3798
J. Phys. Chem. B 2006, 110, 3798-3807
Anharmonicity of Amide Modes† Jianping Wang and Robin M. Hochstrasser* Department of Chemistry, UniVersity of PennsylVania, Philadelphia, PennsylVania 19104-6323 ReceiVed: June 6, 2005; In Final Form: July 13, 2005
The principal contributions to the anharmonic coupling of amide vibrations are explored with the objective of comparing recent experiments with density functional theory and evaluating simple models of mode coupling. Experimental information obtained by means of two-dimensional infrared spectroscopy (2D IR) is reasonably well predicted by the computed one- and two-quantum anharmonic modes of amide-A, -I, and -II types in mono-, di- and tripeptides. The expansion of the vibrational energy up to the cubic and quartic coupling of harmonic modes suggested criteria to assess how localized are the forces determining the anharmonicity. The off-diagonal anharmonicity between an amide-A and one other amide mode was shown to be mainly determined by forces involving only these two modes, whereas the off-diagonal anharmonicity of two amide-I modes in peptides depended significantly on forces due to motions other than those of the amide-I type. Both the diagonal and off-diagonal anharmonicities exhibit sensitivity to peptide structures. These results should prove useful in linking 2D IR experimental results to secondary structure. Further, the results are used to evaluate the vibrational exciton model for the mixed-mode anharmonicities of the amide-I transitions.
Introduction The recently developed technique of two-dimensional infrared spectroscopy (2D IR)1-3 enables the direct measurement of the overtone and combination band frequencies of complex molecules, even those that have very congested spectra. The high mode selectivity of this method arises from its double resonance character: The effects of pumping one fundamental are sensed through measurements on other modes to which it is coupled. The applications of this approach to vibrations corresponding to different amide units of peptides and proteins have been particularly useful, because the anharmonicities are closely related to the coupling between different spatial regions of the molecule and hence can serve as probes of secondary structure.2,4-11 The double resonance signals rely on the existence of anharmonic coupling.1 With the recent availability of automated computational methods that include cubic and quartic contributions to the potential,12,13 it has become possible to examine the anharmonicities of peptides in more detail from an ab initio standpoint and, in turn, to use the recent results of 2D IR to test these theoretical predictions. To show significant 2D IR effects, the anharmonicities must be at least in the range of approximately one-tenth of the infrared line widths, which are typically 15-30 cm-1. The recent 2D IR studies of peptides have revealed that the diagonal anharmonicity ranges from 10 to 20 cm-1 for amide-I modes,1,7,8,14,15 which are predominantly CdO stretching vibrations, to ca. 160 cm-1 for amide-A modes,16,17 which are predominantly NsH stretching motions. The off-diagonal (or mixed-mode) anharmonicity can also be significant fractions of the line width: For example, the mixed-mode anharmonicity of the neighboring amide-I modes of an R-helix is ca. 9 cm-1 8 and that of the amide-A/I combination mode is ca. 1-5 cm-1.16 At the simplest level of theoretical description of coupled mode spectra, the †
Part of the special issue “Michael L. Klein Festschrift”. * Corresponding author. Email:
[email protected]. Tel: 215-8988203. Fax: 215-898-0590.
diagonal anharmonicity of the different amide modes can be introduced empirically and the bilinear coupling between the separated amide modes calculated on the basis of localized amide vibrations.1,18 This approach yields both diagonal and off-diagonal anharmonic shifts in the overtone region that are dependent on the same coupling parameters that give rise to the exciton-like splittings that are often seen in the linear amide-I mode infrared spectrum. This exciton model for the two-particle states assumes that the couplings between the amide-I and other modes can be neglected in describing the splittings exhibited by the fundamentals and the overtones.18 Both transition dipole and transition charge schemes have been used to evaluate the electrostatic part of the coupling between modes responsible for these splittings,2,19-21 and the through-bond part has been estimated from ab initio calculations.21-26 This type of modeling gives qualitative agreement with the experimental results but has not provided any deeper assessment of the validity of the approximations that go into it, as is the intention of this paper. Quantum chemical calculations of vibrational frequencies often assume a harmonic potential, and good performance of the Kohn-Sham formulation of density functional theory (DFT) for predicting molecular geometries and harmonic vibrational frequencies has been established. The hybrid density functional B3LYP yields fundamental vibrational frequencies for amide modes in simple peptides in better agreement with measurements in the gas phase than strictly ab initio approaches, such as second-order Moller-Plesset perturbation theory (MP2) with similar basis sets.27 However, a frequency scaling factor close to unity is still required to match theory and experiment precisely. Because of the anharmonic nature of polyatomic vibrational modes, it has been important to explore the extent to which incorporating anharmonicity will improve the agreement with experimental results. Two well-known methods are the vibrational self-consistent field (VSCF) and its perturbative extension28-32 and the second-order perturbative vibrational treatment (PT2).13,33-42 The VSCF approach requires a huge number of individual single-point calculations, because no
10.1021/jp0530092 CCC: $33.50 © 2006 American Chemical Society Published on Web 08/23/2005
Anharmonicity of Amide Modes
J. Phys. Chem. B, Vol. 110, No. 8, 2006 3799
analytical second derivatives of the total electronic energy are used. The PT2 treatment has the advantage of being less computationally intensive than VSCF, because these analytical second derivatives are employed. A full cubic and a semidiagonal quartic force field can be obtained by central numerical differentiation of analytical second derivatives.13 DFT also offers a reasonable starting point for the theoretical prediction of unscaled vibrational frequencies using the PT2 approach. Recent studies41-43 have shown that the performance of B3LYP density functional with medium-sized basis sets in computation of anharmonic vibrational frequencies of semirigid molecules including DNA bases is quite satisfactory. In addition, Fermi resonances that may occur frequently as molecular size increases have been taken into account in the PT2 approach.13,44 From a technical point of view, the calculation of cubic and quartic force constants can be efficiently parallelized in clustered computers; thus, the DFT plus PT2 approach offers the opportunity to obtain accurate vibrational frequencies at reasonable computational cost. However, because the force fields are isotope-dependent in the normal mode basis, quadratic, cubic, and quartic force constants have to be computed for each isotopomer. In this paper, we initiate a study of the anharmonic vibrational frequencies of small peptides using the DFT/PT2 approach. We address several questions concerning the amide vibrations, such as the factors that principally determine the amide mode anharmonicities, the relationship between the mode anharmonicities and mode delocalization, the relationship between the mixed-mode anharmonicity and the intermode coupling, and the reliability of the vibrational exciton model in predicting the mixed-mode anharmonicities of the amide-A and amide-I transitions. We focus on examining the diagonal and offdiagonal anharmonicities of the amide-A, -I, and -II modes and compare the computations with experimental results from 2D IR. Ab Initio Calculations. The anharmonic vibrational frequencies of mono- and dipeptides including dimers were calculated at the B3LYP level of theory with the 6-31+G** basis set. A tight convergence was forced during the geometry optimization. The calculation of AcProNMe was carried out at a lower level of basis set (6-31+G*) because of the CPU time limit in our computer cluster. However, on the basis of the results for a small peptide, trans-N-methylacetamide (NMA), we found that the addition of the polarization p-functions to hydrogen does not significantly change the calculated anharmonicities of the amide modes. Basis set superposition error is known to be important in dealing with weakly bound systems.45,46 However, the error is much smaller at the level of DFT than that of MP2.27,47 We neglect this error in the calculations of the hydrogen-bonded dimers. In comparison, the calculation of some di- and tripeptides were performed at the Hartree-Fock level of theory with the 6-31+G** basis set, as indicated in the text. All calculations were carried out using Gaussian 03.48 The vibrational energy of fundamentals, overtones, and pair combination bands in the normal-mode basis for a polyatomic molecule can be approximated as
E(ni, nj) ) 3N-6
E0 + hc
∑i
( )
ωi ni +
1
2
3N - 6
+ hc
( )( ) 1
xij ni + ∑ 2 i ej
nj +
1
2
(1)
where ωi is the harmonic frequency, ni is the vibrational quantum number of the ith mode, and xij is the anharmonic correction term. A second-order perturbation treatment of the
Figure 1. Structures of peptide monomers and dimers considered in the study. These structures are relevant to systems studied by 2D IR technique. Additional peptides investigated in this study are mentioned later in the text.
vibrational energy13,33,34 can be used to obtain xij in terms of the cubic and quartic force constants. The anharmonic fundamental, the overtone, and the combination band frequencies are given by νi ) ωi + 2xii + 1/2Σj*i xij, ν2i ) 2νi + 2xii, and νij ) νi + νj + xij. Therefore, the diagonal anharmonicity ∆ii and the mixed-mode anharmonicity ∆ij can be written as ∆ii ) -2xii and ∆ij ) -xij. Results and Discussion Energy-Minimized Structures. Structures were chosen to be representative of peptide conformations for which anharmonic potential parameters of the amide modes are available from 2D IR experiments. Figure 1 shows the DFT optimized geometries for six peptide monomers and two hydrogen-bonded dimers. For the formamide monomer and dimer and the trans-NMA monomer and the AcAlaOMe C5 carbonyl conformers, the structures are fully optimized. For the AcAlaOMe C5 ester conformer and AcProNMe, the hydrogen bond distances were set to rOH ) 1.958 Å and 1.976 Å, respectively. For AcAlaOMe, three conformations (the R-helix, the 310-helix, and the polyproline-II-like) are obtained by fixing the dihedral angles to (φ ) -58°, ψ ) -47°), (-50°, -25°), and (-75°, +135°), respectively. For the formamide dimer, the hydrogen bond length is found to be rOH ) 1.887 Å. For the trans-NMA dimer, the hydrogen bond distance is fixed at rOH ) 1.970 Å. For the hydrogen-bonded dimers, only one conformation is considered for each case. The chosen formamide dimer has C2h symmetry. The two monomers in the trans-NMA dimer have their backbone axis roughly perpendicular to each other. Calculated Harmonic and Anharmonic Frequencies of the Amide Modes. Table 1 shows the calculated harmonic (ω) and anharmonic (ν) frequencies of the amide-A, -I, and -II modes, their overtones, and the combination bands of the A/I and I/II pairs for several mono- and dipeptides. These values are for a single amide unit, which is the one at the acetyl end in all cases except for formamide and AcProNMe (Figure 1). It is found
3800 J. Phys. Chem. B, Vol. 110, No. 8, 2006
Wang and Hochstrasser
TABLE 1: Calculated Harmonic (ω) and Anharmonic (ν) Frequencies (in cm-1) of the Amide-A, -I, and -II Modes, Their Overtones, and the A + I, I + II Combination Bands for Several Mono- and Dipeptides peptide
formamide
trans-NMA
AcAlaOMe(H)
conformation 3733.4a
ωA ωI ωII νA ν2A νI ν2I νII ν2II νA I νI II a
1797.3 1621.5 3535.9 6988.8 1763.6 3509.7 1580.8 3157.6 5297.8 3346.1
3652.4 1751.3 1565.6 3490.5 6833.6 1721.1 3424.3 1516.4 3025.7 5208.1 3239.4
AcProNMe
C5 carbonyl
C5 ester
C5 carbonyl
C7
3608.8b
3614.5 1738.0 1540.7 3442.5 6735.2 1706.5 3397.3 1497.1 2981.0 5146.3 3203.6
2645.5 1736.2 1435.8 2550.2 5019.3 1703.7 3391.8 1390.7 2776.6 4253.3 3097.2
2650.1 1733.8 1439.8 2559.5 5039.6 1703.4 3387.6 1393.9 2782.5 4260.3 3097.1
3488.8c 1744.2 1600.2 3309.7 6431.0 1712.7 3405.5 1552.6 3076.2 5018.5 3261.0
1741.2 1547.7 3427.6 6701.7 1709.1 3400.4 1499.8 2992.0 5133.3 3212.0
The amide-A mode is the asymmetric stretching of NH2.
AcAlaOMe(D)
C5 ester
b
For the amide unit on the acetyl end. c For the amide unit on the amino end.
TABLE 2: Calculated Amide Anharmonicities (in cm-1) for Mono- and Dipeptides in Comparison with the Measured Values ∆A A
species formamide trans-NMA AcAlaOMe(H) C5 ester AcAlaOMe(H) C5 carbonyl AcAlaOMe(D) C5 ester AcAlaOMe(D) C5 carbonyl AcProNMe C7 a
exptl calcd exptl calcd exptl calcd exptl calcd exptl calcd exptl calcd exptl calcd
129a 83.0 147.5 144 ( 7d 153.6 144 ( 7d 149.8 110 ( 7d 81.1 110 ( 7d 79.4 165 ( 15d 188.4
∆I I
∆II II
∆A I
17.5 16.0b 18.0
4.1
1.7
7.1
17.9
7.5
15.6
13.1
15.6
4.9
19.2 13 ( 2e 19.9
5.3 13 ( 2e 29.1
3.5 1.4 ( 0.4d 3.4 2.6 ( 0.8d 2.7 1.6 ( 0.4d 0.6 4.6 ( 0.7d 2.6 3.5d 3.9
∆ I II -1.8 3.5 ( 0.5c -1.9 -3.1 -0.1 -2.8 0.18 4.1 ( 0.6e 4.4
Ref 17. b Ref 1. c Ref 52. d Ref 16. e Values were obtained from AcProNH2.14
that the amide-A transition frequency decreases by 95-200 cm-1 and those of the amide-I and -II modes decrease by ca. 30 cm-1 and 40-50 cm-1, respectively, when anharmonicity is incorporated. The H/D isotope effect on the amide transition frequency is predicted. For example, the replacement of H by D causes the frequencies of the amide-A, -I, and -II modes to decrease, as seen in each conformation of AcAlaOMe. Furthermore, it is found that the transition frequencies of the amide-A mode for AcAlaOMe(H) is different in two conformations, which is also the case for those of the amide-I and -II modes. A similar result is also seen in the D-form isotopomer. The amide-I transition frequency in the C5 ester conformer is slightly higher than that in the C5 carbonyl conformer for both the Hand D- isotopomers. The calculated anharmonic frequencies are in reasonable agreement with gas-phase measurements. For example, for formamide, gas-phase experimental results are νI ) 1755 and νII ) 1580 cm-1.49 The calculated frequencies are νI ) 1763.6 and νII ) 1580.8 cm-1. For trans-NMA, the amide-A, -I, and -II transition frequencies were reported to be 3498, 1708, and 1511 cm-1 in the gas phase,50 whereas the calculated values are 3490.5, 1721.1, and 1516.4 cm-1, respectively. In aqueous solution, the amide-I and -II frequencies are found to be 1622 and 1580 cm-1.51 However, it is not expected that the transition frequencies in the solution phase can be reproduced without a proper treatment of the solvent. In addition, the calculated AcAlaOMe C5 carbonyl conformer has a lower amide-I transition frequency than the C5 ester conformer, which is presumably due to an increased influence of the hydrogen bond on the amide-I transition frequency in the carbonyl conformer. This result is in agreement with experiment.16 A much lower νA is predicted for AcProNMe because of the formation of a C7 intramolecular hydrogen-bonded ring.
Calculated Anharmonicities of the Amide Modes within the Same Amide Unit. Table 2 summarizes the calculated and measured anharmonicities of the amide-A, -I, and -II modes for mono- and dipeptides listed in Table 1. The mixed-mode anharmonicity between the amide-A and -I modes, and between the amide-I and -II modes, within the same amide unit are also given. The measured values are from recent single- and dualfrequency 2D IR studies in the 6- and 3-µm wavelength.1,14,16,17,52 In general, the calculated anharmonicities at this level of theory correlate reasonably with experimental values. This is evidenced by the diagonal anharmonicities of three types of amide vibrational modes shown in Table 2. For the amide-A modes, the calculated values follow the trend of the measured ones in different peptides. The computed H/D isotope effect causes a reduced anharmonicity as shown in Table 2 for the AcAlaOMe conformers, which is in agreement with experiment. The ratio ∆AA(H)/∆AA(D) was calculated to be ∼1.89 in each AcAlaOMe conformer, which is very close to the expected value from a Morse potential. In addition, the calculation also shows an increase of the amide-A mode anharmonicity when a strong hydrogen bond is formed, e.g., in the case of AcProNMe. For the amide-I mode, the calculated anharmonicities range between 15.6 and 19.9 cm-1, which are in the vicinity of the measured values (ca. 16 cm-1). For the amide-II modes, the calculated anharmonicities range from 4.1 to 29.1 cm-1, with the smallest value for formamide and the largest one for AcProNMe. In the latter case, the large anharmonicity is probably due to the intramolecular hydrogen bond. For the same reason, the anharmonicity is smallest in the C5 carbonyl form of the AcAlaOMe conformer. These results indicate that the anharmonicity of the amide-II mode is sensitive to the hydrogen bonding of the amide unit. Although there are not many experimental values with which to compare, the calculations
Anharmonicity of Amide Modes
J. Phys. Chem. B, Vol. 110, No. 8, 2006 3801
TABLE 3: Calculated Transition Frequencies and Mixed-Mode Anharmonicities of the Amide-A and -I Modes through Hydrogen Bond (in cm-1) formamide dimer trans-NMA dimer
AcProNMe
conformation
C2h
perpendicular
C7 H-bonded ring
ωA ωI νA νI ∆A I ∆A I exptl
3339.6 1780.9 3147.5 1742.4 8.0
3523.0 1723.5 3359.9 1697.2 4.0
3488.8 1690.6 3309.7 1659.2 4.1 1.4 ( 0.4b
6.0 ( 2.0a
Pyrrole-N,N-dimethylacetamide dimer in CCl4.53 CHCl3.16 a
b
AcProNMe in
TABLE 4: Calculated Transition Frequencies and Mixed-Mode Anharmonicities of Two Neighboring Amide-I Modes at Two Different Dihedral Angles (in cm-1) alanine dipeptide (φ, ψ) ωi ωj νi νj ∆ij ∆ij (exptl) a
(-58°, -47°) 1771.4 1757.9 1737.2 1726.3 15.2 9.0 ( 1.0b
(-75°, +135°)a 1738.7 1703.5 1714.0 1675.0 1.4 0.2 ( 0.2c
AcProNMe (-50°, -25°) 1754.4 1751.3 1732.5 1750.9 10.4
(-83°, +71°) 1744.2 1690.6 1712.7 1659.2 3.9 1.5 ( 0.4d
trans-NMA dimer 1736.9 1723.5 1705.6 1697.2 8.2
Ac-[13C]Ala-NMe. b Ref 8. c Ref 15. d Ref 16.
show that the anharmonicity of the amide-II mode is predicted within a factor of 2 of experiment. Reasonable agreement between theory and experiment is also seen for the off-diagonal anharmonicities. The calculated amideA/I mixed-mode anharmonicities for the peptides listed in Table 2 fall in the range of 0.6 to 3.9 cm-1, close to the measured values in each case. For the amide-I/II anharmonicities, the calculated value in AcProNMe agrees well with the reported value in AcProNH2 (4.1 ( 0.6 cm-1).14 However, in most other cases, the amide-I/II mixed-mode anharmonicities are predicted by the calculation to be negative, whereas no negative anharmonicity was yet reported for an amide mode. Calculated Anharmonicities of the Amide Modes from Different Amide Units. Table 3 lists the calculated harmonic and anharmonic frequencies of the amide-A and -I modes and their mixed-mode anharmonicity in three systems where two amide units are hydrogen-bonded. Experimental values are also given for comparison. The calculated anharmonicities show the same trends as the experimental values. For the formamide dimer, the calculated hydrogen-bonded IR-active NsH mode, which is the asymmetric NH2 stretching motions, has its anharmonic frequency at 3339.6 cm-1. The calculated IR-active CdO mode, which is the asymmetric combination of two CdO stretching motions, has its anharmonic frequency at 1780.9 cm-1. The amide-A/I mixed-mode anharmonicity is found to be 8.0 cm-1. For the trans-NMA dimer, the hydrogen-bonded amide-A and -I modes have a mixed-mode anharmonicity ∆AI ) 4.0 cm-1. Experimentally, the mixed-mode anharmonicity across a NsH‚‚‚OdC hydrogen bond was found to be 6 ( 2 cm-1 for a pyrrole-N,N-dimethylacetamide hetero dimer in CCl4.53 For AcProNMe, the amide-A/I mixed-mode anharmonicity across an intramolecular hydrogen bond is computed as 4.1 cm-1, a value similar to that of the trans-NMA dimer. The measured value in this case is 1.4 ( 0.4 cm-1 in CHCl3.16 While the agreement is not perfect, the theory and experiment show the same trends. Table 4 shows the calculated harmonic and anharmonic amide-I mode frequencies of two neighboring peptide units and their mixed-mode anharmonicity, compared with experimental
values. These quantities are important indicators of structure. The existence of this mixed-mode anharmonicity indicates that excitation of one of the modes influences the other frequency, and hence, it tells whether the modes are coupled. Two dipeptides with three different sets of dihedral angles are investigated here. The calculated anharmonicities compare well with the experimental values. For the alanine dipeptide in the conformation of a perfect R-helix with Ramachandran angles (φ ) -58°, ψ ) -47°), two amide-I modes have a mixedmode anharmonicity as ∆ij ) 15.2 cm-1, somewhat larger than the measured value (9.0 ( 1.0 cm-1) in the alanine-rich R-helix.8 For the alanine dipeptide in the conformation of a perfect 310helix (-50°, -25°), the mixed-mode anharmonicity is predicted to be ∆ij ) 10.4 cm-1. For the alanine dipeptide with dihedral angles (-75°, +135°), our calculation yields two anharmonic amide-I frequencies at 1714.0 and 1675.0 cm-1 for the Ac[13C]Ala-NMe(D) isotopomer, in which 13C is present on the alanine CdO group. The calculated mixed-mode anharmonicity is ∆ij ) 1.4 cm-1, to be compared with the experimental result for the same isotopomer of ca. 0.2 ( 0.2 cm-1.15 For AcProNMe at a dihedral angle (-83°, +71°), the calculated mixed-mode anharmonicity is 3.9 cm-1, whereas the experimentally observed value is 1.5 ( 0.4 cm-1.16 Composition of the Diagonal and Mixed-Mode Anharmonicities. A detailed analysis of the factors that influence the anharmonicities was also sought from the computation. For the overtone states, the anharmonic shift can be written in terms of a second-order perturbation formula13,33,34
∆ii )
[
1 8
-Φiiii +
5Φiii2 3ωi
+
3N-7 Φ
∑
k*i)1
2 iik
]
(8ωi2 - 3ωk2)
ωk(4ωi2 - ωk2)
(2)
The first two terms on the right-hand side of eq 2 exclusively incorporate energy derivatives with respect to the mode i whose diagonal anharmonicity is being considered. We denoted these terms by ∆′ii. The coefficients Φijk and Φijkl are the third and fourth derivatives of the potential that involve other normal coordinates. The third term in eq 2 is denoted as ∆′′ii, so that ∆ii ) ∆′ii + ∆′′ii. For any two vibrators i and j, the part of the mixed-mode anharmonicity that is dependent on the interaction between vibrators i and j can also be expressed in terms of the cubic and quartic force constants and harmonic frequencies
∆′ij )
(
)
ΦiiiΦijj ΦjjjΦiij 2Φiij2ωi 2Φjji2ωj 1 -Φiijj + + + + 4 ωi ωj 4ω 2 - ω 2 4ω 2 - ω 2 i
j
j
i
(3)
Similarly, ∆′′ij, the contribution to the mixed-mode anharmonicity made by the interaction of the identified pair of modes i and j with the remaining 3N - 8 modes, can be written as ∆′′ij )
[
1
3N-8
∑
4k*j*i)1
ΦiikΦjjk ωk
-
× 2Φijk2(ωi2 + ωj2 - ωk2)ωk
]
ωi4 + ωj4 + ωk4 - 2(ωi2ωj2 + ωi2ωk2 + ωj2ωk2) (4)
and the total mixed-mode anharmonicity is ∆ij ) ∆′ij + ∆′′ij. The composition of ∆ii has been examined for the amide modes to investigate the localization of the anharmonic forces.
3802 J. Phys. Chem. B, Vol. 110, No. 8, 2006
Wang and Hochstrasser
TABLE 5: Harmonic Frequencies, Cubic and Quartic Force Constants (in cm-1), and Their Contributions to the Diagonal Anharmonicities of Three Major Amide Modes in trans-NMA vibrator i
ωi
Φiiii
Φiii
∆′ii
∆′ii/∆ii
amide-A amide-I amide-II
3652.4 1751.3 1565.6
1386.4190 158.9834 63.6187
2368.9961 492.5756 -61.6128
146.8 9.0 -7.4
0.996 0.500
As will be discussed later, simple exciton models of the spectra of peptides need to assume an empirical value for the diagonal anharmonicity of amide transitions as input in constructing the two-particle Hamiltonian matrix. Therefore, we have explored some properties of the diagonal anharmonicity. The simplest case is a monopeptide. Table 5 shows the harmonic frequencies, the cubic and quartic force constants, and their contributions to the diagonal anharmonicities of three major amide modes in trans-NMA. For the amide-A mode, it is seen that 99.6% of ∆ii corresponds to ∆′ii, meaning that the interaction between this normal mode and other normal modes only weakly influences its diagonal anharmonicity. However, for the amide-I mode, it is found that half of the diagonal anharmonicity is due to the cubic and quartic force constants from interactions with other modes. These other modes are mostly at lower frequency, e.g., the NsH bending; CsN, CdO, and CsC stretching; and the CsNsC bending. For the amide-II mode, a negative ∆′ii is obtained (-7.4 cm-1). In this case, interaction with the amide-A mode contributes to ∆′′ii quite substantially, ca. 15.0 cm-1, as a result of a large cubic force constant (Φiik ) -275.6 cm-1). Therefore, the overall ∆ii is positive. The magnitude of ∆′ij for four different vibrator pairs, namely, the amide-A/I, amide-A/A, amide-I/I, and amide-I/II modes, are listed in Table 6. The ratio ∆′ij/∆ij is also given in each case using ∆ij values from Tables 2-4. When the terms in eq 3 are restricted to i and j being the amide-A/I modes within the same amide unit (CONH), the calculated cubic and quartic terms of the potential accounts for 96.2% (in trans-NMA) and 87.9% (in AcProNMe) of the total anharmonic shift of the combination band. When the vibrators i and j are both the amide-A modes, as illustrated in the case of glycine dipeptide (GDP) with the R-helical conformation (-58°, -47°) at the level of HF/6-31+G**, the anharmonicity ∆ij is dominated by ∆′ij (0.926). For the parallel β-sheet conformation (-119°, +113°) at the level of HF/6-31+G**, the anharmonicity ∆ij is also dominated by ∆′ij (0.980). The value of ∆′ij/∆ij decreases to ca. 0.50 or less in the case of the amide-I/I pairs regardless of whether the two peptide units are hydrogen-bonded or covalently connected. In the conformation of the R-helix, this ratio is the largest (ca. 0.495) out of the five cases studied, and the two amide-I modes contribute ∼7.5 cm-1 to the total mixed-mode anharmonicity (∆ij ) 15.2 cm-1). The smallest value of ∆′ij/∆ij is found to be 0.063, as seen in the alanine dipeptide with dihedral angles (-75°, +135°). However, the uncertainty may be larger in this case, because ∆ij is so small (1.4 cm-1). A calculation of the amide-I combination band of glycine tripeptide (GTP) in the R-helical conformation at the level of HF/6-31+G** predicts a mixedmode anharmonicity of 14.1 cm-1. In this example, the two amide-I modes are mostly localized on two amide units that are not directly bonded. It is found that ∆′ij/∆ij ) 0.215, which falls into the same range as that found for two covalently bonded amide units. The ratio ∆′ij/∆ij is larger when the vibrators i and j are the amide-A/I modes from two amide units that are hydrogenbonded; it is found that this ratio is between 0.628 and 0.849
for the three cases studied, the trans-NMA dimer, AcProNMe, and the formamide dimer. The lowest value is found in the formamide dimer, which is a special case, because as will be explained later, some of the cubic force constants are zero due to molecular symmetry. We also examined the case when the vibrators i and j are the amide-I and -II modes within the same amide unit: For transNMA, ∆′I II ) 4.8 cm-1, whereas the total anharmonicity ∆I II is negative (see Table 2); for AcProNMe, ∆′I II ) -5.7 cm-1, whereas the total ∆I II is positive. These results indicate that the anharmonic shift of the amide-I/II combination band depends not only on the interaction of the specific pair but also on the interaction of each mode with the remaining 3N - 8 normal modes. The influence of the different modes on the diagonal and offdiagonal anharmonicities can be examined using eqs 2 and 4. As one example, ∆′′ii and ∆′′ij as a function of the calculated anharmonic normal-mode frequency are shown in Figure 2 for the modes i ) 13 (ν ) 1737.2 cm-1) and j ) 14 (ν ) 1726.3 cm-1), which are the two amide-I modes of the alanine dipeptide with dihedral angles (-58°, -47°). In this example, ∆ii ) 11.5 cm-1 (see Table 7) and ∆ij ) 15.2 cm-1 (see Table 4). The interaction between mode i and other modes contributes 54.9% of the magnitude of ∆ii. The sum of the ∆′′ij in Figure 2B is 7.7 cm-1, implying that the interaction between the pair of normal modes i and j and the remaining 3N - 8 modes contributes 50.7% of ∆ij. Figure 2 also shows that the significant anharmonic contributions to both ∆′′ii and ∆′′ij mostly come from modes 1440, which include the other amide-I mode (mode 14, ν ) 1726.3 cm-1), CH3 rocking (mode 23, ν ) 1435.3 cm-1), CRsH bending, CsC stretching (mode 27, ν ) 1306.4 cm-1), amideIII modes (modes 29 and 30, ν ) 1233.0 and 1165.8 cm-1), and CsN/CsC stretching modes (modes 38 and 40, ν ) 933.3 and 837.4 cm-1). Note that the amide-III mode is a combination of the NsH in-plane bending plus CsC, CsN, and CsC stretching and CdO in-plane bending motions. For the glycine tripeptide in the R-helical conformation, the ∆′′ij composition analysis for two amide-I modes, which are localized mostly on two peptide units at the ends, shows that 38.1% of ∆′′ij is due to the interaction with the third amide-I mode (localized mostly on the peptide unit in the middle) and 61.9% to the interaction with all other modes. These contributions are likely to be conformation-dependent. Similarly, it is found in other conformations that the amide-I normal mode overtones and combinations are significantly coupled to other modes. There are two main consequences of these results. First, the occurrence of offdiagonal forces involving other modes suggests that simple, local models of the anharmonicity are likely to miss some essential structural content. Second, the complexity of the anharmonic coupling matrix in the normal-mode basis also suggests a complex set of relaxation pathways for these modes. Equation 3 is often simplified because of molecular symmetry, which causes the cubic and quartic terms of the potential function to vanish unless the direct product of the representations of the three or four normal coordinates is totally symmetric. Considering the case of the amide-A/I interaction in the formamide dimer, the amide-A mode (mode i) is chosen to be the antisymmetric combination of the NH2 stretching of each monomer, whereas the amide-I mode (mode j), is the asymmetric combination of the CdO stretch. Because of symmetry, Φjjj, Φiij, and Φijj are zero, so that ∆′ij ) -Φiijj/4. These parameters for the formamide dimer are shown in Table 6. Anharmonicity, Delocalization, and Coupling. It is of interest to establish experimental signatures of the delocalization
Anharmonicity of Amide Modes
J. Phys. Chem. B, Vol. 110, No. 8, 2006 3803
TABLE 6: Harmonic Frequencies, Cubic and Quartic Force Constants (in cm-1), and Their Contributions to the Mixed-Mode Anharmonicity of Two Vibrators vibrator i/j pair
species
ωi
ωj
Φiijj
Φjjj
Φiij
Φijj
amide-A/Ia
trans-NMA AcProNMe glycine dipeptide (-58°, -47°) trans-NMA dimer alanine dipeptide (-58°, -47°) alanine dipeptide (-50°, -25°) alanine dipeptide (-75°, +135°) AcProNMe (-83°, +71°) trans-NMA dimer AcProNMe (-83°, +71°) formamide dimer trans-NMA AcProNMe (-83°, +71°)
3652.4 3488.8 3898.9
1751.3 1744.2 3894.5
-10.0318 -10.9234 27.3784
2368.9961 2407.9597 2327.0013
492.5756 430.4880 2331.9213
-30.3836 34.9625 264.6661
-0.7541 -2.1005 -214.8682
3.6556 3.3827 5.6831
0.962 0.879 0.926
1736.9 1771.4
1723.5 1757.9
41.9129 64.2943
-316.8537 380.1054
-382.9548 236.8079
-227.9208 -69.7297
89.5035 307.3860
3.8387 7.5274
0.469 0.495
1754.4
1751.3
33.7583
382.9288
-422.8800
-204.5098
-88.3983
3.7960
0.365
1738.7
1703.5
-0.0975
-519.9987
472.3398
0.1130
-0.2834
0.0887
0.063
1744.2
1690.6
6.9568
430.4880
-445.7393
-83.0377
-54.4184
1.3204
0.340
3523.0 3488.8
1723.5 1690.6
-5.6632 -2.7988
-2413.7035 2407.9597
-382.9548 -445.7393
-56.6293 -62.1182
7.1552 -8.6642
3.3737 3.3601
0.849 0.826
3339.6 1751.3 1744.2
1780.9 1565.6 1600.2
-20.1256 2.5254 5.2009
0.7789 492.5756 430.4880
0.0000 -61.6128 -4.2767
0.0000 8.1680 -19.4619
0.0000 -64.6437 -84.7919
5.0314 4.7656 -5.6859
0.628
amide-A/Ab amide-I/Ib
amide-A/Ic
amide-I/IIa
Φiii
∆′ij
∆′ij/∆ij
a Within the same amide unit (CONH); for AcProNMe, it is the amide-I at the amino end. b For two nearby amide units; for the NMA dimer, two units are hydrogen-bonded (NsH‚‚‚OdC). GDP: at the level of HF/6-31+G**. c For two neighboring amide units forming the NsH‚‚‚OdC hydrogen bond.
TABLE 7: Anharmonic Local Mode Frequency (in cm-1), Coupling (in cm-1), Diagonal Anharmonicity (in cm-1), Wave Function Mixing Angle (in deg), and Mode Delocalization Factor for two Nearest-Neighboring Amide Units in Various Conformationsa amide-I mode i and j (φ, ψ) ν0i
alanine dipeptide
(-58°, -47°) (-75°, +135°) (-50°, -25°) (-83°, +71°) 1734.2 1714.0 1734.8 1711.4
1729.3 ∆ii 11.5 11.7 ∆jj 4.9 βij ξ 31.8 |βij/(ν0i - ν0j )| 1.0 ν0j
AcProNMe
1675.0 21.3 19.3 -0.1 0.1 2.5 × 10-3
1748.7 14.9 21.9 6.0 69.5 0.4
1660.5 19.9 17.3 -8.2 8.9 0.2
trans-NMA dimer 1704.2 1698.6 8.7 13.3 -3.1 24.2 0.6
a Diagonal anharmonicities were obtained from the ab initio calculations.
Figure 2. Contributions of the cubic force constants to diagonal and off-diagonal anharmonicities as a function of the calculated anharmonic frequency. Two amide-I modes (nos. 13 and 14) in the alanine dipeptide with the dihedral angles (-58°, -47°) are investigated: (A) the anharmonicity of mode 13, and (B) the off-diagonal anharmonicity of mode 13/14.
of vibrational excitations in peptides. Therefore, the relationships between the diagonal anharmonicity and mode delocalization were examined for the amide-I transitions. When the amide unit is most localized, e.g., in trans-NMA, the calculated ∆I I is 18.0 cm-1 (see Table 2), but the value is sensitive to the presence of other amide units. This is shown in Table 7 for several cases of dipeptides in which two units are either covalently bonded or hydrogen-bonded. For the alanine dipeptide, the anharmonicities of both amide-I modes depend on the dihedral angles. For example, they are smaller in the R-helical conformation (-58°, -47°) than in the isolated unit, whereas in the conformation (-75°, +135°), both are comparable with the value calculated in the single amide unit. For AcProNMe, the calculated ∆I I decreases significantly when the two amide units are hydrogenbonded. This is true also for the trans-NMA dimer where the diagonal anharmonicities are 8.7 cm-1 and 13.3 cm-1 for the
free and CdO hydrogen-bonded amide-I modes, respectively. In addition, for GTP in the R-helical conformation, the three amide-I mode diagonal anharmonicities at the level of HF/631+G** are 14.0, 10.6, and 9.8 cm-1 in the order of high- to low-frequency modes. These findings correlate reasonably well with experiments: An anharmonicity of 16.0 cm-1 was reported for NMA;1 smaller values, 9-11 cm-1, were reported for two nearby amide units in the R-helix8 and 13 ( 2 cm-1 in the amino-end amide in AcProNH2.14 The two amide units in an alanine dipeptide with a negligible coupling constant showed the high-range value of ca. 16.0 cm-1.15 In addition, the CdO groups in the acetic acid dimer where there is delocalization also showed diagonal anharmonicities smaller than that in the monomer.54 These results suggest that the magnitude of the amide-I mode diagonal anharmonicity may be useful as a peptide structure probe. Previously, only the off-diagonal anharmonicity was used to estimate interactions. For the coupled amide I modes, the degree of delocalization of each local mode is reflected in their wave function mixing, which can be evaluated by means of a wave function demixing approach.21 Briefly, the amide-I normal modes are analyzed as the linear combinations of local (or uncoupled) amide-I modes, and the normal mode eigenvectors are used to obtain the local mode mixing angle ξ, which is defined as ξ ) 1/2‚tan-1[2|βij|/
3804 J. Phys. Chem. B, Vol. 110, No. 8, 2006
Wang and Hochstrasser
TABLE 8: Off-Diagonal Anharmonicities (in cm-1) Obtained from the Two-Vibrator Modeling and ab Initio Calculation for Two Amide-I Modes in Several Dipeptides amide-I mode i and j (φ, ψ) ∆ij (two vibrator) ∆ij (ab initio)
alanine dipeptide (-58°, -47°) 15.1 15.2
(-75°, +135°) 3.7 × 10-4 1.4
(νi0 - νj0)] where βij is the coupling of two local states with frequencies νi0 and νj0. For the amide-I modes in a dipeptide system, such a calculation is straightforward if the mode is assumed to be mainly a CdO stretching vibration. Through such a procedure, the transition frequency of the local modes, the wave function mixing angle, and an intermode coupling are obtained simultaneously. The results for several cases of dipeptides, with two nearest-neighboring peptide units either covalently bonded or hydrogen-bonded, are given in Table 7. For the alanine dipeptide with dihedral angles of (-58°, -47°), it is found that the wave function mixing angle between two local states is ξ ) 31.8° and the interamide-I coupling is 4.9 cm-1 (using the anharmonic frequency ν) or 6.0 cm-1 (using the harmonic frequency ω). Previously, the coupling constants were determined from harmonic frequencies.21,23,55 Here, the results with anharmonic frequencies are also evaluated. We find for the dipeptide with dihedral angles of (-50°, -25°) that the mixing angle is ξ ) 69.5° and the interamide-I coupling is 6.0 cm-1 (using ν), yet -1.0 cm-1 (using ω). However, for dihedral angles of (-75°, +135°), the mixing angle is ξ ≈ 0.1°, and the coupling is very small regardless of what type of frequencies are used (between -0.01 and -0.08 cm-1). For AcProNMe, the wave function mixing angle is ξ ) 8.9° and the coupling is found to be ca. -8.2 cm-1 using either the harmonic or anharmonic frequencies. In the trans-NMA dimer, the mixing angle between two local states is ξ ) 24.2°. The intermode coupling is found to be -3.1 cm-1 (using ν) or -5.0 cm-1 (using ω). These results expose a correlation between the diagonal anharmonicity and the wave function mixing angle: the smaller the anharmonicity, the larger the mixing angle. The relationship between the diagonal anharmonicity and delocalization has been noted previously for the CdO stretching modes of the acetic acid dimer.54 A higher degree of delocalization does not necessarily imply a stronger coupling. If |tan 2ξ| , 1, the coupled states are predominantly localized on the individual amide units. As shown in Table 7, the amide-I modes are highly localized except for the alanine dipeptide with dihedral angles of (-58°, -47°), in which case, a significant delocalization occurs in each of the amide-I modes. For AcProNMe, even though the coupling is significant, the modes are still localized because of a relatively large energy separation in two zero-order frequencies. The Exciton Model of the Amide Two-Particle Transitions. In a molecular complex consisting of N atoms, the 3N 6 vibrational normal modes couple with one another to yield the anharmonic modes. However, it has proven useful to consider that there may be subsystems of modes that are isolated even in low-symmetry molecules. For example, it has been suggested that the spectrum of the overtone region can be deduced by knowing the bilinear coupling constants that determine the exciton splittings in the linear spectrum.1,2,56 For an isolated subsystem consisting of a pair of modes, there are two zero-order frequencies, two diagonal anharmonicities, and one bilinear coupling constant. The Hamiltonian matrices for the one exciton (2 × 2) and two exciton (3 × 3) states can be constructed on the basis of the uncoupled local modes with zero mixed-mode anharmonicity. A harmonic approximation is
(-50°, -25°) 15.8 10.4
AcProNMe
trans-NMA dimer
(-83°, +71°) 1.8 3.9
12.3 8.2
usually assumed so that the zero-order overtone and combination transitions are coupled by a factor of x2 of that between two local fundamental transitions.18 A zero-order diagonal anharmonicity is introduced empirically into the overtone states. Matrix diagonalization generates five energy levels. By fitting these energy levels to the fundamentals, the overtones, and the combination bands observed in linear IR and 2D IR measurements, one can determine the magnitude of the coupling between the two local modes. The coupling is important, because it can be related to the distance and the angular relationship between two chemical bonds or groups, for example, through a transition dipole-dipole or transition charge-transition charge interaction scheme. This so-called two-vibrator model is simple and easy to use, and it has been used recently in analyzing the 2D IR spectra of the 13Cd18O and 13Cd16O double isotopic labeled amide-I states in a 25-membered R-helix7,8 and 13Cd16O labeled and unlabeled dipeptides.4,15 However, the criteria for the validity of such a simple two-vibrator model needs to be further assessed. A significant separation between the frequencies of the pair of modes and the remaining normal modes can certainly be one of the criteria, as seen in double-labeled amide-I/I states in the R-helix; however, it is not the case for a pair of the amide-A and amide-I modes. Such an assessment is needed also, because the computational results presented above indicate that both the diagonal and off-diagonal anharmonicities of the amide modes are influenced to some extent by the interaction between different types of normal modes. We evaluated the two-vibrator model for the amide-I transitions in dipeptides from the criterion that it yields an off-diagonal anharmonicity in agreement with the ab initio result. The zeroorder anharmonic fundamental transition energies for the modes i and j, νi0 and νj0, and the intermode coupling βij are obtained from the wave function demixing of the anharmonic normal modes, as described above. The zero-order diagonal anharmonicities for the modes i and j are set to be ∆ii0 ) ∆jj0 ) 18 cm-1 for two amide-I modes, based on the theoretical result for a monopeptide trans-NMA presented in Table 2. We neglect other sources of variation in the zero-order diagonal anharmonicity. Of course, different values should be used for different types of modes. Using the values of νi0, νj0, and βij given in Table 7, we obtained the amide-I off-diagonal anharmonicities for several dipeptides. The results are given in Table 8 in comparison with the ab initio results. A good correlation between the magnitudes of ∆ij (two vibrator) and ∆ij (ab initio) is observed. These two parameters are plotted against one another in Figure 3 in which additional dipeptide conformations are considered. This includes the π-helix (-57°, -70°), lefthanded helix (+60°, +60°), antiparallel β-sheet (-139°, +135°), parallel β-sheet (-119°, +113°), C7 (+82°, -69°), extended chain (+180°, +180°), as well as extended chain having two cis-amide units in C2 symmetry. We also examined the amide-I mixed-mode anharmonicity of two amide groups that are separated by one peptide unit (j ) i + 2), which is a GTP in the conformation of the R-helix. The calculations are carried out at the HF/6-31+G** level, and the anharmonic frequencies are used to construct the Hamiltonian matrices. As can be seen
Anharmonicity of Amide Modes
J. Phys. Chem. B, Vol. 110, No. 8, 2006 3805 13Cd18O
Figure 3. Amide-I mixed-mode anharmonicities obtained from ab initio calculation and the two-vibrator modeling for two covalently or hydrogen bonded amide units in various conformations. (a) ADP (-75°, +135°); (b) AcProNMe (-83°, +71°); (c) trans-NMA dimer; (d) ADP 310-helix (-50°, -25°); (e) ADP R-helix (-58°, -47°); (f) GDP π-helix (-57°, -70°); (g) GDP left-handed helix (+60°, +60°); (h) GDP antiparallel β-sheet (-139°, +135°); (i) GDP parallel β-sheet (-119°, +113°); (j) GDP C7 (+82°, -69°); (k) GDP extended (+180°, +180°); (l) GDP extended chain with two cis-amide units with C2 symmetry; and (m) GTP (-58°, -47°) with j ) i + 2.
in Figure 3, all the points are distributed within a few wavenumbers of the diagonal, confirming a reasonable correlation between the ∆ij values obtained by the two methods in a wide range of conformation sampling. These results suggest that the two-vibrator modeling of the amide-I transition is generally credible in the two-quantum region. In addition, in the twovibrator model, the sum of the diagonal and off-diagonal anharmonicities equals the sum of the zero-order anharmonicity, i.e., Σiej)1,2 ∆ij ) ∆ii0 + ∆jj0. This relationship is also found to be approximately correct in the ab initio calculated anharmonicities. For example, in the case of the alanine dipeptide with dihedral angles (-58°, -47°), it is found that Σiej)1,2 ∆ij ) 38.4 cm-1, which is about twice the anharmonicity of the amide-I mode of trans-NMA (∆ii0 ) 18.0 cm-1). This relationship also indicates that, as the two eigen-mode diagonal anharmonicities decrease, the off-diagonal anharmonicity increases. This is also observed for the ab initio calculated anharmonicities for each dipeptide (see Tables 7 and 8). Further, these results suggest that the diagonal anharmonicity of the amide-I mode of a monopeptide can be used as an “empirical” zero-order anharmonicity for the two-vibrator model, even though this parameter is influenced by other modes (Table 5) and may change as the chemical environment of an amide unit changes. In summary, these results demonstrate that the Vibrational exciton approach of simulating the amide-I band envelope is generally consistent with ab initio results. In applying this model, N vibrators (N g2) are assigned to N amide units, and the diagonalization of an N × N matrix for the single excitations and an N(N + 1)/2 × N(N + 1)/2 matrix for the double excitations is required. To improve the accuracy and still retain a simple model, one could take into account those low-frequency modes that influence both ∆ii and ∆ij as a secondorder excitonic effect on the exciton Hamiltonians. Calculations for the formamide dimer have shown that the exciton model gives reasonable agreement with the ab initio results only if the zero-order anharmonicity is chosen to be much smaller (9.5 cm-1) than that of the formamide monomer (17.5 cm-1, Table 2). The diagonal anharmonicities of the singly
labeled formamide dimer were computed to be 10.4 cm-1 (mainly 13Cd18O) and -4.9 cm-1 (mainly 12Cd16O) at the B3LYP/6-31+G** level, illustrating how sensitive are these values to the mode coupling through the hydrogen bonds. The low value 10.4 cm-1 for a 13Cd18O amide-I mode in an oligomer is consistent with measurements of diagonal anharmnonicites of 13Cd18O substituted residues in helices that were found to be smaller (ca. 10 cm-1) than for isolated peptide (ca. 16 cm-1).57 We also examined the two-vibrator model for the amide-A/I transitions. The value of ∆ii for the amide-A mode in a monopeptide is independent of other normal modes, as shown in Table 5. The ratio ∆′ij/∆ij is always more than 0.8 when the amide-A mode is involved, as shown in Table 6. These properties, arising from the local-mode character of the amide-A mode, are apparently favorable conditions for the applicability of the two-vibrator model to the amide-A/I transitions. Assuming the two-vibrator model, one should be able to obtain the zeroorder frequencies, zero-order diagonal anharmonicities, and intermode coupling constant based on the ab initio results given in Table 1. These variables were determined in the case of the amide-A/I mode pair in trans-NMA, an example where the coupled modes involve some displacements on the same atoms. The one- and two-exciton Hamiltonian matrices were constructed, and a matrix diagonalization leads to energy levels of two fundamentals, two overtones, and one combination band. A least-squares fitting of these energies to those from the ab initio calculation was carried out. Two zero-order frequencies were determined to be 3474.9 and 1736.5 cm-1 for the amide-A and -I modes, with their anharmonicities being 150.1 and 18.1 cm-1, respectively. These anharmonicities are very close to the ab initio values of the amide-A and amide-I modes in transNMA (Table 2). However, the coupling needed for the model to reproduce the small off-diagonal anharmonicity (3.5 cm-1) was found to be 164.8 cm-1 with an undetermined sign, which is several times larger than the ab initio calculation derived bilinear coupling of -31.2 cm-1 in trans-NMA. The latter was obtained by evaluating the total energy derivative with respect to the two normal coordinates at the level of B3LYP/6311++G**. Finally, we examined the exciton model of two-particle states for the amide-A transitions. For two nearest-neighboring amide-A modes of the dipeptides sampled selectively in the Ramachandran conformation space, the off-diagonal anharmonicity is found to be less than 6.0 cm-1 from the ab initio calculations. This value is comparable to or somewhat smaller than that found for the amide-I transitions. The intermode coupling constant was computed to be varying from -1.0 to +2.5 cm-1 for the amide-A modes in various dipeptides by the wave function demixing approach. The exciton model utilizing these coupling constant results in an extremely small and negative off-diagonal anharmonicity for these dipeptides, because the diagonal anharmonicity is much larger than both the coupling and the zero-order energy shifts. Conclusion In this study, we evaluated the anharmonic potential parameters of the amide modes in peptides. We used a second-order perturbative vibrational treatment to calculate the vibrational frequencies of mono-, di-, and tripeptides in the fundamental, overtone, and combination regions. The introduction of anharmonicity to the amide-A, -I, and -II modes was found to improve the agreement with gas-phase results. The diagonal and offdiagonal anharmonicities were calculated and compared with
3806 J. Phys. Chem. B, Vol. 110, No. 8, 2006 experiments. The results reveal that the higher-order parameters of the anharmonic potential surfaces of peptides derived from ab initio calculations are in general agreement with 2D IR experimental results inasmuch as they show the trends. General agreement between theory and experiments was also observed in the mixed-mode anharmonicities in small intra- or intermolecular NsH‚‚‚OdC hydrogen-bonded peptide complexes. These parameters are intrinsic properties of the peptide backbone and/or local structures and might not be affected too greatly by the surrounding solvent molecules or extension of the structure. Therefore, theory can be used to guide the experimental data interpretation. This study provides some insights into the intrinsic characteristics of amide-mode anharmonicity in peptides, through a detailed composition analysis of the anharmonicity obtained by the ab initio computations. Theory suggests that the diagonal anharmonicity of the amide-I transitions depends strongly on the interaction between the amide-I mode with other normal modes, which are identified to be mostly the low-frequency amide vibrations and peptide backbone motions. On the contrary, the diagonal anharmonicity of the amide-A transitions was found to be much less sensitive to such intermode interactions. When two amide modes are of the same type (e.g., both amide-I or amide-A), the off-diagonal anharmonicity show similar properties as their diagonal anharmonicities. For two different mode types, the off-diagonal anharmonicity was not sensitive to the interactions with other modes, when one of them is mainly localized (e.g., the amide-A mode). It is found that, in peptide oligomers, the amide-I mode anharmonicity and amide-I mode delocalization are closely related to each other. In comparison with the value of a single isolated amide unit, delocalization results in a change in the diagonal anharmonicity: the larger the delocalization, the smaller the diagonal anharmonicity. However, more delocalization is not necessarily associated with a stronger intermode coupling, because the transition energy separation also plays an important role. Since the diagonal anharmonicity of the amide-I modes was related to the mode delocalization, and the latter is structure sensitive, the anharmonicity can be a useful parameter to directly probe the extent of the amide-I mode delocalization and the peptide backbone structure. The Hamiltonian matrix diagonalization-based two-Vibrator model, or more generally, the vibrational exciton model, for describing the amide band envelope of polypeptides in both linear and nonlinear IR spectra, was examined by comparing it with ab initio computational results. The model is applicable to either a collection of the amide-I modes or a mixture of the amide-A and -I modes. However, the model requires “good” empirical values for the diagonal anharmonicities as zero-order input in constructing the one- and two-exciton Hamiltonians. Computations can apparently give a useful estimate of the parameters required for the exciton modeling. The zero-order frequencies are also among these parameters. Moreover, the model should also incorporate a few other low-frequency amide and backbone vibrational motions that are involved in determining the diagonal and off-diagonal anharmonicities of the amide modes. We would guess that these strongly coupled lowerfrequency modes shown in Figure 2A would also be involved in the ultrafast relaxation1 of the amide-I mode. Acknowledgment. This research was supported by grants from NIH (GM12592 and RR01348) and NSF to R.M.H. Computational support was mainly provided by the National Science Foundation CRIF Program, grant CHE-0131132. Most of the anharmonic frequency calculations were carried out on a
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