Ritz assignment and Watson fits of the high-resolution ring-puckering spectrum of oxetane

Journal of Molecular Spectroscopy 219 (2003) 152–162 www.elsevier.com/locate/jms

Ritz assignment and Watson fits of the high-resolution ring-puckering spectrum of oxetaneq Giovanni Moruzzi,a Marc Kunzmann,b,c Brenda P. Winnewisser,b,d,* and Manfred Winnewisserb,d a

c

Dipartimento di Fisica ‘‘Enrico Fermi’’ dellÕUniversit
a di Pisa and INFM, Via F. Buonarroti 2, Pisa I-56127, Italy b Physikalisch-Chemisches Institut, Justus-Liebig-Universit€at, Heinrich-Buff-Ring 58, Giessen D-35392, Germany Institut f€ur Physikalische Chemie der Georg-August-Universit€at G€ottingen, Tammanstrasse 6, G€ottingen D-37077, Germany d Department of Physics, The Ohio State University, 174 West 18th Avenue, Columbus, OH 43210-1106, USA Received 4 November 2002

Abstract Oxetane is a four-membered ring molecule exhibiting a large-amplitude ring-puckering motion. In order to analyze this vibration we recorded a rotationally resolved far-infrared spectrum between 50 and 145 cm
1 . The analysis of the ring-puckering fundamental band with the assignment of 1108 lines, has been presented in a previous paper. In the present work we present a list of further 6531 assigned transitions between the five lowest excited ring-puckering states. The 4983 term values involved in the transitions assigned in this and in the preceding work have been evaluated by the ‘‘Ritz’’ program, and are now available. An A-reduced Watson Hamiltonian in any of the three representations Ir , IIr , and IIIr was used to perform a fit of the assigned transitions. Precise rotational constants and quartic as well as a full set of sextic centrifugal distortion constants were obtained for the investigated ringpuckering states. For the first time, high-resolution values for the vibrational Gv parameters have been obtained, and we have added terms in x6 and x8 to the double minimum-potential well describing the ring-puckering motion, in order to reproduce their values within the experimental accuracy. The same potential still reproduces the lower resolution values of the Q-branch origins involving higher ring-puckering states up to vrp ¼ 14 found in the previous literature. Ó 2003 Elsevier Science (USA). All rights reserved. Keywords: Ring puckering; High-resolution far-infrared spectroscopy; Large-amplitude motion; Oxetane

1. Introduction The oxetane molecule, sketched in Fig. 1, is a fourmembered ring compound with an essentially planar CCCO ring. The rotationally resolved Fourier transform infrared (FTIR) spectrum of the ring-puckering band system of oxetane was recently reported with an analysis of the fundamental and the first hot band [1]. Since this very low-lying mode represents an anharmonic, large-amplitude vibration, the band system is spread over a wide region in the far-infrared (FIR). The q Supplementary data for this article are available on ScienceDirect. * Corresponding author. E-mail addresses: [email protected] (G. Moruzzi), [email protected] (B.P. Winnewisser).

present paper is concerned with the investigation of the entire spectrum of oxetane between 52 and 145 cm
1 , where the ring-puckering fundamental as well as its first five hot bands can be observed. We extend the analysis to the hot bands involving the lowest five excited ringpuckering states. The line assignment of the present work has been done by the Ritz program, to which two new important options, discussed in a later section, have been added. The Ritz program was first introduced for the analysis of the high-resolution FIR spectra of the methanol isotopomers [2]. The program is based on the idea of evaluating term values from the wavenumbers of the assigned transitions by the Rydberg–Ritz combination principle, and then searching for new assignments by extrapolating term values rather than transition wavenumbers. This method has already led to the

0022-2852/03/$ - see front matter Ó 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0022-2852(03)00021-3

G. Moruzzi et al. / Journal of Molecular Spectroscopy 219 (2003) 152–162

Fig. 1. The oxetane molecule, C3 H6 O. The four-membered ring is quasi-planar, with the a and b axes lying in the molecular plane, and the a axis bisecting the COC angle. The permanent electric dipole moment lies along the a axis.

compilation of databases containing more than 138 000 assigned lines and more than 35 000 term values of various methanol isotopomers [2–5]. Recently [6] the program has been rewritten in order to do away with the limitation to methanol and its isotopomers and to make it of more general use. As a first test on a molecule different from methanol, the multimolecule version of the Ritz program has already been successfully used for the analysis of the cyanamide spectrum, and the assignments of 19 600 lines and 3900 term values have been presented in [6]. In the Ritz database developed in the course of this work, we have inserted microwave data available in the literature [7] and also the transitions of the fundamental band investigated in [1,8]. The term values evaluated by the Ritz program have been fitted with Watson Hamiltonians in A reduction, Ir , IIr , and IIIr representations for each investigated ring-puckering state, including the microwave data. The possibility of fitting the assigned line wavenumbers, or term values, to Watson Hamiltonians is a new option of the Ritz package. This new option has been extremely useful for extending the assignments, because a Watson Hamiltonian can predict term values belonging to new level sequences of a given ring-puckering state, while the term-value predictions of a polynomial extrapolation are limited to one level sequence. As described in detail in the preceding publication [1], the spectrum was recorded at 203 K at a pressure of ca. 0.9 mbar (estimated vapor pressure at 203 K) in a 3 m glass cell. As will be seen in Fig. 3, which shows the Q branch of the first hot band, the cold spectrum is rotationally resolved to a large degree even in the Q-branch region. The analysis of the present data provides highly precise values for the band origins, rotational constants,

153

quartic, and all sextic centrifugal distortion constants for levels with ring-puckering quantum number 0 6 vrp 6 5. These data allow us to define more precisely the ring-puckering potential function, determined most recently by Jokisaari and Kauppinen [9] from the positions of the fundamental and hot bands up to vrp ¼ 11 10. The full set of constants allows a convincing confirmation of the models proposed by Creswell and Mills [7] and later by Lesarri et al. [8], which describe the dependence of the quartic constants on the ring-puckering excitation. The effect of the ringpuckering excitation on the full set of sextic distortion constants is seen for the first time. The detailed analysis presented in this work and the preceding paper [1] was undertaken with the intention of providing data for interstellar searches for oxetane. So far, only three-membered ring molecules have been identified in the interstellar medium. Unfavorable partition functions and low transition moments have been obstacles to the detection of most ring molecules, and the terahertz transitions of the spectrum investigated in this work may provide detectable signals under favorable excitation conditions.

2. The oxetane molecule and the term-value notation Since oxetane is a near oblate rotor molecule, Kc is an ‘‘almost good’’ quantum number for labeling the energy levels. Kc -doubling is observable up to relatively high Kc values (Kc / 10 for our FTS resolution), and the members of each doublet are denoted by the corresponding Ka value. Energy levels are further labeled by the total angular momentum quantum number J and by the quantum numbers vi of the vibrational modes. A fourmembered ring is the smallest ring capable of displaying ring-puckering, which is the only large-amplitude vibrational mode present in oxetane. The ring-puckering fundamental is located at approximately 53 cm
1 . All the transitions investigated in the present work involve levels belonging to the ground small-amplitude vibrational state, so that the notation jvrp ; Ka ; Kc ; J i;

ð1Þ

where vrp is the ring-puckering quantum number, is sufficient to label each investigated level unambiguously. Further, the label Ka can be omitted for unresolved doublets. The ring-puckering motion has been described since the work of Chan et al. [10] by the one-dimensional coordinate x, defined as half of the distance between the axis joining the O atom with its opposite C atom and the axis joining the remaining two C atoms. In accord with our usual Ritz terminology, we define a level sequence of oxetane as the set of all levels sharing all the quantum numbers except J and Ka (it would be except J and Kc for near-prolate rotors). In the context

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of the present work a level sequence is thus defined by the two quantum numbers ðvrp ; Kc Þ:

ð2Þ

3. The assignment procedure The assignment procedure of the multimolecule Ritz program has been thoroughly described in [6]. We shall confine ourselves here to discussing a few important features. The basic idea of the program is to determine the term values involved in the assigned transitions by the Rydberg–Ritz combination principle. That is, we evaluate the term values T~ ¼ E=hc which minimize the expression v2 ¼

X ðT~i
T~j
m~ij Þ2 ; 2ij i;j

ð3Þ

where T~i and T~j are the term values of the ith and jth investigated level, respectively, m~ij is the experimental wavenumber of the corresponding transition, and ij its estimated experimental precision. The levels are then divided into level sequences, a sequence being defined by the quantum numbers of Eq. (2). While in the case of the methyl-symmetric methanol isotopomers the term values of a sequence could, in most cases, be fitted very well by a Taylor expansion in J ðJ þ 1Þ truncated at the 3rd or 4th power, the case of a level sequence which can be fitted well by a Taylor expansion is, unfortunately, not very frequent for oxetane, which has considerably greater inertial asymmetry. However, we had already noticed [2,6] that, even when a level sequence strongly deviates from any possible reasonable Taylor expansion (i.e., in the presence of perturbations), it can still be followed by predicting the first unassigned term value, labeled by the quantum number J , by parabolic extrapolation from the assigned term values labeled by the quantum numbers J
1, J
2, and J
3. This usually leads to a reasonable approximation of the new term value. Then, if lines corresponding to transitions from the new level to assigned levels are found in the spectrum, minimization of Eq. (3) determines the ‘‘experimental’’ term value of the new level. This value is then used for finding the level labeled by J þ 1, so that the procedure can be iterated [2]. Thus, although the whole perturbed sequence cannot be fitted by a parabola, a power series fit of the whole sequence is never needed as part of the assignment process. The method used by the Ritz program for searching for a new level sequence in the absence of accurate predictions by a suitable model Hamiltonian has been extensively described in [6], and has been of vital importance in the present work. However, a proper and reliable operation of the Ritz program requires that the selection rules of the observed

transitions are not too restrictive, so that allowed, and observable, transitions connect each level to as many other levels as possible. In the case of the methanol isotopomers, for instance, the number of transitions starting from a single level is between 20 and 30 in most cases. This large number provides both a powerful check of the reliability of the assignment process, and very precise calculated term values. Unfortunately, this is not the case for oxetane. The Dvrp ¼ 1 transitions between different ring-puckering states investigated in the present work obey the c-type selection rule DKc ¼ even;

ð4Þ

and, in practice, only DKc ¼ 0 transitions are observed. Thus, each jvrp ; Kc ; J i energy level is connected to other levels only by the three downward transitions and the three upward transitions jvrp ; Kc ; J i $ jvrp 1; Kc ; J i; jvrp 1; Kc ; J 1i:

ð5Þ

And, of course, for the assignment of each new level belonging to a new vrp state, we can use only the three downward transitions connecting it to the immediately lower, already assigned, vrp
1 state. This, combined with the weakness of many P lines and the usual line overlappings occurring in most Q-branches, made the oxetane molecule a real challenge to investigation by the Ritz method. We have solved this problem by introducing two further options into the Ritz package: (i) The Loomis–Wood procedure [11]. The regularities displayed by the spectrum plotted on the computer monitor by the Loomis–Wood option have allowed us to identify many R branches corresponding to new level sequences, and the Ritz procedure could then easily find the corresponding unblended P and Q transitions. (ii) The possibility of fitting simultaneously the whole set of assigned lines to a set of Watson Hamiltonians in A reduction, with the further option of selecting any of the three representations Ir , IIr , and IIIr (one Watsonian for each involved ring-puckering state). The program of course offers also the possibility of fitting a subset of assigned lines connecting two or more selected ring-puckering states, or to fit directly the Ritz term values of a selected ring-puckering state to the corresponding Watsonian. The fits are optimized by a Levenberg–Marquardt procedure. This possibility was not present in the preceding versions of the Ritz program because a Watsonian usually cannot fit satisfactorily the spectrum of a molecule displaying large-amplitude vibrations, like methanol or cyanamide. However, Creswell and Mills [7] had shown that their microwave data could be fitted with separate Watsonians for each ring-puckering state. Thus, the Watsonian parameters obtained by fitting transitions occurring within a given ringpuckering state can be used for extrapolating the

G. Moruzzi et al. / Journal of Molecular Spectroscopy 219 (2003) 152–162

155

(a)

(b)

Fig. 2. Investigated level sequences and transitions of oxetane: (a) from Kc ¼ 0 to Kc ¼ 20, and (b) from Kc ¼ 20 to Kc ¼ 40. Each box corresponds to a level sequence, and lines connecting the boxes represent the observed transitions. Inside each box are reported the label for the small-amplitude vibrational quantum number (always gr, for ground state, in our case), and, separated by a comma, the ring-puckering quantum number vrp , ranging from 0 to 5 in our investigated transitions. The last number in the upper line of the label reports the highest observed J value. In the lower line of each label the extrapolated J ¼ 0 term value of the level sequence is given; the vertical coordinate of each box is proportional to this value. The quantum number Kc of each sequence can be read on the abscissae axis. Because of the DKc ¼ 0 selection rule valid for the Dvrp ¼ 1 transitions, only the vertical lines shown in the figure correspond to observed FIR transitions. Nearly horizontal lines corresponding to Dvrp ¼ 0, DKc ¼ 1 transitions represent the microwave transitions inserted into the fit.

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G. Moruzzi et al. / Journal of Molecular Spectroscopy 219 (2003) 152–162

term values of a new level sequence belonging to the same ring-puckering state (i.e., a sequence of levels labeled by the same vrp , but by a Kc value not yet occurring in the assignments). This has been particularly useful for the assignment of the low Kc sequences (0 6 Kc 6 2) of vrp ¼ 3; 4, and 5, whose corresponding transitions are very weak, strongly affected by asymmetry, and occur in very crowded spectral regions. We have investigated the oxetane spectrum between 52 and 145 cm
1 . Six different Dvrp ¼ 1 rotation-puckering transitions, involving 0 6 vrp 6 6, occur in this region. However, we have not been able to assign the vrp ¼ 6 5 transitions because the corresponding R branches are weak and lie mostly beyond the spectral range measured in the present work.

4. Results

Table 1 Watson Hamiltonian (Ir representation) parameters for the vrp ¼ 0–5 ring-puckering states of oxetane and their standard errors Constant

vrp ¼ 0

A B C DJ DJK DK dJ dK UJ UJK UKJ UK uJ uJK uK Gv

12045.4607 11734.0562 6730.6608 4.9542 )1.4031 4.5658 1.6433 )0.8332 18.8 24.0 37.08 )66.0 3.0 )26.9 )5.6 0.0

(16) (15) (15) (32) (81) (57) (13) (30) (30) (190) (260) (137) (18) (79) (61)

vrp ¼ 2

An overview of the level sequences investigated in the present work and of the observed transitions connecting them is plotted in Fig. 2. In this figure, each box corresponds to a level sequence, and the lines connecting the boxes represent sets of observed transitions. For each level sequence, the highest observed J value and the energy extrapolated at J ¼ 0 are given. Our Ritz database comprises 10 235 assigned transitions and 4983 levels divided into 194 sequences. The 10 235 transitions consist of the 6531 FIR transitions assigned in the present work, the 1108 FIR transitions assigned in [1], 468 microwave [7] and millimeter wave [8,12] transitions, and 2128 calculated microwave lines used to connect the whole ground state and the first excited ring-puckering state up to Kc ¼ 10. Our 7639 assigned FIR transitions actually correspond to only 6180 different wavenumbers in the FTIR spectrum because of the frequent line overlappings, particularly affecting the Q branches. We have simultaneously fitted a set of 7946 transitions to six Watson Hamiltonians in all three representations (one Watson Hamiltonian for each ring-puckering state). Thus, we have fitted the Ritz database excluding the 2128 calculated transitions, one microwave and 160 FIR lines which lead to a ratio j~ mexp
m~calc j= > 5, where m~exp and m~calc are the experimental and calculated wavenumbers, and  is the experimental accuracy. The parameter G0 was constrained to be zero, while all other parameters were left free. The parameters obtained from the fits, and their standard errors, are shown in Tables 1 and 3 for the Ir and IIr representations, respectively. The energy levels determined in this study and the transitions used in the fits, including the microwave lines, are listed together with their deviations in the supplementary material available from the journal website. The A, B, and C parameters of Tables 1 and 3 show the vrp -dependent changes reported in [7]. As already

A B C DJ DJK DK dJ dK UJ UJK UKJ UK uJ uJK uK Gv

A B C DJ DJK DK dJ dK UJ UJK UKJ UK uJ uJK uK Gv

12058.9679 11718.8936 6789.0463 4.8348 )4.5300 7.8450 1.6715 0.9663 14.6 )0.5 )9.9 6.4 2.5 )1.41 29.8 142.579784

vrp ¼ 1

Unit

12057.9482 (16) 11726.5485 (15) 6772.2114 (15) 4.9850 (30) )6.4696 (79) 9.6678 (57) 1.6132 (13) 1.7846 (29) 16.1 (31) )53.7 (19) 58 .0 (190) )11.9 (130) 2.9 (18) 16.2 (78) 26.1 (60) 52.920348 (13)

MHz MHz MHz kHz kHz kHz kHz kHz mHz mHz mHz mHz mHz mHz mHz cm
1

vrp ¼ 3 (16) (15) (15) (31) (83) (63) (13) (31) (32) (19) (28) (14) (19) (80) (65) (16)

12060.3363 (16) 11709.9775 (16) 6809.7213 (16) 4.8479 (37) )4.5593 (99) 7.8789 (73) 1.6486 (16) 1.0683 (36) 12.2 (39) )17.1 (234) 24.0 (340) )7.0 (160) 2.7 (22) )6.1 (11) 17.8 (79) 247.044598 (18)

vrp ¼ 4

vrp ¼ 5

12059.4845 (13) 11700.5373 (13) 6827.1429 (13) 4.8394 (37) )4.398 (12) 7.7350 (89) 1.6390 (20) 1.0873 (45) 22.4 (46) )30.8 (295) 40.8 (426) )11.2 (200) 1.4 (28) )1.3 (122) 13.0 (98) 364.953822 (20)

12057.1342 (23) 11690.5859 (23) 6842.9180 (23) 4.8061 (90) )4.229 (19) 7.576 (14) 1.6300 (32) 1.1109 (74) 12.0 (144) )15.0 (480) 53.0 (800) )52.0 (550) 2.7 (46) )5.1 (200) 19.5 (180) 493.856243 (23)

MHz MHz MHz kHz kHz kHz kHz kHz mHz mHz mHz mHz mHz mHz mHz cm
1

MHz MHz MHz kHz kHz kHz kHz kHz mHz mHz mHz mHz mHz mHz mHz cm
1

noted in [10,13], these parameters can be fitted to an expression of the form Xv ¼ b0 þ b2 hvjn2 jvi þ b4 hvjn4 jvi;

ð6Þ

where Xv stands for the A, B, or C value for the ringpuckering state v ¼ vrp , and the matrix elements

G. Moruzzi et al. / Journal of Molecular Spectroscopy 219 (2003) 152–162

157

Table 2 The parameters X ¼ ðA þ BÞ=2 and Y ¼ X
C in MHz, which determine the degradation and the distances between the Q branches, for the investigated ring-puckering states vrp

X

Y

DXvrp

DYvrp

0 1 2 3 4 5

11889.757578 11892.247510 11888.930251 11885.155613 11880.006489 11873.859526

5159.097555 5120.036896 5099.884387 5075.435433 5052.867976 5030.942028

2.489932 )3.317259 )3.774638 )5.149124 )6.146963

)39.060659 )20.152509 )24.408437 )22.567457 )21.925948

represent the average values of the powers of the dimensionless ring-puckering coordinate n, defined later in Eq. (16), over the corresponding ring-puckering state. Lesarri et al. [8] showed that a term in n6 needed to be included in this expression to fit the rotational constants they obtained from their millimeter wave data. A section of the Q-branch region of the first hot band is shown in Fig. 3. A comparison with the Q branch region of the fundamental band, [1, Fig. 6], shows that the Q branches are less dispersed than for the fundamental, and that the individual subband Q branches are degraded strongly in the opposite direction. A similar picture is presented by the Q-branch regions of the higher hot bands, which are even more compact and show a similar but successively greater degradation. These differences are a direct consequence of the strong and non-linear vrp -dependence of the rotational constants. If we define the two parameters X ¼ ðA þ BÞ=2 and Y ¼ X
C, the wavenumbers of the DJ ¼ 0, DKc ¼ 0 transitions between two ring-puckering states with vrp ¼ v and vrp ¼ v
1 are approximately m~v;v
1 ðJ ; Kc Þ ðXv
Xv
1 ÞJ ðJ þ 1Þ
ðYv
Yv
1 ÞKc2

ð7Þ

so that DXv ¼ Xv
Xv
1 determines the degradation of the Q branches, while DYv ¼ Yv
Yv
1 determines the distances between Q branches for successive values of Kc . The experimental values of Xv , Yv , DXv , and DYv for the investigated ring-puckering states are shown in Table 2. The dependence of our quartic distortion constants DJ , DJK , DK , dJ , and dK on the ring-puckering quantum number vrp is plotted for the fit in Ir representation in Fig. 4. The extensive data analyzed here, allowing the simultaneous determination of the sextic constants, make our values considerably more accurate than the values reported in [7,8]. Creswell and Mills [7] developed a model for the behavior of the quartic constants. According to their model, DJ and dJ are practically independent of vrp . Fig. 4 shows that this prediction is well verified by our data. The model of Creswell and Mills further predicts for DK , DJK , and dK in a IIr representation

DK ¼ D0K þ 2chv ; DJK ¼ D0JK
2chv ; dK ¼

d0K

ð8Þ


cjhv ;

where D0K , D0JK , and d0K are the values for vrp ¼ 0, including the contributions of all vibrations except ring puckering, j ¼ 0:8829 is RayÕs asymmetry parameter for C3 H6 O, and c¼

h4 oIac ; 4hIa2 Ic2 ox

ð9Þ

x being the ring-puckering coordinate. The parameter hv is defined by Creswell and Mills as X hvjxjv0 ihv0 jxjvi ; ð10Þ hv ¼ ð Ev
E v0 Þ v0 where Ev and Ev0 are the eigenvalues of the ring-puckering Hamiltonian, with v ¼ vrp . Thus, the deviations of DK , DJK and dK from their values at vrp ¼ 0 are expected to be similar, but in the ratios +2:)2:
j. In the Ir representation, it can be shown that the ratio of the ringpuckering contributions derived from this model will have the identical form, but be in the ratios +2:)2:+1. The observed dependence of DK , DJK , and dK is in good accord with these predictions, with small but consistent deviations, as shown in Fig. 4. This figure also reports the puckering dependances of 2chv , chv , and
2chv , represented by the dashed lines, where hv now is X hvjnjv0 ihv0 jnjvi ; ð11Þ hv ¼ ðWv
Wv0 Þ v0 where n is the dimensionless ring-puckering coordinate, defined later in Eq. (16), and Wv and Wv0 are the eigenvalues of the dimensionless Schr€ odinger equation given later in Eq. (18). The matrix elements of Eq. (11) have been evaluated using the numerically calculated eigenfunctions of Eq. (18). Our function hv is thus dimensionless, and the agreement with the experimental data is obtained by setting c ¼ 13:7 kHz. The zig–zag dependence of the parameters on vrp , observed in Fig. 4, and particularly visible at low vrp quantum numbers, is due to the fact that at n ¼ 0, the ring-puckering eigenfunctions corresponding to even vrp quantum numbers have

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G. Moruzzi et al. / Journal of Molecular Spectroscopy 219 (2003) 152–162

Table 3 Watson Hamiltonian (IIr representation) parameters for the vrp ¼ 0–5 ring-puckering states of oxetane and their standard errors Constant

vrp ¼ 0

A B C DJ DJK DK dJ dK UJ UJK UKJ UK uJ uJK uK Gv

12045.4634 11734.0533 6730.6610 4.8922 )1.2169 4.5656 )1.6123 0.9182 13.1 74.1 )72.5 10.1 )0.2 8.0 24.5 0.0

A B C DJ DJK DK dJ dK UJ UJK UKJ UK uJ uJK uK Gv

A B C DJ DJK DK dJ dK UJ UJK UKJ UK uJ uJK uK Gv

(15) (16) (16) (32) (83) (57) (14) (26) (33) (214) (288) (116) (197) (79) (57)

vrp ¼ 1

Unit

12057.9606 (15) 11726.5353 (15) 6772.2122 (15) 4.9709 (30) )6.4270 (82) 9.6675 (57) )1.6062 (13) )1.3695 (26) 9.3 (32) )10.1 (199) )19.6 (268) 42.4 (109) 0.4 (18) )25.9 (75) )4.7 (54) 52.920348 (13)

MHz MHz MHz kHz kHz kHz kHz kHz mHz mHz mHz mHz mHz mHz mHz cm
1

vrp ¼ 2

vrp ¼ 3

12058.9770 (15) 11718.8837 (15) 6789.0470 (15) 4.8208 (31) )4.4878 (88) 7.8448 (63) )1.6645 (14) )0.6507 (27) 10.1 (33) 32.0 (207) )73.5 (282) 50.9 (117) )0.3 (20) 3.2 (77) )10.0 (57) 142.579784 (16)

12060.3456 (16) 11709.9674 (16) 6809.7220 (16) 4.8591 (37) )4.5929 (99) 7.8787 (73) )1.6542 (16) )0.7340 (32) 9.1 (40) 7.2 (248) )26.7 (340) 28.0 (142) )1.1 (23) )1.6 (91) )3.7 (69) 247.044598 (18)

vrp ¼ 4

vrp ¼ 5

12059.4937 (13) 11700.5274 (13) 6827.1436 (13) 4.8690 (37) )4.487 (13) 7.7350 (89) )1.6538 (20) )0.7473 (39) 20.4 (48) )12.0 (311) )2.1 (43) 18.9 (175) )0.4 (29) )4.6 (114) )1.6 (83) 364.953822 (20)

12057.1433 (23) 11690.5760 (23) 6842.9187 (23) 4.8498 (89) )4.359 (20) 7.576 (13) )1.6519 (32) )0.76634 (65) 2.5 (127) 55.7 (602) )84.0 (818) 43.5 (343) 2.2 (58) )16.0 (242) 14.2 (220) 493.856243 (23)

MHz MHz MHz kHz kHz kHz kHz kHz mHz mHz mHz mHz mHz mHz mHz cm
1

MHz MHz MHz kHz kHz kHz kHz kHz mHz mHz mHz mHz mHz mHz mHz cm
1

a maximum, while the odd eigenfunctions have a node (see Fig. 6). The two constants showing the least dependence on vrp , DJ and dJ , were sufficiently well determined by Lesarri et al. [8] so that they presented a model for the small, but detectable, ring-puckering dependence of

those constants. Our values for the vrp -dependent contributions to these constants are fully consistent with theirs. This is the first work reporting complete sets of sextic distortion parameters for 0 6 vrp 6 5. Their dependence on the ring-puckering quantum number is plotted for the fit in Ir representation in Fig. 5. Several of these constants were determined by Lesarri et al., but we were able to determine all seven of them. Although some are not well determined, we have chosen to include them. The errors on all the constants are slightly larger than if we were to omit several of the sextic terms, but the standard deviations show that the present fits give the best reproduction of the data. This choice provides a consistent treatment of the data for each state, and provides upper limits on the magnitudes of the values of the constants. Perhaps most important, the quartic constants are not contaminated with the effects of omitted sextic constants. To test this, we ran a fit with all our data, but with the same selection of sextic constants adjusted as that used by Lesarri et al. [8]. The resulting quartic constants are within the standard errors of the values obtained in [8], whereas the differences between those constants and the constants in Table 3 are as much as 10 times larger. Although the errors on some of the constants are large, it can be seen from Fig. 5 that the sextic constants vary roughly as do the quartic constants as a function of vrp . High-resolution values of Gv are given for the first time in this work. As first noted in [14], a ring-puckering motion can in general be described by a double minimum potential of the form V ðxÞ ¼ a2 x2 þ a4 x4 ;

ð12Þ

with a2 negative and a4 positive. The analysis of the data previously available had confirmed this type of potential for oxetane [9,15,16]. However, Jokisaari and Kauppinen [9] noticed that the potential of Eq. (12) did not give a satisfactory agreement between the calculated and their observed wavenumbers. Thus, the experimental results of [9] where fitted with a potential including also a term in x6 . Analogously, Lesarri et al. [8] found that they needed such a term in their modeling of the ring-puckering dependence of the rotational constants. In fact, a look at the experimental data of [9,15] shows that the energy Ev of the ring-puckering level corresponding to vrp ¼ v increases faster than aðv þ 12Þ4=3 (a being a constant), which is the asymptotic behavior of the energy eigenvalues of an anharmonic oscillator described by a potential function of the type V ðxÞ ¼ kx4 . For this potential function ðE4 =E3 Þ=½ð4 þ 12Þ4=3 =ð3 þ 12Þ4=3  ¼ 0:9989, and the ratio tends to unity for increasing vrp . The presence of the quadratic term in Eq. (12) makes the calculated levels increase even more slowly than aðv þ 12Þ4=3 . A simple semiclassical treatment, based on a Bohr– Sommerfeld-type integral, shows that at high quantum

G. Moruzzi et al. / Journal of Molecular Spectroscopy 219 (2003) 152–162

159

Fig. 3. The Q branch of the first ring-puckering hot band of oxetane. The FTIR spectrum recorded at 203 K and at ca. 0.9 mbar is shown in the lower trace. The corresponding Fortrat diagram is shown in the upper part of the figure. The continuous lines show the line positions predicted by the Watson parameters of Table 1; at the high-J end of each line the value of the corresponding Kc value is reported. Filled circles correspond to the positions of transitions which could be identified in the FT spectrum, while empty circles correspond to transitions which could be obtained as differences of two Ritz term values.

numbers v the energy eigenvalues of an anharmonic oscillator with a potential function of the type V ðxÞ ¼ kjxjm tend to aðv þ 12Þ2m=ðmþ2Þ . For a potential of the type V ðxÞ ¼ x8 , already ðE6 =E5 Þ=½ð6 þ 12Þ8=5 = ð5 þ 12Þ8=5  ¼ 0:9987. Thus, higher powers than the fourth are certainly needed in the potential describing the ring-puckering motion of oxetane. We found out that even powers up to x8 are needed for a satisfactory fit of our high-resolution experimental Gv values. We thus used a potential of the form V ðxÞ ¼ a2 x2 þ a4 x4 þ a6 x6 þ a8 x8 :

ð13Þ

The Schr€ odinger equation for the ring puckering motion can thus be written ! 4 X h2 d2 2k
w ðxÞ þ a2k x wv ðxÞ ¼ Ev wv ðxÞ; ð14Þ 2l dx2 v k¼1 where l is the effective reduced mass for the ringpuckering motion, x is the ring-puckering coordinate,

and wv ðxÞ and Ev are the eigenfunction and the energy eigenvalue corresponding to the ring-puckering quantum number vrp , respectively. Analogously to the case of the harmonic oscillator, a comparison of the eigenvalues Ev of Eq. (14) to our experimental Gv values cannot lead to a separate determination of the quantities l, a2 , a4 , a6 , and a8 , but only of the ratios A2k ¼

a2k lk

ð15Þ

with k ¼ 1; 2; 3; 4. It is possible to rewrite Eq. (14) in dimensionless form by introducing the dimensionless quantities qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 3
ðkþ1Þ ; n ¼ ; and a2k ¼ A2k 22
k h2k
4 A4 q Wv ¼

Ev T~v ¼ ; e s

ð16Þ

160

G. Moruzzi et al. / Journal of Molecular Spectroscopy 219 (2003) 152–162

where the A2k coefficients are defined in Eq. (15), T~v is the term value in cm
1 , (T~v ¼ Ev =hc) and the quantities sffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffi 4 3 A  6 h2 e 4h ð17Þ q¼ ; and s ¼ ; e ¼ 3 hc 2l A4 16 have the dimensions of a length, an energy and a wavenumber, respectively. The change of variables has been chosen so that a4 ¼ 1. Introducing the quantities of Eq. (16) into Eq. (14) we obtain the dimensionless Schr€ odinger equation   d2 w ðnÞ ¼ a2 n2 þ n4 þ a6 n6 þ a8 n8
Wv wv ðnÞ: 2 v dn

Fig. 4. The deviation of the quartic distortion constants DJ , DJK , DK , dJ , and dK in Ir representation from their vrp ¼ 0 value as a function of the ring-puckering quantum number. The constant c multiplying the function hv is c ¼ 13:7 kHz.

ð18Þ

Being one-dimensional, Eq. (18) can easily be integrated numerically by, for instance, a fourth order Runge– Kutta method. For this, we impose the boundary conditions wv ð0Þ ¼ K;

dwv ð0Þ ¼0 dn

for vrp even;

ð19aÞ

wv ð0Þ ¼ 0;

dwv ð0Þ ¼K dn

for vrp odd;

ð19bÞ

where K is an arbitrary constant, for instance K ¼ 1, which will be adjusted a posteriori by the normalization condition on wv ðnÞ. The eigenvalues Wv are then determined by the condition lim wv ðnÞ ¼ 0:

ð20Þ

n! 1

We have determined the a2k and s coefficients of Eqs. (16) and (17) by minimizing the expression h i2 0 ¼14 vX m~v0 v00
sðWv0
Wv00 Þ v2 ¼ ; ð21Þ 2v0 v00 v0 ;v00

Fig. 5. The deviation of the experimentally determined sextic distortion constants in Ir representation from their vrp ¼ 0 value as a function of the ring-puckering quantum number.

where v0 v00 are the experimental accuracies. We put v00rp ¼ 0 and m~v0 0 equal to the Gv0 values evaluated by the Watsonian fits reported in Table 1 for 0 6 v0 6 5, v00rp ¼ v0rp
1 and m~v0rp ;v0rp
1 equal to the observed origins of the Q branches of [9] for 6 6 v0rp 6 11, and [15] for 12 6 i 6 14. We obtained the values

Table 4 Eigenvalues of the dimensionless ring-puckering Hamiltonian and their conversion to wavenumbers v

Wv

sWv (cm
1 )

v

Wv

sWv (cm
1 )

0 1 2 3 4 5 6 7

0.414799 2.285879 5.455906 9.149449 13.318273 17.875831 22.771251 27.967806

11.731920 64.652323 154.311350 258.777142 376.685481 505.588524 644.047428 791.023448

8 9 10 11 12 13 14 15

33.437887 39.159888 45.116403 51.293089 57.677906 64.260600 71.032318 77.985345

945.735713 1107.573103 1276.043338 1450.740752 1631.324837 1817.505498 2009.032435 2205.687375

G. Moruzzi et al. / Journal of Molecular Spectroscopy 219 (2003) 152–162

161

Fig. 6. The potential well for the ring-puckering motion. The abscissa is the dimensionless coordinate n, while the vertical axis gives the term values in cm
1 . The quantum numbers labeling the ring-puckering eigenstates are indicated on the right side of the frame.

where a2 and a6 turn out to be negative, and a4 and a8 positive. It is interesting to note that the a6 coefficient of [9] was positive. The eigenvalues Wv of Eq. (18) and the corresponding term values sWv obtained for the a2k and s coefficients of Eq. (22) are reported in Table 4. The ring-puckering ground state turns out to be 11.732 cm
1 above the central potential barrier, to be compared to the value of 11.86 (5) cm
1 found in previous work [9]. Fig. 6 shows the potential well, the positions of the eigenvalues with 0 6 vrp 6 5 and the corresponding numerically evaluated eigenfunctions. A comparison of the calculated values with our experimental results, and with the experimental results found previously are shown in Table 5. For consistency with the presentation of the experimental data of previous work, which are given as Q-branch origins, our Gv values are labeled by the corresponding vrp quantum number as upper state (v0 ), and by 0 as quantum number for the lower state (v00 ). With the accuracy of the present data, the limits of usefulness of the power series representation of the potential function have been reached. The term in x8 , introduced into the potential in order to reproduce our high-resolution values with vrp 6 5, leads to a slight but systematic deviation of the predicted values to higher wavenumber at high vrp , as shown by the last column of Table 5.

a2 ¼
1:49204ð8Þ;

5. Conclusions

a4 ¼ 1:0

ðfixedÞ;

a6 ¼
6:7ð2Þ  10
3 ;

ð22Þ

a8 ¼ 7:5ð2Þ  10
4 ; s ¼ 28:118807 cm
1 ;

This work, and the previous work of [1], report the first high-resolution investigation of the ring-puckering spectrum of oxetane in the far-infrared. The spectrum consists of c-type transitions, 7639 of which, interconnecting the lowest six ring-puckering states (including

Table 5 Experimental and calculated wavenumbers of the ring-puckering bands v0

v00

:
1 m~exp v0 v00 (cm )

1 2 3 4 5 6 7 8 9 10 11 12 13 14

0 0 0 0 0 5 6 7 8 9 10 11 12 13

52.92035431 142.57979450 247.04460784 364.95384172 493.85624927 138.42 (6)b 146.94 (5)b 154.65 (5)b 161.73 (7)b 168.25 (7)b 174.45 (10)b 180.4 (2)c 185.5 (5)c 190.1 (5)c

a

This work. Ref. [9]. c Ref. [15]. b

(15)a (17)a (39)a (22)a (25)a

sðWv0
Wv00 Þ (cm
1 )

(Obs. ) calc.) 1000 (cm
1 )

52.92040 142.57943 247.04522 364.95356 493.85660 138.45890 146.97602 154.71226 161.83739 168.47023 174.69741 180.58408 186.18066 191.52693

)0.046 0.36 )0.61 0.28 )0.35 )38.90 )36.02 )62.26 )107.39 )220.23 )247.41 )184.08 )680.66 )1426.93

162

G. Moruzzi et al. / Journal of Molecular Spectroscopy 219 (2003) 152–162

the ground state), have been assigned. The Ritz term values of 4983 energy levels have been obtained. Computer files in ASCII form, containing the wavenumbers of the assigned lines and the term values of the assigned levels, are available via E-mail from one of the authors or from the journal website. From an analysis using a Watson Hamiltonian the spectroscopic constants for the six states considered have been obtained. The dominant effect of the ring-puckering mode on the quartic centrifugal distortion constants, observed in earlier analyses based on less complete data of Creswell and Mills [7] and Lesarri et al. [8], has been reviewed. The ability to determine all sextic constants from the present data, for all six states, has allowed us to determine quartic constants effectively unpolluted by sextic centrifugal distortion contributions. This means that we can now see exactly how good these models of Creswell and Mills and Lesarri et al. are—and they are very good. The small but well defined deviations of the experimental values from the predictions of the models are definitely not due to experimental limitations, and can now be attributed to the incompleteness of the models. This means that one or more of the other normal modes have small admixtures of the ring-puckering motion, which thus also contribute, though weakly, to the centrifugal distortion constants. As was to be expected, an analogous effect of the ring-puckering excitation on the sextic distortion constants has been determined. High-resolution values for the first six Gv parameters are given for the first time, and a potential well for the ring puckering motion containing even powers of the ring-puckering coordinate n up to the eighth power is needed to reproduce their values within the experimental accuracy.

Acknowledgments The authors are indebted to Dr. M. Lock and G. Mellau for help in the operation of the Bruker spectrometer in Giessen. The laboratory work in Giessen was supported in part by the Deutsche Forschungs-

gemeinschaft and Fonds der Chemischen Industrie. G.M. is indebted to the Alexander von Humboldt Foundation for financial support of his visits to Giessen. G.M. feels also indebted to the Free Software Foundation, Inc., for the online availability of the Linux operating system and its C/C++ compiler, which supported all the software used in the present work. G.M. is grateful to A. Di Giacomo, G. Paffuti, and E. DÕEmilio for very interesting and fruitful discussions.

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