Band gap opening from displacive instabilities in layered covalent-organic frameworks $^\dag$

Covalent organic frameworks (COFs) offer a high degree of chemical and structural flexibility. There is a large family of COFs built from 2D sheets that are stacked to form extended crystals. While it has been common to represent the stacking as eclipsed with one repeating layer ("AA"), there is growing evidence that a more diverse range of stacking sequences is accessible. Herein, we report a computational study of layer stacking in two prototypical COFs, Tp-Azo and DAAQ-TFP, which have shown high performance as Li-ion battery electrodes. We find a striking preference for slipped structures with horizontal offsets between layers ranging from 1.7 \r{A} to 3.5 \r{A} in a potential energy minimum that forms a low energy ring. The associated symmetry breaking results in a pronounced change in the underlying electronic structure. A band gap opening of 0.8 - 1.4 eV is found due to modifications of the underlying valence and conduction band dispersion as explained from changes in the $\pi$ orbital overlap. The implications for the screening and selection of COF for energy applications are discussed.