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 0021-9606/2005/122͑13͒/134305/21/$22.50122, 134305-1 © 2005 American Institute of Physics 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 polariza- tion is measured at time t4 and position r4, while the r inte- grations 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 combina- tion 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.