Importance of d-p Coulomb interaction for high T$_C$ cuprates and other oxides

Current theoretical studies of electronic correlations in transition metal oxides typically only account for the local repulsion between d-electrons even if oxygen ligand p-states are an explicit part of the effective Hamiltonian. Interatomic interactions such as Upd between d- and (ligand) p-electrons, as well as the local interaction between p-electrons, are neglected. Often, the relative d-p orbital splitting has to be adjusted"ad hoc"on the basis of the experimental evidence. By applying the merger of local density approximation and dynamical mean field theory (LDA+DMFT) to the prototypical case of the 3-band Emery dp model for the cuprates, we demonstrate that, without any"ad hoc"adjustment of the orbital splitting, the charge transfer insulating state is stabilized by the interatomic interaction Upd. Our study hence shows how to improve realistic material calculations that explicitly include the p-orbitals.


Introduction
A primary effect of electronic correlations is to turn materials, which according to band-theory would be metallic, into insulators classified as Mott-Hubbard and charge transfer insulators [1]. Experimentally, various transition metal oxides were intensively investigated [2]. The huge interest in this subject goes back to the unexpected discovery [3] of high temperature superconductivity, by doping the charge-transfer insulating cuprates. On the theoretical side, great efforts to solve the Hubbard model [4] have been made since this model captures, in a fundamental way, the competition between electronic mobility and localization, generated by a purely local interaction.
No exact solution of the Hubbard model in two or three dimension is known hitherto, but a breakthrough was achieved by dynamical mean field theory (DMFT) [5,6,7]. DMFT extends the concept of a mean field theory to quantum mechanics and takes into account, non perturbatively, a major part of the electronic correlations, i.e., the local electronic correlations in time. A further, and significant, advantage of DMFT is the possibility to combine it with ab-initio density functional calculations, e.g., with the local density approximation (LDA) in the so-called LDA+DMFT [8] approach, which has already evolved to a considerable maturity. By means of LDA+DMFT, realistic calculations of correlated materials with a high degree of accuracy became possible, as it is well exemplified by the successful treatment of the Mott-Hubbard transition in V 2 O 3 [9].
The high-temperature superconducting cuprates obviously represent a much more challenging case for DMFT-based methods. In fact, the extremely anisotropic (quasi two-dimensional) layered structure of these compounds enhances the importance of nonlocal spatial correlations. Hence, if aiming at the low-temperature physics of cuprates, non-local correlations need to be taken into account, which could be achieved by cluster [10] or diagrammatic extensions [11] of DMFT. Here, we are not focusing on if, or up to what extent, the Hubbard model describes the unconventional superconductivity of the cuprates -a question on which no consensus has been reached yet. Instead, we will focus on the analysis of the high-temperature regime of the undoped cuprates. The effect of non-local correlations will be gradually mitigated by temperature [12,13,14], employing LDA+DMFT is more justified. LDA+DMFT was already successfully applied to the cuprates for analyzing the anomalies in the restricted optical sum rules [15,16,17,18] and the fingerprints of the Zhang-Rice physics in photoemission spectra [22,23]. Here, we will employ LDA+DMFT ‡ to investigate the fundamental nature of the high-temperature insulating state in the undoped compounds. This insulating state, classified as a charge transfer insulator [2], is in fact common for many transition metal oxides. A crucial step in the LDA+DMFT method is the derivation of the low energy Hamiltonian from ab initio calculations by renormalization (downfolding [24]) to certain basis states around the Fermi energy. The decision on the basis, in fact, is crucial. For the cuprates this question bascially boils down to include oxygen 2p ligand-states explicitly or implicitly. The simplest possible Hamiltonian with both Cu 3d and oxygen 2p states is the Emery model which includes, besides the Cu x 2 − y 2 band, the strongest hybridizing p x and p y states in the xy plane (an extensive discussion on the downfolding procedure for cuprates can be found in Ref. [32]).
In the renormalization sense, physical observables of different effective Hamiltonians should coincide at low energies, albeit the larger effective Hamiltonians might be more accurate than those defined on narrower energy windows. By extending the basis set, one (i) enlarges the energy range and the physical processes explicitly included in the downfolding procedure, and (ii) obtains a better localization of the more correlated dorbitals. Such a general statement is however premature if the extension exclusively relies on the one-particle ("kinetic") part of the Hamiltonian. One should not forget that the Coulomb interaction needs to be renormalized as well.
The material best studied in LDA+DMFT within different basis-sets is arguably LaNiO 3 , and heterostructures thereof. Here, including the p states in a dp-model [26] leads to qualitatively different results compared to the d-only model [27]. As pointed out in Ref. [28], the physics is very different in both cases: in the dp-model the doccupancy is larger with a tendency of two d-electrons to form a spin-1 due to Hund's exchange. On the contrary, the d-only model is built with a (fixed) d 1 configuration. In such a situation, it is pretty unclear, whether the dp-model does actually provide a more accurate description of the physics, as one might naively expect due to the larger number of degrees of freedom explicitly taken into account within the larger basis-set of the downfolded model [29].
Hitherto, the additional oxygen p-orbitals in the extended basis-sets have been considered as not interacting in state-of-the-art LDA+DMFT calculations, i.e., only a local d-d interaction has been taken into account. While the oxygen p-orbitals are, due to their nodeless radial wave function also quite localized, their typical filling in transition metal oxides renders the onsite p-interaction to be non-crucial. Yet, at the same time, our numerical results show that the effect of the interatomic 2p-3d interaction, in the following coined as U pd , does have a large effect. U pd is an essential parameter that controlls the metal-to-insulator transition, as well as the size of the charge transfer gap.
The paper is organized as follows: In Section 2 the construction of the Emery model is described starting from the full LDA bandstructures. Section 3 and the Appendix are dedicated to the DMFT and Hartree treatment of the various Coulomb interactions. Section 4 presents the results obtained, in particular the opening of the charge transfer gap and the importance of U pd to this end. Finally, Section 5 summarizes the paper.

The Emery model revisited
A popular tight-binding Hamiltonian for the high T C cuprates, including oxygen pstates, is a 3-band model suggested by Emery [30] in 1987. The Emery model consists +Sin(3k y /2)Sin(k x /2))) of one planar Cu x 2 − y 2 band, two oxygen p x and p y bands and takes into account a d-p hopping, see Table 1. It is thus the minimal model in order to describe the charge transfer insulating state and, moreover, the physics of a Zhang-Rice singlet [31]. These features are obviously beyond a description of the cuprates within an effective single band model. However, Andersen et al. [32] concluded from downfolding the ab initio LDA bandstructure that the original model as described in Ref. [30] should be extended. These extensions originate basically from the inclusion of an axial degree of freedom (including mostly Cu 4s states and apical oxygen states) which was seen to describe the material-dependence of the electronic structure in the cuprates [33]. The derivation starts from eight bands which are separated into a 4 × 4 σ-bonding block of Cu 3d x 2 −y 2 , O 1 2p x , O 2 2p y , as well as Cu 4s (with some Cu 3d 3z 2 −r 2 character), and another 4 × 4 π-bonding block of Cu-3d xz , Cu 3d yz , O 1 2p z , and O 2 2p z . The generic Hamiltonian for the CuO 2 planes is the 4 × 4 σ-block, since it contains the conduction band. Due to symmetry reasons, the σ-and the π-block do not hybridize in the limit of flat planes [32]. Starting from the four band σ-Hamiltonian one can arrive either at i) a two-band  Table 2: Hopping integrals for the Emery model Table 1 as obtained by 0MTO and 1MTO in [34,35]. Figure 1: (color online) Bandstructure and single particle density of states for the 0MTO and 1MTO downfolded model [34,35]. The color codes the orbital character: 3d x 2 − y 2 is black, whereas the p-states are summarized and plotted in red (grey) -color mixing corresponds to a mixed orbital state. The LDA bands are shown as thin (blue) lines.
Hamiltonian with planar ("dressed" Cu x 2 − y 2 ) and axial ("dressed" Cu 4s) degrees of freedom §, or at ii) the three band Emery model. The latter is obtained by folding the axial degrees of freedom down to the oxygen bands, resulting in an additional (material dependent) p-p hopping t pp and renormalization of the onsite p-energy.
Quantitative Hamiltonians can be obtained by means of N th order Muffin Tin Orbitals (NMTO) [24] based on a Löwdin downfolding scheme with subsequent "Nization" (N = 0: linearization). This technique has been already established and used in several LDA+DMFT calculations [25]. In our paper, we will consider two different low-energy Hamiltonians [34,35]: The first one consists of N = 0 muffin tin orbitals which have been linearized around the Fermi energy. The 0MTO basis gives correct LDA wave functions at the Fermi level and, by virtue of the variational principle, the correct LDA Fermi surface and Fermi velocities. The 1MTO basis gives, in addition to this, the correct LDA wave functions at an energy E1 chosen at the bottom of the pdbonding band, and hence, the correct E(k) = E1 surface and velocities. The resulting Hamiltonian can be written analytically as in Table 1 with hopping parameters given in Table 2.
In Fig. 1 we show these NMTO downfolded bands for La 2 CuO 4 from Kent et al. [35] for N = 0 (linearization around the Fermi energy -left hand side) and N = 1 (with E 1 chosen to be at the bottom of the pd-bonding band at (π, π) -right hand side). The corresponding hopping parameters can be found in Table 2. The difference of the N=0 and N=1 models is the energy range in which they reproduce the cuprate LDA bandstructure (light blue lines): While the N = 0 model only has one fixed energy, namely the Fermi energy E F , the N = 1 model was fixed to ε F and to the energy of the bottom of the bonding pd-bands at around −8 eV. By having to span a wider energy range, the N = 1 orbitals are somewhat less localized, and consequently have longerranged hoppings than the N = 0 orbitals. A thorough discussion on the relations and trends of the hopping parameters can be found in [35] and [34]. In our study we will consider first the N = 0 and then the N = 1 model in order to address the following issue of central interest: The most problematic step in recent studies [37,38] was the value of the d-p splitting ε d − ε p = ∆ dp . While the NMTO downfolding [35] yields a value of ∆ dp = 0.45 eV (N = 0) or ∆ dp = 0.96 eV (N = 1), it turns out that the many-body treatments, which include correlation effects fail to reproduce the insulating behavior of the undoped LSCO. In order to fix this problem ∆ dp was increased "by hand" to values of the order of ∆ dp ≈ 3 eV [35,38] or it was chosen as a variable parameter [37]. Further, Kent et al. [35] pointed out that previous justifications of such enhancement of ∆ dp by means of constrained LDA calculations for La 2 CuO 4 are problematic due to a misleading assumption of the electron count. Our analyis in the subsequent sections will clarify, how these problems are actually related to the assumption of negligible Coulomb interactions between d and p electrons in the many-body calculations.

Methods: dp-models in DMFT+Hartree
We supplement the one-particle extended Emery model from Table 1 by the following two-particle interactions Here i and j sum over all Cu and O sites, respectively; p j ∈ {p x , p y } denotes the particular p orbital we include on a given O site j in the Emery model; ij denotes the restriction to nearest neigbors, i.e., a Cu site i and its four surrounding O sites j; and n ilσ = c † ilσ c ilσ . This way the most important (largest) interaction parameters are included: the local Coulomb interactions on the Cu (U dd ) and O sites (U pp ) and the nearest neighbor Cu-O interaction U pd .
Since part of H U is contained in the NMTO one-particle energies, we need a socalled double counting corrections (DC). For the dp-models, the DC effect is much more important than for d-only models where it often corresponds to a simple energy shift that can be "absorbed" by the chemical potential. We employ the Anisimov DC formula [39] (also coined as fully localized limit) and extend it straightforwardly to the dp-basis: Here, n d(p) are the LDA density for the d(p) orbitals (Cu (O) sites have four O (Cu) neighbors). So the DC corresponds to a relative shift between the d-and p-states, which would be reversed by a simple mean field treatment of U dd U pd , and U pd . Let us emphasize that, in contrast to standard LDA+DMFT implementations [28], we treat all density-density interactions within the dp basis in DMFT+Hartree. Specifically, U dd is treated in DMFT(CTQMC) where we define a local Anderson impurity model from the local Cu d Green function and self energy. The interactions U pp and U pd are treated on the Hartree level, i.e., a self energy contribution for each d and p site is calculated as U pp (U pd ) times the density of the other orbitals involved. Please note that if n d(p) remained at its LDA value the DC term would cancel the effect of U pd in DMFT+Hartree exactly. However, due to the electronic correlations, in particular due to the splitting of the d-orbitals, n d is reduced and n p enhanced.
For U pd this treatment is exact in the DMFT limit since non-local interactions reduce to their Hartree contribution, see Ref. [40]. In the case of U pp , a DMFT solution for the O sites would be in principle required, exactly as for the Cu site. However, since the p-orbitals are almost completely filled, electronic correlations are actually weak and the simpler Hartree treatment well reproduces the DMFT solution, as we explicitly show in the Appendix. Our approach incorporates therefore the advantages of the DMFT and explicitly keeps the p-degrees of freedom also in the interaction part of the many body Hamiltonian. The extended basis-set allows us not only to capture Mott-Hubbard physics, but also that of charge transfer insulators and the concomitant d-p interplay. The latter is, in fact, the relevant one for the physics of undoped/underdoped cuprates.
Before turning to the results let us mention that the analytic continuation of the numerical LDA+DMFT results is more complicated for the extended dp basis-set. This issue is discussed in the Appendix as well.

Results
Our DMFT+Hartree results for both 0MTO and 1MTO basis sets can be summarized in two conclusions: (i) The d-p interaction U pd , which leads to a self consistently determined level shift, drives the system insulating within DMFT+Hartree and opens the charge transfer gap.
(ii) The critical interactions for this metal-to-insulator transition in 0MTO agree with estimates of the interaction strength; for 1NMTO they are at the upper borderline or somewhat larger than what can be expected.
0MTO model For the interaction parameters of the 0MTO calculation we choose U dd = 10 eV, U pp = 5 eV, and take different values of U pd ranging from U pd = 0 eV to U pd = 3.5 eV. The choice of U dd value is motivated from constraint random phase (cRPA) calculations, which gives U cRPA dd ≈ 9.0 eV [44]. Such cRPA values are typically smaller than constraint LDA values, which in some cases yields better results in LDA+DMFT calculations. The reason for this is the frequency dependence of the U cRPA dd which can be actually translated to a Bose factor renormalization of the bandwidth [45]. To the best of our knowledge, for U pd and U pp no reliable estimates are yet available in the literature, but it is certainly a reasonable assumption that these interactions need to be smaller than U dd .
The obtained spectral functions are shown in Fig. 2. As in the bandstructure plots, the color of the spectral function codes the orbital character. Let us start from the spectrum for U pd = 0 eV which is plotted in the top left panel. Although the parameters U dd = 10 eV and U pp = 5 eV are by no means small compared to the bandwidth, we observe a rather uncorrelated spectral function which resembles the non-interacting DOS (Fig. 1) except for the LHB located below -10eV. The main reason for this is that, due to the d-p hybridization, the filling of each band and, in particular of the d-band, is far from an integer value. Upon increasing the value for U pd this hybridization decreases, as can be seen in the spectra, since the charge transfer from d-states to p-states is now connected with a potential shift of the respective states of the order of U pd . Eventually, for values U pd ≈ 1 2 U pp , we observe a rather sudden metal-to-insulator transition between U pd = 2 eV and U pd = 2.5 eV.
In Fig. 2 we observe, that for U pd ≤ 2eV the spectra show a gap between d-states: an "upper Hubbard band" with some p-hybridization above the Fermi energy ε F and a mixed d-p peak around −2 eV (for U pd = 2 eV) or −3 eV (for U pd = 2.5 eV). The "lower Hubbard band" is quite broad and located below ∼ −10 eV, whereas most of the p-spectral weight is located in a large peak around −4 eV (U pd = 2 eV) and −5 eV (U pd = 2.5 eV). For U pd = 3.5 eV, the p bands are shifted to such low energies that they approach the lower Hubbard bands and start hybridizing with them. Let us emphasize that the formation of the narrow peak at E F and the transfer of spectral weight to the lower and upper Hubbard bands occurring between 2 and 2.5eV is a dynamical effect beyond static mean-field, as witnessed by the strong ω-dependence of Σ (see Fig. 5 below). In summary, DMFT+Hartree yields an insulating state for the original 0MTO parameters without the artificial enhancement of the d-p splitting hitherto employed in the literature. Instead, we assumed a finite value of U pd which, in a self consistent way, leads to a suppression of d-p hybridization driving the metal-to-insulator transition. Yet, we should remark two issues: Since 0MTO was designed to reproduce the cuprate bands only around the Fermi energy [35], it is questionable if the 0MTO really yields a good basis for a study of excitations on an energy scale of some eV above and below the Fermi energy such as the d-p interplay. We hence present DMFT+Hartree results of the 1MTO model in the following.
1MTO model As mentioned above, the N=1 orbitals are less localized, and consequently have somewhat longer-ranged hoppings than the N=0 orbitals in order to reproduce the band structure on a larger energy range. Hence, also the corresponding 1MTO values of the local U dd and U pp interaction parameters should be reduced. Concurrently, we find that the 1MTO yields much more metallic solutions. The system remains metallic even for the same interaction parameters as we used for the N=0 model in Fig. 2; to obtain an insulating ground state one needs a larger U dd or U pd .
In Fig. 3 we show the DMFT+Hartree spectral functions for the N=1 model at U dd = 13 eV, U pp = 7 eV, and varying U pd . Increasing U pd beyond 3 eV opens a gap in the spectral function However, U dd = 13 eV is at the upper edge or larger than the physically reasonable parameter range.
Nature of the mixed d-p peak As shown in Figs. 2, 3 a peak of mixed d-and p-orbital character develops in the k-integrated spectral functions around −3 eV (for 0MTO at U pd = 2.5 eV and 1MTO at U pd = 3.5 eV). This peak can be interpreted [46] as a Zhang-Rice state in agreement with previous LDA+DMFT studies [47,48,49]. Figure 3: (color online) DMFT spectra for the 1MTO basis with interaction parameters U dd = 13eV, U pp = 5eV, and varying U pd . We have not broadened the δ-peaks with an additional Lorentzian broadening. The only broadening is due to the intrinsic finite lifetime from ImΣ. Please note that there is spectral weight down to −20eV in the lower two panels which we cut off.
In order to see whether or not this spectral weight is also associated to a coherent excitation we calculate A( k, ω) just before and after the metal-insulator transition, i.e., U pd = 2 − 2.5 eV for 0MTO and U pd = 3 − 3.5 eV for 1MTO. The results are shown in Fig. 4, and the corresponding local d-self-energies are shown in Fig. 5. Interestingly, the coherence of the spectral feature around −3 eV changes dramatically across the metalinsulator transition for both 0MTO (the two lower panels of Fig. 4) and 1MTO (the two upper panels). In the two left panels of Fig. 4, i.e., before the metal-insulator transition, the dispersive band between −3 and −1 eV is very incoherent whereas in the two right panels, i.e., just after the metal-insulating transition, a very well defined quasiparticle excitation emerges. The double-occupancies of the d-orbital and the spectral weight at the Fermi level as a function of U pd shows the following: The values of U pd for which the double-occupancies (not-shown) get suppressed are those for which most of the spectral weight gets shifted away from the Fermi level. In this strongly-correlated Figure 4: (color online) k-resolved spectral function of 0MTO in the lower and the 1MTO in the upper row. The two left panels are before the charge-transfer metal-insulator transition (i.e., for U dd = 10, U pp = 5, U pd = 2 eV and U dd = 13, U pp = 7, U pd = 3 eV in the lower and upper row, respectively). The two right columns are after the metalinsulator transition (obtained by increasing U pd -parameter by 0.5 eV; keeping the other parameters fixed). For both models the mixed pd-state becomes suddenly coherent as soon as we enter the insulating state. metallic solution an additional peak of mixed dp character is formed when a residual finite fraction of itinerant electrons is still present. This spectral feature, which can be interpreted as the emergent Zhang-Rice excitation, becomes more coherent only when the spectral weight at the Fermi level is completely depleted. This depletion, i.e., the opening of the gap, occurs in a similar way as in the Mott-Hubbard transition of the Hubbard model: a pole in the real part of the self-energy develops, see the blue (dashed) lines in Fig. 5. At the same time, the imaginary part of the self energy develops a peak. Since the gap is between d-states above and p-states below the Fermi level, we have however a charge-transfer insulator [1] with a Zhang-Rice singlet.

Conclusions
Presently, within the state-of-the art LDA+DMFT calculations, the splitting between the d-and p-states in cuprates [35] and other transition metal oxides is adjusted  for obtaining agreement with experiment, see e.g. Ref. [50], or new double counting corrections are introduced, see Ref. [51]. This is quite unsatisfactory since it basically destroys the often claimed ab initio character of the calculation. The aim of our work was hence (i) to understand the physical origin behind such adjustments, and (ii) to perform calculations based on the original parameters from NMTO downfolding without artificially changing them. In particular, if the orbital overlap is large, the non-local oxygen-copper interaction U pd can be strong. We have shown that the inclusion of this interatomic interaction is indeed extremely important, because it allows for the description of a self consistently determined d-p level splitting. This mechanism can explain why previous LDA+DMFT calculations that only included the d-d interaction yielded -in some cases-poorer results than corresponding studies in smaller d-only basisset. The artificial p-d shifts that were employed in the literature actually mimic the effect of U pd . The DMFT+Hartree approach with the explicit inclusion of U pd , adoped in this work, is much more satisfactory from both a practical and a conceptual point of view. The DMFT+Hartree treatment can be based entirely on interaction parameters determined ab-initio for the downfolded basis-set chosen, e.g., by means of cRPA [52] or its recently developed locally unscreened version [53]. For the specific case of undoped cuprates, the two downfolded models considered, 0MTO and 1MTO, qualitatively yield the same physics, see Figs. 2-4. Quantitatively, the results for the 1MTO model require somewhat larger interaction parameters than 0MTO to open an insulating gap. For 0MTO we have a charge transfer insulator for plausible interaction parameters, whereas for 1MTO these are at the borderline or even somewhat larger than expected [44]. Let us note in this respect that nonlocal interactions, beyond the realm of our DMFT+Hartree treatment, also play an important role for cuprates and will further stabilize the insulating solution. Also in the paramagnetic phase these non-local correlations will reduce the critical interactions for the metal-to-insulator transition [54,17]. Let us emphasize that the impact of our analysis goes well beyond the particular, though significant case of the insulating phase in undoped cuprates. In fact, for other transition metal oxides with a more three dimensional crystal and electronic structure, non-local correlations beyond DMFT are less important, except for low temperatures [13,14]. Hence, one can expect a DMFT+Hartree calculation including ab-initio the U pd interaction to be sufficient for capturing the physics of downfolded d-p models which are nowadays fixed by hand [50]. In practice, similarly as for the Emery model, the U pd interactions will turn the physics of the dp-model towards that of the d-only model, because the number of d electrons is reduced towards an integer value, entailing stronger electronic correlations. Our paper hence paves the way for better treatment of oxygen p-orbitals in LDA+DMFT calculations. e.g. H k and the Hartree self-energy Σ H can be treated exactly on the real axis. In the practical implementation, we measure the Greens function from a converged run directly in Matsubara frequencies iω n to avoid binning errors as would be the case when measuring the Greens function in imaginary time τ . Then we calculate the self-energy Σ(iω n ) using the Dyson equation Σ(iω n ) = G −1 0 (iω n ) − G −1 (iω n ), assuming that the bath Greens function G −1 0 does not contain statistical errors. To calculate the error for the self-energy we employ the bootstrap method on 300 independent Greens function measurements. The self-energy has a different asymptotic behavior from that of the Greens function. The latter is however needed for maxent-like methods. Therefore a manipulation of the input is needed. To this end one performs a high-frequency expansion of the CTQMC self-energy which for the single band Hubbard model reads: Following the argument above and with the knowledge that the Greens function has a 1/(iω n ) asymptotic behavior, we subtract the Hartree term for the d band, U dd n −σ and divide the remaining part of the self-energy by the prefactor of the 1/(iω n ) term. This new object, Σ(iω n ), can then be analytically continued to the real axis by means of the stochastic maxent method [42,43]. In this respect, by inspecting the correlation matrix of the self-energy we checked that different Matsubara frequencies are uncorrelated, otherwise a transformation to a non-correlated basis would have been necessary. We finally calculated the spectral function, A(k, ω), from Σ(ω) and all other terms, namely the double counting correction DC, the k-dependent Hamiltonian H k , the chemical potential µ and the Hartree selfenergy of Σ H : To test the validity of our approach we also compared data of an ED-calculation with data obtained by our CTQMC implementation, which resulted in very good agreement.
DMFT(CTQMC) for the p-electrons Here, we validate that a Hartree treatment of U pp is sufficient. To this end we performed for some test calculations where both U dd and U pd are treated in DMFT, and U pd in Hartree, which actually is the correct (full) DMFT treatment of the Emery model. Fig. 6 shows the obtained DMFT self energy, which is basically the same as the Hartree self energy. For U pd = 1.5 eV, very minor deviations are discernible at low frequencies, but the Hartree self energy is still an accurate approximation of the DMFT self energy. Therefore, we conclude that a Hartree treament of U pp is absolutely sufficient.