First hyperpolarizability of water in the bulk phase: long-range electrostatic effects included via the second hyperpolarizability

The molecular first hyperpolarizability β contributes to second-order optical non-linear signals collected from molecular liquids. For the Second Harmonic Generation (SHG) response, the first hyperpolarizability β (2ω,ω,ω) often depends on the molecular electrostatic environment. This is especially true for water, due to its large second hyperpolarizability γ(2ω,ω,ω,0). In this study we compute the electronic γ(2ω,ω,ω,0) and β (2ω,ω,ω) for water molecules in their bulk phase using QM/MM calculations. The average value of γ(2ω,ω,ω,0) is smaller than the one for the gaz phase, and its standard deviation among the molecules is relatively small. In addition, we demonstrate that the average second hyperpolarizability ⟨γ(2ω,ω,ω,0)⟩ can be used to describe the electrostatic effects of the neighborhood on the β (2ω,ω,ω) ; but only the more distant neighbors can be accounted this way. This study paves the way towards QM/MM calculations of the first hyperpolarizability of molecules in complex environments, in which long-range electrostatic effects can be crucial, for example nearby charged interfaces.