Molecular Orbitals and Microsymmetry

G R O U P theory tells us the conditions under which an energy level is invariant under changes of the co-ordinate system (rotations, etc.) which do not change the physical properties of the electron system considered. Thus, the (2L + l)-fold degeneracy of multiplet terms and (21 + l)-fold degeneracy of orbitals is closely connected with the assumption of spherical symmetry, surrounding a gaseous atom or ion. In spherical symmetry, the co-ordinate axes can have any direction without introducing any physically significant differences. In a molecule, containing more than one nucleus, the symmetry is degraded from the extremely high spherical one, to a lower symmetry. We shall mainly consider here three relatively high symmetries, the cubic-octahedral Oh, the linear symmetry with centre of inversion Dooh, and the linear symmetry without centre of in version Ooo ν · These three cases can be exemplified by a regular octahedral complex ML 6, by a homonuclear, diatomic molecule A 2, and a heteronuclear, diatomic molecule AB. Unfortunately, it is not possible to describe here the theory of character tables and irreducible representations which is admirably presented in several text-books on group theory. We shall be content to mention that the quantum number L in the RussellSaunders coupling in spherical symmetry is substituted by Γ η in the lower symmetries, and that / is substituted by Tj , while S preserves its denotation. The orbitals, characterized by η and / in spherical symmetry, are substituted by molecular orbitals (M.O.) having a file number η and a definite value of y n having the same possibilities as Γη. Bethe discussed (1929) the behaviour of terms L and orbitals / in Oh and many other, lower symmetries. Any given entity L or / corresponds to a set of one or more values of Γ η or y n. On the other hand, a given wave function Ψ in the low symmetry, having a definite Γ η, does not necessarily correspond to a definite value of L belonging to the spherical symmetry, which is no longer present in