Terms Of Curvilinear Internal Coordinates Research Articles

Aspects of the Eckart frame ro-vibrational kinetic energy operator.

A most decisive part of developing the Eckart frame ro-vibrational kinetic energy operator (KEO) in terms of curvilinear internal coordinates is the introduction of vibrational displacements obeying the Eckart conditions. It is examined here, how such displacements may be constructed by orthogonal and nonorthogonal transformations, and how they can be employed in developing the KEO. In particular, it is shown analytically, by term by term comparison, that the Eckart KEOs obtained by employing Pickett's method and the gateway Hamiltonian method are equivalent. That is, although it may appear in the form of very different expressions, the general, exact ro-vibrational Eckart frame KEO is unique. As side result, it has been shown that the simple formula of the pseudo-potential derived by Watson for the exact KEO given in normal coordinates applies to any Eckart frame KEO, irrespective of the choice of the vibrational coordinates.

Automatic differentiation method for numerical construction of the rotational-vibrational Hamiltonian as a power series in the curvilinear internal coordinates using the Eckart frame.

We present a new numerical method to construct a rotational-vibrational Hamiltonian of a general polyatomic molecule in the Eckart frame as a power series expansion in terms of curvilinear internal coordinates. The expansion of the kinetic energy operator of an arbitrary order is obtained numerically using an automatic differentiation (AD) technique. The method is applicable to molecules of arbitrary size and structure and is flexible for choosing various types of internal coordinates. A new way of solving the Eckart-frame equations for curvilinear coordinates also based on the AD technique is presented. The resulting accuracy of the high-order expansion coefficients for the kinetic energy operator using our numerical technique is comparable to that obtained by symbolic differentiation, with the advantage of being faster and less demanding in memory. Examples for H2CO, NH3, PH3, and CH3Cl molecules demonstrate the advantages of the curvilinear internal coordinates and the Eckart molecular frame for accurate ro-vibrational calculations. Our results show that very high accuracy and quick convergence can be achieved even with moderate expansions if curvilinear coordinates are employed, which is important for applications involving large polyatomic molecules.

Eckart-Sayvetz conditions revisited.

It is shown that vibrational displacements satisfying the Eckart-Sayvetz conditions can be constructed by projection of unconstrained displacements. This result has a number of interesting direct and indirect ramifications: (i) The normal coordinates corresponding to an electronic state or an isotopologue of a molecule are transformed to those of another state or isotopologue by a linear and, in general, non-orthogonal transformation. (ii) Novel interpretation of axis switching. (iii) One may enhance the separation of rotational-large-amplitude internal motions and the vibrational motions beyond that offered by the standard use of the Eckart-Sayvetz conditions. (iv) The rotational-vibrational Hamiltonian given in terms of curvilinear internal coordinates may be derived with elementary mathematical tools while taking into account the Eckart conditions with or without enhancement.

Canonical force field theory in terms of curvilinear internal coordinates: Application to the methane molecule

This Letter generalizes the Kuczera’s theory of the vibrational canonical force field in order to obtain an unambiguous and uniquely defined expression for the anharmonic force field in terms of curvilinear internal coordinates. By using this generalization it has been shown mathematically that the canonical force constants could be transferred between molecules with the same geometrical structure. As illustration, the theory is used to obtain the expressions for the quadratic and cubic canonical force constants of the methane molecule.

A local–normal description of vibrational excitations of pyramidal molecules in terms of Morse oscillators

A general description of vibrational excitations of pyramidal molecules in both local and normal representations is presented. This study is restricted to the case when no tunneling motion is allowed. The Hamiltonian is first written in terms of curvilinear internal coordinates. The Wilson’s G matrix as well as the potential are expanded in terms of Morse variables, which allows the identification of a set of six Morse oscillators as zeroth-order Hamiltonian. An algebraic realization of the Hamiltonian is obtained by introducing a linear expansion of the coordinates and momenta in terms of creation and annihilation operators of Morse functions. This algebraic realization provides in natural form the representation of the Hamiltonian in terms of local interactions. The normal interactions are constructed by successive couplings of tensors defined as linear combinations of the ladder operators. The matrix transformation between the local and normal interactions is obtained for the complete Hamiltonian. This analysis provides the spectroscopic parameters in both local and normal schemes in explicit form as functions of the force constants and structure parameters. To exemplify, the analysis of the vibrational excitations of stibine and arsine is presented. Force constants as well as the corresponding x , K relations are given. A comparison with the results obtained using the U ( ν + 1 ) unitary group approach is included.

A local description of vibrational excitations of stibine in the framework of a unitary group approach

A local description of vibrational excitations of stibine in terms of the unitary group approach U( ν + 1) is presented. Based on the recent reformulation of this algebraic method the Hamiltonian is first expressed in the space of coordinates and momenta and thereafter translated into an algebraic realization. The Hamiltonian proposed is given in terms of curvilinear internal coordinates, where the stretching and bending modes are coupled with each other by Fermi resonances. In the new formulation a one to one correspondence with the Hamiltonian in configuration space is given, with the additional advantage that anharmonicities are taken into account from the outset. All experimental vibrational energies of 121SbH 3 have been reproduced well with an rms deviation of 1.27 cm −1. From the spectroscopic parameters the force constants are obtained. A discussion comparing our approach with previous results using the same model is also included.

Calculation of the τ dependence of the vibration–internal rotation–overall rotation interactions in CH3OH from molecular structure and molecular dynamics

The Guan and Quade theory for vibration-large-amplitude internal-motion-rotation interactions has been applied to the internal rotation problem in CH(3)OH. Through the molecular dynamics, the cos 3tau and sin 3tau dependence of the torsional-rotational coefficients in the effective Hamiltonian have been calculated from molecular structure. The internal rotation coordinate tau(') for the vibrationally distorted molecule is shown to have the necessary threefold symmetry for all values of tau('). For the methyl deformation modes, the vibrational dependence of the internal rotation potential energy is shown to have a threefold symmetry. The S(t) and S(t)S(t) dependence of the inertia tensor and Coriolis coupling coefficients has been developed in terms of curvilinear internal coordinates. The T transformation separating rotation from vibrations in zeroth order is then applied, the kinetic-energy tensor inverted to momentum space, and finally the effective torsion-rotation coefficients are calculated by Van Vleck perturbation theory. When compared to the empirical results, the kinetic-energy contributions to the cos 3tau and sin 3tau dependence of the coefficients are as follows: 54% of P(a)(2) is accounted for, 28% of P(a)P(b), 16% of P(a)P(c), and 91% of the asymmetry. The calculation is inadequate to account for the P(b)(2),P(c)(2), and P(b)P(c) coefficients, ranging from factors of 20-70, even with the incorrect sign for some of the terms. Anharmonic force contributions from the vibrations have not been used in the calculation since these forces are not known at this time.

Internal coordinate Hamiltonian model for Fermi resonances and local modes in methane

A vibrational model which is based on a Hamiltonian expressed in terms of curvilinear internal coordinates is applied to the overtone spectrum of methane, CH4. Symmetrized internal coordinates and their conjugate momenta are used as the bending variables. The stretching part of the Hamiltonian is expressed in an unsymmetrized form. Both the kinetic operator and the potential energy function are expanded as Taylor series around the equilibrium configuration. Symmetrized local mode basis functions for the stretches and symmetrized two- and three-dimensional harmonic oscillator basis functions in the Cartesian representations for bending degrees of freedom are used. Only resonance couplings are taken into account. Apart from some standard diagonal contributions harmonic oscillator matrix elements have been employed. This results in a simple block diagonal Hamiltonian model. The nonlinear least squares method is used to optimize model parameters for 12CH4. Observed vibrational term values up to 6050 cm−1 are included as data. Potential energy parameters obtained from the Hamiltonian parameters agree well with a previously published anharmonic force field calculation. A unitary transformation between internal coordinate and normal coordinate representations is found to provide simple interpretations for the standard normal mode theory based spectroscopic parameters.

Fermi resonances and local modes in pyramidal XH3 molecules: An application to arsine (AsH3) overtone spectra

A simple vibrational Hamiltonian expressed in terms of curvilinear internal coordinates has been used to model both stretching and bending vibrations in pyramidal XH3 molecules. The pure stretching part is expressed as harmonically coupled anharmonic oscillators and the pure bending part as harmonically coupled harmonic oscillators. The stretching and the bending modes are coupled with each other by Fermi resonance terms which contain contributions both from the kinetic and potential energy part of the full vibrational Hamiltonian. Only resonance couplings are included. The Hamiltonian matrices are symmetry factorized by employing symmetrized basis functions which consist of products of stretching oscillator and valence angle bending oscillator functions. This model is applied to observed vibrational term value data of arsine (AsH3). The least squares method is used to optimize potential energy parameters. The results obtained are in good agreement with ab initio calculations. All observed arsine overtone and combination bands have been assigned.

Reaction-path potential and vibrational frequencies in terms of curvilinear internal coordinates

We present a general formulation that allows physically intuitive curvilinear internal coordinates to be used for the calculation of potential energy expansions and generalized normal-mode vibrational frequencies in reaction-path calculations. The reaction path is defined, as usual, as the minimum-energy path in the mass-scaled Cartesian coordinate system, and curvilinear coordinates are used for vibrational frequency calculations at nonstationary points. The method is well adapted for use in variational transition state theory with semiclassical multidimensional tunneling (VTST/MT) approximations to calculate thermal rate constants. We present VTST/MT calculations for five reactions, H+H2→H2+H, O+H2→OH+H, CH3+H2→CH4+H, H+O2→HO2, and Cl+HBr→HCl+Br, to illustrate the use of the new curvilinear coordinates, and we compare the results to calculations employing rectilinear coordinates. We make detailed comparisons not only of the calculated rate constants but also of the vibrationally adiabatic ground-state potential energy curves and bound-state vibrational frequencies as functions of the reaction coordinate.

The rotation–vibration coupling equations for polyatomic molecules in internal coordinates

AbstractAn exact vibration–rotation kinetic energy operator for polyatomic molecules has been obtained on the basis of Sutcliffe's method, in terms of curvilinear internal coordinates and rotational angular moment operators. This operator is derived from the kinetic energy operator in Cartesian coordinates by the successive transformations using the chain rule. This kinetic energy operator can be used not only for the system of any triatomic and tetraatomic molecules and common polyatomic molecules in chemistry, but also for the investigation of the collision problems between two molecules after some modifications. Finally, using this Hamiltonian, the rotation–vibration coupling equations of polyatomic molecules have been derived and discussed. © 1992 John Wiley & Sons, Inc.