Exchange Character of the Anderson Hamiltonian

The exchange character of the Anderson model Hamiltonian has been studied. We find that for determining the conduction-electron Green's function, this Hamiltonian can be replaced by an effective Hamiltonian in which the strong $d-d$ exchange term $U{n}_{d\uparrow }{n}_{d\downarrow }$ has been treated exactly and the $s-d$ mixing term $V_{\mathrm{mix}}$ has been modified so that it exhibits explicitly exchange character. Hence, $V_{\mathrm{mix}}$ gives rise to a Kondo-like divergence in the conduction-electron self-energy. Extending the Anderson Hamiltonian by adding an $s-d$ exchange interaction, we find this exchange interaction is left unchanged in going to our effective Hamiltonian.