  We propose a staggered mesh method for correlation energy calculations of
periodic systems under the random phase approximation (RPA), which generalizes
the recently developed staggered mesh method for periodic second order
M{\o}ller-Plesset perturbation theory (MP2) calculations [Xing, Li, Lin, JCTC
2021]. Compared to standard RPA calculations, the staggered mesh method
introduces negligible additional computational cost. It avoids a significant
portion of the finite-size error, and can be asymptotically advantageous for
quasi-1D systems and certain quasi-2D and 3D systems with high symmetries. We
demonstrate the applicability of the method using two different formalisms: the
direct ring coupled cluster doubles (drCCD) theory, and the
adiabatic-connection (AC) fluctuation-dissipation theory. In the drCCD
formalism, the second order screened exchange (SOSEX) correction can also be
readily obtained using the staggered mesh method. In the AC formalism, the
staggered mesh method naturally avoids the need of performing "head/wing"
corrections to the dielectric operator. The effectiveness of the staggered mesh
method for insulating systems is theoretically justified by investigating the
finite-size error of each individual perturbative term in the RPA correlation
energy, expanded as an infinite series of terms associated with ring diagrams.
As a side contribution, our analysis provides a proof that the finite-size
error of each perturbative term of standard RPA and SOSEX calculations scales
as $\mathcal{O}(N_{\mathbf{k}}^{-1})$, where $N_{\mathbf{k}}$ is the number of
grid points in a Monkhorst-Pack mesh.
