Spin-orbit coupling in an $f$-electron tight-binding model: Electronic properties of Th, U, and Pu

We extend a tight-binding method to include the effects of spin-orbit coupling and apply it to the study of the electronic properties of the actinide elements Th, U, and Pu. These tight-binding parameters are determined for the fcc crystal structure using the equivalent equilibrium volumes. In terms of the single-particle energies and the electronic density of states, the overall quality of the tight-binding representation is good and of the same quality as without spin-orbit coupling. The values of the optimized tight-binding spin-orbit coupling parameters are comparable to those determined from purely atomic calculations.

Figure 1

TB fit quality in terms of the cumulative rms errors at various steps of the optimization procedure. Step 1 relaxes the spin-orbit parameters $\left({\xi }_{nl}\right)$, step 2 relaxes the remaining on-site parameters, and step 3 is a full relaxation of all parameters. Open symbols at step 3 indicate the original scalar-relativistic fit quality. Note that the cumulative rms error is over all of the fitted bands (20 bands for Th, U, and Pu). Although the spin-orbit coupling is an atomic quantity, the improvement of our results in step 3 (which relaxes intersite parameters) that indicates some environmental effects should also be taken into account.

Figure 2

(Color online) TB energy bands for Th $\left(a=9.61\right)$, U $\left(a=8.22\right)$, and Pu $\left(a=8.14\right)$ shown in comparison with FLAPW valence energy bands (dotted lines). Note the good agreement. The abscissa for each calculation has been shifted such that the Fermi energy is at zero. Higher valence states (above the first 20) are not fit. In addition, such states would require additional basis states to be represented properly. Hence the fit quality is reduced in the energy range well above the Fermi level.

Figure 3

(Color online) TB energy bands (dotted lines) for Pu semicore $6p$ states, compared with FLAPW values (solid lines).

Figure 4

(Color online) TB (dotted lines) and FLAPW (solid lines) total DOS, including spin-orbit coupling. Note that the TB calculation is in quite good agreement with the FLAPW results despite using a different BZ integration method. The abscissa for each calculation has been shifted such that the Fermi energy is at zero.