The rotational spectrum of the SH+ radical in its X3Σ− state

Journal of Molecular Spectroscopy 255 (2009) 68–71

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The rotational spectrum of the SH+ radical in its X3R state John M. Brown a,*, Holger S.P. Müller b a b

The Physical and Theoretical Chemistry Laboratory, Department of Chemistry, South Parks Road, Oxford, Oxfordshire, England OX1 3QZ, United Kingdom I. Physikalisches Institut, Universität zu Köln, 50968 Köln, Germany

a r t i c l e

i n f o

Article history: Received 10 January 2009 Available online 27 February 2009 Keywords: Free radical Rotational spectroscopy Laser magnetic resonance Infrared spectroscopy Sulfoniumylidene

a b s t r a c t All the available data on the rotational energy levels of the SH+ (sulfoniumylidene) radical in the v = 0 and 1 levels of the X3R ground state have been subjected to a single, weighted least-squares fit to determine an improved set of molecular parameters for this molecule. The results have been used to calculate the rotational spectrum of the SH+ radical in the v = 0 and 1 levels up to the N = 4–3 transition. Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction The open-shell molecular ion SH+ is a likely candidate for detection in the interstellar medium. It is predicted to have an abundance similar to that of CH+ in regions where magnetohydrodynamic shocks have occurred [1]. In such a gas, it is suggested that SH+ is formed in endothermic reactions that can proceed because of the high translational temperatures arising from the shock:

Sþ þ H2 ! SHþ þ H þ

þ

H2 S þ H ! SH þ H2 :

ð1Þ ð2Þ

+

SH is also predicted to be abundant in UV or X-ray photon-dominated regions where the hot, highly ionised gas drives its formation by reactions such as [2]:

S2þ þ H2 ! SHþ þ Hþ

ð3Þ

in addition to reactions (1) and (2). Accurate frequencies of transitions in the rotational spectrum of SH+ in its X3R state are required to confirm the presence of the molecule in astrophysical sources. With a rotational constant of 9.13 cm1, these transitions fall in the sub-millimetre and farinfrared regions. Although four of the fine and hyperfine components of the N = 1–0 transition have been measured directly by Savage et al. [3], the major study of its rotational spectrum has been carried out by Hovde and Saykally [4] using the laser magnetic resonance (LMR) technique. In these experiments, individual

MF transitions are tuned into resonance with a fixed-frequency, far-infrared laser by application of a variable magnetic field. Rotational transitions of SH+ in the v = 0 and 1 vibrational levels up to N = 3–2 were detected in this manner. However, despite the data being fitted to a full effective Hamiltonian for a molecule in a 3  R state, the extrapolation to zero-field frequencies using this model was not carried out. Realising this, we have fitted all the available data on the rotational transitions of SH+ in its X3R state together to determine an improved set of molecular parameters. The LMR study covered SH+ in both v = 0 and v = 1 levels. Accordingly, we have included the measurements of the fundamental band recorded in the infrared spectrum by Brown et al. [5] and by Civis et al. [6] using velocity modulation diode laser spectroscopy, in the fit. In early work on SH+, Rostas et al. [7] recorded and analysed the A3P–X3R emission spectrum. Since these measurements are significantly less accurate than the other spectroscopic studies, we have not included them in the fit. We have used the molecular parameters determined in the fit to predict the zero-field frequencies of the individual hyperfine transitions, along with estimated uncertainties in these calculated quantities. 2. Procedure and results All the available data on the rotational energy levels of SH+ in the v = 0 and v = 1 levels of the X3R state have been fitted with the standard effective Hamiltonian [8–10]:

Heff ¼ Hrot þ Hss þ Hsr þ Hhfs þ Hzeem * Corresponding author. Tel.: +44 1865 275403; fax: +44 1865 275410. E-mail addresses: [email protected], [email protected] (J.M. Brown). 0022-2852/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jms.2009.02.017

ð4Þ

where Hrot represents the rotational kinetic energy including centrifugal distortion, Hss is the electron spin–spin coupling term,

69

J.M. Brown, H.S.P. Müller / Journal of Molecular Spectroscopy 255 (2009) 68–71 Table 1 Molecular parameters for SH+ in the X3R state. Parametera B0 B1  B0 D0 D1  D0 H0 k0 k1  k0 kD0

c0 c1  c0 cD0 bF (1H) bF1  bF0 c (1H) gS gl gr gN (1H)

m10 a b c d e f g h

Present work b

273 809.454 (65) 8547.44 (70) 14.672 2 (71) 0.148 0 (77) 0.4587  103 g 171 239.14 (76) 609.4 (64) 0.68 (25) 4975.08 (20) 128.8 (17) 0.555 (44)  103 56.904 (93) 2.7 (12) 33.44 (75) 2.002 09 (12) 0.866 (14)  102 0.583(51)  103 5.585 695 h 2449.238 41 (55)

Brown et al.c

Civis et al.d

Hovde and Saykallye

Savage et al.f

273 818(7) 8543.5 (113) 15.38 (42) 0.36 (53) – 171 180 (100) 639 (137) – 4997 (30) 144 (40) – – – – – – – – 2449.237 1 (21)

273 813.3 (39) 8548.6 (49) 14.654 (33) 0.165 (37) – 171 289 (72) 636 (96) – 4953.8 (165) 132.2 (229) – – – – – – – – 2449.238 4 (12)

273 808.4 (7) 8550.8 (15) 14.502 (38) 0.144 (48) – 171 235.5 (39) 577.5 (117) – 4972.7 (13) 137.6 (31) – 56.83 (24) 2.81 (28) 33.09 (39) 2.001 70 (5) 0.908  102 g 0.70(11)  103 0.0 2449.221 7 (8)

273 808.4 g – 14.502 g – – 171 234.04 (11) – – 4972.52 (5) – – 56.87 (8) – 33.4 (7) – – – – –

Value in MHz where appropriate; the vibrational interval is given in cm1. The number in parentheses is one standard deviation of the least-squares fit, in units of the last quoted decimal place. Ref. [5]. Ref. [6]. Ref. [4]. Ref. [3]. Parameter constrained to this value in the fit. Nuclear spin g-factor in nuclear magnetons.

Hsr describes the electron spin–rotation coupling, Hhfs represents the proton hyperfine interactions and Hzeem the effects of the external magnetic field. The Hamiltonian matrix was constructed with all non-zero elements in a basis set truncated at DN = ±2; extension to DN = ±4 produced changes on the order of 1 kHz in the calculation of the magnetic resonance frequencies. The data set in the least-squares fit consisted of the sub-millimetre wave measurements by Savage et al. [3], the far-infrared LMR measurements by Hovde and Saykally [4] and the velocity modulation infrared measurements by Brown et al. [5] and by Civis et al. [6]. Each data point was weighted as the inverse square of the estimated experimental uncertainty; the uncertainties are 75 kHz for the sub-millimetre wave frequencies, 1.5 MHz for the far-infrared LMR measurements and 60 MHz for the infrared data. Values for all the parameters listed in Table 1 were determined in the least-squares fit with one exception. The sextic centrifugal distortion parameter H0 was only barely determined. It was therefore estimated from the expression [11]:

i  h H0  He ¼ 2De =3x2e 12B2e  ae xe

ð5Þ

to be 0.4587  103 MHz and constrained to this value in the fit; the harmonic vibrational wavenumber xe was determined from the work of Rostas et al. [7] to be 2546.69 cm1. The parameter H0 was not included in the fits by previous workers but its participation improved the overall standard deviation of fit because the infrared data [5,6] contain moderately high N values (up to 10). The data set was subjected to a single, weighted least-squares fit (the hyperfine structure was not fitted separately as it was in Refs. [3,4]). The quality of the fit was completely satisfactory with a standard deviation of fit relative to the estimated experimental uncertainties equal to 1.050. The standard deviations of the individual sections of the data were 12.9 kHz (sub-millimetre), 1.16 MHz (far-infrared) and 67.8 MHz (mid-infrared). These values correspond reasonably well to the estimated uncertainties. The sub-millimetre data perhaps fit better than deserved; there are

only four transition frequencies in this set that determine four spectroscopic parameters very well. The values for the parameters determined in the fit are given in Table 1 together with those from Brown et al. [5], Civis et al. [6], Hovde and Saykally [4] and Savage et al. [3] for comparison. It can be seen that the values obtained in the present fit are a significant improvement over those reported earlier simply because all the data have been fitted together. The fit of the (1, 0) vibration–rotation data is considerably improved by the inclusion of two additional centrifugal distortion parameters, kD and particularly cD, which were not considered in previous work. Consequently, the values determined for k and c are somewhat different from those published earlier. Even though kD has not been determined with great significance, it is probably of the right order of magnitude (kD/k is usually considerably smaller than D/B). In addition, we have been able to determine a value for the anisotropic correction to the electron spin g-factor, gl, of 0.866(14)  102, very close to the value of 0.908  102 predicted by Curl’s formula [12]. This parameter was constrained in the previous fit of the LMR data. The values obtained for the various parameters of SH+ are close to those reported by Hovde and Saykally [4]. Their interpretation of the rotational constant and the hyperfine parameters in terms of molecular structure therefore remain valid and will not be repeated here. It is interesting to compare the molecular parameters of SH+ with those of the isoelectronic species PH in its X3R state. For example, when the spin–spin parameter k of SH+ is scaled in proportion to the square of the spin–orbit coupling constants, a value of 88.69 GHz is obtained; the value obtained experimentally for PH is 66.26 GHz [13,14]. Again, the value for the proton dipolar hyperfine parameter c, when scaled in proportion to the 1/ria3 values, is 29.6 MHz; the actual value of c for PH is 19.41 MHz [13,14]. It can be seen from these results that there is a significant difference in the electronic wave functions of these two molecules, probably attributable to the residual charge on SH+. The parameter set given in Table 1 was used to calculate the zero-field frequencies of the rotational transitions (with the proton hyperfine structure included) of SH+ in the v = 0 and v = 1 levels up

70

J.M. Brown, H.S.P. Müller / Journal of Molecular Spectroscopy 255 (2009) 68–71

Table 2 Calculated rotational transition frequencies of the SH+ radical in the m = 0 level of the X 3R state. N

J

1

0 2

1

0 2

1 3

2

1 2

1

3

2 4

3

2

4

3 5

4

3 5 4

3

a b c d e

F0

1e 2½ 1½ 1½ 1e 1½ 1½ ½ ½ 1 ½ ½ 2 3½ 2½ 2½ 1 2½ 1½ 1½ e 0 1½ ½ 2e 2½ 1½ 2½ 1½ 1 1½ ½ 1½ ½ 3e 4½ 3½ 3½ 2 3½ 2½ 2½ 1 2½ 1½ 1½ 4 5½ 4½ 4 4½ 3 4½ 3½ 3½ 2 3½ 2½ 3½

F00 a

m (MHz)

Obs–calc (MHz)

1½ ½ 1½ ½ 1½ ½ 1½ 1½ ½ 2½ 1½ 2½ 1½ ½ 1½ ½ ½ 2½ 1½ 1½ 2½ 1½ ½ ½ 1½ 3½ 2½ 3½ 2½ 1½ 2½ 1½ ½ 1½ 4½ 3½ 4½ 3½ 2½ 3½ 2½ 1½ 3½

526 047.947 (77)c, d 526 038.722 (78)d 526 124.975 (78)d 683 336.51 (53) 683 422.76 (51) 683 358.06 (56) 683 448.72 (58) 345 929.810 (80)d 345 843.573 (136) 1 082 911.91 (43) 1 082 908.87 (43) 1 082 985.88 (44) 1 094 767.24(68) 1 094 755.71 (69) 1 094 781.67 (74) 1 230 626.09 (112) 1 230 565.82 (114) 1 252 142.03 (61) 1 252 079.45 (63) 1 252 065.02 (61) 1 252 156.46 (61) 893 133.16 (85) 893 046.92 (86) 893 107.19 (95) 893 072.88 (94) 1 632 518.0 (10) 1 632 516.5 (10) 1 632 590.5 (11) 1 641 268.2 (13) 1 641 263.9 (13) 1 641 278.3 (13) 1 662 774.3 (16) 1 662 771.9 (16) 1 662 711.7 (16) 2 179 367.2 (20) 2 179 366.3 (20) 2 179 438.8 (20) 2 186 715.2 (23) 2 186 712.9 (23) 2 186 723.0 (23) 2 198 559.8 (31) 2 198 558.5 (31) 2 198 495.9 (31)

0.019 0.010 0.001

0.006

Line strengthb

F0

F00 a

m (MHz)

2½ 1½ 1½ 1½ 1½ ½ ½ ½ ½ 3½ 2½ 2½ 2½ 1½ 1½ 1½ ½ 2½ 1½ 2½ 1½ 1½ ½ 1½ ½ 4½ 3½ 3½ 3½ 2½ 2½ 2½ 1½ 1½ 5½ 4½ 4½ 4½ 3½ 3½ 3½ 2½ 3½

1½ ½ 1½ ½ 1½ ½ 1½ 1½ ½ 2½ 1½ 2½ 1½ ½ 1½ ½ ½ 2½ 1½ 1½ 2½ 1½ ½ ½ 1½ 3½ 2½ 3½ 2½ 1½ 2½ 1½ ½ 1½ 4½ 3½ 4½ 3½ 2½ 3½ 2½ 1½ 3½

509 339.51 (54)c 509 329.80 (62) 509 420.11 (162) 666 144.0 (35) 666 234.3 (32) 666 171.4 (32) 666 261.7 (34) 329 832 (14) 329 742 (14) 1 048 965.8 (14) 1 048 962.6 (14) 1 049 043.2 (20) 1 060 582.4 (30) 1 060 570.2 (30) 1 060 597.6 (31) 1 196 346.9 (19) 1 196 284.0 (20) 1 217 477.18 (99) 1 217 411.82 (108) 1 217 396.6 (14) 1 217 492.4 (13) 859 944.6 (150) 859 854.3 (151) 859 917.1 (150) 859 881.7 (150) 1 581 439.2 (24) 1 581 437.7 (24) 1 581 515.0 (28) 1 589 999.6 (41) 1 589 995.1 (41) 1 590 010.3 (41) 1 611 120.0 (56) 1 611 117.5 (56) 1 611 054.6 (57) 2 111 193.2 (33) 2 111 192.2 (33) 2 111 268.0 (35) 2 118 373.7 (48) 2 118 371.2 (48) 2 118 381.9 (48) 2 129 979.4 (65) 2 129 978.0 (65) 2 129 912.7 (66)

Line strengthb SF 0 F

0

2

1

3

2

1

1

0

1

3

2

2

1

1

0

2

2d

1

1

4

3

3

2

2

1

5

4

4

3

5 4

4 3

3

2

c d

ð7Þ

1d

3

a

ð6Þ

2

4

b

where the quantity on the right hand side of the equation is the reduced matrix element of the rotation matrix [10,16] and g stands for subsidiary quantum numbers. The intensity of a line in absorption can be obtained by multiplying the line strength by the square of the dipole moment l for SH+ (1.285 Debye [17]), by the transition frequency and by the population difference between the lower and upper states. The Einstein A-coefficient for spontaneous emission from state i to j can also be calculated from the line strengths by use of the relation 3

J

1

to N = 4–3. The variance–covariance matrix from the fit was used to calculate the estimated standard error (1r) of each transition [15]. The results are given in Tables 2 (v = 0) and 3 (v = 1). The computed line strengths SF0 F, which are also included in the tables, can be used to assess the relative intensities of individual transitions. The line strength is defined by

Ai!j ¼ ð16p3 m3ij =3e0 hc Þ1 Sij l2

N

SF 0 F 2.0561 1.1421 0.2286 0.1890 0.9441 0.3778 0.1887 0.5780 0.2889 3.2222 2.2555 0.1612 1.8001 1.0001 0.1998 0.7554 0.3778 0.5026 0.3233 0.0359 0.0359 0.7225 0.2889 0.1444 0.1445 4.2975 3.3152 0.1229 3.0477 2.1334 0.1522 2.1192 1.1774 0.2356 5.3406 4.3516 0.0990 4.1667 3.2144 0.1189 3.2474 2.2732 0.1625

Coupling scheme: J = N + S; F = J + I where I is the 1H nuclear spin. For definition, see Eq. (6). Estimated standard error (1r) in units of the last quoted decimal place. Transition measured by Savage et al. [3]. Transition studied directly in the far-infrared LMR experiment [4].

 2 SF 0 F ¼ jhg0 F 0 jjDð1Þ :q ðxÞ jjgFij ;

Table 3 Calculated rotational transition frequencies of the SH+ radical in the v = 1 level of the X 3R state.

2.0577 1.1430 0.2288 0.1880 0.9390 0.3758 0.1878 0.5821 0.2909 3.2229 2.2560 0.1613 1.8001 1.0002 0.1998 0.7513 0.3758 0.5009 0.3222 0.0358 0.0358 0.7277 0.2909 0.1454 0.1455 4.2979 3.3154 0.1229 3.0477 2.1334 0.1522 2.1178 1.1766 0.2354 5.3408 4.3517 0.0990 4.1667 3.2144 0.1189 3.2468 2.2728 0.1624

Coupling scheme: J = N + S; F = J + I where I is the 1H nuclear spin. For definition, see Eq. (6). Estimated standard error (1r) in units of the last quoted decimal place. Transition studied directly in the far-infrared LMR experiment [4].

It can be seen that the rotational frequencies are now known to better than a few MHz for the v = 0 transitions and a slightly larger figure for v = 1. Such a precision is high enough to permit a definitive identification of SH+ in astronomical sources. Predictions of the rotational transitions of SH+, similar to those presented here, are also provided in the Cologne Database for Molecular Spectroscopy, CDMS [18,19]. The database already contained a prior entry (tag 33505) but with less accurately predicted transition frequencies than those given in the present paper. References [1] T.J. Millar, N.G. Adams, D. Smith, W. Lindinger, H. Villinger, Mon. Not. R. Astron. Soc. 221 (1986) 673–678. [2] N.P. Abel, S.R. Federeman, P.C. Stancil, Astrophys. J. 675 (2008) L81–L84. [3] C. Savage, A.J. Apponi, L.M. Ziurys, Astrophys. J. 608 (2004) L73–L76. [4] D.C. Hovde, R.J. Sayally, J. Chem. Phys. 87 (1987) 4332–4338. [5] P.R. Brown, P.B. Davies, S.A. Johnson, Chem. Phys. Lett. 132 (1986) 582–584. [6] S. Civiš, C.E. Blom, P. Jensen, J. Mol. Spectrosc. 138 (1989) 69–78. [7] J. Rostas, M. Horani, J. Brion, D. Daumont, J. Malicet, Mol. Phys. 52 (1984) 1431– 1452.

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