Spectroscopy and potential energy surface of the H₂–CO₂ van der Waals complex: experimental and theoretical studies

Abstract

A 4-D ab initio potential energy surface is calculated for the intermolecular interaction of hydrogen and carbon dioxide, using the CCSD(T) method with a large basis set. The surface has a global minimum with a well depth of 212 cm⁻¹ and an intermolecular distance of 2.98 Å for a planar configuration with both the O–C–O and H–H axes perpendicular to the intermolecular axis. Bound state calculations are performed for the H₂–CO₂ van der Waals complex with H₂ in both the para and ortho spin states, and the binding energy of paraH₂–CO₂ (50.4 cm⁻¹) is found to be significantly less than that of orthoH₂–CO₂ (71.7 cm⁻¹). The surface supports 7 bound intermolecular vibrational states for paraH₂–CO₂ and 19 for orthoH₂–CO₂, and the lower rotational levels with J ≤ 4 follow an asymmetric rotor pattern. The calculated infrared spectrum of paraH₂–CO₂ agrees well with experiment. For orthoH₂–CO₂, the ground state rotational levels allowed by symmetry are found to have (K_a, K_c) = (even, odd) or (odd, even). This somewhat unexpected fact enables the previously observed experimental spectrum to be assigned for the first time, in good agreement with theory, and indicates that the orientation of hydrogen is perpendicular to the intermolecular axis in the ground state of the orthoH₂–CO₂ complex.

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I. Introduction

Carbon dioxide, nitrous oxide, and carbonyl sulfide form an isovalent “CO₂ family” of linear molecules. The van der Waals (vdW) complexes containing these molecules have received much attention in both experimental and theoretical investigations.^1–8 It is found that the vdW complexes containing the CO₂ family have similar intermolecular interactions and spectra, as illustrated, for example, by He–CO₂,⁶ He–N₂O,⁷ and He–OCS.⁸ These three complexes all have a T-shaped structure and a rotational energy level structure which can be explained with an asymmetric rotor Hamiltonian, at least for lower energies. Ab initio potential energy surfaces (PES) for these complexes have been constructed and tested by bound state calculations, usually including two degrees of freedom (the vdW stretch and bend) with the linear molecule (CO₂, N₂O, OCS) taken as a rigid rotor. In general the experimental rotational spectra could be explained with the accuracy of about 10⁻² cm⁻¹ in these calculations. More accurate models have also been developed to account for the other experimental data, for example, Xie et al.⁹ have considered the asymmetric vibrational mode of N₂O in the Hamiltonian to explain the blue shift of the band origin for He–N₂O.

vdW complexes containing hydrogen molecules are of particular interest, in part due to hydrogen’s cosmic abundance and the possibility of complex formation in interstellar space.^10–19 Recently, McKellar and co-workers have reported the infrared spectra of vdW complexes formed by the CO₂ family and hydrogen. For complexes containing OCS²⁰ and N₂O²¹ attached to the experimentally accessible hydrogen species (paraH₂, orthoH₂, orthoD₂, paraD₂, and HD), rotation-vibration spectra were recorded in the region of the asymmetric stretch mode of the triatomic molecules and fitted to a model Hamiltonian. It was found that all these spectra corresponded to those of T-shaped asymmetric rotors with predominantly a-type transitions (ΔK_a = 0) (since the a inertial axis is approximately parallel to the O–C–S or N–N–O molecular axis). Pure rotational microwave spectra have also been obtained for H₂–OCS.²² To date, the only theoretical study of these systems was reported by Paesani and Whaley for H₂–OCS.¹⁵ In their work, a PES was calculated with explicit dependence on the OCS ν₃ vibrational mode, and predicted transition frequencies agreed with the observed values within 0.09 cm⁻¹ or better. Qualitative aspects of the energy level patterns of weakly bound complexes relevant to systems containing H₂ were discussed some time ago in a paper by Nesbitt and Naaman.²³

Due to the spin statistics of the indistinguishable ¹⁶O nuclei, half the rotational energy levels of CO₂ are missing in the observed spectra of the normal isotope. This is a key difference between complexes containing CO₂ and those containing OCS or N₂O. In the recent experimental study of paraH₂–CO₂,²⁴ 8 transitions were observed for the ¹²C¹⁶O₂ complex and 10 transitions for ¹³C¹⁶O₂ and ¹²C¹⁸O₂. These were fitted using an asymmetric rotor Hamiltonian, with results that were very similar to paraH₂–OCS²⁰ and –N₂O.²¹ However, the orthoH₂–CO₂ spectrum was found to be completely different, and no reasonable assignments could be made. It was therefore inferred that orthoH₂–CO₂ must be unlike orthoH₂–OCS and orthoH₂–N₂O, whose spectra resemble the corresponding paraH₂ complex. To our knowledge there is no modern H₂–CO₂ PES available in the literature for rovibrationally bound state calculations. Theoretical studies based on a 4-D PES are clearly desirable to help explain the ‘mystery’ of the orthoH₂–CO₂ infrared spectrum.²⁴ The construction of such a potential surface and bound state calculations for H₂–CO₂ form the main object of the present paper, which also includes further experimental data on orthoH₂–CO₂.

This paper is organized as follows. The theoretical determination of the ab initio PES and bound state energy levels are described in section II. Details of the experimental infrared spectra of H₂–CO₂ are reported and compared with theory in section III. Finally, a discussion and brief conclusions are presented in sections IV and V.

II. Ab initio potential energy surface and bound state calculations

A. Ab initio PES

The 4-D intermolecular PES is described by the Jacobi coordinate system (R, θ₁, θ₂, φ). The molecular axes of the linear molecules H₂ and CO₂ are denoted as r₁ and r₂, respectively. As shown in Fig. 1, R is the vector connecting the H₂ and CO₂ centers of mass, θ₁ is the enclosed angle between R and r₁, θ₂ is the enclosed angle between R and r₂, and φ is the torsional angle between the (R − r₁) and (R − r₂) planes. The H₂ and CO₂ molecules are approximated as rigid rotors, with their bond lengths fixed at the vibrationally averaged values of 0.7666 ¹⁷ and 1.1615 ⁶ Å, respectively.

Fig. 1 Jacobi coordinates for the H₂–CO₂ complex. R denotes the distance between the H₂ and CO₂ centers of mass, θ₁ (θ₂) is the enclosed angle between the R vector and the H₂ (CO₂) axis, and φ denotes the torsion angle of the H₂ axis anticlockwise from the plane formed with the R vector and CO₂ axis.

The intermolecular potential energy for a given geometry of the H₂–CO₂ complex is calculated using the supermolecular approach, with the potential energy expressed as the difference between the energy of the complex, E(H₂–CO₂) and the sum of the monomer energies, E(H₂) + E(CO₂). The full counterpoise method²⁵ is applied for basis set superposition error (BSSE) correction. Molecular energies are calculated with the CCSD(T) method,²⁶ using Dunning’s cc-pVQZ basis sets²⁷ with g functions omitted. Tao and Pan’s 3s3p2d bond functions²⁸ are also used in this work , and this bond function (Bf) has proven to be effective in the calculation of intermolecular interactions. By comparing with other larger basis sets (avqz + Bf, v5z(f) + Bf), the current one has been chosen by considering the balance of the accuracy and the cost, with an estimated error of 3% for the well depth. All the calculations are carried out using the Molpro 2002.6 software package.²⁹

The intermolecular potential energies are calculated for geometries with 24 grid points of intermolecular distance ranging from 2.2 to 8.0 Å, with angles θ₁ and θ₂ varied from 0 to 180° in steps of 15°, and with angle φ varied from 0 to 360° in steps of 30°. By considering symmetry, a total of 6360 discrete data points for the intermolecular potential energy are obtained. The PES is constructed by a combined scheme using the cubic spline interpolation method for the angular part, (θ₁, θ₂, φ), and the interpolating moving least square (IMLS) method^30,31 for the intermolecular distance. A Fortran program to generate the intermolecular potential, and the original ab initio data, are available as supplementary material (see ESI‡).

This potential has a global minimum with a well depth of −211.932 cm⁻¹ at the configuration R = 2.978 Å, θ₁ = 90.0°, θ₂ = 90.0°, φ = 0.0°, with an equivalent minimum at φ = 180.0°. Saddle points are found between the two global minima corresponding to the configuration R = 2.978 Å, θ₁ = 90.0°, θ₂ = 90.0°, and φ = 90.0° or φ = 270.0°, with a barrier height of 89.386 cm⁻¹. The H₂–CO₂ PES appears to be similar to that of H₂-OCS,¹⁵ which also has a co-planar global minimum with R = 3.23 Å, θ₁ = 87.0°, θ₂ = 109.4°, φ = 0.0°, and a well depth of 201.67 cm⁻¹.

B. Methodology of the bound state calculations

By approximating the linear molecules H₂ and CO₂ as rigid rotors, the Hamiltonian of the H₂–CO₂ system may be written as^32,33where μ is the reduced mass of the H₂–CO₂ dimer, Ĵ is the total angular momentum operator, ĵ₁ and ĵ₂ are the rotational angular momentum operators for H₂ and CO₂, respectively, and ĵ₁₂ = ĵ₁ + ĵ₂. B_H2 and B_CO2 denote the appropriate monomer rotational constants: B_H2 = 59.322 cm⁻¹ and B_CO2 = 0.3902 or 0.3871 cm⁻¹ for CO₂ in its ground or excited ν₃ state, respectively.

The wave function is expanded as a linear combination of products of the intermolecular distance and angular basis functions,

where φ_i(R) is the basis function for the vdW stretch, chosen as a sine function in this work. ε is the index of the space-inverse parity and Y_j1j2j12^JMKε is the total symmetry-adapted angular basis function, which has the following explicit form,The total angular basis function is expressed in a body-fixed frame, where D^J_MK is a Wigner rotation matrix which is used to describe the overall rotation of the H₂–CO₂ system and Y_j1j2^J₁₂K takes the formwhere the y_j,m are spherical functions describing the rotation of the two linear molecules H₂ and CO₂, and 〈j₁m₁j₂K − m₁j₁₂K〉 is a Clebsch–Gordan coefficient. Further details of calculating Hamiltonian matrix elements can be found in ref. 33.

The ARPACK software package³⁴ is applied to solve the eigenvalues and eigenfunctions of the bound states. By considering the symmetry of (ε, j₁, j₂), there exist 8 symmetry blocks: (+1, para, even), (+1, para, odd), (−1, para, even), (−1, para, odd), (+1, ortho, even), (+1, ortho, odd), (−1, ortho, even), (−1, ortho, odd). Each block can be solved separately. A total of 55 sine basis functions ranging from 3.0 to 15.0 Bohr are used for the R basis set expansion. The size of the rotational basis functions is controlled by the parameters, j_1max = 5, j_2max = 17, and the energy levels involved in spectroscopic calculation show convergence to 0.001 cm⁻¹ with this basis set.

C. Rovibrational energy levels

The calculated intermolecular vibrational energy levels of para and orthoH₂–CO₂ are listed in Table 1. As our calculated results showed that the rotation of the H₂ in the complex is completely dominated by j₁ = 0 and 1 in the para and ortho species, respectively, the intermolecular vibrational energy levels could be labeled with the indices (n_s, n_b, j₁, p_m1) by analyzing the wavefunction, where p_m1 is a composite index giving the parity of the state and the value of m₁, which is the projection of j₁ (the H₂ angular momentum) on the vector R. When the complex is in its lowest energy equilibrium configuration, R is perpendicular to the O–C–O axis and corresponds to the b inertial axis of the asymmetric rotor model (see below). The quantum number n_s denotes the vdW stretching mode. In place of j₂ (the CO₂ rotation), which is not a very good quantum number, we use a bending label n_b, similar to an atom–diatom complex like CO₂–He.² Then for paraH₂–CO₂ there exist 7 bound vibrational states, labeled as (n_s,n_b,0,0) in Table 1, with the lowest state bound by 50.383 cm⁻¹, corresponding to a total zero-point energy of 161.549 cm⁻¹. This zero-point energy is over half of the global well depth, similar to other H₂-containing vdW complexes.¹⁵ Most states correspond to excitations of n_b which have relatively low energy consistent with the small rotational constant of CO₂. It is interesting to note that label n_b has the same parity (even or odd) as j₂. The highest vibrational state is assigned to the first excited vdW stretch mode, giving an intermolecular stretching frequency of 47 cm⁻¹.

Table 1 Calculated bound intermolecular vibrational state energies (in cm⁻¹) of H₂–CO₂ for CO₂ in its vibrational ground state, with assigned vibrational quantum labels given in parentheses, (n_s, n_b, j₁, p_m1)

Even +1	Even −1	Odd +1	Odd −1
ParaH2–CO2
−50.383 (0,0,0,0)		−28.316 (0,1,0,0)
−23.544 (0,2,0,0)		−20.235 (0,3,0,0)
−15.116 (0,4,0,0)		−7.107 (0,5,0,0)
−3.335 (1,0,0,0)
OrthoH2–CO2
−46.472 (0,1,1,1)	−24.993 (0,1,1,−1)	−71.704 (0,0,1,1)	−51.068 (0,0,1,−1)
−35.693 (0,3,1,1)	−9.226 (1,1,1,−1)	−37.837 (0,2,1,0)	−16.172 (1,0,1,−1)
−25.547 (0,3,1,0)		−31.789 (0,2,1,1)	−4.991 (0,2,1,−1)
−18.368 (0,1,1,0)		−19.099 (1,0,1,0)
−9.725 (0,5,1,0)		−12.831 (0,4,1,1)
−5.742 (1,1,1,1)		−7.137 (1,0,1,1)
−1.786 (1,1,1,0)		−1.215 (1,2,1,0)

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For orthoH₂–CO₂, the vibrational states dominated by (j₁ = 1,m₁ = 0) are labeled (n_s,n_b,1,0), and those dominated by (j₁ = 1,m₁ = ±1) labeled (n_s,n_b,1,1) for parity = +1 and (n_s, n_b, 1, −1) for parity = −1, respectively. We note that the ground vibrational state (0, 0, 1, 1) belongs to the (odd, +1) symmetry block, so that n_b and j₂ have different parity in the ortho species. Labeling of the excited orthoH₂–CO₂ intermolecular modes is difficult since they are strongly anharmonic and mixed, so only the lower energy orthoH₂–CO₂ states in Table 1 could be assigned with complete certainty. There are 19 bound intermolecular vibrational states for orthoH₂–CO₂, almost 3 times more than for the para species. It is important to recall that orthoH₂–CO dissociates into the j_H2 = 1 channel, so that all states with energies lower than 2B_H2 = 118.644 cm⁻¹ are bound. The orthoH₂–CO₂ ground state is bound by 71.7 cm⁻¹, and so is considerably more stable than that of paraH₂–CO₂, as also observed in many other vdW complexes containing hydrogen.

Since the available infrared spectra of H₂–CO₂ involve the CO₂ν₃ asymmetric stretch mode, rotational energy levels of the complex with CO₂ in both the ground state and the v₃ state were calculated (differing in the value used for B_CO2). Rovibrational energy levels for J > 0 were calculated and it was found that no bound states exist with J beyond 12 and 14 for the para and ortho species, respectively. Tables 2 and 3, for paraH₂- and orthoH₂–CO₂, list all the calculated levels with J = 0 to 4 which lie within 20 cm⁻¹ of the lowest level. More complete lists of calculated rovibrational levels up to dissociation are given in supplementary Tables S1 and S2 (see ESI ‡).

Table 2 Calculated bound rotational energy levels (in cm⁻¹) of paraH₂–CO₂ for CO₂ in its vibrational ground state and excited v₃ state, with assigned asymmetric rotor quantum labels given in parentheses, (J, K_a, K_c)

J	v 3	Even +1	Even −1	Odd +1	Odd −1
0	0	−50.383 (000)
1	0		−49.385 (111)	−49.230 (110)	−49.727 (101)
2	0	−48.456 (202)	−47.768 (211)	−48.232 (212)	−46.747 (221)
2	0	−46.706 (220)
3	0	−44.804 (322)	−46.523 (313)	−45.604 (312)	−46.638 (303)
3	0		−42.702 (331)	−42.695 (330)	−44.616 (321)
4	0	−44.324 (404)	−42.782 (413)	−44.278 (414)	−42.238 (423)
4	0	−41.745 (422)	−40.047 (431)	−40.093 (432)	−37.245 (441)
4	0	−37.244 (440)
0	1	−50.402 (000)
1	1		−49.405 (111)	−49.252 (110)	−49.757 (101)
2	1	−48.487 (202)	−47.801 (211)	−48.259 (212)	−46.770 (221)
2	1	−46.730 (220)
3	1	−44.840 (322)	−46.660 (313)	−45.652 (312)	−46.679 (303)
3	1		−42.727 (331)	−42.721 (330)	−44.657 (321)
4	1	−44.375 (404)	−42.849 (413)	−44.327 (414)	−42.292 (423)
4	1	−41.811 (422)	−40.094 (431)	−40.138 (432)	−37.273 (441)
4	1	−37.272 (440)

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Table 3 Calculated bound rotational energy levels (in cm⁻¹) of orthoH₂–CO₂ for CO₂ in its vibrational ground state and excited v₃ state, with assigned asymmetric rotor quantum labels given in parentheses, (J, K_a, K_c)

J	v 3	Even +1	Even −1	Odd +1	Odd −1
0	0			−71.705 (000)
1	0	−70.546 (110)	−71.082 (101)		−70.663 (111)
2	0	−69.536 (212)	−67.932 (221)	−69.860 (202)	−69.188 (211)
2	0			−67.912 (220)
3	0	−67.162 (312)	−68.072 (303)	−66.078 (322)	−67.860 (313)
3	0	−63.689 (330)	−65.976 (321)		−63.691 (331)
4	0	−65.640 (414)	−63.616 (423)	−65.764 (404)	−64.491 (413)
4	0	−61.200 (432)	−57.926 (441)	−63.334 (422)	−61.185 (431)
4	0			−57.927 (440)
0	1			−71.732 (000)
1	1	−70.576 (110)	−71.114 (101)		−70.691 (111)
2	1	−69.571 (212)	−67.963 (221)	−69.899 (202)	−69.228 (211)
2	1			−67.943 (220)
3	1	−67.215 (312)	−68.120 (303)	−66.120 (322)	−67.905 (313)
3	1	−63.722 (330)	−66.021 (321)		−63.724 (331)
4	1	−65.697 (414)	−63.674 (423)	−65.824 (404)	−64.561 (413)
4	1	−61.250 (432)	−57.961 (441)	−63.400 (422)	−61.235 (431)
4	1			−57.961 (440)

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Due to the interchange symmetry of the indistinguishable zero-spin ¹⁶O nuclei, the normal isotope of carbon dioxide, ¹²C¹⁶O₂, can only have even values of the angular momentum, j₂, in its ground vibrational state. The same is true for other symmetric isotopomers containing ¹⁶O or ¹⁸O (e.g.¹³C¹⁶O₂ or ¹²C¹⁸O₂), but not for asymmetric forms (e.g.¹⁶O¹²C¹⁸O) or for symmetric ¹⁷O forms (e.g.¹⁷O¹²C¹⁷O). According to our classification, the allowed symmetry blocks (for ν₃ = 0) in Tables 1–3 are therefore: (ε, j₁, j₂) = (+1, para, even), (−1, para, even), (+1, ortho, even), (−1, ortho, even). Analyzing the labels of the paraH₂–CO₂ rotational levels in Tables 2 and 3, it is found as expected that the allowed rotational levels have (K_a, K_c) = (even, even) or (odd, odd). For orthoH₂–CO₂ the allowed levels are however found to be (K_a, K_c) = (even, odd) or (odd, even) for the ground state. Thus the lowest allowed level for orthoH₂–CO₂ is J_KaKc = 1₀₁, rather than 0₀₀, which is forbidden. This turns out to be the key to assigning the orthoH₂–CO₂ spectrum, as discussed below. When the CO₂ asymmetric stretch vibration, ν₃, is excited, odd values of j₂ are required. In this case, the allowed blocks in Tables 1–3 become: (ε, j₁, j₂) = (+1, para, odd), (−1, para, odd), (+1, ortho, odd), (−1, ortho, odd).

III. Experimental results and comparison with theory

The only available high resolution spectra of the weakly-bound complex H₂–CO₂ were recently obtained²⁴ in the mid-infrared region of the CO₂ν₃ fundamental band, around 2350 cm⁻¹. This experiment was performed using a pulsed supersonic jet expansion and a rapid-scan tunable diode laser probe.³⁵ Both paraH₂–CO₂ and orthoH₂–CO₂ transitions were observed, but only the former were analyzed because the correct assignments for the orthoH₂ species were not evident. The present theoretical results now enable the orthoH₂ assignments to be made successfully, as shown below in section B. However, first we compare the experimental and theoretical results for paraH₂–CO₂ in the following section.

A. paraH₂–CO₂

A total of 8 transitions were experimentally observed and assigned²⁴ for paraH₂–¹²C¹⁶O₂, as listed here in the first two columns of Table 4. In order to obtain theoretical values for these transitions from the energy levels in Table 2, we used the expressionwhere E₀ is a vibrational origin. However, our intermolecular potential was calculated for fixed monomer (CO₂ and H₂) geometries, and so does not vary with CO₂ vibrational state. The CO₂ vibrational dependence of the H₂–CO energy levels was partly included by varying the CO₂ rotational constant, B_CO2. This approximation has proven to be rather effective for rotational spacings,^6,17 but it cannot accurately represent the possible vibrational shift of the complex origin, E₀, relative to the monomer (CO₂) origin. In the present case, we obtain a value for E₀ by simply comparing a well-defined observed transition with the theoretical result and use this for the remaining theoretical transition frequencies.

Table 4 Comparison between observed and calculated transition frequencies in the infrared spectrum of paraH₂–¹²C¹⁶O₂ (in cm⁻¹)

J′Ka′Kc′–J″Ka″Kc″	Observed	Calculated	Obs-Calc
a The calculated band origin was established by fixing the calculated 101 − 000 transition at the observed value (see text).
10 1–00 0	2349.599	2349.599	—^a
11 0–11 1	2349.094	2349.099	−0.005
212–11 1	2350.096	2350.092	+0.004
10 1–20 2	2347.664	2347.672	−0.008
30 3–20 2	2350.754	2350.744	+0.010
11 0–21 1	2347.480	2347.482	−0.002
31 2–21 1	2351.082	2351.082	−0.000
41 4–31 3	2351.173	2351.162	+0.011

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Specifically, we use the J_KaKc = 1₀₁–0₀₀ transition to determine a value of E₀ = 2348.966 cm⁻¹ for the para spectrum.³⁶ The third and fourth columns of Table 4 show the resulting theoretical transitions and the deviations between experiment and theory for paraH₂–CO₂. These deviations are mostly within 0.01 cm⁻¹, with a maximum discrepancy of 0.011 cm⁻¹ for the transition 4₁₄–3₁₃. This excellent agreement shows that our ab initio PES accounts very successfully for the paraH₂–CO₂ spectrum.

In order to produce spectroscopic parameters from the theoretical levels, we fit them with the same Hamiltonian used for the experimental data, a Watson asymmetric rotor expression employing the A-type reduction in the Ir representation,²

There is a question as to which theoretical levels should be fit in order to compare with experiment. Should it be all the calculated levels with J ≤ 4 from Table 2, or should it be only the allowed rotational levels, or should it be only those levels which were experimentally observed? We chose to fit all levels with J ≤ 4, so that the resulting parameter set would represent the calculated levels as accurately as possible. The fitted parameters are given in Table 5, and they show good overall agreement with the corresponding experimental values (taken from Table 2 of ref. 24), especially considering that the theoretical fit includes a wider range of rotational levels and two additional parameters (δ_K and δ_J), compared to the limited experimental fit.

Table 5 Comparison of observed and calculated spectroscopic parameters for paraH₂–CO₂ and orthoH₂–CO₂ in the ground and excited CO₂ ν₃ states (in cm⁻¹)^a

	paraH2–^12C^16O2		orthoH2 –^12C^16O2
	Experiment	Theory	Experiment	Theory
a Quantities in parentheses correspond to 1 σ from the least-squares fits. b Some parameters were fixed in the fits because of limited data (see text and ref. 24).
v 0	2348.9452(1)		2349.0478(2)
A″	0.7558(14)	0.749 106(39)	0.7940(55)	0.789 999(93)
B″	0.403 96(8)	0.406 477(57)	0.373 85(10)	0.369 33(15)
C″	0.253 70(7)	0.249 485(49)	0.251 86(11)	0.252 41(13)
Δ″K	0.000 011^b	−0.000 518(5)	0.0015(14)	−0.000 114(12)
Δ″JK	0.002 03(9)	0.000 969(6)	0.000 25(2)	0.000 487(15)
Δ″J	−0.000 047(4)	0.000 016(2)	0.000 051(2)	−0.000 016(4)
δ″K	0^b	0.000 638(9)	0.001 59(4)	0.000 174(34)
δ″J	0^b	0.000 004(1)	−0.000 003(2)	−0.000 006(3)
A′	0.7558^b	0.749 127(38)	0.7935(149)	0.790 061(92)
B′	0.401 48^b	0.403 202(57)	0.370 90(11)	0.366 64(15)
C′	0.252 27^b	0.248 240(48)	0.250 47(12)	0.251 13(14)
Δ′K	0.000 011^b	−0.000 508(5)	0.0015^b	−0.000 107(12)
Δ′JK	0.002 03^b	0.000 959(6)	0.000 25^b	0.000 481(15)
Δ′J	−0.000 047^b	0.000 015(1)	0.000 051^b	−0.000 015(4)
δ′K	0^b	0.000 632(9)	0.001 59^b	0.000 173(35)
δ′J	0^b	0.000 004(1)	−0.000 003^b	−0.000 006(3)

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B. orthoH₂–CO₂

Transitions due to orthoH₂–CO₂ were clearly observed in the experiments of ref. 24 by recording spectra both with normal H₂ (75% ortho, 25% para) in the supersonic expansion gas mix, as well as samples with enriched paraH₂ (>99% para). As noted previously for other complexes,^20,21 normal H₂ gave predominantly orthoH₂–CO₂ transitions, while the enriched paraH₂ sample gave both the para and ortho complexes. This dominance of orthoH₂ complexes occurs due to their larger binding energy relative to the corresponding paraH₂ complex (e.g. 71.7 vs. 50.4 cm⁻¹ for orthoH₂–CO₂vs. paraH₂–CO₂ in the present calculation). However, the orthoH₂–CO₂ transitions were not rotationally assigned in ref. 24 because they seemed to have a completely different pattern than that observed for paraH₂–CO₂.

The key to the correct assignment is found in the present calculations. Tables 2 and 3 clearly indicate that the allowed ν₃ = 0 levels (in the even, +1 and even −1 columns) have (K_a, K_c) = (even, odd) and (odd, even) for orthoH₂ as compared to (even, even) and (odd, odd) for paraH₂. It is thus no wonder that the orthoH₂–CO₂ spectrum is different: its transitions are the very ones which are missing from the paraH₂–CO₂ spectrum (and vice versa)! With this revelation, the observed orthoH₂–CO₂ transitions can be assigned easily, as given in Table 6. For example, the two most prominent observed lines, at 2350.266 and 2350.399 cm⁻¹, are due to the J_KaKc = 2₀₂–1₀₁ and 2₁₁–1₁₀ transitions, respectively. Table 6 includes 10 lines from ref. 24, plus 6 additional weaker lines which were located and assigned with the help of the present calculations. One previously reported²⁴ line (at 2349.460 cm⁻¹) still cannot be assigned, and is likely due to something other than H₂–CO₂. Part of the orthoH₂–CO₂ spectrum is illustrated here in Fig. 2.

Fig. 2 Part of the observed spectrum of orthoH₂–¹²C¹⁶O₂ complexes, showing 7 of the 16 assigned transitions. The line marked ‘p’ is due to paraH₂–CO₂, the line marked ‘h’ is due to He–CO₂, and the remaining weak lines are due to carbon dioxide dimers, (CO₂)₂.

Table 6 Comparison between observed and calculated transition frequencies in the infrared spectrum of orthoH₂–¹²C¹⁶O₂ (in cm⁻¹)

J′Ka′Kc′–J″Ka″Kc″	Observed	Calculated	Obs-Calc
a The calculated band origin was established by fixing the calculated 000–101 transition at the observed value (see text).
0 0 0–10 1	2348.423	2348.423	—^a
20 2–10 1	2350.266	2350.256	+0.010
11 1–11 0	2348.930	2348.927	+0.003
21 1–11 0	2350.399	2350.391	+0.008
11 1–21 2	2347.912	2347.918	−0.006
21 1–21 2	2349.381	2349.381	+0.000
31 3–21 2	2350.717	2350.704	+0.013
22 0–22 1	2349.060	2349.061	−0.001
32 2–22 1	2350.894	2350.884	+0.010
20 2–30 3	2347.236	2347.246	−0.010
32 2–30 3	2351.008	2351.024	−0.016
40 4–30 3	2351.339	2351.320	+0.019
21 1–31 2	2346.999	2347.007	−0.008
41 3–31 2	2351.688	2351.674	+0.014
42 2–32 1	2351.644	2351.648	−0.004
31 3–41 4	2346.795	2346.808	−0.013

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Calculated values for the orthoH₂–CO₂ transition frequencies are given in Table 6. These were determined in a similar manner to paraH₂–CO₂, with the prominent J_KaKc = 0₀₀–1₀₁ line taken as the standard to establish the vibrational origin. The agreement between observed and calculated frequencies, though slightly worse than for the para species (Table 4), is still excellent, with a maximum deviation of 0.019 cm⁻¹. The parameters resulting from asymmetric rotor fits to the observed spectrum and to the calculated energy levels are given in the last two columns of Table 5 for orthoH₂–CO₂. In the fit to experiment, the upper and lower state centrifugal distortion parameters were constrained to be equal because of the limited data available. In the fit to the theoretical data (Table 3), all levels with J ≤ 4 were included, both allowed and forbidden, as in the case of paraH₂–CO₂. There is good agreement between the experimental and theoretical parameters in Table 5, which is not surprising since we know that the observed and calculated line positions agree well. Most trends in Table 5 match well between experiment and theory: that is, there are similar parameter changes between the para and ortho species and between the lower (ν₃ = 0) and upper (ν₃ = 1) states.

The experimental band origin determined here for orthoH₂–CO₂ is only slightly red-shifted (−0.096 cm⁻¹) relative to the origin of the free CO₂ molecule. Interestingly, the vibrational shifts for H₂–CO₂ complexes are almost identical to those of H₂–OCS complexes, as shown in Table 7. All three family members, CO₂, OCS, and N₂O show a similar trend, in that the orthoH₂ complex is less red-shifted (or more blue-shifted) than the paraH₂ complex.

Table 7 Observed vibrational frequency shifts (in cm⁻¹) in the infrared spectra of H₂–X complexes, with X = CO₂, OCS, and N₂O

X	paraH2–X complex	orthoH2–X complex
a Present work and ref. 24. b ref. 20. c ref. 21.
CO2^a	−0.198	−0.096
OCS^b	−0.205	−0.094
N2O^c	+0.226	+0.624

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IV. Discussion

A. Average structural parameters

Theoretical average structural parameters, 〈R〉 and 〈θ₂〉, determined from the ground state wavefunctions are shown in Table 8. The average value of θ₂ is defined as P₂(cos 〈θ₂〉) = 〈P₂(cosθ₂)〉, where P₂ is the 2nd-order Legendre function. 〈P₂(cosθ₂)〉 is the average value of the Legendre function for the state of interest.

Table 8 Comparison of calculated and observed structural parameters for paraH₂–CO₂ and orthoH₂–CO₂

	〈R〉/Å	〈θ2〉
paraH2–CO2 Theory	3.502	78.37°
paraH2–CO2 Experiment	3.514	80.5°
orthoH2–CO2 Theory	3.384	79.53°
orthoH2–CO2 Experiment	3.522	76.6

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It is interesting to observe that the intermolecular distances deviate greatly from the global minimum in the PES, R = 2.978 Å, reflecting the effects of large amplitude motion and zero point energy. Similarly, the deviation of 〈θ₂〉 from its equilibrium value of 90° can be thought of as an effect of the zero point librational motion of CO₂ within the complex. Estimates of the same structural parameters obtained from the experimental rotational parameters using a simple ‘stick and ball’ model,³⁷ are also shown in Table 8. In the present case, we consider that the theoretical structures are more meaningful than the experimental ones, which rely on a rather crude model. In particular, the vibrationally averaged intermolecular separation 〈R〉 of the ground state of orthoH₂–CO₂, is found to be somewhat shorter than that of the para species, 3.38 vs. 3.50 Å.

B. Energy level pattern

In this paper, we have focused on the rotational energy levels of the ground intermolecular vibrational state of the H₂–CO₂ complex. For moderate values of excitation, J < 5, these levels are found experimentally and theoretically to correspond rather well to an asymmetric rotor pattern. This is interesting in itself for a floppy and weakly-bound van der Waals complex, and particularly so for orthoH₂–CO₂. The lowest excited intermolecular state of paraH₂–CO₂ is calculated to lie almost 22 cm⁻¹ above the ground state (Table 1), and it is assigned as the first excited CO₂ librational state, n_b = 1. The J = 0 level of this state is forbidden, but excited rotational levels with (K_a, K_c) = (even, odd) and (odd, even) are allowed.

The analogous n_b = 1 excited state of orthoH₂–CO₂ appears at 25.2 cm⁻¹, and is dominated by m₁ = ±1 with parity = +1. However the first excited state of orthoH₂–CO₂ is calculated to lie slightly lower in energy, at 20.6 cm⁻¹ (Tables 1 and S2, see ESI‡). This 20.6 cm⁻¹ state is dominated by m₁ = ±1 with parity = −1 and may be a ‘tunneling partner’ of the ground state. The ground and first excited states are thus labeled as (0, 0, 1, 1) and (0, 0, 1, −1), respectively. The fact that all the excited states are separated from the ground state by at least 20 cm⁻¹ probably helps to explain why the ground state levels can be represented by an asymmetric rotor Hamiltonian. It would be very interesting to observe the excited intermolecular vibrations of H₂–CO₂, but experimental access to these states may be difficult.

Since H₂–CO₂ is made up of nonpolar components, we do not expect to observe strong pure rotational transitions. However, with sufficient experimental sensitivity it should be possible to detect such transitions, as shown already⁴ in the case of He–CO₂. For the paraH₂–CO₂ complex, our theoretical results predict the fundamental J_KaKc = 1₁₁–0₀₀ transition to occur in the region of 29931 MHz. For orthoH₂–CO₂, the most accessible transition may be 1₁₀–1₀₁, predicted to occur around 16075 MHz. Other possible microwave transitions can be calculated from Tables 2 and 3.

V. Conclusions

In the present work a new 4-D ab initio intermolecular potential energy surface for the H₂–CO₂ complex has been presented. Energies were calculated using the CCSD(T) method with a large basis set. The PES has a global minimum at two equivalent co-planar configurations (R = 2.978 Å, θ₁ = 90°, θ₂ = 90°, φ = 0° and 180°) with a well depth of 211.932 cm⁻¹ and a barrier between these two minima with a height of 89.386 cm⁻¹. Bound state calculations have been performed for both para and orthoH₂–CO₂ species up to high J values. The orthoH₂–CO₂ ground state is considerably more stable than that of the para species (71.7 vs. 50.4 cm⁻¹). The potential energy surface supports 7 pure intermolecular vibrational states for paraH₂–CO₂ and 19 for orthoH₂–CO₂. Calculated infrared transition frequencies are in very good agreement with experimental values for both paraH₂- and orthoH₂–CO₂. For orthoH₂–CO₂, the allowed ground state energy levels are found to correspond to (K_a, K_c) = (even, odd) or (odd, even) levels, allowing the experimental spectrum to be assigned for the first time, and indicating that the orthoH₂ is perpendicular to the intermolecular axis in the complex.

Acknowledgements

Minghui Yang was supported by Natural Science Foundation of China (No. 20403029) and Outstanding Overseas Scholar Foundation of Chinese Academy of Sciences. Part of the computations were done on Dawning4000 A of the Shanghai supercomputer center. We are grateful to P. R. Bunker for helpful discussions.

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Footnotes

† The HTML version of this article has been enhanced with colour images.

‡ Electronic supplementary information (ESI) available: The ab initio data, a Fortran program to generate the intermolecular potential surface, and a more complete listing of the calculated H₂–CO₂ bound rovibrational levels. See DOI: 10.1039/b614849b

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