By Per Olov Lowdin
Content material: unique vols.: v. forty-one (2002): A tribute to the lifestyles and paintings of Per-Olov Lowdin.-v.42 (2003): see analytic access 118469.-v.44 (2003): see analytic access 88803.-vols. 45,46 (2004): see analytic access 90077.-v. forty eight (2005): Jens Oddershede -- adventurer in quantum chemistry
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This group acts in the fivedimensional (5D) space provided by an h orbital, in analogy to Racah’s use of SO(5) for the d shell . From the work of Backhouse and Gard , we know that, in addition to SOð5Þ . A6 ; the full orthogonal group O(5) (which includes reflections as well as 5D rotations) contains the symmetric group S6 as a subgroup. The six permuted objects of S6 can be taken as the six axes passing through opposite vertices of the icosahedron. Some of the permutations correspond to rotations that send the icosahedron into itself, but others twist it out of shape.
Both could be usefully introduced if we needed to consider electrons in inequivalent h orbitals. However, for a single h shell only one is required, and we can simply represent an h state by a d electron. This allows us to bring in the apparatus of angular-momentum theory again, as well as any Lie groups that have proved useful in studies of the atomic d shell. Thus the states of the icosahedral configuration h N can be found by first finding those of d N and then using the branching rules for SOð4Þ !
E. Stedman is a convention rather than a physical decision, when the physical states involved are all well-known shell states . However, in this search we realized that C~ S is an unacceptable choice of particle – hole conjugation operator. Because complex conjugation Z acts only on the coefficients of a basis function, the definition of C~ S is basis-dependent. For example, Z changes a spherical tensor such as l1; 1l ¼ Y11 / lxl þ ilyl in a Cartesian basis, but not in a spherical basis. As another example, the definitive anticommutation of C~ S with the quasispin generators Q holds only in a Cartesian basis.
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