Separation and Group Function Formalism

In the preceding section, the a—n separation occurs as a direct result of the independent-particle model. The derivation is straightforward, but not entirely satisfactory, because the independent-particle model is a simplification of the actual quantum-mechanical situation. In fact, the a—n separation can be introduced in the frame of more general treatments, e.g. the separated-group function formalism b2,3). 

A quantum-mechanical -electron system is said to consist of M separated groups (A, B, etc.) containing a, bb, etc. electrons, respectively, if it can be described by a wave function of the form 

This condition implies that the natural orbitals of the different groups are mutually orthogonal and exclusive, i.e. no two groups have any (occupied) NO in common, and the NO s of different groups are orthogonal to each other a . 

Physical systems cannot be rigorously described by a separated-group wave function, but that description may often be a rather good approximation. If this is the case, an important simplification of the quantum- 

According to Lykos and Parr 3 , an unsaturated molecule can be described in this manner by taking for S a (in principle complete) linear combination of Slater determinants built from a orbitals only and for n a similar combination built from n orbitals only. Such a description is more general than the independent-particle model, as it includes the latter as a special case (namely where both and 77 are single Slater determinants). 

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