Abstract
It has long been argued that the minimal model to describe the lowenergy physics of the high T_{c} superconducting cuprates must include copper states of other symmetries besides the canonical one, in particular the orbital. Experimental and theoretical estimates of the energy splitting of these states vary widely. With a novel ab initio quantum chemical computational scheme we determine these energies for a range of copperoxides and oxychlorides, determine trends with the apical Cu–ligand distances and find excellent agreement with recent Resonant Inelastic Xray Scattering measurements, available for La_{2}CuO_{4}, Sr_{2}CuO_{2}Cl_{2} and CaCuO_{2}.
Introduction
It is generally accepted that the lowenergy physics of the layered Cu oxide compounds in their normal state is reasonably well described by models which incorporate the “inplane” Cu and O 2p_{x} /2p_{y} orbitals. However, the energy window over which such models provide a qualitatively correct picture is a matter of active research. One additional ingredient which is often invoked is the Cu orbital, perpendicular onto the CuO_{2} layers and the apical O 2p_{z} functions having σtype overlap with the Cu . Recent multiorbital calculations using dynamical meanfield theory^{1} show indeed that some of the features of the optical, xray absorption and photoemission spectra can be better reproduced when the Cu orbitals are explicitly included in the manybody treatment. At finite doping, the inclusion of the Cu functions makes a difference even for the lowenergy states close to the Fermi level^{1}.
The offdiagonal coupling between states of x^{2}−y^{2} and z^{2} symmetry was actually found to substantially affect the dispersion of the lowenergy bands and the shape of the Fermi surface in earlier semiphenomenological models^{2,3}, densityfunctional calculations^{4} and quantum chemical studies^{5}. Moreover, Ohta et al.^{6} and recently Sakakibara et al.^{7} suggested that a direct relation exists between the magnitude of T_{c} and the size of the splitting. The splittings within the Cu 3d shell are also relevant to excitonic models for pairing and highT_{c} superconductivity^{8,9}. Even if the importance of the Cu state is stressed in this considerable body of work, the actual experimental and theoretical estimates of the energy of this state vary widely.
Sharp features at about 0.4 eV in early optical measurements on La_{2}CuO_{4} and Sr_{2}CuO_{2}Cl_{2} were initially assigned to crystalfield Cu to charge excitations^{10}. A different interpretation in terms of magnetic excitations was proposed by Lorenzana and Sawatzky^{11} and latter on confirmed by analysis of the resonant inelastic xray scattering (RIXS) spectra at the Cu K and L_{3}edge^{12,13}. The RIXS experiments also show that in La_{2}CuO_{4} and Sr_{2}CuO_{2}Cl_{2} the Cu to transitions occur at 1.5–2.0 eV^{14}, which is substantially larger than the outcome of earlier wavefunctionbased quantum chemical calculations, 1.0–1.2 eV^{15}, or densityfunctional estimates, 0.9 eV^{7}.
With the aim to settle this point we employ a recently developed ab initio quantum chemical computational scheme to extract the splittings within the Cu 3d shell in several layered copper oxides. Excellent agreement is found for La_{2}CuO_{4}, Sr_{2}CuO_{2}Cl_{2} and CaCuO_{2} with recent RIXS measurements^{14}. Further, the to excitation energies computed here for La_{2}CuO_{4}, YBa_{2}Cu_{3}O_{6} and HgBa_{2}CuO_{4} are relevant to models which attempt to establish a direct relation between the relative energy of the outofplane level and the critical temperature T_{c}^{7}. In particular, the large difference between the critical superconducting temperatures of doped La_{2}CuO_{4} and HgBa_{2}CuO_{4} was directly attributed to a large difference between the to excitation energies^{7}.
Results
To study bound, excitoniclike states such as the dd charge excitations in copper oxides, we rely on realspace ab initio methods. In the spirit of modern multiscale electronicstructure approaches, we describe a given region around a central Cu site by advanced quantum chemical manybody techniques while the remaining part of the solid is modeled at the HartreeFock level. The completeactivespace selfconsistentfield (CASSCF) method was used to generate multireference wavefunctions for further configurationinteraction (CI) calculations^{16}. In the CASSCF scheme, a full CI is carried out within a limited set of “active” orbitals, i.e., all possible occupations are allowed for those active orbitals. The active orbital set includes in our study all 3d functions at the central Cu site and the functions of the Cu nearest neighbor (NN) ions. Strong correlations among the 3d electrons are thus accurately described. The final CI calculations incorporate all single and double excitations from the Cu 3s,3p,3d and O 2p orbitals on a given CuO_{4} plaquette and from the orbitals of the Cu NN's. Such a CI treatment is referred to as SDCI. The CASSCF and SDCI investigations were performed with the molpro quantum chemical software^{17}.
Both SDCI and RIXS results for the Cu dlevel splittings are listed in Table 1. The relative energies of the peaks observed between 1 and 3 eV in the Cu L_{3}edge RIXS spectra^{14} are the sum of a crystalfield contribution, i.e., an onsite crystalfield splitting E_{cf} and a magnetic term ΔE_{mgn}. The quantum chemical calculations have been performed to extract E_{cf} for a ferromagnetic (FM) arrangement of the Cu d spins. A SDCI treatment for an antiferromagnetic (AF) alignment of the Cu spins in the embedded cluster of five Cu sites is computationally not feasible (see Methods for details). ΔE_{mgn} accounts for AF order in the groundstate configuration of the Heisenberg antiferromagnet and is determined as follows. First the value for the NN exchange coupling constant J is computed by considering an embedded cluster consisting of two CuO_{4} plaquettes. For CaCuO_{2}, for example, we find J = 0.13 eV, in good agreement with the theoretical results reported in Ref. [18] and with values from experimental data^{11,13,14}. With this value of J in hand we return to the cluster with five Cu sites and flip the spin of the central Cu ion. This corresponds to an energy increase ΔE = zJ/2 = 2J, where z = 4 is the number of NN's and we neglect the quantum fluctuations. For the crystalfield excited states, the superexchange with the NN Cu spins is much weaker for a hole excited into the orbital and zero by symmetry for a hole into a t_{2g} orbital. This contribution due to intersite superexchange, ΔE′ = 2J′, is not included either in the quantum chemical calculations but in a first approximation we can neglect the weak intersite AF interaction J′ involving a hole. From overlap considerations, J′ is only a small fraction of the groundstate superexchange J. For a meaningful comparison between the SDCI and RIXS data, we subtracted in Table 1 from the relative RIXS energies reported in Ref. [14] the term ΔE_{mgn} = ΔE – ΔE′ ≈ 2J representing the magnetic stabilization of the groundstate configuration with respect to the crystalfield excited states. Since J ≈ 0.13 eV, ΔE_{mgn} ≈ 0.26.
The agreement between our SDCI excitation energies and the results from RIXS is remarkable. As shown in Table 1, the differences between the SDCI and RIXS energies are not larger than 0.15 eV. The only exception is the splitting between the x^{2}−y^{2} and xz/yz levels in CaCuO_{2}, where the SDCI value is 0.3 eV larger than in the RIXS measurements. That an accurate description of neighbors beyond the first ligand coordination shell is crucial is clear from the comparison between our and earlier quantum chemical data. In the calculations described in Ref. [15], only one CuO_{6} octahedron or one CuO_{5} pyramid was treated at the allelectron level. Farther neighbors were described by either atomic model potentials or point charges. Deviations of 0.4 and 0.6 eV (up to 50%) for the z^{2} levels in La_{2}CuO_{4} and Sr_{2}CuO_{2}Cl_{2}^{15}, for example, are mainly due to such approximations in the modeling of the nearby surroundings. The dlevel splittings depend after all on the charge distribution at the NN ligand sites. The latter is obviously sensitive to the manner in which other species in the immediate neighborhood are modeled. The quality of the results reported here is directly related to the size of the clusters, i.e., five CuO_{4} plaquettes, all apical ligands plus the NN closedshell metal ions of the central polyehdron.
Superconductivity has not been observed in Sr_{2}CuO_{2}Cl_{2} and CaCuO_{2}. The dlevel splittings for three representative cuprate superconductors, i.e., La_{2}CuO_{4}, HgBa_{2}CuO_{4} and YBa_{2}Cu_{3}O_{6}, are listed in Table 2. The maximum T_{c} 's achieved by doping in these three materials are 35, 95 and 50 K, respectively. For the YBa_{2}Cu_{3}O_{6} compound, we here refer to the maximum T_{c} which can be achieved by Ca doping^{19}. The large difference between the critical temperatures in La_{2}CuO_{4} and HgBa_{2}CuO_{4} was assigned in Ref. 7 to a large difference between the relative energies of the z^{2} states in the two materials. The densityfunctional results for the splittings between the x^{2}−y^{2} and z^{2} levels in La_{2}CuO_{4} and HgBa_{2}CuO_{4} are 0.91 and 2.19 eV, respectively^{7}. RIXS data are not available for HgBa_{2}CuO_{4} and independent estimates for the energy separation between the x^{2}−y^{2} and z^{2} states are therefore desirable. While we find a rather similar value for HgBa_{2}CuO_{4}, of 2.09 eV, the quantum chemical and RIXS results^{14} for La_{2}CuO_{4} are substantially larger, about 1.4 eV. This makes the difference between the dlevel splittings in the above mentioned compounds less spectacular, i.e., is reduced from 1.3 eV in Ref. [7] to 0.7 eV in the present study, which suggests that the model constructed and the conclusions drawn in Ref. [7] at least require extra analysis.
The distance between the Cu and apical ligand sites increases from 2.40 Å in La_{2}CuO_{4}^{20} to 2.78 Å in HgBa_{2}CuO_{4}^{21}. The effect of this growth of the apical Cu–O bond length on the relative energy of the z^{2} hole state can be understood by using simple electrostatic arguments: when the negative apical ions are closer to the Cu site, less energy is needed to promote the Cu 3d hole into the z^{2} orbital pointing toward those apical ligands. For HgBa_{2}CuO_{4}, the lowest crystalfield excitation is therefore to the xy level and requires about 1.3 eV, see Table 2, while the z^{2} and xz/yz levels are nearly degenerate and more than 0.5 eV higher in energy. On the other hand, in La_{2}CuO_{4} the lowest crystalfield excitation is to the z^{2} orbital, see Table 1. Our results also reproduce the near degeneracy between the z^{2} and xy levels in La_{2}CuO_{4}, as found in the RIXS experiments. In CaCuO_{2}, there are no apical ligands. The splitting between the x^{2}−y^{2} and z^{2} levels is therefore the largest for CaCuO_{2}, about 2.4 eV, see Table 1.
Discussion
The parameter that plays the major role in determining the size of the dlevel splittings in layered cuprates is clearly the apical Cu–ligand distance. There are, however, few other factors which come into play such as the number and nature of the apical ligands, the inplane Cu–O bond lenghts, buckling of the CuO_{2} planes and the configuration of the farther surroundings. Trends concerning the relative energy of the z^{2} hole state in different cuprates are illustrated in Fig. 1, which includes data for systems having one apical O site (YBa_{2}Cu_{3}O_{6}), two apical O's (La_{2}CuO_{4}, HgBa_{2}CuO_{4}), two apical Cl ions (Ca_{2}CuO_{2}Cl_{2}, Sr_{2}CuO_{2}Cl_{2}) or no apical ligand (CaCuO_{2}). The apical Cu–O distances in La_{2}CuO_{4} and YBa_{2}Cu_{3}O_{6}, for example, are nearly the same, 2.40 vs. 2.45 Å^{19,20,22}. In YBa_{2}Cu_{3}O_{6}, however, there is a single apical O. For this reason the z hole state is somewhat destabilized in YBa_{2}Cu_{3}O_{6} and lies above the xy hole configuration, see Table 2. Yet since the Cu ion is shifted towards the apical ion, out of the basal O plane, the x^{2}−y^{2} hole state is also destabilized such that the splitting between the x^{2}−y^{2} and z^{2} levels is finally close to the value found in La_{2}CuO_{4}. Further, the apical Cu–ligand distances are slightly larger in Sr_{2}CuO_{2}Cl_{2} as compared to HgBa_{2}CuO_{4}, 2.86 vs. 2.78 Å, respectively. The apical ions also have a smaller effective charge in Sr_{2}CuO_{2}Cl_{2}, which should lead to a larger relative energy of the z^{2} hole state in Sr_{2}CuO_{2}Cl_{2} as compared to HgBa_{2}CuO_{4}. The fact that the relative energy of the z^{2} hole state is actually larger in HgBa_{2}CuO_{4}, see Fig. 1, must be related to the smaller inplane Cu–O distances in HgBa_{2}CuO_{4}, 1.94 in HgBa_{2}CuO_{4} vs. 1.99 Å in Sr_{2}CuO_{2}Cl_{2}, which stabilizes the groundstate x^{2}−y^{2} hole configuration in the former compound and to farther structural details. From Ca_{2}CuO_{2}Cl_{2} to Sr_{2}CuO_{2}Cl_{2}, the Cu–Cl separation increases from 2.75 to 2.86 Å^{23,24} and the energy of the z^{2} level from 1.37 to 1.75 eV.
In contrast to the z^{2} orbitals, the relative energies of the xy levels display much smaller variations, in an interval of 1.2–1.5 eV, see Tables 1 and 2. Substantially smaller are also the variations computed for the xz/yz levels, in an energy window between 1.6 and 2.0 eV.
To summarize, we employ state of the art quantum chemical methods to investigate the Cu 3d electronic structure of layered Cu oxides. Multiconfiguration and multireference configurationinteraction calculations are carried out on finite clusters including five CuO_{4} plaquettes plus additional apical ligand and closedshell metal ion NN's. The localized Wannier functions attached to these atomic sites are obtained from prior HartreeFock computations for the periodic system. Excellent agreement is found between our theoretical results and recent Cu L_{3}edge RIXS data for La_{2}CuO_{4}, Sr_{2}CuO_{2}Cl_{2} and CaCuO_{2}. RIXS is a novel experimental tool to investigate both magnetic and charge excitations with high resolution and accuracy. Our computational scheme and present results indicate a promising route for the modeling and reliable interpretation of RIXS spectra in correlated 3dmetal compounds. A next step along this path is the computation of transition probabilities and intensities at the ab initio level, which requires the explicit calculation of the intermediate Cu 3p core hole wavefunctions.
Further, the excitation energies computed here for La_{2}CuO_{4}, YBa_{2}Cu_{3}O_{6} and HgBa_{2}CuO_{4} are relevant to models which attempt to establish a direct relation between the critical temperature T_{c} and the strength of the (x^{2}−y^{2})−z^{2} coupling. For La_{2}CuO_{4}, in particular, the densityfunctional estimate used as input parameter in such models^{7} is about 0.5 eV smaller than our result. Consequently, the difference we find between the splittings in La_{2}CuO_{4} and HgBa_{2}CuO_{4} is less spectacular as compared to the value reported by Sakakibara et al.^{7}, suggesting a reevaluation of the analysis in Ref. [7].
Methods
The first step in our study is a restricted HartreeFock (RHF) calculation for the groundstate configuration of the periodic system. The RHF calculations are performed with the crystal package^{25}. We employed experimental lattice parameters^{14,20,21,22,23,24} and Gaussiantype atomic basis sets, i.e., triplezeta basis sets from the crystal library for Cu, O and Cl plus basis sets of either doublezeta or triplezeta quality for the other species. Post HartreeFock manybody calculations are subsequently carried out on finite clusters, which are sufficient because of the local character of the correlation hole. They consist of five CuO_{4} plaquettes, i.e., a “central” CuO_{4} unit plus the four NN plaquettes. When present, the apical ligands, oxygen or chlorine, are incorporated as well in the finite cluster . Additionally, the finite cluster includes in each case the NN closedshell metal ions around the “central” Cu site. In La_{2}CuO_{4}, for example, there are ten La^{3+} NN's. In YBa_{2}Cu_{3}O_{6}, there are one Cu^{1+} 3d^{10}, four Y^{3+} and four Ba^{2+} NN's.
The orbital basis entering the post HartreeFock correlation treatment is a set of projected RHF Wannier functions: localized Wannier orbitals (WO's) are first obtained with the WannierBoys localization module^{26} of the crystal package and subsequently projected onto the set of Gaussian basis functions associated with the atomic sites of ^{27}. Moreover, the RHF data is used to generate an effective embedding potential for the fiveplaquette fragment . This potential is obtained from the Fock operator in the RHF calculation^{27} and models the surroundings of the finite cluster, i.e., the remaining of the crystalline lattice.
The central CuO_{4} plaquette and the four NN Cu sites form the active region of the cluster, which we denote as . The other ions in , i.e., each ligand coordination cage around the four Cu NN's and the NN closedshell metal ions, form a buffer region whose role is to ensure an accurate description of the tails of the WO's centered in the active part ^{27}. For our choice of , the norms of the projected WO's centered within the active region are not lower than 99.5% of the original crystal WO's. While the occupied WO's in the buffer zone are kept frozen, all valence orbitals centered at O and Cu sites in (and their tails in ) are further reoptimized in multiconfiguration CASSCF calculations. In the latter, the groundstate wavefunction and the lowest four crystalfield excited states at the central Cu site are computed simultaneously in a stateaveraged multiroot calculation^{16}. The dlevel splittings at the central Cu site are finally obtained at the CASSCF+SDCI level of theory as the relative energies of the crystalfield excited states. The virtual orbital space in the multireference SDCI calculation cannot be presently restricted just to the region. It thus includes virtual orbitals in both and , which leads to very large SDCI expansions, ∼ 10^{9} Slater determinants for a FM configuration. For this reason, we restrict the CASSCF+SDCI calculations to FM allignment of the Cu d spins.
The effective embedding potential is added to the oneelectron Hamiltonian with the help of the crystalmolpro interface program^{28}. Although the WO's at the atomic sites of are derived for each of the compounds discussed here by periodic RHF calculations for the Cu 3d^{9} electron configuration, the embedding potentials are obtained by replacing the Cu^{2+} 3d^{9} ions by closedshell Zn^{2+} 3d^{10} species. This is a good approximation for the farther 3dmetal sites, as the comparison between our results and RIXS data shows. An extension of our embedding scheme toward the construction of openshell embeddings is planned for the near future.
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Acknowledgements
We thank V. Bisogni, L. Braicovich, G. Ghiringhelli, K. Wohlfeld and V. Yushankhai for fruitful discussions. L. H. acknowledges financial suport from the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG).
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L.H., P.F. and J.v.d.B. wrote the main manuscript text, L.H. and L.S. prepared figure 1. All authors reviewed the manuscript.
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Hozoi, L., Siurakshina, L., Fulde, P. et al. Ab Initio determination of Cu 3d orbital energies in layered copper oxides. Sci Rep 1, 65 (2011). https://doi.org/10.1038/srep00065
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DOI: https://doi.org/10.1038/srep00065
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