Coherent Third-Order Spectroscopic Probes of Molecular Chirality

THE JOURNAL OF CHEMICAL PHYSICS 122, 134305 ͑2005͒

Coherent third-order spectroscopic probes of molecular chirality Darius Abramavicius and Shaul Mukamel Chemistry Department, University of California, Irvine, California 92697-2025 ͑Received 9 December 2004; accepted 19 January 2005; published online 5 April 2005͒

The third-order optical response of a system of coupled localized anharmonic vibrations is studied using a Green’s function solution of the nonlinear exciton equations for bosonized excitons, which are treated as interacting quasiparticles. The explicit calculation of two-exciton states is avoided and the scattering of quasiparticles provides the mechanism of optical nonlinearities. To first-order in the optical wave vector we find several rotationally invariant tensor components for isotropic ensembles which are induced by chirality. The nonlocal nonlinear susceptibility tensor is calculated for infinitely large periodic structures in momentum space, where the problem size reduces to the ␣ exciton interaction radius. Applications are made to and 310 helical infinite peptides. © 2005 American Institute of Physics. ͓DOI: 10.1063/1.1869495͔

I. INTRODUCTION static interactions. The peptide group parameters are usually computed by semiempirical techniques.10,14–16 Laser pulses are characterized both by their carrier fre- The infrared OA of proteins shows intense peaks and quencies ␻ and wave vectors k. The optical response of typi- high sensitivity to peptide backbone. Vibrational circular di- cal molecules, which are smaller than the wavelength of chroism ͑VCD͒͑Refs.17–19͒ and Raman optical activity light, can be calculated in the dipole approximation where ͑ROA͒, which involves differential inelastic scattering of the the wave vector only provides an overall phase for the elec- circularly polarized visible radiation,1,20–22 have focused tric field and can be set to zero. The relevant molecular in- mostly on the amide I, CO stretch, ͑1600–1700 cm−1͒ and ͑ formation is then extracted from the frequency spectrum or amide II, CN stretch, ͑1500–1600 cm−1͒ transitions; the ͒ time dependence through the eigenstates and transition ROA has also routinely provided protein spectra down to dipole matrix. Structural information enters only indirectly ϳ700 cm−1.23–27 The CD spectrum of ␣-helical peptides in via the dependence of couplings and frequencies on the the amide I band shows a distinct sigmoidally shaped band at geometry. 1658 cm−1 that has its negative lobe blue shifted with respect The wave vector does play an important role in the spec- to its positive lobe; the 1550 cm−1 amide II band shows a troscopy of extended chiral systems ͑these are “handed” sys- negative VCD band that overlaps with a larger negative band tems which are distinct from their mirror images1,2͒, where at 1520 cm−1.26 The other CH and NH stretching modes the phase of the electric field does vary across the molecule. ͑3000–3300 cm−1͒ give a very strong VCD signal at For this reason certain tensor components of the response 3300 cm−1 with a distorted sigmoidal shape.24 3 helices function, which vanish in the dipole approximation, become 10 show similar spectra.23,24,26 Recent studies of ␣-helix, finite to first order in the wave vector. A common example ␤ ͑ ͒ for such chirally-sensitive, wave vector-induced, effects is 310-helix, -sheet, and poly Pro II helix reveal characteristic 25 ␤ circular dichroism ͑CD͒—the difference absorption of left- VCD bandshapes: -sheet has weak negative amide I VCD ͑ ͒ ␣ and right-handed circularly polarized light which is related to band, while poly Pro II resembles an inverted -helix. VCD ␤ ␤ 27,28 the rotation of polarization vector of the propagating optical of -sheet hairpins is similar to -sheets. Advances in ab field—phenomenon known as optical activity ͑OA͒.1,3–5 initio calculations of vibrational electric and magnetic tran- Due to its high sensitivity to microscopic structure, CD sition dipoles and their interferences lead to highly accurate spectroscopy of electronic transitions of proteins ͑ECD͒ in determination of vibrational absorption and VCD for small 29–32 the visible and UV has become an important tool for struc- molecules. These allow the transfer of property tensors 33 ture determination.5–8 ECD of proteins in the 180–260 nm from smaller to larger systems of coupled vibrations. The range shows mainly two transitions: 190 nm ͑␲−␲*͒ and VCD of dipeptides in the amide I region was calculated re- 220 nm ͑n−␲*͒.6–11 Based on the relative amplitudes of cently and a map of VCD amplitudes as a function of dihe- 34 different transitions in the CD spectrum proteins have been dral angles was created which showed very good agree- ␣ ␤ 19 classified as -rich, -rich, and P2 structures. A well- ment with experiment for a tripeptide. developed theoretical framework relates the CD spectrum to Pattern recognition and decomposition algorithms for peptide structure. The matrix method12,13 is based on a set of protein structure determination allow to distinguish between parameters describing each chromophoric group in the pro- ␣-helical and ␤-sheet formations using ECD ͑Refs. 7, 35, tein: the parameters represent the charge distributions of all and 36͒, VCD ͑Ref. 37͒, and ROA ͑Refs. 38 and 39͒. Differ- relevant ground and excited electronic states, the electric and ent regions associated with particular structures were identi- magnetic transition densities between different states, and the fied and reflect geometrical changes ͑peptide folding͒ as a coordinates of each chromophoric group, coupled by electro- function of external perturbations.22

Downloaded 08 Aug 2005 to 128.200.11.135. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp 134305-2 D. Abramavicius and S. Mukamel J. Chem. Phys. 122, 134305 ͑2005͒

͑3͒͑ ͒ ͵ ͵ ͵ ͵ ͵ ͵ P␯ r4,t4 = ͚ dr3 dr2 dr1 dt3 dt2 dt1 4 ␯ ␯ ␯ 3 2 1 ͑3͒ S␯ ␯ ␯ ␯ ͑r t ,r t ,r t ,r t ͒ 4, 3 2 1 4 4 3 3 2 2 1 1

ϫ E␯ ͑r ,t ͒E␯ ͑r ,t ͒E␯ ͑r ,t ͒, ͑1͒ 3 3 3 2 2 2 1 1 1

where ␯=x,y,z denotes cartesian components. The polarization is measured at time t4 and position r4, while the r integrations extend over the volume of a single molecule V ͑͐ ͐ ͐ ͐ ͒ dr¯ = Vdx Vdy Vdz¯ and time integrations are from −ϱ to +ϱ. These integration limits are assumed throughout this paper unless they are otherwise specified. tj with j=1,2,3 is the interaction time with the electric field E␯ . Note that FIG. 1. ͑Color online͒ Third-order four-wave mixing experiment. The three j these time variables do not have a particular time ordering. incoming pulses with wave vectors k1, k2, and k3 generate a signal at k4 ͑ ͒ =±k1 ±k2 ±k3 a . Interaction events and the definitions of time intervals in The response function depends parametrically on both the Eqs. ͑1͒, ͑C4͒, and ͑F3͒–͑F6͒͑b͒. Quasiparticle picture of the coherent re- coordinates of the system and the interaction times. ͑ ͒͑ ͒ sponse with respect to Eq. 26 c . Three interactions with the optical fields It carries all material properties relevant for the optical ͑red points or peaks͒ create one-exciton particles. They evolve according to their Green’s functions. The two excitons are scattered by ⌫ ͑blue dashed response. areas͒. Finally two excitons annihilate and the remaining exciton generates In general the third-order response function has 34 tensor the signal. elements. However, in the dipole approximation there are only three linearly independent components for isotropic 48 ͑ CD is a good probe of molecular chirality: it has oppo- systems: xxyy, xyxy, and xyyx xxxx is a linear combination of these elements͒. These have been successfully used site signs for two enantiomers and vanishes for racemates— for improving of spectral resolution in two-dimensional IR equal mixtures of two mirror structures.1 CD of small mol- ͑2D IR͒ spectroscopies,49–53 however, they are not chirally- ecules is usually described by including magnetic transition 3,4 sensitive. dipoles. However, large biological molecules such as DNA In this paper we go one step beyond the dipole approxi- ͑ and proteins and chromophore aggregates e.g., light harvest- mation and calculate the first- and the third-order response ͒ ing systems are spatially extended. Wave-vector-dependent function and the susceptibility, to first order in the optical terms, which go beyond the dipole approximation and are wave vector. The finite tensor elements of third-order re- related to quadrupole and higher moments, become signifi- sponse tensors which contain an odd number of x ͑ or y,orz cant as the molecular size is increased. Local magnetic tran- such as xxxy, etc.͒ vanish for nonchiral molecules and are sition dipoles can be ignored for large molecules: the ratio of therefore chirally-sensitive. We use the nonlinear exciton magnetic dipole and electric quadrupole moment is propor- equations ͑NEE͒ originally developed by Spano and tional to v/c, where v is the speed of electrons ͑for electronic Mukamel54–56 for four-wave mixing of coupled transitions͒ or nuclei ͑for vibrational transitions͒ and c is the two-level57–60 and three-level61–63 molecules. The NEE were speed of light.40 ECD for molecular aggregates was calcu- later extended to particles with arbitrary commutation rela- 64 lated to first order in wave vectors41,42 by modeling them as tions and to Wannier excitons in semiconductors. The a collection of coupled electric dipoles. The dipole coordi- equations establish an exciton scattering mechanism for the nates enter explicitly, showing high sensitivity to the geom- nonlinear response and avoid the expensive calculation of etry. This model has been successfully applied to biological multiexciton eigenstates, which is a considerable advantage light harvesting antenna system and recently to a large class for large systems. Closed expressions are derived for infinite of cylindrical aggregates.43 A more general description of periodic systems, where translational symmetry may be em- ␣ CD and optical rotation was developed using spatially non- ployed to reduce the problem size. We apply our theory to helical polypeptide in the amide I region. local electric and magnetic optical response tensors.44 Spe- In Sec. II we present the Hamiltonian and the NEE for cific tensor elements of the linear response function ͑xy and vibrational excitons. In Sec. III we introduce the notation and yx͒, which is a second rank tensor, are responsible for CD. parameters and calculate the linear susceptibility, recovering Nonlinear techniques which utilize circularly polarized opti- the well known expressions of linear absorption and circular cal fields, such as two-dimensional pump-probe were studied 45,46 dichroism. We then derive in Sec. IV expressions for the as well. third-order susceptibility of oriented systems using the same Third-order techniques performed with linearly polar- assumptions and approximations, and extend them to isotro- ized light ͑time and frequency resolved͒ are routinely used to pic ensembles in Sec. V using a universal tensor averaging probe isotropic systems. The experiment is depicted sche- procedure. Simplified expressions for periodic systems are ͑ ͒ matically in Fig. 1. The nonlinear response function S 3 is a derived in Sec. VI. The linear and the nonlinear signals are 47 ␣ fourth rank tensor, which relates the third-order nonlinear calculated for and 310 helices in Sec. VII. We pay particu- polarization P͑3͒ to the optical field E͑r,t͒, lar attention to two-exciton resonance and compare the re-

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Yץ sults within and beyond the dipole approximation. Various mn ͑ ͑Y͒ ͒ E ͑ ͒ − i + ͚ h + Vmn,mЈnЈ YmЈnЈ = m t Bn t mn,mЈnЈץ -aspects of nonlinear spectroscopy beyond the dipole approxi mation are discussed in Sec. VIII. mЈnЈ E ͑ ͒ ͑ ͒ Derivations of the expressions used in the main text are + n t Bm. 5 given in the Appendices. Appendix A introduces the exciton ␦ ␧ ͑ ␦ ͒ basis, the exciton Green’s functions, and the scattering ma- Here hm,n = m,n m +Jm,n 1− m,n is an effective single exci- ͑Y͒ ␦ ␦ trix. The reduced expressions for the scattering matrix for ton Hamiltonian, hmn,mЈnЈ= mЈ,mhn,nЈ+ n,nЈhm,mЈ is a two- special models of nonlinearities are given in Appendix B. exciton Hamiltonian, Vmn,mЈnЈ=Umn,mЈnЈ+Unm,mЈnЈ is the an- E ͑ ͒ ͑ ͒␮ The optical response is derived from the NEE in Appendix harmonicity matrix, and m t =E rm ,t m. The C. General expressions for the susceptibilities in periodic nonlinearities in these equations originate from the anharmo- systems are derived in Appendix D. Reduced scattering ma- nicity matrix. When it is neglected the two-exciton variables ͑ ͒ trix expressions for periodic systems are given in Appendix can be factorized as Ymn=BmBn, Eq. 5 becomes redundant, E, and the time ordered optical response functions required Eq. ͑4͒ becomes linear and the nonlinear response vanishes. for time-domain short pulse experiments are presented in The polarization is given by the expectation value of Eq. ͑3͒ Appendix F. ͑ ͒ ␮ ͑ ͑ ͒ * ͑ ͒͒␦͑ ͒ ͑ ͒ P r,t = ͚ m Bm t + Bm t r − rm . 6 m Additional variables are required in the NEE when popu- II. THE VIBRATIONAL EXCITON HAMILTONIAN AND 64 THE NONLINEAR EXCITON EQUATIONS lation transport and pure dephasing are included. We shall calculate the nonlinear polarization and explore different We consider N coupled anharmonic local vibrational limits of parameters using the Green’s function solution of modes described by the exciton Hamiltonian the NEE given in Appendix A. mÞn ˆ ␧ ˆ † ˆ ˆ † ˆ H = ͚ mBmBm + ͚ Jm,nBmBn m mn III. LINEAR ABSORPTION AND CIRCULAR DICHROISM OF EXCITONS IN THE MOLECULAR ˆ † ˆ † ˆ ˆ ͵ ˆ ͑ ͒ ͑ ͒ FRAME + ͚ Umn,mЈnЈBmBnBmЈBnЈ − drP r · E r,t . mn,mЈnЈ To introduce the notation and set the stage for calculat- ͑2͒ ing the nonlinear response we first review the nonlocal linear ͑1͒ response. The nonlocal linear response function, S␯ ␯ ˆ † ˆ 2, 1 The creation, Bm, and annihilation, Bm, operators for mode m ϫ͑ ͒ r2t2 ;r1t1 , is defined by the following relation between the ͓ ˆ ˆ †͔ ␦ ͑ ͒ satisfy the boson Bm ,Bn = mn commutation relation. The 1 ͑ ͒ linear polarization P␯ r2 ,t2 and the optical electric field ␧ 2 first two terms represent the free-boson Hamiltonian: m de- 40,65,66 E␯ ͑r t ͒: notes the harmonic frequency of mode m and the quadratic 1 1 1 intermode coupling, J , is calculated in the Heitler–London m,n ͑1͒͑ ͒ ͵ ͵ ͑1͒ ͑ ͒ ͑ ͒ P␯ r ,t = ͚ dr dt S␯ ␯ r t ;r t E␯ r t . ˆ † ˆ † ˆ ˆ 2 2 1 1 , 2 2 1 1 1 1 1 approximation where we neglect B B and BmBn terms. The 2 ␯ 2 1 m n 1 third term represents a quartic anharmonicity and the fourth ͑7͒ term is the interaction with the optical field E͑r,t͒ where The linear susceptibility relates these quantities in the ˆ ͑ ͒ ␦͑ ͒␮ ͑ ˆ † ˆ ͒͑͒ P r = ͚ r − rm m Bm + Bm 3 frequency/momentum domain, m 1 is the polarization operator and ␮ is the transition dipole ͑1͒͑ ␻ ͒ ͵ ͵ ␻ m P␯ k2, 2 = 4 ͚ dk1 d 1 ͑␮x ␮y ␮z ͒ 2 ͑2␲͒ ␯ moment, a vector with the components m , m , m , for 1 mode m located at rm. ␹͑1͒ ͑ ␻ ␻ ͒ ͑ ␻ ͒ ͑ ͒ ␯ ␯ − k2 − 2;k1 1 E␯ k1 1 , 8 The expectation value of the polarization operator which 2, 1 1 describes the response of the system to the optical field will where we use the following convention for the Fourier trans- 55,57,64 be calculated using the NEE. This hierarchy of equa- form tions of motion for exciton variables may be exactly trun- cated order by order in the field since the molecular Hamil- F͑k,␻͒ = ͵ dr ͵ dtF͑r,t͒exp͑ikr + i␻t͒. ͑9͒ tonian conserves the number of excitons and the optical field creates or annihilates one exciton at a time. By neglecting The susceptibility and the response function are connected pure dephasing, the only required exciton variables for the by third-order optical response are B =͗Bˆ ͑͘one-exciton͒ and m m ␹͑1͒ ͑ ␻ ␻ ͒ ͗ ˆ ˆ ͑͘ ͒ ␯ ,␯ − k2 − 2;k1 1 Ymn= BmBn two exciton . The NEE then assume the 2 1 form55,56 ͵ ͵ ͵ ͵ ͑1͒ ͑ ͒ = dr2 dr1 dt2 dt1S␯ ,␯ r2t2;r1t1 B V V 2 1ץ − i m + ͚ h B = E ͑t͒ − ͚ V B* Y , ͑4͒ m,n n m mlЈmЈnЈ lЈ mЈnЈ ץ t n ϫ ͑ ␻ ␻ ͒ ͑ ͒ lЈmЈnЈ exp ik2r2 − ik1r1 + i 2t2 − i 1t1 . 10

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Using Eqs. ͑C1͒, ͑7͒, and ͑3͒, the linear response func- ␹͑1͒ ͑ ␻ ␻ ͒ ␯ ,␯ − k2 − 2;k1 1 tion for our model is given by 2 1 ␯ ␯ ␲ ␦͑␻ ␻ ͒ 2͑ ͒ 1*͑ ͒ ͑␻ ͒ Ј ͑ ͒ ͑1͒ ␯ ␯ =2 i 2 − 1 ͚ d␰ k2 d␰ k1 I␰ 1 +c.c. , 16 S␯ ␯ ͑r t ;r t ͒ = i͚ ␦͑r − r ͒␦͑r − r ͒␮ 2␮ 1 ␰ 2, 1 2 2 1 1 2 m 1 n m n mn ϫ ͑ ͒ ͑ ͒ where, the nonlocal exciton transition dipoles are trans- Gmn t2 − t1 +c.c., 11 formed to momentum space where G͑t͒ is the single exciton Green’s function ͓Eq. ͑A4͔͒ ͑ ͒ ␯ ␯ and c.c. denotes the complex conjugate. From Eqs. 10 and ͑ ͒ ikrm␮ ␺ ͑ ͒ d␰ k = ͚ e m ␰m 17 ͑11͒ we obtain the susceptibility m ␹͑1͒ ͑ ␻ ␻ ͒ ␯ ␯ − k2 − 2;k1 1 2, 1 and the frequency domain Green’s function ␲ ␦͑␻ ␻ ͒ =2 i 2 − 1 i ␯ ␯ ͑␻͒ ͑ ͒ ϫ ͑ ͒␮ 2␮ 1 ͑␻ ͒ I␰ = 18 ͚ exp ik2rm − ik1rn m n Gmn 1 ␻ − ⍀␰ + i␥␰ mn +c.c.Ј, ͑12͒ is the Fourier transform of Eq. ͑14͒. The linear response function ͑and the susceptibility͒ de- Ј where c.c. stands for complex conjugate with reversing the scribes all linear properties of an ensemble of oriented mol- → ␻→ ␻ signs of momentum and frequency: k −k and − , and ecules in the lab frame. ͑␻͒ G is the frequency domain Green’s function given by Eq. The linear absorption of the field is given by67,68 ͑ ͒ ␻ ␻ A6 . Time translational invariance implies that 2 = 1. For ͑ ͒ P 1 ͑r,t͒ץ uniform systems with space translational symmetry we have ␴ ͵ ͵ ␯ ͑ ͒ ͑ ͒ = dr dt͚ͩ E␯ r,t ͪ, 19 ץ k2 =k1. A ␺ ͑ ␯ t Using the exciton basis ␰m eigenstates of the single exciton block of the molecular Hamiltonian͒ defined in Eq. Transforming to the momentum/frequency domain we obtain ͑A1͒, Eq. ͑11͒ reads ͑ ͒ ␯ ␯ 1 ͑ ͒ 2͑ ͒ 1*͑ ͒ ͑ ͒ − i ͑ ͒ S␯ ,␯ r2t2;r1t1 = i͚ d␰ r2 d␰ r1 I␰ t2 − t1 +c.c., ␴ ͵ ͵ ␻ ␻ ͑ ␻͒ 1 ͑ ␻͒ ͑ ͒ 2 1 ␰ A = 4 dk d ͚ E␯ − k,− P␯ k, . 20 ͑2␲͒ ␯ ͑13͒ We next consider a monochromatic optical field where the sum now runs over the one-exciton eigenstates ␰ 1 ␯ ␯ ␾ ͑ ͒ ͑ ͒ ͑ 1 2 −i 0͒ ͑ ␻ ͒ ͑ ͒ and I␰ t is the single exciton Green’s function, E r,t = 2 E0 e + ae e exp ik0r − i 0t +c.c. 21

I␰͑t͒ = ␪͑t͒exp͑− i⍀␰t − ␥␰t͒, ͑14͒ Here e␯ is a unit vector along ␯=x,y,z, a is a ratio of two ␯ ␯ ␾ where ⍀␰ and ␥␰ are the frequency and dephasing rate of the perpendicular amplitudes along 1 and 2, 0 is a phase dif- ␰ exciton state, respectively, ␪͑t͒ is the Heaviside step func- ference between them and E0 is the overall amplitude. The ␯1 ␯2 tion ͑␪͑t͒=0 for tϽ0 and ␪͑t͒=1 for tജ0͒ and the transition vectors e ,e ,k0 form a right-handed orthogonal axis sys- dipole is tem. Equation ͑21͒ represents linearly polarized light along ␯ e 1 when a=0; right-handed circularly polarized light when ␯͑ ͒ ␦͑ ͒␮␯ ␺ ͑ ͒ d␰ r = ͚ r − rm ␰m. 15 ␾ ␲ ␾ m a=1 and 0 =+ /2 and left-handed with a=1 and 0 m =−␲/2. Combining Eq. ͑8͒, ͑20͒, and ͑21͒, we obtain for the The susceptibility is similarly given by linear absorption

E2␻ ␴ ͑␻ ͒ 0 0 ͓␹͑1͒ ͑ ␻ ␻ ͒ ␹͑1͒ ͑ ␻ ␻ ͒ 2␹͑1͒ ͑ ␻ ␻ ͒ 2␹͑1͒ ͑ ␻ ␻ ͒ A 0 = i ␯ ␯ − k0, 0;k0,− 0 − ␯ ␯ k0,− 0;− k0, 0 + a ␯ ␯ − k0, 0;k0,− 0 − a ␯ ␯ k0,− 0;− k0, 0 4 1 1 1 1 2 2 2 2 i␾ ͑1͒ −i␾ ͑1͒ −i␾ ͑1͒ + ae 0␹␯ ␯ ͑− k ,␻ ;k ,− ␻ ͒− ae 0␹␯ ␯ ͑k ,− ␻ ;− k ,␻ ͒ + ae 0␹␯ ␯ ͑− k ,␻ ;k ,− ␻ ͒ 1 2 0 0 0 0 1 2 0 0 0 0 2 1 0 0 0 0 i␾ ͑1͒ − ae 0␹␯ ␯ ͑k ,− ␻ ;− k ,␻ ͔͒. ͑22͒ 2 1 0 0 0 0

By setting a=0 we obtain the absorption of linearly polarized light. The circular dichroism spectrum is deﬁned as a difference ͑ ␾ ␲ ͒ ͑ ␾ ␲ ͒ of absorption of left-circularly polarized light a=1, 0 =− /2 and right-circularly polarized light a=1, 0 = /2 ,

E2␻ ␴ ͑␻ ͒ 0 0 ͓ ␹͑1͒ ͑ ␻ ␻ ͒ ␹͑1͒ ͑ ␻ ␻ ͒ ␹͑1͒ ͑ ␻ ␻ ͒ ␹͑1͒ ͑ ␻ ␻ ͔͒ CD 0 =−i i ␯ ␯ − k0, 0;k0,− 0 + i ␯ ␯ k0,− 0;− k0, 0 − i ␯ ␯ − k0, 0;k0,− 0 − i ␯ ␯ k0,− 0;− k0, 0 . 2 1 2 1 2 2 1 2 1 ͑23͒

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The linear absorption ͑circular dichroism͒ depends on ␹͑3͒ ͑ ␻ ␻ ␻ ␻ ͒ ␯ ,␯ ␯ ␯ − k4,− 4;k3, 3,k2, 2,k1, 1 diagonal ͑off diagonal͒ tensor components of the response 4 3 2 1 ␯ ␯ z =2␲i␦͑␻ − ␻ − ␻ − ␻ ͒P function. Setting 1 =x, 2 =y giving k0 =e kz, the absorption 4 3 2 1 ␯k␻ of a linearly polarized light is ␯ ␯ ␯ ␯ ϫ ͚ exp͑ik r − ik r − ik r − ik r ͒M 4 3 2 1 4 n4 3 n3 2 n2 1 n1 n4n3n2n1 n4n3n2n1 E2␻ ␴ ͑␻ ͒ ␲ 0 0 ͚ ␮x ␮x ͓ ͑ z ͒ ͑␻ ͔͒ ϫ ⌫ ͑␻ ␻ ͒ ͑␻ ͒ † ͑ ␻ ͒ A 0 =2 m nRe exp − ikzrmn Gmn 0 ͚ nЈnЈ,nЈnЈ 1 + 2 Gn ,nЈ 4 G − 3 4 4 3 2 1 4 4 nЈ,n3 mn Ј Ј Ј Ј 3 n4n3n2n1 ϫG Ј ͑␻ ͒G Ј ͑␻ ͒ +c.c.Ј, ͑26͒ and the circular dichroism signal n2,n2 2 n1,n1 1 where the frequency domain Green’s functions G͑␻͒ and the E2␻ ␴ ͑␻ ͒ ␲ 0 0 ͓␮ ϫ ␮ ͔ scattering matrix ⌫͑␻͒ are given by Eqs. ͑18͒ and ͑A16͒, CD 0 =2 ͚ m n z ␯ ␯ ␯ ␯ ␯ ␯ ␯ ␯ 2 respectively. M 4 3 2 1 =␮ 4␮ 3␮ 2␮ 1 is the orientational mn n4n3n2n1 n4 n3 n2 n1 tensor and P denotes the sum over the six permutations of ϫ ͓ ͑ z ͒ ͑␻ ͔͒ ␯k␻ Re iexp − ikzrmn Gmn 0 . ␯ ␻ ␯ ␻ ␯ ␻ ␹͑3͒ 1k1 1, 2k2 2, and 3k3 3. Since is already symmetric with respect to 1 and 2 we only need to permute 1 with 3 and These results are exact for coupled dipoles with arbitrary 2 with 3. We therefore need only three permutations. geometry and a fixed orientation with respect to the optical Equation ͑26͒ leads to the following physical picture for field. the response.64 Each of the three optical fields interacts with the system at points n1t1, n2t2, and n3t3 as shown in Fig. 1. The two excitons generated at points n1 and n2 have positive Ј Ј IV. NONLOCAL THIRD-ORDER RESPONSE FUNCTION frequency and evolve independently to n1 and n2 where they Ј Ј IN THE MOLECULAR FRAME are scattered, changing their positions to n3 and n4. The ex- Ј citon at n4 evolves to n4 and generates the signal, while the Four-wave-mixing processes are described by the third- exciton generated at nЈ evolves to n and is annihilated by 40 3 3 order polarization, the third field. In this representation the scattering matrix is the only source for the nonlinear polarization. For harmonic ͑3͒͑ ␻ ͒ oscillators the matrix V, the scattering matrix, and, conse- P␯ k4, 4 4 quently, ␹͑3͒ vanish. The susceptibility depends on the orien- 1 tation of the molecule in the laboratory frame and has a ͵ ͵ ␻ ͵ ͵ ␻ = 12 ͚ dk3 d 3 dk2 d 2 ͑2␲͒ ␯ ␯ ␯ nontrivial dependence on the wave vectors. 3 2 1 In the exciton basis ͓Eq. ͑A1͔͒ we substitute Eq. ͑A4͒ ͑ ͒ ͵ ͵ ␻ ␹͑3͒ ͑ ␻ ␻ ␻ ␻ ͒ into Eq. 26 to obtain dk d ␯ ␯ ␯ ␯ − k − ;k ,k ,k 1 1 4, 3 2 1 4 4 3 3 2 2 1 1 ␹͑3͒ ͑ ␻ ␻ ␻ ␻ ͒ ␯ ␯ ␯ ␯ − k4,− 4;k3, 3,k2, 2,k1, 1 ϫ ͑ ␻ ͒ ͑ ␻ ͒ ͑ ␻ ͒ ͑ ͒ 4, 3 2 1 E␯ k3, 3 E␯ k2, 2 E␯ k1, 1 . 24 3 2 1 =2␲i␦͑␻ − ␻ − ␻ − ␻ ͒P 4 3 2 1 ␯k␻ ␯ ␯ ␯ ␯ The susceptibility ͓Eq. ͑24͔͒ and the response function ͓Eq. ϫ 4͑ ͒ 3͑ ͒ 2*͑ ͒ 1*͑ ͒ ͚ d␰ k4 d␰ − k3 d␰ k2 d␰ k1 ͑ ͔͒ ␰ ␰ ␰ ␰ 4 3 2 1 1 are connected by 4 3 2 1 ϫ⌫ ͑␻ ␻ ͒ ͑␻ ͒ * ͑ ␻ ͒ ␰ ␰ ,␰ ␰ 2 + 1 I␰ 4 I␰ − 3 ͑3͒ 4 3 2 1 4 3 ␹␯ ␯ ␯ ␯ ͑− k − ␻ ;k ␻ ,k ␻ ,k ␻ ͒ 4, 3 2 1 4 4 3 3 2 2 1 1 ϫI␰ ͑␻ ͒I␰ ͑␻ ͒ +c.c.Ј, ͑27͒ 2 2 1 1 = ͵ dr ͵ dt ͵ dr ͵ dt ͵ dr ͵ dt ͵ dr ͵ dt 4 4 3 3 2 2 1 1 where the exciton transition dipoles were defined in Eq. ͑17͒,

͑3͒ the exciton Green’s function in the eigenstate basis is given S␯ ␯ ␯ ␯ ͑r t ,r t ,r t ,r t ͒ 4, 3 2 1 4 4 3 3 2 2 1 1 by Eq. ͑18͒ and the scattering matrix ⌫͑␻͒ is obtained by the 3 Fourier transform of Eq. ͑C6͒. ϫ ͫ ͑ ␻ ͒ ͚ ͑ ␻ ͒ͬ ͑ ͒ exp i k4r4 + 4t4 − i klrl + ltl . 25 l V. THE RESPONSE OF ISOTROPIC ENSEMBLES The response function is directly observed in time domain experiments with ultrashort optical pulses where the time The optical ﬁelds, wave vectors, and space coordinates integration in Eq. ͑1͒ can be eliminated whereas the suscep- are deﬁned in the lab frame, while the transition dipoles and tibility is observed in the opposite CW limit when the ␻ their position vectors are given in the molecular frame. Ro- ͑ ͒ ͗ ͘ integrations in Eq. 24 can be eliminated. tational averaging, ¯ , needs to be performed over the rela- A Green’s function expression for S͑3͒ obtained by solv- tive orientation of the two frames to calculate the response ing the NEE is given in Eq. ͑C4͒. Applying the transform of functions for isotropic ͑randomly oriented͒ ensembles of Eq. ͑25͒ we obtain molecules.48 Equation ͑16͒ then becomes

Downloaded 08 Aug 2005 to 128.200.11.135. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp 134305-6 D. Abramavicius and S. Mukamel J. Chem. Phys. 122, 134305 ͑2005͒

͑ ͒ 1 ␬,␯ ␬ ␯ ␹␯ ␯ ͑− k − ␻ ;k ␻ ͒ =2␲i␦͑␻ − ␻ ͒ ␮ ␺ ͑ ͒ 2, 1 2 2 1 1 2 1 d␰ = ͚ rm m ␰,m. 33 m ␯ ␯ ϫ ͗ 2͑ ͒ 1*͑ ͒͘ ͑␻ ͒ ͚ d␰ k2 d␰ k1 I␰ 1 ␰ Proceeding along the same steps we obtain average third-order response. Rotational averaging of Eq. ͑27͒ gives +c.c.Ј, ͑28͒ ␹͑3͒ ͑ ␻ ␻ ␻ ␻ ͒ ␯ ␯ ␯ ␯ − k4,− 4;k3, 3,k2, 2,k1, 1 with 4, 3 2 1 ␯ ␯ ␯ ␯ ␯ ␯ ␲ ␦͑␻ ␻ ␻ ␻ ͒P ͚ ͗ 4͑ ͒ 3͑ ͒ ͗ 2͑ ͒ 1*͑ ͒͘ ␺ ␺* ͗ ik2rn −ik1rn ␮ 2␮ 1͘ =2 i 4 − 3 − 2 − 1 ␯ ␻ d␰ k4 d␰ − k3 d␰ k2 d␰ k1 = ͚ ␰n ␰n e 2 1 n n , k ␰ ␰ ␰ ␰ 4 3 2 1 2 1 4 3 2 1 n2n1 ␯ ␯ 2* 1* ͑ ͒ ϫd␰ ͑k ͒d␰ ͑k ͒͘⌫␰ ␰ ␰ ␰ ͑␻ + ␻ ͒I␰ ͑␻ ͒ 29 2 2 1 1 4 3, 2 1 2 1 4 4 * ␯ ϫI␰ ͑− ␻ ͒I␰ ͑␻ ͒I␰ ͑␻ ͒ +c.c.Ј, ͑34͒ ki, and the tensor components i are lab frame quantities, 3 3 2 2 1 1 ␮ while the transition dipoles n and coordinates rn are molecular properties defined in the molecular frame. This type with of averaging required for ␮m size macromolecules was cal- ␯ ␯ ␯ ␯ 4 3 2* 1* ͗d␰ ͑k ͒d␰ ͑− k ͒d␰ ͑k ͒d␰ ͑k ͒͘ culated by Craig and Thirunamachandran for linear absorp- 4 4 3 3 2 2 1 1 69 tion and for CD. ␺ ␺ ␺* ␺* = ͚ ␰ n ␰ n ␰ ␰ For an isotropic ensemble of molecules we should treat 4 4 3 3 2n2 1n1 n n n n each molecule as identical system with its collection of tran- 4 3 2 1 ␯ ␯ ␯ ␯ ik r −ik r −ik r −ik r 4 3 2 1 sition dipoles having unique orientation with respect to the ϫ͗e 4 n4 3 n3 2 n2 1 n1␮ ␮ ␮ ␮ ͘, ͑35͒ n4 n3 n2 n1 lab frame. The coordinate of each mode in the lab frame depends on the position of the molecule. Therefore, in gen- where R eral we should replace the coordinates rm with +rm, where ␯ ␯ ␯ ␯ ik r −ik r −ik r −ik r 4 3 2 1 R ͑ ͗e 4 n4 3 n3 2 n2 1 n1␮ ␮ ␮ ␮ ͘ is the molecular position an origin of the molecular co- n4 n3 n2 n1 ͒ ordinate system in the lab frame and rm is the coordinate ␯ ␯ ␯ ␯ ␬ ␬ ␯ ␯ ␯ ␯ ͗␮ 4␮ 3␮ 2␮ 1͘ ͗ ␮ 4␮ 3␮ 2␮ 1͘ = n n n n + i͚ k4 rn n n n n of mode m with respect to that origin. Taking into account 4 3 2 1 ␬ 4 4 3 2 1 the positions of molecules in the ensemble, factors ␬ ␬ ␯ ␯ ␯ ␯ ͑ ͒ R͑ 4 3 2 1 such as exp ik2rn −ik1rn will change to exp(i k2 − i͚ k ͗r ␮ ␮ ␮ ␮ ͘ 2 1 3 n3 n n n n −k ͒)exp͑ik r −ik r ͒, where now exp(iR͑k −k ͒) is an ␬ 4 3 2 1 1 2 n2 1 n1 2 1 ␬ ␬ ␯ ␯ ␯ ␯ overall phase factor which leads to phase-matching condition ͗ ␮ 4␮ 3␮ 2␮ 1͘ − i͚ k2 rn n n n n for the optical fields when integrated over an isotropic bulk ␬ 2 4 3 2 1 sample. The second factor, exp͑ik r −ik r ͒, is now re- 2 n2 1 n1 ␬ ␬ ␯ ␯ ␯ ␯ sponsible for the variation of phase within the molecule. − i͚ k ͗r ␮ 4␮ 3␮ 2␮ 1͘. ͑36͒ 1 n1 n4 n3 n2 n1 Hereafter we keep the phase-matching condition and neglect ␬ the exp(iR͑k −k ͒) factor. The coordinates r vary only 2 1 m In the exciton basis we finally obtain within one molecule and for molecules smaller than the ␯ ␯ ␯ ␯ ␯ ␯ ␯ ␯ wavelength of light we have kr Ӷ1, and the exponential ͗ 4͑ ͒ 3͑ ͒ 2*͑ ͒ 1*͑ ͒͘ ͗ 4 3 2* 1*͘ m d␰ k4 d␰ − k3 d␰ k2 d␰ k1 = d␰ d␰ d␰ d␰ functions in Eq. ͑29͒ can be expanded to first order, 4 3 2 1 4 3 2 1 ␬ ␬ ␯ ␯ ␯ ␯ ␬ ␬ ␯ ␯ ␯ ␯ ͗ , 4 3 2* 1*͘ ͗ , 3 4 2* 1*͘ ␯ ␯ ␯ ␯ ␬ ␬ ␯ ␯ + i͚ k4 d␰ d␰ d␰ d␰ − i͚ k3 d␰ d␰ d␰ d␰ ͗ ik2rn −ik1rn ␮ 2␮ 1͘ ͗␮ 2␮ 1͘ ͗ ␮ 2␮ 1͘ 4 3 2 1 3 4 2 1 e 2 1 n n = n n + i͚ k2 rn n n ␬ ␬ 2 1 2 1 ␬ 2 2 1 ␬ ␬ ␯ ␯ ␯ ␯ , 2* 3 4 1* ␬ ␬ ␯ ␯ − i͚ k ͗d␰ d␰ d␰ d␰ ͘ ͗ ␮ 2␮ 1͘ ͑ ͒ 2 2 3 4 1 − i͚ k1 rn n n , 30 ␬ ␬ 1 2 1 ␬ ␬ ␯ ␯ ␯ ␯ ͗ , 1* 3 2* 4͘ ͑ ͒ − i͚ k1 d␰ d␰ d␰ d␰ , 37 where ␬=x,y,z. This expansion allows to transform vectors ␬ 1 3 2 4 back to the exciton basis. Substituting Eq. ͑30͒ in Eq. ͑29͒ and performing the summations over modes we obtain Eqs. ͑31͒ and ͑37͒ require second to fifth rank rotational averagings. The first terms in these equations correspond to ␯ ␯ ␯ ␯ ␬ ␬ ␯ ␯ ͗ 2͑ ͒ 1*͑ ͒͘ ͗ 2 1*͘ ͗ , 2* 1*͘ d␰ k2 d␰ k1 = d␰ d␰ + i͚ k2 d␰ d␰ the dipole approximation. The remaining terms which con- ␬ tain the transition dipole and a coordinate represent a first- ␬ ␬ ␯ ␯ ͗ , 1* 2*͘ ͑ ͒ order correction to the dipole approximation. These terms do − i͚ k1 d␰ d␰ , 31 ␬ not depend on the coordinate origin of the molecular frame provided k4 =k3 +k2 +k1, which is the phase-matching con- where we had defined the transition dipole vector for the zero dition. momentum exciton state ␮ The projection of a molecular vector such as rm or m ␯ ␯ ␯ onto the lab coordinate system can be represented as a scalar d ϵ d ͑k =0͒ = ͚ ␮ ␺␰ , ͑32͒ ␯ ␯ ␯ ␰ ␰ m ,m product e ·r ͑and e ·␮ ͒, where e is a unit vector in the m m m lab frame. These products have simple transformations be- and the tensor tween coordinate systems, and rotational averages can be

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͑ ͒ ⑀ ⑀ ͑␣ ␣ ␣ ͒ TABLE I. Isotropic rotational average tensors Ref. 48 . ␣ ␣ ␣ is the antisymmetric Levi–Civita tensor: ␣ ␣ ␣ , is equal to 1 for 3 , 2 , 1 ͑ ͒ ͑ ͒ ͑ ͒ ͑␣ ␣ ␣ ͒ ͑ ͒ ͑ ͒ ͑ ͒ 3 2 1 3 2 1 = xyz , yzx , zxy ,-1for 3 , 2 , 1 = xzy , yxz , zyx , and 0 otherwise.

Tensor element Value

͑ ͒ 2 1 T␯ ␯ ,␣ ␣ ␦␯ ␯ ␦␣ ␣ 2 1 2 1 3 2 1 2 1

͑3͒ 1 T␯ ␯ ␯ ␣ ␣ ␣ ⑀␯ ␯ ␯ ⑀␣ ␣ ␣ 3 2 1, 3 2 1 6 3 2 1 3 2 1

T ␦␯ ␯ ␦␯ ␯ ␦␣ ␣ ␦␣ ␣ 4 3 2 1 4 −1 −1 4 3 2 1 1 ͑4͒ ␦␯ ␯ ␦␯ ␯ −1 4 −1 ␦␣ ␣ ␦␣ ␣ T 4 2 3 1 ΂ ΃ 4 2 3 1 ␯ ␯ ␣ ␣ 30΂ ΃ ΂ ΃ 4... 1, 4... 1 ␦␯ ␯ ␦␯ ␯ −1 −1 4 ␦␣ ␣ ␦␣ ␣ 4 1 3 2 4 1 3 2

T ⑀␯ ␯ ␯ ␦␯ ␯ ⑀␣ ␣ ␣ ␦␣ ␣ 5 4 3 2 1 3 −1−1110 5 4 3 2 1 ⑀␯ ␯ ␯ ␦␯ ␯ ⑀␣ ␣ ␣ ␦␣ ␣ 5 4 2 3 1 −1 3−1−1 0 1 5 4 2 3 1 ⑀␯ ␯ ␯ ␦␯ ␯ ⑀␣ ␣ ␣ ␦␣ ␣ 1 5 4 1 3 2 −1 −1 3 0 −1 −1 5 4 1 3 2 ͑5͒ T␯ ...␯ ,␣ ...␣ 30 ⑀␯ ␯ ␯ ␦␯ ␯ 1−103−11 ⑀␣ ␣ ␣ ␦␣ ␣ 5 1 5 1 5 3 2 4 1 ΂ ΃ 5 3 2 4 1 ΂⑀ ␦ ΃ 1 0 −1 −1 3 −1 ΂⑀ ␦ ΃ ␯ ␯ ␯ ␯ ␯ ␣ ␣ ␣ ␣ ␣ 5 3 1 4 2 5 3 1 4 2 01−11−13 ⑀␯ ␯ ␯ ␦␯ ␯ ⑀␣ ␣ ␣ ␦␣ ␣ 5 2 1 4 3 5 2 1 4 3

easily calculated. For an sth rank product of any system vec- of the susceptibilities. The linear susceptibility ͓Eq. ͑28͔͒ in- tors a we have the following transformation1 between the volves second and third rank rotational averages, while the molecular and the lab frames: third-order susceptibility ͓Eq. ͑34͔͒ requires fourth and fifth ␯ ␯ ␯ ␯ rank averages. Within the dipole approximation we only ͗a s a 1͘ϵ͗͑e s · a ͒ ͑e 1 · a ͒͘ s ¯ 1 s ¯ 1 need second and fourth rank rotational averages involving ͑ ͒ ␣ ␣ s s 1 ͑ ͒ products of the transition dipoles. Third and fifth rank rota- = ͚ T␯ ...␯ ,␣ ...␣ as a1 , 38 ␣ ␣ s 1 s 1 ¯ tional averages are required when we go beyond the dipole s¯ 1 ͑s͒ approximation to first order in the wave vector. These aver- where T␯ ␯ ␣ ␣ =͗l␯ ␣ ...l␯ ␣ ͘ is the average of the trans- s... 1, s... 1 s s 1 1 ages explicitly include one translational coordinate and, thus, formation tensor where l␯␣ is the cosine of the angle between contain qualitatively new information about the molecular laboratory frame axis ␯=x,y,z and molecular frame axis ␣ geometry. Based on Table I, we find that different ranks in =x,y,z. The averages of ranks two to five transformation the linear and nonlinear susceptibilities have different sym- tensors, which are universal quantities independent of system metry properties: second rank averages in the linear suscep- 48 geometry are given in Table I. tibility are related to a Kronecker ␦ symbol, while third rank Using Table I, the rotational averages of the transition averages are related to a Levi Civita ⑀ symbol; these aver- dipoles in Eq. ͑31͒ are ages and, thus, different orders of the susceptibility in the wave vector can be measured independently with different ␯ ␯ 1 2 1* 2 ͗d␰ d␰ ͘ = ␦␯ ␯ ͉d␰͉ , ͑39͒ field configurations. The third-order optical response shows 2 1 3 similar trends: the susceptibility in the dipole approximation is related to the fourth rank rotational average, which is pro- ␬ ␯ ␯ 1 ␣ ␣ ␣ , 2 1* 3, 2 1* portional to a product of ␦ functions; to first order in the ͗d␰ d␰ ͘ = ⑀␬␯ ␯ ͚ ⑀␣ ␣ ␣ d␰ d␰ . ͑40͒ 6 2 1␣ ␣ ␣ 3 2 1 3 2 1 wave vector the susceptibility is related to the fifth rank rotational average, which contains ⑀. Thus, different orders in ͑ ͒ For the averages in Eq. 37 we give only the first two terms the wave vector appear in different tensor components: the ␯ ␯ ␯ ␯ ͑ ͒ ␣ ␣ ␣ ␣ ͗ 4 3 2 1͘ 4 4 3 2* 1* dipole approximation gives xxyy and other components with d␰ d␰ d␰ d␰ = ͚ T␯ ␯ ␯ ␯ ,␣ ␣ ␣ ␣ d␰ d␰ d␰ d␰ , 4 3 2 1 ␣ ␣ ␣ ␣ 4 3 2 1 4 3 2 1 4 3 2 1 even number of repeating indices, while xxxy and other odd 4 3 2 1 number components only show up beyond the dipole ap- ͑41͒ proximation. Each order in the wave vector can thus be mea-

␬ ␯ ␯ ␯ ␯ ͑ ͒ sured by a specific configuration of field polarizations. ͗ , 4 3 2* 1*͘ 5 4 d␰ d␰ d␰ d␰ = ͚ T␬␯ ␯ ␯ ␯ ,␣ ␣ ␣ ␣ ␣ In general there are 3 tensor components in the-third 4 3 2 1 ␣ ␣ 4 3 2 1 5 4 3 2 1 5... 1 order susceptibility, with parametric dependence on both the ␣ ␣ ␣ ␣ ␣ 5, 4 3 2* 1* ͉͑ ͉ ␻ ϫd␰ d␰ d␰ d␰ . ͑42͒ wave vector direction and amplitude kj = j /c, where c is 4 3 2 1 the speed of light͒. Not all of these components are indepen- The remaining averages differ only by permutation of dent. Table I gives the number of independent tensor com- indices. ponents for each order: there is one component in the linear Our final expressions, Eqs. ͑28͒ and ͑34͒ together with response ͑related either to ␦ or ⑀ tensor͒ and three ͑within the Eqs. ͑31͒ and ͑31͒, allow to calculate all tensor components dipole approximation͒ and six ͑beyond the dipole approxi-

Downloaded 08 Aug 2005 to 128.200.11.135. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp 134305-8 D. Abramavicius and S. Mukamel J. Chem. Phys. 122, 134305 ͑2005͒

͑1͒ ␹␯ ␯ ͑− k − ␻ ;k ␻ ͒ =2␲iV␦͑␻ − ␻ ͒ 2, 1 2 2 1 1 2 1 ␯ ␯ ϫ ͗ 2, 1͑ ͒͘ ͑␻ ͒ ͚ Ј M␭␭ k2,k1 I␭ 2 ␭ +c.c.Ј, ͑43͒ where the prime now indicates a summation over different Davydov’s subbands ͑whose number is equal to the number of sites in the unit cell͒, V is the volume of the molecule and ␯ ␯ ␯ ␯ ␬ ␬ ␯ ␯ ͗ 2, 1͑ ͒͘ ͗ 2 1͘ ͗ , 2 1͘ FIG. 2. ͑Color online͒͑a͒ Cells of a periodic infinite system are shown by M␭␭ k2,k1 = d␭ d␭ + i͚ k2 d␭ d␭ squares. Each cell has an origin R and contains several modes ͑three in the ␬ picture are chosen as an example͒ located at sites m indicated by vectors ␳ ␬ ␬ ␯ ␯ m ͗ , 2 1͘ ͑ ͒ with respect to the cell origin. Each mode is invariant with respect to trans- − k1 d␭ d␭ . 44 ͑ ͒ lation along r=R1 −R. b Scattering of two excitons in the lattice. Two excitons at sites nЈ and mЈ interact with the anharmonic potential ⌬ ͓Eq. For the nonlinear susceptibility we similarly obtain ͑ ͒ ͔ ͓ ͑ ͒ E10 with r2 =RnЈ−RmЈ and create another two particles at n and m Eq. 3 ␹␯ ␯ ␯ ␯ ͑− k − ␻ ;k ␻ ,k ␻ ,k ␻ ͒ ͑E11͔͒. The scattering is translationally invariant in the lattice since the 4, 3 2 1 4 4 3 3 2 2 1 1 scattering potential depends only on the distance between the cells. Note =2␲iV2␦͑␻ − ␻ − ␻ − ␻ ͒P that momentum before and after scattering is conserved ͓Eq. ͑E7͔͒. 4 3 2 1 ␯k␻ ␯ ␯ ␯ ␯ ϫ ͗ 4 3 2 1 ͑ ͒͘⌫ ͑␻ ␻ ͒ ͚ Ј M␭ ␭ ␭ ␭ k4k3k2k1 ␭ ␭ ,␭ ␭ 2 + 1 ␭ ␭ 4 3 2 1 4 3 2 1 mation͒ components for the third-order response. Additional 4... 1 continuous dependence on the wave vectors constitutes a ϫ ͑␻ ͒ * ͑ ␻ ͒ ͑␻ ͒ ͑␻ ͒ ͑ ͒ I␭ 4 I␭ − 3 I␭ 2 I␭ 1 +c.c.Ј 45 multidimensional parameter space. For collinear optical 4 3 2 1 fields propagating along z, only x and y polarizations are with the orientational tensors ␯ ␯ ␯ ␯ ␯ ␯ ␯ ␯ allowed. This gives three independent tensor elements within ͗ 4 3 2 1 ͑ ͒͘ ͗ 4 3 2 1͘ M␭ ␭ ␭ ␭ k4k3k2k1 = d␭ d␭ d␭ d␭ the dipole approximation, xxyy, xyxy, xyyx ͑xxxx is linearly 4 3 2 1 4 3 2 1 ␬ ␬ ␯ ␯ ␯ ␯ dependent on these three: from Table I we find that ͗xxxx͘ ͗ , 4 3 2 1͘ + i͚ k4 d␭ d␭ d␭ d␭ =͗xxyy͘+͗xyyx͘+͗xyxy͒͘. Corrections to the dipole approxi- ␬ 4 3 2 1 mation induce three new components, xxxy, xxyx, and xyxx. ␬ ␬ ␯ ␯ ␯ ␯ ͗ , 3 4 2 1͘ − i͚ k3 d␭ d␭ d␭ d␭ The rotationally-averaged time-domain expressions ␬ 3 4 2 1 when the sequence of interactions is ordered and can be ␬ ␬ ␯ ␯ ␯ ␯ , 2 3 4 1 − i͚ k ͗d␭ d␭ d␭ d␭ ͘ experimentally controlled using short pulses are given in 2 2 3 4 1 Appendix F. ␬ ␬ ␬ ␯ ␯ ␯ ␯ ͗ , 1 3 2 4͘ ͑ ͒ − i͚ k1 d␭ d␭ d␭ d␭ . 46 ␬ 1 3 2 4

The one-exciton Green’s functions ͓Eq. ͑D4͔͒ and the scat- ͓ ͑ ͔͒ ␬ ␯ VI. RESPONSE OF ISOTROPIC ENSEMBLES OF tering matrix Eq. E12 are taken at q=0. and are PERIODIC STRUCTURES cartesian components of the field wave vector and polarization, respectively, and the rotational averages may be calcu- ͑ ͒ ͑ ͒ We consider a periodic D dimensional lattice made of lated using Eqs. 39 – 42 . ND cells with M sites ͑modes͒ per unit cell, a lattice con- stant a and volume V=͑Na͒D as shown in Fig. 2. The posi- VII. APPLICATION TO HELICAL POLYPEPTIDES ␳ tion of the mth site is given by the vector R+ m, where R is We have calculated the nonlinear susceptibility of the ␳ the origin of the unit cell and m is the displacement from ␣ amide I vibrational mode of and 310 helical polypeptides. that origin. The helical periodic structures were created by repeating the The exciton Bloch states are described by two quantum operations of translation along the helix axis z and rotation of numbers: q is the exciton momentum within the band and ␭ D the xy plane around that z axis by placing each peptide resi- is the Davydov subband. In general there are N values of due for both systems according to Table II. The resulting 2.5 momenta and M Davydov subbands. The Bloch states and nm ͑0.5 nm͒ one-dimensional unit cell has 18 ͑3͒ residues for Green’s functions for a periodic system are given in Appen- ␣ ͑ ͒ 310 helix. The parameters for the transition dipole orien- dix D. Expanding the linear and the third-order susceptibili- tations and for the couplings between neighboring modes ties in eigenstates we obtain general expressions for the lin- were taken from our previous work70 with the diagonal an- ͓ ͑ ͒ ͑ ͔͒ ϵ⌬ −1 ear and the third-order susceptibilities Eqs. D5 and D7 . harmonicity Vmm,mm=2Umm,mm =−16 cm . For small periodic systems ͑when Nӷ1 but L=Na We constructed a momentum space Hamiltonian for the Ӷ͉k͉−1͒ we can use the expansion in wave vectors. Using the infinite systems ͓Eq. ͑D2͔͒ and calculated the one-exciton Bloch eigenstates in Eqs. ͑28͒ and ͑34͒ and performing rota- eigenstates. The susceptibilities were calculated using Eqs. tional averagings ͓Eqs. ͑31͒ and ͑37͔͒ we find that only zero ͑43͒ and ͑45͒ with the rotational averages obtained from Eqs. momenta contribute to the optical response and obtain ͑39͒–͑42͒. The Green’s functions in the exciton basis were

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␣ TABLE II. Parameters of and 310 helices for transition dipoles.

␣ helix 310 helix

Translation distance along z ͑nm͒ 0.138 0.183 Rotation angle around z ͑deg͒ 100 120 Initial transition dipolea ͑Ϫ0.271, Ϫ0.309, 0.912͒͑Ϫ0.453,Ϫ0.325,0.830͒ Initial coordinate of the transition dipole ͑nm͒͑1.76,0,0͒͑1.42,0,0͒ Number of sites in the unit cell 18 3

aThe transition dipole is normalized so that the amplitude is 1. calculated using Eq. ͑D4͒. Uniform line broadening ␥␭͑q͒ mation and three additional elements, xxxy, xxyx, and xyxx, =3 cm−1 was assumed for all one-exciton states. The scatter- beyond that approximation. We calculated the signal ob- ͑ ͒ ͑ ͒ ␻ ␻ ing matrix was calculated using Eqs. E8 – E12 . A grid of served in the direction k1 +k2 −k3 with 3 =− 1. N=100 momenta was used to evaluate the integral in Eq. The tensor components xxyy and xyxy of ␣ helix shown ͑E9͒. The relevant energy levels form three well-separated in Fig. 4 originate from the dipole approximation ͑xyyx is manifolds of states as shown in Fig. 5͑a͒: the ground state, very similar to xxyy and is not shown͒. They show one ͑ ͒ ␻ ␻ −1 the one-exciton, and the two-exciton manifold, The band- strong longitudinal diagonal peak at 1 = 2 =1642 cm ͑ ⌬͒ ͑ ͒ ␻ ␻ −1 ␣ widths of these manifolds determined by J and are much and a weak transverse peak at 1 = 2 =1661 cm in smaller than the energy gaps between them. helix. The weak crosspeaks between these diagonal peaks are The absorption lineshapes of linearly polarized light of best seen in xyxy. both helices calculated using Eq. ͑22͒ are presented in Fig. 3. The chirally-sensitive susceptibilities calculated beyond Both spectra show two peaks resulting from three ͑one lon- the dipole approximation ͑xxxy and xxyx in Fig. 4͒ show a gitudinal and two degenerate transverse͒ transitions.70 The similar pattern. The largest difference is in the crosspeaks: corresponding frequencies are 1642 and 1661 cm−1 for the ␣ they are very asymmetric with respect to the diagonal line −1 helix and 1646 and 1677 cm for the 310 helix. The trans- and one of them has an amplitude comparable to the stron- verse peaks are relatively weaker for the ␣ helix. The CD gest diagonal peak. ͑ ͒ spectrum calculated using Eq. 23 has also a two peak struc- The corresponding 310 spectra are also shown. The lon- ␻ ␻ −1 ture corresponding to the two peaks in linear absorption. gitudinal diagonal peak shows at 1 = 2 =1647 cm and the ␻ ␻ −1 Both peaks have equal amplitudes but different signs: the weaker transverse at 1 = 2 =1677 cm . Since both systems longitudinal peak is positive and the transverse is negative. are right-handed helices the information contained is similar: Linear spectroscopy shows only transitions between the the 310 helix shows larger separation between the peaks and ground state and the one-exciton states. Qualitatively new stronger crosspeaks compared to the ␣ helix. The crosspeaks ͑␻ ␻ ͒ information is contained in third-order spectroscopy. We are symmetric with respect to diagonal 1 = 2 in the dipole have calculated the CW signal for a collinear conﬁguration approximation. The corresponding crosspeaks have different assuming that all ﬁelds propagate along z. Taking into ac- amplitudes beyond that approximation. count the dispersion relation between wave vector ampli- Our simulated peak positions correlate well with tudes and the frequencies, the susceptibility becomes a func- experiments26,71–73 and previous calculations.74–76 They ͑␻ ␻ ␻ ͒ tion of three variables 1 , 2 , 3 with three independent show that the crosspeaks carry information about exciton tensor elements, xxyy, xyyx, and xyxy, in the dipole approxi- interactions and distances between them: new terms such as xxxy give an asymmetric crosspeak pattern, with comparable amplitudes to the diagonal peaks. This signal also has a complicated dependence on coordinates and should vanish for achiral systems. However, only one-exciton resonances are seen in this signal. The four one-exciton Green’s functions suppress the two exciton resonances contained in the scattering matrix ͓see Eq. ͑45͔͒. We next consider techniques that reveal two-exciton resonances. We start with the following 2D k1 +k2 −k3 signal:

W␯ ␯ ␯ ␯ ͑␻ ,␻͒ 4 3 2 1 3

ϵ ͵ ␻Ј␹͑3͒ ͑␻ ␻ ␻ ␻ ␻Ј ␻ ␻Ј͒͑ ͒ d ␯ ␯ ␯ ␯ − ;− , /2 + , /2 − 47 4, 3 2 1 3 3

FIG. 3. ͑Color online͒ Linear absorption ͑top͒ and circular dichroism ͑bot- ͑ ͒ ͒ ␣ ͑ ͒ ͑ ͒ with the optical frequencies tuned according to Fig. 5 b tom spectra of the inﬁnite black solid and 310 red dashed helices in the ␻ ␻ amide I region ͑spectra are normalized so that the largest peaks are equally where both /2 and 3 are tuned to the one-exciton reso- strong͒. nances ͑␻ also covers two-exciton resonances͒, while ␻Ј is

Downloaded 08 Aug 2005 to 128.200.11.135. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp 134305-10 D. Abramavicius and S. Mukamel J. Chem. Phys. 122, 134305 ͑2005͒

␹ ͑ ␻ ␻ ␻ ␻ ͒ ␣ ͑ ͒ ͑ ͒ ␻ FIG. 4. Third-order susceptibility tensor, ␯ ,␯ ␯ ␯ − 2 ;− 1 , 2 , 1 ,of helix two left columns and of 310 helix two right columns as a function of 1 ␻ 4 3 2 1 ␹ ␹ and 2. Both real and imaginary parts are shown. Two tensor elements xxyy and xyxy are calculated in the dipole approximation. The chirally sensitive tensor ␹ ␹ elements xxxy and xxyx vanish in the dipole approximation.

much smaller than the one-exciton energy as shown in Fig. 5͑b͒. The resonant terms in the susceptibility for ⌫ ͑␻͒ this conﬁguration are of the form ␭ ␭ ,␭ ␭ * 4 3 2 1 ϫI␭ ͑␻−␻ ͒I␭ ͑−␻ ͒I␭ ͑␻/2+␻Ј͒I␭ ͑␻/2−␻Ј͒. The signal 4 3 3 2 1 3 ␻ ␻ has the frequency 3 − and in the dipole approximation becomes

͑ ͒ ␯ ␯ ␯ ␯ dip ͑␻ ␻͒ ϰ ͗ 4 3 2 1͘ ͑␻ ␻ ͒ W␯ ␯ ␯ ␯ 3, i ͚ Ј d␭ d␭ d␭ d␭ I␭ − 3 4 3 2 1 ␭ ␭ 4 3 2 1 4 4... 1 * ϫI␭ ͑␻ ͒⌫␭ ␭ ␭ ␭ ͑␻͒I␭ ␭ ͑␻͒. ͑48͒ 3 3 4 3, 2 1 2 1

This signal involves the product ⌫␭ ␭ ␭ ␭ ͑␻͒I␭ ␭ ͑␻͒ which 4 3, 2 1 2 1 is the two-exciton Green’s function in the exciton basis ͓see Eq. ͑A15͔͒, showing two-exciton resonances along the ␻ axis. By integrating the signal over ␻ FIG. 5. ͑a͒ Energy level scheme of coupled amide I vibrations. The ground, 3 one-exciton, and two-exciton manifolds are shown for the amide I band. The band widths of the one-exciton and two-exciton manifolds are roughly 50 −1 −1 ͑␻͒ϵ͵ ␻ ͑␻ ␻͒ ͑ ͒ W␯ ␯ ␯ ␯ d W␯ ␯ ␯ ␯ , , 49 and 100 cm , respectively. The energy gaps between them are 1600 cm . 4 3 2 1 3 4 3 2 1 3 Two experimental conﬁgurations are shown: ͑b͒ all ﬁelds are resonant with ͑ ͒ ␻ ␻ ␻ ␻ ␻ excitonic transitions; c 1 and 2 are nonresonant, but = 1 + 2 is resonant. we obtain in the dipole approximation

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͑␻ ␻͓͒ ͑ ͔͒ ␣ ͑ ͒ ͑ ͒ ͑ ͒ FIG. 6. Third-order signal W␯ ␯ ␯ ␯ 3 , Eq. 47 of helix two left columns and of 310 helix two right columns . Wxxxx dipole approximation and ͑ ͒ 4 3 2 1 Wxxxy chirally sensitive are shown.

͑dip͒ ͑␻͒ϵ͵ ␻ ͑dip͒ ͑␻ ␻͒ ͑dip͒ ͑␻͒ϵ͵ ␻ ͑dip͒ ͑␻ ␻͒ W␯ ␯ ␯ ␯ d W␯ ␯ ␯ ␯ , U␯ ␯ ␯ ␯ d U␯ ␯ ␯ ␯ , 4 3 2 1 3 4 3 2 1 3 4 3 2 1 3 4 3 2 1 3

␯ ␯ ␯ ␯ ϰ Ј ͗ 4 3 2 1͘⌫ ͑␻͒I ͑␻͒ 1 i ͚ d␭ d␭ d␭ d␭ ␭ ␭ ,␭ ␭ ␭ ␭ , ϰ Ј ͗ ͘⌫ ͑␻͒ ␭ ␭ 4 3 2 1 4 3 2 1 2 1 ͚ d␭ d␭ d␭ d␭ ␭ ␭ ,␭ ␭ . 4... 1 ␻ − ⍀ ␭ ␭ 4 3 2 1 4 3 2 1 1 4... 1 ͑ ͒ 50 ͑53͒

This signal is proportional to the scattering matrix and is which only shows two-exciton resonances. scaled by ͑␻ −⍀͒−1, which is an average off-resonant detun- A different experiment can be performed by detuning ␻ 1 1 ing. Thus, both the W and U signals carry information on and ␻ off resonance ͑␻ −⍀Ͼ␥͒, but ␻ +␻ =␻Ϸ2⍀ cov- 2 1 2 1 two-exciton resonances: the former shows the two-exciton ers the two-exciton band ͑here ⍀ is the average one-exciton ␥ ͓͒ Greens function, the latter shows the two-exciton scattering excitation energy and is the average dephasing see Fig. matrix. The W signal is resonant and should be stronger than 5͑c͔͒. Again the contribution to this signal comes from terms ⌫ ͑␻͒ ͑␻ ␻ ͒ * ͑␻ ͒ ͑␻ ␻ ͒ ͑␻ ͒ U. such as ␭ ␭ ,␭ ␭ I␭ − 3 I␭ 3 I␭ − 1 I␭ 1 , 4 3 2 1 4 3 2 1 Going beyond the dipole approximation, the rotational where ␻ is fixed. Taking into account that I␭ ͑␻ ͒ is con- 1 1 1 factors depend on the wave vectors and, thus, optical fre- stant and I␭ ͑␻−␻ ͒ is not resonant ͉͑␻−␻ −⍀͉ ӷ␥͒ we 2 1 1 quencies and the integrations do not lead to such simple define the signal expressions. However, the resonances should still reflect the two-exciton states, with different amplitudes and lineshapes. W␯ ␯ ␯ ␯ ͑␻ ,␻͒ displayed in Fig. 6 shows two-exciton 4 3 2 1 3 U␯ ␯ ␯ ␯ ͑␻ ,␻͒ϵ͑␻ − ␻ − ⍀͒ 4 3 2 1 3 1 mixed with one-exciton resonances. The chirally-sensitive ϫ␹͑3͒ ͑␻ ␻ ␻ ␻ ␻ ␻ ͒ components show peaks at the same positions as in the di- ␯ ␯ ␯ ␯ 3 − ,− 3, − 1, 1 , 4, 3 2 1 pole approximation, but different relative amplitudes. This is ͑51͒ a consequence of the fact that the molecular Hamiltonian is not affected by the dipole approximation, thus, the resonances remain the same. However, peak amplitudes, con- ͑␻ ␻ ⍀͒ where the factor − 1 − is included to approximately trolled by rotational factors are different. ͑␻ ␻ ͒ compensate the asymmetry of by I␭ − 1 . By integration W␯ ␯ ␯ ␯ ͑␻͒ for the helices displayed in Fig. 7 show ␻ 2 4 3 2 1 over 3 we have three two-exciton resonances which are better resolved in the ␣ 310 helix compared to the helix. The amplitudes of two- exciton resonances show dramatic differences when going beyond the dipole approximation: the central resonance be- ͑␻͒ϵ͵ ␻ ͑␻ ␻͒ ͑ ͒ U␯ ␯ ␯ ␯ d 3U␯ ␯ ␯ ␯ 3, . 52 4 3 2 1 4 3 2 1 comes stronger. This effect is more pronounced for 310 helix compared to ␣ helix. The U␯ ␯ ␯ ␯ ͑␻ ,␻͒ signals displayed in Fig. 8 show 4 3 2 1 3 In the dipole approximation we obtain one-exciton resonances superimposed with two-exciton reso-

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͑ ͒ ͑␻͓͒ ͑ ͔͒ ␣ ͑ ͒ ͑ ͒ FIG. 7. Color online Third-order signal W␯ ␯ ␯ ␯ Eq. 49 of helix left and of 310 helix right . Four tensor elements are shown: Wxxxx, Wxxyy, Wxyyx ͑ ͒ 4 3 2 1͑ ͒ all three in the dipole approximation , and Wxxxy chirally sensitive . Solid-real, dashed-imaginary, dotted-absolute value. nances. These signals decay very slowly with off-resonance ments of the linear susceptibility. CD spectroscopy is related detuning. Thus, measuring the susceptibility over a broad to the off-diagonal tensor elements, which vanish for isotro- frequency range is required to get this signal, as is clearly pic systems in the electric dipole approximation. The CD seen for U displayed in Fig. 9. These ﬁgures demonstrate the signal is induced by the magnetic dipole or higher electric scattering matrix weighted by the orientational factors and multipoles. The magnetic transition dipoles may be neglected show a complicated picture of two exciton resonances: one for extended systems e.g., macromolecules, as will be shown part of the signal ͑real in the dipole approximation and below. ͒ imaginary beyond it has a background plateau, whose mag- We considered a model systems of coupled electric di- nitude is related to the anharmonicity. This can be easily poles distributed in space. The calculations done for helices shown by considering the scattering matrix of a single anhar- can be repeated to other secondary structure motifs such as ␤ monic vibrational mode. The different distribution of the am- sheets and random coils, and to electronic aggregates. Going ␣ plitudes in the and the 310 helices is much more clearly beyond the dipole approximation, which is equivalent to in- seen in these two-exciton related signals compared with the cluding the positions of the dipoles, one-exciton spectra shown in Fig. 4. is required for CD. The nonlocal susceptibilities were expanded to ﬁrst order in the wave vectors: the expansion of VIII. DISCUSSION molecular property functions in wave vector accounts for The linear absorption is described by the electric dipole higher multipoles. approximation which is related to the diagonal tensor ele- The wave vector expansion requires additional condi-

FIG. 8. Third-order signal U␯ ␯ ␯ ␯ ͑␻ ,␻͓͒Eq. ͑51͔͒ of ␣ helix ͑two left columns͒ and of 3 helix ͑two right columns͒. U ͑dipole approximation͒ U 4 3 2 1 3 10 xxxx xxxy ͑ ͒ ␻ −1 ⍀ −1 chirally sensitive are shown. 1 =1800 cm and =1650 cm .

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͑ ͒ ͑␻͓͒ ͑ ͔͒ ␣ ͑ ͒ ͑ ͒ FIG. 9. Color online Third-order signal U␯ ␯ ␯ ␯ Eq. 52 of helix left and of 310 helix right . Four tensor elements are shown: Uxxxx, Uxxyy, Uxyyx ͑ ͒ 4 3 2 1͑ ͒ all three in the dipole approximation , and Uxxxy chirally sensitive . Solid-real; dashed-imaginary. Parameters are the same as in Fig. 8

Ӷ tions. The system size L is limited from above by k0L 1, approach is valid when the coherence size of excitation is where k0 is an optical wave vector. For larger systems L larger than two residues, which is reasonable at ambient tem- must be replaced by the exciton coherence size.77 The limit, peratures. Several hundreds of residues are required to reach thus, holds even for large molecular systems including pro- the 100 nm length of the ␣ helical structure, which validates teins and electronic aggregates. The system size is also lim- the infinite size assumption. Even when the physical size of ited from below since the magnetic transition dipoles may the system is larger than the wavelength, the exciton coher- not be neglected for small systems. The expansion ence size is typically smaller than the wavelength; this limit exp ͑ikr͒Ϸ1+ikr suggests that the amplitude of the terms is always satisfied for vibrational transitions of polypeptides. linear in k are L͉k͉. The local magnetic transition dipoles are Second-order techniques for probing molecular chirality proportional to the factor v/c,40 where v is the speed of are commonly used for isotropic systems. Second harmonic electron for electronic transitions or the speed of nuclei for generation ͑SHG͒ is not allowed in the dipole approximation vibrational transitions and c is the speed of light. Thus, L by symmetry.79,80 For this reason most studies of quadratic Ͼv/␻; ␻ is the optical frequency. This ratio can be esti- nonlinearities were conducted on macroscopically noncen- mated by assuming classical energy: mv2 =2ប␻, where m is trosymmetric systems.81,82 However, sum frequency genera- the mass of particle and ប␻ is its energy. We can then ex- tion ͑SFG͒ of isotropic systems in a noncollinear configura- press this limit as LϾͱ2ប/m␻. For instance, taking values tion is allowed and is used as a probe of molecular of hydrogen atom with excitation energy corresponding to chirality.83–86 Symmetry of isotropic systems allows one spe- 3000 cm−1 we obtain LϾ0.1 nm for vibrational transitions. cific, xyz, component of the second-order susceptibility ten- Thus, our approach holds for delocalized excitons with co- sor to be finite in the dipole approximation. Noncollinear herence size spanning several units. configuration leads to breaking of phase-matching condition Periodicity reduces the problem size considerably: the and, thus, to weak signals.84,85,87 Even weaker SHG signal necessary sums reduce from the total number of dipoles to was observed, showing the significance of magnetic transi- the number of modes in a unit cell ͑for linear J aggregates78 tion dipoles and of electronic quadrupoles. However this it is just one site͒. We have derived the response function and SHG signal originating from terms beyond the dipole ap- the susceptibility for infinite systems. This seems to conflict proximation is not sensitive to molecular chirality. with the upper bound of the system size. However, the sys- We next consider the symmetry properties of rotational tem can be assumed infinite as long as the number of cells is averages shown in Table III with respect to different spec- very large, Nӷ1, and edge effects can be ignored. We per- troscopies and chirality. Only one independent diagonal ten- formed calculations on amide I transitions of infinite helical sor component, xx, of the linear response survives the isotro- peptide. Since the distance between residues is ϳ0.5 nm, our pic rotational averaging in the dipole approximation; all off

TABLE III. Symmetry properties of rotational averages with respect to chirality tensor.

Response Form of terms in isotropic averaging Independent nonzero elements Probe of chirality?

1st ͑dipole͒ ␦␯ ␯ xx n 2 1 1st ͑ikr related͒ ⑀␬␯ ␯ ͑z͒yx y 2 1 2nd ͑dipole͒ ⑀␯ ␯ ␯ xyz y 3 2 1 2nd ͑ikr related͒ ␦␬␯ ␦␯ ␯ ͑x͒xyy, ͑x͒yyx, ͑x͒yxy n 3 2 1 3rd ͑dipole͒ ␦␯ ␯ ␦␯ ␯ xxyy, xyyx, xyxy n 4 3 2 1 3rd ͑ikr related͒ ␦␯ ␯ ⑀␬␯ ␯ ͑z͒xxxy, ͑z͒xxyx , ͑z͒xyxx, ͑z͒zxyz, ͑z͒zxzy, ͑z͒zzxy y 4 3 2 1

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diagonal components ͑and consequently the CD signal͒ van- NEE, which are based on a standard exciton Hamiltonian in ish. Calculating the linear response to first order in wave the Heitler–London approximation. The parameters of this vector, requires a third rank rotational averaging involving Hamiltonian are excitation energies of different modes, inter- the positions of the dipoles. The tensor is then proportional mode couplings ͑quadratic͒, and anharmonicities ͑quartic 48 to ⑀␬␯ ␯ —a Levi–Civita permutation symbol. Thus, one couplings͒, as well as excitation transition dipole and its co- 2 1 independent, xyz, component is nonzero, relating the propa- ordinate. These parameters may be obtained from ab initio 74,88,89 gation direction ͑for instance, z͒ and polarization ͑x and y͒ of calculations performed on small peptide segments. the optical field and system polarization. They can also be readily obtained from experiment. The ex- The number of scalar products to be averaged is inti- citation energies and intermode couplings are observed in mately related to the chirality of the system. An achiral sys- linear absorption and CD. We assumed only local anharmo- tem can be superimposed on itself after spatial inversion par- nicities, which give a shift of one double excitation energy ity operation,1 thus, the rotational average needs to be the from twice the single excitation energy on the same mode. same for the original and the inverted systems. Odd rank This can be obtained from pump-probe measurements. Off- rotational averages which change sign upon inversion must diagonal anharmonicities may also be important when inter- 60 therefore vanish. A chiral system, in contrast, is converted to mode coupling is weak. its mirror immage by parity operation and the two cannot be The experimental techniques which show two-exciton superimposed, thus, odd rank rotational averages are nonzero resonances are most promising since they carry qualitatively and carry opposite signs for enantiomers. Therefore only chi- new information about the system. They also show very ral systems survive odd rank rotational averagings. Even strong dependence on the structure of the peptide backbone. rank rotational averages are not sensitive to spatial inversion The tensor components induced by deviations from the di- and, therefore, to chirality. Thus, the tensors involving odd pole approximation show strong differences between the two rank rotational averages are chirally-sensitive. helices. Compared to CD, different exciton states are probed, Third rank rotational averaging is responsible for the the signal originates from the anharmonicity and multiple second-order response in the dipole approximation. The iso- transition dipoles with the coordinates define the signal. The tropic average is then proportional to the Levi–Civita tensor frequency integrated susceptibilities used in our definitions ͓ ͑ ͒ ͑ ͔͒ leading to xyz tensor element of second-order susceptibility. of the U and W signals Eqs. 47 and 51 could be ob- This element is not accessible by the collinear configuration, tained in time-domain experiments using short optical pulses required for phase matching. Similar to CD, this rotational which have a broad spectral bandwidth. average survives only for isotropic systems with chiral molecules. Going one step beyond the dipole approximation we obtain a chiraly insensitive signal coming from fourth rank ACKNOWLEDGMENTS rotational averaging.79 This material is based upon work supported by the Na- The third-order nonlinear response in the dipole approxi- tional Science Foundation Grant No. CHE-0132571. The mation has three independent nonzero rotational averages, support of the National Institutes of Health Grant No. 1 RO1 xxyy, xyyx and xyxy; the signal is not chirally sensitive. Go- GM59230-10A2 was gratefully acknowledged. ing beyond the dipole approximation leads to the fifth rank orientational averaging which imposes dependence of the signal on the field wave vectors and includes the positions of the dipoles. Fifth rank averaging gives six independent non- APPENDIX A: EXCITON BASIS SET, GREEN’S FUNCTIONS, AND THE SCATTERING MATRIX zero tensor elements: zxxxy, zxxyx, zxyxx, zzxyz, zzxzy, and zzzxy which participate in noncollinear configurations. These ⍀ ␺ The eigenenergies ␰ and eigenvectors ␰m of the one- elements now carry information about chirality for the same exciton block of the Hamiltonian ͓Eq. ͑2͔͒, ͗0Bˆ ͉Hˆ ͉Bˆ†0͘ reasons as xyz in circular dichroism. m n ϵh =␦ ␧ +J ͑1−␦ ͒, define the one-exciton basis, Phase-matching in the third-order response defines the m,n m,n m m,n m,n ␺ ⍀ ␺ ͑ ͒ signal propagation direction. Unlike SFG, the collinear con- ͚ hm,n ␰n = ␰ ␰m. A1 figuration, which gives the strongest signal is allowed in the n third-order response and has three independent tensor ele- ͑ ͒ ͑ ͒ ͑ ͒ ͑ The evolution of a single exciton following an impulsive ments: z xxxy, z xxyx, and z xyxx fields propagate along excitation is described by the one-exciton Green’s function z͒. The frequency permutation requirement in the suscepti- ͑ ͒ ␻ ␻ ␻ ͑ G t , bility implies that they all vanish when 1 = 2 = 3 i.e., ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ third harmonic generation . Noncollinear configuration Bm t = ͚ Gm,mЈ t BmЈ 0 , A2 ͑which can also satisfy phase matching͒ leads to six nonzero mЈ rotational averages where all components participate in the which satisfies the equation response. Thus, the third-order response beyond the dipole approximation together with CD constitute the best probes of dG ͑t͒ m,n + i͚ h G ͑t͒ = ␦͑t͒. ͑A3͒ chirality and carry more information than conventional linear m,nЈ nЈ,n dt Ј and third-order response within the dipole approximation and n SFG. This equation can be solved using the exciton eigenvalues ␺ We have derived expressions for the response using the ␰m. The one-exciton Green’s function is then given by

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͑ ͒ ͚ ␺ ͑ ͒␺* ͑ ͒ Gm,n t = ␰mI␰ t ␰n, A4 I ͑␻͒ϵ͵ ͑ ͒ ͑ ͒ ͑ ␻ ͒ ␰ ␰␰Ј dtI␰ t I␰Ј t exp i t where i = . ͑A13͒ ␻ − ⍀␰ − ⍀␰Ј + i͑␥␰ + ␥␰Ј͒ I␰͑t͒ = ␪͑t͒exp͑− i⍀␰t − ␥␰t͒͑A5͒ By noting that G͑␻͒=i͑␻−h͒−1 and GY͑␻͒=i͑␻−h−V͒−1, ␥ is a Green’s function in the eigenstate basis, ␰ is a dephas- where h and V are tetradic matrices, ⌫͑␻͒ can be obtained ␪͑ ͒ ͓␪͑ ͒ Ͻ ing rate, and t is the step function t =0 for t 0 and ˆ −1 ˆ −1 ˆ −1͑ ˆ ˆ ͒ ˆ −1 ˆ ␪͑ ͒ ജ ͔ using the operator identity A =B +B B−A A , where B t =1 for t 0 which guarantees causality. ˆ Applying the Fourier transform ͓Eq. ͑9͔͒ to Eqs. ͑A4͒ and A are any two operators. Applying this to the Green’s ˆ ˆ and ͑A5͒ we obtain the frequency domain Green’s function, functions with A=͑1/i͒͑␻−h−V͒ and B=͑1/i͒͑␻−h͒ we get the Dyson equation ͑␻͒ ␺ ͑␻͒␺* ͑ ͒ Gm,n = ͚ ␰mI␰ ␰n A6 Y Y ␰ G ͑␻͒ = G͑␻͒ + G͑␻͒͑− iV͒G ͑␻͒. ͑A14͒ with Comparing Eqs. ͑A11͒ and ͑A14͒ gives

Y i − iVG ͑␻͒ = ⌫͑␻͒G͑␻͒. ͑A15͒ I␰͑␻͒ = . ͑A7͒ ␻ − ⍀␰ + i␥␰ Iterating Eq. ͑A14͒ and using Eq. ͑A15͒ finally gives ͑ ͒ The two-exciton evolution Ymn t is similarly described Y ⌫͑␻͒ ͑ ͒2 G͑␻͒ ͑ ͒3 G͑␻͒ G͑␻͒ by the two-exciton Green’s function G =−iV + − i V V + − i V V V + ¯ ϵ − iV 1+iG͑␻͒V −1. ͑A16͒ ͑ ͒ GY ͑ ͒ ͑ ͒ ͑ ͒ „ … Ymn t = ͚ mn,mЈnЈ t YmЈnЈ 0 , A8 mЈnЈ Thus, the calculation of the scattering matrix requires the inversion of the matrix D=1+iG͑␻͒V with matrix elements, which satisfies the equation D ͑␻͒ = ␦ ␦ + i ͚ G ͑␻͒V . ͑A17͒ dGY mn,ij mi nj mn,mЈnЈ mЈnЈ,ij mn,mЈnЈ ͑ ͑Y͒ ͒GY ␦͑ ͒ mЈnЈ + i ͚ hmn,mЉnЉ + Vmn,mЉnЉ mЉnЉ,mЈnЈ = t . dt Љ Љ m n In general this is a ͑N2 ϫN2͒ matrix, where N is the number ͑A9͒ of modes. Computing D is equivalent to finding all two- exciton states and is costly even for moderate N. However, ͑ ͒ The zero-order noninteracting V=0 two-exciton Green’s in practice the size of the scattering matrix can be reduced G function can be factorized into a product of one-exciton considerably for typical forms of the matrix V as shown in G ͑ ͒ ͑ ͒ ͑ ͒ Green’s functions, mn,mЈnЈ t =Gm,mЈ t Gn,nЈ t . The actual Appendix B. This is the main advantage of this method. Green’s function, GY, is connected to G by the Bethe Salpeter equation

t tЈ APPENDIX B: THE SCATTERING MATRIX FOR GY͑ ͒ G͑ ͒ ͵ Ј͵ G͑ Ј͒⌫͑ Ј ͒G͑ ͒ t = t + dt dt1 t − t t − t1 t1 , SPECIAL CASES 0 0 ͑A10͒ We consider the following form for the anharmonicity: ͑⌬ ͒͑␦ ␦ ␦ ␦ ͒ Umn,mЈnЈ= m,n /4 mmЈ nnЈ+ mnЈ nmЈ so that where we have introduced the two exciton scattering matrix ⌬ ⌫͑ ͒ m,n tЈ−t1 . The two-exciton Green’s function and the scatter- ˆ ˆ ˆ † ˆ † ˆ ˆ ͑ ͒ HS = H0 + ͚ BmBnBmBn, B1 ing matrix are tetradic matrices. The scattering matrix is mn 2 ␪͑ ͒ causal and contains a tЈ−t1 factor. ⌬ ⌬ ⌬ In the frequency domain the Bethe Salpeter equation with m,n = n,m. Intramode anharmonicities, m,m, represent ͑ ͒ ͑ ˆ † ˆ † ͉ ͒͘ A10 reads the shift of the overtone BmBm 0 energy with respect to 2␧ . Intermode anharmonicities, ⌬ with mÞn, shift the Y m m,n G ͑␻͒ = G͑␻͒ + G͑␻͒⌫͑␻͒G͑␻͒, ͑A11͒ ͑ ˆ † ˆ †͉ ͒͘ ␧ ␧ combination band BmBn 0 energies from m + n. Equation ͑A16͒ then gives for the scattering matrix where the noninteracting two-exciton Green’s function G is given by ⌫ ͑␻͒ ⌬ −1͑␻͒ ͑ ͒ mn,ij =−i m,n„D …mn,ij. B2 G ͑␻͒ ␺ ␺ I ͑␻͒␺* ␺* ͑ ͒ We assume that the anharmonic potential ⌬ is nonzero mn,mЈnЈ = ͚ ␰m ␰Јn ␰␰Ј ␰ ␰ , A12 m,n mЈ ЈnЈ ͉ ͉ Ͻ ␰␰Ј only for short distances m−n lc, where lc is an interaction ⌬ length of the anharmonicity. The presence of i,j in Eq. ͑ ͒ ⌬ ͑ ͒ and A17 and m,n in Eq. B2 implies that i− j and m−n are

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limited by the interaction length l . Therefore, the effective ϱ ϱ ϱ ϱ ϱ c ͑3͒͑ ͒ ͵ ␶Љ͵ ␶Ј͵ ͵ ͵ matrix size of the scattering matrix reduces to Nl ϫNl .We Bn t4 =2i d d dt3 dt2 dt1 c c 4 ϱ ϱ ϱ ϱ ϱ may then deﬁne a reduced matrix D, − − − − − ␪͑ ͒⌫ ͑␶Љ ␶Ј͒ ͑␻͒ϵ ͑␻͒ ␦ ␦ ͚ ͚ t2 − t1 nЈnЈ,nЈnЈ − Dml ,il Dm,m+l ;i,i+l = mi l l 4 3 2 1 1 2 1 2 1 2 n1n2n3 Ј Ј Ј Ј n1n2n3n4 + iG ͑␻͒⌬ , ͑B3͒ † m,m+l1;i,i+l2 i,i+l2 ϫ ͑ ␶Љ͒ ͑␶Љ ͒ Gn ,nЈ t4 − G Ј − t3 4 4 n3,n3 where the indices l1 and l2 take the values from interval ϫ ͑␶ ͒ ͑␶ ͒ GnЈ,n Ј − t2 GnЈ,n Ј − t1 ͓ ͔ 2 2 1 1 −lc ,...,lc . The complete scattering matrix can now be re- cast in terms of reduced matrix D, ϫE ͑t ͒E ͑t ͒E ͑t ͒, ͑C3͒ n3 3 n2 2 n1 1 ⌫ ͑␻͒ϵ⌫ ͑␻͒ ⌬ ͑ −1͒ m,m+l ;i,i+l m,l ;i,l =−i m,m+l D ml ,il . 1 2 1 2 1 1 2 where we used Eq. ͑A15͒ to change ͑B4͒ ͚ GY ͑␶Љ ͒ ͚ ͐ϱ ␶Ј⌫ −i nЈnЈ VnЈnЈ,nЈnЈ Ј Ј −t2 into nЈnЈ −ϱd nЈnЈ,nЈnЈ 2 1 4 3 2 1 n2n1,n2j 2 1 4 3 2 1 ⌫ ϫ͑␶Љ−␶Ј͒G Ј Ј ͑␶Ј−t ͒, then we factorized G Ј Ј ͑␶Ј All other elements of vanish. n2n1,n2j 2 n2n1,n2j As a special case we consider a local anharmonicity −t ͒ϵG Ј ͑␶Ј−t ͒G Ј ͑␶Ј−t ͒ and performed a summation 2 n2,n2 2 n1,j 2 where we set ⌬ =⌬ ␦ ͑soft–core boson over index j: ͚ G ͑␶Ј−t ͒G ͑t −t ͒=G ͑␶Ј−t ͒␪͑␶Ј m,n m m,n j nЈ,j 2 j,n1 2 1 nЈ,n1 1 ͒ 57 ͒␪͑ ͒ 1 ␶ ␶ 1 approximation . Then −t2 t2 −t1 . The variables Ј and Љ denote the times of the ͑␻͒ ␦ G ͑␻͒⌬ ͑ ͒ ﬁrst and the last exciton-exciton interaction, respectively, as Dm0,i0 = mi + i mm,ii i. B5 shown in Fig. 1͑b͒. and the scattering matrix We use third-order NEE variables and their Green’s functions to calculate the response function. Using Eqs. ͑6͒, ⌫ ͑␻͒ ⌬ ͑ −1͒ ͑ ͒ ͑ ͒ ͑ ͒ mm,nn =−i m D m0,n0. B6 1 , and C3 , we obtain the response function

Thus the excitons interact and scatter only when they occupy ͑3͒ S␯ ␯ ␯ ␯ ͑r t ;r t ,r t ,r t ͒ NϫN 4, 3 2 1 4 4 3 3 2 2 1 1 the same site. The interaction radius is lc =0 and matrix D has the size as of the one-exciton basis. = iP ͚ ␦͑r − r ͒␦͑r − r ͒␦͑r − r ͒␦͑r − r ͒ For large anharmonicities each mode becomes effec- ␯rt 4 n4 3 n3 2 n2 1 n1 n4n3n2n1 tively a two level system and the two excitons then cannot ϱ ϱ ͑ ͒ ␯ ␯ ␯ ␯ reside on the same site hard-core bosons . This can be de- ϫM 4 3 2 1 ͚ ͵ d␶Љ͵ d␶Ј⌫ Ј Ј Ј Ј͑␶Љ − ␶Ј͒ 57,90 n4n3n2n1 n4n3,n2n1 scribed by taking the limit ⌬→ϱ leading to Ј Ј Ј Ј −ϱ −ϱ n4n3n2n1 ⌫ ͑␻͒ =− G͑␻͒ −1 . ͑B7͒ ϫ ͑ ␶Љ͒ † ͑␶Љ ͒ ͑␶Ј ͒ mm,nn „ …mm,nn Gn ,nЈ t4 − G Ј − t3 GnЈ,n − t2 4 4 n3,n3 2 2 ϫG Ј ͑␶Ј − t ͒ +c.c., ͑C4͒ n1,n1 1

␯ ␯ ␯ ␯ ␯ ␯ ␯ ␯ APPENDIX C: TIME DOMAIN GREEN’S FUNCTION where M 4 3 2 1 =␮ 4␮ 3␮ 2␮ 1, c.c. denotes the complex EXPRESSIONS FOR THE THIRD-ORDER n4n3n2n1 n4 n3 n2 n1 conjugate, and P denotes permutation of interaction OPTICAL RESPONSE ␯rt ␯ ͑ ͒ events, which are defined by jrjtj with j=1,2,3: 3,2,1 The NEE can be solved by order-by-order expansion of +͑2,3,1͒+͑1,2,3͒. This makes the response function sym- the variables in the field using the exciton Green’s functions metric with respect to this permutation ͑there are only three ͑Appendix A͒ where the optical field and the lower-order terms since the function is inherently symmetric with respect variables serve as the sources. The first-order variable, to the permutation of 1 and 2͒. ͑ ͒ B 1 ͑t͒, is obtained from Eq. ͑4͒, The response function in the exciton basis is obtained by m ͑ ͒ ͑ ͒ ϱ plugging Eq. A4 into Eq. C4 , ͑ ͒ B 1 ͑t͒ = i͵ dtЈ͚ G ͑t − tЈ͒E ͑tЈ͒. ͑C1͒ m m,n n ͑3͒ −ϱ n S␯ ␯ ␯ ␯ ͑r t ;r t ,r t ,r t ͒ 4, 3 2 1 4 4 3 3 2 2 1 1 ͑2͒ ␯ ␯ ␯ ␯ The second-order variable, Y ͑t͒, is then obtained from Eq. P 4͑ ͒ 3͑ ͒ 2*͑ ͒ 1*͑ ͒ mn = i ␯ ͚ d␰ r4 d␰ r3 d␰ r2 d␰ r1 ͑ ͒ rt␰ ␰ ␰ ␰ 4 3 2 1 5 , 4 3 2 1 ϱ ϱ ϱ ͑2͒ Y ͑1͒ ϫ͵ ␶Љ͵ ␶Ј⌫ ͑␶Љ ␶Ј͒ ͑ ␶Љ͒ ͑ ͒ ͵ Ј G ͑ Ј͒͑E ͑ Ј͒ ͑ Ј͒ d d ␰ ␰ ,␰ ␰ − I␰ t4 − Ymn t = i dt ͚ mn,mЈnЈ t − t mЈ t BnЈ t 4 3 2 1 4 ϱ −ϱ −ϱ − mЈnЈ * ͑1͒ ϫI␰ ͑␶Љ − t ͒I␰ ͑␶Ј − t ͒I␰ ͑␶Ј − t ͒ + c.c., ͑C5͒ E ͑ Ј͒ ͑ Ј͒͒ ͑ ͒ 3 2 2 1 1 + nЈ t BmЈ t . C2 3

The third-order variable then is ﬁnally obtained from Eq. ͑4͒ where the nonlocal exciton transition dipoles are given by in terms of the exciton Green’s functions and the exciton Eq. ͑15͒ and we have transformed the exciton scattering ma- scattering matrix trix into the exciton basis,

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⌫ ͑␶͒ ␺* ␺* ⌫ ͑␶͒ 1 ␰ ␰ ,␰ ␰ = ͚ ␰ ␰ m m ,m m ␯ ik͑R+␳ ͒−iqR ␯ 4 3 2 1 4m4 3m3 4 3 2 1 d␭͑k,q͒ = ͚ ͚ Јe m ␺␭ ͑q͒␮ . ͑D6͒ m m m m ͱ m m 4 3 2 1 V R m ϫ␺ ␺ ͑ ͒ ␰ m ␰ m . C6 2 2 1 1 The R summation ͑a discrete coordinate of cell͒ runs over the cells, q is also a discrete momentum, while k is a continuous vector. ͑ ͒ APPENDIX D: EIGENSTATES AND SUSCEPTIBILITIES The third-order susceptibility is obtained from Eq. 26 OF PERIODIC SYSTEMS and is given by, ͑3͒ ␹␯ ␯ ␯ ␯ ͑− k ,− ␻ ;k ,␻ ,k ,␻ ,k ,␻ ͒ We consider the system shown in Fig. 2. We use periodic 4, 3 2 1 4 4 3 3 2 2 1 1 boundary conditions to represent translational invariance and =2␲i␦͑␻ − ␻ − ␻ − ␻ ͒P ignore edge effects. Since the system is translationary invari- 4 3 2 1 ␯k␻ ͑ Ј ͒ ␯ ␯ ant, the intermode coupling JRm,RЈn =Jm,n R −R now de- ϫ 4͑ ͒ 3͑ ͒ ͚ ͚ Јd␭ k4,q4 d␭ − k3,q3 pends on the distance between cells RЈ−R and on the sites ␭ ␭ 4 3 q4...q1 4... 1 inside each cell, m and n. Note the difference in notation ␯ ␯ 2* 1* ϫd␭ ͑k ,q ͒d␭ ͑k ,q ͒ with the preceding section: now each mode is represented by 2 2 2 1 1 1 a pair of indices, Rm. ϫ⌫ ͑ ␻ ␻ ͒ ͑ ␻ ͒ * ͑ ␻ ͒ ␭ ␭ ␭ ␭ q4q3q2q1, 1 + 2 I␭ q4, 4 I␭ q3,− 3 The one-exciton states of this system are the Bloch 4 3 2 1 4 3 ␰ ϫ ͑ ␻ ͒ ͑ ␻ ͒ ͑ ͒ states. Each eigenstate is represented by a pair of quantum I␭ q2, 2 I␭ q1, 1 + c.c.Ј, D7 numbers q␭, where ␭ denotes different Davydov’s subbands 2 1 in the one-exciton band ͑there are M different subbands͒ where the scattering matrix is given by with momentum q. The momentum q assumes the values ͑ ␲ ͑␲ ͒ ͒ ⌫␭ ␭ ␭ ␭ ͑q ,q ,q ,q ;␻͒ − /a,..., /a −dq including 0 in each dimension, with 4 3 2 1 4 3 2 1 ␲ the step dq =2 /L and L=Na is the length of the system. The 1 one-exciton Bloch states are given by = ͚ exp͑iq R + iq R − iq R − iq R ͒ V2 4 4 3 3 2 2 1 1 R ...R 1 4 1 ␺͑␭͒ ͑ ͒ ͑ ͒␺ ͑ ͒ ͑ ͒ q = exp − iqR ␭m q , D1 * * Rm ͱ ϫ ͚ Ј␺␭ ͑q ͒␺␭ ͑q ͒⌫ ͑␻͒ V 4m4 4 3m3 3 R4m4,...,R1m1 m4...m1 where ␺␭ ͑q͒ are the translationally invariant eigenstates of m ϫ␺␭ ͑q ͒␺␭ ͑q ͒. ͑D8͒ the cell given by 2m2 2 1m1 1 ⌫ ͑␻͒ is the real space scattering matrix identical ͚ ЈJ ͑q͒␺␭ ͑q͒ = ⍀␭͑q͒␺␭ ͑q͒. ͑D2͒ R4m4,...,R1m1 m,mЈ mЈ m ͑ ͒ mЈ to Eq. A16 , where site indices n were changed into the pairs Rm. The reduced expression of the scattering matrix is ͑ ͒ ͚ −iqr ͑ ͒ Here Jm,mЈ q = re Jm,mЈ r ; the prime in the sum over m given by Eq. ͑E7͒. These are the most general expressions denotes the summation over sites inside one cell, while the for arbitrary oriented periodic system. The directions are sum over r runs over cells including r=0; ␭ takes values given in the molecular frame. The scattering matrix can be M ͑ ͒ MϫM from 0 to −1. Jm,mЈ q is a matrix of the size reduced considerably as shown in Appendix E. with indices m and mЈ denoting different sites; each matrix When the system size is much larger than the optical element depends parametrically on the vector q. ͉ ͉ ӷ wavelength k0 L 1, R and q may be treated as continuous In the frequency domain the one-exciton Green’s func- variables and the summations over R and q in Eqs. ͑D5͒ and tion is obtained from Eqs. ͑A6͒ and ͑A7͒, ͑D7͒ can be changed into integrations. In this case the equa- 1 tions are considerably simpliﬁed. The linear susceptibility is ͑␻͒ iq͑RЈ−R͒␺ ͑ ͒␺* ͑ ͒ ͑ ␻͒ GRm,RЈmЈ = ͚ ͚ Јe ␭m q ␭ q I␭ q, , V mЈ ͑1͒ 4 q ␭ ␹␯ ␯ ͑− k − ␻ ;k ␻ ͒ = ͑2␲͒ i␦͑␻ − ␻ ͒␦͑k − k ͒ 2, 1 2 2 1 1 2 1 2 1 ͑D3͒ ␯ ␯ ϫ 2͑ ͒ 1*͑ ͒ ͑ ␻ ͒ ͑ ͒ ͚ Јd␭ k2 d␭ k2 I␭ k2, 2 + c.c.Ј, D9 ͚Ј indicates the sum over different Davydov’s subbands with ␭ the exciton Green’s function, ␯ ␯ ␳ ␯ ͑ ͒ϵ ͑ ͒ ͱV͚Ј ik m␺ ͑ ͒␮ where d␭ k d␭ k,k = me ␭m q m are the transi- i tion dipoles of inﬁnite system between different continuous I␭͑q,␻͒ = . ͑D4͒ ␻ − ⍀␭͑q͒ + i␥␭͑q͒ exciton bands. I␭͑q,␻͒ is the exciton Green’s function in the exciton basis ͓Eq. ͑D4͔͒. Using the eigenstates and Green’s functions of periodic For the third-order susceptibility we obtain systems in Eq. ͑12͒ we obtain the linear susceptibility, ͑3͒ ͑1͒ ␹␯ ␯ ͑− k ,− ␻ ;k ,␻ ,k ,␻ ,k ,␻ ͒ =2␲i␦͑␻ − ␻ − ␻ ␹␯ ␯ ͑− k − ␻ ;k ␻ ͒ =2␲i␦͑␻ − ␻ ͒ 4... 1 4 4 3 3 2 2 1 1 4 3 2 2, 1 2 2 1 1 2 1 ␯ ␯ ␯ * ␯ * ␯ ␯ * − ␻ ͒P␯ ␻ ͚ Јd 4͑k ͒d 3͑− k ͒d 2 ͑k ͒d 1 ͑k ͒ ϫ͚ ͚ Јd 2͑k ,q͒d 1 ͑k ,q͒I␭͑q,␻ ͒ + c.c.Ј, ͑D5͒ 1 k ␭ 4 ␭ 3 ␭ 2 ␭ 1 ␭ 2 ␭ 1 2 ␭ …␭ 4 3 2 1 q ␭ 4 1

ϫ⌫␭ ␭ ͑k ,− k ,k ,k ,␻ + ␻ ͒ where 4... 1 4 3 2 1 1 2

Downloaded 08 Aug 2005 to 128.200.11.135. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp 134305-18 D. Abramavicius and S. Mukamel J. Chem. Phys. 122, 134305 ͑2005͒

* ϫI␭ ͑k ,␻ ͒I␭ ͑− k ,− ␻ ͒I␭ ͑k ,␻ ͒I␭ ͑k ,␻ ͒ + c.c.Ј D ͑q,␻͒ = ␦ ␦ ␦ 4 4 4 3 3 3 2 2 2 1 1 1 r1,m,n;r2,mЈ,nЈ r1,r2 m,mЈ n,nЈ ͑D10͒ + iG ͑q,␻͒⌬ ͑r ͒. r1,m,n;r2,mЈ,nЈ mЈ,nЈ 2 This limit may be achieved for pure oriented semicon- ͑E5͒ ductors or molecular crystals at cryogenic temperatures where the exciton coherence size is larger than the wave- Note that we are using a mixed momentum and real space length. The opposite limiting case of the small systems is representation, which allows us to control the expressions discussed in Sec. VI. using the interaction distance lc. The transformation with respect to r1 and r2 does not simplify the expression since these coordinates are not translationally invariant in our ex- APPENDIX E: THE EXCITON SCATTERING MATRIX IN pressions of D. The exciton scattering matrix in this mixed MOMENTUM SPACE representation now reads

Computing the general scattering matrix ͓Eq. ͑D8͔͒ is ⌫ ͑ ␻͒ ⌬ ͑ ͒ ͑ ␻͒ −1 r ,m,n;r ,mЈ,nЈ q, =−i m,n r1 D q, r ,m,n;r ,mЈ,nЈ. very costly for a large systems. Using a finite interaction 1 2 „ … 1 2 ͑ ͒ radius lc and periodicity we can reduce the problem size E6 N below lc. ͑ ͒ ͑ ͒ ͑ ͒ This expression is illustrated in Fig. 2 b . Here the initial Equations B2 and B4 define the exciton scattering two-exciton pair mЈ and nЈ is separated by the distance r matrix by the reduced matrix D which depends on G ͓see Eq. 2 =RnЈ−RmЈ within interaction radius, which is established by ͑B3͔͒. Using the representation of Fig. 2, the two-exciton ⌬ ͑ ͒ ⌫ mЈnЈ r2 . The pair is scattered by the matrix to create a Green’s function in the coordinate representation has the G ͑␻͒ new pair of excitons at m and n separated by r1 =Rn −Rm. form R m,R n;RЈ mЈ,RЈnЈ , where m... are indices of sites in m n m n The distance between two excitons after scattering is also cells and Rm... labels different cells. The coupling and the ⌬ ͑ ͒ within the interaction distance as required by mn r1 . The anharmonicity matrix are now translationally invariant and distance between the two exciton pairs is not important be- ͑ ͒ ⌬ ͑ ͒ can be given as Jm,mЈ r and m,mЈ r , respectively, where r cause of translational invariance. defines the distance between the cells. We are interested in For the scattering matrix in the eigenstate representation Ј Ј the Green’s function when Rn =Rm +r1 and Rn =Rm +r2 we obtain ͑ when r1 and r2 are within interaction radius lc otherwise the quartic coupling V is zero͒. Thus, we define the reduced ⌫␭ ␭ ␭ ␭ ͑q ,q ,q ,q ;␻͒ Green’s function G ͑␻͒ 4 3 2 1 4 3 2 1 Rmm,r1n;RЈ mЈ,r2nЈ G ͑␻͒ m = R m,͑R +r ͒n;RЈ mЈ,͑RЈ +r ͒nЈ and expand it in the basis of = ␦͑ ͒ ͑ ͒ ͚ exp͑iq rЉ − iq rЈ͒ m m 1 m m 2 q4+q3 , q2+q1 3 1 ͑ ͒ Ͻ Ͻ one-exciton eigenstates given by Eq. D1 , −lc rЈrЉ lc * * 1 ͑ Ј͒͑ Ј ͒ Ј͑ ͒ ϫ ␺ ͑ ͒␺ ͑ ͒⌫ ͑ ␻͒ G ͑␻͒ i q+q Rm−Rm −iq r1−r2 ͚ ␭ m q4 ␭ m q3 rЉm m ;rЈm m q2 + q1, R m,r n;RЈ mЈ,r nЈ = 2 ͚ e 4 4 3 3 4 3 2 1 m 1 m 2 V m4...m1 qqЈ ϫ␺ ͑ ͒␺ ͑ ͒ ͑ ͒ ϫ ͑ ␻͒ ͑ ͒ ␭ m q2 ␭ m q1 . E7 gm,n;mЈ,nЈ q,qЈ, , E1 2 2 1 1 where the unit cell’s Green’s function The total momentum before interaction and after interaction ͑ Ј ␻͒ ͚ Ј␺ ͑ ͒␺ ͑ Ј͒I ͑ Ј ␻͒ is conserved. gm,n;mЈ,nЈ q,q , = ␭m q ␭Јn q ␭␭Ј q,q , The calculation of the response function ͓Eq. ͑45͔͒ re- ␭␭Ј quires the scattering matrix for zero momentum. The final ϫ␺* ͑ ͒␺* ͑ Ј͒͑͒ ␭mЈ q ␭ЈnЈ q E2 expressions are: and ͑ ␻͒ ␺ ͑ ͒␺ ͑ ͒ gm,n;mЈ,nЈ q,− q, = ͚ Ј ␭m q ␭Јn − q i ␭␭Ј I␭␭Ј͑q,qЈ,␻͒ = ␻ − ⍀␭͑q͒ − ⍀␭Ј͑qЈ͒ + i␥␭͑q͒ + i␥␭Ј͑qЈ͒ ϫI ͑ ␻͒␺* ͑ ͒␺* ͑ ͒ ␭␭Ј q,− q, ␭mЈ q ␭ЈnЈ − q , ͑ ͒ E3 ͑E8͒ is a two-exciton Green’s function in the frequency domain. Taking into account the translational invariance with re- Ј 1 −iq͑r −r ͒ spect to Rm and Rm we can transform the two-exciton G ͑␻͒ = ͚ e 2 1 g ͑q,− q,␻͒, r1,m,n;r2,mЈ,nЈ V m,n;mЈ,nЈ Green’s function to momentum space, q ͑ ͒ 1 ͑ ͒͑ ͒ E9 G ͑q,␻͒ = ͚ ei q−q1 r2−r1 r1,m,n;r2,mЈ,nЈ V q1 ͑␻͒ ␦ ␦ ␦ ϫ ͑ ␻͒ ͑ ͒ Dr ,m,n;r ,mЈ,nЈ = r ,r m,mЈ n,nЈ gm,n;mЈ,nЈ q1,q − q1, . E4 1 2 1 2 G ͑␻͒⌬ ͑ ͒ ͑ ͒ M2 ϫM2 + i r ,m,n;r ,mЈ,nЈ mЈnЈ r2 , E10 We can now write the transformed lc lc matrix D, 1 2

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−1 ⌫ ͑␻͒ ⌬ ͑ ͒ ͑␻͒ ͑ ͒ SkIII ͑ ͒ ͗␦͑ ͒ ␦͑ ͒ r ,m,n;r ,mЈ,nЈ =−i mn r1 D r ,m,n;r ,mЈ,nЈ, E11 ␯ ␯ r4t4, ...,r1t1 =2i ͚ r4 − rn r1 − rn 1 2 „ … 1 2 4,..., 1 4 ¯ 1 n4...n1 ␯ ␯ and the Green’s function of the unit cell were given by Eq. ϫ 4... 1͘J ͑ ͒ M n n n n t43,t42,t41 , ͑E2͒. The scattering matrix of the eigenstates is n4...n1 4 3 2 1 ͑F5͒

⌫␭ ␭ ␭ ␭ ͑␻͒ = ͚ ͚ Ј ␺␭ ␺␭ ⌫ ͑␻͒ 4 3 2 1 4m4 3m3 rЉm4m3;rЈm2m1 and we have defined an auxiliary function −l ϽrЈrЉϽl m4 m1 c c ¯ ␶ 2 ␶Љ ϫ␺ ␺ ͑ ͒ J ͑␶ ␶ ␶ ͒ ͵ ␶Љ͵ s ␶Ј ␭ ␭ . E12 , , = ͚ d d 2m2 1m1 n4n3n2n1 3 2 1 s s Ј Ј Ј Ј 0 0 n4n3n2n1 M2͑ ͒ϫM2͑ ͒ The matrix sizes are 2lc +1 2lc +1 , the same ⌫ ͑␶Љ ␶Ј͒ nЈnЈ,nЈnЈ − size as for the inversion problem. These are significantly 4 3 2 1 s s reduced compared to nonperiodic systems. ϫ ͑␶Ј͒ † ͑␶ ␶Ј͒ Gn nЈ s G Ј 3 − s 4 4 n3n3 ϫG Ј ͑␶ − ␶Љ͒G Ј ͑␶ − ␶Љ͒. ͑F6͒ n2n2 2 s n1n1 1 s APPENDIX F: TIME-DOMAIN OPTICAL RESPONSE FUNCTIONS Here tij=ti −tj is the time delay between two consecutive ␶Ј ␶Љ interactions. s and s are the delay times between exciton ␶Ј Time-domain expressions are useful for experiments per- scattering events and the polarization measurement: s =t4 ␶Љ ␶Љ ␶Ј ͑ ͒ formed with ultrafast, well separated, optical pulses. The se- − , s =t4 − as shown in Fig. 1 b . Since the time vari- quence of interactions can then be defined and different tech- ables can be rearranged as t43, t42=t43+t32, t41=t43+t32+t21, niques can be determined by their time ordering.40 The time each response function now depends on three time delays ordered optical response is defined by between different interactions: t43, t32, and t21. The time intervals of Eqs. ͑F3͒–͑F5͒ and their relation to ͑3͒ P␯ ͑r ,t ͒ the actual interaction times are shown in Fig. 1͑b͒. The signal 4 4 4 can be interpreted similarly to the frequency domain suscep- t t t ͵ ͵ ͵ ͵ 4 ͵ 3 ͵ 2 tibility: there are three interactions with the optical fields at = ͚ dr3 dr2 dr1 dt3 dt2 dt1 ␯ ␯ ␯ ϱ ϱ ϱ times t ഛt ഛt which generate three quasiparticles repre- 3 2 1 − − − 1 2 3 sented by one-exciton Green’s functions G , G , and nЈ,n1 nЈ,n2 † 1 2 ͑ ͒ ͑ S 3 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ GnЈ,n two having positive oscillation frequency and one ␯ ,␯ ␯ ␯ r4t4;r3t3,r2t2,r1t1 E␯ r3,t3 E␯ r2,t2 E␯ r1,t1 , 3 3 4 3 2 1 3 2 1 having negative͒. These quasiparticles evolve in time and ͑F1͒ ⌫ two of them with the same phase are scattered by nЈnЈ,nЈnЈ 4 3 2 1 shifting the particles to the positions nЈ and nЈ. The scatter- where t now stands for the first interaction, t for the sec- 3 2 1 2 ing is such that one of the two new excitons corresponds to ond, and t for the third in chronological order. 3 the exciton generated by the field. The other exciton gener- Equation. ͑C4͒ cannot be used directly since the time ates the optical response. arguments have to be rearranged in chronological order. To The time evolution is followed explicitly and the reso- that end we separate the expression into three terms: nant terms can be selected according to wave vectors of op- ͑ ͒ S 3 ͑ ͒ SkI ͑ ͒ tical fields and time ordering of the interaction. The three ␯ ␯ r4t4, ...,r1t1 = ␯ ␯ r4t4, ...,r1t1 4,..., 1 4,..., 1 different time evolutions for the time ordered response func- SkII ͑ ͒ tions are showed in Fig. 10. + ␯ ,...,␯ r4t4, ...,r1t1 4 1 The dependence on the wave vector of the optical field kIII + S␯ ␯ ͑r t , ...,r t ͒ + c.c . , can be obtained by applying spatial Fourier transform to the 4,... 1 4 4 1 1 ͑ ͒ response function. The expressions are considerably simpli- F2 fied in eigenstate basis where we get ␶ where 2 ␶Љ J ͑␶ ␶ ␶ ͒ ͵ ␶Љ͵ s ␶Ј⌫ ͑␶Љ ␶Ј͒ ␰ ␰ ␰ ␰ , , = d d ␰ ␰ ␰ ␰ − 4 3 2 1 3 2 1 s s 4 3, 2 1 s s 0 0 SkI ͑ ͒ ͗␦͑ ͒ ␦͑ ͒ ␯ ␯ r4t4, ...,r1t1 =2i ͚ r4 − rn r1 − rn 4,..., 1 4 ¯ 1 * n4...n1 ϫI␰ ͑␶Ј͒I␰ ͑␶ − ␶Ј͒I␰ ͑␶ − ␶Љ͒ 4 s 3 3 s 2 2 s ␯ ␯ ϫM 4... 1͘J ͑t ,t ,t ͒, n ...n n4n1n3n2 41 43 42 ϫI␰ ͑␶ − ␶Љ͒͑F7͒ 4 1 1 1 s ͑F3͒ and

SkI ͑ ͒ ␯ ,...,␯ k4t4, ...,k1t1 SkII ͑ ͒ ͗␦͑ ͒ ␦͑ ͒ 4 1 ␯ ␯ r4t4, ...,r1t1 =2i ͚ r4 − rn r1 − rn 4,..., 1 4 ¯ 1 ␯ ␯ ␯ ␯ n4...n1 ͗ 4͑ ͒ 3*͑ ͒ 2*͑ ͒ 1͑ ͒͘ =2i ͚ d␰ k4 d␰ − k3 d␰ − k2 d␰ k1 ␯ ...␯ ␰ ␰ 4 3 2 1 ϫM 4 1͘J ͑t ,t ,t ͒, 4... 1 n4...n1 n4n2n3n1 42 43 41 ϫJ␰ ␰ ␰ ␰ ͑t ,t ,t ͒, ͑F8͒ ͑F4͒ 4 1 3 2 41 43 42

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FIG. 10. ͑Color online͒ The Green’s functions and the scattering matrix representing three contributions to the time-domain third-order response ͑adopted from Ref. 64͒.

kII ͑ ͒ S␯ ␯ ͑k t , ...,k t ͒ and in kIII technique k4 =−kIII 4,..., 1 4 4 1 1 ␯ ␯ ␯ ␯ ͑ ␻ ␻ ͒ 4 3* 2 1* W␯ ,␯ ␯ ␯ k4,k3,k2,k1; 2, 3 =2i ͚ ͗d␰ ͑k ͒d␰ ͑− k ͒d␰ ͑k ͒d␰ ͑− k ͒͘ 4 3 2 1 4 4 3 3 2 2 1 1 ␰ ...␰ ␯ ␯ ␯ ␯ 4 1 ͗ 4͑ ͒ 3͑ ͒ 2*͑ ͒ 1*͑ ͒͘ =2i ͚ d␰ k4 d␰ − k3 d␰ − k2 d␰ − k1 ␰ ␰ 4 3 2 1 ϫJ␰ ␰ ␰ ␰ ͑t ,t ,t ͒, ͑F9͒ 4... 1 4 2 3 1 42 43 41 ϱ ␻ SkIII ͑ ͒ d Ј ␯ ,...,␯ k4t4, ...,k1t1 ϫ ͑␻ ͒͵ ⌫ ͑␻Ј͒I ͑␻Ј͒ I␰ ␰ ␰ ␰ ␰ ␰ ␰ 4 1 4 3 ␲ 4 3, 2 1 2 1 −ϱ 2 ␯ ␯ ␯ ␯ 4 3 2* 1* =2i ͚ ͗d␰ ͑k ͒d␰ ͑k ͒d␰ ͑− k ͒d␰ ͑− k ͒͘ 4 4 3 3 2 2 1 1 * ␰ ...␰ ϫI␰ ͑␻Ј − ␻ ͒␪͑␻ − ␻Ј͒. 4 1 3 3 2

ϫJ␰ ␰ ␰ ␰ ͑t ,t ,t ͒, ͑F10͒ −1 4 3 2 1 43 42 41 The function ␪͑␻͒=i͑␻+i␥Ј͒ is taken in the limit ␥␰ ӷ␥Ј Ͼ0. J with k4 =−k3 −k2 −k1 which reduces from fourfold summation to none. The dependence on the wave vectors is also 1L. D. Barron, Molecular Light Scattering and Optical Activity, 2nd ed. expressed clearly. The rotational averages of these quantities ͑Cambridge University Press, Cambridge, 2004͒. 2 were given in Sec. V. Chirality Physical Chemistry, edited by J. M. Hicks ͑Oxford University Press, Oxford, 2002͒. Three possible techniques, kI =−k1 +k2 +k3, kII=k1 −k2 3A. Moscowitz, Adv. Chem. Phys. 4,67͑1962͒. 4 +k3, kIII=k1 +k2 −k3, are now defined by the response func- I. Tinoco, Adv. Chem. Phys. 4,113͑1962͒. tions SkI, SkII, and SkIII, respectively. The parameters of the 5Circular Dichroism. Principles and Applications, 2nd ed., edited by N. ͑ ͒ response functions are the three time delays t , t , t . Con- Berova, R. W. Woody, and K. Nakanishi Wiley, New York, 2000 . 43 32 21 6N. J. Greenfield, Anal. Biochem. 235,1͑1996͒. sider the techniques kI =−k1 +k2 +k3 and kIII=k1 +k2 −k3. 7N. Sreerama and R. W. Woody, Anal. Biochem. 287, 252 ͑2000͒. 8 For kI we set the second time delay t32=0 and perform the N. Sreerama and R. W. Woody, Protein Sci. 12, 384 ͑2003͒. Fourier transforms with respect to the first and third time 9N. J. Greenfield and G. D. Fasman, Biochemistry 8,4108͑1969͒. 10 ͑ ͒ delays: t →␻ and t →␻ . For k we set the first time J. D. Hirst, J. Chem. Phys. 109, 782 1998 . 21 1 43 3 III 11D. H. Chin, R. W. Woody, C. A. Rohl, and R. L. Baldwin, Proc. Natl. delay t21=0 and perform the Fourier transforms with respect Acad. Sci. U.S.A. 99, 15416 ͑2002͒. →␻ →␻ 12 to the second and third time delays: t32 2 and t43 3. R. W. Woody, J. Chem. Phys. 49, 4797 ͑1968͒. The signals become also a function of the directions of op- 13P. M. Bayley, E. B. Nielsen, and J. A. Shellman, J. Phys. Chem. 73,228 ͑1969͒. tical wave vectors in first-order approach we are using. Ex- 14 K. A. Bode and J. Applequist, J. Phys. Chem. 100, 17825 ͑1996͒. pression especially useful for numerical simulations for the 15K. A. Bode and J. Applequist, Macromolecules 30, 2144 ͑1997͒. ͑ ͒ 16 2D signal in the kI technique k4 =−kI is G. Kurapkat, P. Kruger, A. Wolimer, J. Fleischhauer, B. Kramer, A. Zobel, A. Koslowski, H. Botterweck, and R. W. Woody, Biopolymers 41,267 ͑1997͒. W␯ ␯ ␯ ␯ ͑k ,k ,k ,k ;␻ ,␻ ͒ 4, 3 2 1 4 3 2 1 1 3 17P. J. Stephens and M. A. Lowe, Annu. Rev. Phys. Chem. 36,213͑1985͒. 18 ␯ ␯ ␯ ␯ L. A. Nafie, Annu. Rev. Phys. Chem. 48, 357 ͑1997͒. ͗ 4͑ ͒ 3*͑ ͒ 2*͑ ͒ 1͑ ͒͘ =2i ͚ d␰ k4 d␰ − k3 d␰ − k2 d␰ − k1 19F. Eker, X. Cao, L. A. Nafie, Q. Huang, and R. Schweitzer-Stenner, J. ␰ ␰ 4 3 2 1 4... 1 Phys. Chem. B 107, 358 ͑2003͒. 20 * L. Hecht and L. D. Barron, Raman Optical Activity in Modern Techniques ϫI␰ ͑− ␻ ͒I␰ ͑␻ ͒ 1 1 4 3 in Raman Spectroscopy, edited by J. J. Laserna, p. 265 ͑Wiley, New York, 1996͒; L. D. Barron, L. Hecht, I. H. McColl, E. W. Blanch, Mol. Phys. ϱ ␻Ј d * 102,731͑2004͒. ϫ͵ ⌫ ͑␻Ј͒I ͑␻Ј͒ ͑␻Ј ␻ ͒ 21 ␰ ␰ ,␰ ␰ ␰ ␰ I␰ − 3 ␲ 4 1 3 2 3 2 1 L. D. Barron, E. W. Blanch, I. H. McColl, C. D. Syme, L. Hecht, and K. −ϱ 2

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