Vibration-inversion-rotation spectra of ammonia: A vibration-inversion-rotation Hamiltonian for NH2D and ND2H

The model Hamiltonian developed previously for ammonia NH₃ has been used to study the vibration-inversion-rotation energy levels of the isotopic species of ammonia NH₂D and ND₂H. In this model the inversion motion is removed from the vibrational problem by allowing the molecular reference configuration to be a function of the large amplitude motion coordinate.The ground state inversion-rotation energy levels of NH₂D and ND₂H have been calculated with the use of the zeroth-order inversion-rotation Hamiltonian, and the calculated transition frequencies have been compared with the experimental data.     

Hamiltonian description and 6D calculations on the ammonia vibrational levels

In this work, a Hamiltonian formalism and a 6D vibrational calculation procedure is described and implemented, designed for the exploration of vibrational motion in ammonia (and any XH₃ molecule). The 6D potential energy surface of ammonia was modelled in simple analytical form (including the inversion potential) at the planar, totally symmetric (D_3h) reference configuration. Using the described method (which is an adaptation of the formalism, previously developed and applied to benzene), 6D calculations were carried out on the vibrational level system of ammonia ¹⁴NH₃, at the lower levels of vibrational excitation. On the basis of the satisfactory agreement between the calculated and the experimentally measured vibrational frequencies, the values of some important harmonic and anharmonic force constants, characterizing the ammonia PES were determined.     

The torsional-wagging tunneling problem and the torsional-wagging-rotational problem in methylamine

A theoretical formalism is presented for fitting rotational energy levels in isolated (unperturbed) vibrational states of methylamine. This formalism is obtained by recasting and extending theoretical studies in the earlier literature, which were undertaken to help analyze the methylamine microwave spectrum. The present formalism is applicable when both the NH₂ umbrella (wagging) motion and the CH₃ internal rotation (torsion) motion take place near the high-barrier limit and leads to the usual Fourier sine and cosine series expansions for molecular energy levels. The derivation is separated into two parts, one treating the large-amplitude vibrational problem (the torsional-wagging problem) by itself, the other treating the torsional-wagging-rotational problem. In both treatments, permutation-inversion group and extended group ideas are used to determine the allowed terms in an effective rotational-tunneling Hamiltonian operator and to block diagonalize the matrix representation of this operator for a near-prolate symmetric top. The resulting energy levels and selection rules are discussed, but application of the method in detail to the methylamine spectrum is planned for a later paper.     

Analysis of high temperature ammonia spectra from 780 to 2100 cm

A recently-recorded set [Hargreaves et al., Astrophys. J., in press] of Fourier transform emission spectra of hot ammonia is analyzed using a variational line list. Approximately 3350 lines are newly assigned to mainly hot bands from vibrational states as high as v₂ = 2. 431 new energy levels of these states are experimentally determined, considerably extending the range of known rotationally-excited states. Comparisons with a recent study of high J levels in the ground and first vibrational states [Yu et al., J. Chem. Phys., 133 (2010) 174317] suggests that while the line assignments presented in that work are correct, their energy level predictions suffer from problems associated with the use of very high-order perturbation series in the effective Hamiltonian. It is suggested that variational calculations provide a more stable method for analyzing spectra involving highly-excited states of ammonia.     

Vibrational spectroscopy of SnBr4 and CCl4 using Lie algebraic approach

The stretching and bending vibrational energies of SnBr₄ and CCl₄ are calculated in the one-dimensional framework. The dynamical symmetry group of tetrahedral molecule was taken into consideration to construct the model Hamiltonian in this framework. Casimir and Majorana invariant operators were also determined accordingly. Using the model Hamiltonian so constructed, we reported the vibrational energy levels of SnBr₄ and CCl₄ molecules accurately.      

VPT2+K spectroscopic constants and matrix elements of the transformed vibrational Hamiltonian of a polyatomic molecule with resonances using Van Vleck perturbation theory

Vibrational levels of polyatomic molecules are analysed with Van Vleck perturbation theory to connect experimental energy levels to computed molecular potential energy surfaces. Vibrational matrix elements are calculated from a quartic potential function via second-order Van Vleck perturbation theory, a procedure that treats both weak and strong interactions among vibrational states by approximately block-diagonalising the vibrational Hamiltonian. A clear and complete derivation of anharmonic and resonance constants as well as general expressions for both on- and off-diagonal matrix elements of the transformed Hamiltonian is presented. The equations are written in partial fraction form and as a constant multiplied by a harmonic oscillator matrix element to facilitate removing the effect of strongly interacting resonant states both in analytical formulae and in computer code. The derived equations are validated numerically, and results for the isotopomers of formaldehyde (H₂CO, HDCO, D₂CO) are included. The implications of the equations on zero-point energy calculations and experimental fits are discussed. The VPT2+K method is defined by these results for use in fitting and calculating vibrational energy levels.     

Spectroscopy and dynamics of the isolated sp2 CH chromophore in trideuteroacetaldehyde CD3CHO as derived from extrapolated SDCI ab initio calculations

Ab initio CI potential energy (PES) and dipole moment (DMS) surfaces have been calculated with singles and doubles excitation configuration interaction (SDCI) for the 2-dimensional isolated sp² CH chromophore subspace of trideuteroacetaldehyde (CD₃CHO). Different extrapolation schemes to full-CI have been applied after the extrapolation to full-SDCI. Vibrational band centres and absolute intensities are determined variationally on six extrapolated surfaces. The band centres are analysed within the effective Hamiltonian model for isolated CH chromophores and the effective spectroscopic parameters are compared for the different extrapolation schemes. Time dependent population evolution for vibrational quantum motion with a pure CH stretching state being initially populated is calculated for some of the extrapolated surfaces within the effective Hamiltonian model.     

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.     

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).     

The vibrational energy pattern in ethylene (12C2H4)

All known vibrational energies in ¹²C₂H₄([Xtilde]¹ A_g ) are collected. A block-diagonalized vibrational matrix Hamiltonian is set up, combining a Dunham expansion with the 2/10,10, 11/2,12, and the 5,5/9,9, 1,1/11,11,1,9/5,11 and 1,11/5,9 anharmonic resonances, with the normal modes numbered according to the most usual conventions in the previous literature. A step by step fitting procedure is applied. Vibrational constants are obtained, and vibrational assignments are provided and discussed. A slightly modified ensemble of vibrational constants is also produced by taking care of a more extended set of resonances between the CH-type vibrations. A new constant of the motion N = 3ν₁ + 3ν₅ + 3ν₉ + 3ν₁₁ + 2ν₂ + ν₁₀ + ν₁₂ is demonstrated to support the latter vibrational energy pattern.     

Vibration-rotation interaction in HCNO caused by accidental resonances and enhanced by the quasilinearity of the molecule

The recent assignment of the vibrational spectrum of the quasilinear molecule HCNO revealed several near coincidences between vibrational energy levels involving the two bending modes, ν₄ (skeletal bending mode) and ν₅ (HCN-bending mode), and the lowest-lying stretching mode, ν₃ (NO stretching mode). By considering the correlation between the energy levels of a linear and a bent molecular model of HCNO, it is seen that resonance interactions which would be of third or higher order in a linear molecule Hamiltonian would be of first or second order in the Hamiltonian of a bent molecule, and thus might be significant in the quasilinear molecule HCNO. In this way we were able to identify the type of observed interaction occurring between three pairs of nearly coincident levels, (00010, 00002), (00020, 00012), and (00100, 00004). Anomalous centrifugal distortion effects had been observed and reported earlier for the pure rotational transitions arising from molecules in the 00010, 00020, and 00002 levels. Rotational transitions arising from molecules in the 00004 and 00100 vibrational states of HCNO and the 00100 state of DCNO are reported here for the first time. For two pairs of levels, (00010, 00002) and (00100, 00004), we could determine the magnitude of the coefficients of the interaction matrix elements from an analysis of the centrifugal distortion effects.     

A General Variational Algorithm to Calculate Vibrational Energy Levels of Tetraatomic Molecules

A general variational method for calculating vibrational energy levels of tetraatomic molecules is presented. The quantum mechanical Hamiltonian of the system is expressed in a set of coordinates defined by three orthogonalized vectors in the body-fixed frame without any dynamical approximation. The eigenvalue problem is solved by a Lanczos iterative diagonalization algorithm, which requires the evaluation of the action of the Hamiltonian operator on a vector. The Lanczos recursion is carried out in a mixed grid/basis set, i.e., a direct product discrete variable representation (DVR) for the radial coordinates and a nondirect product finite basis representation (FBR) for the angular coordinates. The action of the potential energy operator on a vector is accomplished via a pseudo-spectral transform method. Six types of orthogonal coordinates are implemented in this algorithm, which is capable of describing most four-atom systems with small and/or large amplitude vibrational motions. Its application to the molecules H₂CO, NH₃, and HOOH and the van der Waals cluster He₂Cl₂ is discussed.     

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 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.     

采用二维坐标和变分方法计算磷烷阳离子的伞型振动

提出一个新的二维变分方法计算PH₃⁺(X²A₂")的对称伸缩振动(v1)和伞形振动(v2). 因为采用了对称化的笛卡尔坐标,所以动能项变得简单,同时伞形振动模式也能得到很好的反映. 相比采用经常使用的一维模型计算伞形振动,这个二维模型不需要约化质量的假设,同时也考虑了v1和v2振动模式之间的相互作用. 用二维模型对PH₃⁺首次进行了计算, 前七个能级的理论值和实验值的平均相对误差小于3 cm^{-1. 用相同的方法也计算了NH₃,结果没有PH₃+理想,说明这个方法有一定的局限性.}     

A reduced form of the spin-rotation Hamiltonian for asymmetric-top molecules,with applications to HO2 and NH2

The spin-rotational Hamiltonian for an asymmetric-top molecule in a given vibrational level of an open-shell electronic state may contain more parameters than can be determined from the observed energy levels. This paper describes the reduction of the Hamiltonian by means of a unitary transformation to a form suitable for fitting to observed energies. It is shown that, for molecules of lower than orthorhombic symmetry, there are fewer determinable quadratic spin-rotation parameters than have been used previously. For example, for a molecule belonging to the group C₈, there are four, not five, determinable spin-rotation constants, ?_αβ. Similar indeterminacies exist among the quartic terms of the spin-rotation Hamiltonian. The case of a molecule of orthorhombic symmetry, for which there are six determinable quartic parameters, is considered in detail. The results are applied to the experimental data available on the spin-rotation splittings of the HO₂ and NH₂ radicals in their ground vibrational and electronic states.     

A rotational Hamiltonian for the ground vibrational state of hydrazine

A Hamiltonian matrix suitable for fitting rotational energy levels of the hydrazine molecule in its ground vibrational and electronic states was obtained. This matrix is of dimension 16 × 16, where the 16 functions labeling the rows and columns consist of the two members of a near-prolate asymmetric rotor doublet (with given | K_a |) for the eight different, but chemically equivalent, conformers which the molecule can reach by various combinations of -NH₂ inversion at either end of the molecule, and internal rotation about the NN bond. The matrix is derived in a phenomenological fashion, by applying group theoretical arguments to a model in which tunneling among the various frameworks is assumed to be very slow compared with the vibrational frequencies. A comprehensive treatment of the large-amplitude vibrational potential surfaces and associated tunneling pathways has not been carried out, nor have quartic (J⁴) centrifugal distortion effects been considered in a systematic fashion. Preliminary fits indicate that the model developed can be used to fit the hydrazine microwave data in a consistent fashion, and a full treatment of such data has been undertaken.     

A general rigorous quantum dynamics algorithm to calculate vibrational energy levels of pentaatomic molecules

An exact variational algorithm is presented for calculating vibrational energy levels of pentaatomic molecules without any dynamical approximation. The quantum mechanical Hamiltonian of the system is expressed in a set of orthogonal coordinates defined by four scattering vectors in the body-fixed frame. The eigenvalue problem is solved using a two-layer Lanczos iterative diagonalization method in a mixed grid/basis set. A direct product potential-optimized discrete variable representation (PO-DVR) basis is used for the radial coordinates while a non-direct product finite basis representation (FBR) is employed for the angular variables. The two-layer Lanczos method requires only the actions of the Hamiltonian operator on the Lanczos vectors, where the potential-vector products are accomplished via a pseudo-spectral transform technique. By using Jacobi, Radau and orthogonal satellite vectors, we have proposed 21 types of orthogonal coordinate systems so that the algorithm is capable of describing most five-atom systems with small and/or large amplitude vibrational motions. Finally, an universal program (PetroVib) has been developed. Its applications to the molecules , and the van der Waals cluster He₃Cl₂ are also discussed.     

The microwave spectrum of germyl isocyanate,a quasi-symmetric top

The microwave spectrum of germyl isocyanate in the K, P, and R bands (12.4–40.0 GHz) has been investigated using natural isotopic samples and samples containing specific germanium isotopes, mass numbers 72 and 74, and samples containing GeD₃ groups. The very complex spectrum is analyzed as that of a quasi-symmetric top, in which the skeletal bending motion has an energy maximum at the linear configuration; the height of the maximum is some 339 cm⁻¹. Many vibrational states of the two-dimensional bend lie close in energy, resulting in strong interactions between rotational levels of different vibrational states. The program used for the analysis has been used previously to account for similar spectra of methyl and silyl isocyanates and methyl isothiocyanate; the behavior of the germyl compound is more like that of the methyl isocyanate than that of the silyl compound, where the height of the maximum is much less. Although we did not include centrifugal distortion operators in the Hamiltonian the program successfully predicts k = 0 line positions in J + 1 ← J bunches with J up to 9, using only lines in the J = 3 bunch to define the parameters of the model, which include the height of the hump, the bend angle of the skeleton at the energy minimum (142.18(5)°), the force constant for the bending motion at the minimum (0.1094(7) mdyn Å), and the GeN bondlength at equilibrium (1.8257(5) Å). The threefold barrier to internal rotation of the GeH₃ group at the equilibrium position is about 3 cm⁻¹, much lower than the value found for methyl isocyanate. The fit to the observations, about 100 lines in the 4 ← 3 bunch, is improved by allowing for a variation in the GeN bondlength as the bend angle changes. The quality of the least-squares fit corresponds to an estimated standard deviation for an observation of about 2 MHz, but not all observed lines can be accounted for precisely, showing that additional interactions are involved that are not yet included in the model.     

Variational calculation of low-lying and excited vibrational levels of the water molecule

The variational method of calculating vibrational energy levels for triatomic molecules has been extended to determine the lowest 22 levels for H₂O, using an analytic form for the potential surface developed by Sorbie and Murrell. In addition, complete ab initio calculations of the fundamental frequencies for H₂O have been carried out using extensive configuration interaction forms for the potential. They are predicted to be 3720, 1629, and 3807 cm^?1, to be compared with experimental values of 3656, 1597, and 3754 cm^?1. Also discussed is how the variational method may be extended to allow the calculation of vibrational energy levels for larger molecules.