Linear Triatomic Molecules Research Articles

Cartesian Dynamics of Simple Molecules II Non-Centrosymmetric Linear Triatomics

Abstract A simple spring model for molecular vibrations, which uses Cartesian co-ordinates for both longitudinal and transverse displacements, is applied to non-centrosymmetric linear triatomic molecules. Analytical expressions for the stretching and bending mode frequencies are obtained, which are equivalent to those derived by conventional methods. For most molecules, the effects of the interaction between the outside atoms are shown to be negligible, but for N2O, complex solutions are obtained unless this is included. The validity of the model is demonstrated by the satisfactory agreement between calculated and observed frequencies of isotopic species. For N2O, frequency shifts resulting from isotopic substitution are explained by reference to the calculated eigenvectors.

The use of scaled moments of inertia in experimental structure determinations of polyatomic molecules

Recently developed procedures for determining near-r e structures (r m p )from scaled ground-state moment of inertia data have been extended to general polyatomic molecules. A simple and general empirical method for correctingl m p moments for the large vibration-rotation effects of deuterium substitution has been proposed and illustrated for a variety of polyatomic molecules. The correction requires utilization of a bond elongation parameter,δr D ,whose experimental value is found to be approximately 0.003 A. A theoretical justification of the approximation has been provided for linear triatomic molecules.

Interfragment effect in the predissociative excitation of linear triatomic molecules

A collinear model is used to estimate the vibrational interfragment effect for the predissociation of CO 2 and COS leading to the formation of CO(A), in comparison with the results from the predissociative excitation processes of CS 2 and N 2O. From the systematic trends observed, the interfragment effect was found to depend primarily on the change of bond length in the diatomic oscillator and on the strength of the dissociating bond.

Rotational dependence of Fermi resonance in linear molecules

Formulas in terms of basic molecular parameters are presented for the constants δ stt′ and δ′ stt′ that describe the rotational dependence of the Fermi interaction between a stretching fundamental ω s and an overtone 2 ω t ( t = t′) or a combination ω t + ω t′ of bending modes of a linear molecule. Numerical comparisons are made with existing calculated or observed values of δ 122 and δ′ 122 for several linear triatomic molecules.

An electronic-vibrational interpretation of algebraic hamiltonians

A physical model of diatomic and linear triatomic molecules based on vibrational and (some restricted) electronic degrees of freedom can be transformed into an effective, algebraic Hamiltonian, such that its parameters possess a clearly defined physical meaning: some parameters are interpreted geometrically, as minima on diabatic potential surfaces, others are related to coupling strengths between electronic and vibrational motions.

Electron-impact excitation of atoms in high-lying doubly excited states: Correlation effects in collision dynamics.

Based on the hyperspherical approach, we have calculated the generalized oscillator strengths and the electron-impact-excitation Born cross sections from the initial $^{1}\mathrm{S}^{\mathrm{e}}$ high-lying doubly excited states to the final $^{1}\mathrm{P}^{\mathrm{o}}$ and $^{1}\mathrm{D}^{\mathrm{e}}$ high-lying doubly excited ones. Together with the previous results on the $^{1}\mathrm{S}^{\mathrm{e}}$${\mathrm{\ensuremath{-}}}^{1}$${\mathrm{S}}^{\mathrm{e}}$ excitation processes, we have found a simple propensity rule that the excitation processes with \ensuremath{\Delta}${n}_{2}$=0 are most likely to take place within each manifold, i.e., the S-S, S-P, or S-D excitation process where ${n}_{2}$ is the radial bending quantum number, i.e., the number of the nodes of the bending vibrational wave functions on the body fixed frame based on the rovibrator model of the doubly excited states. In addition to the propensity rule, \ensuremath{\Delta}${n}_{2}$=0, we have also found a set of propensity rules, i.e., \ensuremath{\Delta}v=\ensuremath{\Delta}T=0 for angular correlation and \ensuremath{\Delta}A=0 for radial correlation according to the rovibrational model where v is the bending vibrational quantum number, T is the vibrational angular momentum, and A is the radial correlation quantum number.This may be interpreted as a result of isomorphism of the surface density plot of the squared channel functions between the initial and final states. The radial propensity rule \ensuremath{\Delta}A=0 seems to dominate over the angular one, i.e., \ensuremath{\Delta}v=\ensuremath{\Delta}T=0 and modifies the latter in the $^{1}\mathrm{S}^{\mathrm{e}}$${\mathrm{\ensuremath{-}}}^{1}$${\mathrm{P}}^{\mathrm{o}}$ excitation processes where both types of the propensity rules are incompatible with one another because of the Pauli exclusion principle for two atomic electrons. This leads to the angular propensity rule \ensuremath{\Delta}v=\ensuremath{\Delta}T=1 for the $^{1}\mathrm{S}^{\mathrm{e}}$${\mathrm{\ensuremath{-}}}^{1}$${\mathrm{P}}^{\mathrm{o}}$ excitation processes, though the propensity rule \ensuremath{\Delta}${n}_{2}$=0 and \ensuremath{\Delta}A=0 remains unchanged. The propensity rule for the S-P excitation is equivalent to the selection rule for photoabsorption by He, as is expected. These propensity rules theoretically demonstrate that the two atomic electrons show remarkable collective behaviors as the result of strongly correlated motion between them, while an incident electron interacts with an atom in the high-lying doubly excited state perturbatively. It has been found that the propensity rule obtained here, i.e., \ensuremath{\Delta}v=\ensuremath{\Delta}T=1 for the S-P excitation is not in agreement with the selection rule observed in the threshold excitation spectra of He by electron impact [P. J. M. van der Burgt et al., J. Phys. B 19, 2015 (1986)] and with the selection rule for the infrared photon absorption by a linear triatomic molecule. The reason for these discrepancies is also discussed.

Application of finite‐element method to the two‐dimensional Schrödinger equation

AbstractAn algorithm to solve the two‐dimensional Schrödinger equation based on the finite‐element method is proposed. In our scheme, the molecular Hamiltonian with any arbitrary internal coordinate system can be solved as easily as with the Cartesian coordinate system. The efficient computer program based on the algorithm was developed on the HITAC S‐810 supercomputer. The performance of the program is examined in the following test problems; (1) free particle in a box, (2) linear triatomic molecule with two equal force constants, (3) coupled Morse oscillators, (4) Henon‐Heiles potential, (5) H3O+ symmetric vibrational modes. The algorithm is demonstrated to be very suited for the vector‐type supercomputer.

Hamiltonian theory for vibrational dephasing rates of small molecules in liquids

A technique is developed for solving the generalized Langevin equation (GLE) describing anharmonic oscillators in the weak coupling limit. The GLE is rewritten as a Hamiltonian with a nonlinear system coupled to an infinite bath of harmonic oscillators. A normal mode transformation followed by a perturbation technique is used to obtain the fluctuating system frequency. When the method is applied to a single oscillator with cubic anharmonicity, both the classical and quantal dephasing rates are shown to be equal to the well-known result of Oxtoby. The technique is also applied to a system with more than one vibrational degree of freedom (linear triatomic molecules) to obtain the dephasing rates for the symmetric and antisymmetric normal modes. The effects of system anharmonicity on frequency shifts are investigated.

Measurement of the Renner-Teller effect inX2Π CO2+by diode laser absorption

Four vibronic components of the (v 1, v 2, v 3) = (020) - (010) hot band of CO2 + have been measured by infrared diode laser absorption spectroscopy. The CO2 + molecules were generated in a hollow cathode discharge of a dilute mixture of CO2 in helium at a total pressure of a few hundred mTorr. The hamiltonian operator for a linear triatomic molecule in a 2Π electronic state undergoing bending vibrational motion is described and the vibronic matrix elements given. The theory is used to fit the observed data together with previously reported infrared measurements of the bending fundamental and combination differences extracted from the electronic spectrum of the ion. The analysis yields rotational, spin-orbit, and vibronic parameters including those describing the Renner-Teller and Fermi resonance effects, and the bending potential function including anharmonicity.

Projection of accurate configuration-interaction wave functions for He

To quantify the extent of validity of the molecular model of atoms with correlated electrons, we project well-converged configuration-interaction wave functions for doubly excited states of He and for bound states of alkaline-earth-metal atoms onto simple rotor-vibrator wave functions for the ``linear triatomic molecule'' e-core-e. The three independent vibrational frequencies and the equilibrium electron-nucleus separation are treated as parameters which are varied to maximize the overlap with the well-converged functions. For intrashell states, the overlaps with harmonic normal-mode functions are generally quite large, about 85--95 % for the alkaline-earth-metals, and the corresponding optimized parameters are in rough agreement with the energy-level separations. The overlaps are improved by using an anharmonic local-mode representation, particularly for intershell states (corresponding to excited stretching vibrations), though the optimized parameters appear to lose some of the consistency shown by those in the harmonic normal-mode representation. A comparison is made of plots of the conditional probability densities based on rotor-vibrator functions and on independent-particle functions with those based on accurate functions. This shows clearly the superiority of the molecular picture over the independent-particle picture as a zero-order model for intrashell states of two-electron atoms.

Model calculations for the predissociative excitation of CS 2

A collinear model for predissociation of linear triatomic molecules is presented. A very steep repulsive potential for CS (A) + S is obtained by fitting the experimental results from the dissociation of CS 2 at various excitation energies. Qualitative agreement has also been obtained for the rotational energy distribution of the product fragment. The role of the interfragment effect in determining the CS (A) vibrational distributions is discussed.

Velocity modulation infrared laser spectroscopy of negative ions: The ν3, ν3+ν1−ν1, ν3+ν2−ν2, and ν3+2ν2−2ν2 bands of cyanate (NCO−)

We have measured 132 transitions in the ν3 (CN stretching) fundamental and the corresponding bending and stretching hotbands of the cyanate anion, ranging from P(52) to R(66), with a tunable diode laser using the velocity modulation technique. The spectra were fit to the standard linear triatomic molecule rotation–vibration Hamiltonian, yielding effective molecular parameters for the (0000), (0110), (0200), (1000), (0001), (0111), (0201), and (1001) states. The equilibrium rotational constant was determined to be Be =0.385 933(116) cm−1. A comparison with condensed phase results is presented. As previously oberved for N− 3, cyanate is detected with high vibrational excitation in the bending mode, most likely as a consequence of the formation mechanism.

Three-dimensional analytical quantum mechanical theory for triatomic photodissociation: Role of angle dependent dissociative surfaces on rotational and angular distributions in the rotational infinite order sudden limit

An analytical quantum mechanical model is developed for calculating fragment energy distributions in photodissociation of linear triatomic molecules when the repulsive potential energy surface is anisotropic. The translational continuum function is taken to be given by the infinite order sudden approximation, but an equivalent adiabatic interpretation leads to a natural choice for the diatomic angular momentum j and for the retention of asymptotic rotational energy differences. Normal coordinates are used for the initial bound state before dissociation, while scattering coordinates are applied for wave functions on the dissociative surface. These natural choices lead to three-dimensional nonseparable bound–continuum transition amplitudes. The translational continuum wave function is further approximated using Airy functions, and additional approximations are introduced based on the presence of small amplitude vibrations in the initial bound state. The three-dimensional transition amplitudes are then analytically reduced to sums of one-dimensional quadratures. The theory has been applied to the photodissociation of several rovibronic states of N2O+(Ã 2∑+) (predissociation) and ICN(C̃ 1A′) (direct photodissociation), and the rotational distributions for J=0 are in good agreement with three-dimensional close-coupled calculations except when the potentials become highly anisotropic. Our photodissociation infinite order sudden approximation is tested against various versions of the rotational infinite order sudden approximation for N2O+ and are found to be in good agreement with previous results. The present theory readily permits calculations for J>0 and may be used for the calculation of rotational distributions for excited rotational and/or vibrational states. In the limit of isotropic potentials the remaining integrals are evaluated to provide analytical approximations for the transition amplitudes.

Energy transfer processes in linear triatomic molecule-solid surface collisions

Abstract The semiclassical stochastic trajectory method is extended to the study of rotational and vibrational transitions for linear triatomic molecules colliding with non-rigid solid surfaces. Rotational and vibrational motion are treated by quantum mechanics, translational motion by classical mechanics, and surface atom motion by the classical generalized Langevin equation. Self-consistent coupling of all motions is enforced via Ehrenfest's theorem. Calculations of the kinetic energy and gas temperature dependence of trapping probabilities, vibrational relaxation probabilities and final vibrational state distributions are presented for the CO2-Ag(111) system at surface temperatures of 0 and 600 K. The trapping probabilities are greatly enhanced by the rotational motion and also vary to some degree with the initial vibrational state of the CO2. Total vibrationally inelastic probabilities are on the order of 10−2 for a single collision event with an initial state (00°1). For the initial state (0110) these are much larger, ~ 10−1, due to the nature of bending mode motion. In conjunction with the large trapping probabilities, the mechanism of vibration to vibration, rotation, translation, phonon energy transfer can provide vibration relaxation probabilities in the range of those measured experimentally. A pseudo-selection rule for conservation of vibrational angular momentum is found.

Energy transfer processes in linear triatomic molecule-solid surface collisions

The semiclassical stochastic trajectory method is extended to the study of rotational and vibrational transitions for linear triatomic molecules colliding with non-rigid solid surfaces. Rotational and vibrational motion are treated by quantum mechanics, translational motion by classical mechanics, and surface atom motion by the classical generalized Langevin equation. Self-consistent coupling of all motions is enforced via Ehrenfest's theorem. Calculations of the kinetic energy and gas temperature dependence of trapping probabilities, vibrational relaxation probabilities and final vibrational state distributions are presented for the CO 2-Ag(111) system at surface temperatures of 0 and 600 K. The trapping probabilities are greatly enhanced by the rotational motion and also vary to some degree with the initial vibrational state of the CO 2. Total vibrationally inelastic probabilities are on the order of 10 −2 for a single collision event with an initial state (00°1). For the initial state (01 10) these are much larger, ~ 10 −1, due to the nature of bending mode motion. In conjunction with the large trapping probabilities, the mechanism of vibration to vibration, rotation, translation, phonon energy transfer can provide vibration relaxation probabilities in the range of those measured experimentally. A pseudo-selection rule for conservation of vibrational angular momentum is found.

The vibration-rotation hamiltonian

A method is given for deriving a hamiltonian for a polyatomic molecule in generalized coordinates. By this technique, one may treat linear and nonlinear molecules in the same way. It is shown that the proof which asserts that the determinant of the moment of inertia tensor for a linear molecule vanishes is fallacious and, as an example, a hamiltonian for a linear triatomic molecule is derived and discussed.

The potential in a two-electron atom.

Potential-energy surfaces V(${r}_{2}$,${\mathrm{theta}}_{12}$\ensuremath{\Vert}${r}_{1}$=\ensuremath{\zeta}) representing the potential energy seen by one electron given that the other electron is fixed at \ensuremath{\zeta} are presented graphically for He and ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ in both Cartesian and cylindrical polar representations. These pictures are an aid in visualizing the attractive and repulsive forces which influence the dynamics of the electrons in these systems. They also help to draw the distinction between the potentials around negative ions and neutral atoms. Similar plots of the pseudopotentials of the alkaline-earth atoms show why these quasi-two-electron species tend to behave more like linear triatomic molecules than like a system of two independent electrons in orbits about a nucleus. Finally, conditional probability distributions \ensuremath{\rho}(${r}_{2}$,${\mathrm{theta}}_{12}$\ensuremath{\Vert}${r}_{1}$=\ensuremath{\zeta}) are presented along with their corresponding V(${r}_{2}$,${\mathrm{theta}}_{12}$\ensuremath{\Vert}${r}_{1}$=\ensuremath{\zeta}) to illustrate these points.

Limitations on the usefulness of the angular median and related potentials

The use of the angular median and related effective spherical potentials to predict thermodynamic properties of nonpolar homonuclear diatomic liquids has recently been shown to be efficient and accurate. Here we compare the results obtained from median-like methods for some other molecular liquids with simulation data. We find impressive agreement for linear triatomic molecules but results for tetrahedral molecules and for the overlap potential are very poor. The characteristic shape of potential energy frequency distributions at fixed separations is suggested as a criterion for the success or otherwise of the median potential.

Coriolis-induced intramolecular vibrational energy flow between anharmonic normal modes

The classical flow of energy between anharmonic normal vibrational modes induced by Coriolis coupling is studied for a simple model of a linear triatomic molecule rotating in a plane. The dynamics are analytically solved via a mapping onto a hindered rotor representation. The theory is confirmed by trajectory calculations.

L-Type doubling hyperfine structure of N 2O +

The hyperfine structure in the A 2Σ +- X 2Π transition of N 2O + is complicated by unequal contributions from the two nitrogen atoms. However, using isotopic substitution and the appropriate selection of the quantum numbers N and J, we report here a complete interpretation of the HFS using fast ion beam high-resolution spectroscopy. By comparison between transitions involving the ν 1 and ν 2 vibrational modes, we have observed for the first time l-type doubling hyperfine structure in a 2Π vibronic state of a linear triatomic molecule.