Bridging constrained random-phase approximation and linear response theory for computing Hubbard parameters

Abstract

The predictive accuracy of popular extensions to density-functional theory (DFT) such as DFT+U and DFT plus dynamical mean-field theory (DFT+DMFT) hinges on using realistic values for the screened Coulomb interaction U. Here, we present a systematic comparison of the two most widely used approaches to compute this parameter, i.e. linear response theory (LRT) and the constrained random-phase approximation (cRPA), using a unified framework based on the use of maximally localized Wannier functions. We show that the U in LRT and cRPA can differ as much as 30%. We demonstrate that this discrepancy arises from two main differences: neglecting the response of the exchange-correlation potential in cRPA and additional excitation channels in LRT. By taking these differences into account, we can achieve near perfect agreement between the two techniques. Moreover, we show that in cases with strong hybridization between interacting and screening subspaces, the application of cRPA becomes ambiguous and can lead to unrealistically small U values, while LRT remains well-behaved. Our work formally connects both methods, sheds light on their strengths and limitations, and emphasizes the importance of using a consistent set of Wannier orbitals to ensure transferability of U values between different implementations.

Figures (2)

• Figure 1: (a) Crystal structures of cubic KCuF$_3$ and tetragonal Sr$_2$FeO$_4$. (b) and (c) Band structure and projected densities of states (DOS) of (b) KCuF$_3$ and (c) Sr$_2$FeO$_4$. Zero energy corresponds to the Fermi level. Open and closed circles indicate the bands recalculated from the localized $d$-$p$ and frontier orbital basis sets, respectively. The red and blue insets in (b) show isosurface plots of the Cu $d_{x^2-y^2}$ MLWF in the localized $d$-$p$ and frontier orbital basis sets, respectively.
• Figure 2: Interaction parameters obtained from LRT and cRPA for (a) KCuF$_3$ and (b) Sr$_2$FeO$_4$ using different choices for the Hubbard projectors (frontier, $d$-$p$, and $d$-only). Dark blue and dark red bars correspond to interaction parameters computed from a standard LRT and cRPA calculation respectively. The light blue bar corresponds to a modified LRT calculation, i.e. by removing the contribution of the xc kernel, while the light red bar corresponds to the modified cRPA calculation, i.e. obtained by using the coarse graining. For Sr$_2$FeO$_4$, O$^{xy}$ refers to the oxygen within the Fe planes, while O$^{z}$ refers to the apical oxygen. The coarse-grained cRPA results for the $d$-only model are not physically meaningful and are thus not included here.