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Alleviating Projection-Space Sensitivity in DFT+U via Renormalized U

Manjula Raman, Kenneth Park

Abstract

Although the DFT+U method significantly improves the description of correlated electronic systems, its accuracy is known to depend strongly on the input parameters including local projection space used for the Hubbard correction. As a result, calculations performed with different projection sizes can yield quantitatively different -- and sometimes divergent -- results. In this work, we investigate the dependence of the effective Coulomb interaction $U_{\mathrm{eff}}$ on projection size using constrained DFT calculations for rutile TiO$_2$ and $β$-MnO$_2$. We find that as the projection size increases, the self-consistently calculated $U_{\mathrm{eff}}$ decreases significantly -- by as much as $33$\%. This trend is attributed to renormalization of the Coulomb interaction through orbital relaxation and enhanced screening. When $U_{\mathrm{eff}}$ values recalculated for each projection size are employed, the results for lattice parameters, electronic structure, and relative phase stability become consistent across different projection sizes. These findings can provide a practical route to alleviate projection-size dependence in DFT+U calculations.

Alleviating Projection-Space Sensitivity in DFT+U via Renormalized U

Abstract

Although the DFT+U method significantly improves the description of correlated electronic systems, its accuracy is known to depend strongly on the input parameters including local projection space used for the Hubbard correction. As a result, calculations performed with different projection sizes can yield quantitatively different -- and sometimes divergent -- results. In this work, we investigate the dependence of the effective Coulomb interaction on projection size using constrained DFT calculations for rutile TiO and -MnO. We find that as the projection size increases, the self-consistently calculated decreases significantly -- by as much as \%. This trend is attributed to renormalization of the Coulomb interaction through orbital relaxation and enhanced screening. When values recalculated for each projection size are employed, the results for lattice parameters, electronic structure, and relative phase stability become consistent across different projection sizes. These findings can provide a practical route to alleviate projection-size dependence in DFT+U calculations.
Paper Structure (11 sections, 4 equations, 7 figures)

This paper contains 11 sections, 4 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Methods and Computational Details
  3. LAPW and APW+lo approaches
  4. Control of Projection Space via Muffin-Tin Radii
  5. Estimating ab initio $U_{\mathrm{eff}}$ with constrained DFT
  6. Results & Discussion
  7. Dependence of $U_{\mathrm{eff}}$ on Projection Size
  8. Calculated Properties of TiO$_2$ and MnO$_2$ using Renormalized $U_{\mathrm{eff}}$ Scheme
  9. Dependence of Total Energy on Projection Space
  10. Conclusions
  11. Acknowledgements

Figures (7)

  • Figure 1: (a) the self-consistently determined on-site Coulomb interaction parameter $U_{\mathrm{eff}}$ as a function of Ti muffin-tin radius $R_{\mathrm{MT}}$ for TiO$_2$ using the cDFT method. (b) $U_{\mathrm{eff}}$ values obtained using linear-response theory for rutile TiO$_2$ in literature plotted against different Ti valence sets (open symbols for bottom and left axes) and charge configurations (filled symbols for top and left axes). Ti, Ti_pv, and Ti_sv represent standard, p-- and s--semicore inclusive pseudopotentials, respectively. The dashed lines are only to guide the eyes. References a: curnan2015investigating, b: Orhan2020, c: Mattioli2008, d: xu2015linear, e: farnesi2011ideal.
  • Figure 2: Radial probability distribution function of Ti $d$ orbitals in rutile TiO$_2$ calculated for three different muffin-tin radii, using the PBE exchange-correlation functional.
  • Figure 3: Calculated lattice parameters (a) $a$ and (b) $c$ of TiO$_2$ as a function of Ti muffin-tin radius with two computational schemes. For fixed $U_{\mathrm{eff}}$ scheme (diamond markers), $U_{\mathrm{eff}}$ values are constant and fixed at $4.5$ eV. For renormalized $U_{\mathrm{eff}}$ scheme (square markers), $U_{\mathrm{eff}}$ values from the Fig.\ref{['fig:UvsRmt']} are used at each $R_{\mathrm{MT}}$. As for the reference values, the lattice parameters computed without the Hubbard correction (light blue) are also shown for comparison.
  • Figure 4: Projected density of states (PDOS) for Ti $d$ and O $p$ orbitals in rutile TiO$_2$ using (a) a single fixed $U_{\mathrm{eff}}$$= 4.5$ eV and (b) renormalized values $U_{\mathrm{eff}}$ for different Ti $R_{\mathrm{MT}}$ : 1.91 (solid black), 2.1 (dashed blue), and 2.3 (dash-dotted green) $a_B$. The PDOS for O 2p is in red with the same line patterns. (c) and (d) panels contrast the difference in the calculated CFS between the two different schemes.
  • Figure 5: (a) Self-consistently determined $U_{\mathrm{eff}}$ at selected Mn muffin-tin radii ($R_{\mathrm{MT}}$), (b) and (c) the calculated lattice parameters $a$ and $c$ (d) the magnetic moment as a function of Mn $R_{\mathrm{MT}}$ for $\beta$-MnO$_2$ in AFM ordering, respectively. In the panels (b) through (d), diamond markers correspond to the results using the fixed $U_{\mathrm{eff}}$ value of 5.6 eV, and square markers to those obtained under the renormalized $U_{\mathrm{eff}}$ scheme. Dashed lines are guides to the eye.
  • ...and 2 more figures