# A Water molecule first hyperpolarizability

1. Mar 8, 2016

### Konte

Hello everybody,

My question is about the water molecule. Is the isolated H2O molecule has a non-vanishing element $\beta_{ijk}$ in its first-order hyperpolarizability tensor $\beta$ ?

Thank you everybody.

2. Mar 8, 2016

### Staff: Mentor

Yes.

G. Maroulis, J. Chem. Phys. 94, 1182 (1991); http://dx.doi.org/10.1063/1.460025

3. Mar 8, 2016

### Konte

Thank you Dr Claude,

Indeed, they mention $\beta_{zxx}$, $\beta_{zyy}$ and $\beta_{zzz}$.

Does someone know how to demonstrate that only theses three tensor elements survive for the water molecule H2O ?

Thank you.

4. Mar 8, 2016

### DrDu

Handwaving maybe like this: The tensor is symmetric in the last two indices, so it can be brought to main axis form where only the xx, yy and zz components survive. The polarisation is antisymmetric while the square of the electric field is symmetric under reflections. Hence matrix elements where P is perpendicular to some symmetry plane must vanish. This leaves only zxx, zyy, and zzz.