Final Vibrational Quantum Numbers Research Articles

Quantum mechanical study of the high-temperature H+ + HD → D+ + H2 reaction for the primordial universe chemistry

ABSTRACT We use the time-independent quantum-mechanical formulation of reactive collisions in order to investigate the state-to-state H+ + HD → D+ + H2 chemical reaction. We compute cross-sections for collision energies up to 1.8 eV and rate coefficients for temperatures up to 10 000 K. We consider HD in the lowest vibrational level v = 0 and rotational levels j = 0–6, and H2 in vibrational levels v′ = 0–3 and rotational levels j′ = 0–9. For temperatures below 4000 K, the rate coefficients strongly vary with the initial rotational level j, depending on whether the reaction is endothermic (j ≤ 2) or exothermic (j ≥ 3). The reaction is also found less and less probable as the final vibrational quantum number v′ increases. Our results illustrate the importance of studying state-to-state reactions, in the context of the chemistry of the primordial universe.

Converged quantum dynamics calculations for the F+H2 reaction on the well-studied M5 potential-energy surface

We present converged quantum dynamics calculations for the reaction F+H2(vi=0)→HF(vf=0,1,2,3)+H, where vi and vf denote initial and final vibrational quantum numbers, on potential-energy surface no. 5 of Muckerman.

Comparison of classical simulations of the H + H2 reaction to accurate quantum mechanical state-to-state partial cross sections with total angular momenta J = 0-4 and to experiment for all J

Quantum mechanical calculations are reported for probabilities and partial cross sections for the reaction H + p-H{sub 2} ({nu} = 0, j = 0.2, E{sub rel} = 1.1 eV, J = 0-4) {yields} o-H{sub 2} ({nu}{prime} = 0, 1) + H, where {nu}, j, and {nu}{prime} are initial vibrational, initial rotational, and final vibrational quantum numbers, respectively, e{sub rel} is the initial relative translations energy, and J is the conserved total angular momentum quantum number. The calculations involve three arrangements and 468-780 coupled channels, and they are converged to 0.1-1%. The corresponding quantities are also calculated by the quasiclassical trajectory method, and comparing these results provides a detailed test of the trajectory method. For most final states, the trajectory results agree with the quantal ones within a factor of 1.5 to 2, and the trajectory value for the ({nu}{prime} = 1)/({nu}{prime} = 0) branching ratio is too high by a factor of 1.6. The authors also report trajectory results that are converged with respect to increasing J, and the converged value of the branching ratio is found to be 2.5 times larger than experiment. 49 refs., 5 figs., 9 tabs.

Vibrational/rotational infinite-order sudden approximation applied to helium + molecular iodine

Vibrationally and rotationally elastic and inelastic collisions between He and I/sub 2/ in the total energy range of 0.044-0.5 eV are treated under the vibrational/rotational infinite-order sudden approximation (VRIOSA). In this method, one simultaneously makes the vibrational energy sudden, rotational energy sudden, and centrifugal sudden approximations and generates a completely uncoupled set of equations. The solution of these equations is discussed and the integral and differential cross sections, the integral and differential cross sections summed over final rotational states, the integral cross sections summed over final rotational and vibrational states, and a type of cross section which recently has been measured experimentally are presented and compared with vibrational close-coupled infinite-order sudden approximation results. The dependence of the accuracy of the VRIOSA upon the choice of the arbitrary vibrational quantum number parameters, n vector, is also investigated. Setting n vector equal to the final vibrational quantum number gives good results for the integral and experimental cross sections, while picking n vector equal to the average of the initial and final vibrational quantum numbers gives good results for the differential cross sections. The present comparisons involving a full three-dimensional system and quantities, such as the differential cross section, which are sensitive tomore » the phase of the S matrix extend earlier, more limited studies and show that the VRIOSA should be considered convenient and reliable for light atom-heavy diatomic molecule collisions at total energies up to a few electronvolts.« less

An improved potential energy surface for F+H2→HF+H and H+H′F→HF+H′

We present an improved analytic potential energy surface for the F+H2→FH+H and H+FH′→HF+H′ reactions. The final surface is obtained in two stages. First we create a surface, called No. 4, which is based in the F–H–H barrier region on a previous partly empirical and partly theoretical fit and is based on the F–H⋅⋅⋅H exit channel and H–F–H barrier regions on new large-basis-set configuration interaction calculations. The final surface, called No. 5 incorporates more empirical information for collinear geometries in both the F–H–H and FH⋅⋅⋅H regions but remains a good representation of the ab initio calculations for bending potentials and in the strong-interaction regions. Variational-transition-state theory rate constants and WKB adiabatic barrier heights indicate that the final surface is more accurate than previous surfaces for thermal rate constants and overall reaction thresholds for F+H2→HF+H, F+D2→DF+D, and F+HD→HF+D and for product-state thresholds for HF (n′=3) and DF(n′=4), where n′ is the final vibrational quantum number.

Quasiclassical trajectory calculation of the state-specified differential cross sections and opacity functions for F+H2 → HF(v′)+H

We report well-converged quasiclassical trajectory calculations of the state-specified differential cross sections dσ/dΩ(v′) and opacity functions P(b, v′) for F+H2 → FH(v′)+H, where the final vibrational quantum numbers v′ are assigned by the histogram method and P(b, v′) is the probability of reaction into a specified v′ level at initial impact parameter b. The differential cross sections are calculated by the Legendre moment method using stratified sampling. At a relative translational energy of 0.233 eV, all dσ/dΩ(v′) are backwards peaked, and P(b,v′ = 2) and P(b,v′ = 3) both peak at b = 0. This provides confirmatory evidence that the sideways scattering and high-b maximum in P(b, v′) that have been observed in previous calculations by Redmon and Wyatt and by Jellinek, Baer, and Kouri are indeed quantal effects.

Vibrationally inelastic integral cross sections for the scattering of He from H2. II. Transitions to the ν=2 level

Integral cross sections for the scattering of He from H2 on the Gordon–Secrest potential are reported. Cross sections for excitations to the second excited vibrational level ν=2 were calculated at 11 energies in the range of 2.5–3.0 hωe. Both types of transitions (0, j) →(2,j′) and (1,j)→(2,j′) are shown to have the characteristic structures previously reported for (0,j)→(1,j′). Its detail appears to depend on both Δν and the final vibrational quantum number. Analyses of the cross sections σ2j′←νj and σ1j′←0j where ν?1;j,j′? 4 bring out several interesting observations in the dynamics of collisional excitation.

Quantum vibrational transition probabilities from real classical trajectories: Collinear atom–diatom collisions

Quantum vibrational transition probabilities may be estimated from single, real classical trajectories by exploiting the approximate correspondence between the classical and quantal motions. In this paper we explore the range of validity of several such correspondence methods for collinear atom–diatom collisions by comparing the results with those from rigorous quantum mechanical calculations. The model systems considered here include: (a) a harmonic oscillator with (1) a repulsive exponential, (2) a Landau–Teller, and (3) a Lennard-Jones interaction potential, and (b) a Morse oscillator with a repulsive exponential interaction potential. The range of the dimensionless mass parameter M is 0.00628 to 3.737 and the total system energy ranges from 1.55 to 10.0 in units of h/ωe. For both Morse and harmonic oscillator excitation the semiclassical results are generally more accurate for small M and for small initial and final vibrational quantum numbers. Quantitatively, the results are consistently better for a harmonic oscillator than for a Morse oscillator. The threshold behavior for harmonic oscillator excitation can be predicted quantitatively using a classical trajectory in which the arithmetic mean of the initial and final speeds is equal to the arithmetic mean of the translational speeds for the corresponding quantum transition. By detailed balancing, the probabilities for vibrational de-excitation in slow collisions are also predicted accuretaly.

Analytical Theory of Multiquantum Vibrational Excitation in Molecular Collisions

Multiquantum vibration–translation energy transfer is considered within the framework of the distorted-wave approximation. The restriction to harmonic-oscillator selection rules is removed by treating any transition as a sequence of virtual one-quantum transitions, permitting retention of all the vibrational states, but only the initial and final translational states, in the eigenfunction expansion. Vibrational matrix elements are the same as those of Rapp and Sharp, but translational wavefunctions include the effect of energy transfer; consequently, excitation probabilities display more complicated dependence on final vibrational quantum number than do the classical results. Cross sections for CO compare favorably with Marriott's numerical studies. Relaxation times compiled from the cross sections exhibit a three-part temperature dependence: (1) at very low temperature, the relaxation time is exponential in 1 / T, (2) in a large range of intermediate temperatures, relaxation is described by a multiquantum generalization of the Landau–Teller theory, exact within the restriction of weakly coupled states, (3) at very high temperature, the relaxation time is given by an impulsive formula.

De-excitation of Electronically Excited Sodium by Nitrogen

A semiquantitative calculation is made of the cross section for the quenching of Na(32P) by molecular nitrogen, as a function of initial kinetic energy and of final vibrational quantum number, υf, of the nitrogen molecule. The large observed cross section, which is of gas-kinetic order, can be explained in terms of an intermediate ionic state, involving Na+ and N2− (υ = υ−). This state is unstable at infinite separation of Na and N2, but because of the Coulomb attraction it becomes stable at collision distances below about 3 Å. As a result of the vibrational structure of both the intermediate and final states, we treat the reaction in terms of a diffusion of the probability flux through a two-dimensional network of potential-energy curves parametrized by both the electronic state and also the vibrational quantum numbers υ− and υf. At each potential-energy curve crossing we compute the transition matrix element for insertion into a Landau–Zener type of transition probability. The transition matrix element is represented as the product of an electronic interaction function (obtained from a correlation, due to Hasted and Chong, of results obtained from charge-transfer processes involving multiply charged ions) and a vibrational overlap integral or Franck–Condon factor. Results are also presented on the quenching of Na(42P) by N2, and on the quenching of Na(32P) by CO. All the results have the same general character: The total cross section is of gas-kinetic order and depends only weakly on kinetic energy. The partial cross sections for excitation of the different final vibrational levels vf show a rather broad distribution, with somewhat more than half the energy of electronic excitation ending up as vibrational excitation.

The ultra-violet band-system of carbon monosulphide and its relation to those of carbon monoxide ( the “4th positive” bands) and silicon monoxide

In 1913 L. C. Martin discovered an ultra-violet band-system in the tube discharge through carbon disulphide vapour and in the carbon arc fed with sulphur, and ascribed it, on experimental evidence, to the CS molecule. The bands are degraded towards the red, and the majority of them are double headed. Martin arranged all but seven of the observed bands into seven groups (A, B, ... G) each formed by closely neighbouring bands. The present com­munication records the quantum analysis of the band-heads, and the sub­stantial similarity of structure between this CS system, the ultra-violet systemof SiO and the “4th Positive” system of CO. A brief examination of the photographs reproduced by Martin and of the wave-numbers derived from his tabulated wave-lengths leads to a Deslandres scheme for the band-heads and to a plausible assignment of the initial and final vibrational quantum numbers, n ', n ". In this scheme each of Martin’s seven groups is a sequence of bands corresponding to a constant change of vibrational quantum number, n "— n ', except that groups B and C together with one band of group A form only one sequence, n "— n ' = — 2. It further appears that of the pair of heads (bracketed in Martin’s table of data) of each double­ headed band, the less refrangible head is formed by a Q branch, the corresponding P branch having, of course, no head, and therefore remaining undetected with the dispersion employed by Martin (about 7.8 A./mm. at λ 2420 to 12.7 A./mm. at λ 2850).

The more refrangible band system of cyanogen as developed in active nitrogen

In 1917 Lord Rayleigh and Prof. A. Fowler observed that when the vapour of a carbon compound is introduced into the afterglow of active nitrogen, the two CN systems—the so-called "red" and "violet" systems—are developed, but their appearance in this source is strikingly different from the more familiar appearance of the same bands in the carbon arc in air. The "violet" system as ordinarily observed in the arc, comprises the four well-known groups of bands which degrade towards the further ultra-violet from prominent heads at λλ 4606, 4216, 3883 and 3590 respectively, and four groups of weaker bands, the so-called "tail" bands, degrading towards the red. The modifications of the λ 4216 and λ 3883 groups are shown in figs. 4 and 5 of Plate 6, in Rayleigh and Fowler's paper. The description of these modifications will be much facilitated by reference to Table I, which shows the wave-lengths and wave-numbers of the band-heads, and the initial and final vibrational quantum numbers ( n', n" ) for the bands, as well as other particulars to which frequent reference will be made later. The modifications are conveniently described as two effects:— First Effect.—In the typical case of the λ 4216 ( n" - n' = +1) group in the afterglow, there is a modified development of the lines of the first (0, 1) and the third (2, 3) bands, and a partial suppression of the heads of the second (1, 2) and fourth (3, 4) bands. The lines near the head of the (0, 1) band in the afterglow are apparently identical with lines in the corresponding part of the same band in the arc, but at a short distance, about 9 A. U., from the head there is a conspicuous gap of about 3 A. U., beyond which the lines recover in intensity and overrun the weakened (1, 2) head. The (2, 3) band, though not nearly so strong as the (0, 1), shows a similar gap and regain of intensity, its lines over-running the weakened (3, 4) head. A similar effect occurs in the λ 3883 ( n" - n' = 0) group.

The Isotope Effect in Band Spectra, III. The Spectrum of Copper Iodide as Excited by Active Nitrogen

Excitation of spectra by active nitrogen.---Band spectra of CuF, CuCl, CuBr, and CuI have been obtained by the action of active nitrogen on the vapors of Cu${\mathrm{F}}_{2}$, CuCl and Cu${\mathrm{Cl}}_{2}$, Cu${\mathrm{Br}}_{2}$, and CuI, using the method of Rayleigh and Fowler, as previously applied by them to CuCl. Certain additional heads are present in the red, probably due to CuO formed as a result of slight oxygen impurity in the active nitrogen, and then excited by the latter. The halide bands are unusually sharp. Probably excited ${\mathrm{N}}_{2}$ molecules, in "impacts of the second kind," put halide molecules into the various excited electronic states necessary for the emission of visible band spectra. Many impacts, however, result in dissociation of the halide molecule, as the iodine are line $\ensuremath{\lambda}2062$ and over 80 lines of the copper arc spectrum were identified. The list of copper lines is exactly the same for CuCl as for CuI. Analogous results have been obtained with Pb${\mathrm{I}}_{2}$, Hg${\mathrm{I}}_{2}$, and Hg${\mathrm{Br}}_{2}$. Other reactions of active nitrogen are discussed.Emission of electronic band spectra by polar molecules.---It is suggested that the absence of electronic band spectra for the hydrogen, silver and alkali halides may be associated with the non-occurrence of higher valence compounds of type Na${\mathrm{Cl}}_{2}$, and that the emission of any one of the CuX band spectra follows the transfer of a ${\mathrm{Cu}}^{+}$ electron in the polar ${\mathrm{Cu}}^{+}$${\mathrm{X}}^{\ensuremath{-}}$ molecule from its normal state to one of a group of low lying excited states, whose existence can be correlated with the occurrence of the compounds Cu${\mathrm{X}}_{2}$, such easily excited electrons being absent in ions such as ${\mathrm{Na}}^{+}$. The above relation may be accounted for by supposing that polar molecules cannot carry electronic energy in excess of their heat of dissociation into atoms. By analogy with the observed absence of electronic band spectra for compounds of the NaCl type, the band spectra of the alkaline earth halides should not be due to compounds Me${\mathrm{X}}_{2}$, since the ${\mathrm{Me}}^{++}$ion contains no easily excited electron. The real emitter is probably MeX, which must contain a loosely bound valence electron like that in ${\mathrm{Me}}^{+}$ or in Na.Analysis of band spectrum of CuI, and confirmation of vibrational isotope effect.---The CuI bands, excited by active nitrogen, were photographed under moderate dispersion (see Plate I). The wave-lengths of over 260 heads between $\ensuremath{\lambda}5650$ and $\ensuremath{\lambda}3890$ were measured (Tables I-V). No other CuI bands are present between $\ensuremath{\lambda}1900$ and $\ensuremath{\lambda}7000$. Each ${\mathrm{Cu}}^{63}$I band is found to be accompanied, wherever resolution and intensity are sufficient, by a weaker ${\mathrm{Cu}}^{65}$I band, as expected. The ${\mathrm{Cu}}^{63}$I bands can all be arranged in five systems, the vibrational isotope effect serving as an almost indispensable key to this analysis. One system lies in the green; the other four, with their bands closely intermingled, lie in the blue and violet. Equations are given representing the positions of all the ${\mathrm{Cu}}^{63}$I heads within experimental error, in terms of the initial and final vibrational quantum numbers and their squares. From these equations, the corresponding theoretical equations for ${\mathrm{Cu}}^{65}$I are calculated. The observed ${\mathrm{Cu}}^{65}$I heads fall consistently within experimental error in the calculated positions. Near-equality of the coefficients (of both linear and quadratic terms) concerned with the final state of the molecule, shows that the five systems all correspond to a common final electronic state. This is almost certainly the normal state of the CuI molecule. Analogous relations probably hold for CuCl and CuBr, although the analysis is not yet completed. In all cases the vibration frequency is less and the interatomic distance greater for the excited states than for the normal state. The five excited states of the CuI molecule correspond to electronic energies equivalent to 2.44, 2.68, 2.70, 2.83, and 2.96 volts. An energy-level diagram for CuI is given, showing all the electronic and vibrational energy levels, and the observed transitions with their intensities. It is shown that the initial distribution of CuI molecules with respect to vibrational energy is of a thermal type and probably corresponds to the existing low temperature, thus differing markedly from the high-temperature non-thermal distribution characteristic of such compounds as BO and CN as excited by active nitrogen. Intensity distribution in the CuI bands. A marked tendency is noted here, as previously in the case of BO, for numerically large values of $\ensuremath{\Delta}n$ to be associated with large values of the initial vibrational quantum number ${n}^{\ensuremath{'}}$. This causes the maximum of intensity in band sequences (a sequence contains bands for all of which $\ensuremath{\Delta}n$ has the same value) to shift from the first to higher members of the sequence, as one proceeds to bands at increasing distances from the system-origin. Both the above effect and a preference for positive values of $\ensuremath{\Delta}n$ are explained in terms of Lenz's theory.