Spatial symmetry in the graph
So far the spaces we have described were invariant under the action of one or two operators. Description of molecules or crystals requires the adaptation of many-electron functions to a certain representation Γ of a point group G (cf Hammermesh 1962; Hochstrasser 1966; Bunker 1980). In this case we do not have a single operator but a whole set of group operators to deal with. The situation is simplified very much if the ‘basic building bricks’, i.e. one-electron states, are symmetrized orbitals. The problem of choosing the Γ-adapted subspace is than reduced to the selection of orbital products, such that the desired representation Γ is included in the product of representations for each symmetry orbital, i.e.
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