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The effect of resonant vibrational coupling on the frequency of the ν2 mode of liquid methyl iodide in Raman scattering and infrared absorption
Volume 172, number 6
CHEMICAL PHYSICS LETTERS
21 September 1990
The effect of resonant vibrational coupling on the frequency of the v, mode of liquid methyl iodide in Raman scattering and infrared absorption G. D&e, D. Lindner InstitutflrPhysikalischeund TheoretischeChemieder TechnischenUniversitiit, O-3300Braunschweig,FederalRepublicof Germany
W. Richter and D. Schiel Physikulisch-Technic Bundesanrtalt, D-3300 Braunschweig,FederalRepublicof Germany Received 17 January 1990; in final form I June 1990
The v2 mode of methyl iodide has been measured in isotropic and anisotropic Raman scattering, and in infrared absorption in the neat liquid and in isotopic dilution with the deuterated compound. The local field affecting the shape and position of the intense infd profde has been taken into account by a continuum dielectric approximation. The isotopic dilution shifts observed indicate the action of resonant intermolecular vibrational coupling which presumably is caused by transition dipole interaction. The results are, in principle, in agreement with current theories. For a comparison, certain approximations arc necessary (MSM, etc.), which seem to be responsible for the fact that quantitative agreement is not yet very good.
1. Introduction The position and the shape of Raman scattering and infrared absorption profiles of the vibrational modes of molecules in the liquid state are in certain cases well-suited probes for studying the structure and molecular dynamics in liquids. In this connection it is sometimes observed that in neat liquids, anisotropic Raman bands have higher peak frequencies than the corresponding isotropic bands. This effect, usually called the non-coincidence effect, is caused by resonant intermolecular vibrational coupling (RIVC) if there is a certain amount of molecular orientational order. In recent years, the problem of the different kinds of influence of RIVC on isotropic and anisotropic Raman scattering and infrared absorption has been taken up again in experimental [l-4] and theoretical [S-9] investigations. The results obtained by several authors do not always agree. The only experimental method which can show if RIVC affects the position and shape of a certain vibrational band is the so-called isotopic dilution method [ 10 1. The 0009.2614/90/$03.50
band parameters are observed in the neat liquid and in dilution with an isotopically substituted species of the same compound. In this way, it is possible to decouple the vibrations of neighbouring molecules (if the corresponding vibrational bands in the two isotopic species do not overlap). In anisotropic Raman scattering such experiments suffer from the low intensity. This scattering is very weak in most cases, making it difficult to obtain good results in dilution experiments. This is particularly true if there are perturbations by hot bands or other neighbouring bands, or if the reorientational broadening is large. Published data are not therefore always reliable, and what is more, there are also some discrepancies in the theories. In general such problems are treated by studying the vibrational relaxation function C,(t), which is obtained by a Fourier transformation of the frequency-dependent intensity function Z(w) of the band, because it is often easier to treat timedependent effects in this time representation. In this paper we wish to study the effect of RIVC on band positions (first moments). Such effects (shifts) are a
0 1990 - Elsevier Science Publishers B.V. (North-Holland)
453
Volume 172. number 6
CHEMICAL PHYSICS LETTERS
measure of an ensemble average of that part of the intermolecular interaction which acts on the vibrational energy. This average is time independent and describes the modulation of C,(t) with a constant frequency. It is therefore unnecessary to transform the spectroscopic results into the time domain if we are dealing only with band shifts due to RIVC, but we do need the time representation for the treatment of the other band shape or vibrational correlation parameters, and these will be the subject of following papers.
2. Experimental A Coderg triple-monochromator Raman spectrometer was used with an argon ion laser. For the pressure-dependent measurements we used a pressure cell manufactured by Nova Swiss. As the band shifts upon isotopic dilution sometimes are very small (especially in anisotropic scattering), some atmospheric-pressure measurements were performed using a special technique: To ascertain accurate wavenumber differences for the band maxima obtained with the neat liquid and the isotopic dilution, both bands were recorded in the same run. The principle of this technique has been described by Laane and Kiefer [ 11,12 1. Two sample cells are mounted on a rotating cell holder which can rotate around 180” and back into the primary position. The laser beam passes through the neat liquid cell in one of the two positions, and through the isotopically diluted solution in the other. A microcomputer controls the counting time of the scattered photons, the motion of the cells, the monochromator drive and also collects the counting rates. With this arrangement we studied the isotopic dilution effect in anisotropic scattering and also repeated earlier measurements of this effect in isotropic scattering. The band maximum observed in W scattering geometry (polar scattering) was identified with the band maximum of isotropic scattering. This is possible because the depolarisation ratio of the vz band of iodomethane is very small. The infrared absorption profile of the v2 mode of the neat liquid was measured by the two-thickness method according to 454
21 September 1990
(1) where z1 and 7, are the transmittances of the absorption path lengths d, and d2. For the latter, values of 4 and 12 m were used in a variable cell with KBr windows. In this way the absorption coefficient of the strong band is obtained in good approximation without disturbances from the frequency-dependent reflection losses. For the low-temperature measurements, small cuvettes made of thin CaFz windows with fixed spacings were used which fitted into a toploading liquid-nitrogen-cooled cryostat. In the case of the isotopic dilution measurements, a single thickness only was used (except with the highest concentration).
3. Results Table 1 shows the results for the isotropic and anisotropic scattering experiments at two different temperatures. The values in table 1 are averaged over four experimental runs. The band maxima have been used instead of the first moments since the measured profiles are symmetrical if the “hot” component in the low-frequency wing is eliminated. The appearance of the non-coincidence effect (different band maxima in isotropic and anisotropic scattering) in the case of the neat liquid, together with the band shift upon isotopic dilution, indicates the action of RIVC. In additional measurements, we have found that the mole-fraction dependence of the isotopic dilution shift of the isotropic band is exactly linear. Table 2 shows the corresponding results for the inTable 1 Shifts of the u2band in isotropic (IS) and auisotropic (AS) scattering upon isotopic dilution to a mole fraction of CHJ of 0.1. Uncertainty of relative frequencies: 0.1 cm-’ Temp. (IQ
213 283 313
Baud maximum in neat liquid (cm- ’)
shift (cm-‘)
IS
AS
IS
AS
1240.3 1240.7
1241.0 1241.0
1.7 1.2 1.0
0.6 0.5
Isotopic dilution
Table2 Shifts of the vz band in infixed absorption upon isotopic dilution. Uncertainty of fresuencies: 0.1 cm-’ Temp. (IQ
Band maaimum in neat liquid (cm-‘) measured
correctedfor internal field
Isotopic dilution shit? of the correc@dband maximum for mole fraction x of CHJ in CDS x
shift (cm-‘)
296
1239.5
1239.7
0.05 0.45 0.83
1.8 0.8 0.3
223
1238.1
1238.6
0.05
2.0
frared absorption experiments. As the yz absorption
is rather strong, it is necessary to correct the band profiles obtained for the internal field. For this correction we followed the proposal of Clifford and Crawford [ 13 ] which is based on a continuum dielectric approximation. This correction results in a frequency increase of 0.2 cm-’ at 296 K and OS cm-’ at 223 K in the neat liquid.
4. Diaeussion and comparisonwith theory There are only a few theoretical papers which deal specifically with the non-coincidence effect [ 5,8,9]. These papers are confined to band shifts due to RIVC and do not treat band-shape effects. RlVC band shifts can be described by the changes of the first moments. The fundamental equation, first derived by Wang [ 41, can be written as &I”C =N< GM))
21 September 1990
CHEMICALPHYSICSLETTERS
Voiume 172,number 6
in the directions of the molecular axes. L!,, is the pair coupling energy, given as a circular frequency, a,,=A-‘(llV,*12}.
{ 1) describes a state in which molecule 1 is in the first excited state of the considered vibrational mode whereas molecule 2 is in the ground state. 12) denotes the opposite situation. Vi, is the intermoleo ular interaction operator expanded in a series of the normal coordinates of both pair molecules. The term which gives a non-vanishing matrix element ( 11V,, 12) contains the second mixed derivative of Vi, with respect to the normal coordinates of molecules 1 and 2. Neighbouring vibrationally coupled molecules can vibrate in-phase or anti-phase [ 1,lo]. The coupling energy has opposite signs in these cases. The Legendre factor in (2) describes the effective coupling energy which results from different contributions of the two-phase relations averaged over all orientations of the vector that points from molecule 1 to molecule 2 of the pair. An instructive description of this is given in ref. [ I]. To obtain the averages of eq. (2), it is necessary to known the dominant vibrational-coupling mechanism and the bath Hamiltonian which governs the structure of the liquid, e.g. the pairdistribution function (PDF), g(R, B1, 0,). The fact that v2 of CHJ is very intense in infrared absorption gave rise to the assumption that transition dipole interaction is the main coupling mechanism. This assumption was conlkned by a comparison of experimental and theoretical second moments [ 3, lo]. Logan presented an expression for the IS isotopic dilution shift, extrapolated to infinite dilution, for this kind of interaction
M *s= _ Pn(fll
w
>
(2)
if Logan’s notation [ 7,8 ] is used. The P,, are Legendre polynomials. For the isotropic Raman experiment n=O, for infrared absorption n = 1 and for anisotropic Raman scattering n=2. The 0 are unit vectors in the directions of the transition moments in infrared absorption or the directions of the main axes of the polarizabilityderivative tensors in anisotropic Raman scattering In the case of axial molecules and symmetrical vibrations, these vectors lie
(3)
2(adaq)i & 7c~&Jcr3
T) I
.
(4)
(@lag), is the dipole-moment derivative with respect to the normal coordinate including the mass term. ~0 is the circular frequency of the normal mode ’ under consideration in the free molecule. c is the molecular diameter. t(p, T) depends on the intermolecular interactions that are responsible for orientational order (if RIVC is due to transition dipole interactions, no isotopic dilution shift is expected in the case of random 455
Volume 172,number 6
CHEMICALPHYSICSLETTERS
molecular orientations). The expression for the {(p, T) values, given by Logan, is based on Wertheim’s calculation
[ 141; this author uses the mean
spherical model (MSM) and the assumption that the deviation from random orientation is due to the permanent point-dipole interaction. If one uses the dipole-moment derivative obtained from Dickson, Mills and Crawford [ 15 ] and the molecular diameter 0.458 nm, one has isotopic dilution shifts for IS of 0.37 cm-’ at 313 K and 0.51 cm-’ at 213 K. A comparison with the values of table 1 shows that eq. (4) gives isotopic dilution shifts for IS that are markedly smaller than the ones observed and the temperature dependence is much less in this theory than in experiment. Logan also predicted that the RIVC shift in anisotropic scattering is only l/25 of that in isotropic scattering and should disappear completely in IR absorption. Both predictions are in contrast to our experimental results. The discrepancies between theory and experiment can be summarized as follows: (i) The band shifts of tr2of CHJ observed in isotopic dilution (ID) are larger throughout and show larger temperature dependence than theoretical RIVC first-moment predictions for isotropic Raman scattering. (ii) In anisotropic Raman scattering, ID band shifts are larger than l/25 of the shift in isotropic scattering. (iii) In IR absorption, ID band shifts are observed and are particularly large. One possible reason for these discrepancies can be the fact that the theory assumes only point-dipole interactions to be responsible for orientational order, the degree of which determines the RIVC first moment if the coupling is due to transition dipole interactions. (i) indicates that orientational order is larger than results from MSM calculations. It must probably be taken into account that molecules possess real dipoles (of finite length), and that anisotropic dispersion and repulsive interactions which ‘may lead to orientational order must therefore be considered. The transition dipoles are also real dipoles, and although they cannot form measurable orientational forces, they can enhance vibrational coupling. The experimental results can give no indication of the main reason for the discrepancies. Some authors assume that the transition dipole 456
21 September 1990
moment, which we took from gas-phase measurements [ 151 for the comparison of our results with theory, increases upon condensation. This is also a possible reason. Some additional information can be received by splitting the temperature effect of the RIVC shift into a density and a kinetic energy contribution. Table 3 shows RIVC shifts at 293 K at two densities and also at 2 13 K for the higher density. A comparison of the first and second row of the table shows the pure density contribution, and the difference of the second and third row describes that of the kinetic energy. The data show that in both experiment and theory, the kinetic energy effect is larger than the density effect. What is of particular importance is the fact that both are larger than predicted by the MSM. The reasons are probably that (i) the MSM cannot describe the temperature and density dependence of the order parameter < very well, and (ii) the use of the point-dipole approximation: interactions are, in general, enhanced by real dipoles ( averaged over all orientations ) , and in addition, the multipole expansion of real dipoles leads to terms with a higher radius dependence than with dipoles. It is very likely that (ii ) has the same origin. In the MSM theory, the PDF can be expanded to a complete set of rotational invariants containing only three terms (seeeq. (8) inref. [14]).Theuseofthisthreeterm set to calculate the average of eq. (2) leads to the prediction that &vc almost disappears for anisotropic scattering (n=2). This set is obviously unable to describe the PDF if interactions like those mentioned above contribute to the structure of the liquid, and in such cases the prediction cannot be maintained. Formally, more general expansions could be used for the PDF, such Table 3 Density and kinetic energy contribution to the temperature effeet of theoretical and experimental ID shift extrapolated to infinite isotopic dilution Temp. (K) 293 293 213
Density (s em-‘) 2.280 2.504 2.504
Pressure @Pa) 0.1 168 0.1
RIVC shift (cm-‘) exp.
theory
1.2 1.3 .’ 1.9
0.44 0.47 0.57
.) Interpolated from experiments at 50, 100,200, 300 and 400 MPa.
Volume 172, number 6
CHEMICAL PHYSICS LETTERS
as that presented by Blum [ 161. But this is doomed to failure in most cases as a consequence of a lack of information on real intermolecular interactions. In principle, the reasons given for (ii) can also be put forward for (iii), but as the ID shifts are rather large in the infrared experiment, it seems very likely that there are other secondary reasons. One is surely that the MSM theory uses a linear approximation for the Boltzmann factor. Simple model calculations, which we carried out using realistic dipole moments and temperatures, show that the linear approximation causes only small errors in both Raman experiments. Applied to the infrared experiment, the calculations confirm Logan’s prediction: no ID band shift in IR absorption. But leaving the approximation aside, the calculations indicate ID band shifts of about 10% to 20% of the shifts in isotropic Raman scattering. The experimentally observed shifts are even larger. As has already been mentioned, this could be due to the same reasons as given for (ii). However, in IR absorption, it should be borne in mind that problems may arise due to the strong fi-equency dependence of the refractive index in the frequency region of the absorption band in the neat liquid. This raises the question of whether the dispersion correction used is suitable for comparison with Raman results. As pointed out by Biittcher and Bordewijk [ 171 and as Schrtier’s recent molecular statistic treatment of dielectric fluids [ 181 indicates, this correction only eliminates the effect of the frequencydepcndent field at a reference molecule due to the induced dipoles of neighbouring molecules averaged over all possible orientations and positions. It cannot describe effects due to orientational order. But those effects seem to be negligible in a fast approximation: First, the extent of orientational order is small in ordinary liquids like iodomethane, because orientation-dependent intermolecular interactions (dipole-dipole, etc.) are well below the thermal kinetic energy. Furthermore, the polarizability of iodomethane is nearly isotropic, so the magnitude of the induced dipoles does not greatly depend on orientation. This even holds true in the case of collective excitation in the v2 mode, because the polar-
21 September 1990
izability derivative with respect to this normal coordinate is also nearly isotropic. Finally, one should take into consideration that the entire dispersion correction is small compared with the observed shift, particularly if one considers the ID shift of isotopic mixtures at mean concentrations with respect to very high isotopic dilution. Thus, it seems justified to say that orientational order can scarcely contribute to the field at a reference molecule within the frequency range of interest, and that Crawford’s correction accounts for the dispersion effects to a good approximation.
Acknowledgement Financial support from the Deutsche Forschungsgemeinschaft and the Fends der Chemischen Industrie is gratefully acknowledged. We are most grateful to Professor W. Schr&r, Bremen, for helpful discussions.
References [l] D. Scheibe, J. RamanSpectry. 13 (1982) 103. [ 21 K.L. Oehme, G. Rudakoff and K. Klostermann, J. Chem. Phys. 83 (1985) 1499. [ 31 G. Dirge, R Arndt and J. Yarwood, Mol. Phys. 52 (1984) 399. [4] T. Zyungaud R.E. Wilde, J. Chem. Phys. 86 (1987) 5940. [ 51C.H. Wan8 and J. McHale, J. Chem. Phys, 72 (1980) 4039. [ 61 S. Bratos and G. Tarjus, Phys. Rev. A 24 ( 1981) 159I. 171D.E. Logan, Mol. Phys. 58 (1986) 97. [ 81 D.E. Logan, Chem. Phys. 103 (1986) 215. [ 91 D.E. Logan, Chem. Phys. 131 (1989) 199. [IO] G. D@e, Z. Naturforsch. A 28 (1973) 919. [ 11I J. Laane and W. Klefer, J. Chem. Phys. 72 (1980) 5305. [ 12) J. Laane and W. Kiefer, J. Chem. Phys. 73 ( 1980) 497 1, [ 131A.A. Clifford and B. Crawford Jr., J. Chem. Phys. 70 (1966) 1536. [141 MS. Wertheim, J.Chem. Phys. 55 (1971) 4291. [ I 51 A.D. Dickson, I.M. Mills and B. Crawford Jr., J. Chem. Phys. 27 ( 1957) 445. [ 161L. Blum, J. Chem. Phys. 57 (1972) 1862. [ 171C.J.F. Biittcher and P. Bordewijk, Theory of electric polarization, 2nd Ed., Vol. 2 (Else+%, Amsterdam, 1978) ch. 12. [ 181W. S&r&r, to he published.
457
CHEMICAL PHYSICS LETTERS
21 September 1990
The effect of resonant vibrational coupling on the frequency of the v, mode of liquid methyl iodide in Raman scattering and infrared absorption G. D&e, D. Lindner InstitutflrPhysikalischeund TheoretischeChemieder TechnischenUniversitiit, O-3300Braunschweig,FederalRepublicof Germany
W. Richter and D. Schiel Physikulisch-Technic Bundesanrtalt, D-3300 Braunschweig,FederalRepublicof Germany Received 17 January 1990; in final form I June 1990
The v2 mode of methyl iodide has been measured in isotropic and anisotropic Raman scattering, and in infrared absorption in the neat liquid and in isotopic dilution with the deuterated compound. The local field affecting the shape and position of the intense infd profde has been taken into account by a continuum dielectric approximation. The isotopic dilution shifts observed indicate the action of resonant intermolecular vibrational coupling which presumably is caused by transition dipole interaction. The results are, in principle, in agreement with current theories. For a comparison, certain approximations arc necessary (MSM, etc.), which seem to be responsible for the fact that quantitative agreement is not yet very good.
1. Introduction The position and the shape of Raman scattering and infrared absorption profiles of the vibrational modes of molecules in the liquid state are in certain cases well-suited probes for studying the structure and molecular dynamics in liquids. In this connection it is sometimes observed that in neat liquids, anisotropic Raman bands have higher peak frequencies than the corresponding isotropic bands. This effect, usually called the non-coincidence effect, is caused by resonant intermolecular vibrational coupling (RIVC) if there is a certain amount of molecular orientational order. In recent years, the problem of the different kinds of influence of RIVC on isotropic and anisotropic Raman scattering and infrared absorption has been taken up again in experimental [l-4] and theoretical [S-9] investigations. The results obtained by several authors do not always agree. The only experimental method which can show if RIVC affects the position and shape of a certain vibrational band is the so-called isotopic dilution method [ 10 1. The 0009.2614/90/$03.50
band parameters are observed in the neat liquid and in dilution with an isotopically substituted species of the same compound. In this way, it is possible to decouple the vibrations of neighbouring molecules (if the corresponding vibrational bands in the two isotopic species do not overlap). In anisotropic Raman scattering such experiments suffer from the low intensity. This scattering is very weak in most cases, making it difficult to obtain good results in dilution experiments. This is particularly true if there are perturbations by hot bands or other neighbouring bands, or if the reorientational broadening is large. Published data are not therefore always reliable, and what is more, there are also some discrepancies in the theories. In general such problems are treated by studying the vibrational relaxation function C,(t), which is obtained by a Fourier transformation of the frequency-dependent intensity function Z(w) of the band, because it is often easier to treat timedependent effects in this time representation. In this paper we wish to study the effect of RIVC on band positions (first moments). Such effects (shifts) are a
0 1990 - Elsevier Science Publishers B.V. (North-Holland)
453
Volume 172. number 6
CHEMICAL PHYSICS LETTERS
measure of an ensemble average of that part of the intermolecular interaction which acts on the vibrational energy. This average is time independent and describes the modulation of C,(t) with a constant frequency. It is therefore unnecessary to transform the spectroscopic results into the time domain if we are dealing only with band shifts due to RIVC, but we do need the time representation for the treatment of the other band shape or vibrational correlation parameters, and these will be the subject of following papers.
2. Experimental A Coderg triple-monochromator Raman spectrometer was used with an argon ion laser. For the pressure-dependent measurements we used a pressure cell manufactured by Nova Swiss. As the band shifts upon isotopic dilution sometimes are very small (especially in anisotropic scattering), some atmospheric-pressure measurements were performed using a special technique: To ascertain accurate wavenumber differences for the band maxima obtained with the neat liquid and the isotopic dilution, both bands were recorded in the same run. The principle of this technique has been described by Laane and Kiefer [ 11,12 1. Two sample cells are mounted on a rotating cell holder which can rotate around 180” and back into the primary position. The laser beam passes through the neat liquid cell in one of the two positions, and through the isotopically diluted solution in the other. A microcomputer controls the counting time of the scattered photons, the motion of the cells, the monochromator drive and also collects the counting rates. With this arrangement we studied the isotopic dilution effect in anisotropic scattering and also repeated earlier measurements of this effect in isotropic scattering. The band maximum observed in W scattering geometry (polar scattering) was identified with the band maximum of isotropic scattering. This is possible because the depolarisation ratio of the vz band of iodomethane is very small. The infrared absorption profile of the v2 mode of the neat liquid was measured by the two-thickness method according to 454
21 September 1990
(1) where z1 and 7, are the transmittances of the absorption path lengths d, and d2. For the latter, values of 4 and 12 m were used in a variable cell with KBr windows. In this way the absorption coefficient of the strong band is obtained in good approximation without disturbances from the frequency-dependent reflection losses. For the low-temperature measurements, small cuvettes made of thin CaFz windows with fixed spacings were used which fitted into a toploading liquid-nitrogen-cooled cryostat. In the case of the isotopic dilution measurements, a single thickness only was used (except with the highest concentration).
3. Results Table 1 shows the results for the isotropic and anisotropic scattering experiments at two different temperatures. The values in table 1 are averaged over four experimental runs. The band maxima have been used instead of the first moments since the measured profiles are symmetrical if the “hot” component in the low-frequency wing is eliminated. The appearance of the non-coincidence effect (different band maxima in isotropic and anisotropic scattering) in the case of the neat liquid, together with the band shift upon isotopic dilution, indicates the action of RIVC. In additional measurements, we have found that the mole-fraction dependence of the isotopic dilution shift of the isotropic band is exactly linear. Table 2 shows the corresponding results for the inTable 1 Shifts of the u2band in isotropic (IS) and auisotropic (AS) scattering upon isotopic dilution to a mole fraction of CHJ of 0.1. Uncertainty of relative frequencies: 0.1 cm-’ Temp. (IQ
213 283 313
Baud maximum in neat liquid (cm- ’)
shift (cm-‘)
IS
AS
IS
AS
1240.3 1240.7
1241.0 1241.0
1.7 1.2 1.0
0.6 0.5
Isotopic dilution
Table2 Shifts of the vz band in infixed absorption upon isotopic dilution. Uncertainty of fresuencies: 0.1 cm-’ Temp. (IQ
Band maaimum in neat liquid (cm-‘) measured
correctedfor internal field
Isotopic dilution shit? of the correc@dband maximum for mole fraction x of CHJ in CDS x
shift (cm-‘)
296
1239.5
1239.7
0.05 0.45 0.83
1.8 0.8 0.3
223
1238.1
1238.6
0.05
2.0
frared absorption experiments. As the yz absorption
is rather strong, it is necessary to correct the band profiles obtained for the internal field. For this correction we followed the proposal of Clifford and Crawford [ 13 ] which is based on a continuum dielectric approximation. This correction results in a frequency increase of 0.2 cm-’ at 296 K and OS cm-’ at 223 K in the neat liquid.
4. Diaeussion and comparisonwith theory There are only a few theoretical papers which deal specifically with the non-coincidence effect [ 5,8,9]. These papers are confined to band shifts due to RIVC and do not treat band-shape effects. RlVC band shifts can be described by the changes of the first moments. The fundamental equation, first derived by Wang [ 41, can be written as &I”C =N< GM))
21 September 1990
CHEMICALPHYSICSLETTERS
Voiume 172,number 6
in the directions of the molecular axes. L!,, is the pair coupling energy, given as a circular frequency, a,,=A-‘(llV,*12}.
{ 1) describes a state in which molecule 1 is in the first excited state of the considered vibrational mode whereas molecule 2 is in the ground state. 12) denotes the opposite situation. Vi, is the intermoleo ular interaction operator expanded in a series of the normal coordinates of both pair molecules. The term which gives a non-vanishing matrix element ( 11V,, 12) contains the second mixed derivative of Vi, with respect to the normal coordinates of molecules 1 and 2. Neighbouring vibrationally coupled molecules can vibrate in-phase or anti-phase [ 1,lo]. The coupling energy has opposite signs in these cases. The Legendre factor in (2) describes the effective coupling energy which results from different contributions of the two-phase relations averaged over all orientations of the vector that points from molecule 1 to molecule 2 of the pair. An instructive description of this is given in ref. [ I]. To obtain the averages of eq. (2), it is necessary to known the dominant vibrational-coupling mechanism and the bath Hamiltonian which governs the structure of the liquid, e.g. the pairdistribution function (PDF), g(R, B1, 0,). The fact that v2 of CHJ is very intense in infrared absorption gave rise to the assumption that transition dipole interaction is the main coupling mechanism. This assumption was conlkned by a comparison of experimental and theoretical second moments [ 3, lo]. Logan presented an expression for the IS isotopic dilution shift, extrapolated to infinite dilution, for this kind of interaction
M *s= _ Pn(fll
w
>
(2)
if Logan’s notation [ 7,8 ] is used. The P,, are Legendre polynomials. For the isotropic Raman experiment n=O, for infrared absorption n = 1 and for anisotropic Raman scattering n=2. The 0 are unit vectors in the directions of the transition moments in infrared absorption or the directions of the main axes of the polarizabilityderivative tensors in anisotropic Raman scattering In the case of axial molecules and symmetrical vibrations, these vectors lie
(3)
2(adaq)i & 7c~&Jcr3
T) I
.
(4)
(@lag), is the dipole-moment derivative with respect to the normal coordinate including the mass term. ~0 is the circular frequency of the normal mode ’ under consideration in the free molecule. c is the molecular diameter. t(p, T) depends on the intermolecular interactions that are responsible for orientational order (if RIVC is due to transition dipole interactions, no isotopic dilution shift is expected in the case of random 455
Volume 172,number 6
CHEMICALPHYSICSLETTERS
molecular orientations). The expression for the {(p, T) values, given by Logan, is based on Wertheim’s calculation
[ 141; this author uses the mean
spherical model (MSM) and the assumption that the deviation from random orientation is due to the permanent point-dipole interaction. If one uses the dipole-moment derivative obtained from Dickson, Mills and Crawford [ 15 ] and the molecular diameter 0.458 nm, one has isotopic dilution shifts for IS of 0.37 cm-’ at 313 K and 0.51 cm-’ at 213 K. A comparison with the values of table 1 shows that eq. (4) gives isotopic dilution shifts for IS that are markedly smaller than the ones observed and the temperature dependence is much less in this theory than in experiment. Logan also predicted that the RIVC shift in anisotropic scattering is only l/25 of that in isotropic scattering and should disappear completely in IR absorption. Both predictions are in contrast to our experimental results. The discrepancies between theory and experiment can be summarized as follows: (i) The band shifts of tr2of CHJ observed in isotopic dilution (ID) are larger throughout and show larger temperature dependence than theoretical RIVC first-moment predictions for isotropic Raman scattering. (ii) In anisotropic Raman scattering, ID band shifts are larger than l/25 of the shift in isotropic scattering. (iii) In IR absorption, ID band shifts are observed and are particularly large. One possible reason for these discrepancies can be the fact that the theory assumes only point-dipole interactions to be responsible for orientational order, the degree of which determines the RIVC first moment if the coupling is due to transition dipole interactions. (i) indicates that orientational order is larger than results from MSM calculations. It must probably be taken into account that molecules possess real dipoles (of finite length), and that anisotropic dispersion and repulsive interactions which ‘may lead to orientational order must therefore be considered. The transition dipoles are also real dipoles, and although they cannot form measurable orientational forces, they can enhance vibrational coupling. The experimental results can give no indication of the main reason for the discrepancies. Some authors assume that the transition dipole 456
21 September 1990
moment, which we took from gas-phase measurements [ 151 for the comparison of our results with theory, increases upon condensation. This is also a possible reason. Some additional information can be received by splitting the temperature effect of the RIVC shift into a density and a kinetic energy contribution. Table 3 shows RIVC shifts at 293 K at two densities and also at 2 13 K for the higher density. A comparison of the first and second row of the table shows the pure density contribution, and the difference of the second and third row describes that of the kinetic energy. The data show that in both experiment and theory, the kinetic energy effect is larger than the density effect. What is of particular importance is the fact that both are larger than predicted by the MSM. The reasons are probably that (i) the MSM cannot describe the temperature and density dependence of the order parameter < very well, and (ii) the use of the point-dipole approximation: interactions are, in general, enhanced by real dipoles ( averaged over all orientations ) , and in addition, the multipole expansion of real dipoles leads to terms with a higher radius dependence than with dipoles. It is very likely that (ii ) has the same origin. In the MSM theory, the PDF can be expanded to a complete set of rotational invariants containing only three terms (seeeq. (8) inref. [14]).Theuseofthisthreeterm set to calculate the average of eq. (2) leads to the prediction that &vc almost disappears for anisotropic scattering (n=2). This set is obviously unable to describe the PDF if interactions like those mentioned above contribute to the structure of the liquid, and in such cases the prediction cannot be maintained. Formally, more general expansions could be used for the PDF, such Table 3 Density and kinetic energy contribution to the temperature effeet of theoretical and experimental ID shift extrapolated to infinite isotopic dilution Temp. (K) 293 293 213
Density (s em-‘) 2.280 2.504 2.504
Pressure @Pa) 0.1 168 0.1
RIVC shift (cm-‘) exp.
theory
1.2 1.3 .’ 1.9
0.44 0.47 0.57
.) Interpolated from experiments at 50, 100,200, 300 and 400 MPa.
Volume 172, number 6
CHEMICAL PHYSICS LETTERS
as that presented by Blum [ 161. But this is doomed to failure in most cases as a consequence of a lack of information on real intermolecular interactions. In principle, the reasons given for (ii) can also be put forward for (iii), but as the ID shifts are rather large in the infrared experiment, it seems very likely that there are other secondary reasons. One is surely that the MSM theory uses a linear approximation for the Boltzmann factor. Simple model calculations, which we carried out using realistic dipole moments and temperatures, show that the linear approximation causes only small errors in both Raman experiments. Applied to the infrared experiment, the calculations confirm Logan’s prediction: no ID band shift in IR absorption. But leaving the approximation aside, the calculations indicate ID band shifts of about 10% to 20% of the shifts in isotropic Raman scattering. The experimentally observed shifts are even larger. As has already been mentioned, this could be due to the same reasons as given for (ii). However, in IR absorption, it should be borne in mind that problems may arise due to the strong fi-equency dependence of the refractive index in the frequency region of the absorption band in the neat liquid. This raises the question of whether the dispersion correction used is suitable for comparison with Raman results. As pointed out by Biittcher and Bordewijk [ 171 and as Schrtier’s recent molecular statistic treatment of dielectric fluids [ 181 indicates, this correction only eliminates the effect of the frequencydepcndent field at a reference molecule due to the induced dipoles of neighbouring molecules averaged over all possible orientations and positions. It cannot describe effects due to orientational order. But those effects seem to be negligible in a fast approximation: First, the extent of orientational order is small in ordinary liquids like iodomethane, because orientation-dependent intermolecular interactions (dipole-dipole, etc.) are well below the thermal kinetic energy. Furthermore, the polarizability of iodomethane is nearly isotropic, so the magnitude of the induced dipoles does not greatly depend on orientation. This even holds true in the case of collective excitation in the v2 mode, because the polar-
21 September 1990
izability derivative with respect to this normal coordinate is also nearly isotropic. Finally, one should take into consideration that the entire dispersion correction is small compared with the observed shift, particularly if one considers the ID shift of isotopic mixtures at mean concentrations with respect to very high isotopic dilution. Thus, it seems justified to say that orientational order can scarcely contribute to the field at a reference molecule within the frequency range of interest, and that Crawford’s correction accounts for the dispersion effects to a good approximation.
Acknowledgement Financial support from the Deutsche Forschungsgemeinschaft and the Fends der Chemischen Industrie is gratefully acknowledged. We are most grateful to Professor W. Schr&r, Bremen, for helpful discussions.
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