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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 1996! 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 x(r,r8,v). 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 ~dipole! approximation k–r!1 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. @S0021-9606~96!01541-3# I. INTRODUCTION ing the local field approximation ~LFA! which accounts for intermolecular electrostatic interactions via a local field Circular dichroism ~CD!, 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 ~LA!. 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 ~OA!, i.e., optical rotation In all models discussed since then, the spatial variation ~OR! 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 ' 1 1 ikr 1 {{{ . Usually the series is trun- attributed the origin of optical activity to the spatial variation cated after the second ~dipole! or third ~quadrupole! 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 ~ORP! 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- approximation! 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 rotation! and absorptive ~linear absorption, ally feasible procedure for the investigation of OA which circular dichroism! 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 J. Chem. Phys. 105 (18), 8 November 1996 0021-9606/96/105(18)/7995/16/$10.00 © 1996 American Institute of Physics 7995 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 interaction! can be found in Ref. 28. Finally there are numerous textbooks covering theory and experiment ]H v exhaustively.29–32 mv5a~v!Ev1g~v!Hv2b~v! , ~1! ]t In order to develop a unified microscopic approach to OA which applies to a broad class of molecular systems we ]Ev m 5x~v!H 1g~v!E 2b~v! , ~2! shall employ nonlocal response tensors, which relate the in- v v v ]t duced linear polarization P˜(r,v) and magnetization where a,k are the polarizability and magnetizability of the M˜ (r,v) to the driving electric and magnetic fields molecule and Ev ,Hv are monochromatic fields with fre- Eext(r,v),Hext(r,v) via the linearized relationships quency v. The coefficient g has no effect on OA, and b which is responsible for OA, can be separated into real and ˜ imaginary parts according to P~r,v!5 E dr8 @a~r,r8;v!•Eext~r8,v! b5b 1ib , ~3! 1 2 1b~r,r8;v!•Hext~r8,v!#, ~9! which are related to each other via the Kramers Kro¨nig rela- tions @Eq. ~52! in Ref. 21#. For transparent regions one ob- ˜ tains M~r,v!5 E dr8 @g~r,r8;v!•Eext~r8,v! 4pc R i0 1k~r,r8;v!•Hext~r8,v!#. ~10! b15 ( 2 2 , ~4! 33\ i vi02v In this nonlocal response formulation ~NLRF!, the tensors where the central quantities describing optical activity are x(r,r8;v),x5a,b,g,k 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 u & u & chemical details of different systems enter through their ef- v . 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- R0i5Im~^0umui&•^iumu0&!, ~5! pole, quadrupole, octopole, etc. moments. If so desired, the ˆ ˆ where m, m denote the electric, and magnetic dipole mo- latter can be calculated by integrating Eqs. ~9,10! 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 a(r,r8,v) for investigating the anisotropy of the linear ab- n012 4pv f5N b , ~6! 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 k(r,r8,v) and the factor for the surrounding medium with refraction index n0 cross response tensors b(r,r8,v), g(r,r8,v) 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 f denotes the angle between the approximations applied for their calculation.
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