Origin of Jahn-Teller Distortion and Orbital Order in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>LaMnO</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math>

The origin of the cooperative Jahn-Teller distortion and orbital order in ${\mathrm{LaMnO}}_{3}$ is central to the physics of the manganites. The question is complicated by the simultaneous presence of tetragonal and ${\mathrm{GdFeO}}_{3}$-type distortions and the strong Hund's rule coupling between ${e}_{g}$ and ${t}_{2g}$ electrons. To clarify the situation we calculate the transition temperature for the Kugel-Khomskii superexchange mechanism by using the local density $\mathrm{\text{approximation}}+\mathrm{\text{dynamical}}$ mean-field method, and disentangle the effects of superexchange from those of lattice distortions. We find that superexchange alone would yield ${T}_{\mathrm{KK}}\ensuremath{\sim}650\text{ }\text{ }\mathrm{K}$. The tetragonal and ${\mathrm{GdFeO}}_{3}$-type distortions, however, reduce ${T}_{\mathrm{KK}}$ to $\ensuremath{\sim}550\text{ }\text{ }\mathrm{K}$. Thus electron-phonon coupling is essential to explain the persistence of local Jahn-Teller distortions to $\ensuremath{\gtrsim}1150\text{ }\text{ }\mathrm{K}$ and to reproduce the occupied orbital deduced from neutron scattering.