Introducing graphical representation

Abstract

What does it mean to represent a many-particle model space graphically? It means, that we should be able to identify and label each basis state of that space. For fermionic systems these basis states should be antisymmetric - a very strong requirement, immediately invoking the Pauli principle. Adding spin states |α) and |β) to each orbital state that belongs to V_n we obtain 2n spin-orbital states. These spin-orbital states are ordered and identified in some way, for example

$$|{\phi _1}\rangle = |{\phi _1}\rangle |\alpha \rangle ,|{\overline \phi _1}\rangle = |{\phi _1}\rangle |\beta \rangle ,...|{\phi _n}\rangle ,|{\overline \phi _n}\rangle$$

In the construction of an antisymetric N-particle state at each spin orbital appears at most once, therefore ( ²ⁿ_N ) spin-orbital configurations or different states are possible, provided that no other restriction than antisymmetry is imposed. Description of a non-symmetric molecule in Born-Oppenheimer approximation requires such states when a strong spin-orbit interaction is present. Graphical representation of the states that do not posses any symmetry other that being antisymmetric (corresponding to determinants) is particularly simple.