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Volume 67, Issue 12, 15 December 1977

Comparison of nonperturbative variational approximations to the scattering amplitude for a Yukawa potential
View Description Hide DescriptionVariational methods for calculating approximations to the scattering amplitude of a spinless particle from a Yukawa potential are compared. These arise from the variation of functionals that depend on trial forms of the Heitler operator K or the transition operator T. The trial forms are nonperturbative, and the variational principle is applied in the free particle and partial wave basis to obtain independent approximations. In addition, no artificial separability constraint is imposed on the momentum space representation of either the trial T or K. The full free particle Green’s function is utilized without basis set truncation in all of the calculations. The different approximations provide a choice of technique in the event that one or the other variational method is unstable. The results at two representative energies are in good agreement with an exact calculation.

Model perturbation theoretic calculations with finite continuum basis sets
View Description Hide DescriptionTo examine various aspects of the methods employed in diagrammatic perturbation theory applications the model problem of a hydrogenic atom of nuclear charge Z perturbed by the potential −Z′/r is analyzed. The use of a discrete quadrature for the continuous energy spectrum and a finite coordinate space integration cutoff are tested by comparison with analytic methods. It is found that accurate results are obtained if a physically reasonable choice for the spatial cutoff is used, namely, the maximum significant extent of the zero order wavefunction. For larger values the use of the discrete continuum basis results in a spurious logarithmically divergent contribution to the energy which can be of practical significance. A detailed examination is made of the summation techniques commonly employed in diagrammatic perturbation theory. For the physical choice of cutoff the geometric summation of higher order terms involving excited states in the continuum, as well as bound states, is shown to yield meaningful and accurate results. To illustrate the analysis numerical results are given for Z=1, Z′=1 and Z=4, Z′=1.

Host spin–lattice relaxation narrowing in the EPR of Mn^{2+} in CoSO_{4}⋅7H_{2}O single crystals
View Description Hide DescriptionElectron paramagnetic resonance(EPR) of Mn^{2+}doped in CoSO_{4}⋅7H_{2}O single crystals has been studied from 300 to 77 K and at ∼9.3 GHz. Two different types of crystals of CoSO_{4}⋅7H_{2}O have been obtained, which grow in two different temperature ranges about 298 K. The two forms exhibit different types of spectra: one form [to be referred to as CoSH(I)] shows two 30 line Mn^{2+} spectra with D=459×10^{−4} cm^{−1}, ‖E‖=134×10^{−4} cm^{−1} and the other form [to be referred to as CoSH(II)] shows only a single 30 line spectrum with D=234×10^{−4} cm^{−1}, ‖E‖=19 ×10^{−4} cm^{−1}. The observation of resolved Mn^{2+} spectra at 300 K and its broadening on lowering the temperature in CoSO_{4}⋅7H_{2}O:Mn^{2+} crystals has been discussed in terms of ’’host spin–lattice relaxation narrowing.’’ The observed shift in the Mn^{2+} g _{z} value from that in diamagnetic hosts is probably due to the Co^{2+} ions, which produce a local static magnetic field at the site of Mn^{2+} ions. From the observed linewidth in the EPR spectrum of Mn^{2+} in single crystals of CoSO_{4}⋅7H_{2}O the T _{1} of Co^{2+} in CoSO_{4}⋅7H_{2}O has been estimated to be ∼1.1×10^{−1}1 s.

A b i n i t i o configuration interaction study of the valence states of O_{2}
View Description Hide DescriptionConfiguration interaction calculations have been performed for the 62 electronic states of O_{2} arising from O atoms in the lowest ^{3} P, ^{1} D, and ^{1} S states. The calculations used an extended one‐particle basis set, and included internal and semi‐internal electron correlation effects. Numerical values for potential curves are reported for all states. Detailed comparison with experiment and other calculations is made for the seven lowest bound states:X ^{3}Σ_{ g } ^{−}, a ^{1}Δ_{ g }, b ^{2}Σ_{ g } ^{+}, c ^{1}Σ_{ u } ^{−}, C ^{3}Δ_{ u }, A ^{3}Σ_{ u } ^{+}, and B ^{3}Σ_{ u } ^{−}. For these seven states the maximum error in the calculated spectroscopic constants R _{ e }, D _{ e }, T _{ e }, and ω_{ e } are 0.04 Å, 0.4 eV, 0.2 eV, and 120 cm^{−1}, respectively.

Chemiluminescence from Al+O_{3}: Perturbations, populations, and vibrational analysis for the AlO A ^{2}Π–X ^{2}Σ^{+} transition
View Description Hide DescriptionAs a complement to an earlier study of the AlO B ^{2}Σ^{+}–X ^{2}Σ^{+} emission system, the A ^{2}Π_{ i }–X ^{2}Σ^{+}chemiluminescence resulting from the reaction Al+O_{3}→AlO^{*}+O_{2} has been investigated. It is found that the constants extracted for the A ^{2}Π_{ i } state by McDonald and Innes adequately describe the vibrational structure of this state to high (∼29) vibrational levels, the behavior being indicative of a Morse oscillator. Perturbations previously observed in the X ^{2}Σ^{+} state are explicitly analyzed in terms of A ^{2} Π_{1/2}–X ^{2}Σ^{+} spin–orbit interactions, and a comparison with A ^{2}Π_{1/2}–B ^{2}Σ^{+} perturbations is made. Approximate populations N _{ v }′ for the A ^{2}Π_{ i } vibrational levels are determined; the population distribution is dramatically non‐Boltzmann, peaking at an energy just below that corresponding to the v=0 level of the B ^{2}Σ^{+} state. This peaking is explained in terms of efficient collisional transfer from the A state to the B state.

Variation of electronic polarizabilities and sizes of ions in crystals under hydrostatic pressure
View Description Hide DescriptionA model, based on the energy level analysis performed by Ruffa and empirical relation between polarizability and ionic radius, has been developed to study the variation of electronic polarizabilities and sizes of ions in crystals under hydrostatic pressure. The model has been applied to investigate the photoelastic behavior of alkali halides and MgO. The strain derivatives of electronic polarizabilities and dielectric constants thus evaluated present a good agreement with experimental data and other theoretical investigations.

Quantum theory of scattering of atoms from solid surfaces
View Description Hide DescriptionWe present a quantum theory of scattering of an atom from a solid surface. The formalism is constructed by assuming that the solution of two simpler problems is known; one of them is scattering of the atom by the average lattice potential; the other is the motion of the lattice in the absence of the scattering atom. The theory includes the effect of the thermal motion of the surface atoms as well as multiphonon energy transfer processes. The derivation assumes that the thermal fluctuations of the projectile–lattice potential and the energy of the distortion of the lattice by the projectile can be treated as perturbations.

Nonlinear pressure dependence of fluorescence quenching, intermolecular intersystem crossing
View Description Hide DescriptionA more general theory of collision induced intersystem crossing is described which contains the treatment of Thayer and Yardley as a special case. The fluorescence quenching depicts a strong nonlinear pressure dependence. It is shown that intermolecular intersystem crossing (intermolecular energy transfer) can provide a large and significant contribution to the overall rate of S–T crossing and can provide an explanation of the large self‐quenching. A measure of success is achieved when the theory is applied to the pressure dependence of the fluorescence quenching of quinoxaline and benzophenone. The discrepancy between the experimental and theoretical results is attributed to the neglect of the ground state continuum. A similar problem is encountered in the interpretation of the fluorescence decay of intermediate case molecules when a restricted two state model is considered.

The angular momentum, creation, and significance of quantized vortices
View Description Hide DescriptionThe flux of probability density corresponding to a complex wavefunction, ψ=ψ_{ r }+iψ_{ i }, can form vortices which rotate about a nodal region of ψ, and have integer values of the circulation numbers. For one particle wavefunctions (or for approximate natural spin molecular orbitals), the vortices are either a x i a l or t o r o i d a l. The a x i a lvortices have angular momentumdipole moments which are usually quantized only if the wave function is an eigenfunction of L _{ z }. The a x i a lvorticesinteract with homogeneous magnetic fields whereas the t o r o i d a linteract with inhomogeneous magnetic fields.T o r o i d a lvortices are easily created or annihilated by a perturbation which moves the nodal surface of ψ_{ i } relative to the nodal surface of ψ_{ r } so that these surfaces intersect or become separated. Thus, t o r o i d a lvortices are unstable and may be important only near resonance. However, a x i a lvortices are stable since their creation requires (a) photon absorption or emission, (b) a perturbation (such as a magnetic field) which converts a real into a complex wavefunction, or (c) conversion of a toroidal into an axial vortex by expanding the nodal loop until a part of it reaches the boundary of configuration space. Thus, axial vortices should play an important role in energy transfer, photochemical processes, etc., and their circulation numbers should be good quantum numbers. A method for determining generalized first‐order density matrices and natural spin–orbitals for use in scattering problems is proposed.

Calculation of associative and Penning ionization of H and D by He(2 ^{1} S) and He(2 ^{3} S)
View Description Hide DescriptionAssociative and Penning ionization cross sections are calculated for collisions of H and D with He(2 ^{1} S) and He(2 ^{3} S) in the energy range 0.03–1.0 eV. A b i n i t i o configuration–interaction potential curves V* and V _{+} for these systems are used. For the autoionization width Γ, the recent results of Hickman, Isaacson, and Miller for He(2 ^{3} S) ‐H are employed, and their calculations are extended to obtain the width for He(2 ^{1} S) –H. For the latter system, the calculated associative ionization fraction and the relative energy dependence of the total ionization cross section are in excellent agreement with the recent experiments of Fort, Laucagne, Pesnelle, and Watel. Comparison with the experiments of Morgner at thermal energies, and of Howard, Riola, Rundel, and Stebbings, and Neynaber and Magnuson at higher energies suggests that Γ (R) for He(2 ^{3} S) –H is fairly accurate, and that Γ (R) for He(2 ^{1} S) –H is low by about a factor of 2.

Time and temperature dependence of aftereffects in [^{57}Co(phen)_{3}] (ClO_{4})_{2}⋅2H_{2}O from time‐differential Mössbauer emission spectroscopy
View Description Hide DescriptionUsing the time‐differential Mössbauer coincidence technique we have measured emission spectra of [^{57}Co(phen)_{3}] (ClO_{4})_{2}⋅2H_{2}O (phen=1, 10‐phenanthroline) in the temperature range 10–80 K versus K_{4}[Fe(CN)_{6}]⋅3H_{2}O as absorber. The lifetimes of the unstable ^{57}Fe(II) high‐spin states (^{5} T _{2}), which are formed as a consequence of the EC process of ^{57}Co in the source matrix, have been derived. Including the results from earlier time‐integral Mössbauer emission studies on the same system we find lifetimes ranging from 23 ns at 223 K to 393 ns at 10 K. The temperature dependence of the lifetime τ (^{5} T _{2}) is discussed assuming two possible decay mechanisms: (a) the tunneling of an electron to recombine with a ligand radical formed by autoradiolysis, and (b) the radiationless transition of an electronically excited spin state.

Pressure dependence of the lattice parameters of naphthalene up to 5.5 kbar and a re‐evaluation of the elastic constantssa)
View Description Hide DescriptionThe pressure dependence, up to 5.5 kbar, of the lattice parameters of naphthalene was determined, using neutron diffraction techniques. The variations were fitted to straight lines, the slopes of which are the linear compressibilities. The volume compressibility at 5 kbar was found to be in good agreement with the values obtained by Bridgman and by Vaidya and Kennedy. A set of elastic constants for naphthalene was calculated using the experimental values of the linear compressibilities. This set is found to be in close agreement (except for C _{11}) with a set calculated by Pawley, who used the general intermolecular potential of Kitaigorodskii.

Ion–molecule reactions of N^{+} with CO: Integral reactive cross sections in the collision energy range 0.2–13 eV
View Description Hide DescriptionThe guided beam technique, which allows the precise measurement of integral reactive cross sections of ions, has been used to measure the products of the collision of N^{+} with CO in the energy range 0.2–20 eV (lab). All five possible ionic reaction products CO^{+}, NO^{+}, C^{+}, CN^{+}, and O^{+} have been observed. The cross sections of the exothermic CO^{+} and NO^{+} channels show the usual decrease with E at low E (below 3 eV lab, i.e., 1 eV c.m.), then increase again to a flat maximum (near 12 eV lab, i.e., 5 eV c.m.). The CN^{+} and O^{+} channels show the expected thresholds, then a maximum (near 8 eV lab, i.e., 6 eV c.m.), whereas the C^{+} channel seems to have an activation energy and is still increasing at 13 eV c.m. There is no indication, in the form of kinks, of excited product channels except for dissociation. In addition, time‐of‐flight measurements have been done, giving some information on the forward component of the product velocity. The results are discussed in the light of simple theories, and of correlation diagrams supported by partial knowledge of the potential energy surface of (NCO)^{+} from CI calculations by A. A. Wu. This molecule has three stable linear isomers NCO^{+}, NOC^{+}, and CNO^{+}, which have also linear low lying excited states. So the discussion of the reaction as a collinear collision seems justified. The well in the ^{3}Π states can probably explain complex formation at low energies. The products CO^{+} and O^{+} cannot be formed adiabatically (even in C _{ s } symmetry). This is in accord with the small cross section for O^{+} formation, but not with the large one for CO^{+}, where one has to postulate the existence of nonadiabatic surface ’’jumps’’. Additional experiments, especially differential cross sections, are needed for a more comprehensive discussion of the molecular dynamics, which, after all, is not singular but rather prototypical for triatomic chemical reactions.

Initial slips and transient effects in relaxation phenomena
View Description Hide DescriptionFor a set of observables, it is shown how the solution of a master equation can approximate the solution of the generalized master quation at long times, but with an aproximate initial slip. The slip phenomena is discussed theoretically and then made more concrete by the exact and approximate solution of a simple but realistic system, namely the initial time dependence for the relaxation of a nuclear spin.

Gas phase infrared spectra of oxonium hydrate ions from 2 to 5 μ
View Description Hide DescriptionInfrared spectra of oxonium hydrate ions, generated by pulse radiolysis of argon containing small amounts of water vapor, have been measured using a signal averaging technique. The symmetric and antisymmetric OH stretch frequencies of the H_{3}O^{+} part of H_{9}O_{4} ^{+} are found at 3000 and 2660 cm^{−1}. H_{11}O_{5} ^{+} frequencies are found at 2180, 2860, and 3200 cm^{−1}. In addition, bands at 3620 and 3710 cm^{−1}, due to the outer water molecules are observed. Bands found during the growth of H_{9}O_{4} ^{+} at low water pressures are at 3490, 3290, 3170, 2850, and 2700 cm^{−1}. The 3490 cm^{−1} frequency is attributed to H_{3}O^{+}.

A b i n i t i o studies of the hydrated H_{3}O^{+} ion. II. The energetics of proton motion in higher hydrates (n=3–5)
View Description Hide DescriptionA b i n i t i o molecular orbital calculations on the higher hydrates of H_{3}O^{+}, using a split valence level basis set (4–31G), have led to the following results. (1) Energies for successive hydration are in good accord with gas‐phase thermochemical data. (2) Hydrogen‐bonded OH and O⋅⋅⋅O distances in H_{9}O_{4} ^{+} are in excellent agreement with condensed phase diffraction data. (3) Large variations in OH (≲0.08 Å) and O⋅⋅⋅O (∼0.25 Å) distances caused by strong hydrogen bonding are monotonically correlated with OH stretching force constants and frequencies, which cover a range of 1500 cm^{−1}. (4) Excellent quantitative correlation is obtained between calculated and observed infrared gas‐phase frequencies for six intense OH stretching bands, as represented by the least‐squares fit, ν_{exp}=−704+1440 (F _{ S } G _{ S })^{1/2}±15 cm^{−1}. This least squares relationship is used to assign some of the other experimental absorption bands from the gas phase. The only major uncertainty is in the case of the symmetric H_{3}O^{+} mode in H_{3}O^{+}(H_{2}O)_{3}. (5) The relative magnitudes of frequencies for symmetric (ν_{1}) and antisymmetric (ν_{3}) stretching modes are found to switch upon passing from H_{3}O^{+} (ν_{1}<ν_{3}) to H_{3}O^{+}(H_{2}O)_{3} (ν_{1}≳ν_{3}). (6) The isoelectronic species H_{3}O^{+}(H_{2}O)_{3} and OH^{−}(H_{2}O)_{3} are predicted have very similar frequencies for symmetric stretching of hydrogen‐bonded OH groups, in accord with Raman data for aqueous acid. (7) Addition of a fourth water molecule to the first solvation shell of H_{3}O^{+} does not lead to any significant stabilization, a result consistent with x‐ray and neutron diffraction results from aqueous HCl. (8) The most stable isomer of H_{13}O_{6} ^{+} is found to be an H_{9}O_{4} ^{+} moiety, somewhat perturbed from three‐fold symmetry by two second shell water molecules. (9) The tetrahydrate of the diaquahydrogen ion yields an H_{13}O_{6} ^{+} structure 2.2 kcal/mole higher in energy and provides a possible model for the symmetric intermediate in aqueous proton transfer (observed activation energy, 2.4 kcal/mole). This model emphasizes the importance of concerted reorganization of OH and O⋅⋅⋅O bond lengths in the vicinity of the excess proton, with proton tunneling not expected to be a major factor, since the postulated intermediate offers a symmetric s i n g l e‐w e l l potential for the excess proton.

Picosecond dynamics of conformational changes in 1,1′ binaphthyl
View Description Hide DescriptionOptically induced conformational changes in 1,1′ binaphthyl in different liquids have been studied on a picosecond time scale with subpicosecond optical pulses from a passively modelocked cw dye laser. The technique used was to measure the risetime of an excited state absorption induced by a 307.5 nm pulse and monitored with a 615 nm probing pulse. The risetime was found to be nonexponential with a 2.5±0.5 psec rapid initial rise followed by a slower component which varied from 11 to 22 psec depending on solventviscosity. The orientational relaxation of the molecule in its equilibrium excited state configuration was also obtained.

Diagrammatic perturbation theory applied to the ground state of the water molecule
View Description Hide DescriptionThe diagrammatic many‐body perturbation theory is applied to the ground state of the water molecule within the algebraic approximation. Using four different basis sets, the total energy, the equilibrium OH bond length, and the equilibrium HOH bond angle are examined. The latter is found to be a particularly sensitive test of the convergence of perturbation expansions. Certain third‐order results, which incorporate all two‐, three‐, and four‐body effects, show evidence of good convergence properties.

Stability and relative stability in reactive systems far from equilibrium. I. Thermodynamic analysis
View Description Hide DescriptionWe present in this article an analysis of the stability and relative stability of multiple stationary states in chemical systems driven far from equilibrium. Given the macroscopic kinetic (nonlinear) equations of motion, we postulate that there exists an entropy state function which is maximum for the (time‐dependent) solutions (concentrations of chemical species as a function of time) of the kinetic equations. We derive thermodynamicequations of motion for multivariable systems which follow from a thermodynamicvariational principle for the power dissipation and are consistent with the kinetic equations. From the thermodynamicequations of motion we obtain conditions for stability, marginal stability, and relative stability of stationary states. The point of coexistence of stationary states, being a special case of relative stability, is identified. We discuss examples of one and two variable systems. The thermodynamicanalysis for a one‐variable system agrees with a stochastic solution, and with a kinetic analysis presented in the next article.

Stability and relative stability in reactive systems far from equilibrium. II. Kinetic analysis of relative stability of multiple stationary states
View Description Hide DescriptionWe analyze the relative stability of multiple stationary states in chemically reactive systems far from equilibrium from a stochastic point of view. We take fluctuations into account and invoke the concept of mean first passage time, which gives a measure for the tendency of transition from one stationary state to the other. For one variable (and effectively one‐variable) systems we find analytic results that are in agreement with those of relative stability found in the previous article (Paper I) with a thermodynamicanalysis. The extension of the analysis to multivariable systems is indicated. We show that in the thermodynamic limit relative stability is determined solely by the properties of the stationary states.