Perimetric scale-shape coordinates for triatomic molecules

Perimetric nuclear coordinates of a triatomic molecule treat all three nuclei equivalently and are not subject to the triangle conditions. Through an appropriate orthogonal transformation they can be separated into one scale coordinate, viz., the circumference, and two shape coordinates, which are determined by the angles. The parameter space of the shape coordinates is an equilateral triangle. The basic formulas are given and the relationship between points in coordinate space and molecular shapes are elucidated. Received: 10 January 1996 / Accepted: 2 January 1997     

A spline-fitted potential surface for bent triatomic molecules

The rotated-Morse curve cubic spline method developed previously is extended to bent triatomic molecules by using 2D cubic spline. The Yates—Lester potential for bent H₃ is shown to be accurately fitted over the entire range of the three internal coordinates, with a standard deviation of less than 1 kcal mol^?1. The spline method compares favorably in computational speed with the analytical potential.     

Rovibrational interactions in linear triatomic molecules: a theoretical study in curvilinear vibrational coordinates

Variational solution of the rovibrational problem in curvilinear vibrational coordinates has been implemented and used to investigate the nuclear motions in several linear triatomic molecules, like HCN, OCS, and HCP. The dependence of the rovibrational energy levels on the rotational quantum numbers and the l-doubling has been studied. Two approximations to the rovibrational Hamiltonian have been examined, depending on the level of truncation of the potential energy operator. It turns out that the truncation after the fifth order in the potential is sufficient to produce vibrational energies of high accuracy. An interesting feature of the present formulation of the problem in terms of the curvilinear vibrational coordinates is the explanation of the l-doubling of the rovibrational levels, which in this picture is interpreted as the result of the inequivalency of the average rotational constants in mutually perpendicular planes, rather than as the effect of the Coriolis-type interactions between the vibrational and rotational motions. The present theoretical results are compared with the available experimental data from high-resolution spectroscopy, as well as with other ab initio calculations.     

A general treatment of vibration-rotation coordinates for triatomic molecules

An exact, within the Born–Oppenheimer approximation, body-fixed Hamiltonian for the nuclear motions of a triatomic system is presented. This Hamiltonian is expressed in terms of two arbitrarily defined internal distances and the angle between them. The body-fixed axis system is related to these coordinates in a general fashion. Problems with singularities and the domain of the Hamiltonian are discussed using specific examples of axis embedding. A number of commonly used coordinate systems including Jacobi, bond-length-bond-angle, and Radau coordinates are special cases of this Hamiltonian. Sample calculations on the H₂S molecule are presented using all these and other coordinate systems. The possibility of using this Hamiltonian for reactive scattering calculations is also discussed.     

Routes to vibrational chaos in triatomic molecules

Analysis of both classical and quantum calculations on LiNC and LiCN shows the onset of vibrational chaos is closely associated with the degree of bending excitation. Conversely quasiperiodic stretching states persist above the barrier to isomerization. Classical studies on O₃ give similar results. In the light of these results we re-interpret the high-energy vibrational data on O₃ and HCN and suggest that the observed regular stretching states probably are embedded in the chaotic region. We discuss the importance of mode coupling by the potential.     

Hypervirial SCF treatment for vibrational energy levels of triatomic molecules

The SCF method for the calculation of energy levels of triatomic molecules is applied using the hypervirial perturbative procedure for solving the coupled equations. This treatment allows to obtain in a recursive way the energy corrections and the expectation values required in the SCF treatment, avoiding the explicit calculation of the wave functions. A numerical application is made to the SO₂ and O₃ molecules, comparing our results with those obtained by other methods.     

Characterizing vibrational motion beyond internal coordinates

We present a procedure for the decomposition of the normal modes of a composite system, including its rotations and translations, into those of fragments. The method permits—by the cross-contraction of dyads of mass-weighted displacement vectors, without recourse to valence coordinates—the direct comparison of nuclear motions of structurally similar but otherwise arbitrary fragments of molecules, and it leads to a quantitative definition of the similarity and the overlap of nuclear motions. We illustrate its usefulness by the quantification of the mixing of the normal modes of formic acid monomers upon the formation of a dimer, by the comparison of the overlap of the intermolecular normal vibrations of the water dimer computed with different ab initio schemes, and by the comparison of similarity and overlap of vibrations of (4S,7R)-galaxolide and (4S)-4-methylisochromane. The approach is expected to become a standard tool in vibrational analysis.     

Variational calculation of vibrational energies of triatomic molecules using SCF optimized modes

A self-consistent field optimization of the vibrational coordinates for nonlinear triatomic molecules is presented. The optimal coordinates are obtained by making a three-dimensional rotational transformation of the normal modes and determining the rotation angles as those for which the SCF energy is stationary. The utility of the optimized coordinates in full variational calculations of vibrational energies is studied for the molecules of H₂O, O₃, H₂D⁺, H₂T⁺, and D₂T⁺. For H₂O and O₃, the optimization procedure leads to the local mode representation. It is shown that the use of the optimal coordinates in variational calculations allows a large reduction of the dimension of the Hamiltonian matrix to be diagonalized in order to reach convergence.     

A vibrational Hamiltonian model for triatomic molecules based on the Kratzer and Poschl Teller potentials

A Hamiltonian model to describe molecular vibrations of triatomic molecules is proposed. The Hamiltonian is based on the use of the Kratzer potential variable for the stretching motions and a perturbed Poschl Teller potential for the bending one. The perturbation and variational treatments to compute the vibrational energies of this Hamiltonian can be developed using a zero-order system that includes part of the couplings between the stretching and bending motions. All the matrix elements involved in these calclations can be then evaluated in closed form. A numerical application to the HCN molecule is made. © 1994 John Wiley & Sons, Inc.     

Formulation of the vibrational theory in terms of redundant internal coordinates

Using the generalized inverse of matrices and the method of canonical matrices, this paper develops an unambiguous formulation of the theory of small vibrations of molecules which is valid when redundant internal coordinates are used. Such treatment includes the attainment of unambiguous relationships relating the compliance matrix (the generalized inverse of the canonical force constants matrix) with experimental data (vibrational frequencies, Coriolis coupling constants, centrifugal distortion constants and mean-square amplitudes). Moreover, expressions for the elements of the Jacobian matrices of the above magnitudes with respect to the compliance constants have been also obtained as well as some sum rule relationships.     

Application of adiabatic theory of vibrational predissociation to T-shaped triatomic van der Waals' molecules

An expression for the rate constant of the vibrational predissociation (VPD) of T-shaped triatomic van der Waals' (vdW) molecules is derived on the basis of the adiabatic separation between the high-frequency intramolecular and low-frequency vdW modes. The intramolecular and vdW modes are assumed to be characterized by the Morse-type interaction potentials. The dependence of the VPD line width on the intramolecular vibrational quantum number of a T-Shaped I₂-He vdW molecule is calculated by using the expression derived. The magnitudes of the calculated VPD line width are of the same order as those of the experimental. It is shown that the Condon approximation is insufficient and the non-Condon treatment is necessary to evaluate quantitatively the VPD rate constant within the adiabatic theory.     

Optimal internal coordinates, vibrational spectrum, and effective Hamiltonian for ozone

In this paper the authors use the optimal internal vibrational coordinates previously determined for the electronic ground state of the ozone molecule to study the vibrational spectrum of the molecule employing the second empirical potential energy surface calculated by Tyuterev et al. [Chem. Phys. Lett. 316, 271 (2000)]. First, the authors compute variationally all the bound vibrational energy levels of the molecule up to the dissociation limit and state the usefulness of the optimal coordinates in this respect, which allows us to converge all the bound levels using relatively small anharmonic basis sets. By analyzing the expansion coefficients of the wave functions, they show then that a large portion of the vibrational spectrum of O3 can be structured in nearly separable polyadic groups characterized by the polyad quantum number N=n1+n2+n(theta) corresponding to the optimal internal coordinates. Accordingly, they determine an internal effective vibrational Hamiltonian for O3 by fitting the effective Hamiltonian parameters to the experimental vibrational frequencies, using as input parameters in the fit those extracted from an analytical second-order Van Vleck perturbation theory calculation. It is finally shown that the internal effective Hamiltonian thus obtained accurately describes the vibrational spectrum of ozone in the low and medium energy regimes.     

Theoretical study of highly vibrational states of nonlinear triatomic molecules using Lie algebraic approach

The vibrational excitations of bent triatomic molecules are studied by using Lie algebra. The RMS error of fitting 30 spectroscopic data is 1.66 cm-1 for SO2. The results show that the expansion of a molecular algebraic Hamiltonian can well describe the experimental data. And the total vibrational levels can be calculated using this Hamiltonian. At the same time, the potential energy surface can also be obtained with the algebraic Hamiltonian.     

Simple electrostatic models for vibrating unsymmetrical triatomic molecules and triatomic ions

Bond charge, point-dipole models are used to derive simple universal relations,K =0.0435 (K₁₁ K ₂₂)^1/2 –0.0086 (K ₁₁ ⁺ K ₂₂), andK = 0.11 (K ₁₁ K ₂₂)^1/2 –0.0055 (K ₁₁ +K ₂₂), between the bond stretching force constantsK ₁₁ andK ₂₂ and bond bending force constantK for linear and bent unsymmetrical triatomic molecules respectively. The relations are shown to be approximately valid for a number of molecules. An extension of the models to include charged species (ions) in triatomic molecules is also presented and tested, giving good results.Based on part of a thesis submitted by José L. Gázquez in partial fulfillment of the requirements for the degree of Doctor of Philosophy, The Johns Hopkins University, 1976, and aided by research grants to the Johns Hopkins University and University of North Carolina from the National Institutes of Health and the National Science Foundation.     

Separation of the vibrational,rotational, and translational motions of polyatomic molecules using curvilinear vibrational coordinates

A new method is suggested for separating the vibrational, rotational, and translational motions of polyatomic molecules using curvilinear vibrational coordinates that are linear with respect to the natural vibrational coordinates. It is shown that, in this case, Coriolis interactions between the vibrational and rotational motions are absent. The solutions of the anharmonic vibrational-rotational problems in the curvilinear and linear vibrational coordinates are compared. The absence of Coriolis interactions between the vibrational and rotational motions in the curvilinear vibrational coordinates is proved numerically. The same conclusion is additionally supported by calculations of the anharmonic vibrational energy levels for the H₂O, H₂S, NO₂, SO₂, and ClO₂ molecules in the linear and curvilinear vibrational coordinates using the Hamiltonian designed in the curvilinear vibrational coordinates with and without Coriolis vibrational-rotational interactions. Volgograd Pedagogical University. Translated fromZhurnal Strukturnoi Khimii, Vol. 36, No. 2, pp. 239–254, March–April, 1995. Translated by I. Izvekova     

On the use of “effective harmonic potentials” and “effective normal coordinates” in computing the vibrational energy levels of polyatomic molecules

The purpose of this paper is to discuss the possibility of improving the physical significance of the internuclear potential function derived from the energy levels observed in rotation—vibration spectra. In performing the perturbation calculation, it might be efficient to use normal coordinates and zeroth-order harmonic hamiltonians which depend upon the range of vibrational energies or upon the values of the vibrational quantum numbers. A limited application of this method is made in the case of CO₂, in order to overcome the difficulties linked to the large value of the coefficient k₁₃₃ in the cubic anharmonic potential.     

Treating singularities present in the Sutcliffe-Tennyson vibrational Hamiltonian in orthogonal internal coordinates

Two methods are developed, when solving the related time-independent Schrodinger equation (TISE), to cope with the singular terms of the vibrational kinetic energy operator of a triatomic molecule given in orthogonal internal coordinates. The first method provides a mathematically correct treatment of all singular terms. The vibrational eigenfunctions are approximated by linear combinations of functions of a three-dimensional nondirect-product basis, where basis functions are formed by coupling Bessel-DVR functions, where DVR stands for discrete variable representation, depending on distance-type coordinates and Legendre polynomials depending on angle bending. In the second method one of the singular terms related to a distance-type coordinate, deemed to be unimportant for spectroscopic applications, is given no special treatment. Here the basis set is obtained by taking the direct product of a one-dimensional DVR basis with a two-dimensional nondirect-product basis, the latter formed by coupling Bessel-DVR functions and Legendre polynomials. With the basis functions defined, matrix representations of the TISE are set up and solved numerically to obtain the vibrational energy levels of H3+. The numerical calculations show that the first method treating all singularities is computationally inefficient, while the second method treating properly only the singularities having physical importance is quite efficient.