Visualization of restricted model spaces
In the previous sections I have discussed how to visualize spaces adapted to various operators and spatial point group symmetries. All eigenfunctions of the desired operators and of the desired symmetry that can be built from a set of given one-particle states were taken as the bases of model spaces. Symmetry was thus the only selection criterium employed so far. For practical purposes this is usually not sufficient: full symmetry-adapted many-particle space grows very rapidly with the number of primitive states and although the graphs can always be drawn and their general properties analyzed for practical calculations the dimension of the full space becomes quickly too large to handle even for the best computers. The traditional way of reducing the size of a model space is based on the concept of reference states. Coulomb forces are rather weak, therefore in atomic and molecular problems the independent-particle approximation works rather well and it is possible to choose the one-particle states in such a way, that among all many-particle symmetry-adapted states there is one state or a combination of a few states clearly dominating the exact solution. In such cases one may argue that the model space with a basis composed of these few states (called ‘the reference states’ or simply ‘references’) plus the states that are ‘not far’ from them allow to form a very good approximation to the exact state (wave function) of our system. ‘Not far’ usually means configurations built from almost the same orbitals as the references are, with at most one or two orbitals changed. Matrix elements of a two-particle operator between the reference and the singly or doubly substituted states are in general different from zero, justifying the choice of these states to our model space by their contributions in the second-order perturbation theory.
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