Relationship between the barrier (\(\Delta G^{\ddagger}\)) to thermal @E02011@, the energy of a corresponding optical @C01008@ (\(\Delta E_{\text{op}}\)), and the overall change in standard Gibbs energy accompanying thermal @E02011@ (\(\Delta G^{\,\unicode{x26ac}}\)). Assuming a quadratic relation between the energy of the system and its distortions from equilibrium (harmonic oscillator model) the expression obtained is: \[\Delta G^{\ddagger} = \frac{\Delta E_{\text{op}}^{2}}{4\ (\Delta E_{\text{op}}\,-\,\Delta G^{o})}\] The simplest form of this expression obtains for degenerate @E02011@ (\(\Delta G^{\,\unicode{x26ac}}\)) in e.g. symmetrical mixed @V06588@ systems: \[\Delta G^{\ddagger} = \frac{\Delta E_{\text{op}}}{4}\] Note that for this situation the @M03702@ reads: \[\Delta G^{\ddagger} = \frac{\lambda }{4}\]