Abstract
Symmetry is often exploited in physics and chemistry to derive fundamental principles, reduce computational workloads or obtain physical insights and information without the need for elaborate calculations. Also in the quantum chemical study of luminescent systems and their spectroscopy, group theory, which is the mathematical framework that enables one to exploit symmetries, is ubiquitous and indispensable to keep calculations feasible. While no expert knowledge is required, a selection of basic ingredients from group and representation theory and symmetry aspects from crystal or ligand field theory are reviewed here that are useful in drafting program input and to correctly interpret and process intermediate and final program output. These concepts are applied to generate the input and analyze the output needed to obtain the potential energy curves of cubic Pr\(^{3+}\) defects in BaF\(_2\) from multiconfigurational ab initio calculations. The latter section serves as a support for the symmetry handling in the tutorial of Chap. 5.
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Notes
- 1.
Named after the Norwegian, 19\({\mathrm {th}}\) century mathematician Sophus Lie.
- 2.
All possible rotations along a fixed axis form a Lie group, parametrized by the rotation angle. For all rotations in 3D space, three parameters are required, e.g. the Euler angles.
- 3.
\(\Gamma ' = S^{-1}\Gamma S\), with S an invertible matrix.
- 4.
SO(3) is the matrix group of all orthogonal (i.e. \(O^{-1}=O^T\)) matrices for which \(\mathrm {det}(O)=1\).
- 5.
SU(2) is the matrix group of all unitary (i.e. \(U^{-1}=U^\dagger \)) matrices for which \(\mathrm {det}(U)=1\).
- 6.
The dimension of an irrep is given by the character belonging to the unit operation, \(l_\Gamma = \chi ^{(\Gamma )}(E)\).
- 7.
According to the Mulliken notation, one-dimensional irreps are denoted by A or B, two-dimensional irreps by E and three-dimensional irreps by T.
- 8.
Atomic perturbation sequence where \(\hat{H}_\mathrm {Coulomb}\) is treated first, followed by \(\hat{H}_\mathrm {so}\). This corresponds to physical reality for most elements where the effect of spin-orbit coupling (multiplet splitting) is small compared to the effect of electron repulsion (term splitting). The opposite case is referred to as jj coupling. In general, both Hamiltonians must be diagonalized simultaneously according to the atomic intermediate coupling scheme.
- 9.
\(\hat{\mathcal {P}}\boldsymbol{r}=-\boldsymbol{r}\).
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Barandiarán, Z., Joos, J., Seijo, L. (2022). Symmetry Handling. In: Luminescent Materials. Springer Series in Materials Science, vol 322. Springer, Cham. https://doi.org/10.1007/978-3-030-94984-6_4
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DOI: https://doi.org/10.1007/978-3-030-94984-6_4
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