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Ring puckering potential functions for normal and deuterated trimethylene oxides
JOURNALOF MOLECULAR SPECTROSCOPY 44, 14-17 (1972)
Ring
Puckering Potential Functions Deuterated Trimethylene
for Normal Oxides
and
R. A. KYDD, H. WIESER AND M. DANYLUK Department
of Chemistry,
University
of Calgary, Calgary
44, Alberta,
Canada
Calculations of the potential function for the ring-puckering vibration of trimethylene oxide and four deuterated derivatives have been carried out. It was found that although the spectrum of each molecule could be reproduced by using a mixed quadratic-quartic function, no one function would fit all of the molecules. Barrier heights were found to lie between the values of 15.1 f 0.5 cm-* for the normal molecule and 11.2 f 0.5 cm-1 for the perdeuterated molecule. INTRODUCTION
The success achieved in recent years in explaining the observed transitions involving puckering levels of small ring molecules by using quadratic-quartic potential functions has indeed been remarkable (1). In all cases, experimental information has been more readily available for the normal isotopic species, and although it is generally accepted that the potential function may change with isotopic substitution, there have not been enough data available to demonstrate this clearly. Most workers have assumed that the potential function for the normal molecule would carry over exactly to isotopically substituted molecules, with only a change in reduced mass required to explain (or predict) the new frequencies. In most cases this approach has worked quite well. In fact, in one case where an attempt was made to calculate independently the potential funetions for a molecule (cyclobutane) and for its deuterated analog (a), the agreement with observed data is no better than that obtained by assuming the functions are the same for both molecules (3). The low frequency infrared spectra of trimethylene oxide (TMO) and four of its deuterated analogs which we have recently presented (4) provide an excellent opportunity to investigate more rigorously the change in shape of such potential functions with deuteration. Chan et al. (6) have calculated the potential function for normal TMO, and have found that the same function apparently gave good agreement with the spectrum of the perdeuterated molecule, making the appropriate change in reduced mass. Our results, however, showed quite clearly that the value they accepted for the v = O-l transition for TMO-de was too low by more than 2 cm-l, or about 5 %. As the energy of this transition is more sensitive 14 Copyright All rights
0
1972 by Academic
of reproduction
in sny
Press, form
Inc. reserved.
TRIMETHYLENE
OXIDE PUCKERING POTENTIAL
15
than any other to the barrier height, we thought it worthwhile, in view of the new experimental data, to re-investigate the potential functions of the various trimethylene oxides. RESULTS AND DISCUSSION
The method of calculating the energy levels of a mixed quadratic-quartic potential in a set of harmonic oscillator basis functions is well established. The only disagreement among earlier authors is over which reduced coordinate to use. We found that the transformation described by Laane (1) was most convenient for systematic comparison of observed and calculated frequencies. The puckering potential V = ax4 + bx2 is transformed (see Ref. (1) for details of the transformation) to V = A(z4 + Bz2) where x is the puckering coordinate and z is the reduced coordinate. A program was written to calculate the eigenvalues of the puckering Hamiltonian for any given value of B; each eigenvalue difference was then multiplied by the scaling factor A, chosen to make the first difference equal to the w = O-l transition frequency. For one value of B, it was thus possible to compare, with only one matrix diagonalization, the observed and calculated frequencies for all molecules of interest. We found that if we used 70 basis functions, the first 15 energy levels would not change appreciably with the addition of more basis functions. The data available fix the values of A and B quite well. It is also interesting to express the potentials in terms of the puckering coordinate X, as it is in this coordinate that any difference between the potential functions for the different molecules will appear, unobscured by the effects of reduced mass changes. The conversion from A and B to a and b [see Ref. (I)] requires knowledge of the reduced mass. We have chosen to use for normal TM0 the reduced mass (p) calculated from the usual puckering model (4). For the deuterated molecules, we have used the reduced masses calculated from the observed isotopic ratios for the O-l transitions (4) and the value of p for the normal molecule. Of course a and b will change for different reduced masses; the quoted values, however, give some indication of how much a and b might be expected to vary from molecule to molecule. The barrier height equals A B2/4 (or b2/4a) and is independent of the reduced mass. Table I contains the observed frequencies and the differences between observed and calculated values for each of the five molecules. We have also included in Table I a comparison of our observed frequencies and those calculated for n-TM0 and &-TM0 by Chan et al. (5). Their results and ours for the normal molecule are virtually identical. The comparison for the deuterated molecule, however, gives an indication of the magnitude of the inaccuracy which results from simply transferring the potential function from one molecule to another. While it is clear that the simple transfer gives results which are more than adequate for most purposes (assigning spectra of partially deuterated isomers, for example) we feel that any attempt to analyze the molecular motion in
16
KYDD,
WIESER,
AND
TABLE
DANYLUK
I
OBSERVED WAVENUM~IGRS~AND DIFFERENCES (Obsd n-TM0
a-dz TM0
-
Calcd)
LY ,a’-da TM0
/3-dz TM0
dg TM0
T
A
Obsd
Ab
Obsd
53.5 90.0 105.0 118.2 129.2 138.6 147.1
0.0 0.0 0.1 -0.1 0.0 -0.1 0.0 0.2 0.2 0.4 1.0
48.7 81.7 95.1 107.2 117.0 125.5 133.0 139.9 146.3 152.0 157.4
154.8 161.8 168.4 -175
0.0 0.2 0.1 0.1 0.0 -0.1 -0.2 -0.1 0.0 -0.1 -0.1
Obsd
A
Obsd -__
49.6 83.5 97.2 109.6 119.7 128.5 136.3 143.4 149.7 -155.7 -
0.0 0.1 0.0 -0.1 -0.1 -0.1 0.0 0.0 -0.1 0.0
43.7 73.4 85.3 96.1 104.9 112.6 119.2 125.3 130.9 136.4 141.0
-
L
A 0.0 0.1 0.0 0.0 -0.1 -0.1 -0.3 -0.3 -0.4 -0.1 -0.3
Obsd
A
41.2 69.2 80.5 90.7 99.1 106.2 112.7 118.3 123.7 128.7 133.3
AC
0.0 0.2 0.1 0.1 0.1 0.0 0.0 -0.1 0.0 0.0 ~ 0.1
-
-
2.5 0.6 0.6 0.2 0.1 -0.2 -0.3 -0.6 -0.7 -0.7 -0.7
l-
B Observed wavenumbers (in cm-l) from Ref. (4). b A = Observed - calculated wavenumber. c Calculated wavenumbers taken from Chan et nl. (6). TABLE
II
POTENTIAL FUNCTION CONSTANTS
I
n-TM0 this work
A (cm-l) B P (u) a (lo6 cm+ _Xe4) b (lOa cm-l A--Z) barrier (cm-l) Ha (cm-l) separationb (A)
28.12 -1.465 95.7 7.16 -6.58 15.1 12.2 0.135
a-d2 TM0
Ref. (5) 28.15 -1.474 97.2 7.422 -6.738 15.3 12 0.135
dg TM0
,“,“a m,a'da TM0
this work 25.46 -1.445 110.4 7.07 -6.13 13.3 11.3 0.132
26.10 -1.465 107.3 7.19 -6.34 14.0 11.3 0.133
22.85 -1.445 129.8 7.07 -5.81 11.9 10.2 0.128
21.54 -1.445 141.8 7.07 -5.64 11.2
Ref (5) 21.83 -1.674 142.3 7.422 -6.738 15.3
9.6 0.126
0.135
a H is the separation between the top of the barrier and the lowest energy level b separation between minima.
more detail will require the more precise information about the potential functions which is presented here. Table II contains the values of A and B from which the frequencies were calculated, and also the P, a, b, barrier heights and separation of the minima. We have also included, for comparison, the potential function of Ghan et al. (6) with A and B expressed in our coordinates. The difference between the functions for the ds molecule caused by the need to fit the new value for the v = O-l transition
TRIMETHYLENE
OXIDE PUCKERING
POTENTIAL
17
is most obvious in the values of A and B. The changes in a and b reflect not only the different functions, but also the fact that the reduced masses for both molecules that Chan et al. used differ from ours [see discussion in Ref. (4)]. The uncertainty in the observed frequencies (particularly for the v = O-l transition) governs the uncertainty in the barrier heights, and we estimate that the barriers are accurate to ~~0.5 cm-l. Although we have presented in Table II those potential functions which were scaled to reproduce the v = O-l transition exactly, we did not make such an exact fit a restriction while attempting to find one function to fit all the spectra. Although slightly different values of A and B could be found which would give almost as good agreement with the data, the barrier heights were always within 0.5 cm-l of those reported here. It is clear that this change in the barrier height is real, and to the best of our knowledge, this is the first Gme that the change in potential function with isotopic substitution has been so conclusively demonstrated. ACKNOWLEDGMENTS We would like to thank Professor I. M. Mills for providing a copy of an ALGOL quadratic-quartic potential function program written by J. M. R. Stone. R. A. K. would like to acknowledge financial support in the form of a University of Calgary Teaching Postdoctoral Fellowship. This work was supported by the National Research Council of Canada. RECEIVED:
March 17, 1972 REFERENCES
1. J. LUNE, Appl. Spectrosc. 24, 73 (1970) and references therein. 2. F. A. MILLER AND R. J. CAPWELL, Spectrochim. Aeta A 27,947 (1971). S, J. M. R. &ONE I. M. MILLS. Mol. Phys. 18.631 (1970). 4. H. WIESER, M. DANYLIJK AND R. A. KYDD, J. Mol. Spectry. 43,382-392 (1972). 6. S. I. CHIN, T. R. BORQERS, J. W. RUSSELL, H. L. STRAUSS AND W. D. GWINN, J. Chem. Phys. 44, 1103 (1966).
Ring
Puckering Potential Functions Deuterated Trimethylene
for Normal Oxides
and
R. A. KYDD, H. WIESER AND M. DANYLUK Department
of Chemistry,
University
of Calgary, Calgary
44, Alberta,
Canada
Calculations of the potential function for the ring-puckering vibration of trimethylene oxide and four deuterated derivatives have been carried out. It was found that although the spectrum of each molecule could be reproduced by using a mixed quadratic-quartic function, no one function would fit all of the molecules. Barrier heights were found to lie between the values of 15.1 f 0.5 cm-* for the normal molecule and 11.2 f 0.5 cm-1 for the perdeuterated molecule. INTRODUCTION
The success achieved in recent years in explaining the observed transitions involving puckering levels of small ring molecules by using quadratic-quartic potential functions has indeed been remarkable (1). In all cases, experimental information has been more readily available for the normal isotopic species, and although it is generally accepted that the potential function may change with isotopic substitution, there have not been enough data available to demonstrate this clearly. Most workers have assumed that the potential function for the normal molecule would carry over exactly to isotopically substituted molecules, with only a change in reduced mass required to explain (or predict) the new frequencies. In most cases this approach has worked quite well. In fact, in one case where an attempt was made to calculate independently the potential funetions for a molecule (cyclobutane) and for its deuterated analog (a), the agreement with observed data is no better than that obtained by assuming the functions are the same for both molecules (3). The low frequency infrared spectra of trimethylene oxide (TMO) and four of its deuterated analogs which we have recently presented (4) provide an excellent opportunity to investigate more rigorously the change in shape of such potential functions with deuteration. Chan et al. (6) have calculated the potential function for normal TMO, and have found that the same function apparently gave good agreement with the spectrum of the perdeuterated molecule, making the appropriate change in reduced mass. Our results, however, showed quite clearly that the value they accepted for the v = O-l transition for TMO-de was too low by more than 2 cm-l, or about 5 %. As the energy of this transition is more sensitive 14 Copyright All rights
0
1972 by Academic
of reproduction
in sny
Press, form
Inc. reserved.
TRIMETHYLENE
OXIDE PUCKERING POTENTIAL
15
than any other to the barrier height, we thought it worthwhile, in view of the new experimental data, to re-investigate the potential functions of the various trimethylene oxides. RESULTS AND DISCUSSION
The method of calculating the energy levels of a mixed quadratic-quartic potential in a set of harmonic oscillator basis functions is well established. The only disagreement among earlier authors is over which reduced coordinate to use. We found that the transformation described by Laane (1) was most convenient for systematic comparison of observed and calculated frequencies. The puckering potential V = ax4 + bx2 is transformed (see Ref. (1) for details of the transformation) to V = A(z4 + Bz2) where x is the puckering coordinate and z is the reduced coordinate. A program was written to calculate the eigenvalues of the puckering Hamiltonian for any given value of B; each eigenvalue difference was then multiplied by the scaling factor A, chosen to make the first difference equal to the w = O-l transition frequency. For one value of B, it was thus possible to compare, with only one matrix diagonalization, the observed and calculated frequencies for all molecules of interest. We found that if we used 70 basis functions, the first 15 energy levels would not change appreciably with the addition of more basis functions. The data available fix the values of A and B quite well. It is also interesting to express the potentials in terms of the puckering coordinate X, as it is in this coordinate that any difference between the potential functions for the different molecules will appear, unobscured by the effects of reduced mass changes. The conversion from A and B to a and b [see Ref. (I)] requires knowledge of the reduced mass. We have chosen to use for normal TM0 the reduced mass (p) calculated from the usual puckering model (4). For the deuterated molecules, we have used the reduced masses calculated from the observed isotopic ratios for the O-l transitions (4) and the value of p for the normal molecule. Of course a and b will change for different reduced masses; the quoted values, however, give some indication of how much a and b might be expected to vary from molecule to molecule. The barrier height equals A B2/4 (or b2/4a) and is independent of the reduced mass. Table I contains the observed frequencies and the differences between observed and calculated values for each of the five molecules. We have also included in Table I a comparison of our observed frequencies and those calculated for n-TM0 and &-TM0 by Chan et al. (5). Their results and ours for the normal molecule are virtually identical. The comparison for the deuterated molecule, however, gives an indication of the magnitude of the inaccuracy which results from simply transferring the potential function from one molecule to another. While it is clear that the simple transfer gives results which are more than adequate for most purposes (assigning spectra of partially deuterated isomers, for example) we feel that any attempt to analyze the molecular motion in
16
KYDD,
WIESER,
AND
TABLE
DANYLUK
I
OBSERVED WAVENUM~IGRS~AND DIFFERENCES (Obsd n-TM0
a-dz TM0
-
Calcd)
LY ,a’-da TM0
/3-dz TM0
dg TM0
T
A
Obsd
Ab
Obsd
53.5 90.0 105.0 118.2 129.2 138.6 147.1
0.0 0.0 0.1 -0.1 0.0 -0.1 0.0 0.2 0.2 0.4 1.0
48.7 81.7 95.1 107.2 117.0 125.5 133.0 139.9 146.3 152.0 157.4
154.8 161.8 168.4 -175
0.0 0.2 0.1 0.1 0.0 -0.1 -0.2 -0.1 0.0 -0.1 -0.1
Obsd
A
Obsd -__
49.6 83.5 97.2 109.6 119.7 128.5 136.3 143.4 149.7 -155.7 -
0.0 0.1 0.0 -0.1 -0.1 -0.1 0.0 0.0 -0.1 0.0
43.7 73.4 85.3 96.1 104.9 112.6 119.2 125.3 130.9 136.4 141.0
-
L
A 0.0 0.1 0.0 0.0 -0.1 -0.1 -0.3 -0.3 -0.4 -0.1 -0.3
Obsd
A
41.2 69.2 80.5 90.7 99.1 106.2 112.7 118.3 123.7 128.7 133.3
AC
0.0 0.2 0.1 0.1 0.1 0.0 0.0 -0.1 0.0 0.0 ~ 0.1
-
-
2.5 0.6 0.6 0.2 0.1 -0.2 -0.3 -0.6 -0.7 -0.7 -0.7
l-
B Observed wavenumbers (in cm-l) from Ref. (4). b A = Observed - calculated wavenumber. c Calculated wavenumbers taken from Chan et nl. (6). TABLE
II
POTENTIAL FUNCTION CONSTANTS
I
n-TM0 this work
A (cm-l) B P (u) a (lo6 cm+ _Xe4) b (lOa cm-l A--Z) barrier (cm-l) Ha (cm-l) separationb (A)
28.12 -1.465 95.7 7.16 -6.58 15.1 12.2 0.135
a-d2 TM0
Ref. (5) 28.15 -1.474 97.2 7.422 -6.738 15.3 12 0.135
dg TM0
,“,“a m,a'da TM0
this work 25.46 -1.445 110.4 7.07 -6.13 13.3 11.3 0.132
26.10 -1.465 107.3 7.19 -6.34 14.0 11.3 0.133
22.85 -1.445 129.8 7.07 -5.81 11.9 10.2 0.128
21.54 -1.445 141.8 7.07 -5.64 11.2
Ref (5) 21.83 -1.674 142.3 7.422 -6.738 15.3
9.6 0.126
0.135
a H is the separation between the top of the barrier and the lowest energy level b separation between minima.
more detail will require the more precise information about the potential functions which is presented here. Table II contains the values of A and B from which the frequencies were calculated, and also the P, a, b, barrier heights and separation of the minima. We have also included, for comparison, the potential function of Ghan et al. (6) with A and B expressed in our coordinates. The difference between the functions for the ds molecule caused by the need to fit the new value for the v = O-l transition
TRIMETHYLENE
OXIDE PUCKERING
POTENTIAL
17
is most obvious in the values of A and B. The changes in a and b reflect not only the different functions, but also the fact that the reduced masses for both molecules that Chan et al. used differ from ours [see discussion in Ref. (4)]. The uncertainty in the observed frequencies (particularly for the v = O-l transition) governs the uncertainty in the barrier heights, and we estimate that the barriers are accurate to ~~0.5 cm-l. Although we have presented in Table II those potential functions which were scaled to reproduce the v = O-l transition exactly, we did not make such an exact fit a restriction while attempting to find one function to fit all the spectra. Although slightly different values of A and B could be found which would give almost as good agreement with the data, the barrier heights were always within 0.5 cm-l of those reported here. It is clear that this change in the barrier height is real, and to the best of our knowledge, this is the first Gme that the change in potential function with isotopic substitution has been so conclusively demonstrated. ACKNOWLEDGMENTS We would like to thank Professor I. M. Mills for providing a copy of an ALGOL quadratic-quartic potential function program written by J. M. R. Stone. R. A. K. would like to acknowledge financial support in the form of a University of Calgary Teaching Postdoctoral Fellowship. This work was supported by the National Research Council of Canada. RECEIVED:
March 17, 1972 REFERENCES
1. J. LUNE, Appl. Spectrosc. 24, 73 (1970) and references therein. 2. F. A. MILLER AND R. J. CAPWELL, Spectrochim. Aeta A 27,947 (1971). S, J. M. R. &ONE I. M. MILLS. Mol. Phys. 18.631 (1970). 4. H. WIESER, M. DANYLIJK AND R. A. KYDD, J. Mol. Spectry. 43,382-392 (1972). 6. S. I. CHIN, T. R. BORQERS, J. W. RUSSELL, H. L. STRAUSS AND W. D. GWINN, J. Chem. Phys. 44, 1103 (1966).