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Orientational Time Correlation Functions for Vibrational SumFrequency Generation. 2. Propionitrile Shule Liu†,# and John T. Fourkas*,†,‡,§,∥ †

Department of Chemistry & Biochemistry, University of Maryland, College Park, Maryland 20742, United States Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, United States § Maryland NanoCenter, University of Maryland, College Park, Maryland 20742, United States ∥ Center for Nanophysics and Advanced Materials, University of Maryland, College Park, Maryland 20742, United States ‡

S Supporting Information *

ABSTRACT: Molecular dynamics (MD) simulations of propionitrile have been performed to assess the influence of reorientation on vibrational sum-frequencygeneration (VSFG) spectra at the liquid/vapor (LV) and liquid/silica (LS) interfaces. Orientational time−correlation functions (TCFs) are derived for the VSFG spectroscopy of the symmetric and asymmetric stretches of functional groups such as methylene groups and rotationally hindered methyl groups. The MD simulations are used to compute VSFG orientational TCFs for the methyl, methylene, and cyanide groups of propionitrile at the LV and LS interfaces. Although propionitrile exhibits relatively fast reorientation in the bulk liquid, we find that for symmetric stretching modes at these interfaces, reorientation only plays a significant role in VSFG spectra under SPS polarization conditions. For asymmetric stretches, reorientation affects the VSFG spectra significantly under all polarization conditions. Azimuthal dynamics tend to dominate the orientational TCFs.

I. INTRODUCTION Vibrational sum-frequency generation (VSFG) spectroscopy1−11 allows for the direct probing of molecular organization at an interface between two media. Although VSFG has been used broadly to determine molecular orientations at interfaces, there is a growing appreciation that this technique is also sensitive to dynamic processes at interfaces.7,12−15 One such process that can affect the VSFG signal is reorientation, which has been implicated as the cause for complex line shapes in VSFG spectra at liquid/vapor16−21 (LV), solid/vapor16 and solid/liquid22 interfaces. Although expressions describing the maximum possible influence of reorientation on VSFG spectra have been presented previously,16 extracting detailed information on orientational dynamics from VSFG spectra requires an understanding of the relevant orientational time correlation functions (TCFs). To play a significant role in a VSFG spectrum, reorientation must change the magnitude of the relevant response function by a substantial amount on a time scale that can compete with those of population relaxation and pure dephasing (i.e., a few picoseconds at most). In a previous paper23 (hereafter called paper 1) we derived orientational TCFs for symmetric stretches of high-symmetry species, such as cyanide groups, acetylenic C−H groups, and freely rotating methyl groups in symmetric environments. We also derived orientational TCFs for the asymmetric methyl stretch of freely rotating methyl groups in symmetric environments. By combining these TCFs with experimental Raman depolarization ratios,24 in paper 1 we were able to use molecular dynamics (MD) simulations of acetonitrile to assess © 2014 American Chemical Society

the impact of reorientation on the VSFG spectrum of this liquid at the LV and liquid/silica (LS) interfaces. Our simulations showed that, for the vibrational modes studied in acetonitrile, reorientation plays the greatest role in VSFG spectra of the asymmetric methyl stretch. We demonstrated that, due to the unusual ordering of liquid acetonitrile at the silica interface,25−29 reorientation can also lead to unexpected effects such as the growth of the VSFG signal with time at this interface.23 These predictions of our simulations were in good agreement with the corresponding experimental VSFG spectra for this liquid.25 Here we derive orientational TCFs for more complex situations, including the symmetric and asymmetric stretches of methylene groups and methyl groups that are rotationally hindered and/or in asymmetric environments. As a test case for the evaluation of these TCFs, we have performed MD simulations on propionitrile (ethyl cyanide) at the LV and LS interfaces. Propionitrile is not much larger than acetonitrile, and in the bulk liquid its orientational dynamics are only slightly slower than those of acetonitrile.30 Simulations indicate that, similarly to acetonitrile, propionitrile takes on a lipid-bilayerlike intermolecular structure at the LS interface.31 Both molecules have symmetric and asymmetric methyl stretches as well as a CN stretch. However, propionitrile has a Special Issue: James L. Skinner Festschrift Received: March 21, 2014 Revised: April 30, 2014 Published: April 30, 2014 8406

dx.doi.org/10.1021/jp502847f | J. Phys. Chem. B 2014, 118, 8406−8419

The Journal of Physical Chemistry B

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1 (2) R xzx (τ ) ∝ − μ′Z α′XX ⟨sin θ sin 2Θ cos2 Φ cos χ cos X⟩ 2 1 − μ′Z α′YY ⟨sin θ sin 2Θ sin 2 Φ cos χ cos X⟩ 2 1 + μ′Z α′ZZ ⟨sin θ sin 2Θ cos χ cos X⟩ (3) 2

methylene group, giving it lower symmetry than acetonitrile. In addition, although the rotation of the methyl group in acetonitrile is barrierless, the rotation of the methyl group in propionitrile is hindered. Comparison of the orientational TCFs for acetonitrile and propionitrile allows us to gain further insight into the factors that influence the contribution of reorientation to the VSFG spectra of small molecules.

Here the orientational TCFs depend on both θ and the azimuthal angle χ. The dependence on φ, however, is only as an orientational average at a single time. We next consider the zzz tensor element of the second-order response. In case 3, eq 7 of paper 1 reduces to

II. THEORY The general theory of VSFG spectroscopy and the scheme for determining orientational TCFs were described in detail in paper 1.23 Our expressions were based on describing the xxz, xzx, and zzz tensor elements of the second-order response function for VSFG (R(2)(τ)) in terms of the derivative of the dipole moment (μ′) and the derivative of the polarizability tensor (α′) with respect to the vibrational coordinate of interest. In paper 1 we treated two special cases. Case 1 applies to the symmetric stretches of symmetric functional groups, such as freely rotating methyl groups, CN stretches, and acetylenic CH stretches. Case 2 applies to the asymmetric stretch of freely rotating methyl groups. Here we extend this treatment to the orientational TCFs for two additional special cases that correspond to the symmetric stretches (case 3) and asymmetric stretches (case 4) of hindered methyl or methylene groups. Case 3. In this special case, the α′ tensor is diagonal and μ′ lies along the Z axis of the functional group (capital Cartesian axes here refer to the molecular frame of reference), but all φ are not equally likely. This situation is relevant to symmetric vibrations of rotationally hindered methyl groups or trihalomethyl groups, as well as to symmetric methylene stretches. Under these circumstances, eq 5 of paper 1 for the xxz tensor element of the second-order response reduces to (2) R xxz (τ ) ∝

(2) R zzz (τ ) ∝

This expression can be rewritten as (2) R zzz (τ ) ∝

1 μ′Z (α′XX + α′YY )⟨cos θ − cos θ cos 2Θ⟩ 4 1 + μ′Z (α′XX − α′YY )⟨cos θ cos 2Φ − cos θ cos 2Θ cos 2Φ⟩ 4 1 + μ′Z α′ZZ ⟨cos θ + cos θ cos 2Θ⟩ (5) 2

As in the case of the xxz term, the zzz term contains timeindependent orientational averages as well as orientational TCFs that depend on the angle θ at both time zero and time τ. The PPP signal includes a weighted difference between the xxz tensor element and the zzz tensor element (see eq 3 of paper 1). On the basis of eqs 2 and 5, in this special case the PPP signal deemphasizes the terms that depend on ⟨cos θ⟩ and emphasizes the terms that depend on ⟨cos θ cos 2Θ⟩ and the terms that depend on α′XX − α′YY. Case 4. In this special case, the infrared transition moment is along the X and/or Y axes of the functional group and the α′ tensor has off-diagonal elements in the XZ and/or YZ plane, but all φ are not equally likely. This situation applies, for instance, to the asymmetric stretch of a methylene group or the asymmetric stretch of a methyl group or trihalomethyl group that is not freely rotating. In this special case, eq 5 of paper 1 for the xxz tensor element of the second-order response reduces to

1 μ′Z α′XX ⟨cos θ cos2 Θ cos2 Φ + cos θ sin 2 Φ⟩ 2 1 + μ′Z α′YY ⟨cos θ cos2 Θ sin 2 Φ + cos θ cos2 Φ⟩ 2 1 + μ′Z α′ZZ ⟨cos θ − cos θ cos2 Θ⟩ (1) 2

1 (2) R xxz (τ ) ∝ − μ′X α′XZ ⟨sin θ sin 2Θ cos φ cos Φ⟩ 2 1 − μ′Y α′YZ ⟨sin θ sin 2Θ sin φ sin Φ⟩ 2

Here the angular brackets denote an orientational average. In the shorthand notation used here, lowercase Greek letters denote the angle at time 0 and uppercase Greek letters denote the angle at time τ. This relation can be rewritten as (2) R xxz (τ )

1 μ′Z α′XX ⟨cos θ cos2 Φ − cos θ cos 2Θ cos2 Φ⟩ 2 1 + μ′Z α′YY ⟨cos θ sin 2 Φ − cos θ cos 2Θ sin 2 Φ⟩ 2 1 + μ′Z α′ZZ ⟨cos θ + cos θ cos 2Θ⟩ (4) 2

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For modes of this symmetry, the SPS signal is influenced by an orientational TCF that depends on both θ and φ. Furthermore, reorientation in φ can cause the signal to decay completely. We next consider the xzx tensor element of the second-order response. In the special case considered here eq 6 of paper 1 becomes

1 ∝ μ′Z (α′XX + α′YY )⟨3 cos θ + cos θ cos 2Θ⟩ 8 1 − μ′Z (α′XX − α′YY )⟨cos θ cos 2Φ − cos θ cos 2Θ cos 2Φ⟩ 8 1 + μ′Z α′ZZ ⟨cos θ − cos θ cos 2Θ⟩ (2) 4

(2) R xzx (τ ) ∝ μ′X α′XZ ⟨cos Θ sin φ sin Φ sin χ sin X

Equation 2 contains orientational averages that are independent of time as well as orientational TCFs that depend on the angle θ at both time zero and time τ. We next turn to the xzx tensor element. In the special case considered here eq 6 of paper 1 reduces to

+ cos θ cos 2Θ cos φ cos Φ cos χ cos X⟩ + μ′Y α′YZ ⟨cos Θ cos φ cos Φ sin χ sin X + cos θ cos 2Θ sin φ sin Φ cos χ cos X⟩ 8407

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The SPS signal is affected by reorientation in all three angles, with reorientation in φ and χ being able to cause the signal to decay completely. Finally, we consider the zzz tensor element of the secondorder response. In case 4, eq 7 of paper 1 reduces to (2) R zzz (τ ) ∝ μ′X α′XZ ⟨sin θ sin 2Θ cos φ cos Φ⟩

+ μ′Y α′YZ ⟨sin θ sin 2Θ sin φ sin Φ⟩

(8)

For this tensor element, the second-order response is sensitive to reorientation in θ and φ.

Figure 1. Molecular coordinate system used to calculate orientational TCFs for the methyl group (black axes), methylene group (red axes), and cyanide group (blue axes) of propionitrile.

III. SIMULATION DETAILS The MD simulations used the same system as in our previous work.31 The silica surface was placed at z = 0 and was based on Lee and Rossky’s model of an idealized ß-cristobalite crystal.32 The simulation box had dimensions of Lx = 45.605 Å, Ly = 43.883 Å, and Lz = 200.000 Å and contained 960 propionitrile molecules. The propionitrile potential parameters were based on the OPLS force field developed by Jorgensen and coworkers.33 DL_POLY 2.1834 was used to perform simulations in the NVT ensemble with a time step of 1 fs. Electrostatic interactions were treated with slab-corrected Ewald 3D sums.35 The system was first equilibrated at 298 K for 500 ps. Configurations were then saved every 15 ps over a total trajectory of 450 ps. Orientational TCFs were computed as previously,23 to a time of 12 ps, which is significantly longer than typical time scales for vibrational relaxation and pure dephasing. Our previous work has shown that, as is the case with acetonitrile,25−29 liquid propionitrile forms a lipid-bilayer-like structure at the silica interface.31 We have therefore computed orientational TCFs for the regions of the simulation including the bulk, the LV and LS interfaces, and the sublayers of the LS interface. The locations of these regions are given in Table 1.

methyl stretch of CH3CD2CN, we find that the value of the relevant root of eq 26 in paper 1 (R+) is 1.34, which implies that α′I/α′ZZ is 1.2 and α′A/α′ZZ is 0.34 for this mode. Here α′I is the isotropic portion of the Raman polarizability tensor and α′A is the anisotropic portion. These results are summarized in Table 2. Table 2. Approximate Values of α′I/α′ZZ and α′A/α′ZZ for the Symmetric Stretching Modes of Propionitrile As Determined from Raman Depolarization Ratios

region

ρ

α′I/α′ZZ

α′A/α′ZZ

0.005 0.05 0.06

1.2 0.86 0.60

0.34 0.77 −0.60

The use of α′I/α′ZZ and α′A/α′ZZ to determine the orientational TCFs for the symmetric methyl stretch of propionitrile must be approached with some care. This mode belongs to case 3, for which α′XX and α′YY can differ from one another. Thus, unlike the case for a freely rotating methyl group, α′A is not given by α′XX − α′ZZ. Instead, we have37

Table 1. Boundary Locations along z in the Propionitrile/ Silica Simulation LS interface first sublayer second sublayer bulk LV interface

mode symmetric methyl stretch symmetric methylene stretch CN stretch

α′ A =

location

1 1 1 (α′XX − α′YY )2 + (α′XX − α′ZZ )2 + (α′YY − α′ZZ )2 2 2 2

(9)

0−4.9 Å 0−2.5 Å 2.5−4.9 Å 30−50 Å 56−63 Å

Although the values determined for α′I/α′ZZ and α′A/α′ZZ from experimental Raman data are correct for this mode, eqs 2, 3, and 5 require knowledge of α′XX, α′YY, and α′ZZ. For the xxz tensor element, eq 2 includes a term that is proportional to α′XX − α′YY. However, our simulations show that the orientational TCF in this equation that is multiplied by α′XX − α′YY is vanishingly small for the symmetric methyl stretch because ⟨cos 2Φ⟩ is near zero (Supporting Information). We can therefore rewrite eq 2 as 1 (2) R xxz (τ ) ∝ μ′Z (α′XX + α′YY )⟨3 cos θ + cos θ cos 2Θ⟩ 8 1 + μ′Z α′ZZ ⟨cos θ − cos θ cos 2Θ⟩ (10) 4

IV. RESULTS The coordinates used to describe the vibrations of each of the functional groups in propionitrile are shown in Figure 1. The coordinate Z in each case is chosen to be the axis of maximum symmetry of the functional group. We will first consider the symmetric methyl stretch. In liquid propionitrile, the Raman depolarization ratio (ρ) is 0.07 for this mode.36 However, for CH3CD2CN the Raman depolarization ratio is 0.005 for the symmetric methyl stretch,36 which raises the possibility that the unusually large value in CH3CH2CN may be due largely to a coincidental overlap of the symmetric stretch with a weakly Raman-active mode that is slightly polarized or completely depolarized. Each value of ρ can correspond to two possible values of α′XX/α′ZZ. For a symmetric methyl stretch, we expect α′XX to be greater than α′ZZ. Thus, using the value of ρ for the symmetric

In analogy with eq 11 of paper 1, we can recast this equation in the form (2) R xxz (τ ) ∝ μ′Z α′I ⟨cos θ ⟩ ⎛1 ⎞ 1 + μ′Z α′a ⎜ ⟨cos θ ⟩ + ⟨cos θ cos 2Θ⟩⎟ ⎝ 12 ⎠ 4

(11)

Here the quantity α′a is given by 8408



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Table 3. Shorthand Notation Used for the Orientational TCFs Calculated

α′a =

notation

time correlation function

C1(τ) C2(τ) C3(τ) C4(τ) C5(τ) C6(τ) C7(τ) C8(τ) C9(τ) C10(τ)

⟨cos θ cos 2Θ⟩ ⟨sin θ sin 2Θ cos χ cos X⟩ ⟨sin θ sin 2Θ cos φ cos Φ⟩ ⟨(cos Θ + cos θ cos 2Θ) cos φ cos Φ cos χ cos X⟩ ⟨cos θ cos 2Φ − cos θ cos 2Θ cos 2Φ⟩ ⟨sin θ sin 2Θ cos 2Φ cos χ cos X⟩ ⟨sin θ sin 2Θ sin 2Φ cos χ cos X⟩ ⟨sin θ sin 2Θ sin φ sin Φ⟩ ⟨cos Θ cos φ cos Φ sin χ sin X + cos θ cos 2Θ sin φ sin Φ cos χ cos X⟩ ⟨cos Θ sin φ sin Φ sin χ sin X + cos θ cos 2Θ cos φ cos Φ cos χ cos X⟩

α′XX + α′YY − 2α′ZZ 2

expect any variation of α′XX/α′YY from unity to be small, so α′a should be quite close to α′A. The orientational TCF C1(τ) is plotted in Figure 2a for the symmetric methyl stretch of propionitrile in the bulk and at the

(12)

We will refer to this quantity henceforth as the modified polarizability anisotropy. The orientational TCF ⟨cos θ cos 2Θ⟩ was referred to by the shorthand notation C1(τ) in paper 1,23 and we retain this notation here. A complete list of the shorthand notations for the orientational TCFs used in paper 1 and here is given in Table 3. In the case of the xzx tensor element (eq 3), α′XX and α′YY are multiplied respectively by the orientational TCFs ⟨sin θ sin 2Θ cos 2Φ cos χ cos X⟩ (which we denote as C6(τ)) and ⟨sin θ sin 2Θ sin 2Φ cos χ cos X⟩ (which we denote as C7(τ)). Our simulations show that these orientational TCFs are indistinguishable from one another for the symmetric methyl stretch of propionitrile (Supporting Information). These TCFs must therefore also each be half as large as ⟨sin θ sin 2Θ cos χ cos X⟩. We can thus rewrite eq 3 as 1 (2) R xzx (τ ) ∝ − μ′Z α′a ⟨sin θ sin 2Θ cos χ cos X⟩ 2

(13)

in analogy with eq 12 in paper 1. The orientational TCF ⟨sin θ sin 2Θ cos χ cos X⟩ was denoted as C2(τ) in paper 1. In the case of the zzz tensor element, the orientational TCF that is multiplied by α′XX − α′YY is identical to that in the xxz tensor element, so this TCF can also be neglected. We can thus recast eq 5 as (2) R zzz (τ ) ∝ μ′Z α′I ⟨cos θ ⟩ ⎛1 ⎞ 1 − μ′Z α′a ⎜ ⟨cos θ ⟩ + ⟨cos θ cos 2Θ⟩⎟ ⎝6 ⎠ 2

Figure 2. Orientational TCFs (a) C1(τ) and (b) C2(τ) for the symmetric methyl stretch of propionitrile in the bulk liquid and at the LV and LS interfaces.

(14)

in analogy with eq 15 of paper 1. Thus, for the symmetric methyl stretch of propionitrile the results for case 3 are identical to those for case 123 except that the polarizability anisotropy is replaced by the modified polarizability anisotropy. Unfortunately, the modified polarizability anisotropy cannot be determined uniquely from the Raman depolarization ratio. However, so long as the correct value of R can be determined, for a given ρ there is a unique value of α′YY/α′ZZ for each value of α′XX/α′ZZ. We can therefore determine the ratio of the modified polarizability anisotropy to the polarizability anisotropy as a function of α′ XX/α′YY (Supporting Information). The modified polarizability anisotropy is less than or equal to the actual polarizability anisotropy. As α′XX/α′YY is varied by ±20% from unity, α′a/α′A is never less than about 0.7. We can therefore use the experimentally measured polarizability anisotropy as an upper bound for the influence of reorientation on the VSFG signal. Furthermore, we

LV and LS interfaces (errors bars for all TCFs calculated here are given in the Supporting Information). This TCF is zero in the bulk liquid, as expected, but for both of the interfaces it is positive at short times before tending toward its equilibrium value, which is negative. As expected, the time scale of this relaxation is faster at the LV interface than at the LS interface. For the LS interface, the methyl groups in both sublayers of the surface bilayer have a tendency to point away from the silica surface.31 Thus, in contrast to acetonitrile, C1(τ) for both sublayers of propionitrile exhibits similar behaviors, although the orientational TCF second sublayer actually exhibits less time dependence than the orientational TCF for the first sublayer (Supporting Information). C2(τ) for the symmetric methyl stretch of propionitrile is shown in Figure 2b. This TCF is zero in the bulk and takes on 8409



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0.36 and 1.21, respectively. The corresponding values of α′I/α′ZZ and α′A/α′ZZ for this mode are given in Table 2. We plot C1(τ) for the symmetric methylene stretch in Figure 4a. This TCF is zero in the bulk liquid. At the LV interface,

positive values at the LS and LV interfaces. At the LV interface this TCF decays to zero over a time scale of roughly 2 ps. After some rapid initial dynamics, the decay is considerably slower for the LS interface than for the LV interface. Both sublayers at the LS interface exhibit similar behaviors for this orientational TCF (Supporting Information). On the basis of these orientational TCFs, we can estimate the time dependence of the different tensor elements of the second-order response for the symmetric methyl stretch of propionitrile. The results for the LV interface are shown in Figure 3a, and the results for the LS interface are shown in

Figure 4. Orientational TCFs (a) C1(τ) and (b) C5(τ) for the symmetric methylene stretch of propionitrile in the bulk liquid and at the LV and LS interfaces.

C1(τ) approaches its equilibrium value of approximately 0.025 over the course of a few picoseconds, after an initial rapid decay in which it changes sign. At the LS interface, this TCF is essentially independent of time after some rapid initial dynamics that do not have a large effect on its magnitude. The existence of preferred orientations of the methyl group of propionitrile at the LV and LS interfaces (i.e., the fact that ⟨cos θ⟩ is not zero for the methyl group at these interfaces) leads to ⟨cos 2Φ⟩ not being zero for the methylene group. Thus, unlike the case for the methyl group, the orientational TCF ⟨cos θ cos 2Φ − cos θ cos 2Θ cos 2Φ⟩ (which we denote as C5(τ)) is of substantial magnitude at interfaces for the symmetric stretch of the methylene group. We plot this orientational TCF in Figure 4b. As expected, the TCF is zero in the bulk liquid. At the LV interface C5(τ) rises from near zero initially to a maximum value of approximately 0.033 in about 1 ps, before decaying away over the course of a few picoseconds. The rapid initial dynamics are a feature of both terms in this TCF and are indicative of a strong coupling between θ and φ that causes the value of the latter angle at time τ to depend strongly on the value of the former angle at time 0. At the LS interface this TCF has a small negative value initially but grows in time to a maximum value that approaches 0.05. This growth is the result of a loss of cancellation between the contributions of the two sublayers (Supporting Information). In the case of the methyl symmetric stretch we found above that the TCFs C6(τ) and C7(τ) were equal to one another. The preferred methyl orientation at interfaces causes C6(τ) and C7(τ) to have different magnitudes and different time

Figure 3. Tensor elements of the second-order response function for the symmetric methyl stretch of propionitrile at (a) the LV interface and (b) the LS interface.

Figure 3b. At both interfaces the influence of reorientation is relatively small for the xxz and zzz tensor elements. However, reorientation causes the xzx tensor element to decay to zero over the course of about 2 ps at the LV interface. At the LS interface the effect of reorientation on this tensor element of the response is smaller and occurs on a considerably longer time scale. We next consider the symmetric methylene stretch. The depolarization ratio for this mode is 0.05.36 Based on this value of ρ, |α′I/α′A| is equal to approximately 1.1.23 For this mode, α′XX, α′YY, and α′ZZ should all take on different values, and we cannot extract unique numbers for these elements of the polarizability based on ρ alone. However, as is the case for the symmetric methyl stretch, α′YY should be greater than α′ZZ. We further expect that α′XX should be significantly smaller than α′YY and α′ZZ (see Figure 1 for the coordinate system used). If we further assume that the polarizability of this mode is bond additive, we can use ρ along with the fact that the change in polarizability along each bond should be considerably greater than the change in polarizability perpendicular to each bond to determine approximate values of α′XX/α′ZZ and α′YY/α′ZZ of 8410



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dependences for the symmetric methylene stretch. It is therefore necessary to determine C2(τ), C6(τ), and C7(τ) to calculate the xzx tensor element of the response function for this mode. These TCFs are plotted in Figures 5a, 5b, and 5c,

Figure 6. Tensor elements of the second-order response function for the symmetric methylene stretch of propionitrile at (a) the LV interface and (b) the LS interface.

they do for the symmetric methyl stretch (Figure 3). At both the LV and LS interfaces these two tensor elements change in opposite directions with time. This phenomenon may play a role in the PPP line shape, particularly at the LV interface. The xzx tensor element is quite small at both interfaces, in agreement with experiment,38 but does decay with time as well. We next turn to the CN stretch. For liquid propionitrile, ρ for this mode is 0.06.36 The corresponding R− value is 0.40, implying that α′I/α′ZZ is 0.60 and α′A/α′ZZ is −0.60.23 These results are summarized in Table 2. C1(τ) for the CN stretch is plotted for the bulk liquid, the LV interface, and the LS interface in Figure 7a. As expected, this TCF is zero for the bulk liquid. At the LV interface C1(τ) exhibits rapid dynamics, changing from an initial value of roughly −0.015 to its longtime value of 0.0293 over the course of about 4 ps. However, this TCF exhibits virtually no time dependence at the LS interface (or in its two sublayers; Supporting Information). In Figure 7b we plot C2(τ) for the CN stretch for the bulk liquid, the LV interface and the LS interface. This TCF is also zero in the bulk liquid. At the LV interface C2(τ) decays to zero over the course of about 10 ps. At the LS interface, on the other hand, the magnitude of C2(τ) grows over the full 12 ps period plotted. This phenomenon is the result of the faster decay of the TCF for the second sublayer of propionitrile than for the first sublayer (Supporting Information). In Figure 8a we use these results to determine the time dependence of the tensor elements of the response for the CN stretch at the LV interface. At this interface all of the tensor elements have a significant time dependence. The magnitude of the xxz tensor element increases to some extent with time, whereas the zzz tensor element decays partially and the xzx tensor element decays completely. We therefore expect reorientation to play a measurable role in the VSFG spectrum of the propionitrile CN stretch at the LV interface. As in the

Figure 5. Orientational TCFs (a) C2(τ), (b) C6(τ), and (c) C7(τ) for the symmetric methylene stretch of propionitrile in the bulk liquid and at the LV and LS interfaces.

respectively. All of these TCFs are zero in the bulk liquid. At the LV interface they are all negative and decay to zero over the course of a few picoseconds. At the LS interface all of the TCFs are negative as well, but after an initial decay over a few picoseconds they decay much more slowly and do not reach asymptotic values over the 12 ps period plotted. For each of these TCFs the contributions of the two sublayers exhibit similar behaviors, although they are of different magnitudes from one another (Supporting Information). Based on the values of α′XX/α′ZZ and α′YY/α′ZZ derived above for the symmetric methylene stretch and on the TCFs in Figures 4 and 5, we can evaluate the tensor elements of the response function of this mode directly. The tensor elements for the LV interface are shown in Figure 6a, and those for the LS interface are shown in Figure 6b. As can be seen in these plots, the xxz and zzz tensor elements have a more significant time dependence for the symmetric methylene stretch than 8411



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interface. The xxz and zzz tensor elements are nearly independent of time at this interface. The xzx tensor element grows slightly with time due to the decay of cancellation of the contributions of the two sublayers (Supporting Information), and so in this case reorientation serves to enhance the SPS signal rather than to make it decay. We next turn to the asymmetric methyl stretch of propionitrile. This mode belongs to Case 4. However, as shown in the Supporting Information, our simulations demonstrate that ⟨sin θ sin 2Θ cos φ cos Φ⟩ (which we denote as C3(τ)) and ⟨sin θ sin 2Θ sin φ sin Φ⟩ (which we denote as C8(τ)) are identical for this mode in the bulk and at the LV and LS interfaces, as is also the case for the orientational TCFs ⟨cos Θ cos φ cos Φ sin χ sin X + cos θ cos 2Θ sin φ sin Φ cos χ cos X⟩ (which we denote as C9(τ)) and ⟨cos Θ sin φ sin Φ sin χ sin X + cos θ cos 2Θ cos φ cos Φ cos χ cos X⟩ (which we denote as C10(τ)). This mode can therefore be treated as if it belongs to case 2, which means that the xxz and zzz tensor elements of the response function are proportional to C3(τ) (but with opposite sign) and the xzx tensor element is proportional to ⟨(cos Θ + cos θ cos 2Θ) cos φ cos Φ cos χ cos X⟩ (which we denote as C4(τ)).23 In Figure 9a we plot C3(τ) for the asymmetric methyl stretch of propionitrile in the bulk and at the LV and LS interfaces. Figure 7. Orientational TCFs (a) C1(τ) and (b) C2(τ) for the CN stretch of propionitrile in the bulk liquid and at the LV and LS interfaces.

Figure 9. Orientational TCFs (a) C3(τ) and (b) C4(τ) for the asymmetric methyl stretch of propionitrile in the bulk liquid and at the LV and LS interfaces. Figure 8. Tensor elements of the second-order response function for the CN stretch of propionitrile at (a) the LV interface and (b) the LS interface.

This TCF is zero in the bulk liquid, decays to zero within about 5 ps at the LV interface, and decays relatively slowly at the LS interface (reaching half of its initial value in about 4 ps). At the latter interface, these TCFs are similar for both sublayers (Supporting Information). We plot C4(τ) for the asymmetric methyl stretch of propionitrile in Figure 9b for the same three cases. The behavior of this TCF is similar to that of C3(τ); it is zero in the bulk, decays rapidly at the LV interface (where most of the

case of acetonitrile, we might expect to see some difference in the line width of the CN stretch between the SSP and PPP spectra. In Figure 8b we plot the time dependence of the tensor elements of the response for the CN stretch at the LS 8412



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contributions of the two sublayers are similar in shape, although they are opposite in sign (Supporting Information). On the other hand, C9(τ) (which determines the time dependence of the xzx tensor element) decays quite slowly after the initial rapid decay. This leveling off is a consequence of the contributions of the two sublayers having opposite behaviors (Supporting Information). On the basis of these TCFs, we expect reorientation to have a substantial effect on the contribution of the methylene asymmetric stretch to the VSFG spectrum of propionitrile. The rapid orientational decay and low magnitudes of the response functions for the different tensor elements are likely to be responsible for the fact that there is not a clear contribution from the asymmetric methyl stretch to the experimental VSFG spectrum at either interface.38 Many of the orientational TCFs that we have calculated depend on multiple degrees of freedom. As we have shown previously for acetonitrile,23 comparison of full TCFs to products of TCFs for the individual coordinates can yield important clues regarding the coupling among the dynamics of the different degrees of freedom. There are a number of reasons why the factorizability of TCFs for propionitrile might be different than was found for acetonitrile. First, in acetonitrile the axes of the functional groups are the same as the axes of the diffusion tensor, whereas this is not the case in propionitrile. Second, there is a barrier to internal rotation of the methyl group of propionitrile, which is not the case for acetonitrile. Third, the bilayer that propionitrile forms at the LS interface is entangled,31 as opposed to the interdigitated bilayer formed by acetonitrile.25−29 We begin by considering the symmetric methyl stretch. The only TCF for this mode for which factorization could be applicable is C2(τ). The unfactorized and factorized versions of this TCF (⟨sin θ sin 2Θ cos χ cos X⟩ and ⟨sin θ sin 2Θ⟩⟨cos χ cos X⟩, respectively) are plotted for the LV interface in Figure 11a and for the LS interface in Figure 11b. In both cases the unfactorized and factorized TCFs have similar behaviors (in terms of magnitude and decay time), but they are clearly not identical. In the case of the LS interface, the TCFs for the two surface sublayers decay in the same direction (Supporting Information), so there is not strong cancellation between them. However, the unfactorized and factorized versions of C2(τ) match better for the full surface layer than they do within each individual sublayer (Supporting Information). These results suggest that there is coupling between the dynamics in θ and χ for the methyl group at both interfaces. The vast majority of the relaxation in the factorized TCF arises from dynamics in χ at both interfaces (Supporting Information). The coupling between dynamics in θ and χ further indicates that the rate of azimuthal relaxation is dependent on the value of θ, as would be expected. Next we consider the symmetric methylene stretch. We begin by examining the factorization of C5(τ). The unfactorized and factorized versions of this TCF are shown in Figure 12a for the LV interface and Figure 12b for the LS interface. For both interfaces the unfactorized and factorized TCFs differ remarkably. This behavior is another manifestation of the strong coupling between the dynamics in θ and φ for this group that was discussed above. We examine the factorization for C2(τ), C6(τ), and C7(τ) for the symmetric methylene stretch at the LV interface in Figure 13a and at the LS interface in Figure 13b. At both interfaces, the unfactorized and factorized TCFs exhibit the same

relaxation is complete within 2 ps), and decays more slowly at the LS interface. As shown in the Supporting Information, this TCF has distinctly different contributions from the two sublayers at the LS interface. On the basis of these TCFs, it is apparent that at the LV interface the orientational decay of all of the tensor elements of the response for the asymmetric methyl stretch is nearly complete within 4 ps (Supporting Information). Reorientation therefore plays a substantial role in the appearance of the asymmetric methyl stretch in the VSFG spectrum of propionitrile at this interface. For the LS interface, Figure 9 shows that all of the tensor elements of the response undergo an initial decay on a subpicosecond time scale, followed by a much slower decay (Supporting Information). Reorientation is therefore expected to play a measurable role in the VSFG spectrum of the asymmetric methyl stretch of propionitrile at this interface under all polarization conditions. The final vibrational mode we consider is the asymmetric methylene stretch of propionitrile. Although this mode belongs to case 4, the asymmetric stretch takes place entirely in what we have defined as the YZ plane of the methylene group. We therefore need only consider the second terms of eqs 6−8, which contain the orientational TCFs C8(τ) and C9(τ). These TCFs are plotted in Figure 10a,b, respectively. On the basis of

Figure 10. Orientational TCFs (a) C8(τ) and (b) C9(τ) for the asymmetric methylene stretch of propionitrile in the bulk liquid and at the LV and LS interfaces.

these TCFs, it is apparent that, at the LV interface, all of the tensor elements of the response undergo substantial orientational relaxation on a subpicosecond time scale and decay nearly completely within 4 ps (Supporting Information). In the case of the LS interface, there is a substantial subpicosecond orientational decay for all tensor elements (Supporting Information). After this initial decay, C8(τ) (which determines the behavior of the xxz and zzz tensor elements) continues to decay, eventually changing sign after 3 ps. For this TCF, the 8413



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Figure 11. Unfactorized (black) and factorized (red) versions of C2(τ) for the symmetric methyl stretch of propionitrile at (a) the LV interface and (b) the LS interface.

Figure 13. Unfactorized (black) and factorized (red) versions of C2(τ), C6(τ), and C7(τ) for the symmetric methylene stretch of propionitrile at (a) the LV interface and (b) the LS interface. Note that the factorized versions of C6(τ) and C7(τ) are nearly identical at the LV interface.

differently than it is in the case of C5(τ) for this group. The fact that relaxation of each of these factorized TCFs is dominated by dynamics in χ at both interfaces (Supporting Information) is another manifestation of the behavior seen for the methyl symmetric stretch dynamics. This conclusion is supported by the fact that the two sublayers at the LS interface exhibit similar behaviors for all three of the factorized TCFs (Supporting Information). We next consider factorization of C2(τ) for the cyanide group. The unfactorized and factorized versions of this TCF are plotted for the LV interface in Figure 14a and for the LS interface in Figure 14b. These two versions of the TCF are in good agreement at the LV interface. In this case factorization holds because there is virtually no relaxation in θ, which effectively decouples the degrees of freedom (Supporting Information). On the other hand, the unfactorized and factorized TCFs differ considerably at the LS interface. This behavior arises from significant differences in the two versions of the TCF for both sublayers (Supporting Information). The dynamics of these sublayers are presumably influenced by a combination of the hydrogen bonding in the first sublayer and the entanglement of the sublayers with one another. At both interfaces, essentially all of the relaxation of the factorized version of this TCF arises from the dynamics in χ, which is consistent with the results presented above for other vibrational modes. Next we consider the asymmetric methyl stretch. In this case we examine factorization of C3(τ) and C4(τ). In the former case we examine whether ⟨sin θ sin 2Θ cos φ cos Φ⟩ can be factorized into ⟨sin θ sin 2Θ⟩⟨cos φ cos Φ⟩ and in the latter we

Figure 12. Unfactorized (black) and factorized (red) versions of C5(τ) for the symmetric methylene stretch of propionitrile at (a) the LV interface and (b) the LS interface.

qualitative behavior, but the factorized TCFs nevertheless differ substantially from the unfactorized ones. This behavior is again a consequence of the coupling among the degrees of freedom for the methylene group. However, because all of these TCFs depend on sin θ rather than cos θ, this coupling is manifested 8414



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Figure 14. Unfactorized (black) and factorized (red) versions of C2(τ) for the CN stretch of propionitrile at (a) the LV interface and (b) the LS interface.

Figure 15. Unfactorized (black) and factorized (red) versions of C3(τ) (solid lines) and C4(τ) (dashed lines) for the asymmetric methyl stretch of propionitrile at (a) the LV interface and (b) the LS interface.

examine whether ⟨(cos Θ + cos θ cos 2Θ) cos φ cos Φ cos χ cos X⟩ can be factorized into ⟨cos Θ + cos θ cos 2Θ⟩⟨cos φ cos Φ⟩⟨cos χ cos X⟩. The unfactorized and factorized TCFs are plotted for the LV interface in Figure 15a and for the LS interface in Figure 15b. We find that factorization holds well for C3(τ) at both interfaces, which indicates that the dynamics of θ and φ are not coupled strongly for the methyl group. This lack of coupling arises in part because ⟨sin θ sin 2Θ⟩ is itself only weakly time dependent (Supporting Information). The dynamics of the factorized version of this TCF are again dominated by relaxation in χ at both interfaces (Supporting Information). At the LS interface, the dynamics of this factorized TCF for each of the sublayers are virtually identical (Supporting Information), despite the differences in the average orientation of the methyl groups in these two sublayers. For C4(τ), factorization holds reasonably well for about 500 fs at each interface and then breaks down, which indicates that there is dynamical coupling among the orientational degrees of freedom. The relaxation of this TCF at both interfaces is dominated by dynamics in φ and χ (Supporting Information), although dynamics in θ do make some contribution to the decay at early times. At the LS interface, the behaviors of the factorized TCF are similar for both sublayers (Supporting Information). Finally, we consider the asymmetric methylene stretch. The unfactorized and factorized TCFs C8(τ) and C9(τ) for this mode are shown for the LV interface in Figure 16a and for the LS interface in Figure 16b. In the case of C8(τ) we compare ⟨sin θ sin 2Θ sin φ sin Φ⟩ with the product ⟨sin θ sin 2Θ⟩⟨sin φ sin Φ⟩, and in the case of C9(τ) we compare ⟨cos Θ cos φ cos Φ sin χ sin X + cos θ cos 2Θ sin φ sin Φ cos χ cos X⟩ with the sum of products ⟨cos Θ⟩⟨cos φ cos Φ⟩⟨sin χ sin X⟩ + ⟨cos θ cos 2Θ⟩⟨sin φ sin Φ⟩⟨cos χ

Figure 16. Unfactorized (black) and factorized (red) versions of C8(τ) (solid lines) and C9(τ) (dashed lines) for the asymmetric methylene stretch of propionitrile at (a) the LV interface and (b) the LS interface.

cos X⟩. At both interfaces the unfactorized and factorized versions of both TCFs are qualitatively similar but still differ significantly. This behavior is again an indication that there is substantial dynamical coupling among the different degrees of orientational freedom for the methylene group. The relaxation 8415



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of the factorized version of C8(τ) is dominated by dynamics in φ at both interfaces, whereas dynamics in both φ and χ are important at both interfaces for C9(τ) (Supporting Information). For both factorized TCFs at the LS interface, the contributions of the sublayers are similar in shape (although they are of opposite sign for C9(τ), because this TCF depends on cosines of θ rather than sines of θ).

the increase in magnitude with time reflects the fast dynamics of the second sublayer at this interface.25 In the case of the asymmetric methyl stretch, the time dependences of C3(τ) and C4(τ) differ substantially for acetonitrile and propionitrile. In acetonitrile these TCFs decay completely within 2 ps or less at both the LV and LS interfaces, due to free rotation of the methyl group.23 The hindered rotation of the methyl group in propionitrile causes these TCFs to decay considerably more slowly. The significant difference in the time scale of the dynamics for C3(τ) and C4(τ) for the asymmetric methyl stretch in going from the LV interface to the LS interface (Figure 9) indicates that reorientation of the entire molecule is involved in the decay of these TCFs. For reorientation to have a significant impact on the VSFG spectrum for a given vibrational mode, it must cause the response function to decay substantially on a time scale that is comparable to or faster than the time scale of other decay mechanisms, such as vibrational relaxation and pure dephasing. For symmetric vibrational modes, our results indicate that the effects of reorientation are most likely to be seen in SPS spectra, particularly at the LV interface. The xxz and zzz response functions for these modes tend to have only a small portion that is time dependent, and this portion typically decays significantly more slowly at the LS interface than at the LV interface. Reorientation can cause the VSFG spectrum of an asymmetric vibrational mode to decay completely under any set of polarization conditions. As shown in Figures 9a and 10a, reorientation at the LV interface causes the TCFs for asymmetric vibrations to decay on a time scale that is likely to be competitive with other dephasing mechanisms under all polarization conditions. Reorientation causes each response function to decay to nearly zero on a time scale of approximately 4 ps. At the LS interface, reorientation again makes a notable contribution to the decay of each TCF for asymmetric vibrations. However, as shown in Figures 9b and 10b, the time scales of each these decays is considerably longer and the fractional decay in 4 ps is considerably smaller than is observed in the corresponding TCFs at the LV interface. On the basis of these data, we can expect reorientation to play a measurable role in the VSFG spectra of the asymmetric stretches of the methyl and methylene groups under all polarization conditions. The predicted magnitudes and orientationally induced decay times of the tensor elements of the second-order response for the LV and LS interfaces of propionitrile are in good qualitative agreement with experiment.38 As was the case for acetonitrile,23 reorientation plays the most significant role in SPS spectra and for asymmetric vibrational modes. However, reorientation has less of an influence on the VSFG spectra of the asymmetric vibrational modes in propionitrile than in acetonitrile because neither the methyl nor the methylene group in propionitrile can undergo free rotation. In acetonitrile, the two sublayers of molecules at the LS interface point in opposite directions, and so there is typically significant cancellation between their contributions to orientational TCFs.23 The situation is more complicated for propionitrile, as the organization of the two sublayers is considerably more complex. The contributions of the two sublayers for the cyanide stretches to orientational TCFs are of opposite sign, leading to cancellation. The case is not so simple for the methyl and methylene stretches, however. For these

V. DISCUSSION Acetonitrile and propionitrile are both small-molecule liquids that have a relatively low viscosity at room temperature. As a result, in the bulk they exhibit similar orientational dynamics, with propionitrile reorienting somewhat more slowly due to its larger hydrodynamic volume. For instance, optical Kerr effect39−41 measurements at 295 K reveal collective orientational correlation times for tumbling that differ by less than a factor of 2 (1.60 ps for acetonitrile42 and 2.98 ps for propionitrile38,43). The orientational TCFs for VSFG emphasize different aspects of orientation and dynamics than do the orientational TCFs for techniques used in bulk liquids. Thus, our results show that in many cases orientational dynamics influence the VSFG response function for propionitrile on a time scale that is comparable to, or even faster than, what we found for acetonitrile. For instance, C1(τ) and C2(τ) for the symmetric methyl stretch have larger magnitudes and decay more rapidly for propionitrile than for acetonitrile at both the LV and LS interfaces. The difference in magnitudes between the two liquids is indicative of their different interfacial ordering. The fact that the interfacial orientational correlation times are more similar for the two liquids than they are in the bulk likely has its origins in both the organization and the dynamics of the liquids. For instance, we have found that acetonitrile has a stronger tendency to dipole pair at the LV interface than in the bulk,25 which may serve to makes its orientational correlation time larger than might be expected otherwise. Propionitrile does not exhibit the same increase in dipole pairing at the LV interface,31 and so it can reorient on a time scale that is similar to that of acetonitrile. At the LS interface, OKE experiments reveal that the collective orientational correlation time for propionitrile38 (13.3 ps) is considerably smaller than that for acetonitrile44 (25 ps). The inhibition of the dynamics of acetonitrile as compared to propionitrile at this interface presumably arises from the substantial differences in the interfacial organization of the two liquids. The same effects are seen in C1(τ) and C2(τ) for the symmetric methyl stretches of the two liquids at the LS interface. We can also compare the dynamics of the opposite end of the molecules by considering C1(τ) and C2(τ) for CN stretch. At the LV interface the time scales of the decays of C1(τ) and C2(τ) for the CN stretch are similar for acetonitrile23 and propionitrile, although the initial magnitude of both TCFs is larger for the latter liquid. The difference between the dynamics for the CN stretch and the symmetric methyl stretch for propionitrile at the LV interface probably arise from the fact that the cyanide group tends to be buried in the liquid.31 At the LS interface, neither liquid shows a significant time dependence for C1(τ) for the CN stretch. C2(τ) does exhibit a time dependence for this mode, increasing in magnitude with time for both liquids. In this case the TCF for acetonitrile exhibits faster dynamics, presumably because 8416



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antisymmetric stretches of rotationally hindered methyl groups or methylene groups. Because the Raman polarizability tensors of these groups typically have three independent diagonal elements, the Raman depolarization ratio of the symmetric vibrational modes does not define the ratios of the tensor elements uniquely (in contrast to the case of the symmetric and asymmetric stretches of freely rotating methyl groups23). However, in the case of the symmetric methyl stretch we were able to use ρ to determine an upper bound for the influence of reorientation on the different tensor elements of the second-order response for these modes. In the case of the symmetric methylene stretch we were able to combine ρ, knowledge of the relative magnitudes of the tensor elements of the polarizability, and a bond-additive polarizability model to estimate these tensor elements directly. As we found previously for acetonitrile, reorientation has little effect on the VSFG spectra of symmetric vibrational modes in propionitrile, except for the xzx tensor element. For asymmetric stretches, all of the tensor elements of the secondorder response are influenced by reorientation, although this effect is less significant than for molecules that have freely rotating methyl groups. Nevertheless, it is clear from our results that for molecules in the size range of propionitrile, the effects of reorientation should be considered for VSFG spectra under SPS polarization conditions and for asymmetric vibrations in general. Our results also indicate that there is typically enough coupling among the orientational degrees of freedom of propionitrile that factorization does not work well for most of its orientational TCFs.

modes the contributions of the sublayers can have the same sign or opposite signs depending on the orientational TCF in question (Supporting Information). Thus, interfacial structure can play a crucial role in determining the overall influence of reorientation on the VSFG signal. Vinaykin and Benderskii have recently analyzed the influence of reorientation on the VSFG line shape using a small-step orientational diffusion model.21 They examined two special cases, a weak-confinement model and a wobbling-in-a-cone model. The approach taken here is complementary. We are able to obtain model-independent TCFs that quantify the effects of reorientation on VSFG spectra. Our approach is most useful for exploring general trends in orientational behavior or, when specific systems are considered, for analyzing simulations that supplement experimental data. The approach of Vanaykin and Benderskii can allow for information on orientational dynamics to be determined directly from VSFG spectra even in the absence of simulations. However, their approach does require that specific assumptions be made, and these assumptions are often most easily tested with simulations. We have found that the contribution of the orientational dynamics of propionitrile to VSFG spectra at the LV and LS interfaces is dominated by azimuthal relaxation, and that there is a correlation between the value of θ for a molecule and its rate of azimuthal relaxation. The same picture holds for acetonitrile on the basis of our previous simulations.23 It is clear that reorientation is not isotropic in any of these interfacial systems. We can also assess whether the dynamics observed here are consistent with a wobbling-in-a-cone model, in which the value of θ is constrained to lie within a certain range but reorientation is otherwise unbiased.45,46 The relatively minimal contribution of dynamics in θ to the factorized TCFs studied here (Supporting Information) and previously for acetonitrile23 indicates that, if orientational relaxation is described by a wobbling-in-a-cone model, then the half-angle of the cone must be small. However, simulations indicate that there is a broad distribution of θ at the LV and LS interfaces in both acetonitrile26 and propionitrile,31 which is not consistent with wobbling in a cone. The fact that relaxation in θ is relatively slow despite the broad distribution of this angle probably arises from a combination of hydrodynamic volume effects and the highly anisotropic density distributions at the LV and LS interfaces. It appears from our results for acetonitrile23 and propionitrile that the wobbling-in-a-cone model is not likely to be an accurate description of the orientational dynamics at most LV interfaces, as there is no reason to have a hard limit to the cone angle and the dynamics in χ are likely to depend significantly on θ. In the absence of strong directional interactions, the same probably holds true for most solid/liquid interfaces. That being said, the wobbling-in-a-cone model is attractive for its tractability, and a worthwhile topic for future exploration is whether it can capture the essence of the dynamics at such interfaces even if the parameters it yields should not be taken literally. If not, our results suggest that a new model may be needed to describe the orientational diffusion at most liquid interfaces. Until such a model is available, simulations will continue to be invaluable for assessing the role of interfacial reorientation in VSFG.

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ASSOCIATED CONTENT

S Supporting Information *

Comparison of the polarizability anisotropy and the modified polarizability anisotropy, orientational TCFs and response functions for tensor elements not shown in the main text, contributions of the two sublayers at the liquid/silica interface to orientational TCFs, time dependence of the individual factorized orientational TCFs, the factorization of the orientational TCFs for the sublayers at the LS interface, and error bars for all unfactorized TCFs. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*J. Fourkas: e-mail, [email protected]. Present Address #

Department of Chemistry, University of Chicago, 5735 S. Ellis Ave., Chicago, IL 60637. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Science Foundation Collaborative Research in Chemistry program, grant CHE0628178. We thank John Weeks for helpful discussions.

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REFERENCES

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VI. CONCLUSIONS We have derived orientational TCFs for VSFG spectroscopy in two special cases that are relevant to the symmetric and 8417



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(46) Gengeliczki, Z.; Rosenfeld, D. E.; Fayer, M. D. Theory of Interfacial Orientational Relaxation Spectroscopic Observables. J. Chem. Phys. 2010, 132, 244703.

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