Energy Description of AIM

To write the energy expression, which characterizes an atom in a molecule, for simplicity, a diatomic molecule AB is considered for which the molecular wave function P is known. This way, we can write the first-order density matrix (Rychlewski Parr, 1986 Li Parr, 1986)  

Under these conditions, the molecular kinetic energy will be written (for simplicity will be skipped the above notation for optimized state, being implicitly if not otherwise specified)  

The corresponding functional energy of molecular electronic repulsion will be written as  

In the same manner, the total nucleus-electron molecular energy will be formed by the terms  

This way the density functional character for the terms involved in the molecular energies is emphasized. For example, the energy of the atom A in the considered molecule associates with the density functional energy 

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