Application of the contact transformation method to torsional problems in methyl silane

The effect of the torsional degree of freedom on redundancies in the Hamiltonian and on the dipole operator has been investigated for methyl silane-like molecules. By applying a rotational contact transformation to the torsion-rotation Hamiltonian H_TR for the ground vibrational state, a systematic method is demonstrated for treating the redundancies that relate different terms in H_TR. In general, with this method, the experimentally accessible molecular parameters in the reduced Hamiltonian can be related to the physically significant molecular parameters in the untransformed Hamiltonian. It is shown that H_TR contains a new term which has matrix elements with selection rules (ΔK = ±3), (Δσ = 0), and Δv_T arbitrary, where v_T and σ label the torsional levels and sublevels, respectively. As a result of this term, the distortion dipole constant μ_D which characterizes (ΔK = ±3) matrix elements in C_3v molecules cannot, in systems like CH₃SiH₃, be ascribed entirely to centrifugal distortion but can contain a significant contribution from torsional effects. Furthermore, new transitions can appear in the pure torsional bands which may be strong enough to observe in low barrier molecules. By applying a vibrational contact transformation, the form is derived of the leading torsional terms in the dipole moment expansion. The four dipole distortion constants μ₀^T, μ₂^T, μ_|;^T, and μ_Λ^T which characterize these terms are related to the molecular parameters that enter the Coriolis, centrifugal distortion, and anharmonicity contributions to the vibration-torsion-rotation Hamiltonian.     

The large-amplitude motions in quasi-symmetric top molecules with internal C3v rotors: The vibration-torsion-rotation Hamiltonian of a two-top molecule

An expression for the vibration-torsion-rotation Hamiltonian of a molecule with triatomic nonrigid frame and two internal C_3v rotors has been derived. Three large-amplitude motions, namely skeletal bending and two torsions, are removed from the vibrational problem and are considered together with the rotational problem. The Hamiltonian obtained is applicable to any two-C_3v-top molecule with triatomic linear or bent frame. The zeroth-order skeletal bending-torsion-rotation Hamiltonian is derived and the method of solving the corresponding Schrödinger equation is discussed. The Hamiltonian obtained with only slight modifications is shown to be applicable to any single-C_3v-top molecule with a quasi-linear tetratomic nonrigid frame or to the problem of the large-amplitude bending motion in a pentatomic quasi-linear molecule.     

A new torsion-rotation fitting program for molecules with a sixfold barrier: Application to the microwave spectrum of toluene

A new program is described for fitting rotation-torsion energy levels in molecules like toluene, in which the frame (C₆H₅) has C_2v symmetry and the methyl top has C_3v symmetry, i.e., for molecules where the internal rotation barrier is expanded in cos6nα, where α is the internal rotation angle and n = 1,2,…. The program is based on the theoretical framework developed by Sørensen and Pedersen in their application of the Longuet-Higgins permutation-inversion group G₁₂ to the microwave spectrum of CH₃NO₂. It is specifically designed for sixfold barrier molecules, and allows the user to select almost any symmetry-allowed torsion-rotation term for inclusion in the fitting Hamiltonian. This program leads to a very successful fit of transitions in the microwave spectrum of toluene characterized by J ? 30, K_a ? 12, and by the free-rotor quantum number ∣m∣ ? 3. In these fits we included both published and rather extensive unpublished new measurements, for which fits using other torsion-rotation programs have not been very successful. The fit presented here uses 28 parameters to give an overall standard deviation of 7.4 kHz for 372 line frequencies, and results in a much improved value for the sixfold barrier for toluene, V₆ = 13.832068(3) cal mol⁻¹.     

Reduced effective Hamiltonian for Coriolis-interacting νn and νt fundamentals of C3v molecules

Reduction of the effective Hamiltonian for Coriolis-interacting ν_n(A₁) and ν_t(E) fundamentals in C_3v molecules is caried out. Formulas for the parameters of the reduced Hamiltonian in terms of the parameters of the unreduced one are derived. The reduction procedure allows interperation of the disagreement between the results reported by various authors on simultaneous fitting of interacting ν₂ and ν₅ bands of CH₃F. It is found also that the reduction of the effective Hamiltonian for the isolated degenerate band ν_t(E) should be carried out in order to provide a correct interpretation of the fitted parameters in terms of molecular constants.     

Millimeter-wave spectrum and internal rotation of 1-silylpropyne,CH3CCSiH3

Rotational transitions of CH₃CCSiH₃ have been observed in the millimeter-wave region using a computer-controlled source-frequency modulation spectrometer with a 1.8-m-long free space absorption cell. The observed spectrum clearly showed the effect of internal rotation with a small potential barrier. It has been analyzed by calculating the torsion-rotation energies on the basis of torsional wave functions obtained by diagonalizing the torsional part of the Hamiltonian. The least-squares analysis has yielded the rotational constant B = 2068.2817(4) MHz and a few centrifugal distortion constants. The barrier height to internal rotation has been determined to be 3.77(70) cm^?1 from the contour map of the standard deviation. Also, the A rotational constant of the silyl group around the symmetry axis has been estimated by fixing the A constant of the methyl group to the value of CH₃CCH.     

Tensorial development of the rovibronic Hamiltonian and dipole moment operators for XY3Z molecules with a degenerate electronic state: Preliminary application to the CH3O radical

We present a development of the Hamiltonian and transition moment operators of XY₃Z (C_3v) symmetric tops molecules in a degenerate electronic state with the aid of a tensorial formalism developed in a recent paper [A. El Hilali, V. Boudon, M. Loëte, J. Mol. Spectrosc. 239 (2006) 41-50]. Electronic operators are defined from group theory properties. They provide a new approach to build an effective rovibronic Hamiltonian as well as an effective dipole moment operator for rovibronic transition of XY₃Z molecules. This model is studied qualitatively thanks to the tensorial algebra properties. Expressions of the matrix elements are derived for these operators. A first simple application to the ground electronic state of CH₃O is proposed as an illustrative example.     

Analysis and fit of the Fourier-transform microwave spectrum of the two-top molecule N-methylacetamide

The jet-cooled Fourier-transform microwave spectrum of N-methylacetamide (CH₃NHC(O)CH₃), a molecule containing two methyl tops with relatively low barriers to internal rotation, has been recorded and fit to nearly experimental uncertainty. Measurements were carried out between 10 and 26 GHz, with the nitrogen quadrupole splittings resolved for many transitions. The permutation-inversion group for this molecule is G₁₈ (not isomorphic to any point group), with irreducible representations A₁, A₂, E₁, E₂, E₃, and E₄. One of these symmetry species and the usual three asymmetric rotor quantum numbers J_KaKc were assigned to each torsion-rotation level involved in the observed transitions. F values were assigned to hyperfine components, where . Transitions involving levels of A₁ and A₂ species could be fit to an asymmetric rotor Hamiltonian. The other transitions were first fit separately for each symmetry species using a Pickett-like effective rotational Hamiltonian. Constants from these fits show a number of additive properties which can be correlated with sums and differences of effects involving the two tops. A final global fit to 48 molecular parameters for 839 hyperfine components of 216 torsion-rotation transitions involving 152 torsion-rotation levels was carried out using a newly written two-top computer program, giving a root-mean-square deviation of observed-minus-calculated residuals of 4 kHz. This program was written in the principal axis system of the molecule and uses a free-rotor basis set for each top, a symmetric-top basis set for the rotational functions, and a single-step diagonalization procedure. Such an approach requires quite long computation times, but it is much less prone to subtle programming errors (a consideration felt to be important since checking the new program against precise fits of low-barrier two-top molecules in the literature was not possible). The two internal rotation angles in this molecule correspond to the Ramachandran angles ψ and φ often defined to describe polypeptide folding. Barriers to internal rotation about these two angles were found to be 73 and 79 cm⁻¹, respectively. Top-top coupling in both the kinetic and potential energy part of the Hamiltonian is relatively small in this molecule.     

Ground and first excited torsional states of acetamide

We present a first global study involving rotational levels in the lowest three torsional states of acetamide (CH₃CONH₂). New measurements of this spectrum, consisting of approximately 1600 lines and involving torsion-rotation transitions with J up to 20 and K_a up to 11, have been carried out between 49 and 149 GHz using the millimeter-wave spectrometer in Kharkov. After removing the observed quadrupole hyperfine splittings, the new data were combined with previously published measurements and fitted using a rho-axis-method (RAM) torsion-rotation Hamiltonian in conjunction with a new computer-automated v_t and K labeling algorithm. The final fit used 48 parameters to give an overall weighted standard deviation of 0.72 for 759, 587, and 265 lines belonging, respectively, to the ground, first, and second excited torsional states and 95 lines corresponding to Δv_t=1 transitions falling in the millimeter wave range. Separate root-mean-square (rms) deviations for the A (25 kHz) and E (27 kHz) species, as well as for the v_t=0 state (26 kHz), v_t=1 (25 kHz), v_t=2 (22 kHz), and Δv_t=1 transitions (35 kHz) indicate a similar quality of the fit for the two symmetry species and for the three torsional states. The RAM Hamiltonian was found to have rather good predictive ability when fits of only v_t=0 (or only v_t=0 and 1) transitions were used to calculate v_t=1 (or v_t=2) lines. In addition, the combination of this Hamiltonian and the computer-automated v_t and K labeling algorithm seems to provide a rather powerful tool for the precise assignment and fitting of torsion-rotation transitions in C_3v internal-rotor molecules with low torsional potential barriers.     

Effective correlation-free reduced Hamiltonian for the νt(E), νn(A1) Coriolis-interacting states of C3v symmetric-top molecules

The effective correlation-free vibrational-rotational Hamiltonian for the Coriolis-interacting ν_t(E) and ν_n(A₁) states in C_3v molecules has been derived. The Hamiltonian includes the terms describing the x-y Coriolis interaction up to the fourth-order, and several useful reduction schemes for the Hamiltonian are suggested.     

A new scheme of K-labeling for torsion-rotation energy levels in low-barrier molecules

The problem of attaching K rotational quantum number labels after the second diagonalization step in the ρ-axis-method treatment of methyl-top internal rotation problems is considered. A new partially computer-automated labeling scheme for K-labeling is proposed. The scheme is rather simple and does not require any information other than that provided by the numerical eigenvectors obtained after diagonalization of the torsion-rotation Hamiltonian matrix. It assumes that within a given K stack, torsion-rotation eigenfunctions vary slowly when J changes by unity. The basic idea is thus to search for similarities in basis-set composition in torsion-rotation eigenvectors belonging to adjacent J values. In such a way, torsion-rotation states of adjacent J values having the same value of K are connected. This allows one to transfer a given K-label from lower J values, where it can be determined easily (either from eigenvector composition or from energy-ordering considerations), to higher J values, which are characterized by extensive basis-set mixing. The approach was successfully applied to the K-labeling problem of prolate (acetaldehyde, methanol, and ethyl acetamidoacetate) and oblate (acetic acid, acetamide) rotors characterized by significant torsion-rotation interactions in their spectra. The scheme gives correct K-labels for eigenvectors in the majority of cases. The problems with the remaining cases are mainly caused by localized avoided crossing interactions, which can be fixed relatively easily in manual mode using a graphical visualization of the torsion-rotation energy level diagram.     

Study of the torsion-rotation Hamiltonian for symmetric tops using the millimeter wave spectrum of methyl silane

The torsion-rotation Hamiltonian for symmetric tops has been tested in methyl silane by combining recent anticrossing molecular beam measurements in the ground torsional state (v = 0) with pure rotational spectra taken for v as high as 4. The earlier microwave data set which consisted of J = 1 ← 0 and 2 ← 1 has been greatly extended by studying millimeter transitions for J = 4 ← 3, 5 ← 4, and 13 ← 12. An analysis of the 72 rotational frequencies for v ≤ 2 and the 15 anticrossing data for v = 0 yielded an excellent fit using 14 rotational, torsional, and distortion constants including the effective values for the A rotational constant and the barrier height V₃. No satisfactory fit could be obtained when the data set was extended to include measurements for (v = 3) or (v = 4). For each of these higher torsional levels, the difference between the observed frequencies and the predictions based on the best (v ≤ 2) constants can be expressed in terms of a shift δB_v in the B rotational constant, where δB_v is a smooth function of the torsional energy. This disagreement is of particular interest because it may result from the fact that the molecule passes from hindered to free rotation as v is increased from 2 to 4. The possibility of perturbation by a low-lying vibrational level is considered briefly. The information contained in the different types of spectra is discussed; the redundancy relations are treated and a Fourier expansion of the diagonal torsional matrix elements is introduced. For ¹²CH₃²⁹SiH₃, ¹²CH₃³⁰SiH₃, and ¹³CH₃²⁸SiH₃ pure rotational spectra for v = 0 were studied briefly in natural abundance. The results were combined with existing data for two deuterated symmetric rotors to obtain a structure based only on symmetric top rotational constants.     

Vibration-internal rotation-rotation Hamiltonian of an asymmetric top molecule with C3v internal rotor

An expression for the kinetic energy part of the vibration-torsion-rotation Hamiltonian of an asymmetric top molecule containing a C_3v internal rotor has been derived. The terms for various interactions in the molecule, viz. Coriolis interaction between rotation (both overall and internal rotation) and vibration, centrifugal distortion and anharmonicity of molecular vibrations induced by the internal, and overall rotation of the molecule, have been formulated. For a planar molecule with C_s symmetry we have obtained the vibrationally averaged rotation-internal rotation Hamiltonian. Diagonalization of this Hamiltonian for a particular vibrational state will yield the rotation-internal rotation energy levels and hence the transition frequencies. These data will be useful for analysis of high-resolution infrared spectra obtained by laser or Fourier transform spectroscopy of nonrigid molecules with internal rotor. We also present a set of quartic centrifugal distortion coefficients associated with rotation and internal rotation. These data will be helpful for evaluation of vibrational potential constants of the orthorhombic asymmetric top molecules.     

The millimeter-wave spectrum of perdeuterated methanol, CD3OD

In order to provide accurate rest frequencies for astronomical searches, the spectrum of perdeuterated methanol, CD₃OD, has been measured in the frequency range 62-233 GHz. A total of 379 lines was measured from rotational states up to J=20 and K=10 within the ground and first excited torsional states (v_t=0 and 1). Using a one-dimensional torsion-rotation Hamiltonian, the lines were fitted to measurement accuracy (<30 kHz).     

Effective Hamiltonian for degenerate vibrational states in symmetric top molecules

A mixed matrix-operator form of the effective rotational Hamiltonian has been discussed for the degenerate vibrational states of symmetric top molecules. In this scheme, a rotational contact transformation can be applied to the effective Hamiltonian such that the operators of the “2, +2,” “2, −2,” and “2, −1” l-type interactions as well as the operators of the Δk = ±3 and ±4 interactions are eliminated from the first-order terms of the expansion of the rotational Hamiltonian in terms of the small parameter λ. The results have been used to discuss the correlation between various interaction parameters in the effective rotational Hamiltonian for the doubly degenerate fundamental vibrational levels of semirigid symmetric top molecules. For example, for C_3v or D₃ molecules, the parameter of the “2, −1” interaction is correlated with other parameters and cannot be determined separately by fitting the experimental data (unless there are certain accidental resonances between vibrational-rotational levels).     

A Combined Frequency Analysis of the ν3, ν9, 3ν4 and the Far-Infrared Bands of Ethane: A Reassessment of the Torsional Parameters for the Ground Vibrational State

The lowest frequency parallel fundamental band ν₃ of ethane is Raman active. A stimulated Raman spectrum of the Q branch for this band at a resolution of 0.0055 cm⁻¹ has been measured by D. Bermejo et al. (1992, J. Chem. Phys.97, 7055). The torsion-rotation series in this band with σ=3, where σ=0, 1, 2, and 3 labels the torsional sublevels, is perturbed by over 1 cm⁻¹. The lowest frequency-degenerate fundamental ν₉ is infrared active. A high-resolution (0.0014 cm⁻¹) Fourier transform spectrum of this band has been measured by N. Moazzen-Ahmadi et al. (1999, J. Chem. Phys.111, 9609). The observed torsional splittings for this band are substantially larger than expected from the observed barrier height. Because of a near-degeneracy of the upper level in the ν₉ band with its interacting partner (v₉=0, v₄=3) a perturbation allowed band 3ν₄ has also been observed. We have carried out a combined analysis of ν₃, ν_9, and 3ν₄ together with the far-infrared torsional spectra in the ground vibrational state (gs). A fit to within the experimental error was achieved using 37 parameters. The large torsional splittings in the ν₉ band are attributed to Coriolis-type interactions between the torsional stacks of gs and v₉=1 whereas the large shift for the torsion-rotation series with σ=3 in the ν₃ band is attributed to Fermi-type interactions between the torsional stacks of the gs and v₃=1. The introduction of the Fermi-type interactions causes a considerable change in the leading terms in the torsional Hamiltonian for the gs. These changes are quantitatively explained.     

Hamiltonian for a symmetric-top molecule in an electric field

A method is presented by which one can systematically derive the types of terms, with regard to rotational as well as electric-field dependence and vibronic selection rules, that can occur in the effective Hamiltonian for a symmetric-top molecule placed in a homogeneous electric field. Rotation and molecular point groups somewhat modified from the ordinary ones are used for the derivation. The method is applied to the case of a C_3v molecule as an example, yielding a list of the operators occurring in the effective Hamiltonian with an electric-field dependence up to the second order.     

Torsion—rotation interactions in monodeuterated acetaldehyde

This paper presents an analysis of previously observed rotational transitions up to J = 17 of the sym and asym rotamers of monodeuterated acetaldehyde using a three-level Hamiltonian incorporating connections between rotational states of the v_sym = 0, v_asym = 0⁺ and v_asym = 0 ^? torsional energy level manifolds. Rotational transitions observed to be significantly perturbed in earlier studies, where the spectrum was fitted using a single-level Hamiltonian for the v_sym = 0 torsional state and a separate two-level Hamiltonian for the non-rigid asym states, are now fitted satisfactorily. It is confirmed that the perturbations detected in the earlier studies and tentatively attributed there to torsion-rotation interactions between the energetically distinct sym and asym manifolds do indeed arise as a consequence of such interactions. Rotational and quartic centrifugal distortion constants are given for each torsional state, together with torsion-rotation coupling constants primarily responsible for the perturbations noted in the earlier studies. Tunnelling between the energetically distinct sym and asym potential wells of the torsional potential energy function is also discussed.     

Development of the Hamiltonian and transition moment operators of symmetric top molecules using the O (3) ⊃ C∞v ⊃ C3v group chain

We present a development of the Hamiltonian, dipole moment, and polarizability operators for XY₃Z molecules. These rovibrational operators are written with the aid of a tensorial formalism derived from the one already used in Dijon and adapted to the XY₃Z symmetric tops in a recent paper [A. El Hilali, V. Boudon, M. Loëte, J. Mol. Spectrosc. 234 (2005) 166-174]. We use the O (3) ⊃ C_∞v ⊃ C_3v group chain. Expressions for the matrix elements are derived for these operators.     

Second-order corrections to correlations in muon capture

Muon capture by a nucleus with an arbitrary spin is considered. Second-order terms in 1/M in the effective weak-interaction Hamiltonian are taken into account. New terms in the Hamiltonian associated with the nucleon-nucleus potential are found. A general expression for the angular distribution of neutrinos (recoil nuclei) is derived for polarized muons and oriented target nuclei. Second-order contributions to the amplitudes M _u (k) are obtained. This allows one to calculate second-order corrections to any integral and correlation characteristics in muon capture that are expressed in terms of M _u (k).     

The rα structure,chemical shift anisotropy,and the abnormal orientation of methyldichlorophosphine from the NMR spectrum

NMR studies of methyldichlorophosphine have been undertaken in the nematic phase of mixed liquid crystals of opposite diamagnetic anisotropies. The r_α structure is derived. The proton chemical-shift anisotropy has been determined from the studies without the use of a reference compound and without a change of experimental conditions. It is shown that the molecule orients in the liquid crystal with positive diamagnetic anisotropy in such a way that the C₃ symmetry axis of the CH₃P moiety is preferentially aligned perpendicular to the direction of the magnetic field, unlike other similar systems. This is interpreted in terms of the formation of a weak solvent-solute molecular complex. The heteronuclear indirect spin-spin coupling constants are determined. The sign of the two-bond J_PH is found to be positive.