We present domain-based local pair natural orbital Møller–Plesset second-order perturbation theory (DLPNO-MP2) with Born–von Kármán boundary (BvK) conditions. The approach is based on well-localized Wannier functions in an LCAO formalism and extends the molecular DLPNO-MP2 implementation in the Turbomole program package to periodic systems. The PNOs are formed through a projected atomic orbital (PAO)–orbital specific virtual (OSV)–PNO cascade, using BvK PAOs and OSVs as intermediaries in an analogous manner to the molecular scheme. Our chargeless and surface-dipole corrected local density fitting approach is shown to be numerically stable and to ensure convergent lattice summations over the periodic images for the two- and three-center Coulomb integrals. Through careful benchmarking, we show that the DLPNO approximations in the BvK-DLPNO-MP2 methods are entirely consistent with those of molecular DLPNO-MP2 calculations and with an alternative periodic approach, Megacell-DLPNO-MP2, reported in Paper II of this series Zhu et al., J. Chem. Phys. 163 (2025). The method exhibits a smooth convergence to the canonical correlation energy upon tightening the PNO truncation threshold. Reference MP2 correlation energies are provided for a set of 2D and 3D periodic systems using a triple-zeta basis and supercell sizes up to 13 × 13 and 7 × 7 × 7, respectively.
Nejad et al. (Tue,) studied this question.