Exchange-overlap Densities and the Chemical Bond

This section aims to illustrate the origin of the quantum mechanical exchange-overlap densities and their different behaviour in the case of the chemical bond in ground state H2 and the Pauli repulsion in He2. We choose as starting point for the 1Xg+ ground state of the systems the normalized Heitler-London (HL) wave functions (Magnasco, 2008)  

If x = rs denotes the space-spin variable, we recall from first principles (Magnasco, 2007,2009a) that, for a normalized N-electronwavefunction satisfying the Pauli antisymmetry principle, the one-electron density function is defined as  

12It should be remarked that, while the Heitler-London function (1.93) for H2 is a two-determinant wave function, the Heitler-London function (1.94) for He2 is a single determinant wave function, so that in this case HL and MO approaches coincide. 

The electron densities for the X, states of H2 and He2 resulting from these Heitler-London wave functions are then  

We give in detail below the calculation of the electron density for the Heitler-London wavefunction (1.93) of ground state H2, when a(r), b(r), a(s), (3(s) are all normalized to one  

The spinless 1-electron density (Equation 1.99) satisfies the conservation relation  

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EXCHANGE-OVERLAP DENSITIES AND THE CHEMICAL BOND so that ... 

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