Tunneling Splitting Research Articles

How symmetry helps to improve the estimation of the hyperfine splitting of torsional levels due to tunneling. The case of the HSSSH molecule

In this study, we analyze a series of molecules belonging to the C2V(M) molecular symmetry group which are characterized by several conformers. The use of molecular symmetry at each stage of calculating the energy of stationary torsional states is demonstrated. In particular, the importance is shown of preliminary symmetrization of physical characteristics of the molecules obtained by quantum chemical calculations. For the first time, symmetry-adapted basis functions for the diagonal kinetic coefficients are presented, which for the analyzed molecules do not satisfy all symmetry operations of the C2V(M) group. Using the HSSSH molecule as an example, it is shown how the full or partial accounting for molecular symmetry influences the calculated values of ultra-small tunneling splittings of the ground torsional states of the trans- and cis-conformers. It has also been established that the Hamiltonian matrix is characterized by symmetry which, when taken into account, makes it possible to halve the time of calculation of its elements.

Exchange-biased quantum tunneling of magnetization in a nanomagnetic dimer via a Josephson current

We investigate a Josephson junction where the weak link consists of a nanomagnetic dimer, featuring two large, antiferromagnetically coupled spins under simple easy-axis anisotropy. In this setup, the exchange coupling constant serves as a transverse anisotropy constant, and the quantum tunneling of the magnetization is influenced by the supercurrent, which behaves like an external magnetic field applied along the hard axis. Our results demonstrate that the superconducting current induces non-Kramers freezing and unfreezing of the magnetic moment’s tunneling within a biaxial spin model driven by exchange interaction. We derive an exact expression for the quenching condition caused by the exchange coupling. Additionally, we extend our analysis to tunneling phenomena in two antiferromagnetically or ferromagnetically coupled spins biased by the anisotropic exchange interaction, revealing more varied cases of tunnel splitting.

Intermolecular Bending States and Tunneling Splittings of Water Trimer from Rigorous 9D Quantum Calculations: I. Methodology, Energy Levels, and Low-Frequency Spectrum.

We present the computational methodology that enables the first rigorous nine-dimensional (9D) quantum calculations of the intermolecular bending states of the water trimer, as well as its low-frequency spectrum for direct comparison with experiment. The water monomers, treated as rigid, have their centers of mass (cm's) at the corners of an equilateral triangle, and the intermonomer cm-to-cm distance is set to a value slightly larger than that in the equilibrium geometry of the trimer. The remaining nine strongly coupled large-amplitude bending (angular) degrees of freedom (DOFs) enter the 9D bend Hamiltonian of the three coupled 3D rigid-water hindered rotors. Its 9D eigenstates encompass excited librational vibrations of the trimer, as well as their torsional and bifurcation tunneling splittings, which have been the subject of much interest. The calculations of these eigenstates are extremely demanding, and a sophisticated computational scheme is developed that exploits the molecular symmetry group of the water trimer, G48, in order to make them feasible in a reasonable amount of time. The spectrum of the low-frequency vibrations of the water trimer simulated using the eigenstates of the 9D bend Hamiltonian agrees remarkably well with the experimentally observed far-infrared (FIR) spectrum of the trimer in helium nanodroplets over the entire frequency range of the measurements from 70 to 620 cm-1. This shows that most peaks in the experimental FIR spectrum are associated with the intermolecular bending vibrations of the trimer. Moreover, the ground-state torsional tunneling splittings from the present 9D calculations are in excellent agreement with the spectroscopic data. These results demonstrate the high quality of the ab initio 2 + 3-body PES employed for the DOFs included in the bound-state calculations.

Observation of Pairwise Level Degeneracies and the Quantum Regime of the Arrhenius Law in a Double-Well Parametric Oscillator

By applying a microwave drive to a specially designed Josephson circuit, we have realized a double-well model system: a Kerr oscillator submitted to a squeezing force. We have observed, for the first time, the spectroscopic fingerprint of a quantum double-well Hamiltonian when its barrier height is increased: a pairwise level kissing (coalescence) corresponding to the exponential reduction of tunnel splitting in the excited states as they sink under the barrier. The discrete levels in the wells also manifest themselves in the activation time across the barrier which, instead of increasing smoothly as a function of the barrier height, presents steps each time a pair of excited states is captured by the wells. This experiment illustrates the quantum regime of Arrhenius’s law, whose observation is made possible here by the unprecedented combination of low dissipation, time-resolved state control, 98.5% quantum nondemolition single shot measurement fidelity, and complete microwave control over all Hamiltonian parameters in the quantum regime. Direct applications to quantum computation and simulation are discussed. Published by the American Physical Society 2024

Floquet engineering of electronic states and optical absorption in laterally-coupled quantum rings under a magnetic field.

Tuning electronic and optical properties of low-dimensional quantum systems in a flexible way is of particular importance in designing semiconductor-based devices. Semiconductor quantum rings (QRs) are nanoscopic structures that have become promising systems for physical and technological applications due to their unique electronic and optical properties. Here, we explore the fundamental electronic and optical properties of laterally-coupled QRs by taking into account the combined effects of applied magnetic and non-resonant terahertz intense laser fields. The laser-dressed electronic states are solved using the Floquet theory of periodically driven quantum systems in high-frequency limits within the framework of effective mass approximation. We demonstrate that increasing the laser field parameter leads to reduced tunnel splitting in the energy spectrum and more pronounced Aharonov-Bohm oscillations for the ground state energy, due to the attenuation of the tunnel coupling. Furthermore, depending on the position of the avoided crossings in Aharonov-Bohm oscillations of laterally-coupled quantum rings, the evolution of the avoided crossings with increasing the laser field parameter can be elucidated as the attenuated tunnel coupling or the competition between the attenuated tunnel coupling and the strengthening anisotropy of the laser-dressed QR potential well. The intensity of the intraband optical transition of laterally-coupled QRs can be effectively tuned and reaches a larger value by manipulating the laser parameter and magnetic field. Optical Aharonov-Bohm oscillations in laterally-coupled QRs are still exhibited by changing laser field parameters. Our findings offer a novel approach to manipulate the electronic and optical performances as well as the Aharonov-Bohm oscillations based on the laterally-coupled QRs by using an intense terahertz laser field.

Microwave spectroscopy and large amplitude motion of chlorosulfonic acid (ClSO2OH)

The high-resolution rotational spectrum of chlorosulfonic acid (ClSO2OH) has been studied using both broadband and cavity-based Fourier transform microwave spectrometers over the frequency range of 5–18 GHz. a-, b-, and c-type transitions have been recorded for both the 35Cl and 37Cl isotopologues. The observation of c-type lines establishes that the molecule lacks a plane of symmetry and suggests that the OH group can undergo large amplitude motion between equivalent structures. Interconversion between these structures can be achieved via internal rotation through two inequivalent barriers occurring at Cl-S-O-H torsional angles of 0 or 180degrees. As in previous work on triflic and methanesulfonic acids, two states are observed and are treated as tunneling states which are presumed to arise primarily due to motion through the lower of the two barriers. The a- and c-type transitions occur within each of these states while the b-type transitions cross between them. Rotational, centrifugal distortion, and chlorine nuclear quadrupole coupling constants, as well as the energy difference between the two tunneling states and associated coupling constants, have been determined. The experimental tunneling energies, ΔE, for the 35Cl and 37Cl isotopologues are 52.6926(16) MHz and 52.6397(46) MHz, respectively. Quantum chemical calculations were carried out using MP2 and B3LYP density functional theory (DFT) methods with an aug-cc-pVTZ basis set. The rotational constants from the optimized structures were in good agreement with the experimental values. The lowest energy barrier for OH motion was calculated to be 2.63 kcal/mol at the B3LYP/aug-cc-pVTZ level. The effects of the large amplitude motion are similar to those recently reported for triflic acid (CF3SO2OH) and methanesulfonic acid (CH3SO2OH). However, while the tunneling splittings in chlorosulfonic and triflic acids are virtually identical, they differ significantly from that of methanesulfonic acid.

Nuclear-Electronic Orbital Time-Dependent Configuration Interaction Method.

Combining real-time electronic structure with the nuclear-electronic orbital (NEO) method has enabled the simulation of complex nonadiabatic chemical processes. However, accurate descriptions of hydrogen tunneling and double excitations require multiconfigurational treatments. Herein, we develop and implement the real-time NEO time-dependent configuration interaction (NEO-TDCI) approach. Comparison to NEO-full CI calculations of absorption spectra for a molecular system shows that the NEO-TDCI approach can accurately capture the tunneling splitting associated with the electronic ground state as well as vibronic progressions corresponding to double electron-proton excitations associated with excited electronic states. Both of these features are absent from spectra obtained with single reference real-time NEO methods. Our simulations of hydrogen tunneling dynamics illustrate the oscillation of the proton density from one side to the other via a delocalized, bilobal proton wave function. These results indicate that the NEO-TDCI approach is highly suitable for studying hydrogen tunneling and other inherently multiconfigurational systems.

Fourier transform microwave spectroscopy of the 13C- and 18O-substituted tropolone. Proton tunneling effect for the isotopic species with the asymmetric potential wells.

Fourier-transform microwave spectroscopy has been applied for the 13C/18O-substituted tropolone to observe tunneling-rotation transitions as well as pure rotational transitions. The tunneling-rotation transitions were observed between the 13C-4 and -6 forms as well as between 13C-3 and -7, between 13C-1 and -2, and between 18O-8 and -9 (we denote these tunneling pairs as 13C-46, etc., below) although they have an asymmetric tunneling potential due to the difference in the zero point energy (ZPE). From the observed tunneling splittings ΔEij (0.9800-1.6824 cm-1), the differences in ZPE Δij for the 13C-46, -37, -12, and 18O-89 species are derived to be -0.1104, 0.5652, -1.3682, and 1.3897 cm-1 to agree well with the DFT calculation. The state mixing ratio of the tunneling states decreases drastically from (44%:56%) to (8.7%:91.3%) for 13C-46 and 18O-89 with an increase in the asymmetry Δij of the tunneling potential function. The observed tunneling-rotation interaction constants Fij decrease from 16.001 to 9.224 cm-1 as the differences in ZPE Δij increase, while the diagonal tunneling-rotation interaction constants Fu increase from 1.767 to 13.70 cm-1, explained well by the mixing ratio of the tunneling states.

The Far-Infrared Spectrum of Methoxymethanol (CH3-O-CH2OH): A Theoretical Study.

Methoxymethanol (CH3OCH2OH), an oxygenated volatile organic compound of low stability detected in the interstellar medium, represents an example of nonrigid organic molecules showing various interacting and inseparable large-amplitude motions. The species discloses a relevant coupling among torsional modes, strong enough to prevent complete assignments using effective Hamiltonians of reduced dimensionality. Theoretical models for rotational spectroscopy can improve if they treat three vibrational coordinates together. In this paper, the nonrigid properties and the far-infrared region are analyzed using highly correlated ab initio methods and a three-dimensional vibrational model. The molecule displays two gauche-gauche (CGcg and CGcg') and one trans-gauche (Tcg) conformers, whose relative energies are very small (CGcg/CGcg'/Tcg = 0.0:641.5:792.7 cm-1). The minima are separated by relatively low barriers (1200-1500 cm-1), and the corresponding methyl torsional barriers V 3 are estimated to be 595.7, 829.0, and 683.7 cm-1, respectively. The ground vibrational state rotational constants of the most stable geometry have been computed to be A 0 = 17233.99 MHz, B 0 = 5572.58 MHz, and C 0 = 4815.55 MHz, at ΔA 0 = -3.96 MHz, ΔB 0 = 4.76 MHz, and ΔC 0 = 2.51 MHz from previous experimental data. Low-energy levels and their tunneling splittings are determined variationally up to 700 cm-1. The A/E splitting of the ground vibrational state was computed to be 0.003 cm-1, as was expected given the methyl torsional barrier (∼600 cm-1). The fundamental levels (100), (010), and (001) are predicted at 132.133 and 132.086 cm-1 (methyl torsion), 186.507 and 186.467 cm-1 (O-CH3 torsion), and 371.925 and 371.950 cm-1 (OH torsion), respectively.

Tunneling splittings using modified WKB method in Cartesian coordinates: The test case of vinyl radical.

Modified WKB theory for calculating tunneling splittings in symmetric multi-well systems in full dimensionality is re-derived using Cartesian coordinates. It is explicitly shown that the theory rests on the wavefunction that is exact for harmonic potentials. The theory was applied to calculate tunneling splittings in vinyl radical and some of its deuterated isotopologues in their vibrational ground states and the low-lying vibrationally excited states and compared to exact variational results. The exact results are reproduced within a factor of 2 in most states. Remarkably, all large enhancements of tunneling splittings relative to the ground state, up to three orders in magnitude in some excited mode combinations, are well reproduced. It is also shown that in the asymmetrically deuterated vinyl radical, the theory correctly predicts the states that are localized in a single well and the delocalized tunneling states. Modified WKB theory on the minimum action path is computationally inexpensive and can also be applied without modification to much larger systems in full dimensionality; the results of this test case serve to give insight into the expected accuracy of the method.

The tunneling splittings of the ground state and some excited vibrational states for the inversion motion in H3C− anion and H3Si[rad] radical

Splitting of the ground state and some excited symmetric bending vibrational states due to inversion tunneling in the H3C− anion and H3Si radical is analyzed by numerically solving the vibrational Schrödinger equation of restricted (2D) dimensionality. We used the following two vibrational coordinates for the H3X structure (X = C, Si): the distance of the X atom from the plane of a regular triangle formed by three hydrogen atoms (1) and a symmetry coordinate composed of three distances between chemically non-bonded hydrogen atoms (2). The kinetic energy operator in this case takes the simplest form. The 2D potential energy surface (PES) in the given coordinates was calculated for H3C− at the CCSD(T)/aug-cc-pVTZ, CCSD(T)/aug-cc-pVQZ, CCSD(T)/aug-cc-pV5Z, CCSD(T)/CBS(TQ5), CCSD(T)/d-aug-cc-pVTZ, CCSD(T)/t-aug-cc-pVTZ, and CCSD(T)/q-aug-cc-pVTZ levels of theory, based on recommendations from recently published work [M.C. Bowman, B. Zhang, W.J. Morgan, H.F. Schaefer III, Mol.Phys., 117 (2019) 1069–1077]. The same 2D PES for the H3Si radical was calculated at the CCSD(T)/aug-cc-pVDZ CCSD(T)/aug-cc-pVTZ, CCSD(T)/aug-cc-pVQZ, and CCSD(T)/CBS(D,T,Q) as well as at the CCSD(T)/d-aug-cc-pVTZ, CCSD(T)/un-aug-cc-pVTZ, CCSD(T)/un-aug-cc-pVQZ levels of theory. The tunneling splittings for the D3C− anion and D3Si radical were calculated as well. The results of calculations demonstrate good agreement with available experimental data on umbrella vibration frequencies and inversion splittings for the title molecules.

Identification of Two Stable Side-Chain Orientations of Valine Methyl Ester by Microwave Spectroscopy.

The rotational spectra of two valine methyl ester (ValOMe) conformers have been measured using a cavity-based Fourier-transform microwave spectrometer in the range of 9-18 GHz. Ten conformers of ValOMe were modeled using the ωB97XD/6-311++G(d,p) level of theory, and separate spectra arising from two lowest-energy conformations were observed and assigned. 44 rotational transitions were assigned to conformer I, the lowest-energy configuration, and were fit to Watson's A-reduced Hamiltonian: A = 2552.0145(5) MHz, B = 1041.8216(3) MHz, and C = 938.54890(22) MHz. 14N nuclear quadrupole hyperfine splittings were resolved, and 231 hyperfine components were fit to χaa = -4.187(7) MHz, and χbb-χcc = 1.269(5) MHz. The spectrum of conformer I also reveals tunneling splittings arising from the methyl rotor. XIAM was used to fit the barrier to the internal rotation of the methyl rotor, and the best-fit V3 barrier was found to be 401.64(19) cm-1. 47 rotational transitions were assigned for conformer II (ΔE = 2.08 kJ mol-1), and the fitted rotational constants are A = 2544.2837(3) MHz, B = 1092.3654(15) MHz, and C = 896.3131(12) MHz. 264 hyperfine components were fit to χaa = -4.187(7) MHz and χbb-χcc = 1.518(6) MHz, and the best-fit V3 barrier was found to be 409.74(16) cm-1.

Spectra of the D2O dimer in the O-D fundamental stretch region: The acceptor symmetric stretch fundamental and new combination bands.

The O-D stretch fundamental region of the deuterated water dimer, (D2O)2, is further studied using a pulsed supersonic slit jet and a tunable optical parametric oscillator infrared source. The previously unobserved acceptor symmetric O-D stretch fundamental vibration is detected, with Ka = 0 ← 0 and 1 ← 0 sub-bands at about 2669 and 2674cm-1, respectively. The analysis indicates that the various water dimer tunneling splittings generally decrease in the excited vibrational state, similar to the three other previously observed O-D stretch fundamentals. Two new (D2O)2 combination bands are observed, giving information on intermolecular vibrations in the excited O-D stretch states. The likely vibrational assignments for these and a previously observed combination band are discussed.

Formic Acid-Ammonia Heterodimer: A New Δ-Machine Learning CCSD(T)-Level Potential Energy Surface Allows Investigation of the Double Proton Transfer.

The formic acid-ammonia dimer is an important example of a hydrogen-bonded complex in which a double proton transfer can occur. Its microwave spectrum has recently been reported and rotational constants and quadrupole coupling constants were determined. Calculated estimates of the double-well barrier and the internal barriers to rotation were also reported. Here, we report a full-dimensional potential energy surface (PES) for this complex, using two closely related Δ-machine learning methods to bring it to the CCSD(T) level of accuracy. The PES dissociates smoothly and accurately. Using a 2d quantum model the ground vibrational-state tunneling splitting is estimated to be less than 10-4 cm-1. The dipole moment along the intrinsic reaction coordinate is calculated along with a Mullikan charge analysis and supports the mildly ionic character of the minimum and strongly ionic character at the double-well barrier.

Doping dependent intrinsic magnetization in silicon in Ni/Si heterostructures

This investigation delves into the complex interaction at metal-semiconductor interfaces, highlighting the magnetic proximity effect in Ni/Si interfaces through systematic X-ray magnetic circular dichroism (XMCD) studies at Ni and Si edges. We analyzed two Ni/Si heterostructures with differing semiconductor doping, uncovering a magnetic proximity effect manifesting as equilibrium magnetization in the semiconductor substrate induced by the adjacent Ni layer. Our results display distinct magnetization signs corresponding to the doping levels: low-doped samples show parallel alignment to the Ni layer, while high-doped samples align antiparallel, indicating a nuanced interplay of underlying magnetization mechanisms. These findings pinpoint the roles of electron tunneling and exchange splitting modification in the magnetization behavior. The study enriches the understanding of ferromagnetic-semiconductor interface behavior, setting a precedent for the design of advanced spintronic devices that leverage the nuanced magnetic properties of these hybrid systems.

Tunneling splittings in the vibrationally excited states of water trimer.

Tunneling splitting (TS) patterns in vibrationally excited states of the water trimer are calculated, taking into account six tunneling pathways that describe the flips of free OH bonds and five bifurcation mechanisms that break and reform hydrogen bonds in the trimer ring. A tunneling matrix (TM) model is used to derive the energy shifts due to tunneling in terms of the six distinct TM elements in symbolic form. TM elements are calculated using the recently-developed modified WKB (Wentzel-Kramers-Brillouin) method in full dimensionality. Convergence was achieved for the lowest six excited vibrational modes. Bifurcation widths of the pseudorotational quartets are shown to be of comparable size to the ground-state widths, obtained using instanton theory, or increased for some particular modes of vibration. The largest increase is obtained for the excited out-of-phase flip of two adjacent water monomers with free OH bonds pointing in opposite directions relative to the ring plane. Bifurcation widths in (D2O)3 are found to be two orders of magnitude smaller than in (H2O)3. Geometrical arguments were used to explain the order of states in some TS multiplets in vibrationally excited water trimers.

Conformational adaptation and large amplitude motions of 1-phenyl-2,2,2-trifluoroethanol with two water molecules: a rotational spectroscopic and ab initio investigation.

The 1 : 2 adduct of 1-phenyl-2,2,2-trifluoroethanol (PhTFE), a chiral fluoroalcohol, with two water molecules (PhTFE⋯2H2O) was investigated via chirped pulse Fourier-transform microwave (CP-FTMW) spectroscopy and theoretical calculations. A systematic search of the PhTFE⋯2H2O conformational landscape identified 38 stable minima at the B3LYP-D3BJ/def2-TZVPPD level of theory, 27 of which are within an energy window of 10 kJ mol-1 after applying zero-point energy corrections. Rotational spectra of a single PhTFE⋯2H2O conformer along with eight deuterated and three oxygen-18 isotopologues were assigned. Interestingly, the observed PhTFE⋯2H2O conformer contains PhTFE II, the second most stable monomer conformer, and the most stable PhTFE I dihydrate is ca. 4 kJ mol-1 higher in energy. In contrast, PhTFE I⋯H2O was identified experimentally and theoretically as the most stable 1 : 1 conformer. Furthermore, the observed dihydrate structure experiences large amplitude motions connecting three theoretical minima which differ only in which water oxygen lone pairs are involved in the hydrogen-bonds, i.e., the free OH pointing directions. Additionally, the ortho and para-H2O tunnelling splittings were detected and attributed to the interchange water hydrogen atoms which interact with the aromatic part of PhTFE but not for the water interacting with PhTFE hydroxy group. Extensive theoretical modelling was carried out to gain insight into the associated large amplitude motions including tunnelling, supported by the experimental isotopic and tunnelling splitting data.

Approaching the free rotor limit: extremely low methyl torsional barrier observed in the microwave spectrum of 2,4-dimethylfluorobenzene.

The microwave spectrum of 2,4-dimethylfluorobenzene was recorded using a molecular jet Fourier transform microwave spectrometer in the frequency range from 2.0 to 26.5 GHz. The spectral assignment and modeling were challenging due to the large tunnelling splittings resulting from the very low barrier to internal rotation of the p-methyl group that approaches the free rotor limit. Internal rotation splittings arising from two inequivalent o- and p-methyl groups were observed, analysed and modelled using the modified version of the XIAM code and the BELGI-Cs-2Tops code, giving a root-mean-square deviation of 549.1 kHz and 4.5 kHz, respectively, for a data set of 885 rotational lines. The torsional barriers of the o- and p-methyl groups were determined to be 227.039(51) cm-1 and 3.23(40) cm-1, respectively. The V3 barrier observed for the p-methyl group is lower than in any other para-methyl substituted toluene derivatives with coupled internal rotations, becoming the lowest value ever observed to date. The barrier to internal rotation of the o-methyl group next to a fluorine atom is consistently around 220 cm-1, as confirmed by comparing it to barriers observed in other toluene derivatives. The experimental rotational constants were compared to those obtained by quantum chemical calculations.

Probing the structure and dynamics of the heterocyclic PAH xanthene and its water complexes with infrared and microwave spectroscopy.

To assess the presence of oxygen-containing polycyclic aromatic hydrocarbons (OPAHs) in the interstellar medium and understand how water aggregates on an OPAH surface, we present a comprehensive gas-phase spectroscopy investigation of the OPAH xanthene (C13H10O) and its complexes with water using IR-UV ion dip spectroscopy and chirped-pulse Fourier transform microwave spectroscopy. The infrared spectrum of xanthene shows weak features at 3.42, 3.43, and 3.47 μm, which have been suggested to partly originate from vibrational modes of PAHs containing sp3 hybridized carbon atoms, in agreement with the molecular structure of xanthene. The high resolution of rotational spectroscopy reveals a tunneling splitting of the rotational transitions, which can be explained with an out-of-plane bending motion of the two lateral benzene rings of xanthene. The nature of the tunnelling motion is elucidated by observing a similar splitting pattern in the rotational transitions of the singly-substituted 13C isotopologues. The rotational spectroscopy investigation is extended to hydrates of xanthene with up to four water molecules. Different xanthene-water binding motifs are observed based on the degree of hydration, with O-H⋯π interactions becoming preferred over O-H⋯Oxanthene interactions as the degree of hydration increases. A structural comparison with water complexes of related molecular systems highlights the impact of the substrate's shape and chemical composition on the arrangement of the surrounding water molecules.

Vibrational energy transfer in ammonia-helium collisions.

While the rotational energy transfer of ammonia by rare gas atoms and hydrogen molecules has been the focus of many studies, little is known about its vibrational relaxation, even though transitions involving the umbrella bending mode have been observed in many astrophysical environments. Here we explore the vibrational relaxation of the umbrella mode of ammonia induced by collisions with helium atoms by means of the close-coupling method on an ab initio potential energy surface. We compute cross sections up to kinetic energies of 1500 cm-1 and rate coefficients up to a temperature of 300 K for vibrational, rotational, and inversion transitions involving the lowest two vibrational states. We show that vibrational relaxation is much less efficient than rotation-inversion relaxation, although the rate coefficients for vibrational relaxation strongly increase with the temperature. We also observe important differences for vibrationally-elastic transitions within the lowest two vibrational states, i.e., for rotation-inversion transitions. These are a direct consequence of the difference in the tunnelling splitting of the lowest inversion levels.