Low energy electron diffraction experiments have superb sensitivity to
surface structure, but rely on sophisticated theory for their interpr
etation. Advances in computer power, and developments in the theory it
self, enable us to handle surface structures of moderate complexity. F
or future advances we must look to a completely new approach and the c
ase is made for order-N methods which follow the time evolution of a p
oint source of electrons to generate all beams for all angles of incid
ence and all energies in one shot.