  We derive the effective Hamiltonian of the Nonrelativistic Quantum
Electrodynamic up to $m\alpha^8$ by using scattering matching approach. At
$m\alpha^6$ order, these results are coincide with Pachucki's, which is
obtained by applying Foldy-Wouthuysen transformation. And by using the NRQED
Hamiltonian, we derive the photon-exchange interaction in non-retarded
approximation and the retardation correction up to $m\alpha^8$. The energy
shift of the photon-exchange interaction is obtained by studying the pole of
the total Green function.
