arXiv subjects

Guillaume Dhont

Publications and source records attributed to Guillaume Dhont.

3 recordsLinked to original sources

arXiv2021

Molien generating functions and integrity bases for the action of the SO(3) and O(3) groups on a set of vectors

The construction of integrity bases for invariant and covariant polynomials built from aset of three dimensional vectors under the SO(3) and O(3) symmetries is presented. Thispaper is a follow--up to our previous work that dealt with a set of two dimensional vectorsunder the action of the SO(2) and O(2) groups [G. Dhont and B. I. Zhilinskiı, J. Phys. A:Math. Theor., 46, 455202 (2013)]. The expressions of the Molien generating functions asone rational function are a useful guide to build integrity bases for the rings of invariantsand the free modules of covariants. The structure of the non--free modules of covariants ismore complex. In this case, we write the Molien generating function as a sum of rationalfunctions and show that its symbolic interpretation leads to the concept of generalizedintegrity basis. The integrity bases and generalized integrity bases for O(3) are deducedfrom the SO(3) ones. The results are useful in quantum chemistry to describe the potentialenergy or multipole moment hypersurfaces of molecules. In particular, the generalizedintegrity bases that are required for the description of the electric and magnetic quadrupolemoment hypersurfaces of tetratomic molecules are given for the first time.

math.AC↗

arXiv2017

Topological Phase Transition in a Molecular Hamiltonian with Symmetry and Pseudo-Symmetry, Studied through Quantum, Semi-Quantum and Classical Models

The redistribution of energy levels between energy bands is studied for a family of simple effective Hamiltonians depending on one control parameter and possessing axial symmetry and energy-reflection symmetry. Further study is made on the topological phase transition in the corresponding semi-quantum and completely classical models, and finally the joint spectrum of the two commuting observables $(H=E,J_z)$ (also called the lattice of quantum states) is superposed on the image of the energy-momentum map for the classical model. Through these comparative analyses, mutual correspondence is demonstrated to exist among the redistribution of energy levels between energy bands for the quantum Hamiltonian, the modification of Chern numbers of eigenline bundles for the corresponding semi-quantum Hamiltonian, and the presence of Hamiltonian monodromy for the complete classical analog. In particular, as far as the band rearrangement is concerned, a fine agreement is found between the redistribution of the energy levels described in terms of joint spectrum of energy and momentum in the full quantum model and the evolution of singularities of the energy-momentum map of the complete classical model. The topological phase transition observed in the present semi-quantum and the complete classical models are analogous to topological phase transitions of matter.

quant-ph↗

arXiv2014

A Fast Algorithm for the Construction of Integrity Bases Associated to Symmetry-Adapted Polynomial Representations. Application to Tetrahedral XY4 Molecules

Invariant theory provides more efficient tools, such as Molien generating functions and integrity bases, than basic group theory, that relies on projector techniques for the construction of symmetry--adapted polynomials in the symmetry coordinates of a molecular system, because it is based on a finer description of the mathematical structure of the latter. The present article extends its use to the construction of polynomial bases which span possibly, non--totally symmetric irreducible representations of a molecular symmetry group. Electric or magnetic observables can carry such irreducible representations, a common example is given by the electric dipole moment surface. The elementary generating functions and their corresponding integrity bases, where both the initial and the final representations are irreducible, are the building blocks of the algorithm presented in this article, which is faster than algorithms based on projection operators only. The generating functions for the full initial representation of interest are built recursively from the elementary generating functions. Integrity bases which can be used to generate in the most economical way symmetry--adapted polynomial bases are constructed alongside in the same fashion. The method is illustrated in detail on XY4 type of molecules. Explicit integrity bases for all five possible final irreducible representations of the tetrahedral group have been calculated and are given in the supplemental material associated with this paper.

math-ph↗