Information : Free energy from stationary implementation of the DFT + DMFT functional

We used the implementation of LDA+DMFT of Ref.1, which is based on Wien2K package [2]. The exchangecorrelation functional Exc of LDA was utilized, in combination with nominal double-counting (DC) [1, 3], which was shown to be closest to the exact form of DC [4]. We checked (on the example of Cerium) that the exact-DC gives very similar free energy, as expected for a stationary functional. The convergence of LDA+DMFT results is much faster using nominal DC, hence most of results in this publication are obtained by this simplification. The impurity model is solved using the hybridization expansion version of the numerically exact continuous time QMC method [5, 6]. Of the order of 300 LDA and 30 DMFT iterations were required for precision of 1 meV per formula unit, and between 109−1010 Monte Carlo moves were accepted per impurity iteration for precise enough impurity solution. The resources of Titan supercomputer were used. To construct the projector, the atomic-like local orbitals are used 〈r|φlm〉 = ul(r) r Ylm(r̂). The radial part of the local orbital ul(r) is the solution of the scalar relativistic Dirac equation inside the muffin-tin sphere, linearized at the Fermi level. The muffin-tin spheres are set to touch at the lowest volume. We tested a few other forms of the projectors defined in Ref. 1. The stationary F (V ) is quite insensitive to the precise choice of projector, however, E(V ) changes much more. For SrVO3 calculations, we treated dynamically all five 3d orbitals of Vanadium. The muffin-thin radius of Vanadium was set to Rmt = 1.83 aB , and U = 10 eV was used, which was previously shown to give good spectra [3, 4] for this localized orbital (see spectra below). The Yukawa form of screening interaction than gives J ≈ 1 eV (see note below). Brillouin zone integrations were done over 15 × 15 × 15 k-point in the whole zone in the selfconsistent calculations, and for calculation of the impurity entropy, the hybridization is computed on more precise 36 × 36 × 36 k-points mesh. We mention in passing that impurity entropy is very sensitive to the precise frequency dependence of the hybridization, and requires very dense momentum mesh. For FeO, all five 3d orbitals are treated by DMFT and the muffin-thin radius of iron is set to Rmt = 2.11 aB , and the Coulomb repulsion to previously determined U = 8eV [7], which requires J ≈ 1eV in Yukawa form. In Ce metal, all seven 4f orbitals are treated by DMFT and the muffin-thin sphere is Rmt = 2.5 aB , the k-point mesh is 21× 21× 21, and the Coulomb U = 6 eV [8–10], leads to J = 0.72 eV in Yukawa form. The spin-orbit coupling is included in Cerium, but neglected in SrVO3 and FeO.