The benzo-15-crown-5 molecule may be
approximated by C_{2v} symmetry (figure 9). This leads to a
large reduction in the number of independent non-zero
hyperpolarisability components. In addition, as there is no
delocalisation in the direction perpendicular to the molecular plane
a further simplification occurs leaving only 3 non-zero, independent
tensor elements for SHG: β_{xzx},
β_{zxx} and β_{zzz}^{
21}. Comparisons with results from other benzene species
indicates that the dominant components of the hyperpolarisablity
tensor will be *β*_{zxx}
and *β*_{zzz} ; approximate
calculations with Gaussian 90 confirm these predictions.
*Figure 9. Assumed symmetry of the benzo-15-crown-5 . O*_{z} is the molecular axis adn the plane formed by O_{z} and O_{x} is the molecular plane.
The
relationship between the surface and the molecular properties can be
derived assuming the relevant averaging about the azimuthal angle and
internal molecular axis of the chromophore. Once the dominant
hyperpolarisability components are known, the susceptibility
components may be used to derive the orientation parameter,
D = 〈cos^{3}Θ〉/〈cosΘ〉 . If the orientational distribution is narrow, then *D* may be
taken as being equal to cos^{2}Θ. |