4790
J. Phys. Chem. B 2008, 112, 4790-4800
Conformational Dependence of Anharmonic Vibrations in Peptides: Amide-I Modes in Model Dipeptide Jianping Wang* Beijing National Laboratory for Molecular Sciences, Molecular Reaction Dynamics Laboratory, Institute of Chemistry, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China ReceiVed: NoVember 6, 2007; In Final Form: January 28, 2008
The conformational dependence of a set of anharmonic vibrational parameters for the amide-I modes in peptides has been examined at the level of Hartree-Fock theory. By using glycine dipeptide as a model molecule, the φ-ψ maps of diagonal and off-diagonal anharmonicities, local mode mixing angle, zero-order local mode frequencies, as well as intermode coupling, were calculated, and all were found to exhibit certain conformational sensitivities. The characteristics of the φ-ψ maps of the diagonal and off-diagonal anharmonicities were found to be complementary to each other, and the latter was found to correlate well with that of the mixing angle, reflecting the fact that these anharmonic parameters are interconnected and all determined by the same set of underlying anharmonic force field. The mean values of the diagonal and off-diagonal anharmonicities were found to be 15.7 cm-1 and 9.9 cm-1 respectively. The mean value for the two local mode frequency differences was estimated to be 9.7 cm-1, showing a nondegenerate local mode picture. For significant peptide conformations, the calculated anharmonic parameters were found to be in reasonable agreement with values obtained at the level of density functional theory as well as values obtained with recent two-dimensional infrared experiments.
Introduction Two-dimensional infrared (2D IR) spectroscopy is very sensitive to molecular structures.1-12 It can reveal anharmonic properties of molecular vibrational modes (the so-called vibrators) and the vibrational interactions (the so-called couplings) among them. These vibrators are intrinsically anharmonic, and anharmonicity is one of the key ingredients in 2D IR. Diagonal anharmonicities and mixed-mode off-diagonal anharmonicities form the foundation of 2D IR spectroscopy.13 Recent 2D IR experimental studies of polypeptides have shown that the amide-I vibrators in R-helices14 and in β-sheets15 have different diagonal anharmonicities, suggesting that the anharmonicity is structural-dependent. On the other hand, it is advantageous to use vibrational spectroscopy to examine molecular structures because vibrations are ubiquitous in molecules. Many vibrators in a given molecular system can be regarded as structuresensitive “chromophores” upon IR light excitation. For example, a localized vibrator has its vibrational wave function confined onto a limited number of nuclei and thus can be used to probe local structures and/or local structural distributions. A delocalized vibrator, on the other hand, may reveal regional or globular structural information. Therefore, the 2D IR technique allows one to characterize steady-state molecular structures and monitor dynamical structures in condensed phases by using the vibrators as intrinsic structural probes, without the need of externally introduced molecular chromophores as often required by other spectroscopic techniques. Owing to its strong infrared (IR) transition dipole and its unique frequency location in the mid-IR region, the amide-I mode (mainly the CdO stretching) has long been recognized as a useful structural probe for polypeptides and proteins. Since * To whom the correspondence should be addressed. E-mail: jwang@ iccas.ac.cn. Phone: +86-010-62656806. Fax: +86-010-62563167.
early days, efforts have been made to develop empirical force field in order to simulate conventional IR absorption spectra of peptides.16 The IR spectra of globular proteins in the amide-I region17 have been simulated with the aid of transition dipole coupling.16 On the basis of these studies and also on Davydov’s electronic exciton theory,18 a molecular vibrational exciton (vibron) model was developed to interpret the one-dimensional (1D) IR and 2D IR spectra of the amide-I modes in polypeptides.1 On the other hand, ab initio computations and semiempirical modelings of amide-I mode couplings, spectral line widths, as well as transition frequencies and their distributions, have become the subjects of recent studies.1,19-32 There have also been extensive experimental studies of the amide-I vibrations by IR methods in recent years,33-36 especially by the 2D IR method.12,14,37-43 Experimental studies have shown that sitespecific local structures and dynamics can be extracted from 2D IR spectra of the amide-I modes.11,15,44,45 More recently, 2D IR experiments of peptides with 13C-labelings have revealed a nondegenerate zero-order frequency picture in a 27-membered transmembrane R-helix45 as well as in a 12-membered β-hairpin.44 Without the knowledge of local mode frequency, a set of degenerate zero-order local mode frequencies was often assumed when modeling the 1D and 2D IR spectra of the amide-I modes using the exciton model. In this regard, the connection between these anharmonic vibrational parameters and 2D IR spectroscopy observables is not well-established for the amide-I modes. Therefore, theoretical and experimental investigations of anharmonicities as well as local mode frequencies of the amide-I vibrational modes are of great importance. Conventionally molecular vibrations have been treated mostly as harmonic vibrators. In harmonic approximation, cubic and higher terms of the vibrational potential function are ignored. It is often the case that computed harmonic frequencies at a certain high level of theory for a molecule can be made
10.1021/jp710641x CCC: $40.75 © 2008 American Chemical Society Published on Web 03/26/2008
Amide-I Modes in Model Dipeptide comparable to the gas-phase experimental results only with the aid of a frequency scaling factor, suggesting that the anharmonic correction needs to be considered. Indeed, recent studies46,47 have shown that, by taking into account anharmonicity, reasonable agreement between computed frequencies and gas-phase experimental values can be obtained for molecules with moderate sizes. Such anharmonic correction is also needed in solution phase because vibrational modes and their energy levels in a polyatomic system are often influenced by chemical and solvent environment so as to cause the anharmonicity. A complete set of the 3N-6 anharmonic normal-mode frequencies in an N-atom molecule can be calculated by using a second-order vibrational perturbation theory (PT2)-based approach.48 Briefly, in the PT2 approach, by using analytical second derivatives of the total energy, a full set of cubic force constants and a set of semi-diagonal quartic force constants can be computed via central numerical differentiation formulation. Such computations can be easily parallelized in computer cluster systems. Using the PT2 approach, the anharmonicities of the amide-I, -II, and -A modes in typical di- and tri-peptides have been examined recently.49 The anharmonicities of the 3N-6 modes in a sugar molecule have also been examined very recently.13 These studies have shown that theoretical predictions of the diagonal and off-diagonal anharmonicities of the amide modes as well as the C-H(D) stretching modes in peptides and sugars are consistent with 2D IR experimental observables. Alternatively, one may utilize the vibrational self-consistent field method (VSCF)50 to evaluate the anharmonic vibrational frequencies of a polyatomic system. A very recent study51 using the VSCF approach at the level of MP2/DZP has demonstrated the presence of coupling between the amide-I and the amide-II modes in N-methylacetamide, confirming an earlier experimental result.52 It is of great interest to further examine the structural dependence of these anharmonic parameters and find out how the picture looks for certain types of vibrational modes, for example, the amide-I modes in polypeptides. In this paper, using glycine dipeptide, a widely used model molecule for investigations into the vibrational properties of the peptide amide group, we systematically examine the harmonic and anharmonic vibrational parameters of the two amide-I modes throughout the entire backbone conformation space, which usually consists of (φ, ψ) dihedral angles. We first examine the harmonic and anharmonic properties of the two amide-I modes in several representative conformations using Hartree-Fock (HF) theory and density functional theory (DFT) with B3LYP functional, from which we establish our methodology to compute the anharmonic parameters in the φ-ψ space. We then examine the harmonic and anharmonic normal-mode frequencies and local mode frequencies. We obtain and examine the φ-ψ maps of diagonal and mixed-mode off-diagonal anharmonicities of the two modes and their interconnections with each other and with that of the local mode wave function mixing angle. We then obtain the φ-ψ map of the intermode coupling not only by decomposing the two normal modes into local modes but also by applying an analytical formula in the weak coupling limit. We focus on examining and understanding the sensitivities of these anharmonic parameters to the peptide backbone conformation. We also discuss the computational results and compare them with recent 2D IR experimental observables. II. Computational Methods The vibrational energy of fundamental, overtone, and combination states of a polyatomic molecule in the normal mode
J. Phys. Chem. B, Vol. 112, No. 15, 2008 4791 basis can be approximated anharmonically as:53 3N-6
E(ni,nj) ) E0 + hc
∑ω i
i
( ) ni +
1
2
3N-6
+ hc
∑x iej
ij
( )( ) ni +
1
2
nj +
1
2
(1)
Here, ωi is the harmonic frequency; ni is the vibrational quantum number of the ith mode; xii and xij are the first-order anharmonic correction terms; E0 is zero-point energy; h and c are Plank’s constant and speed of light, respectively. By using the second-order perturbation approach,48,53,54 xii and xij can be obtained with the cubic and quartic terms of the vibrational potential function. N is the total number of atoms in a molecular system. By using eq 1, the anharmonic fundamental, the overtone and the combination transition frequencies are given by νi ) ωi + 2xii + 1/2 ∑j*i xij, ν2i ) 2νi + 2xii, and νij ) νi + νj + xij. Therefore, the diagonal anharmonicity of the ith mode (∆ii) and the off-diagonal anharmonicity of the ith and jth mode (∆ij) can be obtained in a straightforward way: ∆ii ) - 2xii and ∆ij ) - xij. By using these equations, the anharmonicities of the amide-I modes in the glycine dipeptide with various conformations on the basis of normal mode can be evaluated. It should be noted that because the cubic and quartic terms are included in the vibrational potential function, the anharmonicities ∆ii and ∆ij contain the so-called cubic and quartic anharmonic contributions. Vibrational Coupling. Quantum chemical computations can be used to obtain vibrational couplings of the amide-I modes. The vibrational coupling is derived from the harmonic or anharmonic normal-mode frequencies on a local mode basis. This is done by assuming that the coupling of the amide-I modes is in a weak coupling regime and coupling with all of the other normal modes can be ignored, and the amide-I vibration is predominantly localized on the CdO bond. The protocol has been used by several groups in recent years in dealing with the amide-I couplings21,24,26,49,55,56 on the basis of harmonic normalmode frequencies. It has been called a Hessian matrix reconstruction method or wave function demixing method. Our recent study showed that such an operation can be performed by using anharmonic normal-mode frequencies.49 Basically, a reduced wave function (2 × 2 in dimension) for two amide-I eigenstates in a dipeptide can be obtained from the original normal mode wave functions (2 × 3N in dimension). Each element of the reduced matrix is either a sine or cosine function of the wave function mixing angle, thus from which the mixing angle can be easily obtained. A simple matrix operation yields local mode frequencies (harmonic or anharmonic zero-order transition energies) and an intermode coupling constant. It is the aim of this work to fully explore the vibrational coupling of the anharmonic amide-I modes of two adjacent amide groups in the entire backbone conformational space. The vibrational coupling can also be evaluated by treating the anharmonicity and the coupling perturbatively.38,57 The amide-I mode vibrational coupling in the weak coupling limit of a two-vibrator system can be deduced from diagonal and off-diagonal anharmonicities: |βij| ) |-
∆ij(νi - νj + ∆ii)(νj - νi + ∆jj) 1/2 | 2(∆ii + ∆jj)
(2)
Here, ∆i and ∆j are diagonal anharmonicities, and νi and νj are anharmonic frequencies for the ith and jth normal mode respectively. Similar formulas were derived in recent studies.14,58 This formula is useful to evaluate vibrational couplings based on vibrational frequencies and anharmonicities. It should be
4792 J. Phys. Chem. B, Vol. 112, No. 15, 2008
Wang III. Results and Discussions
Figure 1. Molecular structure of glycine dipeptide, with two backbone dihedral angles defined as φ (∠CNCRC) and ψ (∠NCRCN).
pointed out that eq 2 was initially obtained for a nondegenerate two-level system,38,57 in which νi and νj represent uncoupled local mode frequencies. However, eq 2 can be used approximately on a normal mode basis since normal-mode frequencies can be experimentally measured by FTIR technique, and the coupling between the amide-I modes are assumed to be in the weak coupling regime.38 The anharmonicities can be obtained by using conventional pump-probe technique and/or recently developed 2D IR technique. Equation 2 is used in the present work to approximately evaluate the conformation dependence of the vibrational coupling between the amide-I modes of two adjacent amide groups on the basis of the normal modes. Model Molecule, Theory Level, and Basis Set. The harmonic and anharmonic vibrational frequencies of glycine dipeptide in gas phase in various conformations were calculated using Gaussian03.59 It is known that anharmonic frequency is expected to be in better agreement with gas-phase experimental results than harmonic frequency. Our main concern is the anharmonicity, which is the difference between different anharmonic vibrational energy levels. Our early work has shown that, given a reasonable basis set, HF theory could produce diagonal anharmonicities of the amide-I mode for various dipeptide conformations. However, to examine the effect of electron correlation on the calculation of the amide-I anharmonic frequencies, we also perform high-level computations, for example, Moller-Plesset perturbation theory (MP2) calculations for smaller peptide model such as formamide (data not shown), as well as DFT with B3LYP functional for the dipeptide with certain conformations. The molecular structure of the glycine dipeptide (Ac-Gly-NMe or CH3CONHCRH2CONHCH3) is given in Figure 1 with dihedral angle defined as φ (∠CNCRC) and ψ (∠NCRCN), respectively. The 10 representative conformations we selected are 1, RL2-helix (φ ) +90°, ψ ) -90°); 2, π-helix (-57°, -70°); 3, polyproline-II (PPII, -75°, +135°); 4, RL1helix (+60°, +60°); 5, C7 conformation (+82°, -69°); 6, extended structure (180°, -180°); 7, R-helix (-58°, -47°); 8, 310-helix (-50°, -25°); 9, antiparallel β-sheet (-139°, +135°); and 10, parallel β-sheet (-119°, +113°). Full conformation space (-180° e φ e +180°, -180° e ψ e +180°) for the peptide backbone is explored with a total of 169 samplings of partially optimized structures (∆φ ) ∆ψ ) 30°). The 6-31+G* basis set, with a set of diffuse and polarization functions included, is believed to be acceptable for polypeptide structure optimization as well as a harmonic frequency calculation60 using an empirical density functional EDF1.61 In addition, it is also known that the harmonic amide-I frequency could be overestimated by 20-80 cm-1 when using the MP2 theory or B3LYP functional. Considering the nature of our computations and the computational cost, we chose to use the 6-31+G* basis set. At the level of HF/6-31+G*, to complete an anharmonic frequency computation for a partially optimized glycine dipeptide, it requires approximately 18 h (wall clock) computation time of a dual CPU computer (2.33 GHz). In the following sections, we first compare and discuss the computed anharmonic parameters using DFT and HF theories.
A. Anharmonic Properties of Significant Peptide Conformations. The vibrational parameters of the two amide-I modes obtained by using DFT and HF approaches are found to correlate well for the selected conformations. As shown in Figure 2 (left column), two amide-I normal-mode frequencies are obtained: one is usually the asymmetric stretching motion of the CdO bond that has a higher frequency (panel A), and the other is usually the symmetric stretching motion of the CdO bond that is located in the low-frequency region (panel C). The HF frequencies (squares) are subtracted by 150 cm-1 to allow an easy comparison with the DFT frequencies (circles). These two panels show that the calculated harmonic frequencies of the two modes using the HF approach closely follow the trend of those using the DFT approach. Similar behavior is shown for the anharmonic frequencies (Figure 2 right column, panel B and D). Further, the frequency difference between the high- and lowfrequency modes is also obtained (panel E for the harmonic case and panel F for the anharmonic case). It is seen that the frequency differences obtained by HF and DFT are in very good agreement, being true in both the harmonic and the anharmonic cases. This suggests that the HF method is reasonable to use in order to obtain DFT-level results in terms of frequency differences. Furthermore, it is predicted that on the average the DFT anharmonic frequency of the high-frequency mode is 31.7 cm-1 lower than that of the harmonic frequency, and the value is 28.7 cm-1 for the low-frequency mode. It is also observed that on the average the HF anharmonic frequency of the highfrequency mode is 29.0 cm-1 lower than that of the harmonic frequency, and the value is 26.0 cm-1 for the low-frequency mode, indicating a similar anharmonic energy drop for the amide-I modes using different level of theories. Such a drop usually results from the change of the potential surface from the harmonic oscillator to the anharmonic oscillator. In addition, a slightly smaller splitting is predicted for the two modes for certain structures in the anharmonic picture (panel F) than in the harmonic picture (panel E), implying a smaller intermode coupling when these modes are viewed as anharmonic oscillators (see below). However, since the coupling is on the order of half of the energy separation, the change of the coupling is expected to be subtle. The diagonal anharmonicities of the two modes and the mixed-mode anharmonicities for these selected structures are calculated by using the HF and DFT methods, and the results are given in Figure 3. It is found that the diagonal anharmonicity of the amide-I mode is between 10 and 20 cm-1 (panel A and B), whereas the off-diagonal anharmonicity ranges from 0 to 20 cm-1 (panel C). It is also seen that for certain conformations perfect agreement is found between the obtained HF and the DFT results. These results are in agreement with a recent study49 in which the amide-I mode anharmonicities of a few representative dipeptides with a handful of conformations were examined at HF/6-31+G** and B3LYP/6-31+G** levels. All together, these results suggest that the anharmonicities of the amide-I modes predicted by the two theories are consistent when a midsized basis set is chosen. Recent 2D IR studies have yielded experimental values for the amide-I mode anharmonicities: an alanine rich R-helix has diagonal anharmonicity to be between 13.7 and 15.6 cm-1 and has the mixed mode off-diagonal anharmonicity ∆ij ) 9.0 ( 1.0 cm-1 for two nearest neighboring amide-I modes i and j;14 and ∆ii is determined to be 13.0 ( 2.0 cm-1 for a dipeptide AcProNH2 in a C7 (+82°, -69°) conformation.62 Our computation-predicted results are ∆ii ) 12.6 cm-1 (DFT) and 9.8 cm-1 (HF); ∆ij ) 13.7 cm-1 (DFT) and
Amide-I Modes in Model Dipeptide
J. Phys. Chem. B, Vol. 112, No. 15, 2008 4793
Figure 2. Calculated harmonic and anharmonic frequencies, frequency difference (in cm-1) of the two amide-I modes of selected secondary structures, using DFT (circle) and HF (square) approaches. 1: RL2-helix (+90°, -90°). 2: π-helix (-57°, -70°). 3: PPII (-75°, +135°). 4: RL1helix (+60°, +60°). 5: C7 (+82°, -69°). 6: extended structure (+180°, -180°). 7: R-helix (-58°, -47°). 8: 310-helix (-50°, -25°). 9: antiparallel β-sheet (-139°, +135°). 10: parallel β-sheet (-119°, +113°). (A): ωi (high-frequency mode). (B): νi (high-frequency mode). (C): ωj (lowfrequency mode). (D): νj (low-frequency mode). (E): ωi - ωj. (F): νi - νj. Here, the HF frequencies are subtracted by 150 cm-1.
Figure 3. Calculated diagonal anharmonicities, off-diagonal anharmonicities, and vibrational couplings (in cm-1) between the two amide-I modes of the selected secondary structures using DFT (circle) and HF (square) approaches. Sampling of the structures is given in the caption of Figure 2. (A): diagonal anharmonicity of the high-frequency mode. (B): diagonal anharmonicity of the low-frequency mode. (C): mixmode off-diagonal anharmonicity of the two modes.
19.7 cm-1 (HF) for the R-helix; and ∆ii ) 16.8 cm-1 (DFT) and 16.6 cm-1 (HF) for the C7 conformation. Therefore, all of these computed values are in reasonable agreement with experimental results. The local mode frequencies and vibrational coupling between the two amide-I modes have been evaluated using the wave function demixing protocol based on harmonic and anharmonic frequencies. The results are given in Figure 4. In the harmonic approximation (left column), it is seen that the obtained local mode frequencies are somewhat different from the normal-mode frequencies (Figure 2, left column, panel A and B): the frequency of the high-frequency mode (ω0i ) increases slightly, and the frequency of the low-frequency mode (ω0j ) decreases slightly from the normal to local mode pictures for these peptide structures. It is also observed that the HF zero-order frequencies
follow the DFT frequencies quite well in all cases examined. It is for this reason that the intermode couplings obtained using harmonic frequencies for various structures (βωij , panel C) show almost identical results from the HF and DFT methods. The anharmonic local mode frequencies (ν0i and ν0j ) obtained using the HF and DFT methods are given on the right column (panels D, E, and F). It is seen that the situation is quite similar to that on the harmonic basis: the anharmonic local mode frequencies behave similar to their counterparts in the harmonic picture. Further, the intermode couplings for various structures obtained on the anharmonic basis (βνij, panel F), like those from the left column (panel C), also show consistency in the HF method and DFT method. Although it has been known that the local mode frequencies are important in understanding and simulating both the 1D IR and the 2D IR results,44 measurements of the local mode frequencies turned out to be experimentally difficult. On the other hand, recent 2D IR studies have yielded experimental values for the amide-I mode couplings quite easily. For example, the coupling between two nearest neighboring amide units in the R-helix14 was determined to be 8.5 ( 1.8 cm-1, while in the present work it is predicted to be no smaller than 4.9 cm-1 (anharmonic, HF result) and no bigger than 6.4 cm-1 (harmonic, DFT result) for a perfect R-helix, as shown in Figure 4 (panels C and F, conformation no. 7). The coupling in the alanine dipeptide in the PPII conformation40was measured to be 1.5 ( 0.5 cm-1, while in the present work it is predicted to be between 0.2 and 1.1 cm-1, taken with four data points together in Figure 4. These values demonstrate that computed coupling values correlate well with experimental observables. The above benchmark measurements demonstrate that the HF and DFT methods with a proper choice of the basis set can be used to predict the anharmonic parameters for the amide-I modes of the glycine dipeptide quite successfully in several representative conformations. The obtained results are also in reasonable agreement with the available experimental measurables. With the information at hand, we further explore the anharmonic parameters for the glycine dipeptide in the backbone conformational space using the HF method at the level of HF/631+G*. Results are presented and discussed in the following sections.
4794 J. Phys. Chem. B, Vol. 112, No. 15, 2008
Wang
Figure 4. Calculated local mode frequencies and coupling constants (in cm-1) using DFT (circle) or HF (square) approaches. (A-C): harmonic approximation. (D-F): anharmonic approximation. (A) and (D): high-frequency mode. (B) and (E): low-frequency mode. (C) and (F): couplings based on the wave function mixing approach.
Figure 6. Statistics of the harmonic and anharmonic normal-mode frequencies (in cm-1) for the two amide-I modes. (A): harmonic frequency for the high-frequency mode. (B): harmonic frequency for the low-frequency mode. (C): anharmonic frequency for the highfrequency mode. (D): anharmonic frequency for the low-frequency mode. Figure 5. Conformation-dependent harmonic and anharmonic normalmode frequencies (in cm-1) for the two amide-I modes. (A): harmonic frequency (ωi) for the high-frequency mode. (B): harmonic frequency (ωj) for the low-frequency mode. (C): anharmonic frequency (νi) for the high-frequency mode. (D): anharmonic frequency (νj) for the lowfrequency mode. All frequencies were subtracted by 1900 cm-1.
B. Harmonic and Anharmonic Frequencies of the Glycine Dipeptide and Their Conformational Dependence. We first examine the harmonic normal-mode frequencies of the dipeptide; the results are given in Figure 5 (panels A and B). It is observed that the two amide-I frequencies are conformation dependent and show completely different dependences on the (φ, ψ) dihedrals. For the high-frequency mode (ωi, panel A), which is usually the asymmetric CdO stretching motion, a lowfrequency blue region is anti-diagonally located (from lower right to upper left). However, the blue region for the lowfrequency mode (ωj, panel B), which is usually the symmetric CdO stretching motion, is much smaller. It is also observed that ωi and ωj individually have C2 symmetry with the C2 axis perpendicular to the φ-ψ map and centered at (φ ) 0°, ψ ) 0°). The statistics of the transition energy of these two modes are given in Figure 6 (panels A and B, respectively), which shows the characteristics of the distributions of the two normalmode frequencies in the entire backbone conformation space. It is clearly seen that a higher density is shown in the lowfrequency region for ωi with a mean value of ω ˜ i ) 1949.9 cm-1,
however, a higher density is shown in the high-frequency region for ωj with a mean value of ω ˜ j ) 1928.4 cm-1. The profiles of the anharmonic normal-mode frequencies of the two amide-I modes closely resemble those of the harmonic normal-mode frequencies. The conformational dependence of the anharmonic frequencies for the high-frequency mode (νi) and the low-frequency mode (νj) is shown in Figure 5 (panels C and D), and the distributions of the two frequencies are shown in Figure 6 (panels C and D). The anharmonic picture is very much like the harmonic one in terms of the frequency distributions, also showing C2 symmetry in the φ-ψ space. However, a detailed analysis shows subtle difference in mean value and distribution of the harmonic and anharmonic frequencies. From Figure 6, it is clearly seen that a higher density is shown in the low-frequency region for νi with a mean value of ν˜ i ) 1919.8 cm-1, and a higher density is shown in the high-frequency region for νj with a mean value of ν˜ j ) 1899.4 cm-1. Therefore, on the average, the anharmonic frequencies are 30 cm-1 lower than their corresponding harmonic frequencies. In addition, it is seen from Figure 5 that values of ωi and ωj change more dramatically along ψ in the region of (-90° e φ e 90°), similar behavior has been observed in a recent DFT study by Stock and his coworkers.26 Our results show that such a pattern still remains in the conformational dependence of the anharmonic frequencies νi and νj.
Amide-I Modes in Model Dipeptide
Figure 7. Conformation-dependent anharmonicities (in cm-1) for the two amide-I modes. (A): diagonal anharmonicity for the high-frequency mode. (B): diagonal anharmonicity for the low-frequency mode. (C): mixed-mode off-diagonal anharmonicity for the two modes. (D): wave function mixing angle of two local modes.
Figure 8. Statistics of the anharmonicities (in cm-1) for the two amide-I modes. (A): diagonal anharmonicity for the high-frequency mode. (B): diagonal anharmonicity for the low-frequency mode. (C): mixedmode off-diagonal anharmonicity for the two modes. (D): sum of the diagonal and off-diagonal anharmonicities (∆ii + ∆jj + ∆ij). A normal distribution function is given as an approximate fit.
C. Anharmonicities of the Glycine Dipeptide and Their Conformational Dependence. The computed diagonal anharmonicities of the two amide-I modes show strong conformational dependence. This is demonstrated in Figure 7, panel A for the diagonal anharmonicity of the high-frequency mode (∆ii), and panel B for that of the low-frequency mode (∆jj). It can be seen clearly that ∆ii and ∆jj are more or less C2 symmetric in the φ-ψ space. It is found that ∆ii, ranging from 4.9 cm-1 to 26.4 cm-1, shows several domains with large values (red areas). The distribution of ∆ii is shown in Figure 8A, and the mean value is found to be 15.5 cm-1. On the other hand, it is found that ∆jj ranges from 8.5 cm-1 to 24.0 cm-1, also showing several domains with large values. However, it is clearly seen that the conformational dependence of ∆jj is quite different from that of ∆ii. The distribution of ∆jj is shown in Figure 8B, and the mean value is found to be 15.9 cm-1. The mean of the two mean values is 15.7 cm-1. Variation of the values of the diagonal anharmonicity in peptides has been documented in recent 2D IR experimental results, for example, in proline dipeptide (13 ( 2 cm-1),62 in the alanine-rich R-helix (14.2 cm-1),63 in the β-hairpin (between 13.8 and 14.6 cm-1),44 as well as in glycophorin A, a transmembrane helical dimer (11.6 cm-1).64
J. Phys. Chem. B, Vol. 112, No. 15, 2008 4795 These values are in general agreement with the predictions of the φ-ψ maps of ∆ii and ∆jj shown here. These experimental and computational results suggest that the diagonal anharmonicity is very sensitive to peptide local conformation. The computed mixed-mode off-diagonal anharmonicity (∆ij) of the dipeptide also shows strong conformational dependence and is found to be very sensitive to the (φ, ψ) angles. The result is given in Figure 7 (panel C). The value of ∆ij varies from -1.0 cm-1 to 20.3 cm-1, demonstrating several domains of larger values and also several domains of smaller values. It is found that ∆ij is also basically C2 symmetric in the φ-ψ space. In addition, red and blue regions are more clearly defined in ∆ij than in ∆ii or in ∆jj. The HF theory predicts that ∆ij is mostly positive with small negative value localized in the region where ψ and large |φ| values are small (blue region). The distribution of ∆ij is shown in Figure 8C, and the mean value is found to be 9.9 cm-1. The off-diagonal anharmonicity of the two amide-I modes in peptides has been studied by 2D IR spectroscopy very recently. For example, a negligible ∆ij was obtained (0.2 ( 0.2 cm-1) for alanine dipeptide with a PPII conformation,40 and ∆ij ) 9.0 ( 1.0 cm-1 for two nearest neighboring amide-I modes was found for alanine-rich R-helix.14 These values fall into the range of the predicted values in the glycine dipeptide, indicating that ab initio calculation at a reasonable level of theory is able to predict the anharmonic vibrational parameters for the amide-I modes with certain fidelity. Although the diagonal and off-diagonal anharmonicities change dramatically in the φ-ψ space, a detailed examination shows that the maps of ∆ii and ∆jj are somewhat similar. More interestingly, it is found that the profile of ∆ii (and ∆jj as well) seems to be complementary to that of ∆ij, meaning, for a given (φ, ψ) domain, if a red region (larger values) is shown in the map of ∆ii (or ∆jj), most likely, a blue region (smaller values) is shown in the map of ∆ij. Similar characteristics of the anharmonicities are also seen in Figure 3: for the 10 selected conformations, the larger the diagonal anharmonicities, the smaller the off-diagonal anharmonicities. These findings suggest that their sum, ∑i,j ∆ij ) ∆ii + ∆jj + ∆ij, might be invariant for all of the dihedral angles. To verify this, in Figure 8D, the distribution of the sum of the diagonal and off-diagonal anharmonicities in the entire φ-ψ space is calculated. A normal distribution function (with variance σ ) 5 cm-1) is also plotted to act as an approximate fit. This figure shows that the distribution of ∑i,j ∆ij in the φ-ψ space can be fitted roughly with a normal distribution function that is centered at ≈41.3 cm-1 with σ ≈ 5 cm-1. It is well-known that, in the vibrational exciton model of a two-vibrator system, the sum of the diagonal and off-diagonal anharmonicities is a constant. Our results here show that, roughly speaking, ab initio computed anharmonicities of the amide-I modes behave like those in the vibrational exciton model. In other words, the exciton model should be generally applicable to the amide-I modes. This result supports our early findings that the amide-I states are exciton-like.49 On the other hand, in the exciton model, going from uncoupled states to eigenstates usually accompanies the redistribution of the anharmonicities. Although the current study does not predict zeroorder diagonal anharmonicity for the amide-I mode needed for the exciton model, one can speculate that assuming a single (conformationally independent) diagonal anharmonicity, as done in simple models of 2D IR spectral calculations, could still be a valid approximation. This is supported by the results shown earlier in this section that the mean value of the ∆ii and ∆jj is 15.7 cm-1 for the two eigenstates, which is not much different from the value initially obtained from an isolated peptide unit,
4796 J. Phys. Chem. B, Vol. 112, No. 15, 2008
Figure 9. Conformation-dependent harmonic and anharmonic frequencies (in cm-1) for the two local amide-I modes. (A): harmonic frequency for the high-frequency mode. (B): harmonic frequency for the low-frequency mode. (C): anharmonic frequency for the highfrequency mode. (D): anharmonic frequency for the low-frequency mode. All frequency values were subtracted by 1900 cm-1.
N-methylacetamide (16 cm-1).1 This value is often used as the zero-order anharmonicity in spectral simulations. In addition, Figure 7D shows that the mixing angle between two local modes’ wave functions has almost identical conformational dependence as that of ∆ij. The significance of this result is discussed and explained in section E. D. Local Mode Frequencies of the Glycine Dipeptide and Their Conformational Dependence. Local frequencies of the two amide-I modes, obtained after the wave function demixing, are found to be nondegenerate and show strong conformational dependence. The results are shown in Figure 9 in both the harmonic and the anharmonic pictures. For a given peptide conformation, after decoupling two amide-I modes, the NMe side, or the C-terminus side (see Figure 1), generally has a higher zero-order frequency (ω0i ); whereas the Ac side, or the Nterminus side, generally shows a lower zero-order frequency (ω0j ). The dihedral angle dependence of the harmonic local mode frequencies is quite different for the two amide sites, which is clearly shown in Figure 9 (panel A and B). The statistics of the transition energy of these two local modes are given in Figure 10 (panels A and B), showing the characteristics of the distributions of the two frequencies in the backbone conformation space. Figure 9A depicts a relatively larger blue area (low in frequency) for ω0i , with its mean value as ω ˜ 0i ) -1 1943.8 cm , confirmed by its statistics shown in Figure 10A. Figure 9B shows a relatively smaller blue area (low in frequency) for ω0j , also confirmed by its statistics given in Figure 10B. In this case, it is found that a higher density is shown in the low-frequency region for ω0j with a mean value as ω ˜ 0j ) 1934.3 cm-1. By comparing Figure 6 and Figure 10, it is found that ω ˜i - ω ˜ 0i ) 6.1 cm-1 ()1949.9 - 1943.8), and 0 -1 ω ˜j - ω ˜ j ) -5.9 cm ()1928.4 - 1934.3). Thus the average frequency splitting between the two amide-I normal modes due to coupling is 12.0 cm-1 ()6.1 + 5.9). In addition, in the harmonic local mode picture, the average peak splitting (ω ˜ 0i 0 ω ˜ j ) was found to be 9.5 cm-1, representing the extent of local mode non-degeneracy. Further, the conformational dependence of the harmonic local mode frequencies bears a strong resemblance to that of the normal-mode frequencies. This is evident by comparing Figure 9 and Figure 5. The harmonic local mode
Wang
Figure 10. Statistics of the harmonic and anharmonic local mode frequencies (in cm-1) for the two amide-I modes. (A): harmonic frequency for the high-frequency mode (ω0i ). (B): harmonic frequency for the low-frequency mode (ω0j ). (C): anharmonic frequency for the high-frequency mode (ν0i ). (D): anharmonic frequency for the low-frequency mode (ν0j ).
picture shown in Figure 9A,B is also quite similar to that reported in a recent work,56 in which B3LYP functional and 6-31+G* basis set were used. The conformational dependence of the anharmonic local mode frequencies differs only slightly from the harmonic pictures. The anharmonic local mode picture is shown in Figure 9, panel C for the zero-order transition energy (ν0i ) of the amide-I unit at the C-terminus side, and panel D for the zero-order energy ( ν0j ) of the amide-I unit at the N-terminus side. The characteristics of the fluctuations of ν0i and ν0j in the φ-ψ space are analyzed, and the results are shown in Figure 10 panels C and D, respectively. It is found that their mean values are ν˜ 0i ) 1914.3 cm-1 and ν˜ 0j ) 1904.5 cm-1, thus ν˜ i - ν˜ 0i ) 5.5 cm-1 and ν˜ j - ν˜ 0j ) -5.1 cm-1 (ν˜ i and ν˜ j are taken from Figure 6). This means that the average frequency splitting between the two amide-I modes due to coupling is 10.6 cm-1(anharmonic picture), which is slightly smaller than that of the harmonic case (12.0 cm-1). In addition, in the anharmonic local mode picture, the average peak splitting (ν˜ 0i - ν˜ 0j ) was found to be 9.8 cm-1, representing the extent of local mode nondegeneracy in this case. Considering harmonic and anharmonic pictures all together, the average local mode frequency splitting is 9.7 cm-1. The HF theoretical calculation predicts that the zero-order frequencies are nondegenerate in either the harmonic or the anharmonic picture (Figure 9). Local amide-I mode nondegeneracy is believed to be an intrinsic property of polypeptide chains. Recent 1D and 2D IR studies of the β-hairpin44 with 13C labelings have shown that, indeed, two amide-I modes on the same peptide chain have different zero-order frequencies. Presumably, the nondegeneracy of the local states is due to the variation of local chemical and solvent environment.65 Such a nondegeneracy picture is very useful in understanding as well as simulating both 1D IR and 2D IR spectra of peptides, and should be taken into account when initializing a set of zeroorder transition energies for the amide-I modes of polypeptides and proteins according to their backbone dihedrals (φ, ψ) by which the corresponding amide units are arranged. Thus, these local mode frequencies bear site-specific local structure characteristics and are sensitive to environmental changes as well. On the contrary, strictly speaking, the normal modes in the polypeptide bear no direct local structure identities. This is because, when mode mixing occurs, the modes are delocalized and thus local mode characteristics fades away. Under such circumstances, the IR frequency-structure relationship might not be well-defined especially because the sign of the coupling
Amide-I Modes in Model Dipeptide
Figure 11. Conformation-dependent vibrational couplings (in cm-1) between the two local amide-I modes. (A): coupling obtained from the wave function demixing using harmonic frequencies. (B): coupling obtained from the wave function demixing using anharmonic frequencies. (C): perturbative coupling estimated from anharmonicities and harmonic frequencies. (D): perturbative coupling estimated from anharmonicities and anharmonic frequencies.
may flip. It is for these reasons that peak assignment in conventional infrared spectroscopy could become troublesome. The conformational dependence of the local amide-I mode frequencies in the harmonic and anharmonic pictures appears to be quite similar, suggesting that it is possible to scale the frequency from the former to the latter. This seems to be a reasonable choice when no anharmonic frequencies are available. However, the frequency scaling scheme is useful only when a very limited number of modes are considered (two amide-I modes in this case), because it is well-known that the frequency scaling is not applicable simultaneously to all the 3N-6 modes in a molecule in comparison with gas-phase infrared measurements. Obviously, even for a small subset of normal modes, the frequency scaling approach is limited to the gasphase case. To obtain solution-phase vibrational frequencies, one needs to use solution-phase frequency models, and several of which have been made available especially for the amide-I modes recently.30-32 The scaled harmonic frequencies or the unscaled anharmonic frequencies in this study may serve as modified zero-order (unshifted) frequencies, on which the solvent effect may be further considered. E. Intermode Couplings and Wave Function Mixing of the Glycine Dipeptide and Their Conformational Dependence. The computed couplings between two amide-I modes have been examined, and the results are shown in Figure 11 (panel A for the harmonic frequency-based wave function demixings and panel B for the anharmonic case). Overall, it is found that the coupling varies from -13.2 to +17.8 cm-1. The couplings are very sensitive to the peptide conformations. The profile of the couplings is C2 symmetric with respect to (φ ) 0°, ψ ) 0°). Strong negative couplings (blue region) mostly occur in the neighborhood of the (0°, 0°) dihedrals, and the negative coupling region is anti-diagonally distributed. This is found to be the case in both the harmonic and the anharmonic couplings. It is also found that the coupling changes more dramatically along the ψ axis in the region of (-90° e φ e 90°). The profile of the couplings shown in Figure 11 panel A is in agreement with previous studies, including a map obtained at the level of HF/6-31+G**,19 two maps obtained at the level of B3LYP/6-31+G*,21,26 and a very recent map56 obtained at
J. Phys. Chem. B, Vol. 112, No. 15, 2008 4797 the level of Perdew-Burke-Ernzerhof exchange correlation functional (PRPE)66 with TZ2P basis set. Our results here show that the couplings obtained from an anharmonic approximation yield a quite similar picture as the harmonic case. To further demonstrate this, we plot several slices of the couplings along the ψ axis at several fixed φ values (Figure 12, left column) and several slices along the φ axis at several fixed ψ values (Figure 12, right column). It is seen from these samplings that indeed the couplings obtained using the anharmonic frequencies follow quite closely to those obtained using the harmonic frequencies. A similar conclusion can also be drawn from Figure 4 for the sampled conformations. All of these results suggest that harmonic frequency calculations followed by the wave function demixings work well in predicting vibrational couplings. In addition, the conformational dependence of the amide-I coupling cannot be explained simply by using transition dipole interaction16 because of the coexistence of the well-known through-bond and through-space contributions to the coupling for two nearby amide units. Couplings obtained via eq 2 using the diagonal and offdiagonal anharmonicities and normal-mode frequencies, which we refer to as perturbative couplings, have also been examined. The results are shown in Figure 11, panels C and D, for the cases of using the harmonic and anharmonic frequencies, respectively. In these two panels, only the strength of couplings is contour plotted because no information of the sign of the coupling can be obtained by this approach. Quite surprisingly, the magnitude of the perturbative coupling and its profile as a function of the (φ, ψ) angle, including its symmetry, are quite similar to the ab initio computed results (panels A and B). Further analysis of the data in Figure 11, panels C and D, shows that the strength of the perturbative couplings varies from 0 to 20 cm-1, and the mean of the couplings is 6.6 cm-1. These results suggest that a simple analytical formulation in the weak coupling limit works reasonably well in quick estimating the strength of the vibrational couplings between amide-I modes in peptides, provided anharmonicities from nonlinear IR measurements and normal-mode frequencies from linear and/or nonlinear IR spectra are available. However, in order to get the correct sign of the coupling, one should either rely on molecular structure-linear IR analysis, perhaps even with the aid of isotope labeling, or turn to ab initio computations. The reason that the perturbative couplings can grasp the main signature of the ab initio coupling is simple: first, the ab initio obtained coupling is closely related to the energy separation of normal modes; second, the normal-mode frequencies are conformation dependent and so are the diagonal and off-diagonal anharmonicities. Therefore, it is not a surprise that a parameter, that is, the perturbative coupling, calculated using these ab initio-based conformation-dependent parameters, is also conformation sensitive. Here, the mixed-mode off-diagonal anharmonicity ∆ij is presumably the key factor. The degree of wave function mixing of two local amide states is found to correlate well with the off-diagonal anharmonicity ∆ij. This is shown in Figure 7D (mixing angle ξij) and Figure 7C (∆ij). The mixing angle ξij was computed according to recent works:21,49 ξij ) 1/2 tan-1[2|βij|/(ω0i - ω0j )]. Here, βij is the ab initio obtained coupling, and ω0i and ω0j are local mode harmonic frequencies, which can be replaced by anharmonic frequencies. The mixing angle can also be obtained directly from the reduced wave function matrix. Since the maximum mixing occurs at ξij ) 45°, the angle ξij presented in Figure 7D was converted in such a way so that ξij ) ξij (for ξij e 45° ) and ξij ) 90° - ξij (for 45° < ξij < 90°). In this way, the larger the
4798 J. Phys. Chem. B, Vol. 112, No. 15, 2008
Wang
Figure 12. Slices of couplings obtained from harmonic (squares) and anharmonic (hollow circles) based on wave function demixing approach. (A-C): ψ-dependence of the couplings at fixed φ. (D-F): φ-dependence of the couplings at fixed ψ. (A): φ ) -150°. (B): φ ) -60°. (C): φ ) +30°. (D): ψ ) -150°. (E): ψ ) -60°. (F): ψ ) +30°.
angle, the greater the extent that the wave function mixes. For the amide-I modes in peptides, it has been speculated previously that the coupling increases as the mixing angle increases.49 Our results here show that the relationship is valid throughout the entire conformational space, because the similarity between ∆ij and ξij is quite striking in Figure 7, panels C and D. In addition, as has been pointed out earlier,49 two localized states having a small off-diagonal anharmonicity only mix slightly; however, they may not necessarily have a small coupling constant. This is clearly shown in Figure 7 and Figure 11; the profile of ∆ij does not seem to correlate with those of the couplings. Smaller mixing occurs when |βij / (ω0i - ω0j )| , 1, in which case the two coupled states are highly localized on each of the individual amide unit. When the coupling is small, two states could still be largely delocalized if the energy separation were even smaller. These computational results are of great importance in understanding the relationship between molecular structure and 2D IR spectrum in biological context. First, anharmonicities and couplings are on the order of 10-15 cm-1, which can be easily measured with the aid of 2D IR technique. For a larger system, two amide-I modes could still be of measurable coupling strength (e.g., 1 cm-1) at a distance of 10 Å, a conclusion drawn from a simple calculation of the transition dipole-dipole coupling through space. Second, these anharmonic parameters are predicted to be conformational-dependent. The parameters include diagonal and off-diagonal anharmonicities, couplings, mixing angles, and local mode frequencies as well. Thus, they offer a variety of structure probes in reporting local structures and micro-environments of peptides and proteins in condensed phases. Third, anharmonic parameters can be easily computed as well for other modes, such as the amide-A mode (which is mainly the N-H stretching), amide-II modes (which is mainly the N-H in-plane-bending plus C-N stretching), as well as the C-H stretching modes. The characteristics of the anharmonic parameters for these modes and the influences upon isotopic substitution can also be examined easily by the approach outlined in the present work. In fact, the anharmonic properties of the entire normal mode collection (3N-6 modes) of a molecular complex can be obtained provided the system is of a size that is quantum chemically treatable and computationally inexpensive at a reasonable level of theory. Solvent effect can also be taken into account easily either by using the polarizable continuum model, a well-known implicit solvent model,67 or
by using molecular dynamics simulations to sample solutesolvent interactions explicitly. Conclusion In this paper, using the glycine dipeptide as a model molecule, a set of harmonic and anharmonic parameters for up to overtone and combination bands were obtained by sampling the backbone conformation space (-180° e φ e +180°, -180° e ψ e +180°) with sampling steps as ∆φ ) ∆ψ ) 30°. The secondorder perturbative vibrational treatment of the transition energy in the dipeptide was carried out at the level of HF/6-31+G*, allowing a complete set of the 3N-6 anharmonic normal-mode frequencies to be computed inexpensively and reasonably in comparison with the results obtained at the theory level of B3LYP/6-31+G*. The obtained vibrational parameters include harmonic and anharmonic normal-mode frequencies and diagonal and off-diagonal anharmonicities, as well as normal mode wave functions. We focus on the amide-I transitions in the dipeptide. For the two amide-I modes, by size-reducing and wave function demixing the normal modes, vibrational coupling, and local mode frequencies, as well as the wave function mixing angles can be easily obtained. The zero-order transition energies for the local modes and the couplings have been examined on the basis of computed harmonic and anharmonic normal-mode frequencies. We also evaluated the couplings on the basis of anharmonicities and normal-mode frequencies. These anharmonic parameters are generally useful in describing the behavior of two nearest neighboring anharmonic amide-I vibrational modes in peptides and proteins of any size. These parameters were found to be very sensitive to the peptide backbone conformation; therefore, they are potentially structure parameters and may serve as 2D IR probes for peptide conformations in condensed phases. Furthermore, computationpredicted values were found to be in reasonable agreement with the results from recent 2D IR experiments. Although at this moment it seems that only first-principles computation is able to predict large amounts of anharmonic parameters, soon enough it will be the case for experimentalists with the development of new 2D IR techniques, such as polarized 2D IR68 and dualfrequency 2D IR,52 as well as recent pulse-shaped 2D IR.69 Thus, the computational results in this work should prove useful in exploring and revealing the structure-spectra relationship of peptides and proteins.
Amide-I Modes in Model Dipeptide Different types of anharmonic parameters explored in the present work were found to be interconnected with each other and sometimes were even closely correlated. For example, the conformational dependence of the anharmonic frequencies closely resemble those of the harmonic frequencies; the magnitudes of the diagonal and off-diagonal anharmonicities were found to be complementary to each other; and the profile off-diagonal anharmonicity showed correlation to that of the wave function mixing angle. The reason behind this is that these parameters were determined by the same set of anharmonic force fields of the molecular system, and they reflect the anharmonic, delocalized, and weakly coupled nature of amide-I modes, which is presumably due to the intrinsic structural properties of peptides. In addition, the C2 symmetry was seen for all of the φ-ψ maps of the anharmonic parameters, suggesting that maps may be constructed at a higher level of quantum chemical theory within a reasonable amount of computation time by only sampling half of the φ-ψ space. The present work coincides well with the ongoing interest of the multidimensional infrared spectroscopy community in understanding and finding effective tools for assessing anharmonic parameters at a high level of theories. Apparently firstprinciples quantum chemical computations have already become a useful tool in describing or predicting the anharmonic parameters needed for modeling or analyzing 1D IR as well as 2D IR experimental results. Moreover, all the normal modes are potentially useful in characterizing molecular structures in condensed phases: localized modes may reflect site-specific local structural properties, and delocalized modes may reflect global or regional structural properties. Dynamical structural information can be predicted with the aid of ab initio and/or classical molecular dynamics simulations and samplings. One may expect to utilize a variety of vibrational modes in either tracking the dynamical structures of a given molecular system or tracking the chemical reactions which occur upon external triggering without the need of introducing additional chromophores as required by some other spectroscopic methods. Experimentally, developing a time-resolved broadband 2D IR spectroscopic technique seems to be very useful yet very challenging. Finally, our results show that HF theory level with a proper choice of basis set (HF/6-31+G*) is reasonable in predicting anharmonic parameters of peptide amide-I modes. Moreover, our approach can produce a full set of the 3N-6 anharmonic normal-mode frequencies and their overtone and combination transition frequencies in an N-atom molecule. The investigation of other (amide) modes and their isotopic substitutions in peptide oligomers is under way, and the results will be published in the near future. Acknowledgment. This research was supported by the National Natural Science Foundation of China (Grant 20773136), the National Basic Research Program (973) of China (Grant 2007BC815205), and the Chinese Academy of Sciences (Hundred Talent Fund). References and Notes (1) Hamm, P.; Lim, M.; Hochstrasser, R. M. J. Phys. Chem. B 1998, 102, 6123. (2) Asplund, M. C.; Zanni, M. T.; Hochstrasser, R. M. Proc. Natl. Acad. Sci. U.S.A. 2000, 97, 8219. (3) Mukamel, S. Annu. ReV. Phys. Chem. 2000, 51, 691. (4) Zanni, M. T.; Hochstrasser, R. M. Curr. Opin. Struct. Biol. 2001, 11, 516. (5) Woutersen, S.; Mu, Y.; Stock, G.; Hamm, P. Proc. Natl. Acad. Sci. U.S.A. 2001, 98, 11254. (6) Ge, N.-H.; Hochstrasser, R. M. PhysChemComm 2002, 5, 17.
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