Simonetta Cavalli

Hyperquantization algorithm. I. Theory for triatomic systems

In this paper we present the theoretical concepts and methodology of the hyperquantization algorithm for the three body quantum mechanical problem. Within the framework of the hyperspherical approach to reaction dynamics, we use angular momentum algebra (or its generalization, e.g., including Hahn coefficients which are orthonormal polynomials on a set of grid points which span the interaction region) to compute matrix elements of the Hamiltonian operator parametrically in the hyperradius. The particularly advantageous aspects of the method proposed here is that no integrals are required and the construction of the kinetic energy matrix is simple and universal: salient features are the block tridiagonal structure of the Hamiltonian matrix and a number of symmetry properties. The extremely sparse structure is a further advantage for the diagonalization required to evaluate the adiabatic hyperspherical states as a function of the hyperradius. Numerical implementation is illustrated in the following paper by...

Hyperspherical coordinates for molecular dynamics by the method of trees and the mapping of potential energy surfaces for triatomic systems

Some results on hyperspherical coordinates and harmonics for the representation of the many‐body problem are presented, extensive use being made of the method of trees. Properties of these trees are examined: a lemma on the simplification of trees possessing a particular symmetry is proven, and used to discuss the internal coordinates for a system of three particles and the mapping of potential energy surfaces. A framework is provided for relating different couplings of particles by rotations on hyperspheres and alternative hyperangular parametrizations by orthogonal basis transformations. Extensions to nonzero angular momentum or to more than three particles are shown not to be trivial, and the possible role of developments of the tree method, leading to more general hyperspherical coordinates, is briefly considered.

Coordinates for molecular dynamics: Orthogonal local systems

Systems of orthogonal coordinates for the problem of the motion of three or more particles in classical or in quantum mechanics are considered from the viewpoint of applications to intramolecular dynamics and chemical kinetics. These systems, for which the kinetic energy of relative motion is diagonal, are generated by making extensive use of the concept of kinematic rotations, which act on coordinates of different particles and describe their rearrangements. An explicit representation of these rotations by mass dependent matrices allows to relate different particle couplings in the Jacobi scheme, and to build up alternative systems (such as those based on the Radau–Smith vectors or variants thereof): this makes it possible to obtain coordinates which, while being rigorously orthogonal, may approximate closely the local ones, which are based on actual interparticle distances and are in general nonorthogonal. It is also briefly shown that by defining as variables the parameters describing the kinematic rot...

The d-dimensional hydrogen atom: hyperspherical harmonics as momentum space orbitals and alternative Sturmian basis sets

Abstract Hydrogenoid orbitals, i.e. the solutions to the Schrodinger equation for a central Coulomb field, are considered in mathematical dimensions d = 2 and d > 3 different from the physical case, d = 3. Extending known results for d = 3, Sturmian basis sets in configuration (or direct) space — corresponding to variable separation in parabolic coordinates — are introduced as alternatives to the ordinary ones in spherical coordinates: extensions of Fock stereographic projections allow us to establish the relationships between the corresponding momentum (or reciprocal) space orbitals and the alternative forms of hyperspherical harmonics. Properties of the latter and multi-dimensional Fourier integral transforms are exploited to obtain the matrix elements connecting the alternative basis sets explicitly in terms of Wigners rotation matrix elements for d = 2 and generalized vector coupling (or Hahn) coefficients for d > 3. The use of these orbitals as complete and orthonormal expansion basis sets for atomic and molecular problems is briefly commented.

The He + H2+ reaction: a dynamical test on potential energy surfaces for a system exhibiting a pronounced resonance pattern

Quantum mechanical calculations on three potential energy surfaces for the prototype ion-molecule reaction He + H2+ → HeH+ + H have been performed in order to test the influence of their accuracies on reaction probabilities and cross sections. The ab initio points of McLaughlin and Thompson (1979) fitted by two different functional forms, and a fit of a new set of ab initio points have been used. Dynamical results, in particular the rich resonance pattern, illustrate the dependence both on the nature of the potential energy surface, and on the type of functional form used to fit the same ab initio data.

Hyperquantization algorithm. II. Implementation for the F+H2 reaction dynamics including open-shell and spin-orbit interactions

This work focuses on numerical aspects and performances of the hyperquantization algorithm, presented in the preceding paper, for a prototypical atom–diatom reaction. Here we provide also the extensions which allow the treatment of excited electronic surfaces. Test calculations have been carried out on the reaction F+H2 at a total nuclear angular momentum equal to zero, the fine structure of the fluorine atom being also explicitly taken into account. The technique presented is shown to be simple and effective for applications to reactive scattering problems, and the results are competitive with those obtained applying other current methods, especially in the strong triatomic interaction region.

Ab initio dynamics of the He + H+ 2 → HeH+ +H reaction: a new potential energy surface and quantum mechanical cross-sections

The reaction He + H+ 2(v,j = 0) → HeH+(v′ = 0, j′) for v = 0, 1,2 and 3 and for scattering energies near the threshold (0.95–1.15 eV) has been studied by calculating ab initio points at MRCI level and ‘exact’ integral quantum reactive cross-sections. More than 1400 nuclear geometries have been chosen to cover the most important regions for the dynamics, an extended set of points being taken directly on a hyperspherical coordinate grid. A many-body expansion with a large number of terms permits an accurate analytical representation of the potential energy surface with a root-mean-square deviation <12meV. The hyperquantization algorithm has been extended to obtain quantum mechanical integral cross-sections which are compared with previous calculations and with experimental results.

Hyperspherical adiabatic description of interference effects and resonances in collinear chemical reactions

Abstract Single-channel semiclassical scattering calculations on hyperspherical adiabatic potentials for symmetrical collinear chemical reactions are shown to describe accurately interference effects in reactions such as I + Hi. I + MuI, and resonances in H + MuH. Criteria for the validity of this adiabatic approach, and indications on how to introduce non-adiabatic corrections, are briefly discussed.

Hyperspherical harmonics as Sturmian orbitals in momentum space: a systematic approach to the few-body Coulomb problem

To exploit hyperspherical harmonics (including orthogonal transformations) as basis sets to obtain atomic and molecular orbitals, Fock projection into momentum space for the hydrogen atom is extended to the mathematical d dimensional case, higher than the physical case d = 3. For a system of N particles interacting through Coulomb forces, this method allows us to work both in a d = 3( N - 1) dimensional configuration space (on eigenfunctions expanded on a Sturmian basis) and in momentum space (using a ( d + 1)-dimensional hyperspherical harmonics basis set). Numerical examples for three-body problems are presented. Performances of alternative basis sets corresponding to different coupling schemes for hyperspherical harmonics have also been explicitly obtained for bielectronic atoms and H 2 + (in the latter case, also in the Born-Oppenheimer approximation extending the multicentre technique of Shibuya and Wulfman). Among the various generalizations and applications particularly relevant is the introduction of alternative expansions for multidimensional plane waves, of use for the generalization of Fourier transforms to many-electron multicentre problems. The material presented in this paper provides the starting point for numerical applications, which include various generalizations and hierarchies of approximation schemes, here briefly reviewed.

Lifetime of reactive scattering resonances: Q-matrix analysis and angular momentum dependence for the F+H2 reaction by the hyperquantization algorithm

We report a study on the behavior with total angular momentum J of several resonances occurring at collision energies below or slightly above the reaction barrier in the F+H2-->HF+H reaction. Resonance positions and widths are extracted from exact time-independent quantum mechanical calculations using the hyperquantization algorithm and Smiths Q-matrix formalism which exploits complete S-matrix information. The results confirm previous work but provide much greater insight. Identification of quasi-bound states responsible for the resonances based on adiabatic models for the long-range atom-molecule interactions both in the entrance and exit channels, is successful except for the feature occurring at the lowest energy, which is found to overlap with an exit-channel resonance for J approximately 7. The two features are analyzed as overlapping resonances and their excellent Lorentzian fits, with well-behaved J-dependences of positions and widths, support the interpretation of the low-energy feature as a resonance to be associated to the triatomic transition state of the reaction. Resonance role on the reactive observables (integral cross sections and angular distributions) is investigated. The mechanism leading to forward scattering in the reactive differential cross section is commented, while the effects on rate constants, as well as the sensitivity of the resonance pattern to modification of the potential energy surface, are fully discussed elsewhere.