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.     

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 Theorems and SCF Wavefunctions for Coupled Oscillators

The SCF formalism is analysed in connection with hypervirial theorems. Coupled oscillator models are chosen in order to illustrate the formal results.     

On the use of optimal internal vibrational coordinates for symmetrical bent triatomic molecules

The use of generalized internal coordinates for the variational calculation of excited vibrational states of symmetrical bent triatomic molecules is considered with applications to the SO2, O3, NO2, and H2O molecules. These coordinates depend on two external parameters which can be properly optimized. We propose a simple analytical method to determine the optimal internal coordinates for this kind of molecules based on the minimization with respect to the external parameters of the zero-point energy, assuming only quadratic terms in the Hamiltonian and no quadratic coupling between the optimal coordinates. The optimal values of the parameters thus obtained are shown to agree quite well with those that minimize the sum of a number of unconverged energies of the lowest vibrational states, computed variationally using a small basis function set. The unconverged variational calculation uses a basis set consisting of the eigenfunctions of the uncoupled anharmonic internal coordinate Hamiltonian. Variational calculations of the excited vibrational states for the four molecules considered carried out with an increasing number of basis functions, also evidence the excellent convergence properties of the optimal internal coordinates versus those provided by other normal and local coordinate systems.     

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.     

Perturbative virtual SCF CI treatment for energy levels of coupled oscillator systems

A perturbative SCF CI treatment to obtain energy levels of coupled oscillator systems is proposed. The method uses the virtual SCF basis set, and the SCF equations are solved by means of a perturbative treatment that provides the diagonal matrix elements involved in the CI calculation. The off-diagonal matrix elements are calculated using a commutation relationship derived from exact quantum theorems. Numerical results for several systems are obtained and compared with those from others SCF, SCF CI , and variational treatments.     

Potential energy surface for linear triatomic molecules: An algebraic method

Recently, we proposed a new transformation between the angle of canonical coordinates and the bond angle to describe the bending motion in Potential Energy Surfaces (PES) of bent triatomic molecules. In this work we extend the transformation to include linear triatomic molecules. Results for the linear triatomic molecule N₂O are reported.     

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.     

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.     

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.     

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.     

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     

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.     

Calculation of mean-square vibrational amplitudes and shrinkage for linear triatomic molecules of AB2 type

Formulae for the calculation of shrinkage and mean-square amplitudes for linear molecules of AB₂ type have been derived without the use of small harmonic vibration theory. The probability density of the internuclear distance distribution being calculated with allowance for the rotation—vibration interaction. The formulae obtained agree, within experimental error, with electron diffraction data for molecules characterized by large displacements of the nuclei from their equilibrium positions. These data cannot be described by the relations usually employed in gas electron-diffraction investigations.     

Density of levels in vibrational spectra of molecules

Density distribution of the discrete spectrum of a Hamiltonian which represents a system of N-coupled oscillators and, hence may describe molecular vibrations in the local mode approximation, is analyzed. The spectral density moments are expressed as linear combinations of products of coefficients which depend on the molecular topology (analogs of the propagation coefficients in the statistical theory of nuclear and atomic spectra) and of one-particle moments describing individual bonds and interactions between them. The dependence of the first three moments of the energy-level density on the structural parameters of the molecule is discussed. Detailed expressions for several special cases are derived. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63 : 835–842, 1997     

Semiclassical three-dimensional model for vibrational energy transfer in diatomic molecules

A semiclassical model for calculation of rate constants for vibrational excitation in diatomic gases at low temperatures (below 1000 K) is suggested. The model has been tested by its ability to predict the relaxation times of hydrogen (τ_H1 in the temperature region 40–1000 K. The agreement with experimental values is excellent. The isotopic ratio τ_D2/τ_H2 as a function of temperature is predicted.     

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.