Orientational Time Correlation Functions for Vibrational Sum

Article pubs.acs.org/JPCA

Orientational Time Correlation Functions for Vibrational Sum-Frequency Generation. 1. Acetonitrile Shule Liu† and John T. Fourkas*,†,‡,§,∥ †

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

∥

S Supporting Information *

ABSTRACT: Orientational time correlation functions (TCFs) are derived for vibrational sum-frequency generation (VSFG) spectroscopy of the symmetric and asymmetric stretches of high-symmetry oscillators such as freely rotating methyl groups, acetylenic C−H groups, and cyanide groups. Molecular dynamics simulations are used to calculate these TCFs and the corresponding elements of the second-order response for acetonitrile at the liquid/vapor and liquid/silica interfaces. We find that the influence of reorientation depends significantly on both the functional group in question and the polarization conditions used. Additionally, under some circumstances, reorientation can cause the VSFG response function to grow with time, partially counteracting the effects of other dephasing mechanisms.

I. INTRODUCTION Vibrational sum-frequency generation1−11 (VSFG) is a powerful and broadly applicable method for studying the organization of molecules at interfaces. This technique, which offers both interfacial and molecular selectivity, is commonly used to obtain detailed information regarding orientational distributions at interfaces. VSFG can be described in terms of an infrared vibrational transition followed by a Raman vibrational transition, although the expressions that are typically used to derive orientational information from VSFG spectra assume that these transitions occur simultaneously.11 However, a number of groups have pointed out that the interactions need not occur at the same time, and that interactions that are not time-coincident can complicate the extraction of orientational information from VSFG spectra.11−16 We have previously discussed how the contributions of different types of vibrational modes to VSFG spectra are affected by reorientation.16 We analyzed these contributions in two limits. In the conventional limit, the infrared and Raman transitions that lead to the VSFG signal occur simultaneously. In the “infinite time” limit, orientational diffusion completely randomizes the orientations of the molecules between the infrared and Raman transitions. This latter limit gives a qualitative picture of how reorientation can influence VSFG spectra. To obtain a more quantitative measure of the effects of reorientation, it is necessary to consider the orientational time correlation functions (TCFs) that contribute to the VSFG signal, and how the time scales of these TCFs compare to those for other dephasing mechanisms. In this paper we derive orientational TCFs for functional groups of some of the symmetries most commonly probed in VSFG spectroscopy. These TCFs quantify the sensitivity of the VSFG signal to reorientation about different molecular axes for the different sets of polarization conditions used in this spectroscopy. We use molecular dynamics (MD) simulations of the liquid/vapor (LV) and liquid/silica (LS) interfaces of acetonitrile to evaluate these orientational TCFs. Our simulations allow us to © XXXX American Chemical Society

assess the magnitude and relaxation time scale of the portion of the VSFG signal that is sensitive to reorientation in these representative cases, as well as to explore the degree of coupling among different orientational degrees of freedom at these interfaces. Our results are used to interpret the role of reorientation in experimental VSFG spectra of liquid acetonitrile17 at the LV and LS interfaces.

II. THEORY In VSFG, an infrared field that is resonant with a vibrational transition in a molecule generates a vibrational coherence. The coherence is then interrogated by a probe field that is typically in the visible or near-infrared region of the spectrum, generating a signal at frequency ωsig = ωIR + ωprobe. In the simplest molecular picture, this process requires that the vibrational mode have both infrared and Raman activity. VSFG is a second-order nonlinear optical (NLO) technique. In other words, VSFG depends on the second-order NLO susceptibility χ(2) (or, in a time-domain picture, on the secondorder NLO response function R(2)). χ(2) and R(2) are third-rank tensor quantities that depend on three light polarizations. In the case of VSFG, these polarizations (in the conventional order) correspond to the signal, the probe and the infrared beams. Due to symmetry constraints, within the electric dipole approximation χ(2) and R(2) vanish in an isotropic medium.18 Because symmetry is broken at an interface between two isotropic media, a VSFG signal can be generated in such an environment, even though the interfacial molecules are vastly outnumbered by the molecules in the surrounding media. However, only tensor elements of χ(2) or R(2) in which the direction of the Special Issue: Prof. John C. Wright Festschrift Received: June 26, 2012 Revised: August 31, 2012

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dx.doi.org/10.1021/jp306296s | J. Phys. Chem. A XXXX, XXX, XXX−XXX

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normal to the interface (which is conventionally labeled z) appears an odd number of times can be nonzero. At an azimuthally isotropic interface, this requirement leads to four independent, non(2) (2) (2) (2) (2) (2) zero tensor elements: χ(2) xxz = χyyz, χxzx = χyzy, χzxx = χzyy and χzzz. (2) Furthermore, under typical experimental conditions χ(2) = χ , xzx zxx so that there are only three unique tensor elements. VSFG data are typically collected under SSP, SPS, or PPP polarization conditions, where S polarization is parallel to the interface and P polarization is in the plane of incidence and reflection (i.e., with a projection along the interface normal). The effective second-order response function under these different polarization conditions can be expressed as19

results in the limit of simultaneous infrared and Raman transitions are well-known.8,11 We have previously shown how these averages change for commonly studied vibrational modes in the limit that complete orientational randomization occurs between the infrared and Raman transitions.16 Here we extend this treatment to derive orientational TCFs that span the full time range between these limits. Although this type of TCF can be expressed in a compact manner using spherical harmonics,20,21 here instead we employ Euler angles to gain direct insight into orientational diffusion from the molecular perspective. We denote laboratory-frame Cartesian coordinates with lowercase letters and molecular-frame Cartesian coordinates with capital letters. We choose the Z axis of the molecule to be the symmetry axis of the vibrating group. We begin with the molecular and laboratory reference frames being identical. The Euler angle φ represents rotation about the Z axis. The Euler angle θ represents subsequent rotation about the y axis and the Euler angle χ represents rotation around the z (azimuthal) axis. The form of the orientational average for an arbitrary tensor element of the second-order response at an arbitrary time τ is given by

(2) R eff,SSP (τ ) = Lyy(ωsig ) Lyy(ωprobe) Lzz(ωIR ) sin θIRR (2) yyz(τ )

(1) (2) R eff,SPS (τ )

= Lyy(ωsig ) Lzz(ωprobe) Lyy(ωIR ) sin

θprobeR (2) yzy(τ ) (2)

and (2) R eff,PPP (τ ) (2) (τ ) = Lxx(ωsig ) Lxx(ωprobe) Lzz(ωIR ) cos θsig cos θprobe sin θIR R xxz (2) (τ ) + Lxx(ωsig ) Lzz(ωprobe) Lxx(ωIR ) cos θsig sin θprobe cos θIR R xzx

3 (2) R ijk (τ )

(2) (τ ) + Lzz(ωsig ) Lxx(ωprobe) Lxx(ωIR ) sin θsig cos θprobecos θIR R zxx (2) (τ ) + Lzz(ωsig ) Lzz(ωprobe) Lzz(ωIR ) sin θsig sin θprobe sin θIR R zzz

β=1

(3)

3

γ=1 δ=1

(4)

In these equations, τ is the time between the creation of the coherence with the infrared field and the Raman probing of the coherence, Lii represents the nonlinear Fresnel factor for polarization i, and θj is the angle that beam j makes with the interface normal. Each angle can be positive or negative, depending on the experimental geometry. Regardless of whether the infrared and probe beams approach the sample from the same side of the surface normal (a copropagating geometry) or from opposite sides (2) (a counter-propagating geometry), the R(2) xxz(τ) and Rzzz(τ) terms in R(2) (τ) will be of opposite sign. This factor is important in eff,PPP analyzing the differences between SSP and PPP spectra. The tensor elements of the response function are determined by taking angular averages over components of the infrared transition dipole moment and of the Raman tensor for the particular vibration of interest in an individual molecule. The (2) R xxz (τ ) ∝

3

′ )⟩ ∝ ⟨( ∑ Rβ , k(0)μβ′ )(∑ ∑ R γ , i(τ )Rδ , j(τ )αγδ

Here Rζ,n(t) is the rotation matrix that takes molecular axis ζ to laboratory-frame axis n on the basis of the Euler angles at time t, μ′ζ is the component of (∂μ/∂q)q0 in the ζ direction (where q is the vibrational coordinate and q0 is its equilibrium value), α′ξζ is the ξζ element of the tensor (∂α/∂q)q0, and the brackets indicate an orientational average. Although the indices of the second-order susceptibility and the second-order response are conventionally given in reverse temporal order, the time arguments of TCFs are typically given in temporal order. Thus, all of the angular averages here are expressed in terms of the infrared transition followed by the Raman transition. We begin by considering the xxz tensor element of the second-order response. The resultant expression (see Supporting Information for details) is

1 1 ′ ⟨sin θ sin 2 Θ cos ϕ cos2 Φ − sin θ cos ϕ⟩ − μX′ αYY ′ ⟨sin θ sin 2 Θ(cos ϕ cos2 Φ − cos ϕ) + sin θ cos ϕ⟩ μX′ αXX 2 2 1 1 1 ′ ⟨sin θ sin 2 Θ cos ϕ⟩ + μX′ αXY ′ ⟨sin θ sin 2 Θ cos ϕ sin 2Φ⟩ − μX′ αXZ ′ ⟨sin θ sin 2Θ cos ϕ cos Φ⟩ − μX′ αZZ 2 2 2 1 1 ′ ⟨sin θ sin 2Θ cos ϕ sin Φ⟩ − μY′ αXX ′ ⟨sin θ cos2 Θ sin ϕ cos2 Φ − sin θ sin ϕ sin 2 Φ⟩ − μX′ αYZ 2 2 1 1 ′ ⟨sin θ sin 2 Θ sin ϕ sin 2 Φ − sin θ sin ϕ⟩ − μY′ αZZ ′ ⟨sin θ sin 2 Θ sin ϕ⟩ + μY′ αYY 2 2 1 1 1 ′ ⟨sin θ sin 2 Θ sin ϕ sin 2Φ⟩ − μY′ αXZ ′ ⟨sin θ sin 2Θ sin ϕ cos Φ⟩ − μY′ αYZ ′ ⟨sin θ sin 2Θ sin ϕ sin Φ⟩ + μY′ αXY 2 2 2 1 1 ′ ⟨cos θ cos2 Θ cos 2 Φ + cos θ sin 2 Φ⟩ + μZ′ αYY ′ ⟨cos θ cos2 Θsin 2 Φ + cos θ cos2 Φ⟩ + μZ′ αXX 2 2 1 1 1 ′ ⟨cos θ − cos θ cos2 Θ⟩ − μZ′ αXY ′ ⟨cos θ sin 2 Θ sin 2Φ⟩ + μZ′ αXZ ′ ⟨cos θ sin 2Θ cos Φ⟩ + μZ′ αZZ 2 2 2 1 ′ ⟨cos θ sin 2Θ sin Φ⟩ + μZ′ αYZ (5) 2

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azimuthally symmetric). Note that the orientational averages include some terms in which a given angle appears at only time 0 or time τ and other terms in which the angle appears at both of these times. The latter terms are orientational TCFs. We next consider the xzx tensor element of the second-order response. The orientational average in this case yields

Here we have used lowercase Greek letters to denote Euler angles at zero time (i.e., the time of the infrared transition) and uppercase Greek letters to denote the Euler angles at time τ after the infrared transition. For the xxz tensor element of R(2)(τ), there are no terms that contain both χ and X, and so all of the azimuthal averages have been performed in the above expression (under the assumption that the interface is

1 (2) ′ ⟨sin Θ sin ϕ sin 2Φ sin χ sin X + cos θ sin 2Θ cos ϕ cos2 Φ cos χ cos X ⟩ R xzx (τ ) ∝ − μX′ αXX 2 1 1 ′ ⟨sin Θ sin ϕ sin 2Φ sin χ sin X − cos θ sin 2Θ cos ϕ sin 2 Φ cos χ cos X ⟩ + μX′ αZZ ′ ⟨cos θ sin 2Θ cos ϕ cos χ cos X ⟩ + μX′ αYY 2 2 1 ′ sin Θ sin ϕ cos 2Φ sin χ sin X − cos θ sin 2Θ cos ϕ sin 2Φ cos χ cos X + μX′ αXY 2 ′ ⟨cos Θ sin ϕ sin Φ sin χ sin X + cos θ cos 2Θ cos ϕ cos Φ cos χ cos X ⟩ + μX′ αXZ ′ ⟨cos θ cos 2Θ cos ϕ sin Φ cos χ cos X − cos Θ sin ϕ cos Φ sin χ sin X ⟩ + μX′ αYZ 1 ′ ⟨sin Θ cos ϕ cos 2Φ sin χ sin X − cos θ sin 2Θ sin ϕ cos2 Φ cos χ cos X ⟩ + μY′ αXX 2 1 1 ′ ⟨sin Θ cos ϕ sin 2Φ sin χ sin X + cos θ sin 2Θ sin ϕ sin 2 Φ cos χ cos X ⟩ + μY′ αZZ ′ ⟨cos θ sin 2Θ sin ϕ cos χ cos X ⟩ − μY′ αYY 2 2 1 ′ sin Θ cos ϕ cos 2Φ sin χ sin X + cos θ sin 2Θ sin ϕ sin 2Φ cos χ cos X − μY′ αXY 2 ′ ⟨cos θ cos 2Θ sin ϕ cos Φ cos χ cos X − cos Θ cos ϕ sin Φ sin χ sin X ⟩ + μY′ αXZ 1 ′ ⟨sin θ sin 2Θ cos2 Φ cos χ cos X ⟩ μ ′ αXX 2 Z 1 1 1 ′ ⟨sin θ sin 2Θ sin 2 Φ cos χ cos X ⟩ + μZ′ αZZ ′ ⟨sin θ sin 2Θ cos χ cos X ⟩ − μZ′ αXY ′ ⟨sin θ sin 2Θ sin 2Φ cos χ cos X ⟩ − μZ′ αYY 2 2 2 ′ ⟨sin θ cos 2Θ cos Φ cos χ cos X ⟩ + μZ′ αYZ ′ ⟨sin θ cos 2Θ sin Φ cos χ cos X ⟩ + μZ′ αXZ (6) ′ ⟨cos Θ cos ϕ cos Φ sin χ sin X + cos θ cos 2Θ sin ϕ sin Φ cos χ cos X ⟩ − + μY′ αYZ

In this case, every nonzero term in R(2)(τ) contains an orientational TCF for either sin χ or cos χ. Thus, orientational diffusion about the azimuthal axis can cause the xzx tensor element to decay completely to zero, as has been discussed previously.16

We next consider the orientational averages for the zzz tensor element. In this case we find that

(2) ′ ⟨sin θ sin 2 Θ cos ϕ cos2 Φ⟩ − μX′ αYY ′ ⟨sin θ sin 2 Θ cos ϕ sin 2 Φ⟩ − μX′ αZZ ′ ⟨sin θ cos2 Θ cos ϕ⟩ R zzz (τ ) ∝ −μX′ αXX

′ ⟨sin θ sin 2 Θ cos ϕ sin 2Φ⟩ + μX′ αXZ ′ ⟨sin θ sin 2Θ cos ϕ cos Φ⟩ + μX′ αYZ ′ ⟨sin θ sin 2Θ cos ϕ sin Φ⟩ − μX′ αXY ′ ⟨sin θ sin 2 Θ sin ϕ cos2 Φ⟩ − μY′ αYY ′ ⟨sin θ sin 2 Θ sin ϕ sin 2 Φ⟩ − μY′ αZZ ′ ⟨sin θ cos2 Θ sin ϕ⟩ − μY′ αXX ′ ⟨sin θ sin 2 Θ sin ϕ sin 2Φ⟩ + μY′ αXZ ′ ⟨sin θ sin 2Θ sin ϕ cos Φ⟩ + μY′ αYZ ′ ⟨sin θ sin 2Θ sin ϕ sin Φ⟩ − μY′ αXY 1 1 ′ ⟨cos θ cos2 Φ − cos θ cos 2Θ cos2 Φ⟩ + μZ′ αYY ′ ⟨cos θ sin 2 Φ − cos θ cos 2Θ sin 2 Φ⟩ + μZ′ αXX 2 2 1 ′ ⟨cos θ + cos θ cos 2Θ⟩ + μZ′ αXY ′ ⟨cos θ sin 2 Θ sin 2Φ⟩ − μZ′ αXZ ′ ⟨cos θ sin 2Θ cos Φ⟩ + μZ′ αZZ 2 ′ ⟨cos θ sin 2Θ sin Φ⟩ − μZ′ αYZ

(7)

equally likely. This situation is relevant for symmetric vibrations of cylindrically symmetric functional groups, such as the CN stretch of nitriles and acetylenic CH stretches, as well as for the symmetric stretch of freely rotating methyl groups. Under these conditions, for the xxz element of the secondorder response eq 5 reduces to 1 (2) ′ ⟨3 cos θ + cos θ cos 2Θ⟩ R xxz (τ ) ∝ μZ′ αXX 4 1 ′ ⟨cos θ − cos θ cos 2Θ⟩ + μZ′ αZZ (8) 4

As in the case for the xxz tensor element, all of the azimuthal angles that appear are at the same time, and so the azimuthal average has been performed in the above expression. Once again, some terms do not carry a time dependence whereas others lead to orientational TCFs. The above results for the orientational averages of tensor elements of the second-order response are general. We now consider the two special cases that are relevant to the symmetric methyl stretch, the asymmetric methyl stretch, and the CN stretch in acetonitrile. Case 1. In this special case, the α′ tensor is diagonal, αXX ′ = αYY ′ , μ′ lies along the Z axis of the molecule, and all φ are C



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For interpretation of experimental data, it is convenient to reexpress this result in terms of rotationally invariant combinations of the elements of the polarizability tensor that are commonly used in Raman spectroscopy. When αXX ′ = αYY ′ , the isotropic portion of the polarizability is given by

1 ′ + αZZ ′ ) αI′ = (2αXX 3

(2) (τ ) ∝ μZ′ αI′⟨cos θ ⟩ R zzz

⎛1 ⎞ 1 − μZ′ αA′ ⎜ ⟨cos θ ⟩ + ⟨cos θ cos 2Θ⟩⎟ ⎝6 ⎠ 2

As in the case of the xxz tensor element, the part of this response function that depends on the isotropic portion of the polarizability is not sensitive to reorientation, whereas the part that depends on the polarizability anisotropy has a piece that is not sensitive to reorientation and another piece that is sensitive to reorientation. According to eq 3, the response function under PPP polarization conditions depends on a weighted difference between eqs 11 and 15. Thus, PPP spectra emphasize the anisotropic portion of the Raman tensor and deemphasize the isotropic portion. The emphasis of the SSP and PPP spectra on different aspects of the Raman tensor for modes of this special case can be of great value in assigning features in VSFG spectroscopy. It is also important to note that for such modes, the SSP and PPP spectra are sensitive only to reorientation in θ, whereas SPS spectra are also sensitive to azimuthal reorientation. Case 2. In this special case, the infrared transition moment is along the X and/or Y axes of the functional group, the α′ tensor is off-diagonal in the XZ and/or YZ plane, all φ are equally likely, and α′XZ = α′YZ. This case applies to the asymmetric stretch of freely rotating methyl groups. For the xxz tensor element of the second-order response, eq 5 becomes

(9)

and the anisotropic portion is given by ′ − αZZ ′ αA′ = αXX

(10)

In terms of these quantities, eq 8 can be written as (2) R xxz (τ ) ∝ μZ′ αI′⟨cos θ ⟩

⎛1 ⎞ 1 + μZ′ αA′ ⎜ ⟨cos θ ⟩ + ⟨cos θ cos 2Θ⟩⎟ ⎝ 12 ⎠ 4

(11)

As would be expected, the part of this expression that depends on the isotropic portion of the polarizability is not influenced by reorientation. The part of the expression that depends on the polarizability anisotropy also has a piece that is not influenced by reorientation, plus another piece that depends on an orientational TCF of the form ⟨cos θ cos 2Θ⟩. On the basis of the coefficients of the two terms in eq 11, the xxz tensor element (and thus the VSFG signal under SSP polarization conditions) tends to emphasize the isotropic portion of the polarizability. This situation is especially true for modes that are strongly polarized (i.e., that have a large isotropic portion of the polarizability), such as symmetric methyl stretches. For the xzx tensor element of the second-order response, eq 6 in this case reduces to

(2) ′ ⟨sin θ sin 2Θ cos ϕ cos Φ⟩ R xxz (τ ) ∝ −μX′ αXZ

(16)

For acetonitrile, orientational relaxation in φ can be expected to be much faster than orientational relaxation in θ and can cause this response function to decay to zero. If the rate of orientational relaxation in φ is independent of θ, this expression can be written as (2) ′ ⟨sin θ sin 2Θ⟩⟨cos ϕ cos Φ⟩ R xxz (τ ) ∝ −μX′ αXZ

1 (2) ′ − αZZ ′ )⟨sin θ sin 2Θ cos χ cos X ⟩ R xzx (τ ) ∝ − μZ′ (αXX 2 1 = − μZ′ αA′ ⟨sin θ sin 2Θ cos χ cos X ⟩ 2

(17)

If this factorization holds, it implies that the dynamics in φ are independent of the dynamics in θ. Whether the dynamics in these degrees of freedom are truly independent will depend on the system in question. For instance, at the LV interface, acetonitrile molecules with a greater tilt may experience stronger interactions of their methyl groups with other molecules, which could couple φ and θ. For the xzx tensor element of the second-order response function, eq 6 in this case reduces to

(12)

This expression depends only on the anisotropic portion of the polarizability tensor. It is sensitive to reorientation in θ and in χ. Both of these factors tend to lead to weak SPS signals for symmetric methyl stretches. If the dynamics in θ and in χ are independent of each other, then we can use a factorized TCF of the form 1 (2) R xzx (τ ) = − μZ′ αA′ ⟨sin θ sin 2Θ⟩⟨cos χ cos X ⟩ 2

(15)

(2) R xzx (τ )

′ ⟨(cos Θ + cos θ cos 2Θ)cos ϕ cos Φ cos χ cos X ⟩ ∝ 2μX′ αXZ

(13)

(18)

In many situations we would expect the dynamics in χ to depend on the tilt angle θ, and comparing the unfactorized and factorized TCFs therefore gives us a means to assess the degree of coupling between these degrees of freedom. For the zzz tensor element of the second-order response, eq 7 in this case reduces to

Here we have used the fact that for both φ and χ, the orientational TCF for the sine of the angle is identical to the orientational TCF for the cosine of the angle under these conditions. Reorientation in φ and χ can cause this tensor element of the response function to decay completely. If all three coordinates are independent, this TCF factorizes into

(2) R zzz (τ ) ∝

1 ′ ⟨cos θ − cos θ cos 2Θ⟩ μ ′ αXX 2 Z 1 ′ ⟨cos θ + cos θ cos 2Θ⟩ + μZ′ αZZ 2

(2) R xzx (τ )

′ ⟨cos Θ + cos θ cos 2Θ⟩⟨cos ϕ cos Φ⟩⟨cos χ cos X ⟩ ∝ 2μX′ αXZ

(19)

(14)

Here again it is not obvious whether factorization should hold, as for instance one can readily imagine the dynamics in χ being dependent on θ.

In terms of the isotropic and anisotropic portions of the polarizability this expression becomes D



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of α′I and α′A. The Raman depolarization ratio of a vibrational mode is an experimentally measurable quantity that is defined as

For the zzz tensor element in this special case, eq 7 becomes (2) ′ ⟨sin θ sin 2Θ cos ϕ cos Φ⟩ R zzz (τ ) ∝ 2μX′ αXZ

(20)

ρ=

This is the same orientational TCF found for the xxz tensor element of the second-order response in this case.

3αA′ 2 45αI′2 + 4αA′ 2

(22)

Rearranging this equation yields an expression for the absolute value of the ratio of the isotropic portion of the polarizability to the anisotropic portion:

III. SIMULATION DETAILS We have performed MD simulations of acetonitrile on a hydrophilic silica surface using the same system described in our previous work.17,22 The hydrophilic silica surface, which was placed at z = 0, was constructed on the basis of the idealized β-cristobalite (C9) crystal from previous work of Lee and Rossky.23 The simulation box contained 864 acetonitrile molecules. Its dimensions were Lx = 45.605 Å, Ly = 43.883 Å, and Lz = 150 Å. The acetonitrile potential parameters were taken from previous work by Nikitin and Lyubartsev.24 MD simulations in the NVT ensemble were performed using the DL_POLY 2.18 package25 with a time step of 1 fs. Slabcorrected Ewald 3D sums26 were used for the treatment of electrostatic interactions in model systems. After the system had been equilibrated at T = 298 K for 500 ps, configurations were stored every 15 fs. The total sampling lasted for 450 ps and gave a trajectory file with a total of 30 000 configurations for analysis. As has been shown in our previous work17,22,27 and that of other groups,28−30 by looking at the molecular density profiles of alkyl cyanides (as defined by the position of the center of mass) as a function of distance from the silica wall, we can identify the regions of the simulated liquid that correspond to the LS interface, the LV interface, and the bulk. The boundaries of these regions for acetonitrile are given in Table 1. Once the

αI′ = αA′

3 − 4ρ 45ρ

(23)

This function is plotted in Figure 1a for values of ρ ranging from 0 (corresponding to a completely isotropic mode) to 0.75

Table 1. Boundary Locations along z in the Acetonitrile/Silica Simulation region

location

LS interface first sublayer second sublayer bulk LV interface

0−5.0 Å 0−2.5 Å 2.5−5.0 Å 25−30 Å 35−42 Å

Figure 1. (a) Dependence of the Raman depolarization ratio (ρ) on the absolute value of the ratio between the isotropic and anisotropic contributions to the Raman polarizability. (b) Dependence of the Raman depolarization ratio on the VSFG depolarization ratio, R = ′ /αZZ ′ . Most values of ρ correspond to two possible values of R. αXX

regions have been identified, we can calculate density/orientation correlation functions and time correlation functions in them. Furthermore, as demonstrated in our previous work,17,22 the LS interface in this system can be divided into two sublayers. The sublayer closest to the silica surface has cyanide groups that point toward the surface, on average. However, the cyanide groups in the second sublayer tend to point away from the surface. For the calculation of a TCF in a specific region, which has a general form ⟨A(0) B(τ)⟩, we used the so-called “initial position” method30 for sampling the TCF in a given region over n time steps. The equation for calculating the TCF is ⟨A(0) B(τ )⟩ =

1 n

n

∑ t0= 1

1 N *(t0)

(corresponding to a completely depolarized mode). For symmetric methyl and methylene stretches, the depolarization ratio is typically on the order of 0.01 or less, although it can be as large as 0.1 in some cases.31−35 At the higher end of this range, the magnitude of the anisotropic portion of the polarizability is comparable to or slightly larger than that of the isotropic portion. At the lower end of this range, the magnitude of the isotropic portion of the polarizability is nearly a factor of 3 greater than that of the anisotropic portion of the polarizability. VSFG spectroscopy has been shown to be a useful means of determining the relative signs of αI′ and αA′ .36 For a symmetric vibrational mode, the relative signs of the xxz and xzx tensor elements can be related to the VSFG depolarization ratio

N *(t 0)

∑ i=1

Ai (t0) Bi (t0+τ ) (21)

where N*(t0) is the number of molecules in the specific region at reference time t0. We computed each TCF out to a time of 12 ps.

IV. RESULTS To evaluate many of the above expressions for the time dependence of tensor elements, we must know the magnitudes and relative signs

R= E

′ αXX ′ αZZ

(24) dx.doi.org/10.1021/jp306296s | J. Phys. Chem. A XXXX, XXX, XXX−XXX


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The Raman depolarization ratio can be expressed in terms of R as ρ=

3(R − 1)2 5(2R + 1)2 + 4(R − 1)2

direction to the vector used for the symmetric methyl stretch (Figure 2). As a point of reference, the P1(cos θ) orientational correlation time for molecular tumbling in this simulated system is 3.4 ps in the bulk, 1.9 ps at the LV interface, and 24 ps at the LS interface.17 Thus, reorientation would be expected to have the greatest effect on the VSFG signal at the LV interface. For the xxz tensor element, the TCF that we need to calculate for this mode is ⟨cos θ cos 2Θ⟩, which we will denote C1(τ) (a full list of the shorthand notation used for the TCFs is given in Table 3). This TCF is plotted in Figure 3a for the

(25)

This function is plotted in Figure 1b. It can be seen from this plot that there are two values of R that can lead to any value of the Raman depolarization ratio other than 0 or 0.75. These values are given by R± =

2ρ + 1 ±

−ρ(20ρ − 15) 1 − 8ρ

Table 3. Shorthand Notation Used for the Orientational TCFs Calculated

(26)

For relatively small values of the Raman depolarization ratio ( 1 for this mode.36 Using R+, we find that α′I = α′ZZ is 1.28 and αA′ = αZZ ′ is 0.42 for this mode. For the CN stretch of liquid acetonitrile, the Raman depolarization ratio is 0.092,37 which gives possible R values of 0.32 or 8.65. In this case experiments show that R < 1.36 The value of R− implies that αI′/αZZ ′ is 0.54 and αA′ /αZZ ′ is −0.68 for this mode. These results are listed in Table 2.

notation

time correlation function

C1(τ) C2(τ) C3(τ) C4(τ)

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

Table 2. Approximate Values of α′I /α′ZZ and α′A/α′ZZ for the Symmetric Stretching Modes of Acetonitrile As Determined from Raman Depolarization Ratios mode

αI′/αZZ ′

αA′ /αZZ ′

symmetric methyl stretch CN stretch

1.28 0.54

0.42 −0.68

We begin by considering the symmetric methyl stretch of acetonitrile, which belongs to Case 1. The 3-fold molecular axis is defined as Z, and θ is the angle that the vector from the cyanide carbon to the methane carbon makes with the normal vector pointing out of the silica surface (Figure 2). The same Figure 3. Orientational TCFs (a) C1(τ) and (b) C2(τ) for the symmetric methyl stretch of acetonitrile in the bulk liquid and at the LV and LS interfaces.

symmetric methyl stretch at the LV and LS interfaces, with the bulk TCF (which, as expected, is zero) shown for reference. From our simulations, the equilibrium value of ⟨cos θ⟩⟨cos 2θ⟩ is −1.11 × 10−2 for the LV interface and 1.73 × 10−4 for the LS interface. C1(τ) for the LV interface is slightly positive initially and then becomes negative at about 1 ps. This sign change arises because ⟨cos θ cos 2θ⟩ and ⟨cos θ⟩⟨cos 2θ⟩ have opposite signs. The TCF approaches its equilibrium value within about 6 ps, so its time scale is comparable to that for other vibrational dephasing processes. The TCF for the LS interface is positive and grows initially, before starting to decrease after about half of a picosecond. This behavior is a result of the bilayer structure of acetonitrile at this

Figure 2. Molecular coordinates used to calculate orientational TCFs for the methyl group (black axes) and cyanide group (blue axes) of acetonitrile.

TCFs apply to the CN stretch of acetonitrile, albeit with the sign flipped because the CN vector points in the opposite F



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interface.17,22,28−30 The two sublayers of the bilayer have TCFs that are of opposite sign, and the TCF of the entire bilayer reflects the differences in dynamics between these sublayers (see Supporting Information). Additionally, the TCF at the LS interface does not approach its equilibrium value on the time scale shown in Figure 3a, which is an indication that dynamics are much slower at the LS interface than at the LV interface. In addition, we have defined the TCFs here on the basis of the position of a molecule at the initial time only, which is the most spectroscopically relevant choice. However, at long delay times a molecule that started at one interface may no longer be there. As a result, if orientational dynamics are slow, a TCF may not reach the “equilibrium” value, which is calculated only on the basis of the molecules at a given interface. For the xzx tensor element of the symmetric methyl stretch, the relevant TCF is ⟨sin θ sin 2Θ cos χ cos Χ⟩, which we denote C2(τ). This TCF is plotted for the LV and LS interfaces as well as for the bulk liquid in Figure 3b. As expected, this TCF is zero in the bulk and decays to zero in a few picoseconds at the LV interface. Surprisingly, however, this TCF grows with time at the LS interface, and at a time scale that is considerably faster than the 24 ps time of the P1(cos θ) orientational TCF. This behavior again arises from an interference between the two sublayers of the surface bilayer (see Supporting Information). Neither sublayer has a C2(τ) that relaxes completely to the equilibrium value over 12 ps, but the second sublayer relaxes considerably more rapidly than the first sublayer. We note additionally that ⟨cos χ⟩ = 0.019 at the LS interface, which suggests that the area of our simulated silica surface may not be large enough to average out any correlations among the azimuthal orientations of neighboring molecules. However, the time scale of the decay of this TCF is still long compared to time scales for other mechanisms for vibrational dephasing at this interface, which means that reorientation should generally have a relatively small influence on the VSFG spectrum. Given these TCFs, we can now determine the various tensor elements of the response function for the symmetric methyl stretch. These elements are plotted as a function of delay time for the LV interface in Figure 4a and the LS interface in Figure 4b. For the LV interface, even though reorientation is relatively fast, it has a relatively small effect on all of the tensor elements except for xzx and its equivalents, which are small to begin with and decay to near zero in approximately 2 ps. These results are in excellent accord with experiment, in which the SPS spectrum of the symmetric methyl stretch at the LV interface of acetonitrile has not been observed.17 For the LS interface, reorientation again has little effect on any tensor element of the response except for xzx and its equivalents. Once more this response function is small compared to the others, but our simulation shows that reorientation causes it to grow with time, which may allow it to be observed in an experiment that is sensitive enough. We next consider the CN stretch. In this case the correlation functions C1(τ) and C2(τ) are identical to those for the symmetric methyl stretch except that they change sign, because the coordinate systems for the cyanide and methyl groups point in opposite directions (Figure 2). However, the sign of R for the CN stretch is different than that for the symmetric methyl stretch, which has a significant impact on the tensor elements of the response function. The different tensor elements of the response function for the CN stretch are shown in Figure 5a for the LV interface and in Figure 5b for the LS interface. At the LV interface, the xzx tensor element and its

Figure 4. Tensor elements of the second-order response function for the symmetric methyl stretch of acetonitrile at (a) the LV interface and (b) the LS interface.

Figure 5. Tensor elements of the second-order response function for the CN stretch of acetonitrile at (a) the LV interface and (b) the LS interface.

equivalents again decay to zero within a couple of Picoseconds. The zzz tensor element grows by about 20% over the time scale of a few picoseconds, and the xxz tensor element decays by about the same amount over this time scale. We might therefore expect to see some difference in the linewidths between SSP and G



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PPP spectra of this mode at the LV interface. At the LS interface, the major influence of reorientation is again on the xzx tensor element and its equivalents. This tensor element grows with time over a few picoseconds and becomes nearly as large in magnitude as the xxz tensor element, which suggests the CN stretch might be able to be observed under SPS polarization conditions at this interface. We now consider the asymmetric methyl stretch, which corresponds to Case 2. Again, for reference, in our simulations the P1(cos φ) orientational correlation function has a 1/e decay time of 0.35 ps in the bulk, 0.22 ps at the LV interface, and 0.48 ps at the LS interface.17 Thus, spinning reorientation might be expected to have a significant influence on the VSFG signal at both the LV and LS interfaces. For the xxz and zzz tensor elements of the response, we need the TCF ⟨sin θ sin 2Θ cos φ cos Φ⟩, which we denote C3(τ). As can be seen from Figure 6a, this TCF is zero in the

Figure 7. Tensor elements of the second-order response function for the asymmetric methyl stretch of acetonitrile at (a) the LV interface and (b) the LS interface.

decay completely within about 2 ps. Of these tensor elements, xzx and its equivalents are the largest. This result suggests that it may be possible to observe an SPS signal for this mode at the LS interface, although it would be expected to be weak due to the rapid relaxation of the response function due to reorientation. We now turn to the issue of factorization. We begin with the TCF C2(τ), which is relevant to the xzx tensor element of the second-order response for the symmetric methyl stretch. This TCF is displayed for the LV interface in Figure 8a and for the LS interface in Figure 8b. The factorized TCFs are smoother than the unfactorized versions, because obtaining a good average over a single degree of freedom requires fewer configurations than obtaining a good average over multiple degrees of freedom. In the case of the LV interface there is a significant degree of resemblance between the unfactorized and factorized versions of C2(τ). Although the differences between the two versions appear to be larger than can be accounted for by noise in the averaging alone, we should also note that the overall magnitudes of these TCFs are quite small. In the case of the LS interface, the unfactorized and factorized versions of C2(τ) differ significantly. The magnitudes of these TCFs are again relatively small, but they are more than an order of magnitude larger than the corresponding versions of C2(τ) for the LV interface. We can gain more insight into the differences between the unfactorized and factorized versions of C2(τ) at the LS interface by examining the behavior of the individual sublayers, as shown in Figure 9. C2(τ) is of opposite sign for the two sublayers, and so the overall TCF for the LS interface is the result of a considerable degree of cancellation between the contributions of these sublayers. It is clear from Figure 9 that for each individual sublayer the unfactorized and factorized TCFs are quite similar. The small differences between the two versions are accentuated in C2(τ) for the full surface population due to the high degree of cancellation of

Figure 6. Orientational TCFs (a) C3(τ) and (b) C4(τ) for the asymmetric methyl stretch of acetonitrile in the bulk liquid and at the LV and LS interfaces.

bulk liquid. It takes on a small initial value at the LV and LS interfaces, and decays to near zero within about 2 ps. For the xzx tensor element, the TCF required is ⟨(cos Θ + cos θ cos 2Θ) cos φ cos Φ cos χ cos Χ⟩, which we denote C4(τ). This TCF is zero in the bulk, as seen in Figure 6b. At the LV interface it has a small initial value and decays to near zero in less than 1 ps. At the LS interface the initial value is somewhat larger than at the LV interface, and the decay is nearly complete within about 2 ps. The tensor elements of the response for the asymmetric methyl stretch are shown for the LV interface in Figure 7a and the LS interface in Figure 7b. At the LV interface all of the elements of the response decay in a few hundred femtoseconds, suggesting that the VSFG signal from this mode should be very weak (in agreement with experiment, in which it has not been reported17). At the LS interface all tensor elements of the response H



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Figure 10. Unfactorized (black) and factorized (red) versions of the orientational TCFs (a) C3(τ) and (b) C4(τ) for the asymmetric methyl stretch of acetonitrile at the LV interface.

Figure 8. Unfactorized (black) and factorized (red) versions of C3(τ) for the symmetric methyl stretch of acetonitrile at (a) the LV interface and (b) the LS interface.

and factorized TCFs bear resemblance to one another, with similar initial magnitudes and a rapid decay over a few hundred femtoseconds. Though the factorized TCFs show a clear slower decay as well, the unfactorized TCFs are too noisy to distinguish whether such a decay exists. We next turn to the LS interface. The unfactorized and factorized versions of C3(τ) and C4(τ) are plotted for this interface in Figure 11. As was the case for the LV interface, the factorized TCFs are much smoother than the unfactorized TCFs. The magnitudes and decay times of the unfactorized and factorized TCFs are once again similar. We can again get a better sense of the applicability of the factorization approximation at this interface by considering the two interfacial sublayers, as shown in Figure 12. For both C3(τ) and C4(τ) the factorized TCFs closely resemble the unfactorized versions. The poorer agreement between the unfactorized and factorized TCFs in Figure 11 once more results largely from noise that remains after the cancellation of the contributions from the two sublayers. Interestingly, in the case of C4(τ) the difference between the unfactorized and factorized TCFs is greater in the second sublayer than in the first sublayer, which is the opposite of what was observed for C2(τ). These results suggest that although coupling between θ and χ is modest, in the first sublayer this coupling is most pronounced for molecules with a significant tilt angle. Such molecules make a larger contribution to C2(τ), which depends on sinθ, than they do to C4(τ), which depends on cos θ. On the basis of these results, we can conclude that for acetonitrile at the LV and LS interfaces, orientational relaxation in φ is largely decoupled from orientational relaxation in θ. Given that relaxation in φ can occur through internal rotation, this result is well within reason. Additionally, on the basis of our results for C2(τ) and C4(τ), it appears that relaxation in χ is coupled slightly to relaxation in θ for acetonitrile at these interfaces.

Figure 9. Unfactorized (solid lines) and factorized (dashed lines) versions of C2(τ) for the symmetric methyl stretch of acetonitrile for the two sublayers at the LS interface.

the TCFs from the two sublayers. The difference between the unfactorized and factorized versions of C2(τ) is larger for the first sublayer than for the second sublayer, and the disparity between the two versions grows with time. This phenomenon may be a result of the molecules in the first sublayer being hydrogen bonded to the surface. Hydrogen bonding tethers the molecules on the time scale of reorientation,38 which may promote coupling between θ and χ in the form of “wobbling-ina-cone” reorientation whose rate depends on θ. However, on the basis of the strong resemblance between the unfactorized and factorized TCFs for the surface sublayers, this coupling must be relatively small. We next consider whether the orientational TCFs for the asymmetric methyl stretch can be factorized for the interfaces studied here. We begin by considering the LV interface. The unfactorized and factorized TCFs C3(τ) and C4(τ) are plotted for this interface in Figure 10. In both cases the unfactorized I



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dynamics in φ dominate the decay of C3(τ) and C4(τ) at both interfaces, with orientational dynamics in χ making some additional contribution to C4(τ). Thus, we can expect the orientational contribution to the SPS spectrum of the asymmetric methyl stretch of acetonitrile at these interfaces largely to reflect the dynamics of the internal rotation of the methyl group.

V. DISCUSSION Reorientation is only one of several mechanisms that can affect the intensities and line widths of features in VSFG spectra. Other phenomena that can influence these spectral parameters include population relaxation, pure dephasing, inhomogeneous broadening, and resonant energy transfer. For reorientation to play a measurable role in VSFG spectroscopy, it must occur on a time scale that is comparable to or faster than those for these other processes. However, even when reorientation does occur on such a time scale, we have seen that it is not guaranteed to have a significant influence on the features in a VSFG spectrum. Acetonitrile is in many ways an ideal liquid with which to test the influence of reorientation on VSFG spectra. Acetonitrile is a small molecule whose vibrational spectroscopy has been studied with many advanced techniques.17,39−44 Liquid acetonitrile has a relatively low viscosity, so that both tumbling and spinning reorientation occur on time scales that are competitive with other dephasing mechanisms at the LV interface. At the LS interface, on the other hand, the P1 orientational correlation time for tumbling is much longer than typical vibrational dephasing times, whereas spinning orientational relaxation remains rapid. Thus, reorientation has the potential to play a large role in VSFG spectra at the LV interface and a significant role in the VSFG spectra of some modes under some polarization conditions at the LS interface. Few if any other liquids would be expected to have orientational dynamics that are significantly faster than those of acetonitrile at ambient density. Because the orientational TCFs relevant to the VSFG spectroscopy of acetonitrile are different from the P1(cos θ) and P1(cos φ) TCFs, orientational dynamics can influence the second-order response on a different (and typically faster) time scale than that of the P1 orientational correlation times. This effect is particularly relevant for the tumbling reorientation of acetonitrile, which leads to a VSFG orientational correlation time of a few picoseconds even at the LS interface. However, the overall influence of reorientation is also highly dependent on the structure of the liquid at an interface, as well as on the properties of the Raman tensor for the vibrational mode in question. The symmetric methyl stretch is a highly polarized mode, and so reorientation plays a relatively minor role in its VSFG spectroscopy for acetonitrile (except for the xzx tensor elements and its equivalents, which are sensitive only to the anisotropic portion of α′). The CN stretch, on the other hand, has a substantial depolarized component, and so reorientation plays a larger role in the time dependence of the second-order response (particularly at the LV interface). As would be expected, VSFG spectra of the asymmetric methyl stretch are highly sensitive to spinning reorientation. As can be seen in Figure 7, although the different tensor elements of the second-order response are small to begin with for this mode, reorientation causes them to decay very quickly. Even at the LS interface, the tensor elements of the second-order response for this mode decay nearly completely within 2 ps. From our results, it appears likely that methyl rotation is the dominant dephasing mechanism for the asymmetric methyl stretch in VSFG spectra of acetonitrile at these interfaces, which

Figure 11. Unfactorized (black) and factorized (red) versions of the orientational TCFs (a) C3(τ) and (b) C4(τ) for the asymmetric methyl stretch of acetonitrile at the LS interface.

Figure 12. Unfactorized (solid lines) and factorized (dashed lines) versions of the orientational TCFs (a) C3(τ) and (b) C4(τ) for the asymmetric methyl stretch of acetonitrile for the two sublayers at the LS interface.

Additionally, the magnitudes of the individual factor TCFs, the overall degree of relaxation, and the time scale of dynamics are quite different for φ, θ, and χ in C3(τ) and C4(τ) (see Supporting Information). As would be expected, orientational J



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observed behavior to be rather general for species for which the orientational TCFs derived here are applicable. In a future paper we will derive the orientational TCFs relevant to rotationally hindered methyl groups and methylene groups, and will use simulations of propionitrile at the LV and LS interfaces to examine the behavior of these TCFs.

probably explains why this mode has not been reported in spectra from either interface. Another recurring theme in our results is the sensitivity of the orientational contribution in the spectrum to interfacial ordering. For acetonitrile, this sensitivity is especially apparent at the LS interface due to the bilayer structure adopted by the liquid. Differences in the orientational TCFs for the two sublayers, which are typically of opposite sign, lead to complex dynamic effects in second-order response. Of particular interest is the fact that the loss of some of the cancellation between the contributions of these sublayers due to different rates of reorientation can cause the second-order response to increase with time. This increase can partially counteract the influence of other dephasing mechanisms, potentially leading to narrower spectral features. On the basis of our results, reorientation should affect both the intensity and the line width of features in the VSFG spectrum of liquid acetonitrile at the LV and LS interfaces. For the symmetric methyl and CN stretches, these effects are relatively modest. In comparing the VSFG spectra of these modes under different polarization conditions, it may be difficult to distinguish these from the influence of other dynamic processes or from differences in nonlinear Fresenel factors. However, for the asymmetric methyl stretch, spinning reorientation likely plays a major role in determining both the line width and the intensity of the VSFG spectrum. Given sensitive enough experiments it may be possible to detect these spectra and thereby to observe the influence of reorientation directly.

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ASSOCIATED CONTENT

S Supporting Information *

Details of the derivation of eq 5, 6, and 7, plots of C1(τ) and C2(τ) for sublayers at the LS interface, and plots of the individual components of the factorized versions of C2(τ), C3(τ), and C4(τ) at the LV and LS interfaces. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. 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 are grateful to John D. Weeks for helpful discussions and to Zhonghan Hu for the development of the simulation model used here.

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VI. CONCLUSIONS We have derived the orientational TCFs relevant to the VSFG spectrum of the symmetric and asymmetric methyl stretches and the CN stretch of acetonitrile under different polarization conditions. We have used MD simulations to evaluate these orientational TCFs at the LV and LS interfaces, as well as to calculate the time dependence of the different tensor elements of the second-order response function. Although reorientation plays a relatively small role in the VSFG spectra of the symmetric methyl and CN stretches, it has a major influence on the spectra of the asymmetric methyl stretch. Our results are in good agreement with experimental VSFG spectra obtained at these interfaces.17 Because the LS interface of acetonitrile is composed of separate sublayers, we have also observed some effects that initially seem counterintuitive. Due to cancellation between the contributions of these sublayers to some orientational TCFs, the faster dynamics in the second sublayer can lead to an increase in the second-order response with time. Thus, although reorientation leads to broader lines in most spectroscopic contexts, in some situations it may in fact lead to line narrowing in VSFG spectra. In the case of the asymmetric methyl stretch, we have found that factorization of the orientational TCFs into a product of TCFs that each depend on a single coordinate is a reasonable approximation. This result implies that there is not a high degree of coupling among the orientational degrees of freedom for acetonitrile at these interfaces. We further found that the orientational contribution to the VSFG spectrum for this mode is dominated by the dynamics of internal methyl group rotation. Some of the phenomena observed here are linked to the special nature of acetonitrile (and particularly its formation of a lipid-bilayer-like structure at a silica interface17,22). However, the orientational TCFs derived are applicable to many different molecules of interest for VSFG studies. We expect much of the

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Orientational Time Correlation Functions for Vibrational Sum

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