Introduction

It is generally accepted that the low-energy physics of the layered Cu oxide compounds in their normal state is reasonably well described by models which incorporate the “in-plane” Cu and O 2px /2py 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 CuO2 layers and the apical O 2pz functions having σ-type overlap with the Cu . Recent multi-orbital calculations using dynamical mean-field theory1 show indeed that some of the features of the optical, x-ray absorption and photoemission spectra can be better reproduced when the Cu orbitals are explicitly included in the many-body treatment. At finite doping, the inclusion of the Cu functions makes a difference even for the low-energy states close to the Fermi level1.

The off-diagonal coupling between states of x2−y2 and z2 symmetry was actually found to substantially affect the dispersion of the low-energy bands and the shape of the Fermi surface in earlier semiphenomenological models2,3, density-functional calculations4 and quantum chemical studies5. Moreover, Ohta et al.6 and recently Sakakibara et al.7 suggested that a direct relation exists between the magnitude of Tc and the size of the splitting. The splittings within the Cu 3d shell are also relevant to excitonic models for pairing and high-Tc superconductivity8,9. 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 La2CuO4 and Sr2CuO2Cl2 were initially assigned to crystal-field Cu to charge excitations10. A different interpretation in terms of magnetic excitations was proposed by Lorenzana and Sawatzky11 and latter on confirmed by analysis of the resonant inelastic x-ray scattering (RIXS) spectra at the Cu K and L3-edge12,13. The RIXS experiments also show that in La2CuO4 and Sr2CuO2Cl2 the Cu to transitions occur at 1.5–2.0 eV14, which is substantially larger than the outcome of earlier wavefunction-based quantum chemical calculations, 1.0–1.2 eV15, or density-functional estimates, 0.9 eV7.

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 La2CuO4, Sr2CuO2Cl2 and CaCuO2 with recent RIXS measurements14. Further, the to excitation energies computed here for La2CuO4, YBa2Cu3O6 and HgBa2CuO4 are relevant to models which attempt to establish a direct relation between the relative energy of the out-of-plane level and the critical temperature Tc7. In particular, the large difference between the critical superconducting temperatures of doped La2CuO4 and HgBa2CuO4 was directly attributed to a large difference between the to excitation energies7.

Results

To study bound, excitoniclike states such as the d-d charge excitations in copper oxides, we rely on real-space ab initio methods. In the spirit of modern multi-scale electronic-structure approaches, we describe a given region around a central Cu site by advanced quantum chemical many-body techniques while the remaining part of the solid is modeled at the Hartree-Fock level. The complete-active-space self-consistent-field (CASSCF) method was used to generate multireference wavefunctions for further configuration-interaction (CI) calculations16. 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 CuO4 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 software17.

Both SDCI and RIXS results for the Cu d-level splittings are listed in Table 1. The relative energies of the peaks observed between 1 and 3 eV in the Cu L3-edge RIXS spectra14 are the sum of a crystal-field contribution, i.e., an on-site crystal-field splitting Ecf and a magnetic term ΔEmgn. The quantum chemical calculations have been performed to extract Ecf 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). ΔEmgn accounts for AF order in the ground-state 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 CuO4 plaquettes. For CaCuO2, for example, we find J = 0.13 eV, in good agreement with the theoretical results reported in Ref. [18] and with values from experimental data11,13,14. 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 crystal-field 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 t2g 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 ground-state 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 ΔEmgn = ΔE – ΔE′ ≈ 2J representing the magnetic stabilization of the ground-state configuration with respect to the crystal-field excited states. Since J ≈ 0.13 eV, ΔEmgn ≈ 0.26.

Table 1 CASSCF+SDCI versus RIXS results for the Cu d-level splittings in La2CuO4, Sr2CuO2Cl2 and CaCuO2 (eV). The ground-state Cu configuration is taken as reference. A 2J term was here substracted from each of the RIXS values reported in Ref. [14], see text.
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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 x2−y2 and xz/yz levels in CaCuO2, 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 CuO6 octahedron or one CuO5 pyramid was treated at the all-electron 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 z2 levels in La2CuO4 and Sr2CuO2Cl215, for example, are mainly due to such approximations in the modeling of the nearby surroundings. The d-level 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 CuO4 plaquettes, all apical ligands plus the NN closed-shell metal ions of the central polyehdron.

Superconductivity has not been observed in Sr2CuO2Cl2 and CaCuO2. The d-level splittings for three representative cuprate superconductors, i.e., La2CuO4, HgBa2CuO4 and YBa2Cu3O6, are listed in Table 2. The maximum Tc 's achieved by doping in these three materials are 35, 95 and 50 K, respectively. For the YBa2Cu3O6 compound, we here refer to the maximum Tc which can be achieved by Ca doping19. The large difference between the critical temperatures in La2CuO4 and HgBa2CuO4 was assigned in Ref. 7 to a large difference between the relative energies of the z2 states in the two materials. The density-functional results for the splittings between the x2−y2 and z2 levels in La2CuO4 and HgBa2CuO4 are 0.91 and 2.19 eV, respectively7. RIXS data are not available for HgBa2CuO4 and independent estimates for the energy separation between the x2−y2 and z2 states are therefore desirable. While we find a rather similar value for HgBa2CuO4, of 2.09 eV, the quantum chemical and RIXS results14 for La2CuO4 are substantially larger, about 1.4 eV. This makes the difference between the d-level 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.

Table 2 Cu d-level energy splittings for La2CuO4, HgBa2CuO4 and YBa2Cu3O6 (eV). CASSCF+SDCI calculations for 5-plaquette FM clusters.
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The distance between the Cu and apical ligand sites increases from 2.40 Å in La2CuO420 to 2.78 Å in HgBa2CuO421. The effect of this growth of the apical Cu–O bond length on the relative energy of the z2 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 z2 orbital pointing toward those apical ligands. For HgBa2CuO4, the lowest crystal-field excitation is therefore to the xy level and requires about 1.3 eV, see Table 2, while the z2 and xz/yz levels are nearly degenerate and more than 0.5 eV higher in energy. On the other hand, in La2CuO4 the lowest crystal-field excitation is to the z2 orbital, see Table 1. Our results also reproduce the near degeneracy between the z2 and xy levels in La2CuO4, as found in the RIXS experiments. In CaCuO2, there are no apical ligands. The splitting between the x2−y2 and z2 levels is therefore the largest for CaCuO2, about 2.4 eV, see Table 1.

Discussion

The parameter that plays the major role in determining the size of the d-level 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 in-plane Cu–O bond lenghts, buckling of the CuO2 planes and the configuration of the farther surroundings. Trends concerning the relative energy of the z2 hole state in different cuprates are illustrated in Fig. 1, which includes data for systems having one apical O site (YBa2Cu3O6), two apical O's (La2CuO4, HgBa2CuO4), two apical Cl ions (Ca2CuO2Cl2, Sr2CuO2Cl2) or no apical ligand (CaCuO2). The apical Cu–O distances in La2CuO4 and YBa2Cu3O6, for example, are nearly the same, 2.40 vs. 2.45 Å19,20,22. In YBa2Cu3O6, however, there is a single apical O. For this reason the z hole state is somewhat destabilized in YBa2Cu3O6 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 x2−y2 hole state is also destabilized such that the splitting between the x2−y2 and z2 levels is finally close to the value found in La2CuO4. Further, the apical Cu–ligand distances are slightly larger in Sr2CuO2Cl2 as compared to HgBa2CuO4, 2.86 vs. 2.78 Å, respectively. The apical ions also have a smaller effective charge in Sr2CuO2Cl2, which should lead to a larger relative energy of the z2 hole state in Sr2CuO2Cl2 as compared to HgBa2CuO4. The fact that the relative energy of the z2 hole state is actually larger in HgBa2CuO4, see Fig. 1, must be related to the smaller in-plane Cu–O distances in HgBa2CuO4, 1.94 in HgBa2CuO4 vs. 1.99 Å in Sr2CuO2Cl2, which stabilizes the ground-state x2−y2 hole configuration in the former compound and to farther structural details. From Ca2CuO2Cl2 to Sr2CuO2Cl2, the Cu–Cl separation increases from 2.75 to 2.86 Å23,24 and the energy of the z2 level from 1.37 to 1.75 eV.

Figure 1
figure 1

Relative energy of the Cu z 2 level as function of the distance h between the Cu and apical ligands in different cuprates.

There are two apical O sites in La2CuO4 and HgBa2CuO4, two apical Cl sites in Ca2CuO2Cl2 and Sr2CuO2Cl2 and one apical O ion in YBa2Cu3O6. In CaCuO2 the apical ligands are absent.

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In contrast to the z2 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 configuration-interaction calculations are carried out on finite clusters including five CuO4 plaquettes plus additional apical ligand and closed-shell metal ion NN's. The localized Wannier functions attached to these atomic sites are obtained from prior Hartree-Fock computations for the periodic system. Excellent agreement is found between our theoretical results and recent Cu L3-edge RIXS data for La2CuO4, Sr2CuO2Cl2 and CaCuO2. 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 3d-metal 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 La2CuO4, YBa2Cu3O6 and HgBa2CuO4 are relevant to models which attempt to establish a direct relation between the critical temperature Tc and the strength of the (x2−y2)−z2 coupling. For La2CuO4, in particular, the density-functional estimate used as input parameter in such models7 is about 0.5 eV smaller than our result. Consequently, the difference we find between the splittings in La2CuO4 and HgBa2CuO4 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 Hartree-Fock (RHF) calculation for the ground-state configuration of the periodic system. The RHF calculations are performed with the crystal package25. We employed experimental lattice parameters14,20,21,22,23,24 and Gaussian-type atomic basis sets, i.e., triple-zeta basis sets from the crystal library for Cu, O and Cl plus basis sets of either double-zeta or triple-zeta quality for the other species. Post Hartree-Fock many-body calculations are subsequently carried out on finite clusters, which are sufficient because of the local character of the correlation hole. They consist of five CuO4 plaquettes, i.e., a “central” CuO4 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 closed-shell metal ions around the “central” Cu site. In La2CuO4, for example, there are ten La3+ NN's. In YBa2Cu3O6, there are one Cu1+ 3d10, four Y3+ and four Ba2+ NN's.

The orbital basis entering the post Hartree-Fock correlation treatment is a set of projected RHF Wannier functions: localized Wannier orbitals (WO's) are first obtained with the Wannier-Boys localization module26 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 five-plaquette fragment . This potential is obtained from the Fock operator in the RHF calculation27 and models the surroundings of the finite cluster, i.e., the remaining of the crystalline lattice.

The central CuO4 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 closed-shell 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 ground-state wavefunction and the lowest four crystal-field excited states at the central Cu site are computed simultaneously in a state-averaged multiroot calculation16. The d-level splittings at the central Cu site are finally obtained at the CASSCF+SDCI level of theory as the relative energies of the crystal-field 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, ∼ 109 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 one-electron Hamiltonian with the help of the crystal-molpro interface program28. 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 3d9 electron configuration, the embedding potentials are obtained by replacing the Cu2+ 3d9 ions by closed-shell Zn2+ 3d10 species. This is a good approximation for the farther 3d-metal sites, as the comparison between our results and RIXS data shows. An extension of our embedding scheme toward the construction of open-shell embeddings is planned for the near future.