Direct calculation of resonant states in reactive scattering. Application to linear triatomic systems

A direct method for calculating resonant states in reactive scattering is suggested, permitting us to obtain the characteristics of multichannel resonances (partial width amplitudes). The method is based on the construction of a Laurent expansion of the scattering matrix S(? ?iΓ/2) in the complex plane. The position of the poles of the S matrix are derived by solving the dynamical problem with complex energy values. The residue at the pole gives all the information concerning the partial widths. The method is applied to a linear triatomic reactive scattering problem. The properties of the resonant states in the H + H₂ system are calculated as an example. Two broad resonances are found which have not been reported in previous calculations. The interference of overlapping resonances is shown to have a profound effect on the energy dependence of the transition probabilities.     

Role of resonances in building cross sections: Comparison between the Mittag–Leffler and the T‐matrix Green function expansion approaches

Peaks in collision cross sections are often interpreted as resonances. The complex dilation method, as well as other methods relying on analytic continuation of the scattering formalism, can be used to clarify whether these structures are true resonances in the sense that they are poles of the S‐matrix and the associated Green function. The performance of the Mittag–Leffler expansion and T‐matrix Green function expansion methods are formally and computationally compared. The two methods are applied to two model potentials. Eigenenergies, s‐wave residues, and cross sections are computed with both methods. The resonance contributions to the cross sections are further analyzed by removing the residue contributions from the Mittag–Leffler and Green function expansion sums, respectively. It is suggested that the contribution of a resonance to a cross section should be defined through its S‐matrix residue. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Quantum-mechanical rearrangement processes: Off-shell vibrorotational sudden approximation to the dynamics

A vibrorotational sudden approximation for atom—diatom reactive scattering is developed, based on a T-matrix formulation of the process. The theory characteristically involves off-shell T-matrices that have been recently applied with remarkable success to the simpler atom—diatom subreactive regime. From the computational point of view, the scattering calculations must be performed in two successive steps. In the first, the scattering wavefunction, in the vibrorotational sudden approximation, must be evaluated and, for this purpose, standard computer packages of the inelastic regime can be immediately applied. This wavefunction must be used, next, to evaluate the off-shell T-matrix of the theory, by numerical quadrature. For this purpose, an efficient computational scheme is developed, that combines kinematical constraints and partial wave analysis requirements. The simplicity of the formalism and the off-shell constraint on the T-matrix suggest that the theory could be usefully applied for a rationalization of a large body of data in chemical reactions.     

Complex coordinate calculations of resonances in N-electron atoms

We present a discussion of the functional dependences of the wave functions for bound, resonant, and scattering states on the radial coordinate ρ and the rotation angle α in the complex coordinate method. We conclude that for bound states and resonances, ρ and α are constrained to appear in the wave functions only in the for ρ exp(iα). On the other hand, this constraint is not obtained for the scattering states since the energy of the scattering states depends on α. In addition we suggest a partitioning of the resonant wave function into two parts—a boundlike or “Q-space” part and a scattering like or “P-space” part. With these concepts one can incorporate physical insight into the choice of configurations as one does in other methods and can apply the complex coordinate method to many electron systems with an expected rate of convergence similar to other techniques. Its advantages are that a single calculation yields the position and width of the resonance, only square integrable functions are used, only a solution of a straightforward eigenvalue problem is required unlike some methods, arbitrarily accurate target states are easily incorporated, and polarization terms can easily be explicitly included. Variational calculations for the position and width of the lowest ²S resonance in the negative helium ion are reported using trial wave functions containing 39, 43, 55, 24, and 32 “P-space” configurations, respectively. Values of 19.387 eV and 12.13 meV are obtained for the position and width, respectively, for the resonance over a range in the rotation angle of almost two orders of magnitude. One also finds that inclusion of free-particle-like basis functions improves the representation of the scattering states.     

A re-examination of the justification of neglect of differential overlap approximations in terms of a power series expansion inS

Neglect of differential overlap methods are treated as approximations to calculations in a symmetrically orthogonalized basis. The accuracy of this approximation is investigated in terms of a power series expansion of the overlap matrix. TheS-matrix can be transformed into a matrix which will give a convergent series, and this series is used in the examination. The only approximation having any justification from this point of view is the NDDO method and even this neglects certain important three-electron integrals. Corrected expressions for the repulsion integral scaling factors introduced by Chandrasekharet al. are also derived. On leave from The Chemistry School, University of Western Australia.     

The non-N-representability of the Colle–Salvetti second-order reduced density matrix

The Colle–Salvetti second-order reduced density matrix (2-matrix) is an approximation to the 2-matrix obtained from a wave function that is a product of a reference wave function containing little or no correlation times a product of correlation factors that are functions of the coordinates of pairs of electrons. A formal proof is given for the non-N-representability for the Colle–Salvetti 2-matrix using the nonnegativity condition of the 2-matrix. The nonnegativity condition of the particle-hole overlap matrix (G matrix) is also not satisfied. The proof is valid for Colle–Salvetti 2-matrices obtained from both the Hartree–Fock and small multiconfigurational-self-consistent-field wave functions. Even though the Colle–Salvetti 2-matrix is not N-representable, it does satisfy the Pauli principle component of the G-matrix condition because it reduces to an N-representable first-order reduced density matrix. © 1993 John Wiley & Sons, Inc.     

Dynamical and Mechanical Insights into the Li(2S)+ HCl( ) Reaction: A Detailed Quantum Wavepacket Study

Quantum wave packet dynamics of the Li(²S)+HCl( ) reaction in its electronic ground state is studied. The initial state-selected and energy-resolved dynamical attributes such as reaction probability, integral cross section, and thermal rate constant for the Cl-abstraction and H-abstraction pathways are reported. All partial wave contributions of J up to 120 were found to be necessary for the title reaction up to the collision energy of ∼1.0 eV. The dynamical results reveal that the Cl-abstraction is more favored over the H-abstraction for the different rovibrational (v, j) excitations. Due to the existence of an early barrier in the potential energy surface, the cross sections increase with increasing collision energy. The rate constants also monotonously increase with temperature for both channels. Resonances are identified and characterized in terms of eigenfunctions and lifetimes. Nearly 120 well-resolved eigenstates are reported for the LiHCl complex, and they are categorized as van der Waals (vdW), barrier and product states according to the nodal progressions along (R, r, γ). The vdW resonances reveal a local-mode behavior of quasibound type at low energies and extended progressions at high energies. Further, the single-quantized periodic orbit type is also observed in the barrier region, which decays very fast. Finally, the lifetime analysis reveals that the vdW resonances can survive as long as ∼2.2 ps, which is much longer than the lifetime of the resonances in the barrier region.     

Relations Between Atomic Transport Coefficients in Simple Liquids and Moments of Neutron Scattering Functions

Abstract

A possible alternative to Kubo theory is discussed for relating atomic transport coefficients in simple liquids. The arguments used result in a new expansion, in which the density-density response function x is expanded in a power series in the self response function X_s. The coefficients of this expansion can be determined, in principle, to all orders, from the moments of the neutron scattering functions S(qω) and S _s(qω). It is then proposed, by appeal to the hydrodynamic equations, that the radius of convergence of this series can be used to relate diffusion to the sound wave attenuation coefficient. Finally, this x-x_s expansion allows a direct comparison of the exact theory presented in this paper with earlier approximate theories relating incoherent and coherent neutron scattering from liquids.     

Structure factor of a self-avoiding polymer chain at high momentum transfer

The molecular weight of a Gaussian polymer chain can be obtained from the intercept of the asymptote in a plot of the inverse structure factor 1/S(q) as a function of the square of the wave vector q². Using an ϵ expansion and scaling arguments, we show that in a good solvent the situation is more complex and that the molecular weight determination is difficult to justify. The corrections to the asymptotic behavior of the structure factor at large wave vector involve several terms that are difficult to separate experimentally. This is qualitatively explained by the nonuniform swelling of a polymer chain in a good solvent due to the existence of the chain ends.     

Di­chloro­[(6R,7S,8S,14S)‐(–)‐sparteine‐κ2N,N′]­nickel(II)

In the title compound, [NiCl₂(C₁₅H₂₆N₂)], the chiral alkaloid (6R,7S,8S,14S)‐(−)‐l ‐sparteine acts as a bidentate ligand, with two chloride ligands occupying the remaining coordination sites, producing a slightly distorted tetrahedron. The N—Ni—N plane in the title complex is twisted by 81.31 (11)° from the Cl—Ni—Cl plane. Other distortions of the tetrahedron are discussed.     

Accurate modeling of the intramolecular electrostatic energy of proteins

The (?, ψ) energy surface of blocked alanine (N-acetyl–N′-methyl alanineamide) was calculated at the Hartree-Fock (HF)/6-31G* level using ab initio molecular orbital theory. A collection of six electrostatic models was constructed, and the term electrostatic model was used to refer to (1) a set of atomic charge densities, each unable to deform with conformation; and (2) a rule for estimating the electrostatic interaction energy between a pair of atomic charge densities. In addition to two partial charge and three multipole electrostatic models, this collection includes one extremely detailed model, which we refer to as nonspherical CPK. For each of these six electrostatic models, parameters—in the form of partial charges, atomic multipoles, or generalized atomic densities—were calculated from the HF/6-31G* wave functions whose energies define the ab initio energy surface. This calculation of parameters was complicated by a problem that was found to originate from the locking in of a set of atomic charge densities, each of which contains a small polarization-induced deformation from its idealized unpolarized state. It was observed that the collective contribution of these small polarization-induced deformations to electrostatic energy differences between conformations can become large relative to ab initio energy differences between conformations. For each of the six electrostatic models, this contribution was reduced by an averaging of atomic charge densities (or electrostatic energy surfaces) over a large collection of conformations. The ab initio energy surface was used as a target with respect to which relative accuracies were determined for the six electrostatic models. A collection of 42 more complete molecular mechanics models was created by combining each of our six electrostatic models with a collection of seven models of repulsion + dispersion + intrinsic torsional energy, chosen to provide a representative sample of functional forms and parameter sets. A measure of distance was defined between model and ab initio energy surfaces; and distances were calculated for each of our 42 molecular mechanics models. For most of our 12 standard molecular mechanics models, the average error between model and ab initio energy surfaces is greater than 1.5 kcal/mol. This error is decreased by (1) careful treatment of the nonspherical nature of atomic charge densities, and (2) accurate representation of electrostatic interaction energies of types 1—2 and 1—3. This result suggests an electrostatic origin for at least part of the error between standard model and ab initio energy surfaces. Given the range of functional forms that is used by the current generation of protein potential functions, these errors cannot be corrected by compensating for errors in other energy components. © 1995 by John Wiley & Sons, Inc.     

Electronic transitions in collinear H+ + D2 (ν = 0) → HD+ (ν = 0,1) + D via classical trajectories in complex time and complex phase space

A semiclassical treatment of electronic transitions in the collinear rearrangement H⁺ + D₂ (ν = 0) → HD⁺ (ν = 0,1) + D is presented. The treatment represents an extension of Stueckelberg's method for a single nuclear degree of freedom to collisions involving several nuclear degrees of freedom. The classical limit of scattering amplitudes (S-matrix elements) is calculated for the transition between the two adiabatic potential energy surfaces corresponding to the two lowest singlet states of HD⁺₂. S-matrix elements are constructed from trajectories propagating in complex time and complex phase space, which make localized transitions between the two surfaces by crossing their complex line of intersection. The action along each trajectory acquires an imaginary part, which contributes exponential damping to the corresponding amplitude for electronic transition.     

A mathematical model of realistic constitutional chemistry. A synthon approach. I: An algebraic model of a synthon

A mathematical model of a synthon is suggested. The synthon is modelled by a special so-calledS-matrix. The notion of isomeric synthons on the set of atoms A and that of a Family of Isomeric SynthonsFIS (A) is introduced. The chemical reaction is represented by a matrix equation and it is modelled by the so-calledSR-matrix. The notion of the reaction distance (RD) between two isomeric synthons is defined. A mathematical theory of the S andSR-matrices is developed.     

Energies of Resonances in Atoms and Ions with Three Electrons

The energies of resonances of ² Ssymmetry were calculated for atoms and ions with three electrons, H^2?, He^?, and Li, using the stabilization technique. The variational wave function was expanded over a large number (up to 2030) of normalized three-electron basis functions. The basis functions that made significant contributions to the energies were selected using a special procedure. Many high-energy resonances were found with energies of H^2? resonances ranging from 1 to 12 eV. The calculated energies of some He^? and Li resonances were compared with the experimental data and calculations of other authors. The selected basis set of three-electron wave functions was found to be efficient in calculations of high-energy resonances.     

Ultrasonic velocity and apparent isentropic compressibilities in mixtures of nonelectrolytes

Ultrasonic velocities have been determined for binary mixtures of pyridine + n-alkanol (C₁-C₁₀) over the whole composition range at 25‡C. The excess isentropic compressibilities K _S ^E and apparent molar isentropic compressibilities K_Φ,s are estimated from these measurements. The K _S ^E values are negative for all the systems over the complete mole fraction range except pyridine + decanol for which small positive values are obtained. The standard partial molar isentropic compressibilitiesK‡ of the alkanols are positive and increase linearly with the chain length of the alkanol molecules. It indicates that a methylene functional group makes a positive contribution to the expansion coefficient of a solute in these mixtures.     

Dichloro­[(6R,7S,8S,14S)‐(–)‐sparteine‐κ2N,N′]mercury(II)

In the title compound, [HgCl₂(C₁₅H₂₆N₂)], the chiral alkaloid (6R,7S,8S,14S)‐(−)‐l ‐sparteine acts as a bident­ate ligand, with two Cl⁻ ligands occupying the remaining coordination sites, producing a distorted tetra­hedron. The N—Hg—N plane is twisted by 81.1 (2)° from the Cl—Hg—Cl plane. The mid‐point of the N⋯N line does not lie exactly on the Cl—Hg—Cl plane but is tilted towards one of the N atoms by 0.346 Å. Similarly, the mid‐point of the Cl⋯Cl line is tilted toward one of the Cl atoms by 0.163 Å. The packing structure shows that the complex is stabilized by two inter­atomic Cl⋯H contacts involving both Cl atoms and the methyl­ene or methine H atoms of the (−)‐sparteine ligand.     

The geometry and spectra of hyperbolic manifolds

This paper is a self-contained discussion of the relationship between spectral and geometric properties of a class of hyperbolic manifolds. After a review of the fundamentals of hyperbolic manifolds, aspects of the theory for the compact case and the finite-volume case are discussed. The main emphasis of this work is on a class of infinite-volume hyperbolic manifolds ℳ which arise as quotients of hyperbolic spaceH ⁿ by discrete subgroups Г, i.e. ℳ =H ⁿ/Г. This paper describes joint work with R G Froese and P A Perry. For these infinite-volume hyperbolic manifolds, there are very few eigenvalues, so most of the spectral information in carried by the generalized eigenfunctions of the Laplacian. These eigenfunctions can be constructed from the asymptotics of the Green’s function. It is shown how the asymptotic geometry of the manifold determines the asymptotic behavior of the Green’s function, and hence the eigenfunctions, near infinity. This information is used to construct anS-matrix for the manifold which is a pseudo-differential operator acting on sections of a fibre bundle over the boundary of the manifold at infinity. The meromorphic properties of this operator and its inverse, as a function of the spectral parameter, are described. A functional relation between theS-matrix and the generalized eigenfunctions is derived. An important consequence of this relation and the meromorphicity of theS-matrix and its inverse is the existence of the meromorphic continuation of the Eisenstein series associated with the discrete group Г. Finally, an overview of recent progress and some open problems are presented, including a discussion of the asymptotic behavior of the counting function for the scattering poles. Research supported in part by NSF grant DMS93-07438     

A one‐dimensional platinum mixed‐valence complex with bridging thiocyanate S atoms: [[PtII(en)2](μ‐SCN)[PtIV(en)2](μ‐SCN)](ClO4)4 (en is ethane‐1,2‐diamine)

A new one‐dimensional platinum mixed‐valence complex with nonhalogen bridging ligands, namely catena‐poly[[[bis(ethane‐1,2‐diamine‐κ²N,N′)platinum(II)]‐μ‐thiocyanato‐κ²S:S‐[bis(ethane‐1,2‐diamine‐κ²N,N′)platinum(IV)]‐μ‐thiocyanato‐κ²S:S] tetrakis(perchlorate)], {[Pt₂(SCN)₂(C₂H₈N₂)₄](ClO₄)₄}_n, has been isolated. The Pt^II and Pt^IV atoms are located on centres of inversion and are stacked alternately, linked by the S atoms of the thiocyanate ligands, forming an infinite one‐dimensional chain. The Pt^IV—S and Pt^II...S distances are 2.3933 (10) and 3.4705 (10) Å, respectively, and the Pt^IV—S...Pt^II angle is 171.97 (4)°. The introduction of nonhalogen atoms as bridging ligands in this complex extends the chemical modifications possible for controlling the amplitude of the charge‐density wave (CDW) state in one‐dimensional mixed‐valence complexes. The structure of a discrete Pt^IV thiocyanate compound, bis(ethane‐1,2‐diamine‐κ²N,N′)bis(thiocyanato‐κS)platinum(IV) bis(perchlorate) 1.5‐hydrate, [Pt(SCN)₂(C₄H₈N₂)₂](ClO₄)₂·1.5H₂O, has monoclinic (C2) symmetry. Two S‐bound thiocyanate ligands are located in trans positions, with an S—Pt—S angle of 177.56 (3)°.     

On the behaviour of the wave function for the 23S state of the two-electron atom in the region where both electron–nuclear distances are small

The expansion of the wave function for the 2³S state of the two-electron atom in the neighbourhood of the singularity at r₁ = r₂ = 0 is considered. The restrictions imposed on the variational functions by this expansion are discussed. For the 2³S state of He, Li⁺, N⁵⁺ the behaviour of the variational function based on the Fock expansion in the neighbourhood of this singularity is investigated. The agreement of the variational coefficients with the theoretical coefficients is satisfactory. The calculated values of E and 〈δ(r₂)〉 for He, Li⁺, N⁵⁺ are given.     

anti-Bis(μ-2-ammonio­ethane­thiol­ato-κ2S:S)­bis­[di­chloro­palladium(II)] dihydrate

Each of two square-planar Pd^II ions in the title compound, [Pd₂Cl₄(μ-Haet-S)₂]·2H₂O (Haet = 2-ammonio­ethane­thiol­ate, C₂H₇NS), which was obtained by rearrangement of [Pd₂{Pd(aet-N,S)₂}₄]⁴⁺ in acidic solution, is coordinated by two bridging S atoms from two Haet ligands and by two terminal Cl atoms, forming the dinuclear structure. Since the complex is situated on a center of symmetry, the two monodentate Haet arms are located on opposite sides of the central Pd₂S₂ square plane, i.e. the present complex is the anti isomer. The S—C—C—N torsion angle is 177.3 (6)° and some intermolecular hydrogen bonds are observed.