摘要:

Femtosecond two-dimensional infrared spectroscopy: IR-COSY and THIRSTY Nien-Hui Ge and Robin M. Hochstrasser* Department of Chemistry, University of Pennsylvania, Philadelphia, PA 19104-6323, USA. E-mail: hochstra@sas.upenn.edu; Fax: 215-898-0590; Tel: 215-898-8410 Received 31st October 2001, Accepted 13th December 2001 Published on the Web 9th January 2002 The femtosecond two-dimensional infrared (2D IR) spectroscopy that was recently developed shows great promise for determining the dynamics of molecular structure at ultrafast time scales that are hard to access by NMR and X-ray diffraction methods. This Perspective focuses on the IR-COSY and THIRSTY methods that are based on heterodyned three pulse photon echoes. The relationships of 2D spectral properties coupling, structure, correlated fluctuations, energy transfer, and orientational motions to experimental results on small peptides and molecules are described.The status of this exciting new field is also discussed. Introduction An ideal experimental tool for obtaining a complete understanding of chemical and biological processes would provide snapshots of atomic positions of participating molecules at any time throughout the course of chemical reactions or conformational changes. Although no single experimental technique is likely to yield all of this information, exciting new experiments have demonstrated that the idea of using 2D IR spectroscopy to obtain molecular structures with sufficient time resolution to resolve barrier crossing processes and conformational dynamics, now appears to have exceptional promise. One potential of the technique is its intrinsically high time resolution, which is on the order of picoseconds.Thus 2D IR spectroscopy naturally accesses the time scales that are hard to reach by 2D NMR and X-ray diffraction methods in solutions. Exciting biological applications include structural dynamics of nucleotides, peptides, and perhaps small proteins. Other application in photochemistry, polymer dynamics, and liquids will be possible. This field of 2D IR has evolved from the dynamic hole burning methods originally introduced in 19981ƒƒ to the more recent achievement of the infrared version2ƒƒ of NMR COSY and NOESY. Numerous applications of these methods to peptides3ƒƒ,4,5ƒ,6,7ƒ,8,9ƒƒ,10,11ƒƒ,12ƒƒ and small molecules13ƒ have now been reported.Such optimizations of vibrational spectroscopy require the ability to control the responses of particular vibrational transitions depending on their coupling to one another. In NMR the disentangling of complex spectra is accomplished by multiple pulse sequences that manipulate the spin coherences and populations. Therefore in IR spectroscopy multiple IR pulses having well defined spectral bandwidth, phase and amplitude will be required. Vibrational spectra could then be spread into a number of dimensions and the coupling between modes at different spatial locations within the molecule could be determined. Thus our strategy has been to regard the molecule, polymer, liquid, peptide, or small protein as a network of coupled vibrators. The couplings between the separated excitations of this network are obtained from multiple pulse IR experiments and then used to find three-dimensional structures based on knowledge of vibrational dynamics, chemical connectability and the inter-mode potential functions.The successful implementation of such procedures will result in a significant advance in our ability to observe the time evolution of struc- DOI: 10.1039/b109935c PhysChemComm, 2002, 5(3), 17–26 17 tural changes and of the couplings between different pieces of macromolecules. The nonlinear response for a network of coupled vibrators in isotropic media involves a multi-level system consisting of the set of coupled v ~ 0, v ~ 1 and v ~ 2 quantum states of the vibrators (Fig.1). To determine the coupling between vibrators, both IR and Raman transitions can be utilized. The original 2D IR method1 was introduced as a frequency domain experiment in which the vibrational populations were excited by a narrow band IR pulse and probed by a broad band of IR frequencies (Fig. 1a). The 2D IR spectra were assembled by measuring the transient difference probe spectrum as a function of the peak frequency of the pump pulse, analogous to NMR double resonance experiments. Later, a mixed timedomain–frequency-domain 2D experiment was introduced.4 It corresponds to IR pump/probe experiments in which the probe signal is obtained as a function of pulse delays and frequencies, and the Fourier transform of the time delay gives the second Fig.1 Energy level diagrams (top) and representative rephasing Feymann diagrams (bottom) for a two-vibrator system probed by different nonlinear vibrational spectroscopies: (a) and (b) 2D IR; (c) combined IR-Raman method; (d) fifth order Raman spectroscopy. Single quantum states are represented by i and j, overtone states by iz i, j z j, and combination state by i z j. Solid (dash) arrows between energy levels denote transitions on the ket (bra) side. See text for a comparison of these methods. This journal is # The Royal Society of Chemistry 2002 PerspectiveFig. 2 Typical pulse sequences and phase matching geometry used in IR-COSY and THIRSTY spectroscopy.The three pulses are separated by the time delays t and T, and the free induction decay is measured versus t. In heterodyned 2D IR, the three infrared laser beams and the emitted FID signal beams all travel in different directions, and thus are spatially separated from one another. The electric field of the signal beam cannot be measured directly as it is in NMR, so it is overlapped with a fourth local-oscillator beam (shown in dash) that heterodynes it. When T ~ 0, the 2D IR experiment is analogous to the NMR COSY experiment, otherwise it is similar to NOESY. frequency dimension. The more recently developed 2D IR method,2 which forms the focus of this Perspective, is an all time domain experiment in which three weak IR excitation pulses interact with the sample and generate a freely decaying field (Fig.1b). The time-dependent amplitude and phase of the generated field are obtained by heterodyned detection with a fourth IR pulse. By measuring the signal as a function of the time-delays between the infrared pulses, a multi-dimensional time grid is assembled, and the Fourier transform of this grid gives a multi-dimensional frequency spectrum. The pulse sequences used to perform these 2D IR correlation experiments are the direct analogs of the sequences used in the COSY and NOESY methods of pulsed NMR (Fig. 2). In IR-COSY the time delay between the second and third infrared pulses interacting with the system is set equal to zero. The first infrared pulse creates a vibrational coherence, and after some delay a pair of infrared pulses interrupts this initial coherence evolution and generates a particular set of Bohr frequencies in a FID from the transferred coherence.When the third infrared pulse follows some time after the second one,9 we have the THIRSTY (for THree InfraRed pulse STimulated echo spectroscopY) experiment. This is analogous to the three consecutive pulses of NMR NOESY or stimulated spin echo. The finite delay between the second and third pulses allows time for vibrational coherence and population transfer to occur, analogous to the spin transfer that occurs in magnetic resonance. Recently, combined optical-infrared approaches or all optical approaches have also been used to measure the coupling between molecular vibrations. In the combined IR-Raman approach,14 two infrared beams and a visible beam are used to excite and probe vibrational coherences (Fig.1c). An enhancement in the visible output coherence occurs when two infrared beams are scanned into resonance with a fundamental vibration and a combination band of a particular mode. This signal only appears when the transition between a fundamental and a combination band is at a different frequency than the sum of the two frequencies composing the combination. The combination band must undergo a linear IR absorption for this method to be applicable. In the fifth order Raman spectroscopy,15,16 two 18 PhysChemComm , 2002, 5(3), 17–26 pairs of visible pulses are used to nonresonantly excite and transfer vibrational coherences of Raman modes (Fig.1d). The coherence is then probed by the fifth pulse. The visible output is measured as a function of the delay time between the pulse pairs and the probe pulse. The fifth order Raman is an impulsive technique so the spectral bandwidth of the pulses limits the response to vibrational frequencies encompassed by the bandwidth. Due to the nonresonant nature of the experiments, this technique has so far been applied mostly to the study of neat liquids. Theories of multidimensional spectroscopy were originally reported for optical and Raman processes17ƒ and later for combined optical–infrared approaches,18 and for infrared spectroscopy.19,20ƒ,21 Comparisons between NMR and methods that use optical and IR pulses have also been described,22ƒ and a recent issue of Chemical Physics contains articles about many of the new multidimensional spectroscopies currently being developed.23ƒ One important advance in the 2D IR-COSY and THIRSTY methods is the use of heterodyned photon echo method to provide optimal spectral resolution by eliminating the inhomogeneous broadening in the infrared spectrum.Both two-pulse24 and three-pulse IR photon echoes25 of vibrations have been studied previously, but in these reported echo experiments the time-integrated intensity of the generated third order field is measured using a square law detector. This approach has three disadvantages. First, the signal is insensitive to the phase of the field which provides optimized spectral information.Second, the signal decays twice as fast as the heterodyned signal, making an accurate deconvolution of vibrational dynamics from the finite pulse width more difficult. Third, the individual contributions of various Liouville paths to the signal are less discernible. Spectral resolution of the generated vibrational echo field26–28 permits some phase relations to be obtained, but does not yield line-narrowed spectra. However the complete generated field amplitude and phase, free from any static inhomogeneous broadening, can be obtained in principle by means of heterodyne echoes or spectral interferometry techniques. Such realization towards complete characterization of generated fields has been demonstrated for optical transitions29ƒ and for infrared transitions.2 1) The dynamics of vibrational excitations are naturally a key part of the interpretation of 2D IR spectra, much in the same way that spin dynamics have an essential role in NMR and EPR.However, the population relaxation times (T and dynamics of inhomogeneous distributions of vibrational frequencies, constituting the pure dephasing, are in the picosecond and subpicosecond regime at ambient temperatures and therefore ultrafast methods are needed to examine them. There has been considerable work on both the theory and experimental measurement of T1 and pure dephasing relaxation of molecules in solutions. In as much as these processes limit the spectral resolution of 2D IR spectroscopy, it is necessary that they are understood as well as possible.There is still not a reliable predictive theory for T1 relaxations, particularly when the energy is induced by solvent fluctuations to flow through both solvent modes and internal modes of the molecule under consideration.30–32 In peptides containing a number of amide units, the spatial distributions of vibrational relaxation times and energy transfer between the various quasidegenerate modes are not yet widely studied. The relaxation between different peptide units is caused by the fluctuations in the coupling between them according to recent Redfield theory calculations based on molecular dynamics simulations.10,33 Recent measurements10,34 are consistent with the fact that cross relaxation does not dominate the vibrational dynamics, at least in two cases studied.The pure dephasing processes in such molecules will also depend on the spatial location in the molecule of the relaxing group, and on the couplings between the sets of modes. Clearly a wide range ofchallenging questions are waiting to be answered by these new two dimensional techniques. In the present Perspective we describe the principle and application of the heterodyned 2D IR method to illustrate its capability in determining vibrational dynamics and structures. The heterodyned 2D IR experiments In the IR-COSY and THIRSTY experiments,2,9 three phase locked IR pulses with wavevectors k1, k2 and k3 arrive at the sample with time intervals t, between the first and second, and T, between second and third (see Fig.2). They create a macroscopic polarization P(t;t,T) in the sample for times t after the third pulse which in turn generates an electric field pulse E(t;t,T). The emitted IR photon echo field in the direction 2k1 z k2 z k3 is heterodyne detected by the local oscillator pulse traveling in the same direction with wavevector kLO. The measured signal Sabcd(t; t,T) is a function of the time intervals and depends on the polarization condition (a,b,c,d) of the pulses, labeled by the subscripts in the order that the pulses follow in the experiment (k1,k2,k3,kLO). The origin of this freely decaying oscillatory echo field for a particular set of pulse delays between the three phase locked IR pulses is usually represented by a set of Feynman diagrams or Liouville pathways35 signifying the excitation of and transfer between various vibrational populations and coherences. By judicious choice of pulse sequence the type of vibrational coherences that contribute to the generated field can be controlled.When the interaction of the set of vibrators with pulse k1 occurs before the interaction with pulse k2, the coherence evolution involves frequencies of opposite signs during the t and t period, and the so-called rephasing pathways contribute to the signal. When the ordering of pulses k1 and k2 is reversed, the frequencies of two coherences involved in the coherence evolution period have the same sign and the so-called nonrephasing diagrams contribute to the signal.Acquiring 2D spectra using both pulse sequences can provide information on correlated molecular vibrations.12 In addition to time ordering, the polarization of each laser pulse can be individually varied. The polarization dependence of the echo fields is determined by the angles between the transition dipoles that couple to the pulses in the sequence.36,37 More detailed examples will be given later. In an ideal heterodyne experiment, the signal field interferes on the detector with a short local oscillator field pulse and the heterodyne signal remaining after subtraction of the local oscillator intensity is a measure of Re[E(t;t,T)].All three time delays, t, T and t, can be scanned independently. In general the real generated field consists of sine and cosine parts (i.e. it has a phase) which can be separately measured for a given (t, T) from the real and imaginary parts of the Fourier transform of the signal along the t axis. When t and t are scanned in an experiment, the complex 2D IR spectrum S(vt,vt;T) is obtained from the Fourier transform of the Re[E(t;t,T)], along t and t, yielding frequency axes vt and vt. Alternatively, one can scan T and t, and obtain S(vt,vT;t). When all three times are scanned, a 3D IR spectrum S(vt,vt,vT) can be obtained. Fig. 3a shows a typical example of the heterodyned echo signal S(t;t,T ~ 0) for two values of t for a liquid acetone sample containing an equal volume mixture of natural acetone and acetone-2-13C.The signal shows oscillations with a period of 20 fs, corresponding to the vibrational frequency of ketone CLO stretch at y1700 cm21. There are also larger lobes that appear to be separated by y800 fs. This is due to the interference between the two oscillations at 1715 cm21 and 1675 cm21 corresponding to 12CLO and 13CLO stretches. When this signal is Fourier transformed along the t delay, we obtain the spectra in Fig. 3b, which show the signals from both of the CLO oscillators and their decay as a function of t. When 13 Fig. 3 Heterodyned photon echo signals for a 50 : 50 mixture of acetone and acetone-2-13C at several values of time t as a function of the time t.(a) Time-dependent interferograms of the heterodyned echo signal. (b) Fourier transform along the t axis showing the spectra of the signal. The fast oscillations in (a) have a y20 fs period that is determined by the fundamental frequency of the CLO stretch, and the modulation in the signal is a result of the frequency separation between the 12C and C labeled CLO groups (N.-H. Ge and R. M. Hochstrasser, in process of publication). the time variable t is varied at fixed t, an oscillatory signal is also generated. The complete two-time signal involves scanning over all t for each possible value of t. The Fourier transform, in both time axes, of this two-time signal then gives the complex 2D IR spectra. A related technique to the measurement of heterodyned echoes is spectral interferometry of the echo signals.29 Here instead of measuring the frequency integrated signal at a number of local oscillator delay times, the frequency dependent signal at a single value of the local oscillator delay time is measured by dispersing the signal plus local oscillator fields in a monochromator onto an array detector.The signal after subtracting the intensity spectrum of the local oscillator is then the product of the echo spectrum, the local oscillator spectrum, and an oscillation whose frequency depends on the time delay of the local oscillator relative to the echo field. If the local oscillator field is independently measured, the Fourier transform of S(t;t,T) along the t-axis can be deduced.Assuming that the monochromator and array detector are ideal this approach is economical since it does not require a scan over t to obtain the multidimensional spectrum. If a single element detector is used instead of an array detector, the time saving advantage of spectral interferometry is not realized because it requires a scan over the monochromator grating positions to obtain the spectral interferogram. The interferometric signal obtained in this manner is shown in Fig. 4 for a dipeptide acetylproline- ND2 in D2O.2 Fourier analysis of the oscillatory part of the signal recovers the spectrum of the echo field as also shown in Fig. 4. This spectrum shows peaks at 1610 and 1670 cm21, the two amide I’ frequencies of the acetylproline-ND2 dipeptide.The same detection principle has also been used in recent 2D IR studies of metal carbonyls.13 PhysChemComm, 2002, 5(3), 17–26 19Fig. 4 Spectral interferometry of the echo signal for acetylproline-ND2 in D2O with the local oscillator delayed 1.5 ps from the echo signal.2 Shown are both the measured interferometric signal, and the processed signal, which represents the emitted electric field of the echo. This signal shows peaks at 1610 and 1670 cm21 corresponding to the frequencies of the two amide units. The origin of the 2D IR spectrum The simplest example of the heterodyned 2D spectra is a molecule that has only one mode (say, i) in the spectral region of interest. Consider a rephasing pulse sequence. Initially pulse 1 excites the vibrational coherence (0,i) which evolves for time kt.In the second step pulses k2 and k3, with T ~ 0, transfer this coherence into its conjugates, (i,0) or into coherence between v ~ 1 and v ~ 2 states, i.e. (i z i,i). Therefore, the system responds with a frequency of 2vi during the time t, and with frequencies of either vi or vi 2 Di during the time t. The two dimensional Fourier transform of the total signal will have a single peak in the vt dimension, from the vi term, but will have two peaks in the vt dimension, separated by the anharmonicity. Note that the coherences evolve with frequencies of different signs during the t and t periods. Therefore the 2D spectrum S(vt,vt) will show two peaks near the diagonal at the fourth quadrant (vt, 2 vt) ~ (vi,vi) and (vi 2 Di,vi).Quantum mechanics predicts that the fields at vi and vi 2 Di should be out of phase by p so that the corresponding real parts of S(vt,vt) have different signs. When the system consists of two or more coupled vibrators, it is helpful to utilize Feynman diagrams to include contributions from all possible excitation pathways. The 2D IR spectrum obtained by a rephasing pulse sequence can be schematically written as the superposition of contributions as follows, Xi=j Sabcd(vt,vt;T)~(2DFT) fSaibicidiT =b(0ij00ji0)z=0b(0ijiiji0){=a(0ijiijizi,i) (j z j,i), are also possible and they are indicated as orange zSa {Sa {Saibjcjdf T=0e(0ijjijjzj,i) ibicjdjT½ =d (0ij00jj0){=c(0ijiijizj,i) zSaibjcidjT =0d (0ijjijj0){=0c(0ijjijizj,i) contributions are much weaker because they involve transitions (1) ibicf df T=e(0ijiijjzj,i) between the two quantum states of one mode and the one quantum state of another, which are forbidden in the extreme weak coupling limit.For the nonrephasing pulse sequence, the 2D IR spectrum is given by12 ibjcf diT=0a(0ijjijizi,i)g {Sa A more general form of this equation is given in refs. 9 and 20. Each pulse in the IR sequence and the generated field can in principle induce transitions on a different mode of the molecule so up to four different transition dipoles may be 20 PhysChemComm , 2002, 5(3), 17–26 involved in any one term. The I terms represent the various ways (Liouville paths) the infrared pulses can excite the vibrational states, with the three pairs of indices taken from left to right corresponding to the coherences created by the first, second, and third pulses.Each term includes the transition dipole amplitude factors that determine the strengths of the transitions in the sequence, the dynamics (i.e. spectral diffusion and T1 relaxation), and frequency factors for that pathway. Ia, Ib, and Ib’ involve all the IR pulses successively interacting with only a single mode of the system and the dynamics is for only that mode; Ic, Id, Ic’, and Id’ arise from the pulses interrogating different modes and the dynamics involves both modes. Ie, Ie’, and Ia’ involve at least one forbidden transition. Other paths would be needed if pulses other than k2 and k3 were to overlap, which is not the case for the spectra reported here.These factors are all discussed in previous reports.12 The contributions of each term to the signal is dependent on the naibjckdlm factor, which depends on the polarization of the IR pulses, the angles between the transition dipoles associated with different modes of the molecule, and the rotational dynamics.36,37 The n....m symbolizes an average over the orientations of the transition dipoles involved in each path whose modes are labeled with the indices i, j and f. States that do not participate in transitions unless there is some coupling between modes are labeled f for ‘‘forbidden’’. The internal and overall motions are considered to be independent.Manipulating these factors can enhance or suppress specific terms in eqn. (1) and will be discussed below. Fig. 5a demonstrates the predicted rephasing spectrum in the fourth quadrant (vt, 2 vt) for the case of two weakly coupled vibrators, i and j, separated in frequency by more than their coupling. The out-of-phase peaks on the diagonal shown in blue at (vi,vi) and (vi 2 Di,vi) represent single oscillator spectra as discussed above and correspond to Ib, Ib’, and Ia in eqn. (1). The out-of-phase peaks shown in pink occur at points (vj,vi) and (vj 2 Dij,vi) (i | j) which are in the cross peak region and correspond to Id, Id’ and Ic, Ic’. The origin of these cross peaks can be understood by examining the excitation pathways.Consider the vibrational coherent superpositions (0,i) and (0,j) generated by pulse 1 and evolved for time t. In addition to the pathways that give rise to the diagonal peaks, the (0,i) coherence can be transferred by pulses 2 and 3 into coherences such as (j,0) or (i z j,i) that will evolve with frequencies of either vj or vj 2 Dij during the time t (see Fig. 1b for an illustration of the latter excitation pathway). These joint excitations produce the pair of cross peaks below the diagonal in Fig. 5a. They have opposite signs and are separated by the mixed mode anharmonicity Dij in vt. Similarly, the transfer of (0,j) coherence to (i,0) or (i z j,j) gives rise to the pair of cross peaks above the diagonal. These cross peaks will only be manifest if the i and j modes are coupled in some way, and the 2D IR spectrum provides a quantitative measure of this coupling.In addition to the coherence transfer pathways discussed above, other pathways, such as transfer from (0,i) to dashed peaks in Fig. 5a and correspond to Ie, Ie’ and Ia’. For the systems considered below in this Perspective, theirFig. 5 The real part of 2D IR spectra in the rephasing (a) and nonrephasing (b) quadrants predicted by eqn. (1) and (2) for two vibrators i and j. The peaks for vibrator i along the vt axis at vt ~ vi are labeled a–l, corresponding to the subscript of I in the equations. The I, I’, and I@ paths with the same subscript produce 2D IR peaks at the same frequencies. Peaks are labeled with ‘‘z’’ and ‘‘2’’ to designate the sign.The diagonal peaks in blue are separated by the diagonal anharmonicity Di or Dj. The cross peaks in pink are separated by the off-diagonal anharmonicity Dij. The ‘‘forbidden’’ peaks in the extremely weak coupling limit are shown in dashed orange. Xi=j Sabcd(vt,vt;T)~(2DFT) fSaibicidiT =b(i0j00ji0)z=0b(i0jiiji0){=a(i0jiijizi,i) single feature is observed whose center frequency lies between (vi,vi) and (vi 2 Di,vi) and depends on the 0–1 and 1–2 relaxation times. However, the 0–1 and 1–2 transitions can ibjcjdiT =00 b(i0jijji0){=g(i0jijjizj,j) be at least partially resolved in the complex portions of the (2) zSaibicjdjT½ =d (i0j00jj0){=c(i0jiijizj,i) zSa {Saibicf df T=e(i0jiijjzj,i) {Saibjcidf T=l (i0jijjizi,j) ibjcf djT=h(i0jijjjzj,j)g e, Ih, and Il) are different from the rephasing case.{Sa Fig. 5b shows the predicted 2D IR spectrum obtained by the nonrephasing pulse sequence for the same two coupled vibrators. The diagonal peaks shown in blue again represent the single oscillator spectra, involving coherence transfer from (i,0) to (i,0) and (i z i,i), and correspond to Ib, Ib’ and Ia in eqn. (2). Because the coherence evolution during the t and t periods involves frequencies of the same signs, the peaks are located in the first quadrant (vt,vt) of the 2D spectrum. In contrast to the rephasing case in Fig. 5a, some of the cross peaks, corresponding to Ib@ and Ig and shown in pink, appear along the diagonal but still are separated by the mixed mode anharmonicity.Also, the locations of the weak contributions in orange (I spectra since their third order responses have different signs (Figs. 6b–c). This demonstrates one of the advantages in retaining spectral phase information in 2D IR experiments. The slight elongation of the spectrum in Fig. 6a indicates the presence of a small inhomogeneous frequency distribution. Using the multipoint correlation function obtained from stimulated photon echo experiments and the vibrational relaxation times measured from IR pump–probe methods, the 2D IR spectrum of NMAD has been successfully simulated.7 It was found that most of the NMAD vibrational frequency distribution is motionally narrowed with a pure dephasing time of 1.12 ps.This suggests that the energy distribution of amide transitions due to amide structure fluctuations appears to be averaged out very rapidly when the amide is free in D2O. The power of 2D IR-COSY in structural determination was demonstrated on the acetylproline-NH2 dipeptide.9 Fig. 7a The existence of cross-peaks in the 2D IR spectrum implies a coupling between the oscillators. However, it is important to note that cross-peaks can be present in the spectrum when Dij ~0: this is because the coupling can appear indirectly, through interactions with the solvent or other modes of the molecule. For example, if a joint excitation, such as | i z j m, has energy fluctuations that are not identical to the sum of the fluctuations of the parts composing it, there will be cross-peaks in the 2D IR spectrum. This will be true even if the energies of the parts add to give the energy of the joint state (i.e.if Dij~0). In such a case the coupling arises through the interactions of the oscillators with the bath composed of the solvent and the other modes of the molecule. Such indirect coupling, although very interesting, may not be so simply related to the structure of the molecule. This situation will occur when the dynamical parts of Ic and Id are not equal, or Ic’ and Id’ are not equal in eqn. (1). These conditions are analogous to those responsible for the appearance of the diagonal peaks in the 2D IR spectra of a single oscillator when the diagonal anharmonicity is zero.There can still be differences in the coupling of the 0–1 and 1–2 transitions of a single oscillator to the bath modes, which would result in changes in the vibrational relaxation and fluctuations in the anharmonicity, hence a difference in the dynamical parts of Ia and Ib, I’b. Therefore knowledge of the vibrational dynamics of the modes is needed to determine the coupling. Single vibrator: NMA The first demonstration of the heterodyned 2D photon echo method is used to examine responses from the single peptide vibrator, N-methylacetamide-D [NMAD; CH3(CO)- ND(CH3)],2,7 which is the simplest model compound for the linkage between peptide units. It has a single COND unit that gives rise to the so-called amide-I’ transition which is primarily CLO stretch.Shown in Fig. 6a–c are the absolute magnitude, real, and imaginary parts of the complex 2D IR spectrum for NMAD in D2O.7 The magnitude spectrum is centered slightly off the diagonal and has a nearly circular profile with a slight elongation along the diagonal. The shift of the diagonal peaks in the 2D IR spectra to slightly lower frequencies along vt is quite general. Since the FWHM of the 0–1 and 1–2 transitions are comparable to the anharmonicity 16 cm21, the individual contributions are not observed separately in the magnitude spectrum (Fig. 6a). Instead, a Two coupled vibrators: acetylproline PhysChemComm, 2002, 5(3), 17–26 21Fig. 6 Complex 2D IR rephasing spectrum for NMAD in D2O: (a) absolute magnitude; (b) real part; and (c) imaginary part of the spectrum.7 The magnitude spectrum has solely postitive features, whereas the real and imaginary portions have both positive and negative features labeled with ‘‘z’’ and ‘‘2’’.shows the absolute value rephasing 2D spectrum for acetylproline-NH2 in chloroform. The diagonal peaks are labeled A, B, and C. Peaks A and C are from the amide II and I vibrational modes located on the amino end of the molecule. Peak B is the amide I band on the acetyl end. Six cross-peaks are observed above the diagonal labeled D through I. Each of these consists of a pair of unresolved and out-ofphase peaks and result from coupling between the amide I and II modes. It has been determined that acetylproline-NH2 adopts two configurations in CDCl3.The six peaks in the magnitude spectrum result from each configuration having three cross-peaks on either side of the diagonal. The crosspeaks D and E correlate the two amide II bands with the two acetyl amide I bands; cross-peaks F and G correlate the two amide II bands with the two amino amide I bands; and crosspeaks H and I correlate the two acetyl and amino amide I bands. Their positions indicate that the two configurations have degenerate amino amide I frequencies, but different acetyl amide I and amino amide II frequencies. The coupling between the vibrational modes of each peptide unit can be measured from the intensities and frequencies of the cross peaks, which depend on the angular and spatial relations between the vibrational displacements. The angles can be measured independently by varying the polarization of the infrared pulses.For Fig. 7a, the excitation and local oscillator pulses all have the same laboratory polarization, 22 PhysChemComm , 2002, 5(3), 17–26 Fig. 7 The absolute magnitude 2D IR rephasing spectrum for acetylproline-NH2 in CDCl3 taken at three polarization conditions: (a) (0,0,0,0); (b) (0,p/2,p/2,0); and (c) (p/4,2p/4,p/2,0).9,11 Peaks labeled A and C are from the amide II and I vibrational modes on the amino end of the molecule. Peak B is the amide I band on the acetyl end. Peaks D–M are cross peaks. namely at angles (0,0,0,0) with respect to a laboratory fixed axis. For Fig. 7b, the polarization of the k2 and k3 excitation beams have been rotated, and are perpendicular to the polarization of k1 and kLO, namely (0,p/2,p/2,0), and the magnitude of the spectrum has been normalized with respect to that of Fig.7a. This makes the intensity of peaks A–C equal in the two polarizations. However, the intensities of the offdiagonal peaks after normalization vary. For example, the intensity of peak D decreases by 30% between Fig. 7a and 7b, while the intensity of peak F increases. The intensity of peak I does not appreciably change. Although the intensity changes are most easily observed in the magnitude spectra, the peaks may also change phase, which can only be determined by comparison of the real parts of the spectra in the two polarizations.By analyzing the polarization dependence of the intensity and phase for each cross peak, the angle between each coupled pair of transition dipoles has been determined. Because the amide-I modes of peptide units have transition dipoles whose directions are given by the positions of the amide atoms,38,39 structural information can be extracted from the measured angles between transition dipoles. From the polarization dependence, it was concluded that acetylproline- NH2 in CDCl3 adopts at least two conformations: the transition dipoles of the acetyl and amino amide I bands ofone conformation are at an angle of v20u with respect to one another, and in the other conformation this angle is 35u. The angles were related to possible structures by comparison to proposed structures generated from a normal mode analysis. The results indicate that the dihedral angles are close to those of an a-helix for one of the distributions, and to an extended structure for the other.Polarization dependences have been used to measure the angles between transition dipoles in peptides in double resonance 2D IR3,5 and heterodyned 2D IR9 experiments and yield structure information. However, extraction of structural information can be difficult when the cross-peaks overlap with the more intense diagonal peaks. In heterodyned and double resonance 2D IR experiments, the measurements in the two polarization conditions can be subtracted to yield difference spectra with the diagonal peaks removed, if the molecule does not rotate.36 In practice there is always some overall or internal rotation, so the method contains intrinsic uncertainties.Furthermore it can be difficult to normalize the two spectra recorded in different polarizations, especially when the cross-peaks and diagonal peaks overlap. These difficulties have been circumvented by a recent discovery of polarization conditions for the IR laser pulses that completely eliminate the diagonal peaks from the 2D IR spectra.11 These conditions allow significantly more information to be gathered about the coupling between different spatial regions of the molecule and about the dynamics of the structural changes for these regions. Fig. 7c shows the application of this technique to acetylproline-NH2 in chloroform.11 The polarization condition for k1, k2, k3 and kLO is (p/4,2p/4,p/2,0).For this condition, each of the orientation factors naibjckdlm becomes36 ncoshikcoshjl 2 coshilcoshjkm which vanishes for i~j~k~l corresponding to the diagonal peaks. In the case where two of the transition dipole directions in the molecule frame are involved in the interaction with the pulses, this signal vanishes when the two transition dipoles are parallel (P2 ~ 1). Under this ‘‘magic’’ polarization condition, the diagonal peaks A–C are suppressed completely in principle but by only about 90–95% because of the imperfections of the current polarizers: the spectral discernibility of the cross peaks is nevertheless substantially improved.Clearly J and K are a set of cross-peaks, as are L and M. The relative cross-peak intensities change because of their differing angular dependencies. For instance, cross-peak F, which is very weak in Fig. 7a taken with parallel polarization, has significant intensity in Fig. 7c as peak L because the two transition dipoles involved are almost at 90u. The cross-peak G in Fig. 7a also contributes to peak L. The transition dipoles that form peak H, which is not very intense in Fig. 7c, are at 35u. The signal corresponding to peak I is not visible in Fig. 7c because the transition dipoles involved are nearly parallel. Removing the intense diagonal peaks allows a more clear observation for the phase of the cross peaks in the real 2D spectrum. A simulation of the cross peak to account for the observed phase leads to the conclusion that the lower amide II band is correlated to the higher acetyl amide I band, and the cross-peaks G, H, and E belong to one conformer, while F, I, and D belong to another.These results help to distinguish the different conformers. Vibrational dynamics: effects of correlations Traditionally, the pulse sequence in photon echo experiments is designed to look at the system responses that arise from the rephasing diagrams. These are the processes that would result in the rephasing of any induced dipoles that are present in the inhomogeneous distribution of vibrational frequencies, and hence will result in an echo or equivalently line-narrowed spectra. However, we have shown that this conventional picture of photon echoes applies strictly only to two level systems.12 A vibrator or set of vibrators is in principle a multiple level system characterized by a number of Bohr frequencies, two in the case of a single oscillator, and N(N2 z N z 2)/2 for N oscillators, assuming a third order nonlinear optical process.It is essential to consider the statistical correlations in the inhomogeneous distributions associated with these Bohr frequencies to predict the rephasing (echo-like behavior) and hence line narrowing properties of the system. Let us first consider the diagonal peaks that give rise to single oscillator spectra in Fig. 5. For the blue diagonal peaks located at (vi,vi), the same vibrational coherence is involved in both the t and t periods.The dephasing of vibrational coherence during time t due to spectral inhomogeneity can be exactly rephased during time t (apart from relaxation) because the frequency fluctuation of the same vibration coherence is strictly correlated during the two time periods. The absolute value 2D lineshape is elongated along the diagonal axis of the rephasing spectrum, showing an inhomogeneously broadened width and is line-narrowed when slicing through the antidiagonal. There is, however, no line narrowing in the nonrephasing spectrum and hence the peak height is reduced. This corresponds to the traditional echo in a simple two-level system. For the blue diagonal peaks located at (vi 2 Di,vi) in Fig.5, however, two different coherences associated with one oscillator are involved, the degree of line narrowing in the rephasing and nonrephasing spectra depends on the correlation between the fluctuations of the 1–2 and 0–1 transitions, so that the frequency correlation between the anharmonicity Di and vi determines the spectral lineshape. For zero correlation, the axis of elongation in the rephasing spectrum lies along the diagonal if the two transitions have similar total dephasing rates. Positive and negative correlations make the elongation axis lie at an angle counter-clockwise and clockwise from that of zero correlation, respectively. The nonrephasing spectrum exhibits no line-narrowing as before, but the linewidth along the vt axis, compared to that of the zero correlation case, changes from being narrower to broader when the correlation changes from positive to negative.When the linewidths of the two peaks at (vi,vi) and (vi 2 Di,vi) are broader than the diagonal anharmonicity Di, the 2D lineshape can be complicated because the composite effects from both peaks are not resolved. Fig. 8a shows the experimental absolute magnitude 2D IR spectra in the rephasing and nonrephasing quadrants for acetone in ethylene glycol. The rephasing spectrum of acetone in ethylene glycol is a lot more elongated along the diagonal, indicating line narrowing, while the nonrephasing spectrum is four times less intense and not elongated. These are characteristics of inhomogeneous broadening.Because ethylene glycol is a viscous solvent, it is expected that the CLO stretching mode of acetone molecules may sample different slowly varying solvent environments that form a fixed inhomogeneous distribution. If, in addition, we assume a very rapid spectral diffusion, the conditions of Bloch dynamics prevail. Fig. 8b shows a global fit to the experimental 2D spectra. The model takes into account the motionally narrowed dephasing rates of the 0–1 and 1–2 transitions, the Gaussian inhomogeneous distributions of the 0–1 transition and diagonal anharmonicity, and the correlation between them. The overall lineshapes are well reproduced. It is found that the lineshape is sensitive to the combination of the fluctuation of the anharmonicity and its correlation to the 0–1 transition frequency but not sensitive to the individual values.Although this is the case for this particular system, there may exist other systems that exhibit significant anharmonicity fluctuation and strong correlation, and a unique determination of individual parameters could become possible. We emphasize the significant differences between the spectra in the two quadrants. The result for acetone in ethylene glycol can be contrasted to that of NMAD in D2O. In Fig. 6a the peak is only slightly PhysChemComm, 2002, 5(3), 17–26 23Fig. 8 Experimental absolute magnitude 2D rephasing (a) and nonrephasing (b) spectra for acetone in ethylene glycol.12 The lineshape and intensity of the two spectra show the characteristics of inhomogeneous broadening.The simulated absolute magnitude 2D rephasing (c) and nonrephasing (d) spectra were obtained by global fitting assuming Bloch dynamics. The correlation between the frequency distributions of the 0–1 transition and diagonal anharmonicity has been taken into account. elongated along the diagonal, suggesting the inhomogeneous frequency distribution for NMAD is small. However, the 2D IR spectrum was acquired only in the rephasing quadrant. A comparison to the nonrephasing spectrum is expected to give a clearer picture of the underlying vibrational dynamics. For coupled vibrators, the cross peaks in 2D spectra (pink peaks in Fig. 5) not only yield information on the geometric arrangements of the transition dipoles but also reveal the frequency correlation between the frequency distributions of the two vibrators because coherences on different vibrators are involved in the t and t evolution.Fig. 9 illustrates the rephasing and nonrephasing absolute magnitude 2D spectra for two coupled vibrators with cij ~ 1, 0, and 21 where cij is the statistical correlation coefficients for the frequency fluctuations of the two vibrators. The case of cij ~ 0 may be applicable to cases such as the dipeptides that have been studied9,33 where the two coupled vibrators are at different ends of the peptide and might each independently interact with surrounding solvents. The lineshape and intensity of the cross peaks clearly depend on the frequency correlation between the two vibrators.In the rephasing spectra the cross peaks are pronounced and linenarrowed for positive correlation (Fig. 9a), but broadened and diminished as the correlation becomes zero and negative (Fig. 9c and e). On the contrary, in the nonrephasing spectra the cross peaks are line-narrowed in the direction perpendicular to that in the rephasing spectra for negative correlation (Fig. 9f), but broadened as the correlation becomes zero and positive (Fig. 9d and b). The results imply that the coupling strength between vibrators cannot be determined merely based on the cross peak intensities in a rephasing spectrum because a low intensity may result from negative correlations rather than from small coupling. Measuring rephasing and nonrephasing spectra separately is expected to provide more complete information on coupling and correlation.In addition to providing information on correlation, comparison of the rephasing and nonrephasing spectra also helps to resolve closely spaced spectral features.12 For instance, when the separation in frequencies between two uncoupled vibrators is less than the linewidth, the two peaks may 24 PhysChemComm , 2002, 5(3), 17–26 constructively interfere into a single peak that is elongated along the diagonal in the rephasing spectrum. It is not straightforward to tell the existence of two vibrators. The destructive interference in the overlapping region between vibrators in the nonrephasing spectrum, however, may give rise to a valley appearing between two peaks.This behavior was observed for acetylproline-ND2 in D2O and led to the conclusion that this dipeptide adopts multiple conformations in D2O as well as in chloroform.12 Vibrational dynamics: T dependence In addition to inhomogeneous distributions and their correlations, the 2D IR lineshapes are influenced by other factors, such as homogeneous lifetime and population/coherence transfer. Information on these factors is obtained from the T-dependence of the signal. As shown in Fig. 5a, the vibrational dynamics of the two amide I bands of acetylproline-NH2 appears to be qualitatively similar in CDCl3 according to their 2D IR profiles at T ~ 0. The 2D spectrum obtained in a THIRSTY experiment at T ~ 1000 fs also exhibits similar profiles for peaks B and C.9 However, the intensity of peak B decays more rapidly with T than does peak C, indicating that the T dependence of the I terms in eqn.(1) is different for the two amide I bands. This difference could arise from differences in spectral diffusion or from population relaxation between the two amide I bands. From simulations, it was concluded that the accelerated decay of peak B compared to peak C is due to faster population relaxation. The differences in relaxation rates are also apparent in the Fourier transform of the photon echo signal as a function of t delay. This conclusion was confirmed later by pump–probe measurements in which the T1 of the acetyl amide I mode was found to be half of that of the amino amide I mode.34 For the cross-peaks each Liouville path depends on the vibrational dynamics at both the acetyl and amino ends of the molecule.At T ~ 1000 fs, the cross-peaks H and I appear to have become better resolved and shifted to higher frequencies along vt, and peak H depolarizes.9 These effects were0 Fig. 9 Calculated absolute magnitude 2D spectra for two coupled vibrators at vi ~1700 cm21 and v 0~1660 cm21.12 The dynamical parameters are identical for the two vibrators. Frames (a), (c), and (e) are rephasing spectra with the inhomogeneous frequency distributions of the two vibrators being strictly correlated, not correlated, and strictly anticorrelated, respectively. Namely, the statistical correlation coefficent cij ~ 1, 0, and 21, respectively.Frames (b), (d), and (f) are the corresponding nonrephasing spectra. attributed to the occurrence of coherence and/or population transfer occurs during the waiting time T in the absence of the infrared fields. If, for example, population transfer occurred between states |im and |jm during time T, the first three terms of eqn. (1), which normally give rise to the diagonal peaks, will now produce cross-peaks. If this occurs, the polarization would be affected because naibicjdjm does not have the same angular dependence as naibjcidjm, and the signal intensity would be affected as well. Pump–probe measurements showed that the energy transfer between amide I and amide II states significantly contribute to the relaxation of the amino end amide I mode.34 Conclusions There are significant advantages in using 2D IR nonlinear methods, such as heterodyned echoes, rather than FTIR.These advantages include: (1) for N oscillators there are N(N z 1)/2 times more observables in the nonlinear infrared experiment compared with FTIR; (2) coupling of modes can be obtained directly and free from models in 2D IR; (3) properties of potentials such as the diagonal and off-diagonal anharmonicities are measured directly in the 2D IR experiment but not in FTIR. The complexity of linear IR spectra in the overtone and combination band regions is greatly reduced in 2D IR; (4) the various dynamical contributions to the vibrational spectra are directly measured in 2D IR while linear IR measures only the j dipole correlation function from which it is difficult to extract unique properties of the dynamics of the frequency distribution.The inhomogeneous distributions and the correlations between them are observables of 2D IR while FTIR cannot adequately dissect the lineshapes. Population and coherence transfer can be measured in 2D IR but not generally in FTIR. However, there are currently some limitations of nonlinear IR, the most important of which is the difficulty of generating phase controlled femtosecond pulses of IR radiation over the complete range needed to study all modes of interest. We expect that this limitation will soon be removed by advances in laser technology and experimental design. So far 2D IR has been applied to small molecules, peptides and liquids.Extension to larger systems, such as DNA, membranes, polymers, or even cells, should be straightforward considering the spectral line narrowing capability of this technique and the ease of isotope editing40 different pieces of the structures. The 2D IR spectroscopy provides a combination of structure sensitivity and time resolution. It promises to become a powerful technique in determining distributions of structures that are averaged out in NMR, such as the random coil state of proteins or the transient opening of membrane channels. The methods are immediately employable as structural probes of kinetic processes that can be initiated with light pulses such as protein folding or chemical reactions.There are many variants of 2D IR and higher dimensional IR to be explored. The information content available to multidimensional IR is sufficiently large to make it a method of PhysChemComm, 2002, 5(3), 17–26 25choice for monitoring distributions of complex molecular assemblies. Therefore, the field looks very promising indeed. Acknowledgement We gratefully acknowledge Dr Martin Zanni for his essential contribution to many of the experiments that are discussed and referenced here. The research was supported by grants toRMH from NIH (GM12592), NSF (CHE99-88138), with instrumentation developed in the NIH Research Resource RR-13456. References 1 P. Hamm, M. H. Lim and R. M. Hochstrasser, J. Phys. Chem. B, 1998, 102, 6123.ƒƒ The first example of 2D IR spectroscopy using the double resonance technique. It looks at three globular proteins, and simulates their results with an excitonic model that is also described. 2 M. C. Asplund, M. T. Zanni and R. M. Hochstrasser, Proc. Natl. Acad. Sci. USA, 2000, 97, 8219. ƒƒ The first publication of heterodyned 2D IR spectroscopy. 3 P. Hamm, M. Lim, W. F. DeGrado and R. M. Hochstrasser, Proc. Natl. Acad. Sci. USA, 1999, 96, 2036. ƒƒ This paper quantitatively simulates the 2D IR spectra of a pentapeptide, and addresses some of the fundamental issues of how to determine structures from a 2D spectra for the first time. 4 P. Hamm, M. Lim, W. F. DeGrado and R. M. Hochstrasser, J. Chem. Phys., 2000, 112, 1907.5 S. Woutersen and P. Hamm, J. Phys. Chem. B, 2000, 104, 11316. ƒ Another example of how polarization measurements can be used to facilitate structure determination. It also addresses the importance of including through-bond couplings when interpreting 2D-IR spectra. 6 S. Woutersen and P. Hamm, J. Chem. Phys., 2001, 114, 2727. 7 M. T. Zanni, M. C. Asplund and R. M. Hochstrasser, J. Chem. Phys., 2001, 114, 4579. ƒ A quantitative study on the lineshapes of the diagonal peaks in heterodyned 2D IR spectra for a model peptide unit. 8 S. Woutersen, Y. Mu, G. Stock and P. Hamm, Chem. Phys., 2001, 266, 137. 9 M. T. Zanni, S. Gnanakaran, J. Stenger and R. M. Hochstrasser, J. Phys. Chem. B, 2001, 105, 6520. ƒƒ IR-COSY and THIRSTY study of a dipeptide. It includes a model to explain the cross-peaks observed in the 2D IR spectra, requirements for cross-peaks in 2D IR, and a method to determine angles between vibrational dipoles from the polarized spectra.10 S. Woutersen, Y. G. Mu, G. Stock and P. Hamm, Proc. Natl. Acad. Sci. USA, 2001, 98, 11254. 11 M. T. Zanni, N.-H. Ge, Y. S. Kim and R. M. Hochstrasser, Proc. Natl. Acad. Sci. USA, 2001, 98, 11265. ƒƒ A novel polarization condition is described that eliminates diagonal peaks in 2D spectra. 12 N.-H. Ge, M. T. Zanni and R. M. Hochstrasser, J. Phys. Chem. A, in press. ƒƒ This paper shows that 2D IR spectra collected separately in rephasing and nonrephasing quadrants can provide optimal information needed to reveal correlated molecular vibrations. 13 O.Golonzka, M. Khalil, N. Demirdoven and A. Tokmakoff, 26 PhysChemComm , 2002, 5(3), 17–26 ƒPhys. Rev. Lett., 2001, 86, 2154. ƒ This paper presents and discusses a 2D IR spectrum with well-resolved peaks. 14 W. Zhao and J. C. Wright, Phys. Rev. Lett., 2000, 84, 1411. 15 G. R. Fleming, D. A. Blank, M. Cho and A. Tokmakoff, Pract. Spectrosc., 2001, 26, 437. 16 V. Astinov, K. J. Kubarych, C. J. Milne and R. J. D. Miller, Chem. Phys. Lett., 2000, 327, 334. 17 Y. Tanimura and S. Mukamel, J. Chem. Phys., 1993, 99, 9496. This paper introduces the multidimensional approach to optical spectroscopy. 18 S. Hahn, K. Kwak and M. Cho, J. Chem. Phys., 2000, 112, 4553. 19 P. Hamm and R. M. Hochstrasser, in Ultrafast Infared and Raman Spectroscopy, ed. M. D. Fayer, Marcel Dekker Inc., New York, 2001 p. 273. 20 W. M. Zhang, V. Chernyak and S. Mukamel, J. Chem. Phys., 1999, 110, 5011. ƒ A theoretical description of 2D four-wave-mixing techniques and their application to electronic and infrared spectroscopy. 21 A. Piryatinski, S. Tretiak, V. Chernyak and S. Mukamel, J. Raman Spectrosc., 2000, 31, 125. 22 D. Keusters, H. S. Tan and W. S. Warren, J. Phys. Chem. A, 1999, 103, 10369. ƒ This paper presents comparisons between NMR and 2D optical spectroscopies that highlights the role that the phases of the pulses and the beam directions play in determining the spectra. 23 Chem. Phys., 2001, 266, 137–350. ƒ This issue of Chemical Physics (#’s 2 and 3) contains articles about the recent advances being made in optical and infrared multidimensional spectroscopies. 24 A. Tokmakoff and M. D. Fayer, Acc. Chem. Res., 1995, 28, 437. 25 P. Hamm, M. Lim and R. M. Hochstrasser, Phys. Rev. Lett., 1998, 81, 5326. 26 (a) M. C. Asplund, M. Lim and R. M. Hochstrasser, Chem. Phys. Lett., 2000, 323, 269; (b) M. C. Asplund, M. Lim and R. M. Hochstrasser, Chem. Phys. Lett. 2001, 340, 611. J. Phys. Chem. A, 2001, 105, 8025. 27 K. A. Merchant, D. E. Thompson and M. D. Fayer, Phys. Rev. Lett., 2001, 86, 3899. 28 N. Demirdoeven, M. Khalil, O. Golonzka and A. Tokmakoff, 29 J. P. Likforman, M. Joffre and V. Thierry-Mieg, Opt. Lett., 1997, 22, 1104. ƒ Spectral interferometric detection of optical echoes. 30 A. Seilmer and W. Kaiser, in Ultrashort laser pulses, ed. W. Kaiser, Springer-Verlag, New York, 1993 p. 279. 31 J. C. Owrutsky, D. Raftery and R. M. Hochstrasser, Ann. Rev. Phys. Chem., 1994, 45, 519. 32 L. K. Iwaki, J. C. Deak, S. T. Rhea and D. D. Dlott, Pract. Spectrosc., 2001, 26, 541. 33 S. Gnanakaran and R. M. Hochstrasser, J. Am. Chem. Soc, 2001, 123, 12886. 34 I. Rubtsov and R. M. Hochstrasser, to be published. 35 S. Mukamel, Principles of nonlinear spectroscopy; Oxford University Press, New York, 1995. 36 R. M. Hochstrasser, Chem. Phys., 2001, 266, 273. 37 O. Golonzka and A. Tokmakoff, J. Chem. Phys., 2001, 115, 297. 38 S. Krimm and J. Bandekar, Adv. Protein Chem., 1986, 38, 181. 39 H. Torii and M. Tasumi, J. Chem. Phys., 1993, 96, 3379. 40 J. Torres, P. D. Adams and I. T. Arkin, J. Mol. Biol., 2000, 300, 677.

ISSN:1460-2733    

DOI:10.1039/b109935c

出版商:RSC    

年代:2002

数据来源: RSC