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Improved treatment of relativistic effects in linear augmented plane wave (LAPW) method: application to Ac, Th, ThO2 and UO2

A. V. Nikolaev, U. N. Kurelchuk, E. V. Tkalya

TL;DR

This work tackles the challenge of accurately incorporating relativistic effects in the full-potential LAPW method for heavy elements, notably actinides. The authors introduce Dirac-based averaged radial functions for Bloch-type LAPW states ($P^{av}_{\ell}$, $\dot{P}^{av}_{\ell}$), correct the spherical-potential matrix elements, and implement a refined spin-orbit coupling (SOC) treatment—especially for the semicore $6p$ states by using the $6p_{3/2}$ component—while also accounting for small radial components $Q$ near the nucleus. The combined approach yields sizable shifts in structural parameters (up to $\sim$0.15 Å in lattice constants and up to $\sim$26 GPa in bulk moduli) and a substantial increase in valence electron density at the nucleus (factors of $\sim$2.3–4.3), with UO$_2$ showing a small SOC-induced band gap of about $0.2$–$0.4$ eV rather than metallic behavior. These results demonstrate the importance of a fully relativistic, nucleus-aware treatment in FLAPW for actinides and suggest improved accuracy for related spectroscopic analyses and materials predictions.

Abstract

We examine the influence of the relativistic effects within the linear augmented plane wave method (LAPW) with the potential of general shape for solids and suggest a few ways to account them more accurately: (1) we introduce new radial dependencies for LAPW (Bloch-type) basis functions, based on two actual radial solutions of the Dirac equation for j=l-1/2 and j=l+1/2 one-electron states; (2) the canonical LAPW matrix elements for the spherically symmetric component of the potential, assuming non-relativistic radial wave functions, should be corrected; (3) we argue that for a realistic spin-orbit energy splitting of the semicore 6p-states the spin-orbit interaction constant zeta(p) should be calculated with the 6p3/2 radial component; (4) in cases when two j=l +/- 1/2 components are occupied (for example for the 6p states of actinides) the electron density, associated with the small components of valence electrons, can be taken into the calculation scheme. We demonstrate that the new treatment for the relativistic effects is capable to change the equilibrium lattice constant up to 0.15~Å and the bulk modulus up to 26 GPa. We find that the electron density of valence electrons at the nucleus increases by 2.3-4.3 times due to the inclusion of small components, which can be essential for precise description of the potential and density close to the nuclear region, important for nuclear spectroscopies. In contrast to the common believe that in plain band structure treatment UO2 is a metal, we show that in the presence of the spin-orbit coupling UO2 has a small gap of forbidden states (0.2-0.4 eV) at the Fermi level, where the highest occupied and the lowest unoccupied $5f$ bands slightly overlap, as in calculations of the conduction and valence band in solid Ge.

Improved treatment of relativistic effects in linear augmented plane wave (LAPW) method: application to Ac, Th, ThO2 and UO2

TL;DR

This work tackles the challenge of accurately incorporating relativistic effects in the full-potential LAPW method for heavy elements, notably actinides. The authors introduce Dirac-based averaged radial functions for Bloch-type LAPW states (, ), correct the spherical-potential matrix elements, and implement a refined spin-orbit coupling (SOC) treatment—especially for the semicore states by using the component—while also accounting for small radial components near the nucleus. The combined approach yields sizable shifts in structural parameters (up to 0.15 Å in lattice constants and up to 26 GPa in bulk moduli) and a substantial increase in valence electron density at the nucleus (factors of 2.3–4.3), with UO showing a small SOC-induced band gap of about – eV rather than metallic behavior. These results demonstrate the importance of a fully relativistic, nucleus-aware treatment in FLAPW for actinides and suggest improved accuracy for related spectroscopic analyses and materials predictions.

Abstract

We examine the influence of the relativistic effects within the linear augmented plane wave method (LAPW) with the potential of general shape for solids and suggest a few ways to account them more accurately: (1) we introduce new radial dependencies for LAPW (Bloch-type) basis functions, based on two actual radial solutions of the Dirac equation for j=l-1/2 and j=l+1/2 one-electron states; (2) the canonical LAPW matrix elements for the spherically symmetric component of the potential, assuming non-relativistic radial wave functions, should be corrected; (3) we argue that for a realistic spin-orbit energy splitting of the semicore 6p-states the spin-orbit interaction constant zeta(p) should be calculated with the 6p3/2 radial component; (4) in cases when two j=l +/- 1/2 components are occupied (for example for the 6p states of actinides) the electron density, associated with the small components of valence electrons, can be taken into the calculation scheme. We demonstrate that the new treatment for the relativistic effects is capable to change the equilibrium lattice constant up to 0.15~Å and the bulk modulus up to 26 GPa. We find that the electron density of valence electrons at the nucleus increases by 2.3-4.3 times due to the inclusion of small components, which can be essential for precise description of the potential and density close to the nuclear region, important for nuclear spectroscopies. In contrast to the common believe that in plain band structure treatment UO2 is a metal, we show that in the presence of the spin-orbit coupling UO2 has a small gap of forbidden states (0.2-0.4 eV) at the Fermi level, where the highest occupied and the lowest unoccupied bands slightly overlap, as in calculations of the conduction and valence band in solid Ge.

Paper Structure

This paper contains 8 sections, 22 equations, 5 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Method
  3. Explicitly averaged radial basis wave functions
  4. Correction of LAPW matrix elements
  5. Calculated spin-orbit coupling constants, special treatment for $6p-$states
  6. Averaged account of small radial components
  7. Application to actinides
  8. Conclusions

Figures (5)

  • Figure 1: Radial Dirac functions $P_{j=1/2}(r)$ and $P_{j=3/2}(r)$ of the $6p_{1/2}$ and $6p_{3/2}$ semicore states of thorium, (a) close to the nuclear region, and (b) inside the MT-sphere.
  • Figure 2: Radial basis function $P^{av}_{\ell=1}(r)$ (avD) and the canonical KH radial basis function $P^{KH}_{l=1}(r)$KHMPK of the semicore $6p$ states of thorium, (a) close to the nuclear region, and (b) inside the MT-sphere.
  • Figure 3: Averaged radial functions $\bar{P}_{\ell}(r)$ (a) for $6d$ ($\ell=2$), and (b) for $5f$ ($\ell=3$) states and the corresponding canonical KH radial function $\bar{P}_{l}(r)$KHMPK close to the nuclear region.
  • Figure 4: Radial function (avD) $P_{\ell=1}(r)$ of $6p$-states constructed from two large components ($P_1$ and $P_{-2}$, Eq. (\ref{['i1a']})) of the $6p_{1/2}$ and $6p_{3/2}$ states of thorium, and two associated small components $Q_{-1}$ and $Q_2$ (a) close to the nuclear region, and (b) on a larger scale.
  • Figure 5: The upper panel of the calculated band structure of UO$_2$ with the spin-orbit coupling, PBE calculation. The highest occupied $5f$ electron band is shown by solid line, lowest unoccupied $5f$ bands by dashed lines. The vertical gap with $\triangle E$ from $0.2-0.4$ eV is visible. (The zero energy here corresponds to the position of the first $5f$ band at the $\Gamma$ point, $E_F = -0.16$ eV.)