Introduction

After many years of intensive experimental and theoretical studies, the water dimer is now a textbook example of hydrogen bonding. Not only its structure with a near-linear hydrogen bond and its binding energy are important, but also the observation that it has eight equivalent hydrogen-bonded equilibrium structures separated by relatively small energy barriers. This implies that the dimer can quantum-mechanically tunnel between the corresponding minima in the potential energy surface (PES). The tunneling frequencies of H2O dimers and larger clusters between multiple equivalent equilibrium structures, as well as the frequencies of various intermolecular vibrations, have been the subject of numerous experimental studies by high-resolution molecular beam spectroscopy1,2,3,4,5,6. These tunneling and vibrational frequencies can also be accurately computed from a given intermolecular potential surface; they depend very sensitively on its shape7,8,9. Therefore, comparison with the measured data provides a very critical check of the quality of ab initio calculated water pair potentials9,10,11,12,13,14,15,16. The pair potential is the leading term in a many-body potential for liquid water and ice and accounts for 80–90% of the interactions in the bulk phase, which triggered the extensive search for precise intermolecular potential energy surfaces17. Previously, we reported the low-frequency spectrum of water dimer in helium nanodroplets. This spectrum served as a sensitive test of the water dimer potential and dipole moment surfaces18.

Sulfhydryl (SH) compounds are well known for their weak non-covalent interactions. Cysteine, for instance, plays a role in establishing side chain conformations and stability of secondary structures in peptides through inter- and intramolecular interactions19,20,21. The simplest SH-containing compound, hydrogen sulfide (H2S), acts as a biological signaling molecule22,23,24 and as a superconductor precursor25,26. H2S is the simplest sulfur-bearing molecule detected in the interstellar medium (ISM) and plays an important role in astrochemistry27,28. However, there remains a significant knowledge gap compared to its isovalent water (H2O). Despite the significance of studying the pair interaction potential of H2S in detail, so far no high-quality potentials for (H2S)2 were available. The few IR studies reported so far, mostly in cryogenic matrices, were suffering from the overlap of spectra with higher cluster sizes and the small transition dipole moment of the dimer. In previous molecular beam studies29, four broad IR bands have been reported in the frequency range between 2590 and 2620 cm−1.

However, only a single vibrational transition in (H2S)2 is predicted to be intensive enough for observation in this frequency range. Thus, the assignment of the IR spectra was up to now a puzzle to be solved. Here, we present an intermolecular potential surface and use it to predict the different tunneling energy levels of the H2S dimer as well as its far-infrared spectrum, and compare them to those of the water dimer. We used our helium nanodroplet machine in Bochum to record the first mass-selective measurement of IR spectra of small H2S clusters. Based on these measurements, we can unambiguously assign three bands to the H2S dimer, while in the same frequency range also trimer bands are observed. After increasing the signal-to-noise ratio, we were able to resolve a rotational substructure of each of the dimer bands. As explained in the Methods section, H2S (just as H2O) has two different nuclear spin isomers, called para and ortho H2S (pH2S and oH2S), which have different rotational states. Based on the observed nuclear spin multiplicitiess of alternating rotational levels, see the table in the Methods section, we could assign the observed IR absorption peaks at 2598.2, 2602.0 and 2604.2 cm−1 to the donor bound S-H stretch mode in ortho-ortho, ortho-para, and para-para H2S dimers and additional bands to larger clusters. So far, only transitions in one of these species —ortho-para—had been observed in microwave studies30, which provided the rotational constants of its lowest two states. In contrast to the tetrahedral structure of ice, in the condensed form of H2S each molecule is surrounded by twelve neighbors31,32, indicating a more isotropic and less directional intermolecular potential. The binding energy De was calculated to be about 7 kJ/mol33, which is roughly one third of the De of the water dimer, indicating a shallower potential energy surface.

Based on the structure, as determined by microwave spectroscopy30 and subsequently supported by ab initio calculations33, the H2S dimer is known to be hydrogen bonded. However, its hydrogen-bonded geometry shows differences from that of the H2O dimer, which can be understood by considering the monomer structures. The S-H bonds in H2S are nearly perpendicular, with an HSH angle of 92∘, while the HOH angle in H2O is about 104∘, which is nearly the tetrahedral angle. This implies that both the OH binding orbitals and the lone electron pairs in H2O can be regarded as sp3 hybrids, while in H2S the nearly perpendicular SH bonds are formed with the S atom p orbitals, and the lone pairs are sp hybrids pointing in opposite directions perpendicular to the H2S plane. Hence, the acceptor plane in the H2S dimer is nearly perpendicular to the SH bond of the donor involved in the hydrogen bond, in contrast with the H2O dimer where the OH bonds in the acceptor and the donor OH bond point to the corners of a tetrahedron.

Previously, low-resolution IR spectroscopic studies on the H2S monomer and dimer have been reported in cryogenic matrices34,35,36,37,38,39,40, as well as in molecular beams29 and in liquids41. However, information on the intermolecular potential energy surface is lacking so far. Furthermore, the assignment of the IR transitions was ambiguous, due to the small IR transition dipole of H2S and the spectral overlap of the IR bands of H2S clusters (dimers, trimers, or oligomers) in low-temperature matrices.

Here, we report an IR spectrum of H2S dimers in superfluid helium droplets in the frequency range of the bound S–H stretch mode. We were able to resolve a rotational structure and to assign the three bands observed at 2598.2, 2602.0, and 2604.2 cm−1 to a single vibrational band with resolved tunneling states belonging to the ortho-ortho, ortho-para, and para-para nuclear spin isomers of the H2S dimer. This is in line with our calculations, which predict well separated energy levels for ortho-ortho, ortho-para, and para-para H2S dimers, but only a single strong vibrational transition in this frequency range. Bands observed at 2588 and 2620 cm−1 were assigned to the trimer, based on the pick-up curves. Thus, we reassign the previously observed bands at 2590 and 2618 cm−1 in molecular beams also to the trimer and the band at 2605 cm−1 to the dimer.

Accompanying calculations based on a newly developed ab initio potential surface were carried out to predict the dimer vibration-rotation-tunneling (VRT) states. These take into account the large amplitude intermolecular vibrations of different molecular symmetries and allow us to predict also the far-infrared spectrum at 0.4 K, calculated with the new ab initio intermolecular potential and dipole function. The frequencies of the intermolecular vibrations are much lower than in the H2O dimer. Also, the tunneling splitting pattern of the VRT levels is very different: the donor-acceptor interchange splitting is much larger than in the water dimer, the acceptor switch splitting is substantially smaller. We discuss the nature of the various tunneling processes and of the large amplitude intermolecular vibrations in comparison with those of the water dimer. The zero-point energies of the three nuclear spin isomers are obtained from the VRT calculations, and the zero-point energy of the para-para species also from Diffusion Monte Carlo calculations in full dimensionality. From Vibrational Self-Consistent Field and Configuration Interaction42,43 (VSCF/VCI) calculations we obtain the fundamental frequencies of the four SH-stretch modes, with a focus on the donor-bound SH mode which is observed experimentally. The calculations are in agreement with the conclusion from the experiment that the a-axis component of the transition intensity is the dominant one.

Results

Vibration-rotation-tunneling states

The lower bound vibration-rotation-tunneling VRT levels, calculated with the pseudospectral method are listed for pH2S-pH2S, oH2S-oH2S, and oH2S-pH2S in Supplementary Tables 3, 4, 5. The tunneling levels are displayed graphically in Fig. 1 and the levels including intermolecular vibrations in Supplementary Fig. 2. The energy levels are characterized by the total angular momentum J, which is a good quantum number, and an approximate quantum number K, which is the projection of the total angular momentum J on the intermolecular axis. We use the absolute value of K, because the VRT wave functions are even/odd combinations of functions with  + K and  − K. From the energies with the same K and J = 0, 1, 2 we derive the dimer end-over-end rotational constant (B + C)/2 and from the splitting between levels with the same K > 0 and different  ± parity we extract B − C, see Table 1. If the dimer were a rigid rotor, the differences between the energies for different K would yield the rotational constant A, but actually it is quite floppy, K affects its internal motions, and thus A is not a simple rotational constant.

Fig. 1: Tunneling energy levels for J = K = 0 and J = K = 1.
figure 1

(a) H2O-H2O levels, (b) H2S-H2S levels.

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Table 1 Rotational constants of the lowest E + and E − states, corresponding to oH2S-pH2S
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The three tunneling processes that connect the eight equivalent global minima in the potential for water dimer are acceptor switch, donor-acceptor interchange, and bifurcation (or donor) tunneling. For H2O–H2O they give rise to the typical energy level pattern shown in Fig. 1a with a large acceptor tunneling splitting of about 11 cm−1, considerably smaller splittings due to interchange, and very small shifts from bifurcation tunneling8,9,18. The corresponding picture for H2S-H2S in Fig. 1b is very different, however. We can identify the tunneling processes in H2S-H2S that give rise to this different picture by looking at the VRT wave functions in Supplementary Figs. 3 and 4. The acceptor switch tunneling splitting is 5.06 cm−1, which is considerably smaller than the splitting of about 11 cm−1 in H2O-H2O. This is probably related to the acceptor plane being nearly perpendicular to the hydrogen bond in H2S-H2S, which implies that the acceptor tunneling pathway is longer than in H2O-H2O. The interchange tunneling splitting, i.e., the energy difference between \({A}_{1}^{+}\) and \({B}_{1}^{+}\) is 13.29 cm−1. This is much larger than the interchange splitting of about 0.7 cm−1 in H2O-H2O, so the associated energy barrier is smaller. These differences cause the energy level picture for H2S-H2S to look very different from the corresponding picture for H2O-H2O.

The mixed oH2S-pH2S states with symmetries E ± are the only states that have a non-zero dipole moment because of the asymmetry in their wave functions, see Supplementary Fig. 5. The \({A}_{1,2}^{\pm }\) and \({B}_{1,2}^{\pm }\) states of pH2S-pH2S and oH2S-oH2S are symmetric or antisymmetric under PAB, they all have equal weights for donor-acceptor and acceptor-donor structures, and their dipole moment averages to zero. Therefore, in the microwave spectrum30 of H2S-H2S one can only observe the transitions from E + to E − states and vice versa. In Table 1, we list our calculated rotational constants for the E + and E − states, together with the measured data30. We can now identify the lower and upper states mentioned in ref. 30 as the E + and E − states, respectively, which are separated in energy by 1.23 cm−1. The maximum deviation of our calculated rotational constants from the values extracted from the microwave spectrum30 is 1.27 %. This confirms the accuracy of the H2S-H2S potential used in our VRT calculations.

In the H2O dimer, there is just one excited intermolecular vibrational state with energy below 100 cm−1, while H2S-H2S has many excited states below 65 cm−1, see Supplementary Fig. 2 and Supplementary Tables 3, 4, 5. This illustrates that the H2S dimer is much floppier than the H2O dimer. The wave functions shown in Supplementary Fig. 6 have nodal planes that depend on at least two intermolecular coordinates. Due to the strong coupling of these coordinates the excited states are mostly of mixed character.

It is well known and illustrated by our results below that the vibrational frequencies of molecular dimers in He nanodroplets agree with the gas phase values to within a few cm−1. While rotational levels of larger complexes with small energy splittings can couple to the phonon bath of helium droplets larger energy spacings are mostly unaffected. For a direct comparison of experimentally observed intermolecular modes in the gas phase and in helium droplets, we refer to our papers on water dimer and water trimer18,44, with the frequencies of the predicted and measured intermolecular modes agreeing to within the experimental uncertainty.

Theoretical far-infrared spectrum

The far-IR spectrum of H2S-H2S at temperature T = 0.37 K, which is the temperature of He droplets in the molecular beam, is shown in Fig. 2. It is calculated with the Lanczos-based recursive residue generator method45,46 and the six-dimensional (6D) dipole function described in the Methods section below. Due to nuclear spin conservation, the lowest states of each permutational symmetry are populated. Since dipole transitions are only allowed between states with the same permutational symmetry and opposite parity, the spectrum contains contributions from transitions starting from the occupied states for each of the symmetries with weights given by their nuclear spin multiplicities.

Fig. 2: Far-infrared spectrum of H2S-H2S at T = 0.37 K calculated with the ab initio intermolecular potential and dipole function.
figure 2

The lines in the spectrum were assigned with the use of the transition frequencies and line strengths calculated separately for each of the symmetries.

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The lines in the spectrum of Fig. 2 were assigned with the use of the transition frequencies and line strengths calculated for each of the symmetries with the analytical method. The assignments are indicated in Fig. 2 and are listed in Table 2. The higher peaks are not specified, because the corresponding excited states are more and more complex.

Table 2 Assignment of the lines in the theoretical far-infrared spectrum at T = 0.37 K
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The experimental mid-infrared spectrum of H2S-H2S presented in this paper involves excitation of the bound S-H stretch mode in the donor. Although the dimer with one internally excited monomer formally has the same permutation-inversion symmetry group G16, either monomer can be excited and the number of dimer excited states is larger than the number of ground states. Effectively, states in which the donor is excited differ from those in which the acceptor is excited, but these states are connected by interchange tunneling. However, the interchange tunneling splitting is expected to be substantially smaller than in the ground state, because the equivalence of excited donor-acceptor and acceptor-donor states not only requires a geometry change but also hopping of the excitation from one monomer to the other. Both measurements and calculations on HF and HCl dimers in their ground and monomer excited states47,48,49,50,51,52,53,54,55 support this assumption.

Although the explicit calculation of the monomer excited states of H2S-H2S would require 12-dimensional (12D) calculations, as reported15,56 for H2O-H2O, we can give some indications on the basis of our 6D calculations that are useful to assign the measured mid-IR spectrum. At a temperature of T = 0.37 K in the He droplets in which the dimer was embedded and equilibrated, only the lowest states of each symmetry are populated. Given the nuclear spin weights in the table in the Methods section, this implies that lines in the spectrum due to excitations from the lowest \({B}_{2}^{-}\) states with even J and \({A}_{2}^{+}\) states with odd J alternate in intensity since the nuclear spin weights of these states are 3 and 6, respectively. Such an intensity alternation will not occur for excitations from the lowest E + states with even J and E − states with odd J which have the same nuclear spin weight 3. For the lowest \({A}_{1}^{+}\) states with even J the situation is different again: the corresponding \({B}_{1}^{-}\) states with odd J have nuclear spin weight zero. This implies that transitions from these lowest \({A}_{1}^{+}\) states are only allowed for even initial J = 0, 2, etc.

Dissociation energy

The dissociation energy, D0, is obtained using an extrapolated value of De of 583 cm−1 at the CCSD(T) complete basis set (CBS) limit (details in Supplementary Table 6), plus twice the zero-point energy (ZPE) of the isolated H2S monomer minus the zero-point energy of the H2S dimer. The latter is obtained from standard unconstrained diffusion Monte Carlo (DMC) calculations, using the present 12D PES, following the protocol used previously for the calculation of D0 for the water dimer57. The resulting ZPE of the H2S dimer is 6822 cm−1 with an uncertainty of several cm−1. The ZPE of the monomer is taken from the literature58, which reported exact variational calculations on an accurate ab initio PES, which yield excellent agreement with experimental values for vibrational transitions. The H2S ZPE reported there is 3294.3 cm−1 and thus for two monomers the ZPE is 6588.6 cm−1. From these numbers we obtain 349 cm−1 for D0 with a conservative estimated uncertainty of 10 cm−1. Note this D0 corresponds to total angular momentum J = 0 of the dimer and in DMC, the wave functions are assumed to be nodeless, so this D0 value refers to pH2S-pH2S.

As already pointed out, three different dimer nuclear spin isomers exist: pH2S-pH2S, oH2S-pH2S, and oH2S-oH2S. They have different D0 values because nuclear spin is conserved upon dissociation, and they dissociate into the corresponding para and ortho monomers. From CCSD(T) computations near the CBS limit with the rigid-monomer geometries used in the 6D calculations of the VRT states it is found that De = 579 cm−1. In order to obtain the D0 values, one needs the ZPE’s of each of the three species given by its lowest VRT level, as well as the energies of the lowest allowed rotational states of the para and ortho monomers.

The ZPE’s of pH2S-pH2S, oH2S-pH2S, and oH2S-oH2S given in Supplementary Tables 3, 4, and 5 are 241, 247, and 246 cm−1, respectively. The ground rotational state of pH2S is the \({j}_{{k}_{a}{k}_{c}}={0}_{00}\) state with energy zero and with experimental H2S rotational constants59 one finds that the lowest allowed rotational state 101 of oH2S has an energy of 13.7491 cm−1. The resulting D0 values of pH2S-pH2S, oH2S-pH2S, and oH2S-oH2S are 338, 346 and 361 cm−1, respectively. The value for pH2S-pH2S agrees with the value from DMC calculations on the 12D potential to about the estimated statistical uncertainty of 10 cm−1 in the latter value.

Experimental Results

In Fig. 3, we show the experimentally observed IR spectrum of H2S clusters in helium nanodroplets in the frequency range of the bound donor stretch, i.e., between 2570 cm−1 and 2630 cm−1 at the mass-to-charge ratio m/z = 34 corresponding to \({({{{{{\rm{H}}}}}_{2}}^{32}{{{\rm{S}}}})}^{+}\). IR transitions are observed at 2583.8, 2588.4, 2598.2, 2602.0, 2604.2, and 2619.7 cm−1. At each of these peak frequencies, we recorded so-called pick-up curves for assignment to a specific cluster, see Supplementary Note 6 for more details. Based on these pickup curves, the bands at 2598.2, 2602.0, and 2604.2 cm−1 could be assigned to H2S dimers, while the bands at 2583.8, 2588.4, and 2619.7 cm−1 are attributed to H2S trimers. In the dimer bands we clearly observe P and R branches. The substructure of these branches could be reproduced to experimental uncertainty when taking into account the correct spin statistics: alternating spin weights (3:6) for subsequent J levels for the ortho-ortho transition, (3:3) for the ortho-para transition, and (1:0) for the para-para transition. We were unable to observe any monomer bands, even at lower H2S pressures. Note that also in the gas phase, only weak bands have been reported previously at 2615 and 2625 cm−1 41.

Fig. 3: Experimentally recorded IR spectrum of H2S clusters in helium nanodroplets at 0.37 K, m/z = 34 and partial H2S pressure of 4.5 × 10−6 mbar.
figure 3

We observe three bands of H2S-H2S centered at 2598.21, 2601.95, and 2604.17 cm−1, which could be assigned to three different vibrational-rotational-tunneling transitions. In the zoom, we show a comparison of the experimentally recorded rotational line shape (black) with the adapted predicted spectrum (colored sticks) assuming alternating or non-alternating intensity ratios of subsequent rotational levels according to the respective spin statistics. Black arrows show the wavenumber where pick-up curves were recorded (see Supplementary Information).

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In Table 3 we present harmonic and VSCF/VCI excitation energies and intensities for the indicated four S-H stretches. Given the strong evidence from calculations that the wave functions sample the three low-energy minima, we present results that are averaged over these minima along with standard deviations. The corresponding structures labeled Minimum I, II, and III are shown in Supplementary Fig. 7.

Table 3 VSCF/VCI frequencies (cm−1), intensities (arbitrary units) and percent contributions from a, b, and c principal axis components for different SH-stretch modes with symmetries \({A}^{{\prime} }\) and A″ for Minimum I, II, and III of the H2S dimer
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The first indications for feasible tunneling were found in the microwave spectrum of the H2S dimer measured by Das et al.30 in a molecular beam. They reported rotational transitions from two distinct ground states. These two states are the E + and E − states, which we predict to be separated by 1.23 cm−1. Using a Boltzmann distribution it can be estimated that at the molecular beam temperature of 4 K the populations of the E + and E − levels are 0.61 and 0.39, whereas at the superfluid helium droplet temperature of 0.37 K these are 0.992 and 0.008. Therefore, only the E + state is expected to be populated in superfluid helium droplets.

While in previous studies29,35,36,37,39 the IR bands observed in molecular beams and in matrices were assigned to the ν1 acceptor and ν3 donor/acceptor modes, all observed H2S dimer bands are now assigned to ro-vibrational tunneling transitions involving the bound donor SH stretch mode, see Table 4. If we assume that all transition dipole moments are nearly the same, the intensity ratio of the various lines in the spectrum is determined by the nuclear spin multiplicities given in the table in the Methods section. We propose the following assignments: \({A}_{1}^{-}({J}_{{{{\rm{odd}}}}},{K}_{a}=0)\leftarrow {A}_{1}^{+}({J}_{{{{\rm{even}}}}},{K}_{a}=0)\) at 2598.2 cm−1, E −(Jodd)/E +(Jeven), (Ka = 0) ← E +(Jeven)/E −(Jodd), (Ka = 0)at 2602.0 cm−1, and \({B}_{2}^{+}({J}_{{{{\rm{odd}}}}})/{A}_{2}^{-}({J}_{{{{\rm{even}}}}}),{K}_{a}=0\leftarrow {B}_{2}^{-}({J}_{{{{\rm{even}}}}})/{A}_{2}^{+}({J}_{{{{\rm{odd}}}}}),{K}_{a}=0\) at 2604.2 cm−1.

Table 4 Observed bands (frequencies in cm−1) of H2S clusters in helium nanodroplets (our measurements), in molecular beams, and in matrices
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To ensure an unambiguous assignment of the IR spectra we tested different spin statistical weight ratios of the rotational lines in the P and R branches of the band at 2604.2 cm−1. It was found that a spin statistical weight ratio of 3:6 for Jeven: Jodd yields the smallest mean square deviation between measurement and fit. As explained in the last paragraph of the section “Theoretical far-infrared spectrum”, this intensity ratio of 3:6 indicates that this band belongs to oH2S-oH2S with the lower states in the transitions having \({B}_{2}^{-}\) symmetry for even J and \({A}_{2}^{+}\) symmetry for odd J. The fitted rotational constant of this band is B = (0.0259 ± 0.0003) cm−1 and the difference between the vibrational ground and excited state values is ΔB = ( − 0.0009 ± 0.0001) cm−1. In the band at 2602.0 cm−1 no intensity alternation was observed. This agrees with an intensity ratio of 3:3, indicating that this band belongs to oH2S-pH2S, with lower states of E + symmetry for even J and E − symmetry for odd J. The fitted rotational constants of this band are B = (0.02230 ± 0.0001) cm−1 and ΔB = ( − 0.00067 ± 0.00005) cm−1. No unambiguous fit of the band at 2598.2 cm−1 was possible due to the low signal-to-noise ratio. The nearest lines in the P and R branches of this band are separated by 0.14 cm−1, while for the other two bands, we find gaps of 0.09 and 0.10 cm−1. This is in agreement with an assignment of the 2598.2 cm−1 band to pH2S-pH2S, with lower levels of A1 symmetry for even J and B1 symmetry for odd J. Since B1 states have spin statistical weight zero, see the table in the Methods section, the odd J levels are missing. So the spacing should be 6B, while for the ortho-ortho and ortho-para dimers it is 4B, which nicely agrees with the observed gaps and confirms our assignment. The rotational constants in the helium droplets are smaller by a factor of 2.3 compared to the gas phase, which is in good agreement with previous studies60.

In the vibrationally excited state, the energy level spacings are expected to differ from the ground state spacings, since upon the vibrational excitation of the donor, the barrier in any tunneling path involving donor-acceptor interchange is expected to be significantly increased. Based on the experimentally observed band center frequencies and the predicted tunneling splittings in the ground state, we can deduce the tunneling splittings in the vibrationally excited states shown in the energy level diagram in Fig. 4. Indeed, upon vibrational excitation, the donor-acceptor interchange splitting becomes much smaller. It is notable that the resulting splitting pattern in the energy level scheme of the excited state of the H2S dimer resembles remarkably well the tunneling splitting pattern in the ground state of the water dimer.

Fig. 4: Energy level schemes of the H2O and H2S dimers.
figure 4

(a) H2O dimer in the vibrational ground state, (b) H2S dimer in the ground and vibrationally excited state. The energy scheme in the ground state corresponds to our VRT levels calculated on a high-quality ab initio intermolecular potential. The arrows represent experimentally observed bands. The stars mark the enery levels populated at 0.37 K.

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Table 3 gives calculated VSCF/VCI frequencies and intensities using the 12D PES at the three minima. Recall that these calculations are restricted to the six intramolecular modes, and do not reflect the delocalized nature of the exact ground vibrational wavefunction. The frequencies are  ≈100 cm−1 downshifted relative to the harmonic frequencies (given at Minimum I in Supplementary Table 2). VSCF/VCI stick spectra, shown in Supplementary Fig. 7, span a range from 1000 to 4000 cm−1 and include combination and overtone transitions.

A summary of experimental results from a variety of experiments, including the present He nanodroplet one, is given in Table 4. From these, it can be concluded that the VSCF/VCI results are in good agreement with experiment, i.e., within about 10 cm−1. Note the VSCF/VCI calculations are “vibrational only”, i.e., for J = 0 and no attempt was made to determine the experimental energies of the triplet reported here.

Discussion

The question whether the H2S dimer is hydrogen bonded and how its properties compare with those of its smaller sibling, the H2O dimer, attracted the interest of a broad chemical community61,62. It triggered several infrared spectroscopic studies of the H2S dimer in molecular beams29 and in various rare gas and other matrices36,37,38,39,40. The spectra showed several partly overlapping bands in the frequency range of the S-H stretch mode, but their assignment remained uncertain due to the low resolution, the possible effects of different matrix sites, and the possibility that some of the bands belong to trimers and larger clusters. The microwave spectrum of the H2S dimer measured in a molecular beam30 revealed clearly that this dimer has a hydrogen-bonded geometry, which evoked much attention61,62. A hydrogen bonded geometry was also found in subsequent ab initio calculations of the dimer’s equilibrium structure33.

The vast amount of studies of the H2O dimer not only addressed its structure, but also the different tunneling processes connecting eight equivalent equilibrium structures and the intermolecular vibrations. High-resolution far-infrared spectra1,2,3,4,5,6, in combination with accurate calculations of its VRT states, yielded interesting information on the tunneling dynamics and the low-frequency intermolecular vibrations, which served as a very critical test of the available ab initio water pair potentials9,10,11,12,13,14,15,16. Such information was still lacking for the H2S dimer, but it is provided here. We present mass-selected IR spectra of the H2S dimer in helium nanodroplets in a molecular beam, accompanied by calculations of its VRT states with the use of a new accurate ab initio pair potential and dipole function. The low temperature of 0.37 K and the superfluid nature of the helium droplets made it possible to resolve and analyze the rotational substructure of the observed infrared bands and, with the help of the calculated tunneling splittings in the vibrational ground state and the symmetry relations, to unambiguously assign the three bands in the measured spectrum. All three experimentally observed dimer bands are attributed to the stretch mode of the donor SH group directly involved in the hydrogen bond; this assignment is supported by VSCF/VCI calculations on the intramolecular vibrations which show that only this mode has sufficient intensity to be observed. Just as H2O, H2S has two nuclear spin isomers, ortho and para, which have different rotational states. The splitting of the donor SH stretch transition into three distinct bands is caused by different tunneling splittings in ortho-ortho, ortho-para, and para-para H2S dimers. Also the different dissociation energies D0 of the ortho-ortho, ortho-para, and para-para H2S dimers into the corresponding monomers are reported here. The microwave spectrum30 only addressed the mixed ortho-para dimer, because it is the only dimer species with a non-vanishing dipole moment. The rotational constants calculated for the lowest two tunneling states of this ortho-para dimer agree well with the values deduced from the microwave spectrum30, which confirms that our ab initio H2S dimer potential is accurate.

The calculations presented provide a full 12-dimensional potential and dipole function of the H2S dimer depending on all intra- and intermolecular degrees of freedom. DMC calculations on this 12D potential yield the vibrational ground state of the dimer, while rigorous variational quantum calculations on the 6-dimensional intermolecular potential yield the tunneling and low-frequency intermolecular vibrational states. Moreover, we predict the H2S dimer’s far-infrared spectrum, with the use of the 6D dipole function.

The tunneling splitting pattern in the H2S dimer is found to be very different from the well-known pattern in the H2O dimer, which is mainly caused by more facile donor-acceptor interchange tunneling and more strongly hindered acceptor switch tunneling. Also the intermolecular vibrations differ strongly from those of the H2O dimer, which is reflected by the predicted far-infrared spectrum. They have much lower frequencies, larger amplitudes, and different intermolecular modes are more strongly mixed. All of this indicates that the H2S dimer is much floppier than the H2O dimer. Calculations by symmetry-adapted perturbation theory, see Supplementary Table 7, indicate that the less directional character of the hydrogen bond in the H2S dimer, as compared to the H2O dimer, is caused by a less dominant role of the electrostatic dipole and quadrupole interactions and relatively larger dispersion interactions, associated with the larger polarizability of H2S. In summary, we conclude that the experimental spectrum and its analysis, and the calculations presented in this paper clearly show that the H2S dimer, although hydrogen bonded, shows very different dynamical properties than the isovalent H2O dimer. The insights gained in this study of the H2S dimer may also lead to a better understanding of the nature of non-covalent bonds involving sulfhydryl (SH) groups in biological systems, as compared to bonds involving OH groups.

Methods

Experimental details

The Bochum helium nanodroplet machine has been described in detail before63,64. It consists of five differentially pumped vacuum chambers interconnected by conflate flanges. Helium droplets were generated in the first chamber (expansion chamber), where pressurized helium gas passes through the precooled nozzle of 5 micron diameter into vacuum. The nozzle was operated at 10–20 K, and a backing pressure of 40–50 bar to form the different sizes of droplets (1000–10,000 He atoms). The pressure in the expansion chamber is kept at about 10−5 mbar. The helium droplets are collimated by a skimmer of 0.5 mm diameter before they enter the first pickup chamber which is kept at a pressure of 10−6 mbar; with a partial pressure of 6.5 × 10−6 mbar for pick up of H2S. The last chamber (detection chamber) hosts a quadrupole mass spectrometer (Pfeiffer QMG 422) to ionize the droplet beams. The mass spectrometer has two alternative modes for detection, either mass selective (at a given m/z) or via a high pass filter (detection of all ionized helium clusters and fragments with m/z > 4) The pressure in the detection chamber was kept in the range of 10−10 mbar. As a radiation source served a continuous wave optical parameter oscillator (OPO, Lockheed-Martin Aculight ARGOS 2400SF15 Module C). A 15 mW Yb-doped fiber laser at 1064 nm, which was amplified to 15 W using a diode-pumped fiber, served as a seed laser. The pump laser is split into an idler and a signal beam. The OPO laser source can be scanned in the wave length range between 3.2 μm and 3.9 μm, the average idler output power is 1 W, the frequency resolution is 30 MHz. A Bristol 621A wavemeter is used for frequency calibration. For phase-sensitive detection, the laser was chopped with a frequency of 25-35 Hz before entering the detection chamber.

Theory

Potential and dipole moment surfaces

Here we report a new full twelve-dimensional (12D) potential energy surface that makes use of Δ-Machine Learning to bring an MP2-based PES to near CCSD(T) quality, using a limited number of explicitly correlated CCSD(T) energies65,66. In the present case, these are CCSD(T)-F12b/haQZ-F12 calculations, as described in detail in Supplementary Note 1. In addition, an MP2-based dipole moment surface (DMS) is reported. Both the PES and DMS are represented by permutationally invariant polynomials67,68. The new PES and DMS are used in diffusion Monte Carlo calculations of the ground vibrational state, extensive rigorous rigid-monomer quantum calculations of the VRT states and transition line intensities, and approximate VSCF/VCI calculations for intramolecular mode excitations.

Rigid-monomer quantum calculations

The rigid-monomer calculations of the VRT states use a 6D intermolecular potential obtained from the full 12D potential by fixing the geometry of both H2S monomers at an SH bond length of 1.3384 Å and an HSH angle of 92. 5°. This monomer geometry is close to the geometry of the hydrogen bond acceptor, which is the least perturbed monomer, at the global minimum of the 12D potential for the H2S dimer. It must be the same for both monomers in order to reproduce all tunneling effects including donor-acceptor interchange. The 6D intermolecular potential depends on the monomer-monomer center-of-mass distance R and five Euler angles defining the orientations of the monomers. The VRT states of H2S-H2S are calculated with two different methods, which were applied earlier to the H2O dimer7,9. The first method is the split Wigner pseudospectral approach7 that employs a grid basis in all six intermolecular coordinates in combination with an analytic basis. The second method9 uses only the analytic angular basis and expands the 6D potential in angular expansion functions. Details are given in the Supplementary.

In addition, we predicted the far-infrared spectrum with two different methods. The first method uses recursive residue generation45,46 to directly get the transition dipole moments as the projections of eigenstates multiplied with the dipole function onto the initial Lanczos vectors without actually computing all final eigenstates, which can be efficiently done in the grid basis of the split Wigner pseudospectral approach. The second method expands not only the potential but also the dipole moment function in angular basis functions and analytically calculates the transition dipole moment over the initial and final states. The dipole moment function in the six intermolecular coordinates is obtained from the 12D dipole moment surface in the same way as the 6D rigid-monomer intermolecular potential is derived from the 12D potential surface, with the same fixed H2S monomer geometry. The symmetry group of the H2S dimer in its equilibrium geometry is the point group Cs. But the dimer has eight equivalent equilibrium structures with rather low energy barriers between the corresponding global minima in the potential, so the molecular symmetry used in our calculations is the permutation-inversion (PI) or molecular symmetry69 group G16. This group has 10 irreducible representations (irreps) listed in Table 5. We note here that for the H2S monomer, just as for H2O, one denotes rotational states with the asymmetric rotor quantum numbers \({j}_{{k}_{a}{k}_{c}}\). We used the most abundant 32S isotope with nuclear spin I = 0. States that are even under the permutation P12 that interchanges the two H atoms have even ka + kc, are called para-H2S (pH2S), and have total nuclear spin I = 0 (multiplicity 1), and states odd under P12 have odd ka + kc, are called ortho-H2S (oH2S), and have total nuclear spin I = 1 (multiplicity 3). These monomer nuclear spin weights, combined with the permutation PAB that interchanges the monomers A and B, cause the H2S dimer states for different irreps to have the nuclear spin multiplicities given in Table 5. The symbol E* denotes the inversion operation. In both methods to compute the VRT states the calculations were performed separately for each irrep with a symmetry-adapted basis, as explained in Refs. 7,9.

Table 5 Permutation-inversion group G16, irrep definitions
Full size table

The dipole moment function is invariant under all permutations, just as the Hamiltonian, but it changes sign under inversion. So for the overall vibration-rotation-tunneling transition it carries the irrep \({A}_{1}^{-}\). Electric dipole transitions are only allowed between irreps with the same permutation symmetry and opposite  ± parity under inversion.

VSCF/VCI and diffusion Monte Carlo calculations

The VSCF/VCI calculations employed the code MULTIMODE70 and are not at the same level of rigor as the rigid-monomer quantum calculations described in the preceding section. They do not take into account the delocalized nature of the VRT states, but instead they are done at three reference configurations which are nearly isoenergetic. Moreover, they do not include rotation of the dimer (J = 0)33. The Diffusion Monte Carlo calculations of the zero-point energy in 12D are exactly as reported previously for the water dimer57. Details of these calculations are given in the Supplementary.