We present ab initio studies of the conductance in the ballistic limit for relaxed monatomic Cu wires. Our approach is based on the density functional theory in the frame of the Korringa-Kohn-Rostoker Green's function method. Atomic relaxations are calculated by means of ab initio based many body potentials. We confirm the parity oscillation of the conductance ond demonstrate that the effect is robust with respect to structural relaxation of the wire.