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The anisotropic hyperfine interaction of Cu atoms caused by nearby 3d‐impurity atoms has been extensively studied some years ago by the Slichter group performing NMR measurements on single crystals. A first principles description of these interesting and challenging experiments is given in terms of self‐consistent calculations of the electronic structure of the corresponding 3d‐impurity systems. Our theoretical approach is based on the Korringa‐Kohn‐Rostoker‐Green’s function method of band structure calculations and takes perturbations of Cu‐host atoms up to the fourth neighboring shell around the impurity into account. In addition to our former work [B. Drittler et al., Phys. Rev. B 39, 6334 (1989)] dealing only with the dominating isotropic Fermi contact interaction, we also treat the dipolar spin part of the hyperfine interaction, which we assume to be the most important source of the anisotropic Knight shift deduced from single‐crystal measurements.