First-principles DFT+DMFT calculations of structural properties of actinides: Role of Hund's exchange, spin-orbit coupling, and crystal structure

We utilize a combination of an ab initio calculation of effective Coulomb interactions and a DFT+DMFT calculation of total energy to study the structural properties of pure actinides. We first show that the effective direct Coulomb interactions in plutonium and americium are much smaller than usually expected. Secondly, we emphasize the key role of Hund's exchange in combination with the spin-orbit coupling in determining the structural parameters of $\delta$-plutonium and americium. Thirdly, using this ab initio description, we reproduce the experimental transition from low volume early actinides (uranium, neptunium, $\alpha$-plutonium) to high-volume late actinides ($\delta$-plutonium, americium, and curium) without the need of an artificial magnetism. Finally, we compare the energies and structural properties of $\alpha , \gamma , \epsilon$, and $\delta$ phases of plutonium to experimental data.

Figure 1

Calculation scheme used in this study. The first part of the calculation is a self-consistent cRPA scheme as implemented by us and discussed in Ref. [37] and in Sec. 2a. The self-consistent values of $U$ and $J$ are used in a fully self-consistent DFT+DMFT calculation as implemented by us in abinit [39] using the same Wannier functions [39, 40], and a continuous time quantum Monte Carlo solver [41] to solve the Anderson impurity model. At full self-consistency, the total internal energy is computed following Refs. [39, 42], and the $f$ spectral function is computed using the maximum entropy method (see also text). Notations in this scheme are the same as in Refs. [37, 39, 40].

Figure 2

Ab initio bare and effective interactions in actinides. Bare direct Coulomb interaction $v$ [on part (a)], Hubbard $U$, and Hund exchange $J$ [on part (b)] interactions for U, Np, Pu, Am, and Cm.

Figure 3

Theoretical volumes for actinides compared to experiment. (d) Theoretical volumes as obtained in DFT/PBE (NM) and in DFT/PBE+DMFT—with calculated values of the effective direct Coulomb and Hund interactions (see Fig. 2) in comparison to experiment (extrapolated to 0 K for $\gamma$-, $\epsilon$-, and $\delta$-Pu [32]), (a) GGA (AFM) [8, 11], (b) GGA+OP (AFM) [12, 80], and (c) LDA+GA (NM) [32]. Experimental data are taken from Refs. [81, 82, 83, 84, 85, 86, 87, 88]. NM refers to a nonmagnetic calculation without any local, or global static magnetic moment. AFM refers to antiferromagnetic calculations. GGA+DMFT, with the ab initio determination of $U$ and $J$ is able to describe the equilibrium volumes of all elements and all phases of Pu without any magnetic moment, in agreement with experiment. See Sec. 4a for a detailed discussion.

Figure 4

Role of spin-orbit coupling (SOC) and exchange. Volumes (per atom) of $\alpha$-Pu, $\delta$-Pu, and Am for the full DMFT calculations (with $J$ and SOC), for calculations without exchange $J$ and for calculations without SOC, in comparison to experiment. The nonmagnetic calculation for curium is not stable without spin-orbit coupling. These calculations show that the combination of both the spin-orbit coupling and exchange $J$ is necessary to recover the equilibrium volume of $\delta$-Pu and Am.

Figure 5

Sketch of the role of interactions for two different phases. The parabolic curves represent total energy vs volume, and the minima of the curves represent the equilibrium volume. We call $\alpha$ a delocalized phase with an important chemical bonding, and a low symmetry and volume (such as $\alpha$-U, $\alpha$-Np, or $\alpha$-Pu), whereas $\delta$ is a compact phase (such as fcc Pu, or dhcp Am), with large interatomic distances. (a) and (b) are two effects induced by electronic interactions (see discussion in Sec. 5b1). As discussed in the text (see Secs. 5b1 and 5d), we emphasize that even with the same $U$ and $J, \alpha$-Pu, and $\delta$-Pu have very different equilibrium volumes in DFT+DMFT because of their different chemical bonding.

Figure 6

Role of crystal structure on theoretical volumes. Comparison of DFT+DMFT calculations of volumes (per atom) for actinides with the fcc structure, $\gamma$-Pu structure, $\delta$-Pu structure, and experimental structures. Arrows show the direction of the decrease of DFT+DMFT energy towards the most stable phase (in agreement with experiment). For example, for plutonium, we indeed found that the $\alpha$ phase is more stable than the $\gamma$ phase, whereas the $\delta$ fcc phase is less stable, in agreement with experiment.

Figure 7

DFT+DMFT calculations of volumes (per atom) for actinides, with the same imposed structure (the $\gamma$ phase of plutonium). The graph compares calculations with computed $U$ and $J$ for each actinides, with calculations with a fixed $U$ and $J$. For reference, DFT+DMFT calculations for experimental structures are also given.

Figure 8

DFT+DMFT $5f$ spectral functions of $\delta$-Pu compared to the GGA and GGA+$U 5f$ projected density of states (DOS) used for the cRPA calculation of $U$. Calculations are done at 800 K and GGA+$U$ and GGA DOS are widened (with a 0.5 eV broadening) to be easily compared to DFT+DMFT spectral function.

Figure 9

Experimental photoemission spectra (left) of $\alpha$-uranium (398 K) [106], $\alpha$-neptunium (80 K) [107], plutonium ($\alpha$ at 423 K and $\delta$ at 77 K) [108], americium (at 298 K) [109], and curium (298 K) [91], compared to our DFT+DMFT spectral functions (right) computed with ab initio effective interactions computed with the self-consistent cRPA scheme. The full spectral function, and theoretical direct photoemission spectra are presented in dashed and full lines, respectively. Results are nearly identical at 800 and 200 K. Magnetic calculations for curium are indicated by dotted lines.