Hi there! You can have a look at this 1998 paper describing the generalized GB potential for biaxial ellipsoids. There, you have the dimensionless pre-exponential strength coefficient \varepsilon_{AB}(\omega_1,\omega_2), which depends on the dimensions of the two ellipsoids and their orientations, and the anisotropic distance function \sigma_{AB}(\omega_1,\omega_2,\vec{r}_{12}), both depending on the symmetric overlap matrix \mathbf{A}_{AB}(\omega_1,\omega_2).
Here is a graphical demonstration of the variation of the GB potential w.r.t. the orientation, courtesy of Matteo Ricci.