Hubbard $U$ parameters for transition metals from first principles (2024)

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  Figure 1

  The Hubbard $U$ parameters for $3d,4d$, and $5d$ transition metals. The different symbols denote the results obtained with: the cLDA method (red filled and open squares for nonmagnetic and magnetic metals, respectively), the $d$-band center shifts between HF and DFT methods (green circles for nonmagnetic metals), the cRPA method by Şaşıoğlu etal. [15] (blue filled and open diamonds for nonmagnetic and magnetic metals, respectively), the cLDA method by Nakamura etal. [26] (open circles). The black symbols represent the experimental values (the measured correlation energy deduced from Auger and XPS spectroscopy) of de Boer etal. [53], Sawatzky and Post [54], Antonides etal. [55] (stars) and Kaurila etal. [56] (crosses, with uncertainty at the level of $\pm 0.4\mathrm{eV}$). We note, however, that an exact correspondence between the measured and computed values is not expected and we provide these values only for qualitative comparison. All the reported data are also provided in the ESI, Table S2 [57].

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  Figure 2

  The projected $d$ orbitals density of states for fcc Cu metal: (a)computed with the standard DFT ($U=0$), hybrid functional (PBE0) and Hartree-Fock methods, as well as measured with XPS by Ref.[46], and (b)computed with $\mathrm{DFT}+U$ and different projections of the $d$ states occupancy: atomic orbitals (AO), Wannier functions (WF). Gaussian smearing of $0.03\mathrm{Ry}$ has been used to match the experimental band broadening. The XPS data are scaled vertically to match the intensity of the computed $d$-band.

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  Figure 3

  The $d$-bandwidths $W$ (derived considering: only the occupied part of the $d$-band (left panels) and the entire $d$-bandwidth (right panels)), derived as described in the text. Magnetic states of Cr, Mn, Fe, Co, and Ni metals are considered. Computed reference data (ref. calc.) are taken from Şaşıoğlu etal. [15]. Experimental reference data are taken from Hüfner etal. [66] for Ni, Cu and Ag and from Smith etal. [82] for Rh, Pd, Ir, Pt, and Au. AO and WF indicate calculations performed with atomic orbitals and Wannier functions as projectors, respectively. Uncertainty of the measured values is in the order of $\pm 1\mathrm{eV}$, and arises mainly due to unclear definitions of the band limits. All the reported data are also provided in the ESI, Table S4 [57].

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  Figure 4

  The $U/W$ ratio computed with the cLDA and cRPA [15] Hubbard $U$ parameter values. The bandwidths $W$ are those from standard DFT calculations ($U=0$), to allow a straightforward comparison to the cRPA reference data [15] computed in such way. These were derived considering: only the occupied part of the $d$-band (left panels) and the entire $d$-bandwidth (right panels). Magnetic states of Cr, Mn, Fe, Co, and Ni metals are considered.

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  Figure 5

  The position of the $d$-band center (dbc) with respect to the Fermi level, derived considering: only the occupied part of the $d$-band (left panels) and the entire $d$-bandwidth (right panels). Magnetic states of Cr, Mn, Fe, Co, and Ni metals are considered. Experimental XPS data (with maximal uncertainty of $\pm 0.1\mathrm{eV}$), calculated reference data (standard (ref. calc.) and refined (ref. calc. refi., see the text for explanation)) are taken from Hofmann etal. [46] for Fe, Co, Ni, Cu, Pd, Ag, Pt, and Au and from Smith etal. [82] for Rh and Ir metals. Refined DFT calculations of Hofmann etal. [46] are given by blue stars. More details are provided in the ESI, Table S5 [57].

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  Figure 6

  The projected density of states for fcc Cu computed with the $\mathrm{PBEsol}+U$(cLDA) method (atomic orbitals as projectors, $U=11.7\mathrm{eV}$), with Gaussian smearing of $0.015\mathrm{Ry}$. The results show a clear hybrid $spd$ state between the main $d$-band and the Fermi level. XPS data are taken from [46] and scaled vertically to match the intensity of the computed $d$-band.

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  Figure 7

  The projected $d$ orbitals density of states for selected metals, computed with different projections of $d$ states occupancy: atomic orbitals (AO), Wannier functions (WF). Gaussian smearing of $0.03\mathrm{Ry}$ has been applied to match the experimental band broadening. Magnetic and non-magnetic states of Fe, Cr, and Ni metals are considered. XPS data are taken from [46] and are rescaled vertically to match the intensity of the computed $d$-bands. The reference experimental spectra agree well with earlier measurements by Höchst etal. [101], Hüfner etal. [66], Smith etal. [82]. Our computed DOS for $U=0$ (DFT approach) agree well with the calculations by Hofmann etal. [46].

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