Hyperangular momenta and energy partitions in multidimensional many-particle classical mechanics: The invariance approach to cluster dynamics (original) (raw)
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Quantum dynamics of kinematic invariants in tetra- and polyatomic systems
Physical Chemistry Chemical Physics, 1999
For the dynamical treatment of polyatomic molecules or clusters as n-body systems, coordinates are conveniently broken up into external (or spatial) rotations, kinematic invariants, and internal (or kinematic) rotations. The kinematic invariants are related to the three principal moments of inertia of the system. At a Ðxed value of the hyperradius (a measure of the total moment of inertia), the space of kinematic invariants is a certain spherical triangle, depending on the number of bodies, upon which angular coordinates can be imposed. It is shown that this triangle provides the 24-element (group O) octahedral tesselation of the sphere for n \ 4 and the 48-element (group octahedral tesselation for n P 5. Eigenfunctions describing the O h ) kinematics of systems with vanishing internal and external angular momentum can be obtained in closed form in terms of Bessel functions of the hyperradius and surface spherical harmonics. They can be useful as orthonormal expansion basis sets for the hyperspherical treatment of the n-body particle dynamics.
Statistics of energy partitions for many-particle systems in arbitrary dimension
In some previous articles, we defined several partitions of the total kinetic energy T of a system of N classical particles in R d into components corresponding to various modes of motion. In the present paper, we propose formulas for the mean values of these components in the normalization T = 1 (for any d and N) under the assumption that the masses of all the particles are equal. These formulas are proven at the "physical level" of rigor and numerically confirmed for planar systems (d = 2) at 3 N 100. The case where the masses of the particles are chosen at random is also considered. The paper complements our article of 2008 [Russian J Phys Chem B 2(6):947-963] where similar numerical experiments were carried out for spatial systems (d = 3) at 3 N 100.
Geometry and symmetries of multi-particle systems
Journal of Physics B: Atomic, Molecular and Optical Physics, 1999
The quantum dynamical evolution of atomic and molecular aggregates, from their compact to their fragmented states, is parametrized by a single collective radial parameter. Treating all the remaining particle coordinates in d dimensions democratically, as a set of angles orthogonal to this collective radius or by equivalent variables, by-passes all independent-particle approximations. The invariance of the total kinetic energy under arbitrary d-dimensional transformations which preserve the radial parameter gives rise to novel quantum numbers and ladder operators interconnecting its eigenstates at each value of the radial parameter.
Invariant energy partitions in chemical reactions and cluster dynamics simulations
Computational Materials Science, 2006
A separation of intra and intermolecular degrees of freedom, based on an invariant phase-space hyperangular momentum approach, is proposed as useful alternative to the usual roto-vibrational mode analysis. Two partitions (hyperspherical and projective) of the kinetic energy are here illustrated for simulations of chemical reactions and shown interesting to analyze caloric curves of clusters. Case studies are Ar 13 and Ar 38 .
Hyperspherical harmonics for polyatomic systems: Basis set for kinematic rotations
International Journal of Quantum Chemistry, 2002
As a continuation of our previous work for the construction of expansion basis sets for the quantum mechanical treatment of N -body problems of interest for intermolecular and intramolecular and reactive dynamics of polyatomic molecules and clusters, we develop here a group-theoretical procedure which allows us to obtain explicitly the hyperspherical harmonics for the description of the collective motions. The coordinates involved are related to the invariants of the N -body system, which are referred to the moments of inertia. Although this work is limited to the case where both external and internal (kinematic) rotations are zero, and the example of N ¼ 4 is explicitly worked out, this method, which gives hyperspherical harmonics as linear combinations of ordinary spherical harmonics, can be extended to cover the general N -body case.
Theoretical Chemistry Accounts, 2007
The hyperspherical method is a widely used and successful approach for the quantum treatment of elementary chemical processes. It has been mostly applied to three-atomic systems, and current progress is here outlined concerning the basic theoretical framework for the extension to four-body bound state and reactive scattering problems. Although most applications only exploit the advantages of the hyperspherical coordinate systems for the formulation of the few-body problem, the full power of the technique implies representations explicitly involving quantum hyperangular momentum operators as dynamical quantities and hyperspherical harmonics as basis functions. In terms of discrete analogues of these harmonics one has a universal representation for the kinetic energy and a diagonal representation for the potential (hyperquantization algorithm). Very recently, advances have been made on the use of the approach in classical dynamics, provided that a hyperspherical formulation is given based on "classical" definitions of the hyperangular momenta and related quantities. The aim of the present paper is to offer a retrospective and prospective view of the hyperspherical methods both in quantum and classical dynamics. Specifically, regarding the general quantum hyperspherical approaches for three-and four-body systems, we first focus on the basis set issue, and then we present developments on the classical formulation that has led to applications involving the implementations of hyperspherical techniques for classical molecular dynamics simulations of simple nanoaggregates.
Hyperspherical harmonics for polyatomic systems: basis set for collective motions
Theoretical Chemistry Accounts: Theory, Computation, and Modeling (Theoretica Chimica Acta), 2004
As a continuation of our previous work for the construction of expansion basis sets for the quantum mechanical treatment of N -body problems of interest for intermolecular and intramolecular and reactive dynamics of polyatomic molecules and clusters, we develop here a group-theoretical procedure which allows us to obtain explicitly the hyperspherical harmonics for the description of the collective motions. The coordinates involved are related to the invariants of the N -body system, which are referred to the moments of inertia. Although this work is limited to the case where both external and internal (kinematic) rotations are zero, and the example of N ¼ 4 is explicitly worked out, this method, which gives hyperspherical harmonics as linear combinations of ordinary spherical harmonics, can be extended to cover the general N -body case.