Canonical Van Vleck Perturbation Theory Research Articles

Infrared vibrational spectra and absolute intensities of fundamental bands of bis(trifluoromethyl)ketene: Ab initio interpretation using the second order operator Van Vleck perturbation theory

Bis(trifluoromethyl)ketene (BTFMK, (CF3)2CCO) is an important chemical compound for the organic synthesis and in polymer chemistry. In this work, IR spectra of gaseous BTFMK were recorded in the range of 2280–400 cm−1 with a resolution of 1 cm−1. The experimental absolute IR intensities of 17 fundamental bands of BTFMK were measured and juxtaposed with ab initio anharmonic ones. The molecular structures, semi-diagonal quartic potential energy surfaces (PES), and cubic surfaces of dipole moment components (μ^α) were calculated at the MP2/cc-pVTZ level of theory. The predicted frequencies were refined by substituting the harmonic part of PES with the MP2/cc-pVQZ counterparts. The second-order operator canonical Van Vleck perturbation theory (CVPT2) was employed for predicting anharmonic energy levels, IR integral intensities and full experimental (re)-assignments. The deviation of predicted IR intensities from observations for ν1−ν4, ν6, ν7, ν9, ν14, ν15, ν22 was ≤12% on average. Three predicted Fermi resonances were experimentally identified: ν14/ν6+ν20, ν17/ν9+ν21 and ν18/ν7+ν25.

Rigorous vibrational Fermi resonance criterion revealed: two different approaches yield the same result

Molecular vibrational resonances are manifest by abnormally small energy differences in denominators of anharmonic constants derived within the second-order vibrational perturbation theory (VPT2), or in abnormally large coefficients Ξ of unitary transformation generators in the canonical Van Vleck perturbation theory (CVPT). A quantitative measure of the vibrational resonance can be derived by assuming the perturbation parameter to be a complex variable , and the numerical analysis of the diverging high order Rayleigh-Schrödinger perturbation theory (RSPT) expansions for closely spaced states. The location of branch points for complex values of within the unit circle serves as the definitive evidence for a resonance. In practice, RSPT series for diverging resonant states can be treated by resummation using quadratic Padé-Hermite approximants. The critical values of can be further found and checked against the condition . A comparative analysis of selected resonances for some molecules (HO, HDO, HS and HCO) revealed a threshold value of that is equivalent to the RSPT parameter . The fundamentally different approach based on the polyad analysis of the resonance vectors spanning the ()-dimensional subspace for the ethylene molecule led to nearly the same value for the resonance criterion.

The combination of multipartitioning of the Hamiltonian with canonical Van Vleck perturbation theory leads to a Hermitian variant of quasidegenerate N-electron valence perturbation theory.

Many recent developments in the area of multistate multireference perturbation theories focused on methods that use a state-averaged 0th order Hamiltonian. We recently found that the dynamic correlation dressed complete active space method fails in describing ligand field and charge transfer states in a balanced way precisely because it uses a state-averaged 0th order Hamiltonian [L. Lang and F. Neese, J. Chem. Phys. 150, 104104 (2019)]. The multipartitioning idea allows the use of state-specific 0th order Hamiltonians in a multistate framework and could therefore alleviate the mentioned problem. However, the effective Hamiltonian is non-Hermitian in the traditional formulation of multipartitioning, which can lead to unphysical behavior, especially for nearly degenerate states. In order to achieve a more balanced treatment of states with different physical character and at the same time have a Hermitian effective Hamiltonian, we combine in this work multipartitioning with canonical Van Vleck perturbation theory. At the 2nd order, the result is a Hermitian variant of multipartitioning quasidegenerate N-electron valence state perturbation theory. The effect of model space noninvariance of the method is discussed and the benefit of a Hermitian formulation is highlighted with numerical examples. The method is shown to give good results for the calculation of electronic transitions of the [CuCl4]2 -complex and for the calculation of electron paramagnetic resonance parameters, which are two examples where the balance between ligand field and charge transfer configurations is of utmost importance.

Absolute IR vibrational band intensities of hexafluoroacetone: Comparison of experiment and anharmonic ab initio calculation using the second-order operator canonical Van Vleck perturbation theory

Hexafluoroacetone (HFA, OC(CF3)2) is widely utilized in organic synthesis, chemistry of heat resistant polymers and biochemistry. In this work, infrared (IR) spectra of gaseous HFA were recorded in the range 2000–340 cm−1 with a resolution of 1 cm−1. For the first time, experimental absolute IR intensities of fourteen fundamental bands of HFA were measured and compared with intensities predicted by anharmonic ab initio calculations. Theoretical molecular structures, semi-diagonal quartic potential energy surfaces (PES), and cubic surfaces of dipole moment (DM) components of HFA were calculated using MP2/cc-pVTZ ab initio quantum-mechanical model. In addition, harmonic frequencies were also calculated at MP2/cc-pVQZ level of theory and used for refining the PES. The numerical-analytic implementation of the second-order operator canonical Van Vleck perturbation theory (CVPT2) was employed for predicting anharmonic IR spectra of HFA. The deviation from observations of predicted IR intensities for ten fundamentals (ν1, ν2, ν8, ν13–ν21) was less than 25% on average. The significant difference between ab initio harmonic intensities calculated with MP2/cc-pVTZ and MP2/cc-pVQZ models for some normal modes indicates that an inaccuracy of the quantum-mechanical model is probably the major source of errors in theoretical intensities. An interpretation of the IR experimental spectra and associated literature data for HFA is accomplished; the ν4, ν7, ν10 and ν15 fundamentals were reassigned. Three predicted Fermi resonances were identified in experimental spectra, ν3/ν5 + ν8, ν17/ν10 + ν19 and ν20/ν9 + ν23. An empirical fit of harmonic frequencies within anharmonic model assisted in a reliable identification of a number of overtones and binary combination tones. The fitted harmonic frequencies were compared with MP2/cc-pVQZ and a simplified complete basis set (CBS) extrapolation. It was found that MP2/cc-pVQZ frequencies were more reliable. The efficiency and convenience of the theoretical model employed and associated techniques for interpretation of complex spectra of HFA are demonstrated.

Anharmonic vibrational analysis of s-trans and s-cis conformers of acryloyl fluoride using numerical-analytic Van Vleck operator perturbation theory

A new gas-phase infrared (IR) spectrum of acryloyl fluoride (ACRF, CH2CHCFO) with a resolution of 0.1cm−1 in the range 4000–450cm−1 was measured. Theoretical ab initio molecular structures, full quartic potential energy surfaces (PES), and cubic surfaces of dipole moments and polarizability tensor components (electro-optical properties, EOP) of the s-trans and s-cis conformers of the ACRF were calculated by the second-order Møller-Plesset electronic perturbation theory with a correlation consistent Dunning triple-ζ basis set. The numerical-analytic implementation of the second-order operator canonical Van Vleck perturbation theory was employed for predicting anharmonic IR and Raman scattering (RS) spectra of ACRF. To improve the anharmonic predictions, harmonic frequencies were replaced by their counterparts evaluated with the higher-level CCSD(T)/cc-pVTZ model, to form a “hybrid” PES. The original operator representation of the Hamiltonian is analytically reduced to a quasi-diagonal form, integrated in the harmonic oscillator basis and diagonalized to account for strong resonance couplings. Double canonical transformations of EOP expansions enabled prediction of integral intensities of both fundamental and multi-quanta transitions in IR/RS spectra. Enhanced band shape analysis reinforced the assignments. A thorough interpretation of the new IR experimental spectra and existing matrix-isolation literature data for the mixture of two conformers of ACRF was accomplished, and a number of assignments clarified.

Quasi-degenerate perturbation theory using matrix product states.

In this work, we generalize the recently proposed matrix product state perturbation theory (MPSPT) for calculating energies of excited states using quasi-degenerate (QD) perturbation theory. Our formulation uses the Kirtman-Certain-Hirschfelder canonical Van Vleck perturbation theory, which gives Hermitian effective Hamiltonians at each order, and also allows one to make use of Wigner's 2n + 1 rule. Further, our formulation satisfies Granovsky's requirement of model space invariance which is important for obtaining smooth potential energy curves. Thus, when we use MPSPT with the Dyall Hamiltonian, we obtain a model space invariant version of quasi-degenerate n-electron valence state perturbation theory (NEVPT), a property that the usual formulation of QD-NEVPT2 based on a multipartitioning technique lacked. We use our method on the benchmark problems of bond breaking of LiF which shows ionic to covalent curve crossing and the twist around the double bond of ethylene where significant valence-Rydberg mixing occurs in the excited states. In accordance with our previous work, we find that multi-reference linearized coupled cluster theory is more accurate than other multi-reference theories of similar cost.

Anharmonic Vibrational Analysis of the Infrared and Raman Gas-Phase Spectra of s-trans- and s-gauche-1,3-Butadiene.

A quantum-mechanical (hybrid MP2/cc-pVTZ and CCSD(T)/cc-pVTZ) full quartic potential energy surface (PES) in rectilinear normal coordinates and the second-order operator canonical Van Vleck perturbation theory (CVPT2) are employed to predict the anharmonic vibrational spectra of s-trans- and s-gauche-butadiene (BDE). These predictions are used to interpret their infrared and Raman scattering spectra. New high-temperature Raman spectra in the gas phase are presented in support of assignments for the gauche conformer. The CVPT2 solution is based on a PES and electro-optical properties (EOP; dipole moment and polarizability) expanded in Taylor series. Higher terms than those routinely available from Gaussian09 software were calculated by numerical differentiation of quadratic force fields and EOP using the MP2/cc-pVTZ model. The integer coefficients of the polyad quantum numbers were derived for both conformers of BDE. Replacement of harmonic frequencies by their counterparts from the CCSD(T)/cc-pVTZ model significantly improved the agreement with experimental data for s-trans-BDE (root-mean-square deviation ≈ 5.5 cm(-1)). The accuracy in predicting the rather well-studied spectrum of fundamentals of s-trans-BDE assures good predictions of the spectrum of s-gauche-BDE. A nearly complete assignment of fundamentals was obtained for the gauche conformer. Many nonfundamental transitions of the BDE conformers were interpreted as well. The predictions of multiple Fermi resonances in the complex CH-stretching region correlate well with experiment. It is shown that solving a vibrational anharmonic problem through a numerical-analytic implementation of CVPT2 is a straightforward and computationally advantageous approach for medium-size molecules in comparison with the standard second-order vibrational perturbation theory (VPT2) based on analytic expressions.

Criteria for first- and second-order vibrational resonances and correct evaluation of the Darling-Dennison resonance coefficients using the canonical Van Vleck perturbation theory.

The second-order vibrational Hamiltonian of a semi-rigid polyatomic molecule when resonances are present can be reduced to a quasi-diagonal form using second-order vibrational perturbation theory. Obtaining exact vibrational energy levels requires subsequent numerical diagonalization of the Hamiltonian matrix including the first- and second-order resonance coupling coefficients. While the first-order Fermi resonance constants can be easily calculated, the evaluation of the second-order Darling-Dennison constants requires more complicated algebra for seven individual cases with different numbers of creation-annihilation vibrational quanta. The difficulty in precise evaluation of the Darling-Dennison coefficients is associated with the previously unrecognized interference with simultaneously present Fermi resonances that affect the form of the canonically transformed Hamiltonian. For the first time, we have presented the correct form of the general expression for the evaluation of the Darling-Dennison constants that accounts for the underlying effect of Fermi resonances. The physically meaningful criteria for selecting both Fermi and Darling-Dennison resonances are discussed and illustrated using numerical examples.

Polyad quantum numbers and multiple resonances in anharmonic vibrational studies of polyatomic molecules

In the theory of anharmonic vibrations of a polyatomic molecule, mixing the zero-order vibrational states due to cubic, quartic and higher-order terms in the potential energy expansion leads to the appearance of more-or-less isolated blocks of states (also called polyads), connected through multiple resonances. Such polyads of states can be characterized by a common secondary integer quantum number. This polyad quantum number is defined as a linear combination of the zero-order vibrational quantum numbers, attributed to normal modes, multiplied by non-negative integer polyad coefficients, which are subject to definition for any particular molecule. According to Kellman's method [J. Chem. Phys. 93, 6630 (1990)], the corresponding formalism can be conveniently described using vector algebra. In the present work, a systematic consideration of polyad quantum numbers is given in the framework of the canonical Van Vleck perturbation theory (CVPT) and its numerical-analytic operator implementation for reducing the Hamiltonian to the quasi-diagonal form, earlier developed by the authors. It is shown that CVPT provides a convenient method for the systematic identification of essential resonances and the definition of a polyad quantum number. The method presented is generally suitable for molecules of significant size and complexity, as illustrated by several examples of molecules up to six atoms. The polyad quantum number technique is very useful for assembling comprehensive basis sets for the matrix representation of the Hamiltonian after removal of all non-resonance terms by CVPT. In addition, the classification of anharmonic energy levels according to their polyad quantum numbers provides an additional means for the interpretation of observed vibrational spectra.

Anharmonic Vibrational Analysis of the Gas-Phase Infrared Spectrum of 1,1-Difluoroethylene Using the Operator Van Vleck Canonical Perturbation Theory

Anharmonic vibration frequencies of 1,1-difluoroethylene (11DFE) in the gas phase are predicted by means of the numerical-analytic operator version of the canonical van Vleck perturbation theory in the second and fourth orders (CVPT2 and CVPT4). The full quartic and "semi-diagonal" sextic rectilinear normal coordinate potential energy surfaces, needed for CVPT2 and CVPT4, respectively, were obtained with the MP2/cc-pVTZ quantum-mechanical model. CVPT2 is superior to the traditional second-order vibrational perturbation theory approach (VPT2) because of the uniform general treatment of the Fermi and second-order Darling-Dennison resonances. The fourth-order version, CVPT4, provides a more refined solution and proves convergence of the perturbative treatment. Labeling of the basis functions by polyad numbers breaks down the infinite Hamiltonian matrix into a block-diagonal form. The polyad expression for 11DFE has been determined as P = 14(ν1 + ν7) + 8ν2 + 6(ν3 + ν8) + 4(ν4 + ν9) + 3(ν6 + ν11 + ν12) + 2(ν5 + ν10), where the νi are quantum numbers. The theoretical prediction of anharmonic infrared absorption intensities corroborated an assignment of the majority of observed gas-phase bands up to 3500 cm(-1). The solution was refined by iteratively fitting harmonic frequencies, until predicted fundamental anharmonic frequencies matched the observed values. The average error for about 90 observed frequencies after fitting only fundamental frequencies is ∼1.05 cm(-1). The fitted "semi-experimental" harmonic frequencies agree very well the quantum-mechanical predictions based on the CCSD(T)/cc-pVTZ and CCSD(T)/cc-pVQZ models.

Numerical-Analytic Implementation of the Higher-Order Canonical Van Vleck Perturbation Theory for the Interpretation of Medium-Sized Molecule Vibrational Spectra

Anharmonic vibrational states of semirigid polyatomic molecules are often studied using the second-order vibrational perturbation theory (VPT2). For efficient higher-order analysis, an approach based on the canonical Van Vleck perturbation theory (CVPT), the Watson Hamiltonian and operators of creation and annihilation of vibrational quanta is employed. This method allows analysis of the convergence of perturbation theory and solves a number of theoretical problems of VPT2, e.g., yields anharmonic constants y(ijk), z(ijkl), and allows the reliable evaluation of vibrational IR and Raman anharmonic intensities in the presence of resonances. Darling-Dennison and higher-order resonance coupling coefficients can be reliably evaluated as well. The method is illustrated on classic molecules: water and formaldehyde. A number of theoretical conclusions results, including the necessity of using sextic force field in the fourth order (CVPT4) and the nearly vanishing CVPT4 contributions for bending and wagging modes. The coefficients of perturbative Dunham-type Hamiltonians in high-orders of CVPT are found to conform to the rules of equality at different orders as earlier proven analytically for diatomic molecules. The method can serve as a good substitution of the more traditional VPT2.

Assignment and Extracting Dynamics from Experimentally and Theoretically Obtained Spectroscopic Hamiltonians in the Complex Spectral and Classically Chaotic Regions

An analysis of existing algebraic multiresonance spectroscopic Hamiltonians, derived by fitting to experimental data or from classical canonical or quantum Van Vleck perturbation theory, allows without any significant further classical or quantum calculation the assignment of quantum numbers and motions to states observed in spectra that were previously thought to be irregular or just unassignable. In such cases, inspection of the amplitude and phase of eigenfunctions previously calculated in the Hamiltonians derivation process but now transformed to a reduced dimension semiclassical action-angle representation reveals extremely simple albeit unfamiliar topologies that give quantum numbers by simply counting nodes and phase advances. The topology allows these simple wave functions to be sorted into dynamically different excitation ladders or classes of states which are associated with different regions of phase space. The rungs of these ladders or the states in the classes intersperse in energy causing the spectral complexity. No experimental procedure allows such sorting. The success of the work stems from (1) the qualitative insights of nonlinear dynamics, (2) the conversion of the quantum problem in full dimension to a semiclassical one in reduced dimension by use of a canonical transform that takes advantage of the polyad and other constants of the motion, and (3) the judicious choice of the reduced angle variables to reflect rational ratio resonance frequency conditions. This leads to localization of those semiclassical wave functions, which are affected by the particular resonance. In reverse, the localized appearance of the reduced dimension wave function reveals which resonances govern it and makes sorting visually simple. The success of the work also stems from (4) the revealing use of plots of phase advances as well as the usual densities of the eigenstates for sorting and assignment purposes. Even in classically chaotic regions, organizing trajectories, which correspond to averages over regional phase space structures that need not be computed, can easily be drawn as the structure about which eigenfunction localization takes place. The organizing trajectories when transformed back to the full dimensional configuration space reveal the internal molecular motions. The complexity of the usual quantum stationary and propagated wave functions and associated classical trajectories forbids most often such assignments and sorting. This procedure brings the ability to interpret complex vibrational spectra to a degree previously thought possible only for lower excitations. The new methodology replaces and extends the computationally more difficult parts of a procedure used by the authors that was applied successfully to several molecules during the past few years. The new methodology is applied to DCO, CHBrClF, and the bending of acetylene.

A perturbative calculation of the rovibrational energy levels of methane

The rovibrational energy levels of methane are determined from a quartic ab initio potential energy force field where the expansion coordinates are the Morse coordinates for the stretches and extension coordinates for the bends. Energies are calculated using canonical Van Vleck perturbation theory. Results are obtained for both rotation–vibration Hamiltonians expressed as functions of curvilinear and rectilinear normal coordinates. Second, fourth, and sixth order curvilinear results are compared with experimental results, and fourth order results for the rectilinear and curvilinear Hamiltonian are compared to each other. The calculated rovibrational levels are in good agreement with the experimental values for low J levels. The calculated rotational level splittings are in even better agreement with the experiment. In particular, the ground state tetrahedral splittings, which are as small as 10 −4 cm −1, are well reproduced by our calculations at sixth order.

The vibrational overtones of SiH4 isotopomers: experimental wavenumbers, assignment, ab initio dipole moment surfaces and intensities

The overtone spectra of SiHD3, SiH2D2 and SiH3D in the range 2000–9000 cm−1 have been measured up to 720000 Pa m using a Bruker 120HR interferometer. The transition frequencies and intensities are reported. The observed band centers are found to be in good agreement with the predictions from high order canonical Van Vleck perturbation theory (CVPT) calculations. A four-dimensional Si–H stretching dipole moment surface (DMS) has been determined from coupled cluster [CCSD(T)] methods and expanded using a bond dipole model and a Cartesian model. The band intensities of the Si–H and Si–D stretching transitions have been computed employing these expansions and a local mode force field optimized to the observed band centers. The calculations agree reasonably well with observations for most bands. The vibrational intensity anomaly in the third local mode manifolds is partly reproduced and is attributed to the cancellation of linear and quadratic terms in the DMS.

A nine-dimensional high order perturbative study of the vibration of silane and its isotopomers

The vibrations of silane isotopomers with Td, C3v, and C2v symmetry are studied by means of high order canonical Van Vleck perturbation theory (CVPT). Transforming the quartic ab initio force field of Martin, Baldridge, and Lee [Mol. Phys. 95, 254 (1999)] into curvilinear normal coordinates, CVPT is used to calculate energies that agree well with experimental data. Both low energy stretch–bend combination bands and high energy stretch local mode bands of silane up to 12 000 cm−1 are well reproduced. The choice of polyad quantum number is discussed with respect to different molecules. Comparing sixth- to eighth-order level of theory, most of the levels agree to within 0.1 cm−1. Spectroscopic constants are given for all the major isotopomers. The construction of the full cubic and quartic resonance operators for symmetric top species are summarized. The Si–H and Si–D stretch modes of Td and C3v symmetry species are studied with a 4D stretch variational model using both the fitted three-parameter potential and the quartic ab initio stretch potential. Large energy level differences between these two calculations indicate that the fitted potential constants of the stretch variational model are different from those derived directly from the ab initio surface. By comparing with the nine-dimensional CVPT calculation, it is shown that the direct and indirect Fermi resonances are responsible for the inability of the four-dimensional model to fit some stretch levels.

Anharmonic force field and vibrational frequencies of tetrafluoromethane (CF4) and tetrafluorosilane (SiF4)

Accurate quartic anharmonic force fields for CF4 and SiF4 have been calculated using the CCSD(T) method and basis sets of spdf quality. Based on the ab initio force field with a minor empirical adjustment, the vibrational energy levels of these two molecules and their isotopomers are calculated by means of high order Canonical Van Vleck Perturbation Theory (CVPT) based on curvilinear coordinates. The calculated energies agree very well with the experimental data. The full quadratic force field of CF4 is further refined to the experimental data. The symmetrization of the Cartesian basis for arbitrary combination bands of Td group molecules is discussed using the circular promotion operator for the doubly degenerate modes, together with tabulated vector coupling coefficients. The extraction of the spectroscopic constants from our second order transformed Hamiltonian in curvilinear coordinates is discussed, and compared to a similar procedure in rectilinear coordinates.

High resolution spectroscopic study of CH3D in the region 5900–6100 cm−1

The vibrational overtone band of monodeuterated methane (CH3 in the region 5900–6100 cm−1 was recorded by a high resolution Fourier transform spectrometer. Rotational analyses were performed separately for the parallel 2v 4(A1) and perpendicular 2v 4(E) bands, whose origins were found to be 5980.41 cm−1 and 6022.21 cm−1, respectively. The A1 band displays J-resolved rotational structure while the K structure is resolved only at higher J values. A resonance Hamiltonian with the polyad number P = 2n 1 + 2n 4 + n 5 is derived using the fourth order Canonical Van Vleck Perturbation Theory (CVPT) to show that the 2v 4(A1 and 2v 4(E) are vibrationally perturbed by Fermi and Darling—Dennison resonances. The observation of rotational l-resonance in the 2v 4(E) band is a consequence and evidence of the mixing of the vibrational states with different l values.

A combined use of perturbation theory and diagonalization: Application to bound energy levels and semiclassical rate theory

A new method, mixed diagonalization, is introduced in which an effective Hamiltonian operator acting on a reduced dimensional space is constructed using the similarity transformations of canonical Van Vleck perturbation theory (CVPT). This construction requires the characterization of modes into two categories, global and local, which in the bound vibrational problem are tantamount to the large and small amplitude vibrations, respectively. The local modes in the Hamiltonian are projected out by CVPT, and the resulting Hamiltonian operator acts only on the space of global modes. The method affords the treatment of energy levels of bound systems in which some vibrational assignments are possible. In addition, it systematically provides a reduced dimensional Hamiltonian which is more amenable to exact numerical solution than the original full-dimensional Hamiltonian. In recent work, a semiclassical transition state theory (SCTST) rate expression has been written in terms of a Hamiltonian operator parameterized by the imaginary action along the local reaction path in the transition state region [Chem. Phys. Lett. 214, 129 (1993)]. We show that the Hamiltonian constructed by mixed diagonalization has this form, and can be used to obtain more accurate semiclassical rate expressions.

An algebraic approach to calculating rotation-vibration spectra of polyatomic molecules

An approach for calculating electric dipole intensities of transitions to highly vibrationally excited states in polyatomic molecules is presented. We apply canonical Van Vleck perturbation theory (CVPT) to the space fixed dipole moment operator, where both the rotational and vibrational coordinate dependencies are expressed in terms of boson creation and annihilation operators. This approach is general and can be applied to a large class of polyatomic molecules. Theoretical spectra are presented for H2O with T rot = 300 K and H2CO with T rot = 10 K.

Rotation–vibration interactions in highly excited states of SO2 and H2CO

Canonical Van Vleck perturbation theory (CVPT) is used to investigate rotation–vibration mixing of highly excited vibrational states of SO2 and H2CO. For SO2 we find a nearly complete separation of the rotational and vibrational degrees of freedom, even for J=12 and Evib=11 000 cm−1. In contrast, for H2CO we observe extensive mixing between rotational and vibrational degrees of freedom at similar rotational excitation but with Evib=8000 cm−1. Although a-axis Coriolis coupling is pronounced, b- and c-axis Coriolis couplings play an important additional role in mixing states with different Ka quantum numbers. The implementation of CVPT, the choice of internal coordinates, and the convergence of the results are discussed in detail.