  Despite recent advances, systematic quantitative treatment of the electron
correlation problem in extended systems remains a formidable task.
Systematically improvable Green's function methods capable of quantitatively
describing weak and at least qualitatively strong correlations appear promising
candidates for computational treatment of periodic systems. We present a
periodic implementation of temperature-dependent self-consistent 2nd-order
Green's function method (GF2), where the self-energy is evaluated in the basis
of atomic orbitals. Evaluating the real-space self-energy in atomic orbitals
and solving the Dyson equation in $\mathbf{k}$-space are the key components of
a computationally feasible algorithm. We apply this technique to the 1D
hydrogen lattice - a prototypical crystalline system with a realistic
Hamiltonian. By analyzing the behavior of the spectral functions, natural
occupations, and self-energies, we claim that GF2 is able to recover metallic,
band insulating, and at least qualitatively Mott regimes. We observe that the
iterative nature of GF2 is essential to the emergence of the metallic and Mott
phases.
