A. A. CLIFFORD AND B. CRAWFORD, JR.
1536
Vibrational Intensities.
XIV.
The Relation of Optical Constants to
Molecular Parameters
by A. A. Clifford and Bryce Crawford, Jr. Molecular Spectroscopy Laboratory, Department of C h i s t r y , University of Minnesota, Minneapolis, Minnesota 66466 (Received November 1, 1966)
A method is described in which optical constants for condensed phases can be corrected for dielectric effects. This involves transformation to the complex “local susceptibility,” which is defined as the ratio of the polarization to the “internal field.” The local susceptibility can then be fitted to a Van Vleck and Weisskopf band shape, using a simple graphical technique, and molecular parameters obtained. The results of this procedure, applied to the attenuated total reflection (ATR) data from the preceding paper, are given.
It has long been recognized that the absorption of light by molecules in the condensed phase is significantly affected by electric fields due t o the surrounding absorbed particles. This dielectric effect has given rise both to difficulties in rationalizing band shapes and to methods for making dielectric corrections to absorption maximal and intensitiesa2 Now that it is possible by various methods, including attenuated total reflection (ATR), to make accurate measurements of both real and imaginary parts of the complex rei~)],a further attempt to fractive index, A [ = n ( l fit experimental data to theoretically predicted behavior can be made. A number of models for absorption have been previously discussed. These include the damped harmonic oscillator, the Lorentz oscillator,3 and a modification of the latter, translated into quantum mechanical terms, by Van Vleck and Wei~skopf.~All these models give rise directly to expressions relating the polarization of the dielectric, P, to the field felt by the molecules, the “local field’’ or “internal field,” E i . We may usefully define their ratio as the “local susceptibility,” 6 ( = C ’ iC”). For example, the Van Vleck and Weisskopf model gives
+
+
where S is the strength factor and the function in is the vacuum the shape fact”r’ brackets we can wavenumber of the incident radiation and Y O is the waveThe JOuTnd of Physical Chemistry
number corresponding to the oscillator’s “natural frequency.’’ y represents a damping constant also expressed as a wavenumber which is related to the mean collision or relaxation time, 7 , thus y = 1/27rcy
The strength factor is given by S = N/3hvoclpifl
(2)
with N the concentration of oscillators per cubic centimeter and pu the electric-moment matrix element for the transition.
Calculating the Local Susceptibility I n the gas phase the refractive index, n, is essentially unity; hence, to a good approximation, the absorption coefficient is proportional to the imaginary part of the local susceptibility and can be used directly in bandshape considerations. I n condensed phases, however, we need to calculate the local susceptibility more rigorously. The problem is one of obtaining a value for the local field; for this we naturally first turned to the Lorentz-Lorenx equation and have as yet found it (1) J. G. Kirkwood, cited by W. West and R. T. Edwards, J . Chem. Phys., 5 , 14 (1937); E. Bauer and M. Magat, J . Phys. Radium, 9, 319 (1938). (2) P. N. Schatz, Spectrochim. Acta, 21, 617 (1965); 9. R. Polo and M. K. Wilson, J. C h m . Phys., 23, 2376 (1955). (3) H.A. Lorentz, “The Theory of Electrons,” G. E. Stechert, New York, N. Y., 1923. (4) J. H.Van Vleck and V. F. Weisskopf, Rev. Mod. Phys., 17, 227 (1945).
OPTICAL CONST.4NTS AND MOLECULAR PARAMETERS
1537
Table I : Optical Constants and Local Susceptibilities, Experimental and Calculated, for the 678-Cm-l Band of Liquid Benzene Fre.C’
quency, cm -1
nr
n
630 632 634 636 638 640 642 644 646 648 650 652 654 656 658 660 662 664 666 668 670 672 674 676 678 680 682 684 686 688 690 692 694 696 698 700 702 704 706 708 710 712 714 716 718 720
0.011 0.012 0.012 0.013 0.015 0.017 0.020 0.020 0.023 0.025 0.028 0.033 0.041 0.060 0.069 0.093 0.137 0.194 0.280 0.512 0.782 1.117 1.307 1.189 0.986 0.768 0.487 0.341 0.214 0.124 0.087 0.063 0.051 0.042 0.036 0.029 0.024 0.021 0.019 0.017 0.016 0.014 0.013 0.012 0.012 0.011
1.619 1.626 1.628 1.635 1.643 1.649 1.655 1.678 1.694 1.719 1.728 1.754 1.772 1,762 1.819 1.865 1.905 1.980 2.088 2.211 2.256 2.064 1.750 1.277 1.042 0.885 0.848 0.840 0.925 1.033 1.099 1.148 1.181 1.213 1.238 1.261 1,278 1.295 1.310 1.314 1.317 1.335 1.339 1.345 1.350 1.362
Exptl
Calcd
Exptl
Calcd
0.0838 0.0845 0.0847 0.0855 0.0863 0.0870 0.0876 0.0900 0.0917 0.0942 0.0951 0.0977 0.0995 0.0986 0.1040 0.1083 0.1122 0.1190 0.1285 0.1419 0.1535 0.1613 0.1641 0,1264 0.0645 0.0028 -0.0211 0.0250 -0.0112 0.0057 0.0157 0.0230 0.0279 0.0325 0.0361 0.0393 0.0416 0.0440 0.0460 0.0466 0.0470 0.0494 0.0499 0.0507 0.0514 0.0530
0.0841 0.0848 0.0854 0.0862 0.0870 0.0879 0,0889 0.0900 0.0912 0.0926 0.0942 0.0960 0.0981 0.1005 0.1034 0.1068 0.1109 0.1160 0.1224 0.1304 0.1404 0.1517 0.1595 0.1430 0.0707 -0.0020 -0.0192 -0.0118 -0.0007 0.0092 0.0171 0.0234 0.0285 0.0326 0.0360 0.0389 0.0413 0.0434 0,0452 0.0468 0.0481 0.0494 0.0505 0.0514 0.0523 0.0531
0.0012 0.0013 0.0013 0.0014 0.0016 0.0018 0.0021 0.0021 0.0024 0.0025 0.0028 0,0032 0.0039 0.0058 0.0064 0.0083 0.0118 0.0156 0.0205 0.0331 0.0464 0.0712 0.1018 0.1538 0.1694 0.1462 0.0865 0.0583 0.0352 0.0196 0.0133 0.0094 0.0075 0.0061 0.0051 0.0041 0.0033 0.0029 0.0026 0.0023 0.0022 0.0019 0.0017 0.0016 0.0016 0.0014
0.0011 0.0012 0.0013 0.0014 0.0016 0,0018 0.0020 0.0022 0.0025 0.0029 0.0033 0,0038 0.0045 0.0053 0.0064 0.0078 0.0098 0.0127 0.0169 0.0234 0.0341 0.0528 0.0868 0.1413 0.1788 0.1421 0.0878 0.0537 0.0349 0.0240 0.0175 0.0132 0.0103 0.0083 0.0068 0.0057 0,0048 0.0041 0.0036 0.0031 0.0028 0.0025 0.0022 0.0020 0.0018 0.0016
-
quite adequate in explaining our results. (An unsuccessful attempt to improve upon it is outlined later.) This equation relates the local field, Ei, to the macroscopic field, E; thus Ei
=
E
=
ji being the usual macroscopic electric susceptibility; or, in more familiar form:
(3)
(47r/3)€3
and hence we have 1/e = l/? - (47r/3)
+
where 2 (= e’ id’)is the complex dielectric constant. Separating into real and imaginary parts Volume 70,Number 6 May 1966
A. A. CLIFFORD AND B. CRAWFORD, JR.
1538
C’
=
[
3/4n 1 -
(E’
+ ++
3(e’2)2 2,
d‘2
1
2.0
I
I
I-
.os2
0
So, by using Maxwell’s relation, 2 = A2, and then eq 4 experimental refractive index data can be transformed into values of the (Lorentz-Lorenz) local susceptibility. An example is given for the 678-cm-’ band of benzene in Table I and Figure 1, using the ATR data from the preceding paper. Transformation to local susceptibility apparently produces similar though changed traces. In particular the plot of C” against frequency is more symmetrical than the corresponding plot for m. Also the frequency which gives the maximum value of C” is seen to be slightly different from the absorption “band center.”
A Graphical Method for Comparison of Experimental Results with the Van Vleck and Weisskopf Band Shape Basic Equations. Having transformed our data into variables which are more relevant to theoretical considerations we may now attempt to understand the band shape. For the infrared region the Van Vleck and Weisskopf expression is likely to be the most useful. Equation 1 gives the contribution of a single transition to the local susceptibility. In practice, we must take into account other transitions and also a contribution due to nonresonant or Debye absorption. Fortunately, if the other bands are sufficiently remote and we choose frequencies high enough so that nonresonant absorption is negligible, the local susceptibility resulting from them will be a constant real quantity which we can call K . Equation 1 now becomes
+ + +v) iy+ ir1 - v) - ir -
YO
C=K+S[ (YO
(YO
C’ = K
+ 2S[ + 7 4
C”
+
=
2 4
Y4
I
I
700
6 50
v (cm’)
Figure 1. Optical constants and local susceptibilities for the 678-cm-1 band of liquid benzene. For the local susceptibilities, the line represents experimental data and the circles are calculated points.
to frequencies in the far-infrared and higher, y2 > y2. vo is thus obtained from the C” maximum and an effective dielectric correction is easily made. This correction will not, of course, take into account the change of the potential function of the molecule as it moves into the condensed phase. What it will do is to give a corrected frequency shift
OPTICALCONSTANTS AND MOLECULAR PARAMETERS
1539
Table 11: Band Parameters Obtained from ATR Data Using the Plotting Technique“
Benzene Benzene Chloroform Main band Carbon tetrachloride Subsidiary Ped
Frequency of nx maximum, om-’
om-1
cm-1
seo
K
674 1035 762 786
678 1035 769 792
3.91 5.56 6.95 5.84
1.36 0.96 0.76 0.91
0.0691 0.0647 0.0604 0.0235
108 98 15,400 4.37 4.32 990 206 190 30,100 182 135 30,100
(762)
(766)
(10.3)
(0.52)
(0.106)
(69)
r YO,
y,
lo%,
loss’
loss”
(91)
(gas
rc
(14,800)
I
rliq
rPw
phase)
21,000 1,100 37,200
15,800 850 30,000
12,600’ 820‘ 36,000”
58,000
44,400
49,500d
are given in square centimeters per mole. * J. Overend in “Infra-red Spectroscopy and Molecular Structure,” M. Davies, Ed., Elsevier Publishing Co., Amsterdam, 1963, Chapter X. ’ J. Morcillo, J. Herranz, and J. F. Biarge, Spectrochim. Acta, 15, 110 (1959). L. P. Lindsay and P. N. Schatz, ibid., 20, 1421 (1964).
’ Intensities (1’)
from which a more precise evaluation of intermolecular forces in the condensed phase can be made. The extent of this correction in some bands can be seen by comparing columns 2 and 3 of Table 11. The figures in Table I1 have been calculated from the ATR data for the four bands published in the preceding paper. In the case of the weak benzene 1035-cm-l band the correction is negligible. In the other cases it amounts to a shift of a few wavenumbers to higher frequencies. The Co-C’ Plots. Values of C” can now be calculated throughout a band and plots of C” against C‘ can be drawn. According to eq 10 this should give a straight line of slope y . Plots for the four bands mentioned are given in Figure 2. Acceptable straight lines are obtained in all cases with, however, partial deviations produced by the “hot” band on the benzene 1035cm-’ figure and by the minor peak of the carbon tetrachloride doublet. These off -line points are therefore not included in the calculation of the values of y which are shown in T:rble 11. Calculation of Intensities. The absorption cross section r, calculated in the gas phase from the BouguerLambert absorption coefficient cy
with C, the molar concentration of the gas,s is physically significant since it bears the simple relationship
r
= 8a3~o/3h~lpr,12
(13)
to the electric-moment matrix element but,. For the condensed phase, our model indicates that the same cross section, for which relation 13 will hold,
can be calculated from the strength factor S of the local susceptibility; we find
rc = 8 ? r a Y , ( ~ / 4 ~
(14)
where vo is the wavenumber and N and d are the molecular weight and the density. I’” is, of course, the intensity corrected for the dielectric effect, which is otherwise obtained using the Polo-Wilson equation.2 Values of S can be obtained now that vo and y are known by plotting C’ and C” against the real and imaginary parts of the shape factor. As can be seen from eq 6 and 7, both these plots should give a slope of 2s. Figure 3 gives an example of such plots for the benzene 678-cm-l band. The straight-line behavior again represents a test of the approach. In this way two slightly different values of S, as shown in Table 11, are obtained from each band. S’ is the value obtained from the real part and S” from the imaginary part of the local susceptibility. Then from the average of X‘ and SI’, rCcan be obtained using eq 14. These are shown in Table I1 alongside values @‘Iiq) obtained by the conventional technique of eq 12. Comparison of Experimental and Theoretical Local Susceptibilities. Plots of the real part of the local susceptibility against the shape factor will also give values for the contribution to the local susceptibility from outside the band, K , shown also in Table 11. Now that all the parameters in the Van Vleck and Weisskopf equation have been determined, theoretical values of the local susceptibility can be calculated and compared with experimental values. These are shown for the 678-cm-l benzene band in Table I and Figure 1. (5)
B. L. Crawford, Jr., J. Chem. Phys., 29, 1042 (1958). Volume 70, Number 6
M a y 1966
A. A. CLIFFORD AND B. CRAWFORD, JR.
1540
w
VI V
7
Y
0.4 a
-0.41
/
c’ +
4.01
-
I
0.00
LOCAL
0.04
i
630
I
I
0.01
0.10
SuSCEPTlBlLtTY
(RE4L
1
0
0
I
.Z 0
0.15
P4RT)
LOCAL
1
Malm bond ’Hot‘‘ band
20
E
-
&
1
1
-0.04I
I
1
I
I
o Maln band
‘
0
0
0
Subsidiary ptak
0 0
‘
e..,
The region at the lower frequency side of the band center, where the biggest deviations occur, is also the region where the experimental results are the least reliable. The agreement at the edges of the band is particularly gratifying.
Comparison with Other Methods of Obtaining Intensities Our present approach offers very real advantages, aside from the pleasure of finding such a nice fit to a simple model, since the value of is not obtained by an integration and hence avoids those problems of “integration limits’’ and “wing corrections” which have long plagued the student of intensities. It does, of course, assume a particular model, subject to a “straight-line” test and this may not be possible in all cases. It is worth mentioning, therefore, that a generalized. I?’ can be obtained from the integral, over the band, of the imaginary part of the local susceptibility using the following expression The Journal of Physical Chemistry
(REAL PART)
0.0
=!
Y
SUSCEPTIBILITY
I”
= 87r2(M/d)JC”dv
(15)
which is independent of a part,icular band-shape model ; this reduces appropriately to the gas phase r of eq 12 and the Van Vleck-Weisskopf rCof eq 14. Also it has been argued by Schatz2 that the PoloWilson equation will adequately correct for dielectric effects, on the basis of the Lorentz-Lorenz field, in spite of the variation of the refractive index, n, throughout the band, provided the appropriate value o n is used. Thus, our graphical method and the PoloWilson equation should give the same result. This is seen to be essentially true by comparing the values of rc and rPw(obtained by a Polo-Wilson correction to 171iq) in Table 11. We attribute the discrepancy in the benzene 1035-cm-’ band case to integration difficulties, because of the associated hot band. We also feel that, as well as integration problems, the Polo-Wilson method suffers from lack of precision due to the dif’
OPTICALCONSTANTS AND MOLECULAR PARAMETERS
1541
I
SHAPE
-20 FACTOR
0 20 (REAL PART1
40
I
Y
p
(ern-’)
Figure 4. Comparison of experimental and theoretical local susceptibilities for the carbon tetrachloride 792-cm-1 doublet. The line represents experimental data and the circles are calculated points.
o
Treatment of Data for Adjacent Bands
0.00 SHAPE
FACTOR.
(IMAGINARY
PART1
Figure 3. Plots of the local susceptibility against the shape factor for the liquid benzene 678-cm-l band.
ficulty which arises in practice of assigning the correct value to n. Values for gas phase intensities are also given in Table 11, although these are not expected to equal the corrected liquid phase intensities in all cases because of nondielectric or “chemical” actione6 We have not needed to rely here on theoretical considerations to predict when both intensities hhould be the same and hence to decide whether the Lorentz-Lorenz correction was adequate. Instead, our criterion has been “straight-line” behavior: a point which is amplified later.
The method outlined in the last section only applies, strictly speaking, to single remote bands. Thus, the carbon tetrachloride doublet main band did not give good plots. However, some approximate parameters were obtained both for the main band and the subsidiary peak. Data for the subsidiary peak are given in parentheses in Table 11. These values were later refined by least squares to fit the local susceptibility data. The final parameters are shown in Table 111, and Figure 4 shows how the experimental and calculated values compared at the end of the refinement process. As mentioned in the preceding paper, these data were not obtained by the later techniques and are not so reliable as the benzene data. We are therefore satisfied by the agreement obtained. Again we note an advantage of the present approach over integration procedures, in that overlapping bands can be analyzed in terms of their separate intensities, without the subjective curve sketching which has been resorted
Improvement of the Method The four bands studied so far have obeyed the as(6) W.B. Person, J . Chem. Phys., 28, 319 (1958); A. D.Buckingham, Proc. Roy. Soe. (London), A248, 169 (1958). (7) E.g., A. D. Dickson, I. M. Mills, and B. L. Crawford, Jr., J . Chem. Phys., 27,445 (1957).
Volume 70, Number 6 M a y 1966
1542
A. A. CLIFFORD AND B. CRAWFORD, JR.
Thus, to a first approximation, the slope of a Co-C’ at any point will be
Table 111: Band Parameters Obtained for the Carbon Tetrachloride 792-Cm-‘ Doublet by Leasesquares Refinement” om-’
om-’
10**r, sea
IPS
Fa
791.2 766.6
5.2 10.3
1.02 0.52
180 107
34,000
Y.
YO.
Main band Subsidiary peak
19,600
K
i
0.0596
” Intensities (r)are given in square centimeters per mole. sumptions of the Lorentz-Lorenz field and the Van Vleck and Weisskopf band shape to within the reliability of the experimental data. However, two possible improvements were explored. Jmprovemenls to the Lorentz-Lorenz Field. For an isotropic medium, the local field at a point in a dielectric can in general be expressed by3 Ei
=
E - a(4s/3)P
(16)
At the center of the band, as v --t vo, C(y)” 2S(r)v/y. The slope will therefore be
At the edges of the band C(y)” v2)/(vo2- v2) and the slope becomes Nr)rdr SS(r)dr
+
+
2S(y)vy(vo2
+
J
From Schwarz’s inequality it follows that
where a is a scalar which depends on the arrangement of charged particles around the point under consideration. In the cases where the Lorentx-Lorenx equation holds, a, of course, becomes unity. Using eq 16, the local susceptibility is given by
This was used to calculate C with values of a slightly different from unity. This, however, gave C” - C’ plots which were worse straight lines than those obtained previously. The Efect of a Range si Relaxation Times. Models of dielectrics in which molecules are assumed to have a range of relaxation times are often used to explain nonresonant absorption data.8 Application of this idea to resonant absorption was mentioned by Van Vleck and Wei~skopf.~Equation 5 would become in this case
e=K +
e(r)dr
range of y
(18)
where
and S(y) has only positive values. Separating into real and imaginary parts, one obtains instead of (lo)
C’ = K
+ 2j’S(r)dr +
The J o u d of Physical Chmietry
and thus the slope of the Co-C‘ plot should be greater at the edges of the band than in the band center, with the straight line becoming a figure eight. Actually the deviations from linearity are, if anything, in the opposite direction and the data do not therefore suggest a range of relaxation times.
Physical Significance of the Results Physically speaking, our fit of the Van Vleck and Weisskopf band shape indicates that we are observing single transitions and that rotational motion is effectively suppressed. These systems therefore are more significantly described in terms of the angular position of the molecules rather than their angular momenta and they approximate to Shimizu’s “Debye limit. ” Recently some interesting work has been done in which band shapes have been discussed in terms of time-correlation functions. In these terms, the treatment outlined here is equivalent to the application of a damped-oscillator correlation function to the local -
~~
~
(8) H. Frohlich, “Theory of Dielectrics,” 2nd ed, Oxford University Press, London, 1958; C. P. Smith, “Dielectric Behaviour and Structure,” McGraw-Hill Book Co., Inc., New York, N. Y., 1955; C. J. F. Battcher, “Theory of Electric Polarisation,” Elsevier Publishing Co., Amsterdam, 1952; K. 5. Cole and R. H. Cole, J . Chem. Phys., 9, 341 (1941) (9) R. G.Gordon, ibid., 43, 1307 (1965); H.Shimim, ibid., 43,2453 (1965).
PROTON RESONANCE SPECTRUM OF BUTATRIENE
susceptibility. (Compare Frolich's derivation of equations for the dielectric constant and loss in the region of resonant absorption.6) The small deviations from the Van Vleck and Weisskopf band shape of the data presented here compared with the probable errors do not justify transformation for further investigation of a correlation function. However, this may be worthwhile with other ATR data, especially where partial rotation appears to be significant.I0 I t seems likely that correlation functions derived from the local susceptibility, when this can be adequately calculated, rather than from the absorption coefficient, will prove to be more fruitful for strong bands in condensed phases.
Acknowledgments. We gratefully acknowledge the financial support for this work obtained from the National Science Foundation through Grant GP-3411 and would also like to thank our colleagues in this
1543
laboratory, especially Dr. John Overend, for many helpful discussions.
Addendum It has been brought to our attention that the idea of transforming a spectrum to correct for dielectric effects is not a new one. Bakhshiev, Girin, and Libovll have used expressions for the internal field to calculate what they call true absorption spectra. Their method of obtaining the true absorption spectra using the Lorentz-Lorenz field is identical, except for a constant factor, with that for the imaginary part of the (Lorent,z-Lorenz) local susceptibility as described in this paper. (10) E.B., W. J. Jones and N. Shepperd, Trana. Faraday Soc., 56, 625 (1959). (11) N. G. Bakhshiev, 0. P. Girin, and V. S. Libov, Opt. Spectry. (U.S.S.R.), 14, 255, 336, 395 (1963); 16, 549 (1964).
Proton Resonance Spectrum of Butatriene'a
by Stephen G. Frankisslb and Ikuo Matsubara Mellon Inatitute, Pittsburgh, Penmylvania
(Received November 8, 1966)
The proton resonance spectrum of butatriene a t -55" is reported. The long-range trans (H,H') and cis (H,H') coupling constants are equal to 8.95 cps, which agrees well with Karplus' calculated value (7.8 cps). The proton chemical shifts of ethylene, allene, and butatriene are briefly discussed.
The long-range coupling constants (JHH~) between protons separated by a-electronic structure have been the subject of several recent experimental2-* and theor e t i ~ a studies. l ~ ~ ~ ~Karplus, in particular, has emphasized the importance of the contributions of a-electron to these coupling constants.10 He has terms (JHHJ[T]) estimated the values Of J"J[T] in a few Systems and has predicted a large (7.8 cps) value for J"![a] in butatriene. Although the observation of J"' in buta-
triene provide check Of his the only reported attempt to observe its Spectrum Was
thwarted by its rapid polymerization.l1 We have found, however, that a t low temperatures (below (1) (a) This work was supported by the National Science Foundation under Grant GP-1628; (b) Department of Chemistry, University College, Gower St., London, W.C.1, England, to whom all communications should be sent. (2) D. F. Koster and A. Danti, J. Phys. Chem., 69, 486 (1965), and references quoted therein' (3) M. W. Hanna and J. K. Harrington, ibid., 67, 940 (1963). (4) R. K. Kullnig and F. C. Nachod, ibid., 67, 1361 (1963). (5) A. A. Bothner-By and R. K. Harris, J . A m . C h m . Soc., 87, 3451 (1965), and references quoted therein.
Volume 70, Number 6 May 196%
