Theoretical calculation of a full-dimensional ab initio potential energy surface and prediction of infrared spectra for Xe–CS₂

Abstract

An effective four-dimensional (4D) ab initio potential energy surface (PES) for Xe–CS₂ which explicitly involves the intramolecular Q₁ symmetric stretching and Q₃ antisymmetric stretching vibrational coordinates of CS₂ is constructed. The computations are carried out employing single- and double-excitation coupled-cluster theory with a non-iterative perturbation treatment of triple excitations [CCSD(T)] method with a large basis set. Two vibrationally averaged potentials at the ground and ν₁ + ν₃ (ν₁ = 1, ν₃ = 1) excited states are obtained by integrating the 4D potentials over the Q₁ and Q₃ coordinates. The potentials have a T-shaped global minimum and two equivalent linear local minima. The radial discrete variable representation/angular finite basis representation and the Lanczos algorithm are employed to calculate the rovibrational energy levels for Xe–CS₂. The infrared band origin shift associated with the fundamental band of CS₂ is predicted, which is red-shifted by −1.996 cm⁻¹ in the ν₁ + ν₃ region. In addition, we further predict the spectroscopic parameters for the ground and the ν₁ + ν₃ excited states of Xe–CS₂. Compared with the previous Rg–CS₂ (Rg = He, Ne, Ar, Kr) complexes, we found that the complexes of the rare gas atoms with CS₂ display obvious regularities.

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I. Introduction

The linear CX₂ (X = O, S) molecules play an important role in the investigation of atmosphere. While complexes consisting of small linear molecules (CO₂, OCS, CS₂) bound to a rare gas (Rg) have had a remarkable growth in interest in spectroscopic studies. Such studies are aimed at advancing our understanding of the weak intermolecular forces and dynamics of these bound molecules. As we know, CO₂ exists extensively in nature and absorbs infrared radiation in the ground. Accordingly, complexes containing CO₂ have been widely investigate both experimentally^1–7 and theoretically.^8–25 As an analog of CO₂, CS₂ is one of the sulfur compounds in the Earth’s atmosphere and has also been widely studied because sulfur is a key element in the spectroscopy of giant planets. In addition, some interesting differences were explored for Rg–CS₂ compared to Rg–CO₂.

Experimentally, the first infrared spectra for the Rg–CS₂ complexes²⁶ with Rg = He, Ne, Ar were studied in the CS₂ ν₃ region. Meanwhile, Mivehvar et al. also reported the high resolution spectra of the He–CS₂ complex in the ν₁ + ν₃ region (2185 cm⁻¹). In their work, the structures were T-shaped and the spectroscopic parameters were presented. Theoretically, a number of ab initio potential energy surfaces (PESs) for Rg–CS₂ were constructed at different levels of theory. For instance, Farrokhpour and co-workers²⁷ first determined the PESs for the Rg–CS₂ complexes using CCSD(T) theory with the aug-cc-pVDZ basis set. Soon after, Zang et al.²⁸ reported the two-dimensional PES for the Rg–CS₂ (Rg = He, Ne, Ar) complexes by using the aug-cc-pVTZ basis set at the CCSD(T) level. However, these theoretical studies have obtained the PESs that encompass only two intermolecular vibration coordinates and a fixed CS₂ geometry. The results showed that the spectral pattern in the infrared region cannot be properly identified. Thus, construction of the PESs that involve the intramolecular vibration modes of CS₂ is needed. Recently, we calculated a 3D PES of the Rg–CS₂ complex^29,30 including the antisymmetric stretching Q₃ coordinate of the CS₂ monomer, and showed a good agreement with the experimental infrared data. In order to explore more detailed spectral information for Rg–CS₂, we further carried out the calculation of a four-dimensional PES of the Rg–CS₂ (ref. 31–33) complexes, which incorporates the Q₁ symmetric and Q₃ anti-symmetric stretching coordinates of CS₂. We have successfully reported the 4D PESs for the complexes of the lighter rare gas atoms with CS₂ by this method. To our knowledge, the experimental and theoretical information in the ν₁ + ν₃ region are absent for Xe–CS₂.

In order to discover the trends and diversities among the Rg–CS₂ complexes, we present a reliable 4D PES for Xe–CS₂ with CS₂ in the ν₁ + ν₃ region by applying the CCSD(T) method with the aug-cc-pVQZ basis set. This paper is arranged as follows: in Section II, we show the computational details that include ab initio and rovibrational energy levels. The discussion of the PES, the calculated rovibrational bound states, and the predicted infrared spectra are presented in Sections III–IV. Finally, a brief conclusion is given in Section V.

II. Computational details

A. Ab initio calculations

For the Xe–CS₂ dimer, the Jacobi coordinates (R, θ, Q₁, Q₃) are employed to describe the geometry. In the geometric variables, R denotes the intermolecular distance between the center of mass of CS₂ to the Xe atom, the angle of the vector R with respect to the CS₂ molecule is defined as θ. Q₁ and Q₃ are the normal mode coordinates, which describe the ν₁ symmetric stretching vibration and ν₃ antisymmetric stretching vibration of CS₂, respectively. Here, the intramolecular vibrational coordinates (Q₁, Q₃) can be simply defined aswhere r_CS1 and r_CS2 mean the two C–S bond lengths of CS₂, r_e is the average bond length derived from experimental spectra data.³⁴ The two-dimensional Q₁ and Q₃ potential curves were computed at the CCSD(T) level to determine the energy levels and wave functions for the Q₁ and Q₃ modes. The coordinate scaling method³⁵ was employed to adjust the two-dimensional potential in order to reproduce the experimental frequencies for the fundamental band.³⁴ We generated 25 potential optimized discrete variable representation (PODVR)^36,37 grid points corresponding to Q₁ = −0.122411, −0.044046, −0.027968, −0.101203, −183772a₀, Q₃ = −0.245743, −0.116813, 0.0, 0.116813, 0.245743a₀ for the ground state, and Q₁ = −0.139155, −0.060791, 0.011220, 0.084454, 0.167022a₀, Q₃ = −0.247934, −0.117859, 0.0, 0.117859, 0.247934a₀ for the ν₁ + ν₃ (ν₁ = 1, ν₃ = 1) excited state.

The ab initio potential energies were computed for a total of about 9000 discrete points. A relatively dense grid was calculated with 27 values of R ranging from 5.50a₀ to 24.00a₀ and 13 points of θ from 0° to 180° at intervals of 15°. The four dimensional PES for Xe–CS₂ was performed using the CCSD(T)³⁸ method. The aug-cc-pVQZ basis set of Woon and Dunning³⁹ was used for carbon and sulfur atoms, and the quasirelativistic pseudopotential aug-cc-pVQZ-PP⁴⁰ basis set was used for the Xe atom, supplemented with an additional set of bond functions (3s3p2d1f1g).⁴¹ The full counterpoise procedure (FCP)⁴² was selected to correct the basis set superposition error (BSSE). The vibrationally averaged 2D potentials V_ν1+ν3(R,θ) were obtained by averaging the 4D potential over the Q₁ and Q₃ vibrational coordinates, which can be written as

The cubic spline interpolation was employed to generate the averaged PESs for R and θ coordinates. The root-mean-square (rms) deviation is about 0.08 cm⁻¹ in the final fit. All calculations were carried out using the MOLPRO package.⁴³

B. Calculations of rovibrational energy levels

With the Born–Oppenheimer approximation, the vibrational averaged 2D intermolecular Hamiltonian of the Xe–CS₂ complex can be written as^44,45where μ is the reduced mass of the Xe–CS₂ complex, Ĵ and ĵ are the angular momentum operators for the total and monomer rotations. I_ν1+ν3 represents the vibrationally averaged rotational moment of the inertia of CS₂, which can be defined by the following equation,

Based on the PES, the rovibrational Hamiltonian and wave functions were calculated with the radial DVR/angular FBR method.^46,47 In our work, 120 sine-DVR⁴⁸ grid points were used for the radial coordinate R. For the angular coordinate, we selected 90 DVR grids and 89 basis functions of associated Legendre polynomials. The Lanczos algorithm method^49,50 was used to diagonalize the Hamiltonian matrix, and we selected 10 000 Lanczos iterations to obtain the eigenvalues and eigenvectors. For the angular part, the parity-adapted rotational basis for the three Euler angles (α,β,γ) can be written as

where were the normalized rotational functions for Xe–CS₂, the space-inversion parity p = 0 or 1 illustrates that the basis is even or odd under inversion. The total parity was given by (−1)^J+P.

III. Potential energy surface

The contour plot of the vibrationally averaged 2D ν₁ + ν₃ excited state PES is showed in Fig. 1, which clearly displays that the global minimum is a T-shaped configuration at R = 3.97 Å and θ = 90.0° with a well depth of 413.875 cm⁻¹. In addition, there are two equivalent linear local minima at R = 5.56 Å and θ = 0° or 180° with a depth of 248.045 cm⁻¹. In our PES, between the two minima, there are two saddle points with an energy barrier of 187.997 cm⁻¹ relative to the global minimum, located at R = 5.08 Å and θ = 45° or 135°. In order to compare the Xe–CS₂ complex with the other Rg–CS₂ complexes, the differences in the geometries and well depths between the Rg–CS₂ complexes^{31–33,51,52} are listed in Table 1. One can see that the Rg–CS₂ complexes have similar shapes, such as the T-shaped global minimum, two linear local minima, and two saddle points. However, there are some deviations among the Rg–CS₂ complexes. For example, the depth of the global and local minima become deeper from He–CS₂ to Xe–CS₂, which indicates the intermolecular interactions become stronger with the increasing mass of the rare gas atom. For another, the well depth of Xe–CS₂ is much larger than the other Rg–CS₂ complexes, which means the Xe atom is strongly hindered from free motion around the CS₂ molecule. In addition, the minimum energy distance gradually become larger from He–CS₂ to Xe–CS₂. Compared with the previous theoretical study of the potential,⁵¹ the contour plots look almost the same as those for our work, and the positions and energies of the stationary points are shifted only slightly.

Fig. 1 Contour plot (in cm⁻¹) of the averaged intermolecular potential energy surface for Xe–CS₂ with CS₂ at the ν₁ + ν₃ excited state.

Table 1 The stationary geometries (Å and deg) and well depths (cm⁻¹) of Rg–CS₂ (Rg = He, Ne, Ar, Kr, Xe)

Rg–CS2	Global minimum	Local minimum	Saddle point
He–CS2 (ref. 31)	(3.36, 90, −52.68)	(5.00, 0/180, −32.11)	(4.56, 59/121, −21.39)
Ne–CS2 (ref. 33)	(3.44, 90, −104.35)	(5.03, 0/180, −67.15)	(4.56, 44/136, −51.12)
Ar–CS2 (ref. 32)	(3.68, 90, −279.77)	(5.26, 0/180, −173.47)	(4.92, 39/141, −146.75)
Kr–CS2 (ref. 52)	(3.76, 90, −367.06)	(5.39, 0/180, −226.16)	(4.92, 45/135, −203.49)
Xe–CS2 (ref. 51)	(3.97, 90, −414.77)	(5.56, 0/180, −243.66)	(5.08, 46/134, −224.37)
Xe–CS2	(3.97, 90, −413.88)	(5.56, 0/180, −248.04)	(5.08, 45/135, −225.88)

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IV. Bound states of rovibrational energy levels and infrared spectra

Based on the potential, the energy levels of the bound states were calculated. In order to simplify our writing, we label the ν₁ + ν₃ of CS₂ as the ν₅ vibrational state for Xe–CS₂. In Table 2, we display the first twenty pure vibrational bound states of the Xe–CS₂ complex for the ground and ν₅ states of CS₂. As seen in Table 2, the first bound state for the ν₅ state is −374.239 cm⁻¹, which reveals the zero-point energy is 39.636 cm⁻¹ relative to global minimum, only about one-tenth of the global well depths. Due to the deep well and high barrier for Xe–CS₂, the bound states are much deeper than the other Rg–CS₂ complexes.^31–33,52 The ν₁ + ν₃ band origin shift is determined by Δv = E₀⁵ − E₀⁰,⁵³ where E₀^v are the ground state energies of Xe–CS₂ with the CS₂ monomer in the corresponding ν vibrational state. Meanwhile, we further predicted the infrared band origin shift of Xe–CS₂, which is −1.996 cm⁻¹ indicating that is negative (red-shift) in sign and large in magnitude. In addition, the band origin shifts of the other Rg–CS₂ complexes^{26,29–33,51,52} are listed in Table 3, together with the Rg–CO₂ complexes.^2,6 One can see that the tendency of the CS₂ shifts are similar to CO₂. Besides, the results indicate the lighter complexes (He, Ne) tend to positive (blue shifts), while the larger complexes (Ar, Kr, Xe) tend to negative (red-shifts). The predicted ν₁ + ν₃ band shift for Xe–CS₂ (−1.996 cm⁻¹) is larger than that for Xe–CO₂ (−1.447 cm⁻¹), revealing that Xe–CS₂ has a larger weakening of the vdW bond upon vibrational excitation in the ν₁ + ν₃ region.

Table 2 The calculated energy levels (in cm⁻¹) for the first twenty vibrational bound states of Xe–CS₂ with CS₂ at the ground and the ν₁ + ν₃ excited states

Ground state				ν5 state
N		N		N		N
0	−372.243	10	−276.947	0	−374.239	10	−278.046
1	−340.903	11	−275.624	1	−342.887	11	−276.971
2	−330.927	12	−273.955	2	−335.591	12	−275.602
3	−312.666	13	−266.022	3	−314.627	13	−266.029
4	−308.886	14	−258.545	4	−310.894	14	−259.381
5	−308.813	15	−257.516	5	−310.755	15	−259.247
6	−298.008	16	−256.517	6	−299.059	16	−256.907
7	−295.275	17	−252.014	7	−296.373	17	−253.608
8	−284.470	18	−251.523	8	−285.564	18	−253.276
9	−283.164	19	−245.973	9	−284.769	19	−246.196

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Table 3 The vibrational shifts of band origin for Rg–CS₂, compared with Rg–CO₂ (in cm⁻¹)

	CO2 ν3	CS2 ν3	CS2 ν1 + ν3
He	+0.095 (ref. 6)	+0.108 (ref. 26)	+0.228 (ref. 31)
Ne	+0.130 (ref. 2)	+0.180 (ref. 26)	+0.251 (ref. 33)
Ar	−0.470 (ref. 2)	+0.067 (ref. 26)	−0.049 (ref. 32)
Kr	−0.884 (ref. 2)	−0.787 (ref. 51)	−1.236 (ref. 52)
Xe	−1.447 (ref. 2)	−1.066 (ref. 51)	−1.996

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In Fig. 2, we present the contour plots of the wave functions of Xe–CS₂ with CS₂ at the ground state. It is clear from Fig. 2 that the vibrational ground state is localized around the T-shaped global minimum. The first two vibrational excited states are characterized predominantly by bending and stretching vibrations, respectively, while the corresponding wave functions are distributed and exhibit a strong mixing between the bending and stretching vibration modes for the higher vibrational excited states. The calculated average distance 〈R〉 and the average angle 〈θ〉 for the ground state are 3.99 Å and 87.85°, respectively, which is very close to the global minimum (R = 3.97 Å and θ = 90°). The radial dispersion and angular dispersion are 0.14 Å and 2.15°, respectively. Therefore, the ground state is rigid.

Fig. 2 Contour plots of the wave functions for the lowest six vibrational states of Xe–CS₂ with CS₂ at the ν₁ + ν₃ excited state.

The rovibrational energies of Xe–CS₂ are labeled by the asymmetric rotor quantum numbers J_KaKc, where J denotes the total angular momentum, K_a and K_c represent the projections of J onto the a and c principal axes of inertia. The rovibrational energy levels consist of four blocks, (even/even), (even/odd), (odd/even), and (odd/odd) for different combination parity of (j/p). The rovibrational energies of Xe–CS₂ for the vibrational ground and ν₁ + ν₃ excited states are shown in Table 4, and the total angular momentum J ranges from 0 to 5.

Table 4 The rovibrational energy levels (in cm⁻¹) of Xe–CS₂ with CS₂ at the ground and ν₁ + ν₃ excited states

Level	J	E (JKaKc)
		Even/Even	Even/Odd	Odd/Even	Odd/Odd
Ground state	0	−372.243 (000)
	1		−372.203 (101)	−372.113 (110)	−372.116 (111)
	2	−372.122 (202)	−371.770 (221)	−372.040 (212)	−372.028 (211)
		−371.769 (220)
	3	−371.649 (322)	−372.002 (303)	−371.902 (312)	−371.925 (313)
			−371.648 (321)	−371.207 (330)	−371.208 (331)
	4	−371.843 (404)	−371.488 (423)	−371.772 (414)	−371.734 (413)
		−371.487 (422)	−370.430 (441)	−371.047 (432)	−371.046 (431)
		−370.430 (440)
	5	−371.288 (524)	−371.644 (505)	−371.525 (514)	−371.580 (515)
		−370.229 (542)	−371.284 (523)	−370.846 (532)	−370.846 (533)
			−370.229 (541)	−369.436 (550)	−369.436 (551)
ν5 state	0	−374.239 (000)
	1		−374.199 (101)	−374.110 (110)	−374.114 (111)
	2	−374.119 (202)	−373.770 (221)	−374.037 (212)	−374.026 (211)
		−373.770 (220)
	3	−373.650 (322)	−373.999 (303)	−373.900 (312)	−373.923 (313)
			−373.649 (321)	−373.213 (330)	−373.214 (331)
	4	−373.839 (404)	−373.489 (423)	−373.770 (414)	−373.732 (413)
		−373.487 (422)	−372.442 (441)	−373.053 (432)	−373.052 (431)
		−372.442 (440)
	5	−373.288 (524)	−373.641 (505)	−373.522 (514)	−373.579 (515)
		−372.241 (542)	−373.284 (523)	−372.851 (532)	−372.852 (533)
			−372.241 (541)	−371.456 (550)	−371.456 (551)

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The rovibrational energy levels within J ≤ 3 are used to fit to a Watson asymmetric top Hamiltonian⁵⁴ employing the a-type reduction in the I^r representation,

The fitted molecular spectroscopic constants for the Xe–CS₂ complex are given in Table 5. Because the energy levels for Xe–CS₂ at the ground state are almost equal to the ν₁ + ν₃ excited state, the molecular parameters of the two states are very similar. The calculated inertial defects Δ₀ of this complex with CS₂ at the ground state and the ν₁ + ν₃ excited state are around 1.07 amu Ų and 1.13 amu Ų, respectively. For Rg–CS₂ (Rg = He,³¹ Ne,³³ Ar,³² Kr⁵²), the inertial defects Δ₀ in the ground state were found to be 6.14, 3.09, 2.93, 2.32 amu Ų, respectively. The decrease in the inertial defects reveals that the vdW complexes are more rigid with the increase in the mass of the rare gas atom. The asymmetry parameter κ, κ = (2B − A − C)/(A − C) (equal to −1 for the symmetric prolate top) is −0.917 and −0.916 for this complex with CS₂ at the ground and the ν₁ + ν₃ excited states, respectively.

Table 5 The calculated spectroscopic constants (in cm⁻¹) and the inertial defects Δ₀ (in amu Ų) for the Xe–CS₂ complex with CS₂ at the ground and the ν₁ + ν₃ excited states

	Ground state	ν5 state
A	0.108321	0.107674
B	0.022038	0.022023
C	0.018291	0.018261
ΔK	−2.443 × 10^−7	−3.110 × 10^−7
ΔJK	3.748 × 10^−7	3.258 × 10^−7
ΔJ	4.563 × 10^−6	8.778 × 10^−6
δK	2.050 × 10^−6	2.225 × 10^−6
δJ	9.176 × 10^−9	2.993 × 10^−9
Δ0	1.07	1.13

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V. Conclusions

In this paper, a new four-dimensional potential energy surface for the Xe–CS₂ complex including the Q₁ and Q₃ normal modes was constructed. We calculated the intermolecular potential with five PODVR grid points for the Q₁ and Q₃ normal modes, respectively, using the CCSD(T) method with the aug-cc-pVQZ basis set plus bond functions. Based on the ab initio potential points, two vibrationally averaged PESs of the Xe–CS₂ complex were generated. We found that each potential energy surface has a T-shaped global minimum and two equivalent local linear minima. The bound rovibrational energy levels of Xe–CS₂ were obtained by employing the radial DVR/angular FBR method and the Lanczos algorithm. The predicted band origin shift is −1.996 cm⁻¹. Meanwhile, we found that the Rg–CS₂ complexes have very similar features. For example, each PES is characterized by a T-shaped global minimum. For another, the regularities were represented among the complexes from He–CS₂ to Xe–CS_2. It is expected that the work on the Rg–CS₂ complexes in the ν₁ + ν₃ region of CS₂ should be useful for further theoretical and experimental studies.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 21705011).

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Footnote

† Miao Qin and Xiuchan Xiao contributed equally to this work.

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