  We provide the first rigorous study of the finite-size error in the simplest
and representative coupled cluster theory, namely the coupled cluster doubles
(CCD) theory, for gapped periodic systems. Assuming that the CCD equations are
solved using exact Hartree-Fock orbitals and orbital energies, we prove that
the convergence rate of finite-size error scales as
$\mathscr{O}(N_\mathbf{k}^{-\frac13})$, where $N_{\mathbf{k}}$ is the number of
discretization point in the Brillouin zone and characterizes the system size.
Our analysis shows that the dominant error lies in the coupled cluster
amplitude calculation, and the convergence of the finite-size error in energy
calculations can be boosted to $\mathscr{O}(N_\mathbf{k}^{-1})$ with accurate
amplitudes. This also provides the first proof of the scaling of the
finite-size error in the third order M{\o}ller-Plesset perturbation theory
(MP3) for periodic systems.
