In many applications there is more than one operator that commutes with the Hamiltonian of a system. Eigenstates of angular momentum usually have also some spin symmetry. In the simplest case the states should be adapted to (L̂ z , Ŝ z ) operators. One way of representing the space of such states is to use a fagot graph describing L̂ z -adapted configurations, as in Fig 14a, and classify the final states for each fagot function (subspace) according to a subgraph of Fig 14b, as already mentioned in the previous section. Several other representations may also be useful. The simplest non-fagot representation is obtained when α and ß-type orbitals are separated, with 1/2N + M S orbitals of the α-type at top and 1/2N − M S orbitals of the ß-type at bottom of a graph. In Fig 15 full space for states with M L = 1 and M S = 1 of N = 5 electrons distributed among 2s, 2p, and 3p orbitals, is represented for two orbital orderings. Although it is not possible to find a planar graph representing this space the graph of Fig 15b, with orbitals grouped according to their m l values, is more legible than the graphs corresponding to other orbital orderings. In Fig 16 more complicated case, with 3s, 3p and 3d orbital basis, is represented. The graph is almost planar, with only a few arc lines that cross off vertices. Each vertex in the (L̂z, Ŝz)-adapted graphs is uniquely labeled by: the number of electrons e, the intermediate projection of angular momentum m l , the spin projection m s and the level o corresponding to some orbital, i.e. it may be designated v(o: e,m l ,m s ).
KeywordsAngular Momentum Planar Graph Model Space Spin Orbital Spin Projection
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