Infrared intensities of liquids. Part XXIII. Infrared optical constants and integrated intensities of liquid benzene-d 1 at 25 C

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1 Journal of Molecular Structure (2000) Infrared intensities of liquids. Part XXIII. Infrared optical constants and integrated intensities of liquid benzene-d 1 at 25 C J.E. Bertie*, Y. Apelblat, C.D. Keefe 1 Department of Chemistry, University of Alberta, Edmonton, Alberta, Canada, T6G 2G2 Received 28 July 1; accepted 22 September 1 Abstract This paper presents the first absolute infrared absorption intensities of liquid benzene-d 1,C 6 H 5 D, at 25 C. It also presents what, surprisingly, seems to be the first complete assignment of an infrared spectrum of liquid benzene-d 1 recorded with post- 150 instrumentation. The spectra of the real and imaginary refractive indices are given as graphs and tables between 6200 and 500 cm 1, and a table is given of the peak wavenumbers, absolute integrated intensities and vibrational assignments between 4800 and 500 cm 1. Errors in the imaginary refractive index, k, values are estimated to be 5 7% for peaks and 5 20% for the baseline between 6200 and 4700 cm 1, % for both peaks and baseline between 4700 and 825 cm 1, 3 4% for both peaks and baseline between 825 and 620 cm 1, and 40 50% between 620 and 500 cm 1 where the very strong peak near 600 cm 1 was too intense for us to measure accurately. Errors in the real refractive indices, n, are estimated to be 0.25% at 8000 cm 1 increasing to 0.5% near 800 cm 1 and, due to the uncertain intensity of the peak near 600 cm 1, to range up to 10% between 710 and 500 cm 1. The refractive index spectra were converted to spectra of the real and imaginary dielectric constants, e 0 and e 00, the molar absorption coefficient, E m, and the real and imaginary molar polarizabilities under the Lorentz local field, a 0 m and a 00 m. The peak heights and wavenumbers in the spectra of the different absorption quantities are compared for the most intense bands. Integrated intensities were determined as C j, the area under bands in the ~na 00 m spectrum, for all bands between 4800 and 500 cm 1. The contributions from the different bands were separated by fitting the spectrum with classical damped harmonic oscillator bands. The estimated errors in the integrated intensities range from 2 to 10% for most bands, although they may reach 100% for very weak bands and shoulders. The integrated intensities of the fundamentals and the corresponding transition dipole moments are summarized and compared with literature values for the gas. Crawford s F-sum rule shows that the measured integrated intensities of C 6 H 5 D are nicely consistent with those reported recently for C 6 H 6 and C 6 D 6. The total integrated intensity of the first overtone of the CH stretches is 20 times smaller than that of the fundamentals Elsevier Science B.V. All rights reserved. Keywords: Benzene-d 1 ; Infrared intensities; Optical constants; Refractive indices; Dielectric constants Dedicated to Professor James R. Durig on the occasion of his 65th birthday. * Corresponding author. Tel.: ; fax: address: john.bertie@ualberta.ca (J.E. Bertie). 1 Current address: Department of Physical and Applied Sciences, University College of Cape Breton, Sydney, Nova Scotia, Canada B1P 6L /00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S (00)

2 136 J.E. Bertie et al. / Journal of Molecular Structure (2000) Introduction This paper continues our report of absolute infrared absorption intensities of liquids measured by transmission spectroscopy, by presenting quantitative infrared intensities of liquid benzene-d 1,C 6 H 5 D, at 25 C. Similar measurements of the intensities of liquid benzene [1], toluene [2], chlorobenzene [3], dichloromethane [4] and benzene-d 6 [5] have been published recently and the first four of these have formed the basis of intensity standards [6]. This paper also presents the assignment of most of the features in the infrared spectrum of liquid C 6 H 5 D. A number of studies of the vibrations of C 6 H 5 D have been reported in the literature. In 146 Bailey et al. [7] reported and fully assigned the complete infrared spectrum of the gas between 3400 and 380 cm 1, as well as the Raman spectrum, presumably of the liquid, over the same range. Since then, the majority of the reports have presented only complete [8 13] or partial [14,15] assignments of the 30 fundamental vibrations, and only three reports [10,14,16] have contained assignments of some combination and overtone bands. In particular, the complete assignment of spectra obtained with post-150 instrumentation has not been reported previously. There are no previous reports of the quantitative infrared intensities of liquid C 6 H 5 D. Three groups [17 1] have reported quantitative infrared intensities of gaseous benzene-d 1. In the present work experimental absorbance spectra [1] of liquid benzene-d 1 at 25 C were measured. From these spectra, spectra of the optical constants, i.e. of the real, n, and imaginary, k, refractive indices, were calculated by methods described previously [1,20]. The refractive index spectra were converted to spectra of the real and imaginary dielectric constants, the molar absorption coefficient and, under the Lorentz local field, the molar polarizability, as described elsewhere [21]. To the extent that the Lorentz local field is valid, molecular properties and behavior are more directly reflected in the spectrum of the imaginary molar polarizability, am, 00 than in the spectra of the imaginary refractive index, the imaginary dielectric constant, or the molar absorption coefficient [21]. Thus, the imaginary molar polarizability spectrum is the absorption quantity of greatest interest in the study of liquid-phase molecules. Correspondingly, the integrated intensity of band j, C j, is defined as the area under the band in the spectrum of ~na 00 m [21] Z C j ˆ ~na 00 m ~n d ~n 1 Under the assumption that all of the hot transitions of the fundamental contribute to the fundamental band, C j is related to the dipole transition moment, R j, through Eq. (2) [21 24] C j ˆ NAp 3hc 0 g j ~n j R j 2 2 Under the assumptions of mechanical and electrical harmonicity, C j can also be related to the square of the dipole moment derivative with respect to the jth normal coordinate, m 2 j ˆ 2m=2Q j 2 ; through [21] C j ˆ NA 24pc 2 g j m 2 j 0 3 In these equations, N A is the Avogadro constant, h is Planck s constant, c 0 is the velocity of light in vacuum, and g j is the degeneracy of the jth vibration. For C 6 H 5 D, g j ˆ 1 for all vibrations. In order to calculate the integrated intensities, the am 00 spectrum must be separated into contributions from the different bands. This is not trivial when the spectrum contains adjacent or overlapping bands. However, Eqs. (2) and (3) result from both quantum theory and the classical damped harmonic oscillator (CDHO) model [21 24], so the separation can be attempted by fitting the am 00 spectrum with CDHO bands. When this is successful, as for benzene-d 6 [5], methanol [25] and benzene-d 1, the integrated intensity C j may be obtained directly from the parameters of the CDHO band without numerical integration, as is described elsewhere [5,25]. 2. Experimental Benzene-d 1, labeled 8%, was purchased from

3 J.E. Bertie et al. / Journal of Molecular Structure (2000) Cambridge Isotope Laboratory and was used as is except that it was kept over molecular sieve to ensure dryness. The samples were checked by gas chromatography and by infrared spectroscopy of the liquid. Non-benzene impurities were determined to be less than 0.06%. The samples were also analyzed by mass spectrometry and were found to contain C 6 H 6, 13 CC 5 H 5 D and 0.45% C 6 D 6. The intensity of the n 4 band of C 6 H 6 at 67 cm 1 in the infrared 2 spectrum of the liquid benzene-d 1 indicated 3.5% of C 6 H 6. We were unable to distinguish between 12 C 6 H 5 D and 13 CC 5 H 5 D in the infrared spectrum of liquid benzene-d 1, although its natural abundance suggests the presence of about 6% of 13 CC 5 H 5 D. The experimental and instrumental details of this work have been described [1,20] and are summarized briefly here. The infrared spectra were measured with a Bruker IFS 113 V spectrometer. A Globar source, a 10 mm aperture, and deuterated triglycine sulfate (DTGS) detector were used. The interferograms were recorded with cm s 1 optical retardation velocity and 1 cm 1 nominal resolution. Trapezoidal apodization, multiplicative phase correction, and one level of zero-filling were used in the Fourier transform. Experimental absorbance spectra of liquid benzene-d 1 were measured in fixed path length cells with KBr windows and path lengths between 11 and 1500 mm. To determine the linear absorption coefficients at the anchor points [1,20], spectra were also measured in KBr cells with fixed path lengths of 500 and 1500 mm, and in a NaCl cell with variable path lengths up to 3.7 mm. The path lengths of the cells were determined as described previously [1 5]. To assist in the assignment of the bands in the spectrum of the liquid, an infrared spectrum of gaseous benzene-d 1 was recorded on the Bruker IFS 113 V in order to observed the band contours, and Raman spectra were measured to observe the wavenumber shifts and polarizations. Accurate Raman wavenumber shifts were obtained from an unpolarized spectrum recorded at 2 cm 1 nominal resolution on a Bruker FT-Raman spectrometer with Nd : YAG 2 The n 4 band is at 673 cm 1 in the absorbance spectrum of neat C 6 H 6 (l) but shifts to 67 cm 1 on dilution. excitation. In the software, the HeNe wavenumber was set to its vacuum value, cm 1, and that of Nd:YAG was set to 34.2 cm 1. Measurement of both Stokes and anti-stokes Raman shifts of four bands of chlorobenzene and one of dichloromethane showed that the Stokes wavenumber shifts are accurate to ^0.1 cm 1. Parallel- and perpendicular- polarized Raman spectra were recorded with 0 excitation on a dispersive SPEX spectrometer with laser excitation at nm, 380 mw power, slit width of 2 cm 1 and step size 0.5 cm 1. The intense parallel-polarized bands allowed wavenumber calibration by comparison with the accurate though unpolarized FT spectra. 3. Infrared intensities 3.1. Imaginary refractive index spectrum The k spectrum was determined by the procedures described in previous papers [1 5], and the presentation in this section follows the logic described in those papers. The values of the linear absorption coefficient, K, required to correct the baseline at the anchor points [1,20] are given in Table 1 with the cell pathlengths used, the corresponding k values, the 5% confidence limits of the values of K and k and the approximate value of the real refractive index, n, used at each anchor point. The baseline absorption is extremely weak above 4500 cm -1 and could be determined to only t10% precision with the available path lengths up to 3.7 mm. The approximate n values were determined from a preliminary calculation of n and k by program RNJ46A from an uncorrected spectrum recorded through an 11 mm cell. The experimental absorbance, EA, spectra from cells of many thicknesses were converted to imaginary refractive index spectra by program RNJ46A, using the anchor point information. The k spectra were only used in regions where the corresponding EA peak maxima were between 0.2 and 2.0 absorbance units. These regions, the path lengths used, and the number of spectra averaged to give the average k spectrum for the region, are given in Table 2. The average k spectra for the different regions were merged to give the k spectrum from 6200 to 620 cm 1.

4 138 J.E. Bertie et al. / Journal of Molecular Structure (2000) Table 1 (Decadic) linear absorption coefficients for benzene-d 1 at 25 C ~n (cm 1 ) Path length (mm) n ~n a K ~n (cm 1 ) b k ~n c ( 5) d ( 15) (43) (13) (20) (6) (14) (47) (12) (43) (11) (41) (25) (10) (10) (4) (30) (13) (13) (6) (5) (2) (4) (2) (4) (2) (6) (3) (51) (32) () (6) (3) (2) (3) (2) (3) (2) (27) (22) (8) (7) (3) (3) (3) (3) (28) (27) () () (24) (25) (17) (18) (20) (22) (16) (1) (34) (4) (18) (25) (15) (21) (24) (4) (8) (1) (11) (20) (5) (10) (83) (17) () (20) (7) (17) (16) (44) (37) (13) a This column gives the approximate value of n used to calculate the reflection from the cell windows during the calculation of K from experimental absorbance spectra. b K values with their 5% confidence limit in the last digit (given in parentheses), are given to the precision used in further computations. c k values and Dk, their 5% confidence limits in the last digit (given in parentheses) were calculated from k ˆ 2:303K= 4p ~n : d This value was set to zero because the experimental absorbance was less than 0.02 when measured through 3.7 mm path, the longest cell used. The 5% confidence limit was estimated from the maximum possible error in the absorbance (0.02) in these cells.

5 J.E. Bertie et al. / Journal of Molecular Structure (2000) Table 2 Pathlengths and number of spectra recorded for the regions processed Region (cm 1 ) Pathlength (mm) Number of spectra The region of the most intense band, cm 1 was not well determined, because in our thinnest cell, 11 mm, the peak EA was 2 and varied significantly between spectra. Initially the average of the four k spectra between 620 and 500 cm 1 that were calculated from these EA spectra was merged with the k spectrum for the higher-wavenumber region. The following steps were taken in an attempt to improve the reliability of the k spectrum below 620 cm 1. The n spectrum was calculated from the k spectrum, as described later, and was used with the k spectrum to calculate the spectrum of the imaginary molar polarizability, am 00 [5,21,24 26]. Three CDHO bands were then fitted to the wings of the am 00 band at 600 cm 1, two for the strong peak near 610 cm 1 and one for the weak shoulder near 625 cm 1. The need for two bands to fit the strong peak is obvious in the original spectra. The fit was very good. The sum of the CDHO bands was then converted back to a k spectrum and merged with the k spectrum for the higher wavenumber region to give the final k spectrum of liquid benzene-d 1 from 6200 to 500 cm 1. The Fig. 1. The 605 cm 1 band of liquid C 6 H 5 Dat25 C. Top: The original average imaginary refractive index, k, band and the final k band obtained as described in the text. Bottom: The bands in the dielectric loss, e 00, and imaginary molar polarizability, a 00 m, spectra. The e 00 band has been multiplied by 3 for clarity. The units of a 00 m are cm 3 mol 1. improved k spectrum near 600 cm 1 is compared with the average of the original four k spectra in the top box of Fig. 1. The k spectrum of C 6 H 5 Dat25 Cis shown in Fig. 2, and is tabulated in Table 3 in the Compact Table format [27] Precision and accuracy of k The accuracy of a k value is described by the absolute error in the value. This error cannot be known exactly, so can only be estimated. Here the error is estimated [1 5] as the sum of the precision of the k value, expressed by its 5% confidence limit (5% CL), and the systematic error that arises from fixing the baseline at the anchor points. This systematic error

6 140 J.E. Bertie et al. / Journal of Molecular Structure (2000) reproducibility. Between 620 and 500 cm 1 the error is 40%, which is the 5% CL of the peak value; as discussed above, this large error results from our inability to measure precisely the very intense band near 600 cm Real refractive index spectrum Fig. 2. Imaginary refractive index spectrum, k ~n ; of liquid C 6 H 5 D at 25 C. Top box, cm 1 ; middle box cm 1 ; bottom box, cm 1. Divide the ordinate scale labels in the top, middle and bottom boxes by 10, 20 and 60, respectively, for the upper curve in the box. is taken to be the average of the 5% CLs of the anchor points (Table 1) at the two ends of the region. Previous studies have shown [1 4] that this method of estimating the accuracy of intensities measured by the Bruker FT spectrometer in this laboratory is supported by measurements on other instruments in other laboratories. The errors in the cm 1 region, where the absorption is very weak, come mainly from the anchor points. Thus, the errors in the peak and baseline k values are 5 7% and 5 20%, respectively. Below 4700 cm 1 the absorption is much stronger and the errors from the two sources are about equal. The errors in k are % between 4700 and 825 cm 1, for both the peaks and the baseline absorption. Between 825 and 620 cm 1, the errors are larger, 3 4%, mainly due to unusually poor The n spectrum of C 6 H 5 D was calculated by Kramers Kronig (KK) transformation of the final k spectrum with the assumption that k is zero between 6200 and 8000 cm 1. The KK transform requires a value of n, that was taken to be the refractive index at 8000 cm 1 that is due only to electronic polarization [28], n el (8000 cm 1 ). The value was used, and was obtained in the following way. Fits of the literature values of n of C 6 H 6 at seven different visible wavelengths and 25 C yielded [1,28] n el ˆ 1:4804 at 8000 cm 1. For C 6 H 5 D the visible-wavelength-dependence of n is not known, and the only reported values are for the Na D line [2 31], where n ˆ 1:5006 [30] at 20 C and [31] at 25 C. 3 These values are smaller than the values at the Na D line for C 6 H 6, n ˆ 1:5010 [2,32] at 20 C and 1.47 [32] at 25 C. The difference between n of C 6 H 5 D and n of C 6 H 6 was assumed to be the same at 8000 cm 1 as at the Na D line, so n el (8000 cm 1 ) of C 6 H 5 Dat25 C was taken to be The accuracy of the n values can be estimated as follows. The value of n at wavenumber ~n i was calculated by adding the value Dn ~n i that is calculated by KK transform of the k spectrum to the value n el 8000 cm 1 ˆ1:4800: Thus, the uncertainty in n was estimated as the sum of the 0.05% uncertainty inherent in our KK transform [33], the 0.03% uncertainty in n el (8000 cm 1 ) plus the % uncertainty in Dn ~n i that results from the uncertainty in the k values. Thus, above 710 cm 1 the uncertainty in the n values is ^ 0.25%. Because the strong absorption near 610 cm 1 was poorly measured, the n values are uncertain by 1% at 640 cm 1 increasing to 10% near the 610 cm 1 peak and decreasing to 1% at 575 cm 1 and 0.5% at 500 cm 1. 3 Ref. [31] actually gives for C 6 H 5 Dat20 C, but it also gives for C 6 H 6 compared with the accepted value , so we have reduced their value for C 6 H 5 D by Note also that Aldrich [2] give n ˆ 1:480 for C 6 H 5 Dat20 C, which is assumed to mean at 25 C.

7 J.E. Bertie et al. / Journal of Molecular Structure (2000) Table 3 Imaginary refractive indices, k, between 6200 and 500 cm 1 of liquid benzene-d 1 at 25 C a,b,c cm 1 XE YE

8 142 J.E. Bertie et al. / Journal of Molecular Structure (2000) Table 3 (continued) cm 1 XE YE

9 J.E. Bertie et al. / Journal of Molecular Structure (2000) Table 3 (continued) cm 1 XE YE a The column headed cm 1 contains the wavenumber of the first n or k value in the row. The column headed XE and YE contain the X- and the Y-exponent for the row, respectively. The columns headed 0,1,2, 6, contain the ordinate values, and the headings give the indices of the ordinate values in the row. In a row which starts with ~n 0 ; the wavenumber corresponding to the ordinate indexed J is ~n J ˆ ~n :002=16384 J2 XE : The k values in the row are the ordinate value times 10 YE. Thus the entries indexed 16 in the first row shows that: k ˆ and n ˆ 1:475 at ~n ˆ 6200: :002= ˆ 6138:31 cm 1 : b The k values in the table can be interpolated via the 4-point spline interpolation program Trecover [27] to the original wavenumber spacing, cm 1, to give the original k values with errors 1% below 4500 cm 1, 2% between 4500 and 5000 cm 1, and 5% above 5000 cm 1, except for isolated spikes. The n values can be recovered via Trecover with errors 0.1% in nearly all cases. c The n values are based on n el 8000 cm 1 ˆ1:4800. One other source of error must be addressed. The Dn ~n i values from the KK transform should be added to the values n el ~n i ; instead of being added to n el (8000 cm 1 ) [28]. For C 6 H 6 n el is at 8000 cm 1, at 4000 cm 1 and at 0cm 1, so an error that increases with decreasing wavenumber from 0% at 8000 cm 1 to 0.3% at 0cm 1 exists for C 6 H 6 when n el (8000 cm 1 ) is used [28]. The correct numbers are not known for C 6 H 5 D, but the n values probably also contain errors of this magnitude. Thus, the total uncertainty in the n values is 0.25% near 8000 cm 1 increasing to 0.5% near 800 cm 1, below which the uncertainty in the 610 cm 1 peak causes the larger errors discussed above. The final n spectrum of C 6 H 5 Dat25 C is shown in Fig. 3 and is tabulated in the Compact Table format [27] in Table 4. The final k and n spectra are available in digital form through J.E.B. s web site The programs Comptab and Trecover for creating Compact Tables and recovering spectra from them [27], are also available through this site.

10 Spectra of other intensity quantities J.E. Bertie et al. / Journal of Molecular Structure (2000) Fig. 3. Real refractive index spectrum, n ~n ; of liquid C 6 H 5 Dat 25 C. Top box, cm 1 and n ˆ 1:478 to 1.480; middle box, cm 1 and n ˆ 1:458 to 1.45; bottom box, cm 1 and n ˆ 1:10 to The top spectrum in the bottom box was offset by multiplying the original spectrum (bottom) by 5 and subtracting 5.6. again by program Dequant. The am 00 spectrum is shown in Fig. 5. The molar concentration and molar volume equaled mol l 1 and 8.21 cm 3 mol 1 in these calculations, calculated from the density [35] g cm 3 at 25 and molecular weight g mol 1. For the 27 most intense bands in the k spectrum, Table 5 lists the peak wavenumber and full-width-athalf-height in the k spectrum and the peak heights in the spectra of these different intensity quantities. The peak wavenumbers and shapes are different in the spectra of these different intensity quantities for very strong absorption bands. For C 6 H 5 D, the difference between the peak wavenumbers exceeds 0.1 cm 1 for only the three strongest bands, those at 77, 6 and 606 cm 1. For these three bands Table 5 shows the peak wavenumbers in the e 00, am 00 and ~na 00 m spectra in parentheses underneath the peak heights. The difference in peak shape is detectable only for the strongest band, that at 606 cm 1. Fig. 1 includes this band in the e 00 and am 00 spectra, with the e 00 values multiplied by 3 in order to give roughly the same peak height for both. The E m band essentially coincides with the k band when scaled to the same peak height, and the ~na 00 m band essentially coincides with the am 00 band. The (negligible) difference between the am 00 and ~na 00 m peak wavenumbers is due to the mechanical anharmonicity of the vibration and the differences between the k, e 00 and am 00 bands are due to the long-range dielectric effects in the interaction of the infrared light with the liquid [21]. The spectra of the molar absorption coefficient, 4 E m, and the real and imaginary dielectric constants, e 0 and e 00, were calculated from the spectra of n and k by program Dequant, which is available through J.E.B. s web site. The molar absorption coefficient spectrum is shown in Fig. 4. These intensity quantities are all macroscopic properties of the liquid. The spectrum of a microscopic molecular property, the imaginary molar polarizability [21,24,26], am, 00 was calculated under the approximation of the Lorentz local field, 4 The symbol E m is used instead of the recommended [34] e for the molar absorption coefficient, in order to avoid confusion with the dielectric constants. 4. Vibrational integrated intensities As was discussed previously [5] for C 6 D 6, in order to separate the contributions to the intensity from the different bands, the am 00 spectrum was fitted between 4800 and 500 cm 1 with CDHO bands. For C 6 H 5 D, 28 bands were fitted to the am 00 spectrum. Each of these fitted bands extended from 4800 to 500 cm 1, and we call their sum the fitted spectrum. The integrated intensity C j of each of these bands was determined analytically through C j ˆ S j p=2; where S j is a fitting parameter that equals the peak height of the corresponding ~na 00 m band multiplied by the full-widthat-half-height [5,21,25]. In the terminology of Ref. [21], S j is N A m 2 j = 12p 2 c 2 : The peak wavenumbers,

11 J.E. Bertie et al. / Journal of Molecular Structure (2000) Table 4 Real refractive indices, n, between 6200 and 500 cm 1 of liquid benzene-d 1 at 25 C a,b,c cm 1 XE

12 146 J.E. Bertie et al. / Journal of Molecular Structure (2000) Table 4 (continued) cm 1 XE

13 J.E. Bertie et al. / Journal of Molecular Structure (2000) Table 4 (continued) cm 1 XE a The column headed cm 1 contains the wavenumber of the first n or k value in the row. The column headed XE and YE contain the X- and the Y-exponent for the row, respectively. The columns headed 0,1,2, 16, contain the ordinate values, and the headings give the indices of the ordinate values in the row. In a row which starts with ~n 0 ; the wavenumber corresponding to the ordinate indexed J is ~n J ˆ ~n :002=16384 J2 XE : The n values are given directly with the decimal point implicitly after the first digit. Thus the entries indexed 16 in the first row of the tables show that: k ˆ and n ˆ 1:475 at ~n ˆ 6200: :002= ˆ 6138:31 cm 1 : b The k values in the table can be interpolated via the 4-point spline interpolation program Trecover [27] to the original wavenumber spacing, cm 1, to give the original k values with errors 1% below 4500 cm 1, 2% between 4500 and 5000 cm 1, and 5% above 5000 cm 1, except for isolated spikes. The n values can be recovered via TRECOVER with errors 0.1% in nearly all cases. c The n values are based on n el 8000 cm 1 ˆ1:4800. ~n j ; full-widths-at-half-height, G j, and C j of these fitted bands are listed in Table 6, together with the wavenumbers of features in the experimental am 00 spectrum and the Raman spectrum of the liquid. The quality of the fit is shown graphically in Fig. 5, which includes the fitted spectrum as well as the experimental am 00 spectrum, and in Fig. 6 which shows more detail in two regions, one of which includes the region of greatest misfit above 800 cm 1, near 1056 cm 1. Each curve in Fig. 5 and the upper curve in each box of Fig. 6 consist of both the experimental am 00 spectrum and the fitted spectrum, which essentially overlap even in the expanded views in Fig. 5. The lower curves in Fig. 6 show the individual bands required for the fit, truncated for clarity to extend only three full-widths-at-half-height from the band center. The quality of the fit can be described in several ways. Of first importance is that the presence of nearly all of the 28 bands is obvious in the experimental am 00 spectrum, either as peaks, shoulders, changes in slope, or asymmetric tails, as is illustrated in Fig. 6. Second,