# (L̂_{z},Ŝ_{z})—adapted graphs

## Abstract

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 1/2*N* + *M*_{ S } orbitals of the α-type at top and 1/2*N* − *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, 2*p*, and 3*p* 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 3*s*, 3*p* and 3*d* orbital basis, is represented. The graph is almost planar, with only a few arc lines that cross off vertices. Each vertex in the (L̂_{z}, Ŝ_{z})-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 }).

### Keywords

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