J. Phys. Chem. 1984, 88, 593-600 593

static C-H bond dipoles. However, we have shown that bending modes are far from simple, and that the dipole moment derivatives for these modes are dominated by bent bond moments rather than bond dipoles. The magnitude of orbital following will be related to the percent s character in the bond, and this may be a major contributor to the linear relations shown in Figure 1 of ref 37.

The dipole moments for the C-H stretching modes of methane and ethylene correspond to an apparent C+-H- bond dipole. In order to change this to C--H+, previous workers have invoked charge flows during the vibration which are so large that they mask the static bond dipole. This is unreasonable, and, as we have shown, the charge flows are in reality fairly small. Thus, the previous conclusions concerning the sign of the C-H bond dipole have been based on erroneous models, and are incorrect.

It is interesting to note that the observed anharmonic vibrational frequencies stand in almost constant ratio to the calculated harmonic frequencies. The harmonic frequencies estimated from the experimental data do not give this constant ratio, and bring the C H stretching frequencies into better agreement while leaving the bending modes relatively unchanged. This is surprising since one might expect that electron correlation, which is neglected in the R H F calculations, would have a larger effect on the stretching modes than on the bending modes,38 and yet it is the latter which show the larger deviation after the anharmonicity correction has been made. It is possible that the correction for stretching modes

(37) Gussoni, M.; Jona, P.; Zerbi, G. J . Chem. Phys. 1983, 78, 6802. (38) DeFrees, D. J.; Raghavachari, K.; Schlegel, H. B.; Pople, J. A. J . Am.

Chem. SOC. 1982, 104, 5576. DeFrees et al. found that the calculated equilibrium bond lengths were significantly affected by the inclusion of electron correlation, but that the calculated bond angles were essentially unchanged.

has been overestimated, and that for be.nding modes has been underestimated, but it is also possible that the bent bonds formed in the bending modes may lead to a larger correlation energy correction. The level of calculation used in the study of the force fields for MH4 species by Yamaguchi and S ~ h a e f e r ~ ~ and of ethylene by Pople et aL40 extended to include anharmonicity will be required to resolve this question for molecules having larger numbers and types of vibrational modes.

Calculations The calculations were carried out by using the program GAM-

ESS4’ and the 6-31G** basis set.13 The vibrational frequencies were calculated in Cartesian coordinates as described by Gwinn.,O The wave functions were integrated numerically as previously described.12 As an indication of the accuracy of the integration, the valence electron population of a methyl group of ethane was found to be 6.9997 e as compared to the expected value, 7.0000 e.

Acknowledgment. This investigation was supported by the Office of Basic Energy Sciences, Department of Energy. The computer was obtained with the aid of grants from the National Science Foundation (CHE-80-23366) and the Dreyfus and Hey1 Foundations.

Registry No. Ethane, 74-84-0; ethene, 74-85-1; ethyne, 74-86-2; methane, 74-82-8; ammonium, 14798-03-9; silane, 7803-62-5.

(39) Yamaguchi, Y.; Schaefer, H. F., 111 J . Chem. Phys. 1980, 73, 2310. (40) Pople, J. A.; Krishnan, R.; Schlegel, H. B.; Binkley, J. S. Int. J .

(41) Dupuis, M.; Spangler, D.; Wendoloski, J. J. National Resource for Quantum Chem., Quantum Chem. Symp. 1979 13, 225.

Computing in Chemistry, Program QGO1, 1980.

Conformational Dependence of Fermi Resonances in n -Alkanes. Raman Spectra of 1 ,I ,1,4,4,4-Hexadeuteriobutane

S. Abbate, CNR, Units di Spettroscopia presso il Politecnico di Milano dell’lstituto di Chimica Applicata dei Materiali di Genova, Milano, Italy

S . L. Wunder, Polymer Sciences and Standards Division, National Bureau of Standards, Washington, D. C. 20234

and G. Zerbi* Dipartimento di Chimica Industriale e Ingegneria Chimica, Politecnico di Milano, Milano, Italy (Received: June 13, 1983)

The Raman spectrum of molten 1,1,1,4,4,4-hexadeuteriobutane in the C-H stretching region shows three main peaks near 2940,2900, and 2850 cm-I. These bands vary in the temperature range 36-300 K analogously to what happens in polyethylene and longer paraffins. Fermi resonance calculations have been performed and supplemented with intensity evaluations of the unperturbed fundamentals, through electrooptical parameters. Both trans-CD3CH2CH2CD3 and gauche-CD,CH2CH2CD3 have been considered; the calculations show that the antisymmetric CH stretching fundamental, which in principle should contribute to the Fermi resonance in gauche-CD,CH2CH2CD3, is very weakly perturbed. The increase with temperature of the ratio of the intensities of the bands at 2940 and 2850 cm-’ is here reanalyzed.

Introduction It is now common knowledge that the vibrational Raman

spectrum of polymethylene chain molecules (polyethylene, n-al- kanes, fatty acids, phospholipids, etc.) is a useful tool for probing their complex and yet unsettled structure.l Indeed these molecular

systems can exist in several phases and undergo peculiar inter- and intramolecular motions (molecular flexibility) which are still the subject of active study in several group^.^,^ The main con- clusion of these studies is that, while in the solid these chain molecules exist mostly in the trans planar conformation: they are,

(1 ) G. Zerbi, ‘Advances in Infrared and Raman Spectroscopy”, R. J. H. Clark and R. E. Hster, Eds., Heyden, London, in press; “Probing the Real Structure of Chain Molecules by Vibrational Spectroscopy. Some Recent Aspects”, Adv. Chem. Ser., American Chemical Society, New York, 203,487 (1983).

I 0022-3654/84/2088-0593$01.50/0

(2) G. Zerbi, R. Magni, M. Gussoni, K. Holland Moritz, A. B. Bigotto,

(3) M. Maroncelli, Song Ping Qi, H. L. Strauss, and R. G. Snyder, J . Am. and S. Dirlikov, J . Chem. Phys., 75, 3175 (1981).

Chem. Sac., 104, 6237 (1982).

0 1984 American Chemical Society

594

however, able to host small conformational distortions (defects) in the various solid phase^.^,^ Such defects become dominant in the melt or solution states.

The vibrational Raman spectrum can be divided into a few structurally relevant ranges, each one providing detailed and independent information on the conformational structure and dimensions of the polymethylene segments.5 Of particular interest is the C-H stretching region near 2900 cm-I, which, in principle, contains much useful information but has not yet been fully exploited because a complete understanding of its features has not yet been achieved. After many correlative type works by several authors on various molecules containing alkyl residue^,^,' important progress has been made since Snyder et al. pointed out the existence of several Fermi resonances in the trans conformation which mainly determine the frequency and spectral shape in the 2900-cm-' r a r ~ g e . ~ . ~ An analysis similar to that presented by Snyder et aL9 was carried out in our laboratory.I0J1 We did extend, however, our interpretation both to the case of perfect tridimensional orthorhombic lattices as well as to the case when chains twist in distorted conformations containing several gauche (G) structures. We reported the case of molten polyethylene,I0J1 for which we proposed that the ratio R = 12940/12850 could be taken as a relative measure of G content. The Raman spectra of n- hexadecane in the melt and in solution were also reported.I2 Some disagreement regarding the use of R has recently been ex- pressed13J4 based on the fact that a nonnegligible contribution to the observed spectrum of n-alkanes in the C-H stretching region is given by the scattering due to the C-H stretching of the CH3 group which has been shown to be sensitive to the en~i ronment . '~ ,~~

The present work is an attempt to contribute toward a better understanding of this spectral region of the Raman spectrum. We study n-butane, the shortest hydrocarbon molecule which can exist, in the trans (T) or gauche (G) conformation and substitute the six hydrogens of the methyls with deuterium atoms in order to remove the perturbations by the CH3 groups. While this molecule in the solid phase is T, in the melt an equilibrium between T and G structures exists16 similar to what happens in longer normal paraffins.

In general, the experimental observations to be accounted for in the Raman spectrum near 2900 cm-' of molecules containing n-alkyl residues are the following:

(1) As shown in ref 10 (Figure 15) the spectrum of trans crystalline polyethylene shows two main peaks near 2850 (strong, broad) and 2890 (very strong and sharp) cm-l and two weak broad bands near 2900 and 2940 cm-I. The generally accepted as- signment is as follows: the bands near 2850,2900, and 2940 cm-' are the fundamental symmetric CH2 stretch (d,) and overtones of A, species in Fermi resonance. The band near 2890 cm-' is the B,, antisymmetric CH2 stretching (d-) floating on top of a broad background of A, species.

(2) Upon melting (ref 10, Figure 17) the band near 2850 cm-' becomes the strongest feature, and the d- collapses and shifts about 10 cm-l upward with increasing temperature becoming, seemingly, a very broad feature. The overtone level near 2940 cm-' increases

The Journal of Physical Chemistry, Vol. 88, No. 3, 1984 Abbate et al.

(4) G. Minoni and G. Zerbi, J . Phys. Chem., 86, 4791 (1982); G. Zerbi,

(5) R. G. Snyder in "Methods of Experimental Physics", Vol. 16A, Aca-

(6) B. P. Gaber and W. L. Peticolas, Biochim. Biophys. Acta, 260,4650

(7) R. C. Spiker and I. W. Levin, Biochim. Biophys. Acta, 361, 3880

(8) R. G. Snyder, S. L. Hsu, and S. Krimm, Spectrochim. Acta, Part A,

G. Minoni, M. P. Tulloch, J . Chem. Phys., 75, 5853 (1983).

demic Press, New York, 1980.

( 1967).

(1975).

34, 395 (1978): (9) R. G. Snyder and J. R. Scherer, J . Chem. Phys., 71, 3221 (1979). (10) S. Abbate. S. L. Wunder, and G. Zerbi, J . Phys. Chem., 85, 3140

(1982). (11) G. Zerbi and S. Abbate, Chem. Phys. Lett., 80, 455 (1981). (12) S. L. Wunder and S. D. Merajver, J . Chem. Phys., 74, 5341 (1981). (13) R. G. Snyder, J . Chem. Phys., 76, 3342 (1982). (14) R. G. Snyder, H. L. Strauss, and C. A. Elliger, J . Phys. Chem., 86,

(15) R. A. MacPhail, R. G. Snyder, and H. L. Strauss, J . Chem. Phys.,

(16) L. Colombo and G. Zerbi, J . Phys. Chem., 73, 2013 (1980).

5145 (1982).

77, 1128 (1982).

a I

b I

Figure 1. (a) trans- 1 , l , 1,4,4,4-Hexadeuteriobutane: structure and Cartesian axes. (b) gauche-l,1,1,4,4,4-Hexadeuteriobutane: structure and Cartesian axes.

in intensity with increasing temperature. For normal paraffins and for molecules containing n-alkyl residue the collapsing and the broadening of d- is verified" and is further supported by the depolarized spectra.14 The upward shift of d- was first noticed by Gaber and Peticolas,6 who interpreted it qualitatively in terms of change of "lateral interactions". The broadening is also ob- served, as a function of temperature, in the depolarized spectra of small model molecules such as n-propane, n-butane, n-pentane, and several deuterio derivatives.I8 The shift is clearly observed in our spectra of CD3CH2CH2CD3.

The interpretation that we first suggestedl03" for such spectral changes in going from T to G structures was that in the all-T structure the overtone peak near 2940 cm-' is weak because it shares its intensity, borrowed from the d+(A,) fundamental, with a manifold of overtone (two-phonon) states of the CH2 bending modes around 2900 cm-'. When G structures are generated, overtone T levels near 2900 cm-' are removed, thus giving back intensity to the 2940-cm-I overtone band. As already mentioned, we proposed that the ratio of the intensities I2940/I2850 could be taken as proportional to the G content of the sample. For n- alkanes Snyder has shown that the sensitivity of the vibrations of the methyl r+ band at 2938 cm-I to the environment can affect the above ratio,l33l4 thus casting some doubts on the applicability of our method to the study of conformational changes in n-alkanes. The idea instead seems to work when CH3 end groups are removed by deuteration14 or in the case of polyethylene, which has prac- tically no CH3 groups.I0J1

In the analysis of the situation we then became concerned with the possible contribution from the d- band in this spectral range. To our knowledge, however, there are not yet interpretations of the fact that d- broadens and shifts upward when G structures are generated. It is not even clear whether d- simply broadens or whether a multiplicity of bands is generated through some mechanism when G structures are introduced.

The fact that the natural breadth of the d- band increases is clearly d e m ~ n s t r a t e d ' ~ ~ ' ~ * ' ~ and its interpretation is the subject of current interest in our group. Another possible contribution to the intensity and breadth of d- in terms of molecular mobility has been suggested by Snyder et al." In this paper, however, we wish to focus on whether new Fermi resonances from distorted molecules place spectral features in the vicinity of the d- trans band.

Theory The two conformations of butane are shown in Figure 1, a (trans

= T) and b (gauche = G), respectively. trans-Butane belongs ~~~ ~~

(17) R. G. Snyder, J. R. Scherer, and B. Gaber, Biochim. Biophys. Acta,

(18) S. L. Wunder and G. Zerbi, to be submitted for publication. 601, 47 (1980).

Conformational Dependence of Fermi Resonances

TABLE I: Calculated Eigenvectors for Three Normal Modes of ~ ~ u ~ ~ - C D , C H , C H , C D , ~

The Journal of Physical Chemistry, Vol. 88, No. 3, 1984 595

symmetry d,' d.' 61 d," d_I1 6 I1 b species

-1.022 0 0.096 - 1.022 0 0.096 e++ A, 0.028 0 0.950 0.028 0 0.95 0 9+ A, 0.028 0 0.95 1 -0.028 0 -0.95 1 9 - BZLl

a Q,' corresponding to symmetric CH, stretching, 9+ to symmetric HCH bending, 9 - to antisymmetric HCH bending. The eigenvectors are in terms of unnormalized group coordinates of the two CH, units, I and I1 (see text). described by the eigenvectors in terms of group coordinates such as d+I , etc. The notation of the type d++ is used in the text for a qualitative assignment of the band. Such an assignment is fairly close to what is exactly described by dynamical calculations (Le., e++, etc.).

The notation in this column refers to normal modes

TABLE 11: Calculated Eigenvectors for Six Normal Modes of gauche-CD,CH,CH,CD, a

symmetry d+' d_" 61 d+I1 d_'I 611 b species

1.022 -0.029 -0.098 1.022 0.029 -0.098 e++ A -1.023 -0.039 0.094 1.023 0.039 -0.094 e+- B -0.035 -1.049 0.00 2 -0.035 -1.049 0.002 Q-+ A

-0.031 0.00 1 -0.948 -0.031 0.001 -0.948 9 A -0.025 -0.003 -0.950 0.025 0.003 0.950 9' B

-0.038 1.048 0.002 0.038 - 1.048 -0.002 Q; - B

a e++, symmetric in-phase CH, stretching; Q+-, symmetric out-of-phase CH, stretching; Q.+, antisymmetric in-phase CH, stretching; antisymmetric out-of-phase CH, stretching; 9+, in-phase HCH bending; 9- , out-of-phase HCH bending. The eigenvectors are given in terms of internal unnormalized group coordinates of the two CH, units, I and I1 (see text). The notation in this column refers to normal modes de- scribed by the eigenvectors in terms of group coordinates such as d:, etc. The notation of the t y p e d + + is used in the text for a qualitative assignment of the band. Such an assignment is fairly close to what is exactly described by dynamical calculations (i.e., e++, etc.).

to the DZh symmetry point group, for which the Raman-active species are A,, B,,, BZg, and B3r The C H stretching modes which are active in the Raman are the CH, symmetric stretching in phase in the two CH2 units (u(d++), A, species) and the antisymmetric CH2 stretching in phase in the two CHI units (u(d-+)) which is of B3g species. (The Cartesian axis system choice made here does not coincide with that of our previous work.1° The species B3* here is B,, in ref 10 and BzU here is B3u in ref 10.) The scissoring normal modes are of symmetry Ag for the in-phase ~(6') and Bzu for the out-of-phase u(6-) . 2u(6+) and 2 4 6 9 both belong to the A, species and thus can interact with u(d++) whereas ~(6') + u ( 6 ) is active in the infrared (BzU species). u(d-+) is not involved in any Fermi resonance scheme. Thus, in trans-butane there will be only one Fermi m a t r i ~ ~ ~ ~ ' ~ for the Raman case to be considered, involving u(d++), 2 4 6 9 , and 2 4 6 7 .

gauche-Butane belongs to the C2 symmetry point group, which has only two representations, referred to as A and B species. In-phase symmetric CH2 stretching, u(d++), and in-phase anti- symmetric CH2 stretching, v(d-+), belong to the A species, while the corresponding out-of-phase symmetric CH2 stretching, u(d+-), and out-of-phase antisymmetric CH, stretching, v(d--), belong to the B species. ~(6') has A symmetry; u(6-) has B symmetry; 246') and 2 4 6 7 are of A symmetry; u(S+) + v(6-) is of B sym- metry species. Both A and B species are Raman active. In gauche-butane two Fermi matrices have then to be considered: the first, of A species, involving u(d++), u(d-+), 2u(6+), and 2467, and the second of B species, involving u(d+-), v(d--), and u(S+)

The anharmonic potential that we assume for both trans- and gauche-butane is of the form previously reported for polyethylene and n-alkanes:

+ u(6-) .

In eq 1, dll and d,' denote the two C H stretchings of the first CH2 unit; aI is the H C H bending of the same unit. d+' = dll + d,' is the symmetric, unnormalized CHI symmetric stretching group coordinate. We define coordinates of the second unit (11) analogously. The value 0.201 mdyn rad-2 for fdha has been suc- cessfully used for CD3CH2CD3, and per-

TABLE 111: Fermi Resonance Matrices for Case i u s b

(a) trans-CD, CH, CH ,CD, u(d++) 2 4 s +) 2 4 6 -)

2880.000 -23.319 -23.134 -23.319 2898.000 0 -23.134 0 2934.000

(b) gauche-CD,CH,CH,CD,, A Species

v(d_+) u(d++) 2u(6+) 2u(6-)

2910.000 0 -0.811 -0.814

-0.811 23.053 2918.000 0 2905.000 23.053 23.243

-0.814 23.243 0 291 0.000

(c) gauche-CD,CH,CH,CD,, B Species

u(d.-) U(d+.) Y ( 6 + ) + u ( 6 - )

( O

) 2910.000 0 -1.198

( O -1.198 -32.970 2914.000 2902.000 -32.970

a See text. b Units: cm-1.

deuteriopolyethylene.20 It will be used here also. In order to evaluate the anharmonic interaction elements between the fun- damental and overtone states mentioned above, we have to transform group internal coordinates into normal coordinates. These transformations have been derived through a normal-co- ordinate calculation based on Snyder's force field.,, The transformation is shown in Table I for trans-CD3CH2CH,CD3 and in Table I1 for gauche-CD3CHzCH2CD3 for all the pertinent normal modes. In the latter case, it should be noticed that not only u(d++) and u(d+-) but also u(d-+) and v(d--) contain con- tributions from d+I, d_", aI aI1 contrary to the T case. This allows the participation of u(d_$) and u(d--) in the Fermi resonance scheme. This fact has so far been overlooked. We test in this work the effect on the Fermi resonances of the participation of these new levels. Inserting the values of Table I and I1 into eq 1 and following the method presented in ref 10, we obtain the anharmonic coefficients of the Fermi resonance matrices shown in Table 111. In trans-butane u(6-) was taken as 1472 cm-', and u(6+) as 1454 cm-I; in gauche-butane ~(6') = 1464 cm-' and u(6-)

(19) G. Dellepiane, S . Abbate, P. Bosi, and G. Zerbi, J . Chem. Phys., 73, 1040 (1980).

(20) S. Abbate and G. Zerbi, to be submitted for publication. (21) R. G. Snyder, J . Chem. Phys., 47, 1316 (1967).

596

TABLE IV: Case i Eigenvalues (cm-') and Normalized Eigenvectors Resulting from the Diagonalization of the Matrices in Table 111'

The Journal of Physical Chemistry, Vol. 88, No, 3, 1984

(a) trunsCD,CH,CH,CD, U ( d _ + ) u(d++) 2v (6 +) 2u (6 -)

2944 (0, 0.394, .-0.199, -0.897) 2909 (0, -0.394, 0.845, -0.361)

2859 (0, 0.830, 0.496, 0.255)

(b) guiiche-CD,CH,CH,CD,, A Species

2900 ( I , 0, 0, 0)

v ( d _ + ) v(d+') 2u (6 *) 2u ( 6 -)

2943 (0.026, -0,650, -0.605, -0.406) 2914 (0.009, -0.121, 0.681, -0.722) 2910 (-0.999, -0.035, 0.000, -0.007) 2876 (0.002, -0.750, 0.414, 0.516)

(c) gauche-CD,CH,CH,CD,, B Species Y ( d . - ) . v ( d + - ) V(S+) t v ( K )

2941 (0.029, 0.640, 0.022) 29 10 (-0.999, 0.035, -0.009) 2875 (0.022, 0.768, 0.640)

a The values of u(d.+) at 2900 cm-' was added only for reasons of understanding, but in trans-CD,CH,CH,CD, is uncoupled from the other modes. We italicize the contributions from fundamental levels.

Abbate et al.

= 1460 cm-'; these values are the ones that we calculated from our normal-coordinate analysis on CD3CH2CH2CD3. In the evaluation of overtone and combination levels, an anharmonicity of 10 cm-' was assumed, as in ref 10. The choice of the unper- turbed frequencies in the C-H stretching region can be made in different ways. We discuss in this paper two cases.

(i) v(d++) = 2880 cm-I for trans-butane as was done previously for trans-polyethylene and v(d++) = 2905 cm-' and v(d+-) = 2902 cm-' for gauche-butane similar to that of CD3CH2CD3 and used for disordered polyethylene; 'O," v(d-+) = 2900 cm-I for trans- butane and v(d-+) = v(d--) = 2910 cm-I for gauche-butane. This corresponds to assuming some dispersion in the calculated values of the C H stretching frequencies, due to some kind of interaction between neighboring CH2 units. Indeed, normal-coordinate calculations based on Snyder's force fieldZ2 show some dispersion due only to geometrical factors. However, even accepting the idea of a dispersion, the shape of the dispersion curve described by our values differs from that calculated with the force field by Snyder.

(ii) Alternatively we assumed v(d++) - 2880 cm-' as if there were no dispersion and in going from T to G the frequency does not change. We have also tried the above values for both trans- and gauche-CD3CH2CH2CD3 and accordingly we also did not modify the v(d-+) from the value of 2900 cm-'.

Diagonalization of the matrices in Table I11 gives the fre- quencies and eigenvectors presented in Table IV for case i. As can be seen v(d-+) at 2900 cm-', which is uncoupled from other Fermi resonant modes in trans-butane, is only weakly involved in Fermi resonance in gauche-butane. The overtone levels are mixed predominantly with the d++ level in both T and G, but to a different extent. d-+ instead does not give significant contri- butions to the overtone levels.

To evaluate the intensities corresponding to the frequencies in Table IV we proceeded as follows. We assumed that no change in polarizability occurs for pure overtone or combination mod- es; 10~19 we calculated the derivatives of the polarizability with respect to fundamentals by the set of electrooptical parameters from ref 23 for the CH2 unit and from ref 24 for CD3. The values for d++ and d-+ for trans-CD,CH2CH2CD, are given in Table

(22) L. Piseri and G. Zerbi, J . Chem. Phys. 48, 3561 (1968). (23) S. Abbate, M. Gussoni, and G. Zerbi, J . Chem. Phys., 73, 4680

(24) S. Abbate, M. Gussoni, and G. Zerbi, Indian J . Pure Appl. Phys., (1980).

16, 199 (1978).

TABLE V: Case i Calculated Derivatives of the Polarizability Tensors with respect to Unperturbed Harmonic Normal Modes for trans-CD,CH,CH,CD, and gauche-CD,CH,CH,CD,a

(a) vans-CD,CH,CH,CD,

v(d.+) 0 0 0 az = 0 0 0 2.65 ,) y 2 = 21.02 0 2.65 0

-1.30 0 0 0 -2.43 0

u(d++) a* = 5.67 y2 = 3.36

0 -3.42

(b) gauche-CD,CH,CH,CD, v(d.+) A species

.-0.05 -0.66 1.15 a'= 0.01 -0.66 0.83 1.63 ) y 2 = 15.90 1.15 1.63 -1.05

-0.19 -0.24 2.85

0 -0.68 -0.39 -0.68 0.95 -0.55 -0.39 -0.55 -0.95

v(d:) B species

-0.21 -0.27 -0.48

a Units in tensors: A' amu-'", units for CY' and y 2 A4 amu-'

TABLE VI: Case i Calculated Frequencies and Intensities (Polarized and Depolarized) for trans-CD,CH,CH,CD, and gauche-CD,CH,CH,CD,'

symmetry U, cm-' species IPOl Idepol

(a) f~ans-CD,CH,CH,CD, 41.67 2.12 41.04 2.08 63.07 84.09

183.07 9.30

(b) gauch e-CD, CH,CH , CD 2942 A 108.96 2.85 2941 B 2.11 2.82 2914 A 3.80 0.1 1 2910 A 47.09 62.78 2910 B 15.76 21.01 2876 A 144.45 3.64 2875 B 2.86 3.81

2944 A, 2908 A, 2900 B3,

2859 A,

a Units for Intensities: A 4 a m - '

Va; those for d++ and d-+ for gauche-CD3CH2CH2CD3 (A species), and for d+- and d-- for gauche-CD3CH2CH2CD3 (B species), are given in Table Vb. Also

are given. We then constructed the polarizability derivatives relative to the modes in Table IV as linear combinations of the tensors in Table V using the coefficients of the fundamentals in the eigenvectors of Table IV (these values are italicized). In Table VI we report the corresponding intensities for the T case (Table VIa) and the G case (Table VIb). Since the actual spectra are those of the liquid (as must be the case in order to produce G structures), the relative population of conformers is taken from

Conformational Dependence of Fermi Resonances

! a i

I

2950

2950 28M en-’

2900 2850 I cm-1

b

depolarlzed

2950 2900 28501cm”

Figure 2. (a) Calculated Raman spectrum of trans-1,1,1,4,4,4-hexa- deuteriobutane (polarized case). Assumed bandshape: Lorentzian; half-bandwidth = 5 cm-I. (b) Calculated Raman spectrum of trans- 1,1,1,4,4,4-hexadeuteriobutane (depolarized case). Assumed bandshape: Lorentzian; half-bandwidth = 5 cm-’.

TABLE VII: Case ii Calculated Frequencies and Intensities (Polarized and Depolarized) for gauche-CD,CH,CH,CD, a

I>, cm-’ species IPOl Idepol

2935 A 66.96 1.85 2934 B 1.46 1.95 2914 A 3.62 0.11 2900 A 41.21 62.92 2900 B 15.86 21.14 2860 A 186.60 4.52 2860 B 3.4 1 4.54

symmetry

a Units for intensities: k4 amu-’

previous works.I6 At - 200 K T = 65% and G = 35%. The polarized and depolarized intensities are defined respectively as

IF, = 45a2 + 3y* Idepol = 4y2

The spectra which we obtained, assuming Lorentzian band- shapes with 5-cm-I half-bandwidth, are shown in Figures 2a,b

The Journal of Physical Chemistry, Vol. 88, No. 3, 1984 591

l a n

t 2950

b

I t depolarized

(trans-butane), 3a,b (gauche-butane), and 4a,b (mixture) for case i. For case ii we only report the calculated spectra in Figure 5a,b for G and Figure 6a,b for the mixture. The calculated frequencies and intensities are given in Table VII. These spectra have to be compared with the experimental Raman spectra of Figures 7 (polarized case) and 8 (depolarized case) of liquid CD3CH2C- H2CD3 taken at three different temperatures. (For the description of experimental details see ref 18.)

Discussion Our calculations and the comparison with experimental data

allow us to understand a number of facts for this molecule and to point out features common to longer normal paraffins and polyethylene.

(a) In going from T to G Fermi resonance interactions can change, allowing d+ and d- to mix. This may cause some re- distribution of intensities whose extent is estimated in this paper. Such redistribution depends on many factors, of which the most important ones are the dynamical coupling (eigenvectors) and the

598 The Journal of Physical Chemistry, Vol. 88, No. 3, 1984

l a n Abbate et al.

t 2950 28W lor'

i I

I depolarized I depolarized

i

fl 1 :

choice of the unperturbed frequencies. In this work we tried two sets of unperturbed frequencies, which hopefully encompass what happens in reality.

(b) Let us consider the depolarized spectrum. The lines at -2940 and -2860 cm-I, calculated weak for the T case, are predicted to show a very slight increase for the G case (Figures 2b, 3b, 4b, 5b, and 6b). The experimental spectrum (Figure 8) does show an increase of the corresponding lines; however, a firm experimental assessment cannot be made, because the spectrum is not free from polarized scattering.

(c) Let us consider the polarized case for which the most im- portant experimental observations for CD,CH2CH2CD, to be accounted for (Figure 7) are (I) the decrease in intensity and upper shift of the d- line, (11) the increase with temperature of the ratio R = Z2940/Z2860, and (111) the slight upper shift of d+ band at -2860 cm-'.

(I) Calculations indicate that the feature centered near 2900 cm-' is composed of two transitions, namely, the d- fundamental

a

2950 2900 2850 I om-?

b

I

'bDI 2110) 2 8 5 0 7

Figure 5. (a) Case ii: Calculated Raman spectrum of gauche- 1,1,1,4,4,4-hexadeuteriobutane (polarized case). v(d-+) and v(d:) par- ticipate to Fermi resonance. Assumed bandshape: Lorentzian; Half- bandwidth = 5 cm-'. (b) Case ii: Calculated Raman spectrum of gauche-l,l,1,4,4,4-hexadeuteriobutane (depolarized case). v(d_+) and v(d--) participate to Fermi resonance. Assumed bandshape: Lorentzian; half-bandwidth : 5 cm-I.

and 26'. From T to G the intensity of 26' collapses, while the intensity of d- decreases slightly (see Table VI). Then the overall appearance of the feature near 2900 cm-' must decrease. This fact occurs for both calculations i and ii (see Tables VI and VI1 and Figures 2a, 3a, 4a, 5a, and 6a). The upper shift from T to G is clearly seen in going from Figure 2a to Figures 3a and 4a for case i. However, this fact may occur only because the un- perturbed frequency of d- in the G form has been chosen at 2910 cm-'. The other choice of case ii (Figures 5a and 6a) does not seem to give an indication of any upper shift even if the polarized spectrum of the mixture gives an overall asymmetrical broadening and upper shift because of the overlapping of the features of T and G isomers.

(11) The increase of the ratio R is nicely verified in both cases i and ii in going from T to G and in going from T to the mixtures at -70 OC. From integrated areas on calculated spectra we obtain the following: for T (Figure 2a) R = 0.22; for G, case i (Figure 3a) R = 0.79; for G, case ii (Figure sa) R = 0.41; for the mixture, case i (Figure 4a) R = 0.38. Thus, we can conclude that the increase of R in going from T to G is rather independent of the

Conformational Dependence of Fermi Resonances

2950 2900

The Journal of Physical Chemistry. Vol. 88, No. 3, 1984 599

2850Icrn 1

/ a

1 Polarized

B i ! , ,

choice of the unperturbed fundamental frequencies. The origin of the change of R is immediately seen in Table IVa, and b, where it is seen that the level a t 2943 cm-' has more d++ character in G than in T. Experimentally, for two mixtures a t 125 and 300 K R at peak heights ranges from -0.65 to -0.75.

The use of R in real polymethylene molecules was introduced by us as a possible measure of the G content in these In the previous works the conclusions were reached on the ap- proximate basis of Fermi resonance schemes of a gas of CH2 groups ordered for the T case (thus generating dispersion) and noninteracting for the G case. In the case treated in the present paper the molecule is small and it has been treated exactly. Indeed, the two CH, groups are dynamically coupled in both the T and G cases. These couplings have important consequences in the Fermi resonance schemes especially in the case of G.

(111) The choice of the unperturbed C H stretching fundamental frequencies just discussed affects the position of the so-called d+ near 2850 cm-I. With choice i this band shifts upward with

BUTANE-06 MELT 1 POLARIZED A

I I 2950 2900 2850 cm-1

Figure 7. Experimental Raman polarized spectra of liquid CD,CH,C- H2CD, at the following temperatures: 125, 237, and 300 K.

BUTANE - 0 6 MELT

A

I I I 2950 2900 2850 cm-'

Figure 8. Experimental depolarized Raman spectra of liquid CD3CH,- CH,CD, at the following temperatures: 125, 237, and 300 K. Because of polarization scrambling, this spectrum shows contributions from the polarized spectrum, which were not subtracted out.

increasing G content (Figures 2a-4b). Small upper shifts are indeed observed in CD3CH2CH2CD,, in liquid polyethylene, and very noticeably in the spectra of perovskite structures in the disordered phase.25 In calculation ii, where the frequencies are the same for T and G, the changes of the values of 246) do not cause an upward shift of the v(d+) mode (Table VI1 and Figure 5 , a and b).

In section a of this discussion it was pointed out that the d- fundamental could participate in the Fermi resonance scheme of gauche-butane because of the lowering of the symmetry. As far as our calculations go and as indicated by the eigenvectors in Table IV, the extent of mixing of the d- mode into other Fermi resonant modes is so little that d- can be considered decoupled in a first

-

(25) M. Rey-Lafon, private communication

600

approximation, for this molecule.

J . Phys. Chem. 1984, 88, 600-604

with the experimental observations in longer hydrocarbon

Conclusions We refer to the spectroscopic observations mentioned at the

beginning and which require some further study: (1) For the specific case of 1,1,1,4,4,4-hexadeuteriobutane

studied in this work, the calculations, which include Fermi res- onances as well as electrooptical parameters for predicting in- tensities, account satisfactorily for the observed spectra in both the polarized and depolarized cases, for both T and G structures. (2) For the d- band near 2900 cm-’ we neglect, as already

stated, the problem of the natural breadth of the band, which will be treated elsewhere. From our calculations, the d- band has to shift upward in going from T to G structures, if the unperturbed fundamentals are different from T to G. Attempting a gener- alization to longer hydrocarbon chains, the increase of content of G structures may generate a broader multiplet structure in the polarized spectra centered at a frequency above 2900 cm-’, which increases along with the increase of G content. This seems to agree

chains.2,10 (3) With regard to the meaning and use of the ratio R =

I2940/I2850, we consider this work as a step forward, with respect to our previous work, in disentangling the various contributions and refer to later work for a more thorough assessment of its use as structural probe for longer molecules. R has been tested and explained on CD3CH2CH2CD3 and we think it is usable in com- pounds with no CH3 end groups such as polyethylene and com- pounds with CD3 end groups, from which the contribution of CH3, as pointed out by Snyder, has been removed. Moreover, this ratio should be measured in the polarized spectra of liquid samples, where the contribution of the d- mode is smaller.

Acknowledgment. This work was supported by Consiglio Nazionale delle Ricerche, Programma Finalizzato Chimica Fine e Secondaria, Sottoprogetto Polimeri.

Registry No. 1,1,1,4,4,4-Hexadeuteriobutane, 13183-67-0.

Physical Meaning of Electrooptical Parameters Derived from Infrared Intensities

M. Gussoni,

Unit6 di Spettroscopia Molecolare in Milano dell’lstituto CNR, di Chimica Applicata dei Materiali di Genoua, Milano, Italy

C. Castiglioni, and G. Zerbi*

Dipartimento di Chimica Industriale e Ingegneria Chimica del Politecnico di Milano, Milano. Italy (Received: June 3, 1983)

It is shown that infrared intensities, when interpreted in terms of the model of electrooptical parameters (EOP), provide experimental atomic charges q / at the equilibrium geometry and charge fluxes aqa/aRt during the vibrational motion; these parameters account for several physical and chemical properties of molecules. q/ and dq,/dR,, calculated from the vibrational intensities of many molecules, are compared and discussed. Various properties associated with electronic charge on the atoms are defined, classified, compared, and used in predicting the chemical behavior of molecules.

Introduction

Considerable effort has been made in the interpretation of infrared intensities in the last decade.’ Use is generally made of parameters derived from infrared intensities of small molecules both for elucidating the charge distribution within the molecule and for predicting the intensities of larger ones. The parameters used most often are the atomic polar tensors (APT)Z and the electrooptical parameters (EOP).3 The former are derivatives of the molecular dipole moments with respect to atomic Cartesian displacements; the latter are bond dipole moments a t the equi- librium geometry and derivatives of bond dipole moments with respect to internal coordinates (Le., vibrational coordinates as defined in ref 4). The definition of electrooptical parameters2tarts from the assumption that the molecular dipole Foment M can be expressed as a sum of bond dipole moments, M = &.& ( z k = unit vector of the k-th bond). Thus, the derivative of M with respect to the internal vibrational coordinate R, can be written as

ah?/aRt = C(zko(aMk/aR,) -k / .LkO(aak /aR , ) ) k

k k o and apk /aRt are the so-called EOP. Both APT5 and EOP6 have been shown to be useful for predicting spectra; the EOP, due to their localized meaning, have been particularly successful for predicting spectra in sophisticated applications such as the in- terpretation of the spectra of geometrically disordered molecules.6

A particular kind of EOP describes atomic charges qm0 at the equilibrium geometry and their fluxes dq,/aR, during vibrations R,; they were introduced by van Straten and Smit’ as derived from standard EOP, but they are the same as those previously intro- duced in a different way by Decius; hence, we call them equi- librium charges-charge fluxes (ECCF) as proposed by Decius. The ECCF parameters can be derived from EOP by assuming’

qB = qk) and rk is the interatomic distance between atoms A and B. The ECCF of course have the same capability as EOP for predicting intensities and give a more straightforward description of the charge distribution in the molecule, as can be inferred from observed intensities (together with the observed molecular dipole moments). In this paper we will discuss the physical meaning of ECCF.

It is w e l l - k n o ~ n ~ ~ ’ ~ that the reliability of any kind of intensity

/.Lk = qkrk where q k is the bond charge on bond AB (qA = -qk;

(1) W. B. Person and G. Zerbi, Eds., “Vibrational Intensities in Infrared

(2) W. B. Person, ref 1, Chapter 4. (3 ) M. Gussoni, ref 1, Chapter 5. (4) E. B. Wilson, J. C. Decius, and P. C. Cross, “Molecular Vibrations”,

and Raman Spectroscopy”, Elsevier, Amsterdam, 1982.

McGraw-Hill, New York, 1955.

0022-3654 /84/2088-0600$01.50/0

(5) W. B. Person, ref 1, Chapter 14. (6) G. Zerbi, M. Gussoni, S . Abbate, and P. Jona, ref 1, Chapter 15. (7) A. J. van Straten and W. M. A. Smit, J . Mol. Spectrosc., 62, 291

(8) J. C. Decius, J . Mol. Spectrosc., 57, 384 (1975). (1976).

0 1984 American Chemical Societv