Symmetries in Science II pp 265-269 | Cite as

# Classification of Operators in Atomic Spectroscopy by Lie Groups

## Abstract

The use of Lie groups in atomic spectroscopy dates from the 1949 article of Racah.^{1} It was shown there that the Coulomb interaction between f electrons could be described as a linear combination of four operators e_{i} (i = 0, 1, 2, 3) whose descriptions in terms of the irreps W and U of SO(7) and G_{2} are (000)(00) for i = 0 and 1, (400)(40) for i = 2, and (220)(22) for i = 3. Highest weights are used to define W and U, and we should also note that Racah used an acute-angled pair of axes for the two numbers (u_{l}u_{2}) defining U, rather than the obtuse-angled scheme sometimes preferred today.^{2} For the atomic d shell, only three operators of the type e_{i} are required to represent the Coulomb interaction: two of them correspond to the SO(5) scalar irrep (00), the third to the irrep (22) of that group. At the time of Racah’s article on the f shell, all d-electron matrix elements of the Coulomb interaction had already been expressed as linear combinations of the Slater integrals F_{k} involving the radial parts of the eigenfunctions, so there was little incentive to rework the calculation for the d shell. Shortly after the appearance of Racah’s article, the effects of configuration interaction on the configurations d^{N} and d^{N}s began to be studied, but they were analyzed by traditional methods. The effects of two-electron excitations were shown by Trees^{3} and Racah^{4} to be reproducible by changes in the Fk together with just two new effective operators, L^{2} and Q, in the limit where second-order perturbation theory is adequate. The eigenvalues of L^{2} are L(L + 1); those of Q are zero for all terms ^{2S+1}L of d^{2} except for ^{1}S. It is not difficult to show that L^{2} transforms like a mixture of the irreps (00) and (22) of SO(5), while Q belongs to (00) alone.

## Keywords

Coulomb Interaction Azimuthal Quantum Number Orthogonal Operator Slater Integral Multiplicity Label## Preview

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## References

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