Raman intensities of lattice modes and the oriented gas model

The applicability of the oriented gas model to the calculation of Raman intensities of molecular crystals is investigated. The basic implications of the model and its consequences are discussed. The polarized spectra of only 3 out of 11 crystals considered are compatible with the model. For these, quantitative calculations based on calculated eigenvectors give reasonable agreement with experiment, while the others are in clear disagreement with available data. It is shown that assignments of unpolarized spectra based on the model’s predictions may easily be erroneous. Some of the parameters affecting the calculation are analyzed, confirming the conclusion that the oriented gas model is not a reliable tool for assignment of Raman lattice bands of molecular crystals.


INTRODUCTION
Raman bands due to lattice vibrations are unequivocally assigned to their proper factor group species by recording polarized spectra of Single crystals. These spectra, however, are not always available for molecular crystals, since a large number of them offer considerable experimental difficulties. In these cases, additional theoretical considerations might be of great use. For internal modes, comparison of observed relative intensities and those calculated using the oriented gas model give good qualitative agreement. 1-3 For external modes, however, this calculation has been mostly limited by the lack of knowledge of the actual crystal normal modes, and additional approximations assuming uncoupled rotations about molecular prinCipal axes have led to unsatisfactory results. 4 Recently, Elliot and Leroi have used the model in combination with lattice dynamical calculation to assign the unpolarized spectra of benzene 5 and ethylene, 6 using the results for the latter as an important argument to decide on its crystal structure.
In the last few years, several different intermolecular force field models of the atom-atom type have been developed to fit simultaneously statical and dynamical properties of molecular crystals. 7,8 The resulting calculated crystal vibration frequencies have contributed to the assignment of some normal modes and, in addition, provide a quantitative description for external modes in terms of rotation and translation coordinates. As these coordinates are used as a basis for constructing the dynamical matrix, the eigenvectors of the secular equation give a direct description of how the mixing of molecular librations and translations gives rise to an external crystal normal mode. Nevertheless, these calculations, although adequate, may contain inversions regarding the frequency ordering of some normal modes of different symmetry, thus limiting their capacity to define assignments in unpolarized spectra.
In the present work, we explore the possibilities offered by the applications of the oriented gas model in combination with lattice mode frequency calculations to predict Raman band intensities and positions, in order to contribute to the assignment of external modes.
It must be stressed that owing to the uncertainties in experimental intensities, and to the roughness of the oriented gas model, only qualitative agreement could be expected. However, knowledge of relative orders of magnitude should be of great use, allowing, for instance, the prediction of vanishingly low observable intensities.
We have applied the model to a series of 11 molecular crystals including aromatic hydrocarbons and chlorinated and brominated benzenes. For these compounds, the crystal structure is known and a large number of low frequency Raman bandS has been Observed and assigned, making a qualitative discussion possible. On the other hand, except for the brominated benzenes, intermolecular force fields are available, allowing also a quantitative test of the model.

CALCULATION METHOD
The relative intensities in polarized and unpolarized Raman spectra for a nondegenerate normal mode Q n can be calculated by respectively. p and (J label coordinates of an orthogonal set of coordinates fixed to the crystal, vn is the frequency of the normal mode, and (au)pa is the path element of the unit cell pol ariz ability tensor. When crystal modes are expressed in terms of some set of molecular coordinates, the derivatives of the molecular polarizabilities can be written as where q ... is the kth coordinate of the set chosen for the mth molecule in the unit cell. mined by the intramolecular field only and is consequently independent of motions and distortions introduced by other molecules of the crystal. Statement (b) is equivalent to saying that the molecular polarizability does not change while the molecule translates or librates. According to (a), the crystal polarizability tensor au is related to the molecular polarizability a by (3) where 1T", is the matrix relating molecular principal axes corresponding to the mth molecule and the crystal fixed axes.
The derivatives of au with respect to principal molecular axes rotation and translation coordinates R"". and Til". can be calculated, according to (b), as with (MIl)I} = -(Iill, the Levi-Civita tensor. It can be seen that translations do not contribute, in this model, to Raman intensities. The rotation coordinates R"m can be expressed in terms of symmetry coordinates S,,(y). The derivatives of the element (a"),,, with respect to external normal coordinates Qiy) of symmetry y are obtained by In these equations, the derivatives 8R"",/8S,,(y) and the transformation matrices 1T", are obtained from crystal data, and the a tensor is the polarizability expressed in principal axes. If the molecular distortion is not very important, free molecule polarizabilities can be used. The derivatives as,,(y)/8Q,,(y) which express the composition of each crystal mode are determined from the eigenvectors of the dynamical matrix diagonalization.

APPLICATION OF THE ORIENTED GAS MODEL TO EXTERNAL MODES
Intermolecular and intramolecular force fields are involved in the calculation of Raman intensities by the oriented gas model. Nevertheless, this model makes a clear distinction between both fields when it conSiders that the molecular polarizabilities produced by the internal field are unchanged when the crystal field acts on the molecules during the vibrational motion. This assumption implies that the trace of the polarizability tensor, which is invariant with respect to Similarity transformations, is preserved during librations in any orthogonal coordinate system and therefore for the mth molecule in the nth normal mode Qn. From the additivity hypothesiS, it is possible to extend expression (6) to the crystal unit cell: It must be stressed that thiS relationship stems from the basic assumption of the model only and is independent from other factors of the intensity calculation such as crystal structure, pol ariz abilities, and eigenvectors of the secular equation.
Equation (7) establishes relationships between permissible relative intensities in three orthogonal polarizations pp, aa, and TT with p 1 a 1 T, which can be summarized by  Table I).
Even though other invariants of the polarizability tensor do not provide direct relationships between observ-. ables, Eq. (8) constitutes a rough guideline for the applicability of the model. Raman intensities are in general poorly determined experimental data. In particular, intensities in three orthogonal polarizations cannot be compared directly owing to unavoidable modifications in the experimental setup. However, both the guideline expressed in Eq. (8) and the results from more refined calculations can be applied admitting a possible correction factor common to all bands observed in each polarization. The magnitude of this factor may be estimated within the assumptions of the model. In general, owing to the orthogonality of normal modes, The observable relation Cpa/Cp'a' is independent from the mixing of librations in the normal modes and q~m in (9) can be replaced by R"",. The oriented gas model can only predict spectra compatible with values of Cpa calculated according to (9) together with relative intensities restricted by (8).
T ABLE II. Compatibility of the oriented gas model with observed polarized spectra. Crystal Naphthalene1.
For rigid body external motions, the six nonvanishing F pa(Y) vectors are determined, within the assumptions of the model, by molecular polarizabilities and crystal structures only. Even though the space is hexadimensional, for noncentrosymmetric molecular sites, the components of F pa(Y) vanish for the three translational coordinates according to Eq. (4). In general, this description establishes that the maximum number of zero intensity modes Qn(Y) predicted by the model for each symmetry species y is equal to the difference between the dimensions of the y symmetry space and the subspace spanned by the nonvanishing Fpu(Y) vectors. In this sense, the invariance of the trace [Eq. (7)], and the definition in Eq. (10), show that We can take, for example, the space group Cih with two molecules per unit cell located at C i sites, which is a very common case for molecular crystals. 9 The symmetry space 'Y is tridimensional. For Bg modes, as only F ab and Fbe are nonzero, except for accidental de-generaCies, only one mode may have zero intensity. As FaAAg) does not lie in the same plane as Fpp(Ag), the model does not allow zero intensity Ag bands. The accidental degeneracies may stem from molecular or crystal structure. If the molecular polarizability tensor is diagonal [reduced form of Eq. (5)] and has two identical elements, all vectors F pa(Y) have to lie in the degenerate plane. If the crystal unique axis coincides with a molecular principal axis, all F pa(Ag) are parallel to this axis, and there may be up to two Ag zero intensity modes.

CRYSTAL NORMAL MODE CALCULATION
The calculation method used in the present work is discussed in detail in Ref. 10. We have used an atomatom intermolecular potential of the form The eigenvectors provide a direct picture of the degree of mixing of molecular rotations and translations in each normal mode. For the case of benzene, they were taken directly from Ref. 10. In all cases, atom-atom contacts up to a distance of 6 A were taken into account. The rigid body approximation has been used throughout, as it is a negligible source of errors for the eigenvectors. 11

RESULTS AND DISCUSSION
First we will discuss the applicability of the model to a series of Raman spectra of molecular crystals in semiqualitative terms. Table II summarizes the compatibility of the observed spectra with three main features of the model which do not depend on the intermolecular force field.
For naphthalene and anthracene, the intensity relations for the three Ag modes and the values of Cpp in three orthogonal polarizations are acceptable, and all three allowed bands are observed. Values of Cpa (pif-a) are also consistent, and the model would be a priori applicable.
For benzene, although the Ag band intensities relation is preserved, the value of Cae, predicting a very weak spectrum in that polarization, is in contradiction with experiment.
For biphenyl, the band at 54 cm-1 escapes the scheme of   zene, the bands at 57 cm-1 and 48 cm-l, respectively, are observed only with bb polarization, and therefore incompatible with the model. In the case of 1, 2, 4, 5-tetrachlorobenzene, the intensity relations for Ag modes is in doubtful agreement with the model. Also, the magnitude of disagreement between observed and calculated values of C pc> for these crystals cannot be attributed to change in experimental setups only.
For a-p-dichlorobenzene, there are no polarized spectra, while for hexachlorobenzene the data are incomplete. For this last crystal, the model seems to be appropriate.
Even though the qualitative agreement is not particularly encouraging, we have performed quantitative calculations for the crystals under consideration (except for the  brominated benzenes, for which no suitable potential parameters are available) in order to see if some general features of the spectra could be deduced from the application of the model.
For the hydrocarbons we have used atom-atom potential parameters proposed by Williams,7 and for chlorinated benzenes, those obtained by Bonadeo and D' Alessio. 11 The transformation matrices relating molecular principal axes and crystal fixed axes were obtained from crystal structural data. For monoclinic crystals we have chosen a crystal system (a'be) with a'lbe, or (abe ') with e' 1 ab, according to the available experimental polarizations.
In every case we have employed molecular polarizabilities from the iSOlated mOlecules without further refinements. In the same approximation, justified by the smallness of the molecular distortion in the crystal, we have assumed that the polarizability tensor is diagonal in the principal inertia axes system and applied the re-duced expression of Eq. (5) for the calculation. Thus, the molecular pol ariz abilities do not appear in calculations for 1, 3, 5-trichlorobenzene, hexachlorobenzene, and benzene, where only one difference between diagonal elements is nonzero and appears as a scaling factor. In the other cases, the polarizabilities used in the calculations are included in the corresponding table. a a Table III shows the calculated frequencies, intensities, and eigenvectors for each normal mode of the crystals under consideration. Figures 1-10 show the observed spectra, and, indicated by vertical lines, the calculated intensities for the bands, in arbitrary units. Table III show that in most cases it is impossible to think of crystal normal modes as being pure rotations around principal molecular axes. In fact, most modes are mixed more than 20%, and only in the case of anthracene do all modes have less than 10% coupling. This result shows that, except for a few cases where the moments of inertia are widely different, the assumption of uncoupled rotations is unrealistic. On the other hand, calculated intensities may depend stronglyon the eigenvectors. In anthracene, for instance, the assumption of pure rotations leads to a 46: 100 ratio for the intensities of the 121 cm-1 band in the aa and bb polarizations, while our calculation, with an eigenvector containing 90% rotation about the lowest moment inertia axis leads to a 148: 100 ratio. However, reasonable changes in the eigenvectors do not alter the order of magnitude of the calculated intensities. Table IV shows the results obtained for {3-p-dichlorobenzene using three different sets of molecular pol arizabilities. It can be seen that the general pattern is not substantially changed in the unpolarized spectrum, although some relative intensities may change by factors as large as 4 Or 5. The differences in the polarized predictions are much larger. the intensity of some bands changing as much as 20% of the intensity of the strongest band.

The eigenvectors listed in
As discussed before, the oriented gas model is a priori compatible with the polarized spectra of naphthalene and anthracene. It can be seen in Figs. 1 and 2 that the model's prediction of the intensity pattern is acceptable, within the roughness of the model and the uncertainties in the observed spectra. For biphenyl, Fig. 3, if one accepts that the band at 54 cm-1 is the superposition of two Ag bands, as suggested by our calculations, the results are acceptable too.
Elliot and Leroi 5 have performed an oriented gas model intensity calculation for the nonpolarized spectrum of benzene. At the time of their work, the polarized spectra were not available. Although on the basis of calculated intensities they correctly assigned the 61 cm-1 band to the B3g species, they wel'e led to attribute the 100 cm-1 band to a superposition of Bag and B3K modes, while the polarized spectra clearly show that it belongs to the Bu species. Their calculated results do not agree with ours, maybe because we USe different eigenvectors, but both predict a low intensity ac spectrum, which again does not agree with experiment (see Fig. 4).
For hexachlorobenzene, Fig. 5, the ab polarization prediction widely fails, while the rest of the pattern is halfway compatible with observed spectra.
For !3-P-dichlorobenzene, Fig. 6, the superposition of two Ag bandS is not supported by any theoretical or experimental argument. In any case, the polarized spectra do not agree with the model's predictions, the biggest problem being that the lowest lying Ag mode, not observed, is predicted to be the most intense band of the spectrum. Figure 7 shows an unpolarized spectrum and the corresponding calculated intensities. It can be seen that if one would rely on the calculation, the Bg band at 55 cm-1 would be mistakenly asSigned to the Ag species. This type of results is a warning against the use of the model in the assignment of bands.
The calculated pattern agrees well with the unpolarized spectrum of a-p-dichlorohenzene, Fig. 8, but this result is not very significant since only three bands are involved.

CONCLUSIONS
In the present work, we have considered the applicability of the oriented gas model to 11 molecular crystals. Out of these, only two (naphthalene and anthracene) undoubtedly meet the conditions imposed by the basic assumptions of the model, and biphenyl may be included in this category with additional assumptions on the interpretation of the spectra, while for hexachlorobenzene and a-p-dichlorobenzene the available data are incomplete. For the rest of the crystals, a Priori considerations show that the model is inadequate.
Quantitative calculations on nine of these crystals using available force fields show that while for naphthalene, anthracene and biphenyl the mOdel yields results which roughly agree with experiment, the predictions for the other crystals are in clear contradiction with experimen- tal data.
These results are rather discouraging, since the a Priori applicability, which seems to be a good criterion by which to judge calculated intenSities, is verifiable only if polarized spectra are available, in which case the assignment of the bands is experimentally assured. Therefore, the usefulness of the heuristic power of the model is severely limited: we have shown how assignments based on the model may easily be erroneous. On the other hand, we have not found any particular property which could differentiate the crystals for which the model is appropriate from those for which it is not.
The obvious conclusion is that the electronic cloud around the molecules is indeed distorted appreciably during the librational motion, and that further refinements of the model, based on theoretical considerations, will have to come from molecular orbital calculations, which may indicate in more detail the conditions under which the Simple oriented gas model could be an acceptable zeroth order approximation to the problem of the Raman intensity of lattice modes.