Finite-size effects in periodic coupled cluster calculations

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. Given exact Hartree-Fock orbitals and their corresponding orbital energies, we demonstrate that the correlation energy obtained from the approximate CCD method, after a finite number of fixed-point iterations over the amplitude equation, exhibits a finite-size error scaling as O(N-k(-1/3)). Here N-k is the number of discretization points in the Brillouin zone and characterizes the system size. Under additional assumptions ensuring the convergence of the fixed-point iterations, we demonstrate that the CCD correlation energy also exhibits a finite-size 1 error scaling as O(N-k(-1/3)). Our analysis shows that the dominant error lies in the coupled cluster amplitude calculation, and the convergence of the finite-size error in energy calculation can be boosted to O(N-k(-1)) with accurate amplitudes. This also provides the first proof of the scaling of the finite-size error in the third order Moller-Plesset perturbation theory (MP3) for periodic systems.