Evaluation of Point Group Symmetry in Lanthanide(III) Complexes: A New Implementation of a Continuous Symmetry Operation Measure with Autonomous Assignment of the Principal Axis

The structure of molecular systems dictates the physical
properties,
and symmetry is the determining factor for all electronic properties.
This makes group theory a powerful tool in quantum mechanics to compute
molecular properties. For inorganic compounds, the coordination geometry
has been estimated as idealized polyhedra with high symmetry, which,
through ligand field theory, provides predictive capabilities. However,
real samples rarely have ideal symmetry, and although continuous shape
measures (CShM) can be used to evaluate deviation from an ideal reference
structure σideal, this often fails for lanthanide(III)
complexes with high coordination numbers, no obvious choice of principal
axes, and no obvious reference structure. In lanthanide complexes,
the unique electronic structures and associated properties are intricately
tied to the symmetry around the lanthanide center. Therefore, robust
methodologies to evaluate and estimate point group symmetry are instrumental
for building structure–property relationships. Here, we have
demonstrated an algorithmic approach that orients a molecular structure
Q in the best possible way to the symmetry axis of any given point
group G and computes a deviation from the ideal symmetry σsym(G,Q). This approach does not compute the deviation from
an ideal reference system, but the intrinsic deviation in the structure
induced by symmetry operations. If the structure contains the symmetry
operation, there is no deviation and σsym(G,Q) =
0. The σsym deviation is generated from all of the
symmetry operation ÔS in a point group G using the
most correct orientation of the sample structure in each group G.
The best orientation is found by an algorithm that minimizes the orientation
of the structure with respect to G. To demonstrate the methodology,
we have investigated the structure and symmetry of 8- and 9-coordinated
lanthanide(III) aqua complexes and correlated the luminescence from
3 europium(III) crystals to their actual symmetry. To document the
methodology, the approach has been tested on 26 molecules with different
symmetries. It was concluded that the method is robust and fully autonomous.