molecular orientation

Absorption probability (referred to electric dipolar absorption) for a molecular transition with its electric transition (dipole) moment at an angle $θ$ with the electric vector of the light is proportional to $cos 2 θ$ . For the whole sample it is proportional to the orientation factor $K θ = < cos 2 θ >$ , averaged over all sample molecules. This average is $1$ for a sample with all transition moments perfectly aligned along the electric vector of the light, $1/3$ for an isotropic sample and $0$ for a sample where all transition moments are perpendicular to the electric vector.

Notes:

The directional cosines provide, especially for uniaxial samples, a simple description of exactly those orientation properties of the sample that are relevant for light absorption. With the principal coordinate system ( $x$ , $y$ , $z$ ), forming angles $θ = α , β , γ$ with the light electric vector in the $z$ direction, all orientation effects induced by light absorption are contained in $K θ θ = K θ$ . Since the sum of $K θ$ for three perpendicular molecular axes is equal to $1$ , only two independent parameters are required to describe the orientation effects on light absorption.
A related, commonly used description is based on diagonalized Saupe matrices:
$S θ = ( 3 K θ − 1 ) / 2$
The principal (molecular) coordinate system ( $x$ , $y$ , $z$ ) forming angles $θ = α , β , γ$ with the light electric vector should be chosen such that the matrix $K$ and the tensor $S θ$ are diagonal.

To describe processes involving two or more photons, such as luminescence of a uniaxial, aligned sample, an expansion of the directional cosines to the fourth power is required.
Order parameters (related to Wigner matrices) are an alternative to the directional cosine-based description of molecular alignment. Order-parameter methods also work well for non-uniaxial samples and provide a seemingly more complex, but in other ways convenient, description of molecular orientation distributions. Wigner matrices are used as a basis set for an expansion of the orientation–distribution function.

Source:

PAC, 2007, 79, 293 (Glossary of terms used in photochemistry, 3rd edition (IUPAC Recommendations 2006)) on page 371