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The temperature dependence of quantum mechanical tunneling effects in the nmr spectrum of a methyl group
Volume
22. number
2
CHEMICAL
PHYSICS
1 October
LETTERS
1973
THETEMPERATUREDEPENDENCEOF QUANTUMMECHANICALTUNNELINGEFFECTS INTHENMRSPECTRUMOFAMETHYLGROUP* Charles S. JOHNSON Jr.** and Carolyn MOTTLEY Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 2 7514, USA
f
Received
3 July 1973
NMR line shapes have been obtained for CHjCD21 in CDsCD21 from 4.2 to 127°K. At 4.2”K tunneling is evident, at 87°K neither tunneling nor classical rotation can be detected, and at 127°K there is rapid, thermally activated rotation. The apparent tunneling frequency is found to decrease with increasing temperature.
It has recently been shown that the tunneling frequency of the methyl group in CH3CD21 can be determined for the torsional ground state by analysis of the NMR line shapes at 4.2”K [l] . The best fit between theory and experiment is obtained for the tunneling frequency J/pd = 0.9 where 3J is the tunneling splitting, d = a/.& P3, /.I = $yE, and R is the interproton distance. The tunneling frequency turns out to be approximately 21 kHz and the implied barrier height for internal rotation, assuming a sinusoidal potential function, is 4.0 f 0.1 kcal/mole. These numbers, which were obtained with R = 1.8 8, are very sensitive to the interproton distance. We have now observed line shapes for CH,CD,I dissolved in CD,CD,I from 4.2 to 127”K, and in particular have recorded spectra which show the transition from tunneling to the non-rotating case [2]. The derivative spectra (fig. 1) show little change as the temperature is increased from 4.2”K until about 40°K is reached, then the outer peaks move with increasing temperature toward the center. A variety of new line shapes are found from 40 to 80”K, and at
* Supported in part by grants from the National Science Foundation (GP-31383X1) and the Materials Research Center, U.N.C., under Grant Number GH-33632 from the National Science Foundation. ** John Simon Guggenheim Memorial Fellow, 1972-1973.
430
IA
CH3CD21
In CD&D21
Fig. 1. First derivative half-spectra for CH3CD2 I dissolved in CD3CDzI. The concentrations were I :5 by volume for the sample used below 8O“K and 1: 2.3 by volume for the sample m used above 80°K.
about 85°K the spectrum becomes that predicted for a fixed triangle of spins. At still higher temperatures (85-127°K) the onset of thermally activated rotation is evident from the line shapes [3]. Here we are primarily concerned with the initial change in line shape as the temperature is increased
Volume 22, number 2
CHEMICAL PHYSICS LETTERS
1 October 1973
where a! = exp [L(El -E,,)/RT] and Al? = E, - E, we have ln(J,-(J))
Fig. 2. Plot of In [(Jo-tJ))/d] versus T-’ (“K-l). The slope is -AE/R and the intercept is In [Jo(l +x)1, see text.
from 4.2’K. We have not been able to fit these spectra exactly with existing theories; however, fair agreement is obtained using the theory of Apaydin and Clough [4] with an adjustable splitting constant J. The unexpected result of this procedure was that the effective J decreases with increasing temperature. This suggests the following tentative model for the temperature dependence. Suppose that the rate of transitions between the torsional levels is high at all temperatures and that the populations of the levels are described by Boltzmann factors. Under these conditions the observed tunneling frequency may be the weighted average of the frequencies of the populated levels. This, of course, requires that only A ** A and E + E transitions occur and that spin flips are not involved. If J, is the splitting constant for the nth torsional level, then the numerical calculations of Stejskal and Gutowsky [5] show, by extrapolation, that for a barrier height of 4 kcal/mole: JI /Jo = -70, J2/Jo = +2 600, J3/Jo = -56 000. The separations between the torsional levels turn out to be: (in Cal/mole) El - E. = 690, E, - El = 650, E3 .~ E, x 6 10. Using these numbers we find at 50°K that (J) = Jo< 1 -0.067+0.0036 ~ . ..). which agrees reasonably well with experiment. At low temperatures let us assume that
W,
,
= -AE/RT
. Then with J, = -xJ,
t In [Jo(l tx)]
,
so that a plot of In (Jo-(J)) versus T - 1 should permit the determination of both AL? and x. Such a plot, constructed with a limited set of data, is shown in fig. 2. Here Jo is taken to be the observed value of J at 4.2”K, and (J) is the value of J required for the best fit to the experimental spectra at each temperature. The straight line in fig. 2 represents the least squares fit with Af? = 700 Cal/mole and x = 2 15. We think that these numbers are sufficiently close to our prediction to provide support for the simple theory described above; however, quite a lot of uncertainty is present because of the low signal-to-noise ratios as well as the distortions resulting from saturation and modulation. Also, we have found that J,, with the determined x, depends somewhat on the thermal history of the sample. Apparently, quick cooling can lead to a mixture of phases [6] so that the best fit values of J, range from 0.7 to 0.9. In some experiments there is evidence that the main spectrum is superposed with a low intensity spectrum corresponding to a much larger value of Jo. Attempts to understand these effects and to refine the experiment are underway. The procedure of averaging the tunneling frequencies for the populated levels has previously been used by Das [7] and by Stejskal and Gutowsky [5] in attempts to explain the temperature dependence of T, in solids. The conceptional difficulties, which arise from the expected uncertainty broadening of the levels, are now well known [S] . Our calculation must be justified on different grounds since we are dealing with a T, effect. We, in fact, assume that the nuclei sample each torsional level without phase interruption as in a chemical exchange process [9]. The validity of this calculation must, of course, be demonstrated by a more detailed analysis than we have presented. It should be mentioned that the idea of taking into account the magnitudes and signs of the splittings for the excited torsional levels was suggested by Allen in connection with the treatment of tunneling assisted spin-lattice relaxation [lo] . Clough has recently observed a decrease in the apparent tunneling frequency for the methyl group in irradiated methyl malonic acid in an ESR experiment. 431
Volume 22, number 2 His explanation
is quite different
CHEMICAL PHYSICS LETTERS
from that suggested
here [Ill.
References [ 1 ] C. Mottley, T.B. Cobb and C.S. Johnson Jr., J. Chem. Phys. 55 (1971) 5823. [2] E.R. Andrew and R. Bersohn, J. Chem. Phys. 18 (1950) 159; 20 (1952) 294.
432
1 October 1973
[3] T.B. Cobb and C.S. Johnson Jr., J. Chem. Phys. 52 (1970) 6224;53 (1970) 4122. [4] F. Apaydin and S. Clough, J. Phys. Cl (1968) 932. [S] E.O. Stejskal and H.S. Gutowsky, J. Chem. Phys. 28 (1958) 388. [6] H.W. Fenrick and J.E. Willard, J. Am. Chem. Sot. 88 (1966) 412. [7] T.P. Das, J. Chem. Phys. 25 (1956) 896; 27 (1957) 763. [8] J.H. Freed, J. Chem. Phys. 43 (1965) 1710. [9] C.S. Johnson Jr., Advan. Magn. Reson. 1 (1965) 33. [lo] P.S. Alien, private communication. [ 111 S. Clough, private communication.
22. number
2
CHEMICAL
PHYSICS
1 October
LETTERS
1973
THETEMPERATUREDEPENDENCEOF QUANTUMMECHANICALTUNNELINGEFFECTS INTHENMRSPECTRUMOFAMETHYLGROUP* Charles S. JOHNSON Jr.** and Carolyn MOTTLEY Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 2 7514, USA
f
Received
3 July 1973
NMR line shapes have been obtained for CHjCD21 in CDsCD21 from 4.2 to 127°K. At 4.2”K tunneling is evident, at 87°K neither tunneling nor classical rotation can be detected, and at 127°K there is rapid, thermally activated rotation. The apparent tunneling frequency is found to decrease with increasing temperature.
It has recently been shown that the tunneling frequency of the methyl group in CH3CD21 can be determined for the torsional ground state by analysis of the NMR line shapes at 4.2”K [l] . The best fit between theory and experiment is obtained for the tunneling frequency J/pd = 0.9 where 3J is the tunneling splitting, d = a/.& P3, /.I = $yE, and R is the interproton distance. The tunneling frequency turns out to be approximately 21 kHz and the implied barrier height for internal rotation, assuming a sinusoidal potential function, is 4.0 f 0.1 kcal/mole. These numbers, which were obtained with R = 1.8 8, are very sensitive to the interproton distance. We have now observed line shapes for CH,CD,I dissolved in CD,CD,I from 4.2 to 127”K, and in particular have recorded spectra which show the transition from tunneling to the non-rotating case [2]. The derivative spectra (fig. 1) show little change as the temperature is increased from 4.2”K until about 40°K is reached, then the outer peaks move with increasing temperature toward the center. A variety of new line shapes are found from 40 to 80”K, and at
* Supported in part by grants from the National Science Foundation (GP-31383X1) and the Materials Research Center, U.N.C., under Grant Number GH-33632 from the National Science Foundation. ** John Simon Guggenheim Memorial Fellow, 1972-1973.
430
IA
CH3CD21
In CD&D21
Fig. 1. First derivative half-spectra for CH3CD2 I dissolved in CD3CDzI. The concentrations were I :5 by volume for the sample used below 8O“K and 1: 2.3 by volume for the sample m used above 80°K.
about 85°K the spectrum becomes that predicted for a fixed triangle of spins. At still higher temperatures (85-127°K) the onset of thermally activated rotation is evident from the line shapes [3]. Here we are primarily concerned with the initial change in line shape as the temperature is increased
Volume 22, number 2
CHEMICAL PHYSICS LETTERS
1 October 1973
where a! = exp [L(El -E,,)/RT] and Al? = E, - E, we have ln(J,-(J))
Fig. 2. Plot of In [(Jo-tJ))/d] versus T-’ (“K-l). The slope is -AE/R and the intercept is In [Jo(l +x)1, see text.
from 4.2’K. We have not been able to fit these spectra exactly with existing theories; however, fair agreement is obtained using the theory of Apaydin and Clough [4] with an adjustable splitting constant J. The unexpected result of this procedure was that the effective J decreases with increasing temperature. This suggests the following tentative model for the temperature dependence. Suppose that the rate of transitions between the torsional levels is high at all temperatures and that the populations of the levels are described by Boltzmann factors. Under these conditions the observed tunneling frequency may be the weighted average of the frequencies of the populated levels. This, of course, requires that only A ** A and E + E transitions occur and that spin flips are not involved. If J, is the splitting constant for the nth torsional level, then the numerical calculations of Stejskal and Gutowsky [5] show, by extrapolation, that for a barrier height of 4 kcal/mole: JI /Jo = -70, J2/Jo = +2 600, J3/Jo = -56 000. The separations between the torsional levels turn out to be: (in Cal/mole) El - E. = 690, E, - El = 650, E3 .~ E, x 6 10. Using these numbers we find at 50°K that (J) = Jo< 1 -0.067+0.0036 ~ . ..). which agrees reasonably well with experiment. At low temperatures let us assume that
W,
,
= -AE/RT
. Then with J, = -xJ,
t In [Jo(l tx)]
,
so that a plot of In (Jo-(J)) versus T - 1 should permit the determination of both AL? and x. Such a plot, constructed with a limited set of data, is shown in fig. 2. Here Jo is taken to be the observed value of J at 4.2”K, and (J) is the value of J required for the best fit to the experimental spectra at each temperature. The straight line in fig. 2 represents the least squares fit with Af? = 700 Cal/mole and x = 2 15. We think that these numbers are sufficiently close to our prediction to provide support for the simple theory described above; however, quite a lot of uncertainty is present because of the low signal-to-noise ratios as well as the distortions resulting from saturation and modulation. Also, we have found that J,, with the determined x, depends somewhat on the thermal history of the sample. Apparently, quick cooling can lead to a mixture of phases [6] so that the best fit values of J, range from 0.7 to 0.9. In some experiments there is evidence that the main spectrum is superposed with a low intensity spectrum corresponding to a much larger value of Jo. Attempts to understand these effects and to refine the experiment are underway. The procedure of averaging the tunneling frequencies for the populated levels has previously been used by Das [7] and by Stejskal and Gutowsky [5] in attempts to explain the temperature dependence of T, in solids. The conceptional difficulties, which arise from the expected uncertainty broadening of the levels, are now well known [S] . Our calculation must be justified on different grounds since we are dealing with a T, effect. We, in fact, assume that the nuclei sample each torsional level without phase interruption as in a chemical exchange process [9]. The validity of this calculation must, of course, be demonstrated by a more detailed analysis than we have presented. It should be mentioned that the idea of taking into account the magnitudes and signs of the splittings for the excited torsional levels was suggested by Allen in connection with the treatment of tunneling assisted spin-lattice relaxation [lo] . Clough has recently observed a decrease in the apparent tunneling frequency for the methyl group in irradiated methyl malonic acid in an ESR experiment. 431
Volume 22, number 2 His explanation
is quite different
CHEMICAL PHYSICS LETTERS
from that suggested
here [Ill.
References [ 1 ] C. Mottley, T.B. Cobb and C.S. Johnson Jr., J. Chem. Phys. 55 (1971) 5823. [2] E.R. Andrew and R. Bersohn, J. Chem. Phys. 18 (1950) 159; 20 (1952) 294.
432
1 October 1973
[3] T.B. Cobb and C.S. Johnson Jr., J. Chem. Phys. 52 (1970) 6224;53 (1970) 4122. [4] F. Apaydin and S. Clough, J. Phys. Cl (1968) 932. [S] E.O. Stejskal and H.S. Gutowsky, J. Chem. Phys. 28 (1958) 388. [6] H.W. Fenrick and J.E. Willard, J. Am. Chem. Sot. 88 (1966) 412. [7] T.P. Das, J. Chem. Phys. 25 (1956) 896; 27 (1957) 763. [8] J.H. Freed, J. Chem. Phys. 43 (1965) 1710. [9] C.S. Johnson Jr., Advan. Magn. Reson. 1 (1965) 33. [lo] P.S. Alien, private communication. [ 111 S. Clough, private communication.