻͒ϫH͑r,␻͒. ͑A1͒ ͑45–48, 61–62͔͒. As demonstrated in Refs. 42, 44 only very 4␲ few oscillators contribute significantly to the optical response -which leads to a drastic reduction in computational cost, as Integrating ٌS ϭ (c/4␲)(H•ٌϫEϪE•ٌϫH) over a vol can be inferred from the dimension of the matrices ume V with surface O and making use of Maxwells’ equa- ˜␣ and ˜␣ . Moreover, the latter quantities carry tions ͑15, 16͒ leads to aia j ,bkbl a␯ ,b␯Ј information about the coupling between oscillators of differ- 1 d Ќ 2 Ќ 2 ent molecules and thus provide most valuable insight into Ϫ dF •Sϭ dr͓͑H ͒ ϩ͑E ͒ ͔ Ͷ 8␲ dt͵V intermolecular interactions. O MЌץ PЌץ -The well known Tinocco formula Eq. ͑81͒ for the rota tional strengths was recovered by introducing the local field ϩ dr E• ϩH• . ͑A2͒ ͬ tץ tץ ͫ V͵ approximation, the long wavelength approximation, and an approximative matrix inversion (1Ϫ␣T)Ϫ1Ϸ1ϩ␣T to our In this energy balance equation the left-hand side is the de- (0) general result for ⌬␴A (␻). crease of energy per unit time, due to the flux through the Finally we compared our results with those obtained surface O , and the right hand side has been separated into the from Tinoccos’ formulae by application to naphthalene time derivative of the energy of the fields E, H, and a dis- dimers and trimers. The aggregate geometries were taken sipation term due to the induced electric and magnetic di- from Ref. 46, and we first chose a damping rate small poles. We now assume V to be the entire space, and apply enough to reveal the involved electronic transitions. The re- this result to an isolated aggregate. The absorbed energy per sulting spectra then reflect the structure of the oscillator and unit time is then given by the time average of Eq. ͑A2͒. rotational strengths. For the monomer, the average frequency Dividing by the time averaged incident energy flux dependent absorption coefficient ¯⑀ of circularly polarized 1 T c light is characterized by one absorption peak at 5.58 eV ͓see ⌽inϭ dt ͉EextϫHext͉ ͑A3͒ Fig. 2͔. This peak splits up into two ͑three͒ peaks for the T͵0 4␲ dimer ͑trimer͒ when calculated with the LFA, which can be yields the effective absorption cross section ␴ . For mono- understood with a simple two ͑three͒ dimensional exciton A chromatic fields with frequency ␻ϭ2␲/T the time average Hamiltonian. The exact NLRF yields four absorption peaks of the first term on the rhs of Eq. ͑A2͒ vanishes ͑as the fields for the dimer and the trimer, which are not as much separated have the same values at tϭ0 and tϭT), and using as the LFA indicates. Each absorption peak has a corre- ͐drfr gЌ r ϭ͐drfЌ r g r we arrive at sponding peak in the CD spectrum ͑Fig. 4͒. This figure also „ …– „ … „ …– „ … contains the sign information of the rotational strengths, T Mץ P Ќץ Ќ 1 1 which is essential for differentiating between right or left ␴Aϭ dt dr E • ϩH • . ͑A4͒ ͬ tץ tץ ͫ ⌽ T ͵0 ͵V in handed geometries. The increased number of transitions reflects the corrections due to nonlocal interactions between The transverse electric and magnetic Maxwell fields EЌ and spatially extended molecules. We finally demonstrated the HЌ are solutions of the wave equations breakdown of the LFA for intermolecular separations com- ץ ץ 4␲ 2ץ 1 parable to molecular dimensions in Fig. 6. For intermolecu- Ќ Ќ Ќ E Ϫ 2 2 E ϭ 2 P ϩcٌϫM , ͑A5͒⌬ ͬ tץͫ tץ t cץ lar distances smaller than 10 Å the NLRF has to be em- c ץ ץ 4␲ 2ץ ployed. 1 HЌϪ HЌϭϪ cٌϫPϪ MЌ , ͑A6͒⌬ ͬ tץ ͫ tץ t2 c2ץ c2 which follow from the Maxwell Eqs. ͑15, 16͒. These linear equations can be solved using Greens’ functions. The Max- ACKNOWLEDGMENTS well fields are given to lowest order by the external fields Eext(r,t),Hext(r,t), and corrections are due to multiple scat- One of us ͑Th. W.͒ holds an E. Schro¨dinger Stipendium tering of the incoming wave by the charge distribution: ͑Grant No. J-01023-PHY͒ from the Fonds zur Fo¨rderung der Ќ Wissenschaftlichen Forschung, Austria, which is gratefully E ͑r,t͒ϭEext͑r,t͒ϩEЈ͑r,t͒, ͑A7͒ acknowledged. We gratefully acknowledge the support of HЌ͑r,t͒ϭH ͑r,t͒ϩHЈ͑r,t͒. ͑A8͒ the Air Force Office of Scientific Research, the National Sci- ext ence Foundation and cpu time on a C90 granted by Air Force The primed fields can be obtained from Eqs. ͑A5, A6͒ after major shared resource centers ͑MSRC͒. transforming to k,␻ space via

J. Chem. Phys., Vol. 105, No. 18, 8 November 1996

E f ͑k,␻͒ϭ dr dt f͑r,t͒eϪi͑krϪ␻t͒, ͑A9͒ 0 i͑krϪ␻t͒ Ϫi͑krϪ␻t͒ ͵ ͵ Eext͑r,t͒ϭ e1͑e ϩe ͒, ͑A16͒ ͱ2 resulting in H 2 0 i͑krϪ␻t͒ Ϫi͑krϪ␻t͒ Ϫ4␲␻ Hext͑r,t͒ϭ e2͑e ϩe ͒, ͑A17͒ EЈ͑k,␻͒ϭ ͓PЌ͑k,␻͒ϪkˆϫM͑k,␻͔͒, ͑A10͒ ͱ2 ␻2Ϫk2c2 with H0ϭnE0, and e2ϭekϫe1. Upon substitution into Eq. Ϫ4 2 ␲␻ Ќ 20 we ﬁnd ϭ (cE H /4 ), which results in Eq. 23 HЈ͑k,␻͒ϭ ͓kˆϫP͑k,␻͒ϩM ͑k,␻͔͒, ͑A11͒ ͑ ͒ ⌽in 0 0 ␲ ͑ ͒ 2Ϫk2c2 ˆ ez, this assumes the form؍ex and k؍␻ Choosing e1 where kˆϭk/͉k͉. Equations ͑9, 10͒ become 4␲␻ ␴͑0͒͑␻͒ϭ Im dr drЈ eϪik͑r؊rЈ͒ ͑A18͒ A nc ͵ ͵ ˜ 1 P͑k,␻͒ϭ dkЈ͓␣͑k,ϪkЈ;␻͒ Eext͑kЈ,␻͒ ͑2␲͒3 ͵ • 1 ϫ ␣xx͑r,rЈ,␻͒ϩn␬yy͑r,rЈ,␻͒ ͫn ϩ␤͑k,ϪkЈ;␻͒•Hext͑kЈ,␻͔͒, ͑A12͒

1 yx xy M˜ ͑k,␻͒ϭ dkЈ͓␥͑k,ϪkЈ;␻͒ E ͑kЈ,␻͒ ϩ␥ ͑r,rЈ,␻͒ϩ␤ ͑r,rЈ,␻͒ . ͑A19͒ ͑2␲͒3 ͵ • ext ͬ Circularly polarized ﬁelds can be written as ϩ␬͑k,ϪkЈ;␻͒•Hext͑kЈ,␻͔͒, ͑A13͒ E and the transverse vector components in k–space are ob- Ϯ 0 i͑krϪ␻t͒ Ϫi͑krϪ␻t͒ Eext͑r,t͒ϭ ͑eϮe ϩeϯe ͒, ͑A20͒ tained by multiplication with the projector (1Ϫkˆkˆ). Substi- ͱ2 tuting Eqs. ͑A7, A8͒ into Eq. ͑A4͒ yields to the separation H Eq. ͑21͒ Ϯ 0 i͑krϪ␻t͒ Ϫi͑krϪ␻t͒ Hext͑r,t͒ϭϮi ͑eϮe Ϫeϯe ͒, ͑A21͒ ͑0͒ ͑1͒ ͱ2 ␴Aϭ␴A ϩ␴A , ͑A14͒ 32 (0) where the complex unit vectors are deﬁned as where ␴A is given by Eq. ͑22͒, and P͑؊k,t͒ exϪieyץ T 1 1 ͑1͒ eϩϭ , ͑A22͒ ␴A ϭ dt dk EЈ͑k,t͒• t ͱ2ץ ͫ ͵ ⌽ T͵0 in

M͑؊k,t͒ exϩieyץ ϩHЈ͑k,t͒ . ͑A15͒ eϪϭ . ͑A23͒ t ͬ ͱ2ץ •

A linearly polarized plane wave with wavevector k ϭ (n␻/ We ﬁnd again ⌽in ϭ (c/4␲)E0H0 and the difference in ab- c) ek and polarization direction e1 perpendicular to ek can be sorption cross sections for right and left circularly polarized represented by light Eq. ͑22͒ becomes

͑0͒ ͑0͒ ͑0͒ ⌬␴A ͑␻͒ϭ␴AϩϪ␴AϪ ͑A24͒ 8␲␻ 8␲␻ ϭ Im dr drЈ͑e* ␣͑r,rЈ,␻͒ e Ϫe* ␣͑r,rЈ,␻͒ e ͒eϪik„r؊rЈ…ϩ Re dr drЈ nc ͵ ͵ ϩ• • ϩ Ϫ• • Ϫ c ͵ ͵ 8␲␻ *ϫ͑e* ␤͑r,rЈ,␻͒ e ϩe* ␤͑r,rЈ,␻͒ e ͒eϪik„r؊rЈ…Ϫ Re dr drЈ͑e* ␥͑r,rЈ,␻͒ e ϩe ϩ• • ϩ Ϫ• • Ϫ c ͵ ͵ ϩ• • ϩ Ϫ 8␲␻n r,rЈ,␻͒ e ͒eϪik„r؊rЈ…ϩ Im dr drЈ͑e* ␬͑r,rЈ,␻͒ e Ϫe* ␬͑r,rЈ,␻͒ e ͒eϪik„r؊rЈ…. ͑A25͒͑␥ • • Ϫ c ͵ ͵ ϩ• • ϩ Ϫ• • Ϫ

4 Switching back to the real basis vectors ex ,ey we recover Eq. R. W. Woody and I. Tinocco Jr., J. Chem. Phys. 46, 4927 ͑1967͒. ͑28͒. 5 Craig W. Deutsche, J. Chem. Phys. 52, 3703 ͑1969͒. 6 F. M. Loxsom, J. Chem. Phys. 51, 4899 ͑1969͒. 7 V. Devarajan and A. M. Glazer, Acta Crystallogr. A 42, 560 ͑1986͒. 8 1 M. M. Bouman et al., Mol. Cryst. Liquid Cryst. 256, 439 ͑1994͒;M.M. A. M. Glazer and K. Stadnicka, J. Appl. Cryst. 19, 108 ͑1986͒. Bouman and E. W. Meijer, Adv. Mater. 7, 385 ͑1995͒. 9 K. Sauer and L. A. Austin, Biochem. 17, 2011 ͑1978͒; R. J. Cogdell and 2 W. C. Johnson, Jr. and I. Tinocco, Jr., Biopolymers 7, 727 ͑1969͒. H. Scheer, Photochem. Photobiol. 42, 669 ͑1985͒; K. Sauer et al. ͑unpub- 3 C. L. Cech, W. Hug, and I. Tinocco, Jr., Biopolymers 15, 131 ͑1976͒. lished͒.

J. Chem. Phys., Vol. 105, No. 18, 8 November 1996

Downloaded¬07¬Mar¬2001¬to¬128.151.176.185.¬Redistribution¬subject¬to¬AIP¬copyright,¬see¬http://ojps.aip.org/jcpo/jcpcpyrts.html 8010 T. Wagersreiter and S. Mukamel: Optical activity of molecular aggregates

10 A. Scherz et al.,inChlorophylls, edited by H. Scheer ͑Chemical Rubber, 30 L. D. Barron, Molecular Light Scattering and Optical Activity ͑Cambridge Boca Raton, 1991͒, p. 237. University, Cambridge, 1982͒. 11 R. J. Gogdell and H. Scheer, Photochem. Photobiol. 42, 669 ͑1985͒. 31 D. J. Caldwell and H. Eyring, The Theory of Optical Activity, ͑Wiley– 12 K. D. Philipson and K. Sauer, Biochem. 11, 1880 ͑1972͒. Interscience, New York, 1971͒. 13 M. Born, Phys. Z. 16, 251 ͑1915͒; Optik ͑Springer, Berlin, 1930͒. 32 D. P. Craig and T. Thirunamachandran, Molecular Quantum Electrody- 14 C. W. Oseen, Ann. Phys. 48,1͑1915͒. namics ͑Academic, New York, 1984͒. 15 L. Rosenfeld, Z. Phys. 52, 161 ͑1928͒. 33 16 J. K. Jenkins and S. Mukamel, J. Chem. Phys. 98, 7046 ͑1993͒. M. J. Stephen, Proc. Cambridge. Philos. Soc. 54,81͑1958͒. 34 L. Bellegui and S. Mukamel, J. Chem. Phys. 101, 9719 ͑1993͒. 17 F. Gray, Phys. Rev. 7, 472 1916 . ͑ ͒ 35 Th. Wagersreiter and S. Mukamel, J. Chem. Phys. 104, 7086 ͑1996͒. 18 R. de Mallemann, Rev. Gen. Sci. Pure Appliq. 38, 453 ͑1927͒. 36 Th. Wagersreiter and S. Mukamel, Chem. Phys. ͑in press͒. 19 S. F. Boys, Proc. R. Soc. London, Ser. A 144, 655, 675 ͑1934͒. 37 S. Mukamel, Principles of Nonlinear Optical Spectroscopy ͑Oxford Uni- 20 J. G. Kirkwood, J. Chem. Phys. 5, 479 ͑1937͒. versity Press, Oxford, 1995͒. 21 W. Mofﬁtt and A. Moscowitz, J. Chem. Phys. 30, 648 ͑1959͒. 38 Landau Lifshitz, Electrodynamics of Continuous Media Pergamon, Ox- 22 J. Applequist, J. R. Carl, and K. K. Fung, J. Am. Chem. Soc. 94, 2952 ͑ ford, 1960͒, Vol. 8. ͑1971͒. 39 23 J. Applequist, J. Chem. Phys. 58, 4251 1973 . A. D. Buckingham and H. C. Longuet-Higgins, Mol. Phys. 14,63͑1968͒. ͑ ͒ 40 24 J. Applequist, K. R. Sundberg, M. L. Olson, and L. C. Weiss, J. Chem. A. D. Buckingham and M. B. Dunn, J. Chem. Soc. A 1971, 1988. 41 Phys. 70, 1240 ͑1979͒. I. Tinocco Jr., Adv. Chem. Phys. 4, 113 ͑1962͒. 42 25 Howard De Voe, J. Chem. Phys. 41, 393 ͑1964͒. A. Takashashi and S. Mukamel, J. Chem. Phys. 100, 2366 ͑1994͒. 43 26 Howard De Voe, J. Chem. Phys. 43, 3199 ͑1965͒. M. Hartmann, V. Chernyak, and S. Mukamel, Phys. Rev. B. 52, 2528 27 E. U. Condon, Rev. Mod. Phys. 9, 432 ͑1937͒. ͑1995͒. 28 J. A. Schellman, Chem. Rev. 75, 323 ͑1975͒. 44 G. Chen and S. Mukamel, Chem. Phys. Lett. 240, 296 ͑1995͒. 29 Circular Dichroism, Principles and Applications, edited by K. Nakanishi, 45 D. M. Hanson, J. Chem. Phys. 52, 3409 ͑1970͒. N. Berova, and R. W. Woody ͑VCH, New York, 1994͒. 46 J. E. Wessel and J. A. Syage, J. Phys. Chem. 94, 737 ͑1990͒.

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