The interpretation of the circular dichroism (c. d.) of coordination compounds is discussed with particular reference to the ligand field transitions of
d
3
and low-spin
d
6
systems. The experimental crystal spectra indicate by their large intensities that the solution spectra have to be interpreted on the basis of large cancellations caused by the overlapping of positive and negative c. d. contributions from closely lying energy levels. Some quantitative consequences of this have been derived. Symmetry considerations and the angular overlap model have been applied to tris(bidentate) and
cis
-bis(bidentate) chromophores which in most cases have been considered ortho-axial except for the perturbation due to the chelation. This perturbation and the chirality caused by the chelation have been described in terms of the small angular parameters (
δ
and
ϵ
) which represent a displacement of the ligating atoms, the ligators, away from the ortho-axial positions. The molecular orbital orientation of the angular overlap model has been demonstrated, and the ligand field perturbation within this model has been given as a sum of a σ and two different
π
contributions, corresponding to ligator
π
orbitals vertical and parallel to the plane of the chelating ligands. For the σ part of the perturbation, which is considered the most important part, the matrix elements connecting orbitals within each cubic subset (
e
and
t
2
), for some matrix elements in contrast with the results of the electrostatic model, do not depend on
δ
and
ϵ
to first order. However,
e
and
t
2
orbitals are connected by σ terms, first order in
δ
and
ϵ
. The perturbation energies can also be separated in a different way, also in order of decreasing importance, the regularly octahedral perturbation, the non-octahedral orthoaxial perturbation and finally the perturbation due to chelation. It is recommended to treat
d
n
systems by considering first and together the effect of the octahedral part of the perturbation and that caused by the interelectronic repulsion, and diagonalize with respect to these two perturbations before the smaller perturbation contributions are considered. This can be done within the expanded radial function model, which considers the interelectronic repulsion parametrizable as in spherical symmetry. With the purpose of illuminating this the field strength series of ligands, ordering the ligands according to their values of
∑
= ∆/
B
Racah
, has been given.
∑
is the parameter of the expanded radial function model which determines the extent of the mixing of pure cubic subconfigurations. The symmetry restrictions imposed upon ligand field operators in order to make them able to contribute to rotational strengths are discussed on the basis of a rotational strength pseudo tensor. When this is expressed with respect to our standard basis functions it can be written as a symmetrical matrix with the same symmetry properties as the corresponding energy matrix except for sign changes by improper rotations. The parentage problem for interrelating absolute configurations is discussed also on the basis of the tensor. A comparison between the results of the angular overlap model and those of the electrostatic model is made. Throughout the usual real
d
-functions have served as our limited basis set, and these functions together with the real
p
-functions define the standard octahedral irreducible representations. Functions belonging to these standard octahedral bases are generally not symmetry adapted with respect to our whole gerade perturbation, but they are symmetry adapted to the main part of it, the (holohedrized) octahedral part. Re-diagonalization of the whole perturbation with respect to functions which are diagonal for the combined perturbations of the holohedrized octahedral ligand field and the interelectronic repulsion, has the advantage of moving by far most of the gerade lower symmetry perturbation into the diagonal. This means that the energy levels become described almost completely by linear combinations of our standard cubic basis functions which belong to the same irreducible representation of the octahedral group, but which are symmetry adapted to the whole perturbation. These functions will, in general, be connected by small non-diagonal elements which mix the purely gerade-cubic levels. Since the polarization properties of the c. d. are governed by the directions of the magnetic dipole transition moments involved, they can be directly obtained for the linear combinations mentioned, on the basis of the very simple polarization properties of the standard cubic basis components.
Analytical angular solutions for the atom–diatom interaction potential in a basis set of products of two spherical harmonics: two approaches