Screened Coulomb interaction in the maximally localized Wannier basis

We discuss a maximally localized Wannier function approach for constructing lattice models from first-principles electronic structure calculations, where the effective Coulomb interactions are calculated in the constrained random-phase approximation. The method is applied to the $3d$ transition metals and a perovskite $\left(\mathrm{Sr}\mathrm{V}{\mathrm{O}}_3\right)$. We also optimize the Wannier functions by unitary transformation so that $U$ is maximized. Such Wannier functions unexpectedly turned out to be very close to the maximally localized ones.

Figure 1

(Color online) Band structure of paramagnetic nickel in LDA (solid thin lines) and projected bands (dotted thick lines).

Figure 2

(Color online) On-site screened Coulomb interaction of paramagnetic nickel in the maximally localized Wannier basis as a function of frequency: fully screened interaction (closed circles), result from cRPA (open circles), and bare Coulomb interaction (dotted line).

Figure 3

(Color online) On-site screened Coulomb interaction $U$ for transition metals from cRPA (open circles) and fully screened $U$ from RPA (closed circles). Results when the matrix elements of $U=W_r(\omega =0)$ are taken (a) in the maximally localized Wannier function basis and (b) in the truncated partial waves or the heads of the LMTO-ASA basis.

Figure 4

(Color online) On-site exchange interaction from fully screened interaction (closed triangles), cRPA (open triangles), and from bare interaction (dotted line) in (a) Ni, (b) Fe, and (c) Cu.

Figure 5

(Color online) On-site screened exchange interaction $J$ (triangles) and off-site screened Coulomb interaction $U^{\prime }$ (open circles).

Figure 6

(Color online) Diagonal elements of Coulomb interaction of Ni as a function of $R=\mid \mathbf{R}\mid$ (in units of lattice constant). (a) Bare Coulomb interaction and (b) screened Coulomb interactions in cRPA (open circles) and in RPA (closed circles).