A theoretical treatment of the Ã 2Σ+state of the Ar⋯HS/Ar⋯SH van der Waals complex

We present an ab initio potential energy surface for the Ã 2Σ+ state of the Ar⋯HS van der Waals molecule.

The surface represents a fit to 196 points calculated at the RCCSD(T) level with an aug-cc-pV5Z basis set and encompasses both the Ar⋯SH and Ar⋯HS isomers.

We have further calculated vibrational levels on this surface using a discrete variable representation (DVR) approach.

The vibrational levels supported within the Ar⋯HS minimum are compared with levels derived from high-resolution laser induced fluorescence spectra.

Predictions of vibrational levels for the Ar⋯SH isomer, which have not been observed experimentally, are also presented.


There is currently much interest in the spectroscopy and structure of van der Waals complexes of rare gas atoms with free radical species OH and SH.

Carter et al1. have recently reviewed studies of the complexes with Ne, Ar and Kr concentrating on the analysis of the rotational and vibrational structure of the electronic transition between the X̃ 2Π3/2 and Ã 2Σ+ states.

For each of these species, high quality empirical potential energy surfaces have been derived from high resolution laser induced fluorescence (LIF) spectra obtained in the Miller group.

Ab initio potential energy surfaces have been calculated for the OH and SH complexes with He and Ne.2,3

In these studies, basis sets of aug-cc-pVTZ or aug-cc-pVQZ quality4–6 were augmented with sets of bond functions7,8 at the mid-point of the vector connecting the rare gas atom with the centre-of-mass of the diatomic radical.

The ab initio calculations were made with the RCCSD(T) method.9,10

For both the empirical and ab initio surfaces the rotation–vibration energy levels were calculated by a discrete variable representation (DVR) method derived from that originally proposed by Choi and Light.11

A number of spectroscopic studies of the Ar⋯HS complex have been reported by Miller and coworkers: The laser induced fluorescence spectrum for the Ã 2Σ+–X̃ 2Π transition was first reported in .199312

Subsequent papers reported improved data for the vibrational spectrum of Ne⋯HS, Ar⋯HS and Kr⋯HS13 and a high resolution rotationally resolved spectrum of Ar⋯HS.14

In a further paper, Korambath et al15. used these spectroscopic data to derive empirical potential energy surfaces for the Ã 2Σ+ states of Ar⋯HS and Kr⋯HS.

The parameters of a functional form suggested by Bowman and co-workers16,17 were varied to reproduce the observed transition wavenumbers and rotational constants and a DVR method was used to calculate the energy levels and rotational constants.

For Ar⋯HS, 11 parameters were varied to fit 7 vibrational energy levels and 8 rotational constants.

As shown by Miller et al., the experiment samples only the portion of the potential energy surface for Ar⋯HS angles between 0 and 90° (i.e., from the Ar⋯H–S linear geometry to the T-shaped structure).

Hence the optimization of the potential function did not vary parameters relevant to 180° (i.e., Ar⋯SH isomer).

Agreement was within ±0.3 cm−1 for the energy levels.

The empirical potential energy function of Korambath et al15. has been used subsequently by McCoy18 in calculations of the lifetime of Ar⋯HS Ã 2Σ+ state levels and by Firsov et al19. in calculations of the vibrational energy levels of Ar⋯HS by a vibration self-consistent field method.

In most experimental studies, Ar⋯HS complexes have been produced by photolysis of a mixture of H2S co-expanded with Ar.12–14

A novel alternative method for the generation of the open-shell complexes has been described by Mackenzie et al.,20,21 in which small ArnH2S clusters (n ≤ 2) are first formed in a pulsed supersonic expansion and then photolyzed downstream from the expansion.

It was shown that bound Arn⋯SH species can be produced efficiently by this method.

Further, analysis of LIF spectra of the complexes generated indicated extensive van der Waals stretch/bend and overall rotational excitation.

Quasi-classical trajectory calculations to simulate the cluster photofragmentation dynamics have been made by Fair and Nesbitt.22

In the absence of ab initio potentials, this necessitated the construction of qualitative potential energy surfaces from Ar⋯H2O, Ar⋯OH, H2S and Ar2 surfaces.

Sumiyoshi et al23. have obtained rotational spectra by Fourier transform microwave spectroscopy (FTMW) for the X̃ 2Π ground state of Ar⋯SH.

They derived a two-dimensional potential energy surface by a least-squares procedure in which they fitted the observed rotational transitions.

Since the experimental data were insufficient to determine all of the potential parameters, some parameters were derived from an ab initio potential surface calculated by the RCCSD(T) method with an aug-cc-pVQZ basis set.

The same authors have since revised their surface to satisfactorily reproduce all of the experimentally obtained rotational frequencies for both Ar⋯SH and Ar⋯SD.24

Their ab initio surfaces were obtained from RCCSD(T) calculations with an aug-cc-pVTZ basis set augmented with bond functions.

An effective correction term was introduced to account for a small difference between the potentials for the SH and SD complexes.

In this article we report an ab initio potential energy surface for the Ã 2Σ+ state of Ar⋯SH.

Our calculations consider the full range of geometries from the linear Ar⋯HS configuration to linear Ar⋯SH.

The ab initio energies have been fitted to a functional form which we have used in a DVR calculation for the vibrational energy levels.

Rotational constants calculated for levels on our surface agree extremely well with the experimental data of Korambath et al.

In common with other ab initio calculations the energy levels spacings calculated on our surface are systematically lower than those observed.

By extending the surface to encompass all angles, we are able to predict wavenumbers for Ã-state levels in the Ar⋯SH isomer transitions to which, to the best of our knowledge, have not been identified thus far.

Computational details

The potential energy surface

The ab initio surface calculations were made with the MOLPRO suite of programs.25

The aug-cc-pV5Z basis set4–6 was used and the molecular orbitals for the 2 2A′ state calculated by the RHF method.

In all cases care was taken to ensure that the electronic configuration of the HS fragment corresponded to ⋯4σ241 and at no point were any convergence problems encountered.

Electron correlation was accounted for with the RCCSD(T) method9,10 and the counterpoise method of Boys and Bernardi was used to correct for basis set superposition errors.26

The geometry of the complex is defined in terms of Jacobi coordinates (R, γ) where R defines the distance of Ar from the center-of-mass of the HS radical and γ defines the angle between the vector R and the HS molecular axis.

The angle γ is defined such that γ = 0° corresponds to the Ar⋯HS configuration.

The angle γ was varied from 0 to 180° in steps of 15°.

For most angles R ranged from 2.75 to 10 Å but for γ = 180° the range was extended to 2.4 Å to give a more complete description of the linear Ar⋯SH configuration.

Ab initio energies were calculated for a total of 196 points.

In all calculations the SH distance was kept fixed at 1.423 Å, the re value for the Ã 2Σ+ state of SH.27

To fit the ab initio energies we have used an extended form of the model potential proposed by Bukowski et al28. which has been used successfully to fit a number of ab initio potential surfaces for rare gas complexes.8

The potential function is written as the sum of short range (Vsr) and asymptotic (Vas) termsV(R, γ) = Vsr(R, γ) + Vas(R, γ).The short-range interaction is given byVsr(R, γ) = G(R, γ)exp[D(γ) − B(γ)R],where D(γ), B(γ) and G(R, γ) are expansions in terms of Legendre polynomials, Pl(cos γ).

The functions B(γ), D(γ) are given by the expressions

andin which bl, dl and gnl are adjustable parameters.

The asymptotic term in eqn. (1) is given bywhere Cnl are adjustable parameters and the function fn(x) is the Tang–Toennies damping function,29In previous formulations,28 in which there were 40 adjustable parameters, the summations over l were restricted to l = 0–5 and the polynomial in R in the expression for G(R, γ) included terms up to and including R3.

We were not able to obtain a satisfactory fit with 40 adjustable parameters.

With this more flexible function involving 60 parameters, we were able to fit 196 ab initio points such that for the vast majority of points the deviation from the calculated energy was less than 1 cm−1.

Table 1 contains details of the parameters obtained in our fit.

Vibrational energy levels

Vibrational energy levels on the final surface were calculated by a DVR method in which the eigenvalues of the Hamiltonianwere obtained.

In eqn. (7), is the total angular momentum operator of the complex and ĵ is the angular momentum operator for the HS diatomic molecule.

The projections of J and j along the body-fixed z-axis (along R, the vector connecting the Ar atom to the center of mass of HS) are the same and both are represented by projection quantum number K. bv is the vibrationally averaged rotational constant for the HS radical, and the reduced mass of the atom–diatom complex is represented by μR.

The calculations were performed using the DVR code of McCoy30 in which the Hamiltonian is set up and diagonalized in three stages for a given value of J and the total parity of the system.

Only the v = 0 state of the HS moiety, with b0/hc = 8.289 cm−1, was considered.

For each value of R and a given value of K, the eigenvalues of the one-dimensional bend Hamiltonian are determined.

In this stage 30 zero-order bend states are used and the 10 most important are retained following diagonalization.

The R dependence is introduced by taking 120 DVR points along the vector R using the form of Colbert and Miller31 for the DVR.

After this diagonalization the 20 most important stretch/bend eigenstates are retained for each value of K.

Results and discussion

A contour diagram of our potential energy surface is shown in Fig. 1.

The global minimum corresponds to the linear Ar⋯HS isomer (γ = 0°) but there is a second minimum almost as deep for linear Ar⋯SH.

The two minima are separated by a >600 cm−1 barrier which prevents internal rotation for the lowest vibrational levels.

Cuts through the potential energy surface for γ = 0° and γ = 180° are shown in Fig. 2 and details of the geometries of the minima and the saddle point are given in Table 2.

In Table 3 we report the energy levels and rotational constants obtained from the DVR calculations.

The vibrational levels are labeled by convention as (vSH, nK, vvdw) where vSH is the number of vibrational quanta in the SH moiety, nK is the quantum number describing the bending motion (with projection K on the internuclear axis) and vvdw is the number of quanta in the van der Waals vibrational mode.

In Fig. 3(a) we show the square of the vibrational wavefunction, |ψ2|, for the (0, 00, 0) zero point levels of the Ar⋯HS and Ar⋯SH isomers superposed on the potential energy surface.

However, due to the polar nature of the (R, γ) coordinate system used here, |ψ2| does not accurately represent the probability density function.

As discussed by Carter et al.,1 for this we need to plot |ψ2|sin(γ) as shown in Fig. 3(b).

Fig. 4 shows the probability density functions calculated for the (0, 00, 1) and (0, 00, 2) van der Waals stretch levels of each isomer superposed on the potential energy surface.

The wavefunctions for the pure stretching levels of each isomer are confined to their respective potential minima but sample a considerable range of R values: e.g., the two (0, 00, 2) wavefunctions each sample a range of R > 1 Å.

In Fig. 5 we show the calculated probability density functions for the (0, 11, 0) and (0, 11, 1) levels.

Interestingly, the potential does not support bending levels for the Ar⋯SH isomer.

These would lie energetically above the saddle-point in which regime Ar⋯SH and Ar⋯HS geometries would be considered as merely different conformations of the same species.

At this point it is illustrative to compare the ab initio surface with the empirical surface of Korambath et al15. derived from the high resolution laser induced fluorescence spectra.

We discuss this comparison for each of the two minima on the surface.

The Ar⋯HS potential minimum

All lines in the Ar⋯HS Ã ← X̃ LIF spectrum were assigned to levels supported in the Ar⋯HS isomer minimum (−92° < γ < 92°) and the empirical surface15 was therefore fitted in this region of (R, γ) space only.

Our ab initio surface has a minimum of −742.5 cm−1 in this region which is considerably less deep than the −877.2 cm−1 on the empirical surface of Korambath et al.15

Interestingly, however, our value for the experimental dissociation energy, D0 (446.7 cm−1) is significantly closer to the measured value (458 cm−1) than that obtained from the empirical surface (572 cm−1).

Table 3 shows very good agreement between the rotational constants, Bv, calculated for levels on the ab initio surface and those calculated on the empirical surface.

The agreement is less satisfactory, however, for the vibrational level spacings also reported in Table 3.

The spacings calculated on the ab initio surface are consistently ∼5% smaller than the experimentally derived separations.

Unsurprisingly, the levels calculated on the empirical surface reproduce the measured levels typically to within 0.3 cm−1.

Our level of agreement is comparable with that obtained in other ab initio calculations for similar van der Waals molecules: In the ab initio study of NeSH by Cybulski et al.,3 the calculated energy level separations deviated from experiment by about 4%; Chakrovarty et al32. computed the vibrational energy levels for the Ã 2Σ+ state of Ar–OH using the CEPA surface of Esposti and Werner33 and determined vibrational spacings which were too small by about 15%.

The Ar⋯SH minimum

In the absence of experimental data for this region, Korambath et al. did not attempt to fit their empirical potential surface beyond the saddle point (i.e., for γ > 90°).

For completeness, however, they approximated the surface in this region to that of the Ar⋯OH potential which itself was only known in this geometry from ab initio studies.

It is no surprise then that our ab initio surface differs significantly from the empirical surface in this region.

Our surface shows a minimum only 10% less deep than the global minimum (see Table 2).

The Ar⋯SH minimum is substantially less localized in γ than the corresponding minimum on the empirical surface.

As Firsov et al. note,19 on the empirical surface, the energy reaches half-way to the dissociation limit at an Ar⋯S–H angle only 5° from the Ar⋯SH linear geometry.

By contrast, on the ab initio surface the molecule must bend 20° from linear before this is the case.

There is still a large barrier to internal rotation (greater than 600 cm−1 from either linear extreme) but our results suggest that the molecule samples a larger range of γ space than previously thought.

We calculate that the Ar⋯SH minimum supports several vibrational levels, the spacings between which, for van der Waals stretch levels, are comparable with those between levels supported in the global minimum (see Table 3).

In the light of the above it is reasonable to ask why, if well-defined bound levels exist for the Ar⋯SH isomer (i.e., γ ∼180°), transitions to these levels have not been identified in the LIF spectrum.

The reasons are probably two-fold.

Firstly when Korambath et al. started to fit the experimental data, it may have been thought that the Ar⋯HS X̃ 2Π ground state wavefunction was localized in a single minimum around the Ar⋯HS global minimum as it is in the isovalent Ar⋯HO (where the barrier to internal rotation is 44 cm−1, significantly above the van der Waals zero point energy.34)

In such a case the Franck–Condon factors for transitions from the X̃ state would indeed be dominated by transitions to the Ar⋯HS Ã state geometry.

The other minimum would be spectroscopically invisible and it would be sensible to look to fit the spectrum to transitions to levels in the Ar⋯HS minimum.

Secondly, the à state of the mercapto radical undergoes rapid electronic predissociation as a result of interaction with a nearby a4 Σ state.1

The effect is to reduce the lifetime of even the v = 0 level to ∼1 ns.

The van der Waals complexes of SH with rare-gas atoms live significantly longer due to the dramatic stabilizing effect of the complexation which lowers the à state potential to such an extent that the Ar⋯HS à state fluoresces with a lifetime of up to 600 ns.12,20,21

Experimentally, this lifetime enhancement has the major advantage that the fluorescence detection window can be delayed with respect the laser pulse thereby discriminating against the large background fluorescence signals of the SH radical which is present in large concentrations in all experiments of this type.

However, as discussed by Carter et al. in their review,1 this “caging of predissociation” can only be observed in the Ar⋯HS isomer since only in this geometry is the repulsive interaction between the H atom and the a4 Σ state significant.

Ar⋯SH isomer levels will, in all probability, fluoresce at a sub-ns rate comparable with the SH monomer.

Thus any experiment introducing a delay before the fluorescence is measured – including the experiments by both Yang et al12. and Mackenzie et al20,21. – is insensitive to fluorescence from the Ar⋯SH levels.

The FT microwave spectroscopy of the Ar⋯SH ground state by Sumiyoshi et al23,24. has shown that, far from being a hindered rotor, the zero point level lies some 20 cm−1 above the barrier to internal rotation and thus its wavefunction samples the full bending coordinate.

Indeed, this work has shown that the global minimum on the ground state potential lies in the Ar⋯SH minimum (some 7 cm−1 lower than the Ar⋯HS configuration) and the ground level wavefunction has local maxima at both linear geometries.

There should therefore be non-zero Franck–Condon overlap between the ground-state Ar⋯SH geometry and levels of the corresponding geometry in the à state.

The Ar⋯SH isomer is expected to predissociate at a rate comparable with the SH radical and thus in order to observe such transitions, it will be necessary to collect the early part of the fluorescence profile in the same way that the SH radical fluorescence is collected.

The search for transitions in the Ar⋯SH isomer is currently taking place in our laboratory.

The presence of bound levels in two different isomeric forms on the same potential surface implies that the levels in each minima should exhibit tunnelling splittings.

These splittings should be exquisitely sensitive to the height and shape of the barrier to internal rotation raising the prospect that high resolution microwave spectroscopy could be used to elucidate the finer points of the potential.

In this light we note that Fawzy et al. have calculated the rotational levels and microwave transition intensities in exactly this class of molecule.35,36

Finally, it is appropriate to discuss the suitability of the single determinant reference coupled cluster approach used here.

There is a danger in this type of calculation on excited states that the reference wavefunction in question can be contaminated by other states of the same symmetry.

In the present case the most likely cause of such contamination would be the A′ component of the ground state.

Ideally one would perform multi-reference coupled cluster calculations for such cases.

We chose the method used here on account of its success in previous treatments of the equivalent states in the Ne⋯OH, He⋯OH, Ne⋯SH and He⋯SH complexes.2,3

In our calculations we find that SCF solutions for the different components are so different that the program finds the correct one every time.

We experienced no problems in the convergence of the CCSD(T) and the T1 diagnostic, a measure of the quality of the SCF wavefunction, is reasonable37 (≤0.0216 across the interesting regions of the potential).

There clearly is some contamination of the surface but this value of T1 is not out of line with calculations on other open-shell species.

Hence, whilst this single configuration reference approach is not ideal, we believe the results are valid.

This conclusion is further supported by the good agreement with the empirical potential surface in the region accessible to experiment.

One obvious future extension of this work, besides performing multi-reference calculations, would be to use a basis-set extrapolation method.

Our decision to use the aug-cc-pV5Z basis was made following preliminary investigations with different basis set sizes at the Ar⋯HS geometry, which indicated that the convergence was sufficient as to render the extra cost of the extrapolation to the infinite basis set limit unjustifiable.

We note that ours is already the largest basis set to have been used for such systems.


We have calculated an ab initio potential energy surface for the à state of the Ar⋯SH van der Waals complex at the RCCSD(T) level with an aug-cc-pV5Z basis set.

Some 196 ab initio points were calculated and subsequently fitted using the model potential proposed by Bukowski et al.

Low-lying vibrational levels supported in this potential have been calculated using the DVR method.

The ab initio surface is compared with the experiments of the Miller group and is shown to reproduce well the experimental rotational constants although the calculated energy level spacings are systematically low by up to 5%.

In contrast to the empirical surface we have calculated the potential across the full (R, γ) configuration space.

This has allowed us to calculate the bound levels in the Ar⋯SH linear configuration potential well.

Transitions to these levels have not, the best of our knowledge, been assigned in any available spectrum and the reasons for this are discussed.