Frequency-dependent screened interaction in Ni within the random-phase approximation

First-principles calculations of the screened interaction used in Hubbard Hamiltonians (Hubbard $U)$ are valuable to bridge the gap between model Hamiltonians and real systems. Choosing Ni as an example, we have calculated the screened interaction within the random-phase approximation. It is found that the value of the statically screened interaction in the 3$d$ orbital is 2.2 eV, which is significantly smaller than the values obtained from the constrained local-density approximation method (3.7–5.4 eV). We analyze the possible origins of the difference and reconcile the discrepancy by taking into account the energy dependence of the screened interaction. To undertand our numerical results, we have also studied the screened potential in real space.