Physical interpretation of orthogonal Hilbert space transformations in tight-binding systems with nonorthogonal bases

We study a recently introduced Hilbert space transformation from a nonorthogonal basis to an orthogonal basis, with an emphasis on the physical interpretation of the new Hilbert space. We find that the new Hilbert space may be interpreted as an orthogonal basis in the same physical space, wherein the basis overlap is formally transferred to the hopping matrix elements in the orthogonal system. The result is a standard tight-binding system in an orthogonal basis, with long-range hopping. The procedure is illustrated using an infinite one-dimensional hopping system with nearest-neighbor overlap. We also use the formal procedure to solve for the transmission characteristics of an impurity site coupled with semi-infinite leads, as an example of the interpretive power of the ordinary tight-binding formalism in the case of transport in the presence of basis-overlap effects: the antiresonances are produced, in the orthogonal space, by the presence of second-nearest-neighbor hopping.