Analysis of some combination-overtone infrared bands of 32S16O3

Journal of Molecular Spectroscopy 225 (2004) 109–122 www.elsevier.com/locate/jms

Analysis of some combination-overtone infrared bands of

S O3q

32 16

Arthur Maki,a Thomas A. Blake,b Robert L. Sams,b John Frieh,c Jeffrey Barber,c Tony Masiello,c Engelene t.H. Chrysostom,c Joseph W. Nibler,c,* and Alfons Weberd,e b

a 15012 24th Ave., S.E. Mill Creek, WA 98012-5718, USA Pacific Northwest National Laboratory, P.O. Box 999, Mail Stop K8-88, Richland, WA 99352, USA c Department of Chemistry, Oregon State University, Corvallis, OR 97332-4003, USA d National Science Foundation, Arlington, VA 22230, USA e National Institute of Standards and Technology, Gaithersburg, MD 20899, USA

Received 5 November 2003; in revised form 27 January 2004 Available online 16 March 2004

Abstract Several new infrared absorption bands for 32 S16 O3 have been measured and analyzed. The principal bands observed were m1 þ m2 (at 1561 cm
1 ), m1 þ m4 (at 1594 cm
1 ), m3 þ m4 (at 1918 cm
1 ), and 3m3 (at 4136 cm
1 ). Except for 3m3 , these bands are very complicated because of (a) the Coriolis coupling between m2 and m4 , (b) the Fermi resonance between m1 and 2m4 , (c) the Fermi resonance between m1 and 2m2 , (d) ordinary l-type resonance that couples levels that differ by 2 in both the k and l quantum numbers, and (e) the vibrational l-type resonance between the A01 and A02 levels of m3 þ m4 . The unraveling of the complex pattern of these bands was facilitated by a systematic approach to the understanding of the various interactions. Fortunately, previous work on the fundamentals permitted good estimates of many constants necessary to begin the assignments and the fit of the measurements. In addition, the use of hot band transitions accompanying the m3 band was an essential aid in fitting the m3 þ m4 transitions since these could be directly observed for only one of four interacting states. From the hot band analysis we find that the A01 vibrational level is 3.50 cm
1 above the A02 level, i.e., r34 ¼ 1:75236ð7Þ cm
1 . In the case of the 3m3 band, the spectral analysis is straightforward and a weak Dk ¼ 2, Dl3 ¼ 2 interaction between the l3 ¼ 1 and l3 ¼ 3 substates locates the latter A01 and A02 ‘‘ghost’’ states 22.55(4) cm
1 higher than the infrared accessible l3 ¼ 1 E0 state. Ó 2004 Elsevier Inc. All rights reserved.

1. Introduction We have been engaged in obtaining a complete understanding of the available infrared and Raman spectra of sulfur trioxide, SO3 [1–7]. Thus far this study has concentrated on all the fundamental vibrational states of the most abundant isotopomer, 32 S16 O3 , and of the three other isotopomers that have the same D3h molecular symmetry, 32 S18 O3 , 34 S16 O3 , and 34 S18 O3 . The earlier papers of this series showed that a complete understanding of the m1 CARS spectrum required an understanding of the interactions among the states q

Supplementary data associated with this article can be found in the online version, at doi:10.1016/j.jms.2004.02.008. * Corresponding author. Fax: 1-541-737-2062. E-mail addresses: [email protected], [email protected] (J.W. Nibler), . 0022-2852/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jms.2004.02.008

1000 00 ; 0000 22 ; 0000 20 ; 0100 11 , and 0200 00 . These states are coupled by two Fermi resonance interactions due to the two vibrational potential constants k144 and k122 . Additional important interactions involve the Coriolis interaction between m2 and m4 , and the l-type resonance interaction, due to q4 , between the 0000 20 A01 , and 0000 22 E0 levels. Generally, the results obtained have been in remarkable agreement with parameters predicted from high-level ab initio calculations by Martin [8]. Until now, the strongest band of SO3 that has not been studied is the m1 þ m3 band, ð1011 00 Þ1
ð0000 00 Þ0 , near 2450 cm
1 . As expected from our experience with the CARS spectrum of m1 [1,3], the ð1011 00 Þ1 state is obviously perturbed and the perturbing states must be the states ð0011 20;2 Þ1;1;3 , ð0111 11 Þ0;2 , and ð0211 00 Þ1 . The various interactions involved are the same as were found in the m1 study [3], plus at least one more interaction, the

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vibrational l-type resonance, which occurs when both v3 6¼ 0 and v4 6¼ 0. The band structure is consequently quite complex and the analysis of m1 þ m3 , when completed, will be reported separately. The vibrational l-type resonance is more easily studied in the present analysis of the m3 þ m4 band near 1920 cm
1 . Originally we anticipated that the m3 þ m4 ðA01 þ A02 þ E0 Þ states would be free from any vibrational resonances that do not involve vibrational angular momentum. That expectation has been verified. However, we expected that the Coriolis coupling between m2 and m4 would still be a complicating factor so that the effect of the nearby m2 þ m3 (E00 ) state should also be considered in the analysis even though the the transition ð0111 00 Þ1 –ð0000 00 Þ0 is not infrared allowed and therefore cannot be directly observed. Similarly the A01 and A02 vibrational levels ð0011 11 Þ0 could not be observed as transitions from the ground state. These difficulties were overcome by the observation of the hot band transitions ð0011 11 Þ0 A01 –ð0000 11 Þ1 E0 , ð0011 11 Þ0 A02 –ð0000 11 Þ1 E0 , ð0011 11 Þ2 E0 –ð0000 11 Þ1 E0 , and ð0111 00 Þ1 E00 –ð0100 00 Þ0 A002 near 1390 cm
1 . Finally, the analysis of the m1 þ m2 and m1 þ m4 bands was expected to involve no new interactions and so they should be the simplest new bands to be analyzed after the 3m3 band. This was only partially true.

2. Experimental details Fig. 1 gives an overview of the three bands analyzed in this work. The experimental conditions for obtaining the spectra were similar to those described in [2]. Briefly, the spectra were recorded with a White cell at conditions listed in the caption using a Bruker IFS 120 HR1 Fourier transform spectrometer at the Pacific Northwest National Laboratory facilities. The cell was fitted with AgCl windows and was flushed with SO3 (Aldrich, 99.5%) several times before spectra were recorded. For each measurement, 256 scans were averaged. The entire optical path, except for that of the absorption cell, was evacuated during the measurements. H2 O lines that were used for calibration below 2000 cm
1 [9,10] are due to the residual amounts of this gas present in the evacuated optical path in the spectrometer. The more intense of these can be identified by the negative absorption dips below the baseline, e.g., as in the lower panel of Fig. 1. Similarly, wavenumbers of

1

Certain commercial equipment, instruments, and materials are identified in the paper to adequately specify the experimental procedure. Such identification does not imply recommendations or endorsements by the National Institute of Standards and Technology, or the Pacific Northwest National Laboratory, or the National Science Foundation, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.

absorption lines of residual CO2 gas [10] were used to calibrate the higher frequency region. The calibration uncertainties in the measured wavenumbers range from 0.0001 cm
1 at the lowest frequency to 0.0003 cm
1 at the highest frequency.

3. Analysis of the spectra 3.1. Some general considerations In accord with the usual convention, we shall use upper case L and K as unsigned quantum numbers and lower case as signed quantum numbers. We note that all of the various interactions discussed below couple only levels that have the same values of J and of jk
lj. Also, if the notation or an equation uses  and/ or , then the upper (or lower) sign should be used throughout. In this paper, the analysis includes the m3 þ m4 state, for which the different values for the two vibrational angular momentum quantum numbers, l3 and l4 , are very important. We also wish to set the stage for the eventual analysis of such states as m3 þ 2m4 . In both cases, the state designations to be used for different combinations of l3 and l4 can be confusing. For that reason we shall use a common notation for the different vibrational levels that assumes k P 0. This notation, (v1 ; v2 ; vl33 ; vl44 Þl , allows us to use signed quantum numbers for vibrational angular momentum with the understanding that if the sign of k were reversed, then the signs of all the vibrational angular momentum quantum numbers should also be reversed. If unambiguous, we may leave out the parentheses and the value of l, which is given by l ¼ l3 þ l4 . We may also use unsigned values of the vibrational angular momentum quantum numbers when the sign is either irrelevant or obvious, such as when only one degenerate vibration is involved. When useful, we give the symmetry designation of the vibrational level. With this notation the l ¼ 2 E0 components of the m3 þ m4 state become ð0011 11 Þ2 , or simply ð0011 11 Þ2 . The allowed infrared transition from the ground state is ð0011 11 Þ2 –ð0000 00 Þ0 , or ð0011 11 Þ2 –ð0000 00 Þ0 . The notation would be ð0011 11 Þ0 for the l ¼ 0 A01 and A02 states of m3 þ m4 . Note that there remains some ambiguity because these components represent different linear combinations of l3 and l4 , e.g., ð001þ1 1
1 Þ0  ð001
1 1þ1 Þ0 , where the  sign depends upon the sign of k and whether J is odd or even. With the additional complications encountered in this work we have also changed our notation for the matrix elements from that of our earlier papers. For this work the diagonal matrix elements are represented by the symbol Ei ðk; l3 ; l4 Þ where the subscript i stands for a specific vibrational state and E is given by Eq. (1) below.

A. Maki et al. / Journal of Molecular Spectroscopy 225 (2004) 109–122

111

Fig. 1. FTIR Spectra of SO3 at room temperature in a 19.2 m pathlength White cell. (Top panel) 3m3 band, 430 Pa (3.2 Torr), 0.004 cm
1 Res. (Middle panel) m3 þ m4 band, 270 Pa (2.0 Torr), 0.002 cm
1 Res. (Bottom panel) m1 þ m2 ; m1 þ m4 bands, 130 Pa (1.0 Torr), 0.002 cm
1 Res. In the bottom trace, features accompanied with negative dips below the baseline are due to residual water vapor in the spectrometer.

There are three types of off-diagonal matrix elements. The first type represents purely vibrational coupling matrix elements (cubic or quartic Fermi resonance couplings) and is designated Wmno or Wmnop with no commas separating the subscripts, which indicate the vibrations coupled such as 122 for the coupling between the m1 and 2m2 . The second type of matrix element Wm;n;o has commas separating the subscripts and stands for a Coriolis coupling in which the first subscript indicates the change in k, the second subscript indicates the change in l3 , and the third subscript indicates the change in l4 . Although the subscripts could indicate either a positive or a negative change in k, e.g., W2;1;1 , for simplicity in notation, we have chosen to always indicate a positive change in k. Then the appropriate sign is used

for the changes in the other quantum numbers, e.g., W2;1;1 in the example above. If there is still an ambiguity such as for a zero change in k, then the change in l3 is taken as positive. This will be clearer as each matrix element is defined below. Similar notation will be used for a few higher order Coriolis interactions involving different vibrational states. These interactions are indicated by the third matrix term Cm;n;o , where C is used to distinguish these elements from the more significant and well-established Wm;n;o elements. There is no single reference that gives the Hamiltonian matrix elements and theory for the interactions that are needed for this analysis. We have found to be very useful the papers by Di Lauro and Mills [11] and Cartwright and Mills [12]. The very old exposition by

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Nielsen [13] is also useful. To our knowledge this is the first example for D3h molecules of an analysis of the spectrum involving a state with two different quantum numbers for vibrational angular momentum. 3.2. The m1 þ m2 and m1 þ m4 bands The strong Coriolis interaction that couples the m2 and m4 states also couples the ð1100 00 Þ0 A002 and ð1000 11 Þ1 E0 states. In addition, the Fermi resonances that act through the k122 and k144 vibrational potential constants couple the ð1100 00 Þ0 and ð1000 11 Þ1 states to the ð0000 31 Þ1 , ð0100 20 Þ0 , ð0200 11 Þ1 , and ð0300 00 Þ0 states. Furthermore, there is also the l-type resonance that couples, for instance, the ð0000 31 Þ1 and ð0000 33 Þ3 levels. Fig. 2 shows the relative positions of the various vibrational levels in this region and the different interactions connecting the levels. In principle the Hamiltonian matrix could be very large for all these interacting levels but in the leastsquares analysis it was found that the largest matrix needed in the final fit was the 13  13 matrix displayed in

Fig. 3. Of course, the K 6 J rule resulted in smaller matrices for fitting some low J transitions. For the analyses the diagonal elements of the matrix had the usual form hv1 ; v2 ; v3; v4 ; J ; k; l3 ; l4 jH =hcjv1 ; v2 ; v3 ; v4 ; J ; k; l3 ; l4 i ¼ Eðv; J ; k; l3 ; l4 Þ ¼ Gðv; lÞ þ Bv J ðJ þ 1Þ þ ðCv
Bv Þk 2
2ðCf3 Þv kl3 þ gJ3 v J ðJ þ 1Þkl3 þ gK3 v k 3 l3
2ðCf4 Þv kl4 þ gJ4 v J ðJ þ 1Þkl4 þ gK4 v k 3 l4 2 K 4
DJv J 2 ðJ þ 1Þ2
DJK v J ðJ þ 1Þk
Dv k 3

2

þ HvJ J 3 ðJ þ 1Þ þ HvJJK J 2 ðJ þ 1Þ k 2 þ HvJKK J ðJ þ 1Þk 4 þ HvK k 6 þ splitting terms:

ð1Þ

Here we have used both l3 and l4 because they will be needed for the analysis of the m3 þ m4 band system. The band centers, in the absence of any vibrational resonance, are given by m0 ¼ Gðv; lÞ0
Gðv; lÞ00 :

Fig. 2. Vibrational energy levels interacting directly and indirectly with the 1000 11 state.

ð2Þ

A. Maki et al. / Journal of Molecular Spectroscopy 225 (2004) 109–122

113

Fig. 3. Hamiltonian matrix for m1 þ m2 and m1 þ m4 .

If there is a vibrational resonance involved in the analysis, then the perturbed band center can only be determined by solving the Hamiltonian matrix. In Tables 1–3 we give the vibrational energy levels in wavenumbers with respect to a zero ground state so that Gð0; 0Þ ¼ 0:0 cm
1 and m0 ¼ Gðv; lÞ0 is the ‘‘deperturbed’’ band center. We also give in Table 1 the position of the perturbed band center, mc , calculated for the J ¼ 0, K ¼ 0 transition from the ground state. For all the observed transitions the lower state has been extensively studied and so the fits made use of the lower state constants taken from [6] and given in Table 4. As described in [2,3,6], the K > 0 levels are doubly degenerate, but this degeneracy is lifted by a small coupling between them. This is achieved with the small splitting term added to the diagonal term value expression in Eq. (1). For the K ¼ 3 levels the splitting contribution is given by d3K Dv ½J ðJ þ 1Þ½J ðJ þ 1Þ
2½J ðJ þ 1Þ
6 where dij is the Kronecker symbol and the upper sign is for even J while the lower sign is for odd J . Similarly, the splitting term for K ¼ 2 is given by d2K tv ½J ðJ þ 1Þ½J ðJ þ 1Þ
2 with the same sign convention. The K ¼ 1 splitting, which is required for the m3 þ m4 (L ¼ 2) levels, is given by d1K pv ½J ðJ þ 1Þ: Here again the upper sign applies to even J and the lower sign to odd J . The K ¼ 1 splitting was earlier observed for 2m3 but was accounted for there by means of a Dk ¼ 2, Dl ¼
4 matrix element [2,6]. For the present instance we prefer the use of a simple diagonal

splitting constant because it allows us to use a smaller Hamiltonian matrix. Splittings of levels with K > 3 were not observed in the present work. The off-diagonal matrix elements in the interaction matrix are divided into three categories: vibrational, Coriolis between vibrational states, and Coriolis within a vibrational state. The vibrational off-diagonal matrix elements were: hv1 ;v4 ;J ;k;l4 jH =hcjv1
1;v4 þ 2;J ;k;l4 i ¼ W144 2

1=2

J K ¼ 12½k144 þ k144 J ðJ þ 1Þ þ k144 k 2 fv1 ½ðv4 þ 2Þ
l24 g

ð3Þ

and hv1 ; v2 ; J ; k; l4 jH =hcjv1
1; v2 þ 2; J ; k; l4 i ¼ W122 J ¼ 2
1=2 ½k122 þ k122 J ðJ þ 1Þfv1 ðv2 þ 1Þðv2 þ 2Þg1=2 :

ð4Þ

For the analysis of the m1 þ m4 and m1 þ m2 bands the Coriolis terms that coupled states with different vibrational quantum numbers were the same terms that coupled m2 with m4 , namely: W1;0;1 ¼ hv2 ; v4 ; J ; k  1; l4  1jH =hcjv2 þ 1; v4
1; J ; k; l4 i ¼  ½ðBf24 Þ þ zJ24 J ðJ þ 1Þ þ zK24 kðk  1Þ  ½ðv2 þ 1Þðv4  l4 þ 1Þ

1=2

1=2

½J ðJ þ 1Þ
kðk  1Þ : ð5Þ

For the m1 þ m2 and m1 þ m4 system the only Coriolis terms that were needed for the coupling of different k and l levels within a vibrational state were of the type, W2;0;2 , where W2;0;2 ¼ hv4 ; J ; k; l3 ; l4 jH =hcjv4 ; J ; k  2; l3 ; l4  2i 2

2

¼ 14q4 f½ðv4 þ 1Þ
ðl4  1Þ ½J ðJ þ 1Þ
kðk  1Þ  ½J ðJ þ 1Þ
ðk  1Þðk  2Þg

1=2

:

ð6Þ

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A. Maki et al. / Journal of Molecular Spectroscopy 225 (2004) 109–122

Table 1 Rovibrational constants in cm
1 for levels related to m1 þ m2 and m1 þ m4 Constants m0 mc DC  103 DB  103 DDJ  109 DDJK  109 DDK  109 Cf4 gJ4  107 gK4  107 q4  104 Bf24 zJ24  107 zK24  107 p244  106 t4  109 D2  1013 k144 J k144  105 K  105 k144 k122 K k122  105 c1;0;1  105 No. trans. rms dev. a b

(1000 11 )1

(0000 31 )1 a

1593.6909(21) 1595.9224d )0.57338(12)e )0.984 2(37) 2.74(19) )7.24(43) 4.55(31) )0.084 285(2) )13.3(3) 12.9(3) 5.77(8) 0.20258(14) )5.65(7) 2.52(12)

1589.8110(11) 1588.9871 )0.47510(17) )0.198(4) 2.72(34) )13.9(10) 11.6(7) )0.084838(2) )8.30(10) 7.83(10) 4.26(3) 0.201 07(6) )4.89(9) 3.48(12)

)0.69(12)

4.56(56)

(0000 33 )3 1591.09679(18) 1591.0968 [)0.47510] [)0.198] [2.72] [)13.9] [11.6] [)0.084838] [)8.30] [7.83]

(1100 00 )0

(0100 20 )0

(0100 22 )2

(0200 11 )1

(0300 00 )0

1560.5961(73) 1565.0866 )0.2882(3) 0.048(8) 3.81(30) )4.9(9) 1.2(7)

1557.8772(5) 1557.5105 )0.1881(8) 0.6294(25) 6.2(7) )13.2(8) [6.26]

1558.5186(8) 1558.5186 [)0.1881] [0.6294] [6.2] [)13.2] [6.26] )0.084700(10) )9.14(18) 8.37(21)

1525.6052(30) 1524.1976 0.09390(25) 1.516(5) 4.87(11) [)8.5] [4.1] )0.084411(4) )7.50(17) 7.32(17) 4.27(6) 0.20089(6) )5.37(17) [2.52]

[1492.35]c [1488.23] [0.3747] [2.4369] [5.5] [)7.0] [1.88]

4.600(24) 0.201 00(5) )5.26(4) 3.22(8)

[0.201 07] [)4.89] [3.48]

[0.201 00] [)5.26] [3.22] 1.6(5)

)0.27(72) 0.61(64) )1.6762(10) [)2.7] [2.6] 9.971(9) [)2.51]

)1.604 0(9) )2.63(3) 2.58(3) 9.885(10) [)2.51] 1348 0.00031

b

)7.6(5) 704 0.00033

165f 0.00036

552 0.00024

[)0.45]

150 0.00035

110 0.00035

480 0.00026

Twice the standard deviation is given in parentheses. ‘‘Deperturbed’’ value for A01 . The difference A01
A02 ¼ 0:00024ð15Þ, because of the large uncertainty, should be considered undetermined, see

text. c

Constants given in square brackets were fixed. In many cases the constants were fixed at the same value for different values of l. The mc values are observed (or perturbed) band centers for the transition from the ground state. e The changes in C, B, etc. are changes from the ground state constant, i.e., DC ¼ C(vib. state) ) C(ground state). f Because of the high degree of mixing, it is difficult to give a precise division of which upper state goes with each transition. d

In addition to the terms already used in our previous papers on SO3 [1–7], we considered two new terms in an effort to better fit the data. One of these was a vibrational resonance term, W2244 , that couples the E0 levels (0000 31 )1 and (0200 11 )1 and the A002 levels (0100 20 )0 and (0300 00 )0 W2244 ¼ hv2 ; v4 ; J ; k; l4 jH =hcjv2
2; v4 þ 2; J ; k; l4 i ¼ 2
3=2 k2244 fv2 ðv2
1Þ½ðv4 þ 2Þ2
l24 g1=2 :

ð7Þ

It was found however that inclusion of W2244 decreased the value of the k122 term from about 9.9 to 8.9 cm
1 and it also had a similar but somewhat smaller effect on the k144 term. All three constants were strongly correlated and it was found that eliminating the k2244 term only increased the weighted standard deviation from about 0.59 to about 0.64. This elimination also diminished the uncertainty in the m0 constants. For these reasons, W2244 was not included in the final fits. More important was the second new term, a Coriolis interaction, C1;0;1 , between the levels (0000 31 )1 and (1100 00 )0 . This is represented by C1;0;1 ¼ hv1 ; v2 ; v4 ; J; k; l4 jH =hcjv1
1; v2
1; v4 þ 3; J ; k  1; l4  1i ¼  c1;0;1 ½J ðJ þ 1Þ
kðk  1Þ1=2 :

ð8Þ

Since this matrix element involves a net change of three more vibrational quanta than is the case for the W1;0;1 matrix element, the constant c1;0;1 should be, and is, much smaller than Bf24 . Despite the complexity of the interaction matrix, the initial assignments of the transitions for both m1 þ m2 and m1 þ m4 were easy to make. The fit, however, required a relatively good knowledge of the various interaction constants and the approximate positions of the interacting vibrational levels. Table 1 gives the constants determined from the analysis. In this table we give the changes in the constants from the ground state values so that, for instance, DDJv ¼ DJv
DJ0 . Since the DH terms could not be determined, they were fixed at zero. In the course of the analysis, it became apparent that a number of nominal m1 þ m2 and m1 þ m4 transitions were to upper state levels that actually had a composition with greatest contribution from 3m4 , or m2 þ 2m4 , or 2m2 þ m4 . This was a consequence of mixing of states that was particularly strong in regions of level crossings. Fig. 4 shows a reduced energy plot ðEJ0 ;k
l
EJ00;k
l Þ of the various rovibrational levels versus J value. Following the convention of Herzberg [14], we indicate with a þl label those levels for which kl > 0 and with a
l label those levels with kl < 0.

A. Maki et al. / Journal of Molecular Spectroscopy 225 (2004) 109–122 Table 2 Rovibrational constants in cm
1 for the m2 þ m3 and m3 þ m4 levels 1 1 2

1 1 0

1 0 1

Table 4 Ground state constants (in cm
1 ) for

115

32 16

S O3

Constant

(001 1 )

(001 1 )

(011 0 )

Constant

Value

m0 ð¼ mc Þ r34 J r34  105 DC  103 DB  103 DDJ  109 DDJK  109 DDK  109 Cf3 gJ3  107 gK3  107 Cf4 gJ4  107 gK4  107 q3  104 qJ3  109 q4  104 Bf24 zJ24  107 zK24  107 c1;2;
1  103 p34  106 t3  109 D34  1013 No. trans. rms dev.

1917.67613(2)a

1918.22879(5) 1.75236(7) 0.25(2) )0.75654(14) )1.129(12) 2.23(8) )7.39(20) 5.18(13) [0.0824919]c [)0.724] [1.172] [)0.085481] [)6.20] [5.90]

1884.57459(5)

C B DJ  107 DJK  107 DK  107 HJ  1012 HJK  1012 HKJ  1012 HK  1012 D0  1014

0.17398813(3)a 0.34854333(5) 3.1086(5) )5.4922(6) 2.5688(3) 0.68(1) )2.63(2) 3.24(3) )1.27(2) )0.92(14)

)0.75858(7)b )1.176(12) )0.11(4) )1.18(11) 1.39(8) 0.0824919(8) )0.724(19) 1.172(19) )0.085481(6) )6.20(68) 5.90(68) 1.3653(28) 2.56(12) 3.57(25) 0.19811(52) )4.41(13) 3.34(7) )2.17(4) 8.45(16)

)0.47019(17) )0.433(25) 2.38(9) )2.32(24) 0.32(18) 0.0834449(13) 3.34(4) )2.89(4)

1.453(6) [2.6]

)2.60(4) 2151 0.00019

1.9(11) 615 0.00024

623 0.00025

a

Twice the standard error is given in parentheses. The changes in C, B, etc. are changes from the ground state constant, i.e., DC ¼ C(vib. state) ) C(ground state). c Constants given in square brackets were fixed. In many cases the constants were fixed at the same value for l ¼ 0 and l ¼ 2. b

Table 3 Rovibrational constants in cm
1 for the 3m3 band and comparison with values for m3 and 2m3 Constant

m3

2m3

3m3

m0 ð¼ mc Þ DC  103 DB  103 DDJ  109 DDJK  109 DDK  109 Cf3 gJ  107 gK  107 q3  104 qJ3  109 t3  109 No. trans. rms dev.

1391.52025(3)a )0.59914(3) )1.13061(4) 0.34(1) )0.64(2) 0.37(1) 0.0835290(5) 0.638(6) )0.187(6) 1.361(2) 2.59(5) 2.057(8) 2526 0.00027

2777.87142(7)b )1.19955(8) )2.26333(16) 0.59(7) )1.16(15) 0.68(8) 0.0830205(6) 0.630(9) )0.162(9) 1.238(32) 4.4(16)

4136.38766(8)c )1.80512(15) )3.40029(26) 1.19(18) )2.48(35) 1.42(20) 0.0822946(18) 0.65(4) )0.15(4) 1.139(5) 2.5(4) 4.66(17) 676 0.00040

1139 0.00032

a

Twice the standard error is given in parentheses. Value for m0 ðl ¼ 2Þ. The difference m0 ðl ¼ 2Þ
m0 ðl ¼ 0Þ ¼ 11:466ð17Þ cm
1 . c Value for m0 ðl ¼ 1Þ. The difference m0 ðl ¼ 3Þ
m0 ðl ¼ 1Þ ¼ 22:55ð4Þ cm
1 . b

(Note that this means, for example, degenerate states of m2 þ 2m4 such as k ¼ 4, l ¼
2 and k ¼
4, l ¼ þ2 both occur in the region labeled l ¼
2, i.e., the sign

a

Twice the standard deviation is given in parentheses.

of the label l is not to be taken literally but corresponds to that for positive k.) In the figure the different states are distinguished by color and symbol and the sequences for each jk
lj value are displayed. Only states of common J and jk
lj value can interact and the sequences for jk
lj ¼ 12 are highlighted. Similar reduced energy plots for the m3 þ m4 region, discussed later, are shown in Fig. 5. In both plots, levels shown in red are infrared inaccessible from the ground state but reveal themselves as perturbations to accessible levels, as well as in hot band transitions. Such plots give clear evidence of the many avoided crossings of the energy levels. For instance in the highlighted series for jk
lj ¼ 12 in Fig. 4, at low J the k ¼ 11, l ¼
1 levels of m1 þ m4 ð1000 1
1 Þ
1 are above the 3m4 k ¼ 9, l ¼
3 ð0000 3
3 Þ
3 levels but this order switches for the jk
lj ¼ 15 sequences. As seen in Fig. 2 these two states are not directly coupled but at the crossover point near J ¼ 12, a repulsion exists between them due to two combined interactions; the W144 Fermi resonance between the m1 þ m4 and 3m4 (l ¼
1) E0 levels and the W2;0;2 l-type resonance mixing of the l ¼
1 and
3 levels of 3m4 . The consequence is that these states are mixed and we see some transitions from the ground state to 3m4 l ¼
3 even though these are normally forbidden. A similar situation applies to the pair of states m1 þ m2 and m2 þ 2m4 l ¼
2 where an avoided crossing is apparent at about J ¼ 24 for the highlighted jk
lj ¼ 12 sequence in the bottom part of Fig. 4. The order of levels actually switches between jk
lj ¼ 15 and 18 levels and in this region the states are again heavily mixed, due to the combination of Fermi resonance and l-type resonance. The mixing results in the observation of transitions from the ground state to the ‘‘forbidden’’ 0100 22 E00 levels. We also see over 400 transitions that are primarily due to transitions from the ground state to the 2m2 þ m4 level. These probably borrow intensity from the m1 þ m4 transition because of the W122 Fermi resonance. Due to this and other sources of level mixing, we were able to

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0 00 Fig. 4. Reduced energy ðEJK
EJK Þ diagram showing the states interacting in the region of m1 þ m2 and m1 þ m4 for 32 S16 O3 . The display is of jk
lj progressions vs J for each state; the progressions for jk
lj ¼ 12 are highlighted to show the effect of the interactions. States of at least 50% unique character are labeled and identified with a characteristic symbol and color. Transitions to levels shown in red are not infrared allowed from the ground state. States that do not have any component greater than 50% are denoted as mixed states (+). Not shown are two lower states, 2m2 þ m4 at 1525 cm
1 and 3m2 at 1489 cm
1 , that push the displayed levels upwards due to Fermi interactions.

obtain quite a few transitions involving the 3m4 ; m2 þ 2m4 , and 2m2 þ m4 states. However, there was only a very minor mixing involving the 3m2 levels; this state was therefore not directly observed. Instead, we used the constants previously found [3] for m2 and 2m2 to estimate the constants for 3m2 . 3.3. The m3 þ m4 band A schematic diagram showing the levels and transitions studied as part of this band analysis is given in Fig. 6. The m3 þ m4 vibrational state is in Coriolis resonance with the m2 þ m3 state but the only allowed infrared transition from the ground state is ð0011 11 Þ2 E0
ð0000 00 Þ0 A01 . The three other levels that interact with the ð0011 11 Þ2 level are the ð0011 11 Þ0

0 00 Fig. 5. Reduced energy ðEJK
EJK Þ diagrams for m2 þ m3 and m3 þ m4 . The top panel shows the regular trends in the jk
lj sequences when all off-diagonal interactions are turned off. Most of the m2 þ m3 levels lie below those displayed for m3 þ m4 . The middle panel shows that half the m3 þ m4 levels, those with l3 ¼
1, are pushed up by the nearby m2 þ m3 l3 ¼
1 levels when the W1;0;1 Coriolis interaction is turned on. The bottom panel shows the levels when all interactions are turned on. The result is a splitting of the ð0011 11 Þ0 state into A01 and A02 states separated by about 3.5 cm
1 due to the r34 interaction, hence the avoided crossings in the region of 1918 cm
1 . These states, shown in red, are not infrared accessible from the ground state but can be seen as hot band transitions.

A01 and A02 levels and the ð0111 0Þ1 E00 level. Those four levels are coupled through the Coriolis interaction given by Eq. (5), the l-type resonance given by Eq. (6), a similar l-type resonance involving l3 and q3 , and a vibrational l-type resonance matrix element given by hv1 ;v2 ;v3 ;v4 ;J ; k; l3 ;l4 jH =hcjv1 ;v2 ;v3 ;v4 ;J ; k;l3  2; l4  2i J K J ðJ þ 1Þ þ r34 K 2 ¼ W0;2;
2 ¼ 14½r34 þ r34

 ½ðv3  l3 Þðv3  l3 þ 2Þ ðv4  l4 Þðv4  l4 þ 2Þ1=2 :

ð9Þ

A01

This matrix element results in a separation of the and A02 vibrational levels for ð0011 11 Þ0 . The analysis shows

A. Maki et al. / Journal of Molecular Spectroscopy 225 (2004) 109–122

117

A01 and A02 levels. After many of the allowed ð0011 11 Þ2 –ð0000 00 Þ0 transitions had been fit, it was possible to also fit some of the constants for the A01 and A02 levels and search for them as hot band transitions among the m3 lines near 1390 cm
1 . Eventually we were able to measure over 600 of the allowed hot band transitions ð0011 11 Þ0 –ð0000 11 Þ1 , both the A01 –E0 and A02 –E0 transitions. The hot band transitions could also be verified by means of the lower state combination differences. Although the 0111 00 level is lower than the m3 þ m4 levels by more than 30 cm
1 , its inclusion is crucial to a good fit of the latter due to the strong Coriolis coupling indicated by Eq. (5). Until those levels were measured, the least-squares fits did not give constants that agreed with our expectations. The m2 þ m3 level was eventually observed by means of more than 600 hot band transitions, ð0111 00 Þ1 –ð0100 00 Þ0 . Although the data could be fit quite well without adding any more matrix elements, the values found for q3 better matched the expected value when an additional matrix element was added to the Hamiltonian matrix. Since the value of Bf24 was so well determined, it seemed reasonable to add the similar but higher order term C1;2;
1 ¼hv2 ; v3 ; v4 ; J ; k  1; l3  2; l4  1jH =hcjv2 þ 1; v3 ; v4
1; J ; k; l3 ; l4 i ¼  c1;2;
1 ½J ðJ þ 1Þ
kðk  1Þ1=2 : Fig. 6. Energy level diagram showing the measured transitions used to determine the m2 þ m3 and m3 þ m4 levels.

that r34 is 1.75 cm
1 . This results in a separation of the A01 and A02 levels of 2r34 ¼ 3.5 cm
1 , so its effect must not be underestimated. We find that the A01 levels are above the A02 levels regardless of the sign of r34 .

ð10Þ

Because this matrix element involves a change in l3 as well as l4 , it should be much smaller than the one given by Eq. (5) even though the changes in k and l are the same in both cases. In our least-squares fits the Hamiltonian matrix was truncated at a maximum size of 6  6 although a few lowJ levels had smaller matrices because of the K 6 J rule. The Hamiltonian matrix used in the fit had the form

E1 ðk; l3 ; l4 Þ W2;0;2 W2;2;0 W0;2;
2 C1;2;
1 W1;0;1 W2;0;2 E ðk
2; l ; l
2Þ 0 W 0 W 1 3 4 2;2;0 1;0;1 W2;2;0 0 E ðk þ 2; l þ 2; l Þ W W 0 1 3 4 2;0;2 1;0;1 W0;2;
2 W2;2;0 W2;0;2 E1 ðk; l3 þ 2; l4
2Þ W1;0;1 C1;2;
1 C1;2;
1 0 W1;0;1 W1;0;1 E2 ðk þ 1; l3 þ 2; l4
1Þ W2;2;0 W1;0;1 W1;0;1 0 C1;2;
1 W2;2;0 E2 ðk
1; l3 ; l4
1Þ The initial assignments of the ð0011 11 Þ2 –ð0000 00 Þ0 transitions were quite easy. The strong r R; p P ; r Q, and p Q transitions were obvious and could be verified through ground state combination differences. For K < 18 we also observed the r P and p R transitions. In addition, we were able to assign more than 500 hot band transitions to ð0011 11 Þ2 –ð0000 11 Þ1 , especially near 1400 cm
1 where the m3 spectrum is least cluttered. None of the forbidden ð0011 11 Þ0 –ð0000 00 Þ0 transitions were observed. Initially we had to estimate the location of the

which is appropriate if l3 ¼
1 and l4 ¼ þ1. For different values of l3 and l4 the matrix would look slightly different but would still be 6  6. Here E1 stands for the 0;2 1 ð0011 11 Þ state while E2 stands for the ð0111 00 Þ state. 3.4. The 3m3 band This is a rather straightforward example of a perpendicular band for which the assignments were relatively easy to make, especially since the fundamental and

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Fig. 7. The Q-branch region near the band center for the m3 , 2m3 , and 3m3 regions. The axis for 3m3 is shown at the bottom and the spectra for m3 , 2m3 are at the same scale but with the band centers shifted to be coincident with that of 3m3 .

first overtone had already been analyzed [2,6]. Fig. 7 shows the Q-branch region near the band center for the m3 , 2m3 , and 3m3 regions. Due to nuclear spin statistics, the odd J lines are missing in the K ¼ 0 Q-branch and it is seen that the DB spacings between J , J þ 2 lines increase by 2 and 3 as one goes from m3 to 2m3 to 3m3 , respectively. It may also be noted that, due to the selection rules, the m3 and 3m3 Q-branches are better separated and less congested than is the case for 2m3 . The one complication for 3m3 was the interaction between the ð0031 00 Þ1 E0 and ð0033 00 Þ3 A01 and ð0033 00 Þ3 A02 vibrational levels. That interaction was represented in the Hamiltonian matrix by an off-diagonal term given by the q3 , or W2;2;0 , counterpart of Eq. (6). Since the earlier work allowed us to predict that the ð0031 00 Þ1 E0 level should be about 22 cm
1 below theother two levels, and since we had a good estimate of the expected value of q3 , one could predict ahead of time that the kl < 0 levels would be relatively unperturbed. Even the kl < 16 and kl > 25 levels are not noticeably perturbed. Since any perturbations due to the W2;2;0 term must be very small at low J , the pattern of the perpendicular band transitions could easily be followed for all values of K at low J -values. The analysis of the interaction between the l ¼ 1 and l ¼ 3 levels was aided by the fact that the same constant, q3 , is involved in the interaction between the k ¼ þ1; l ¼ þ1 and the k ¼
1, l ¼
1 levels. As can be seen in Eq. (6), the vibrationally dependent term is different, but the constant q3 is the same. Thus, from our analysis of the r R0 , r P0 , and r Q0 transitions we have an accurate value for q3 which is used in the matrix element

that couples the l ¼ 1 and l ¼ 3 levels and gives the position of the l ¼ 3 sub-state. In the case of the 2m3 band such a direct determination of q3 was not possible, hence the uncertainty in q3 obtained from the 2m3 analysis was larger even though more and better data were available.

4. Results 4.1. The m1 þ m2 and m1 þ m4 bands Most of the constants needed to fit the combination bands between 1500 and 1640 cm
1 had values within 10% of the constants determined earlier from the analyses given in [1–7]. With the aid of Fig. 2, one can see that the ‘‘deperturbed’’ m0 and observed mc band centers show shifts that are in the directions expected from the repulsion of levels subject to a Fermi resonance interaction. The ð1100 00 Þ0 level is perturbed the most, increasing by 4.49 cm
1 because of the push by the lower ð0100 20 Þ0 and ð0300 00 Þ0 states. As expected, m0 and mc are identical for the ð0000 33 Þ3 and ð0100 22 Þ2 states since, as seen in Fig. 2, these levels are not perturbed by any vibrational Fermi interaction, only by rotational perturbations (that are zero for these J ¼ K ¼ 0 reference states). The changes in the various rotational and centrifugal distortion constants are close to the values expected from the appropriate combination of constants for the fundamental levels. For example, DC1þ4 ¼
0:000573 cm
1 for the m1 þ m4 state, only 0.3% smaller

A. Maki et al. / Journal of Molecular Spectroscopy 225 (2004) 109–122

than the sum DC1 þ DC4 ¼
0:000575 cm
1 . A similar calculation for DDJK gives values differing by 22%. Such differences are about what one would expect for the vibrational dependence beyond that given in the above equations. The l-type resonance constant q4 and the Coriolis coupling constant Bf24 are also in good agreement with the values found earlier even though they are highly correlated. The centrifugal corrections to these are more variable and some of the small splitting constants are found to be an order of magnitude larger than the values found for other vibrational states. Even the signs of the splitting constants are different in some cases. This is perhaps not surprising since, as a general rule these higher order constants are more sensitive to the limited selection of higher J and k transitions. They also tend to be more sensitive to omissions in the model used to fit the data, specifically the omission of some weak interactions with levels outside the range included in the Hamiltonian matrix. The J and K dependence of the Fermi resonance terms were poorly determined so the values found for m1 and given in our earlier paper [3] were used in the present analysis. The analysis also assumed that the higher order centrifugal distortion constants, the H terms, were the same in the ground state and in all the other states involved in this analysis. That is to say, in all cases DHJ ¼ HJ0
HJ00 ¼ 0, etc. The DD terms were also assumed to be independent of l, so that, for instance, the fit required that DDJ for the 3m4 state was the same for l4 ¼ 1 and l4 ¼ 3. The A01 and A02 level separation of (0000 33 Þ3 is expected to be much smaller than the A01 –A02 separation of 3.5 cm
1 seen for the ð0011 11 Þ0 levels. Indeed, if we include this separation in the fitting of 3m4 constants, the least squares fit gives a value 0.00024(15) cm
1 ; hence, though not well determined, the separation is essentially zero. 4.2. The m3 þ m4 band Initially the assignments were made on the basis of the constants extrapolated from the earlier work. The fit gave ð0011 11 Þ2 and ð0011 11 Þ0 DC ¼
0:000759 and
0:000757 cm
1 , respectively, nearly identical to the sum
0:000757 cm
1 of the DCÕs for m3 and m4 . The observed values for DB were
0:001176 and
0:001130 cm
1 compared with the sum
0:001206 cm
1 for m3 and m4 . The Cf3 and Cf4 constants were also close to what was expected but the different g terms were not. Because some level crossings are sensitive to the values of B assigned to the two states, ð0011 11 Þ2 and ð0011 11 Þ0 , it was necessary to allow those two states to have different B values. The difference seems inconsequential, 0.000046(18) cm
1 , but the uncertainty is smaller than the difference and the data cannot be fit very well with both states having the same B value. This difference is equivalent to having a c34 term in

119

the expansion of the B rotational constant. In fact, most of the constants were allowed to be different for the L ¼ 0 and 2 states of m3 þ m4 and it is surprising how different the DD terms are. The major constants for the 0111 00 state were also quite close to what was expected, based on the values obtained for m2 and m3 . For both m2 þ m3 and m3 þ m4 bands, the data did not extend to high enough values of J and K to determine any difference between the Hv and H0 constants, consequently the H constants were assumed to have the same values for both the upper and lower states of the transitions. The vibrational l-type resonance constant, r34 ¼ 1:752 cm
1 , is 30% smaller than the value 2.528 cm
1 calculated by Martin [8]. However we are not certain how he defined his r34 ; it could be defined to be the A01 –A02 vibrational separation in which case our r34 would be 28% larger than his ab initio value. From the m0 values, we calculate anharmonic vibrational constants x34 ¼
3:646 cm
1 and x14 ¼
1:348 cm
1 that are quite close to the values calculated by Martin,
3:553 and
1:249 cm
1 respectively. Closely related to the x34 constant is the x34 constant that depends on the l quantum numbers. We find that x34 is
0:2633 cm
1 which is close to the value
0:279 cm
1 calculated by Martin for his constant G34 which we believe is the same as x34 . Experimentally the values of r34 and x34 are not strongly correlated since x34 is given by the average position of the A01 and A02 vibrational levels and r34 is given by the difference in the A01 and A02 vibrational levels. Our value
1:951 cm
1 for x12 has the opposite sign from what Martin calculated, 1.113 cm
1 . This discrepancy may be because we have corrected for the effect of the Fermi resonance or it may just be a consequence of our omission of a k2244 Fermi resonance term. One would expect that the values determined for k122 and k144 , and hence the x12 and x14 constants, should be sensitive to the values used for the Fermi resonance constants. As discussed in Section 4.4 below and also shown in Table 5 the signs of certain off-diagonal constants were not determined by fitting the transitions. In our earlier paper [2] the sign of q3 was chosen to be negative to indicate that for m3 , K ¼ 1, the A01 rotational levels are above the hypothetical A02 rotational levels for even J values and below for odd J values. In the present paper Table 5 Permissible sign combinations for m2 þ m3 and m3 þ m4 parameters Constant

Sign permutations

q3 q4 r34 Bf24 c1;2;
1

+ + + + )

+ + + ) +

) + ) + +

) + ) ) )

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A. Maki et al. / Journal of Molecular Spectroscopy 225 (2004) 109–122

Fig. 8. Q-branch region of the m3 þ m4 band.

we give both q3 and r34 positive signs because of some preliminary work on other bands where those signs cannot be changed. Nevertheless, for m3 þ m4 , K ¼ 0, l ¼ 0, we find that the A01 rotational levels are above the hypothetical A02 rotational levels for even J , and below for odd J . The splitting constant, t3 , is opposite in sign to what was found for m3 . This means that the A01 and A02 rotational levels are reversed, possibly because the vibrational symmetry is E00 rather than E0 . Reduced energy diagrams for the m3 þ m4 region are offered in Fig. 5. The top panel shows the simple patterns expected for the ð001þ1 1þ1 Þþ2 , ð001
1 1
1 Þ
2 (E0 ) and ð001þ1 1
1 Þ0 , ð001
1 1þ1 Þ0 (A01 and A02 ) levels when no off-diagonal interactions of these states with the lower m2 þ m3 levels are included. To the right in the top panel are seen some of the encroaching levels of the nearby, lower ð011
1 0Þ
1 states of m2 þ m3 . The middle panel shows the dramatic effect of turning on the W1;0;1 Coriolis interaction of the m3 þ m4 and m2 þ m3 states. Since the l3 ¼
1 component of the latter is closest to m3 þ m4 , it is seen that the main result is that m3 þ m4 levels with l3 ¼
1 are repelled upwards, i.e., ð001
1 1
1 Þ
2 (E0 ) and ð001
1 1þ1 Þ0 levels move up. In contrast, levels with l3 ¼ þ1 are only mildly perturbed by the W1;0;1 interaction because the l3 ¼ þ1 levels of m2 þ m3 and m3 þ m4 are widely separated. A second dramatic change is seen when all off-diagonal interactions are included (lower panel). The largest effect comes from the r34 coupling which mixes the ð0011 11 Þ0 levels and produces a region of avoided crossings at about 1918 cm
1 . For the jk
lj ¼ 0 sequence indicated with + symbols, the A01 levels with even J values are pushed up and the A02 levels with odd J values are lowered. The effect of the various perturbations on the upper E0 levels can be seen in the r Q3 ; r Q0 ; and p Q3 branches observed for m3 þ m4 (Fig. 8). The r Q3 branch corresponds to transitions from k ¼ 3, l ¼ 0 in the lower

state to k ¼ 4, l ¼ 2 (so jk
lj ¼ 6) in the upper state. The observed shading to higher wavenumber value with increasing J value is because, as mentioned above, upper state levels (labeled l ¼
2 in Fig. 5) are pushed upward by the Coriolis interaction with m2 þ m3 l ¼
1 levels. There is also an interaction with the jk
lj ¼ 6 levels of the ð0011 11 Þ0 states which are both above and below the ð0011 11 Þ2 state. The level pattern for the upper k ¼ 1, l ¼ 2 ðjk
lj ¼ 3Þ levels of the r Q0 branch is ‘‘flatter’’ because of the conflicting interactions. The result is the observed compression of the r Q0 branch. It should be noted that this branch consists of only even J transitions since the odd J levels are missing in the lower k ¼ 0 state. Finally, a very unusual pattern is seen for the p Q3 branch, for which the upper state levels have k ¼ 2, l ¼ 2 (so jk
lj ¼ 0) and the states are indicated with + symbols in Fig. 5. These levels are mixed due to the q3 and q4 coupling between the ð0011 11 Þ2 E0 state and the ð0011 11 Þ0 A01 , A02 states. In particular, the k ¼ 2, l3 ¼ 1, l4 ¼ 1 levels are coupled through q3 to the k ¼ 0, l3 ¼ 1, l4 ¼ 1 levels. For even J the k ¼ 0, l3 ¼ 1, l4 ¼ 1 levels are above ð0011 11 Þ2 and therefore the even J levels are pushed down in wavenumber. For odd J the k ¼ 0, l3 ¼ 1, l4 ¼ 1 levels are below ð0011 11 Þ2 and therefore the odd J levels are pushed up in wavenumber. The q4 coupling has a similar effect. The result is the band heading seen in the J even lines of this Q-branch, as indicated in Fig. 8. In this figure, the calculated positions of the lines agree very well with the observations but the intensity calculations are only approximate since mixing of states was ignored for the latter. The irregular structure made the assignment of this band quite challenging but the virtually exact replication of the transition wavenumbers from the determined parameters offers assurance that the analysis is correct.

A. Maki et al. / Journal of Molecular Spectroscopy 225 (2004) 109–122

4.3. The 3m3 band The parameters determined for the 3m3 band are given in Table 3. The crossing of the ð0031 00 Þ1 E0 and ð0033 00 Þ3 A01 and A02 levels occurs, by chance, almost exactly half-way between the jk
lj ¼ 18 and 21 levels of the l ¼ þ1 and +3 components of these states. This is shown in Fig. 9. For jk
lj ¼ 18, the two levels interacting through the W2;2;0 term are about 1.5 cm
1 apart; for jk
lj ¼ 21 they are about the same distance apart, but in the reverse order. With such a relatively large separation compared to the value of W2;2;0 , this interaction term causes the levels to be displaced by less than 0.02 cm
1 at the highest observed J (J ¼ 44), see the top panel of Fig. 9. The perturbation is, however, fairly obvious since the jk
lj ¼ 18 levels are displaced downward and the jk
lj ¼ 21 levels are displaced upward and the displacements get systematically larger as J increases. The calculation of the interaction uses for the constant q3 an accurate value derived from the K ¼ 1 splitting and therefore gives a good estimate of the rather large separation of the two vibrational energy levels, ð0031 00 Þ1 and ð0033 00 Þ3 . In contrast, the separation of the A01 and A02 vibrational levels is poorly determined; a preliminary least-squares fit gave 1.7 cm
1 , but with an uncertainty of 1.3 cm
1 , which we take as an indication that the separation cannot be determined. In fact, we believe that the separation is small because, as mentioned earlier, the similar separation for the

121

(0000 33 Þ3 A01 and A02 levels was smaller than 0.001 cm
1 . It should be noted that we are here referring to the vibrational separation of the levels which applies even for J ¼ 0 and K ¼ 0. With the analysis of 3m3 we now have a nice series of constants for the fundamental, and first and second overtones of a relatively unperturbed vibrational state. Table 3 gives the constants for all three vibrational states, m3 , 2m3 , and 3m3 . As expected, the parameters show monotonic and nearly equal changes with increasing values of v3 . 4.4. Signs of the off-diagonal constants In this paper, as in our earlier papers, the signs of some of the off-diagonal constants are fixed by the mathematics of the least-squares analysis and the signs of others are undetermined. In the latter case the relative signs of two or more constants may be determined even though their signs can be reversed in unison. For instance, in the analysis of the 3m3 band the signs of q3 and qJ3 may be either positive or negative but they both must have the same sign. This was also found to be true in our earlier analysis of the m3 and 2m3 bands. On the other hand, in all the cases that we have studied, the value of q4 must be positive and this has nothing to do with any convention we might have used in assigning the transitions. The difference between m3 and m4 is, of course, due to the Coriolis

Fig. 9. Effect of perturbations for 32 S16 O3 . (Top figure) The effect of turning off the W2;2 coupling of the l ¼ 1 and l ¼ 3 states of 3m3 . The figure applies to the observed transitions for kl > 0. The Obs. ) Calc. values are negative and increase up to jk
lj ¼ 18 and are positive for jk
lj > 18. (Bottom figure) Reduced energy level diagram for the J ¼ 44 levels of 3m3 . The level crossing for kl > 0 occurs between jk
lj ¼ 18 and 21.

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A. Maki et al. / Journal of Molecular Spectroscopy 225 (2004) 109–122

coupling between m2 and m4 , which can never be ignored. Surprisingly, however, the Coriolis coupling terms Bf24 ; zJ24 , and zK24 do not have fixed signs, but in all cases studied thus far, their signs may be reversed in unison without affecting the rms deviation of the fit and without affecting the values or signs of any other constants, including q4 . In the present work, we have used a positive value for Bf24 simply to be consistent with earlier papers. Similarly, in the analysis of the m1 þ m4 state and the other associated states as described above, the signs of k144 and k122 are not fixed although the fit shows that they must be opposite in sign. In this paper as well as in our earlier papers we have chosen to use the signs that Martin calculated [8], especially since the values agree so well with his ab initio values. When both v3 6¼ 0 and v4 6¼ 0 the situation is more complicated with regard to the signs of q3 and r34 . In the analysis of the data for the m3 þ m4 and m2 þ m3 states the signs of q3 and r34 are not fixed, but they must both have the same sign. If the sign of one is changed, then the sign of the other must also be changed. Table 5 gives the four sign permutations, out of a possible 25 , that have the lowest, and same, rms deviation for the least-squares fit of the present measurements. In this table it is assumed that the higher order J and K dependent terms such as zJ24 and zK24 maintain the same sign relationship to the main term, Bf24 , as given in Table 2. For K ¼ 0, l ¼ 0, regardless of the sign given to the r34 constant, the analysis shows that the A01 rotational levels of ð0011 11 Þ0 are above the hypothetical A02 rotational levels for even J values and below for odd J values. This means that the A01 vibrational level is above the A02 vibrational level for the ð0011 11 Þ0 state. In our past papers we have given q3 a negative sign to be consistent with the convention we used for the signs of the different splitting constants. However, in our preliminary fits of the m1 þ m3 band, which will be reported later, it appears that the sign of q3 should be positive. In that case r34 should also be positive and so we have tentatively used positive signs for those constants in Tables 2 and 3. We should like to emphasize, however, that the present analysis does not allow us to give either of these constants a sign.

Acknowledgments The research described here was performed, in part, in the Environmental Molecular Sciences Laboratory, a national scientific user facility sponsored by the Department of EnergyÕs Office of Biological and Environmental Research and located at Pacific Northwest National Laboratory.

References [1] E.t.H. Chrysostom, N. Vulpanovici, T. Masiello, J. Barber, J.W. Nibler, A. Weber, A. Maki, T.A. Blake, J. Mol. Spectrosc. 210 (2001) 233–239. [2] A. Maki, T.A. Blake, R.L. Sams, N. Vulpanovici, J. Barber, E.t.H. Chrysostom, T. Masiello, J.W. Nibler, A. Weber, J. Mol. Spectrosc. 210 (2001) 240–249. [3] J. Barber, E.t.H. Chrysostom, T. Masiello, J.W. Nibler, A. Maki, A. Weber, T.A. Blake, R.L. Sams, J. Mol. Spectrosc. 216 (2002) 105–112. [4] J. Barber, E.t.H. Chrysostom, T. Masiello, J.W. Nibler, A. Maki, A. Weber, T.A. Blake, R.L. Sams, J. Mol. Spectrosc. 218 (2003) 197–203. [5] J. Barber, E.t.H. Chrysostom, T. Masiello, J.W. Nibler, A. Maki, A. Weber, T.A. Blake, R.L. Sams, J. Mol. Spectrosc. 218 (2003) 204–212. [6] S.W. Sharpe, T.A. Blake, R.L. Sams, A. Maki, T. Masiello, J. Barber, N. Vulpanovici, J.W. Nibler, A. Weber, J. Mol. Spectrosc. 222 (2003) 142–152. [7] T. Masiello, J. Barber, E.t.H. Chrysostom, J.W. Nibler, A. Maki, A. Weber, T.A. Blake, R.L. Sams, J. Mol. Spectrosc. 223 (2004) 84–95. [8] J.M.L. Martin, Spectrochim. Acta Part A 55 (1999) 709–718. [9] R.A. Toth, J. Opt. Soc. Am. B 8 (1991) 2236–2255. [10] HITRAN 96 listing in L.S. Rothman, C.P. Rinsland, A. Goldman, S.T. Massie, D.P. Edwards, J.-M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J.-Y. Mandin, J. Schroeder, A. McCann, R.R. Gamache, R.B. Watson, K. Yoshino, K.V. Chance, K.W. Jucks, L.R. Brown, V. Nemtchinov, P. Varanasi, J. Quant. Spectrosc. Radiat. Transfer 66 (1998) 665–710. [11] C. Di Lauro, I.M. Mills, J. Mol. Spectrosc. 21 (1966) 386–413. [12] G.J. Cartwright, I.M. Mills, J. Mol. Spectrosc. 34 (1970) 415–439. [13] H.H. Nielsen, in: S. Fl€ ugge (Ed.), Handbuch der Physik, vol. 37, Part I, Springer, Berlin, 1959, pp. 173–313. [14] G. Herzberg, Infrared and Raman Spectra of Polyatomic Molecules, Van Nostrand, New York, 1945, p. 403.