Vibrational Spectroscopic Determination of Local Solvent Electric Field

ARTICLE pubs.acs.org/JPCA

Vibrational Spectroscopic Determination of Local Solvent Electric Field, SoluteSolvent Electrostatic Interaction Energy, and Their Fluctuation Amplitudes Hochan Lee,† Gayeon Lee,† Jonggu Jeon,† and Minhaeng Cho*,†,‡ † ‡

Department of Chemistry and Research Institute for Natural Sciences, Korea University, Seoul 136-701, Korea Multidimensional Spectroscopy Laboratory, Korea Basic Science Institute, Seoul 136-713, Korea ABSTRACT: IR probes have been extensively used to monitor local electrostatic and solvation dynamics. Particularly, their vibrational frequencies are highly sensitive to local solvent electric field around an IR probe. Here, we show that the experimentally measured vibrational frequency shifts can be inversely used to determine local electric potential distribution and solutesolvent electrostatic interaction energy. In addition, the upper limits of their fluctuation amplitudes are estimated by using the vibrational bandwidths. Applying this method to fully deuterated N-methylacetamide (NMA) in D2O and examining the solvatochromic effects on the amide I0 and II0 mode frequencies, we found that the solvent electric potential difference between O(dC) and D(N) atoms of the peptide bond is about 5.4 V, and thus, the approximate solvent electric field produced by surrounding water molecules on the NMA is 172 MV/cm on average if the molecular geometry is taken into account. The solutesolvent electrostatic interaction energy is estimated to be 137 kJ/mol, by considering electric dipoleelectric field interaction. Furthermore, their root-mean-square fluctuation amplitudes are as large as 1.6 V, 52 MV/cm, and 41 kJ/mol, respectively. We found that the water electric potential on a peptide bond is spatially nonhomogeneous and that the fluctuation in the electrostatic peptidewater interaction energy is about 10 times larger than the thermal energy at room temperature. This indicates that the peptidesolvent interactions are indeed important for the activation of chemical reactions in aqueous solution.

nonuniform solvent electric potential (EP) distribution, ϕsolvent(R), which is created by surrounding polar solvent molecules or other neighboring chemical groups. One of the most thoroughly investigated molecular systems is N-methylacetamide (NMA) in polar solvents.49,27,29,3546 It has been demonstrated that, as long as the antenna points are sufficiently many and well-positioned on the amide, the distributed interaction site model, regardless of specific designs and positions of interaction sites, works quantitatively well in predicting solvatochromic frequency shifts of amide vibrational frequencies. On the basis of the distributed multipole interaction site model, the solvatochromic vibrational frequency shift of the jth normal mode exposed to an EP, ϕ(R), if compared with that in the reference EP, ϕ0(R), is determined by the difference EP, ϕ(R) ϕ0(R), on the solute molecule:12,47

I. INTRODUCTION Interaction between solute and solvent molecules is responsible for most of the chemical and biological processes in condensed phases. Various spectroscopic methods have been used to directly probe such solutesolvent interaction energies and dynamics. In particular, vibrational spectroscopy has provided intricate details on molecular structures and conformational changes that are strongly affected by magnitude and time scales of solutesolvent interactions. Since normal-mode frequencies of a solute are sensitive to nuclear configurations determined by the solute electron density, they provide critical information on site-specific hydrogen bonding and electrostatic interactions with surrounding water molecules. Such vibrational frequency shift induced by the local electric field produced by inhomogeneously distributed solvent molecules is known as a vibrational solvatochromic phenomenon,114 whereas that induced by external static electric field is known as vibrational electrochromism or Stark effect.1,2,1023 Recently, we showed that the distributed multipole analysis method2426 can be used to quantitatively describe the solvatochromic vibrational frequency shifts of infrared-active modes of a solute in polar solvents.57,1014,22,2734 Here, the multiple interaction sites distributed over the solute molecule can be viewed as a solute antenna system capable of sensing spatially r 2011 American Chemical Society

Δωj ðϕ, ϕ0 Þ ¼ ωj ðϕÞ ωj ðϕ0 Þ ¼

N

∑ Γ^ x½ϕðR x Þ ϕ0 ðR x Þ x¼1

ð1Þ

Here the number of interaction sites (fragments of solute charge distribution) is N, and Rx is the position vector of the center of Received: October 10, 2011 Revised: November 14, 2011 Published: November 16, 2011 347

dx.doi.org/10.1021/jp209709e | J. Phys. Chem. A 2012, 116, 347–357

The Journal of Physical Chemistry A

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the xth fragment. Detailed expression for the operator^Γx acting on the difference EP function ϕ(Rx) ϕ0(Rx) depends on how high multipole moments of the fragmented solute charge distribution are taken into consideration to describe the frequency shift. In the case that the reference system is the solute molecule in the gas phase, the reference EP ϕ0(R) is 0 and the frequency ωj(ϕ0) corresponds to that of an isolated molecule. Over the past decade, the distributed interaction site model, where only the distributed transition charge contributions to the vibrational frequency shift are taken into account with the distributed higher multipole terms being ignored, has been proven to be sufficiently accurate and useful to estimate the solvatochromic effects on a variety of peptide vibrational modes and other IR probes. Within this distributed charge approximation, eq 1 is given as a linear combination of the solvent difference EP values at the distributed interaction sites: Δωj ðϕ, ϕ0 Þ ¼ ωj ðϕÞ ωj ðϕ0 Þ ¼

N

∑ ljx ½ϕðRx Þ ϕ0 ðR x Þ x¼1

estimated by any means. In this paper, we present a novel procedure to estimate the solvent electric field, interaction energy, and their fluctuation amplitudes by combining IR spectroscopy and the distributed interaction site model for vibrational solvatochromism. We shall apply it to an aqueous NMA solution.

II. SEMIEMPIRICAL DETERMINATION OF LOCAL SOLVENT ELECTRIC FIELD Vibrational Solvatochromism. A general framework to describe vibrational solvatochromism has been presented and discussed recently,12 where the charge distribution of the solute interacting with solvent electric field is divided into multiple fragments. In the present paper, we shall assume that the reference system is an isolated solute molecule so that ϕ0(R) = 0. Therefore, the frequency shift is defined as the difference between the vibrational frequency of solute in solution and that in the gas phase. According to this theory, the frequency shift of the jth vibrational mode in reference to the gas-phase value is given by the following expansion in terms of distributed multipole moments of solute charge distribution,

ð2Þ

The susceptibility ljx parameter of the xth site can be viewed as a measure of how strongly the xth site transition charge density changes upon vibration of the jth normal mode. If the corresponding l parameter is large, that site is highly susceptible to local change of solvent EP and strongly contributes to the frequency shift. These l parameters were obtained from quantum chemistry vibrational analyses of a large number of solute solvent clusters. Then the next step was to determine the solvent EP, ϕ(R), by using molecular dynamics (MD) trajectories.79,22,2931,4850 The instantaneous solvent configurations and detailed solvation structures were used to obtain the timedependent solvent EP values at all the interaction sites. The frequency shifts and fluctuations of IR probe modes were directly compared with experimentally measured peak frequencies and bandwidths of the corresponding IR spectra. The agreements were found to be quantitative. Despite the success of this theoretical approach, the inverse of the procedure has not been attempted. That is to say, it has been difficult to obtain the solvent EP and electric field directly from the vibrational spectroscopic data without relying on MD trajectories and local solvent configurations. We note that more simplified approaches based on the linear vibrational Stark effect, where the multiple chargeEP interactions in eq 2 are replaced by a point dipoleelectric field interaction in terms of an empirically measured Stark tuning rate Δμ, have been reported.21,51,52 In favorable cases, as shown by Boxer, Corcelli, and co-workers,51,52 this method can be used to extract the local electric field component of vibrational chromophores along a single representative direction, sometimes with an extra measure to account for specific H-bond interactions. However, it is believed that the present method has the potential to provide detailed information on solute electrostatic environment by systematically improving the distributed multipole model and taking into account multiple IR bands. Moreover, even though there exist quite a few experimental methods used to estimate the solvation enthalpy and free energy,5356 it has been difficult to selectively measure the ensemble average solvent electric field and the resulting solutesolvent electrostatic interaction energy. Furthermore, the fluctuation amplitudes of such solvent electric field around a solute and of the interaction energy, even though knowledge of such quantities are critical in all the thermally driven processes in condensed phases, could not be directly

Δωj ðϕÞ ¼

¼

∑x Γ^ x ϕðR x Þ ¼ ∑ ½lxj Lxj 3 ∇ L~ xj : ∇∇ þ x

3 3 3 ϕðR x Þ

∑x lxjϕðR x Þ þ ∑x Lxj 3 EðR x Þ þ ∑x L~ xj : ∇EðR x Þ þ

333

ð3Þ The definition of the operator^Γx, which was mentioned in eq 1, is given in the second line of eq 3. The three coefficients in the last line of eq 3 are NL AN lxj ¼ ½F^j qx ðQ ÞQ 0 þ ½F^j qx ðQ ÞQ 0 NL AN Lxj ¼ ½F^j px ðQ ÞQ 0 ½F^j px ðQ ÞQ 0 1 NL ~ 1 AN ~ ~ Lxj ¼ ½F^j Θx ðQ ÞQ 0 ½F^j Θx ðQ ÞQ 0 6 6

ð4Þ

where NL F^j

1 ∂2 , 2Mj ωj ∂Q j 2

AN F^j

1 2Mj ωj

g

∂

∑i Miωijj i2 ∂Q i ð5Þ

Here, Mj, ωj, and Qj are respectively the reduced mass, frequency, and coordinate of the jth normal mode, and gijj is the cubic anharmonic coefficient. The partial charge, dipole moment, and quadrupole moment of the fragment x are denoted as qx, px, ~x, respectively. The partial derivatives are evaluated at the and Θ equilibrium structure Q0 of an isolated molecule. The first terms in eq 4 describe the contributions from the nonlinear coordinate dependencies of distributed multipoles. The second terms in eq 4, on the other hand, are related to the mechanical anharmonicity contributions to the vibrational solvatochromism.12 Once the multipole (except for monopole) terms in eq 3 are truncated, the solvatochromic frequency shift is highly simplified as ϕðRÞ Δωj ðϕÞ ¼ Bl j 3 ~

ð6Þ

where the xth vector elements ofBl j and ϕB(R) are ljx and ϕ(Rx), respectively. Here, due to the charge neutrality condition, the 348

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sum of theBl j vector elements is 0: N

∑ ljx ¼ 0 x¼1

ð7Þ

Therefore, the number of independent l parameters is N 1, where N is again the number of distributed interaction sites. Solvent Electric Potential. For a given jth normal mode, the frequency shift induced by solvent EP was described by eq 6. It is, however, impossible to inversely calculate the distribution of solvent EP, ϕ(Rx), by using a single experimentally measurable quantity Δωj(ϕ). However, let us assume that N 1 different vibrational frequency shifts are all described by the same linearized expression in eq 6 with different sets of l parameters: ϕðRÞ, Δω1 ðϕÞ ¼ Bl 1 ~

Figure 1. Permanent dipole moment and atomic movements in the amide I0 and II0 modes of NMA-d7. The permanent dipole moment direction is indicated by the gray arrow in panel a. The atomic movements in the amide I0 and II0 modes are depicted as black arrows in panels a and b, respectively. In the DFT-optimized geometry, the atomic positions are as follows (in angstroms): RC = (0.5790, 0, 0.3606), RO = (0.5790, 0, 1.5822), RN = (0.5790, 0, 0.3606), RH = (0.5337, 0, 1.3656), RCO = (0.5790, 0, 0.9714), RNH = (0.5564, 0, 0.8631), and RCN = (0, 0, 0). In the same reference frame, the direction of the permanent dipole moment is given as (0.2697, 0, 0.9629) and its magnitude is 3.97 D. The angle between the CO bond and the permanent dipole moment vector is about 164.

Δω2 ðϕÞ ¼ Bl 2~ ϕðRÞ , :::,

ΔωN1 ðϕÞ ¼ Bl N1~ ϕðRÞ

ð8Þ

Then, the N 1 equations in eq 8 can be combined into a single expression as 0 1 Δω1 ðϕÞ B B Δω2 ðϕÞ C C B C B C l @ A ΔωN1 ðϕÞ 0

B B ¼B B @

l1, 1 l2, 1 l

l1, 2 l2, 2 l

333 333 ⋱

lN1, 1

lN1, 2

333

return to this issue on how to calculate solvent electric field later in this paper. In summary, if the number of independent l parameters for each vibrational frequency shift is identical to the number of vibrational modes, the solvent EP values at the distributed interaction sites can be semiempirically determined. Here, this procedure is semiempirical in nature, because the frequency shifts [Δωj(ϕ)] are to be experimentally measured and the (N 1)2 l parameters should be independently determined by carrying out quantum chemistry calculations for a number of solutesolvent clusters or solutesolvent configurations sampled from the corresponding MD trajectories. The present approach to the calculation of solvent EP or electric field is quite general and can be applied to a variety of vibrational degrees of freedom of any arbitrary polyatomic molecules as long as the solvatochromic vibrational frequency shifts are quantitatively described by using the distributed interaction site theory. In this paper, we shall consider the amide I0 and II0 modes of NMA-d7 (N-methylacetamide with all seven H atoms replaced with D atoms), which is an excellent prototype model for peptide. The reason that the fully deuterated NMA-d7 is considered instead of NMA is because the amide vibrations tend to delocalize over the CH bend local modes in normal NMA. Therefore, to avoid such complications, to reduce possible mixings of the OdCN-D local modes with CH bends in NMA, and to make the amide I vibration uncoupled from solvent H2O bending modes, we here consider NMA-d7 dissolved in D2O. Amide I0 and II0 Mode Frequency Shifts. Among a number of peptide vibrations, the amide I and II modes have been extensively studied over the years because their line shapes and peak frequencies are highly sensitive to polypeptide secondary structures and local H-bonding interactions. The amide I0 and II0 modes specifically denote the corresponding amide vibrations of peptides in D2O solution, where the amide NH group is deuterated. In this case of NMA-d7, the amide I0 mode is mainly localized on the CdO stretch local mode and the amide II0 mode is mainly a CN stretching vibration (see Figure 1). For a number of NMA-d7n(D2O) clusters, we carried out density functional theory (DFT)/B3LYP (6-31++G**) geometry optimizations and vibrational analyses to obtain the amide I0 and II0 normal-mode frequencies and eigenvectors. All electronic structure calculations were performed with the Gaussian 03

1 ϕðR 1 Þ ϕðR N Þ C C C CB B CB ϕðR 2 Þ ϕðR N Þ C C CB l A A@ lN1, N1 ϕðR N1 Þ ϕðR N Þ l1, N1 l2, N1 l

10

ð9Þ Here, we chose the Nth site as the reference site and used the following equality (see eq 7): ljN ¼

N1

∑

x¼1

ljx

ð10Þ

Next, by calculating the inverse matrix of the N 1 by N 1 square l-matrix on the right-hand side of eq 9, the difference EP values can be rewritten as 0 1 ϕðR 1 Þ ϕðR N Þ B B ϕðR 2 Þ ϕðR N Þ C C B C B C l @ A ϕðR N1 Þ ϕðR N Þ 0 11 0 1 l1, 1 l1, N1 l1, 2 Δω1 ðϕÞ 3 3 3 B C B C B l2, 1 B C l2, 2 l2, N1 C 333 C B Δω2 ðϕÞ C ¼B B l C B C l ⋱ l l @ A @ A lN1, 1 lN1, 2 3 3 3 lN1, N1 ΔωN1 ðϕÞ ð11Þ Once the N 1 vibrational frequency shifts appearing on the right-hand side of eq 11 are experimentally measured and the corresponding l parameters are predetermined by using the procedure discussed in refs 6,7, and 47, one can directly calculate the difference EP values on the left-hand side of eq 11. Using the relative distances between distributed interaction sites, one can approximately obtain the local electric fields, 3ϕ. We shall 349



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percentages of THF from 0% to 80%, the amide I0 and II0 spectra were measured and are shown in Figure 2. Cases with THF higher than 80 vol % are not considered in the present work because the NMA-d7 molecules have a strong tendency to form a dimer and even larger multimers in such nonpolar solvents. Even in the cases where D2O in the mixed solvent is 20, 30, 40, and 50 vol %, the amide I0 and II0 IR spectra show a shoulder band at around 1650 and 1475 cm1, respectively. These shoulder subbands mostly originate from NMA-d7 dimers in these mixed solvents. In order to separate the amide I0 and II0 spectra of monomeric NMA-d7 molecules from the whole spectra, we performed fitting analyses with two Gaussian line-shape functions. In Table 1, the resulting peak frequencies and standard deviations of the amide I0 and II0 bands of the monomeric NMAd7 molecules in solutions are presented. In the same table, the amide I0 and II0 mode frequency shifts compared to the gas phase values, ωI0 (ϕ0 = 0) = 1696.5 cm1 and ωII0 (ϕ0 = 0) = 1405.5 cm1, are also given (see Figure.2b). Determination of l Parameters. The l parameters connecting the site EP and the vibrational frequency shift in eq 6 is a central quantity of the present method. The accuracy of the method hinges upon the accuracy of these coefficients in the expansion of the vibrational frequency shifts in terms of the electric potentials at the interaction sites. Here, they are determined from a series of quantum chemistry calculations as follows. First, from the DFT calculations of various NMAd7n(D2O) clusters, we have the scaled amide I0 and II0 mode frequencies as well as detailed coordinates of surrounding D2O molecules. We then use the distributed interaction site models for the two amide vibrations as ΔωI0 ðϕÞ ΔωII0 ðϕÞ

! ¼ QM

l I0 , 1 lII0 , 1

lI0 , 2 lII0 , 2

!

ϕðR 1 Þ ϕðR 3 Þ ϕðR 2 Þ ϕðR 3 Þ

!

ð12Þ QM

Note that the number of interaction sites is 3. As long as the three sites are placed anywhere on the OdCND group, we found that the multivariate least-squares fitting results are in excellent agreement with DFT calculation results and the Pearson correlation coefficients are larger than 0.9 in most cases. This indicates that the distributed interaction site model is quite robust and does not strongly depend on detailed locations of the distributed sites in these cases. Among different choices of three sites, we considered three representative cases here in detail. Case I is that the three sites are at O(dC), D(N), and the midpoint of the CN bond. Case II is that they are at C(dO), N(D), and the midpoint of the CN bond. Case III is that the three sites are at the midpoint of the CdO bond, the midpoint of the ND bond, and the midpoint of the CN bond. The total number of NMAd7n(D2O) clusters considered here is 35. Therefore, we have 35 pairs of amide I0 and II0 mode frequency shifts and also 35 values of ϕ(R1) ϕ(R3) and ϕ(R2) ϕ(R3) obtained from the NMA-d7n(D2O) clusters. Thus, the multivariate leastsquares analyses provided the three l parameters in e, where e is the elementary charge. They are summarized in Table 2. Let us examine the first case in more detail, where the three sites are assumed to be at O, D, and the midpoint of the CN bond. For the amide I0 mode, the l parameter of the carbonyl oxygen atom is 6.846 103 e, whereas that of the D(N) atom is 0.746 103 e, which is an order of magnitude smaller than that of the O atom. This indicates that the local solvent EP change at the O atomic site contributes to the amide I0 mode

Figure 2. (a) Experimental IR spectra of NMA-d7 in a series of mixed D2O/THF solvents. The two extreme cases of 100% and 20% D2O by volume are indicated by arrows. The two bands in the spectrum correspond to amide I0 and II0 modes, respectively. (b) Experimental peak shifts of amide I0 and II0 bands from the isolated case. Amide I0 and II0 peaks are red- and blue-shifted with increasing D2O concentration. (c) Widths of the amide I0 and II0 bands.

program.57 The frequency scaling factor used is 0.9803, which was obtained by directly comparing the amide I and II mode frequencies of NMA monomer, trans-NMA dimer, and cis-NMA dimer with the experimentally measured frequencies of NMA monomer and dimers in N2 and Ar. In the case of the NMA monomer in N2 and Ar, the experimentally measured amide I mode frequencies are 1706 and 1708 cm1, respectively.58 The corresponding amide II mode frequency in low-temperature N2 matrix is 1511 cm1.59 Since there are no experimental results for NMA-d7, we used the DFT calculation results with the empirically determined scaling constant (=0.9803) mentioned above. From this procedure, the amide I0 and II0 mode frequencies of NMA-d7 in the gas phase are calculated to be 1696.5 and 1405.5 cm1, respectively, which are the reference frequencies, ωI0 (ϕ0 = 0) and ωII0 (ϕ0 = 0). We next carried out the Fourier transform infrared (FTIR) studies of NMA-d7 in 100% D2O and a variety of D2O/ tetrahydrofuran (THF) mixed solvents. With varying volume 350



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Table 1. Amide I0 and II0 Mode Frequencies and Corresponding IR Absorption Bandwidthsa vol % D2O

ωI0 , cm1

σ[ωI0 ] (SD)

ωII0 , cm1

σ[ωII0 ] (SD)

ΔωI0 , cm1

ΔωII0 , cm1

100

1606.4

12.4

1493.7

7.4

90.2

88.2

90

1607.9

12.8

1493.6

6.9

88.7

88.1

80

1609.9

13.0

1490.7

8.2

86.7

85.2

70

1611.8

13.2

1489.0

8.7

84.8

83.5

60

1613.8

13.8

1487.0

9.2

82.7

81.5

50

1616.0

13.9

1485.0

9.7

80.5

79.5

40

1617.6

14.2

1483.9

9.7

79.0

78.5

30 20

1619.4 1624.1

14.2 15.2

1482.1 1479.6

10.0 9.5

77.1 72.5

76.6 74.1

a Two Gaussian functions were used to fit to the experimentally measured spectra. Amide I0 and II0 mode frequencies of isolated NMA-d7 are 1696.5 and 1405.5 cm1, respectively. Corresponding peak shifts in reference to the gas-phase values (ΔωI0 , ΔωII0 ) are given in the last two columns.

Table 2. l Parameters for Amide I0 and II0 Modesa case I

a

case II

case III

amide modes

lO, 103 e

lD, 103 e

lCN, 103 e

lC, 103 e

lN, 103 e

lCN, 103 e

lCO, 103 e

lND, 103 e

lCN, 103 e

I0 II0

6.8456 4.6077

0.7456 0.4423

7.5912 4.1654

43.994 27.290

14.434 3.6531

58.428 30.943

15.001 9.3698

4.704 0.57589

19.705 9.9456

Detailed discussions on the three cases of the three-site model are provided in the main text.

frequency shift 10 times more strongly than that at the amide D atomic site does. For the amide II0 mode, the l parameter of the carbonyl oxygen atom is 4.608 103 e and that of the D(N) atom is 0.442 103 e. Again, the relative weight of the D atomic site is small in comparison to that of the O atomic site. Therefore, we found that the amide I0 and II0 mode frequency shifts are strongly affected by the local EP change at the carbonyl group in this case. This conclusion remains valid even in cases II and III, as can be seen in Table 2. Solvent Electric Potential and Field. Since we have experimentally measured amide I0 and II0 frequency shifts and also we have the l parameters for these two modes, it is now immediately possible to use the inverse expression, that is, for case I, ! ! !1 ϕðR O Þ ϕðR CN Þ ΔωI0 ðϕÞ lI0 , 1 lI0 , 2 ¼ lII0 , 1 lII0 , 2 ϕðR D Þ ϕðR CN Þ ΔωII0 ðϕÞ theory

from the DFT geometry-optimized structure of NMA-d7, we could also calculate the magnitude of solvent electric field to be about 172 MV/cm. Such solvent electric field experienced by a small organic molecule like NMA has not been experimentally measured before. Noting that typical electric field used in vibrational Stark effect spectroscopy is about a few megavolts per centimeter, so the solvent electric field is significantly large. In this work, since we have all the amide I0 and II0 mode frequency shifts for varying volume percentages of THF solvent, we could obtain the local solvent EP and electric field magnitudes (see Table 3). As the volume percent of THF in the D2O/THF mixed solvents increases, the solvent polarity decreases. Consequently, the amide I0 and II0 mode frequencies become blue- and red-shifted in comparison to those of NMA-d7 in 100% D2O solution. As expected, the solvent EP values and electric field magnitudes decrease as the solvent polarity decreases. SoluteSolvent Coulomb Interaction Energy. Once the solvent electric field is determined in a semiempirical manner, one can directly calculate the solutesolvent electrostatic interaction energy by using the solute dipolesolvent electric field interaction:

exp

ð13Þ to estimate the solvent difference EP values, ϕ(RO) ϕ(RCN) and ϕ(RD) ϕ(RCN). Here RO, RD, and RCN are the position vectors of the O atomic site, the D atomic site, and the midpoint of the CN bond, respectively. In the case of NMA-d7 in 100% D2O solution, the experimentally measured amide I0 and II0 frequency shifts are 90.2 and 88.2 cm1, respectively. Then, eq 13 with the case I distributed interaction site model (see Table 2) provides the water EP differences between the O atomic site and the midpoint of the CN bond and between the D atomic site and the midpoint of the CN bond as 16 200 and 27 500 cm1/e, respectively. These values can be converted to volts and they are 2.0 and 3.4 V, respectively (see Table 3). Then, the potential difference between the O and D atomic sites, which is denoted as ϕ(RO) ϕ(RD), is 5.4 V. By use of this estimated electric potential difference and the distance between the O and D atoms, which is 3.15 Å, obtained

EC ¼ μNMA 3 Esolvent

ð14Þ

In the literature, the permanent dipole moment of a normal NMA molecule is found to be about 4.4 D in 1,4-dioxane at 298 K.60,61 In addition, Hiramatsu and Hamaguchi61 used infrared electroabsorption spectroscopic method to determine the angle between the permanent dipole moment and the amide I transition dipole moment of NMA in 1,4-dioxane. They showed that the two dipole moments are parallel to each other. A DFT calculation on isolated NMA-d7 yielded the dipole moment of 3.972 D nearly in parallel with RD RO. Combining this with the empirical |Esolvent| determined above, we obtain the electrostatic interaction energy EC of 137 kJ/mol for case I at the experimental temperature of 22 C. 351



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Table 3. Solvent EP Differences and Solvent Electric Field Magnitudes Determined from Experimental IR Spectraa case I

vol % D2O 100

ΔϕD,CN,

|Esolvent|,

EC,

ΔϕC,CN,

ΔϕN,CN,

|Esolvent|,

EC,

ΔϕCO,CN,

ΔϕND,CN,

|Esolvent|,

EC,

(V)

(V)

(MV/cm)

(kJ/mol)

(V)

(V)

(MV/cm)

(kJ/mol)

(V)

(V)

(MV/cm)

(kJ/mol)

2.004

3.409

171.8

136.6

0.490

0.718

88.6

52.2

1.244

1.592

131.5

100.8

1.981 (0.168)

80

1.954 (0.161)

70

1.914 (0.168)

60

1.867 (0.176)

50

1.820 (0.182)

40

1.792 (0.183)

30

1.750 (0.187)

20

case III

ΔϕO,CN,

(0.168) 90

case II

1.675 (0.188)

(1.625) 3.444 (1.629) 3.531 (1.577) 3.479 (1.636) 3.395 (1.717) 3.322 (1.767) 3.324 (1.784) 3.251 (1.819) 3.328 (1.836)

(51.8) 172.2 (52.0) 174.1 (50.3) 171.1 (52.2) 167.0 (54.8) 163.2 (56.4) 162.4 (56.9) 158.7 (58.0) 158.8 (58.6)

(41.2) 136.9 (41.3) 138.4 (40.0) 136.0 (41.5) 132.8 (43.6) 129.7 (44.8) 129.1 (45.2) 126.2 (46.1) 126.2 (46.6)

(0.074) 0.487 (0.073) 0.486 (0.068) 0.476 (0.072) 0.465 (0.075) 0.454 (0.079) 0.449 (0.079) 0.438 (0.082) 0.428 (0.079)

(0.279) 0.723 (0.279) 0.736 (0.269) 0.724 (0.280) 0.707 (0.294) 0.691 (0.303) 0.69 (0.306) 0.674 (0.312) 0.682 (0.313)

(21.2) 88.7 (21.2) 89.5 (20.4) 88.0 (21.2) 85.9 (22.2) 83.9 (23.0) 83.4 (23.2) 81.6 (23.6) 81.4 (23.7)

(12.5) 52.2

(0.152) 1.235

(12.5) 52.7

(0.150) 1.228

(12.0) 51.8

(0.138) 1.204

(12.5) 50.6

(0.146) 1.175

(13.1) 49.4

(0.154) 1.146

(13.5) 49.1

(0.162) 1.132

(13.6) 48.0

(0.162) 1.106

(13.9) 47.9

(0.167) 1.074

(14.0)

(0.159)

(0.629) 1.602 (0.629) 1.631 (0.607) 1.606 (0.630) 1.567 (0.662) 1.532 (0.682) 1.529 (0.688) 1.495 (0.702) 1.514 (0.706)

(30.0) 131.5 (30.0) 132.5 (28.8) 130.2 (30.0) 127.1 (31.5) 124.2 (32.5) 123.4 (32.8) 120.6 (33.5) 120.0 (33.5)

(23.0) 100.8 (23.0) 101.6 (22.1) 99.8 (23.0) 97.4 (24.1) 95.2 (24.9) 94.6 (25.1) 92.4 (25.7) 92.0 (25.7)

Here Δϕm,n represents the solvent EP difference between sites m and n, that is, Δϕm,n = ϕ(Rm) ϕ(Rn). With the assumption that the NMAwater interaction is described as the permanent dipolesolvent electric field interaction, we calculated the electrostatic (Coulombic) interaction energy, EC. Standard deviations estimated from amide I0 and II0 IR absorption bandwidths are given in parentheses. a

Although the solvation enthalpy of NMA in water was measured by using calorimetry and reported by Della Gatta et al.,62 the electrostatic NMAwater interaction energy has not been reported before because there is no experimental method used to measure such quantity directly. Graziano63 showed that the hydration enthalpy change is about 81.0 kJ/mol at 25 C. Summing up the van der Waals and H-bonding interaction energy, he also showed that the overall NMAwater interaction energy is 121 kJ/mol at 25 C. This value was compared with the computational result 116 kJ/mol by Jorgensen and Swenson,37 where the OPLS potentials and TIP4P water model were used. Torii et al.36 performed HartreeFock calculations for NMA with three H-bonded water molecules, where the solvation effect was approximately taken into account by using the self-consistent reaction field method. The H-bond energies were estimated to be 75.6 kJ/mol. These experimental and theoretical data contain, in addition to the electrostatic interaction EC, some of the extra contributions to the solutesolvent interaction energy, such as the electronic polarization energy of the solute and solvent, cavitation and reorganization energy of the solvent, and electronic dispersion interaction, which are all beyond the scope of this work. Despite this difference and the pointdipole approximation used in eq 14, the electrostatic NMA water interaction energy in the present work is quite comparable to these reported values. Even though we can pursue more accurate calculations of the solutesolvent interaction energy using atomic partial charges and solvent EPs obtained here, we shall pay attention to the more important issue of how large the fluctuation amplitudes of solvent EP, electric field, and solutesolvent electrostatic interaction energy are. Fluctuation Amplitudes. We have discussed the relationship between the vibrational frequency shift and local solvent EP

distribution. Another important piece of information that can be extracted from the vibrational spectrum is related to the bandwidth. Line broadening of the IR band is caused by a few different contributions such as lifetime, rotational relaxation, pure dephasing, and inhomogeneous line broadening. In the present work, we shall simply assume that the bandwidths of the amide I0 and II0 spectra are determined by the inhomogeneous line broadening mechanism. Certainly this is a problematic assumption, not only because the lifetimes of the amide I0 and II0 mode excited states are on the order of picosecond time scales but also because the pure dephasing processes are not negligibly small in comparison to the inhomogeneous broadening processes. However, since we intend to estimate the upper limits of the fluctuation amplitudes of the solvent EP, electric field, and solutesolvent electrostatic interaction energy, the inhomogeneous limit is assumed for the sake of simplicity. In Table 1, the measured standard deviations of the Gaussian-fitted amide I0 and II0 spectra, denoted as σ[ωI0 ] and σ[ωII0 ], respectively, are given. Then, by taking into consideration the error propagation, it becomes possible to estimate the standard deviations of the solvent difference EP values, that is, σ[ϕ(RO) ϕ(RCN)] and σ[ϕ(RD) ϕ(RCN)]. In the case that NMA-d7 is dissolved in pure D2O and the case I three-site model is chosen, we found that the standard deviations σ[ϕ(RO) ϕ(RCN)] and σ[ϕ(RD) ϕ(RCN)] are 0.17 and 1.63 V, respectively. Since the averages of ϕ(RO) ϕ(RCN) and ϕ(RD) ϕ(RCN) are 2.0 and 3.4 V, we have ϕðR O Þ ϕðR CN Þ ¼ 2:00 ( 0:17 V ϕðR D Þ ϕðR CN Þ ¼ 3:41 ( 1:63 V ϕðR O Þ ϕðR D Þ ¼ 5:41 ( 1:64 V

ð15Þ

The fluctuation amplitude of ϕ(RO) ϕ(RCN) appears to be an order of magnitude smaller than the average value, whereas the 352



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Table 4. Average Peak Shifts of Amide I0 and II0 Modes Predicted from MD Trajectorya

fluctuation amplitude of the solvent EP at the D atomic site is almost comparable to the average solvent EP. We next perform the error propagation calculation to estimate the fluctuation amplitude of the solvent electric field. Since the two standard deviations σ[ϕ(RO) ϕ(RCN)] and σ[ϕ(RD) ϕ(RCN)] were determined, we found that the standard deviation of the electric field amplitude, which is denoted as σE, is about 52 MV/cm: jED2 O j ¼ 172 ( 52 MV=cm

case II

case III

amide I0 , cm

88.2 (18.4)

85.9 (37.7)

91.2 (23.7)

amide II0 , cm1

67.2 (13.3)

67.3 (25.1)

69.5 (16.1)

The reference frequencies are 1696.5 and 1405.5 cm1 for the amide I0 and II0 modes, respectively. Standard deviations are given in parentheses. a

ð16Þ

observation made was that the local EP around the carbonyl group differs from that around the amide ND group. Furthermore, the corresponding fluctuation amplitudes were found to be site-dependent. In order to shed light onto such site-specific information on the local EP, we carried out classical MD simulation of NMA-d7 dissolved in D2O. Quite similar MD simulation studies for NMA in TIP3P water were performed by Hayashi et al.64 They calculated the spatial distributions of the solvent EP and the in-plane components of the solvent electric field vector around the peptide bond. Nevertheless, their composite system differs from ours, and site-specific EP values that are needed to directly compare with the present experimental results were not reported. In the simulation, the NMA-d7 molecule was described by the flexible AMBER ff99SB parameter set and D2O was described by the rigid TIP4P force field. Additional simulations with frozen NMA-d7 configuration produced nearly identical results. The simulation system consisted of one NMA-d7 and 972 D2O in a rectangular box (∼30.8 Å in length each side) under the periodic boundary condition. AMBER 11 package was used in the simulation.65 The long-range electrostatic interactions were calculated with the particle-mesh Ewald method, and the LennardJones interactions were cut off at 12 Å. After equilibration, the trajectory for analyses was accumulated for 1 ns in 10 fs intervals under the constant energy condition with time step size of 0.5 fs. Before we directly compare the semiempirically determined solvent EP values at a few sites of the solute NMA-d7 in D2O with those obtained from MD trajectory, we first discuss simulation results for the amide I0 and II0 IR absorption spectra and compare such simulated spectra with experimentally measured ones. Combining the solvent EP obtained from the MD trajectory with the l parameters determined from DFT calculations (Table 2), we can predict the peak shifts via eq 6, independent of experiments.7 To calculate the solvent EP ϕ(R), we assumed that the atomic partial charges of the O and D atoms of D2O are 0.824 and 0.412 e. Here, it should be noted that the l parameters were obtained from the solutewater clusters with the same partial charges of water used, because the currently available molecular mechanics force fields, for example, TIP4P model for liquid water, were not developed to predict accurate electrostatics. In Table 4, we have summarized the average and standard deviation of these theoretical peak shifts of the amide I0 and II0 modes relative to the reference gas phase values of 1696.5 and 1405.5 cm1, respectively. Compared to the experimental peak shifts of 90.2 and 88.2 cm1 for the amide I0 and II0 modes in Table 1, the theoretical values from the three interaction site models (cases I, II, and III) in Table 4 show excellent agreement for the amide I0 mode, while they underestimate the experimental value by 2025% for the amide II0 mode. The standard deviations of the peak shifts seem to be dependent on the specific interaction site model. As the two peripheral interaction sites

As a matter of fact, this is a very large fluctuation. It should be emphasized that the fluctuation amplitude of local solvent electric field has never been experimentally measured before, even though it is a critical quantity of chemical reaction dynamics in general, directly related to the energetic activation of reactants as well as the dissipation of product excess energies. Once the fluctuation amplitude of the solvent electric field is determined, one can readily calculate the standard deviation of the solutesolvent electrostatic interaction energy, which is found to be about 41 kJ/mol. Therefore, we have, for case I, EC ¼ 137 ( 41 kJ=mol

case I 1

ð17Þ

Noting that the mean formation enthalpy of NH chemical bond is about 400 kJ/mol, it is surprising that the solutesolvent interaction energy fluctuation is quite large and just about an order of magnitude smaller than one of the weak chemical bond energies in NMA. In D2O solutions of peptides, the peptide NH protons are readily exchanged with D-atom in D2O solvent at room temperature. The present work shows that such a hydrogendeuterium exchange reaction can indeed be driven by thermal fluctuation of the local solvent configuration around a solute due to large fluctuation amplitudes of the local solvent electric field and solutesolvent interaction energy. In Table 3, we additionally present the standard deviations of solvent EPs, electric fields, and interaction energies of NMA-d7 in D2O/THF mixed solvents to show their solvent polarity dependencies. In the case of EC (cf. eq 14), DFT-optimized geometry was used to obtain the angle between μNMA and Esolvent. The amide I0 and II0 mode frequencies are approximately proportional to the volume percent of THF in the mixed solvent (see Figure 2b). Therefore, the solvent EP and electric field also change linearly with respect to the THF volume percent in the mixed solvent. Before we close this section, it should be emphasized that the line broadenings of the amide I0 and II0 bands were assumed to be dictated by the inhomogeneity of local solvent configurations. Therefore, the estimated fluctuation amplitudes discussed in this section should be considered as their upper limits. In the near future, by carrying out polarization-controlled IR pumpprobe and 2D IR spectroscopy of the amide I0 and II0 modes in NMA-d7 in D2O, the lifetimes, rotational relaxation rates, and pure and inhomogeneous dephasing widths can all be separately measured. Then it will be possible to obtain refined numbers for the fluctuation amplitudes. Nevertheless, the essential procedure and computational method outlined in this paper would not be altered, and the present data reported here should be considered as the reference values for comparison.

III. COMPARISONS WITH MD SIMULATION RESULTS In the previous section, we presented semiempirically calculated solvent EP, electric field, and resulting solute solvent electrostatic interaction energy in D2O. An interesting 353



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NMA-d7 in D2O. In the figure, the peak positions and heights of all spectra were made to coincide for better comparison of the bandwidth. Considering that the lifetime broadening effect7,67 was not taken into account in the simulated spectra, the spectra obtained with the case I interaction site model show the best agreement with experiment in both amide I0 and II0 modes. On the other hand, case II, which has two peripheral interaction sites located at the amide C and N atoms, overestimates the bandwidth significantly. These results are in line with similar trends of the static frequency shift distribution in Table 4 and indicate that the peripheral atomic sites (O and D) should be included in the distributed interaction site model to accurately predict the inhomogeneous line broadening. Nevertheless, the solvatochromic frequency shifts of amide I0 and II0 modes and bandwidths of simulated spectra appear to be in reasonable agreement with experimental results. Hereafter, we shall directly compare the solvent EP values that are estimated by analyzing MD trajectories with the semiempirically determined values in section II. Among the three different cases of the distributed interaction site model (in Table 2), we choose case I for detailed comparisons, so that we need to calculate the solvent EP values at the O atomic site, the midpoint of the CN bond, and the D atomic site. The averages and standard deviations of the corresponding difference EP values, ϕ(RO) ϕ(RCN) and ϕ(RD) ϕ(RCN), estimated from the MD trajectory are found to be

Figure 3. Simulated IR spectra of (a) amide I0 and (b) amide II0 modes. Cases IIII of the simulated spectra correspond to three different interaction site models defined in the main text. The experimental spectra are also displayed as black dashed lines. For comparison, the peak positions and heights are matched in all spectra.

ϕðR O Þ ϕðR CN Þ ¼ 1:71 ( 0:34 V ðMDÞ ϕðR D Þ ϕðR CN Þ ¼ 1:03 ( 0:32 V ðMDÞ

ð20Þ

The computed ϕ(RO) ϕ(RCN) agrees fairly well with the experimental value of 2.00 ( 0.17 V in eq 15, whereas ϕ(RD) ϕ(RCN) deviates from the experimental value of 3.41 ( 1.63 V in both the average and standard deviation. Then, by using the calculated ϕ(RO) ϕ(RD) and the distance between O and D atoms in NMA-d7, the estimated solvent electric field magnitude is

become closer to the central site at the midpoint of amide CN bond (i.e., when going from case I to III, then to II), the standard deviation becomes larger both in amide I0 and II0 modes. We then calculated the simulated IR spectra of the two modes using the standard linear response function theory.7,66,67 Briefly, the line-shape function I(ω) is proportional to the Fourier transform of the linear response function R(t): Z ∞ dt RðtÞeiωt IðωÞ ∞ ð18Þ 2 RðtÞ ¼ θðtÞIm½jμge j2 eiω̅ eg tgðtÞ p

jED2 O j ¼ 87:4 ( 17:7 MV=cm

ð21Þ

This result differs by a factor of 2 from the experimental value of 172 ( 52 MV/cm. The same procedure was repeated for the other two cases of interaction sites and the results are summarized in Table 5. In Figure 4, the calculated difference EP values from the three interaction site models are plotted together with the semiempirically determined difference EP values for NMA-d7 in D2O. They show that the calculated difference EPs near the CO bond, ΔϕO,CN, ΔϕC,CN, and ΔϕCO,CN, agree fairly well with the semiempirically determined values. However, the difference EPs near the ND bond, ΔϕD,CN, ΔϕN,CN, and ΔϕND,CN, and the electric field magnitude |Esolvent| exhibit much larger deviations (compare Table 3 with Table 5). To further investigate the electrostatic environment around the solute NMA-d7 in D2O solution, we have calculated the EP due to solvent at fixed grid points on the solute molecular plane and plotted the result as a two-dimensional contour map in Figure 5. Due to large negative and positive charges on O and D sites of NMA-d7 (0.568 and +0.272 e, according to the employed AMBER force field), respectively, the EP has an extremum right off each atomic site (Figure 5a). From the contour line intervals near the O and D sites, it is clearly seen that the electric field near the oxygen is stronger than that near the deuterium. The fluctuations in EP displayed in Figure 5b

where θ(t) is the Heaviside step function, μge and ω ̅ eg are the transition dipole moment and average transition frequency in solution, and g(t) is the line broadening function. Under the second-order cumulant expansion approximation, g(t) can be expressed in terms of the frequencyfrequency time correlation function (FFCF) C(t) = Æδω(t)δω(0)æ where δω(t) denotes fluctuations in the transition frequency relative to ω ̅ eg: ~ I ðωÞ i Z ∞ ~ I ðωÞ it Z ∞ Im½C Im½C dω dω sin ωt π 0 ω π 0 ω2 ~ I ðωÞ 1Z ∞ Im½C βpω dω coth þ ð1 cos ωtÞ ð19Þ π 0 ω2 2

gðtÞ ¼

~ I(ω) is the Fourier transform of the imaginary part of where C ~ I(ω)] = tanh C(t). Under a classical approximation, Im[C ~ cl(ω) and eq 18 can be evaluated from the Fourier (βpω/2)C ~ cl(ω) of the classical FFCF Ccl(t) obtained from transform C the MD trajectory. The results are displayed in Figure 3 together with the experimental spectra (dashed lines) of 354



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Table 5. Solvent EP Differences and Electric Field Magnitudes Determined from MD Simulation of NMA-d7 in D2O Solventa case I ΔϕO,CN, V

ΔϕD,CN, V

1.709

1.027

(0.345) a

(0.322)

case II ΔϕC,CN, V

ΔϕN,CN, V

87.4

0.349

0.327

(17.7)

(0.122)

|Esolvent|, MV/cm

(0.118)

case III ΔϕCO,CN, V

ΔϕND,CN, V

50.7

0.960

0.659

(15.7)

(0.219)

|Esolvent|, MV/cm

(0.195)

|Esolvent|, MV/cm 76.0 (16.8)

The quantities have the same meanings as in Table 3. Standard deviations are shown in parentheses.

show that they have roughly the same magnitude near the oxygen and deuterium sites, consistent with the standard deviations of the difference EP in Table 5. However, this result is not fully consistent with either the semiempirical results shown in Table 3 or the widespread notion that the carbonyl oxygen makes a stronger H-bond with solvent water molecules than the amide hydrogen does. Further analysis of the MD trajectory in terms of the H-bonding dynamics (using the H-bond criterion of donoracceptor distance less than 3.0 Å and donorH/D acceptor angle larger than 160) reveals that this is because the amide hydrogen is more than 10 times less likely to make H-bonds than the carbonyl oxygen does, and it spends more than 90% of the time without making H-bonds. Therefore, even though the H-bond of the NH group has a much shorter lifetime than that of the carbonyl oxygen, its time-dependent changes do not contribute appreciably to the fluctuating electrostatic environment because its formation is not a very likely event in the first place.

Figure 4. Comparison of difference EP Δϕm,n obtained from the semiempirical procedure (red) and MD simulation (blue). The solvent EP relative to that at the midpoint of the CN bond is plotted. Site AB denotes the midpoint of atoms A and B.

IV. SUMMARY In this paper, we have presented a novel semiempirical procedure to calculate the site-specific solvent EP, electric field, and electrostatic interaction energy. The method combines the experimental IR spectrum with electronic structure calculation and determines the electrostatic environment of a vibrational chromophore through the distributed interaction site model. As an application, we have studied the electrostatic environment around an NMA-d7 solute in a series of mixed D2O/THF solvents, using the IR spectra of the amide I0 and II0 modes to determine the electric potential at three interaction sites of the solute. The solvent EP and electric field show a nearly monotonic increase with increasing D2O concentration, as expected from increasing solvent polarity. The electrostatic solutesolvent interaction energy estimated from the permanent dipole moment of NMA and calculated solvent electric field is in fair agreement with existing solvation energy data on the NMA in aqueous solution. However, their exact values depend on the choice of the interaction sites in the model. It turns out that the carbonyl oxygen and amide hydrogen atoms should be included in the set of multiple interaction sites of the amide group. The fluctuations in the electrostatic environment have also been determined from the bandwidths of the IR spectra, and they lead to fluctuations in the electrostatic interaction energy as large as 30% of the average value. A comparative MD simulation study shows that the semiempirically determined electrostatic environment near the carbonyl bond of the amide group agrees well with the MD results, while that near the ND bond shows notable deviation. It also indicates that the carbonyl oxygen and amide hydrogen/deuterium sites are essential in capturing the fluctuating solvent electrostatic environment near NMA and accurately predicting the IR spectra

Figure 5. (a) Contour plot of the average EP (in volts) due to D2O solvent on the molecular plane (x = 0) of the NMA-d7 solute, calculated from the MD trajectory. Atomic positions are marked in the plot. The contour lines are drawn at 0.1 V intervals. (b) Contour plot of the fluctuations in EP (in volts) in the same reference frame. The contour lines are drawn at 0.02 V intervals. The empty region in panel b is more accessible by the solvent and therefore exhibits larger fluctuations in solvent EP than the intramolecular region. 355



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of amide I and II modes. Overall, the present method is expected to be useful in investigations of the electrostatic environment of chemical and biological systems by IR spectroscopic means. In the future, the method can be made more robust by systematically taking into account more IR modes and molecular interaction sites in the model.

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’ AUTHOR INFORMATION Corresponding Author

*E-mail [email protected].

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Vibrational Spectroscopic Determination of Local Solvent Electric Field

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