Theory of chemical bonds in metalloenzymes XX: magneto-structural correlations in the CaMn4O5 cluster in oxygen-evolving complex of photosystem II

ABSTRACT Magneto-structural correlations in oxygen-evolving complex (OEC) of photosystem II (PSII) have been elucidated on the basis of theoretical and computational results in combination with available electron paramagnetic resonance (EPR) experimental results, and extended x-ray absorption fine structure (EXAFS) and x-ray diffraction (XRD) results. To this end, the computational methods based on broken-symmetry (BS) UB3LYP solutions have been developed to elucidate magnetic interactions in the active manganese catalyst for water oxidation by sunlight. The effective exchange interactions J for the CaMn(III)Mn(IV)3O5(H2O)3Y(Y = H2O or OH−) cluster (1) model of OEC of PSII have been calculated by the generalised approximate spin projection (GAP) method that eliminates the spin contamination errors of the BS UB3LYP solution. Full geometry optimisations followed by the zero-point energy (ZPE) correction have been performed for all the spin configurations of 1 to improve the J values that are compared with accumulated EPR in the S2 state of Kok cycle and magnetic susceptibility results of Christou model complex Ca2Mn(IV)3O4 (2). Using the calculated J values, exact diagonalisation of the spin Hamiltonian matrix has been carried out to obtain excitation energies and spin densities of the ground and lower excited states of 1. The calculated excitation energies are consistent with the available experimental results. The calculated spin densities (projection factors) are also compatible with those of the EPR results. The calculated spin densities have been used to calculate the isotropic hyperfine (Aiso) constants of 55Mn ions revealed by the EPR experiments. Implications of the computational results are discussed in relation to the structural symmetry breaking (SSB) in the S1, S2 and S3 states, spin crossover phenomenon induced by the near-infrared excitation and the right- and left-handed scenarios for the O–O bond formation for water oxidation.


Introduction
Electron-and spin-correlation effects play important roles for theoretical insight of the nature of chemical bonds in metalloenzymes. Transition-metal clusters confined in protein fields have often been regarded as strong correlation electron systems (SCES), where the orbital, spin and charge degrees of freedom are variable, avoid strong electron repulsions. The HF, KS-DFT and their hybrid models inevitably entail broken-symmetry (BS) pictures of the labile chemical bonds of metalloenzymes. Temperature-dependent paramagnetism has been regarded as an indication of appearance of local spins in SCES, providing several exchange-coupled models such as the Heisenberg, Kondo and Anderson models [1].
In this series of papers [2], we have examined the nature of chemical bonds in exchange-coupled open-shell systems on the basis of BS methods [1,3] followed by approximate spin-projection procedure for finite systems [4][5][6]. Three-centre, four-electron bonds in the exchangecoupled organic systems such as 1,3-diradicals exhibit the so-called triplet instability [1,3], providing BS orbitals obtained by the HOMO-LUMO mixing as illustrated in Figure 1(A). They are more or less localised on the left and right terminal atoms (•R-O-R•), respectively, being responsible for 1,3-diradical states with singlet and triplet spin configurations, where • denotes a local spin. The effective exchange integrals (J) in the spin Hamiltonian model are formally defined as 2J = 1 E − 3 E, where X E denotes the total energy of the singlet (X = 1) or triplet (X = 3) state [7]. The same situation appears in the case of dπ-pπ-dπ bonds of the exchange-coupled transitionmetal (M) oxides; (•M-O-M•) as shown in Figure 1(B) [4][5][6]. The HOMO-LUMO mixing of the dπ-pπ-dπ bonds affords the BS orbitals responsible for singlet (low spin) or triplet (high spin) metal-diradical state. Both orbital and spin degrees of freedom play crucial roles for exchange-coupled systems, providing iso-lobal and iso-spin analogies between organic (•R-O-R•) and inorganic (•M-O-M•) exchange-coupled systems as shown in Figure 1(C), where ↑ and ↓ denote the up-spin and downspin, respectively.
The BS methods are applicable to multinuclear exchange-coupled systems. The HOMO-SOMO-LUMO mixing becomes necessary for equilateral triangle systems such as triangular H 3 radical because of the near degeneracy among HOMO, and degenerated SOMO and LUMO in the BS approach [8], entailing an interesting spin state expressed by so-called triangular spin alignments as illustrated in Figure 2(A). Two-component spinors, namely general spin orbitals (GSO), are necessary for BS descriptions of such non-collinear spin structures [8,9]. Similarly the iso-lobal and iso-spin analogies indicate that triangle spin alignments are feasible for triangular manganese oxides Mn 3 O 4 in Figure 2(C) and cubane-like CaMn 3 O 4 cluster in Figure 2(F) [10]. Moreover, tetrahedral spin alignments are obtained for cubane-like structures such as the tetrahedral H 4 cluster in Figure 2(B) [3,9,11]. Such non-collinear spin alignments are also feasible for Mn 4 O 4 clusters with the cubane-like structure [12] and for Mn 4 O 6 clusters with the adamantane-like structure as shown in Figure 2(D) and 2(E), respectively. Thus, magneto-structural correlations are important and interesting for elucidation of electronic and spin states of exchange-coupled cluster systems. The spin structures in Figure 2 are responsible for pictorial expressions of spin-correlation functions for the systems [1].
In the past few decades, a number of electron paramagnetic resonance (EPR) experiments have been performed to elucidate the electronic structure and function of the catalytic site of water oxidation in the oxygenevolving complex (OEC) of photosystem II (PSII) . In the early 1980s, Dismukes and Siderer [25] first reported the EPR spectrum of the S 2 state of the Kok cycle for water oxidation. The EPR spectrum obtained was consistent with an exchange-coupled tetramer of Mn ions. After this discovery [25], Brudvig et al. [26] proposed a cubane-like structure of the Mn 4 O 4 complex in Figure 2(D) for OEC of PSII, which is reorganised into an adamantine-like Mn 4 O 6 complex in Figure 2(E) after the insertion of molecular oxygen. Ferreira et al. [15] found that the London x-ray diffraction (XRD) structure consisted of cubane-like CaMn 3 O 4 cluster in  [15] were almost equivalent, and therefore, our GSO DFT computations [10] indicated that the non-collinear spin structure was the ground state for the cluster as illustrated in Figure 2(F). On the other hand, the Berlin XRD structure [17] provided the isosceles triangle structure for the Mn 3 framework in OEC of PSII that was consistent with the extended x-ray absorption fine structure (EXAFS) structure with shorter Mn-Mn distances (2.7Å) and a longer Mn-Mn distance (3.3Å). This implies that the triangular spin alignment [10] in the London structure [15] is relaxed into an ordinary collinear spin structure [57][58][59][60][61][62][63][64][65] with the axial up(↑)-down(↓)-up(↑) spin alignment in the Berlin XRD structure [17].
Umena et al. [21] discovered the high-resolution XRD structure of the CaMn 4 O 5 cluster in OEC of PSII at 1.9Å resolution as illustrated in Figure 3. They have elucidated almost symmetrical structure of the catalytic site of water oxidation: the CaMn(III) 2 Mn(IV) 2 O 5 cluster in the S 1 state, where the Mn(III) 4(a) -O (5) and Mn(III) 1(d) -O (5) bond lengths are 2.5 and 2.6Å, respectively, and O (5) is the bridge oxygen for the Mn(III) 4(a) -O (5) -Mn(III) 1(d) bond. The triangle Mn 3 structure of the cubane-like CaMn 3 O 4 fragment in their distorted chair structure CaMn 4 O 5 [21] exhibits the isosceles framework that is responsible for the collinear spin structure. The exchange-coupled fourmanganese cluster affords one highest spin (HS) structure (A), four intermediate-spin (IS) structures (B, C, D and E) and three low-spin (LS) structures (F, G and H) as shown in Figure 4. The IS structure E was calculated to be the ground state [60][61][62] for the high-resolution XRD structure [21] in the S 1 state of the Kok cycle (see Figure S1) for water oxidation (substrate waters are shown in Figure S2 However, the computational result was in contradiction to the parallel polarisation EPR result [27] that was consistent with the paramagnetic spin (S = 1) state of the S 1 state of OEC of PSII. Koulogliotis et al. [28] also performed the EPR studies of the S 1 state, concluding that the ground spin state of the S 1 resting state is diamagnetic, whereas the S 1 active state is paramagnetic. Yamauchi et al. [29] have also performed the parallel polarisation EPR studies of the S 1 state followed by the temperature variation experiment, elucidating that the signal at g = 4.8 originates from an excited state with triplet spin state (S = 1) with separation from the diamagnetic ground state (S = 0) of about 2.5 K (1.74 cm −1 ). Thus, the temperature-dependent paramagnetism is familiar in the exchange-coupled Mn oxides in OEC of PSII (see Figure S1).
The above results indicated the necessity of refinements of the high-resolution XRD structure [21] by quantum mechanical (QM) calculations [64,65]. Therefore, full geometry optimisation followed by the vibrational analysis [23,65] was performed for each spin configuration of the model cluster in the S 1 state of OEC of PSII. The energy gaps among these configurations obtained by vertical, adiabatic and adiabatic plus zero-point energy (ZPE) correction procedures have been used for Figure . Spin vector models for the eight spin configurations in the CaMn  O  cluster of OEC. The eight broken-symmetry (BS) UBLYP solutions are available in conformity with these spin vector models where each Mn ion has the local high-spin configuration. The six effective exchange integrals (J) for the Heisenberg spin Hamiltonian model are also determined by the generalised approximate spin projection (GAP) procedure using total energies and total spin angular momentums obtained by the BS UBLYP calculations. computation of the effective exchange integrals (J) in the spin Hamiltonian model [65]. The J values are calculated by (1) analytical method and (2) generalised approximate spin projection (GAP) method that eliminates the spin contamination errors of UB3LYP solutions. Using J values derived from these methods, exact diagonalisation of the spin Hamiltonian matrix was carried out to obtain exact QM energy spectrum and other physical observables such as spin densities of the ground and lower-excited states of the cluster. The exact diagonalisation results [65] for the right (R)-opened structure in the S 1 state in Figure 3(C) indicated the ground singlet state with the temperature excited triplet state (S = 1) that was consistent with the parallel polarisation EPR results [27][28][29].
In this paper, we have performed the full geometry optimisation of each spin configuration of the S 2 state in Figure 4. The above three procedures developed by our group have been applied to elucidate the effective exchange integral (J) in the Heisenberg spin Hamiltonian. The exact diagonalisation of the spin Hamiltonian matrix has been carried out, yielding excitation energies and spin densities of the ground and lower-excited states of the cluster. The exact diagonalisation results of the spin Hamiltonian by the use of the J values obtained by the GAP procedure are found to be consistent with the EPR results for the S 2 state. Implications of the computational results are discussed in relation to (1) the necessity of the exact diagonalisation for computations of reliable energy levels, (2) magneto-structural correlations in the CaMn 4 O 5 cluster of OEC of PSII, (3) structural symmetry breaking (SSB) in the S 1 , S 2 and S 3 states, and (4) the R-and L-handed scenarios for the O-O bond formation for water oxidation. Implications of the computational results are also discussed in relation to the SSB in the labile Mn a -X-Mn d bond, spin crossover phenomenon induced by the NIR excitation, waterassisted proton-shuttle mechanism for the O-O bond formation for water oxidation.

Structural symmetry breaking of the CaMn 4 O 5 cluster
Past several years, the degree of SSB of the Mn a -X-Mn d bond (X = O or OH − in Figure 3) in the CaMn 4 O 5 cluster of OEC of PSII has been under great debates (see supporting material SII.2). Our previous BS DFT computations [57][58][59][60][61][62][63] have revealed that the SSB via the Jahn-Teller (JT) effects of Mn(III) ions is a key concept for theoretical understanding of possible geometries of the labile Mn cluster, and S 2 (x = 1 and z = 3) states, and have revealed three possible geometrical structures: R-opened (R; Mn a -X ….Mn d ), central (C; Mn a -X-Mn d ) and left-opened (Mn a ….X-Mn d ) ones as illustrated in Figure 3(A), 3(C) and 3(D), respectively [23,24,62,63]. If the potential surface has a singlevalley structure, the central structure (C) may become the true minimum. On the other hand, the R-and Lopened structures may become true local minima in the case of the double-well potential (see Figure S3). Previously, the SSB has been investigated by geometry optimisations based on the HS solution [23]. Moreover, the optimised HS geometry has been assumed for other spin configurations [23]. Therefore, in order to confirm previous magneto-structural correlations based on the vertical approximation [23,53,55], we here perform full geometry optimisations of all the spin configurations to elucidate energy levels and relative stabilities for the R-opened structures, S 2ac (R) and S 2ab (R), and left-opened structures, S 2ac (L) and S 2ab (L), of 1, where the notations '2' , 'a' , 'b' and 'c' in S 2XY (X, Y in Figure 3) represent the S 2 state, Y = O 2− , Y = OH − and Y = H 2 O, respectively, and 'L' and 'R' represent the left-side-and right-sideopened structures in Figure 3, respectively. The X and Z in Figure 3 remain O 2− (= a) and H 2 O in the S 2 state, respectively.

Damage-free X-ray free-electron laser (XFEL) structure
Very recently, Suga and their collaborators [66] performed the X-ray free-electron laser (XFEL) experiments of PSII and obtained a radiation damage-free XRD structure of the dark stable S 1 state of PSII by using the femtosecond X-ray pulses of XFEL provided by at the SPring-8 angstrom compact free-electron laser (SACLA) facility [66] and a huge number of highly isomorphous large crystals. The new XFEL structure at 1.95Å resolution [66] was found to be different from the R-opened structure [67] based on the assumption that the O (5) (= X) site (S 1ab(c) ) in Figure 3 was the oxygen dianion [23,24]. Alternatively, the XFEL structure [66] was topologically similar to the high-resolution XRD structure [21] that was suggested to be X-ray damaged on the experimental and theoretical grounds [2,64]. The XFEL structure was theoretically reproduced on the assumption that the O (5) (= X) site was protonated, namely hydroxide anion (S 1bb(c) ) [67,68]. This, in turn, implies that the Y site in Figure 3 may be hydroxyl anion (OH − )(= b) or water molecule (H 2 O)(= c). Thus, the XFEL results [66] raised a fundamental question, namely how to understand the geometrical structure of the CaMn 4 O 5 cluster, indicating the necessity of re-examination of assumptions employed for several theoretical model structures of OEC of PSII, and also for re-examinations of the EXAFS structures reported previously.

Magneto-structural correlations
No XRD structure is reported for the S 2 state at present, though the high-resolution XRD result is available for the S 1 state [21]. Fortunately, a number of EPR experimental results are presented for the S 2 state , affording precise information to elucidate magneto-structural correlations in OEC. Moreover, accumulated biochemical results have suggested similarity between the S 1 and S 2 structures because of no proton release and one electron release in the S 1 to S 2 transition ( Figure S1) , though the EPR results for the S 1 state [27][28][29] are limited. Figures S4A and S4B illustrate our strategy of elucidations of magneto-structural correlations in OEC of PSII based on theoretical and computational results in combination with available EPR experimental results, the high-resolution XRD [21] and XFEL [66] structures, together with EXAFS. EPR experimental results for the S 2 state have been analysed on the basis of spin Hamiltonian model [69][70][71][72][73][74][75][76][77][78][79][80][81]. The parameters such as effective exchange integrals (J) and spin density (Q) obtained by theoretical calculations are crucial for elucidation of magneto-structural correlations in the S 2 state based on available XRD and EPR results . Therefore, both molecular structure determined by XRD [13][14][15][16][17][18][19][20][21][22] and magnetic structure obtained by EPR  would be self-consistently combined for theoretical elucidations of structural bases for water-splitting reaction in OEC of PSII. Moreover, reliable structural information for the S 2 state can be feedback to the S 1 and S 3 states [23,65].
The spin Hamiltonian models have been employed for theoretical analysis of the magnetic resonance and related experiments . In order to elucidate magnetostructural correlations, we have developed the spin Hamiltonian model for four-site four spin systems [57][58][59][60][61]. Six effective exchange integrals (J) for the systems are determined by using the energy gaps among the eight spin configurations in Figure 4 obtained by the vertical, adiabatic and adiabatic, plus ZPE correction methods. Theoretical formulations and analytical expressions of the effective exchange integrals (J) for 1 in the S 2 state are also shown in supporting material SVIII. Derivations of spin Hamiltonian models and magnetic interaction parameters such as hyperfine constants [52,[54][55][56][57][58] are also shown in supporting material SVI. Theoretical details of the reliability tests of BS DFT approach in relation to the density matrix renormalization group (DMRG) computational results [64] have been described in the supporting materials. The magnetic interaction parameters for the R-and L-opened structures of 1 have been evaluated for comparisons with accumulated EPR results  to confirm the magneto-structural correlations [23,65]. The computational methods and basis sets [82][83][84][85] are the same in previous papers, namely the use of G09 program [85] and our own subroutines for the BS computations [23,65] (see supporting information SIII). Scope and applicability of the QM model without protein environments in this paper are briefly discussed in supporting section SII.2.

Optimised geometries of eight spin configurations
Previously, we have performed the full geometry optimisations of the R-and L-opened structures of the CaMn 4 O 5 cluster by using the HS configuration [23]. The optimised geometry has been assumed for other spin configurations in Figure 4. However, reliability of this vertical approximation [23,53,57] is not examined yet for the CaMn 4 O 5 cluster in the S 2 state of the Kok cycle. As a continuation of the previous work [23,65], we have performed the adiabatic and adiabatic + ZPE correction methods, respectively. The newly optimised geometrical parameters for S 2ac (R), S 2ab (R), S 2ac (L) and S 2ab (L), where a = O 2− , b = OH − and c = H 2 O, are summarised in Tables S1A, S1B, S2A and S2B, respectively. For comparisons, the optimised geometrical parameters for the S 1 state are also summarised in Table S3.
As shown previously [23], there are two different geometrical structures relating to the SSB of the Mn a -X-Mn d bond of the CaMn(IV) 3 Mn(III) (1) in the S 2 state as shown in Figure 3.
First of all, we have performed full geometry optimisations of the S 2 state structure (S 2ac (R)) of 1 in Figure 3 by using the UB3LYP energy gradient method. The supporting Table S1A summarises the optimised Mn-Mn and Ca-Mn distances for the eight spin configurations of S 2ac (R). From the computational results in Table S1A, the geometrical parameters are almost the same among the eight different spin configurations, indicating no serious geometry change. Therefore, the average values for the eight spin configurations are also calculated to elucidate general trends for the geometrical parameters.
The average Mn-Mn distances optimised for S 2ac (R) Figure 3) indicated a general trend (rule Ib) [23,24]: This trend (Ib) revealed by full geometry optimisations of the eight spin configurations is common under the assumption that the O (5) site is the oxygen dianion. In fact, the rule Ib is applicable to the proposed structures for the S 1 (or S 2 ) state by other theoretical groups [86][87][88][89][90][91][92][93][94][95][96], where the O (5) site is assumed to be the oxy-  [97][98][99][100][101][102]. The rule Ib is a general trend for the R-opened structure with O (5) = O −2 (= a) even in the two-electron reduced (S −1ac ) state [24]. On the other hand, the corresponding Mn-Mn distances by SP8 XRD [21] and XFEL [66] This trend referred to as the rule Ia was not altered for the damage-free XFEL structure (Mn-Mn distances are given in parentheses) in the S 1 state [66]. The rule Ia is applicable for the optimised structures where the O (5) site is assumed to be protonated, namely hydroxide anion (OH − ) [23,24]. Some of the Mn-Mn distances for the S 1 state by SP8 XRD [21] are a little longer than the corresponding calculated values. Possible origins of the elongations have already been discussed in our previous papers (see SII.4) [57][58][59][60][61][62][63]. The main conclusion is that the structure of the CaMn 4 O 5 framework confined with protein matrix of PSII is reasonable even if the reductions of Mn ions were induced by X-ray radiation, indicating that the SP8 XRD [21] and SACLA XFEL [66] structures are reliable enough for initial trials of theoretical investigations of magneto-structural correlations in OEC of PSII.
Recent discovery of the L-opened structure in the S 2 state [23,24,55] has been a remarkable contribution of theoretical investigations of OEC, showing the shift of the O (5) atom from the right to left side. The geometry optimisations of 1 by the high-spin UB3LYP solution [23] indeed elucidated two different L-opened structures: one is the intermediary L-opened structure S 2aY (L) (Y = b, c) and the other is the fully L-opened structure S 2aY '(L) (Y = b, c). In order to confirm this finding based on the HS solution [23], full geometry optimisations by using other seven spin configurations have been performed to confirm its reliability. Table 1 summarised the optimised geometrical parameters for the S 2ac '(L) structure. The geometrical parameters are almost the same among the eight different spin configurations. Moreover, we have already pointed out the general trend, named rule Ic [23]: The optimised Mn-Mn distances for the eightdifferent-spin configurations (Table 1)  The general trend Ic is also observed for a deprotonated structure S 2ab '(L) in accord with the L-opened structure [23,55,65] with the closed cubane skeleton.  Table S2A summarises the fully optimised geometrical parameters of the S 2ac (L) in Figure 3 [23], whereas it is obtuse in the XRD S 1 structure [21] because of the JT distortion of Mn(III) d [52]. The nearly equilateral triangle may have the HS ground state or the noncollinear LS configuration in Figure 2(C). Thus, general trends Ia-Ic concluded by using the HS solution [23] are not changed after the full geometry optimisations of all the spin configurations in Figure 4. Table S2B summarises the fully optimised geometrical parameters of S 2ab (L) in Figure 1 The optimised Ca-Mn distances obtained for the eight different spin configurations of the R-opened structure, S 2ac (R), by UB3LYP are quite similar as shown in Table   S1A. The average optimised Ca-Mn distances have elucidated a general tendency: This relationship is the same as the distance rule IIa obtained by the high-spin solution [23]. The average Ca-Mn a , Ca-Mn b , Ca-Mn c and Ca-Mn d distances of S 2ac (R) are 3.78(3.82), 3.47(3.55), 3.33(3.28) and 3.60(3.54),Å, respectively, where the corresponding Ca-Mn distances by the high-resolution XRD structure [21] are given in parentheses. The calculated Ca-Mn distances are consistent with the corresponding XRD values. The average optimised Ca-Mn distances for S 2ab (R) also exhibit the trend IIa as shown in Table S1B.
The Ca-Mn distances for the L-opened structure, S 2ac (L), are almost the same for all the spin configurations as shown in Table S2A. The average optimised Ca-Mn distances have provided the following relationship: The above trend is the same as the distance rule IIb obtained by the high-spin solution [23]. The elongated Mn-O distances revealed by the highresolution XRD structure [21] have been under debates among theoretical groups [86][87][88][89][90][91][92][93][94][95][96]. The optimised Mn a -O (5) and Mn d -O (5) distances were quite similar for the eight spin configurations of the R-opened structure, S 2ac (R), as shown in Table S1A. In fact, the average Mn a -O (5) and Mn d -O (5) distances were 1.77 and 3.37Å, in accord with the R-opened structure. The corresponding values are 1.79 and 3.20Å, respectively, for S 2ab (R) as shown in Table S1B. The Mn d -O (5) distance is contracted by 0.2Å by the deprotonation of Y = H 2 O in Figure 3. (5) )]/2, are 0.80 and 0.71Å, respectively. Thus, the R-opened structures are consistent with those of other groups [53,86,91] and our previous results [23,61,62].
The optimised Mn a -O (5) and Mn d -O (5) distances were almost the same for the eight spin configurations for the L-opened structure, S 2ac (L): the situation was the same for S 2ab (L). The average Mn a -O (5) and Mn d -O (5) distances for S 2ac (L) were 2.89 and 1.85Å, respectively, in accord with the L-opened structure in Figure 3(D). The corresponding values for S 2ab (L) are 2.60 and 1.85Å, respectively. Therefore, the Mn a -O (5) distance is shortened by 0.3Å by the deprotonation of Y = H 2 O in Figure 3. On the other hand, the corresponding values for S 2ac '(L) were 3.40 and 1.85Å, respectively. The Mn a -O (5) distance is elongated by 0.5Å in the transition from S 2ac (L) to S 2ac '(L). Therefore, the SSB parameters for S 2ac (L), S 2ab (L) and S 2ac '(L) are 0.52, 0.38 and 0.78Å, respectively.
The above computational results for the eight spin configurations support the use of the HS configuration for qualitative discussions of the optimised geometrical parameters [23,53,55]. However, it is noteworthy that subtle geometrical differences entail significant differences for magnetic interaction parameters as shown later. The almost symmetrical Mn(III) a -O (5) -Mn(III) d bond of the CaMn 4 O 5 cluster revealed by the high-resolution XRD experiment in the S 1 state [21] in Figure 3(A) is collapsed into the R-and L-opened structures [23] as illustrated in Figure 3(C) and 3(D). The R-opened structures, S 2ac (R) and S 2ab (R), in the S 2 state are consistent with the mixed-valence (MV) configuration: Mn(IV) a -O (5) ….Mn(III) d , where the JT effect for the Mn(III) d ion plays a significant role for the geometrical deformation [23]. On the other hand, the JT effect for the Mn(III) a ion affords the L-opened structure: Mn(III) a ….O-Mn(IV) d , in accord with the S 2ac (L), S 2ab (L) and S 2ac '(L) structures. The SSB [23,24] via the JT effect of Mn(III) is one of the most fundamental concepts for theoretical understanding of the nature of labile chemical bonds in the CaMn 4 O 5 cluster of OEC revealed by the high-resolution XRD [21]. The R-opened structure discovered by previous computations [23,53,86,91] is only one of the BS structures of OEC. Present computational results for the S 2 state are consistent with existence of both R-and L-opened structures even in the S 1 and S 3 states [23] in conformity with the labile nature of the Mn a -O-Mn d bond revealed by the BS DFT computations [57][58][59][60][61], though its main framework revealed by XRD [21] is robust in the S 1 , S 2 and S 3 states.

... Right-opened structure
In our early joint papers between theory and experiment [57][58][59][60][61], we have assumed the XRD structures [21] for eight spin configurations ( Figure 4) to elucidate relative stabilities among them. We have shown that the HS configuration (↑↑↑↑) ( 14 A) based on the XRD structure in the ground state was contradicted to the available EPR results that indicate the LS (S = 1/2) ground state in the S 2 state . We have examined several DFT functionals to reproduce the ground doublet state for the XRD structure [21] and found it was a difficult task as shown in S. Yamanaka, et al. [60]. As a continuation work [23,24,[60][61][62][63], we have performed full geometry optimisations of the eight spin configurations to elucidate the optimised geometries and relative energies among them. The vertical and adiabatic energy gaps, and adiabatic + ZPE values for OEC model at the B3LYP/BSI level, have been calculated, where the total energy of the HS ( 14 A) configuration is used as the reference. Table S4A summarises the calculated energy differences for the R-opened structure S 2ac (R). Figure 5 illustrates the energy gaps among the eight spin configurations of the species.
As shown in Figure 5, the relative energies of the eight spin configurations of S 2ac (R) are variable, depending on the spin coupling modes and the optimised geometries. The energy gaps between the ground (↑↓↓↑) and for S 2ab (R) (see Table S4C). This means that the relative stability between the spin configurations (↑↓↓↑) and (↑↓↑↓) may be variable, depending on deprotonation of W2 (H 2 O) and environmental effects such as pH and temperature.
The doublet spin configuration (↑↓↓↑) ( 2 H) (S total = (3 − 3 − 3 + 4)/2 = 1/2) is the ground state for the R-opened structure S 2ac (R) with the MV configuration: Ca(II)Mn(IV) a Mn(IV) b Mn(IV) c Mn(III) d . This trend is independent on the above computational procedures. The situation is not altered after the refinements by the UB3LYP calculations (Table S4B), where more flexible large basis sets (basis set II) have been used. This tendency is also true for S 2ab (R) generated by the deprotonation of Y = H 2 O (Table S4C). The intermediate spin is the lowest excited configuration for S 2ac (R) in the vertical approximation, whereas antiferromagnetic spin alignment (↑↓↑↓), namely the other doublet configuration ( 2 G) (S total = (3 − 3 + 3 − 4)/2 = −1/2), becomes the lowest excited configuration under the adiabatic and adiabatic + ZPE approximations (see Figure 5), indicating the necessity of full geometry optimisations. On the other hand, the doublet (↑↓↑↓) ( 2 G) configuration becomes the first excited state for S 2ab (R) by all three computational methods ( Figure S5 and Table S4C).

... Left-opened structures
We have calculated the vertical, adiabatic and adiabatic + ZPE energy differences for the intermediately L-opened structure S 2ac (L) as shown in Table S4A. The HS configuration (↑↑↑↑) ( 14 A)(S total = (4 + 3 + 3 + 3)/2 = 13/2) is the ground state under the vertical approximation for S 2ac (L) with the MV configuration: becomes the ground configuration in the adiabatic and adiabatic + ZPE approximations. These tendencies are the same for S 2ab (L) ( Table S4C). Full geometry optimisations are crucial for elucidations of the magneto-structural correlations. For the S 2ac (L) structure, the energy gaps between the IS ( 6 E) and HS ( 14 A) configurations are 0.59 and 0.51 kcal/mol for the adiabatic and adiabatic + ZPE approximations, respectively. The corresponding energy gaps are reduced to 0.16 and 0.26 kcal/mol for S 2ab (L), indicating significant effect of deprotonation of W2. Moreover, the HS ( 14 A) and doublet (↑↓↓↑) ( 2 H) excited configurations for S 2ac (L) and S 2ab (L) are almost degenerated in energy under the adiabatic + ZPE approximation, indicating the necessity of the exact diagonalisation of the spin Hamiltonian model. Figure 6 shows the energy levels for the fully Lopened structure: S 2ac '(L) obtained by the vertical, adiabatic and adiabatic + ZPE procedures. From Figure 6, the IS configuration (↓↑↑↑) ( 6 E) is the ground state [23] irrespective to the computational procedures, indicating the stabilisation of the IS ( 6 E) configuration for S 2a '(L) consisted of the HS configuration (↑↑↑) of the cubane fragment and down spin (↓) of the dangling Mn(III) a , namely (3 + 1) model [48] for the MV configuration: Ca(II)Mn(III) a Mn(IV) b Mn(IV) c Mn(IV) d . This trend is not altered by the UB3LYP computations by the use of the extended basis set II (see Table S4B) [23]. The doublet configuration (↑↓↓↑) ( 2 H) is the lowest excited state for S 2ac '(L). The energy gaps between the ground sextet ( 6 E) and doublet excited ( 2 H) configurations are 0.78, 0.65 and 0.48 kcal/mol, for the vertical, adiabatic and adiabatic + ZPE computational procedures, respectively (Table S4A).
The energy differences between the ground states of S 2ac (R) and S 2ac '(L) are 0.50, 0.95 and 1.07 kcal/mol for the vertical, adiabatic and adiabatic + ZPE computational procedures, respectively (Table S4A). Therefore, the R-opened structure S 2ac (R) is more stable by about 1 kcal/mol than the L-opened structure S 2ac '(L) by the latter two methods. However, the energy differences between them become almost zero by the three methods with the basis set II as shown in Table S4B. On the other hand, the energy differences between the ground states of deprotonated S 2ab (R) and S 2ab '(L) are 2.28, 3.69 and 3.52 kcal/mol for the vertical, adiabatic and adiabatic + ZPE computational procedures, respectively (Table S4C). The large energy difference except the vertical one may reduce the appearance of the g = 4.1 signal as in the case of cyanobacteria PSII. Thus, the multiline g = 2 line of EPR is responsible for the structure S 2ac (R) or S 2ab (R) on both experimental and theoretical grounds . We conclude that fully optimised geometries of the eight spin configurations starting from the XRD structure [21] are reliable enough for qualitative discussions on the ground and lower lying excited spin configurations of the CaMn 4 O 5 cluster in the S 2 state of OEC of PSII. Table . The effective exchange integrals a (cm − ) of the Heisenberg spin Hamiltonian model for the left-opened structure S ac '(L) in the S  state of the Kok cycle for OEC of PSII (corresponding values for S ab '(L) are given in parentheses).

Theoretical calculations of effective exchange integrals
Analytical,  GAP, a Vertical, b Adiabatic, c Adiabatic + ZPC.  Table 2.
The situation is the same for the R-opened structure S 2ab (R) (see parentheses in Table 2) [56]. The J ab and J cd values are negative in sign in accord with the antiferromagnetic (anti-parallel) spin alignment (↑↓↓↑) of the ground state of the R-opened structure, S 2ac (R) and S 2ab (R). On the other hand, the J bc value is positive in sign, in consistent with the ferromagnetic spin alignment between the b-and c-sites in these structures. Such qualitative tendencies are independent on the different energy levels obtained by three computational methods and ligands: Y = H 2 O or OH − (see Figure S5).

... Left-opened structures
As well as the analysis of the R-opened structures, we have performed the analysis of the fully L-opened structure S 2ac '(L). The energy levels in Figure 6 are, therefore, mapped into the effective exchange integrals (J pq ) in the spin Hamiltonian model for S 2ac '(L) as shown in Table 3 (see also   [53] are rather similar to those of the S 2ac (L) structure instead of S 2ac '(L) except for the J bd value. The J bd value [53] is positive in sign in accord with the vertical approximation. Thus, J values are highly sensitive to small geometry changes, indicating sensitive fingerprint parameters for elucidations of magnetostructural correlations. The agreements between the calculated and observed J values for both R-and L-opened structures in the S 2 state indicate that these optimised geometries starting from the XRD [21] and XFEL [66] structures are plausible for OEC of PSII, indicating, in turn, the reliability of the framework structure of XRD and XFEL because of the observed similarity between the S 1 and S 2 structures.

Energy levels by the exact diagonalisations
As shown in preceding computational results, electronand spin-correlation effects in the CaMn 4 O 5 cluster in OEC of PSII are qualitatively grasped under the BS hybrid DFT approximations. Indeed, the energy gaps in Figure 5 provide qualitative picture for relative stabilities among spin configurations at the BS DFT followed by GAP procedure to eliminate spin contaminations. However, the GAP procedure is still insufficient because of lack of the configuration mixing of BS solutions. In this paper, the strong electron-correlation effects involved in the CaMn 4 O 5 cluster are mapped into the effective exchange interactions (J) in the Heisenberg model. Therefore, the energy levels at the exact quantum mechanical level can be obtained by the exact diagonalisation of spin Hamiltonian model for further refinements of the BS DFT computations. The dimension of the spin Hamiltonian matrix becomes 320 = 5 × 4 × 4 × 4 where 5(= 2(4/2) + 1) for Mn(III) and 4(= 2(3/2)+1) for Mn(IV) in the S 2 state. Figure 7 illustrates the energy levels of the ground and lower excited states for the L-opened structure, S 2ac (R). The procedure of the exact diagonalisation of the spin Hamiltonian was described in Section SVII. The results show that the doublet state (S = 1/2) is the ground state S 2ac (R) that is independent on the computational methods of J values. The spin densities of the ground doublet state exhibit the plus(+)-minus(−)-minus(−)-plus(+) topology in consistent with the LS (S = 1/2) configuration (↑↓↓↑) at the BS level computation (see Figure 5), indicating utility of the BS DFT approach as a first step to the CaMn 4 O 5 cluster in OEC of PSII. The energy gap between the ground doublet and lowest excited quartet states is over 30 cm −1 for S 2ac (R). The corresponding experimental excitation energies are 35 [104], 21.7 [50], 23.5 [50] and 26.5 [52] cm −1 , respectively, supporting the computational result based on the optimised geometry starting from the XRD structure [21]. The small energy gaps between the ground and excited states indicate thermal mixing of them at a room temperature: for example, 300 K, kT = 208 cm −1 . Figure 8 illustrates the energy levels of the ground and lower excited states for S 2ac '(L). The results show that the sextet state (S = 5/2) is the ground state that is independent on the computational methods of J values. The spin densities of the ground sextet state exhibit the minus(−)-plus(+)-plus(+)-plus(+) topology in consistent with the IS (S = 5/2) configuration (↓↑↑↑) at the BS level computation (see Figure 6). The IS (S = 9/2) state is the lowest excited state, independent on the computational procedures of J. The energy gap between the IS S = 5/2 and S = 9/2 states is over 25 cm −1 . Furthermore, the energy gap between the S = 5/2 and S = 7/2 states is over 100 cm −1 for S 2ac '(L). Thus, the multiplet states (S = 5/2, 7/2 and 9/2) are not nearly degenerated under the assumption Y = H 2 O. However, the S = 5/2 and S = 9/2 states are thermally mixed at a room temperature (>208 cm −1 ) As well as the above discussions, the energy levels of S 2ab (R) and S 2ab '(L) were illustrated in Figures S5 and  S6, and Figures S7 and S8, respectively. As for S 2ab (R), the doublet state (S = 1/2) is the ground state, independent on the computational procedures of J values (Figures S5 and S6). The spin densities for the ground doublet state also exhibit the plus(+)-minus(−)-minus(−)plus(+) topology (↑↓↓↑) in Figure S5 (see also Figure 4). The energy gap between the ground doublet and lowest excited quartet states is over 20 cm −1 for S 2ab (R) in compatible with the experimental values [50,52,104]. The computational results indicate that the LS doublet (S = 1/2) configuration observed by EPR can be reproduced under both assumptions (Y = H 2 O and OH − ) in the Ropened structure.
As for S 2ab '(L), the ground state is the highest spin at the vertical level of calculation ( Figure S7). However, the sextet (S = 5/2) state is the ground state under the adiabatic and adiabatic + ZPE correction, indicating the importance of full geometry optimisation and the quantum correction. On the other hand, the septet (S = 7/2) becomes the ground state after the exact diagonalisation (see Figure S8). Moreover, the energy gap between the ground (S = 7/2) and the next excited intermediate (S = 9/2) states is smaller than 15 cm −1 as shown in Figure  S8. The multiplet states (S = 5/2, 7/2 and 9/2) are nearly degenerated for the S 2ab '(L) under the assumption Y = OH − . Thus, the sign and magnitude of J bd play an important role for energy levels of these states, indicating discrimination between H 2 O and OH − at the W2(Y) site. The calculated S = 7/2 spin state will be further discussed (see later) in relation to the ESR spectroscopy of the OEC after the irradiation of the NIR light, where the S = 7/2 state has been detected [105,106]. Judging from all the computational results mentioned above, the BS approach followed by the exact diagonalisation is useful enough for elucidation of the excitation energies of the CaMn 4 O 5 cluster in the S 2 state of OEC of PSII. The calculated excitation energies support the assumption of the main cluster skeleton revealed by XRD [21] for the S 2 state, though the XRD does not provide information on discrimination between H 2 O and OH − at the Y-site.

Theoretical calculations of spin densities
Extensive 55 Mn EPR measurements have described by using the projection factors (spin densities) based on the observed hyperfine constants (A iso ) of the Mn ions (see Table S6) in the multiline g = 2 spectra for the doublet ground state of the CaMn 4 O 5 cluster in OEC of PSII. The projection factors obtained by the experiments [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56] are summarised in Table 4. The magnitude of the projection factors (ρ) for S 2ac (R) indicates the following tendency: According to the EPR results [52],  (7). Table 4 summarises the spin densities for the ground doublet state obtained by the exact diagonalisation of spin Hamiltonian models, which include effective exchange integrals (J) obtained by three different procedures [65]. The spin densities on Mn(IV) a and Mn(III) d are almost equivalent in contradiction to the experimental tendency when the J pq values are determined by using the vertical energy levels in Figure 5. The results are independent on the computational procedures of the J pq values. On the other hand, the spin density on Mn(III) d becomes the largest value for all the cases by using J pq values obtained by the adiabatic and adiabatic + ZPE energy levels. Moreover, the general trend in Equation (7) is reproduced by the analytical, fitting and GAP procedures by the use of adiabatic and adiabatic + ZPE energy levels. The full geometry optimisations for the eight spin configurations are not at all trivial for reproducing the general tendency for projection factors of the EPR experiments .
The   (7). This, in turn, implies that discrimination between H 2 O and OH − at the Y-site only by the observed projection factors by EPR is still difficult. The ESEEM [48] and ENDOR [56] results were consistent with Y = H 2 O and OH − , respectively. The spin densities (Q) of the Mn ions for S 2ac '(L) were also obtained by the exact diagonalisations as the same as above. The results are summarised in Table 5. The results show that the magnitude of the spin densities Q indicates the following tendency: Based on the vertical energy levels, the spin densities Q for the Mn p (p = a, b, c, d) ions are −1.32, 1.07, 1.32 and 1.40 by the analytical procedure, and −1.25, 0.97, 1.32 and 1.42 for the GAP procedure, respectively. The signs of the spin projections are consistent with the spin alignment (↓↑↑↑) for the ground state of the S 2ac '(L) structure. Based on the adiabatic + ZPE energy levels in Figure 8, the corresponding spin densities Q for S 2ac '(L) are −1.30, 1.02, 1.35 and 1.40 by the analytical procedure, and −1.02, 0.72, 1.30 and 1.47 for the GAP procedure. The topology of the Q values is also consistent with the spin alignment (↓↑↑↑) (see also Table S7B). Thus, the experimental projection factors for the S 2 state of OEC of PSII by EPR [48] are correctly reproduced by spin densities (or projection factors) obtained for the CaMn 4 O 5 cluster by the exact diagonalisation of the spin Hamiltonian matrices consisting of the calculated J values. This means that the R-and L-opened structures in Figure 3 are plausible models for the S 2 state, supporting the main framework by the high-resolution XRD structure [21] used for starting trials for geometry optimisations of the S 1 and S 2 states.

Comparison between the calculated and experimental hyperfine constants
Extensive CW-EPR and pulsed-ENDOR measurements [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56] have revealed the hyperfine constants of the 55 Mn ions in the multiline g = 2 spectra for the doublet ground state of the OEC of PS II. Britt et al. [43][44][45][46][47][48] [39] have also presented the similar A iso values. However, the notations and assignments of Mn ions are different between the experimental groups [39,48,52]. Therefore, we have reassigned the Mn ions on the basis of the sign and magnitude of the spin densities obtained by the exact diagonalisation of the spin Hamiltonian matrices consisting of J by GAP. Table 6 summarises the isotropic hyperfine constants A iso obtained by the average of the observed A xx , A yy and A zz components (details are given in supporting material SVI).
The results show that the magnitude of the hyperfine components for the multiline g = 2 spectra for the doublet ground state of the OEC of PSII indicates the following tendency: The corresponding A iso values by Britt et al. [43][44][45][46][47][48] Table 6 are consistent with the spin alignment (↑↓↓↑) for the ground state of the S 2ac (R) structure. Thus, the sign and magnitude of the experimental A iso values by the three groups [39,48,52] are consistent within error bars, indicating that the observed A iso value at the Mn(III) d site is the largest among the four Mn sites.
The calculated spin densities (Q) for the R-opened structure, S 2ac (R) or S 2ab (R), are consistent with the sign and magnitude of the hyperfine constants for the S 2 multiline spectra. Therefore, we have calculated the A iso values by scaling the calculated spin densities by using a simple relationship: A iso = − aQ, where the scaling constant a   Tables S4A and 5). The magnitude of the A iso value on the Mn d(1) by these computations becomes larger than that of Mn a(4) , indicating the tendency concluded by the EPR experiments [39,[43][44][45][46][47][48]52]. The calculated A iso value for Mn d(1) is consistent with the observed values (about 400 MHz) for the Mn(III) ion in model complexes [69]. This indicates that full geometry optimisations of each spin configuration are desirable for improvements of subtle energy differences for the eight spin configurations. The same tendency is obtained by all other computations examined later. For example, the corresponding A iso values are, respectively, −279, 204, 185 and −318 by the GAP procedure based on the same adiabatic energy level.
The addition of the ZPE correction to adiabatic energy level has also been examined. The  [40]. The ratio A iso (Mn d )/A iso (Mn a ) becomes 1.25 in accord with the experimental value 1.24 [40]. Thus, the computational results are consistent with the available experimental values for the multiline g = 2 EPR spectra, indicating that the A iso value on the Mn(III) d ion is the largest among those of four Mn ions of the CaMn 4 O 5 cluster of OEC of PSII [39,[43][44][45][46][47][48]52]. Table S8A summarises the hyperfine components (A iso ) for the R-opened structure S 2ab (R). The  Table S4C. The experimental trend in Equation (9) is reproduced for S 2ab (R) even by the vertical energy gap. The corresponding A iso values are, respectively, −204(−251), 168(208), 193(191) and −366(−312) by GAP based on the adiabatic energy gaps, indicating the experimental trend: the experimental values are given in parentheses [52]. Therefore, the ratio A iso (Mn d )/A iso (Mn a ) is 1.79(1.24) [40]. The computational results for the S 2ab (R) structure are also consistent with the accumulated EPR experiments [39,[43][44][45][46][47][48]52]. This means that the A iso values are not conclusive for discrimination between H 2 O and OH − at the Y-site, indicating the necessity of the ENDOR experiment [56].
The CW EPR g = 4.1 spectra is too broad to determine the hyperfine constants for the sextet (S = 5/2) ground state of the OEC of PSII. However, the ammonia-treated sample exhibited the weak hyperfine lines for the g = 4.1 spectra. The simulations of the line by Britt et al. [43][44][45][46][47][48] have shown that the A iso values for the Mn a , Mn b , Mn c and Mn d ions are, respectively, 117, −238, −259 and −327 MHz as shown in Table S8B. Previous [23] and present computational results indicate that the L-opened structure, S 2ac '(L), is responsible for this broadband. We have calculated the A iso values for the S 2ac '(L) structure by using the same procedures for the S 2ac (R) structure. The calculated values are summarised in Table S8B Figure 6. The tendency for the absolute A iso values indicates that the A iso value of the Mn d ion is the largest. The relationship is applicable for all other parameter sets examined here. The calculated A iso values are consistent with the observed values by Britt et al. [43][44][45][46][47][48] except for the small observed A iso value for the Mn a ion. This discrepancy, in turn, suggests that the ammonia may be ligated to the Mn(III) a ion of the L-opened structure (see Figure S3) in the ammonia-treated sample with the g = 4.1 spectra exhibiting no oxygen evolution.
The isotropic hyperfine constants (A iso ) have been regarded as very sensitive fingerprint parameters for magneto-structural correlations for the CaMn 4 O 5 cluster of OEC of PSII . The calculated A iso values for the R-and L-opened structures are consistent with the magnetic interaction parameters revealed by accumulated EPR and related experiments in the S 2 state . This means that the main geometric structures by the high-resolution XRD [21] and XFEL [66] in the S 1 state are reliable enough for refinements of the S 1 and S 2 structures by eliminating the experimental errors with full geometry optimisations that are applicable for the eight spin configurations in Figure 4.

Comparison with the EXAFS structure
Since the reports of XRD with 1.9Å high resolution [21], the structural studies on OEC of PSII have dramatically advanced. However, at present, the reported OEC structure is only in the static S 1 state, not in the S 2 state, where detailed magnetic structural work has been reported. EXAFS results have provided essential distance information for both S 1 and S 2 states. The EXAFS results [97][98][99][100][101][102] [24] are rather close to the S 0 results by EXAFS [97], where the corresponding QM/MM values combined with EXAFS by Luber et al. [90] are also given in parentheses. These results indicated one shorter (2.7Å) and two longer (2.8Å) Mn-Mn distances (see Table S3) in contradiction to the EXAFS results [97]. Situations were the same for the QM computations by other groups [53,89].
Refinements of the present J pq and A iso values by QM/MM calculations will be published elsewhere.

Magneto-structural correlations in OEC
In the past decade, we have investigated magnetostructural correlations in OEC as illustrated in Figure 9. In our early papers [5,6,10,12,[57][58][59][60][61][62][63]65,80,108] [80,108] clusters were precursors for OEC of PSII. The doublet ground state S = 1/2 in the S 0ac state with the XRD structure was resulted in accord with the EPR experiment. On the other hand, the IS (S = 3) and HS (S = 13/2) configurations were predicted in the S 1ac and S 2ac states, respectively, under the assumed XRD geometry [21] in contradiction to the available experimental results, indicating the necessity of refinements of the XRD structure [62,63]. The optimised R-opened structure S 1ac (R) [65] exhibited the thermally excited triplet state that was detected by the parallel EPR spectroscopy [27][28][29]. On the other hand, the optimised structure S 1bb (C R ) with X = O (5) = OH − and Y = OH − (b denotes OH − ) provided large (>25 cm −1 ) triplet excitation energy, indicating the EPR silent state at 4.8 K in accord with the EPR experiments in polar conditions [27][28][29]65]. The optimised S 1ac (R) and S 1bb (C R ) structures [67,68] were found to be compatible with the EXAFS [97] and XFEL [66] structures, respectively (see Section 4.1).
Full geometry optimisations in the S 0 state were performed to elucidate possible structures in the oneelectron reduction state, providing the S 0ac (R) and S 0bc (C R ) structures as shown in Figure 9 [23,62]. Therefore, the degree of contamination of S 0bc (C R ) in the XFEL structure (C R ) in the S 1 state [66] remains as one of the many unsolved problems [67,68]. On the other hand, the R-and L-opened geometrical isomers were obtained by the geometry optimisations of the S 2ac (R) (or S 2ab (R)) and S 2ac (L) (or S 2ab (L)) structures in the S 2 state generated in the one-electron oxidation of the S 1 state [23,62]. The spin structures of these isomers were thoroughly investigated in this paper. The proton release becomes necessary for transition from S 1ac (R) or S 1bb (C R ) to S 2ab (R) or S 2ab (L). However, the proton release from OEC was found to be very small in this step, indicating the necessity of proton trapping by amino acid residues such as Asp61, Glu65 and Glu312 [62,63,107]. The geometrical and spin structures of the S 3ab (R) and S 3ab (L) isomers were also examined previously [65]. The S = 3 ground state with the (↓↑↑↑) spin structure was the ground state for S 3ab (L), whereas S = 3 with the (↑↓↑↑) spin structure and S = 0 with the (↑↓↓↑) spin correlation were nearly degenerated for S 3ab (R) [23,65]. This means that the relative stability between them may be variable, depending on environmental effects, indicating the necessity of detailed experimental and theoretical investigations.

Spin crossover phenomena induced by the NIR irradiation
In cyanobacterial PSII, the g = 4.1 EPR signal has not been reported . Biochemical procedures leading to the formation of this signal in plant PSII do not yield the detectable g = 4.1 signal in cyanobacteria. Boussac et al. [36][37][38][39]105,106] have detected a third signal at g = 10 and g = 6 optimally formed by IR illumination of the S = 1/2 ground state. This signal converges to the g = 4.1 state in the case of plant PSII, whereas it decays directly to the S = 1/2 multiline signal in cyanobacteria PSII. Therefore, they have concluded that the g = 4.1 state in cyanobacteria, should it exist, is unstable and cannot be trapped by the EPR signal. The present computational results may provide a possible explanation of the absence of the g = 4.1 signal in cyanobacteria.
The energy difference between the R-opened S = 1/2 and the L-opened S = 5/2 configurations is not so large in present QM model calculations as shown in Tables  S4A and S4B. However, previous QM/MM computations [24,107] using the protein matrix of the high-resolution XRD structure for Thermosynechococcus (T.) vulcanus [21] have revealed the greater stability (over 10 kcal/mol) of the S = 1/2 R-opened structure than the S = 5/2 Lopened structure. The projection factors and isotropic hyperfine constants obtained for the R-opened structure by the QM/MM method are consistent with the available experiments as shown in Tables 4 and 5. The situation is the same for the L-opened structure as shown in Tables 6 and S8C. Therefore, the large energy difference between the R-and L-opened structures is responsible for the absence of the g = 4.1 EPR signal for cyanobacteria. We have also shown that the intervalence transition (IVT) between Mn(III) and Mn(IV) in the mixed valence Mn complexes is often thermally impossible because of tight ligand fields [63]. The present computational results provide a possible energy diagram for the spin crossover phenomenon induced by the IR illumination of OEC of PSII as illustrated in Figure 10 [36][37][38][39]105,106].
The vertical IVT from Mn(III) d to Mn(IV) a in the doublet (↑↓↓↑) ground state of the R-opened structure affords the doublet (↑↓↓↑) (S = 1/2) excited state: it is noteworthy that the up-spin transfer (spin delocalisation) is feasible between parallel spin alignments [63].  (5) ) is induced by the IVT as illustrated in Figure 10. The newly formed S = 7/2 state with the Lopened structure may be assigned as the S = 7/2 state of the L-opened structure S 2aY (L) (Y = b or c) with the third signal at g = 10 and 6. The excitation energy from the S = 7/2 state of S 2ab (L) to the excited S = 5/2 state is over 10 cm −1 (Table S7B). On the other hand, the S = 5/2 state with g = 4.1 signal becomes the ground state for the L-opened structure S 2ac (L) (Y = H 2 O). The excitation energy from the S = 5/2 state of S 2ac (L) to the first excited S = 9/2 state is over 25 cm −1 . Therefore, the third signal state with g = 10 and 6 observed in both plant and cyanobacteria PSII may collapse to the g = 4.1 signal state with S = 5/2. However, the lifetime of the g = 4.1 state may be too short to be detected by EPR [105,106] because it is a high-energy state in the case of cyanobacteria. The present theoretical model is consistent with the available EPR results [36][37][38][39]105,106,109]. Further EPR and ENDOR experiments on the L-opened structure are expected to discriminate between H 2 O and OH − at the W2(Y) site.

One g2 and two g4 S2 structures
The DFT calculations revealed one R-opened S 2 structure with the g2 (S = 1/2) spin state detected by EPR. The Ropened S 3 (R) structure with S = 3 (or 0) optimised by DFT was similar to that of the S 2ac (R) g2 state. Indeed the SSB parameters were 0.66 and 0.79(0.80) for S 2ac (R) and S 3ac (R) (S 3ab (R)), respectively, indicating the structural similarity among them [23]. On the other hand, two L-opened S 2 structures with the same g4 (S = 5/2) spin state were revealed by the BS DFT calculations [23,55]. The SSB parameters for S 2ac (L) and S 2ac '(L) were 0.47 and 0.77, respectively, showing the different degrees of the SSB. The Mn a -O (5) bond length for S 2ac (L) was shorter than 3.0Å, indicating no possibility of insertion of small molecules such as H 2 O, NH 3 , F − , etc. The Mn a -O (5) bond length for S 2ab '(L) was longer than 3.3Å, indicating vacancy for the insertion of such small molecules. Indeed, the SSB parameter was 0.77 for water-inserted S 2ac '(L) (W = H 2 O in Figure S3). Similarly, the SSB parameters were 0.83 and 0.89, respectively, for the L-opened S 3 structure, S 3ab (L), and water-inserted S 3ac (L) (W = OH − in Figure 3). Judging from the SSB parameters, S 2ab '(L), S 3ab (L) and water-inserted S 3ac (L) exhibited structural similarity.
The two different g4(S = 5/2) L-opened structures revealed by our DFT computations are expected to be responsible for complex behaviours of the EXAFS spectra [97][98][99][100][101][102]. The different behaviours of the Berkeley EXAFS spectra for the S 2 and S 3 states in the Kok cycle of OEC of PSII [97][98][99][100][101][102] are compatible with the geometrical change from S 2ac (R) (or S 2ab (R)) to S 3ab (L) (or water-inserted S 3ac (L)) in the S 2 -S 3 transition. In fact. the  [23,58,65]. On the other hand, S 3ab (R) structure was consistent with the A-model of the latest EXAFS [97]. The XFEL experiments for the S 3 state at high resolution (<2.5Å) will provide clear-cut evidence for threedimensional discrimination between S 3 (R) with R(Mn a -Mn b ) = 2.7Å and S 3 (L) with R(Mn a -Mn b ) = 3.4Å.

Right-and left-handed scenario for water oxidation
The R-and L-opened structures revealed by the BS DFT calculations [23,55] open R-and L-handed scenarios for the O-O bond formations of water oxidation in OEC of PSII. In fact, full geometry optimisations have revealed that water molecule can be inserted into the coordination-unsaturated Mn(III) d site in the R-opened structure, whereas it is inserted into the Mn(III) a site in the L-opened structure generated by the oxygen (O (5) shift) as shown in Figure S3 [23]. Past several years, Siegbahn [86][87][88][89] has theoretically investigated several radical-coupling pathways (for example, O (2) -O (4) coupling [58]) for the water splitting reaction, proposing several R-handed transition structures for water oxidations. Our computational results have shown that the L-handed scenario presented by several groups [9,[57][58][59][60][61][62][63]108,[110][111][112][113][114][115][116] in the past decades still remains as a possible route for water oxidation. We have located a transition state for the O-O bond formation in the L-handed scenario based on the small QM model (QM Model I in our notation) [59]. However, as shown in our recent paper [24], valine185 in front of the O (5) site may play significant role(s) [117] for controlling relative stability between the R-and L-type transition structures: they are supposed to be highly dependent on structures of protein matrix employed for QM/MM modelling [24] of OEC of PSII [21] (see also Figure 12).

Spin Hamiltonian model for radical coupling reactions
Historically transition-metal oxo bonds had been formally regarded as M(m + 2)O 2− , (m = 1-3) indicating the nucleophilic reactivity of oxygen dianions. Therefore, it was so surprising that in the early 1980s, the high-valent (m = 2, 3) transition metal oxo species (M = Fe, Mn, etc.) were found to exhibit the radical and/or electrophilic reactivity as demonstrated in the case of epoxidation reactions of double bonds and hydrogen abstraction reactions from H-C bonds of alkanes, etc. In the early 1980s, we have examined the HOMO-LUMO mixing of the dπpπ bonds of transition metal oxo (M = O) species [4,6], showing the metal diradical character (•M-O•) that can be grasped by spin Hamiltonian models [118]. In this series of papers [79,108], we have examined the scope and applicability of the BS approaches to the species. Developments of our theoretical approaches to the manganeseoxo Mn = O (•M-O•) species were summarised previously [80].
Spin Hamiltonian models examined in this paper have been used for theoretical modelling of radical reactions [118]. The radical abstraction reaction with linear tri-centre system was characterised as exchangeallowed because of smooth bond interchange of covalent bonds. On the other hand, the radical insertion reaction through the triangle geometry in Figure 2(A) is exchange-forbidden because of the necessity of sudden switching of exchange couplings [118]. The radical reaction through planar four-centre transition structure was regarded as exchange-allowed [119] because of no curve crossing, whereas the radical reaction through the perpendicular (D 2d ) transition structure in Figure 2(B) was characterised as exchange-forbidden because of the curve crossing between different spin structures [119]. Thus, the effective exchange interactions (J) were useful for prediction of allowed or forbidden radical coupling modes, providing spin-correlation diagrams for radical reactions [120]. The spin Hamiltonian models [118][119][120] were applied for the derivation of selection rules of radicalcoupling reactions in native and artificial water-oxidation catalysts [108]. They were successfully applied to the Tanaka-type Ru catalyst for water oxidation [121,122]. The spin-correlation diagrams were already presented in our previous papers [6,[57][58][59][60][61][62][63]80,108] for pictorial understanding of variations of exchange coupling modes in the water oxidation.
As an example [108], Figure 11 illustrates the spincorrelation diagrams for water oxidation (see Equation (1)) in a CaMn 4 O 5 model complex that has been proposed before discovery of the high-resolution XRD structure [21] (see Figure 6 in Part XV [108] of this series of  Figure 11. The assumption was supported by the DFT computational [65] and experimental [21,103] results. The local singlet diradical (LSD) configuration is necessary for the exchange-allowed O-O bond formation in an early stage, whereas the spin inversion is necessary after the O-O bond formation to release triplet molecular oxygen (local triplet diradical (LTD) mechanism) as illustrated by the red allows. The near degeneracy between the ground and lower excited states is responsible for such spin crossover. Thus, spin alignments are not at all trivial for pictorial understanding of radical mechanisms for water oxidation in OEC of PSII [108,[118][119][120][121][122].

Proton shuttle mechanism for water oxidation in OEC
In radical coupling mechanisms in Figure 11, the Ca ion and hydrogen-bonding networks for the CaMn 4 O 5 cluster (see Figure S2) play no significant role for the O-O bond formations. Our previous QM/MM computations [24,107] have revealed the channel structures for proton transfer and water inlet for water oxidation in Equation (1) in OEC of PSII. Previously [23], we have proposed a water-assisted proton-shuttle mechanism for the water oxidation based on (1) the Lewis acid character of the Ca(II) ion, (2) proton-accepting (base) ability of the O (5) Figure 12; the latter state corresponds to the ((3444) with (↓↑↑↑) + O•(↑)) structure [98]. After photoinduced one-electron oxidation in the S 4 state, the following reactivity scenar- of the Ca-O (5) bond and hydrophobic environment by Val185 enhance the base character of the O (5) site. On the other hand, mutation of Val185 by amino-acid residues with the OH group has prevented the oxygen evolution [117] because of the protection of the O (5) site with the hydrogen bonding [118]. The W5 in the hydrophilic environment in OEC (see Figure S2) stabilises the generated MnOOH species [23]. The superposed state of the above two extremes (A) and (B) may also be feasible, providing a chameleonic mechanism (C) [23,80] where the weight of the structure (B) may be variable with the environmental conditions. Locations of reaction pathways for water oxidation by realistic QM/MM models [107] are inevitable for further discussions on (A)-(C) mechanisms for water oxidation [110,112,115,116].

Conclusion remarks
As a continuation of previous theoretical studies on the S 1 and S 3 states [65], we have elucidated magneto-structural correlations in the S 2 state of OEC of PSII. Theoretical discovery [23,55] of the L-opened structure in the S 2 state of OEC of PSII based on the BS UB3LYP computations has opened a new structural basis for lucid understandings of accumulated EPR and related experimental results  because previous theoretical computations have provided only the R-opened structure [86][87][88][89][90][91][92][93][94]. The optimised geometrical parameters for the CaMn 4 O 5 cluster model in the S 2 state are consistent with the available EXAFS results of OEC of PSII [97][98][99][100][101][102], though the high-resolution S 2 XRD structures are not available yet. Recently, Kern et al. [123] presented the XFEL structure for the S 2 state, but the resolution was not sufficient for elucidation of subtle geometry change in the S 1 -S 2 transition. The full geometry optimisations have elucidated three different rules (Ia-Ic) for the Mn-Mn distances in the cluster. The calculated magnetic interaction parameters by BS DFT have concluded that the R-and L-opened structures [23,55] in the S 2 state are assigned, respectively, as the g = 2.1 multiline doublet (S = 1/2) and g = 4.1 sextet (S = 5/2) states observed by the EPR spectroscopies . The spin crossover phenomenon from the R(S = 1/2) to L(S = 5/2) state induced by the infrared light illumination [36][37][38][39] in cyanobacteria PSII is also explained based on the computational results for the model clusters with and without deprotonation of water molecule coordinated to the Mn a ion. Thus, the SSB in the labile Mn-O-Mn bond [57][58][59][60][61][62][63] of the CaMn 4 O 5 cluster in Figure 3 is one of the basic features for the active site of water oxidation in the high-resolution XRD structure [21] of OEC of PSII. The discovery of the L-opened structures in the S 3 state [23,55] supported our previous left-handed scenarios [23,58,108,124] for the O-O bond formation in water oxidation in Equation (1) (see Figures 11 and 12).
The energy gaps among the eight spin configurations have been calculated by the vertical, adiabatic and adiabatic + ZPE correction methods based on the geometry optimisations by the BS UB3LYP computations. These energy gaps have been mapped into the effective exchange integrals (J) in the spin Hamiltonian model by the GAP procedure eliminating spin contamination errors of the BS approximation [65]. The exact diagonalisation of the spin Hamiltonian model consisting of the J values has been performed to obtain the excitation energies and spin densities (projection factors) which are utilised for elucidation of the magneto-structural correlations in OEC of PSII. The full geometry optimisations of the eight spin configurations have been necessary for the reproduction of the EPR experimental results . The calculated spin densities (Q) on the Mn ions are mapped into the isotropic hyperfine constant (A iso ) of the 55 Mn d(4) site by using a scaling factor: A iso = −aQ (a = 208 MHz). This simple and practical procedure has supported the magneto-structural correlations [23]. The present theoretical computations of magnetic interaction parameters for the CaMn 4 O 5 cluster have provided reasonable explanations of the EPR  and EXAFS results in the S 2 state of OEC of PSII [97][98][99][100][101][102]. Spin Hamiltonian models examined have been applied for the derivation of spin-correlation diagrams of water-oxidation processes in OEC of PSII [57][58][59][60][61][62][63].
In conclusion, theoretical computations based on QM models [57][58][59][60][61][62][63] for OEC of PSII are crucial for unification of available geometrical, electronic and spin-state information obtained by XRD, XFEL, EXAFS, EPR and optical results for the active site (the CaMn 4 O 5 cluster) of OEC (see Figures S4A and S4B). The BS DFT computations of the cluster models in the S 1 , S 2 and S 3 states of OEC [23,24,[57][58][59][60][61][62][63] have revealed that the CaMn 4 O 5 cluster in OEC of PSII can be regarded as a typical SCES, where the orbital, spin and charge degrees of freedom play important roles for theoretical description of labile Mn-O bonds confined with proteins. In this regard, the mean-field hybrid DFT approach is regarded as a first theoretical step to elucidate the mechanisms of water oxidation in OEC of PSII. Mapping of the BS DFT results to the spin Hamiltonian models followed by the exact diagonalisation provided the energy levels and projection factors that are responsible for the EPR and optical spectroscopies. Post-DFT approaches are crucial for confirmation and refinement of the hybrid DFT results and spin Hamiltonian models.