  We provide a derivation of the tight-binding model that emerges from a full
consideration of a particle bound in a periodic one-dimensional array of square
well potentials, separated by barriers of height $V_0$ and width $b$. We derive
the dispersion for such a model, and show that an effective
next-nearest-neighbor hopping parameter is required for an accurate
description. An electron-hole asymmetry is prevalent except in the extreme
tight-binding limit, and emerges through a "next-nearest neighbor" hopping term
in the dispersion. We argue that this does not necessarily imply
next-nearest-neighbor tunneling; this is done by deriving the transition
amplitudes for a two-state effective model that describes a double-well
potential, which is a simplified precursor to the problem of a periodic array
of potential wells.
