The local geometry, energy stabilization, and pseudolocal ${\mathit{t}}_{1\mathit{u}}$ vibration mode of the ${\mathrm{Cu}}^{+}$ impurity in the alkali halide crystals have been investigated with the ab initio perturbed ion cluster-in-the-lattice methodology. The electronic structure of different clusters, containing up to 179 ions, has been computed for nine Cu:AX systems (A=Li,Na,K;X=F,Cl,Br). The calculations clearly show that the nearest-neighbor relaxations induced by impurity substitution are essentially determined by the substituted cation, the anion playing a rather minor role. In contrast with predictions deducible from empirical ionic radii, we find negligible or very small relaxations for Cu:LiX systems, and inward relaxations of about -0.1 Å for Cu:NaX systems [in very good agreement with recent extended x-ray absorption fine-structure (EXAFS) measurements on Cu:NaCl]. For the Cu:KX family we found inward relaxations as large as -0.3 Å. The stabilization energy associated to the substitution reaction turns out to range from -0.2 to -1.8 eV, with a remarkable dependence upon the substituted cation. The ${\mathit{t}}_{1\mathit{u}}$ frequencies, computed without including the intershell coupling, decrease with increasing cationic size, showing a trend that agrees with the experimental data reported by McClure for Cu:LiCl, Cu:NaF, and Cu:NaCl. Our methodology, in its present form, does not reproduce the off-center equilibrium position of the ${\mathrm{Cu}}^{+}$ ion observed in Cu:NaBr, Cu:KCl, and Cu:KBr.

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