U_i(\theta_i,z_i)=-U_0\left(S+\alpha\sigma\cos \left(\frac{2\pi z_i} d \right) \right)\left(\frac 3 2 \cos^2\theta_i-\frac 1 2 \right ) .
In this formalism, a liquid crystal material is treated as a continuum; molecular details are entirely ignored.