Molecular symmetry group—Schrodinger

It was sometimes believed, in the scientific literature, that the Molecular Symmetry Group of Longuet-Higgins and the Schrodinger Supergroup of Altmann were isomorphic, i.e., both theories were equivalent [19]. We shall see, however, that is not generally true, especially when some symmetry is retained in the molecule. 

Inspection of Fig. 2, shows that the equivalence between the Schrodinger Supergroup and the Molecular Symmetry Group operations, for the CH3—BF2 molecule, is easily verified. In Table 2, the two groups appear to be identical. 

I ble 2 Equivalence between the Molecular Symmetry Group and Schrodinger Supergroup operations in the CH3 — BFj molecule. 

The complete group of the Hamiltonian is the combination of all these possible symmetries. This derivation is directly evident from the mathematical form of the Hamiltonian and expresses fundamental properties of molecular space and time. Yet it took 40 years, from Schrodinger to Longuet-Higgins, to obtain a clear definition of the molecular-symmetry group. Three kinds of symmetries may be identified ... 

The irreducible representations of a symmetry group of a molecule are used to label its energy levels. The way we label the energy levels follows from an examination of the effect of a symmetry operation on the molecular Schrodinger equation. 

The coordinates are expressed in the molecular axes x,y,z, whidi are rigidly attached to the molecule. These coordinates and masses are labelled in some laboratory axes, X,Y,Z, fixed in space. Under this definition, a symmetry operation is a change of axes that leaves the Hamiltonian operator (14) invariant, and the group of all such operations is the Schrodinger subgroup. 

James Rondinelli, a professor of materials sdence and engineering at Drexel University, uses group theory as an aid in designing new materials. The symmetry of a crystal plays a role in its properties, but the nature of the relationship is not always obvious. Rondinelli solves the electronic Schrodinger equation of various crystalline materials and uses the results to uncover connections between the molecular structure and the material properties, For example, perovskite, a mineral composed of calcium titanium oxide, has several possible crystal... 

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