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Volume 67, Issue 4, 15 August 1977

Quantum mechanical theory of a structured atom–diatom collision system: A+BC(^{1}Σ)
View Description Hide DescriptionThe problem of a ^{2} P state atom colliding with a singlet sigma state diatom, which involves multiple potential surfaces, is investigated. Within a diabatic representation for the electronic degrees of freedom (plus spin–orbit interaction), coupled scattering equations are derived in both space‐fixed and body–fixed coordinate systems, and coefficients analogous to Percival–Seaton coefficients are exhibited. Approximations to the exact equations, including angular momenta decoupling approximations, are discussed for both the space‐fixed and body‐fixed formalisms.

Electronuclear basis for three‐dimensional electronic nonadiabatic chemical reactions
View Description Hide DescriptionThe quantum mechnical theory of electronic nonadiabatic atom–diatomic molecule reactive collisions in three dimensions is discussed with emphasis on the reaction F(^{2} P _{3/2},^{2} P _{1/2})+H_{2}(^{1}Σ^{+} _{ g },v,j). It is known that the first two excited electronic adiabatic surfaces of the FH_{2} system are very close to the ground statesurface at large F–H_{2} distances. Only the ground statesurface, however, can lead to reaction at subdissociative energies. The importance of considering the exchange of flux between these surfaces is discussed. An electrorotational basis set {Ω^{ J M } _{α}} is formulated for use in a quantum mechanical, reactive scattering calculation for systems of this type. These basis functions are formed by starting with good total angular momentum (nuclear plus electronic) kets in the space‐fixed frame. Then, these functions are written in terms of the body‐fixed natural collision (NCC) coordinates, and the Euler angles that orient the body frame with respect to the space frame. One feature of the {Ω^{ J M } _{α}} setting them apart from other basis sets typically used in scattering calculations is that they cannot be factored into products of nuclear and electronic parts. It is then shown how the Ω^{ J M } _{α} may be written in terms of adiabatic electronic basis functions used in the ’’diatomics‐in‐molecules’’ (DIM) treatment of atom–diatom systems. By writing the {Ω^{ J M } _{g}a} in terms of DIM basis functions, matrix elements of the type 〈Ω^{ J M } _{α′}‖Ĥ_{el}‖Ω^{ J M } _{α}〉 are evaluated as linear combinations of pure electronic matrix elements of Ĥ_{el} in the DIM basis. Four nonmixing parity types are found for the Ω^{ J M } _{α}. Plots of selected matrix elements 〈Ω^{ J M } _{α′}‖Ĥ_{el}‖Ω^{ J M } _{α}〉 as functions of the reaction coordinate s and vibrational coordinate ρ are presented for F+H_{2}. Matrix elements of T̂_{rot} (the nuclear rotational kinetic energy operator expressed in NCC) are also evaluated in the electrotational basis. The method of conversion from the electrotational basis to an adiabatic electronic basis is formulated. This transformation should be useful after the system passes the zone where electronic transitions are significant.

Hidden potentials in classical theorems
View Description Hide DescriptionSome exact equations are derived which clarify some potentials that are hidden in the classical theorems such as the Hellmann–Feynman (H–F) and the integral Hellmann–Feynman (I‐H–F) theorems. The differential form of the density equation given previously includes not only the classical force operator but also the force operator associated with the quantum‐mechanical potential introduced by Bohm. The latter arises essentially only from the noncommuting property of coordinates and momenta in quantum mechanics. However, after integration only the classical force term survives and results in the H–F theorem. The important role of the quantum force term is completely h i d d e n in the H–F theorem. This fact would be closely related to the nondeterminicity and the classical interpretation of the H–F theorem. A similar role of the quantum potential is also shown for the I‐H–F theorem. We have also investigated the origin of force density (the integrand of the H–F theorem) and isolated the roles of the generalized exchange and correlation effects which are also h i d d e n in the H–F theorem.

Short range antiferromagnetic coupling in bis(N,N‐dialkyldithiocarbamato) iron^{III} halides
View Description Hide DescriptionSeveral bis(N,N‐dialkyldithiocarbamato) iron^{III} have been studied by Mössbauer spectroscopy in the presence of external magnetic fields. The spectra have shown strong evidence of dimeric antiferromagnetic coupling between the molecular units in the crystal lattice. The dimeric coupling occurs through superexchangeinteracton between different molecules. The magnitude of the coupling constant J _{ D }, which was estimated from the magnetically perturbed Mössbauer spectra, was found to be of the order of 0.25–0.50 K for the complexes studied here.

Use of the virial theorem in construction of potential energy functions for diatomic molecules
View Description Hide DescriptionThe nth‐order diatomic potential energy functions W ^{(T)} _{ n } and W ^{(V)} _{ n } are constructed, by the integration of the virial theorem −W−R[d W/d R]=T and 2W+R[d W/d R]=V, respectively, using the nth‐order truncations of the perturbational λ=1−(R _{ e }/R) power series expansions of the kinetic (T) and potential (V) parts of the vibrational potential. The resulting W ^{(T)} _{ n } potential is a linear combination of terms R ^{−1}, R ^{−2},⋅⋅⋅,R ^{−n }, and (lnR)/R; the W ^{(V)} _{ n } potential is a linear combination of terms R ^{−1}, R ^{−2},⋅⋅⋅,R ^{−n }, and (lnR)/R ^{2}. For n=2, predictions of W ^{(T)} _{2}, W ^{(V)} _{2}, and also the generalized two‐logarithmic second‐order potential W ^{(T,V)} _{2} [including both the (lnR)/R and (lnR)/R ^{2} terms] are compared with experiment and the results obtained from the Morse and Clinton potentials. Second‐order logarithmic potentials for the ground states of H_{2}, CO, and HF are given and compared with the Kol/os and Wolniewicz potential for H_{2} and the RKR classical turning points for CO and HF. Convergence properties of the W ^{(T)} _{ n } and W ^{(V)} _{ n } potentials are tested using the ground state of H_{2} as an example.

On the dynamics of exothermic triatomic exchange reactions
View Description Hide DescriptionAnalytic expressions are derived for the population of vibrational states for triatomic exchange reactions. The predictions are in good accord with the so‐called ’’Polanyi rules.’’ The results are expressed in terms of the masses, frequencies, exothermicity, interaction length, and the attractive part of the potential A _{⊥} E*. Comparison with exact quantum‐mechanical calculations is made for a series of light‐heavy‐heavy (LHH) reactions. An almost linear relation between the attractive part of the potential and the product vibrational energy is obtained. For the LHH‐mass combination the slope is almost one. For other mass combinations like HHL, HLL, or HLH the final vibrational energy is relatively insensitive to A _{⊥}, showing always strong population inversion. The width of the vibrational energy distribution is smallest for the HLH mass combination.

Linear response theory and the mechanical energy relaxations of solid high polymer systems
View Description Hide DescriptionLinear response theory is applied to mechanical energy relaxation phenomena of polymer solids below the glass transition temperature. The relaxation modulus is related to time‐correlation functions of ’’forces.’’ The time‐correlation functions are approximately calculated in the paper in the single relaxation time model. The relaxation time is explicitly calculated in the approximation for systems with pendant cyclohexyl groups. The results obtained are examined in the light of experimental data and show that the present linear response theory approach is promising for mechanical spectroscopy of solid polymer systems.

Ionization of xenon atoms in high Rydberg states by collision with molecules
View Description Hide DescriptionIonization of xenon atoms in single highly excited states ‖n f≳ (where 25⩽n⩽40) by collisions with CCl_{4}, CCl_{3}F, and SF_{6} has been investigated. Absolute rate constants for Xe^{+} production are reported together with the identities of the major negatively charged species produced in the collisions. Cross sections determined from these rate constants are also given. The data lend support to the theoreticalmodel, which views such collisions as being dominated by the interaction between the target molecules and the excited electron, with the Xe^{+} ion core playing a minor role.

Nuclear corrections to molecular properties. V. Refinements in a b i n i t i o normal‐coordinate potential energy and property surfaces for water and their effect on the vibrational analysis
View Description Hide DescriptionA b i n i t i osurfaces in reduced normal‐coordinate space for water, for the potential energy and components of the dipole moment and second moment of charge about the center of mass, have been refined for use in calculating energies of vibration and expectation values of of these properties in low‐lying states of vibration. The two procedural refinements consist of (i) direct fitting of the surfaces to values obtained at a grid of points, where the internal coordinates for each point have been transformed to reduced normal coordinates by an iterative procedure, and (ii) (small) rotation of the axes into coincidence with a vibrationally invariant set at each asymmetric configuration on the grid. The first refinement brings the calculated values of the constants of anharmonicity x _{ i j } into much closer agreement with experiment. Both refinements generate significant changes in the fitted propertysurfaces. For several isotopic species of water calculated expansion coefficients are listed for the potential energy in reduced normal coordinates and the dipole moment in internal and reduced normal coordinates; expectation values are also given.

Recoil‐ and field‐enhanced diffusion
View Description Hide DescriptionA quantum mechanical description is presented of diffusion that is enhanced by the presence of a constant force field (field‐enhanced diffusion) or by the recoil energy obtained by collisions with bombarding particles (recoil‐enhanced diffusion). One‐ and three‐dimensional equations are given, the one‐dimensional one being particularly pertinent to channeling situations. Methods for calculating relevant transition rates are examined; these are used to illustrate the similarities between the two enhancements.

Microscopic theory of polymer internal viscosity: Mode coupling approximation for the Rouse model
View Description Hide DescriptionThe internal viscosity or friction coefficient η̂_{κ} for normal mode κ of an N‐monomer Rouse‐type ringpolymer is defined microscopically using the Mori projection technique and is then computed within the simplest (trilinear) mode coupling approximation. One finds that η̂_{κ}=ζγy[(κ/N)^{2}], where y (x) is a nonanalytic function of x, ζ is the monomerfriction coefficient, and γ is proportional to the square of an equilibrium four point polymercorrelation function. This correlation function is nonvanishing for non‐Gaussian chains; e.g., γ=18/25 for freely jointed rigid rods. For κ/N→0, y (x) is found explicitly to yield ^{lim} _{(κ/N)} _{→0}η̂_{κ} = (ζγ/2π√3)(πκ/N)^{2} ln(N/πκ). For 0.45≲κ/N≲0.50 numerical evaluation of y (x) shows η̂_{κ} is approximately independent of κ. For the bulk of the κ range, 0.10≲κ/N≲0.40, η̂_{κ}≃ζΦ [(κ/N)−0.05], where Φ is an order unity constant proportional to γ. For the freely jointed chain Φ≃1.7. This last form for η̂_{κ} agrees well with the Cerf–Peterlin empirical form η̂_{κ}=ζ (κ/N) φ. Experimental data are typically fit with φ∼1−3.

The rotational sudden approximation at low energies
View Description Hide DescriptionThe infinite‐order sudden approximation is studied at low energies for a system with widely spaced energy levels to determine the limits of its validity. It is found that even under these extreme conditions it gives highly accurate results for cross sections summed over final rotational states. The elastic cross sections are also reasonably accurate in this approximation, though, for this worst case situation, the state to state cross sections for inelastic scattering are not even qualitatively correct for high Δj transitions. The high accuracy of the cross sections summed over final rotational states commends the sudden approximation for use in reducing the complexity of carrying rotational channels in the calculation of vibrational transitions. The use of the sudden approximation in studying expansions of interaction potentials is discussed.

Coulomb holes and correlation coefficients for electronic shells: The Be‐like ions
View Description Hide DescriptionAn expression has been obtained for the Coulomb hole associated with any pair of occupied HF spin orbitals for a many‐electron system. The required partitioning of the known correlated second‐order density matrix was achieved here, up to and including the pair‐correlation effects, by using the many‐particle theory proposed by Sinanoğlu. Ground‐state wavefunctions were then analyzed for the Be‐like ions when 3⩽Z⩽8 which, besides validating the partitioning technique, provided insight into the ’’radial’’ and ’’angular’’ components of intrashell correlation effects. The correlation coefficients τ_{ϑ}, τ_{ r } and τ_{1/r }, taken from classical statistics, were also determined and their variation was examined with respect to the atomic number Z. These techniques provided a clear‐cut means for comparing and contrasting, in a quantitative manner, the role of correlation within individual atomic shells. Moreover, the procedure is applicable to the comparative assessment of different correlated wavefunctions for the same state of a given system.

Laser photoelectron spectrometry of C_{5}H^{−} _{5}: A determination of the electron affinity and Jahn–Teller coupling in cyclopentadienyl
View Description Hide DescriptionThe photodetachment electron spectrum of cyclopentadienide (C_{5}H^{−} _{5}) yields a value for the cyclopentadienyl electron affinity of 1.786±0.020 eV. Additionally, the spectrum shows structure corresponding to vibrations in the neutral radical. From this an upper bound is placed on the linear Jahn–Teller coupling constant k ^{2}<0.5. The quadratic coupling is shown theoretically to be absent for this molecule, a special case of the absence of quadratic coupling in D _{ n } or C _{ n } groups when n is not divisible by three. The magnitude of this Jahn–Teller distortion is sufficient to induce nontotally symmetric Jahn–Teller modes in the spectrum, but is insufficient to suppress the symmetric modes and, in particular, a strong 0–0 transition is observed. Thus, the geometries of the ion and the neutral, apart from the Jahn–Teller distortion, are not radically different.

Large amplitude vibrational motion in a one dimensional chain: Coherent state representation
View Description Hide DescriptionA study is made of the quantum mechanical motion of a one dimensional finite chain of anharmonic oscillators with free ends. It is shown that, for states which time evolve as coherent states (minimum uncertainty wave packets) of the normal mode vibrations, the motion is equivalent to a classical system with an effective potential interaction determined by convoluting the quantum wave packet and the potential energy. Some examples are discussed, with particular attention given to the Toda and Morse potentials, which are shown to be invariant in form under this convolution. The similarities between the classical Toda and Morse lattices are then utilized to infer the existence of compressional solitary waves in the Morse lattice from the well known soliton solutions of the Toda lattice. Further, for the Morse lattice an analytic expression is found for the first order perturbative correction to the Toda solitons for large amplitude vibrations. We also discuss the relation between the existence of such solitary waves and the rate of vibrational relaxation in molecular systems.

Concentration dependence of the zero‐field splittings of triplet phenanthrene‐d _{10} oriented in biphenyl single crystals
View Description Hide DescriptionZero‐field splittings and splitting parameters of phosphorescent phenanthrene oriented in single crystal biphenyl are reported as functions of guest content in the range 10^{−6} to 2×10^{−2} mole fraction guest from measurements at ∼77 K. The dependences, which have been fit to equations of the form y=a x ^{ b }, are attributed to molecular crystal field effects. In addition, EPR absorption linewidths have been measured as a function of guest content and are discussed in terms of spin–spin interactions. The liquid–solid phase behavior of the two component phenanthrene–biphenyl system is also reported.

Laser correlation spectroscopy of amorphous polymethylmethacrylate
View Description Hide DescriptionWe have studied the light scattered from amorphous polymethylmethacrylate (PMMA) using the correlation technique for a range of temperatures from 6 to 165 °C encompassing the glass transition temperature (T _{ g }∼120 °C). The data were analyzed in terms of two exponential decays, and the angular dependence of each of the corresponding relaxation frequencies was examined. The results for the high frequency relaxation mode are angular independent and fall reasonably well on two straight lines of different activation energy (∼8 kcal/mole at high temperatures and ∼1 kcal/mole at low temperatures), indicating the presence of two coupled relaxation mechanisms. The low frequency relaxation results are quite sensitive to the inhomogeneities of unannealed samples and have in this case irregular angular dependences. The angular dependence disappears for samples close to T _{ g } and for annealed samples. Below T _{ g }, this relaxation process has a fairly constant frequency of about 3 Hz independent of temperature. Above T _{ g }, its frequency increases very rapidly with temperature to reach 130 Hz at 165 °C and follows the backbone main‐chain relaxation frequency measured by other techniques.

Quantum exchange effects in trimer ground states
View Description Hide DescriptionThe ground state of molecular trimers is studied in three dimensions for these pair potential models: the square well, the exponential, the Yukawa, the Gaussian, and the Lennard‐Jones 12–6. Three spin cases are considered: three spin‐zero bosons and the spin‐ (1/2) and spin‐ (3/2) states of three spin‐ (1/2) fermions. Variational wavefunctions are constructed which satisfy the exchange symmetry requirements for these cases. Bounds are obtained for the threshold coupling constants at which self‐bound trimers occur. Consequences for possible self‐bound trimers of ^{3}He are discussed.

Microwave optical double resonance spectrum of NH_{2}. III. ^{1}H and ^{14}N hyperfine coupling constants and spin rotation constants in X̃ ^{2} B _{1}
View Description Hide DescriptionOver 200 microwave transitions involving 31 rotational levels in the X̃ ^{2} B _{1} state%of NH_{2} have been observed using microwave optical double resonance. The majorityh of the observed lines are magnet magnetic dipole allowed transitions between the two spin components of a given rotational state, J=N−1/2←N+1/2, N _{ k } _{ a } _{ k } _{ c } The observed lines have been analyzed to yield accurate values for the spin rotation (including those describing the effects of centrifugal distortion), ^{14}N hyperfine and ^{1}H hyperfine constants. The three spin rotation constants are determnied to be A _{ s }=−9267.8 (1.4), B _{ s }=−1353.9 (0.5), and C _{ s }=12.0(0.3), where all constants are in MHz and the quoted error in parentheses is three standard deviations. The five centrifugal distortion constants for the spin rotation interaction derived are (in a notation analogous to Watson’s rotational centrifugal distortion constants) Δ^{ s } _{ N }=−0.312(0.007), ΔS _{ k }=−33.9 (0.6), Δ^{ S } _{ N K }=3.51(0.11), δ^{ s } _{ N }=−0.155 (0.004), and δ^{ s } _{ K }=−0.480 (0.077). The ^{14}N hyperfine constants determined in this analysis are (O)_{ I }=28.2 (0.4), (a a)_{ I }=−42.8 (1.3), (b b)_{ I }=−44.7(1.0), and (c c)_{ I }=87.5 (0.5). The ^{1}H hyperfine constants are (O)_{ I }=−67.2(0.4), (a a)_{ I }=18.6(1.3), (b b)_{ I } =−13.4(0.9), and (c c)_{ I }=−5.2(0.6). Quadrupole constants for ^{14}N are also determinable from the data as (a a)_{ Q }=0.1 (1.3), (b b)_{ Q } =−1.6(1.1), and (c c)_{ Q }=1.5(0.6).

Energy distribution in selected fragment vibrations in dissociation processes in polyatomic molecules
View Description Hide DescriptionThe full quantum theory of dissociation processes in polyatomic molecules is converted to a form enabling the isolation of a selected fragment vibration. This form enables the easy evaluation of the probability distribution for energy partitioning between this vibration and all other degrees of freedom that results from the sudden Franck–Condon rearrangement process. The resultant Franck–Condon factors involve the square of the one‐dimensional overlap integral between effective oscillatorwavefunctions and the wavefunctions for the selected fragment vibration, a form that resembles the simple golden rule model for polyatomic dissociation and reaction processes. The full quantum theory can, therefore, be viewed as providing both a rigorous justification for certain generic aspects of the simple golden rule model as well as providing a number of important generalizations thereof. Some of these involve dealing with initial bound state vibrational excitation, explicit molecule, fragment and energy dependence of the effective oscillator, and the incorporation of all isotopic dependence. In certain limiting situations the full quantum theory yields simple, readily usable analytic expressions for the frequency and equilibrium position of the effective oscillator. Specific applications are presented for the direct photodissociation of HCN, DCN, and CO_{2} where comparisons between the full theory and the simple golden rule are presented. We also discuss the generalizations of the previous theory to enable the incorporation of effects of distortion in the normal modes as a function of the reaction coordinate on the repulsive potential energy surface.