A minimal model, consisting of a 4 × 4 Hamiltonian and pertaining to a single Mn-O-Mn bond in the manganites, is introduced and solved to elucidate the physics of the dynamical Jahn-Teller effect on the double-exchange (DEX) interaction. The effect originates from the coupling of the motion of the bridging oxygen atom to the Mn electrons. It is shown that (i) the DEX interaction is considerably reduced by the dynamical Jahn-Teller effect and (ii) this reduction depends on the oxygen mass

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... to an isotope shift in the Curie temperature, T c ∝ M α , with α ∼ 0.2. The isotope shift is found to depend sharply on the electron hopping parameter t. The exact results are also compared to the results obtained from both the adiabatic and the diabatic approximations.