  We present a general procedure for obtaining progressively more accurate
functional expressions for the electron self-energy by iterative solution of
Hedin's coupled equations. The iterative process starting from Hartree theory,
which gives rise to the GW approximation, is continued further, and an explicit
formula for the vertex function from the second full cycle is given. Calculated
excitation energies for a Hubbard Hamiltonian demonstrate the convergence of
the iterative process and provide further strong justification for the GW
approximation.
