We present an atomic-orbital formulation of second-order Møller–Plesset (MP2) theory for periodic systems. Our formulation is shown to have several advantages over the conventional crystalline orbital formulation. Notably, the inherent spatial decay properties of the density matrix and the atomic orbital basis are exploited to reduce computational cost and scaling. The multidimensional k-space integration is replaced by independent Fourier transforms of weighted density matrices. The computational cost of the correlation correction becomes independent of the number of k-points used. Focusing on the MP2 quasiparticle energy band gap, we also show using an isolated fragment model that the long range gap contributions decay rapidly as 1/R5, proof that band gap corrections converge rapidly with respect to lattice summation. The correlated amplitudes in the atomic orbital (AO) basis are obtained in a closed-form fashion, compatible with a semidirect algorithm, thanks to the Laplace transform of the energy denominator. Like for its molecular counterpart, the Laplace quadrature can be accurately carried out by using few quadrature points, 3–7 depending on the application. In particular, MP2 quasiparticle energy band gaps are computed accurately with 3 Laplace quadrature points. All these traits indicate that robust calculations of the correlation correction to the Hartree–Fock (HF) energy and band gap of large systems can be carried out. We present benchmark periodic MP2 calculations on polyacetylene, polyphenylenevinylene, hexagonal boron nitride, and stacked polyacetylene.

Ayala, P. Y., Kudin, K. N., & Scuseria, G. E. (2001). Atomic orbital Laplace-transformed second-order Møller–Plesset theory for periodic systems. The Journal of Chemical Physics, 115(21), 9698–9707.