Gaia Grossi

Angular momentum coupling schemes in the quantum mechanical treatment of P‐state atom collisions

For the multichannel Schrodinger equations which arise in the quantum mechanical close coupling treatment of atomic collisions involving fine structure effects, alternative representations are developed by angular momentum algebra. The various representations are closely related to Hund’s coupling schemes for rotating diatomic molecules. Matrix elements for the electrostatic interaction and for the orthogonal transformations which connect the various representations, are given explicitly for the case when only one atom has internal angular momenta and follows LS coupling. The limit of large angular momenta, of interest under semiclassical conditions, is also considered. Some examples of applications to P atom collisions are discussed.

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.

Decoupling approximations in the quantum mechanical treatment of P‐state atom collisions

Several decoupling schemes are considered, for the effective simplifications of the close coupling equations which arise in the treatment of atomic collisions involving fine structure. Calculations were performed for two models of 2P atoms colliding at thermal energy with a 1S species. From the comparison of exact and approximate results, the relative merits and limits of full or partial decoupling schemes were assessed. Possible improvements and extensions are suggested.

Hyperspherical diabatic and adiabatic representations for chemical reactions

Abstract A diabatic formalism for the treatment of chemical reactions is introduced, using hyperspherical coordinates and expansions for both potentials and wavefunctions in hyperspherical harmonics. The technique is illustrated in the collinear case by the harmonic expansion for the potential energy surface of H 3 . Adiabatic potential energy curves are constructed, alternative diabatic representations considered, and possible exact and approximate treatments indicated.

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.

Alternative Sturmian bases and momentum space orbitals: an application to the hydrogen molecular ion

Abstract The relationship between alternative separable solutions of the Coulomb problem both in configuration and in momentum space is exploited in order to obtain Sturmian orbitals of use as expansion basis sets in atomic and molecular problems. The usual spherical basis is obtained by separation in polar coordinates. The mathematical properties are explored for a type of basis set for quantum mechanical problems with axial symmetry, examples being diatomic molecules, or atoms under the influence of a uniform electric field. Because of its appropriateness for treatment of the Stark effect in atomic physics, this alternative basis set is called Stark basis . The Stark basis corresponds to separation in parabolic coordinates in configuration space and in cylindrical coordinates in momentum space. Focks projection onto the surface of a sphere in the four dimensional hyperspace allows us to establish the connections of the momentum space wave functions with hyperspherical harmonics. As an application, the Shobuya and Wulfman analysis of momentum space molecular orbitals is reformulated and monoelectronic multicenter integrals are calculated by angular momentum algebra. Properties of the Stark basis, which exhibits less coupling at long range, are illustrated with reference to the case of the hydrogen molecular ion in the fixed nuclei approximation.

Coupling schemes for atom–diatom interactions and an adiabatic decoupling treatment of rotational temperature effects on glory scattering

The quantum mechanical theory for scattering of a particle by a rigid rotor is formulated in five alternative diabatic representations, corresponding to alternative coupling schemes. Use is made of a recently introduced procedure for obtaining discrete representations by artificial quantization. In order to develop an efficient computational scheme for obtaining information on the interaction potential from atom–diatom scattering experiments, decoupling approximations are developed. An adiabatic representation in the coupled states framework is applied to the computation of integral cross sections and nonadiabatic coupling effects are analyzed. The approach provides an accurate description of the experimentally observed dependence of glory scattering from the rotational temperature of the diatom.

The origin of chiral discrimination: supersonic molecular beam experiments and molecular dynamics simulations of collisional mechanisms

The target of the present paper is the study of chirality effects in molecular dynamics from both a theoretical and an experimental point of view under the hypothesis of a molecular dynamics mechanism as the origin of chiral discrimination. This is a fundamental problem per se, and of possible relevance for the problem of the intriguing homochirality in Nature, so far lacking satisfactory explanations. We outline the steps that have been taken so far toward this direction, motivated by various experimental studies of supersonic molecular beams carried out in this laboratory, such as the detection of aligned oxygen in gaseous streams and further evidence on nitrogen, benzene and various hydrocarbons, showing the insurgence of molecular orientation in the dynamics of molecules in flows and in molecular collisions. Chiral effects are theoretically demonstrated to show up in the differential scattering of oriented molecules, also when impinging on surfaces. Focus on possible mechanisms for chiral bio-stereochemistry of oriented reactants may be of pre-biotical interest, for example when flowing in atmospheres of rotating bodies, specifically the planet Earth, as well as in vortex motions of celestial objects. Molecular dynamics simulations and experimental verifications of the hypothesis are reviewed and objectives of future research activity proposed.

On the ridge effect in mode transitions: Semiclassical analysis of the quantum pendulum

Abstract A uniform asymptotic analysis of the Mathieu equation provides a semiclassical description of the quantum pendulum as a model for chemical reactions, normal versus local molecular vibrations and several phenomena of atomic and molecular physics which involve transitions between modes at a potential ridge. The ridge effect is described, introducing a matrix of non-adiabatic coupling P whose properties are briefly studied.

On hyperspherical mapping and harmonic expansions for potential energy surfaces

It is noted that the potential energy function, because of rotational invariance can be expressed in terms of six‐dimensional harmonics for J = 0 only. (AIP)