Theory of superexchange for 3dn-Ions (1≤n≤9) involved in natural and artificial magnets II- derivation of the various exchange energies Ji vs key molecular integrals

In part I we have set the fundamental physical bases of a general treatment of superexchange for 3dⁿ-metallic cations A and B (1lesnles9) characterized by sigma-type bonds on both sides of the diamagnetic bridge X without the presence of a pi-type orbital (with here A=B or AneB). In addition we have shown that, due to the electron-hole symmetry, the study may be restricted to the case of 3dⁿ-ions (with 1lesnles5) because the treatment of the corresponding 3d^10-n dual ion characterized by the same spin quantum number is formally strictly similar: in all the expressions obtained for the 3dⁿ-ion (with 1lesnLt5) the charge of electron -e may be replaced by the charge of hole +e. In this part, we start from the intermediate cationic states defined in I, thus allowing to express the collective state describing the molecule AXB. In the last step we also express the exchange energies J_i of the various fragments A_iX_i of the bond A-X (or X-B) vs key molecular integrals characterizing this bond and defined in part I. Finally we remark that this general treatment may be employed to exactly calculate exchange coupling energies in more complex isolated molecules or ionic entities appearing in natural or artificial magnets.