Orthogonal coordinates for the dynamics of four bodies and for the representation of potentials of tetra‐atomic molecules

Abstract

We discuss systems of orthogonal coordinates for the dynamical treatment of four particles, generated by making extensive use of the concept of kinematic rotations, which act on coordinates of the particles and are represented by matrices only dependent on their masses. The explicit representations of the kinetic rotation matrices are given: this allows us to define alternative particle schemes, such as those based on the Jacobi and Radau-Smith vectors, as well as on mixed types of vectors, of possible interest for specific molecules or aggregates. A list is given of relevant formulas connecting these coordinate sets to the geometrical parameters (internuclear distances, bond and dihedral angles) of use for the representation of the potential energy surface of four atomic systems. Applications are indicated for molecular and cluster physics. INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 2873 RAGNI, BITENCOURT, AND AQUILANTI FIGURE 2. Specific rotation matrices K providing local orthogonal vectors for four bodies from scaled position q, according x = qK. The actual lengths of the vectors depend on the of mass scalings, and here are drawn without such scalings to show clearly connections with bond directions and angles.Particles are given the labels i, j, k, and l. To simplify the notation for masses we have used M ij = m i + m j , M ijk = M ij + m k , and m = M ij + M kl = M ijk + m l is the total mass. J VectorsThe J or "Jacobi tree" vectors can be used for example in the case of a weakly bond complex, between a diatomic molecule and an atom, which interacts with another atom. Such a system can be (FH 2 )Ar, where i and j are hydrogen atoms, k is a fluorine atom and l is argon atom.The Jacobi tree (J) vectors, Figure 3, are obtained by

Analysis of vibrational modes of the P4 molecule through hyperspherical variants of the local orthogonal coordinates: The limit of dissociation in dimers

“…These coordinates, together with h 1 , h 3 , and h 2 , represent a good internal coordinate set to describe the geometrical parameters [13]. A particular hyperspherical parametrization [25,26] obtained from the modules of these vectors allows to introduce another type of internal coordinates:…”

“…As it was considered previously, the vector of the center of mass has been neglected. Formally, the V vectors can be built performing an appropriate kinematic rotation on the H vectors that involve only h 2 and h 3 ones [13]. In this case, they can be written as:…”

“…1(a) and (b)] can be obtained from the laboratory position vectors using the kinematic rotation matrices procedure [13,28]. Their mass-scaled form is given by…”

Aligned molecular collisions and a stereodynamical mechanism for selective chirality

“…Quantum calculations of dimensions larger than those characteristics of three-atom systems are still at the edge of the range of current research, but the only molecular dynamics calculations for chiral systems appear feasible within classical mechanics. Indeed, the extension of the quantum approach to many-body system dynamics is tied to the explicit inclusion of a formal treatment of chirality in the theory (Ragni et al 2007).…”

A general rigorous quantum dynamics algorithm to calculate vibrational energy levels of pentaatomic molecules

“…In particular, both new Lanczos algorithms utilized the merits of quantum mechanical Hamiltonian in a set of orthogonal scattering coordinates. That is, the resulting Hamiltonian is very compact with a partitioned structure, and has a common formulation except for the mass-dependent prefactors regardless of types of orthogonal vectors [11,13,25,34,36,49,[66][67][68]. Otherwise, if non-orthogonal coordinates [25, [69][70][71][72][73][74][75][76] are used, the kinetic energy operators obtained are rather complicated.…”