Effective Coulomb interaction in transition metals from constrained random-phase approximation

The effective on-site Coulomb interaction (Hubbard $U$) between localized $d$ electrons in 3$d$, 4$d$, and 5$d$ transition metals is calculated employing a parameter-free realization of the constrained random-phase approximation using Wannier functions within the full-potential linearized augmented-plane-wave method. The $U$ values lie between 1.5 and 5.7 eV and depend on the crystal structure, spin polarization, $d$ electron number, and $d$ orbital filling. On the basis of the calculated $U$ parameters, we discuss the strength of the electronic correlations and the instability of the paramagnetic state toward the ferromagnetic one for 3$d$ metals.

Figure 1

Average bare on-site direct Coulomb matrix elements between the $d$ orbitals for TMs. In the inset we show the results for exchange Coulomb matrix elements.

Figure 2

Left panels: $U$ and $\mathrm{Ũ}$ for the TM series. With open and filled diamonds we show $U$ and $\mathrm{Ũ}$ for the magnetic state of 3$d$ TMs. For comparison, experimental results from Refs. 19 (stars), 20 (pluses), and 21 (crosses) as available are given. Right panels: the same for $J$ and $\mathrm{J̃}$.

Figure 3

(a) $U(\omega )$ for bcc V, Nb, and Ta. The inset shows $J(\omega )$ for the same elements. (b) The same as (a) for fcc Ni, Pd, and Pt.

Figure 4

(a) The ratio $U/W$ of the effective Coulomb interaction $U$ and the $d$ bandwidth $W$ for the TM series. The inset shows the $d$ bandwidths. (b) Density of states $N({\epsilon }_F)$ at the Fermi level for the NM states.