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 σ_ideal, 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 σ_sym(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 σ_sym(G,Q) = 0. The σ_sym deviation is generated from all of the symmetry operation Ô_S 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.