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Yes, this understanding is correct. The Born-Oppenheimer is assumed in the derivation of the Jahn-Teller effect. Also, the scenario you describe in the last paragraph is qualitatively similar to the Dynamic Jahn-Teller effect.


1

If the ions are separated they have no shared band. Bringing them close together results in a bonding state, i.e. a shared electronic state that has a potential energy below that of the ions at infinity. The other questions follow from understanding this.


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This makes sense to me. $l$ and $m$ are two different summation indices. If you used the same index, say $l$, you would have only the $c^*_{lu}$ $c_{lu}$ terms, without the cross terms when $l\neq m$.



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