Water molecule first hyperpolarizability

[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.

 

[DrClaude]

Yes.

A complete description of the electric dipole moment (μ), the dipole polarizability (α), the first dipole (β), and the second dipole (γ) hyperpolarizability tensors is reported for the ground state of the water molecule at its equilibrium geometry. Self‐consistent‐field (SCF) and complete fourth‐order many‐body perturbation theory (MP4) values of the independent components are calculated via a finite‐field method from the perturbed energies of the molecule in the presence of a homogeneous electric field. The dependence of the calculated values on the basis set is studied at both the SCF and the MP4 levels. Electron correlation has a strong effect on the hyperpolarizability. Our best SCF values are calculated with a large (13s10p6d2f/9s6p2d)[9s7p6d2f/6s5p2d] basis set comprising 136 contracted Gaussian‐type functions and are 0.7789 e a₀ for the dipole moment and 8.531 e²a₀² E_h⁻¹, −10.86 e³ a₀³ E_h⁻², and 979 e⁴ a₀⁴ E_h⁻³ for the mean dipole polarizability and first and second dipole hyperpolarizabilities, respectively. The electron correlation correction to these properties is estimated at −0.055±0.005 e a₀, 1.11±0.14 e² a₀² E_h⁻¹, −7.1±1.3 e³ a₀³ E_h⁻², and 749±113 e⁴ a₀⁴ E_h⁻³. Agreement with experiment is very good for the dipole moment and mean dipole polarizability. As regards the hyperpolarizability, satisfactory agreement with the frequency‐dependent values of Ward and Miller may also be deduced, but further experimental and theoretical work on the dispersion of the hyperpolarizability is needed for an effective rapprochement of theory and experiment.

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

 

[Konte]

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

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.

 

[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.