J. Phys. Chem. 1994, 98, 771-776
771
Molecular Polarizability and Polarizability Derivatives in Cyclohexane Analyzed with the Theory of Atoms in Molecules Kathleen M. Gough,' Hemant K. Srivastava, and K a t a h Belohorcovh Department of Chemistry, Brock University, St. Catharines, ON Canada L2S 3AI Received: August 17, 1993"
The molecular polarizability of cyclohexane is obtained in an ab initio calculation with the D95**basis set. The derivatives of the molecular polarizability with respect to the symmetric stretch of the axial and the equatorial CH bonds are obtained by numericaldifference. Wave functions from the molecular orbital calculations are analyzed with the theory of atoms in molecules, to reveal the relocation of charge density which occurs when the molecule is placed in an electric field, at the equilibrium geometry and with the bonds stretched. The molecular polarizability arises from the combined effects of transfer of charge between hydrogen atoms on opposite sides of the molecule and the reorientation of charge around individual atoms (primarily the carbon atom atomic dipoles). Both interatomic charge transfer and intraatomic charge reorientation are affected by the stretching vibration. The calculated results are compared to experimental measurements of molecular polarizability and polarizability derivatives from Rayleigh and Raman scattering. The present method of analysis is compared to other models. Problems associated with the bond polarizability model, frequently used for the interpretation of Raman scattering intensities, are identified.
Introduction Thecalculation of molecular electrostatic propertiesfrom first principles is a subject of continued and growing interest. Recent advances in quantum mechanical methods have made possible the rapid calculation of both infrared Raman vibrational intensities, so that predicted values may rapidly eclipse the amount of experimentallyverified absolute intensity data.'s2 Applications of intensity analysis include the study of structural change on going from ground to excited states3 and of energy flow in intramolecular vibrational energy redistribution? It has been suggested recently that Raman scattering intensities can be used as predictors of nonlinear optical behavior in conjugated polymers? They have also been used in the analysis of polyelectrolyte chain conformation.6 However, the interpretation of these intensities is limited by the lackof a valid model. The present paper addresses this problem. We have been analyzing the origin of Raman trace scattering intensities with the intent of obtaining a simple, quantitative picture of charge reorganization during a molecular ~ibration.~J To achieve this goal, it has first been necessary to obtain reliable estimates of the equilibriumpolarizability. Analytical values for the molecular polarizability and even the polarizability derivative can now be calculated in various commercial molecular orbital (MO) packages.lg2 However, these results are always of a "black box" nature, yielding only the total molecular polarizability or its derivativewith respect to a basis-set-dependent normal mode. The conversion from normal modes to symmetry coordinates or internal coordinates is a simple mathematical step; however, this still does not provide the information we seek. Such results lack the atomic detail that would allow for the identification of the origin and, ultimately, the prediction of intensities on the basis of molecular structure. The present method offers a solution to this dilemma. We obtain an ab initio MO wave function for a molecule in the absence and then in the presence of an electric field and analyze the charge distribution with the theory of atoms in molecules (AIM)? The fundamental principleof the AIM theory is that atoms within a molecule can be identified from the topological properties of the charge distribution obtained from a quantum mechanical calculation. The surface of an atom is defined by the points for e
Abstract published in Aduance ACS Abstracts, January 1 , 1994.
0022-3654/94/2098-0771$04.S0/0
which there is a zero flux in the gradient of the charge density. The complete atom, or atomic basin, is then this surface plus the enclosedvolume. All of the observables which can be determined for the whole molecule can also be determined for an atom which has been so defined. The sum of the atomic polarizabilities produces a quantitative recovery of the total molecular polarizability obtained analytically in the MO calculation, while the AIM analysis enables us to draw a picture of the charge density and its response to the electric field on an atom by atom basis. Molecular polarizability is a measure of the ease with which the electrons in a molecule can be displaced when the molecule is placed in a uniform electric field. This displacement appears in two forms: first, there is actual movement of electric charge from one atomic basin to another (charge transfer, CT); second, the charge within an atomic basin will be skewed by the presence of the field (atomic dipole, AD). It has been showns that the charge-transfer contribution to the equilibrium molecular polarizability increaseswith the length of a molecule and is greatest between the atoms at the extreme ends for a given field orientation. In addition, the AD induced in an interior atom (e.g. carbon in alkanes) also responds to the field produced by the CT between the terminal atoms, typically producing an opposing dipole. Polarizability derivatives are obtained from the numerical difference between the equilibrium polarizability and that for a geometry in which some nuclei have been displaced along a symmetry coordinate. The AIM analysis of the symmetric stretching vibrations in methane, ethane, and propanes showed that the polarizability derivativearises from changes in the amount of charge transferred across the molecule, as well as from changes in the atomic dipoles. The fluctuations in the amount of charge transferred into or out of an atomic basin are not necessarily confined to the atoms of the bonds being stretched. Both positive and negative ACT terms occur. In contrast, changes in the atomic dipoles occur primarily within the displaced hydrogen and the carbon to which it is bonded and make a positive contribution to the polarizability derivative. The method of analysis presented here offers a unique perspective on the charge flows which occur inside a vibrating molecule. We have now applied this analysis to the equilibrium molecular polarizability and the polarizability derivatives for the symmetric CH stretching modes of cyclohexane. Measurement of the absolute Raman trace scattering cross sections for the 0 1994 American Chemical Society
772 The Journal of Physical Chemistry, Vol. 98, No. 3, 1994
symmetricvibrationsof cyclohexanehas revealed that theintensity of the band associated with the stretch of the axial CH bonds is about 30% lower than that of the equatorial CH bonds. As the experimental intensity is dependent on the square of the polarizability derivative, it translates, in this case, to a difference of about 14% in aa/ar. The experimental parameters were obtained following a full-harmonic force field analysis of the experimental spectra and therefore do not carry the basis-set dependence which would occur in purely ab initio calculations. We are interested in determining if the present method of analysis can shed any light on this phenomenon.
Gough et al.
TABLE 1: Comparison of Experimental and Theoretical Values of Mean Molecular Polarbability (10-40 Cm*/V) and Its Derivatives (10-30 Cm/V) for the Axial and Equatorial CH Stretches term experiment Gaussian 90 AIM ~
12.20 12.8 12.8 11.0
EO xx YY
zz
a q a r equatorial
9.71 10.15 10.15 8.83
9.67 10.08 10.09 8.85
1.33b
1.26 1.71 1.71 0.36
1.07 1.30 1.57 0.35
1.17b
1.03 0.36 0.36 2.43
1.20 0.63 0.54 2.51
xx
YY ZZ
Computation Method The approach taken was identical to that in the previous study* and is summarized only briefly here. The geometry of cyclohexane was fully optimized using the D95** basis set,IOwith the Gaussian 90 molecular orbital (MO) program package.' The molecular polarizabilitywas calculated for this geometry and for geometries in which only the axial CH bonds or only the equatorial CH bonds had been extended by 0.0054.040 A. The wave functions were obtained from the Gaussian 90 read-write file and analyzed with the AIMPAC" suite of programs. All calculations were performed on a Silicon Graphics 4D/340 computer. The program SADDLE, part of the AIMPAC suite, is used to identifycritical points in a molecule. These can be bond critical points, Le. the point between two bonded atoms at which there is zero flux in the charge density, or ring critical points, such as that found at the center of the cyclohexane ring. After these values have been obtained, the atomic properties are obtained by integration over the atomic basin. SADDLE was also used to obtain values of the charge density in a plane slicing through the molecule, in order to illustrate more clearly the nature of the charge reorganization. The charge density values for a grid of points in the xy plane were plotted with Mathematica.12 Within the AIM theory, the charge distribution in a molecule can be subdivided into atomic basins, separated by surfaces of zero flux in the gradient of the charge density. By integration over an atomic basin, one obtains all of the atomic properties, such as charge, dipole moment, etc. The atomic polarizability, aij, is calculated from
where Ni and NOare the atomic electron populations when the field is applied in the ith direction and when there is zero field, respectively; ri is the coordinate of the atom along the jth axis; pi and are the atomic dipoles with and without the applied field Qi. The first term represents the CT contribution to the molecular polarizability; the second, the AD contribution. The molecular polarizability is found by summing the atomic polarizabilities. The polarizability derivatives for the axial and equatorial CH stretching modes were then calculated from the numerical difference between the equilibrium molecular polarizability and the polarizability at the stretched geometry: d a / ar 1/ n Aa 1 Ar, where n is the number of equivalent CH bonds stretched. We have investigated the effect of using a central difference, [a(+Ar)- a(-Ar)]/2 Ar, for ethane. There was no measurable change in the computed value, while the computation time was doubled; therefore this procedure was not implemented for cyclohexane.
=
In Table 1, the equilibrium molecular polarizability from the literature is compared to those from the direct analytical calculation in Gaussian 90 and to the results recovered with AIM. The mean molecular polarizability and the anisotropy have been determined for cyclohexane from the refractive index and the
aa/ar axial
~~
xx
YY 2.2
a Reference 13: Values for xx, yy, and zz components are taken from the reported mean polarizability of 12.3 Cm2/V and the anisotropy (all
- a J , -1.93 Cm2/V. Reference 14.
Molecular Polarizability of Cyclohexane, at the Optimized Geometry, from Gaussian 90, with D95** Basis Set and from AIM Analysis of the Wave Function TABLE 2
polafizabilty term
c 1-6 Ha,
H, CT total
c 1-6 HaX
Heq
contribution to polarizability ( 1 w Cm2/V) xx
YY
zz
0.333 1.541 12.651 14.525
0.056 11.477 0.367 11.900
-6.178 1.082 0.662 -4.434
-3.992 0.113 0.824 -3.055
CT 0.396 1.542 12.652 14.590
AD -6.247 1.081 0.661 4.505
AD total Rayleigh scattering depolarization ratio, re~pective1y.l~The anisotropy was determined for three different wavelengths. The data shown are for the 638.2-nm line, a value which is probably still about 5% greater than the static limit. For comparison, the difference between the static and dynamic mean molecular polarizabilities of N2, CO, HCl, and Cl2 ranges from 1.0-2.3% and increaseswith the magnitude of the molecular polarizability.15 The ab initio values from Gaussian 90 are static HF/D95** level (see discussionbelow). The experimentalRaman trace scattering intensity is dependent on the square of the derivativeof the mean molecular polarizability. The experimental Gaussian 90 and AIM values for the polarizability derivatives with respect to the symmetric CH stretches are given in Table 1. Diagonal elements of the derivative tensor are shown for both the Gaussian 90 and AIM calculations. There is poorer agreement between the latter sets of values than was obtained for the smaller alkanes,*because the AIM calculations are approaching the current limits of the integration reproducibility (vide infra). The CT and AD contributions to the molecular polarizability, summed over equivalent atoms, are listed in Table 2. Individual atomic contributions to the molecular polarizabilityare displayed in Figures 1-3, illustrating the patterns of charge transfer versus atomic dipole for electricfields applied in the x ,y , and zdirections. Carbon atom contributions are in italics, and hydrogen atoms are in normal text, while the arrows indicate the direction of the electron transfer (la, 2a, and 3a) or of the induced atomic dipole vector (1b, 2b, and 3b). Not every atomic term is displayed, as some atoms in the figure are eclipsed; however, reproducibility for the atoms was at least as good as the number of significant figures given. Off-diagonal terms are frequently of the same order of magnitude as the diagonal terms; however the molecular orientation has been chosen so as to align the principal axes of the molecular polarizability tensor with the Cartesian axes, and thus, the off-diagonal terms ultimately cancel out on summation.
Molecular Polarizability of Cyclohexane
The Journal of Physical Chemistry, Vol. 98, No. 3, 1994 773 0.05 1
c
0.050
0.172
0.046 Figure 2. (a, Top) Illustration of charge transfer in cyclohexane for an electric field directed along the +y axis (top to bottom in this figure). (b, Bottom) Illustration of induced atomic dipoles at the carbon atoms for the field along the y axis. See caption to Figure 1.
0.172 Figure 1. (a, Top) Illustration of charge transfer in cyclohexane in the presence of an electric field. The CT dipole is taken as positive for the vector pointing from plus to minus along the +x axis (right to left in this figure). Numbers shown give the charge transferred into or out of an atomic basin, in electrons, for an applied field of 0.009 445 V. (b, Bottom) Illustration of the atomic dipoles induced in the carbon atoms by the electric field. Dipole vector is also taken as positive when pointing from plus to minus along the +x axis. Units: 1 V Cm2/V.
magnitude of the bond displacement (ar) and divided by 6, the number of equivalent bonds, to yield the dCT/dr and dAD/dr contributions to acu/dr, again in SI units Cm/V). It is emphasized that these are derivatives of the total molecular polarizability with respect to the stretch of a single CH bond and not “bond polarizability” derivatives (vide infra).
Only diagonal terms are given. The numbers for the CT are the actual amounts of electronic charge transferred into or out of an atomic basin. The AD contributions have been scaled by the magnitude of the applied field, as per the second term in eq 1, and converted to SI units (1W Cm2/V). There is some controversy concerning the relative importance of charge transfer versus changes in atomic first moments in the determination of the molecular polarizability.16 The charge density in the ring plane and the change in this density with the application of an electric field are displayed in Figure 4a and 4b, respectively. Their significance is discussed below. The atomic contributions to the polarizability derivatives for the axial and equatorial CH stretches are given in Table 3. For a given fixed field direction, the responses, at the individualatoms reflect their position relative to the field. The data in Table 3 have been summed over equivalent atoms so that theD3dsy”etry of the molecule is regained. The results were then scaled by the
Molecular Polarizability. Wave Function Quality and Recovery of Data. The direct analytical calculation of the molecular polarizability with the D95**basis set yields a value which is lower than that measured experimentally. The choice of basis set, the omission of electron correlation, and the limitation to a static rather than a dynamic calculation have been discussed in considerabledetail in two earlier p a p e r ~ . ~For J ~ the series ethane, ethylene, and acetylene,I7it has been shown that the magnitudes of the polarizability and of its derivative were reduced slightly with the inclusion of electron correlation via MP2 theory. However, the effect was not significantfor saturated hydrocarbons. Similar results have been observed for other molecules1*and are attributed to the increase in antibonding character. On the other hand, inclusion of the laser frequency would raise the value of the calculated polarizability by a similar a m o ~ n t . ~ JIt~ is J ~not possible to perform such dynamiccalculationswithin the Gaussian program package. We note therefore that since we are restricted
Discussion
774
Gough et al.
The Journal of Physical Chemistry, Vol. 98, No. 3, I994
TABLE 3: Breakdown of Contributions to the Polarizability Change Associated with the Totally Symmetric CH Axial and CH Eauatorial Stretching Modes. Units: 101-30 Cm/V
am/ar
acT/ar xx
YY
zz
xx
YY
0.144 -0.060 0.656 0.740
0.172 -0.068 0.642 0.746
0.240 -0.007 0.246 0.479
0.108 0.134 -0.008 0.234
0.072 0.133 -0.011 0.194
0.264 0.979 -0.078 1.165
ZZ
Equatorial
C Ha,
H, total
1.549 -0.344 -0.660 0.555
1.797 -0.327 -0.651 0.819
-0.157 -0.084 0.121 -0.120
0.359 -0.001 0.003 0.361
0.308 -0.001 0.004 0.311
0.420 0.921 0.011 1.352
Axial
C 0.002
Ha,
0.002 0.002
H, total
r4.661 0.140 I
0.136
0.021- 0.022 0.022 Figure 3. (a, Top) Illustration of charge transfer in cyclohexane for an electric field directed along the +z axis (bottom to top in this figure). (b, Bottom) Illustration of induced atomic dipoles at the carbon atoms for the field along the z axis. See caption to Figure 1.
EO
J
EY-EO
x- 2.5 Figure 4. (a, Top) Charge density in the xy plane of cyclohexane with zero applied field; the z axis represents charge density in atomic units. Maxima correspond to the projection of the nuclei on the xy plane. Grid points are at intervals of 0.5 bohr. The carbon atom lying on the +y axis near the top of the figure corresponds to the carbon at the bottom of Figure 2b. (b)Change in charge density due to electric field applied along they axis. Note only half of the molecule is illustrated in this figure (y1 0). See text for further discussion.
to static SCF level calculations, modifications to the basis set such as inclusion of more polarization functions, in order to obtain values matching experiment, would be somewhat cosmetic. We have shown previously that the trend in the molecular polariz-
abilities is reproduced satisfactorily7~*for the alkanes, as is the trend for the polarizability derivatives. A first test of the reliability of the AIM method is to check the total electron population and the reproducibility for symmetrically equivalent atoms. The sum of the atomic electron populations for any of the calculations here is correct to at least 0.008%. Atomic properties for equivalent atoms are the same to better than six significant figures. The molecular polarizability obtained from the sum of the atomic polarizabilities (eq 1) is always within 1% of that from the analytical Gaussian 90 calculation (Table 1). We therefore conclude that the AIM analysis is reliably recovering the information contained in the wave function. AIM Results. Physical observables, such as molecular polarizabilities and polarizability derivatives, can be reconstructed from the atomic properties obtained by integration of the wave function over atomic basins, as defined by the theory of atoms in molecules. In the past,8 we have shown that a molecule in an electric field exhibits the properties of a dielectricmaterial. While it is not freely conducting, there is some transfer of charge from one end of a molecule to the other, such that a surface charge polarization appears. This induced charge-transfer dipole opposes the applied field and affects the response within the molecule's interior. The local field experienced by an interior atom is the sum of the applied external field and the field produced by the surface charge polarization. This is evidenced by the fact that the induced atomic dipoles of the carbon atoms in the alkanes are parallel to the external field and oppose the surface charge polarization. The patterns of charge transfer and induced atomic dipole in cyclohexane (Table 2) are identical to those observed in methane, ethane, and propane. The molecular polarizability in thexy plane (the ring plane) arises mainly from an end to end transfer of charge between equatorial hydrogens (Figures 1a and 2a). This dipole is opposed by the reorientationof charge within the carbon atom (Figures l b and 2b). When the field is applied along the z axis, the charge transfer occurs between the axial hydrogen atoms, while the charge distribution in the carbon atoms shifts to create an opposing dipole (Figures 3a and 3b). Comparison with Other Models. Several other methods are currently being used to estimate molecular polarizabilities, such as theatomic monopole-dipole interaction modeP and distributed multipole a n a l y s i ~ . ' ~In - ~an ~ investigation of transferable net atomic charges in saturated hydrocarbons,21the latter method was found to provide the more acceptable results, particularly for nonpolar molecules. The atom-centered point-chargemodel did not reproduce the charge distribution accurately. The atomic monopole-dipole interaction model has recently been used to calculate molecular polarizabilities in alkanes, including cyclohexane, and some aromatic compounds.16 The model consists of four parameters; an atomic monopole polarizability parameter (representingcharge transfer) and an atomic dipole parameter, for each of carbon and hydrogen. All atomic polarizabilities
The Journal of Physical Chemistry, Vol. 98, No. 3, 1994 775
Molecular Polarizability of Cyclohexane were assumed to be isotropic. Initial estimatesof these parameters were optimized to reproduce experimental polarizabilities. A physically acceptable fit could not be achieved when CT terms were included; hence they concluded that there was no CT contribution. Their conclusion is in direct contrast to the AIM results presented here and elsewhere.s They have argued that our choice of interatomic surface could be affecting the apparent relative contributions. To investigatethis issue, we have calculated the charge density for a grid of points in the xy plane (Le., slicing through the plane of the ring) with and without the applied field. This twodimensional plane has been selected as illustrative of the threedimensional shift in charge which occurswhen the field is applied. The unperturbed charge density manifests the D3d symmetry of the molecule, with maxima at the points immediately above or below the carbon nuclei and smaller local maxima above and below the equatorial hydrogen atoms (Figure 4a). A density difference map was generated from the (6, - 6,)data (Figure 4b). Only half of the molecule is shown. The carbon nuclei are located approximately above the (f2.5,l.S) and below the (0.0, 3.75) coordinates on the xy plane. The presence of the electric field has shifted the carbon charge density toward they = 0 axis, while an increase in charge density beyond the equatorialhydrogen atoms is evidenced by the rising slope along the x = 0 axis at the far edge 0,= 4.5-5.0). From thePROAIM results, theintegration over the atomic volumes showed that the greatest charge transfer was to the equatorial hydrogen atom at (0,4.0) (Figure 2a). This is accompaniedby the induction of atomic dipoles oriented in the opposite direction at the carbon atoms (Figure 2b and Table 2). The chargedistribution changesin Figure 4b show this alternating increaseand decreaseof charge buildup throughout the molecule, exactly as found in the PROAIM results. Can one define an interatomic surface which would result in a zero charge-transfer component, as was found in the point-charge model?l 6 The charge increase about the equatorial hydrogen in found on the extreme edge of the molecule, with a large trough between the hydrogen and the carbon to which it is bonded. The dipoles about the carbon atoms are oriented in the direction opposite to charge transfer. Overall, one obtains the following pattern across the ring: electric field:
a+)
€(-)
charge pattern:
+ (- + - + - + - +) -
molecule:
Hq (C
c c
C)
Hq
The atomic dipoles centered about the carbon atoms oppose the dipole established through the transfer of charge between the equatorial hydrogens at opposite ends of the ring. Should one wish to draw surfaces which would numerically eliminate a chargetransfer contribution, it would be necessary to put the carbonhydrogen interatomicsurfacemuch closer to the hydrogen nucleus, given the location of the charge maximum at (0,S.O) in Figure 4b. This would still not explain the distribution of charge about the carbon atoms. Thus it appears that the charge transfer calculated through PROAIM is at least a faithful representation of the behavior of the charge density as calculated with this level of quantum mechanics. This combination of surface charge polarization accompanied by internal counterpolarization is observed consistently for a variety of molecular electrostatic phenomena: molecular dipole moments of diatomic hydrides9 and alkanes22 and in the equilibrium molecular polarizability of all molecules studied to date.8*23The choice of the zero flux surface is not arbitrary, as it is only for this unique surface that the topological and quantum mechanical definitionsof an atom coincide? The atomic electron populations are known to exhibit a small basis-set most noticeable on going from a balanced (e.g. 6-31G** set) to
unbalanced (e.g. 4-31G* set). It is not significant for the large, balanced basis set employed here. The change in the atomic dipole of the carbons atoms can only be occurring in response to the intense local field produced by the charge transfer between the hydrogen atoms. Our results are based on an ab initio analysis of the molecular wave function, which incorporates the natural asymmetry of the atoms and which accounts for >99% of the total electron population. We conclude that charge transfer is a very real phenomenon and is the principal source of molecular polarizability in the alkanes studied so far. Polarizability Derivatives. Recouery of Wave Function Znformation. The first concern is that the MO calculationsprovide a good approximationto the experimentalvalues. As was found for the CH stretching modes in methane, ethane, and propane? the da/drvalues with the D95** basis set, from Gaussian 90, are lower than those measured experimentally,but once again they reproduce the ordering observed experimentally. From the breakdown into xx, yy, and zz components (Table l), it is also apparent that the derivative for the axial CH stretch arises mainly from changes in the z direction, while the equatorial CH stretch causes changes in the xy plane, as one might intuitively expect. The polarizability derivatives for cyclohexane which are recovered by the AIM programs are not as good as the results for the straight chain alkanes in the previous paper. This is due to accumulated integration errors for the larger molecule. In the AIM programs, the atomic basin is first identified by the points at which the flux in the gradient vector of the charge density becomes zero. The shape of the hydrogen atom is typically a slightly distorted sphere. The carbons are also quite regular but present a greater technical problem as the overall shape is the volume left over after the hydrogens are removed and thus includes cusplike points. When the molecule is placed in an electric field, this surface is warped slightly. We have found that the integrationsfor equivalent hydrogen atoms are easily reproducible for up to seven or eight significant figures; however, the carbon atoms are less reliable. In the PROAIM setup, it is possible to vary the relevant parameters to increase (in polar coordinates) the number of lp and 6 planes into which space is sliced as well as the number of points along the r axis, thus creating a finer mesh. We have consistently run all integrationsat the maximum limits to obtain the best possible results. Errors in the integrations are directly traceable to the carbon atoms only, some of which are unique because of the direction of the applied field, and thus are not recoverable from other atomic data. The size of the error isabout lO-lS%ofthederivative, whichisjust sufficient toreverse the apparent ordering of the derivatives. AIM Results. While this is a disappointing result, we find from comparison of the Gaussian 90 and AIM data (Table 1) that the orientational dependence of the derivative is recovered. Thus, a qualitative description of the origins of the polarizability derivatives can still be made from the final breakdown of the derivatives into atomic contributions. The changesin the atomic CT and AD contributions to the polarizability associated with the stretch of an axial or equatorial CH bond appear in Table 3. There are striking similarities between these results and those found previously for the alkanes. The equatorial CH bonds are structurally similar to the methyl CH bonds in propane which lie in the plane of the carbon chain. In both molecules, these bonds are the shortest and their stretch produces the greatest change in the molecular polarizability. It has already been noted that the polarizability change is greatest for the field direction which is aligned with that bond’s orientation. Because of the D3d symmetry of cyclohexane, this is equal to the x and y directions for the equatorial CH bonds. For the axial, it is greatest in the z direction. For both bond types, there is a large positive change in the carbon CT, and CT,, terms which is partially offset by a smaller negative change in CT at each of the hydrogen atoms.
Gough et al.
116 The Journal of Physical Chemistry, Vol. 98, No. 3, 1994
In cyclohexane, the axial CT term is about half that of the equatorial, whereas in propane all methyl CH contributions were about equal. The changes in the AD terms are again confined mainly to the atoms of the bonds being stretched and are positive. The axial CH bonds in cyclohexane are nearly aligned with the z axis of the polarizabilitytensor. Individualatomiccontributions to the derivatives are only somewhat similar to the out-of-plane methyl and the methylene C H bonds of propane. Changes in CT and AD terms are greatest when the field is applied to the z direction and are mostly confined to the carbons and the axial hydrogens. The increase in the total CT,, term is slightly larger than the total AD,, term. The equatorial CH bonds are more nearly orthogonal to the field direction and the direction of bond stretch, a fact which probably accounts for their minimal involvement in the derivative. Their contribution is small and negative. Finally, it is useful to note that the equilibrium molecular polarizability is smaller in the z direction than in the xy plane and that the derivatives for the stretch of the axial and equatorial CH bonds mirror this feature, just as in propane. There, the derivative for the in-plane CH bonds, aligned with the long axis of the molecule and the direction of greatest polarizability, is greater than that for the out-of-plane methyl and the methylene CH bonds. In both molecules, it is the increase in the carbon CT term along the principal axis which appears to be the significant factor. Comparison with Other Models. Our principal goal is the establishment of a simple description of the origin of Raman trace scattering intensities,one which will permit the interpretation and prediction of intensities on the basis of molecular structure. The method most widely used for the interpretation of intensities is the bond polarizability model, which postulates that the molecular polarizability can be approximated as a sum of polarizability ellipsoids directed along each bond. A limited set of bond parameters is empirically fit to measured intensities, to provide derivatives of the bond polarizability tensors. These parameters may be transferred to similar molecule^.*^^^^ While it is true that the charge density in a molecule is greatest around the nuclei and along the bond paths which connect them, the model is intentionallyoversimplified in order to make the problem more tractable. The aa/& values reported here give the change in the entire molecular polarizability with respect to the stretch of a single CH bond. Our analyses indicate that the entire molecular charge distributionchanges in a concerted fashionunder the influence of an applied field. The position of a bond within a molecule and its orientation with respect to the principal axes of the molecular polarizability tensor determine both the origin and the magnitudeof thederivativeassociated with a bond stretch. These terms are not generally transferable. Summary
The molecular polarizability and polarizability derivatives for the symmetric CH stretching modes in cyclohexane have been calculated with the D95** basis set. The wave functions from the molecular orbital calculation have been analyzed with the theory of atoms in molecules, and a simple picture of the charge reorientation due to the applied field has been obtained. The molecular polarizability is found to be a combination of two opposing factors. The first is a surface polarization, achieved by transfer of electronic charge density between hydrogen atoms on
opposite sides of the molecule, for a given field direction. The dipole induced by charge transfer is offset by the second factor, the smaller but significant changes in the atomic dipoles of the carbon atoms, which shift to oppose the charge-transfer dipole. The polarizability is greatest in the plane of the ring. The symmetric stretch of the equatorial CH bonds has been found by experiment to produce a greater change in the molecular polarizability than the stretch of the axial. The quantum mechanical calculations reproduce this ordering. The AIM analysis of the polarizability derivatives shows that, just as in the case of the straight chain alkanes, the charge transfer between carbon atoms increases when the CH bonds are stretched, and this term is greatest when the CH bond is more nearly aligned with the carbon chain. This method of analyzing both the equilibrium polarizability and polarizability derivativesmay prove useful in the study of larger molecules, such as conducting polymers, where charge transfer is believed to play the most significant role.
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