  Dynamic second-order nonlinear susceptibilities,
$\chi^{(2)}(2\omega,\omega,\omega)\equiv \chi^{(2)}(\omega)$, are calculated
here within a fully first-principles scheme for monolayered molybdenum
dichalcogenides, $2H$-MoX$_2$ (X=S,Se,Te). The absolute values of
$\chi^{(2)}(\omega)$ across the three chalcogens critically depend on the band
gap energies upon uniform strain, yielding the highest $\chi^{(2)}(0)\sim$ 140
pm/V for MoTe$_2$ in the static limit. Under this uniform in-plane stress,
$2H$-MoX$_2$ can undergo direct-to-indirect transition of band gaps, which in
turn substantially affects $\chi^{(2)}(\omega)$. The tunability of
$\chi^{(2)}(\omega)$ by either compressive or tensile strain is demonstrated
especially for two important experimental wavelengths, 1064 nm and 800 nm,
where resonantly enhanced non-linear effects can be exploited: $\chi^{(2)}$ of
MoSe$_2$ and MoTe$_2$ approach $\sim$800 pm/V with -2\% strain at 1064 nm.

Materials Science