# (L̂_{z},Ŝ^{2})—adapted graphs

## Abstract

Representation of (L̂_{ z }, Ŝ^{2})-adapted spaces does not differ much from the representation of (L̂_{ z }, Ŝ_{ z })-adapted spaces, as the reader has probably guessed by now. S-diagram is made from M-diagram by removing vertices corresponding to negative *M* values. The graph with α, *ß* α… order of spin orbitals as well as four-slope graphs are a kind of extended M-diagrams: their paths, after doubly occupied orbitals are removed and dependence of arc slopes on *m*_{ l } values ignored, are indeed reduced to the paths of M-diagram. Therefore restricting in Fig **17**,**18** values of *S*_{ k } to the positive ones we obtain description of (L̂_{ z },Ŝ^{2})-adapted space. However, the simpler (L̂_{ z },Ŝ_{ z })-adapted graphs with α and *ß*-type orbitals separated (Fig **15**,**16**) can not be easily modified to represent spin eigenfunctions. These graphs, if α-type levels are palced at the top, have for all vertices *M*_{ S } ≥ 0. In this section previously unexplored possibility of representing the (L̂_{ z },Ŝ_{ z }) and (L̂_{ z }, Ŝ^{2})-adapted spaces with the help of a fagot graph is presented. The (L̂_{ z }, Ŝ^{2})-adapted graphs are very convenient in calculations of atomic properties and in nuclear shell model calculations; bearing in mind the complexity of L̂^{2}-adapted states it is better to use larger but simpler bases of (L̂_{ z },Ŝ^{2})-adapted spaces, especially that we may transform whole blocks of matrix elements instead of individual functions from spaces in which matrix elements are easy to calculate to spaces where calculation is more complicated (cf Duch 1986a and Section **2.4** of this volume). In some cases, important for practical calculations, when the space of the highest *S* value is desired description of (L̂_{ z },Ŝ^{2}) and (L̂_{ z },Ŝ^{2})-adapted spaces is equivalent.

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