Main

Although it has long been suspected that spin fluctuations play a crucial role in the extraordinary electronic properties of cuprates, important aspects of the antiferromagnetic interactions in these systems remain unclear. Key questions concern the reasons behind the very large AF couplings observed in cuprates, and how they can be accommodated within the accepted superexchange mechanism1,2, as well as the role of apical oxygens. The strength of the AF coupling can be characterized by the nearest-neighbour (NN) coupling J, which enters in, for example, the Heisenberg model for undoped systems and the t–J model for doped systems3. Superexchange involves (virtual) electron hopping between Cu 3d and O 2p orbitals. Antiparallel spins on neighbouring Cu atoms allow for more hopping possibilities than parallel spins, leading to an AF coupling2,4. While the superexchange is well understood at the model level, the ab initio calculation of J is a major problem. For instance, calculations in a minimal though physically plausible active space underestimate J by almost an order of magnitude. This is therefore a long-standing problem in the ab initio community5,6,7,8,9,10.

An exact wavefunction (WF)-based calculation11 within the NN J model would involve correlating ~100 electrons among ~300 orbitals, leading to an eigenvalue problem in a Hilbert space of 10115 determinants. Since problems on such a scale are out of reach, we use the complete active space self-consistent field (CASSCF) method12,13,14 together with multi-reference perturbation theories to systematically approximate the correlation energy. In the CASSCF(n,m) approach, a subset of n active electrons are correlated in a subset of m active orbitals, leading to a highly multi-configurational (CAS) reference WF. The choice of the active space will be discussed shortly, but let us note that, although the solution of the CAS WF is still an exponentially scaling problem, it is manageable with novel quantum chemistry methods, namely with full configuration interaction quantum Monte Carlo (FCIQMC)15,16 and density-matrix renormalization group (DMRG)14,17, as long as n and m are not too large. Additionally in the SCF process, all orbitals are self-consistently optimized in the field of the CAS WF, to yield the variational minimum. The CAS WF is then augmented using a number of second-order techniques, including n-electron perturbation theory (NEVPT2)18,19, multi-reference linearized coupled cluster (MR-LCC2)19,20 or multi-reference configuration interaction with single and double excitations (MR-CISD)11. These methods capture the remaining (weak) correlation involving electrons and orbitals outside of the active space. Using such approaches, it is possible, for example, to justify that the lowest electron removal state of La2CuO4 is the Zhang–Rice singlet state21 and study it in detail22. We use a variety of second-order methods to gauge their reliability. As the active space is enlarged, the corresponding second-order corrections diminish. The key question that arises is: What is the ‘minimal’ active space necessary to obtain a qualitatively correct reference WF sufficient to compute J reliably? We find that the necessary active space needs to be far larger than previously imagined, including relatively high-energy Cu 4d and O 3p orbitals. Exclusion of these from the active space leads to a dramatic underestimation of J.

We analyse the reason for the strong dependence of J on the active space and, in particular, the importance of 4d orbitals. As mentioned above, the superexchange mechanism depends on O–Cu hopping. The Coulomb energy cost Ueff of this hopping is strongly reduced by an expansion of the Cu 3d orbitals, referred to as breathing23, when an electron hops into a Cu 3d orbital. This breathing effect at the same time increases teff, the Cu–O effective hopping integral24. In a similar way, the O 2p orbital breathes as the O occupancy is changed. In the superexchange mechanism, J depends on both Ueff and teff (Supplementary Note 4) and the breathing effects therefore strongly influence J, as we shall shortly show.

The breathing effect involves a single 3d → 4d excitation, leading to an expansion of the charge density when an electron is added to the d shell. There are also important double 3d3d → 4d4d excitations, which provide radial (in–out) correlations11. For a fixed number of d electrons, these correlations lead to a contraction of the charge density, at least if the basis has sufficient flexibility to satisfy the virial theorem. Correlation and breathing compete, making their simultaneous description complicated. Both effects lead to occupancy of 4d orbitals, but they are otherwise very different, and the ab initio calculations need to have the flexibility to capture both effects in a balanced way.

Ab initio calculations

To study the electronic structure of cuprates, we employ the embedded cluster model. With this approach, accurate high-level calculations are performed for a small representative unit of the solid, while its environment is treated in a more approximate manner25. The details of the employed model are presented in Methods section.

We first perform CASSCF calculations with two singly occupied Cu \(3{d}_{{x}^{2}-{y}^{2}}\) orbitals in the active space CASSCF(2,2), similar to the one-band Hubbard model. Such minimal active-space calculations account for the unscreened Anderson superexchange mechanism (d9−d9 and d8−d10 configurations) and give a qualitatively correct AF J coupling. The value of the J obtained this way is, however, only ~20% of experimental data26,27,28,29 (Table 1). As can be seen, the second-order corrections nearly double J, but are clearly insufficient. The uniform behaviour of the different dynamical correlation methods suggests that the extended Hilbert space (CASSCF(2,2) reference WF plus second-order perturbation) is inadequate to qualitatively describe the system. Enlarging the reference WF represents a natural remedy to this problem. The only exception is the difference-dedicated configuration interaction (DDCI) method, which gives J values very close to experiment on top of CASSCF(2,2) reference8,30. However, the DDCI is essentially a subspace of the MR-CISD, and significant differences of J calculated by these two methods imply that the description of electronic structure given by DDCI is far from being complete.

Table 1 Values of the superexchange parameters J (meV) obtained with different methods
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Because an electron hopping from the bridging O σ-bonding 2py to the Cu \(3{d}_{{x}^{2}-{y}^{2}}\) orbital plays a crucial role in the superexchange (see, for example, ref. 4), this orbital is an obvious candidate to add into the active space. Such CASSCF(4,3) calculations roughly correspond to an unscreened multi-band Hubbard model. However, the obtained magnetic couplings turn out to be less than 1 meV higher compared with CASSCF(2,2). The reason is that, despite the inclusion of important ligand-hole determinants (d9–p5–d10 and d10–p4–d10), their energies are too high to be effective because the orbital optimization is primarily driven by the dominant d9–p6–d9 configuration5,7,30. When we include the effect of further excited determinants at second-order level on top of the CASSCF(4,3) WF, J is still much smaller than the experimental value, indicating that important correlation effects are still missing.

To effectively lower the energy of d9–p5–d10, d8–p6–d10 and d10–p4–d10 determinants, one has to take into account orbital relaxation that comes along with them5,7. This can be done by adding a proper set of orbitals previously kept empty to the active space, namely Cu 4d and O 3p. Having additional d orbitals in the active space has been shown to be necessary to describe multiplet splittings for the late transition metals of the first row (see, for example, refs. 31,32,33).

Because of the variationality of the orbital optimization within the CASSCF procedure, the active orbitals are allowed to change, and a balanced choice of active space is required to ensure convergence. Such a balanced active space can be constructed with Cu 3d and 4d orbitals with eg character plus the bridging oxygen 2py and 3py orbitals7 (Fig. 1, top three rows). Results for such CASSCF(8,10) calculations are shown in the third block of Table 1. The extension of the active space leads to a systematic differential effect with J increasing significantly at all levels of theory. Results close to experiment were reported using this active space together with a different formulation of the perturbation theory7, but our calculation results are about 80% of experimental values. To achieve a balanced description of all relevant effects, we consider all copper 3d and 4d orbitals, together with the bridging oxygen 2p and 3p orbitals, resulting in CASSCF(24,26). This active space yields a diagonalization problem in a space of ~1014 Slater determinants, which we treat with DMRG and FCIQMC as approximate solvers13,14. With the additional many-body contribution from the Cu t2g and π-bonding O orbitals taken into account in the large CASSCF, we find further stabilization of the singlet compared with the triplet. Second-order correction on top of the CASSCF(24,26) reference finally brings J close to the experimental values (see corresponding block in Table 1). The active orbitals are shown in Fig. 1; note that both 3d and 4d orbitals have significant amplitudes at the bridging oxygen p orbitals.

Fig. 1: Orbitals explicitly correlated and optimized in CASSCF(28,30) calculations for the singlet state of La2CuO4 compound.
figure 1

Orbitals active in CASSCF(2,2), CASSCF(8,10) and CASSCF(24,26) calculations are indicated by smaller rectangles. Pink and green (yellow and blue) colours indicate phases of orbitals that are occupied (empty) at CASSCF(2,2) level.

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To verify that J values are converged with respect to the active space size, we performed even larger computations. We further correlated the peripheral O x2 − y2-like 2p and 3p orbitals, the latter being strongly mixed with Cu 4s (Fig. 1). The last block in Table 1 shows the results of these CASSCF(28,30) calculations. The obtained J values are indeed close to the CASSCF(24,26) ones, with a slightly larger fraction being captured by CASSCF itself and less by the perturbation theory correction.

In a simple theory of superexchange1,2, a model of Cu2O is treated with one non-degenerate orbital on each atom. As discussed above, including only these orbitals in CASSCF(4,3) underestimates J by almost one order of magnitude. We now discuss why it is necessary to consider the large active space.

Effective model

It is instructive to discuss the breathing effect in a very simple model, with an effective hopping directly between two Cu atoms that simulates the actual superexchange via bridging O. We show how the radial extent of the Cu 3d orbital is effectively increased in intermediate states with increased 3d occupancy. This has two important consequences. First, the effective energy cost of increasing the occupancy of 3d level is reduced, since the electrons can avoid each other better23. Second, the hopping between the two sites is enhanced, as the Cu 3d orbital expands24.

As in the CASSCF calculations, we use a fixed orthogonal basis set for all intermediate states. Therefore the breathing effect of a 3d orbital is described as a mixing of the 3d and 4d orbitals. The system can effectively expand or contract an effective 3d orbital, being a linear combination of a 3d and a 4d orbital, depending on their relative sign. To illustrate how this happens, we consider a Cu2 dimer, including just one 3d and one 4d level on each atom, as indicated in Fig. 2. The levels have spin but no orbital degeneracy. We use the Hamiltonian

$$\begin{array}{lll}H&=&\mathop{\sum }\limits_{\sigma }\left[\mathop{\sum }\limits_{L}^{\mathrm{A,B}}\,\mathop{\sum }\limits_{i=1}^{2}{\varepsilon }_{i}{n}_{Li\sigma }+\,\mathop{\sum }\limits_{i=1}^{2}\mathop{\sum }\limits_{j=1}^{2}\,{t}_{ij}\,\left({c}_{\mathrm{A}i\sigma }^{{\dagger} }{c}_{\mathrm{B}j\sigma }^{}+\,{c}_{\mathrm{B}i\sigma }^{{\dagger} }{c}_{\mathrm{A}j\sigma }^{}\right)\right]\\ &&+\ \mathop{\sum }\limits_{L}^{\mathrm{A,B}}\left[{U}_{11}{n}_{L1\uparrow }{n}_{L1\downarrow }+{U}_{22}{n}_{L2\uparrow }{n}_{L2\downarrow }+{U}_{12}\mathop{\sum}\limits_{\sigma \sigma ^{\prime} }{n}_{L1\sigma }{n}_{L2\sigma ^{\prime} }\right]\\ &&+\ \mathop{\sum }\limits_{L}^{\mathrm{A,B}}\mathop{\sum}\limits_{\sigma }\left({K}_{1}{n}_{L1\sigma }+{K}_{2}{n}_{L2\sigma }\right)\left({c}_{L1\bar{\sigma }}^{{\dagger} }{c}_{L2\bar{\sigma }}^{}+{c}_{L2\bar{\sigma }}^{{\dagger} }{c}_{L1\bar{\sigma }}^{}\right).\end{array}$$
(1)

The first index on cLiσ refers to the site, and the second labels the orbital. That is, i = 1(2) refers to a 3d (4d) orbital. The hopping between the Cu atoms is described by tij. We include the direct on-site Coulomb integrals U11, U12 and U22, describing 3d–3d, 3d–4d and 4d–4d interaction, respectively. Ki refers to a Coulomb integral with three equal orbitals and the fourth different:

$${K}_{i}={e}^{2}\int {\mathrm{d}}^{3}r\int {\mathrm{d}}^{3}r^{\prime} \frac{{\phi }_{i}{({{{\bf{r}}}})}^{2}{\phi }_{1}({{{{\bf{r}}}}}^{\prime}){\phi }_{2}({{{{\bf{r}}}}}^{\prime})}{| {{{\bf{r}}}}-{{{\bf{r}}}}^{\prime} | }.$$
(2)

These integrals are crucial for the breathing effect. If, for example, the 3d orbital on an atom is doubly occupied, the last term in equation (1) can excite a single electron from the 3d orbital ϕ1 to the 4d orbital ϕ2. The effect of breathing is already evident in the atom (Supplementary Note 3). For simplicity, we here put \({t}_{12}={t}_{21}=-\sqrt{{t}_{11}{t}_{22}}\), \({U}_{12}=\sqrt{{U}_{11}{U}_{22}}\) and \({K}_{1}/{K}_{2}=\sqrt{{U}_{11}/{U}_{22}}\). We have used ε2 − ε1 = 24 eV, U11 = 13 eV, U22 = 10 eV, K1 = −8 eV, t11 = −0.5 eV and t22 = −0.8 eV.

Fig. 2
figure 2

Schematic representation of states in equation (3).

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Table 2 presents the singlet–triplet splitting obtained by solving the Hamiltonian in equation (1). It illustrates how inclusion of the integral K1 strongly increases the splitting, because of breathing effects. To understand these results better, we consider a simpler calculation within only three configurations for the singlet state:

$$\begin{array}{lll}\left|1\right\rangle &=&\frac{1}{\sqrt{2}}\left({c}_{\mathrm{A}1\uparrow }^{{\dagger} }{c}_{\mathrm{B}1\downarrow }^{{\dagger} }+{c}_{\mathrm{B}1\uparrow }^{{\dagger} }{c}_{\mathrm{A}1\downarrow }^{{\dagger} }\right)\left|{{{\rm{vac}}}}\right\rangle \\ \left|2\right\rangle &=&\frac{1}{\sqrt{2}}\left({c}_{\mathrm{A}1\uparrow }^{{\dagger} }{c}_{\mathrm{A}1\downarrow }^{{\dagger} }+{c}_{\mathrm{B}1\uparrow }^{{\dagger} }{c}_{\mathrm{B}1\downarrow }^{{\dagger} }\right)\left|{{{\rm{vac}}}}\right\rangle \\ \left|3\right\rangle &=&\frac{1}{2}\left({c}_{\mathrm{A}1\uparrow }^{{\dagger} }{c}_{\mathrm{A}2\downarrow }^{{\dagger} }+{c}_{\mathrm{A}2\uparrow }^{{\dagger} }{c}_{\mathrm{A}1\downarrow }^{{\dagger} }+\right.\\ &&\hspace{1em}\left.{c}_{\mathrm{B}1\uparrow }^{{\dagger} }{c}_{\mathrm{B}2\downarrow }^{{\dagger} }+{c}_{\mathrm{B}2\uparrow }^{{\dagger} }{c}_{\mathrm{B}1\downarrow }^{{\dagger} }\right)\left|{{{\rm{vac}}}}\right\rangle ,\end{array}$$
(3)

where \(\left|{{{\rm{vac}}}}\right\rangle\) is the vacuum state with no electrons. These configurations are shown schematically in Fig. 2. \(\left|1\right\rangle\) corresponds to the d9–p6–d9 configuration mentioned above, while \(\left|2\right\rangle\) and \(\left|3\right\rangle\) resemble the d8–p6–d10 configuration without and with 4d occupation, respectively. The Hamiltonian in equation (1) within the basis given by equation (3) reads

$$H=\left(\begin{array}{ccc}2{\varepsilon }_{1}&2{t}_{11}&\sqrt{2}{t}_{12}\\ 2{t}_{11}&2{\varepsilon }_{1}+{U}_{11}&\sqrt{2}{K}_{1}\\ \sqrt{2}{t}_{12}&\sqrt{2}{K}_{1}&{\varepsilon }_{1}+{\varepsilon }_{2}+{U}_{12}\end{array}\right).$$
(4)

Diagonalizing this matrix, we obtain the second column of Table 2. These results agree rather well with the full calculation for the model in equation (1), although the basis set in equation (3) is incomplete. The splitting is smaller because the higher-energy configurations have been neglected.

Table 2 Triplet–singlet splitting without (K1 = 0) and with breathing (all values in eV)
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We can now use Löwdin folding, focusing on the upper 2 × 2 corner of (H − z)−1

$${(H-z)}^{-1}={\left(\begin{array}{ll}2{\varepsilon }_{1}-2{t}_{12}^{2}/{{\Delta }}E-z&\ 2{t}_{11}-2{t}_{12}{K}_{1}/{{\Delta }}E\\ 2{t}_{11}-2{t}_{12}{K}_{1}/{{\Delta }}E&\ {U}_{11}-2{K}_{1}^{2}/{{\Delta }}E\end{array}\right)}^{-1},$$
(5)

where ΔE = ε2 − ε1 + U12 and we have introduced the approximation z ≈ 2ε1 in some places. The matrix in equation (5) shows rather clearly that there is an interference between breathing and hopping from the 3d orbital on one site and the 4d orbital on the other site.

The effective value of U has now been reduced,

$${U}_{11}\to {U}_{11}^{{{{\rm{eff}}}}}\equiv {U}_{11}-2\frac{{K}_{1}^{2}}{{{\Delta }}E},$$
(6)

while the effective hopping has been increased,

$${t}_{11}\to {t}_{11}^{{{{\rm{eff}}}}}\equiv {t}_{11}-\frac{{t}_{12}{K}_{1}}{{{\Delta }}E},$$
(7)

since K1 < 0 and t11 and t12 have the same sign. For the triplet case, the basis state \(\left|2\right\rangle\) does not exist, and these renormalization effects are not present. The singlet–triplet splitting is then

$${E}_{\mathrm{T}}-{E}_{\mathrm{S}}\approx \frac{4{\left({t}_{11}-{t}_{12}{K}_{1}/{{\Delta }}E\right)}^{2}}{{U}_{11}-2{K}_{1}^{2}/{{\Delta }}E}\equiv 4\frac{{\left({t}_{11}^{{{{\rm{eff}}}}}\right)}^{2}}{{U}_{11}^{{{{\rm{eff}}}}}}.$$
(8)

This illustrates the importance of the renormalization of both U11 and t11 by the breathing effect.

The presented model only includes non-degenerate d orbitals. Including the full five-fold degeneracy increases the renormalization of U by approximately a factor of five. The model has only 3d–3d hopping, but it illustrates the breathing effects. The more realistic case of 3d–2p–3d hopping results in a more complex expression for J where renormalization of t and U cannot be disentangled (Supplementary Note 4).

Reduction of U

The calculated bare on-site Coulomb integral between two 3d electrons is very large (~28 eV), leading to drastically suppressed charge fluctuations in the simplest model. For this reason, the CASSCF(2,2) and CASSCF(4,3) calculations give a very small J. However, by increasing the active space size, this energy cost can be strongly reduced. Crucial effects are the change of the effective radial extent of the 3d orbital (breathing) and rearrangements of the non-3d charge density as the number of 3d electrons varies (screening)23, which are captured in the CASSCF(24,26) calculation with second-order correction.

To disentangle these different effects, we performed a series of simpler, constrained calculations34. We put all hopping integrals from d (3d or 4d) basis functions on the Cu atoms equal to zero. We can then prescribe the total occupancy of d orbitals on each Cu atoms. We performed two calculations, one with the configuration d9–p6–d9 and one with d8–p6–d10. In both cases, the system is allowed to relax fully, except that hopping to or from d orbitals is suppressed. We then obtain that the energy of the d8–p6–d10 state is higher than the d9–p6–d9 by 10–13 eV depending on initial conditions of constrained calculations (Supplementary Note 2). This means that bare U ≈ 28 eV has been reduced to Ueff ≈ 10 eV. Experimental35 and theoretical estimates36,37,38,39 suggest that Ueff is reduced even further (~8 eV). This may be owing to more long-ranged effects left out in our finite-size cluster calculation.

Figure 3 shows charge differences due to breathing and screening for the d8–p6–d10 calculation, discussed above. A calculation was first performed for the d9–p6–d9 state, then a d electron was moved from one Cu atom to the other, keeping all orbitals unchanged. The corresponding densities at two copper sites are denoted \({\rho }_{({d}^{8})}\) and \({\rho }_{({d}^{10})}\). This d8–p6–d10 state is then allowed to relax self-consistently, giving the densities \({\rho }_{{d}^{8}}\) and \({\rho }_{{d}^{10}}\). The solid red curve in Fig. 3b shows the change in the charge density \({\rho }_{{d}^{10}}-{\rho }_{({d}^{10})}\), illustrating how charge is moved from the inner part of Cu to the outer part (breathing). The dashed red curve shows the radial integral of the charge density difference, revealing that more charge is removed from the inner part than is added to the outer part. Since the number of d electrons is the same in the two calculations, non-d charge has been moved away from the Cu atom with the d10 configurations as a response to the addition of one d electron (screening). Adding a d electron to a Cu atom thus only leads to an increase of the net charge by about half an electron, because of screening, which substantially reduces the energy cost.

Fig. 3: Electron density difference (in atomic units) due to orbital relaxation in the d8–p6–d10 configuration.
figure 3

a, Electron density difference in the plane of CuO4 plaquettes. b, Electron density difference integrated over a sphere centred on one of the Cu atoms (full curves) as a function of the radius shown in a; the result of an additional radial integration (dashed curves) as a function of the upper integration limit.

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Dependence of J on position of apical oxygens

As seen in Table 1, the magnetic coupling in Sr2CuO3 is nearly two times larger than that in La2CuO4. In both cases, the computation of J is done using only two magnetic centres, so this difference should not be attributed to the dimensionality of the two materials. The other structural difference is the presence of apical oxygen ions in La2CuO4, which changes the local multiplet splittings, mainly the position of \(3{d}_{{z}^{2}}\) levels26,40,41,42. The relative energy of the \(3{d}_{{z}^{2}}\) orbital is believed to be connected to the shape of the Fermi surface and the value of the critical temperature in doped cuprates43,44,45,46.

There is experimental evidence that J also changes depending on the local geometry47. However, because different compounds have to be used experimentally, local distortions are accompanied by changes of Cu–O distances and type of adjacent metal ions. Therefore, it is instructive to investigate the dependence of J on the distance to apical oxygen ions in La2CuO4 compound with an accurate computational method. We varied the apical O’s positions within the cluster while keeping the electrostatic potential untouched and computed magnetic couplings using the procedure described above. The results of these calculations are presented in Fig. 4. It can be seen that, with increase of the distance to apical oxygen, the NN J grows. Moreover, the growth is faster when more electron correlation is taken into account. One obvious effect that leads to an increase of J is the lowering of the \(4{d}_{{z}^{2}}\) orbital energy and corresponding enhancement of the orbital breathing. We observe 13% growth of the occupation of \(4{d}_{{z}^{2}}\) orbitals upon 0.8 Å displacement of apical oxygens at the CASSCF(24,26) level.

Fig. 4
figure 4

Dependence of J on the distance to apical oxygens in La2CuO4.

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The computational strategy presented and justified here was recently used to predict the superexchange strength in infinite-layer nickelate compounds48 preceding the consistent experimental studies49,50.

Methods

We use clusters that include two CuO4 (CuO6) units, two (ten) neighbouring Cu2+ ions and all adjacent Sr2+ (La3+) ions, in total [Cu4O7Sr16] and [Cu12O11La16] for Sr2CuO3 and La2CuO4, respectively (Supplementary Fig. 1). The rest of the solid is modelled by an array of point charges fitted to reproduce the Madelung potential in the cluster region51,52,53. Further details of the embedded cluster approach including possible improvements54,55,56,57,58,59,60 are presented in Supplementary Note 2. We employed crystal structures as reported experimentally61,62. The value of the NN superexchange parameter can be easily extracted by mapping the energy spectrum of the two-magnetic-site cluster to two-site Heisenberg model. To make this mapping straightforward, the peripheral Cu ions are represented by total-ion potentials with no associated electrons, such that J can be extracted as the energy difference of the lowest triplet and singles states25. We use all-electron cc-pVDZ and cc-pVTZ basis sets for central Cu and O ions63,64 and large-core effective potentials for other species65,66,67. We utilize several quantum chemistry computational packages68,69,70,71. Small CASSCF calculations up to (8,10) active space were done with OpenMolcas, Molpro and PySCF programs69,70,71. Results by different codes are fully consistent with differences in total energies of no more than 10−6 Hartree. All NEVPT2 and MR-LCC2 calculations were carried out with IC-MPS-PT and StackBlock programs17,19. CASPT2 calculations were performed with OpenMolcas69. MRCI-SD and DDCI calculations were done with the ‘mrcic’72 and ‘mrci’73 modules of Molpro71, respectively. Large CASSCF(24,26) calculations were carried out with OpenMolcas69 using NECI68,74,75 as FCIQMC solver13 and independently with PySCF70 using StackBlock as DMRG solver17. The largest walker number in FCIQMC calculations was set to 107. DMRG calculations were carried out with bond dimension M = 3,000. Bond dimension M = 6,000 was used for the uncontracted part of NEVPT2 and MR-LCC2 calculations. To prevent Cu 3s, 3p and bridging O 2s orbitals from entering the active space in CASSCF(28,30) calculations, we keep all orbitals below O 2p frozen during the SCF procedure. Further details are provided in Supplementary Methods.

Data shown in Fig. 3 were obtained in constrained calculations using the generalized active space SCF (GASSCF) method34 as implemented in OpenMolcas69. To perform them, we divide the starting atomic-like orbitals into three groups: all d orbitals at the first copper ion (15 in cc-pVDZ basis), all d orbitals at the second copper ion (15) and the rest. Any orbital rotation between these groups is forbidden via ‘super-symmetry’ constraint. With GASSCF, we specify two disconnected active spaces, for example, (8,5) and (10,5) for the first and second Cu ion, respectively. This way, it is possible to fix the occupation of d orbitals at each site and perform all possible remaining optimizations. The results of GASSCF calculations were cross-checked using ANO-L-VDZP basis set76,77. More detail is provided in Supplementary Note 1.

The bare on-site Coulomb interaction between two 3d electrons of U ≈ 28 eV was computed as \(U\,=\,2{E}_{{d}^{9}}\,-\,{E}_{{d}^{8}}\,-\,{E}_{{d}^{10}}\) for isolated Cu ion at CASCI level with nine, eight and ten electrons in five orbitals optimized for the d9 state. The corresponding Coulomb integral \(< 3d3d\parallel 3d3d >\) was 29.8 eV in these calculations.

Data for the density plots were obtained using the Multiwfn program78. Molecular orbitals and geometries were plotted using Jmol79 and renmol80.