Selfenergy selfconsistent density functional theory plus dynamical mean field theory
Abstract
We propose a hybrid approach which employs the dynamical meanfield theory (DMFT) selfenergy for the correlated, typically rather localized orbitals and a conventional density functional theory (DFT) exchangecorrelation potential for the less correlated, less localized orbitals. We implement this selfenergy (plus charge density) selfconsistent DFT+DMFT scheme in a basis of maximally localized Wannier orbitals using Wien2K, wien2wannier, and the DMFT impurity solver w2dynamics. As a testbed material we apply the method to SrVO and report a significant improvement as compared to previous + calculations. In particular the position of the oxygen bands is reproduced correctly, which has been a persistent hassle with unwelcome consequences for the  hybridization and correlation strength. Taking the (linearized) DMFT selfenergy also in the KohnSham equation renders the socalled “doublecounting” problem obsolete.
k r
I Introduction
Density functional theory (DFT)dft1 ; dft2 is by far the most widely used method in solid state physics, owing to its immense success in predicting solid state properties such as crystal structures, ionization energies, electrical, magnetic and vibrational properties. However, treating electron correlating within an effectively singleparticle framework makes it inadequate, even with the best available exchange correlation potentials, for an important class of materials: strongly correlated electron systems. This is the realm of dynamical mean field theory (DMFT) Metzner89a ; Georges92a ; Georges96a which incorporates local, dynamic correlations, and has been merged with DFT for the calculation of realistic correlated materials Anisimov97a ; Lichtenstein98a ; dmft0 ; dmft1 ; dmft2 . In DMFT, the electrons can stay at a lattice site or dynamically hop between lattice sites in order to suppress double occupation and hence the cost of the Coulomb interaction, without any symmetry breaking unlike in the static DFT+U approachldau . The method has been successfully applied to transition metals Lichtenstein98a and their oxides Held01a , moleculessbmoldftp , adatomspanda and felectron systemsSAVRASOV ; Held01b , proving its versatility.
The early developments in this direction are “oneshot” DFT+DMFTSavrasov04 ; Minar05 ; millis_csc ; Frank ; Pouroskii07 ; Aichhorn2011 ; haule2010 ; csc_sb calculations. In a “oneshot” calculation, first a DFT calculation is converged for a given material. Subsequently the DFT Hamiltonian is supplemented with a local Coulomb interaction for the correlated orbitals and this problem is subsequently solved within the DMFT framework. The physical properties such as the spectral function, susceptibility or magnetization are calculated from this “oneshot” DMFT solution of a DFT derived Hamiltonian.
Subsequently charge selfconsistent (CSC) DFT+DMFT calculations have been implemented and applied. Here, the total electronic charge density is updated after the DMFT calculation, now including effects of electronic correlations. With this updated charge density the KohnSham equations of DFT are solved, a new Hamiltonian is derived which is again solved by DMFT etc. Both cycles, DFT and DMFT, are converged simultaneously. The physical properties are calculated from the converged solution. The correlationinduced change in the charge density can be significant. Hence for some materials using CSC leads to a major correction; for other materials the corrections are minute. Incorporation of this CSC correction in a sitetosite charge transfer has been studied extensively Savrasov04 ; Minar05 ; millis_csc ; Frank ; Pouroskii07 ; Aichhorn2011 ; haule2010 . More recently, also the effect of the interorbital and momentumdependent charge redistribution has been studied csc_sb .
While DFT provides a reasonable starting point for both “oneshot” and CSC DFT+DMFT, the incompatibility of the DFT and DMFT approach is seen in many occasions, e. g., in socalled “” DMFT calculation for transition metal oxides Held02 ; Wang07 ; haule2010 ; Parragh2013 ; Haule2014 ; Dang2014 ; Hansmann2014 . The reason behind this is that in DFT the bands are too close to the Fermi level. Hence, there is a too strong intermixture of and bands and the orbitals or not strongly correlated. Within the framework of DFT+DMFT, one consequently needs to introduce an adjustment to the  splitting, adjusting it either to the experimental oxygen positionDang2014 ; zhong , adding a  interaction parameterHansmann2014 , or modifying the double counting haule2010 ; Haule2014 or exchangecorrelation potentials Nekrasov2012 ; Nekrasov2013 . For example, in SrVO, the proper renormalization of the t band has been obtained with an additional shift applied of the bands as large as 5eV relative to the t bands zhong , for correcting the position of the  bands to that observed in experiment.
There have been considerable efforts to improve on the exchange part of the exchangecorrelation potential. Approaches in this direction include GWHedin +DMFTGWDMFT1 ; GWDMFT2 ; Tomczak12 ; GWDMFT3 and quasiparticle selfconsistent GW (QSGW)QSGW1 ; QSGW2 +DMFT QSGWDMFT1 ; QSGWDMFT2 ; also hybrid functionals hydmft instead of the more widespread local density approximation (LDA) or generalized gradient approximation (GGA) exchangecorrelation potential can be employed. But all of these approaches do not solve the problem of the wrong position of the oxygen band. In this paper, we propose an alternative selfenergy selfconsistent (SC) DFT+DMFT scheme. For the correlated orbitals, i.e., those that acquire a Coulomb interaction in DMFT, SC DFT+DMFT takes the (linearized) DMFT selfenergy as the exchange correlation potential in a similar way as proposed by Schilfgaarde and Kotani QSGW1 ; QSGW2 for QSGW. That is, when solving the KohnSham equation, these correlated orbitals sense the (linearized) DMFT selfenergy instead of the conventional LDA or GGA exchangecorrelation potential. For the less correlated orbitals, that do not acquire an interaction in the DMFT, the GGA is still employed. The method is selfconsistent, for both electronic charge density and selfenergy, and free from the double counting ambiguity. We employ the approach to SrVO and find that it renders the correct position of the oxygen orbitals.
The outline of the paper is as follows: In Section II, we introduce the SC DFT+DMFT. In this section, we first recapitulate the conventional steps of DFT in Section II.1, the projection onto Wannier functions in Section II.2, and DMFT in Section II.3. Carrying out these three steps constitutes a socalled “oneshot” DFT+DMFT calculation, whereas, as discussed in Section II.4, in a CSC scheme the charge recalculated after the DMFT is feed back to the KohnSham equation to obtain a new oneparticle KohnSham Hamiltonian until selfconsistency is obtained. The decisive step of the present paper, described in Section II.5, is to take not only the charge but also the DMFT selfenergy as the exchangecorrelation potential of the correlated orbitals when going back to the KohnSham equation after the DMFT step. The proper subtraction of the Hartree term to avoid a double counting is discussed in Section II.6. An overview of the method in form of a flow diagram of the individual steps as well as of the full SC DFT+DMFT scheme is provided in Section II.7 and Fig. 1. Section III presents the results for SrVO, and Section IV a summary and outlook.
Ii Methodology
In this section, we present the formalism and implementation of selfenergy selfconsistency (SC). The actual implementation is based on the maximally localized Wannier functions (MLWF) and extends our previous CSC DFT+DMFT csc_sb implementation. Let us emphasize, that the SC is a major improvement on the CSC: not only the charge but—based on the DMFT selfenergy—also the exchangecorrelation potential of the KohnSham equations is changed. Specifically, our starting point is a DFT calculation within Wien2kw2k , followed by a DMFT calculation which is performed with w2dynamicsw2d using continuoustime quantum Monte Carlo (CTQMC) CTQMC as an impurity solver. The identification of localized orbitals in DMFT is done with Wien2wannierwien2wannier , an interface between Wien2kw2k and wannier90wanrev . In SC, the selfconsistency does not only include an update of the charge in the KohnSham equation but further modifies the exchangecorrelation potential on the basis of the linearized DMFT selfenergy. This decisive step is presented in Section II.5. This way, genuine effects of electronic correlations are included in the exchange correlationpotential and a double counting is avoided.
ii.1 DFT cycle
Let us start by defining the central quantities of the SC DFT+DMFT: the electronic charge density as the key quantity in DFT and the Greens function (or the related selfenergy) as the central component of DMFT. The charge density at a given spatial position \br, is given by the equaltime Green’s function or as a sum of all Matsubara frequency:
(1) 
While the local DMFT Green’s function defined at localized Wannier orbitals is given by
(2) 
Here , denote the orbitals on the same site, is the inverse temperature and the factor ensures the convergence of the summation over Matsubara frequencies . The full Greens function for the solid appears in both equations and can be written as
(3) 
Here, is the chemical potential and the oneparticle Hamiltonian of the KohnSham equation consisting of the kinetic energy operator and the effective KohnSham (KS) potential . In a DFT calculation, the KS potential has an explicit dependence on the total electronic charge and consists of an external potential due to the nuclei (ions), a Hartree potential , describing the electronelectron Coulomb repulsion and an exchangecorrelation potential , i.e., . Altogether this yields
(4) 
There are several existing formulation of the latter, such as using LDA lda , GGA gga or hybrid functionalsmbj ; b3lyp . For our calculations on SC, we have employed GGA but this is of little importance as the potential will be replaced by a newly formulated one that is obtained from the selfenergy, . The DFT selfconsistency cycle (“DFT cycle”) hence consists of the following two steps:
(i) The calculation of the exchange correlation potential from the electronic charge distribution .
ii.2 Wannier projection
Our starting point is a selfconsistent DFT calculation with a converged electronic charge density. At this point is calculated with GGA. The next step is to construct a localized orbital basis, in which DMFT is applied. To this end, we employ MLWFs, which are constructed by a Fourier transform of the DFT Bloch waves :
(5) 
Here, is the unitary transformation matrix, the volume of the unit cell, () denotes the band indices of the Bloch waves (Wannier functions). Here and in the following hats denote matrices (operators) in the orbital indices. In Eq. (5), we restrict ourselves to an isolated band window with Bloch waves. This window may, e.g., include the  or orbitals of a transition metal oxide or, as in our example below, plus oxygen orbital. In the scheme of maximally localized Wannier functionswanrev , the spread (spatial extension) of the Wannier functions describing the DFT bandstructure in the given energy window is minimized; and is obtained from this minimization.
In general, the target bands are “entangled” with other, less important bands—at least at a few \tkpoints. These bands are projected out by a socalled “disentanglement” procedure. That is, at each \tkpoint, there is a set of Bloch functions which is larger than or equal to the number of target bands, i.e., . The disentanglement transformation takes the form
(6) 
Here, the band index belongs to the outer window” with Bloch wave functions, while label the target bands. Hence, the disentanglement matrix is a rectangular matrix. A Fourier transformation of leads to the Wannier orbitals in \tkspace
(7) 
and the corresponding Wannier Hamiltonian
(8)  
(9) 
The two equations correspond to the case without and with disentanglement.
ii.3 DMFT cycle
The Hamiltonian is supplemented with a local Coulomb interaction, and the resulting lattice problem is solved in DMFT by mapping it onto an auxiliary impurity problem, which is solved selfconsistently in DMFTGeorges92a ; Georges96a . Here, either the noninteracting Green’s function of the impurity problem or the local selfenergy can be considered as a dynamical mean field. The DMFT formalism consists of the following four steps: (i) The \tkintegrated lattice Dyson equation yields the local interacting Green’s function
(10) 
from the local selfenergy and oneparticle KohnSham Hamiltonian ; \tkpoints are considered in the reducible Brillouin Zone.
(ii) The impurity Dyson equation provides the noninteracting impurity Green’s function
(11) 
(iii) Solving the Anderson impurity problem (AIM) defined by and gives interacting Green’s function
(12) 
This is numerically the most involved step; we employ the continuoustime quantum MonteCarlo method CTQMC in the w2dynamics implementationw2d to this end.
(iv) Applying the impurity Dyson equation a second time once again gives the selfenergy
(13) 
In the DMFT selfconsistency cycle (“DMFT cycle”), the obtained selfenergy is now used again in step (i) to recalculate a new local Green’s function until a convergence is achieved. The “oneshot DFT+DMFT” ends after a full “DFT cycle” and one subsequent “DMFT cycle”. Physical quantities, e. g., spectral function, susceptibility etc. are extracted at this point. Both in a charge CSC and SC DFTDMFT one goes instead back to the DFTpart as discussed in the following.
ii.4 Recalculation of the charge density
For the SCapproach, we now go one step further. We construct a new electronic charge density °(as has been done before) and a new exchange correlation potential for the correlated subspace. The total charge density is separable into two parts; (i) the correlated part, , formed by the correlated orbitals (typically the  or orbitals) and (ii) the noninteracting part, , formed by the rest of the system, i.e.:
(14) 
Including the DMFT correlations, can be calculated from the local DMFT Green’s function as follows:
(15) 
Here, is the expectation value of the occupation operator in the localized Wannier orbitals basis , which can be directly calculated from the equal time (or Matsubara sum) of the corresponding DMFT Green’s function which is again a matrix with respect to the orbitals. For a faster convergence of the Matsubara sum, it is advisable to express as
(16) 
Here, the functional behavior of at higher frequency is considered by a model Green’s function , and provides the analytical frequency sum of .
(17)  
(18)  
(19) 
Note that is, in general, not diagonal in Wannier representation. To calculate the analytical sum, , we diagonalize . If is the ’s eigenvectors and the ’s eigenvalue of , we get
(20)  
(21) 
The operator is then transformed to the Bloch basis utilizing the unitary and the disentanglement matrices, and :
(22)  
(23) 
From this, the correlated charge density is finally obtained as:
(24) 
The remaining density is calculated within DFT and added to to obtain the total electronic charge density.
ii.5 Recalculation of the exchangecorrelation potential from the DMFT selfenergy
The next step is the key aspect of the SC DFT+DMFT approach: recalculating the exchangecorrelation potential for the next iteration step on the DFT side. The Hartree potential, is calculated as usual from the total density, including the effect of electronic correlations on the density. The exchangecorrelation (XC) potential for the next step is however not derived from the total charge density (e.g. using the GGA functional) as in previous CSC DFT+DMFT calculations. Instead, we have adapted the following assumption: If the correlated orbitals are fairly localized, the XC potential can be divided into two parts:
(25) 
Here, correspond to the XC potential for the correlated subspace and accounts for the XC of the rest of the system. To determine these two XC potentials, we first calculate and from the corresponding densities and , respectively, employing the GGA functional for both densities. From these, we obtain also the difference . This procedure has the following advantage: the total XC potential, , calculated from includes the corevalance interaction and the interaction between correlated and uncorrelated subspace . Even after subtraction of , will still possess that part of the interaction. Only the XC potential stemming from the interaction within the correlated subspace is taken out in . Similar subtractions of the contributions to the exchangecorrelation potential have been done before Nekrasov2012 ; Nekrasov2013 ; Haule2015 , but not the next step: taking the DMFT selfenergy for the exchangecorrelation potential instead.
That is, we employ a new XC potential within the correlated subspace, , which is given by the (linearized) DMFT selfenergy, . By construction, is local (in Wannier space) and represented in Matsubara frequencies. Because of frequencydependence cannot be employed directly in the oneparticle KohnSham equation.
As we focus on the low energy part of the spectrum, we linearize the selfenergy around zero frequency
(26) 
This linearized selfenergy is still frequencydependent and still cannot be included in the KohnSham equations which is based on a frequencyindependent Hamiltonian. But thanks to the linearized selfenergy, we can use the fact that the relevant selfenergy, when determining the eigenvalues of the KohnSham equation, is taken for a particular frequency: the frequency that is equal to the KohnSham eigenvalue.
Hence we can approximate Eq. (26) by a Hermitian operator
(27) 
One further technical complication is that we do not have the selfenergy for real frequencies. Hence, we instead estimate the (constant plus) linear behavior as following:
(28)  
(29) 
For the results below we take the limit in Eq. (29) by simply considering the value at the lowest Matsubara frequency, but more complicated fitting procedures may be taken.
We also have to take into account that the DMFT selfenergy contains a Hartree contribution. This is to be subtracted from the XC potential since the same is already included in the effective KSpotential, i.e.,
(30) 
Here, one can deduce the Hartree term of DMFT as
(31) 
from the spinorbitalresolved occupations of the Wannier orbitals, and the equivalent formula for the opposite spin.
Since we need the exchange correlation potential in real space , we now have to transform the (linearized) selfenergy back to the Bloch basis utilizing the preobtained transformation matrices (without and with disentanglement):
(32)  
(33) 
Finally, the exchangecorrelation potential within the correlated subspace can be written on a radial grid as,
(34) 
In the KohnSham equation we henceforth employ the XC potential or the following oneparticle Hamiltonian instead of Eq. (4):
(35) 
ii.6 Exact double counting subtraction
In the SC formalism, the part of the selfenergy used as exchange correlation within the correlated subspace is now explicitly defined through Eq. (35). One can hence subtract this contribution exactly when calculating the DMFT Green’s function in Eq. (10), simply by setting
(36) 
where comes from the previous iteration.
Let us note again that the Hartree term enters only once in form of but not in thanks to the subtraction in Eq. (30); using instead of for the doublecounting warranties that the Hartree term cancels for the selfenergy.
In SC DFT+DMFT, the ambiguity of the double counting term is hence avoided altogether. The correlated orbitals that acquire a Coulomb interaction in DMFT obtain a linearized in the KohnSham equation which is known exactly and can be hence subtracted as when going back to the DMFT side.
Indeed after subtracting in Eq. (10) not even the linearization approximation of the selfenergy enters the DMFT Green’s function any longer—but the full, frequency dependent DMFT selfenergy. The linearization and including it as in the KohnSham potential only serves the purpose that the KohnSham wave functions and eigenvalues include some potential effects of the DMFT selfenergy. On the DMFT side the full selfenergy is taken; and no further XC potential within the correlated subspace.
ii.7 Flow diagram of Sc Dft+dmft
The full SC DFT+DMFT, altogether consists of the following workflow, as depicted schematically in Fig. 1:

A converged charge density is obtained within DFT to have a reasonable electronic structure to start with (upper left part of Fig. 1). The target bands are identified as a prelude for the Wannier projection. In the following SC DFT+DMFT cycles (green arrows in Fig. 1), a single DFT iteration is performed with an updated DFT KohnSham Hamiltonian (i.e., without the orange arrow in the upper left part). The XC potential for the correlated subspace is supplemented with the one obtained from the DMFT selfenergy as discussed in Section II.5. For this step, we employ the modified Wien2k program package.

A single DMFT cycle is performed using w2dynamicsw2d (lower right part of Fig. 1). This provides the selfenergy , local Green’s function , and the DMFT chemical potential , which is fixed to the particle number. It needs to be noted that at this point is used as double counting term and is calculated accordingly. Moreover, for practical purposes, it is beneficial to start with a converged “oneshot” DFT + DMFT calculation. Further, a mixing (underrelaxation) between old and new DMFT selfenergy is employed.

The correlated charge distribution as well as the XC potential are updated (lower left part of Fig. 1). At first, is calculated from the DMFT Green’s functions, as in Eq. (16). As described in Eqs. (23)(23), is transformed back to the DFT eigenbasis to calculate the correlated charge distribution in real space. In a similar fashion, the XC potential in the correlated subspace is calculated from the DMFT selfenergy through Eq. (30) and transformed back to DFT eigenbasis as presented in Eqs. (32), on a radial grid by employing Eq. (34).

The new DFT+DMFT charge density is compared with the old density. If the difference does not match the convergence criteria, the new density is mixed with the old density, serving as the new density. The charge density of the correlated orbitals is then used to calculate , which provides as described in Eq. (25). The exchange correlation potential in the KS Hamiltonian is updated with and according to Eq. (35). At the same time, the DMFT selfenergy is also compared for two consecutive iterations for convergence.
Iii Results
The SC DFT+DMFT scheme is applied to SrVO, a testbed material for methodological developments for strongly correlated electrons systems. The cubic perovskite structure of SrVO results in degenerate t bands near Fermi energy that are singly occupied and unoccupied bands. Bulk SrVO exhibits a strongly correlated metallic behavior and the electronic features are mostly governed by partially filled t bands. In DFT+DMFT schemes, one typically treats isolated t bands with explicit electron correlation in DMFT—coined “only” model. As a consequence of the DMFT correlations, the wide band of DFT are renormalized by factor of about 0.5, yielding a strongly correlated metal. Additional lower and upper Hubbard peaks appears at 1.7 eV and 2.5 eV, respectivelysvoexptheo ; Pavarini03 ; Liebsch03a ; Nekrasov05a ; Nekrasov05b . In the energy range of the latter, also the bands are located. The agreement of the t spectral function with experiment is reasonably goodsvoexptheo . SrVO has also been studied in GW+DMFT by various groups Tomczak12 ; P7:Casula12b ; Taranto13 ; Tomczak14 ; bohenke ; GWDMFT3 . GW+DMFT yields a somewhat better position of the lower Hubbard bandTomczak12 ; Taranto13 ; Tomczak14 but does not solve the problem with the wrong position of the oxygen bands Tomczak12 ; Tomczak14 ; GWDMFT3 .
One can include noninteracting bands within DMFT in a cocalled “+” calculation. However, the energy difference between and bands derived ab initio in DFT is underestimated. Consequently there is a too strong hybridization between and orbitals, and the effective orbitals have a significant contribution. This in turn means that the occupation is much larger than one. A + calculation with interaction in the t bands and no interaction in the uncorrelated bands hence yields only a weakly correlated solution with too wide t bands around the Fermi energy and no Hubbard bands Held02 ; Wang07 ; haule2010 ; Parragh2013 ; Haule2014 ; Dang2014 ; Hansmann2014 .
A  interaction Hansmann2014 or an adhoc “doublecounting” term Dang2014 ; zhong , which corrects the onsite energies of the orbitals to the experimental position, needs to be introduced in order to obtain a proper Hubbard peak below Fermi energy, as observed in experiment. Let us note that the origin of this peak, has been debated. Namely, within a GW+extended DMFT calculationbohenke it has been identified as a plasmon peak, which is however much less pronounced than in experiment, while Backes et al.ovac identified it coming form oxygen vacancy in a GW+DMFT framework. Altogether, this leaves + DFT+DMFT calculations in a quite unsatisfactory state, relying on parameter tuning or adhoc corrections of the level or exchange correlation potential for getting the correct position of the oxygen level.
In our implementation, we employ instead the DMFT selfenergy as the (selfconsistently updated) exchangecorrelation potential for the orbitals of SrVO. That is, the GGA potential is only used for the less correlated oxygen orbitals, whereas for the correlated, localized orbitals the local DMFT selfenergy from a + calculation is used. In principle, this DMFT potential should also be employed for the orbitals, but since these are essentially unoccupied, the DMFT selfenergy would reduce to a Hartree term which is included in the GGA as well.
In Fig.2, we first present the \tkdependent spectral function of SrVO as obtained in a + model within a standard oneshot DFT+DMFT calculation, using =9.5 eV, =0.75 eV, zero and , and room temperature (=40). Fully localized limit (FLL) double counting term is considered here. Let us note that within a  model the impurity orbitals are more localized compared to those in a only model, causing larger values of the interaction parameters than in only calculation. The specific values are chosen following Aryasetiawan et al.clda and will be considered for all the calculations, presented in this article.
The band renormalization is reasonable with 0.48. However, the bands appear around 2 eV to 7 eV, which does not agree with the experimental photoemission spectra morikawa ; svoexptheo ; svo_expt ; svo_arpes . As explained before, the bands have to be adjusted to describe photoemission spectra. In SrVO, the required shift is as large as 5.0 eV zhong , which combined with the large (9.5 eV) of Ref. clda, would even result in an insulating solution.
Next, we turn to the SCDFT+DMFT, which does not necessitate such an adhoc shift and treats SrVO in a completely abinitio manner. As mentioned in section II, we started from a converged ‘oneshot’ DFT+DMFT selfenergy (i.e. from the solution of Fig. 2) . Upon SC selfconsistency we however obtain the solution Fig. 3. The position of the bands with respect to the bands has improved significantly, with an excellent agreement with experimental spectra (without any adjustable parameters as is obtained from Ref. clda, ). In addition, interestingly, over the iteration in SC, also the dispersion of the bands is slightly changed compared to that DFT.
The scenario can be further clarified by inspecting the \tkintegrated spectral function, Fig. 4, which compares our SCspectra with photoemission spectroscopy (PES) by K. Morikawa et al.morikawa . Here, the central t quasiparticle peak is to a minor extent more renormalized than the ‘oneshot’ DFT+DMFT calculation: the factor is 0.4. The lower and upper Hubbard peaks appear around 1.7 eV and 2.2 eV, respectively. These can also be seen in the \tkresolved spectra in Fig.3. The positions of the Hubbard bands well agrees with the PES spectrum. Please keep in mind that more bulksensitive PES has a larger weight in the quasiparticle peak than in the lower Hubbard band, similar as in our SC DFT+DMFT calculation svoexptheo . Further, there is additional spectral weight of the orbitals (not included in our calculations as these are unoccupied) which should be located slightly above our upper Hubbard band, as was already discussed in the very first DFT+DMFT calculations svoexptheo .
The main improvement with respect to previous DFT+DMFT calculations is that we also obtain an excellent description of the position of the oxygen orbitals without any adjustable parameter or adhoc shift. This includes their width and relative weight to the bands and , in particular, their splitting into two subgroups of oxygen orbitals: out of 9 orbitals the first branch consists of 6 orbitals with a peak at 5.2 eV while the rest are peaked at 6.4 eV. A substantial shift of the orbitals in the right direction has already been obtained when taking out the electron contribution from the exchange correlation potential Nekrasov2012 ; Nekrasov2013 ; Haule2015 . But replacing it by the DMFT selfenergy in SC DFT+DMFT is not only more appealing from a fundamental point of view, it also gives a larger shift to the correct experimental position.^{1}^{1}1 To check any sort of dependence of converged result on the starting point, we considered several starting points. The starting selfenergies, in this respect, are obtained from DMFT calculations, performed with Hamiltonians where levels are shifted by 1.75 eV and 5.0 eV. Once SC is started, the shift in the Hamiltonian is taken off and as the double counting is calculated from the selfenergy itself, the method remains double counting free. The dotted lines in Fig. 4 refer to those calculations which essentially produces an indifferent result, confirming the robustness of the method.
Iv Summary and Outlook
We have introduced the SC DFT+DMFT method which is free from any double counting problem, and employed it to SrVO. It yields largely improved results, in particular with regard to the position of the oxygen bands, which has been a major shortcoming of previous DFT+DMFT calculations. The essential step is to take the DMFT selfenergy as the exchangecorrelation potential of the correlated orbitals in the KohnSham equation of the “DFT step”. As the latter is a oneparticle equation, we must employ a linearized selfenergy at the proper quasiparticleenergy in a similar manner as in QSGW QSGW1 ; QSGW2 .
However, when going back to the “DMFT step” this selfenergy is readily replaced by the correct, frequencydependent DMFT selfenergy, using the manybody Dyson equation. Hence, solving the KohnSham equations with the linearized selfenergy can be seen as an intermediate step, only to adjust the oneparticle orbitals to the actuality of electronic correlations. Thereafter the selfenergy with its full frequency dependence is taken again.
This is not fully correct, since for the less correlated orbitals we still take the plain vanilla GGA potential of DFT. One might be tempted to extend the correlated subspace to all orbitals, using a DMFT selfenergy also for these. Indeed, this is what is done in QSGW. However, we believe that in contrast to the QSGW this is not adequate for SC DFT+DMFT since the local DMFT selfenergy should only provide a proper exchangecorrelation potential for the more localized orbitals, typically the  or orbitals of a transition metal oxide, lanthanide or actinide. For these orbitals the local correlations as described in DMFT are prevalent. For the more extended, e.g., orbitals, on the other hand the exchangepart is more important. This can be described to a large extent by the GGA, at least for metals.
Using a combination of DMFT selfenergy for the correlated orbitals and GW for the less correlated orbitals, and feeding both back to the KohnSham equation in a linearized form is at least appealing, and possibly even better than SC DFT+DMFT method, pending extensive further implementations and examination which are beyond the scope of the present paper. An even further step is to include also nonlocal correlations beyond which is possible using the ab initio dynamical vertex approximation (DA) AbinitioDGA ; DGA ; RMP , and to feed the obtained nonlocal selfenergy back to the KohnSham equation in the same way as we do in the present paper for the local DMFT selfenergy. The decisive step has been however already done in the present paper which shows that using a linearized DMFTlike selfenergy in the KohnSham equation does not only work properly but also yields largely improved results compared to previous + calculations.
References
 (1) W. Kohn, Rev. Mod. Phys. 71, 1253 (1999).
 (2) R. O. Jones, O. Gunnarsson, Rev. Mod. Phys. 61, 689 (1989).
 (3) W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62 324 (1989).
 (4) A. Georges and G. Kotliar, Phys. Rev. B 45 6479 (1992).
 (5) A. Georges, G. Kotliar, W. Krauth and M. Rozenberg, Rev. Mod. Phys. 68 13 (1996).
 (6) V. I. Anisimov, A. I. Poteryaev, M. A. Korotin, A. O. Anokhin and G. Kotliar, J. Phys. Cond. Matter 9 7359 (1997).
 (7) A. I. Lichtenstein and M. I. Katsnelson, Phys. Rev. B 57 6884 (1998).
 (8) K. Held, I. A. Nekrasov, G. Keller, V. Eyert, N. Bl ̵̈umer, A. K. McMahan, R. T. Scalettar, T. Pruschke, V. I. Anisimov, and D. Vollhardt (2006), physica status solidi (b) 243 (11), 2599, previously appeared as Psik Newsletter No. 56 (April 2003).
 (9) G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti, Rev. Mod. Phys. 78, 865 (2006).
 (10) K. Held, Advances in Physics 56, 829 (2007).
 (11) V. Anisimov, F. Aryasetiawan, A. I. Lichtenstein, J. Phys.:Condens. Matter 9, 7359 (1997).
 (12) K. Held, G. Keller, V. Eyert, V. I. Anisimov and D. Vollhardt, Phys. Rev. Lett. 86 5345 (2001).
 (13) S. Bhandary, M. Schüler, Patrik Thunström, I. di Marco, B. Brena, O. Eriksson, T. Wehling and B. Sanyal, Phys. Rev. B 93, 155158 (2016).
 (14) S. K. Panda, I. Di Marco, O. Granäs, O. Eriksson and J. Fransson, Phys. Rev. B 93, 140101(R) (2016).
 (15) S. Y. Savrasov, G. Kotliar and E. Abrahams, Nature 410 793 (2001).
 (16) K. Held, A. K. McMahan and R. T. Scalettar, Phys. Rev. Lett. 87 276404 (2001).
 (17) S. Y. Savrasov and G. Kotliar, Phys. Rev. B 69, 245101 (2004).
 (18) J. Minár, L. Chioncel, A. Perlov, H. Ebert, M. I. Katsnelson, and A. I. Lichtenstein, Phys. Rev. B 72, 045125 (2005).
 (19) Hyowon Park, Andrew J. Millis, and Chris A. Marianetti Phys. Rev. B 90, 235103 (2014)
 (20) F. Lechermann, A. Georges, A. Poteryaev, S. Biermann, M. Posternak, A. Yamasaki, and O. K. Andersen Phys. Rev. B 74, 125120 (2006).
 (21) L. V. Pourovskii, B. Amadon, S. Biermann, and A. Georges Phys. Rev. B 76, 235101 (2007).
 (22) M. Aichhorn, L. Pourovskii, and A. Georges Phys. Rev. B 84, 054529 (2011).
 (23) K. Haule, C.H. Yee, and K. Kim Phys. Rev. B 81, 195107 (2010)
 (24) S. Bhandary, E. Assmann, M. Aichhorn, K. Held, Phys. Rev. B 58, 155131(2016)
 (25) P. Hansmann, N. Parragh, A. Toschi, G. Sangiovanni, K. Held, New J. Phys. 16, 033009 (2014).
 (26) K. Held, http://online.kitp.ucsb.edu/online/cem02/held (unpublished).
 (27) X. Wang, M. J. Han, L. de’ Medici, H. Park, C. A. Marianetti, and A. J. Millis, Phys. Rev. B 86, 19513 (2007).
 (28) K. Haule, T. Birol, G. Kotliar, Phys. Rev. B 90, 075136 (2014).
 (29) N. Parragh, G. Sangiovanni, P. Hansmann, S. Hummel, K. Held, A. Toschi, Phys. Rev. B 88, 195116 (2013).
 (30) H. T. Dang, A. J. Millis, and C. A. Marianetti, Phys. Rev. B 89, 161113 (2014).
 (31) Z. Zhong et al.. Phys. Rev. Lett. 114, 246401 (2015).
 (32) I.A. Nekrasov, N.S. Pavlov, M.V. Sadovskii, Pis’ma v ZhETF 95, 659 (2012); arXiv:1204.2361.
 (33) I. A. Nekrasov, N. S. Pavlov, M. V. Sadovskii JETP 116, Issue 4 (2013)
 (34) K. Haule Phys. Rev. Lett. 115, 196403 (2015).
 (35) L. Hedin, Phys. Rev. 139, A796 (1965).
 (36) S. Biermann, S., F. Aryasetiawan, and A. Georges, Phys. Rev. Lett. 90, 086402 (2003).
 (37) P. Sun and G. Kotliar, Phys. Rev. B 66, 085120 (2002).
 (38) J. M. Tomczak,P. Liu, A. Toschi, G. Kresse, and K. Held European Phys. J. Special Topics 226, 2565 (2017).
 (39) J. M. Tomczak, M. Casula, T. Miyake, F. Aryasetiawan, and S. Biermann, Euro. Phys. Lett. 100, 67001 (2012).
 (40) S.V. Faleev, M. van Schilfgaarde, T. Kotani, Phys. Rev. Lett.93, 126406 (2004)
 (41) A.N. Chantis, M. van Schilfgaarde, T. Kotani, Phys. Rev. Lett. 96, 086405 (2006).
 (42) J.M. Tomczak, M. van Schilfgaarde, G. Kotliar, Phys. Rev. Lett. 109, 237010 (2012)
 (43) J. M. Tomczak, J. Phys.: Conference Series 592, 012055 (2015).
 (44) D. Jacob, K. Haule and G. Kotliar EPL 84, 57009 (2008)
 (45) P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka, J. Luitz, WIEN2k, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties (Karlheinz Schwarz, Techn. Universitat Wien, Austria, 2001), ISBN 3950103112.
 (46) N. Parragh, A. Toschi, K. Held, and G. Sangiovanni, Phys. Rev. B 86, 155158 (2012); M. Wallerberger et al. (unpublished).
 (47) E. Gull, A. J. Mills, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer, P. Werner, Rev. Mod. Phys. 83, 349 (2011).
 (48) J. Kunes, R. Arita, P. Wissgott, A. Toschi, H. Ikeda, and K. Held, Comp. Phys. Comm. 181, 1888 (2010).
 (49) A. A. Mostofi, J. R. Yates, Y.S. Lee, I. Souza, D. Vanderbilt and N. Marzari, Comput. Phys. Commun. 178 685 (2008).
 (50) W. Kohn and L.?J. Sham, Phys. Rev. 140, A1133 (1965).
 (51) J.?P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996); 78, 1396 (1997).
 (52) F. Tran and P. Blaha Phys. Rev. Lett. 102, 226401 (2009)
 (53) A. D. Becke, J. Chem. Phys. 98, 5648?5652 (1993).
 (54) K. Morikawa, T. Mizokawa, K. Kobayashi, A. Fujimori, H. Eisaki, S. Uchida, F. Iga, and Y. Nishihara, Phys. Rev. B 52, 13711 (1995).
 (55) A. Sekiyama, H. Fujiwara, S. Imada, S. Suga, H. Eisaki, S. I. Uchida, K. Takegahara, H. Harima, Y. Saitoh, I. A. Nekrasov, G. Keller, D. E. Kondakov, A. V. Kozhevnikov, Th. Pruschke, K. Held, D. Vollhardt and V. I. Anisimov, Phys. Rev. Lett. 93 156402 (2004).
 (56) K. Yoshimatsu, T. Okabe, H. Kumigashira, S. Okamoto, S. Aizaki, A. Fujimori, and M. Oshima, Phys. Rev. Lett. 104, 147601 (2010)
 (57) T. Yoshida, K. Tanaka, H. Yagi, A. Ino, H. Eisaki, A. Fujimori and Z.X. Shen, Phys. Rev. Lett. 95, 146404 (2005).
 (58) E. Pavarini, S. Biermann, A. Poteryaev, A. I. Lichtenstein, A. Georges and O. K. Andersen, Phys. Rev. Lett. 92 176403 (2004).
 (59) A. Liebsch, Phys. Rev. Lett. 90 096401 (2003).
 (60) I. A. Nekrasov, G. Keller, D. E. Kondakov, A. V. Kozhevnikov, T. Pruschke, K. Held, D. Vollhardt and V. I. Anisimov, Phys. Rev. B 72 155106 (2005).
 (61) I. A. Nekrasov, K. Held, G. Keller, D. E. Kondakov, T. Pruschke, M. Kollar, O. K. Andersen, V. I. Anisimov and D. Vollhardt, Phys. Rev. B 73 155112 (2006).
 (62) M. Casula, A. Rubtsov, and S. Biermann, Phys. Rev. B 85, 035115 (2012).
 (63) C. Taranto, M. Kaltak, N. Parragh, G. Sangiovanni, G. Kresse, A. Toschi, and K. Held, Phys. Rev. B 88, 165119 (2013).
 (64) J. M. Tomczak, M. Casula, T. Miyake, and S. Biermann, Phys. Rev. B 90, 165138 (2014)
 (65) L. Boehnke, F. Nilsson, F. Aryasetiawan, P. Werner, Phys. Rev. B 94, 201106 (2016).
 (66) S. Backes et al.., Phys. Rev. B 94, 241110(R) (2016)
 (67) F. Aryasetiawan, K. Karlsson, O. Jepsen, and U. Schönberger, Phys. Rev. B 74, 125106 (2016)
 (68) A. Galler, P. Thunström, P. Gunacker, J. M. Tomczak, and K. Held (2017a), Phys. Rev. B 95, 115107; A. Galler, P. Thunström, J. Kaufmann, M. Pickem, J. M. Tomczak, and K. Held, arXiv:1710.06651; A. Galler, J. Kaufmann, P. Gunacker, M. Pickem, P. Thunström, J. M. Tomczak, and K. Held J. Phys. Soc. Japn. 87 , 041004 (2018).
 (69) A. Toschi, A. Katanin, and K. Held Phys. Rev. B 75, 045118 (2007).
 (70) G. Rohringer, H. Hafermann, A. Toschi, A. A. Katanin, A. E. Antipov, M. I. Katsnelson, A. I. Lichtenstein, A. N. Rubtsov, and K. Held Rev. Mod. Phys. 90, 025003 (2018).