Past exam of the condensed matter physics major option course of the University of Oxford / 2019 / 4 / c

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.

Complete the square:The minimum is stable because $d^{2}E/d{ϵ}^2=\beta >0$. HenceHere the Jahn-Teller stabilization energy is the positive magnitude of the energy reduction.