  We present first-principles calculations for moments of the current up to the
third order in atomic-scale junctions. The quantum correlations of the current
are calculated using the current operator in terms of the wave functions
obtained self-consistently within the static density-functional theory. We
investigate the relationships of the conductance, the second, and the third
moment of the current for carbon atom chains of various lengths bridging two
metal electrodes in the linear and nonlinear regimes. The conductance, the
second-, and the third-order Fano factors exhibit odd-even oscillation with the
number of carbon atoms due to the full and half filled {\pi}* orbital near the
Fermi levels. The third-order Fano factor and the conductance are positively
correlated.
