  We describe a method for calculating, within density functional theory, the
electronic structure associated with typical defects which substitute for Cu in
the CuO2 planes of high-Tc superconducting materials. The focus is primarily on
Bi2Sr2CaCu2O8, the material on which most STM measurements of impurity
resonances in the superconducting state have been performed. The magnitudes of
the effective potentials found for Zn, Ni and vacancies on the in-plane Cu
sites in this host material are remarkably consistent with phenomenological
fits of potential scattering models to STM resonance energies. The effective
potential ranges are quite short, of order 1 A with weak long range tails, in
contrast to some current models of extended potentials which attempt to fit STM
data. For the case of Zn and Cu vacancies, the effective potentials are
strongly repulsive, and states on the impurity site near the Fermi level are
simply removed. The local density of states (LDOS) just above the impurity is
nevertheless found to be a maximum in the case of Zn and a local minimum in
case of the vacancy, in agreement with experiment. The Zn and Cu vacancy
patterns are explained as due to the long-range tails of the effective impurity
potential at the sample surface. The case of Ni is richer due to the Ni atom's
strong hybridization with states near the Fermi level; in particular, the short
range part of the potential is attractive, and the LDOS is found to vary
rapidly with distance from the surface and from the impurity site. We propose
that the current controversy surrounding the observed STM patterns can be
resolved by properly accounting for the effective impurity potentials and
wave-functions near the cuprate surface. Other aspects of the impurity states
for all three species are discussed.
