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The exciton state in an insulator is defined as an excited state of the fully interacting electronic system. Its motion is governed by the homogeneous Bethe-Salpeter equation. From this we derive the effective-mass approximation in the limit of small binding energy, small total momentum, and large spatial extent of the exciton. The corrections to the effective-mass approximation are briefly indicated.
Sham et al. (1966) studied this question.