Quantum tunneling dynamical behaviour on weakly bound complexes: the case of a CO2–N2 dimer

Miguel Lara-Moreno a, Thierry Stoecklin a, Philippe Halvick a and Majdi Hochlaf *b
aUniversité de Bordeaux, ISM, CNRS UMR 5255, 33405, Talence, France
bUniversité Paris-Est, Laboratoire Modélisation et Simulation Multi Echelle, MSME UMR 8208 CNRS, 5 bd Descartes, 77454 Marne-la-Vallée, France. E-mail: hochlaf@univ-mlv.fr

Received 14th July 2018 , Accepted 3rd September 2018

First published on 3rd September 2018


Quantum tunneling is a common fundamental quantum mechanical phenomenon. The dynamics induced by this effect is closely connected to the shape of the potentials. Here we treat the CO2–N2 van der Waals complex dynamics using a first principles treatment where nuclear motions and nuclear spins are fully considered. This dimer is found to exhibit complex spectral and dynamical features that cannot be accounted for using standard experimental and theoretical models. We shed light on some aspects of its quantum tunneling dynamics that remained unexplained since its first evidence 85 years ago. CO2–N2 represents also an important prototype for studying the systematic (as in NH3) lifting of degeneracy due to tunneling effects and large amplitude motions. Vibrational memory and quantum localization effects are evidenced. Plural potential wells separated by potential barriers are commonly found for polyatomic organic and inorganic molecules (e.g., cistrans isomerization and enol–keto tautomerism). The present findings are useful for understanding the complex quantum effects that may occur there.


1 Introduction

Weakly bound molecular clusters formed by abundant molecular species (e.g., H2, CO, N2, O2, and CO2) can be produced in laboratories. They can be studied in extraordinary detail by high-resolution spectroscopic techniques in the frequency or time domains.1–4 The analysis of their spectra allows a quantitative understanding of both fundamental and applied physical as well as chemical phenomena. So far, these spectra represented critical tests for the theories on the molecular structure and reactivity which are being elaborated.

The CO2–N2 complex is formed through favorable interactions between CO2 and N2, which are important constituents of the Earth's atmosphere. Upon complexation, the infrared (IR) inactive vibrational modes of CO2 and N2 become slightly allowed and therefore may participate in the energy redistribution and transfer in the atmosphere (e.g., greenhouse effect). This occurs after elastic and inelastic collisions, where rotational and/or vibrational and/or electronic (de-) excitation processes may take place. In addition, trapping of the complex in the potential wells should lead to unexpected quantum effects which influence the outcomes of these collisions. As discussed in ref. 5, the physical and chemical properties of weakly bound molecular systems may exhibit complex behaviors that could be observed experimentally. Among them, we can cite tunneling, vibrational quantum localization, quantum resonances, and non-Arrhenius law evolution of rate constants. Some of these effects remain not yet fully understood, where non-intuitive physical chemistry is in action. However they are important for understanding the dynamics of complex molecular systems. It is hence worth investigating them using state-of-the-art theoretical and experimental techniques.

To date, several experimental works have investigated the rotational and rovibrational spectroscopy of the CO2–N2 cluster.3,4,6,7 The ground state of the complex has been characterized using IR and Fourier transform microwave (FTMW) spectroscopies. Accurate structural and spectroscopic data for the ground state of the complex and its isotopologues are available in the spectral region around the ν3 band of the isolated CO2. These studies proposed a distorted T-shape structure, hereafter referred to as the R0 structure, with O[double bond, length as m-dash]C[double bond, length as m-dash]O as the cross of the T and the N≡N axis pointing toward the carbon atom. The N2 and CO2 monomers’ internuclear axis deviates by some degrees (5°–20°) from those of a pure C2v molecule. In contrast, theoretical investigations3,8–11 at several levels of theory showed that the equilibrium structure of the most stable form, hereafter referred to as Re, is of C2v symmetry. In 2015, some of us proposed that the deviation between R0 and Re structures may be due to dynamics such as that established for the weakly bound benzene dimer.12

In this work, we incorporated the state-of-the-art potential energy surface (PES) reported previously11 for the CO2–N2 cluster to treat the nuclear motions. We use a variational approach that fully accounts for all angular momenta couplings (cf. the ESI). Our treatment provides the energy levels and their 4D rovibrational wavefunctions for the most abundant isotopologue 12C16O214N2. We analysed both the pattern of the rovibrational levels and the corresponding wavefunctions. A further complication comes from the permutation of identical nuclei which is solved here by finding the correspondence between the C2v point group (which characterizes the CO2–N2 complex at equilibrium geometry) and the G8 permutation group (cf. the ESI). For the most abundant isotope of nitrogen, namely 14N, the non-zero nuclear spin of nitrogen atoms (IN = 1) results in two sets of states: ortho (I = 0, 2) and para (I = 1). Through the analysis of the microwave spectrum of 12C16O214N2, Klemperer and co-workers4 proved, however, that both ensembles do not mix. They will be considered separately here. Our work shows that the analysis of these wavefunctions allows full understanding of the complex dynamics of this dimer.

2 Potential energy surface

An accurate multi-dimensional interaction potential of the CO2–N2 electronic ground state is available.11 It was mapped along the four intermonomer Jacobi coordinates, namely the distance R connecting the CO2 and N2 centers of mass, the polar angles θ1 and θ2 of the rigid rotors CO2 and N2, respectively, and the torsional angle ϕ. This 4D-PES was generated at the CCSD(T)-F12/aug-cc-pVTZ level. This potential is strongly anharmonic. Its main characteristics are depicted in Fig. 1. Two types of minima (denoted as MIN1 and MIN2) and three types of saddle points (denoted as TS1, TS2 and TS3) can be located on this PES. TS1 and TS3 points are first-order saddle points while TS2 points are second-order saddle points (i.e. stationary points with two imaginary frequencies). Let us note that TS3 is determined in the present work and not mentioned previously.11 TS1 connects the two equivalent MIN1 structures, while TS3 connects MIN1 and MIN2 structures. The transformation from one MIN1 structure to the other one is obtained by an out-of-plane rotation of N2 within the complex. Similarly, upon inversion of the two oxygen atoms, there are two equivalent MIN2 structures and four equivalent TS3 saddle points. Geometric data of the stationary points are collected in Table 1.
image file: c8cp04465a-f1.tif
Fig. 1 Main characteristics of the 4D-PES of the CO2–N2 complex as deduced from ref. 11. The energy is given with respect to separated CO2 and N2 monomers. The red double arrow highlights possible systematic lifting of degeneracy due to tunneling.
Table 1 Structural parameters of the CO2–N2 complex as deduced from the 4D-PES and experimentally.4 See the text for the definition of coordinates. For some points, ϕ is not defined (ND). MIN1, MIN2, TS2 and TS3 have two or four equivalent geometries. Only one is reported in this table. Other geometries can be obtained by symmetry operations
R (a0) θ 1 (deg) θ 2 (deg) ϕ (deg) Energy (cm−1)
R e/MIN1 6.98 90 0 ND −321.24
R 0 7.04 92 19.2 0
MIN2 8.47 0 90 ND −158.90
TS1 6.42 90 90 90 −163.59
TS2 9.93 0 0 ND −30.56
TS3 8.41 13.16 79.51 180 −158.12


The depths of the potential wells are De (MIN1) = −321.24 cm−1 and De (MIN2) = −158.90 cm−1 (with respect to separated CO2 and N2 monomers). TS1 is located at −163.58 cm−1, TS2 at −30.56 cm−1 and TS3 at −158.12 cm−1. Since a potential energy barrier of 157.7 cm−1 separates the two equivalent potential wells MIN1, bound states with localized wavefunctions in each potential well are expected. Systematic lifting of degeneracy due to the tunneling effect (as in NH3) should be observed. A splitting of ∼2 × 10−4 cm−1 has been predicted by a crude 1D model.4 In contrast, the MIN2 potential well is so shallow that localized bound states cannot a priori be expected.

3 Rovibrational bound states

Fig. 2 depicts the vibrational normal modes of CO2–N2 for the MIN1 and MIN2 equilibrium structures determined using Wilson's FG method.13 They correspond to the intermonomer stretching, the out-of-plane inversion of N2 around its center of mass and the two in-plane bending motions. Variationally, a total of 1091 bound states were found for J = 0. For the levels up to ∼−83 cm−1, Table 2 lists their energy, symmetry and parity. An attempt to assign vibrational quantum numbers to these levels was made, based not only on the vibrational normal mode rigid rotor model but also through the examination of their 4D rovibrational wavefunctions (cf. the ESI).
image file: c8cp04465a-f2.tif
Fig. 2 Intermonomer vibrational modes of the MIN1 and MIN2 structures. The representations to which each mode belongs are given for the C2v (in black) and the G8 (in red) groups. See the ESI for more details.
Table 2 Lowest energy levels for J = 0 with their assigned vibrational quantum numbers v1, v2, v3, v4 whenever possible, parity ε, representation Γi of the group G8 as well as the number of the level in a given representation. Energies are given in cm−1 with respect to the energy of separated monomers. Levels identified as the two components of a quasi-degenerate pair are listed in the same line. |ΔE| is the energy splitting due to tunneling. When possible, it is also indicated whether the vibrational wavefunctions are mainly localized in MIN1 or MIN2 potential wells
v 1 v 2 v 3 v 4 ε Energy Γ i E| MIN v 1 v 2 v 3 v 4 ε Energy Γ i E| MIN
0 0 0 0 + −224.68 −224.68 A 1′·0 B 2′·0 <0.01 1 2 0 0 0 + −101.51 −101.26 B 2′′·10 A 1′·12 0.25 2
1 0 0 0 −192.44 −192.44 A 2′′·0 B 1′′·0 <0.01 1 −101.29 A 2′′·9
0 1 0 0 + −178.79 −178.79 B 2′′·0 A 1′′·0 <0.01 1 0 0 3 0 + −100.69 −100.66 B 2′·11 A 1′·13 0.03 1
0 0 1 0 + −178.36 −178.36 A 1′·1 B 2′·1 <0.01 1 0 2 0 1 + −98.93 −98.68 A 1′′·9 B 2′′·11 0.25 1
0 0 0 1 + −173.53 −173.53 B 2′′·1 A 1′′·1 <0.01 1 + −98.46 A 1′′·10
2 0 0 0 + −164.00 −163.86 A 1′·2 B 2′·2 0.14 1 0 0 1 2 + −97.57 −97.52 B 2′·12 A 1′·14 0.05 1
1 1 0 0 −150.52 −150.49 B 1′·0 A 2′·0 0.03 1 + −97.17 A 1′·15
1 0 1 0 −149.78 −149.57 A 2′′·1 B 2′′·1 0.21 1 3 0 0 0 −96.94 −96.44 B 1′·7 A 2′′·10 0.50 2
1 0 0 1 −143.49 −143.41 A 2′·1 B 1′·1 0.08 1 + −96.42 B 2′·13
3 0 0 0 −139.97 −138.44 A 2′′·2 B 1′′·2 1.53 1 + −95.20 A 1′·16
0 1 1 0 + −138.79 −138.78 B 2′′·2 A 1′′·2 0.01 1 0 1 0 2 + −95.68 −95.68 B 2′′·12 A 1′′·11 <0.01 1
0 2 0 0 + −138.21 −138.20 A 1′·3 B 2′·3 0.01 1 + −95.06 B 2′′·13
0 0 2 0 + −136.72 −136.70 A 1′·4 B 2′·4 0.02 1 0 0 0 3 + −94.86 −94.40 A 1′′·12 B 2′′·14 0.46 1
0 0 0 2 + −132.95 −130.13 A 1′·5 B 2′·5 2.82 1 + −94.83 B 2′·14
0 1 0 1 + −132.93 −130.19 B 2′·6 A 1′·6 2.74 1 −93.27 B 1′′·8
0 0 1 1 + −130.94 −130.90 A 1′′·3 B 2′′·3 0.04 1 0 0 2 1 + −93.13 −92.79 A 1′′·13 B 2′′·15 0.34 1
+ −129.38 A 1′·7 1 2 1 1 0 + −92.41 −88.63 A 1′′·14 B 2′′·16 3.78 1
+ −125.38 B 2′·7 1 + −92.01 B 2′·15
2 1 0 0 + −127.74 −127.14 B 2′′·4 A 1′′·4 0.60 1 1 0 0 1 −91.91 −91.87 B 1′′·9 A 2′·6 0.04 2
2 0 0 1 + −120.71 −119.26 A 1′′·5 B 2′′·5 1.45 1 + −91.62 A 2′·17
2 0 1 0 + −119.70 −115.11 A 1′·8 B 2′·8 4.59 1 0 2 0 0 + −91.20 −91.05 A 1′·18 B 2′′·17 0.15 2
−116.69 A 2′′·3 + −91.04 A 1′′·15
0 0 0 0 + −116.11 −116.11 B 2′′·6 A 1′·9 <0.01 2 + −90.9 A 1′·19
1 1 1 0 −114.82 −113.96 B 1′·2 A 2′·2 0.86 1 −90.47 A 2′′·11
1 2 0 0 −113.73 −113.26 B 1′′·3 A 2′′·4 0.47 1 + −90.31 B 2′·16
3 1 0 0 −111.76 −109.03 B 1′·3 A 2′·3 2.73 1 + −90.12 A 1′′·16
1 0 2 0 −110.38 −110.23 A 2′′·5 B 1′′·4 0.15 1 −89.96 B 1′·8
1 0 0 2 −107.84 −107.36 B 1′′·5 A 2′′·6 0.48 1 + −89.95 A 1′·20
1 0 0 0 −107.69 −107.49 A 2′′·7 B 1′·4 0.20 2 1 3 0 0 −89.00 −86.28 B 1′·9 A 2′·7 2.72 I
3 0 0 1 −107.28 −97.83 A 2′·4 B 1′·6 9.45 1 + −88.70 A 1′·21
0 3 0 0 + −104.98 −104.79 B 2′′·7 A 1′′·6 0.19 1 + −88.18 B 2′·17
0 2 1 0 + −104.97 −104.81 A 1′·10 B 2′·9 0.16 1 0 0 0 1 + −86.61 −86.55 B 2′·18 A 1′′·14 0.06 2
1 0 1 1 −104.87 −101.54 B 1′·5 A 2′·5 3.33 1 −86.53 B 1′′·10
0 1 0 0 + −104.62 −104.62 B 2′·10 A 1′′·7 <0.01 1 1 2 1 0 −86.24 −85.43 A 2′′·12 B 1′′·11 0.81 1
+ −104.47 B 2′′·8 + −84.99 A 1′′·18
+ −104.19 A 1′·11 + −84.75 B 2′·19
−103.39 A 2′′·8 −84.71 A 2′·8
−102.93 B 2′′·6 + −84.32 B 2′′·18
0 1 2 0 + −102.58 −101.97 B 2′′·9 A 1′′·8 0.61 1 0 0 1 0 + −83.92 −83.83 A 1′·22 B 2′′·19 0.09 2
−101.91 B 1′′·7 + −83.21 A 1′·23


A way to estimate the accuracy of not only the present variational calculations but also the PES itself is to compare the calculated frequencies with the experimental determinations. The FTMW spectra of the CO2–N2 complex have been recently measured by Frohman et al.4 The reported frequencies correspond to pure rotational transitions with Ka= 0 and ΔKa= 0 as well as hyperfine transitions. The lack of transitions with ΔKa ≠ 0 is well explained by symmetry imposed selection rules A1′ ↔ A2′ and B1′ ↔ B2′ (cf. the ESI) which, for the ground state, correspond to transitions of type JeoJee′. The selection rules for pure rotational transitions are detailed in the ESI. A comparison of the aforementioned results (neglecting the hyperfine structure) with those obtained from our variational approach is shown in Table 3. The experimental frequencies are reported for transitions between rotational and hyperfine levels. Since the hyperfine structure has been neglected in the present calculations, we have selected the experimental transitions with I = I′ = 0 (which happen only for ortho states) for comparison with the calculated frequencies. The agreement between the calculated and experimental frequencies is quite good, within approximately 1%, which represents a maximum deviation of 0.006 cm−1. This allows us to conclude that the 4D-PES model provides an accurate description of the interaction energy of the CO2–N2 van der Waals complex.

Table 3 Calculated and experimental frequencies (MHz) for the transitions image file: c8cp04465a-t1.tif. Relative errors from experimental data are given as percentages

image file: c8cp04465a-t2.tif

This work Exp.4 Rel. error%
101–000 3880.484
202− 101 7527.429 7608.377 1.06
303–202 11277.417 11388.436 0.97
404–303 15003.717 15148.195 0.95
505–404 18698.230 18877.125 0.95


The two MIN1 equivalent structures have C2v point symmetry. The transformation between these two structures is obtained by the permutation of the two indistinguishable N atoms. Multiplying the four symmetry elements of the C2v group by the permutation of N atoms, one obtains four new symmetry elements. In total, we have eight symmetry elements which compose the G8 group. The vibrational wavefunctions localized in any of the MIN1 potential wells belong to the representations of the C2v group. If we neglect tunneling, the wavefunctions localized in a MIN1 potential well are identical to those localized in the other equivalent MIN1 potential well, and have the same energy. Symmetrical and antisymmetrical combinations (with respect to the permutation of N atoms) of the degenerate local wavefunctions give global wavefunctions belonging to the representations of the G8 group. The correspondence between the representations of the C2v and G8 groups is indicated in Fig. 2. The representations of G8 to which the wavefunctions belong are listed in Table 2. The same analysis can be done with the two MIN2 equivalent structures, where the permutation of the two indistinguishable O atoms should be, however, considered.

Systematic tunneling effects through the potential barrier connecting two equivalent structures are expected, therefore the global wavefunctions are no longer doubly degenerate. The splittings of the energy levels are listed in Table 2. For the lowest levels, we do not observe the corresponding splitting of the calculated energies. Since the numerical accuracy of our calculations is limited to 0.01 cm−1, any splitting smaller than this limit cannot be calculated. Indeed, energy splitting could be as small as ∼2 × 10−4 cm−1, as shown by a simple 1D model.4 For higher energy levels, the splitting is larger and can reach a few cm−1. For instance, the excitation of the ν1 mode of motion produces most of the largest energy splittings. Indeed, the ν1 mode corresponds to an out-of-plane rotation of N2, which is thus strongly associated with the MIN1 ↔ MIN1 interconversion, as shown in Fig. 2. Table 2 shows that this splitting goes from less than 0.01 cm−1 for the MIN1 (0,0,0,0) level to more than 4 cm−1 for levels above the barrier, e.g. for the MIN1 (3,0,0,0) level. For MIN2, since the barrier is very low, this effect is more pronounced when ν1 is excited, for instance an appreciable splitting (of 0.2 cm−1) is computed for the MIN2 (1,0,0,0) state. In sum, we observe splittings for all the levels of this complex but to a lesser extent (of a few tenths of cm−1) for ν2, ν3, ν4 modes. This is due to the couplings between the ν1 coordinate and the other vibrational mode coordinates as observed for NH3.

Fig. 3 displays the 3D contour plots of the quasi-degenerate ground state wavefunctions A1′ and B2′. By definition, the B2′ wavefunction has a nodal plane at θ2 = 90°. However, Fig. 3 reveals two additional and unexpected nodal planes (schematized by the change from blue to red). This is uncommon for a ground state wavefunction which is usually nodeless. Both wavefunctions are spread out across a large range of the R coordinate. This distance deviates greatly from the equilibrium value of the global minimum MIN1, reflecting the occurrence of large amplitude motions and zero point vibrational energy effect. Furthermore, both polar angles θ1 and θ2 deviate significantly from their values at MIN1 equilibrium. This most likely induces distortions from the T-shape C2v configuration as observed experimentally. Note that our A0, B0 and C0 rotational constants are in close agreement with those deduced by μw spectroscopy4 (Table 4). Such dynamical behaviors were proposed to explain the distorted C2v planar structure of the H2–CO2 complex.14 Our work suggests the examination of the corresponding vibrational wavefunctions for confirmation.


image file: c8cp04465a-f3.tif
Fig. 3 Left panel: 3D plot of the ground state quasi-degenerate wavefunctions. The A1′ wavefunction is the symmetrical combination of the two wavefunctions localized in the equivalent MIN1 structures. The B2′ wavefunction is the corresponding antisymmetrical combination. Right panel: 3D plot of the quasi-degenerate vibrational wavefunction of the MIN1 (0,0,1,0) level. The A1′ wavefunction is the symmetrical combination of the two wavefunctions localized in the equivalent MIN1 structures. The B2′ wavefunction is the corresponding antisymmetrical combination.
Table 4 Spectroscopic constants
Vibrational frequencies (cm−1) MIN1 MIN2
Variational Harmonic Ref. 11 Variational Harmonic Ref. 11
ν 1 32.2 36.9 46 8.4 6.0 10
ν 2 45.9 50.7 32 29.5 30.8 42
ν 3 46.3 58.2 61 32.3 40.3 44
ν 4 51.2 60.5 88 11.5 10.8 17

Rotational constants (MHz) MIN1 MIN2
Variational Inertia tensor Exp.4 Variational Inertia tensor Exp.
A 0 11861.37 11702.98 11885.3 41500.51 60313.50
B 0 2087.97 2089.74 2062.88 1440.98 1306.05
C 0 1780.38 1773.12 1743.86 1240.81 1278.37

Rotation–vibration coupling constants (MHz) α β γ α β γ
ν 1 807.29 20.65 6.01


The variationally determined fundamental frequencies and their harmonic counterparts are given in Table 4. For a given isomer, the differences between both sets of data evidence the strongly anisotropic character of the PES and the obvious importance of anharmonicities in the vibrational motions for this van der Waals complex. Table 2 reveals a high density of vibrational levels. This favors anharmonic resonances to take place between levels belonging to the same representation of the G8 group. Both the mixing of their unperturbed wavefunctions and the displacement of their energies are expected. For illustration, we display in Fig. 3 the wavefunctions of the (0,0,1,0) levels. Fig. 3 shows that the nodal planes are not parallel to the coordinate axes. Note that these levels belong to the same G8 irreducible representation (A1′ and B2′) as the ground state. Both levels are close in energy, thus we expect a mixing among the A1′ levels as well as among the B2′ levels. This may explain the appearance of the unexpected nodal planes in the ground state wavefunctions and the atypical features given above. More generally, these anharmonic resonances make the assignment of quantum numbers to the levels of this complex difficult, which should be viewed as tentative for some of the levels listed in Table 2.

Above the potential barrier of MIN1 ↔ MIN1 interconversion, several phenomena take place: vibrational quantum localization, anharmonic resonances and mixing of the wavefunctions pertaining to the non-equivalent MIN1 and MIN2 potential wells. For illustration, Fig. 4 displays the wavefunctions of the A1′·0, A1′·10, B2′′·6, and B2′′·7 levels. The wavefunction B2′′·6 is located significantly above TS3, but remains remarkably localized in the region of the MIN2 potential well. The A1′·10 and B2′′·7 wavefunctions also lie well above TS3. They are mainly localized in the MIN1 potential well with a small component in the MIN2 potential well. The right panel of Fig. 4 provides some insight into the surprising localization of the wavefunctions. This figure shows the potential energy curve obtained by varying θ1 while θ2 = 0° (the curve between MIN1 and TS2) and the potential energy curve obtained by varying θ2 while θ1 = 0° (the curve between TS2 and MIN2). The extent of the wavefunctions is clearly limited by the potential energy curves resulting from these cuts of the 4D-PES, in spite of the fact that in some other directions the wavefunctions are free to extend as shown in the left panel of Fig. 4. In fact, this vibrational memory effect can be observed for several wavefunctions regardless of whether or not the corresponding rovibrational state's energy is below or above the interconversion barriers MIN1 ↔ MIN1 or MIN1 ↔ MIN2. As a consequence, the assignment of these levels is relatively straightforward. As demonstrated for the [H,S,N] system,15 these states conserve the memory of the equilibrium structure in the respective potential well above which they are located.


image file: c8cp04465a-f4.tif
Fig. 4 Left panel: Minimum energy path (blue line) between MIN1 and MIN2 minima and selected vibrational wavefunctions computed at the points of the minimum energy path. Right panel: Minimum energy path (blue line) between MIN1 and TS2 and between TS2 and MIN2 as well as selected vibrational wavefunctions computed at the points of the minimum energy path.

Localized vibrational states with an energy above the barrier separating two isomers have been observed previously in other molecular systems. This is the case for HCN and its isomer HNC, where Bačić and Light observed that a significant fraction of vibrational wavefunctions with energies above the isomerization barrier remains completely localized on one or the other side of the barrier.16 These wavefunctions are highly excited in one or both stretching motions with little excitation of the bending motion. In contrast, for the delocalized wavefunctions, the bending motion is highly excited. Indeed, the isomerization of HCN into HNC implies a rotation of H around CN, which is essentially triggered by the bending motion while the stretching motions have no effect on this rotation. A more complicated dynamics occurs in the [H,S,N] system15 which has two weakly bound isomeric complexes, namely SH⋯N. and SN⋯H. The rotation of H around SN produces three equilibrium configurations. Localized rovibrational wavefunctions with energy above the isomerization barriers have been observed. Again, excitation of the bending motion triggers isomerization while excitation of the stretching motions prevents isomerization. Let us note that the HCN ↔ HNC and [H,S,N] systems have two or more isomers separated by significant barriers. This means that every isomer corresponds to a potential well deep enough to support one or several bound states with energies below the corresponding barriers. A second class of systems is defined by the case where one isomer has one (or more) very low vibrational frequency and a very low energy barrier, significantly lower than the harmonic zero-point energy. In such a case, the potential well is not deep enough to support even one vibrational level. The acetylene ↔ vinylidene isomerization and the MIN1 ↔ MIN2 isomerization of CO2–N2 belong to this second class. Vibrational states localized in the vicinity of the vinylidene minima have been obtained by accurate quantum calculations.17 The acetylene ↔ vinylidene isomerization is triggered by the rotation of H atoms around the C2 core, while the stretching and torsional motions have no effect on isomerization. The separation of the vibrational modes into two groups was recently illustrated by Baraban et al.18 The first group contains the active modes which trigger the isomerization process, while the modes of the second group are inactive (or spectator) modes. Inactive modes withhold vibrational energy in motions which cannot allow isomerization. The structures represented in Fig. 1 suggest that the stretching motion along R and the torsional motion along ϕ are inactive modes for the MIN1 ↔ MIN2 isomerization. Rotation of N2 or CO2 appears to be the active mode. An investigation of the CO2–N2 PES in the vicinity of the MIN2 point shows, however, that the potential energy is repulsive when N2 or CO2 is rotated. This is in agreement with the harmonic vibrational modes depicted in Fig. 2. A pure rotation of N2 or CO2 alone cannot lead to isomerization. Only a concerted rotation of N2 and CO2, or in other words, only a particular combination of the two rotational motions, can lead to isomerization. Therefore, we may assume that the MIN2 localized states exist because the isomerization process is hindered.

The A1′·0 ground state wavefunction presented in Fig. 4 has two nodes. These nodes are also observed in Fig. 3. The PES along the minimum energy path is also plotted in Fig. 4 and it can be seen that the nodal structure of the wavefunction reflects the ripples of the PES. Let us note that the corrugated nature of the PES in the region of the MIN1 potential well has been already observed.11 More physical insight into this effect is provided by Fig. 5. The two nodal planes are clearly related to the wavy PES.


image file: c8cp04465a-f5.tif
Fig. 5 Left panel: 2D contour plot for the ground state wavefunction of MIN1. The black dashed line is the potential isoline equal to the energy of the ground state. For each point (θ1,θ2), both the wavefunction and the potential energy have been calculated with the values of ϕ and R which minimize the potential energy. Right panel: 2D contour plot for the ground state wavefunction of MIN2. The black dashed line is the potential isoline equal to the energy of the MIN2 ground state. For each point (θ1,θ2), both the wavefunction and the potential energy have been calculated with the values of ϕ and R which minimize the potential energy.

The localization of the B2′′·6 wavefunction in the region of the MIN2 potential well is really surprising, since the energy of this vibrational state is about 42 cm−1 above the TS3 saddle point. This wavefunction corresponds to the ground state of the MIN2 potential well and is quasi-degenerate with the A1′·9 wavefunction. Fig. 5 gives a more detailed view of the latter wavefunction, along with a cut of the PES at the energy of this state. The A1′·9 wavefunction is clearly the symmetric combination of two local wavefunctions pertaining to the MIN2 potential wells and does not spread out across the whole coordinate space energetically allowed. The small energy splitting between B2′′·6 and A1′·9 is also evidence of the localization of these wavefunctions. These levels are unusual and they are pointed out for the first time for a molecular system.

Although the hyperfine structure has been neglected in the present study, the nuclear spins and the consequences of the spin statistics on the rovibrational spectra are of great importance. According to the Bose–Einstein statistics, the total wavefunction must be symmetric under any permutation of identical nuclei. Since the electronic wavefunction is fully symmetric, the total wavefunction can be restricted here to the product of the rovibrational and nuclear spin wavefunctions. Since the nuclear spin of CO2–N2 can be I = 0 or 2 (ortho states) or I = 1 (para state), only rovibrational states of symmetry A1′ and A2′ are allowed for para states and only rovibrational states of symmetry B1′ and B2′ are allowed for ortho states (see the ESI). Rovibrational states of symmetry A1′′, A2′′, B1′′, and B2′′ are, however, forbidden by Bose–Einstein statistics. Furthermore, the ortho intensities are expected to be twice the para intensities.

4 Conclusion

Accurate calculations of the intermonomer vibrational bound states of the CO2–N2 van der Waals system have been presented. The effects of the permutations of indistinguishable atoms have been considered and investigated. The most notable effect is the existence of symmetrically equivalent potential wells leading to systematic lifting of degeneracy induced by tunneling effects. The examination of wavefunctions has revealed unexpected features: the ground state wavefunction has several nodal planes and the wavefunctions of highly excited states remain localized in small regions of the coordinate space. A good agreement with the experimental rotational transition energies has been obtained, thus demonstrating the accuracy of the 4D-PES model and validating the nuclear motion treatment.

In spite of its importance for atmospheric chemistry, our work reveals that the structure and the spectroscopy of the CO2–N2 complex are governed by full quantum effects, including tunneling, large amplitude motions, anharmonic resonances and vibrational quantum localization. Their spectroscopic signatures were already observed experimentally and are explained here for the first time. The plural potential induced complex dynamics could be found in several organic and inorganic molecules such as those presenting several conformers (e.g., cistrans), isomers and tautomers (enol–keto) interacting mutually on the same PES. The present work suggests that their spectroscopy and dynamics cannot be fully understood without considering quantum tunneling.

Conflicts of interest

There are no conflicts to declare.

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Footnote

Electronic supplementary information (ESI) available. See DOI: 10.1039/c8cp04465a

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