Volume 45, Issue 12, 15 December 1966

Collision‐Induced Dissociation of D_{2} ^{+} Ions by Argon and Nitrogen
View Description Hide DescriptionThe reactions D_{2} ^{+}+X→D^{+}+D+X, where X is Ar or N_{2}, have been studied with an angular ion‐scattering apparatus at ion laboratory energies in the range 5–120 eV. Kinetic‐energy distributions as well as angular distributions of the product D^{+} ions have been determined. The experimental method consists of directing a well‐collimated, mass‐analyzed, and energy‐selected ion beam into a scattering region and analyzing the product ions with a 127° electrostatic velocity selector and a radio‐frequency mass filter. The product ion analysis and detection system rotates about the center of the scattering region. The results are discussed in terms of a simple two‐step collision model, where the D_{2} ^{+} ion is first excited electronically to the antibonding ^{2}Σ_{ u } state, and then dissociates. The possibility of direct excitation of the D_{2} ^{+} ion into the vibrational continuum is also discussed. If either model is assumed for the reaction, the experimental results indicate that the most probable reaction channel for the collision‐induced dissociationreaction is one in which primary D_{2} ^{+} ions with large vibrational energies give rise to D and D^{+} products which separate with small relative velocities. These results are consistent with some existing opinion that the collision‐induced dissociation cross section increases rapidly as the vibrational excitation of the primary ion is increased.

Energetic and Angular Studies of ArD^{+} and N_{2}D^{+} Formation
View Description Hide DescriptionThe reactions Ar^{+}+D_{2}→ArD^{+}+D and N_{2} ^{+}+D_{2}→N_{2}D^{+}+D have been studied with an angular ion‐scattering apparatus at ion laboratory energies in the range 2–100 eV. Kinetic‐energy distributions as well as angular distributions of the product ions have been determined. The experimental method consists of directing a mass‐analyzed and velocity‐selected ion beam into a collision chamber containing target gas at low pressure. The product ions are velocity analyzed with a 127° electrostatic velocity selector and mass analyzed in a quadrupole field radio‐frequency mass filter. The product‐ion analysis and detection system rotates about the center of the scattering region. It is found that the observed Q values for these reactions (where Q is the energy transformed from internal to translational) show a marked dependence on the primary‐ion kinetic energy, E _{1}. The Q values for both reactions are endothermic over the entire energy range of these experiments, being approximately zero at the lowest energies and exhibiting endothermic maxima in the region 50<E _{1}<60 eV. It is shown that these maxima correspond to the formation of internally excited product ArD^{+} and N_{2}D^{+} ions, with excitation energies approximately equal to the dissociation energies for ArD^{+}→Ar+D^{+} and N_{2}D^{+}→N_{2}+D^{+}. The product‐ion kinetic‐energy distributions also give evidence of back scattering in the center‐of‐mass system. The energetic and angular features of both reactions are essentially the same, indicating that the collision mechanisms leading to the formation of both product ions do not differ significantly.
The experimental results indicate that both reactions proceed according to a ``pickup'' mechanism at moderate coilision energies. At very low collision energies the results suggest that complex formation may become important.

Born—Oppenheimer Approximation and the Calculation of Infrared Intensities
View Description Hide DescriptionIn evaluating transition moments it is customary to transform from the momentum representation to the dipole representation by making use of the commutator relations between the Hamiltonian and the particle coordinates. In the present paper we show that in the case of infrared, vibrational transitions it makes quite a difference whether we introduce the Born—Oppenheimer approximation before or after this transformation. In the first case the transition moment depends on the nuclear contribution to the molecular dipole moment and in the second case on the total dipole moment of the molecule. In order to dissolve this discrepancy we calculated the first‐order correction to the Born—Oppenheimer approximation. We introduced this correction in the transition moment calculations and we found that the correction terms bring the two different results into agreement with each other. It was found that the correction is quite small in the case where the Born—Oppenheimer approximation is introduced after the momentum—dipole transformation and that the correction is considerable in the case where the Born—Oppenheimer approximation is introduced before the transformation. We conclude that the correct expression for the transition moment contains the derivative of the molecular dipole moment with respect to the normal coordinates of the modes of vibration, in agreement with the expressions that are customarily used.

Explicit Formulas for the Determination of the Exponents for Gaussian Atomic Orbitals
View Description Hide DescriptionExplicit formulas of all the first‐ and second‐order differentials of the energy of the atomic orbital with respect to the linear coefficients and/or the screening parameters of the ellipsoidal Gaussian functions, are given, so that all the exponents and linear coefficients of the atomic Gaussian orbital can be evaluated by the Newton—Raphson method. This method can be simplified to become the solution of several algebraic linear equations when α_{ rx } ^{ p }≡α_{ ry } ^{ p }≡α_{ rz } ^{ p }. The screening exponents are determined in such a way that the calculated Gaussian orbitals resemble Slater orbitals.

Study by Mass Spectrometry of the Decomposition of Ammonia by Ionizing Radiation in a Wide‐Range Radiolysis Source
View Description Hide DescriptionA wide‐range radiolysis source which can be used to approximate an ideal experimental study of a chemical reaction induced by ionization has been designed and constructed for a study of the decomposition of ammonia. The source consists of three separate compartments in series, each with a separate electron beam which can be independently controlled over a wide range of intensity and energy. The pressure in the source can be varied over a range of 10^{−9} to 10 torr and decreases progressively from compartment to compartment. Reactions can be followed in time starting from about 10^{−14} sec after irradiation up to the time for the formation of final stable products. Each compartment has a distinct function and is used to observe reactions which occur at different times after irradiation. Compartment 3 is a high‐sensitivity ion source which ionizes reaction products produced in a preceding compartment for mass identification by a sensitive mass spectrometer to which the apparatus is attached. Also, cross sections for ionization of both positive and negative ions by electrons are measured in this compartment. Reactions which occur from ∼10^{−14} to ∼10^{−3} sec after irradiation are studied in Compartment 3. Compartment 2, the medium‐pressure compartment (10^{−8}−1 torr), is used to measure (a) rate constants for ion—molecule reactions, (b) cross sections for the production of free radicals by electrons, and (c) energy levels of excited states. In Compartment 2, reactions are studied which require times of the order of 10^{−3} to 1 sec. Compartment 1 (pressure to 10 torr) is used for the initial radiation of the sample to produce final products and to measure (a) the threshold energy, (b) the percentage of each due to ion—molecule reactions, and (c) G values.
Irradiation of NH_{3} with 100‐eV electrons at a pressure of 1 torr in the wide‐range radiolysis source produced 74.9% H_{2}, 24.9% N_{2}, and 0.2% N_{2}H_{4} as reaction products with G values of 8.8, 2.9, and 0.03, respectively. Positive‐ion—molecule reactions produced 54% of the H_{2}, 63% of the N_{2}, and 20% of the N_{2}H_{4}. All of the products appeared at a threshold energy of about 4 eV corresponding to the energy necessary to produce H^{−} and NH_{2} ^{−} from NH_{3} by electrons. The abundance of the products increased sharply at about 10 eV corresponding to the ionization potential of NH_{3}.
The primary products of irradiation were 58.8% positive ions, 40.8% free radicals and neutral species, and 0.4% negative ions. Cross sections for the production of each of the species by 100‐eV electrons and appearance potentials of the negative ions are reported. The total cross sections observed were: positive ions, 2×10^{−16}; negative ions, 1.4×10^{−18}; and free radicals, 1.4×10^{−16} cm^{2}/molecule. In addition to the free radicals produced by dissociation of NH_{3}, the secondary radicals NH_{4}, N_{2}H_{2}, N_{2}H_{3}, and N_{2}H_{5} were observed. Appearance potentials were measured and in some cases calculated (by an energy‐calibrated molecular‐orbital scheme) for the free radicals and the following values in electron volts were obtained: experimental, H (13.8), N (14.6), NH (12.8), NH_{2} (11.7), N_{2}H_{2} (9.9), N_{2}H_{3} (7.6), and N_{2}H_{4} (8.8); theoretical: NH_{2} (10.5), N_{2}H_{2} (10.1), and N_{2}H_{3} (7.8). The most abundant positive and negative ions observed at 1 torr pressure were NH_{4} ^{+}, 83% and NH_{3} ^{−}, 51%. Rate constants and cross sections are given for the positive‐ion—molecule reactions; the negative‐ion spectrum is reported for a pressure of 1 torr.
By generalizing from the data collected, an attempt is made to specify the elementary reactions leading to the reaction products. It is shown that NH_{2}, H, and NH_{4} ^{+} are the most important transient species in the reaction mechanism and that essentially all of the primary ions formed react with NH_{3} to produce NH_{4} ^{+} and either H or NH_{2}. In certain respects the conclusions from this study agree with those obtained by electric‐discharge techniques.

Lower‐Bound Procedure for Energy Eigenvalues by the Partitioning Technique
View Description Hide DescriptionA lower‐bound procedure for obtaining energy eigenvalues by use of the partitioning technique and bracketing theorem, which have been developed by Löwdin, is extended to the case of a multidimensional reference manifold and is applied to the ground state of the two‐electron isoelectronic series. Except for H^{−}, the agreement between upper and lower bounds is quite satisfactory. The process of obtaining lower bounds for excited states is considered.

Transition‐Moment Variation in the γ System of NO
View Description Hide DescriptionThe fluorescence lifetime of the v′=0 level of the A ^{2}Σ^{+} state of NO has been measured as 196.5±3.0 nsec. This figure is compared with the most recent literature results and conclusions are drawn for the variation of the transition moment on internuclear separation.

Critical Opalescence: The Rayleigh Linewidth
View Description Hide DescriptionDebye's derivation of the Ornstein—Zernike modification of Einstein's theory of critical opalescence is reviewed. It is shown that Debye's free‐energy function leads to a modification of the Landau theory for the width of the Rayleigh line which is equivalent to the results of Fixman, Botch, and Felderhof. This equivalence is utilized to predict the magnitude of the departure of Rayleigh linewidths from the Landau prediction by comparison with the results of angular‐scattering intensity measurements. It is thus shown that in light‐scattering experiments, pure fluids should obey the Landau theory quite accurately, whereas critical mixtures should show significant departures from the Landau prediction.

Kinetics of Growth of Multicomponent Chains
View Description Hide DescriptionA theory is presented by which the rate of growth and composition (including all pair and higher distributions) of a chain growing in a multicomponent system may be calculated. Each position in the chain may be occupied by any of the components of the system. Only nearest‐neighbor interactions are assumed and the rate constants α^{ ij } for addition of Species j to a chain ending in Species i, and β^{ ij }, for the removal of Species j from a chain ending in i, j are assumed known and independent of chain length, except for those referring to the first step of the chain, which are distinct. The full kinetic equations for the growth of such chains are formulated and a solution obtained for steady‐state conditions. It is shown that when the matrix of α^{ ij }/β^{ ij } is indecomposable and primitive a solution of the equations which is independent of chain length always exists for sufficiently long chains and computational methods for obtaining this solution for a relatively large number of components (of the order of 10) are presented. In addition, the relationship of α^{ ij }/β^{ ij } to the energetics of the system is derived.

Time‐Dependent Behavior of Activated Molecules. High‐Pressure Unimolecular Rate Constant and Mass Spectra
View Description Hide DescriptionA quantum‐mechanical theory of the unimolecular decay of metastable or activated molecules is developed using Fano's treatment of resonance scattering. A resonance state is synonymous with the so‐called activated molecule in unimolecular kinetics, and a set of widths is associated with each state which is a measure of the coupling to the various dissociation continuum channels (each channel designates an ``activated complex''). If the widths are small compared to the spacings between those neighboring states which are coupled to the same continua, then an ensemble of molecules prepared in a given activated state will decay exponentially in time, as a radiating or autoionizing excited atomic state does. However, unimolecular decay is fundamentally a problem in ``overlapping'' resonance widths, and is best considered using Fano's theory, which incorporates proper treatment of the overlap. This paper is particularly concerned with the implications of overlapping widths on the high‐pressure rate constant, and on mass spectra.
There are two effects of overlapping which are most striking. First, the time decay of an ensemble of metastable molecules is no longer a pure exponential, but for the special cases considered is represented by a sum of exponential, oscillatory, and/or linear terms; this can affect the interpretation of mass spectra. Second, the high‐pressure rate constant is related to the initial rate of decay of a canonical ensemble of activated molecules, and proper consideration of overlap imposes an upper bound on the rate of dissociation without any artificially imposed restrictions on the widths. This bound yields the ``universal rate constant,'' kT/h times a transmission coefficient which is a function of the widths and spacings of the activated molecules, and has an upper limit of 1.

Carbon Trioxide: Its Production, Infrared Spectrum, and Structure Studied in a Matrix of Solid CO_{2}
View Description Hide DescriptionReactions of oxygen atoms with CO_{2} molecules to give a species identified as CO_{3} have been observed in three systems: (1) the photolysis of solid CO_{2} at 77°K with vacuum‐ultraviolet light from a xenon resonance lamp; (2) the photolysis of O_{3} in a CO_{2} matrix at 50°—60°K with 2537‐Å light from a mercury arc; and (3) the radio‐frequency discharge of CO_{2} gas followed by trapping of products at 50°—70°K. The infrared spectrum of the species includes absorptions at 568, 593, 972, 1073, 1880, 2045, 3105, and 3922 cm^{−1}. Isotopic studies using CO_{2} enriched with ^{18}O or ^{13}C show that the molecular formula is CO_{3} and provide a basis for determining the molecular structure. Frequency assignments and isotopic product‐rule calculations favor a planar C _{2v } molecule in which the carbon atom is bonded to the unique oxygen atom by a strong carbonyl bond and to two equivalent oxygen atoms by weaker bonds. There is presumably covalent bonding between the two equivalent oxygen atoms. In the proposed frequency assignment it is assumed that the absorption arising from the out‐of‐plane bending mode is unobserved. Results of photolyzing isotopically nonequilibrated ^{18}O‐enriched CO_{2} indicate that a D _{3h } or C _{3v } species with three equivalent oxygen atoms is involved in the reaction mechanism or is a readily accessible excited state. There is evidence that the observed photodecomposition of CO_{3} by visible and ultraviolet light yields oxygen atoms and CO_{2}.

Chemistry of Positive Ions. VI. Positive‐Ion Chemistry in Solid Methane
View Description Hide DescriptionThe polymerization of solid methane at 77°K by γ rays was studied at two dose rates in the dose range 4 to 150 Mrad, corresponding to 0.04% to 1.4% conversion. The polymer contains an average of 20 carbon atoms per molecule and its empirical formula is C_{20}H_{40}, independent of the dose. The yield is 0.32 methane molecules converted to polymer per 100 eV absorbed and any induction dose must be less than 1 Mrad (0.01% conversion). We report a detailed characterization of the polymer based on physical and spectroscopic studies. Products up to heptane are identified. Preliminary studies of solid ethane and propane afforded polymer with lower yields and fewer numbers of monomer units than the methanepolymer.
From an analysis of the dose and dose‐rate dependence of the polymerization of solid methane, we conclude that only two types of mechanism are possible. The most useful mechanism is one in which a carbonium ion is transformed directly into a heavy hydrocarbon ion. This process would require about 6 eV of excitation energy. The other possible mechanism requires an efficient energy‐transfer process leading to repetitive polymer‐building and polymer‐degrading reactions.

On the Analytical Mechanics of Chemical Reactions. Quantum Mechanics of Linear Collisions
View Description Hide DescriptionThe analytical quantum mechanics of chemically reactive linear collisions is treated in the vibrationally near‐adiabatic approximation. The ``reaction coordinate'' in this approximation is found to be the curve on which the classical local vibrational and internal centrifugal forces balance. Expressions are obtained for the calculation of transmission coefficients for these nonseparable systems. Some implications for tunneling calculations in the literature are noted. Expressions for nonadiabatic corrections are derived, the latter being associated with vibrational transitions undergone by the transmitted and reflected waves. When the system does not have enough energy to react, the last results refer to the vibration—translation energy‐transfer problem in linear collisions.
Two novel features are the introduction of an actual coordinate system which passes smoothly from one suited to reactants to one suited to products and the introduction of an adiabatic‐separable method, a method which includes curvilinear effects. Extensions to collisions in higher dimensions are given in later papers.

On the Analytical Mechanics of Chemical Reactions. Classical Mechanics of Linear Collisions
View Description Hide DescriptionThe classical mechanics of chemically reactive linear collisions is investigated for vibrationally near‐adiabatic reactions. A coordinate system which passes smoothly from one suited to the reactants to one suited to the products is used. The Hamilton—Jacobi equation is then solved in the adiabatic approximation by introduction of an ``adiabatic‐separable'' method. Nonadiabatic corrections, which describe the probability of vibrational transitions, are also calculated. They involve the Fourier component of local internal centrifugal and vibration frequency‐change terms. The reaction coordinate for the adiabatic system is shown to be that curve on which local vibrational and internal centrifugal forces balance pointwise. Applications can be made to the role of translational—vibrational energy interchange in reactions, reaction‐cross‐section theory, bobsled effect, and other topics. The results may be compared with electronic computer calculations as they become available.

Spectra of Praseodymium in Yttrium Gallium Garnet and in Yttrium Aluminum Garnet
View Description Hide DescriptionThe absorption and fluorescence spectra of Pr^{3+} in YGaG and in YAlG were measured at temperatures between 16°K and room temperature. Crystal‐field calculations, using a cubic field with A _{4} ^{0}=−350 cm^{−1} and A _{6} ^{0}=90 cm^{−1}, gave a reasonable fit with the experimentally found energy levels. Some manifolds overlap due to the large crystal‐field splitting. The centers of gravity found by fitting the calculated manifolds to the experimental data are close to and well in line with the centers for Pr^{3+} in LaCl_{3} and LaBr_{3}.

Molecular Level‐Crossing Spectroscopy
View Description Hide DescriptionClassical and quantum formulations are presented for the effects of level crossings on the angular distribution of molecular resonance fluorescence. It is shown that a study of the ``molecular Hanle effect,'' or its electric‐field analog, may be used to determine the product of the radiative lifetime ρ_{ v′J′} with the magnetic moment μ_{ m }, or the electric dipole moment μ_{ e }, respectively, for the (v′, J′) excited state. The theory is developed for diatomic or polyatomic molecules characterized by symmetric‐top wavefunctions. Order‐of‐magnitude estimates are performed, and the observation of zero‐field molecular level crossings appears to be quite feasible.

Relativistic Interaction Energies between Atoms in Degenerate States
View Description Hide DescriptionThe interaction of two like atoms in degenerate quantum states of the same energy and the interaction of two unlike atoms in arbitrary states is considered in the Breit—Pauli approximation. For these interactions the calculation of the relativistic long‐range interaction energy, through O(α^{2}), is discussed with specific allowance for degeneracy in the interacting atoms. Possible interactions where relativistic effects may be important are discussed. As a specific example the interaction of two spin‐degenerate atoms (L=0, S≠0) is calculated through O(α^{2}/R ^{6}) (where R is the interatomic separation). The nonrelativistic energy is given by the usual London dispersion energy which varies as 1/R ^{6} while relativistic effects introduce an interaction energy which varies as α^{2}/R ^{3}.

Far‐Infrared Spectra of Mercuric Halides and Their Dioxane Complexes
View Description Hide DescriptionThe far‐infrared spectra (600–60 cm^{−1}) of HgCl_{2}, HgBr_{2}, HgI_{2} (red), and their one‐to‐one p‐dioxane complexes (in the solid state) were investigated. All of the observed spectra, except those of HgI_{2}, were assigned on the base of the symmetry properties of the crystals. High‐pressure effects were investigated for the anti‐symmetric stretching vibration of HgCl_{2} and its dioxane complex. The frequency shifts of all of the compounds in the various states, from the gaseous state through the dioxane complex under high pressure, were elucidated in terms of the distance between the mercury atom and its nearest neighbors.

Vibronic Effect on the Mean‐Square Amplitudes of Internuclear Distances in a Tetrahedral Molecule
View Description Hide DescriptionA quantitative relation has been derived for the mean‐square amplitudes of internuclear distances in a tetrahedral XY_{4} molecule when vibronic interactions are present. The calculation has revealed that, first, the mean‐square amplitude l _{Y—Y} increases by the presence of vibronic interactions and second, the effect is large when the frequency of the degenerate vibration ν_{2} is low.

Vibronic Interactions in Vanadium Tetrachloride by Gas Electron Diffraction
View Description Hide DescriptionThe molecular structures of VCl_{4} and TiCl_{4} have been determined by gas electron diffraction. The mean square amplitude of the Cl–Cl distance in VCl_{4} is found to be abnormally large compared with that of TiCl_{4}. This is attributed to the vibronic interactions in VCl_{4}. The observed mean square amplitude has led to the evaluation of the first‐order vibronic interaction constant a′. Other experimental results, such as bond lengths, the shrinkage effect, and vibrational spectra are discussed in relation to the theory of vibronic interactions.