Classification of Operators in Atomic Spectroscopy by Lie Groups
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 ei (i = 0, 1, 2, 3) whose descriptions in terms of the irreps W and U of SO(7) and G2 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 (ulu2) defining U, rather than the obtuse-angled scheme sometimes preferred today.2 For the atomic d shell, only three operators of the type ei 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 Fk 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 dN and dNs began to be studied, but they were analyzed by traditional methods. The effects of two-electron excitations were shown by Trees3 and Racah4 to be reproducible by changes in the Fk together with just two new effective operators, L2 and Q, in the limit where second-order perturbation theory is adequate. The eigenvalues of L2 are L(L + 1); those of Q are zero for all terms 2S+1L of d2 except for 1S. It is not difficult to show that L2 transforms like a mixture of the irreps (00) and (22) of SO(5), while Q belongs to (00) alone.
KeywordsCoulomb Interaction Azimuthal Quantum Number Orthogonal Operator Slater Integral Multiplicity Label
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