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Vibrational relaxation and frequencies of liquid molecules. II. Comparison of theoretical and experimental results
Chemical Physics 133 (1989) North-Holland, Amsterdam
151-163
VIBRATIONAL RELAXATION AND FREQUENCIES OF LIQUID MOLECULES. II. COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS Takuya FUKUDA
‘, Shun-ichi
IKAWA ’ and Masao KIMURA
Department of Chemistry, Faculty of Science, Hokkaido University, Sapporo 060, Japan Received
17 July 1988
The Raman isotropic band widths and the peak frequencies of totally symmetric vibrational modes were measured for the CH,I, CD,& CH,Br, and CHCI, molecules in liquid phases at various temperatures. The integrated intensities of the Raman scattering and the infrared absorption of the fundamental and overtone bands were measured to obtain the intensity parameters required for calculating the band widths and the gas-to-liquid shifts of the frequency by the theory reported in part I. The agreement between the calculated and the observed values is reasonably good. The phase modulation was the dominant process both in the band broadening and in the frequency shift. It was pointed out, for the first time, that the cross term between the attractive and the repulsive parts of the mechanical collision potential played an important role in the band broadening. Without the cross term, the calculated band widths of the vcH and &,cH modes would be an order of magnitude larger than the observed widths.
1. Introduction
tal and theoretical points of view. However, it is still quite insufficient to understand what the ma.in interaction mechanisms are and what the dominant relaxation processes are. In view of this situation, we developed, in the previous paper [ 11, a band shape theory which was as comprehensive as possible dealing with the dipole-dipole, the dispersion, and the mechanical collision potential all together and allowing for the ED, PM, and REE processes, starting from the generalized Langevin equation [ 81. The isotropic band widths and the vibrational frequencies of liquid molecules were given as functions of the physical parameters of the solute and solvent molecules. The purpose of the present paper is, firstly, to measure the Raman isotropic band widths and peak frequencies of CHJ, CD31, CH-,Br, and CHC13 in the liquid phase at various temperatures, secondly, to estimate all the physical parameters required for the calculation of the band width and the frequency shift, and last, to compare the observed and calculated band widths and the frequency shifts. Thus, for the first time, it becomes possible to compare the contributions from the various potentials and processes to the Raman band shapes. This paper is concerned with the calculation for neat liquids at 293 K and the next paper will discuss the effect of solvents and temperature.
Raman scattering is a useful tool for studying the dynamics and interactions of molecules in the liquid phase. The isotropic and anisotropic band shapes tell how the vibrational and the reorientational relaxations proceed in the liquid phase, and the shift of the peak frequency indicates the average strength of the local field acting on the molecule [ l-71. This dynamical and statical information must, ultimately, be explained by or elucidate the real feature of the intermolecular potential. However, the potential field in the liquid phase is quite complicated involving various mechanisms such as dipole-dipole, dispersive, and repulsive interactions. The fluctuation of the potential caused by the thermal motion of molecules is related to the relaxation phenomena. For the vibrational relaxation, the following three processes have been proposed [ 5-71: ( 1) energy dissipation (ED), (2) phase modulation (PM), (3) resonant energy exchange (REE). Many studies have been reported on the Raman band shapes from both the experimenI Present address: Hitachi Hitachi, Ibaragi, Japan. ’ To whom correspondence
Research
Laboratory,
Hitachi
Ltd,
should be addressed.
0301-0104/89/$03.50 0 Elsevier Science Publishers ( North-Holland Physics Publishing Division )
B.V.
T. Fukuda et al. / Vibrational relaxation andfrequencies
152
2. Experimental procedures and results 2. I. Raman band shapes Raman scattering of liquids was excited by a HeNe 632.8 nm laser line from a NEC GSL-108 laser apparatus operated at 50 mW and the 90” scattering was measured by use of a JASCO R-300s spectrometer with a cooled RCA C31034A photomultiplier. The details of the optics have been described elsewhere [ 9 1. Scanning of the spectrometer and acquisition of the photon counting signal were performed under control by a PANAFACOM U-200 digital computer. As typical examples of the measurements, the C-I stretching Raman bands of a CH31-CD31 mixture are shown in fig. 1. The whole region was divided into several subregions, which were scanned at different conditions depending on the signal intensity and its gradient. The temperature of the liquid samples was controlled within + 1 K in the range from 190 to 3 11 K. The wavenumber calibration was performed with forty lines of the Ne natural emission. Its uncertainty was less than 0.4 cm-‘. The slit function measured with the Ne emission line was close in shape to a Gaussian function. Then the observed Raman bands were simulated by a computer as a convolution of Lorentzian and Gaussian functions, and the width of the former was taken as a true band width. When some bands overlapped, the parameters
ofliquid molecules. II
of the band concerned were obtained by decomposing the whole spectrum into some Voigt components. The example is shown in fig. 2. The background due to the Rayleigh scattering and, sometimes, due to the fluorescence of a small amount of impurity was excluded by drawing a linear base line. The isotropic and anisotropic spectra were obtained from the obscattering served parallel and perpendicular intensities: ziso(“>=r,j(W)-$L
zaniso(“>
(WI
=IJ.
toI
I
(1)
(2)
.
The half band widths corrected for the slit width effect, ri,, and raniso, give the widths owing to the vibrational and reorientational relaxations [ 3,4 ] : rvlb =cso 3 r,,, =
ranis0
(3) -
r,,,.
(4)
The observed values of ri, and peak frequency, wo, are listed in tables 1 and 2, and r,, and the depolarization ratio, p, for neat liquids are listed in table 3. For isotopically diluted solutions, observed band widths and frequencies changed linearly to the mole fraction, as shown by examples in fig. 3. Differences between values at mole fractions 1 and 0 are attributed to the broadening and the frequency shift due to the resonant energy exchange [ 10,111,
x 13175pp
1.0
0.5
::-:: 0
5:lo
_“-A:
560
‘540
‘,.,__/’
520
i
‘.
.... .“--.
500
460
480
‘440 w/m-f-’
x 1779pps
1.0 1
,r.., :
'.
1,
WAm-’
Fig. 1. The I,, and I, scattering control by a digital computer.
intensities in the vc_, region of an isotopic mixtures (CHJ/CD,I= l/4) at 262 K measured under the The maximum count rates were 13 175 and 1799 pps for the measurements of I,, and I,, respectively.
T. Fukuda et al. / Vibrational relaxation andfrequencies
620
Fig. 2. Computer simulation of I,,(w) of the vC_ar mode of liquid CH>Br at 293 K. (0 ) Observed, (-) calculated total, (---) calculated Voigt components corresponding to the CHs79Br and CH,s’Br species, and the hot band, respectively, from left to right.
rREE=r(m f=l)-riSOt(mf=O),
(5)
AL2REE=~O(mf=1)-W~(mf=0),
(6)
2.2. Raman scattering intensities The frequency dependence of sensitivity bf the spectrometer can be neglected in a band shape measurement, which was limited in the range of 100 cm-’ at most. However, this effect becomes considerable
where the values at mf = 0 are obtained by linear exTable 1 Observed
Raman
band widths and peak frequencies
(cm-
153
trapolation. As expected from eq. (36) of ref. [ 11, hereafter designated as eq. (I-36), rREE is always positive. A&,, values are negative for most liquids with an exception of the breathing mode of benzene [ 111, where the contribution of the charge transfer mechanism has been suggested. The residual band width, .isot( mf=O), and the residual frequency shift, obsof(mf=O)-o~, where o, is the gas-phase frequency, are assigned to the contributions of the phase modulation effect, r,, and AS&,,+respectively. The AL& and r,,, values obtained are listed in the last columns in tables 4 and 5.
560 cd
560
600
of liquid molecules. II
’ ) of totally symmetric
modes of CHJ
in the liquid phases
T(K)
Solvent
Cont. (mole fraction)
211
CD,1
1.0 0.5 0.2 0.1
2945.9( 2946.0( 2946.8( 2946.9(
1) 1) 1) 1)
2.3(l) 2.4(2) 2.4(2) 2.4(2)
1238.6( 1) 1239.2(2) 1239.8(l) 1240.2(l)
3.5(l) 2.6(2) 2.3(l) 1.9(l)
523.4( 1) 523.3(2) 523.5( 1) 523.8( 1)
3.0( 1) 2.7(2) 2.6(2) 2.2(l)
262
CD,1
1.0 0.5 0.2 0.1 0.1 0.1
2946.9( 2947.7( 2948.4( 2948.5( 2948.4( 2959.6(
1) 1) 1) 1) 1) 1)
2.3( 1) 2.3(l) 2.3(2) 2.3(l) 2.1(l) 3.6(2)
1239.8(2) 1240.0(2) 1240.6( 1) 1240.7(2) 1238.5(2) 1248.1(6)
2.6(l) 2.0( 1) 1.6(l) 1.4(l) 1.0(l) 2.3(3)
523.9( 1) 524.0( 1) 524.1(l) 524.2( 1) 526.4(3) 524.2(2)
2.5(l) 2.2( 1) 2.2(2) 2.0(l) 1.2( 1) 2.1(l)
1.0 0.1 0.1 0.1 0.1 0.1 0.1
2947.8(2) 2948.6( 1) 2948.5( 1)
2.3(l) 2.4( 1) 2.0(2)
1239.6( 1) 1240.9( 1) 1238.8(l) 1247.7(2) 1244.1(l)
2.2( 1) 1.3(l) 0.9(2) 1.6(3) 1.0(2)
524.1(l) 523.3(2) 526.5(4) 524.0( 1) 525.6(2)
2.1(l) 1.9(2) 1.1(l) 2.4(2) 1.7(2)
2955.8(l)
3.0( 1) 1242.2(l)
0.8(2)
528.6( 1)
1.2(2)
1.0 0.5 0.2 0.1 0.1 0.1
2948.3( 2949.3( 2949.9( 2949.8( 2949.8( 2960.2(
1239.8( 1) 1240.6(2) 1241.1(l) 1241.0( 1) 1239.1(l) 1247.9(5)
2.0( 1) 1.6(l) 1.3(2) 1.2(2) 0.8(2) 1.9(2)
524.3( 1) 524.5(l) 524.5( 1) 524.5( 1) 526.8(2) 524.3(2)
2.0(l) 1.7( 1) 1.6( 1) 1.6(l) 1.1(l) 1.7(2)
cs* CD&N 293
CDJ CS, CHjNO, GH6 GD6 n-C,H,6
311
CDJ
-2 CD&N
1) 1) 1) 1) 1) 1)
2.4(l) 2.3( 1) 2.3( 1) 2.3(2) 2.0(2) 3.4( 1)
T. Fukuda et al. / Vibrational
154 Table 2 Observed
Raman
T(K)
band widths and peak frequencies
Solvent
(cm-’
relaxarion and frequencies
) of totally symmetric
sf liquid molecules. II
modes of CDJ.
Cont. (mole fraction) I- iC”
WJ 210 260 293 CHJ 310 CHJ CS? Solvent
T(K)
190 240 293 CD3Br csz Solvent
T(K)
CH,Br, and CHCl, in the liquid phases
1.0 1.0 1.0 0.1 1.0 0.1 0.1
2136.9(2) 2137.8(2) 2138.5(l) 2139.4(l) 2139.0( 1) 2139.7(i) 2140.7( 1)
1.4(l) 1.3(l) 1.4(l) 1.2(2) 1.4(l) 1.2(l) 0.8(l)
2953.5( I) 2954.7( 1) 2956.1(l) 2957.5( 1) 2952.3(2)
2.2(l) 2.4( 1 ) 2.4( 1 ) 2.3(2) 2.2(4)
937.3( 1 ) 938.4( 1) 938.6( 1) 939.4( I ) 939.1(l) 939.8( 1) 939.6( 1 )
I;,,
2.0( I ) 1.6(I) 1.4(l) 1.1(l) 1.3(l) 1.1(l) 0.5(l)
494.0( 494.5( 494.4( 494.9( 494.4( 494.4( 195.8(
3.0( 1) 2.5( I ) 2.0( 1) 1.0(l) 1.1(2)
594.4( 1) 595.8( 1 ) 597.1(2) 598.0( 1) 600.5(4)
3.3(2) 2.8(2) 2.2(2) 1.7(2) 0.9(3)
(4)
I;,”
668.7(l) 668.5 ( I ) 668.4 668.7(l) 668.6(2)
0.9(l) 1.1(l) 1.0(5) l.?(l)
I.0 1.0 1.0 0.2 0.1
CDCI, 310
1.0 0.1 1.0
3015.7(3) 3016.8(l) 3017.5(3) 3018.1(l) 3018.0(l)
4.7(2) 4.4(2) 4.2(l) 4.1(l)
in the measurements of integrated scattering intensities which were measured relative to that of the v, band of carbon tetrachloride used as an internal standard. The sensitivity of our apparatus measured with a calibrated tungsten lamp decreased by 40% with an increase in the Raman shift from 0 to 3000 cm-‘. The Raman integrated intensity defined as the sum of the parallel and perpendicular integrated intensities,
1, =I,, $1, is given by
3
1) 1)
1295.1(2) 1296.1(3) 1296.8(4) 1298.3(2) 1294.2(2)
Cont. (mole fraction)
1.0 1.0
293
I) 1)
?.4(2) 1.8(2) 1.6(2) 1.3(l) 1.6( 1) 1.3(3) 0.9( I )
Cont. (mole fraction)
r IS0 210 260
I) 1) I)
(7)
368.6(2) 368.0( 1) 367.8(4)
0.5(I) 0.4(2) 0.4(l)
367.8(2)
0.4(
Z,=A(w,-w,)“[ X hS,/45w,
1 -exp(
1)
-fiw,lkT)]-’
,
(8)
where wL and w,~are the angular frequencies of the laser line and the Raman shift of the center of the band concerned, A is a constant characteristic of the experimental apparatus and conditions, and S,, referred to as the scattering activity [ 121 of the molecule, is given by the isotropic and anisotropic transition polarizabilities as &=45(~,/fi)(I
+&Trl
I2 (9)
T. Fukuda et al. / Vibrational relaxation andfrequencies Table 3 Observed
Raman T(K)
reorientational
band widths
(cm-‘)
and depolarization
of liquid molecules. II
ratios of totally symmetric
155
modes
CHSI 6HCH
“CH
VCI
I- reo
P
r reo
P
r reo
P
211 262 293 311
2.4(6) 3.8(8) 5.2( 13) 5.8( 19)
0.011(S) 0.007(S) 0.008 (5) 0.009(3)
2.2( 10) 3.1(13) 5.2(8) 6.2( 15)
0.066(20) 0.046( 17) 0.049(6) 0.053(28)
0.8(2) 2.6(3) 4.0(7) 6.1(7)
0.199( 14) 0.192(6) 0.180(20) 0.189(23)
T(K)
CD,1
210 260 293 310
2.0( 10) 3.7( 10) 3.9(9) 4.1(11)
0.003( 1) 0.007(3) 0.009(2) O.OlO(3)
1.6(l) 3.1(3) 4.0(6) 5.0(2)
0.078(6) 0.076( 5) 0.090( 16) 0.087( 12)
1.2(2) 2.9(2) 4.0(2) 4.8(2)
0.189(7) 0.232(21) 0.194( 16) 0.209(24)
T(K)
CH,Br 6 HCH
VCH
rreo 190 240 293
T(K)
P
rl-e0
P
rreo
P
O.Oll(7) 0.017(6) 0.018(S)
-
0.067 (8) 0.051(7) 0.064( 10)
1.1(2) 3.6(2) 5.7(2)
0.184(6) 0.165(6) 0.164(11)
CHCll 6 xcx
VCH
210 260 293 310
“Ck
“CC1
rreo
P
rreo
P
rwo
P
1.5(9) 2.8(4) 4.6(3) 4.8(3)
0.187(8) 0.150( 14) 0.177(12) 0.170(11)
0.8( 1) 2.5(l) 5.3(l) 4.1(l)
0.144(9) O.llO(l4) 0.114(8) 0.106(4)
0.3(2) 2.7(4) 3.8( 12) 6.4( 15)
0.012( 10) 0.011(8) 0.014(7) 0.014(9)
If the vibration is harmonic, eq. (9) for the fundamental transition is reduced to the well-known form 112,131 (10) (11) where (x(I) and jLl(” are respectively tive of the isotropic and anisotropic
the first derivapolarizabilities
with respect to the normal coordinate. The scattering intensity of the liquid sample, Z,, was measured at room temperature relative to that of the v, mode of carbon tetrachloride, I,., as the internal standard. The scattering activity of the sample was obtained by the following relation:
T. Fukuda et al. / Vibrational relaxation andfrequencies
156
of liquid molecules. II
given by the refractive indices of the sample and carbon tetrachloride, n, and n,, respectively. The S, value used was 1.36~ 10e9 cm4 gg ‘, which was obtained from the scattering cross section of carbon tetrachloride v, mode measured at 632.8 nm excitation by Kato and Takuma [ 141. The S, values obtained are listed in table 6. 1235
2.3. Infrared absorption intensities
mole fraction
mole fraction
Fig. 3. Plots of the peak frequencies and the isotropic band width versus mole fraction for the sHCH mode of CHxI in solutions at 262 K. (0 ) In CD,I, (0) in CS2, (a) in CD,CN.
where f, is the mole ratio of carbon tetrachloride to the sample and is taken to be l/9 in the present experiment, r( co,) and r( 0,) are the relative sensitivities of the spectrometer at Raman shifts w, and o,, respectively. The last factor on the right-hand side of eq. (12) is the correction term for the internal field effect on the intensity of the internal standard, and is
Table 4 Observed
and calculated
frequency
shifts (cm-‘)
The infrared absorption intensities of the fundamentals and the first overtones of the liquid samples were measured at room temperature with two models of Fourier transform spectrometers. A 400-4000 cm-’ range was scanned on a DIGILAB FTS-14A spectrometer with a resolution of 2 cm-‘. Liquid cell with KRS-5 windows was used and sample thicknesses were in the range from 7 to 500 urn. A NICOLET 7199 spectrometer was used for measurement of the 4000-8000 cm-’ range with a resolution of 1.2 cm-‘. The sample thicknesses were 0.5, 1 and 5 mm and the windows used were KBr and fused quartz. The first overtone of the v, vibration of CH3Br was measured in the gas phase with 0.12 cm- ’ resolution using a 10 cm gas cell. The observed intensities (13) where c is the concentration
( molecules/cm3)
at 293 K
Calculated APP CHJI CD,1 CH,Br CHC&
0.33 0.33 1.01 0.06
6 HCH
CHJ CD,1 CHjBr
VCX
CH,I CD31 CH,Br CHCI,
VCH
and 1
Observed A$-pP
A-%,
A@
A@
A-Q&
-5.01 -2.13 - 5.64 -3.24
I.42 0.79 1.63 1.19
-3.59 -1.94 -4.01 -2.05
AQ
A&w
A&Q
AGEE
-7.12
-4.44
-0.29 -0.30 -0.51 -0.07
-23.5 -16.9 -16.9 - 15.4
-1.9 -0.9 - 1.7 -0.6
-3.86 -2.89 -4.80 -2.45
-3.53 -2.56 -3.19 -2.39
-0.07 0.00 0.37
-0.67 -0.98 0.00
-0.74 -0.98 0.37
- 13.10 -9.01 - 13.45
4.37 2.96 4.63
-8.73 -6.05 -8.82
-9.47 - 7.03 -8.45
-0.57 -0.54 -0.41
-11.2 - IO.4 -9.2
-1.4 -0.8 -1.8
-0.64 -0.75 -2.85 -0.11
-4.03 -2.74 -3.32 - 1.1 I
-4.67 - 3.49 -6.17 -1.22
-2.11 -2.05 -2.06 -0.59
0.44 0.43 0.43 0.11
- I.67 -1.62 -1.63 -0.48
-6.34 -5.11 - 7.80 - 1.70
-0.51 -0.38 -1.10 -0.19
-9.1 -6.8 -13.4 -2.7
-0.3 0 - I.1 -0.3
-4.50 - 7.80
T. Fukuda et al. / Vibrational relaxation andfiequencies Table 5 Observed
and calculated
band widths
(cm-‘)
157
of liquid molecules. II
at 293 K
Calculated
Observed
P’P
pdirp
r Clec
“CH
CHJ CD,1 CH,Br CHCl,
0.25 0.28 0.83 0.11
3.84 2.01 3.38 2.67
4.09 2.29 4.2 1 2.88
6 HCH
CHJ CDJ CH,Br
0.44 0.49 0.36
0.11 0.23 0.00
“CX
CHJ CDJ CH3Br CHCl,
0.34 0.79 2.51 0.06
4.00 1.77 1.57 0.57
P
P
CH31
CDJ
0.62 0.17 0.43 0.44
4.71 2.46 4.64 3.32
0.01 0.03 0.02 0.00
2.3 1.4 2.4 4.2
0 0.2 0.1 -
0.55 0.72 0.36
27.89 13.ia 16.95
14.98 6.90 9.73
-40.35 - 18.82 -25.35
2.52 1.25 1.33
3.07 1.97 1.69
0.11 0.10 0.02
2.2 1.4 2.0
1.0 0.3 1.1
4.34 2.56 4.08 0.63
0.52 0.49 0.29 0.06
0.12 0.12 0.06 0.02
-0.47 -0.44 -0.25 -0.05
0.18 0.16 0.10 0.03
4.52 2.72 4.18 0.66
0.04 0.02 0.11 0.04
2.1 1.6 2.2 1.0
0.4 0.3 0.6 -
CH,Br
“CH
456(93) 41(9) 218(46)
“Cl
375(70) 77(15) 155(28)
“CH
478(1 lo)
s HCH
10(3) 126(27)
“C&
CHC&
and infrared
6 HC”
6 DCD
“CH “CCl
rREF.
-5.00 -1.52 -3.72 - 13.40
SS(O-rl)
“CD
rREE r
1.59 0.49 1.20 5.53
Mode
“Cl
rnlech r
4.03 1.20 2.94 a.31
Table 6 Observed Raman scattering activities ( lob9 cm” g-‘) and overtone transitions of totally symmetric modes Compound
TAR
219(49) a7(19)
the sample thickness (cm), were corrected for the internal field effect by the Palo-Wilson formula [ 15 1,
The results are listed in table 6. 3. Parameter values 3.1. Anharmonicity constants The anharmonicity constant of the oscillator takes
integrated
intensities
&(0-+2)
1.9(4) 00
2.4(5) 3.9( 10)
1.2(2) %O
0.69( 12)
(lo-”
cm2 s-’ molecule-‘)
of fundamental
k,(o+l)
r-,(0+2)
274(27) 913(70) 129( 10)
12.3( 10) 36.0(37) 4.5( 14)
154(15) 618(21)
1.2(3) 7.4(3)
73(5)
16(6)
439(44) 426(72) 435(30)
23.7(70) 15.5(45) 7.5(12)
370(30) 315(30)
51.0(6) 2.7(4)
part in the leading terms in the theoretical expressions of both the band width and the frequency shift due to the phase modulation. Besides, the anharmonicity constant is necessary to calculate the intensity parameters from the observed infrared and Raman intensities as will be shown in section 3.2. According to the second-order perturbation calculation, the third-order anharmonicity constant of the vibrational potential of a diatomic molecule [ 11,
(15) is related to the first anharmonicity coefficient of the
vibrational
where I and F indicate the initial and the final states of the transition, respectively, P, and FF are the distribution probabilities of the states, and p is the dipole moment operator, For a transition from the ground state, 0-d U,eq. ( 19 ) is approximated as
energy terms, o,w,, as [ 16 ]
IiT-&= &off&x,
.
(16)
where w, and o,x, can be obtained served frequencies of the fundamental overtone,
from the oband the first
0, =3w, --cl&,
(171
w,x,=(2w,
(18)
-e&)/2.
x I (4plO> I?. The scattering as
These relations for a diatomic molecule can be approximately applied to the total symmetric C-H stretching, the H-C-H bending, and the C-X stretching modes of the molecules studied in the present paper, because the cross terms between these and other modes are small enough to be neglected in the estimation of K,,, [ 17,181. The K,,, values obtained are listed in table 7 which also includes the diagonal elements of the inverse G matrix required for the calculation of the effect of mechanical collisions on the band shape. 3.2. Intemify
activity given by eq. (9) is rewritten
(21) where
I_
@=
r,
The dipole moment or the isotropic polarizability, represented by X, is expanded in the normal coordinate to the second order:
parameters
In part I of this series [ 11, it has been shown that the band broadening and the frequency shift owing to the dipole-dipole interaction and the dispersion interaction can be calculated from the first and the second derivatives of the dipole moment and the polarizabihty with respect to the vibrational normal coordinate. The values of these parameters are obtained from the observed infrared intensities and the Raman intensities. The infrared intensity is given by
X=,YW t__X’I ‘q+ +,pq
‘
(23)
Then the squared matrix element becomes ~(v~x~o>~~=(~~i”“)‘(v~q~o)’ +x”‘x’2’(u/q~o)
(viq2iO>
f~(X”‘)2(v~q’~O)
(241
The wavefunction of a slightly anha~onic oscillator is given by the first-order perturbation theory,
I191 zcor= c -(~Pr-P,_)I(FlPlII)12, 2xwo
(20)
(19)
I./. 3ch
Table 7 Anharmonicity constants ( 1O47cm-’ g”’‘12s-l) and G-’ matrix elements (amu) for totally symmetric modes ijHCH
L’<,H
CW CD,1 CH,Br CHCI,
W%
-K,,
C,’
-L
G;;’
- k;,,
G,’
118
0.98
10.8
11.0
I.% 0.98 0.93
5.83 11.8
0.44 0.76 0.43
I.29
36 128 193
1.04 1.69 1.23
11.0 10.4 17.9
T. Fukuda ef al. / Vibrational relaxation andfrequencies
Iv>=
I~>“-(K,;i16,,‘5)
- [v(u-l)((v-2)]“2~v-3)o},
(25)
where 1u) ’ is the wavefunction of the harmonic oscillator. Using the observed values of K,,; and p, eqs. (20) and (21) were fitted to the observed infrared intensities and the scattering activities, respectively, for the fundamental and the first overtone. The values obtained for ,LL(‘),$*), CX(‘)and cy(*) are listed in table 8, where 01~~)of the C-H stretching mode was assumed to be zero because the Raman scattering of the C-H overtone was too weak to be observed. The above procedure does not give the signs of the parameters, but only the relative signs of ,LL(‘)and pc2), and 01~‘) and (Y(*). Since there has been no established method of finding the sign of p(i), we take notice of the relationship between p(‘) and the frequency shift. The difference in vibrational frequency between dilute solutions in nonpolar and polar solvents will be due mainly to the difference in the dipole-dipole interaction between solute and solvent molecules. According to eq. (I-54), the frequency shift due to the dipole-dipole interaction in a dilute solution, AL&$&,is proportional to the third power of the permanent dipole moment of the solvent mole-
of liquid molecules. II
159
cule and the sign of the shift is given by the factor, -fpj’) +,D.$‘)/~o,. Therefore, the signs of p(‘) and P (*), which are opposite to each other, were determined so as to reproduce the order of the vibrational frequencies in nonpolar and polar solvents. The signs ofp (’ ) determined are negative for the vCH and 6HcH modes and positive for the vex mode. These are consistent with the previous studies on halogenated methanes [20-231. The p(i) and PC*) values obtained for the vcH mode of CHC13 are in good agreement with the values of p(l)= -33.0 esu g-II2 and ~(‘~‘=63.0~ 10Zoesu cm-’ g-l, reported by Boobyer [20].Thesignofa(” was assumed to be positive for all the totally symmetric modes in accordance with the previous studies [ 24,25 1. 3.3. Diffusion coefjcients The rotational diffusion coefficient of the molecular axis of a symmetric top molecule, D,, is given by the reorientational band width of the totally symmetric band [ 3 1, T,,,, D, = f ncT,,
.
(26)
Although this equation claims a constant value of r,,,, actually observed values of r,,, vary from band to band. The reason for this seems not to be understood yet. In the present study, the average of the observed
Table 8 Infrared and Raman intensity parameters of totally symmetric modes Compound CHJ
Mode
&2)
(10-5cm2g-“2)
( 10’5cmg-‘)
9.93 2.81 5.48
- 1.67 -0
-21.0 -41.8 14.7
7.2 26.6 -24.6
9.05 3.12 4.60
- 1.55 -0.72
“CBT
-36.1 - 35.3 34.5
43.5 28.6 -11.7
11.1 1.42 4.36
_ -1.72 ZO
“CH
-33.8
“CD
6DCD VCI
VCH
6HCH CHCI,
(y(1)
34.5 46.5 -9.4
“CI
CH,Br
P (2) ( 10Zoesu cm-’ g-‘)
-28.4 -51.8 19.4
VCH
6HCH
CD,1
(1) P (esu g-l/*)
VCCI
29.6
79.9
5.62
-8.86
4.3 1
-0.16
T Fukuda et ai. / Vibrational relaxation andfrequencies
160 Table 9 Rotational
and translational
diffusion
coefficients
and viscosities
(K)
D ( 10-5cm2
CHJ
211 262 293 311
5.6 10 14 17
1.1 2.5 3.7 4.5
1.29 0.68 0.50 0.44
CD,1
210 260 293 310
4.4 9.4 12 16
CH,Br
190 240 293
3.3 11 18
1.1 3.7 5.8
0.33 b’
210 260 293 310
2.8 8.3 14 16
0.64 1.7 2.7 3.3
1.85 0.88 0.64 0.53
CHCI,
[ 26 ]
s-‘)
” Estimated by eq. (29).
r re0 values
was used, and calculated values of D, are listed in table 9. The translational diffusion coefficient is calculated from the literature value of the viscosity [ 261, q, through the following relation by Gierer and Wirtz v71,
D=k,T/3xclZf;d,,
a) rl (cP)
DC (lO’Os_‘)
T
” Taken from ref.
of liquid molecules. II
(27)
where
(28) and d, and d/, are the molecular radii of the solute and solvent, respectively. Values of D calculated using the molecular radii from table 10 below are listed in table 9. Since the value of q of the CH,Br liquid has not yet been reported, it was estimated from the ratio of the rotational diffusion coefficients for CH31 and CH3Br and the viscosity of CH31 using the following equation,
(291
which is based on the relationship,
D,x 1/qd3.
3.4. Other parameters Table 10 lists other parameters used for calculating the band shape parameters. Values of the permanent dipole moment p(O), polarizability, (Y(O),ionization potential, ZP,and number density, pF, were taken from the standard tables or literature [ 28-301. The molecular diameters, d,, are given by the Lennard-Jones potential parameters [ 3 I] and the radii of atoms in molecules were obtained subtracting the bond lengths from the molecule radii. The ranging parameter of the repulsion potential is given by L = (dJ2 + a) / 12
[Il. 4. Results and discussion The band widths and the frequency of the Raman isotropic scattering of liquids were calculated using the values of the parameters estimated in section 3. Here, we are concerned with the results of the calculation for neat liquids at 293 K. The effect of solvents and temperature on the band width and the frequency will be discussed in the next paper.
T. Fukuda et al. / Vibrational relaxation andfrequencies
Table 10 Parameter values required for the calculation ofrand
AS2
(0) ,u
(ym
I
d,
a
( 10” molecule cm-‘)
(D)
( 10-25cm3)
(ev)
(10-8cm)
(lo-*cm)
9.63 10.94 7.48
1.62 1.84 1.04
73.0 55.5 82.3
9.54 10.54 11.48
4.66 4.46 5.18
6.6 25.0 35.8 55.3
13.60 13.01 11.84 10.45
H Cl Br I
of the vibrationalfrequency
The calculated shift is the sum of the two components arising from the electrical and mechanical interactions, CL-S2=AQ,ec+ALnmec,, .
(30)
The electrical part, AL&,,, consists of the contributions from the dipole-dipole and the dispersion interactions, AL& = A&Pip + A@isp 9
(31)
where ALPip is given by eq. (I-54) and ALPP by eq. (I-64b). The mechanical component, AL&_,,, is divided into the attractive part and the repulsive part, which involve A ,!,I) and R 1(/’ ) 3respectively, in eq. (Ig6), AL!mech=AL2A+AP.
(32)
From a viewpoint of interaction process, AL2is decomposed into two terms due to the resonant energy exchange and the phase modulation, m=
UREE
161
P
CH31 CH3Br CHC13
4.1. Shift
of liquid molecules. II
+ ~Pbl
,
(33)
where AG!REE= A@&
+ mdizR
(34)
is the part of A.@.,,, and the sum of all other terms is AL&. The calculated values of each term of the frequency shift and their total are listed and compared with the observed values in table 4. Agreement between the calculated and experimental values of AL2 is reasonably well for the &., and vex modes but considerably poor for the vcH mode.
0.81 1.01 1.13 1.24
The calculated shifts of the vcH modes are l/2- l/4 of the observed values. This fact suggests that the CH stretching force constant of the liquid molecule is slightly different from the gas phase value, owing to an environment-induced change in the electric charge distribution of the molecule. If the difference between calculated and observed frequencies is attributed to the change in the force constant, the molecule has a 0.5-l % smaller force constant in the liquid than in the gas phase. Both the calculated and observed results show that the phase modulation, or the average static field effect, is the dominant part of the gas-liquid frequency shift. The contribution from the resonant energy exchange, or the vibrational resonant coupling, is in a range of lo-20% of the total shift. The calculation shows that the relative importance of the parts of the intermolecular potential varies among the three modes. The contributions of the mechanical and the electrical interactions to the frequency shift are comparable for the vcH mode, where the dominant terms are APiSp and ALP. For the 6 HCHmode, the mechanical one is overwhelmingly important, while for the vex mode the electrical one is superior to the mechanical one. The main terms for the &+cH mode are AD’ and AQR which have opposite signs and cancel each other to some extent. For the vex mode, APiSp and ALP are the main terms. 4.2. Band broadening The calculated band width is composed trical and the mechanical parts, r= re,ec + rmectl .
of the elec-
(35)
7’. Fukuda et al. / Vibrational relaxation andfrequencies
162
The electrical part is given by r
=rdip+rdlsp l?lCC
rd’p =
rdplfi + rg
rdlsp = r”,a
(36) ,
+ rdis!$ RE
where Tdp;E; , r&, (I-56), (I-57), The mechanical
(37) (38)
>
r$z,
and r$;% are given by eqs.
(I-65b), and (I-65a), part is written as
r mech =rA+rf+rAR ,
respectively.
(39)
where r,,,,,, is given by eq. (I-87 ), and r” consists of the terms involving only A I,’) and A I’:‘, TR is the term involving only R/,” and Rj?‘, and PR is the cross term involving both the A,, and R,, factors. The calculated values are listed and compared with the observed values in table 5. The calculated total band widths, r, are in moderately good agreement with the observed widths. It can be seen from table 5 that the phase modulation is the dominant process for the broadening of the bands as for the shift of the frequencies. For the vCH and vex modes, the contribution of the electrical interaction is larger than that of the mechanical one, but they are in the reverse order for the Z&n mode. It should be noticed that the cross term rARplays an important role. This term has a negative sign and cancels out the self-terms to a large extent. Then the magnitude of rmech is an order of magnitude smaller than its components. Without the cross term, the calculated total band widths of the vC.H and the 6,,:, modes would be an order of magnitude larger than the observed band widths. To the best of our knowledge, this is the first to notice the importance of the cross term which arises from dealing with the repulsive and the attractive parts of the potential simultaneously.
5. Conclusion The present theoretical expressions for the Raman band widths and the peak frequency have been shown to be useful for studying the intermolecular potential and the vibrational relaxation process in the liquid phase. In spite of the simple model used and rather rough estimation of some physical parameters, the calculated band widths and frequency shifts were in
qfllquid molecules. II
moderate agreement with the observed values. It is particularly important to deal with the intermolecular potential including all possible terms and to treat every possible process of the relaxation in the same theoretical scheme. By doing this, we could compare the effects of the potential terms and compare the contributions of relaxation processes to the Raman band shape. For the first time, we found out the important role of the cross term, PK. in the band broadening. Quantitatively, the present results of the calculation are not yet satisfactory. To improve the theory, the first step is to obtain more precise values of the parameters. By means of those, the theory can be tested more quantitatively, and then, it will become possible to find out a way of improving the theory if necessary. Two possible ways of improvement are to include the effect of the intramolecular mode coupling and to allow for the internal motion of solvent molecules [ 1 1.
Acknowledgement The authors thank Dr. Kohji Fukushi suggestions and discussions.
for helpful
References [ I] T. Fukuda,
S. Ikawa and M. Kimura, Chem. Phys. 133 (1989) 137. [2] A.M. Benson Jr. and H.G. Drickamer, J. Chem. Phys. 27 (1957) 1164. [3] F.J. Bartoli and T.A. Litovitz, J. Chem. Phys. 56 (1972) 404,414. [4] L.A. Nafie and W.L. Peticolas, J. Chem. Phys. 57 (1972) 3145. [S] D.W. Oxtoby. .4dvan. Chem. Phys. 40 (1979) I. [6] J.H.R. Clarke, in: Advances in Infrared and Raman Spectroscopy, Vol. 4, eds. R.J.H. Clark and R.E. Hester (Heyden, London. 1980) p. 108. [ 71 K. Fujita, T. Fukuda. K. Fukushi and M. Kimura, J. Raman Spectry. 16 (1985) 377. [8] H. Mori. Progr. Theoret. Phys. 33 ( 1965) 423. [9] K. Fukushi and M. Kimura, J. Raman Spectry. 8 (1979) 125. [lo] G. Diige, R. Arndt and J. Yarwood. Mol. Phys. 52 (1984) 399. [ I I ] S. Ikawa, M. Ito, T. Fukuda and M. Kimura, J. Mol. Liq. 32 (1986) 219.
T. Fukuda et al. / Vibrational relaxation andfrequencies [ 121 J.R. Nestorand E.R. Lippincott, J. Raman Spectry. 1 ( 1973) 305. [ 131 H.W. Schrotter and H.J. Bernstein, J. Mol. Spectry. 12 (1964) 1. [ 141 Y. Kato and H. Takuma, J. Chem. Phys. 54 ( 1971) 5398. [ 151 S.R. Pa1oandM.K. Wilson, J.Chem. Phys. 23 (1955) 2376. [ 161 G. Herzberg, Molecular Spectra and Molecular Structure, Vol. 1. Spectra of Diatomic Molecules (Van Nostrand, Princeton, 1950). [ 171 S. Reichman and J. Overend, J. Chem. Phys. 48 ( 1968) 3095. [ 181 R. Zygan-Maus and S.F. Fischer, Chem. Phys. 63 ( 1981) 475. [ 191 E.B. Wilson Jr., J.C. Decius and P.C. Cross, Molecular Vibrations (McGraw-Hill, New York, 1955). [20] G.J. Boobyer, Spectrochim. Acta 23 A (1967) 335. [21] J.H. Newton and W.B. Person, J. Chem. Phys. 64 (1976) 3036.
of liquid molecules. II
163
[ 221 A.J. van Straten and H.A. Smith, J. Chem. Phys. 67 ( 1977) 470. [23] S. Abbate and M. Gussoni, Chem. Phys. 40 (1979) 385. [ 241 R. Escribano, J.M. Orga, S. Montero and C. Domingo, Mol. Phys. 37 (1979) 361. [ 25 ] M.C. van Hemert, Mol. Phys. 43 ( 198 1) 229. [ 261 Landolt-Bornstein, Numerical Data and Functional Relationships, New Series, II/5 (Springer, Berlin, 1969). [27] A. Gierer and K. Wirtz, Z. Naturforsch. 8a ( 1953) 532. [28] Landolt-Bornstein, Zahlenwerte und Funktionen, I/3 (Springer, Berlin, 195 1). [29] A.L. McClellan, Tables of Experimental Dipole Moments, Vol. 2 (Rahara Enterprises, El Cerrito, 1974). [ 301 T. Yoshino and H.J. Berstein, J. Mol. Spectry. 2 ( 1958) 2 13. [ 311 F.M. Mouritz and F.H.A. Rummers, Can. J. Chem. 55 (1977) 3007.
151-163
VIBRATIONAL RELAXATION AND FREQUENCIES OF LIQUID MOLECULES. II. COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS Takuya FUKUDA
‘, Shun-ichi
IKAWA ’ and Masao KIMURA
Department of Chemistry, Faculty of Science, Hokkaido University, Sapporo 060, Japan Received
17 July 1988
The Raman isotropic band widths and the peak frequencies of totally symmetric vibrational modes were measured for the CH,I, CD,& CH,Br, and CHCI, molecules in liquid phases at various temperatures. The integrated intensities of the Raman scattering and the infrared absorption of the fundamental and overtone bands were measured to obtain the intensity parameters required for calculating the band widths and the gas-to-liquid shifts of the frequency by the theory reported in part I. The agreement between the calculated and the observed values is reasonably good. The phase modulation was the dominant process both in the band broadening and in the frequency shift. It was pointed out, for the first time, that the cross term between the attractive and the repulsive parts of the mechanical collision potential played an important role in the band broadening. Without the cross term, the calculated band widths of the vcH and &,cH modes would be an order of magnitude larger than the observed widths.
1. Introduction
tal and theoretical points of view. However, it is still quite insufficient to understand what the ma.in interaction mechanisms are and what the dominant relaxation processes are. In view of this situation, we developed, in the previous paper [ 11, a band shape theory which was as comprehensive as possible dealing with the dipole-dipole, the dispersion, and the mechanical collision potential all together and allowing for the ED, PM, and REE processes, starting from the generalized Langevin equation [ 81. The isotropic band widths and the vibrational frequencies of liquid molecules were given as functions of the physical parameters of the solute and solvent molecules. The purpose of the present paper is, firstly, to measure the Raman isotropic band widths and peak frequencies of CHJ, CD31, CH-,Br, and CHC13 in the liquid phase at various temperatures, secondly, to estimate all the physical parameters required for the calculation of the band width and the frequency shift, and last, to compare the observed and calculated band widths and the frequency shifts. Thus, for the first time, it becomes possible to compare the contributions from the various potentials and processes to the Raman band shapes. This paper is concerned with the calculation for neat liquids at 293 K and the next paper will discuss the effect of solvents and temperature.
Raman scattering is a useful tool for studying the dynamics and interactions of molecules in the liquid phase. The isotropic and anisotropic band shapes tell how the vibrational and the reorientational relaxations proceed in the liquid phase, and the shift of the peak frequency indicates the average strength of the local field acting on the molecule [ l-71. This dynamical and statical information must, ultimately, be explained by or elucidate the real feature of the intermolecular potential. However, the potential field in the liquid phase is quite complicated involving various mechanisms such as dipole-dipole, dispersive, and repulsive interactions. The fluctuation of the potential caused by the thermal motion of molecules is related to the relaxation phenomena. For the vibrational relaxation, the following three processes have been proposed [ 5-71: ( 1) energy dissipation (ED), (2) phase modulation (PM), (3) resonant energy exchange (REE). Many studies have been reported on the Raman band shapes from both the experimenI Present address: Hitachi Hitachi, Ibaragi, Japan. ’ To whom correspondence
Research
Laboratory,
Hitachi
Ltd,
should be addressed.
0301-0104/89/$03.50 0 Elsevier Science Publishers ( North-Holland Physics Publishing Division )
B.V.
T. Fukuda et al. / Vibrational relaxation andfrequencies
152
2. Experimental procedures and results 2. I. Raman band shapes Raman scattering of liquids was excited by a HeNe 632.8 nm laser line from a NEC GSL-108 laser apparatus operated at 50 mW and the 90” scattering was measured by use of a JASCO R-300s spectrometer with a cooled RCA C31034A photomultiplier. The details of the optics have been described elsewhere [ 9 1. Scanning of the spectrometer and acquisition of the photon counting signal were performed under control by a PANAFACOM U-200 digital computer. As typical examples of the measurements, the C-I stretching Raman bands of a CH31-CD31 mixture are shown in fig. 1. The whole region was divided into several subregions, which were scanned at different conditions depending on the signal intensity and its gradient. The temperature of the liquid samples was controlled within + 1 K in the range from 190 to 3 11 K. The wavenumber calibration was performed with forty lines of the Ne natural emission. Its uncertainty was less than 0.4 cm-‘. The slit function measured with the Ne emission line was close in shape to a Gaussian function. Then the observed Raman bands were simulated by a computer as a convolution of Lorentzian and Gaussian functions, and the width of the former was taken as a true band width. When some bands overlapped, the parameters
ofliquid molecules. II
of the band concerned were obtained by decomposing the whole spectrum into some Voigt components. The example is shown in fig. 2. The background due to the Rayleigh scattering and, sometimes, due to the fluorescence of a small amount of impurity was excluded by drawing a linear base line. The isotropic and anisotropic spectra were obtained from the obscattering served parallel and perpendicular intensities: ziso(“>=r,j(W)-$L
zaniso(“>
(WI
=IJ.
toI
I
(1)
(2)
.
The half band widths corrected for the slit width effect, ri,, and raniso, give the widths owing to the vibrational and reorientational relaxations [ 3,4 ] : rvlb =cso 3 r,,, =
ranis0
(3) -
r,,,.
(4)
The observed values of ri, and peak frequency, wo, are listed in tables 1 and 2, and r,, and the depolarization ratio, p, for neat liquids are listed in table 3. For isotopically diluted solutions, observed band widths and frequencies changed linearly to the mole fraction, as shown by examples in fig. 3. Differences between values at mole fractions 1 and 0 are attributed to the broadening and the frequency shift due to the resonant energy exchange [ 10,111,
x 13175pp
1.0
0.5
::-:: 0
5:lo
_“-A:
560
‘540
‘,.,__/’
520
i
‘.
.... .“--.
500
460
480
‘440 w/m-f-’
x 1779pps
1.0 1
,r.., :
'.
1,
WAm-’
Fig. 1. The I,, and I, scattering control by a digital computer.
intensities in the vc_, region of an isotopic mixtures (CHJ/CD,I= l/4) at 262 K measured under the The maximum count rates were 13 175 and 1799 pps for the measurements of I,, and I,, respectively.
T. Fukuda et al. / Vibrational relaxation andfrequencies
620
Fig. 2. Computer simulation of I,,(w) of the vC_ar mode of liquid CH>Br at 293 K. (0 ) Observed, (-) calculated total, (---) calculated Voigt components corresponding to the CHs79Br and CH,s’Br species, and the hot band, respectively, from left to right.
rREE=r(m f=l)-riSOt(mf=O),
(5)
AL2REE=~O(mf=1)-W~(mf=0),
(6)
2.2. Raman scattering intensities The frequency dependence of sensitivity bf the spectrometer can be neglected in a band shape measurement, which was limited in the range of 100 cm-’ at most. However, this effect becomes considerable
where the values at mf = 0 are obtained by linear exTable 1 Observed
Raman
band widths and peak frequencies
(cm-
153
trapolation. As expected from eq. (36) of ref. [ 11, hereafter designated as eq. (I-36), rREE is always positive. A&,, values are negative for most liquids with an exception of the breathing mode of benzene [ 111, where the contribution of the charge transfer mechanism has been suggested. The residual band width, .isot( mf=O), and the residual frequency shift, obsof(mf=O)-o~, where o, is the gas-phase frequency, are assigned to the contributions of the phase modulation effect, r,, and AS&,,+respectively. The AL& and r,,, values obtained are listed in the last columns in tables 4 and 5.
560 cd
560
600
of liquid molecules. II
’ ) of totally symmetric
modes of CHJ
in the liquid phases
T(K)
Solvent
Cont. (mole fraction)
211
CD,1
1.0 0.5 0.2 0.1
2945.9( 2946.0( 2946.8( 2946.9(
1) 1) 1) 1)
2.3(l) 2.4(2) 2.4(2) 2.4(2)
1238.6( 1) 1239.2(2) 1239.8(l) 1240.2(l)
3.5(l) 2.6(2) 2.3(l) 1.9(l)
523.4( 1) 523.3(2) 523.5( 1) 523.8( 1)
3.0( 1) 2.7(2) 2.6(2) 2.2(l)
262
CD,1
1.0 0.5 0.2 0.1 0.1 0.1
2946.9( 2947.7( 2948.4( 2948.5( 2948.4( 2959.6(
1) 1) 1) 1) 1) 1)
2.3( 1) 2.3(l) 2.3(2) 2.3(l) 2.1(l) 3.6(2)
1239.8(2) 1240.0(2) 1240.6( 1) 1240.7(2) 1238.5(2) 1248.1(6)
2.6(l) 2.0( 1) 1.6(l) 1.4(l) 1.0(l) 2.3(3)
523.9( 1) 524.0( 1) 524.1(l) 524.2( 1) 526.4(3) 524.2(2)
2.5(l) 2.2( 1) 2.2(2) 2.0(l) 1.2( 1) 2.1(l)
1.0 0.1 0.1 0.1 0.1 0.1 0.1
2947.8(2) 2948.6( 1) 2948.5( 1)
2.3(l) 2.4( 1) 2.0(2)
1239.6( 1) 1240.9( 1) 1238.8(l) 1247.7(2) 1244.1(l)
2.2( 1) 1.3(l) 0.9(2) 1.6(3) 1.0(2)
524.1(l) 523.3(2) 526.5(4) 524.0( 1) 525.6(2)
2.1(l) 1.9(2) 1.1(l) 2.4(2) 1.7(2)
2955.8(l)
3.0( 1) 1242.2(l)
0.8(2)
528.6( 1)
1.2(2)
1.0 0.5 0.2 0.1 0.1 0.1
2948.3( 2949.3( 2949.9( 2949.8( 2949.8( 2960.2(
1239.8( 1) 1240.6(2) 1241.1(l) 1241.0( 1) 1239.1(l) 1247.9(5)
2.0( 1) 1.6(l) 1.3(2) 1.2(2) 0.8(2) 1.9(2)
524.3( 1) 524.5(l) 524.5( 1) 524.5( 1) 526.8(2) 524.3(2)
2.0(l) 1.7( 1) 1.6( 1) 1.6(l) 1.1(l) 1.7(2)
cs* CD&N 293
CDJ CS, CHjNO, GH6 GD6 n-C,H,6
311
CDJ
-2 CD&N
1) 1) 1) 1) 1) 1)
2.4(l) 2.3( 1) 2.3( 1) 2.3(2) 2.0(2) 3.4( 1)
T. Fukuda et al. / Vibrational
154 Table 2 Observed
Raman
T(K)
band widths and peak frequencies
Solvent
(cm-’
relaxarion and frequencies
) of totally symmetric
sf liquid molecules. II
modes of CDJ.
Cont. (mole fraction) I- iC”
WJ 210 260 293 CHJ 310 CHJ CS? Solvent
T(K)
190 240 293 CD3Br csz Solvent
T(K)
CH,Br, and CHCl, in the liquid phases
1.0 1.0 1.0 0.1 1.0 0.1 0.1
2136.9(2) 2137.8(2) 2138.5(l) 2139.4(l) 2139.0( 1) 2139.7(i) 2140.7( 1)
1.4(l) 1.3(l) 1.4(l) 1.2(2) 1.4(l) 1.2(l) 0.8(l)
2953.5( I) 2954.7( 1) 2956.1(l) 2957.5( 1) 2952.3(2)
2.2(l) 2.4( 1 ) 2.4( 1 ) 2.3(2) 2.2(4)
937.3( 1 ) 938.4( 1) 938.6( 1) 939.4( I ) 939.1(l) 939.8( 1) 939.6( 1 )
I;,,
2.0( I ) 1.6(I) 1.4(l) 1.1(l) 1.3(l) 1.1(l) 0.5(l)
494.0( 494.5( 494.4( 494.9( 494.4( 494.4( 195.8(
3.0( 1) 2.5( I ) 2.0( 1) 1.0(l) 1.1(2)
594.4( 1) 595.8( 1 ) 597.1(2) 598.0( 1) 600.5(4)
3.3(2) 2.8(2) 2.2(2) 1.7(2) 0.9(3)
(4)
I;,”
668.7(l) 668.5 ( I ) 668.4 668.7(l) 668.6(2)
0.9(l) 1.1(l) 1.0(5) l.?(l)
I.0 1.0 1.0 0.2 0.1
CDCI, 310
1.0 0.1 1.0
3015.7(3) 3016.8(l) 3017.5(3) 3018.1(l) 3018.0(l)
4.7(2) 4.4(2) 4.2(l) 4.1(l)
in the measurements of integrated scattering intensities which were measured relative to that of the v, band of carbon tetrachloride used as an internal standard. The sensitivity of our apparatus measured with a calibrated tungsten lamp decreased by 40% with an increase in the Raman shift from 0 to 3000 cm-‘. The Raman integrated intensity defined as the sum of the parallel and perpendicular integrated intensities,
1, =I,, $1, is given by
3
1) 1)
1295.1(2) 1296.1(3) 1296.8(4) 1298.3(2) 1294.2(2)
Cont. (mole fraction)
1.0 1.0
293
I) 1)
?.4(2) 1.8(2) 1.6(2) 1.3(l) 1.6( 1) 1.3(3) 0.9( I )
Cont. (mole fraction)
r IS0 210 260
I) 1) I)
(7)
368.6(2) 368.0( 1) 367.8(4)
0.5(I) 0.4(2) 0.4(l)
367.8(2)
0.4(
Z,=A(w,-w,)“[ X hS,/45w,
1 -exp(
1)
-fiw,lkT)]-’
,
(8)
where wL and w,~are the angular frequencies of the laser line and the Raman shift of the center of the band concerned, A is a constant characteristic of the experimental apparatus and conditions, and S,, referred to as the scattering activity [ 121 of the molecule, is given by the isotropic and anisotropic transition polarizabilities as &=45(~,/fi)(I
+&Trl
I2 (9)
T. Fukuda et al. / Vibrational relaxation andfrequencies Table 3 Observed
Raman T(K)
reorientational
band widths
(cm-‘)
and depolarization
of liquid molecules. II
ratios of totally symmetric
155
modes
CHSI 6HCH
“CH
VCI
I- reo
P
r reo
P
r reo
P
211 262 293 311
2.4(6) 3.8(8) 5.2( 13) 5.8( 19)
0.011(S) 0.007(S) 0.008 (5) 0.009(3)
2.2( 10) 3.1(13) 5.2(8) 6.2( 15)
0.066(20) 0.046( 17) 0.049(6) 0.053(28)
0.8(2) 2.6(3) 4.0(7) 6.1(7)
0.199( 14) 0.192(6) 0.180(20) 0.189(23)
T(K)
CD,1
210 260 293 310
2.0( 10) 3.7( 10) 3.9(9) 4.1(11)
0.003( 1) 0.007(3) 0.009(2) O.OlO(3)
1.6(l) 3.1(3) 4.0(6) 5.0(2)
0.078(6) 0.076( 5) 0.090( 16) 0.087( 12)
1.2(2) 2.9(2) 4.0(2) 4.8(2)
0.189(7) 0.232(21) 0.194( 16) 0.209(24)
T(K)
CH,Br 6 HCH
VCH
rreo 190 240 293
T(K)
P
rl-e0
P
rreo
P
O.Oll(7) 0.017(6) 0.018(S)
-
0.067 (8) 0.051(7) 0.064( 10)
1.1(2) 3.6(2) 5.7(2)
0.184(6) 0.165(6) 0.164(11)
CHCll 6 xcx
VCH
210 260 293 310
“Ck
“CC1
rreo
P
rreo
P
rwo
P
1.5(9) 2.8(4) 4.6(3) 4.8(3)
0.187(8) 0.150( 14) 0.177(12) 0.170(11)
0.8( 1) 2.5(l) 5.3(l) 4.1(l)
0.144(9) O.llO(l4) 0.114(8) 0.106(4)
0.3(2) 2.7(4) 3.8( 12) 6.4( 15)
0.012( 10) 0.011(8) 0.014(7) 0.014(9)
If the vibration is harmonic, eq. (9) for the fundamental transition is reduced to the well-known form 112,131 (10) (11) where (x(I) and jLl(” are respectively tive of the isotropic and anisotropic
the first derivapolarizabilities
with respect to the normal coordinate. The scattering intensity of the liquid sample, Z,, was measured at room temperature relative to that of the v, mode of carbon tetrachloride, I,., as the internal standard. The scattering activity of the sample was obtained by the following relation:
T. Fukuda et al. / Vibrational relaxation andfrequencies
156
of liquid molecules. II
given by the refractive indices of the sample and carbon tetrachloride, n, and n,, respectively. The S, value used was 1.36~ 10e9 cm4 gg ‘, which was obtained from the scattering cross section of carbon tetrachloride v, mode measured at 632.8 nm excitation by Kato and Takuma [ 141. The S, values obtained are listed in table 6. 1235
2.3. Infrared absorption intensities
mole fraction
mole fraction
Fig. 3. Plots of the peak frequencies and the isotropic band width versus mole fraction for the sHCH mode of CHxI in solutions at 262 K. (0 ) In CD,I, (0) in CS2, (a) in CD,CN.
where f, is the mole ratio of carbon tetrachloride to the sample and is taken to be l/9 in the present experiment, r( co,) and r( 0,) are the relative sensitivities of the spectrometer at Raman shifts w, and o,, respectively. The last factor on the right-hand side of eq. (12) is the correction term for the internal field effect on the intensity of the internal standard, and is
Table 4 Observed
and calculated
frequency
shifts (cm-‘)
The infrared absorption intensities of the fundamentals and the first overtones of the liquid samples were measured at room temperature with two models of Fourier transform spectrometers. A 400-4000 cm-’ range was scanned on a DIGILAB FTS-14A spectrometer with a resolution of 2 cm-‘. Liquid cell with KRS-5 windows was used and sample thicknesses were in the range from 7 to 500 urn. A NICOLET 7199 spectrometer was used for measurement of the 4000-8000 cm-’ range with a resolution of 1.2 cm-‘. The sample thicknesses were 0.5, 1 and 5 mm and the windows used were KBr and fused quartz. The first overtone of the v, vibration of CH3Br was measured in the gas phase with 0.12 cm- ’ resolution using a 10 cm gas cell. The observed intensities (13) where c is the concentration
( molecules/cm3)
at 293 K
Calculated APP CHJI CD,1 CH,Br CHC&
0.33 0.33 1.01 0.06
6 HCH
CHJ CD,1 CHjBr
VCX
CH,I CD31 CH,Br CHCI,
VCH
and 1
Observed A$-pP
A-%,
A@
A@
A-Q&
-5.01 -2.13 - 5.64 -3.24
I.42 0.79 1.63 1.19
-3.59 -1.94 -4.01 -2.05
AQ
A&w
A&Q
AGEE
-7.12
-4.44
-0.29 -0.30 -0.51 -0.07
-23.5 -16.9 -16.9 - 15.4
-1.9 -0.9 - 1.7 -0.6
-3.86 -2.89 -4.80 -2.45
-3.53 -2.56 -3.19 -2.39
-0.07 0.00 0.37
-0.67 -0.98 0.00
-0.74 -0.98 0.37
- 13.10 -9.01 - 13.45
4.37 2.96 4.63
-8.73 -6.05 -8.82
-9.47 - 7.03 -8.45
-0.57 -0.54 -0.41
-11.2 - IO.4 -9.2
-1.4 -0.8 -1.8
-0.64 -0.75 -2.85 -0.11
-4.03 -2.74 -3.32 - 1.1 I
-4.67 - 3.49 -6.17 -1.22
-2.11 -2.05 -2.06 -0.59
0.44 0.43 0.43 0.11
- I.67 -1.62 -1.63 -0.48
-6.34 -5.11 - 7.80 - 1.70
-0.51 -0.38 -1.10 -0.19
-9.1 -6.8 -13.4 -2.7
-0.3 0 - I.1 -0.3
-4.50 - 7.80
T. Fukuda et al. / Vibrational relaxation andfiequencies Table 5 Observed
and calculated
band widths
(cm-‘)
157
of liquid molecules. II
at 293 K
Calculated
Observed
P’P
pdirp
r Clec
“CH
CHJ CD,1 CH,Br CHCl,
0.25 0.28 0.83 0.11
3.84 2.01 3.38 2.67
4.09 2.29 4.2 1 2.88
6 HCH
CHJ CDJ CH,Br
0.44 0.49 0.36
0.11 0.23 0.00
“CX
CHJ CDJ CH3Br CHCl,
0.34 0.79 2.51 0.06
4.00 1.77 1.57 0.57
P
P
CH31
CDJ
0.62 0.17 0.43 0.44
4.71 2.46 4.64 3.32
0.01 0.03 0.02 0.00
2.3 1.4 2.4 4.2
0 0.2 0.1 -
0.55 0.72 0.36
27.89 13.ia 16.95
14.98 6.90 9.73
-40.35 - 18.82 -25.35
2.52 1.25 1.33
3.07 1.97 1.69
0.11 0.10 0.02
2.2 1.4 2.0
1.0 0.3 1.1
4.34 2.56 4.08 0.63
0.52 0.49 0.29 0.06
0.12 0.12 0.06 0.02
-0.47 -0.44 -0.25 -0.05
0.18 0.16 0.10 0.03
4.52 2.72 4.18 0.66
0.04 0.02 0.11 0.04
2.1 1.6 2.2 1.0
0.4 0.3 0.6 -
CH,Br
“CH
456(93) 41(9) 218(46)
“Cl
375(70) 77(15) 155(28)
“CH
478(1 lo)
s HCH
10(3) 126(27)
“C&
CHC&
and infrared
6 HC”
6 DCD
“CH “CCl
rREF.
-5.00 -1.52 -3.72 - 13.40
SS(O-rl)
“CD
rREE r
1.59 0.49 1.20 5.53
Mode
“Cl
rnlech r
4.03 1.20 2.94 a.31
Table 6 Observed Raman scattering activities ( lob9 cm” g-‘) and overtone transitions of totally symmetric modes Compound
TAR
219(49) a7(19)
the sample thickness (cm), were corrected for the internal field effect by the Palo-Wilson formula [ 15 1,
The results are listed in table 6. 3. Parameter values 3.1. Anharmonicity constants The anharmonicity constant of the oscillator takes
integrated
intensities
&(0-+2)
1.9(4) 00
2.4(5) 3.9( 10)
1.2(2) %O
0.69( 12)
(lo-”
cm2 s-’ molecule-‘)
of fundamental
k,(o+l)
r-,(0+2)
274(27) 913(70) 129( 10)
12.3( 10) 36.0(37) 4.5( 14)
154(15) 618(21)
1.2(3) 7.4(3)
73(5)
16(6)
439(44) 426(72) 435(30)
23.7(70) 15.5(45) 7.5(12)
370(30) 315(30)
51.0(6) 2.7(4)
part in the leading terms in the theoretical expressions of both the band width and the frequency shift due to the phase modulation. Besides, the anharmonicity constant is necessary to calculate the intensity parameters from the observed infrared and Raman intensities as will be shown in section 3.2. According to the second-order perturbation calculation, the third-order anharmonicity constant of the vibrational potential of a diatomic molecule [ 11,
(15) is related to the first anharmonicity coefficient of the
vibrational
where I and F indicate the initial and the final states of the transition, respectively, P, and FF are the distribution probabilities of the states, and p is the dipole moment operator, For a transition from the ground state, 0-d U,eq. ( 19 ) is approximated as
energy terms, o,w,, as [ 16 ]
IiT-&= &off&x,
.
(16)
where w, and o,x, can be obtained served frequencies of the fundamental overtone,
from the oband the first
0, =3w, --cl&,
(171
w,x,=(2w,
(18)
-e&)/2.
x I (4plO> I?. The scattering as
These relations for a diatomic molecule can be approximately applied to the total symmetric C-H stretching, the H-C-H bending, and the C-X stretching modes of the molecules studied in the present paper, because the cross terms between these and other modes are small enough to be neglected in the estimation of K,,, [ 17,181. The K,,, values obtained are listed in table 7 which also includes the diagonal elements of the inverse G matrix required for the calculation of the effect of mechanical collisions on the band shape. 3.2. Intemify
activity given by eq. (9) is rewritten
(21) where
I_
@=
r,
The dipole moment or the isotropic polarizability, represented by X, is expanded in the normal coordinate to the second order:
parameters
In part I of this series [ 11, it has been shown that the band broadening and the frequency shift owing to the dipole-dipole interaction and the dispersion interaction can be calculated from the first and the second derivatives of the dipole moment and the polarizabihty with respect to the vibrational normal coordinate. The values of these parameters are obtained from the observed infrared intensities and the Raman intensities. The infrared intensity is given by
X=,YW t__X’I ‘q+ +,pq
‘
(23)
Then the squared matrix element becomes ~(v~x~o>~~=(~~i”“)‘(v~q~o)’ +x”‘x’2’(u/q~o)
(viq2iO>
f~(X”‘)2(v~q’~O)
(241
The wavefunction of a slightly anha~onic oscillator is given by the first-order perturbation theory,
I191 zcor= c -(~Pr-P,_)I(FlPlII)12, 2xwo
(20)
(19)
I./. 3ch
Table 7 Anharmonicity constants ( 1O47cm-’ g”’‘12s-l) and G-’ matrix elements (amu) for totally symmetric modes ijHCH
L’<,H
CW CD,1 CH,Br CHCI,
W%
-K,,
C,’
-L
G;;’
- k;,,
G,’
118
0.98
10.8
11.0
I.% 0.98 0.93
5.83 11.8
0.44 0.76 0.43
I.29
36 128 193
1.04 1.69 1.23
11.0 10.4 17.9
T. Fukuda ef al. / Vibrational relaxation andfrequencies
Iv>=
I~>“-(K,;i16,,‘5)
- [v(u-l)((v-2)]“2~v-3)o},
(25)
where 1u) ’ is the wavefunction of the harmonic oscillator. Using the observed values of K,,; and p, eqs. (20) and (21) were fitted to the observed infrared intensities and the scattering activities, respectively, for the fundamental and the first overtone. The values obtained for ,LL(‘),$*), CX(‘)and cy(*) are listed in table 8, where 01~~)of the C-H stretching mode was assumed to be zero because the Raman scattering of the C-H overtone was too weak to be observed. The above procedure does not give the signs of the parameters, but only the relative signs of ,LL(‘)and pc2), and 01~‘) and (Y(*). Since there has been no established method of finding the sign of p(i), we take notice of the relationship between p(‘) and the frequency shift. The difference in vibrational frequency between dilute solutions in nonpolar and polar solvents will be due mainly to the difference in the dipole-dipole interaction between solute and solvent molecules. According to eq. (I-54), the frequency shift due to the dipole-dipole interaction in a dilute solution, AL&$&,is proportional to the third power of the permanent dipole moment of the solvent mole-
of liquid molecules. II
159
cule and the sign of the shift is given by the factor, -fpj’) +,D.$‘)/~o,. Therefore, the signs of p(‘) and P (*), which are opposite to each other, were determined so as to reproduce the order of the vibrational frequencies in nonpolar and polar solvents. The signs ofp (’ ) determined are negative for the vCH and 6HcH modes and positive for the vex mode. These are consistent with the previous studies on halogenated methanes [20-231. The p(i) and PC*) values obtained for the vcH mode of CHC13 are in good agreement with the values of p(l)= -33.0 esu g-II2 and ~(‘~‘=63.0~ 10Zoesu cm-’ g-l, reported by Boobyer [20].Thesignofa(” was assumed to be positive for all the totally symmetric modes in accordance with the previous studies [ 24,25 1. 3.3. Diffusion coefjcients The rotational diffusion coefficient of the molecular axis of a symmetric top molecule, D,, is given by the reorientational band width of the totally symmetric band [ 3 1, T,,,, D, = f ncT,,
.
(26)
Although this equation claims a constant value of r,,,, actually observed values of r,,, vary from band to band. The reason for this seems not to be understood yet. In the present study, the average of the observed
Table 8 Infrared and Raman intensity parameters of totally symmetric modes Compound CHJ
Mode
&2)
(10-5cm2g-“2)
( 10’5cmg-‘)
9.93 2.81 5.48
- 1.67 -0
-21.0 -41.8 14.7
7.2 26.6 -24.6
9.05 3.12 4.60
- 1.55 -0.72
“CBT
-36.1 - 35.3 34.5
43.5 28.6 -11.7
11.1 1.42 4.36
_ -1.72 ZO
“CH
-33.8
“CD
6DCD VCI
VCH
6HCH CHCI,
(y(1)
34.5 46.5 -9.4
“CI
CH,Br
P (2) ( 10Zoesu cm-’ g-‘)
-28.4 -51.8 19.4
VCH
6HCH
CD,1
(1) P (esu g-l/*)
VCCI
29.6
79.9
5.62
-8.86
4.3 1
-0.16
T Fukuda et ai. / Vibrational relaxation andfrequencies
160 Table 9 Rotational
and translational
diffusion
coefficients
and viscosities
(K)
D ( 10-5cm2
CHJ
211 262 293 311
5.6 10 14 17
1.1 2.5 3.7 4.5
1.29 0.68 0.50 0.44
CD,1
210 260 293 310
4.4 9.4 12 16
CH,Br
190 240 293
3.3 11 18
1.1 3.7 5.8
0.33 b’
210 260 293 310
2.8 8.3 14 16
0.64 1.7 2.7 3.3
1.85 0.88 0.64 0.53
CHCI,
[ 26 ]
s-‘)
” Estimated by eq. (29).
r re0 values
was used, and calculated values of D, are listed in table 9. The translational diffusion coefficient is calculated from the literature value of the viscosity [ 261, q, through the following relation by Gierer and Wirtz v71,
D=k,T/3xclZf;d,,
a) rl (cP)
DC (lO’Os_‘)
T
” Taken from ref.
of liquid molecules. II
(27)
where
(28) and d, and d/, are the molecular radii of the solute and solvent, respectively. Values of D calculated using the molecular radii from table 10 below are listed in table 9. Since the value of q of the CH,Br liquid has not yet been reported, it was estimated from the ratio of the rotational diffusion coefficients for CH31 and CH3Br and the viscosity of CH31 using the following equation,
(291
which is based on the relationship,
D,x 1/qd3.
3.4. Other parameters Table 10 lists other parameters used for calculating the band shape parameters. Values of the permanent dipole moment p(O), polarizability, (Y(O),ionization potential, ZP,and number density, pF, were taken from the standard tables or literature [ 28-301. The molecular diameters, d,, are given by the Lennard-Jones potential parameters [ 3 I] and the radii of atoms in molecules were obtained subtracting the bond lengths from the molecule radii. The ranging parameter of the repulsion potential is given by L = (dJ2 + a) / 12
[Il. 4. Results and discussion The band widths and the frequency of the Raman isotropic scattering of liquids were calculated using the values of the parameters estimated in section 3. Here, we are concerned with the results of the calculation for neat liquids at 293 K. The effect of solvents and temperature on the band width and the frequency will be discussed in the next paper.
T. Fukuda et al. / Vibrational relaxation andfrequencies
Table 10 Parameter values required for the calculation ofrand
AS2
(0) ,u
(ym
I
d,
a
( 10” molecule cm-‘)
(D)
( 10-25cm3)
(ev)
(10-8cm)
(lo-*cm)
9.63 10.94 7.48
1.62 1.84 1.04
73.0 55.5 82.3
9.54 10.54 11.48
4.66 4.46 5.18
6.6 25.0 35.8 55.3
13.60 13.01 11.84 10.45
H Cl Br I
of the vibrationalfrequency
The calculated shift is the sum of the two components arising from the electrical and mechanical interactions, CL-S2=AQ,ec+ALnmec,, .
(30)
The electrical part, AL&,,, consists of the contributions from the dipole-dipole and the dispersion interactions, AL& = A&Pip + A@isp 9
(31)
where ALPip is given by eq. (I-54) and ALPP by eq. (I-64b). The mechanical component, AL&_,,, is divided into the attractive part and the repulsive part, which involve A ,!,I) and R 1(/’ ) 3respectively, in eq. (Ig6), AL!mech=AL2A+AP.
(32)
From a viewpoint of interaction process, AL2is decomposed into two terms due to the resonant energy exchange and the phase modulation, m=
UREE
161
P
CH31 CH3Br CHC13
4.1. Shift
of liquid molecules. II
+ ~Pbl
,
(33)
where AG!REE= A@&
+ mdizR
(34)
is the part of A.@.,,, and the sum of all other terms is AL&. The calculated values of each term of the frequency shift and their total are listed and compared with the observed values in table 4. Agreement between the calculated and experimental values of AL2 is reasonably well for the &., and vex modes but considerably poor for the vcH mode.
0.81 1.01 1.13 1.24
The calculated shifts of the vcH modes are l/2- l/4 of the observed values. This fact suggests that the CH stretching force constant of the liquid molecule is slightly different from the gas phase value, owing to an environment-induced change in the electric charge distribution of the molecule. If the difference between calculated and observed frequencies is attributed to the change in the force constant, the molecule has a 0.5-l % smaller force constant in the liquid than in the gas phase. Both the calculated and observed results show that the phase modulation, or the average static field effect, is the dominant part of the gas-liquid frequency shift. The contribution from the resonant energy exchange, or the vibrational resonant coupling, is in a range of lo-20% of the total shift. The calculation shows that the relative importance of the parts of the intermolecular potential varies among the three modes. The contributions of the mechanical and the electrical interactions to the frequency shift are comparable for the vcH mode, where the dominant terms are APiSp and ALP. For the 6 HCHmode, the mechanical one is overwhelmingly important, while for the vex mode the electrical one is superior to the mechanical one. The main terms for the &+cH mode are AD’ and AQR which have opposite signs and cancel each other to some extent. For the vex mode, APiSp and ALP are the main terms. 4.2. Band broadening The calculated band width is composed trical and the mechanical parts, r= re,ec + rmectl .
of the elec-
(35)
7’. Fukuda et al. / Vibrational relaxation andfrequencies
162
The electrical part is given by r
=rdip+rdlsp l?lCC
rd’p =
rdplfi + rg
rdlsp = r”,a
(36) ,
+ rdis!$ RE
where Tdp;E; , r&, (I-56), (I-57), The mechanical
(37) (38)
>
r$z,
and r$;% are given by eqs.
(I-65b), and (I-65a), part is written as
r mech =rA+rf+rAR ,
respectively.
(39)
where r,,,,,, is given by eq. (I-87 ), and r” consists of the terms involving only A I,’) and A I’:‘, TR is the term involving only R/,” and Rj?‘, and PR is the cross term involving both the A,, and R,, factors. The calculated values are listed and compared with the observed values in table 5. The calculated total band widths, r, are in moderately good agreement with the observed widths. It can be seen from table 5 that the phase modulation is the dominant process for the broadening of the bands as for the shift of the frequencies. For the vCH and vex modes, the contribution of the electrical interaction is larger than that of the mechanical one, but they are in the reverse order for the Z&n mode. It should be noticed that the cross term rARplays an important role. This term has a negative sign and cancels out the self-terms to a large extent. Then the magnitude of rmech is an order of magnitude smaller than its components. Without the cross term, the calculated total band widths of the vC.H and the 6,,:, modes would be an order of magnitude larger than the observed band widths. To the best of our knowledge, this is the first to notice the importance of the cross term which arises from dealing with the repulsive and the attractive parts of the potential simultaneously.
5. Conclusion The present theoretical expressions for the Raman band widths and the peak frequency have been shown to be useful for studying the intermolecular potential and the vibrational relaxation process in the liquid phase. In spite of the simple model used and rather rough estimation of some physical parameters, the calculated band widths and frequency shifts were in
qfllquid molecules. II
moderate agreement with the observed values. It is particularly important to deal with the intermolecular potential including all possible terms and to treat every possible process of the relaxation in the same theoretical scheme. By doing this, we could compare the effects of the potential terms and compare the contributions of relaxation processes to the Raman band shape. For the first time, we found out the important role of the cross term, PK. in the band broadening. Quantitatively, the present results of the calculation are not yet satisfactory. To improve the theory, the first step is to obtain more precise values of the parameters. By means of those, the theory can be tested more quantitatively, and then, it will become possible to find out a way of improving the theory if necessary. Two possible ways of improvement are to include the effect of the intramolecular mode coupling and to allow for the internal motion of solvent molecules [ 1 1.
Acknowledgement The authors thank Dr. Kohji Fukushi suggestions and discussions.
for helpful
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