Supplemental Materials
I. COMPUTATIONAL DETAILS
All CDFT calculations were conducted using the Quantum Espresso (QE) software package.1 For ZnO, the
experimental wurtzite structure (space group
P63 mc ) with lattice constants a=3.2494 Å and c=5.2038 Å was used
in all calculations.2 We adopted a 2×2×2 supercell of the four-atom hexagonal cell to calculate the response matrix
elements, and a 5×5×3 Monkhorst-Pack3 grid was used to sample the reciprocal space. This k-point grid converged
the total energy to within 1 meV/atom and the diagonal elements of the response matrix to within 1%. However,
convergence of final results was not achieved, due to numerical instability of the inversion of a near-singular matrix.
The final response matrices were constructed using a 4×4×4 supercell, for which we used Vanderbilt ultrasoft
potentials from the original QE library for both Zn and O atoms.1 For Cu2O, we used the highly symmetric cubic
structure (space group
Pn3m ),
with the experimental lattice constant of 4.2696 Å. We again adopted a 2×2×2
supercell of the six-atom cubic unit cell to calculate the response matrix elements, and a 3×3×3 Monkhorst-Pack
grid was used to sample the Brillouin zone. The final response matrices were constructed using a 6×6×6 supercell;
this k-point sampling and size of extrapolated cell converged the final results to within 0.2 eV. Projector Augmented
Wave (PAW) all-electron frozen core potentials from the original QE library1 were used for both Cu and O atoms.
For both compounds, we used the PBE exchange-correlation functional and a 680 eV kinetic energy cutoff for the
planewave basis that converged the total energy to within 1 meV/atom. The electron occupations and Brillouin zone
integrations were always determined using the tetrahedron method with Blöchl corrections.4 All response matrix
elements were derived numerically using seven level-shift points, with level-shift values of -0.06, -0.04, -0.02, 0.00,
0.02, 0.04, and 0.06 eV.
All UHF calculations were carried out using a modified version of the GAMESS quantum chemistry package.5, 6
We used the LANL2DZ effective core potentials (ECPs)7 and basis sets developed by Wadt and Hay for all the
metal atoms, and 6-31G(d) all-electron basis sets for all the oxygen atoms. Na + and Mg2+ Hay-Wadt ECPs were used
to represent cores of Cu+ or Zn2+ ions respectively on the cluster boundary surface (nearest neighbors to the
outermost oxygen ions, replacing corresponding +1 and +2 point charges in the point charge array). These ECPs
furnish the correct short-range Pauli repulsion between the outermost cluster oxygen anions and the surrounding
nearest-neighbor cations in the crystal environment. All results were converged to within 0.5 eV with respect to
cluster size. All the clusters (structures can be found in Figure 1, Figure 2, and the attached xyz files) were
embedded in an array of point charges, which represent the formal charges on the ions (-2e for O, +1e for Cu and
+2e for Zn) in the crystal environment. We used up to 4600 point charges for ZnO and 2200 point charges for Cu 2O,
so the results were converged to within 0.3 eV with respect to the size of the embedded charge array. The charges on
the edges and faces of the point charge array were assigned fractional values according the method of Evjen.8
Figure 1. Structure of ZnO cluster model containing 13 Zn ions. Zinc atoms are represented by blue spheres and
oxygen atoms by red spheres.
Figure 2. Structure of Cu2O cluster model containing 13 Cu ions. Copper atoms are represented by cyan spheres and
oxygen atoms by red spheres.
The DFT-PBE and PBE+U calculations for bulk ZnO and Cu2O properties were conducted using the VASP 5
program.9-12 We performed the calculations on the four-atom hexagonal cell for ZnO and the six-atom cubic cell for
Cu2O. A -point-centered 4×4×3 k-mesh for ZnO and a 6×6×6 k-mesh for Cu2O, along with a 700 eV plane-wave
basis kinetic energy cutoff were used for all calculations. Both the reciprocal space sampling and the energy cutoff
ensured total energy convergence within 1 meV/atom. The tetrahedron method with Blochl corrections4 were used
for electron occupations and Brillouin zone integration. Standard PAW potentials were employed from the VASP
library. The cell energies were scanned as a function of cell volume and the bulk moduli were derived via fitting the
energy-volume curves to the Murnaghan equation of state.13
The one-shot G0W0 calculations were conducted using the bands generated by the PBE+U method. The same kpoint sampling and kinetic energy cutoff as the PBE+U calculations were used in the G0W0 calculations. We
utilized 128 bands for both Cu2O and ZnO, and 70 points for the ω-integration for both systems. All these settings
converge the final band gap within 0.1 eV.
II. EXTRA FIGURE
Figure 3. Cu occupancy as a function of
filled d10 configuration.
III. G0W0 BAND STRUCTURE
U in
in Cu2O. With increasing
U in
, the system gradually approaches the
Figure 4. The total and projected density of states (DOS) of ZnO calculated by G0W0, using PBE+U orbitals and
eigenvalues as input.
Figure 5. The total and projected density of states (DOS) of Cu2O calculated by G0W0, using PBE+U orbitals and
eigenvalues as input.
Table 1. Positions of the valence band peaks for ZnO
(eV)
A
B
C
PBE+U/G0W0
a
Exp
a. Reference 14
-7.2
-3.9
-1.0
-8.8
-3.7
-1.0
Table 2. Positions of the valence band peaks for Cu2O
(eV)
A
B
C
D
E
F
PBE+U/G0W0
-6.5
-5.1
-2.8
-2.3
-1.5
-0.6
Expa
a. Reference 15
-7.3
-6.0
-3.5
-3.0
-2.0
-1.0
The calculated band gaps are 2.7 eV and 1.7 eV for ZnO and Cu2O, compared to the experimental value of 3.3 eV
and 2.1 eV, respectively. As shown in Figure 4, Figure 5, Table 1, and Table 2, our calculations decently reproduce
the experimental band gaps and band positions, validating the quality of the PBE+U orbitals and energies. The
remaining errors compared to the experimental results are due to the significant mixing of the O-2p component in
the valence bands, which requires introduction of exact exchange for these electrons as well. Use of much more
expensive hybrid exchange-correlation functionals can remove these errors, but our results show that the much less
expensive DFT+U theory used here, with ab initio determination of U-J via the UHF method described here,
furnishes overall very good properties, especially for ground electronic state properties.
IV. ATTACHED XYZ FILES
The xyz coordinates of all the cluster models used in the UHF calculations can be found in the attached files. All
the files start with the coordinates of the cluster model in Å, followed by the coordinates of ECP sites (represented
by Na and Mg ECPs) on the boundary. Then the point charge positions are also given, as well as the associated
formal charge scaling factor, determined using Evjen’s method (listed in the fourth column).
REFERENCES
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13