N M R STUDY OF
103
ANISOTROPIC kfOLECULAR ROTATION
A Nuclear Magnetic Resonance Study of Anisotropic Molecular Rotation in Liquid Chloroform and in Chloroform-Benzene Solution1
by Wesley T. Huntress, Jr.2 Department of Chemistru, Stanford Uniuersity, Stanford, California 9@05
(Received J u n e $7, 1968)
The nuclear spin relaxation times of the deuteron and chlorine-35 in the symmetric-top niolecules CDCb and CHC18 and in two equimolar mixtures CDCl3-CGHG and CHCl&,DG have been measured as a function of temperature. The reorientation of the molecules in these liquids is shown to be anisotropic. A rotational diffusion equation is assuined for the motion, and the rotational diffusion constants D , and , Di are calculated from the relasation tiines of chloroform in the neat liquid and for the chloroform and benzene in the mixtures. The change in the reorientation of chloroform molecules betwen neat and benzene solution is interpreted in tornis of the formation of a comples between chloroform and benzene.
I. Introduction Within the limit that a molecule in a liquid reorients in random steps of small angular displacement, the rotational motion may be described by a rotational diffusion equation that is isomorphous to the Schroedinger equation for the free rigid rotor, and the reorientation can be dcscribed as rotations about the three principal axes of a diffusion tensor fixed in the molecule analogous to the moment-of-inertia tensor for a free m ~ l e c u l e . ~The frequency of the rotation about any one of the principal diffusion axes is proportional to the component of the diffusion tensor for reorientation about that axis. This particular description has been used extensively for molecular reorientation in liquids, but nearly all applications of the model have assumed isotropic reorientation in which the motion is characterized by a single diffusion constant. Emphasis has iisually been placed on the comparison of an experimentally determined isotropic diffusion constant with that calculated hydrodynamically from the viscosity of the liquid. These comparisons have proven to be poor, owing to the inappropriateness of the hydrodynamic assumption for small molecules, and this appro:ich may obscure the most interesting aspect of the problem. The full rotational diffusion tensor gives much more information about the rotational motion of a molecule in a liquid, and recent work has shown that the reorientation of nonspherical and nonlinear molecules in liquids must in general be regarded as anisotropi~.~ Nmr spectroscopy provides a convenient probe for the study of molecular reorientation, since the nuclear spin relaxation time is dependent on the details of tlie inolccular motion. From an experimental point of view, magnetic nuclei which relax via the quadrupolar mechanism appear the best suited for the study of molecular re~rientation.~I n this paper, the nuclear spin relaxation times of the deuteron and chlorine-35 in neat
deuteriochloroform, chloroform, and mixtures of chloroform and deuteriochloroform with benzene and benzened6 are reported. The full rotational diffusion tensors for the molecules, both in neat and benzene solution, are calculated from the relaxation times, and the motion is shown to be anisotropic. The difference in the motion of the chloroform molecules in neat and in benzenc solution is interpreted in terms of the formation of a complex between the chloroform and benzene molecules in solution. The applicability of the diffusion equation to tlie rotational motion of the molecules in liquid chloroform is discussed in section VI.
11. Theoretical Section Nuclei of spin I > that possess a quadrupole moment relax by the interaction of the nuclear electric quadrupole moment with tlie molecular electric field gradient at the nucleus. The quadrupole moment of the nucleus is oriented in a space-fixed direction due to the polarization of the magnetic nuclei by the external magnetic field. The field gradient at the nucleus due to the electrons in the bond fluctuates in orientation due to the moleculzr rotational motion, imparting a time dependence to the quadrupole Hamiltonian that gives rise to the spin relaxation. For covalently bonded quadrupolar nuclei, the electric field gradient a t the nucleus is usually quite large, and the quadrupolar interaction can become two or three orders of magnitude larger than any other interaction. Hence, relaxation mechanisms other than the quadrupolar interaction do not contribute significantly to the relaxation time (1) Submitted in partial fulfillment for the degree of Doctor of Philosophy, Stanford University. (2) National Institutes of Health Predoctoral Fkllow, 1966-1968; Jet Propulsion Laboratory, Pasadena, Calif. (3) L. D. Favro, PILUS. Rev., 119, 53 (1960); F. Perrin, J . Phys. R a d i u m , 5, 497 (1934); 7, 1 (1936). (4) T. T. Bopp, J . Chem. Phys., 47, 3621 (19G7). (5) W, T. Huntress, ibid., 48, 3524 (1968).
Volume 79, Number 1 J a n u a r y 1969
WESLEYT. HUNTRESS, JR.
104 and may be ignored. This greatly simplifies the expressions for the relaxation time. In addition, the quadrupolar mechanism does not depend on relative translation between molecules in the liquid, and the complication of accounting for relative translation in the expression for the relaxation time is neatly avoided. The equation for the relaxation time of a quadrupolar nucleus in the case of an axially symmetric field gradient and in the limit of extreme narrowing is6$6
where I is the nuclear spin, Q is the quadrupole moment of the nucleus, and q is the field gradient at the nucleus. The quantity e2Qq is the quadrupole coupling constant. J("(0)is the spectral density of the rotational motion at zero frequency, given by the Fourier transform at zero frequency of the correlation function for the molecular reorientation
2
a(z)((t)= Ccie - t / ( d i is0
-
(5)
co = (l/.,)(3 cos3 B 1)2 and ( T ~ = ) ~ (6DJ-l; CI = 3 sin20 cos2 B and (Q)I = (50, D1J-l; and c2 = sin4B and ( T ~ = ) ~(2D1 4DI1)-land B is the angle
+
+
between the symmetry axis of the molecule and the ) ~ the direction of the field gradient. The times ( T ~ are angular correlation times for the symmetric top. Deuteriochloroform is a symmetric-top molecule with the chlorines at an angle of 110" 55' from the symmetry axis along the C-D bond' (see Figure 1). Both chlorine-35 and the deuteron are quadrupolar nuclei with and I = 1, respectively. Assuming that spins I = the diffusion equation is an appropriate equation of motion for deuteriochloroform in the liquid, the equations for the relaxation time of the deuteron and chlorine-35 are, respectively
and where the correlation function is given by W ( t ) = ('/2[3 cos2 W(0)
- 11 x '/2[3
COS2
o(t) - 11) (3)
where Nt) is the angle between the external magnetic field and the direction of the field gradient in the molecule. The fences denote an ensemble average. The first term within the brackets is the tensor projection (hence the superscript 2) of the molecular field gradient onto the space-fixed axis system a t some arbitrarily specified initial time t = 0, and the second term is the tensor projection a t some later time t. In order to calculate the correlation function rigorously, the molecular dynamics must be known. This calculation from first principles is at present intractable, but if it is assumed that the reorientation is characterized by a random walk over small angular displacements, the equation of motion is given by a rotational diffusion equation, which for a symmetric-top molecule is
where 0 = 110" 55'. Since the bonds in deuteriochloroform are all single bonds, it has been assumed that the field gradients are cylindrically symmetric about the C-C1 and C-D bond axes and are oriented along the bond direction. Since the deuteron lies on the symmetry axis of the molecule, rotation of the molecule about the symmet)ry axis does not reorient the field gradient. The relaxat,ion time of the deuteron is therefore independent of the C3 rotation of deuteriochloroform and depends only upon the reorientation about an axis perpendicular to C
L
E 6s'
cL-7c1D CL
OEUTEROCHLOROFORW
The quantity P(O,t) is the probability density that the top will be oriented at 0 at the time t, and Li is the angular momentum operator about the ith axis. 0, 1 is the rotational diffusion constant for reorientation about the z axis, or symmetry axis, and D L is the rotational diffusion constant for reorientation about, an axis perpendicular to the symmetry axis. The diffusion constants are statistical parameters for the motion and are functions of the average interactions of the molecule with its environment in the liquid. The solutJions to the diffusion equation (eq 4 ) are exponentially decaying and give for the correlation function (eq 3)5 The Joumal of Physical Chemistry
OEUTEROCHLOROFORM-BENZENE COMPLEX
Figure 1.
(6) H.Shimiru, J . Chem. Phys., 40, 754 (1964). (7) P. N. Wolfe, ibid., 25, 976 (1956).
ROTATION
105
the symmetry axis, or DL. However, both motions of the molecule are effective in reorienting the field gradient at the chlorines, and the relaxation time of the chlorines depends on both D, and D ,1. The constant D I can be determined directly from the relaxation time of the deuteron in deuteriochloroform, and Dll can be calculated from D, and the relaxation time of chlorine-35 providing that the quadrupole coupling constants for 2D and 35C1in deuteriochloroform are known.
this fast-exchange limit the observed relaxation time is given by
N M R STUDY OF ANISOTROPIC MOLECULAR
111. The Chloroform-Benzene Complex The relaxation times of the chlorines and deuteron in deuteriochloroform were measured both in neat solution and in 1: 1equimolar solution in benzene. Chloroform forms a weak complex with benzenea8b9 Convincing evidence has been given to show that the complex formed is 1:1 in chloroform and benzene.*>g The chloroform proton in C H C ~ ~ - C ~mixtures HO exhibits a large anomalous upfield shift, indicating that the proton lies above the plane of the benzene ring and very nearly over the center of the ring.Q The distance of the chloroform proton from the plane of the ring, 3 A, calculated from this chemical shiftQ indicates that the most probable structure for the complex is one in which the chloroform symmetry axis is perpendicular to the plane of the benzene ring with the chloroform proton (or deuteron) oriented toward the ring (see Figure 1). Therefore, the complex is also a symmetric-top molecule. Two solutions containing various isotopically substituted chloroform and benzene were studied: an equimolar mixture of CDC13 and C6Hs (I) and an equimolar mixture of CHC13 and C6D6(11). From I the relaxation time of the deuteron gives DL for the complex. Assuming that the substitution of deuterons for protons does not alter the rotational motion significantly, the value of D L is the same for both I and 11. The cg motion of benzene in the complex, D,,(C6), can therefore be calculated from the relaxation times of the deuterons in I1 given the value of D, calculated from the relaxation times of the deuteron in I. The ( 2 6 motion of the benzene in the complex may or may not be the same as the C3 motion of the chloroform in the complex, Dll(C3). This latter motion may be calculated from the relaxation time of chlorine-35 in either I or I1 and the value of DL. Not all of the chloroform molecules in the equimolar solutions are complexed with benzene. I n order to obtain the relaxation times for the nuclei in the pure complex, the fraction of chloroform complexed must be determined. Since only one chloroform proton resonance line is observed in the proton high-resolution nmr spectrum of 11, the exchange of chloroform between the complex and bulk solution must be faster than the differencebetween the Larmor frequencies of the chloroform proton in the complex and in bulk solution. I n
where f is the fraction of complexed chloroform. It is assumed that the relaxation time of the nucleus in t,he bulk solution is equal to the relaxation time of the nucleus in neat solution. It is possible to calculate f for the equimolar solutions from the chemical shift of the chloroform proton in 11. In the fast-exchange limit, the observed chemical shift of this proton is given by a similax expression &bad
= f8oomglex
+ (1 -
f)&free
(9)
+
fa Gfree Bcomplex - airee and =
where A = the chemical shifts, 6, are solution chemical shifts (Ssolution = dgas - Usotvent). The quantity u801ventis the shielding due to solvent effects (ussolvent = U A OB uw UE). The shielding constant U B is the shielding due to bulk susceptibility. This effect is eliminated by using an internal reference from which to measure the relative chemical shift. The bulk susceptibility affects the internal reference and solute molecules equally and disappears from the problem. The shielding constant UE arises from the effect of neighboring polarized solvent molecules on polar solute molecules and is negligible for chloroform in benzene.1° The contribution UA is due to the magnetic anisotropy of the solvent and should be the same for molecules of approximately the same shape. The internal reference used was tetramethylsilane (TATS). It is assumed that TAIS and chloroform are close enough in general shape that U A is the same for both. The contribution UW is due to van der Waals forces between solvent and solute. Since chloroform demonstrates a small self-association,9 this effect must be taken into account. The additional chemical shift introduced by chloroform self-association is designated by Sw(f, T ) since the amount of self-association depends on the temperature and concentration of chloroform. Equation 9 for the observed solution chemical shift becomes
+ + +
%bsd(T>
=
fa 4-
Giree
f Gw(f,T)
(10)
The quantity A is independent of temperature, since an internal reference was used, and can be obtained by dilution measurements. At low concentrations of chloroform in benzene the contribution of GW is negligible, and a plot of the observed chemical shift us. concentra(8) M. Tamres, J . Amer. Chem. Soc., 74, 3375 (1952). (9) C. W. Reeves and W. G. Sohneider, Can. J . Chem., 3 5 , 251 (1967). (10) C. J. Creswell and A. L. Allred, J . Phys. Chem., 66, 1469 (1962).
Volume 73,Number 1
January 1980
106
WESLEYT.HUNTRESS, JR.
tion of chloroform yields A from the slope at infinite dilution and the shift a t infinite dilution.ll Chemical shift studies of chloroform i n benzene s o h tion with cyclohexane as an internal reference have shown A to be independent of temperature.I0 If the observed chemical shift in solution is referred to the value in free chloroform a t the same temperature, eq 10 becomes
SA -!-
&b,d(T)
6w(f,!f!')
(11)
-
where Aobsd Sobsd 6free. The self-association of chloroform results in a small downfield shift of the chloroform proton indicative of the usual type of hydrogen bonding.$ This is in contrast to the abnormal large upfield shift for the chloroform proton in the aromatic complex. The chemical shift due to the self-association of chloroform is approximately linear in chloroform concentration,@so that for the equimolar solutions I: and I1 =2
6W(.f>T)=
- f)6W0(T)
(12)
- '/26w0(T)]+ '/&'(T)
(13)
so,,
SOL" (FROM TM81
?OR THE 1.1
and eq 11 becomes &bsd(T)
= f[A
and f can be found from =
[AOb;(T)
I
380 -S 0
- ' -/ d W'/i6s0(T)] 0(~?
-40
I
I
-eo
I 0
I
I
eo
I
I
A
40
T
(14)
The value of A is given by various authors from dilution studies as 1.36,9 1.56,'O and 1.61 ppm,12 and perhaps as high as 1.91 ppm.10 In view of such scatter, 8 value of A = 1.5 is taken as a reasonable and convenient choice. The number A o b s d ( T ) was obtained as a function of temperature by measuring the chemical shift of the proton in solution I1 with 1% added TMS as a. reference on an Varian A-60 nmr ~pectrometer.'~This shift was referred to the chemical shift measured in the same way for neat chloroform also containing 1%TMS as an internal reference. The number Swo(!?') was found by calibrating the measured neat chloroform chemical shift from TAlS with the value 0.3 ppm at 25°.g The results are given in Figure 2. The small temperature dependence of 6@ is indicative of the seli-association of chloroform. The fraction of chloroform complexed In these solutions calculated from this data is given in Figure 3 for the several values of A obtained from the literature.
Figure 2. Chemical shift data for the chloroform proton in benzene-& and in neat solution.
.IO[ 3.10
-330
3.50
3.70
3.90
4.10
130
450
IT. Relaxation Time Measurements A . Chlo~ine-%?. Chlorine-35 line widths in COVE+ lent compounds are on the order of several kilocycles owing to the large quadrupole moment of chlorine-35 arid the consequent large quadrupolar interaction. For such lines, a wide-line spectrometer built largely from Varian DP-60 equipment was used, employing a Varian 12-in. nmr high-resolution electromagnet and power supply, a Varian V-4311 radiofrequency unit and probe operating a t 4.33 Me, and a Varian V-428OA precession The Journal of Physical Chemistry
field-sweep unit. In order to detect lines as broad as several kilocycles, the signal was field modulated to (11) P. J. Berkeley, Jr., and M. W. Hanna, J. Phua. Chsm., 67, 840
(1963). (12) P. D. Groves, P. J. Buck, and J. Homer, Chem. Ind. (London), 916 (1967). (13) The author wishes to acknowledge the assistance of Yoko Kanasawa of the Nmr Laboratories of Stanford Uiiiversity for measuring the proton chemical shifts in these solutions.
NMRSTUDY OF ANISOTROPIC MOLECULAR ROTATION obtain a derivative spectrum and phase detected with a PAR JB-4 phase-sensitive detector. The modulation voltage was obtained from the internal-reference oscillator of the phase-sensitive detector through a General Radio 1206B audioamplifier for fine voltage control to the Helmholtz coils in the probe. A Dynakit Mark I11 audioamplifier was used to match the output impedance of the audioamplifier to the 22-ohm input impedance of the Helmholtz coils. The output of the phase-sensitive detector mas recorded on a Bausch & Lomb VOM5 recorder. The relaxation time Tz is related to the peak-peak separation Avpp in the derivative spectrum by
107 40 t
OML010P01Y IN THE OOYPLCX
IOa//T C K T '
I n order to calculate Aupp from the derivative spectrum, Figure 4. Chlorine-35 relaxation times. the chart paper was calibrated by recording the side bands produced by modulating the narrow chloride the same in deuterated or undeuterated samples within anion signal in a saturated aqueous solution of sodium experimental error, indicating that there is little or no chloride at a frequency W Y >> T2-'. At ambient probe isotope effect in the molecular reorientation by subtemperature the C1- line width is 1s cps. A modulation stitution of deuterons for protons. The relaxation frequency of 1 kcps mas employed, producing two times are shorter for the chloroform in benzene solution. first-order side bands symmetrically displaced about the Therefore, chlorine-35 line widths are broader and the center band separated by 2 kc. rotational motion of the molecule has become slower in Equation 15 for the relaxation time is valid only with benzene solution. This is taken as evidence for the in the following limits: (1) yH1 D L over the entire liquid range. Representative data and calculations for deuteriochloroform at 20" are given in Table 11. Since Table 11: Rotational Data and Calculations for Deuteriochloroform a t 20' 8ec--
31 X 10-6 (W1)1.35 (D) -------e*Qq, MOP 79 (Wl) 0.17(D)
0.96 X 10" 1 . 8 x 10'1 1 . 8 x 10-12 0.92 x lo-'* 159 300 0.81 X 1.10 x 10-12
111 = 1.891', deuteriochloroform in the gas phase tumbles about twice as fast about a perpendicular axis as it rotates about the C3 axis. I n the liquid, however, DL = 0.53011 (at 20") so that the motion about the symmetry axis i s faster by a factor of about 2 than the motion about a perpendicular axis. With respect to the gas-phase motion then, the tumbling motion of the molecule in the liquid is more hindered than the CB motion, presumably due to closer packing and selfassociation through hydrogen bonding along the Ca axis. That molecular association through hydrogen bonding may be at least partially responsible for the relatively large hindrance of the perpendicular motion
'
3.2
I
I
3.4
3.0 IO'/T
9
* R. G. White, Phys. Rev., a R4w is microwave spectroscopy. 94, 789A (1954). R. G. White, J . Chem. Phys., 23,253 (1954). V. W. Weiss and W. H. Flygare, ibid., 45, 8 (1966). e P. Thaddeus, L. C. Krisher, and J. H. N. Loubser, ibid., 40, 257 (1964). W. H. Flygare, ibid., 41, 206 (1966). ' J. C. Rowell, W. D. Phillips, L. R. Illelby, and M. Panar, ibid., 43, 3442 (1965).
-----TI,
.2
e
M W
1
I
3.8
4.0
I 42
I J
4,4
(OK)-'
Figure 7 . Calculated rotational diffusion constants for chloroform and the chloroform-benzene complex.
is suggested by the evidence from the chemical shift measurements that chloroform is somewhat selfassociated in the liquid. As the temperature is lowered, the values for both D I and Dl I become smaller, indicating that the motion becomes slower, The temperature dependences are not the same for both motions. The tumbling motion has a stronger temperature dependence, which may be due to increasing self-association a t lower temperatures. If, as for the relaxation times, the temperature dependence is described by an activation energy, the values for the activation energy, E,, extracted from the Arrhenius plot of Figure 7 are E,' = 1.6 rf: 0.1 l a d / mol and E," = 0.7 i 0.1 lrcal/mol. The relaxation time of the deuteron directly reflects the temperature dependence of D L and has the same activation energy. The relaxation time of the chlorines depends on both D L and DI I and therefore has an activation energy ( E , = 1.2 0.1 kcal/mol) intermediate to E', and E a ' ] . The results for DL in neat deuteriochloroform are compared in Figure 7 with results obtained from dielectric relaxation in chloroform.l 8 The dipole moment in chloroform is directed along the symmetry axis so that the dielectric relaxation time depends only on D I. The dielectric data in Figure 7 have been corrected by a factor
*
where ea and E - are the dielectric constants of chloroform a t zero and infinite frequency, respectively. This factor is introduced to correct for the fact that the macroscopic correlation time associated with the macroscopic polarization measured by dielectric relaxation is somewhat different than the microscopic molecular correlation time. (18) 5. Mallikarjun and N. E. Hill, Trans. Faraday Sac., 61, 1389 (1965). (19) S. H. Glarum, J . Chem. Phys., 33, 1371 (1960). (20) R. H. Cole, ibid., 42, 637 (1965). Volume 73, Number I
Januarg 1969
WESLEYT. HUNTRESS, JR.
110 From Figure 7 it can be seen that if a value of 170 kc is chosen for the value of the deuteron quadrupole coupling constant, the dielectric and nuclear magnetic relaxation results agree almost perfectly. The correction to the dielectric data amounts to approximately 20%. From the uncertainty in the exactness of relation (eq 17) for the correction factor, the coupling coiistant may have values from 155 to 170 kc. From the error in the measured relaxation times and the quadrupole coupling constants for chlorine-35 and the deuteron in deuteriochloroform, the maximum error in the calculated diffusion constants amounts to DI, &20% and D I , , *GO%. The difference in the error limits is due partially to the nonlinearity of eq 7 in the diffusion constants. The diffusion constants for the chloroform-benzene complex are also given in Figure 7 for comparison with neat chloroform. The C3 motion of the chloroform in the complex, D I1 (C8),has slowed somewhat by a factor of approximately 1.3, or within experimental error in differencefrom the neat chloroform value. I n contrast, the tumbling motion, D,, is slowed considerably, by as much as a factor of 4. This is consistent with the complexing occurring along the symmetry axis of the chloroform and perpendicular to the plane of the benzene ring. The temperature dependence for the motions in the complex becomes stronger, corresponding to increases in the activation energies of 0.7-0.9 kcal/mol for the CB motion and of 1.6-2.2 kcal/mol for the tumbling motion. The results for theCg motion of the benzene, D I ,(c6)~ in the complex are also given in Figure 7. Since the deuterons in benzene are a t an angle of 90" from the symmetry axis, the equation for the relaxation time of these deuterons becomes
The value of D, for the complex is that calculated from the deuteron relaxation time in the cDc13-C~Hacomplex. The value of DlI(C6)can be calculated using this value for D,, a quadrupole coupling constant for perdeuteriobenzene of 190 laps from Table I, the relaxation time of perdeuteriobenzene in both the neat liquid and solution 11, and the fraction of benzene complexed in solution 11. The results show that within experimental error the values of Dl (C6) and D11 (C,) are essentially the same a t all temperatures. Therefore, the benzene ring and chloroform tetrahedron more than likely do not slip relative to each other in the complex. The C3 motion of the chloroform is relatively unaffected on complexing with benzene. If the assumption is made that in a like manner the COmotion of the benzene is unaffected on complexing with chloroform, the value of D l ,(C,) is the same for complexed benzene and for benzene in neat solution. The difference beI
The Journal of Physical Chemistry
tween the relaxation time of benzene in neat and chloroform solution is then ascribed to the change in D, on complexing. The value of DL in neat benzene can be Calculated from eq 18 given D ,I (C6) and the relaxation times and quadrupole coupling constant of the deuterons in neat perdeuteriobenzene. The results are given in Table 111and indicate that within experimental error the motion in benzene can be considered isotropic and described by a single rotational diffusion constant.
Table 111: Rotational Diffusion Constants for Benzene-&
10 30 50
1.3
1.2 1.3 1.4
1.4 1.5
The experimenal error in these values and others for the diffusion constants in the complex are comparable with those for neat deuteriochloroform with an additional estimated error of about 15% introduced due to the errors in the chemical shift measurements.
VI. Discussion The nmr relaxation time is dependent only upon the area under the correlation function (see eq 2), and in order to calculate the correlation function (eq 3) it was necessary to assume a model for the molecular dynamics. The functional form of the correlation function, and consequently the details of the molecular dynamics, cannot be determined by the nmr experiment. The applicability of the assumed diffusion equation to the motion in chloroform can, however, be determined by comparison of the angular correlation time calculated from the experimental diffusion constant, ( n ) t = (6Di)-I, with the mean free period of rotation (in radians/sec) for the free rotor, ( ~ 1 )=~ ( I / I C T ) ~ ' ~ If . the time it takes for a molecule in the liquid to rotate , very much through 1 radian about the ith axis, ( r 0 ) ( is shorter than the time i t takes the free molecule in the ) ~ ~ the gas phase to rotate through 1 radian, ( 7 ~ then motion is most likely diffusional in nature
x i = (2) >> 1 Tf
2
Applying this criterion to the data in Table 11, X I = 2.2 and xl = 0.9 at T = 20". Over the temperature rnnge of the measurements, xI ranges between the value of 2 a t high temperatures and the value of 5 a t low teniperatures, and x i ] N 1 a t all temperatures. Inertial contributions to the motion are therefore probably important in chloroform and to the C, motion in particulnr. However, the qualitative interpretation that the CJ
NMR STUDY O F
111
ANISOTROPIChfOLECULAR ROTATION
motion is faster than the tumbling motion is still valid. Unlike the nmr experiment, infrared and Raman band shapes are directly related to the functional form of the rotational correlation function. The band shape is determined by the rotational structure superimposed on the vibrational levels. I n a liquid, the rotational line structure disappears, giving a single line the shape of which is given by the rotational spectral density. I n principle, the exact form of the rotational correlation function can be calculated from the band
where g(w) is the normalized band shape u(u) = I(w)/ fI(w) do. The correlation function in eq 20 is a vectoy correlation function ( W ( t ) = (cos n(0) cos Q(t))). There are problems associated with the infrared and Raman experiments involving the overlap of underlying bands, the assignment of transitions, the determination of transition moments and polarizability tensors for asymmetric molecules, and the frequency shifts that obscure the pure rotational envelope. Nevertheless, if such difficulties can be overcome, a model for the reorientation such as the diffusion equation is not required, and the exact time dependence of the correlation function can be obtained. Chloroform has two types of bands in the infrared spectrum: parallel bands in which the transition dipole is parallel to the symmetry axis and perpendicular bands in which the transition dipole is perpendicular to the symmetry axis. The parallel bands will yield the correlation function for the perpendicular or tumbling motion since for these transitions AK = 0. The perpendicular bands will contain information about both modes Of motion* If the motion is diffusionalJ then the correlation functions become
@ , , ( l ) ( t )=
e- 2 D l t
aL(I)(t) = e- ( D l
-I-Dll)t
(21) (22)
for the parallel and perpendicular bands, respectively. The band shapes given by the Fourier transform of the functions (eq 21 and 22) are therefore Lorentzian. If inertial contributions are important, the band shapes are non-Lorentzian and the experimental correlation functions will not be exponential. The form of the correlation functions and the appearance of the infrared band shapes for motion in the inertial limit, the diffusion limit, and in intermediate cases are given in ref 22 and 23. The infrared spectrum of chloroform at the C-H stretch AI parallel band (vl = 3020 cm-I) and the bending mode E perpendicular band (v4 = 1213 crn-l) have ~ been reported as a function of t e m p e r a t ~ r e . ~The appearance of the band shapes is Lorentzian, but correlation functions calculated from the Fourier transform of the band shapes are not exactly exponential and indicate that inertial contributions are important at short times t < r f but that the behavior is exponential at long times t > r f .
Aclcnozuledgrnents. The author wishes to acltnowledge the assistance and encouragement of Professor John D. Baldeschwieler, and helpful discussions with Dan Wallach and T. T. Bopp. The support of the National Science Foundation under Grant G1’-4924x and a Kational Institutes of Health Predoctoral Fellowship to t he author are gratefully acknowledged. (21) R* Go Gordon, J . Chem. P h ~ s . v
43* 1307 (1965)*
(22) G* Gordon* 429 3658 (1965). (23) H. Shimizu, BUZZ. Chem. SOC.Jap., 39, 2385 (1966). (24) H. Shimizu, J. Chem. Phys., 43, 2453 (1965). (25) M. P. Lisitsa and Yu P. Tsyastiohenlco, Opt. Spectrosk, 9, 229
(1960).
Volume 73, Number 1 Januarg lQ68
