  Nonlinear optical (NLO) responses have garnered tremendous interest for
decades due to their fundamental and technological interests. The theory and
calculations of NLO responses including electron-hole interactions, which is
especially crucial for reduced-dimensional materials, however, remain
underdeveloped. Here, we develop an ab initio approach to calculate
second-order nonlinear responses (such as second harmonic generation (SHG) and
shift current) with excitonic effects in an exciton-state basis, going beyond
the independent-particle approximation. We compute SHG in monolayer h-BN and
MoS2 employing exciton states from GW-Bethe-Salpeter equation (GW-BSE)
calculations and show both materials exhibit huge excitonic enhancement. The
physical origin of the enhancement is directly understood through the coupling
amplitudes among exciton states, assisted with diagrammatic representations.
Our method provides an accurate and ab initio description of second-order NLO
responses, capturing self-energy and electron-hole interaction effects.
