Calculation of Coulomb interaction strengths for 3dtransition metals and actinides

Coulomb interaction strengths ($U_{\mathrm{dd}}$ and $U_{\mathrm{ff}}$) have been calculated from Hartree-Fock-Slater atomic calculations for 3d transition and 5f actinide elements, respectively. By decomposing the different contributions to the response (screening) to the 3d charge fluctuation, we show that a substantial reduction in $U_{\mathrm{dd}}$ arises due to the relaxation of the 3d charge distribution itself. This, combined with the screening due to the response of the 4s charge density, is shown to provide a very compact screening charge comparable to the metallic case, explaining the success of the atomic calculations for estimating U even in the metals. A pronounced dependence of $U_{\mathrm{dd}}$ (or $U_{\mathrm{ff}}$) on the number of electrons $n_d$ ($n_f$) or the electronic configuration is also shown here.