Symmetry measures of the electron density

Abstract

In this communication we define electronic symmetry operation and symmetry group measures, eSOM and eSGM, respectively, develop the basic algorithms to obtain them, and give some examples of the possible applications of these new computational tools. These new symmetry measures based on the electron density have been tested in an analysis of (a) the inversion symmetry for heteronuclear diatomic molecules, for the eclipsed and staggered conformations of ethane and tetrafluoroethane, and for a series of octahedral sulfur halides; (b) the reflection symmetry of three different conformers of tetrafluoroethene; and (c) the loss of C(6) symmetry along the B(2u) distortion mode of benzene and an analysis of rotational symmetry for different six-member ring heterocycles.

Concurrent Symmetries: The Interplay Between Local and Global Molecular Symmetries

“…The procedure for obtaining WA C H T U N G T R E N N U N G (A,R i ) in each case has been described elsewhere and the reader interested in the mathematical details of symmetry operation measures is addressed to the more technical references. [4,10,11] Due to the different description of molecular shape, the actual values of SA C H T U N G T R E N N U N G (A,G) for a given molecule calculated by using its nuclear framework or its valence electron density are not the same. In general, the symmetry contents calculated for the electron density are lower, that is, SA C H T U N G T R E N N U N G (A,G) is larger than when it is calculated for the bare nuclear framework.…”

“…We thus carried out Hartree-Fock calculations to obtain the valence electron density distribution of those molecules, derived 2D density maps at regular intervals along the molecular axis, and calculated the C m and C 3 rotational measures at each height. [11] The results for d 6 complexes…”

A Continuous Symmetry Analysis of Chemical Bonding

“…This leads to a further simplification of the expression for the inversion symmetry of the density since in this case the self-similarity Z of both atomic orbitals [29] must have the same value [Eq. (20)].…”

“…Our previous experience calculating continuous symmetry measures for the electron density using different ab initio computational models has shown that there is not a strong dependence on the calculated CSMs with the computational details. [29] Optimization of the geometry of the HeH + molecule at the HF/ STO-3G level gives an equilibrium distance of d 0 = 0.929 . The canonical molecular orbitals for this geometry are given in Equation (23).…”

“…Although the majority of this work has been based on molecular structures described by the coordinate vectors of their atoms, some extensions of the CSM to electron densities and molecular orbitals have been proposed. [26][27][28][29] A general method for calculating CSMs has been published recently, [30] allowing us to calculate the symmetry content of any system that can be expressed as a point (or assembly of points) in some metric space. We can use this method to calculate the symmetry of an object described by a set of vectors (like the coordinate vectors describing the positions of the nuclei of a molecule), functions (like the wave function, the electron density or the individual molecular orbitals), or matrices (like the representation of an operator in a given basis set).…”