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A high resolution study of l-type resonance in cyclopropane
JOURNAL
OF
MOLECULAR
A High
SPECTROSCOPY
Resolution
41, 177-181
(1972)
Study of l-type Cyclopropane
Resonance
in
ARTHUR G. ~L~AKI National
Bureau of Standards,
Washington,
D. C. ZOS’S4
The ~5+ Y~O band of cyclopropane (C3H6) centered at 2089.52 cm-l has been measured and analyzed taking into account the l-type resonance effect described by Cartwright and Mills. A least-squares fit of more than 300 well-resolved transitions gave a set of band const’ants which reproduced the measurements with a standard deviation of 0.093 cm-l. The analysis confirms the Z-type resonance treatment given by Cartwright and Mills and shows that their band contour fit gave remarkably accurate constants for this band. INTRODUCTION In the past few years computer simulated spectra have been used more and more frequently to aid in the interpretation of unresolved ro-vibronic bands (not only in the infrared region, but also in the visible and ultraviolet spectral regions). These so-called band cont,our analyses have been used to determine, for both t)he upper and lower stat’es, the rotational constants for symmetric top and asymmetric rotor type spectra. Band contour analyses have also been used to determine the importance of various types of perturbations, and to show how some perturbations may affect t’he gross appearance of the band contours. There has been some skepticism regarding the accuracy of the band constants and interaction const’ants obtained from such band contour analyses. Since there are many cases for which such band contour analyses offer the only hope of obtaining certain types of structural information, it is important to check the constants obtained from some band cont,our analyses which could not have been prejudiced by a prior knowledge of the correct const,ants. Since Mills (I, 2) has been one of the foremost exponents of the use of computer simulat,ed spectra for the analysis of unresolved (or part)ially resolved) spectra, we have measured the high resolution spectrum of one of the bands of cyclopropane for which Cartwright and Mills (2) have published a band cont’our analysis. The band chosen for study was the perpendicular band I+, + vlO(I?) of cyclopropane (C&H,). Even in low resolution spectra this band displays in t,he &-branch region a peculiar transmission spike which Cartwright and Mills have
shown t,o be due to l-type resonance effects. 177 Copyright
a
1972 by Academic
Press, Inc.
175
MAKI
Previous high or medium resolution infrared measurements on cyclopropane were made by Giint,hard et al. (S), Duncan et al. (4-G), and McCubbin et al. (7). The rotational Raman spectrum was measured by Mathai et al. (8) and subsequently Jones and Stoicheff (9) determined an improved set of rotational constants. The most accurate values of Bo and D, seem t’o be those given by t,he infrared measurements of XIcCubbin, Withstandley, and Polo (7). EXPERIMENTAL
The measurements were made in a l.!% m absorption cell cooled to dry ice temperat,ure (195 K) over 90% of its length. As an indication of the resolut’ion available for this work, a very compressed spectrum of the 2089 cm-’ band is presented in Fig. 1. On the expanded measurement charts lines separated by 0.025 cm-l were just barely resolved; Cartwright and Mills used much lower resolution [presumably slightly bet,ter than that, indicated in Fig. 1 of Ref. (5)]. The present, spect,ra were t’aken on the NBS high resolution infrared spectrometer recently modified by Dr. W. B. Olson and placed in a vacuum tank to avoid refractive index corrections in the calibration. The spectrometer is a double pass inst,rument, with a focal length of 2.3.5 m. The measurement’s were made using first order of a 5 X X in. grating with 400 lines/mm. Calibrat’ion was achieved by interpolation between thorium lines observed in higher order on the grating [see Ref. (IO)]. ANALYSIS
AND
RESULTS
Since it has been shown that Z-type resonance has a significant effect for many oblate rotor molecules, the spectrum was analyzed by applying the theory of Z-type resonance as described by Mills (I, 2). A computer program was written which uses the energy expressions (including Z-type resonance) given by Cartwright and Mills (2) to fit a set of band constants to the observed data using a leastsquares criterion.
FIG. 1. Infrared absorption due to the ~5 + YIOtransition of cyclopropane pat hlength at, a pressure of 8 Torr and a temperature of 200°K.
with a 1.55 m
l-TYPE
RESONANCE
IN CYCLOPROPANE
179
The program operated according t,o the following sequence of steps: (1) The transit,ion measurements and assignments are given to the computer and an initial guess was made for the band constants. (2) For each observed transition t,he lower energy levels were calculated by I he usual equation
F” (J”, k”) = B,[J” (,I” + 1) -
.J[J”
(AZ”)“I + C,, (/i” )”
(*I” + l)]’ -
(3) The unperturbed
upper
I),,./”
(.I” + 1) (A”“)’ -
state energy
F’ (J’, k’, I’) = V” + B,[J’(J’
levels were calculated
+ -
I),@“)?
1) &[.I
by Eq.
(I 1 (2).
(li’)‘] + c, (A.‘)2 (J’ + 1 )]”
-
2 (C{)&‘I’
-
D,,J’ (J’ + 1) (x:‘)’ - D, (k’)4.
(2)
(The 7 terms were left out of this equation because the data could be fit satirfactorily without them. ) (4) Equation (2) was also used to calculate the energy of that level which interacts wit.h the upper state of the assigned transition through the particular mat,rix element (at, lt + 1, J, k + 1 ( I?/hc /vt, II - 1, .I, k - 1). (5) The above matrix element was evaluated using t(he conventions set, up by Cart.wright and Mills (a), namely: (v, , dt + 1, J, k + 11 A//MY /21L , I, =
(-!/4)Q,[(Ut
1, J, k + I)* -
1)
2 l/Z I, I
. {[.I(.1 + 1) - X-(x:+ l)][J(J
0 ) + 1) - k(k -
1)]1’?
(6) The eigenvalues and eigenvectors resulting from the upper state inter‘ were determined by solving the appropriate 2 X 2 secular determinant. (7) From the eigenvectors and the partial derivst.ives of the elements of the determinant the Jacobian was formed. (8) The Jacobian was then used in a least-squares fit of the residuals (vob - vcai) t’o obtain corrections for the band constants. (9) If the corrections to any of the constants were greater than a tenth of the estimated st,andard error of those constants, another iteration It-as performed by going back to step 2. The computer program used double precision operations throughout and utilized a least-squares subroutine, ORTHO, which has been shown to be very reliable (11). The assignments necessary for step 1 could be made quite unambiguously since the first ‘P or ‘R line of a subband for a given /? was nearly always obvious. Since by chance C-CC-B z s B, t,he &-branch spacing was very nearly equal action
MAKI
180
to one-third the ‘P or RR line spacing expected in each subband, causing the ‘P and RR lines to fall into groups or clumps separated by jJ$ B. The order of appearance of each new K” = J” line showed that the quantum numbers of each successive line within each clump were given by J, K = m, n; m + 1, n + 3; t’he fact m + 2, n + 6; etc., where n and m are integers. This established that the ‘Q transitions were at higher frequencies than the “Q transitions. If the ‘Q had been lower in frequency than the “Q, then the quantum numbers would have been given by J, K = m, n; m - 1, n + 3; m - 2, n + 6; etc. The assignments were also facilitated by the nearly complete absence of ‘R and RP lines. The measurements covered the complete range of K values from K = 0 to K = 31 and an equal range of J values. The analysis included a number of resolved Q-branch lines for ‘Q3 , pQp , “&I , “Qz , and "Qa . A total of 337 tSransit’ions were used in the least-squares fit and Table I contains t,he result,ing constant,s. TABLE P.IHAMETEKS
TO Frr
USED
Parameter
2089.5201
VI3
0.67024b 0.6689351 0.424NnY 0.4171817
(1The rmcertainties
f
0.0010
f
0.0000055
(G). DJ
D JK DK
f
YIOBAND OF CTCLOPROPANE
Parameter
Value (cm-l)*
Ba R” CO c’,
I
THE MEASUREMENTS OF THE YS +
O.OOOOO48d
!l
are three times the estimated
standard
Value (cm-‘)a -0.028269 f 0.000060 8.2b x 10-T -(G.23 f 2.40) x 10’ (1.4 f 2.3) x 1O-7 0.0024589 f 0.0900270 errors;
see text for caution on
the interpretation of these constants because of possible model errors. b Ground state values given in ref. 7. c The value of Ca has not yet been accurately determined. d The correlation between Co and C, is such that C, is more poorly known than Co ; the uncertainty given in t,his table assumes that the value for Co is without error. TABLE B.\ND CONSTANTV Constant
&,, B’-B”
C’-(Cr)‘-B’ Q
OBTAINED
II
FOR THE ~5 +
~1~TRANSITION
This work*
-0.002818 2089.520 ff -0.001305 f -0.22348 f 0.002459 f
O.OOld O.OOOOO5 0.000005 0.00007 0.000027
OF CYCLOPROPlNE
Cartwright and Mills”
-0.0028 2089.4 -0.0012 -0.219 0.0025
f f It f
0.0005 0.1 0.0005 0.02 0.0005
n All constants are given in wavenumbers (cm-‘). b The uncertainties given for the NBS measurements are three times the estimated standard errors. c See Ref. (2). d The band center has an additional uncertainty of about 0.006 cm-’ due to possible syst,ematic errors in the absolute calibration of the spectra.
Z-TYPE
RESONANCE
IN CYCLOPROPANE
1Sl
In the least-squares analysis the values of B0 and L), were cont,ained to agree with the values given by McCubbin, Withstandley, and 1’010 (7). The constants given in Table I were able to reproduce the measured transitions with a standard deviation of 0.003 cm-’ which is the expected magnitude of experimental error. The uncertainties given in Tables I and II are only estimated from statistical considerations and do not reflect, model errors such as the neglect of the 17terms. Olson has found (12) that (at least in the case of CH,D and SiHsD) leaving out the 7 terms and forcing the D terms to be the same in t,he upper and lower states, a:: was done in this analysis, can lead to gross errors in the values obt,ained for the I), and D,, terms. Because of this, the reader is cautioned not to use t,he D, and D,, values given in Table I for force constant calculations, but, rather to consider them simply as effective values needed to fit the data for t,he ~5 + v10band. The true values of t’he ground state centrifugal distortion constants D, and I),, for cyclopropane may be significant,ly different from the values given in Table I. In Table II the band constants given by this analysis are compared with the const,ants given by Cartwright and Mills. The agreement is quite remarkable and shows that, if all significant perturbations are properly t,aken into account, it is possible for careful workers to use the band contour method to obtain quite good con&ants. One cannot stress t,oo much, however, the necessit#y of considering all significant, perturbat,ions when making band contour analyses. It must also be realized that in most cases hot bands will play a more significant role than they do in cyclopropane and it is not, usually possible to allow for the effects of hot bands. RECEIVED
August
30, 1971 ACKNOWLEDGMENTS
The author gratefully acknowledges the assistance of Robert Sams, who measured the spectra, and Dr. J. D. Simmons, who provided a subroutine for calculating the Jacobian. REFERENCES 1. IAN M. MILLS, Pure Appl. Chem. 18,285 (1969); C. DIL.IUKO AND I. M. MIUS, J. .Ilo/. Spectrosc. 21, 386 (1966). 2. G. J. CARTWRIGHT AND I. M. MILLS, J. Mol. Spectrosc. 34,415 (1970). 3. H. H. G~~NTHARD, R. C. LORD, AND T. K. MCCUBBIN, JR., J. Chem. Phys. 26,768 (1956). 4. J. L. DUNCAN, J. Mol. Spectrosc. 26,451 (1968). 6. J. L. DUNCAN AND D. C. MCKEAN, J. Mol. Spectrosc. 27, 117 (1968). 6. J. L. DUNCAN AND D. ELLIS, J. Mol. Spectrosc. 28,540 (1968). 7. T. K. MCCUBBIN, V. WITHSTANDLEY, AND S. It. POLO, J. Mol. Spectrosc. 31,95 (1969). 8. P. M. MATHAI, G. G. SHEPHERD, AND H. L. WELSH, Carl. J. Phys. 34, 1448 (1956). 9. W. J. JONES AND B. P. STOICHEFF, Cara. J. Phys. 42,2259 (1964). 10. A. G. MAKI, W. B. OLSON, AND R. L. S_ms, J. Mol. Spectrosc. 38,433 (1970). 11. R. H. WAMPLER, J. Res. Nat. Bur. Stand. B 73,59 (1969). 12. WM. B. OLSON, private communication.
OF
MOLECULAR
A High
SPECTROSCOPY
Resolution
41, 177-181
(1972)
Study of l-type Cyclopropane
Resonance
in
ARTHUR G. ~L~AKI National
Bureau of Standards,
Washington,
D. C. ZOS’S4
The ~5+ Y~O band of cyclopropane (C3H6) centered at 2089.52 cm-l has been measured and analyzed taking into account the l-type resonance effect described by Cartwright and Mills. A least-squares fit of more than 300 well-resolved transitions gave a set of band const’ants which reproduced the measurements with a standard deviation of 0.093 cm-l. The analysis confirms the Z-type resonance treatment given by Cartwright and Mills and shows that their band contour fit gave remarkably accurate constants for this band. INTRODUCTION In the past few years computer simulated spectra have been used more and more frequently to aid in the interpretation of unresolved ro-vibronic bands (not only in the infrared region, but also in the visible and ultraviolet spectral regions). These so-called band cont,our analyses have been used to determine, for both t)he upper and lower stat’es, the rotational constants for symmetric top and asymmetric rotor type spectra. Band contour analyses have also been used to determine the importance of various types of perturbations, and to show how some perturbations may affect t’he gross appearance of the band contours. There has been some skepticism regarding the accuracy of the band constants and interaction const’ants obtained from such band contour analyses. Since there are many cases for which such band contour analyses offer the only hope of obtaining certain types of structural information, it is important to check the constants obtained from some band cont,our analyses which could not have been prejudiced by a prior knowledge of the correct const,ants. Since Mills (I, 2) has been one of the foremost exponents of the use of computer simulat,ed spectra for the analysis of unresolved (or part)ially resolved) spectra, we have measured the high resolution spectrum of one of the bands of cyclopropane for which Cartwright and Mills (2) have published a band cont’our analysis. The band chosen for study was the perpendicular band I+, + vlO(I?) of cyclopropane (C&H,). Even in low resolution spectra this band displays in t,he &-branch region a peculiar transmission spike which Cartwright and Mills have
shown t,o be due to l-type resonance effects. 177 Copyright
a
1972 by Academic
Press, Inc.
175
MAKI
Previous high or medium resolution infrared measurements on cyclopropane were made by Giint,hard et al. (S), Duncan et al. (4-G), and McCubbin et al. (7). The rotational Raman spectrum was measured by Mathai et al. (8) and subsequently Jones and Stoicheff (9) determined an improved set of rotational constants. The most accurate values of Bo and D, seem t’o be those given by t,he infrared measurements of XIcCubbin, Withstandley, and Polo (7). EXPERIMENTAL
The measurements were made in a l.!% m absorption cell cooled to dry ice temperat,ure (195 K) over 90% of its length. As an indication of the resolut’ion available for this work, a very compressed spectrum of the 2089 cm-’ band is presented in Fig. 1. On the expanded measurement charts lines separated by 0.025 cm-l were just barely resolved; Cartwright and Mills used much lower resolution [presumably slightly bet,ter than that, indicated in Fig. 1 of Ref. (5)]. The present, spect,ra were t’aken on the NBS high resolution infrared spectrometer recently modified by Dr. W. B. Olson and placed in a vacuum tank to avoid refractive index corrections in the calibration. The spectrometer is a double pass inst,rument, with a focal length of 2.3.5 m. The measurement’s were made using first order of a 5 X X in. grating with 400 lines/mm. Calibrat’ion was achieved by interpolation between thorium lines observed in higher order on the grating [see Ref. (IO)]. ANALYSIS
AND
RESULTS
Since it has been shown that Z-type resonance has a significant effect for many oblate rotor molecules, the spectrum was analyzed by applying the theory of Z-type resonance as described by Mills (I, 2). A computer program was written which uses the energy expressions (including Z-type resonance) given by Cartwright and Mills (2) to fit a set of band constants to the observed data using a leastsquares criterion.
FIG. 1. Infrared absorption due to the ~5 + YIOtransition of cyclopropane pat hlength at, a pressure of 8 Torr and a temperature of 200°K.
with a 1.55 m
l-TYPE
RESONANCE
IN CYCLOPROPANE
179
The program operated according t,o the following sequence of steps: (1) The transit,ion measurements and assignments are given to the computer and an initial guess was made for the band constants. (2) For each observed transition t,he lower energy levels were calculated by I he usual equation
F” (J”, k”) = B,[J” (,I” + 1) -
.J[J”
(AZ”)“I + C,, (/i” )”
(*I” + l)]’ -
(3) The unperturbed
upper
I),,./”
(.I” + 1) (A”“)’ -
state energy
F’ (J’, k’, I’) = V” + B,[J’(J’
levels were calculated
+ -
I),@“)?
1) &[.I
by Eq.
(I 1 (2).
(li’)‘] + c, (A.‘)2 (J’ + 1 )]”
-
2 (C{)&‘I’
-
D,,J’ (J’ + 1) (x:‘)’ - D, (k’)4.
(2)
(The 7 terms were left out of this equation because the data could be fit satirfactorily without them. ) (4) Equation (2) was also used to calculate the energy of that level which interacts wit.h the upper state of the assigned transition through the particular mat,rix element (at, lt + 1, J, k + 1 ( I?/hc /vt, II - 1, .I, k - 1). (5) The above matrix element was evaluated using t(he conventions set, up by Cart.wright and Mills (a), namely: (v, , dt + 1, J, k + 11 A//MY /21L , I, =
(-!/4)Q,[(Ut
1, J, k + I)* -
1)
2 l/Z I, I
. {[.I(.1 + 1) - X-(x:+ l)][J(J
0 ) + 1) - k(k -
1)]1’?
(6) The eigenvalues and eigenvectors resulting from the upper state inter‘ were determined by solving the appropriate 2 X 2 secular determinant. (7) From the eigenvectors and the partial derivst.ives of the elements of the determinant the Jacobian was formed. (8) The Jacobian was then used in a least-squares fit of the residuals (vob - vcai) t’o obtain corrections for the band constants. (9) If the corrections to any of the constants were greater than a tenth of the estimated st,andard error of those constants, another iteration It-as performed by going back to step 2. The computer program used double precision operations throughout and utilized a least-squares subroutine, ORTHO, which has been shown to be very reliable (11). The assignments necessary for step 1 could be made quite unambiguously since the first ‘P or ‘R line of a subband for a given /? was nearly always obvious. Since by chance C-CC-B z s B, t,he &-branch spacing was very nearly equal action
MAKI
180
to one-third the ‘P or RR line spacing expected in each subband, causing the ‘P and RR lines to fall into groups or clumps separated by jJ$ B. The order of appearance of each new K” = J” line showed that the quantum numbers of each successive line within each clump were given by J, K = m, n; m + 1, n + 3; t’he fact m + 2, n + 6; etc., where n and m are integers. This established that the ‘Q transitions were at higher frequencies than the “Q transitions. If the ‘Q had been lower in frequency than the “Q, then the quantum numbers would have been given by J, K = m, n; m - 1, n + 3; m - 2, n + 6; etc. The assignments were also facilitated by the nearly complete absence of ‘R and RP lines. The measurements covered the complete range of K values from K = 0 to K = 31 and an equal range of J values. The analysis included a number of resolved Q-branch lines for ‘Q3 , pQp , “&I , “Qz , and "Qa . A total of 337 tSransit’ions were used in the least-squares fit and Table I contains t,he result,ing constant,s. TABLE P.IHAMETEKS
TO Frr
USED
Parameter
2089.5201
VI3
0.67024b 0.6689351 0.424NnY 0.4171817
(1The rmcertainties
f
0.0010
f
0.0000055
(G). DJ
D JK DK
f
YIOBAND OF CTCLOPROPANE
Parameter
Value (cm-l)*
Ba R” CO c’,
I
THE MEASUREMENTS OF THE YS +
O.OOOOO48d
!l
are three times the estimated
standard
Value (cm-‘)a -0.028269 f 0.000060 8.2b x 10-T -(G.23 f 2.40) x 10’ (1.4 f 2.3) x 1O-7 0.0024589 f 0.0900270 errors;
see text for caution on
the interpretation of these constants because of possible model errors. b Ground state values given in ref. 7. c The value of Ca has not yet been accurately determined. d The correlation between Co and C, is such that C, is more poorly known than Co ; the uncertainty given in t,his table assumes that the value for Co is without error. TABLE B.\ND CONSTANTV Constant
&,, B’-B”
C’-(Cr)‘-B’ Q
OBTAINED
II
FOR THE ~5 +
~1~TRANSITION
This work*
-0.002818 2089.520 ff -0.001305 f -0.22348 f 0.002459 f
O.OOld O.OOOOO5 0.000005 0.00007 0.000027
OF CYCLOPROPlNE
Cartwright and Mills”
-0.0028 2089.4 -0.0012 -0.219 0.0025
f f It f
0.0005 0.1 0.0005 0.02 0.0005
n All constants are given in wavenumbers (cm-‘). b The uncertainties given for the NBS measurements are three times the estimated standard errors. c See Ref. (2). d The band center has an additional uncertainty of about 0.006 cm-’ due to possible syst,ematic errors in the absolute calibration of the spectra.
Z-TYPE
RESONANCE
IN CYCLOPROPANE
1Sl
In the least-squares analysis the values of B0 and L), were cont,ained to agree with the values given by McCubbin, Withstandley, and 1’010 (7). The constants given in Table I were able to reproduce the measured transitions with a standard deviation of 0.003 cm-’ which is the expected magnitude of experimental error. The uncertainties given in Tables I and II are only estimated from statistical considerations and do not reflect, model errors such as the neglect of the 17terms. Olson has found (12) that (at least in the case of CH,D and SiHsD) leaving out the 7 terms and forcing the D terms to be the same in t,he upper and lower states, a:: was done in this analysis, can lead to gross errors in the values obt,ained for the I), and D,, terms. Because of this, the reader is cautioned not to use t,he D, and D,, values given in Table I for force constant calculations, but, rather to consider them simply as effective values needed to fit the data for t,he ~5 + v10band. The true values of t’he ground state centrifugal distortion constants D, and I),, for cyclopropane may be significant,ly different from the values given in Table I. In Table II the band constants given by this analysis are compared with the const,ants given by Cartwright and Mills. The agreement is quite remarkable and shows that, if all significant perturbations are properly t,aken into account, it is possible for careful workers to use the band contour method to obtain quite good con&ants. One cannot stress t,oo much, however, the necessit#y of considering all significant, perturbat,ions when making band contour analyses. It must also be realized that in most cases hot bands will play a more significant role than they do in cyclopropane and it is not, usually possible to allow for the effects of hot bands. RECEIVED
August
30, 1971 ACKNOWLEDGMENTS
The author gratefully acknowledges the assistance of Robert Sams, who measured the spectra, and Dr. J. D. Simmons, who provided a subroutine for calculating the Jacobian. REFERENCES 1. IAN M. MILLS, Pure Appl. Chem. 18,285 (1969); C. DIL.IUKO AND I. M. MIUS, J. .Ilo/. Spectrosc. 21, 386 (1966). 2. G. J. CARTWRIGHT AND I. M. MILLS, J. Mol. Spectrosc. 34,415 (1970). 3. H. H. G~~NTHARD, R. C. LORD, AND T. K. MCCUBBIN, JR., J. Chem. Phys. 26,768 (1956). 4. J. L. DUNCAN, J. Mol. Spectrosc. 26,451 (1968). 6. J. L. DUNCAN AND D. C. MCKEAN, J. Mol. Spectrosc. 27, 117 (1968). 6. J. L. DUNCAN AND D. ELLIS, J. Mol. Spectrosc. 28,540 (1968). 7. T. K. MCCUBBIN, V. WITHSTANDLEY, AND S. It. POLO, J. Mol. Spectrosc. 31,95 (1969). 8. P. M. MATHAI, G. G. SHEPHERD, AND H. L. WELSH, Carl. J. Phys. 34, 1448 (1956). 9. W. J. JONES AND B. P. STOICHEFF, Cara. J. Phys. 42,2259 (1964). 10. A. G. MAKI, W. B. OLSON, AND R. L. S_ms, J. Mol. Spectrosc. 38,433 (1970). 11. R. H. WAMPLER, J. Res. Nat. Bur. Stand. B 73,59 (1969). 12. WM. B. OLSON, private communication.