 Solution

The term $-\alpha ϵ$ is the leading electronic energy gain from splitting the initially degenerate orbitals and occupying the lower levels. Its negative sign expresses the lowering of the occupied-electron energy for the chosen elongation branch of the Jahn-Teller effect. The positive coefficient $\alpha$ is a linear electron-lattice coupling in this reduced model.

The term $\beta {ϵ}^2/2$ is the harmonic elastic cost of moving the ligands away from the undistorted reference structure. Mechanical stability requires $\beta >0$. The competition between linear gain and quadratic cost produces a finite equilibrium distortion rather than indefinite elongation.