Properties of the One-Particle Density Matrix

A system with N electrons is fully described by the corresponding wave function and, following the interpretation of Bom,  

If we consider the Hartree-Fock approach of building a Slater determinant from one-particle functions (p, (compare our section on the basics of SCF theory), we can similarly define the one-particle density as 

The density operator is a projector onto the occupied space, which becomes more clear if one considers, for example, an arbitrary function f that is expanded in the basis (p,. as 

By projection with the density operator, the orthonormality causes all components other than those corresponding to the occupied space to disappear  

Projecting a second time gives the same result, leading to the idempo-tency property of projection operators (p = p) in 

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