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

The molecular first hyperpolarizability beta contributes to second-order optical non-linear signals collected from molecular liquids. For the Second Harmonic Generation (SHG) response, the first hyperpolarizability beta(2 omega, omega, omega) often depends on the molecular electrostatic environment. This is especially true for water, due to its large second hyperpolarizability gamma(2 omega, omega, omega,0). In this study we compute the electronic gamma(2 omega, omega, omega,0) and beta(2 omega, omega, omega) for water molecules in liquid water using QM/MM calculations. The average value of gamma(2 omega, omega, omega,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 bulk second hyperpolarizability gamma(2 omega, omega, omega,0) can be used to describe the electrostatic effects of the distant neighborhood on the first hyperpolarizability beta(2 omega, omega, omega). In comparison with more complex schemes to take into account long-range effects, the approximation is simple, and does not require any modifications of the QM/MM implementation. The long-range correction can be added explicitly, using an average value of gamma for water in the condensed phase. It can also be easily added implicitly in QM/MM calculations through an additional embedding electric field, without the explicit calculation of gamma.