  Building on recent developments in electronic-structure methods, we define
and calculate the flexoelectric response of two-dimensional (2D) materials
fully from first principles. In particular, we show that the open-circuit
voltage response to a flexural deformation is a fundamental linear-response
property of the crystal that can be calculated within the primitive unit cell
of the flat configuration. Applications to graphene, silicene, phosphorene, BN
and transition-metal dichalcogenide monolayers reveal that two distinct
contributions exist, respectively of purely electronic and lattice-mediated
nature. Within the former, we identify a key $metric$ term, consisting in the
quadrupolar moment of the unperturbed charge density. We propose a simple
continuum model to connect our findings with the available experimental
measurements of the converse flexoelectric effect.
