Vibrational Transition Probabilities Research Articles

Vibrational exitation of diatomic molecules in high energy ion-molecule collisions

The vibrational transition probability of diatomic molecules in high energy ion-molecule collisions is derived analytically. The interaction energy is formulated as the sum of the Morse function and the polarization energy. Both the linear and quadratic approximations of the interaction energy are employed in the derivation. The O 2 + AR + system is considered for numerical illustration.

Semiclassical Transition Probabilities for Collinear Collisions between an Atom and a Morse Oscillator

The N-state semiclassical approach is used in the calculation of vibrational transition probabilities for atom–Morse-oscillator collinear collisions. Exact semiclassical calculations for several initial classical phases are compared with rigid-body results. The transition probabilities are found to be strongly dependent upon the initial phase. Two procedures are used for matching the Morse potential to a quadratic potential. The resulting transition probabilities indicate that the potentials should be matched on their repulsive sides rather than at their minima.

Quantum Vibrational Transition Probabilities in Atom–Diatomic Molecule Collisions. III. Reactive Scattering

The methods of Diestler and McKoy and Cheung and Wilson are combined to yield a procedure for calculating vibrational transition probabilities for inelastic reactive and nonreactive collisions. The model treated is that of a collinear collision between an atom and a diatomic molecule. The procedure should result in marked reductions in the computer time and core requirements for such calculations. An alternative method based solely on the approach of Cheung and Wilson is also developed; this method requires that the potential-energy surface in the vicinity of the transition state be approximated by a quadratic form and should permit extremely rapid computation of reactive-scattering transition probabilities. A second alternative method based on the use of the Born approximation in the vicinity of the transition state is also discussed.

Quantum Vibrational Transition Probabilities in Atom–Diatomic Molecule Collisions. II Linear and Multistep Intermolecular Potentials

Three methods for the calculation of vibrational transition probabilities in colinear atom–diatomic molecule collisions are outlined. These are based on the approximation of the intermolecular potential by terms which permit the solution of the Schrödinger equation by separation of variables in each of several regions into which the configuration space of the system is divided. Boundary conditions between the regions lead to systems of linear equations the solutions to which yield quantities from which the transition probabilities are easily obtained. Plots of transition probabilities for various repulsive intermolecular potentials are shown and discussed.

Quantum Vibrational Transition Probabilities in Atom–Diatomic Molecule Collisions

A generalization of the method of Shuler and Zwanzig is shown to permit the calculation of the scattering matrix for collinear atom–diatomic molecule systems having rather general intermolecular potentials. The procedure is related to the method of invariant imbedding. It is then applied to the calculation of transition probabilities for a harmonic oscillator molecule interacting with the bombarding atom via a hard-sphere potential with a “step”—i.e., V(y) = ∞, y < σ0; V(y) = V1, σ0 ≤ y < σ1; V(y) = 0, σ1 ≤ y, where y is the distance between the bombarding atom and the nearest end of the molecule. The dependence of the transition probabilities on the height (depth) and width of the potential step is studied. Resonance effects on the transition probabilities were observed; these involve the De Broglie wavelength of the collisional motion and the width of the potential step. Application of this approach to diatomic–diatomic collisions is out-lined.

Electronic and vibrational transition probabilities of isotopic hydrogen molecules H 2, HD, and D 2 based on electron energy loss spectra

Electron energy loss spectra of the isotopic hydrogen molecules H 2, HD, and D 2 have been measured in the spectral region of the Lyman and Werner bands using 34 keV primary electrons. The intensity distributions within the Lyman and Werner band system obtained for the three molecules are compared with each other. The band intensities exhibit a considerable isotope effect. With increasing nuclear mass the maximum intensity within the band system is shifted towards higher vibrational quantum numbers and the number of vibrational bands with appreciable intensity increases. From the experimental band intensities the variation of the electronic transition moments with internuclear distance (0.6-0.95 Å) was derived using theoretical Franck-Condon factors quoted by Halmann and Laulicht and WKB-Franck-Condon factors calculated in this paper. Agreement with theory is stated for the Werner band system. The dependence on internuclear distance for the Lyman band system is found to be weaker than previous experimental results but still stronger than most theoretical dependences.

Vibrational Transition Probabilities for the Morse Oscillator

The method of Shuler and Zwanzig has been used to calculate the vibrational transition probabilities for a Morse oscillator suffering an impulsive, colinear collision with an atom. Results are presented for the system H2 colliding with D.

Franck-Condon Factors for the Ionization of H2O and D2O.

Franck-Condon factors have been calculated for vertical transitions of H2O and D2O involving both bond length and angle changes. It is shown that even in the harmonic oscillator approximation different Franek-Condon factors are obtained for positive and negative angle changes. The results are used to obtain the geometry of the ion ground state. Satisfactory agreement is obtained for the isotope effect on the vibrational transition probabilities. The effects of anharmonicity are discussed semi-quantitatively.

Vibrational transition probabilities and r-centroids for the A-X system of the magnesium fluoride molecule

Franck-Condon factors (vibrational transition probabilities) and r-centroids have been evaluated by an approximate analytical method for the A-X system of the MgF molecule. The r-centroids for a given band (ν′, ν″) have been found to decrease smoothly with increasing wavelengths λ ν′ν″. The value of δr̆ ≡ (r̆ ν′ + 1,ν ″ + 1-r̆ ν′, ν″ in a sequence has also been found to remain constant.

Probability of vibrational excitations in high energy collisions

Vibrational transition probability Pnm for n → m is formulated from the solution of the time-independent Schrödinger equation for colinear collision between an oscillator and a particle (an atom or a molecule). The resulting expression can be used for calculating Pnm in high energy collisions.

The Role of Vibrational, Excitation in the Formation of 4‐Center Transition States

AbstractThe experimental results describing the kinetics of the interaction between hydrogen and deuterium obtained by Farkas and Farkas are discussed in the light of the recently derived rate constants of the reactions: H2 + M → H + H + M and H + D2 → HD + D. It is suggested that in the early study of this exchange reaction, the dissociation of hydrogen occurred under heterogeneous conditions, so that the conclusions regarding the relative rates of the various steps in the exchange scheme are not entirely correct. A technique which provides purely homogeneous conditions at high temperatures is briefly described, and recent experimental results obtained in various exchange studies are presented. It is shown that under homogeneous conditions, at the temperature range over which exchange readily occurs, dissociation processes are slow, and the free radicals are present in too small concentrations for a free radical mechanism to contribute to the observed rate of exchange. The experimental results suggest that exchange occurs via a 4‐center intermediate, but not as a result of a hard collision between the exchanging molecules. A reaction may occur only if one of the molecules is excited to a critical vibrational state before colliding with its exchange partner. In a few diatomic molecules where the efficiency of translation → vibration energy exchange is not very high, the rate of vibrational excitation determines the rate of exchange. For one typical study, an evaluation of the rate of vibrational excitation from available data on translation → vibration transition probabilities is described. The agreement between the calculated and the experimental rates is excellent.

Vibrational transition probabilities of the bands of the BaO (A 1$\Sigma$-x 1$\Sigma$) system

The band spectrum of BaO has been obtained by spraying barium chloride solution into a flame. The integrated intensities of the bands have been determined by photographic photometry. The experimental results along with the theoretically computed Franck-Condon factors have been used to evaluate a relation between Re, the electronic transition moment, and r, the internuclear separation, in the form Re(macronrνprimeνPrime) = constant (1 - 0536r). This relation has been used to obtain improved Franck-Condon factors. The theoretically computed Franck-Condon factors, with and without the inclusion of Re variation, have been compared with the experimental band strengths.

Non-empirical calculations of the probabilities of vibrational transitions in hydrogen halide molecules

Calculations of the probabilities of vibrational transitions P 1.0 have been made for HX molecules (X is F. Cl, Br, I) for their collisions with HX, H 2, He, Ar. The rotational anisotropy of the intermolecular potential was taken into account. A diagram of electronic density for the HF molecules was used in the calculations.

Franck-Condon factors and r centroids for the B-X and C-X systems of the MgF molecule

Arrays of Franck-Condon factors (qνprimeν−or+) and r centroids (rνprimeν−or+), which are closely related to relative vibrational transition probabilities, have been computed upon a Morse model for the B - X and C - X systems of the MgF molecule. This molecule is of importance to astrophysicists as it occurs in the spectra of sunspots. The r centroids for a given (νprime, ν−or+) band have been found to decrease smoothly with increase of corresponding wavelength λνprimeν−or+. The value of Δr identical with rνprime + 1,ν−or+ + 1 -rνprimeν−or+ in a sequence has also been found to remain constant. The computed values of the r centroids indicate that the potentials are neither very anharmonic nor very wide.

Photoelectron spectra and ionic structure of carbon dioxide, carbon disulphide and sulphur dioxide

Photoelectron spectra of carbon dioxide, carbon disulphide and sulphur dioxide excited by 584 Å radiation have been measured at a resolution of 20 mV over the whole energy range. Ionization potentials, vibration frequencies of the ions and relative vibrational transition probabilities are presented for carbon dioxide and carbon disulphide and are used to derived structural information about the ionic states. A vibrational analysis of the photoelectron spectrum of sulphur dioxide leads to estimates of the ionic geometry in different states and to the conclusion that dπ bonding is effective in one of the 4 b 2 or 1 a 2 orbitals.

Most favorable orientation for vibrational transitions in asymmetric diatomic molecules

Dependence of the probability of vibrational transitions on molecular rotations is investigated with a four-center interaction energy. A strongly angle-dependent expression of the vibrational transition probability is obtained for asymmetric diatomic molecules. As an example, the HBr-HBr collision is chosen. For this system it is shown that the molecular vibration is most efficiently excited when the velocity of relative motion is along the colinear direction of Br-H H-Br. The colinear direction H-Br Br-H is shown to be least efficient.

Exact Quantum-Mechanical Calculation of Vibrational Energy Transfer to an Oscillator by Collision with a Particle

In a recent paper,! Secrest and Johnson (SJ) pre­sented an exact quantum-mechanical solution for vibrational transition probabilities in the collinear collision of a particle with a harmonic oscillator. Their approach utilized an elegant Green's function procedure in terms of plane waves. It was found that their results did not reduce to the first-order distorted-wave ap­proximation (FODWA) of Jackson and Mott

Measurement of Threshold Electrons in the Photoionization of H2 and D2*

Measurements of electrons produced in the photoionization of H2 and D2 are reported. The experimental technique employs a differential electron-energy analyzer to detect photoelectrons produced with nearly zero initial kinetic energy (≲0.1 eV). By varying the photon energy a spectrum of “threshold” photoelectrons is recorded exhibiting the vibrational structure of the H2+ and D2+ ions. From these spectra values of I(H2), I(D2), ωe(H2+), ωe(D2+), ωexe(H2+), and ωexe(D2+) are derived. In favorable cases the relative vibrational transition probabilities are also reported and compared with theoretical predictions as well as with other experiments. Some disagreements are noted in comparing the results obtained by this method with theoretical Franck–Condon factors based on potential curves appropriate to nonrotating molecules. It is suggested that the vibration–rotation interaction should be taken into account in calculations pertaining to electronic transitions in light molecules.

WKB Calculation of Vibrational Transition Probabilities in Molecular Collisions

A comparison is made between vibrational transition probabilities calculated by use of the WKB approximation with those obtained from other methods. For Cl2–Cl2, the transition matrix element for a purely repulsive exponential potential is within 1.5% of the Jackson–Mott exact value over the energy range from 1500k–3900k. For a “shifted exponential” potential, the WKB matrix element agrees satisfactorily with the exact value and with the Hartmann–Slawsky approximation at higher energies. It is also shown that the WKB matrix element for a Morse potential is larger than the Lennard-Jones (12–6) matrix element by a factor of 1.3–1.5 over the energy range from 900k–3900k for Cl2–Cl2. Both for the exponential and inverse-law potentials, the inclusion of attractive forces in the calculation leads to significantly larger matrix elements. Application of the WKB method to the calculation of the collision numbers of Cl2–Cl2 and O2–O2 gives results within a factor of about 2 of the experimental data up to temperatures 2000° and 4000°K, respectively, when the effect of molecular orientation on the transition is properly taken into account.

Inelastic molecular collisions with an orientation‐averaged interaction energy

AbstractThe importance of molecular orientations for vibrational‐translational energy transfers between diatomic molecules has been investigated. An angle‐dependent potential function is assumed, and it is averaged over the orientations and vibrations of colliding molecules. For I2I2 and Cl2Cl2, it is found that the calculated average vibrational transition probability for a colinear collision is over‐estimated by large factors (1/γ) compared to that obtained when all possible molecular orientations are considered. At 3000K, 1/γ = 34.4 for I2I2 and 17.6 for Cl2Cl2, while it is 6.8 and 5.9 for N2N2 and O22, respectively. It is also shown that 1/γ decreases rapidly as temperature increases. At 20000K, 1/γ ≈︂ 3 for I2I2, Cl2Cl2, and N2N2, while it is ≈︂ 2.5 for O2O2. In general, when the molecules are large, and when strong attractive forces act between them, 1/γ is very large at low temperatures (<10000K).