DOI: 10.1038/NCHEM.1677 Entangled quantum electronic wavefunctions of the Mn 4 CaO 5 cluster in photosystem II Yuki Kurashige 1 *, Garnet Kin-Lic Chan 2, Takeshi Yanai 1 1 Department of Theoretical and Computational Molecular Science, Institute for Molecular Science, National Institutes of Natural Sciences, Okazaki, Aichi 444-8585, Japan. 2 Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA *Correspondence to: kura@ims.ac.jp Index Section S1. Guide for reading the population diagrams 2 Section S2. Natural orbitals and occupation numbers 2 Section S3. Spin density and spin projections for the S 2 -state 3 Section S4. Interpretation of the pairwise entanglement and assignment of bond character 4 Section S5. Guide for reading the entanglement map 6 Section S6. Entanglement analysis for the XRD and QM/MM structures 7 Section S7. Bond configurational analysis for the XRD structure 7 Figure S1. Mapping of the localized molecular orbitals to the DMRG lattice 11 Figure S2. Population diagrams of the [XRD+H] + model 12 Figure S3. Diagram of the pair-wise orbital entanglement for the S 1 -state 13 Tables S1 S9. Natural orbitals and occupation numbers 14 Table S10. Bonding character of Mn-O 23 Tables S11 S12. Cartesian coordinates of the XRD and XRD+H models 24 Tables S13 S14. Pairwise orbital entanglement matrix 28 NATURE CHEMISTRY www.nature.com/naturechemistry 1
Supporting information: Section S1: Population diagrams (Fig. 1a d). Figure 1a shows that the oxidation state of Mn1 is Mn III, with four singly-occupied 3d orbitals (three Mn1 t 2g orbitals and one Mn1 e g orbital). The other Mn1 e g orbital is almost unoccupied (0.08 electrons). It is covalently coupled to an O3 2p orbital leading to σ bonding and σ * antibonding Mn1-O3 orbitals. In Fig. 1b, Mn2 is assigned the intermediate oxidation state Mn (III )IV, in the sense that all three Mn2 t 2g orbitals are singly occupied and the two Mn2 e g orbitals, which point towards O1,3 and O2,3, respectively, are occupied by 0.37 and 0.22 electrons, respectively. The interaction of these two Mn2 e g orbitals with the ligand O 2p orbitals creates two sets of σ and σ * orbitals between Mn2 and O1 3. We assign the oxidation state of Mn3 as Mn (II )III (Fig. 1c). All three Mn3 t 2g orbitals are singly occupied and the occupation of the two Mn3 e g orbitals pointing to O5 and O2, respectively, are 0.73 electrons and 0.34 electrons, respectively. The two Mn3 e g orbitals, interacting with the coordinated O2 and O5 2p orbitals, respectively, each contribute to forming σ and σ * orbitals (Mn3-O2 and Mn3-O5). Figure 1d indicates that Mn4 is in oxidation state Mn II. All five Mn4 3d are singly occupied orbitals. One of the Mn4 e g orbitals mixes with the O4 2p orbitals perpendicular to Mn4-O4-Mn3 plane, giving rise to π and π * Mn4- O4 orbitals. Section S2: Natural orbitals and occupation numbers. The spinless one-electron reduced density matrix was computed from the DMRG wavefunction during the one-site sweeping procedure and was then diagonalized to obtain natural orbitals and NATURE CHEMISTRY www.nature.com/naturechemistry 2
associated occupation numbers as eigenfunctions and eigenvalues, respectively. The deviation in occupancy from integer values (0, 1, or 2) reflects the strength of electron correlation. In addition to the schematic energy level diagrams of natural orbitals [Fig. 1 ([XRD]), Fig. S2 ([XRD+H] + ), and Fig. 3 (QM/MM)], Tables S1 S5 list the occupation numbers, orbital energies, and bonding character for several models of the OEC; XRD, [XRD+H] +, [XRD] 2-, [XRD+H] 2-, and QM/MM, respectively. We studied these models so as to be able to assign the cluster state of our OEC model. We found that in the S 1 state, some of the O 2p orbitals in the XRD and [XRD+H] + models were not doubly occupied. If we assume that the O 2p orbitals should be doubly occupied at these geometries, this requires us to modify the total cluster charge, and the XRD and [XRD+H] + models then appear to be reduced by two and three electrons (such reduced models correspond to [XRD] 2- and [XRD+H] 2- ). This further supports the hypothesis that at the XRD geometry the cluster is not in the S 1 state. Conversely, if we examine the cluster at the QM/MM geometry, all oxygen 2p orbitals are doubly occupied, confirming that this is the likely candidate structure for the S 1 state. Section S3: Spin density and spin projections for the S 2 -state. The DMRG-CASSCF calculations were performed at several candidate S 2 structures that are different in the protonation states of the waters (W1 and W2) that are ligated to Mn4. These are denoted the W1:H 2 O, W2:H 2 O, W1:OH, W2:H 2 O, W1:H 2 O, W2: OH, and W1: OH, W2: OH model structures. These structures have been proposed earlier in several DFT studies 30,34. The first three structures were proposed in Ref [30] (where they were called models 1, 1d1, and 1d2 ), and the latter structure was proposed in Ref[34]. For the first three structures, we simplified the amino-acid residues to a carboxylate or an imidazole ring (Tables S6 S9). The NATURE CHEMISTRY www.nature.com/naturechemistry 3
def2-tzvpp basis sets for Mn, Ca, O, and N atoms and def2-svp basis sets for C and H atoms were used. 47,51-52 Scalar relativistic effects were included using the Douglas-Kroll-Hess (DKH) Hamiltonian. 53,54 Table 1 shows the spin projections of the DMRG-CASSCF wavefunctions for the S 2 -state (S=1/2, M s =1/2). The spin projections for the each atom were determined by the Mulliken spin population analysis. We also show for comparison the spin projections obtained in Ref[30] by using the 4-site, i.e. Mn1 4, Heisenberg spin model, for which the exchange parameters J were determined by fitting the broken symmetry (BS) DFT energies. Section S4: Interpretation of the pairwise entanglement and assignment of bond character The pairwise orbital entanglement (mutual orbital information) I pq, based on the von Neumann entropy, is a well-known concept in quantum information theory that is used to quantify the entanglement between two (possibly disconnected) regions. This quantity has recently been studied in chemical systems (J. Rissler, R. M. Noack, and S. White, Chem. Phys. 323, 519 (2006); K. Boguslawski, K. H. Marti, O. Legeza, and M. Reiher, J. Chem. Theory Comput. 8, 1970 (2012), K. Boguslawski, P. Tecmer, O. Legeza, and M. Reiher, J. Phys. Chem. Lett. 3, 3129 (2012)). The pairwise orbital entanglement measures the effective number of entangled degrees of freedom between the two orbitals. Some insight into this quantity can be obtained by considering a minimal 2-center 2-electron bond, with atomic orbitals! and!. The maximum value log 4 = 1.39 between two orbitals is achieved in the delocalized covalent molecular orbital type wavefunction! =! 1 +! 1! 2 +! 2 [!!!!!!!! ]. [ MO (covalent-bond) type ] NATURE CHEMISTRY www.nature.com/naturechemistry 4
(I pq = log(4) = 1.39 ) The diradical singlet, valence-bond like wavefunction! = [! 1! 2 +! 1! 2 ][!!!!!!!! ] [ VB (valence-bond) type ] ( I pq = log(2) = 0.69 ) yields a pairwise entanglement of log 2 = 0.69. The wavefunction of an ionic or coordinationbond character! = [! 1! 2 ][!!!!!!!! ] [ionic or coordination-bond type] ( I pq = log(1) = 0 ) yields a pairwise entanglement of log 1 = 0. From the above rigorous argument, when using the localized orbital representation, the magnitude of the orbital entanglement I pq characterizes the bonding type. The full set of pairwise entanglements is given in Tables S13 and S14. Note that the values are smaller than associated with the 2-center 2-electron bonds above, because of the property of entanglement monogamy: if A is entangled with B, that limits the amount of entanglement between B and C. Effectively, the entanglement becomes distributed across the different bonds. Nonetheless, the distinction between different kinds of bonds can still be seen. NATURE CHEMISTRY www.nature.com/naturechemistry 5
The orbital labels refer to the localized orbitals shown in Fig. S1. The entanglement analysis (and indeed any correlation function analysis) depends on the choice of orbitals. We have chosen localized orbitals for our analysis, rather than a delocalized orbital basis (such as the natural orbital basis), because the localized orbitals allows us to identify individual bonds between atoms, rather than delocalized bonds between multiple Mn centers. This is probably the appropriate basis to study a strongly correlated system of localized spins, of which the Mn centers in the OEC are a prime example. Section S5: Guide for reading the entanglement map The diagrams shown in Fig. 5 provide a network-analysis-oriented way of visualizing a matrix I pq. The values of I pq are shown in Tables S13 and S14. As written in the Main text, they represent the strength of the two orbital pair-wise entanglements, computed from reduced density matrices of the high-dimensional DMRG wave function. 1. The diagram shows quantum entanglements of 35 nodes (these look like blobs ). For example, the five nodes labeled Mn1 correspond to the five localized d-type orbitals of the atom Mn1, which are all employed as active space orbitals. The three nodes labeled O1 correspond to the three p-type active-space orbitals of the atom O1. Each single node represents an atomic-like orbital. (It does not directly correspond to the atom itself.) The shapes of the 35 orbitals corresponding to all the nodes are plotted in Fig. S1. 2. The diagrams are drawn using the network exploration and visualization software Gephi [M. Bastian, S. Heymann, and M. Jacomy, International AAAI Conference on Weblogs and NATURE CHEMISTRY www.nature.com/naturechemistry 6
Social Media, 2009; http://gephi.org/]. The thickness of the lines between nodes p and q is proportional to the mutual information matrix element I pq. 3. The placement of the nodes in the diagram of Fig. 5 was determined using the forcedirected layout algorithm Force Atlas implemented in Gephi. 4. The layout algorithm forms clusters when each node has a high connectivity (i.e. many elements of I pq emanating from that node are significant). The cluster of Mn orbitals on each atom are closely connected, as shown in Fig. 5. This arises from quantum spin fluctuations on each Mn atom that are neglected in the usual spin models used to interpret the magnetism. 5. Several of the Mn-O orbitals display rather thick lines and this illustrates significant bonding. Medium lines indicates singlet diradical bonding. The changes in these lines as a function of the Kok cycle is described in the Main Text. Section S6: Entanglement analysis for the XRD and QM/MM structures (the S 1 state) The calculation of the pairwise orbital entanglement (mutual information) matrix I pq in the localized orbital representation was also performed on the XRD and QM/MM structures for the S 1 state. The corresponding entanglement map is shown in Fig. S3. As we can see the map is quite sensitive to the change in moving from the XRD structure to the refined S1 structure of the QM/MM calculation, illustrating the changes in the bonding of the Mn3, Mn4, and O5 atoms. Section S7: Bond configurational analysis for the XRD structure (the S 1 state) The configurational characters of the bonds in Table S10 are computed from the one- and twoparticle density matrix for the S 1 XRD structure. The table displays the bond character of the six NATURE CHEMISTRY www.nature.com/naturechemistry 7
pairs of bonding σ (and π) and anti-bonding σ * (and π * ) orbitals formed between the Mn 3d and O 2p orbitals. The occupancy of each of the orbital pairs is decomposed into spin configurations with a ratio. For example, for a given bonding orbital! we can define a corresponding creation and annihilation operator!!,! respectively. Then the (unnormalized) weight of the bonding orbital configuration corresponds to the two-particle density matrix element!!!!!!. Similarly, creation and annihilation operators can be defined for the anti-bonding orbital and thus the relative weights of all the (two-electron) configurations shown in Table S10 can be found. Note that the natural orbitals are employed for defining the bonding and anti-bonding orbitals,! and!, respectively. The relative weights of the non-two-electron configurations can be determined from the sum of one-particle density!! +!! by assuming that there are more than two electrons in the all configurations. Overall, the main weight is from anti-parallel spin configurations, such as doubly-occupied σ (and σ*) and the excited configuration where σ and σ* are singly-occupied. The former is the standard closed-shell bond, while the latter mixes in if the bond is stretched. Overall, these might be viewed as normal singlet bonding configurations. However, a relatively large weight is observed also in parallel spin configurations ( or ) with two electrons occupying σ and σ*, respectively, e.g. a ratio of 25% for π/π* Mn4 O4. This admixture of anti-parallel and parallel spins, is a many-electron correlation effect. The large weight of the parallel spin component seems to arise from the nontrivial interaction with the other electron-rich Mn orbital sites, which donate the same spin. This spin transfer is not a simple charge transfer in a one-particle picture (since the densities of the Mn atoms are not affected) but can be attributed to many-body (quantum or multi-configurational) effects. These spin interactions cannot be described by NATURE CHEMISTRY www.nature.com/naturechemistry 8
generalized valence bond theory, and require a more highly entangled wave function treatment as we have carried out. NATURE CHEMISTRY www.nature.com/naturechemistry 9
References 51. D. A. Pantazis, X.-Y. Chen, C. R. Landis, F. Neese, All-electron scalar relativistic basis sets for third-row transition metal atoms, J Chem Theory Comput 4, 908 919 (2008). 52. F. Weigend, Hartree Fock exchange fitting basis sets for H to Rn, J Comput Chem 29, 167 175 (2007). 53. G. Jansen, B. A. Hess, Revision of the Douglas-Kroll transformation, Phys. Rev. A 39, 6016 6017 (1989). 54. B. Hess, Relativistic electronic-structure calculations employing a two-component no-pair formalism with external-field projection operators, Phys. Rev. A 33, 3742 3748 (1986). NATURE CHEMISTRY www.nature.com/naturechemistry 10
Fig. S1. Mapping of the localized molecular orbitals to the 1D lattice of the DMRG algorithm. These orbitals are obtained by the DMRG-CASSCF procedure. In these plots, the amino ligands and waters are omitted for clarity. NATURE CHEMISTRY www.nature.com/naturechemistry 11
Fig. S2. Orbital energy levels and electron occupancies of the four Mn sites for [XRD+H] + model: (a) Mn1, (b) Mn2, (c) Mn3, (d) Mn4. The labeling follows Fig. 1 (XRD model). NATURE CHEMISTRY www.nature.com/naturechemistry 12
DOI: 10.1038/NCHEM.1677 Fig. S3. Diagram of the pair-wise orbital entanglement of the XRD structure (left) and QM/MM refined structure (right) for the S1 state. 13 13 NATURE CHEMISTRY www.nature.com/naturechemistry
Table S1. Natural orbitals (NO) in the Mn 3d and µ-oxo 2p active space for the XRD model. The electron occupations, energy levels, and characters are presented. The orbital energies (E h ) are given by the diagonal elements of the Fock matrix. NO occupation energy(e h ) character 1 9 2.00 1.95 0.592 0.524 O1 5 2p 10 1.94 0.585 σ Mn1-O3 11 1.80 0.483 σ Mn2-O2,3 12 1.69 0.415 σ Mn3-O2 13 1.66 0.424 σ Mn2-O1,3 14 1.28 0.268 σ Mn3-O5 15 1.14 0.198 π Mn4-O4 16 29 1.07 0.94 0.213 0.048 Mn1 4 3d 30 0.86 0.092 π * Mn4-O4 31 0.73 0.080 σ * Mn3-O5 32 0.37 0.015 σ * Mn2-O1,3 33 0.34 0.084 σ * Mn3-O2 34 0.22 0.050 σ * Mn2-O2,3 35 0.08 0.209 σ * Mn1-O3 NATURE CHEMISTRY www.nature.com/naturechemistry 14
Table S2. Natural orbitals (NO) in the Mn 3d and µ-oxo 2p active space for the [XRD+H] + model. The electron occupations, energy levels, and characters are presented. The orbital energies (E h ) are given by the diagonal elements of the Fock matrix. NO occupation energy(e h ) character 1 9 2.00 1.98 0.759 0.721 O1 5 2p 10 1.96 0.752 σ Mn1-O3 11 1.92 0.744 σ Mn3-O2 12 1.80 0.616 σ Mn2-O3 13 1.57 0.555 σ Mn2-O1 14 1.35 0.535 π Mn3-O2 15 1.14 0.319 π Mn4-O4 16 29 1.06 0.94 0.361 0.204 Mn1 4 3d 30 0.86 0.211 π * Mn4-O4 31 0.66 0.326 π * Mn3-O2 32 0.44 0.179 σ * Mn2-O1 33 0.21 0.073 σ * Mn2-O3 34 0.09 0.017 σ * Mn3-O2 35 0.04 0.304 σ * Mn1-O3 NATURE CHEMISTRY www.nature.com/naturechemistry 15
Table S3. Natural orbitals (NO) in the Mn 3d and µ-oxo 2p active space for the [XRD] 2- model. The electron occupations, energy levels, and characters are presented. The orbital energies (E h ) are given by the diagonal elements of the Fock matrix. NO occupation energy(e h ) character 1 11 2.00 1.95 0.279 0.126 O1 5 2p 12 1.90 0.242 σ Mn1-O1 13 1.78 0.157 σ Mn2-O3 14 1.67 0.055 σ Mn3-O2 15 1.34 0.025 σ Mn3-O5 16 31 1.05 0.93 0.121 0.237 Mn1 4 3d 32 0.67 0.248 σ * Mn3-O5 33 0.34 0.424 σ * Mn3-O2 34 0.26 0.375 σ * Mn2-O3 35 0.15 0.449 σ * Mn1-O1 NATURE CHEMISTRY www.nature.com/naturechemistry 16
Table S4. Natural orbitals (NO) in the Mn 3d and µ-oxo 2p active space for the [XRD+H] 2- model. The electron occupations, energy levels, and characters are presented. The orbital energies (E h ) are given by the diagonal elements of the Fock matrix. NO occupation energy(e h ) character 1 12 2.00 1.95 0.380 0.191 O1 5 2p 13 1.88 0.137 σ Mn1-O3 14 1.62 0.075 σ Mn3-O4 15 1.42 0.034 σ Mn2-O2 16 32 1.05 0.94 0.187 0.254 Mn1 4 3d 33 0.63 0.341 σ * Mn2-O2 34 0.43 0.370 σ * Mn3-O4 35 0.13 0.556 σ * Mn1-O3 NATURE CHEMISTRY www.nature.com/naturechemistry 17
Table S5. Natural orbitals (NO) in the Mn 3d and µ-oxo 2p active space for the QM/MM model. The electron occupations, energy levels, and characters are presented. The orbital energies (E h ) are given by the diagonal elements of the Fock matrix. NO occupation energy(e h ) character 1 9 1.99 1.94 0.570 0.499 O1 5 2p 10 1.90 0.510 σ Mn3-O2,4 11 1.88 0.486 σ Mn2-O1,3 12 1.85 0.500 σ Mn3,4-O4,5 13 1.84 0.471 σ Mn1-O1,3 14 1.82 0.446 σ Mn2-O2,3 15 1.81 0.411 σ Mn3,4-O4,5 16 29 1.08 0.91 0.177 0.085 Mn1 4 3d 30 0.22 0.174 σ * Mn3,4-O4,5 31 0.20 0.151 σ * Mn3,4-O4,5 32 0.20 0.146 σ * Mn2-O2,3 33 0.18 0.159 σ * Mn1-O1,3 34 0.13 0.196 σ * Mn2-O1,3 35 0.12 0.196 σ * Mn3-O2,4 NATURE CHEMISTRY www.nature.com/naturechemistry 18
Table S6. Natural orbitals (NO) in the Mn 3d and µ-oxo 2p active space for the Siegbahn s S 0 - state model. The electron occupations, energy levels, and characters are presented. The orbital energies (E h ) are given by the diagonal elements of the Fock matrix. NO occupation energy(e h ) character 1 10 2.00 1.97 0.475 0.280 O1 5 2p 11 1.94 0.307 σ Mn4-O5 12 1.93 0.289 σ Mn3-O2,3 13 1.92 0.280 σ Mn2-O1,2 14 1.91 0.286 σ Mn1-O1,3 15 1.82 0.273 σ Mn4-O4 16 30 1.06 0.94 0.128 0.097 Mn1 4 3d 31 0.19 0.375 σ * Mn4-O4 32 0.11 0.430 σ * Mn1-O1,3 33 0.10 0.433 σ * Mn2-O1,2 34 0.10 0.441 σ * Mn3-O2,3 35 0.05 0.424 σ * Mn4-O5 NATURE CHEMISTRY www.nature.com/naturechemistry 19
Table S7. Natural orbitals (NO) in the Mn 3d and µ-oxo 2p active space for the Siegbahn s S 1 - state model. The electron occupations, energy levels, and characters are presented. The orbital energies (E h ) are given by the diagonal elements of the Fock matrix. NO occupation energy(e h ) character 1 9 1.99 1.95 0.345 0.295 O1 5 2p 10 1.92 0.317 σ Mn4-O4 11 1.92 0.299 σ Mn2-O2,3 12 1.90 0.311 σ Mn3-O4,5 13 1.86 0.281 σ Mn3-O2,5 14 1.85 0.248 σ Mn1-O1,3 15 1.83 0.263 σ Mn4-O4,5 16 29 1.09 0.006 Mn1 4 3d 0.94 0.094 30 0.18 0.181 σ * Mn4-O4,5 31 0.16 0.160 σ * Mn1,-O1,5 32 0.16 0.159 σ * Mn3-O2,5 33 0.14 0.138 σ * Mn3-O4,5 34 0.10 0.101 σ * Mn2-O2,3 35 0.06 0.065 σ * Mn4-O4 NATURE CHEMISTRY www.nature.com/naturechemistry 20
Table S8. Natural orbitals (NO) in the Mn 3d and µ-oxo 2p active space for Siegbahn s S 2 -state model. The electron occupations, energy levels, and characters are presented. The orbital energies (E h ) are given by the diagonal elements of the Fock matrix. NO occupation energy(e h ) character 1 8 1.99 1.95 0.490 0.424 O1 5 2p 9 1.93 0.475 σ Mn4-O5 10 1.91 0.432 σ Mn1-O1,3 11 1.90 0.431 σ Mn3-O2,5 12 1.88 0.414 σ Mn2-O1,3 13 1.88 0.425 σ Mn2-O1,2 14 1.87 0.435 σ Mn3-O4,5 15 1.85 0.393 σ Mn4-O4,5 16 28 1.13 0.91 0.143 0.047 Mn1 4 3d 29 0.17 0.235 σ * Mn4-O4,5 30 0.16 0.251 σ * Mn3-O4,5 31 0.14 0.256 σ * Mn2-O1,3 32 0.13 0.270 σ * Mn2-O1,2 33 0.12 0.295 σ * Mn1-O1,3 34 0.12 0.279 σ * Mn3-O2,5 35 0.04 0.498 σ * Mn4-O5 NATURE CHEMISTRY www.nature.com/naturechemistry 21
Table S9. Natural orbitals (NO) in the Mn 3d and µ-oxo 2p active space for Siegbahn s S 3 -state model. The electron occupations, energy levels, and characters are presented. The orbital energies (E h ) are given by the diagonal elements of the Fock matrix. NO occupation energy(e h ) character 1 7 1.99 1.95 0.497 0.448 O1 5 2p 8 1.93 0.506 σ Mn4-O5 9 1.93 0.480 σ Mn1-O3 10 1.91 0.449 σ Mn3-O2,5 11 1.89 0.448 σ Mn2-O1,3 12 1.87 0.456 σ Mn4-O4,5 13 1.86 0.425 σ Mn2-O2,3 14 1.84 0.406 σ Mn3-O4,5 15 1.81 0.417 σ Mn1-O1,3 16 27 1.12 0.92 0.161 0.106 Mn1 4 3d 28 0.21 0.176 σ * Mn1-O1,3 29 0.18 0.226 σ * Mn3-O4,5 30 0.16 0.229 σ * Mn2-O2,3 31 0.15 0.226 σ * Mn4-O4,5 32 0.13 0.241 σ * Mn2-O1,3 33 0.10 0.285 σ * Mn3-O2,5 34 0.05 0.477 σ * Mn1-O3 35 0.04 0.505 σ * Mn4-O5 NATURE CHEMISTRY www.nature.com/naturechemistry 22
Table S10. Bonding character of selected Mn-O σ and π bonds for the XRD structure (the S 1 state). Each bond is decomposed into normalized percentage weights of different local configurations, as computed from the two-particle density matrix in the natural orbital basis. bonding / antibonding orbitals a ratio of spin configurations on bonding and anti-bonding orbitals σ * (π * ) σ (π ) Mn1 O3(σ/σ * ) 90% 1% 4% 3% 2% Mn2 O1,3(σ/σ * ) 65% 8% 11% 13% 3% Mn2 O2,3(σ/σ * ) 76% 3% 9% 9% 2% Mn3 O5(σ/σ * ) 48% 20% 11% 21% 1% Mn3 O2(σ/σ * ) 70% 8% 9% 10% 2% Mn4 O4(π/π * ) 38% 24% 13% 25% 1% or or etc. NATURE CHEMISTRY www.nature.com/naturechemistry 23
Table S11. Cartesian coordinates of the XRD model (Å) x y z Ca - 0.281609-1.940622-0.903371 Mn 1.635468 0.995683-0.675860 Mn 1.506402-0.652080 1.638874 Mn - 1.040318 0.636517 1.198880 Mn - 3.316370 0.313944-0.681368 O 1.714365-0.853219-0.402914 O - 0.556695-1.156426 1.439062 O 0.975830 1.275876 0.986317 O - 2.979566-0.124063 1.350707 O - 0.864448 0.468796-1.167774 C - 3.287487-2.903672-1.030096 O - 2.259139-3.294715-0.438409 O - 3.627865-1.700367-1.077023 C 2.463147-0.049101-3.292887 O 1.672688-0.995878-3.499145 O 2.236950 0.867093-2.473177 C 0.966099 4.976212-1.943047 N 2.280535 5.069253-1.528479 C 0.550227 3.732005-1.595212 C 2.638382 3.939172-0.947737 N 1.608959 3.108833-0.977587 C - 2.535764 3.118598 0.345773 O - 1.633269 2.615479 1.055269 O - 3.198365 2.476389-0.502778 C 4.232536 0.627098 1.081305 O 3.588609-0.156366 1.811915 O 3.706429 1.284691 0.172240 C 1.819713-3.595741 1.339206 O 0.934245-3.698418 0.471348 O 1.923451-2.559430 2.022816 C 0.179304 0.465091 4.097723 O 1.153950-0.172124 3.627025 O - 0.751429 0.926864 3.394500 O 0.994581-3.651335-2.180516 H 0.854301-4.150020-3.013824 H 1.134201-4.245728-1.386811 O - 1.311077-1.832399-3.053557 H - 2.015102-1.139047-3.088334 H - 0.517722-1.513008-3.571293 O - 3.512764 0.654604-2.726306 H - 4.337560 0.425427-3.211600 H - 3.065875 1.462897-3.067240 O - 5.485448 0.583472-0.292978 H - 5.643373 0.109418 0.579104 H - 5.725227 1.534044-0.169083 C - 4.150909-3.928312-1.735590 H - 4.054646-4.903310-1.238024 H - 3.798059-4.038966-2.777558 H - 5.208435-3.622025-1.763566 C 2.803508-4.715230 1.566017 H 3.249364-4.667764 2.569444 H 3.607529-4.589978 0.817702 NATURE CHEMISTRY www.nature.com/naturechemistry 24
H 2.315884-5.688505 1.409582 C 5.721527 0.784030 1.296335 H 5.985991 0.627968 2.352363 H 6.050875 1.777743 0.961793 H 6.251255 0.022371 0.697089 C 3.767087 0.031612-4.034796 H 4.500662-0.595579-3.494271 H 3.665527-0.374910-5.052430 H 4.152207 1.061052-4.072429 C 0.126923 0.672562 5.589556 H - 0.215932-0.258441 6.074418 H 1.130141 0.901270 5.979753 H - 0.574104 1.480174 5.838565 H 2.867132 5.891320-1.638931 H 3.612621 3.693317-0.545786 H 0.464350 5.796289-2.442098 H - 0.392706 3.221169-1.741881 C - 2.847401 4.584076 0.491331 H - 1.924884 5.180757 0.399488 H - 3.577581 4.907129-0.262356 H - 3.252714 4.781574 1.499063 O - 4.935324-0.373757 2.892672 H - 4.054946-0.502001 2.393946 H - 4.815080 0.414585 3.476177 O 3.630154-2.523016-1.228181 H 3.030402-2.908122-1.919105 H 3.069410-1.801266-0.811323 NATURE CHEMISTRY www.nature.com/naturechemistry 25
Table S12. Cartesian coordinates of the [XRD+H] + model (Å) x y Z Ca - 0.281091-1.937627-0.909091 Mn 1.629284 1.002151-0.670757 Mn 1.505125-0.655138 1.637437 Mn - 1.044759 0.629333 1.203799 Mn - 3.320973 0.309013-0.676640 O 1.712563-0.847636-0.405246 O - 0.556905-1.163443 1.436594 O 0.969806 1.274185 0.992834 O - 2.982175-0.136324 1.353509 O - 0.869654 0.471461-1.163589 H - 1.100088 1.111682-1.878656 C - 3.284838-2.907112-1.038227 O - 2.255309-3.298135-0.448588 O - 3.628015-1.704417-1.080184 C 2.458100-0.030270-3.292332 O 1.669728-0.978037-3.501992 O 2.230187 0.882121-2.468865 C 0.950123 4.986149-1.921733 N 2.264541 5.080568-1.507419 C 0.537288 3.739607-1.578668 C 2.625275 3.949007-0.931366 N 1.597755 3.116420-0.964038 C - 2.546339 3.111340 0.361316 O - 1.642343 2.607477 1.068370 O - 3.207869 2.470994-0.489477 C 4.228043 0.632550 1.083685 O 3.586277-0.155308 1.811467 O 3.699981 1.292549 0.177500 C 1.825084-3.596843 1.325876 O 0.939439-3.698098 0.458030 O 1.926769-2.563029 2.013563 C 0.176655 0.449153 4.101355 O 1.152547-0.183924 3.627655 O - 0.755480 0.911580 3.400423 O 0.998366-3.640304-2.193652 H 1.857654-3.329355-2.576713 H 1.197800-4.245399-1.429853 O - 1.311894-1.823216-3.058325 H - 1.505934-2.706217-3.459246 H - 0.525584-1.444584-3.553824 O - 3.519142 0.657376-2.720109 H - 3.369121-0.158193-3.258467 H - 4.113887 1.308445-3.152095 O - 5.490478 0.571982-0.286148 H - 5.688527 0.109598 0.580462 H - 5.880203 1.477450-0.256724 C - 4.146239-3.930917-1.747391 H - 4.696194-4.521876-0.994547 H - 3.506255-4.635812-2.300664 H - 4.877617-3.463915-2.422205 C 2.811572-4.714952 1.547744 H 3.297649-4.644520 2.530452 NATURE CHEMISTRY www.nature.com/naturechemistry 26
H 3.584024-4.635139 0.762126 H 2.305805-5.685538 1.442391 C 5.716772 0.792058 1.298632 H 5.921119 1.059137 2.347998 H 6.123237 1.561024 0.629916 H 6.225377-0.167759 1.106113 C 3.761492 0.056413-4.034532 H 4.467027-0.659045-3.573674 H 3.632202-0.248162-5.083981 H 4.196618 1.063887-3.983314 C 0.124518 0.650548 5.594030 H - 0.188614-0.293226 6.073593 H 1.124512 0.904198 5.975969 H - 0.595181 1.438146 5.851609 H 2.849399 5.905418-1.615358 H 3.601007 3.710855-0.529238 H 0.443815 5.806681-2.415058 H - 0.423666 3.267129-1.721150 C - 2.861284 4.575502 0.512872 H - 1.934898 5.159112 0.622954 H - 3.446425 4.943115-0.340150 H - 3.448789 4.729393 1.435697 O - 4.936609-0.396679 2.895395 H - 4.047691-0.558633 2.444563 H - 4.817752 0.304544 3.579873 O 3.631829-2.509698-1.238066 H 4.555219-2.321714-0.953478 H 3.002506-1.806334-0.885610 NATURE CHEMISTRY www.nature.com/naturechemistry 27
Table S13. The pairwise orbital entanglement matrix Ipq of the DMRG-CASSCF(44e, 35o) many-electron wavefunction at the XRD structure (Fig.S3). ψ 1 ψ 2 ψ 3 ψ 4 ψ 5 ψ 6 ψ 7 ψ 8 ψ 9 ψ 10 ψ 1 0.000 0.140 0.143 0.141 0.143 0.001 0.002 0.030 0.002 0.135 ψ 2 0.140 0.000 0.141 0.142 0.142 0.006 0.000 0.029 0.030 0.125 ψ 13 0.143 0.141 0.000 0.140 0.143 0.000 0.000 0.029 0.003 0.174 ψ 14 0.141 0.142 0.140 0.000 0.141 0.022 0.001 0.030 0.040 0.124 ψ 15 0.143 0.142 0.143 0.141 0.000 0.000 0.001 0.032 0.002 0.137 ψ 16 0.001 0.006 0.000 0.022 0.000 0.000 0.000 0.002 0.000 0.000 ψ 17 0.002 0.000 0.000 0.001 0.001 0.000 0.000 0.041 0.000 0.000 ψ 18 0.030 0.029 0.029 0.030 0.032 0.002 0.041 0.000 0.002 0.032 ψ 19 0.002 0.030 0.003 0.040 0.002 0.000 0.000 0.002 0.000 0.007 ψ 10 0.135 0.125 0.174 0.124 0.137 0.000 0.000 0.032 0.007 0.000 ψ 11 0.001 0.000 0.002 0.000 0.013 0.000 0.001 0.008 0.000 0.001 ψ 12 0.054 0.054 0.053 0.053 0.054 0.004 0.001 0.162 0.004 0.030 ψ 13 0.080 0.080 0.078 0.079 0.080 0.001 0.003 0.107 0.003 0.042 ψ 14 0.089 0.089 0.086 0.088 0.088 0.003 0.002 0.148 0.011 0.048 ψ 15 0.090 0.090 0.088 0.089 0.090 0.000 0.001 0.117 0.002 0.065 ψ 16 0.058 0.057 0.056 0.057 0.058 0.000 0.003 0.155 0.001 0.032 ψ 17 0.012 0.012 0.012 0.012 0.012 0.000 0.001 0.050 0.001 0.014 ψ 18 0.002 0.002 0.002 0.002 0.002 0.000 0.002 0.006 0.000 0.001 ψ 19 0.005 0.005 0.005 0.005 0.005 0.000 0.000 0.003 0.000 0.008 ψ 20 0.002 0.002 0.002 0.002 0.002 0.000 0.001 0.017 0.000 0.002 ψ 21 0.020 0.020 0.020 0.020 0.020 0.000 0.000 0.007 0.000 0.010 ψ 22 0.003 0.003 0.002 0.003 0.003 0.000 0.000 0.002 0.000 0.001 ψ 23 0.042 0.042 0.041 0.041 0.041 0.000 0.000 0.023 0.001 0.035 ψ 24 0.091 0.091 0.089 0.090 0.090 0.000 0.000 0.056 0.002 0.091 ψ 25 0.094 0.094 0.092 0.094 0.094 0.000 0.000 0.058 0.002 0.095 ψ 26 0.061 0.061 0.059 0.060 0.060 0.000 0.000 0.034 0.001 0.055 ψ 27 0.058 0.058 0.056 0.057 0.058 0.000 0.000 0.033 0.001 0.053 ψ 28 0.011 0.011 0.010 0.011 0.011 0.001 0.000 0.005 0.000 0.008 ψ 29 0.098 0.098 0.095 0.097 0.097 0.002 0.000 0.063 0.002 0.103 ψ 30 0.099 0.100 0.097 0.099 0.099 0.001 0.000 0.062 0.002 0.105 ψ 31 0.098 0.098 0.096 0.098 0.098 0.014 0.001 0.061 0.002 0.103 ψ 32 0.102 0.102 0.100 0.102 0.102 0.001 0.000 0.064 0.002 0.108 ψ 33 0.026 0.026 0.026 0.026 0.026 0.000 0.000 0.008 0.001 0.012 ψ 34 0.001 0.001 0.001 0.001 0.001 0.000 0.000 0.000 0.000 0.001 ψ 35 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 NATURE CHEMISTRY www.nature.com/naturechemistry 28
ψ 11 ψ 12 ψ 13 ψ 14 ψ 15 ψ 16 ψ 17 ψ 18 ψ 19 ψ 20 ψ 1 0.001 0.054 0.080 0.089 0.090 0.058 0.012 0.002 0.005 0.002 ψ 2 0.000 0.054 0.080 0.089 0.090 0.057 0.012 0.002 0.005 0.002 ψ 13 0.002 0.053 0.078 0.086 0.088 0.056 0.012 0.002 0.005 0.002 ψ 14 0.000 0.053 0.079 0.088 0.089 0.057 0.012 0.002 0.005 0.002 ψ 15 0.013 0.054 0.080 0.088 0.090 0.058 0.012 0.002 0.005 0.002 ψ 16 0.000 0.004 0.001 0.003 0.000 0.000 0.000 0.000 0.000 0.000 ψ 17 0.001 0.001 0.003 0.002 0.001 0.003 0.001 0.002 0.000 0.001 ψ 18 0.008 0.162 0.107 0.148 0.117 0.155 0.050 0.006 0.003 0.017 ψ 19 0.000 0.004 0.003 0.011 0.002 0.001 0.001 0.000 0.000 0.000 ψ 10 0.001 0.030 0.042 0.048 0.065 0.032 0.014 0.001 0.008 0.002 ψ 11 0.000 0.009 0.014 0.002 0.006 0.065 0.046 0.000 0.001 0.014 ψ 12 0.009 0.000 0.138 0.118 0.109 0.139 0.332 0.005 0.005 0.009 ψ 13 0.014 0.138 0.000 0.125 0.124 0.131 0.155 0.055 0.014 0.020 ψ 14 0.002 0.118 0.125 0.000 0.130 0.110 0.072 0.018 0.004 0.006 ψ 15 0.006 0.109 0.124 0.130 0.000 0.113 0.074 0.015 0.054 0.010 ψ 16 0.065 0.139 0.131 0.110 0.113 0.000 0.230 0.010 0.005 0.064 ψ 17 0.046 0.332 0.155 0.072 0.074 0.230 0.000 0.013 0.026 0.044 ψ 18 0.000 0.005 0.055 0.018 0.015 0.010 0.013 0.000 0.001 0.000 ψ 19 0.001 0.005 0.014 0.004 0.054 0.005 0.026 0.001 0.000 0.006 ψ 20 0.014 0.009 0.020 0.006 0.010 0.064 0.044 0.000 0.006 0.000 ψ 21 0.000 0.011 0.017 0.017 0.030 0.013 0.008 0.001 0.089 0.011 ψ 22 0.000 0.002 0.002 0.006 0.002 0.003 0.001 0.000 0.000 0.014 ψ 23 0.000 0.023 0.032 0.035 0.035 0.025 0.009 0.002 0.096 0.007 ψ 24 0.000 0.052 0.072 0.078 0.077 0.055 0.030 0.003 0.010 0.036 ψ 25 0.000 0.054 0.073 0.080 0.079 0.057 0.024 0.021 0.013 0.004 ψ 26 0.000 0.034 0.047 0.051 0.051 0.036 0.014 0.002 0.065 0.002 ψ 27 0.000 0.033 0.045 0.049 0.049 0.034 0.013 0.002 0.116 0.002 ψ 28 0.000 0.006 0.008 0.010 0.009 0.007 0.003 0.000 0.001 0.031 ψ 29 0.000 0.056 0.076 0.084 0.083 0.059 0.027 0.003 0.004 0.047 ψ 30 0.000 0.057 0.078 0.085 0.084 0.060 0.026 0.003 0.010 0.003 ψ 31 0.000 0.056 0.076 0.084 0.083 0.059 0.025 0.003 0.004 0.004 ψ 32 0.000 0.059 0.080 0.087 0.086 0.061 0.027 0.003 0.004 0.002 ψ 33 0.000 0.014 0.020 0.022 0.025 0.015 0.004 0.001 0.044 0.000 ψ 34 0.000 0.000 0.001 0.001 0.001 0.000 0.000 0.001 0.006 0.000 ψ 35 0.000 0.000 0.000 0.001 0.000 0.001 0.000 0.000 0.001 0.005 NATURE CHEMISTRY www.nature.com/naturechemistry 29
ψ 21 ψ 22 ψ 23 ψ 24 ψ 25 ψ 26 ψ 27 ψ 28 ψ 29 ψ 30 ψ 1 0.020 0.003 0.042 0.091 0.094 0.061 0.058 0.011 0.098 0.099 ψ 2 0.020 0.003 0.042 0.091 0.094 0.061 0.058 0.011 0.098 0.100 ψ 13 0.020 0.002 0.041 0.089 0.092 0.059 0.056 0.010 0.095 0.097 ψ 14 0.020 0.003 0.041 0.090 0.094 0.060 0.057 0.011 0.097 0.099 ψ 15 0.020 0.003 0.041 0.090 0.094 0.060 0.058 0.011 0.097 0.099 ψ 16 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.002 0.001 ψ 17 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 ψ 18 0.007 0.002 0.023 0.056 0.058 0.034 0.033 0.005 0.063 0.062 ψ 19 0.000 0.000 0.001 0.002 0.002 0.001 0.001 0.000 0.002 0.002 ψ 10 0.010 0.001 0.035 0.091 0.095 0.055 0.053 0.008 0.103 0.105 ψ 11 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 ψ 12 0.011 0.002 0.023 0.052 0.054 0.034 0.033 0.006 0.056 0.057 ψ 13 0.017 0.002 0.032 0.072 0.073 0.047 0.045 0.008 0.076 0.078 ψ 14 0.017 0.006 0.035 0.078 0.080 0.051 0.049 0.010 0.084 0.085 ψ 15 0.030 0.002 0.035 0.077 0.079 0.051 0.049 0.009 0.083 0.084 ψ 16 0.013 0.003 0.025 0.055 0.057 0.036 0.034 0.007 0.059 0.060 ψ 17 0.008 0.001 0.009 0.030 0.024 0.014 0.013 0.003 0.027 0.026 ψ 18 0.001 0.000 0.002 0.003 0.021 0.002 0.002 0.000 0.003 0.003 ψ 19 0.089 0.000 0.096 0.010 0.013 0.065 0.116 0.001 0.004 0.010 ψ 20 0.011 0.014 0.007 0.036 0.004 0.002 0.002 0.031 0.047 0.003 ψ 21 0.000 0.017 0.432 0.067 0.047 0.031 0.119 0.008 0.015 0.060 ψ 22 0.017 0.000 0.007 0.002 0.002 0.006 0.006 0.572 0.052 0.006 ψ 23 0.432 0.007 0.000 0.092 0.081 0.060 0.091 0.007 0.043 0.049 ψ 24 0.067 0.002 0.092 0.000 0.135 0.099 0.086 0.010 0.109 0.113 ψ 25 0.047 0.002 0.081 0.135 0.000 0.092 0.087 0.009 0.112 0.117 ψ 26 0.031 0.006 0.060 0.099 0.092 0.000 0.143 0.006 0.064 0.075 ψ 27 0.119 0.006 0.091 0.086 0.087 0.143 0.000 0.006 0.062 0.093 ψ 28 0.008 0.572 0.007 0.010 0.009 0.006 0.006 0.000 0.070 0.015 ψ 29 0.015 0.052 0.043 0.109 0.112 0.064 0.062 0.070 0.000 0.127 ψ 30 0.060 0.006 0.049 0.113 0.117 0.075 0.093 0.015 0.127 0.000 ψ 31 0.014 0.032 0.042 0.107 0.112 0.066 0.062 0.075 0.143 0.127 ψ 32 0.016 0.005 0.045 0.112 0.122 0.067 0.065 0.016 0.131 0.133 ψ 33 0.041 0.005 0.079 0.065 0.051 0.320 0.215 0.002 0.015 0.056 ψ 34 0.001 0.001 0.003 0.002 0.028 0.032 0.013 0.001 0.001 0.003 ψ 35 0.005 0.113 0.001 0.001 0.001 0.015 0.004 0.272 0.032 0.004 NATURE CHEMISTRY www.nature.com/naturechemistry 30
ψ 31 ψ 32 ψ 33 ψ 34 ψ 35 ψ 1 0.098 0.102 0.026 0.001 0.000 ψ 2 0.098 0.102 0.026 0.001 0.000 ψ 13 0.096 0.100 0.026 0.001 0.000 ψ 14 0.098 0.102 0.026 0.001 0.000 ψ 15 0.098 0.102 0.026 0.001 0.000 ψ 16 0.014 0.001 0.000 0.000 0.000 ψ 17 0.001 0.000 0.000 0.000 0.000 ψ 18 0.061 0.064 0.008 0.000 0.000 ψ 19 0.002 0.002 0.001 0.000 0.000 ψ 10 0.103 0.108 0.012 0.001 0.000 ψ 11 0.000 0.000 0.000 0.000 0.000 ψ 12 0.056 0.059 0.014 0.000 0.000 ψ 13 0.076 0.080 0.020 0.001 0.000 ψ 14 0.084 0.087 0.022 0.001 0.001 ψ 15 0.083 0.086 0.025 0.001 0.000 ψ 16 0.059 0.061 0.015 0.000 0.001 ψ 17 0.025 0.027 0.004 0.000 0.000 ψ 18 0.003 0.003 0.001 0.001 0.000 ψ 19 0.004 0.004 0.044 0.006 0.001 ψ 20 0.004 0.002 0.000 0.000 0.005 ψ 21 0.014 0.016 0.041 0.001 0.005 ψ 22 0.032 0.005 0.005 0.001 0.113 ψ 23 0.042 0.045 0.079 0.003 0.001 ψ 24 0.107 0.112 0.065 0.002 0.001 ψ 25 0.112 0.122 0.051 0.028 0.001 ψ 26 0.066 0.067 0.320 0.032 0.015 ψ 27 0.062 0.065 0.215 0.013 0.004 ψ 28 0.075 0.016 0.002 0.001 0.272 ψ 29 0.143 0.131 0.015 0.001 0.032 ψ 30 0.127 0.133 0.056 0.003 0.004 ψ 31 0.000 0.132 0.016 0.001 0.089 ψ 32 0.132 0.000 0.018 0.064 0.002 ψ 33 0.016 0.018 0.000 0.004 0.025 ψ 34 0.001 0.064 0.004 0.000 0.014 ψ 35 0.089 0.002 0.025 0.014 0.000 NATURE CHEMISTRY www.nature.com/naturechemistry 31
Table S14. The pairwise orbital entanglement matrix Ipq of the DMRG-CASSCF(44e, 35o) many-electron wavefunction at the QM/MM structure (Fig.S3). ψ 1 ψ 2 ψ 3 ψ 4 ψ 5 ψ 6 ψ 7 ψ 8 ψ 9 ψ 10 ψ 1 0.000 0.130 0.135 0.131 0.052 0.017 0.052 0.005 0.025 0.016 ψ 2 0.130 0.000 0.135 0.133 0.054 0.021 0.003 0.003 0.055 0.012 ψ 13 0.135 0.135 0.000 0.129 0.043 0.005 0.001 0.004 0.019 0.083 ψ 14 0.131 0.133 0.129 0.000 0.077 0.022 0.001 0.026 0.072 0.017 ψ 15 0.052 0.054 0.043 0.077 0.000 0.284 0.000 0.007 0.484 0.004 ψ 16 0.017 0.021 0.005 0.022 0.284 0.000 0.008 0.025 0.107 0.001 ψ 17 0.052 0.003 0.001 0.001 0.000 0.008 0.000 0.099 0.002 0.000 ψ 18 0.005 0.003 0.004 0.026 0.007 0.025 0.099 0.000 0.011 0.001 ψ 19 0.025 0.055 0.019 0.072 0.484 0.107 0.002 0.011 0.000 0.008 ψ 10 0.016 0.012 0.083 0.017 0.004 0.001 0.000 0.001 0.008 0.000 ψ 11 0.005 0.007 0.005 0.021 0.016 0.014 0.016 0.047 0.028 0.007 ψ 12 0.009 0.009 0.008 0.012 0.003 0.036 0.130 0.379 0.006 0.002 ψ 13 0.117 0.112 0.113 0.108 0.018 0.001 0.059 0.030 0.005 0.014 ψ 14 0.102 0.100 0.111 0.096 0.017 0.001 0.005 0.026 0.008 0.115 ψ 15 0.100 0.103 0.100 0.103 0.017 0.048 0.001 0.011 0.044 0.012 ψ 16 0.015 0.016 0.015 0.015 0.005 0.001 0.023 0.055 0.005 0.004 ψ 17 0.002 0.002 0.002 0.004 0.001 0.007 0.017 0.045 0.000 0.001 ψ 18 0.001 0.001 0.001 0.001 0.000 0.000 0.002 0.000 0.000 0.001 ψ 19 0.009 0.009 0.009 0.008 0.002 0.000 0.001 0.001 0.000 0.002 ψ 20 0.002 0.002 0.002 0.002 0.001 0.000 0.002 0.007 0.001 0.002 ψ 21 0.005 0.005 0.005 0.004 0.001 0.000 0.000 0.000 0.000 0.002 ψ 22 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.000 ψ 23 0.015 0.015 0.015 0.015 0.004 0.000 0.000 0.000 0.001 0.003 ψ 24 0.065 0.066 0.066 0.063 0.015 0.000 0.001 0.003 0.005 0.008 ψ 25 0.091 0.092 0.092 0.088 0.021 0.000 0.001 0.003 0.007 0.010 ψ 26 0.096 0.097 0.097 0.093 0.022 0.000 0.001 0.003 0.007 0.010 ψ 27 0.037 0.037 0.037 0.035 0.008 0.000 0.001 0.001 0.003 0.006 ψ 28 0.041 0.041 0.041 0.040 0.009 0.000 0.000 0.001 0.003 0.004 ψ 29 0.085 0.086 0.086 0.082 0.020 0.000 0.001 0.003 0.006 0.009 ψ 30 0.100 0.100 0.100 0.096 0.023 0.001 0.001 0.003 0.007 0.010 ψ 31 0.105 0.105 0.105 0.101 0.024 0.001 0.001 0.003 0.008 0.011 ψ 32 0.101 0.102 0.102 0.098 0.024 0.001 0.001 0.003 0.007 0.010 ψ 33 0.001 0.001 0.001 0.001 0.000 0.000 0.000 0.000 0.000 0.001 ψ 34 0.003 0.004 0.004 0.003 0.001 0.000 0.000 0.000 0.000 0.000 ψ 35 0.009 0.009 0.009 0.009 0.002 0.000 0.000 0.000 0.000 0.001 NATURE CHEMISTRY www.nature.com/naturechemistry 32
ψ 11 ψ 12 ψ 13 ψ 14 ψ 15 ψ 16 ψ 17 ψ 18 ψ 19 ψ 20 ψ 1 0.005 0.009 0.117 0.102 0.100 0.015 0.002 0.001 0.009 0.002 ψ 2 0.007 0.009 0.112 0.100 0.103 0.016 0.002 0.001 0.009 0.002 ψ 13 0.005 0.008 0.113 0.111 0.100 0.015 0.002 0.001 0.009 0.002 ψ 14 0.021 0.012 0.108 0.096 0.103 0.015 0.004 0.001 0.008 0.002 ψ 15 0.016 0.003 0.018 0.017 0.017 0.005 0.001 0.000 0.002 0.001 ψ 16 0.014 0.036 0.001 0.001 0.048 0.001 0.007 0.000 0.000 0.000 ψ 17 0.016 0.130 0.059 0.005 0.001 0.023 0.017 0.002 0.001 0.002 ψ 18 0.047 0.379 0.030 0.026 0.011 0.055 0.045 0.000 0.001 0.007 ψ 19 0.028 0.006 0.005 0.008 0.044 0.005 0.000 0.000 0.000 0.001 ψ 10 0.007 0.002 0.014 0.115 0.012 0.004 0.001 0.001 0.002 0.002 ψ 11 0.000 0.009 0.011 0.047 0.067 0.389 0.053 0.000 0.000 0.066 ψ 12 0.009 0.000 0.024 0.023 0.024 0.026 0.299 0.002 0.003 0.004 ψ 13 0.011 0.024 0.000 0.111 0.111 0.034 0.011 0.044 0.014 0.004 ψ 14 0.047 0.023 0.111 0.000 0.117 0.075 0.012 0.002 0.023 0.026 ψ 15 0.067 0.024 0.111 0.117 0.000 0.091 0.034 0.003 0.007 0.018 ψ 16 0.389 0.026 0.034 0.075 0.091 0.000 0.158 0.006 0.003 0.142 ψ 17 0.053 0.299 0.011 0.012 0.034 0.158 0.000 0.003 0.024 0.022 ψ 18 0.000 0.002 0.044 0.002 0.003 0.006 0.003 0.000 0.052 0.000 ψ 19 0.000 0.003 0.014 0.023 0.007 0.003 0.024 0.052 0.000 0.001 ψ 20 0.066 0.004 0.004 0.026 0.018 0.142 0.022 0.000 0.001 0.000 ψ 21 0.000 0.001 0.005 0.015 0.004 0.001 0.003 0.004 0.036 0.020 ψ 22 0.002 0.000 0.000 0.001 0.013 0.002 0.000 0.001 0.005 0.005 ψ 23 0.001 0.002 0.014 0.013 0.013 0.004 0.007 0.034 0.262 0.020 ψ 24 0.002 0.007 0.062 0.058 0.057 0.012 0.010 0.013 0.131 0.020 ψ 25 0.002 0.010 0.086 0.080 0.079 0.017 0.010 0.006 0.073 0.005 ψ 26 0.003 0.010 0.092 0.084 0.083 0.017 0.003 0.107 0.026 0.003 ψ 27 0.001 0.005 0.035 0.034 0.032 0.006 0.024 0.017 0.176 0.033 ψ 28 0.002 0.004 0.039 0.036 0.036 0.008 0.001 0.001 0.002 0.015 ψ 29 0.002 0.009 0.081 0.074 0.074 0.016 0.003 0.000 0.004 0.006 ψ 30 0.003 0.010 0.094 0.087 0.087 0.018 0.003 0.000 0.005 0.004 ψ 31 0.003 0.011 0.099 0.091 0.091 0.019 0.003 0.000 0.005 0.004 ψ 32 0.003 0.010 0.096 0.089 0.088 0.018 0.003 0.001 0.005 0.003 ψ 33 0.000 0.000 0.001 0.002 0.001 0.000 0.003 0.004 0.044 0.006 ψ 34 0.000 0.000 0.003 0.003 0.003 0.001 0.000 0.004 0.001 0.000 ψ 35 0.000 0.001 0.009 0.008 0.008 0.002 0.000 0.000 0.001 0.003 NATURE CHEMISTRY www.nature.com/naturechemistry 33
ψ 21 ψ 22 ψ 23 ψ 24 ψ 25 ψ 26 ψ 27 ψ 28 ψ 29 ψ 30 ψ 1 0.005 0.000 0.015 0.065 0.091 0.096 0.037 0.041 0.085 0.100 ψ 2 0.005 0.000 0.015 0.066 0.092 0.097 0.037 0.041 0.086 0.100 ψ 13 0.005 0.000 0.015 0.066 0.092 0.097 0.037 0.041 0.086 0.100 ψ 14 0.004 0.000 0.015 0.063 0.088 0.093 0.035 0.040 0.082 0.096 ψ 15 0.001 0.000 0.004 0.015 0.021 0.022 0.008 0.009 0.020 0.023 ψ 16 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 ψ 17 0.000 0.000 0.000 0.001 0.001 0.001 0.001 0.000 0.001 0.001 ψ 18 0.000 0.000 0.000 0.003 0.003 0.003 0.001 0.001 0.003 0.003 ψ 19 0.000 0.001 0.001 0.005 0.007 0.007 0.003 0.003 0.006 0.007 ψ 10 0.002 0.000 0.003 0.008 0.010 0.010 0.006 0.004 0.009 0.010 ψ 11 0.000 0.002 0.001 0.002 0.002 0.003 0.001 0.002 0.002 0.003 ψ 12 0.001 0.000 0.002 0.007 0.010 0.010 0.005 0.004 0.009 0.010 ψ 13 0.005 0.000 0.014 0.062 0.086 0.092 0.035 0.039 0.081 0.094 ψ 14 0.015 0.001 0.013 0.058 0.080 0.084 0.034 0.036 0.074 0.087 ψ 15 0.004 0.013 0.013 0.057 0.079 0.083 0.032 0.036 0.074 0.087 ψ 16 0.001 0.002 0.004 0.012 0.017 0.017 0.006 0.008 0.016 0.018 ψ 17 0.003 0.000 0.007 0.010 0.010 0.003 0.024 0.001 0.003 0.003 ψ 18 0.004 0.001 0.034 0.013 0.006 0.107 0.017 0.001 0.000 0.000 ψ 19 0.036 0.005 0.262 0.131 0.073 0.026 0.176 0.002 0.004 0.005 ψ 20 0.020 0.005 0.020 0.020 0.005 0.003 0.033 0.015 0.006 0.004 ψ 21 0.000 0.024 0.394 0.068 0.043 0.015 0.073 0.007 0.006 0.012 ψ 22 0.024 0.000 0.040 0.006 0.037 0.002 0.001 0.125 0.051 0.049 ψ 23 0.394 0.040 0.000 0.030 0.053 0.031 0.030 0.006 0.011 0.012 ψ 24 0.068 0.006 0.030 0.000 0.088 0.079 0.197 0.024 0.053 0.064 ψ 25 0.043 0.037 0.053 0.088 0.000 0.112 0.064 0.033 0.083 0.103 ψ 26 0.015 0.002 0.031 0.079 0.112 0.000 0.049 0.035 0.088 0.107 ψ 27 0.073 0.001 0.030 0.197 0.064 0.049 0.000 0.013 0.027 0.033 ψ 28 0.007 0.125 0.006 0.024 0.033 0.035 0.013 0.000 0.179 0.109 ψ 29 0.006 0.051 0.011 0.053 0.083 0.088 0.027 0.179 0.000 0.136 ψ 30 0.012 0.049 0.012 0.064 0.103 0.107 0.033 0.109 0.136 0.000 ψ 31 0.003 0.023 0.013 0.068 0.107 0.113 0.035 0.060 0.115 0.134 ψ 32 0.006 0.010 0.013 0.065 0.104 0.112 0.032 0.079 0.124 0.136 ψ 33 0.070 0.019 0.041 0.133 0.015 0.007 0.234 0.017 0.007 0.015 ψ 34 0.002 0.006 0.002 0.003 0.004 0.049 0.003 0.030 0.034 0.010 ψ 35 0.005 0.091 0.003 0.004 0.027 0.005 0.007 0.426 0.180 0.057 NATURE CHEMISTRY www.nature.com/naturechemistry 34
ψ 31 ψ 32 ψ 33 ψ 34 ψ 35 ψ 1 0.105 0.101 0.001 0.003 0.009 ψ 2 0.105 0.102 0.001 0.004 0.009 ψ 13 0.105 0.102 0.001 0.004 0.009 ψ 14 0.101 0.098 0.001 0.003 0.009 ψ 15 0.024 0.024 0.000 0.001 0.002 ψ 16 0.001 0.001 0.000 0.000 0.000 ψ 17 0.001 0.001 0.000 0.000 0.000 ψ 18 0.003 0.003 0.000 0.000 0.000 ψ 19 0.008 0.007 0.000 0.000 0.000 ψ 10 0.011 0.010 0.001 0.000 0.001 ψ 11 0.003 0.003 0.000 0.000 0.000 ψ 12 0.011 0.010 0.000 0.000 0.001 ψ 13 0.099 0.096 0.001 0.003 0.009 ψ 14 0.091 0.089 0.002 0.003 0.008 ψ 15 0.091 0.088 0.001 0.003 0.008 ψ 16 0.019 0.018 0.000 0.001 0.002 ψ 17 0.003 0.003 0.003 0.000 0.000 ψ 18 0.000 0.001 0.004 0.004 0.000 ψ 19 0.005 0.005 0.044 0.001 0.001 ψ 20 0.004 0.003 0.006 0.000 0.003 ψ 21 0.003 0.006 0.070 0.002 0.005 ψ 22 0.023 0.010 0.019 0.006 0.091 ψ 23 0.013 0.013 0.041 0.002 0.003 ψ 24 0.068 0.065 0.133 0.003 0.004 ψ 25 0.107 0.104 0.015 0.004 0.027 ψ 26 0.113 0.112 0.007 0.049 0.005 ψ 27 0.035 0.032 0.234 0.003 0.007 ψ 28 0.060 0.079 0.017 0.030 0.426 ψ 29 0.115 0.124 0.007 0.034 0.180 ψ 30 0.134 0.136 0.015 0.010 0.057 ψ 31 0.000 0.134 0.004 0.005 0.031 ψ 32 0.134 0.000 0.004 0.060 0.061 ψ 33 0.004 0.004 0.000 0.002 0.025 ψ 34 0.005 0.060 0.002 0.000 0.013 ψ 35 0.031 0.061 0.025 0.013 0.000 NATURE CHEMISTRY www.nature.com/naturechemistry 35