On the Empirical Analysis of Optical Kerr Effect Spectra: A Case for

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On the Empirical Analysis of Optical Kerr Effect Spectra: A Case for Constraint John S. Bender, John T. Fourkas, and Benoit Coasne J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b09751 • Publication Date (Web): 21 Nov 2017 Downloaded from http://pubs.acs.org on November 22, 2017

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OKE Fitting

The Journal of Physical Chemistry

On the Empirical Analysis of Optical Kerr Effect Spectra: A Case for Constraint John S. Bender, † John T. Fourkas,*, †,‡,§,¶ and Benoit Coasne*,║ †

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

Univ. Grenoble Alpes, CNRS, LIPhy, 38000 Grenoble, France

1 Environment ACS Paragon Plus

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ABSTRACT: Ultrafast optical Kerr effect (OKE) spectroscopy is a widely used method for studying the depolarized, Raman-active intermolecular dynamics of liquids. Through appropriate manipulation of OKE data, it is possible to determine the reduced spectral density (RSD), which is the Bose-Einstein-corrected, low-frequency Raman spectrum with the contribution of diffusive reorientation removed. OKE RSDs for van der Waals liquids can often be fit well to an empirical function that is the sum of a Bucaro-Litovitz function and an antisymmetrized Gaussian (AG). Although these functions are not directly representative of specific intermolecular dynamics, the AG fit parameters can provide useful insights into the microscopic properties of liquids. Here we show that fits using the AG function are typically not well-determined, and that equally good results can be obtained with a wide range of fitting parameters. We propose the use of a physicallymotivated constraint on the amplitude of the AG function, and demonstrate that this constraint leads to more intuitive trends in the fit parameters for temperature-dependent RSDs in 1,3,5trifluorobenzene and hexafluorobenzene.


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OKE Fitting


1. Introduction Over the past few decades, the optical Kerr effect1-3 (OKE) has developed into a mainstay in the ultrafast spectroscopy of liquids. In its typical implementation, OKE spectroscopy allows for the measurement of the low-frequency, depolarized, Raman-active dynamics of a transparent liquid. These dynamics can arise from processes such as librations, interaction-induced (II) intermolecular scattering, diffusive reorientation, and low-frequency intramolecular vibrations. In 1990, McMorrow and Lotshaw presented a formalism for transforming time-domain optically-heterodyne-detected OKE data into the frequency domain, providing a means to calculate the Bose-Einstein-corrected, low-frequency (0 to ~500 cm-1), depolarized Raman spectrum from OKE data.4-5 Subtracting well-characterized diffusive reorientation dynamics from the timedomain data and using this Fourier transform deconvolution procedure4-5 yields the so-called reduced spectral density (RSD). In van del Waals liquids the RSD is typically broad and featureless, and extracting information regarding the underlying molecular motions that shape the spectrum is therefore difficult. This problem is made all the more challenging by the overlapping polarizability contributions to the molecular response that gives rise to the spectrum. Nevertheless, OKE RSDs have been invaluable in the investigation of the molecular dynamics of many simple3 and complex1 liquids. To extract information from the RSDs of liquids, fits to empirical line shapes are often used.614

These fits are typically composed of two types of function. The first is a Bucaro-Litovitz (BL)

function,15 𝐴𝐴𝐵𝐵𝐵𝐵 𝜔𝜔𝛿𝛿 𝑒𝑒 −𝜔𝜔⁄𝜔𝜔𝐵𝐵𝐵𝐵 ,

(1)

where 𝐴𝐴𝐵𝐵𝐵𝐵 is an amplitude, 𝛿𝛿 is a fitting parameter, and 𝜔𝜔𝐵𝐵𝐵𝐵 is a characteristic frequency. The BL function is generally used to describe the low-frequency portion of OKE RSDs. This function was


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originally developed to describe depolarized light scattering in atomic fluids. Because the polarizability of an isolated atom is isotropic, any depolarized scattering in such media arises from collisions. Thus, the BL portion of fits to OKE RSDs is often assumed to arise from II scattering. However, it is well established that the BL function overestimates II effects in molecular liquids.6, 16

The second function is the antisymmetrized Gaussian6 (AG) 2

2

𝐴𝐴𝐴𝐴𝐴𝐴 �𝑒𝑒 −(𝜔𝜔−𝜔𝜔𝐴𝐴𝐴𝐴)/2𝜎𝜎𝐴𝐴𝐴𝐴 − 𝑒𝑒 −(𝜔𝜔+𝜔𝜔𝐴𝐴𝐴𝐴)/2𝜎𝜎𝐴𝐴𝐴𝐴 � ,

(2)

where 𝐴𝐴𝐴𝐴𝐴𝐴 is an amplitude, and ±𝜔𝜔𝐴𝐴𝐴𝐴 and 𝜎𝜎𝐴𝐴𝐴𝐴 are the center frequencies and the width parameter,

respectively, of the Gaussian functions comprising the total AG function. The Gaussian nature of this function can be thought of as modeling inhomogeneity, and antisymmetrization ensures that the function goes to zero at zero frequency. The AG function is often assumed to describe librational scattering, i.e scattering that arises from the dynamics of the anisotropic polarizability of individual molecules, even when the dynamics itself is collective. The AG function dominates the high-frequency side of the RSD, where molecular dynamics simulations have shown that librational dynamics are the major contribution to the spectral shape for van der Waals liquids.1720

The sum of the BL and AG line shapes often fits OKE RSDs quite accurately, and is an excellent tool for being able to describe the RSD lineshape using a limited number of parameters. However, extracting meaningful physical interpretations from the fits can be difficult and even misleading. In some cases, empirical fits led to parameters that seemingly contradict both the intuitive picture of temperature effects and the appearance of the spectra. Such was the case, for instance, in some of our previous work on the temperature dependence on the RSDs of 1,3,5trifluorobenzene (TFB) and hexafluorobenzene (HFB).7 In both liquids, ωAG was found to decrease


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and σAG to increase with decreasing temperature.7 As the temperature of a liquid decreases, the number of accessible librational states should decrease, which should translate into a decreased width of the AG function. The stiffening of the intermolecular potential upon cooling similarly should increase the average librational frequency. Another issue with empirical fits is that there is overlap throughout the RSD among the II contribution, the collective molecular (CM) contribution, and the cross term between these two. For a liquid composed of rigid molecules, the CM component arises solely from molecular rotation, and thus is purely librational. The II component arises from both translational and rotational degrees of freedom. These two components are always positive in the time and frequency domains, whereas the cross term between them is negative at most frequencies.18-22 The BL function is always positive, and the AG function is positive at positive frequencies. Parameters extracted from empirical fits therefore cannot be used rigorously to assign individual scattering mechanisms to any portion of the RSD. Despite the above caveats, we have recently demonstrated that, for liquid benzene, the AG parameters can be used to distinguish between the effects of temperature and density on the RSD.16 In particular, ωAG is sensitive to density when temperature is held constant and σAG is sensitive to temperature when density is held constant.16 Thus, empirical fits to OKE RSDs can indeed carry meaningful physical information. A common problem with nonlinear least-squares fitting is that the final fit parameters are sensitive to the initial values used. In such a case, the final parameters are not unique, which is usually a sign that the effects of two or more parameters can compensate for one another in determining the shape of the final function. In the course of our previous work on benzene16 we discovered that in fits of the RSDs to a BL function and an AG function, such an interplay exists


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between the AAG and ωAG. As we show below, because this interplay is so strong, the AG parameters in fits are not well-determined. Indeed, ωAG can compensate for AAG so effectively that the latter parameter can take on any value within a wide range without having a significant influence on the shape of the AG function. A useful approach for alleviating the initial-value problem in nonlinear least-squares fits is to reduce the number of free parameters by placing a constraint on one or more of the parameters. Here we show that to extract meaningful physical information from empirical fits to the sum of a BL function and an AG function, it is essential to impose a physically reasonable constraint on AAG, thus reducing the number of free parameters from six to five. We use the CM, II and cross term components from molecular dynamics (MD) simulations of the OKE RSD of benzene to justify what form this constraint should take: that AAG correspond to the peak height of the AG function. We then apply this constraint to temperature-dependent OKE data for HFB and TFB to show that the constraint leads to the parameter temperature dependences that one would intuitively expect for these liquids.

2. Theory The OKE RSD arises from the low-frequency, depolarized, Raman-active polarizability dynamics of a liquid. The many-body polarizability, 𝚷𝚷, of a liquid arises from a CM component and an II component,

𝚷𝚷 = 𝚷𝚷𝐶𝐶𝐶𝐶 + 𝚷𝚷𝐼𝐼𝐼𝐼 .

(3)

𝚷𝚷 is a second-rank tensor that describes the polarizability of the liquid in Cartesian space. The CM

component is the sum of all the molecular polarizabilities in the liquid. The II component is the


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sum of all intermolecular polarizability interactions. The relevant dynamics are then encoded on the many-body polarizability time correlation function (TCF), 𝐶𝐶𝑥𝑥𝑥𝑥 (𝑡𝑡) = 〈𝚷𝚷𝑥𝑥𝑥𝑥 (0)𝚷𝚷𝑥𝑥𝑥𝑥 (𝑡𝑡)〉 ,

(4)

which tracks an off-diagonal elements of the polarizability tensor, and thus the decay of polarizability anisotropy in the liquid. Insertion of eq 3 into eq 4 shows that there are three components in the polarizability TCF: a CM term, an II term, and a cross term (CMII) 𝐶𝐶𝐶𝐶 (𝑡𝑡) 𝐶𝐶𝐶𝐶 (0)𝚷𝚷 𝐶𝐶𝐶𝐶 (𝑡𝑡)〉 𝐶𝐶𝑥𝑥𝑥𝑥 = 〈𝚷𝚷𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥 𝐼𝐼𝐼𝐼 (𝑡𝑡) 𝐼𝐼𝐼𝐼 (0)𝚷𝚷 𝐼𝐼𝐼𝐼 (𝑡𝑡)〉 𝐶𝐶𝑥𝑥𝑥𝑥 = 〈𝚷𝚷𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥

𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 (𝑡𝑡) 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 (0)𝚷𝚷 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 (𝑡𝑡) 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 (0)𝚷𝚷 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 (𝑡𝑡)〉 𝐶𝐶𝑥𝑥𝑥𝑥 = 〈𝚷𝚷𝑥𝑥𝑥𝑥 + 𝚷𝚷𝑥𝑥𝑥𝑥 . 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥

(5) (6) (7)

𝐶𝐶𝐶𝐶 (𝑡𝑡) For liquids composed of rigid molecules, 𝐶𝐶𝑥𝑥𝑥𝑥 contains only dynamics that depend on

reorientation of the molecule about axes that change the apparent orientation of the polarizability 𝐶𝐶𝐶𝐶 (𝑡𝑡) ellipsoid in the laboratory frame. 𝐶𝐶𝑥𝑥𝑥𝑥 is therefore, dominated by orientational diffusion and

𝐼𝐼𝐼𝐼 (𝑡𝑡) 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 (𝑡𝑡) librational dynamics. 𝐶𝐶𝑥𝑥𝑥𝑥 contains translational dynamics as well as reorientation, and 𝐶𝐶𝑥𝑥𝑥𝑥

describes the coupling between interaction-induced and molecular polarizability dynamics. The RSD can be calculated by starting with the nuclear response function, 𝑑𝑑

𝑅𝑅(𝑡𝑡) ∝ − 𝑑𝑑𝑑𝑑 𝐶𝐶𝑥𝑥𝑥𝑥 (𝑡𝑡) ,

(8)

removing the contribution of orientational diffusion, and then taking the Fourier transform of the resultant function. Orientational diffusion is often removed by assuming an exponential or multiexponential functional form and subtracting this function from 𝑅𝑅(𝑡𝑡). The RSD, 𝐼𝐼(𝜔𝜔), is defined as the imaginary component of the Fourier transform of the resultant function


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∞

𝐼𝐼(𝜔𝜔) = ∫0 �𝑅𝑅(𝑡𝑡) − �𝑒𝑒 −𝑡𝑡⁄𝜏𝜏𝑐𝑐 �1 − 𝑒𝑒 −𝑡𝑡⁄𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 �� sin(𝜔𝜔𝜔𝜔) 𝑑𝑑𝑑𝑑 .

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(9)

Here, 𝜏𝜏𝑐𝑐 is the time constant for collective orientational diffusion and 𝜏𝜏𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 is the assumed rise time

of these dynamics, which is usually on the order of a few hundred femtoseconds. Using molecular dynamics trajectories, 𝐼𝐼(𝜔𝜔) can be calculated for each of the polarizability components

corresponding to eqs 5, 6, and 7.

3. Results and Discussion 3.1 Simulated Spectra The simulated spectra are taken from our previous work,16 and use a distributed polarizability model for benzene in which the atomic polarizabilities within a molecule do not interact with one another.18, 23 Figure 1 shows the simulated RSD for benzene at 293 K. The RSD was fit using the sum of a BL function and an AG function. In a free fit the value of AAG

Figure 1 Fits of BL and AG functions to the simulated RSD for benzene (black line) at 293 K, including the free fit (red dotted line, AAG = 4.59) and constrained fits for AAG = 1 (green dashed line) and AAG = 0.8 (blue dotted-dashed line).

was 4.59, which is much larger than the peak height of the spectrum. However, equally good fits were obtained by constraining AAG to any value between 0.8 and 4.59 (Table 1). The free fit and fits with AAG = 1.0 and 0.8 are shown in Figure 1, and are nearly indistinguishable from one another. Thus, constraining the amplitude of the AG function does not substantially change the shape or the amplitude of the overall fit.


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Inspection of the individual BL and AG functions comprising the

Table 1 Fit parameters for simulated benzene spectrum and several values of AAG. The uncertainties in ABL, δ, and ωBL are approximately 5%. The uncertainties in ωAG and σAG are approximately 2%.

total fit can point us toward a

ABL

δ

ωBL (cm-1) AAG ωAG (cm-1) σAG (cm-1)

0.21 0.78

16.2

0.80

48.7

48.7

0.19 0.84

14.1

1.00

38.4

51.2

0.19 0.87

13.4

1.50

24.6

54.0

0.19 0.88

13.3

2.00

18.2

54.8

subsequent analysis. Comparisons

0.19 0.88

13.3

3.00

12.0

55.4

of the BL and AG portions of

0.19 0.88

13.3

4.59

7.79

55.6

sensible protocol for fitting the OKE RSD and for obtaining realistic parameter values for

empirical fits to simulated OKE data have been performed previously by Kiyohara et al.24 for CS2 and by Ryu and Stratt17 for benzene. In both cases it was concluded that there was not a strong correspondence between the empirical fit components and the actual contributions to the spectrum. Indeed, a fit to two positive functions cannot be expected to capture the generally negative cross term between the II and CM components of the RSD. However, if we view the AG function as being composed of a positive Gaussian and a negative Gaussian, we have a basis for making comparisons to all three contributions to the RSD. Figure 2 shows comparisons between the BL and AG functions from the free and constrained fits in Figure 1 and the individual polarizability components of the simulated RSD for benzene. In Figure 2A, the BL function is compared with the II component of the RSD. Even though the BL function and the II component of the spectrum have similar shapes, the amplitude of the BL function for all of the fits is considerably greater than the amplitude of II component of the spectrum. The BL function is relatively insensitive to the value of AAG, indicating that the BL fitting parameters are well-determined.


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Figure 2 Comparisons of (A) the interaction-induced component of the RSD with the BL function from the fits, (B) the collective molecular component of the RSD with the positive component of the AG function from the fits, (C) the cross term between the interaction-induced and collective molecular components of the RSD with the negative component of the AG function for the fits, and (D) the sum of the cross term and the collective molecular components of the RSD with the total AG function from the fits.

Figure 2B shows a comparison of the positive component of the AG function from the fits with the CM component of the RSD. The free-fit amplitude of the AG function is much larger than the amplitude of the CM spectrum, indicating that the amplitude of the AG function can become unrealistic when the fit is not constrained in a physically realistic manner. When the amplitude of the AG function is constrained in such a manner, the positive component of this function mimics the CM spectrum more closely, especially at high frequencies.


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The cross term between II scattering and CM scattering is negative, and it is natural to compare the negative component of the AG function with this cross term (Figure 2C). It is again clear from this comparison that the free fit gives an amplitude that is much too large. With a physically realistic constraint of the amplitude of the AG function, the negative component of the AG function again becomes comparable to the CMII spectrum. The CMII spectrum exhibits a peak with negative amplitude at ~22 cm-1. This peak plays a significant role in the shape of the total spectrum by flattening its low-frequency side. The fit parameters to the simulated spectrum for several values of AAG are given in Table 1. These data show that as AAG is constrained to smaller values, 𝜔𝜔𝐴𝐴𝐴𝐴 shifts to higher frequency and 𝜎𝜎𝐴𝐴𝐴𝐴 decreases. The data also demonstrate that 𝜔𝜔𝐴𝐴𝐴𝐴 shifts over a much wider range of values than

does 𝜎𝜎𝐴𝐴𝐴𝐴 for the same range of AAG values.

Figure 2D shows the sum of the CM and CMII spectra, along with the total AG function. Even

though the amplitude of the AG function is constrained to different values for the three fits, the shape and position of the total function are largely unaffected. This result demonstrates that the total function can be essentially constant even when the fit parameter values vary substantially. The comparison of the calculated spectrum with the AG function shows that the AG function accurately captures the high-frequency side of the spectrum. The inability of the AG function to reproduce the low-frequency side of this spectrum is the reason that the BL function overestimates the II scattering amplitude in the total RSD. It is clear that the BL function, the positive Gaussian in the AG function, and the negative Gaussian in the AG function do bear a qualitative resemblance to the II, CM and cross term contributions, respectively, in a typical RSD. This resemblance may account for how well empirical fits to these two functions generally work. The actual amplitude of the AG function is


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determined both by AAG and by the difference between the Gaussian functions comprising the total AG function. These competing effects lead to fit parameters that, in the absence of a constraint, are not unique. In cases such as this one, in which the individual components of the simulated RSD have been calculated, a sensible constraint is to set AAG such that the positive Gaussian fits well to the tail of the CM contribution and the negative Gaussian fits well to the tail of the cross term component. However, we do not have the luxury of knowing these components for experimental data, so it is necessary to devise another means of constraining AAG in a physically realistic manner. The best constrained values of AAG, 0.8 and 1.0, are near the peak height of the AG portions of the three different fits in Figure 2D (we note that values of AAG that are smaller than the peak height will lead to lower AG functions, and hence to poorer fits). To test whether these observations can lead us to a reasonable procedure for determining a constraint, we next turn to experimental data.

3.2 Experimental Spectra Figure 3 shows the experimental RSD7 for TFB at 294 K and the individual components of free and constrained fits to the sum of the BL and AG functions. Figure 3A shows the total fits to the spectrum. The total fit is nearly unchanged for the three different values of AAG shown, which range from 0.6 to 4.42 (the value from a free fit). Figure 3B shows the RSD along with the BL functions corresponding to the three fits. Again, the BL function is relatively insensitive to the amplitude of the AG function. Figure 3C shows the RSD and the positive and negative components of the AG function. When the fit is unconstrained, the amplitude of these components is unrealistically large. The center frequency of the Gaussian function in the free fit is also


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Figure 3 Comparison of (A) the total fits, (B) the BL functions, (C) the positive and negative components of the AG function, and (D) the total AG function with the experimental spectrum for 1,3,5-trifluorobenzene (black line) at 294 K. The free fit functions are red dotted lines. The constrained fit functions are represented by green dashed lines and blue dashed-dotted lines for AAG = 1 and AAG = 0.6, respectively.

unrealistically low (2.70 cm-1), given that this function is related to librational dynamics, which dominate the high-frequency side of the RSD in van der Waals liquids. When AAG is constrained to a smaller value, the Gaussian center frequency shifts toward the actual position of the AG peak. Figure 3D shows the RSD and the total AG function. The shapes of the total AG functions are nearly independent of the value of AAG. The height of the total AG function is approximately 0.6, which is why 0.6 was chosen as the lowest constraint value for AAG. The difference between the


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center frequency of the Gaussian functions comprising the total AG function and the peak position of the total AG function decreases as AAG is made smaller. We conclude that the fit parameters give a more realistic description of the total AG function when the fits are constrained. Constraining AAG can potentially give a different physical picture of the evolution of the RSD of a liquid with variables such as temperature and pressure. Figure 4 shows the RSDs for 1,3,5-trifluorobenzene over a wide range of temperatures.7 The high-frequency side of the RSD shifts to higher frequency with decreasing temperature. This shift is due to the increase in librational frequencies

Figure 4 Experimental RSDs trifluorobenzene from 272 K to 344 K.6

of

1,3,5-

within the liquid as the molecular potential stiffens with increasing density.16 In our previous work, unconstrained fits to these data showed a trend in which 𝜔𝜔𝐴𝐴𝐴𝐴 shifted to lower frequency with decreasing temperature, which is the opposite of the apparent behavior.7

Figure 5 shows the temperature dependence of the parameters from fits to these RSDs for three values of AAG. The trend shown in the shift of 𝜔𝜔𝐴𝐴𝐴𝐴 is reversed when AAG is constrained to a small

enough value. Additionally, the evolution of 𝜎𝜎𝐴𝐴𝐴𝐴 with temperature is reversed, although 𝜎𝜎𝐴𝐴𝐴𝐴 is affected to a lesser extent than is ωAG. As the temperature of the liquid increases, the molecules

can access the more anharmonic portion of the molecular potential, resulting in a greater range of librational frequencies in the liquid. Therefore, the width of the function associated with libration


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Figure 5 Dependence of (A) 𝝎𝝎𝑨𝑨𝑨𝑨 and (B) 𝝈𝝈𝑨𝑨𝑨𝑨 on temperature for 1,3,5-trifluorobenzene. The parameters of the free fits are represented by black circles. Constrained fits for AAG = 1 and AAG = 0.6 are represented by red circles and green triangles, respectively. The black lines are linear regressions of the data.

should increase. Thus, the expected behavior for these parameters is displayed by the constrained fits. Furthermore, the correlation of 𝜔𝜔𝐴𝐴𝐴𝐴 with temperature increases when the amplitude of the AG function is constrained to a physically realistic value.

We performed the same type of analysis for temperature-dependent hexafluorobenzene RSDs. Figure 6 shows the comparison between the RSD for HFB7 at 293 K and the BL and AG functions for three values of AAG ranging from 0.6 to 2.55 (the value from a free fit). Figure 6A shows the spectrum and the three fits, all of which are nearly identical. Figure 6B shows that the shape of the BL function is nearly unaffected by constraining AAG. Figure 6C shows the positive and negative components of the AG function corresponding to the fits in Figure 6A. Once again, the free fit amplitude of the AG function is unrealistically large. Figure 6D shows that for the three different values of AAG, the total AG functions are nearly identical, and have a peak height of approximately 0.6. These results again demonstrate that although the values extracted from the fits to the RSD using the BL and AG functions are not mathematically unique, the resultant forms of


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Figure 6 Comparison of (A) the total fits, (B) the BL functions, (C) the positive and negative components of the AG function, and (D) the total AG function with the experimental spectrum for hexafluorobenzene (black line) at 293 K. The free fit functions are red dotted lines. The constrained fit functions are represented by green dashed lines and blue dash-dot lines for AAG = 0.8 and AAG = 0.6, respectively.

the functions are nearly identical, so that the height of the AG function can be used to determine the constrained value of AAG. Figure 7 shows the RSDs for HFB over a wide range of temperatures.7 The high-frequency edge of the spectrum shifts to higher frequency with decreasing temperature, as was the case for TFB. However, we previously found that 𝜔𝜔𝐴𝐴𝐴𝐴 extracted from free fits was nearly constant with

respect to temperature.7 Furthermore, 𝜎𝜎𝐴𝐴𝐴𝐴 decreased with increasing temperature, which is the opposite of the expected behavior.


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Figure 8 shows plots of 𝜔𝜔𝐴𝐴𝐴𝐴 and 𝜎𝜎𝐴𝐴𝐴𝐴 as a

function of temperature for HFB for three different values of AAG. Figure 8A shows that when AAG is constrained to a value near the actual height (AAG = 0.6; see Figure 6D) of the total AG function, the correlation of 𝜔𝜔𝐴𝐴𝐴𝐴 with temperature increases and

the shift to lower frequency with increasing temperature becomes more pronounced. This

Figure 7 Experimental RSDs hexafluorobenzene from 267 K to 349 K.6

for

behavior is the result of decreasing the difference between the Gaussian center frequency and the actual peak position of the AG function. Figure 8B shows a plot of 𝜎𝜎𝐴𝐴𝐴𝐴 as a function of temperature for three values of AAG. The width parameter is

less affected by constraining the amplitude of the AG function than is the center frequency. However, the behavior of the parameter with respect to temperature reverses as AAG is constrained

to smaller values. Again, the behavior of this parameter conforms to expectation when AAG is constrained to a reasonable value.

Figure 8 Dependence of (A) 𝝎𝝎𝑨𝑨𝑨𝑨 and (B) 𝝈𝝈𝑨𝑨𝑨𝑨 on temperature for hexafluorobenzene. The parameters of the free fits are represented by black circles. Constrained fits for AAG = 0.8 and AAG = 0.6 are represented by red circles and green triangles, respectively. The black lines are linear regressions of the data.


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The implications of the constraint of the amplitude of the AG function can be significant. In our previous analysis of the experimental spectra presented here, we proposed a model based on the local structure of the liquids to account for the different shapes of the OKE RSDs for benzene, HFB, and TFB.7 The motivation for this model was the disparate behavior of the parameters corresponding to the AG function for benzene versus HFB and TFB. Given that the three molecules have the same disk-like shape and nearly the same hydrodynamic volume, it was natural to assume that the differing local structures of each liquid may account for the difference in the extracted fit parameters for the temperature dependent spectra. By constraining the amplitude of the AG function, the fit parameters more accurately represent the apparent peak position and width. The temperature evolution of the extracted fit parameters is consistent across the three liquids when AAG is constrained to a value near the height of the total AG function, counter to the model we proposed previously. Our results suggest that in using BL and AG functions to fit OKE RSDs, AAG should be constrained appropriately if one wishes to make any sort of subsequent analysis based on the fit parameters. An initial fit should be performed to determine the actual height of the AG function, and a subsequent fit should be performed in which AAG is constrained to a value near this height. This process ensures that 𝜔𝜔𝐴𝐴𝐴𝐴 is close to the frequency at which the AG function reaches its

maximum. Correspondingly, by constraining AAG, 𝜎𝜎𝐴𝐴𝐴𝐴 becomes more representative of the actual

width of the AG function.

4. Conclusions Empirical fits are a broadly used method of extracting information from OKE RSDs. Our results indicate that the AG function that is commonly used for such fits does not give well-defined


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parameters in the absence of any constraints. In particular, there is a strong interplay between AAG and ωAG, so that it is possible to adjust the latter parameter to obtain virtually identical AG functions over a wide range of AAG values. As a result, fits that accurately reproduce the shape of an RSD may give AG parameters that do not have physical meaning, such as values of ωAG that are far from the peak of the apparent librational contribution to the spectrum. Thus, the AG parameters obtained from unconstrained fits cannot be used for meaningful comparisons of RSDs of different liquids, or of the same liquid under different conditions. Decomposing simulated RSDs into their II, CM, and cross-term components gave important clues as to the appropriate constraints to use in fitting with AG functions. With the insight that, in the context of comparison to the components of the RSD, the AG function is best treated as a positive Gaussian and a negative Gaussian, it becomes clear that AAG should be adjusted such that the positive Gaussian should resemble the CM component of the RSD and the negative Gaussian should resemble the cross-term component. This strategy is not feasible for experimental RSDs, but leads to the idea that the value of AAG should be governed by the peak height of the AG portion of the fit. To test this idea, we refit previously reported, temperature-dependent RSDs for TFB and HFB. The fitting parameters reported originally for these liquids did not match with the physical picture of how temperature should affect librational frequencies. Constrained fits, on the other hand, do match the expectation that ωAG should increase and σAG should decrease with decreasing temperature. Furthermore, the ωAG values obtained from the constrained fits make physical sense based on the appearance of the spectra. We believe that this method of constraining AG fits should be widely applicable to van der Waals liquids, and will enable more meaningful comparisons among RSDs. Hydrogen-bonded


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liquids and ionic liquids tend to feature multiple librational bands, and so the validity of this approach will need to be tested for RSDs of these materials. However, when ωAG is large compared to σAG, the positive-frequency portion of the AG function is essentially an isolated Gaussian. In this case the fit parameters should be well-defined, and so a constraint may only be required for the fitting of the librational band that has the lowest frequency.

AUTHOR INFORMATION Corresponding authors * E-mail: [email protected], [email protected] * Phone: John T. Fourkas Benoit Coasne

(301) 405-7996 +33 04 76 51 47 62

Notes These authors declare no competing financial interest.

ACKNOWLEDGMENTS This work was supported by the National Science Foundation, grant CHE-1362215. J.S.B. was supported by a National Science Foundation Graduate Research Fellowship under grant DGE0750616.


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On the Empirical Analysis of Optical Kerr Effect Spectra: A Case for

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