Transformation of the effective rotational Hamiltonian of nonrigid X 2 Y molecules

The transformation of the effective rotational Hamiltonian H of nonrigid X ₂ Y molecules to the form having a minimum number of diagonals in the basis of rotational functions of a symmetric top is discussed. Such a transformation is a generalization of the reduction transformation performed for the polynomial effective Hamiltonian H. It is shown that in the general case the transformation substantially changes the form of the initial Hamiltonian, which restricts the region of applicability (J<J*) of the reduced Hamiltonian represented in a class of elementary functions in terms of angular momentum operators. The values of the rotational quantum number J* are estimated for the (000) ground and (010) vibrational states of the H₂O molecule.     

A symmetry-based approach to the effective Hamiltonian for symmetric top molecules in degenerate vibronic states

The effective Hamiltonian for a symmetric top molecule in a degenerate vibronic state is obtained. Included in this Hamiltonian are the rotational, spin-rotational, spin-orbit coupling and electronic spin-spin interactions. The terms of the Hamiltonian are expressed as the product of molecular ‘constants,’ rotational angular momentum operators, and symmetry operators. A formalism is derived, and tables included, to determine whether or not a symmetry operator vanishes for a given vibronic state of a particular molecular symmetry. In this way, one can easily obtain all the non-vanishing Hamiltonian terms for a particular application.     

The two-dimensional anharmonic oscillator: Vibration-rotation constants of fulminic acid,HCNO

The lowest-wavenumber vibration of HCNO and DCNO, ν₅, is known to involve a largeamplitude low-frequency anharmonic bending of the CH bond against the CNO frame. In this paper the anomalous vibrational dependence of the observed rotational constants B(v₅, l₅), and of the observed l-doubling interactions, is interpreted according to a simple effective vibration-rotation Hamiltonian in which the appropriate vibrational operators are averaged in an anharmonic potential surface over the normal coordinates (Q_5x, Q_5y). All of the data on both isotopes are interpreted according to a single potential surface having a minimum energy at a slightly bent configuration of the HCN angle (～170°) with a maximum at the linear configuration about 2 cm^?1 higher. The other coefficients in the Hamiltonian are also interpreted in terms of the structure and the harmonic and anharmonic force fields; the substitution structure at the “hypothetical linear configuration” determined in this way gives a CH bond length of 1.060 Å, in contrast to the value 1.027 Å determined from the ground-state rotational constants.We also discuss the difficulties in rationalizing our effective Hamiltonian in terms of more fundamental theory, as well as the success and limitations of its use in practice.     

Analysis of Coriolis perturbations in the high-resolution infrared spectra of arsine (AsH3)

Numerical methods for the analysis of high-resolution infrared spectra of symmetric top molecules perturbed by Coriolis interactions between degenerate and nondegenerate vibrational levels are discussed in the second order of approximation. Application to the high-resolution infrared spectra of the AsH₃ molecule in the region of the fundamentals ν₁, ν₃ and ν₂, ν₄ yields considerably improved values of the molecular constants of AsH₃, including the band origins rotational constants, Coriolis coupling constants, centrifugal distortion constants, and the parameter of the K-type doubling effect.     

Coupling of vibration,rotation, and electron spin in multiplet states

The effective Hamiltonian of Van Vleck (1) for energy calculations of rotating molecules in multiplet states, is extended to include the effect of vibrational angular momentum, within a given nondegenerate electronic state. H_C and H_S operators are defined, to account for vibration-rotation (Coriolis) and spin-vibration interactions, respectively. Such interactions are discussed for degenerate E vibrational states of symmetric tops, as well as for the occurrence of vibrational angular momentum involving different vibrational states, with accidentally low energy separation. Following the classification of Hund's case (b) for the spin-rotation wave function (coupled representation), we find that H_C operators do not mix different F components, whereas H_S operators give also matrix elements between F components differing by one unit in the quantum number N, for given J.     

Third-order theory of the line intensities in the allowed and forbidden vibrational-rotational bands of C3v molecules

A third-order theory of the intensities of the allowed and “forbidden” (perturbation-allowed) transitions to the fundamental vibrational levels of C_3v semirigid molecules has been worked out by using the method of contact transformations applied to the electric dipole moment operator. Explicit expressions have been obtained for the linestrengths of the allowed (Δk = 0, ±1) as well as forbidden (Δk = ±2, ±3, ±4) transitions from the ground vibronic state to the fundamental vibrational levels of C_3v molecules. The treatment takes into account all the important Coriolis and anharmonic interactions in a C_3v molecule, including the effect of the “2, 2” and “2, −1” l-type interactions and the Δk = ±3 interactions on the intensities of the allowed and forbidden vibrational-rotational transitions. The expressions for the linestrengths of the allowed and forbidden transitions are given here in a form suitable to fit the experimental data on the intensities in the vibrational-rotational spectra of C_3v molecules.     

Rotational transitions in degenerate vibrational states of C3v symmetric top molecules,with application to CH3C15N

General algebraic expressions for the vibration-rotation energy levels and the associated rotational transitions of C_3v symmetric top molecules are developed. These expressions are presented in a convenient form for analysing the spectra of a molecule with any degree of excitation of a degenerate vibrational mode and have been applied to the spectrum of CH₃C¹⁵N.     

The vibrational Zeeman effect

The theory of the vibrational Zeeman effect in symmetric top molecules is presented. It is shown that vibrational g factors may be related to rotational g factors and estimates are made of their magnitude for a number of molecules; these calculations indicate that vibrational Zeeman effects should be observable in symmetric top molecules in degenerate vibrational states. In addition, the novel features of the theory for linear molecules are discussed.     

The microwave spectra of ortho-fluorotoluene with the asymmetric internal rotors CH2D and CD2H

The rotational spectra of αd₁- and αd₂-ortho-fluorotoluene in the ground state of the methyl group torsion have been measured. The evaluation of the spectra has been based on the theory for the internal rotation of an asymmetric internal top formulated earlier by several authors. The barrier potential being threefold symmetric (V₃), each torsional level consists of three nondegenerate substates, designated as sy and ±asy. The sy-state is assigned to the conformation with the unique methyl hydrogen isotope within the molecular heavy-atom plane (sy-rotamer), while the ±asy-states belong to the respective out-of-plane conformation (asy-rotamer). In the torsional ground state the level spacing between the ±asy substates is very small and numerous accidental close degeneracies are present between the rotational level systems based on these torsional substates. The rotational levels involved are strongly perturbed by the coupling between molecular overall rotation and internal rotation. Large deviations from a rigid rotor spectrum and (+) ? (?) intersystem (“tunneling”) transitions are observed. The spectrum of the asy-rotamer can be well reproduced by a “two-dimensional” Hamiltonian containing 11 “rotational constants,” 9 of which are determined by a fit to the spectrum. Several are sufficiently barrier-dependent to derive V₃. We obtain (in cal/mole) 567 ± 48 for αd₁-ortho-fluorotoluene, 711 ± 40 for the αd₂-isotope. The deviations from 649 cal/mole for the normal isotope are appreciable, probably indicating shortcomings of the semirigid model. The sy-rotamer presents a rigid rotor spectrum.     

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.     

Simultaneous analysis of the ν1, ν4, 2ν2, ν2 + ν5, and 2ν5 infrared bands of 12CH3F

The Fourier-transform spectrum of CH₃F from 2800 to 3100 cm^?1, obtained by Guelachvili in Orsay at a resolution of about 0.003 cm^?1, was analyzed. The effective Hamiltonian used contained all symmetry allowed interactions up to second order in the Amat-Nielsen classification, together with selected third-order terms, amongst the set of nine vibrational basis functions represented by the states ν₁(A₁), ν₄(E), 2ν₂(A₁), ν₂ + ν₅(E), 2ν₅⁰(A₁), and 2ν₅^±2(E). A number of strong Fermi and Coriolis resonances are involved. The vibrational Hamiltonian matrix was not factorized beyond the requirements of symmetry. A total of 59 molecular parameters were refined in a simultaneous least-squares analysis to over 1500 upper-state energy levels for J ≤ 20 with a standard deviation of 0.013 cm^?1. Although the standard deviation remains an order of magnitude greater than the precision of the measurements, this work breaks new ground in the simultaneous analysis of interacting symmetric top vibrational levels, in terms of the number of interacting vibrational states and the number of parameters in the Hamiltonian.     

The interacting states (020), (100), and (001) of H217O and H218O

A fit of about 350 rotational levels of the (020), (100), and (001) vibrational states has been performed for H₂¹⁷O as well as for H₂¹⁸O leading to the determination of 51 rotational and coupling constants for each isotopic species. The Fermi-type interaction and the two Coriolis-type interactions have been taken into account by appropriate rotation-vibration operators and the v-diagonal part of the Hamiltonian is, for each vibrational state, a Watson-type Hamiltonian. The results are very satisfactory since 87% of the experimental levels are reproduced within 15 × 10^?3 cm^?1.     

The rotational dependence of the Renner-Teller interaction: a new term in the effective Hamiltonian for linear triatomic molecules in Π electronic states

An additional term in the effective Hamiltonian for a molecule in a ^2S+1Π state subject to a small Renner—Teller effect is derived. It takes the form of a rotational dependence of the Renner—Teller operator and is assigned the parameter ?ω_2,D . This term reproduces two characteristics of the spin—rotational levels of Renner—Teller molecules, both of which are well documented by experimental examples but are not explicable in terms of the previous effective Hamiltonian. These are the small difference between the B values of different vibronic components of the same bending vibrational level and the anomalously strong dependence of the spin—rotation parameter γ on ν₂ for molecules with S > 0. These effects have been explained by previous workers using perturbation theory but in a less general and more complicated fashion.     

The influence of Coriolis-coupling on the partition function of spherical top molecules

Using Schwinger's operator perturbation theory for the partition function, it is shown that Coriolis-coupling can be neglected for spherical top molecules. The results contain the symmetric top as special case. Spherical top molecules of the XY₄-type are taken as numerical examples for a wide range of rotational constants and vibrational frequencies.     

Jahn-Teller and L-uncoupling effects on rotational energy levels of symmetric and spherical top molecules

The effects of a small Jahn-Teller effect and of Coriolis L-uncoupling forces on the rotational energy levels of a degenerate electronic state are considered. In general the two types of mechanism make similar contributions to the energy levels, and can only be separated by the use of additional information such as the isotopic variation of the molecular parameters. The theory of the rotational energies is formulated in a way that is applicable to both symmetric and spherical top molecules. The expressions derived for symmetric tops reproduce the results of previous workers. For E or F electronic states of spherical tops, the rotational parameters are equivalent to those employed for the fundamental levels of degenerate vibrations, but they must be interpreted in terms of the form of the appropriate double or triple potential energy surface. Parameters calculated for the 3d²F₂-3s²A₁ transition of the ND₄ molecule from the ab initio potential surface of Havriliak and King do not agree well with the experimental values of Herzberg and Hougen.     

Critical phenomena in the rotational spectra of H2X molecules

Using the method of generating functions for the effective rotational Hamiltonian of H₂X molecules, an expression for estimating the critical value J _c of the rotational quantum number is obtained, starting with which the formation of four-level clusters in a molecule is possible. The analysis made in the harmonic approximation for rotational operators and for a particular form of generating functions showed the impossibility of clusterization for nonrigid H₂X molecules with strong centrifugal effects.     

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.     

Ab initio ro-vibrational Hamiltonian in irreducible tensor formalism: a method for computing energy levels from potential energy surfaces for symmetric-top molecules

A theoretical approach to study ro-vibrational molecular states from a full nuclear Hamiltonian expressed in terms of normal-mode irreducible tensor operators is presented for the first time. Each term of the Hamiltonian expansion can thus be cast in the tensor form in a systematic way using the formalism of ladder operators. Pyramidal XY₃ molecules appear to be good candidates to validate this approach which allows taking advantage of the symmetry properties when doubly degenerate vibrational modes are considered. Examples of applications will be given for PH₃ where variational calculations have been carried out from our recent potential energy surface [Nikitin et al., J. Chem. Phys. 130, 244312 (2009)].