  Nonlinear optical (NLO) phenomena such as harmonic generation, Kerr, and
Pockels effects are of great technological importance for lasers, frequency
converters, modulators, switches, etc. Recently, two-dimensional (2D) materials
have drawn significant attention due to their strong and unique NLO properties.
Here, we describe an efficient first-principles workflow for calculating the
quadratic optical response and apply it to 375 non-centrosymmetric
semiconductor monolayers from the Computational 2D Materials Database (C2DB).
Sorting the non-resonant nonlinearities with respect to bandgap $E_g$ reveals
an upper limit proportional to $E_g^{-4}$, which is neatly explained by two
distinct generic models. We identify multiple promising candidates with giant
nonlinearities and bandgaps ranging from 0.4 to 5 eV, some of which approach
the theoretical upper limit and greatly outperform known materials. Our
comprehensive library of ab initio NLO spectra for all 375 monolayers is freely
available via the C2DB website. We expect this work to pave the way for highly
efficient and compact opto-electronic devices based on 2D materials.
