High resolution GHz and THz (FTIR) spectroscopy and theory of parity violation and tunneling for 1,2-dithiine (C₄H₄S₂) as a candidate for measuring the parity violating energy difference between enantiomers of chiral molecules

Abstract

We report high resolution spectroscopic results of 1,2-dithiine-(1,2-dithia-3,5-cyclohexadiene, C₄H₄S₂) in the gigahertz and terahertz spectroscopic ranges and exploratory theoretical calculations of parity violation and tunneling processes in view of a possible experimental determination of the parity violating energy difference Δ_pvE in this chiral molecule. Theory predicts that the parity violating energy difference between the enantiomers in their ground state (Δ_pvE ≃ 1.1 × 10⁻¹¹(hc) cm⁻¹) is in principle measurable as it is much larger than the calculated tunneling splitting for the symmetrical potential ΔE_± < 10⁻²⁴ (hc) cm⁻¹. With a planar transition state for stereomutation at about 2500 cm⁻¹ tunneling splitting becomes appreciable above 2300 cm⁻¹. This makes levels of well-defined parity accessible to parity selection by the available powerful infrared lasers and thus useful for one of the existing experimental approaches towards molecular parity violation. The new GHz spectroscopy leads to greatly improved ground state rotational parameters for 1,2-dithiine. These are used as starting points for the first successful analyses of high resolution interferometric Fourier transform infrared (FTIR, THz) spectra of the fundamentals ν₁₇ (1308.873 cm⁻¹ or 39.23903 THz), ν₂₂ (623.094 cm⁻¹ or 18.67989 THz) and ν₃ (1544.900 cm⁻¹ or 46.314937 THz) for which highly accurate spectroscopic parameters are reported. The results are discussed in relation to current efforts to measure Δ_pvE.

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1 Introduction

According to the traditional point of view both in the framework of classical molecular structure^1–6 and within the ordinary quantum theory of molecules retaining only the electromagnetic force^7,8 the ground states of the enantiomers of chiral molecules are energetically exactly equivalent by symmetry.^1,5–10 This would result in a reaction enthalpy Δ_RH^⊖₀ = 0 exactly for the stereomutation reaction (1). However, the situation has fundamentally changed with the discovery of parity violation.^11–21 In this case one predicts a small “parity violating” energy difference Δ_pvE₀ between the ground states of the two enantiomers, leading to the prediction of a non-zero parity violating reaction enthalpy Δ_pvH^⊖₀:

P ⇌ M Δ_pvH^⊖₀ = N_AΔ_pvE₀ (1)

where N_A is the Avogadro constant and we use the P and M notation for “axially chiral” molecules. This has important consequences for our understanding of the foundation of stereochemistry¹⁰ and possibly also of the evolution of biomolecular homochirality.^22–29 There has been considerable theoretical work predicting the magnitude of Δ_pvE, first qualitatively,^23,30 then in semiquantitative computations^31,32 all leading to exceedingly small absolute values. Recent progress in the theory of molecular parity violation, starting with our finding³³ that the early estimates of Δ_pvE were too low by up to two orders of magnitude for benchmark molecules such as H₂O₂ and related ones, confirmed by extensive subsequent work,^34–46 led to a new impetus to the field reviewed in ref. 9, 47 and 48, including serious attempts to detect parity violation in chiral molecules, so far unsuccessful, however. While several different experimental schemes concerning molecular parity violation have been proposed in the past^{26,30,49–53} (see also the reviews in ref. 9, 10, 29, 47 and 54–56), only two of these appear to be actively pursued at present, as outlined in Fig. 1. In this figure the solid lines represent measurable energy levels of the chiral molecule under consideration, whereas the dotted lines refer to the potential energy minima including the parity violating potential.⁴⁴ In theory the measurable energy represented by the solid lines would be obtained from the expectation values of the multidimensional parity violating potential in the quantum state considered, added to the energy eigenvalues of that state as computed from parity conserving ordinary molecular quantum mechanics^9,41,42 (see also Section 2).

Fig. 1 Scheme of energy levels including parity violation in 1,2-dithiine (not to scale), see also ref. 44 and 47.

In a first experimental scheme originally proposed by Letokhov and coworkers^30,49 one measures the difference in line positions, say, in the infrared spectrum of the separate enantiomers (hν_M − hν_P in Fig. 1). This provides the difference (Δ_pvE* − Δ_pvE₀) between two parity violating energy differences (Δ_pvE* and Δ_pvE₀) in two different energy levels. Early attempts towards such experiments were reported by Kompanets et al.⁴⁹ and Arimondo, Glorieux and Oka⁵⁰ (both unsuccessful). The work of Kompanets et al. on CHFClBr was further pursued by our group using high resolution microwave and infrared spectroscopy in molecular beams with detailed rovibrational analyses in order to identify suitable lines for detecting parity violation in these spectra.⁵⁷ These results of our group were subsequently used to carry out experiments at very high resolution by the Paris group⁵⁸ achieving about Δν/ν ≈ 10⁻¹³ relative uncertainty, compared to theoretical predictions of shifts of the order Δν/ν ≃ 10⁻¹⁶, ref. 39–42. Recent progress by two groups following this scheme currently can be seen from ref. 59 and 60. One can estimate that successful experiments of this kind might be possible in the near future for chiral molecules involving very heavy atoms which have relatively large values of Δ_pvE because of the well-known scaling with atomic number.^35–48,61 Here it must be noted that atomic spectroscopy has already been successful to detect parity violation for very heavy atoms such as Cs, Bi etc.,^62–67 where a current limitation in the fundamental analysis actually results from uncertainties in the theory for such heavy-atom systems, which will be also a drawback if such experiments are successful on related molecules (see ref. 9 and 29 for a discussion). Another drawback of this first scheme is that it cannot provide Δ_pvE₀ (or Δ_pvE*) separately. Also, this scheme requires the synthesis of separate enantiomers. We shall not discuss this scheme further here.

In the second scheme, proposed in ref. 52, one uses an excited intermediate level of well-defined parity (marked E₊ for positive parity in Fig. 1) which can be reached by electric dipole transitions from both enantiomers (marked hν_P+ and hν_M+ in Fig. 1). Then one can obtain Δ_pvE₀ by a combination difference (hν_P+ − hν_M+). Alternatively, one can use a time dependent variant of this scheme whereby a sequence of two electric dipole transitions to a superposition state of the two enantiomers is prepared. This superposition state has a well-defined parity initially (say, with a probability p₋(t = 0) = 1), which evolves in time according to eqn (2) for the complementary population of the state of positive parity detectable with high sensitivity on a “zero background”^9,52

p₊(t) = 1 − p₋(t) = sin²(πΔ_pvEt/h). (2)

Here, we use the notation Δ_pvE for some general quantum state which may be the ground state (then one has Δ_pvE = Δ_pvE₀), the excited state with Δ_pvE = Δ_pvE* in Fig. 1, or some other selected quantum state. Recent detailed theoretical simulations have shown the feasibility of such experiments, in principle, for the realistic, simple chiral molecule ClOOCl.^68,69 Furthermore the experimental proof of principle has been demonstrated for the simple achiral molecule NH₃, where the current experimental sensitivity indicates that parity violating energy differences on the order of about 100 aeV or (hc) 10⁻¹² cm⁻¹ corresponding to about Δ_pvH^⊖₀ = 10⁻¹¹ J mol⁻¹ should be detectable with the existing set-up.⁷⁰ Such values are predicted for chiral molecules with atoms no heavier than sulfur or chlorine⁷¹ which is a great advantage for a future theoretical analysis of experiments with very high accuracy^9,44 in addition to the advantage of providing separate values for individual Δ_pvE. Another advantage of the second scheme is that experiments can be carried out on racemic mixtures as well as on separate enantiomers.

We should note that a general condition for both experimental schemes is that the parity violating energy difference Δ_pvE is much larger than the tunneling splitting ΔE_± calculated using the hypothetical symmetrical (parity conserving) Hamiltonian, thus

|Δ_pvE| ≫ |ΔE_±|. (3)

Indeed, this condition is a necessary requirement for an energy level scheme as in Fig. 1 to be applicable at all. The inequality (3) does not hold for chiral molecules such as HOOH and HSSH^9,48,72 but has been predicted theoretically to be fullfilled for ClOOCl^68,69 and ClSSCl,⁷¹ for example, and also for the case of 1,2-dithiine to be studied here.

Current spectroscopic techniques result in some limitations on the complexity of the chiral molecules which can be studied. Also spectroscopically accessible states of well-defined parity must be identified by appropriate high resolution analyses. Such states can be found in electronically excited states, for instance, in molecules such as difluoroallene⁷⁶ or they can be found in the electronic ground state at energies near or above the barrier for stereomutation as shown for ClOOCl.^68,69 One can thus identify the following steps in preparing for experiments on Δ_pvE in chiral molecules.

(1) Identify suitable chiral molecules by approximate exploratory theoretical calculations, demonstrating sufficiently large absolute values |Δ_pvE| to be measurable and small absolute values |ΔE_±| for the tunneling splitting such that the inequality (3) holds.

(2) Carry out the necessary synthesis and exploratory high resolution spectroscopic analyses.

(3) Identify spectroscopic lines suitable to carry out the experiment according to the second scheme.

(4) Carry out the experiment to measure p₊(t) according to eqn (2) with significant accuracy to provide Δ_pvE.

(5) Analyse highly accurate experimental results of Δ_pvE by means of highly accurate theoretical calculations to extract fundamental parameters of the standard model of particle physics (SMPP) such as the Weinberg parameter which may be significantly different in experiments of high energy physics and “low energy atomic and molecular physics” spectroscopic experiments, for instance.⁹

In the present work we show that the chiral molecule 1,2-dithiine is a possible candidate for experiments on Δ_pvE by carrying out in essence steps 1 and 2 of the program listed above. In Section 2 we show by theoretical calculations that 1,2-dithiine has a favourable, large absolute value |Δ_pvE| and a very small ground state tunneling splitting |ΔE_±| satisfying the inequality (3). Furthermore the barrier for stereomutation is calculated to be low enough for states above the barrier with well-defined parity to be accessible by the currently available high resolution lasers in the infrared range.⁷⁰

1,2-Dithiine as shown in Fig. 2 is a six membered chiral ring compound of C₂ symmetry which has been synthesized in ref. 77 and 78 and since this early synthesis combined with the finding that 1,2-dithiines occur as natural products^79,80 there have been substantial experimental and theoretical studies, of which we can cite here only a selection^81–89 including also in particular a review⁸² and a recent theoretical study of parity violation in the series of molecules starting with dioxine (C₄H₄O₂) leading to 1,2-dithiine and the higher members of this series with Se, Te, and Po⁸³ and studies of electronic structure and antiaromaticity.^84–89 Disulfides in general have been discussed as candidates for experimental studies of parity violation by us before,^71,72 recently extended to the slightly more complex trisulfane.⁹⁰ 1,2-Dithiine is spectroscopically already quite complex but recent experimental progress has made molecules of similar complexity such as pyridine, pyrimidine, fluorobenzene, chlorobenzene,⁹¹ and phenol as well as naphthalene and azulene accessible to high resolution infrared spectroscopic analyses (see ref. 91, and references cited therein).

Fig. 2 Perspective drawing of the molecular structure of 1,2-dithiine (C₄H₄S₂) and principal axes together with the general atom numbering scheme (P enantiomer, sulfur yellow (1,2), carbon black (3,4,5,6) and hydrogen light grey (7,8,9,10)).

The outline of the present paper is briefly as follows. In Section 2 we present the theory for tunneling and parity violation. In Section 3 we discuss our experiments on the synthesis, high resolution THz (FTIR) and also GHz spectroscopy. In Section 4 we discuss the spectroscopic assignments and in Section 5 an extended rotational analysis of the ground state of 1,2-dithiine as well as of three of the stronger fundamentals ν₂₂ (ν₀ = 623.3121 cm⁻¹), ν₁₇ (ν₀ = 1308.8724 cm⁻¹) and ν₃ (ν₀ = 1544.9004 cm⁻¹) demonstrating the feasibility of such analyses for this chiral molecule, for which the infrared spectra have not been analysed at high resolution before, while some limited microwave data existed already⁷⁵ with a total of only 15 microwave lines for the normal isotopomer. We conclude with a detailed discussion and an outlook towards the feasibility of experiments to measure parity violation in this chiral molecule. A preliminary report on the present research has been given in ref. 92–94.

2 Theory

The interesting non-planar structure of 1,2-dithiine is shown in Fig. 2. The equilibrium structure of the electronic ground state (X̃ ¹A) was optimized at the MP2/cc-pVTZ, MP2/cc-pVQZ and CCSD(T)/cc-pVTZ levels of theory. These computations were carried out using the Gaussian 09⁷³ and Molpro^95,96 program packages. The planar ring structure would formally have 8 π-electrons and thus would be antiaromatic and destabilized. However, this effect is reduced by avoiding the planar geometry. The predicted ring structure possesses C₂ symmetry and the optimized structural parameters are summarized in Table 1 for the equilibrium geometry. The calculated structure agrees well with an experimental r_s-structure from microwave spectroscopy.⁷⁵ The corresponding theoretical rotational constants are compared with the experimental results in Section 4 below in more detail. Again, the overall agreement is remarkably good. The only structural parameter which shows a large difference between experiment and theory is the S–S bond length differing by about 3 pm when compared with the CCSD(T) results, where one must consider also the different physical meanings of r_s and r_e structures. The calculated harmonic wavenumbers [small omega, Greek, tilde]_i, the vibrational fundamentals [small nu, Greek, tilde]_i (with anharmonic corrections) and the predicted integrated band strengths (in km mol⁻¹) are given in Table 2. The most intense fundamental [small nu, Greek, tilde]₂₂ = 623 cm⁻¹ corresponds to a CH “out-of-plane” bending mode, followed by the ring twisting fundamental [small nu, Greek, tilde]₂₄ = 214 cm⁻¹ and the CH “in-plane” bending fundamental [small nu, Greek, tilde]₁₇ = 1303 cm⁻¹ where “out-of-plane” and “in-plane” have obviously only approximate meaning for this non-planar molecule. Further intense absorptions arise from the CC stretching polyads and the CH stretching polyads.

Table 1 Optimized equilibrium structure parameters (r_e in pm, angles in degrees) of the ground electronic state X̃ ¹A₁ of 1,2-dithiine (C₂ symmetry, atom numbering as in Fig. 2). The MP2 calculations were done using the Gaussian 09 package⁷³ and the CCSD(T) calculations using the Molpro program package.⁷⁴ (We indicate: method/basis set as shown)

	MP2/cc-pVTZ	MP2/cc-pVQZ	MP2/cc-pV5Z	CCSD(T)/cc-pVTZ	Exp.^a
a Experimental structural parameters from ref. 75 (rs structure).
r(S1C6)	176.21	175.58	175.17	177.59	175.9(4)
r(S1S2)	206.23	204.77	203.88	208.35	205.1(3)
r(C5C6)	135.08	134.91	134.93	134.89	135.3(3)
r(C4C5)	144.71	144.44	144.36	146.12	145.1(1)
r(C6H7)	108.21	108.16	108.16	108.32
r(C5H10)	108.25	108.19	108.17	108.41
α(C6S1S2)	97.99	98.12	98.19	98.27	98.7(2)
α(S1C6C5)	121.39	121.17	121.07	121.88	121.4(2)
α(C4C5C6)	124.18	124.14	124.07	124.40	124.2(2)
α(S1C6H7)	116.65	116.77	116.90	116.09
α(C6C5H10)	117.85	117.87	117.90	118.14
τ(C6S1S2C3)	54.99	55.31	55.59	53.25	53.9
τ(S2S1C6H7)	141.14	141.07	141.02	141.82
τ(S2S1C6C5)	−42.27	−42.37	−42.51	−41.11	−41.2
τ(S1C6C5H10)	−177.25	−177.18	−177.08	−177.85

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Table 2 Calculated harmonic wavenumbers [small omega, Greek, tilde]_i (in cm⁻¹), IR intensities (in km mol⁻¹), and anharmonic fundamentals [small nu, Greek, tilde]_i (in cm⁻¹) of 1,2-dithiine as calculated using the Gaussian 09 package⁷³

Mode	Symmetry	[small omega, Greek, tilde] i (MP2/cc-pVTZ)	Intensity	[small nu, Greek, tilde] i (MP2/cc-pVTZ)	Approximate description
1	A	3226	8.495	3085	CH stretches
2	A	3204	0.206	3077	CH stretches
3	A	1568	11.063	1522	CC stretches
4	A	1390	4.468	1349	CH in-plane bends
5	A	1181	0.098	1164	CH in-plane bends
6	A	997	2.670	981	C2C3 stretch
7	A	938	0.814	920	CH out-of-plane bends
8	A	780	3.451	767	CH out-of-plane bends
9	A	740	4.977	728	CS sym. Stretches
10	A	573	0.100	565	SS stretch, CH out-of-plane bends
11	A	523	0.068	516	SS stretch, CH out-of-plane bends
12	A	448	0.243	441	CCC in-plane bends
13	A	204	0.001	200	Ring pucker
14	B	3221	5.420	3115	CH stretches
15	B	3197	0.882	3075	CH stretches
16	B	1610	0.210	1563	CC stretches
17	B	1329	14.809	1303	CH in-plane bends
18	B	1184	1.162	1166	CH in-plane bends
19	B	929	0.139	910	CH out-of-plane bends
20	B	858	2.711	847	CCC in-plane bends
21	B	702	1.495	684	CS asym. stretches
22	B	632	92.517	623	CH out-of-plane bends
23	B	315	0.083	312	CSS in-plane bends
24	B	214	12.041	214	CCC out-of-plane bends

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The twisted structure with a dihedral angle τ(C₆–S₁–S₂–C₃) of 53.25 degrees gives rise to P and M enantiomorphism, with a planar structure of C_2v symmetry being the transition state for interconversion between the two enantiomers (Fig. 3 and Table 3). Formally, this corresponds to the cis-transition state of general disulfides with internal rotation.^71,72 The S–S bond length is elongated to 217.8 pm (CCSD(T)/cc-pVDZ level) and one finds as angles α(C₆–S₁–S₂) = 104.43° and α(S₁–C₆–C₅) = 129.72° for the transition state which are the only structural parameters apart from the obvious dihedral angles differing substantially from the equilibrium structures. The transition state as calculated at the CCSD(T)/cc-pVQZ level of theory is found to be about 2500 cm⁻¹ above the potential minimum (without vibrational zero-point energy correction). The planar structure is shown in Fig. 3 which also indicates the atom numbering with C₆, S₁, S₂, C₃, C₄, C₅ and H₇, H₈, H₉, H₁₀. The structural parameters are summarized in Table 3 and the harmonic wavenumbers of the transition state are given in Table 4 with the species in C_2v symmetry.

Fig. 3 Transition state and alternative atom numbering in 1,2-dithiine.

Table 3 Optimized molecular structure parameters (r_e in pm, angles in degrees) of the planar transition state for the stereomutation of 1,2-dithiine (C_2v symmetry). The MP2 calculations were done using the Gaussian 09 package⁷³ and the CCSD(T) calculations using the Molpro program package⁷⁴

	MP2/cc-pVTZ	MP2/cc-pVQZ	CCSD(T)/cc-pVDZ
r(S1C6)	175.36	174.81	177.97
r(S1S2)	212.78	211.46	217.82
r(C5C6)	134.18	133.95	135.75
r(C4C5)	145.37	145.18	147.54
r(C6H7)	108.30	108.24	109.94
r(C5H10)	108.10	108.04	109.75
α(C6S1S2)	104.73	104.88	104.43
α(S1C6C5)	129.57	129.49	129.72
α(C4C5C6)	125.70	125.63	125.85
α(S1C6H7)	109.87	109.85	109.75
α(C6C5H10)	116.67	116.71	117.01

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Table 4 Calculated harmonic wavenumbers [small omega, Greek, tilde]_i (in cm⁻¹) of the planar transition state for the stereomutation of 1,2-dithiine as calculated using the Gaussian 09 program package⁷³

Mode	Symmetry	[small omega, Greek, tilde] i (MP2/cc-VTZ)	[small omega, Greek, tilde] i (MP2/cc-pVQZ)
1	A1	3233	3231
2	A1	3202	3202
3	A1	1628	1626
3	A1	1403	1402
5	A1	1209	1208
6	A1	985	985
7	A1	729	732
8	A1	513	515
9	A1	458	462
10	A2	927	928
11	A2	746	743
12	A2	503	509
13	A2	i162	i168
14	B1	926	926
15	B1	642	640
16	B1	263	264
17	B2	3222	3221
18	B2	3198	3199
19	B2	1645	1644
20	B2	1341	1342
21	B2	1213	1212
22	B2	850	851
23	B2	678	681
24	B2	314	315

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The parity-conserving electronic (V_el) and parity-violating potentials (E_pv) computed along the IRC interconversion path are given in Fig. 4. E_pv was computed with our recently developed^44,97 coupled-cluster singles and doubles linear response approach (CCSD-LR) using the Zurich version of the PSI3 program.⁹⁸E_pv is obtained as a linear response function of the coupled cluster wavefunction with an effective one-electron spin–orbit coupling serving as a static perturbation. We employed a decontracted cc-pVDZ basis set augmented with a set of steep s and p functions. Details of the theoretical approach have been described in ref. 44.

Fig. 4 Parity-violating potential E_pv (CCSD-LR/cc-pVDZ) and the parity-conserving potential V_el (CCSD(T)/cc-pVTZ) computed along the IRC interconversion path for 1,2-dithiine. The P enantiomer has negative values of −τ, the M enantiomer has positive values of −τ (in degrees °).

Based on our calculations, and as can be seen from Fig. 4, we predict the P-enantiomer to be stabilized with respect to the M-enantiomer. The computed value of E_pv for P-1,2-dithiine is E_pv/(hc) = −5.72 × 10⁻¹² cm⁻¹. The signed electronic parity violating the energy difference Δ_pvE^el between the two enantiomers as defined by eqn (4) is obtained to be

Δ_pvE^el(q_e) = E^el_pv(q_M) − E^el_pv(q_P) = (hc) 11.44 × 10⁻¹² cm⁻¹ (4)

where q stands for the reaction coordinate along the IRC path. The value for Δ_pvE^el(q_e) is remarkably large and is related to the parity violating potential having its maximum as a function of the dihedral angle near the equilibrium geometry. It can be compared with the very similar values in the range (11 ± 1) × 10⁻¹² cm⁻¹ obtained with several methods in ref. 83.

In order to estimate the ground-state tunneling splitting (ΔE_±) the τ(C₆–S₁–S₂–C₃) dihedral angle (denoted by τ) was selected to be the reaction coordinate describing the interconversion of the two puckered enantiomers through a planar transition state. For every energy value as a function of τ the remaining internal coordinates were optimized at the MP2/cc-pVTZ level of theory and the resulting structures were then used for single-point CCSD(T)/cc-pVQZ electronic structure computations. This definition of the reaction coordinate was justified by MP2/cc-pVTZ IRC computations, where the resulting IRC path connecting the planar transition state with the two enantiomeric potential minima was found to be almost identical to the 1D MP2/cc-pVTZ reaction path following simply a variation of τ. The structural parameters as a function of the reaction coordinate values q_i and the harmonic wavenumbers as a function of q are given in the ESI.†

We employed the quasi-adiabatic channel reaction path Hamiltonian approach (RPH)^99–101 to compute the tunneling splittings ΔE_±. In conjunction with the Born–Oppenheimer potential (CCSD(T)/cc-pVQZ, see the dashed line in Fig. 5) we used Cartesian coordinates, energy gradients and harmonic force fields (all of these quantities were obtained at the MP2/cc-pVTZ level of theory) along the reaction path to generate the lowest quasi-adiabatic channel potential (Fig. 5, solid line). The barrier heights are 2529 cm⁻¹ (Born–Oppenheimer potential) and 2559 cm⁻¹ (lowest quasi-adiabatic channel potential). The RPH energy levels corresponding to the lowest quasi-adiabatic channel potential are given in Table 5. It can be seen from Table 5 that the tunneling splittings increase as the energy levels approach the barrier. At v = 12, about 2170 cm⁻¹ above the minimum of the channel and still almost 400 cm⁻¹ below the barrier the tunneling splitting of 0.1 cm⁻¹ would be easily measurable in the infrared region. This large value is quite remarkable in view of the essentially pure “heavy atom” motion for the tunneling process.

Fig. 5 Potential energy function corresponding to the 1D reaction path (see the text for discussion). The electronic Born–Oppenheimer (dashed line) and the lowest quasi-adiabatic channel potentials (solid line) are referred to their respective minima being separated by the 23D zero point energy (hc) 14 882 cm⁻¹ and τ_CSSC denotes the C₆–S₁–S₂–C₃ dihedral angle. Using the shift by this large difference in reference energies, the two effective potentials almost coincide.

Table 5 RPH energy levels for the lowest adiabatic channel of 1,2-dithiine as E/(hc) in cm⁻¹. The energy levels with even parity are given, the splittings as separation from the odd parity levels are given in parentheses. All levels are given with reference to the zero-point level (101.53 cm⁻¹ above the minimum of the channel)

v = 0	0.00 (4 × 10^−25)	v = 10	1860.60 (1.4 × 10^−4)
v = 1	202.39 (3 × 10^−22)	v = 11	2019.19 (4.2 × 10^−3)
v = 2	401.84 (7 × 10^−20)	v = 12	2169.37 (0.10)
v = 3	598.16 (7 × 10^−18)	v = 13	2307.68 (1.80)
v = 4	791.14 (4 × 10^−17)	v = 14	2421.75 (17.48)
v = 5	980.51 (8 × 10^−15)	v = 15	2514.78 (52.76)
v = 6	1166.00 (2 × 10^−12)	v = 16	2636.09 (69.83)
v = 7	1347.23 (1.4 × 10^−9)	v = 17	2779.88 (76.93)
v = 8	1523.79 (8.4 × 10^−8)	v = 18	2936.34 (82.13)
v = 9	1695.15 (3.8 × 10^−6)	v = 19	3102.76 (86.59)

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Since the tunneling splittings of the lowest tunneling doublets are very small owing to the high interconversion barrier and large effective tunneling mass, their direct variational computation is not easily possible due to numerical limitations. In order to estimate ΔE_± we invoked an extrapolation technique that has been successfully used for the calculation of small tunneling splittings in ClSSCl.⁷¹ This method resulted in a ground-state tunneling splitting of ΔE_±/(hc) = 4 × 10⁻²⁵ cm⁻¹. We have also calculated the expectation value of Δ_pvE in the torsional ground state and found it to differ by less than 10% from Δ_pvE^el. Thus the parity violating energy difference Δ_pvE clearly exceeds the tunneling splittings for the hypothetical symmetrical potential by many orders of magnitude and 1,2-dithiine is a suitable candidate for measuring Δ_pvE by the technique proposed in ref. 52 from this point of view. We shall discuss further spectroscopic aspects in the following sections.

3 Experimental

3.1 Synthesis of 1,2-dithiine

1,2-Dithiine was synthesized according to the scheme in Fig. 6 using a procedure which is a modification of ref. 81, as described here, because the original procedure from ref. 81, which was tried as well, did not give quite satisfactory results in step c.

Fig. 6 Scheme for the synthesis of 1,2-dithiine.

1,4-Bis(trimethylsilyl)buta-1,3-diyne (2). 1,4-Bis(trimethylsilyl)buta-1,3-diyne (2) was obtained according to the method of ref. 102. The yield was 75% and the purity was 97% as controlled by GC-MS.

1,4-Bis(benzylthio)buta-1Z,3Z-diene (3). 1,4-Bis(benzylthio)buta-1Z,3Z-diene (3) could be obtained following ref. 81. No recrystallisation was carried out. The reaction product was used without further purification. The yield was 80% and the purity was 90% as obtained by NMR. The NMR-shifts correspond to the values given in ref. 81.

1,4-Bis(acetylthio)buta-1Z,3Z-diene (4). 1,4-Bis(acetylthio)buta-1Z,3Z-diene (4) could be obtained by placing 6.5 g (22 mmol) of compound 3 (Fig. 6) in a 3-neck round bottom flask equipped with a mechanical stirrer, a dry ice cooler, a funnel for solids and a 3-way valve (connection to a balloon, the apparatus and a tube to the fume hood) and slowly condensate 300 ml of ammonia at −78 °C into the reaction flask. After adding 4.1 g (186 mmol) of freshly cut sodium the colour turned to dark blue and the reaction mixture was stirred at −78 °C overnight. Adding 20 g of ammonium chloride under a mild stream of nitrogen dried over phosphorous pentoxide the blue colour began to disappear. To remove ammonia from the liquid remaining in the flask, the cooling bath was replaced by a water bath and the gas was allowed to evaporate. After hydrolysing with 200 ml of deionised water the reaction mixture was extracted 4 times with 100 ml of diethyl ether where no hydrolysis product could be detected by GC-MS. The water phase was transferred into a 500 ml round bottom flask, then we added 40 ml (424 mmol) of acetic anhydride. After cold filtration the product was washed twice with cold water to give a pale yellow solid which was dried overnight at 50 °C and 50 mbar. The yield was 78% and the purity was determined by GC-MS⁷⁷ to be 90%.

1,2-Dithiine (5). 1,2-Dithiine (5) could be obtained according to the procedure of ref. 82. Only the workup differs from the method described therein. After extraction the solvent was removed at −78 °C and 10⁻³ mbar. The red oil was then condensed in a trap cooled with liquid nitrogen to give the expected 1,2-dithiine. This product was used without further purification. The substance is light sensitive. The yield was 55% and the purity as determined by GC-MS was about 95%. Due to photochemical and further reactions impurities develop over time in the samples used for the spectroscopy. In particular, thiophene was observed by infrared spectroscopy. However, in the high resolution spectra the identity of the lines assigned to 1,2-dithiine was obvious from the spectra, also confirming the identity of the main substance.

3.2 GHz measurements of 1,2-dithiine

The ground state pure rotational spectra of 1,2-dithiine have been recorded between 65 and 117 GHz using our GHz spectrometer described in detail in ref. 103 and only slightly modified here. It consists of an Agilent synthesizer (E8257D PSG, Agilent Technologies Option 520 UNX) working in the range 250 kHz to 20 GHz with a frequency resolution of 0.001 Hz and frequency locked to a 10 MHz-rubidium standard (FS725, Stanford Research Systems) for short-term stability and to a GPS standard (TM-4, Spectrum Instruments Inc.) for long-term stability. The signal is frequency multiplied by a factor of 6 (S10MS-AG, OML Inc.) and coupled out to free space via a conical W-band antenna (QWH-WCRR, QuinStar Technology Inc.) leading to an operation in the frequency range of 67 to 118 GHz. The beam is focused through Teflon lenses into a 2.5 m path length glass cell equipped with two Teflon windows and mounted in a Brewster angle to the incoming and outcoming beam. Finally the beam is detected with a two-channel indium antimonide (InSb) bolometer [QFI/2(2),QMC Instruments Ltd.] after passing through a second Teflon lens. All transitions were measured using the frequency modulation technique (second harmonic lock-in detection) with a modulation frequency of 50 kHz.

The gas sample was prepared by introducing the vapour at a pressure of approximately 0.025 mbar (measured using two Baratron gauges 627B.1T and 627B11M, MKS Instruments) from a liquid sample to the 2.5 m glass cell, which was covered by an aluminum foil to prevent 1,2-dithiine from photochemical decomposition due to exposure to daylight. A typical measured linewidth (full-width-at-half-maximum) at 100 GHz was close to 200 kHz. Fig. 7 shows a survey spectrum in the GHz range.

Fig. 7 Rotational survey spectrum of the ground state of 1,2-dithiine in the range 3.23–3.42 cm⁻¹ (97–102.5 GHz). The spectrum is frequency modulated. One recognizes the 600 MHz spacing of the strongest adjacent K_a lines. The pressure was p = 0.025 mbar, the path length was l = 2.5 m and the temperature was T = 295 K. The upper trace shows the measured rotational spectrum and the lower trace a simulation using the parameters in Table 9.

3.3 FTIR measurements of 1,2-dithiine

The FTIR spectrum of 1,2-dithiine has been recorded using our Fourier transform spectrometer Bruker IFS 125 HR Zürich prototype (ZP 2001) in the region 600–3200 cm⁻¹. The Zurich FTIR setup is described in detail in ref. 106–109. The effective resolution ranged from 0.001 cm⁻¹ in the range 600–1000 cm⁻¹, close to the Doppler width (FWHM), which ranged from 0.0006 cm⁻¹ at 500 cm⁻¹ to 0.0036 cm⁻¹ at 3000 cm⁻¹. A White-type cell with path lengths ranging from 3.2 m up to 19.2 m and a 3 m glass cell were used for room temperature (295 K) measurements. The sample pressure measured using a MKS Baratron was varied between 0.1 and 1 mbar. All spectra were self-apodized. Table 6 shows the measurement parameters. Apertures from 0.8 mm to 1.0 mm were used. The spectra were calibrated with OCS (600–1300 cm⁻¹)^104,109 and water (1280–1365 cm⁻¹).¹⁰⁵ In general, the calibrated wavenumbers are estimated to be accurate within 0.0002 cm⁻¹. The resolutions specified in Table 6 correspond to nominal bandwidths defined as 0.9 × 1/d_MOPD.¹⁰⁹ The data given in the ESI† correspond to calibrated wavenumbers. Due to the instability of 1,2-dithiine some impurities arising from reaction products such as thiophene have been identified in the spectra. As a survey we also measured further spectra covering the range 800–5000 cm⁻¹ at low resolution.

Table 6 Measurement parameters for the 600–3600 cm⁻¹ region of the infrared spectrum of dithiine

Region/cm^−1	600–1000	1150–2200	2500–3600
Resolution/cm^−1	0.0008	0.001	0.001
Windows	KBr	KBr	KBr
Source	Globar	Globar	Tungsten
Detector	MCT	MCT	InSb
Beam splitter	Ge:KBr	Ge:KBr	Si:CaF2
Opt. filter/cm^−1	550–1000	900–2500	550–5000
Aperture/mm	1.0	0.8	0.8
ν mirror/(kHz)	40	40	40
Calib. gas	OCS^104	H2O^105	OCS^104

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4 Rotational and rovibrational assignments

4.1 General aspects and symmetry considerations

Table 7 includes the character tables for the point group C₂, the molecular symmetry groups M_S2 and M_S4 including the possibility of tunneling following Longuet-Higgins¹¹⁴ and the systematic notation of ref. 110 for the symmetry species indicating parity explicitly by + and −. We also show in Table 7 the notation for the C_2v symmetry group of the transition state and the induced representations Γ(M_S2)↑(M_S4) which indicate the symmetries of the tunneling sublevels. The nuclear spin statistical weights g arising from the 2⁴ = 16 nuclear spin functions of the 4 protons (H₇, H₈, H₉, H₁₀) combined with the motional functions to Pauli allowed states are indicated for each symmetry. The combination of the rotational quantum numbers K_a and K_c for the asymmetric top in a totally symmetric vibronic level is given as well in Table 7, which also includes the number of vibrational modes of each symmetry (n_Γv). According to the C₂ symmetry the modes with A symmetry generate a-type transitions and the modes with B symmetry generate b- and c-type transitions. In a more detailed description the modes with A symmetry can split into A₁ and A₂ symmetry and the modes with B symmetry into B₁ and B₂ symmetry in the planar transition state as shown in Tables 2 and 4. As a consequence the fundamentals with B symmetry generate mainly c-type transitions and weak b-type transitions if they correspond to out-of-plane modes in the planar transition state (B₁) or stronger b-type transitions than c-type transitions if they correspond to in-plane modes (B₂) in the planar transition state. We have also calculated the electric dipole transition moment components μ_a, μ_b and μ_c for the vibrational fundamentals in order to assist the assignments. The fundamentals with A symmetry generate weak a-type transitions if the modes correspond to out-of-plane modes (A₂) in the planar transition state because modes with A₂ symmetry are not infrared active. This is important with respect to the ring puckering mode ν₁₃ which would correspond to an A₂ mode in the planar configuration. For that reason the fundamental ν₁₃ is extremely weak. However, fundamentals with A symmetry generate strong a-type transitions if they correspond to in-plane modes in the planar transition state.

Table 7 Character table for the C₂ and M_S2 symmetry groups (upper table) and character table for the C_2v and S₂*(M_S4) symmetry groups (lower table). For the molecular symmetry group (M_S4) (αβ) corresponds to the combined permutation of the symmetrically equivalent nuclei of the molecule and the upper right index of the symmetry species indicates parity (+ or −)^110–113

C 2		E	C 2
	M S 2	E	(αβ)	μ	J	K a K c	n Γv	n Γs	g	Γ(MS2)↑MS4
A	A	1	1	μ z	J z	ee, eo	13	10	10	A^+ + A^−
B	B	1	−1	μ x , μy	J x , Jy	oo, oe	11	6	6	B^+ + B^−

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C 2v		E	C 2	σ xz	σ yz
	S 2*(MS4)	E	(αβ)	(αβ)*	E*	μ	J	K a K c	n Γv	n Γs	g
A1	A^+	1	1	1	1	μ z		ee	9	10	10
A2	A^−	1	1	−1	−1		J z	eo	4	0	10
B1	B^−	1	−1	1	−1	μ x	J y	oo	3	0	6
B2	B^+	1	−1	−1	1	μ y	J x	oe	8	6	6

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4.2 Assignment procedure

Fig. 7 shows part of the ground state rotational spectrum of 1,2-dithiine in the range 97–102.5 GHz. The assignment of the rotational transitions has been performed using the Pgopher program,^115,116 a Loomis–Wood assignment program, and the WANG program¹¹⁷ up to J = 50. The infrared spectral region between 600 cm⁻¹ and 1000 cm⁻¹ as illustrated in Fig. 8 shows one strong absorption feature, and the region between 1150 cm⁻¹ and 2200 cm⁻¹ shows two strong absorption bands of 1,2-dithiine at 1310 cm⁻¹ and 1590 cm⁻¹. The strong sharp lines visible in Fig. 8 are due to water absorption lines. In the following description the term ‘subband’ or ‘series’ describes the transitions belonging to one K_a or K_c value. The assignment of the observed rovibrational transitions belonging to a particular subband consisting of P and R branches has been carried out efficiently using an interactive Loomis–Wood (LW) assignment program¹¹⁸ (see also ref. 109 and 119) previously designed for linear molecules.^120–122 This limitation is, however, only superficial as the assignment program has been used successfully for several asymmetric top molecules: CHClF₂,^123,124 CHCl₂F,^108,125 CDBrClF,¹⁰⁶ pyridine, and pyrimidine.¹²⁶

Fig. 8 Survey spectrum of 1,2-dithiine in the range 600–1000 cm⁻¹ (upper panel), 1150–2200 cm⁻¹ (middle panel) and 2600–3200 cm⁻¹ (third panel). The 1150–2200 cm⁻¹ range (middle panel) is dominated by strong water absorption lines. The decadic absorbance lg(I₀/I) is shown as a function of wavenumber. The pressure was p = 0.7 mbar, the path length was l = 3 m and the temperature was T = 295 K. The standing wave pattern generates the broad base line waves in the 600–1000 cm⁻¹ range. However, the width of these waves is 100 to 1000 times broader than the effective linewidth and therefore does not disturb the high resolution analysis.

4.3 Ground state of 1,2-dithiine

1,2-Dithiine is an asymmetric rotor with C₂ symmetry, its only permanent electric dipole moment component lies along the a-axis (μ_a = 1.85 D⁷⁵), allowing transitions obeying a-type selection rules (eo ↔ ee and oe ↔ oo) for rotational transitions. The assignment of the rotational transitions observed here was assisted by using previously reported spectroscopic parameters from Gillies et al.,⁷⁵ which included only 15 transitions up to J = 5, K_a = 5. A total of 560 transitions were newly assigned up to J ≤ 50 and K_a ≤ 19 in the range of 65 to 117 GHz in the present work. The overview spectrum in Fig. 7 illustrates the rotational structure of 1,2-dithiine in the ground state such as the stronger ^aR-type transitions progressing nicely with separations between adjacent K_a lines by about 600 MHz, while the ^aQ-type transitions are overshadowed and clustered.

4.4 The ν₂₂ band of 1,2-dithiine

The ν₂₂ band of 1,2-dithiine at 623 cm⁻¹ (Fig. 8) is the strongest band in the IR absorption spectrum of 1,2-dithiine and consists of c- and b-type transitions. A comparison with the FTIR spectra of the already analysed aromatic molecules naphthalene,¹²⁷ azulene¹²⁸ and phenol¹²⁹ illustrates that the ν₂₂ fundamental of 1,2-dithiine corresponds to the strongest out-of-plane fundamentals of these aromatic molecules which are also dominated by strong Q branches due to c-type transitions. However, the planar aromatic molecules show no b-type transitions according to the C_2v symmetry. The c- and b-type transitions are not split until K_c < 11. The c-type series were identified as P, R and Q branches with ee ↔ oe and eo ↔ oo transitions. In addition, some b-type lines (around 20) above K_c ≥ 11 have been assigned with ee ↔ oo and eo ↔ oe transitions. The spacing between two transitions of a c-type series is approximately 2A. The assignments were checked by comparison with ground state combination differences. Loomis–Wood diagrams of this band are shown in Fig. 9. The patterns of P and R branches of the c-type series are clearly visible in Fig. 9. The upper diagram shows the P branch and the middle diagram the R branch. The visible series are transitions where c- and b-types fall together. The lines have been assigned up to J ≤ 78 and K_a ≤ 73 and K_c ≤ 34.

Fig. 9 Loomis–Wood (LW) diagrams of the c-type transitions of the ν₂₂ band. The upper part shows the P branch and the middle part the R branch. The different K_c series are clearly visible which are unsplit b- and c-type transitions. The lower part shows the Loomis–Wood diagram of the b-type transitions of the ν₁₇ band of 1,2-dithiine. The different K_a series of the P and R branches as well as the local resonances (crossings) are clearly visible.

4.5 The ν₁₇ band of 1,2-dithiine

The observed spectrum of the ν₁₇ fundamental of 1,2-dithiine at 1308 cm⁻¹ clearly shows a characteristic b-type contour within the P and R branches and a c-type contour within the sharp center Q branch as shown in Fig. 8 (middle panel) and in more detail in Fig. 13. The shape of the b-type series is very similar to the shape of the b-type series assigned in the ν₁₅ fundamental of fluorobenzene.¹³⁰ However, due to the C_2v symmetry of fluorobenzene the ν₁₅ fundamental shows no c-type transition and therefore no visible Q branch. The ν₁₇ band was chosen to be analyzed first, as a series of regularly spaced spectral patterns can be identified even through preliminary visual inspection. A simulated spectrum was generated using the estimated band center from inspecting the recorded high resolution data and the ground state spectroscopic parameters obtained from the analysis of the rotational spectra measured in the present work. The first attempt to assign the observed spectrum was based on the comparison with this simulated spectrum. The preliminarily assignments of the transitions were then confirmed by comparing the ground state combination differences with the GHz data. The simulated spectrum was refined by newly added assigned transitions and this process was continued in an iterative fashion until a few progressions of transitions were assigned. However, as the Loomis–Wood diagram of the P branch in Fig. 9 (lower part) illustrates, the b-type subbands are perturbed from J = 32 and onwards and therefore have only been assigned up to J ≤ 32. A closer look at the Loomis–Wood diagram indicates that the ν₁₇ state is perturbed from at least two states: one with higher energy and smaller rotational C constant and the second with lower energy and larger rotational C constant. Only b-type transitions were assigned. The c-type transitions are not rotationally resolved in this band.

4.6 The ν₃ band of 1,2-dithiine

The ν₃ band centered at 1544.9 cm⁻¹ exhibits a strong absorption feature in the recorded spectrum as shown in the FTIR survey spectrum in Fig. 8 (middle trace). The strong sharp lines are due to water absorption. The assignment as the ν₃ fundamental is consistent with the predicted intensity of this band being the highest in the region of 1400–1600 cm⁻¹ (Table 2). A few strong a-type transitions with low K_c values were readily assigned with the aid of Loomis–Wood diagrams and simulated spectra using ground state constants and the estimated band center allowed a better prediction of more transitions to be assigned. As for the ν₁₇ band, local perturbations were encountered and the assignment of lines was carried out up to J = 40 only. Also, the relatively noisy background prevented the transitions with K_a larger than 10 from being assigned.

5 Results of the spectroscopic parameters and discussion

The rovibrational analysis has been carried out using Watson's A reduced effective Hamiltonians in the I^r representation (with the definitions of axes according to Fig. 2) up to sextic centrifugal distortion constants:^117,131 The angular momentum operators are given by Ĵ² = Ĵ_x² + Ĵ_y² + Ĵ_z² and Ĵ_± = Ĵ_x ± iĴ_y. The I^r representation was chosen to reduce correlations during the fit. The spectroscopic data were analyzed using the WANG program described in detail in ref. 117 and the SPFIT program.¹³² The spectroscopic constants were fitted for each band separately according to the A-reduction.

5.1 Rotational analysis and parameters of the ground state of 1,2-dithiine

560 rotational transitions in the GHz region were combined with the 15 lines reported in ref. 75 in a joint least squares analysis. Using only the three rotational constants and five centrifugal distortion constants included in the adjustment, the overall d_rms of the fit consisting of 575 rotational transitions is remarkably low (less than 7 kHz, or 0.0000002 cm⁻¹). Table 8 lists all the spectroscopic parameters of the ground state of 1,2-dithiine. The rotational constants A, B and C obtained here compare well with those determined in the earlier microwave work. However, the previous study did not have a sufficient number of transitions (just a total of 15 lines) to accurately determine centrifugal distortion constants. In our work we observed more than 500 transitions and also covered a large range of the rotational spectrum with quantum numbers extending to J ≤ 50 and K_a ≤ 19, which led to a significant improvement in these centrifugal distortion constants. It was possible to observe intensity alternation in the spectra due to the nuclear spin statistical weights. According to Table 7 the expected intensity alternation in the rotationally resolved spectra of 1,2-dithiine is ee : eo : oo : oe = 10 : 10 : 6 : 6. Fig. 10 shows as an example a comparison of the strong R branch lines (upper trace) with simulations using the spectroscopic constants from Table 8 and using nuclear spin statistical weights (middle trace) and simulations without these weights (lower trace). The lower panel displays an enlarged part of the upper panel. Even for congested lines the effects of nuclear spin statistical weights are visible, confirming the assignments. This complete set of ground state rotational parameters (Table 8) plays an important role in the subsequent analysis of the dense rovibrational spectra in the preliminary assignment stage, especially when the upper state is perturbed.

Table 8 Equilibrium rotational constants A_e, B_e, C_e and spectroscopic parameters in cm⁻¹ of the ground state of 1,2-dithiine in the A-reduction. The uncertainties are listed in parentheses in terms of 1σ in units of the last digits

	Theory (MP2/cc-pVTZ), t.w.	Expt. t.w.^a
a Equilibrium rotational constants (Ae, Be, Ce) from vibrational perturbation theory (t.w.) and experimental ground state rotational constants (A0, B0, C0) from our present GHz and FTIR experiments. (t.w.) (“semi-experimental” values). b This work including the lines of ref. 75.
A e/cm^−1	0.111061	0.111544^a
B e/cm^−1	0.103524	0.103980^a
C e/cm^−1	0.058925	0.058852^a

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	Ground state^75	t.w. ground state^b
A 0/cm^−1	0.1109554 (6)	0.110955440 (2)
B 0/cm^−1	0.1034997 (6)	0.103499682 (2)
C 0/cm^−1	0.05857394 (2)	0.058573904 (1)
Δ J /10^−6 cm^−1	0.0007 (2)	0.010663 (3)
Δ JK /10^−6 cm^−1	0.0037 (7)	0.013592 (9)
Δ K /10^−6 cm^−1	−0.002 (1)	−0.005986 (5)
δ J /10^−6cm^−1	0.0002 (1)	0.002249 (1)
δ K /10^−6 cm^−1	0.00023 (3)	0.015793 (1)
N	15	560+15
d rms/cm^−1	0.0000001	0.0000002
J max for fit	6	50
J max assigned	6	50
K a max	3	19
K c max	5	39

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Fig. 10 A comparison of the a-type rotational transitions in the ground state of 1,2-dithiine measured at 295 K (top trace) with a simulation at 295 K (lower trace). The transitions between the lower J′′, K_a′′, K_c′′ and the upper state J′, K_a′, K_c′ are labeled as J′, K_a′, K_c′ ← J′′, K_a′′, K_c′′.

5.2 Rovibrational analysis and parameters of the ν₂₂ band of 1,2-dithiine

According to the calculated transition moments μ_b = −1.33 × 10⁻³ D and μ_c = −2.20 × 10⁻³ D the b-type absorption lines are predicted to have less than 40% of the strength of the corresponding c-type transitions. Around 1500 b-type transitions and around 4000 c-type transitions have been used in the adjustment. However, most of these b-type transitions, as already mentioned, are from non-split c-type and b-type transitions for K_c ≤ 11 (around 1480 lines). Pure b-type transitions (around 20) have been assigned and used near the center of the Q branch region. At higher J levels the b-type transitions are too weak to be detected. Pure c-type transitions (around 2500) have been assigned in the entire spectrum. In general, b- and c-type transitions up to J ≤ 78, K_a ≤ 73 and K_c ≤ 34 were used for the fit of the spectroscopic constants. Global perturbations were observed, perhaps through interactions with the 2ν₂₃ state or 3ν₂₄ state or possibly also a combination with ν₁₃ and even ν₁₂. The “polyad” between 600 and 700 cm⁻¹ comprises 8 levels. The constants of the ground state have been held fixed to the values in Table 8 and the constants of the ν₂₂ have been adjusted using the least squares analysis. Due to the interactions some of the sextic constants could be determined for ν₂₂ as Table 9 shows, although the corresponding spectroscopic parameters have an effective meaning only. Around 4000 transitions were used in the fit with a d_rms = 0.00026 cm⁻¹.

Table 9 Spectroscopic constants in cm⁻¹ of the vibrational states ν₁₇, ν₂₂ and ν₃ of dithiine in the A-reduction. If there are no uncertainties listed (in parentheses in terms of 1σ in units of the last digits), the parameter was fixed during the fit at the value indicated

	ν 17	ν 22	ν 3
[small nu, Greek, tilde] 0	1308.87327 (38)	623.093915 (25)	1544.900370 (51)
A	0.1110980 (30)	0.110782038 (46)	0.1110040 (57)
B	0.1033660 (24)	0.103339231 (78)	0.1032180 (47)
C	0.05857310 (19)	0.05848408 (11)	0.05849980 (16)
Δ J /10^−6	0.000690 (20)	−0.0081798 (81)	0.00910 (12)
Δ JK /10^−6	0.001400 (92)	0.03133 (31)	0.0090 (11)
Δ K /10^−6	−0.0005986	−0.01207 (24)	−0.0130 (27)
δ J /10^−6	0.0002249	−0.002719 (40)	0.0002248
δ K /10^−6	0.0015793	0.01488 (12)	0.0015792
Φ J /10^−12	—	−1.181 (14)	—
Φ JK /10^−12	—	1.410 (44)	—
Φ K /10^−12	—	−0.492 (30)	—
ϕ J /10^−12	—	−0.4683 (57)	—
ϕ JK /10^−12	—	0.218 (15)	—
ϕ K /10^−12	—	−0.421 (19)	—
N	1006	4056	875
d rms/cm^−1	0.000137	0.00026	0.00025
J max for fit	32	78	40
J max assigned	32	78	40
K a max	12	73	10
K c max	30	33	40

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A comparison of the measured spectrum with the simulated spectrum calculated using the parameters listed in Tables 8 and 9 is shown in Fig. 11 and 12 for the R and Q branch regions. The upper panel of Fig. 11 shows the complete ν₂₂ fundamental (upper trace) and a simulation (lower trace). The lower panel of Fig. 11 is an enlarged region of the measured R branch lines (upper trace). The lower traces show simulations with and without nuclear spin statistical weights. As the simulation illustrates the intensity effect due to nuclear spin statistical weights is not visible in this part of the spectrum. The agreement between the measured and the simulated spectra is of comparable quality in both cases. Differences are due to numerous hot band lines not shown in the simulation and the relatively high noise level in the experiment.

Fig. 11 A comparison of the ν₂₂ band of 1,2-dithiine measured at 295 K (upper panel, top trace) with a simulation at 295 K (upper panel, lower trace). The upper trace in each panel displays the observed spectrum and the lower traces display the simulation with and without nuclear spin statistical weights. The middle and lower panels are enlargements of the sections of the R branches outlined in the upper panel. See parameters in the figure. The decadic absorbances lg(I₀/I) are shown as a function of wavenumber, with maximum peak values of lg(I₀/I) = 2.00 for the upper spectrum (cutoff of the Q-branch lines) and 0.7 for the lower spectrum (3 m path length, 0.7 mbar, Δ[small nu, Greek, tilde] = 0.001 cm⁻¹).

Fig. 12 A comparison of the Q branch region of the ν₂₂ band of 1,2-dithiine measured at 295 K (upper panel, top trace) with simulation at 295 K (upper panel, lower trace). The upper trace in each panel displays the observed spectrum and the lower trace displays the simulation. The middle and lower panels are enlargements of the sections of the Q branches outlined in the upper panel. The numerous Q branches resulting from hot bands are visible in the experimental spectrum. See parameters in the figure. The decadic absorbances lg(I₀/I) are shown as a function of wavenumber, with maximum peak values of lg(I₀/I) = 1.20 for the upper and middle spectra (cutoff of the Q-branch lines) and 0.7 for the lower spectrum (3 m path length, 0.7 mbar, Δ[small nu, Greek, tilde] = 0.001 cm⁻¹).

Only 20% of the molecules are in the ground state at 295 K, while 80% molecules generate hot bands. The Q branch region in Fig. 12 (upper trace) illustrates numerous hot bands. The determination of the band center is only possible through the use of a Loomis–Wood assignment program. The other two panels display an enlargement of the Q branch region. As one can see in the third panel there is a remarkably good agreement between the simulation and the experimental spectrum, considering the important hot band contributions. A partial rovibrational analysis was possible for one of the hot bands.

5.3 Rovibrational analysis and parameters of the ν₁₇ band of 1,2-dithiine

As the ab initio calculations shown in Table 2 suggest, the ν₁₇ band is of B symmetry which permits both b- and c-type transitions with an intensity ratio of approximately 8 : 1 according to the calculations. A total of 1006 transitions obeying b-type selection rules (eo–oe and ee–oo) were assigned and analysed in a least squares adjustment varying only the three rotational constants and five centrifugal distortion constants of ν₁₇ with a d_rms = 0.00014 cm⁻¹ but fixing the parameters for Δ_K, δ_k and δ_j to the ground state values. Although perturbations have been encountered from J = 32 onwards, there is good agreement between the simulated spectrum based on the parameters shown in Table 9 and the observed spectra as shown in Fig. 13. Only unperturbed lines up to J ≤ 32 have been used in the least squares adjustment. However, the assignment of the perturbed lines above J > 32 has been checked by ground state combination differences calculated from the perturbed P and R branch lines of ν₁₇.

Fig. 13 A comparison of the ν₁₇ band of 1,2-dithiine measured at 295 K (upper panel, top trace) with a simulation at 295 K (upper panel, lower trace). The upper trace in each panel displays the observed spectrum and the lower trace displays the simulation. The middle and lower panels are enlargements of the sections of the R branches outlined in the upper panel. The Q branch region consists of numerous Q branches resulting from hot bands. See parameters in the figure. The decadic absorbances lg(I₀/I) are shown as a function of wavenumber, with maximum peak values of lg(I₀/I) = 1.60 for the upper spectrum (cutoff of the Q-branch lines) and 0.6 for the two lower panels.

The strong Q branch as shown in Fig. 13 with a calculated |μ_b/μ_c| = 2.8 is due to c-type transitions according to the prediction. However, the Q branch region is more complicated due to numerous hot bands. It was not possible to identify single c-type transitions because of the strong congestion of lines which is in agreement with the predicted transitions. Some of the hot band lines can be grouped into series using the Loomis–Wood diagram, however an adjustment of the spectroscopic parameters is not yet possible. Considering a possible assignment of the band observed at 1308.9 cm⁻¹ to ν₄ predicted at 1350 cm⁻¹ by the ab initio calculations reported in Table 2, we have also conducted an analysis using a-type selection rules in the assignment. While this assignment gives a satisfactory fit as well, the shape of the Q branch in the simulation is much broader and differs significantly from the experimental shape. Therefore the assignment as ν₁₇ seems to be well justified. As in the case of ν₂₂, our data are insufficient to make use of nuclear spin statistical weights in the assignment of ν₁₇. We have also tested fits with a variation of Δ_K (and also other parameters) being systematically fixed at a series of values. The conclusion from these tests is that the shape of the Q branch can be best simulated if all quartic constants are fixed at the values of the quartic constants of the ground state. However, this fixing leads to slightly larger d_rms = 0.0005 cm⁻¹ which is still below the instrumental resolution. Thus the spectroscopic parameters have clearly a meaning as “effective parameters” only which it does, however, not affect our conclusions.

5.4 Rovibrational analysis and parameters of the ν₃ band of 1,2-dithiine

Consistent with A symmetry under C₂, 875 transitions from the ν₃ band were assigned obeying a-type selection rules eo–ee and oe–oo (Fig. 14). These lines were fit with the variation of one more spectroscopic constant than used for the ν₁₇ band. To minimize the possible effects from the observed perturbation in this band, only transitions with J(max) = 40 (onset of local resonances) were assigned. There is good agreement between the observed and simulated spectra in unperturbed regions and the band provides a good example for a totally symmetric fundamental.

Fig. 14 A comparison of the ν₃ band of 1,2-dithiine consisting of a-type transitions measured at 295 K (top trace) with a simulation at 295 K (lower trace). The upper trace displays the observed spectrum and the lower trace displays the simulation. See parameters in the figure. The decadic absorbances lg(I₀/I) are shown as a function of wavenumber, with the maximum peak values of lg(I₀/I) = 0.3 for the spectrum (cutoff of the Q-branch lines).

6 Discussion and conclusions

Designing and selecting molecules towards specific properties for a variety of applications is a well-known task in chemistry.^133,134 The theoretical selection and spectroscopic characterization of a chiral molecule suitable for the measurement of the parity violating energy difference Δ_pvE between the enantiomers by the method proposed by us three decades ago⁵² is a new, very great challenge, given the current technical possibilities.⁷⁰ The present work contributes some major results towards this goal, both theoretical and spectroscopic, for the moderately complex molecule 1,2-dithiine.

From the theoretical point of view, we have shown that in the ground and low energy rovibrational states parity violation clearly dominates over tunneling in the quantum dynamics of 1,2-dithiine by many orders of magnitude. Thus the reaction enthalpy Δ_pvH₀ for the stereomutation reaction (1) is, in principle, a measurable quantity for 1,2-dithiine, similar to the simpler molecule dichlorodisulfane (Cl–SS–Cl) which we had studied before,⁷¹ and quite different from the case of disulfane HSSH.⁷² A detailed study of the conformation dependence of parity violation in 1,2-dithiine, indeed, shows that this is a particularly favorable molecule, as the maximum of the absolute value of the parity violating potential (about 6 × 10⁻¹² (hc) cm⁻¹ corresponding to Δ_pvE^el(q_max) = 12 × 10⁻¹² (hc) cm⁻¹ in Fig. 4) is found to be close to the equilibrium geometry. The expectation value Δ_pvE₀ in the vibrational ground state has been calculated approximately by means of the ground state of the lowest quasiadiabatic channel for torsion and differs by less than 10% from the equilibrium value Δ_pvE^el(q_e) = 11.44 × 10⁻¹² (hc) cm⁻¹ (eqn (4)), very close to the maximum value. Again, 1,2-dithiine might be compared to ClSSCl in this respect where the predicted Δ_pvE₀ is smaller by a factor of about 8 (1.35 × 10⁻¹² (hc) cm^−1 71), because of the relatively unfavorable position of the equilibrium geometry, far removed from the position of the maximum in the parity violating potential as a function of torsional angle with Δ_pvE^el(q_max) ≈ 12 × 10⁻¹² (hc) cm⁻¹, which is actually quite similar to the maximum value in 1,2-dithiine. Also in our recent study of Δ_pvE in trisulfane, the predicted Δ_pvE₀ = 1.6 × 10⁻¹² (hc) cm⁻¹ is found to be much smaller than the maximum value as a function of the torsional (stereomutation reaction) coordinate (Δ_pvE^el(q_max) = 8.2 × 10⁻¹² (hc) cm⁻¹). Another, even less favorable example of this kind, which we encountered in the past, is 1,3-difluoroallene which in many other spectroscopic aspects is quite easily accessible.⁷⁶

Our theoretical results also provide further evidence that simple scaling laws, as repeatedly discussed since the early theoretical work of Bouchiat and Bouchiat^62,63 for atoms (with parity violating effects increasing with about the third power of the atomic number Z), are of limited significance in predicting parity violation in molecules. Many earlier discussions have pointed out that an approximate scaling with Z⁵ can be derived for parity violation in molecules^{31,35,37,61,83} with the implicit assumption that the number of neutrons in a typical nucleus is proportional to Z as well. However, the strong dependence of molecular parity violation upon conformation can easily change the effects by an order of magnitude and more, as seen from the examples given above. Examples are known, where a conformation change in a substituent without changing the stereomutation coordinate can even change the sign of the parity violating potential without changing the enantiomeric nature.^33,35,135 Indeed, in strong contrast to atomic parity violation the molecular parity violation is characterized by a complicated (3N − 6)-dimensional effective parity violating potential hypersurface to be added to the ordinary parity conserving potential hypersurface as a perturbation of different symmetry,^9,61 and our results of 1,2-dithiine in relation to other molecules studied demonstrate the complicated consequences of this. It has been pointed out⁶¹ that a Zⁿ scaling might be applied to a series of homologous molecules either at their individual equilibrium geometries or at identical (non-equilibrium) geometries or else at geometries related to the maximum absolute values of |Δ_pvE^el| of the different molecules, with quite different results in each case. As an alternative to the effective parity violating potential one might also consider the tensor components of the parity violating potential⁶¹ or values corresponding to the main chiral axes³⁶ in relation to a Zⁿ scaling. In practice, simple rules including Zⁿ-scaling and geometrical factors might be derived as the first selection criteria,⁴⁸ but in the end, for individual molecules to be selected for the very demanding experiments, quantitative calculations as carried out here for 1,2-dithiine including the conformation dependence and tunneling processes are required. In this respect, 1,2-dithiine turns out to be quite a favorable candidate for experiments.

Tunneling in the ground state has been shown in the present work to be completely suppressed for 1,2-dithiine with

ΔE_± < 10⁻²⁴ (hc) cm⁻¹ ≪ ΔE_pv ≃ 10⁻¹¹ (hc) cm⁻¹ (6)

similar to ClSSCl, for example.⁷¹ However the barrier for stereomutation in 1,2-dithiine is much lower than in ClSSCl. Thus in 1,2-dithiine tunneling can become important above about 2300 cm⁻¹, making states of well-defined parity accessible in the infrared range, where powerful high resolution lasers are available for parity selection experiments⁷⁰ and where spectroscopic analyses are possible, making the “infrared route” towards parity violation possible. This does not exclude the possibility of also using the route via excited electronic states possibly with rovibrational preselection, as suggested for 1,3-difluoroallene,⁷⁶ but too little is known at present concerning the high resolution electronic spectra of 1,2-dithiine for which we plan a more detailed study as well.

We have shown here that high resolution analyses are possible in the infrared range for 1,2-dithiine, with three strong fundamentals ν₂₂ at 623.094 cm⁻¹, ν₃ at 1544.900 cm⁻¹ and ν₁₇ at 1308.873 cm⁻¹. The latter band might be used as a starting point for a stepwise two-photon excitation to the 2ν₁₇ range at 2600 cm⁻¹, where tunneling can be important, if couplings to tunneling sublevels of well-defined parity can be identified. In the future also high resolution analyses of the 2ν₁₇ absorptions or the CH stretching polyad at around 3000 cm⁻¹ can become possible. Indeed, the overtone of the fundamental ν₃ studied here is predicted to show a strong Fermi-resonance with the CH-stretching fundamentals using our ab initio calculations. In all cases a major remaining challenge will be the analysis of rovibrational couplings to tunneling sublevels of well-defined parity. Some first evidence for rovibrational couplings has been seen in the present analysis of ν₁₇. At higher energies intramolecular rovibrational couplings are very likely,¹¹⁹ given the high density of vibrational states, which is about 50 states per cm⁻¹ interval at 2600 cm⁻¹ and 1300 states per cm⁻¹ interval at around 3000 cm⁻¹. The corresponding spectroscopic analyses of the resulting complex spectra can be successful, however, with existing high resolution infrared laser spectroscopy of molecular beams, achieving even hyperfine resolution in excited vibrational states of ammonia, for example, (i.e. Δν < 1 MHz around ν = 100 THz or Δ[small nu, Greek, tilde] < 0.00003 cm⁻¹ at [small nu, Greek, tilde] = 3300 cm^−1 70). Double resonance with IR-lasers⁷⁰ and GHz spectroscopy¹⁰³ can also be helpful in this context. We have furthermore carried out extensive calculations on various deuterated isotopomers of 1,2-dithiine, where the CD infrared chromophore makes the wavenumber range just above the tunneling switching range accessible. These results are presented in the ESI† as well. The study of the lowest frequency fundamentals by synchrotron-based THz (FTIR) spectroscopy is in progress.

The present work thus clearly opens the route towards successful spectroscopic experiments to measure the parity violating energy difference Δ_pvE in 1,2-dithiine with a rather short time τ_pv = 1.5 s for complete parity change in this molecule; the initial time evolution in the ms range would be clearly detectable by the now available techniques.⁷⁰ While a simpler molecule with fewer atoms would certainly still be desirable to complete the necessary spectroscopic groundwork, 1,2-dithiine has the advantage of a relatively rigid structure apart from the simple spin 1/2 protons with only the rather light C and S nuclei with zero spin, a simple nuclear structure, which is almost ideal for highly accurate theoretical analyses in terms of fundamental parameters of the standard model of particle physics from experimental results, once available.⁹

Acknowledgements

Our work is financially supported by an advanced grant from the ERC, the Schweizerischer Nationalfonds and the ETH Zürich. The research leading to these results had in particular also received funding from the European Union's Seventh Framework Programme (FP7/2007–20013) ERC grant agreement no. 290925. Very substantial help from Guido Grassi and Fabienne Arn in the synthesis work is gratefully acknowledged. We also acknowledge the help from and enjoyed the discussion with Andreas Schneider and Andres Laso and also support from the COST project MOLIM.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6cp01493c

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