Good question. There is no contribution to the energy at first order. The wavefunction (unperturbed) has one electron on each site (either ferro or antiferro ordered). Let us call this initial wavefunction |GS>. The perturbation hops an electron from one site to a neighboring site. Let us write this figuratively as |Excited> = HOP|GS>. In first order perturbation theory the change in energy is
which is zero because |GS> and |Excited> are orthogonal. Perhaps this requires a comment in the book?
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