Analysis of over-magnetization of elemental transition metal solids from the SCAN density functional

Recent investigations have found that the strongly constrained and appropriately normed (SCAN) meta-generalized gradient approximation exchange-correlation functional significantly over-magnetizes elemental Fe, Co, and Ni solids. For the paradigmatic case, bcc Fe, the error relative to experiment is $\gtrsim 20\%$. A comparative analysis of magnetization results from SCAN and its deorbitalized counterpart, SCAN-L, leads to identification of the source of the discrepancy. It is not from the difference between Kohn-Sham (SCAN-L) and generalized Kohn-Sham (SCAN) procedures. The key is the iso-orbital indicator $\alpha$ (the ratio of the local Pauli and Thomas-Fermi kinetic energy densities). Its deorbitalized counterpart ${\alpha }_L$ has more dispersion in both spin channels with respect to magnetization in an approximate region between 0.6 and 1.2 bohrs around an Fe nucleus. The overall effect is that the SCAN switching function evaluated with ${\alpha }_L$ reduces the energetic disadvantage of the down channel with respect to up compared to the original $\alpha$, which in turn reduces the magnetization. This identifies the cause of the SCAN magnetization error as insensitivity of the SCAN switching function to $\alpha$ values in the approximate range $0.5\lesssim \alpha \lesssim 0.8$ and oversensitivity for $\alpha \gtrsim 0.8$.

Figure 1

Fixed spin-moment energy vs magnetic moment for bcc Fe at $a_{\text{exp}}$ from PBE, SCAN, SCAN-L, TPSS, and TPSS-L. Dots show minimum FSM energy values. Experimental value from Ref. [10].

Figure 2

Ratio of angularly averaged $\alpha$'s, $\langle {\alpha }_L\rangle /\langle \alpha \rangle$ for spin-up (above) and spin-down (below) as function of radial distance from Fe nucleus (evaluated with bcc Fe PBE densities) at selected fixed spin moments.

Figure 3

Angularly averaged $\alpha$ (above) and ${\alpha }_L$ (below) as a function of radial distance from a Fe nucleus (evaluated with bcc Fe PBE densities) at selected fixed spin moments. Spin-up plots are in the left column, spin-down in the right.

Figure 4

Left: Angularly averaged SCAN switching function $f_x\left(\alpha \right)$ as a function of radial distance from the Fe nucleus (evaluated with bcc Fe PBE spin densities) at $m_{sp}=2.5{\mu }_B/\text{atom}$. Right: Radial behavior of the angularly averaged SCAN switching function evaluated with ${\alpha }_L$, $f_x\left({\alpha }_L\right)$, for the same densities.

Figure 5

bcc Fe densities of states for SCAN and SCAN-L. Up spin, upper panel; down spin, lower panel.