Edwin Lo | Searcharxiv

arXiv1999

Implications of non-feasible transformations among icosahedral $h$ orbitals

The symmetric group $S_6$ that permutes the six five-fold axes of an icosahedron is introduced to go beyond the simple rotations that constitute the icosahedral group $I$. Owing to the correspondence $h\leftrightarrow d$, the calculation of the Coulomb energies for the icosahedral configurations $h^N$ based on the sequence $O(5) \supset S_6 \supset S_5 \supset I$ can be brought to bear on Racah's classic theory for the atomic d shell based on $SO(5) \supset SO_L(3) \supset I$. Among the elements of $S_6$ is the kaleidoscope operator ${\cal K}$ that rotates the weight space of SO(5) by $π/2$. Its use explains some puzzling degeneracies in d^3 involving the spectroscopic terms ^2P, ^2F, ^2G and ^2H.

physics.atom-ph↗

arXiv1998

Coulomb and spin-orbit interaction matrix elements in d^2d' configuration

The $d^2d'$ configuration is analysed in group-theoretical terms. Starting from the table given by Condon and Odabasi (1980) for the configuration $d^2d'$, we determine a set of convenient group-theoretical basis states, and rewrite the Coulomb matrix elements in terms of this new basis. Linear combinations from the different parts of the Coulomb operators are formed such that they have simple group transformation properties in our scheme. The sequence of groups that we use is $U(20) \supset SO_T(3) \times U(10) \supset SO_T(3) \times SO_S(3) \times U(5) \supset SO_T(3) \times SO_S(3) \times SO(5) \supset SO_T(3) \times SO_S(3) \times SO_L(3)$, where $T$ denotes the {\em isospin} of \v Simonis et al (1984), in which electrons with the same angular momentum $l$ but different principle quantum numbers $n$ are accommodated by introducing the eigenvalue $M_T$ of $T_0$. Using the Wigner-Eckart theorem and selection rules on the higher symmetry groups, the tables of the Coulomb and spin-orbit matrix elements for the reconstituted operators (with simple group transformation properties) are much simplified in terms of these basis states.

physics.atom-ph↗