First determination of isotopically invariant parameters of the negative ion—Hydrogen sulfide anion (SH−)

Journal of Molecular Spectroscopy 249 (2008) 117–120

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First determination of isotopically invariant parameters of the negative ion—Hydrogen sulfide anion (SH) Zdeneˇk Zelinger a,*, Agnes Perrin b, Michal Strˇizˇík c, Pavel Kubát a a

´ Institute of Physical Chemistry, v.v.i., Academy of Sciences of the Czech Republic, Dolejškova 3, 182 23 Prague 8, Czech Republic J. Heyrovsky Laboratoire Interuniversitaire des Systemes Atmospheriques (LISA-UMR7583), CNRS, Universites Paris 12 et Paris 7, 61 Av. du Général de Gaulle, 94010 Creteil cedex, France c ´škovice, 700 30 Ostrava, Czech Republic VŠB – Technical University of Ostrava, Faculty of Safety Engineering, Lumírova 13, Ostrava-Vy b

a r t i c l e

i n f o

Article history: Received 9 December 2007 In revised form 24 March 2008 Available online 29 March 2008 Keywords: Hydrogen sulfide anion Isotopically invariant parameters

a b s t r a c t The first isotopically invariant Dunham analysis of a negative ion (the hydrogen sulfide anion SH) that uses all available infrared and sub-millimeter wave experimental data on 32SH, 33SH, 34SH, and 32SD yields accurate information on Born–Oppenheimer breakdown parameters. The potential constants of expansion of the potential function up to the sixth order were calculated. Differences between the construction of the potential function based on the Morse potential and that on the base of the power series expansion are shown. Ó 2008 Elsevier Inc. All rights reserved.

1. Introduction The energy levels of molecular vib-rotors are often calculated in terms of the Dunham potential energy expression [1]. Born– Oppenheimer-breakdown corrections for diatomic molecules were discussed on this basis [2–4]. The concept of isotopically invariant parameters describing spectroscopic features of a molecule seems to be a very efficient tool, especially when demonstrating the breakdown of Born–Oppenheimer approximation. So far performed studies of isotopically invariant molecular ion parameters were only concerned with positive ions as ArH+, HeH+, KrH+[5–9]. In this paper, we applied calculation of isotopically invariant Dunham parameters to a negative ion—the hydrogen sulfide (SH) anion. Accurate spectroscopic information in terms of Dunham coefficients is presented—it is of importance particularly in view of the recent revival of high resolution anion spectroscopy [10]. Negative molecular ions can be generated in the plasma of a gas discharge. In that case, the concentration of negative ions is mostly of an order of magnitude lower than that of positive ions, because the negative part of the charge is generally carried by free electrons, rather than by molecules capable of binding an additional electron. However, anions can be detected thanks to special modulation techniques. The high-resolution rotational–vibrational spectrum of SH was measured with a diode laser spectrometer using the velocity modulation technique [11]. Consequently, the spectroscopic studies were extended employing an FT spectrometer [12]. We carried out spectroscopic studies of the deuterated species, SD [13,14]. Finally, the spectroscopic studies of SH and SD were performed in a sub-millimeter wave region [15]. All * Corresponding author. Fax: +420 286582307. E-mail address: [email protected] (Z. Zelinger). 0022-2852/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jms.2008.03.010

these spectroscopic studies were based on a difference in directions of motion of positive and negative ions along the electric field of the glow discharge. In fact, this is just the method of the ion velocity modulation often used in spectroscopic studies of small ions [16–18]. Our final analysis includes all the high-resolution spectroscopic data describing SH and SD [11–15]. A simultaneous fit of all the isotopic data (see Table 1) provides information related to the validity of Born–Oppenheimer approximation. By using Dunham parameters Ykl, the vibration-rotation term values are expressed in power series as follows: X Y kl ðv þ 1=2Þk ½JðJ þ 1Þl : ð1Þ Ekl ¼ kl

The Born–Oppenheimer approximation used in [1] assumes ignoring of the inertia of electrons. It means that the electrons instantaneously follow a motion of the nuclei. The inversion of the principal correction terms to this approximation of order me/mn, where me is the mass of the electron and mn is the mass of the nucleus, was discussed by Van Vleck [19]. Based on that, isotopically variant parameters Ykl are written in terms of isotopically invariant parameters according to Watson’s expression [20] as given below: " # me DSkl me DH kl þ Y kl ¼ lðk=2þlÞ U kl 1 þ ; ð2Þ MS MH where U kl ; DSkl ; DHkl are the invariant parameters; DSkl ; DHkl parameters correspond to sulfur and hydrogen, respectively, and the charge-adjusted reduced mass l is given by: l¼

MS MH ; ðM S þ M H  Cme Þ

ð3Þ

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Z. Zelinger et al. / Journal of Molecular Spectroscopy 249 (2008) 117–120

Table 1 Observed transitions and residuals of the overall fit, data and assignments from references [11–13,15], all frequencies are in cm1, residuals in cm1, the first number indicates the isotope (0 for 32SH, 1 for 34SH, 2 for 33SH, 3 for 32SD) I

Observed

o-c

vl

Jl

vu

Ju

Weight

Ref.

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 2 3

18.827087 2522.5024 2522.5030 2503.0941 2483.1164 2483.1160 2462.5813 2441.5010 2441.4990 2419.8877 2419.8880 2397.7534 2397.7530 2375.1103 2351.9706 2351.9690 2328.3466 2304.2503 2279.6938 2559.5631 2559.5540 2577.1916 2594.2031 2610.5860 2626.3287 2626.3220 2641.4196 2655.8475 2669.6011 2682.6693 2695.0413 2706.7062 2717.6534 2717.6520 2727.8725 2737.3534 2746.0859 2754.0602 2418.7839 2399.9690 2380.5847 2360.6430 2340.1560 2319.1357 2297.5943 2297.5960 2275.5437 2275.5440 2252.9961 2229.9637 2206.4584 2454.6581 2471.6937 2471.6900 2488.1127 2503.9034 2519.0543 2519.0560 2533.5541 2533.5500 2547.3914 2547.3880 2560.5552 2573.0343 2584.8180 2595.8955 2606.2564 2439.4940 2460.5330 2623.9890 2639.0560 2680.2510 2418.8720 9.752051

0.000031 0.000 0.001 0.000 0.000 0.000 0.001 0.001 0.001 0.002 0.003 0.003 0.003 0.004 0.003 0.002 0.002 0.001 0.006 0.000 0.009 0.000 0.000 0.001 0.001 0.007 0.001 0.001 0.001 0.001 0.000 0.001 0.001 0.000 0.002 0.002 0.001 0.001 0.000 0.000 0.000 0.001 0.001 0.002 0.002 0.004 0.002 0.003 0.002 0.001 0.003 0.000 0.000 0.004 0.000 0.000 0.001 0.001 0.001 0.005 0.001 0.004 0.001 0.001 0.000 0.000 0.001 0.005 0.003 0.003 0.003 0.001 0.003 0.000081

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0

0 1 1 2 3 3 4 5 5 6 6 7 7 8 9 9 10 11 12 0 0 1 2 3 4 4 5 6 7 8 9 10 11 11 12 13 14 15 1 2 3 4 5 6 7 7 8 8 9 10 11 0 1 1 2 3 4 4 5 5 6 6 7 8 9 10 11 5 4 4 5 8 6 0

0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 0

1 0 0 1 2 2 3 4 4 5 5 6 6 7 8 8 9 10 11 1 1 2 3 4 5 5 6 7 8 9 10 11 12 12 13 14 15 16 0 1 2 3 4 5 6 6 7 7 8 9 10 1 2 2 3 4 5 5 6 6 7 7 8 9 10 11 12 4 3 5 6 9 5 1

10000 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10000

[15] [12] [11] [12] [12] [11] [12] [12] [11] [12] [11] [12] [11] [12] [12] [11] [12] [12] [12] [12] [11] [12] [12] [12] [12] [11] [12] [12] [12] [12] [12] [12] [12] [11] [12] [12] [12] [12] [12] [12] [12] [12] [12] [12] [12] [11] [12] [11] [12] [12] [12] [12] [12] [11] [12] [12] [12] [11] [12] [11] [12] [11] [12] [12] [12] [12] [12] [11] [11] [11] [11] [11] [11] [15]

Table 1 (continued) I

Observed

o-c

vl

Jl

vu

Ju

Weight

Ref.

3 3 3 3 3 3 3 3 3 3 3 3 3

19.501142 1816.2480 1805.8560 1795.2600 1784.4550 1864.9960 1874.0780 1882.9340 1891.5520 1899.9400 1915.9900 1896.4300 1902.5900

0.000042 0.002 0.001 0.003 0.001 0.002 0.000 0.002 0.002 0.000 0.004 0.008 0.008

0 0 0 0 0 0 0 0 0 0 0 1 1

1 3 4 5 6 1 2 3 4 5 7 12 13

0 1 1 1 1 1 1 1 1 1 1 2 2

2 2 3 4 5 2 3 4 5 6 8 13 14

10000 1 1 1 1 1 1 1 1 1 1 1 1

[15] [13] [13] [13] [13] [13] [13] [13] [13] [13] [13] [13] [13]

where MS and MH are atomic masses of sulfur and hydrogen atoms, me stands for the electron mass, and C is the charge number of the ion, which equals to 1 for SH. In Eq. (2), the terms including D parameters demonstrate a breakdown of the Born–Oppenheimer approximation. The rotational transition frequencies from [15] were combined with vibrational–rotational transition frequencies from [11–13] to determine values of the isotope invariant parameters of the negative ion during the least squares fitting. Results are listed in Table 2. The full Dunham analysis was performed by combining available IR data with sub-millimeter wave data for several isotopic forms of the hydrosulfide anion. This new analysis of the hydrosulfide anion yields accurate information on the Born–Oppenheimer breakdown parameters. In order to obtain a simple analytical representation of the SH anion vibration, all available transitions of hydrogen sulfide and deuterium sulfide anions [11–13,15] were fitted to the Morse potential U of a qualitatively correct behavior within the dissociation region: U ¼ Dð1  eax Þ2 :

ð4Þ

This expression introduces the Morse constant a and the dimensional-free x parameter as given below: x¼

r  re ; re

ð5Þ

where r is the internuclear distance, and re the equilibrium value of r. The value D stands for the dissociation energy given by

Table 2 The isotopically invariant parameters of SH a

This work

Previous workb

U10 U20 U30 U01 U11 U21 U02 U12 U03 DS10 DH 10 DH 01

2610.5858(87) 46.6745(76) 0.9430(17) 9.342935(16) 0.286687(33) 0.0000474(77) 0.00047426(27) 0.00000250(27) 0.00000000413(74) 0.173(28) 0.5692(11) 0.3896(35)

2617.549(6) 52.0644(29) 9.34258(20) 0.28667(12) 0.000039(24) 0.0004759(27) 0.00000280(83) 0.000000018(13)

Standard deviation of the fit: 0.0028 cm1 a The unit of Ukl is cm1 (amu)(k/2+l), parameters Dkl are non-dimensional, the numbers in parentheses are the estimated 2 r uncertainties in units of the last quoted digits. b The conversion relation Eq. (2) was used for data from [11].

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D¼

U 210 : 4a2 U 01

70000

ð6Þ

2

3

U ¼ a0 x ð1 þ a1 x þ a2 x þ a3 x þ . . .Þ;

ð7Þ

where

a4 ¼

a1 ¼ a;

31 4 a ; 360

a5 ¼ 

a2 ¼

7 2 a ; 12

126 5 a ; 7!

1 a3 ¼  a3 ; 4 250 6 a : a6 ¼ 8!

50000

U / cm-1

2

a0 ¼ Da2 ;

Morse potential series expansion

60000

The relationships between the potential constants a1, a2, a3, a4, a5, a6 and the Morse constant a are yielded by the potential energy U(x) power series expansion (Eq. (7)) and by considering Eq. (4) [1,21]:

40000 30000 20000 10000

ð8Þ

0 -10000

The reliability and accuracy of the Born–Oppenheimer parameters set achieved during the study are strongly dependent on the number and variety of investigated isotopes (and not only on the number of transitions observed for each isotope). We disposed of lines of isotopes 32SH [11,12], 34SH[11], 32SD [13] but unfortunately only one line of 33SH [11]. In the first step of the least squares analysis, we fitted the available IR transition frequencies together with available sub-millimeter wave data (compare with the CDMS [22]). After adding the sub-millimeter wave data, larger and systematic discrepancies between observed and calculated values for IR data appeared. The large discrepancies were not reduced although more parameters Ukl of higher orders were employed for the calculation. These discrepancies were significantly reduced after an inclusion of the Born–Oppenheimer parameters DSkl ; DHkl . However, our set of experimental data allowed us to determine only DS10 ; DH01 ; DH10 parameters (see Table 2). The parameter DS01 was not determinable; it was fixed at 0 during the fitting. In our analysis, the isotopically invariant parameters H U 10 ; U 20 ; U 30 ; U 01 ; U 11 ; U 21 ; U 02 ; U 12 ; U 03 ; DS10 ; DH 10 ; D01 were fitted for the first time. These isotopically invariant parameters correspond to the ordinary band spectrum constants xe, xexe, xeye, Be, H ae, ce, De, be, He, while the DS10 ; DH 10 ; D01 parameters are high order Born–Oppenheimer parameters. In Table 2, the currently determined parameters are compared with those from previous analyses [11], recalculated to isotopically invariant ones based on the relation Eq. (2). The standard deviation of the fit of all isotopes was 0.0028 cm1. Table 3 contains results of fitting the data to an expansion of the potential function up to the sixth order (terms up to x8) and calculating the coefficients of a Morse expansion in terms of the coefficients of the six order potential function [1]. The U01 value (corresponding to the Be value) derived from this fit is in a good agreement with the value derived from Dunham parameters fit (Table 2). The potential constants a1, a2, a3 were compared with

0.0

0.1

0.2

0.3

0.4

r / nm 

Fig. 1. Potential function of SH , the solid line represents the Morse potential according to Eq. (4), the open circles represent the power series expansion according to Eq. (7).

results of the analysis published in [11]. The potential constants a4, a5, a6 are newly determined (see Table 3). The standard deviation of the fit of these potential constants was 0.0068 cm1 for all isotopes. The potential constants a1, . . . , a6 of the SH anion given in Table 3 were used for the construction of potential functions of the negative ion SH. Differences between the potential function construction on the base of the Morse potential according to Eq. (4) and that on the base of the power series expansion according to Eq. (7) are depicted in the Fig. 1. In this spectroscopic study, the isotopically invariant Dunham parameters were determined for the negative ion—the hydrogen sulfide anion—for the first time. The potential constants of expansion of the potential function up the sixth order were determined for this negative ion. Acknowledgments The authors gratefully acknowledge the financial support from the Czech Science Foundation (No. 202/06/0216) and from the Ministry of Education, Youth and Sports of the Czech Republic: Research programs LC06071 and OC111, OC186 (in the frame of the COST 729 action). Z. Zelinger is grateful to the ‘‘UFR de Physique” of the University Paris Diderot – Paris 7 for a two months position as ‘‘Invited Professor”. The authors are indebted to the European Community for financial support within the ‘‘QUASAAR” (QUAntitative Spectroscopy for Atmospheric and Astrophysical Research) network project (Contract MRTN-CT-2004-512202). References

Table 3 The potential constants a1–a6 of the SH anion a

This work

Previous workb

U10 U01 a1 a2 a3 a4 a5 a6

2611.463(48) 9.346970(63) 2.45525(52) 4.2071(25) 6.412(43) 1.84(16) 4.03(46) 207.03(349)

2615.58(51) 9.3441(5) 2.45103(98) 3.8957(280) 4.702(99)

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

Standard deviation of the fit: 0.0068 cm1 a The unit of Ukl is cm1 (amu)(k/2+l), parameters a1–a6 are non-dimensional, the numbers in parentheses are the estimated 2 r uncertainties in units of the last quoted digits. b The conversion relation Eq. (2) was used for data from [11].

[11] [12] [13]

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