Direct Exchange Interaction in Heitler-London Model
The idea of direct exchange interaction was originally derived in Heitler-London model which explained the binding energy of the hydrogen molecule. This model inspired Heisenberg's theory of ferromagnetism which based on an exchange interaction between the spins of two neighboring atoms. Consider two hydrogen atoms \(a\) and \(b\) with their isolated atomic orbitals represented as \(\phi_{a}(\vec{r})\) and \(\phi_{b}(\vec{r})\), respectively. The hamiltonian of an isolated hydrogen atom is given as: \[ H = - \frac{\hbar^{2}}{2m}\nabla^{2} - \frac{e^{2}}{r}\] The hamiltonian of the hydrogen molecule is given by the sum of those for the isolated atoms: \[ H = - \frac{\hbar^{2}}{2m}\nabla_{1}^{2} - \frac{\hbar^{2}}{2m}\nabla_{2}^{2} - \frac{e^{2}}{r_{a1}} - \frac{e^{2}}{r_{b2}} + \left( \frac{e^{2}}{r_{12}} + \frac{e^{2}}{r_{ab}} - \frac{e^{2}}{r_{b1}} - \frac{e^{2}}{r_{a2}} \right)\] where \(r_{1}\) and \(r_{2}\) represent coordinates of the two electrons in the m...