The effect on the isotropic spin density of the (k,k′) dependence of the exchange coupling $J(\mathbf{k},{\mathbf{k}}^{\prime })$ between a localized magnetic moment and conduction electrons predicted by Ruderman-Kittel-Kasuya-Yosida (RKKY) theory is determined quantitatively. Spherical local moments are employed, viz., $\mathrm{Gd}\left(4\mathrm{}{f}^7\right)$ and $\mathrm{Fe}\left(3\mathrm{}{d}^5\right)$ (which are taken as representative of rare-earth and transition metal moments, respectively). The conduction bands are described by simple orthogonalized plane waves appropriate to a "free-electron" metal with k the wave vector of the incident electron and k′ that of the scattered electron. We find that a $Q$-dependent coupling (where $Q\equiv |\mathbf{k}-{\mathbf{k}}^{\prime }|$) has some justification when dealing with a Gd local moment but has considerably less justification for Fe. Both the (k,k′) and the $Q$-coupling schemes yield a "main" spindensity peak which is more diffuse than that yielded by coupling approximations traditionally applied to RKKY theory. Spin-density results were obtained which are appropriate to the outer reaches of a lattice site and to the nuclear site of either the local moment or neighboring atoms (these involve inclusion of core $s$ terms in the spin density). These results suggest that spin distributions obtained by neutron diffraction and those inferred from hyperfine field measurements may differ significantly.

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