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Supporting material
SI Universal MO-VB approach via BS hybrid DFT calculations
SI.1 Natural orbital analysis of broken-symmetry hybrid DFT solutions
Spin-polarized Hartree-Fock (HF), Hartree-Fock-Slater (UHFS), DFT (UDFT) and hybrid UHF-UDFT
calculations have been performed for transition-metal clusters. These broken symmetry computational
methods provide molecular orbitals concepts that are useful and handy for lucid understanding of chemical
bonds of the transition metal complexes such as CaMn4O5 cluster. However, BS orbitals themselves are
often complex because of spin polarization effects.
In order to obtain a simple MO picture, the first-order
density matrix ( 1(r1,r2 )) of the HF and related BS solutions is diagonalized as
1(r1,r2 )    i†i ds   nii†i ,
i
(s1)
i

where  i denotes the BS MO i, and  i and n i mean, respectively, the natural molecular orbitals
(NMO) and the occupation number. The BS MOs are expressed with the bonding and antibonding NMOs


pair as


*
 i  cosiHOMOi +siniLUMOi
(s2a)
*
 i  cosiHOMOi  siniLUMOi
(s2b)
where i denotes the orbital mixing parameter determined by the SCF calculations. HOMO and LUMO
denote, respectively, the highest-occupied and lowest-unoccupied molecular orbitals, respectively. Since
*
are symmetry-adapted delocalized orbitals and usually belong to different spatial
 HOMO i and LUMOi
symmetries, BS MOs are often spatially symmetry-broken; namely they are more or less localized orbitals,
showing a characteristic feature of the broken-symmetry (BS) methods. The number (i) of the BS MOs in
equation (s2) is over ten, indicating that complete active spece (CAS) for symmetry-adapted (SA) CI-type
treatments of CaMn4O5 cluster are larger than twenty-orbitals twenty electrons {20, 20}. SA CASSCF
{20,20} and CASPT2 are hardly applicable to the active sites of OEC[1], even now.
The orbital overlap Ti between corresponding BS MOs in equation s2 is introduced to express
localizability of broken-symmetry orbitals. It is defined as
Ti = i+ i–  cos2 .
(s3)
The occupation numbers of the natural molecular orbitals (NMO) are expressed by the orbital overlap as
follows:

nHOMO i = 1 + Ti
(s4a)
nLUMO i = 1 - Ti
(s4b)
+
–
The orbital overlap Ti becomes 1.0 in the case of the closed-shell case;  i   i   i ; this means that BS
MOs reduce to the conventional closed-shell MOs at the instability threshold, showing that BS MOs are
natural generalization of the MO approach to non-closed-shell species such as CaMn4O5(H2O)4.

The
closed-shell MO pictures accompanied with cluster symmetry are already used for beautiful explanation of
stable organometallic cluster compounds.
On the other hand, Ti is 0.0 for the complete mixing case (i =
/4); this corresponds to the pure open-shell state: for example, localized d-electrons in CaMn4O5(H2O)4.
Furthermore, the BS MOs with smaller nonzero Ti values describe labile d-p chemical bonds with moderate
diradical character in 1. These bonds play crucial roles for water splitting reaction at OEC of PSII.
In
order to express the decrease of chemical bonding via orbital symmetry breaking, the effective bond order bi
is defined by
bi 
nHOMO i  nLUMO i 1  cos 2i  1  cos 2i 

2
2
(s5a)
 cos 2 i  Ti ,
(s5b)
where nHOMO  i and nLUMOi denote the occupation numbers of the bonding (HOMO-i) and antibonding
(LUMO+i) NMOs, respectively. The effective bond order (bi) is nothing but the orbital overlap between
BS MOs under the generalized MO (GMO) approximation (here hybrid DFT Kohn-Sham orbitals are also
regarded as generalized MO).
The valence-bond concepts are also introduced for theoretical understanding of chemical bonds in
open-shell transition-metal complexes such as 1. In order to extract the VB concepts under the BS DFT
approximation, the localized MO (LMO) are defined as the completely spin polarized BS MOs as
 i+ i   4  
1
*
HOMOi  LUMOi
 LMOa
2
(s6a)
 i i   4  
1
*
HOMOi  LUMOi
 LMOb ,
2
(s6b)




where LMOa and LMOb are mainly localized on the a and b sites of manganese ions of 1, respectively. It
is noteworthy that LMOs are quite different from the atomic orbitals (AO) in the simple VB theory because
LMO are orthogonal and still molecular orbitals in nature. Then the expression of LMOs by AO-like
orbitals in Figs. 1 and 2 is regarded as just symbolic for qualitative and lucid understanding of electronic and
spin structures of 1.
In order to extract the VB configuration interaction (CI) concepts from the BS DFT calculations, the
delocalized MO expression via NMO can be transformed into LMO expression. To this end, the BS MOs
in Equation (3) are re-expressed with LMOs in Equation (s6) as follows
 i  cos  iLMOa +sin iLMOb
(s7a)
 i  cos  iLMOb +sin iLMOa ,
(s7b)
where the mixing parameter i is given by i   4 . The BS MO configuration can be expanded with
using LMOs to the VB CI form as
BSI   i i


 cos LMOa +sin  LMOb  cos LMOb +sin  LMOa


(s8a)

1
 2 cos2SD  2TD  sin2ZWa  ZWb ,
2
(s8b)
Similarly, the low-spin (LS) BSII MO configuration is expressed by
BSII   i– i

1
  2 cos2SD  2TD  sin2 ZWa  ZWb .
2
(s8c)
where the pure singlet (SD) and triplet (TD) covalent (CV) wavefunctions are given by

SD 
1
LMOa LMOb  LMOb LMOa
2

(s9a)
TD 
1
LMOa  LMOb   LMOb LMOa .
2
(s9b)




On the other hand, zwitterionic (ZW) VB configurations are resulted from the charge transfer from LMOa
to LMOb (vice versa) as follows:


ZWa  LMOa LMOa , ZWb  LMOb LMOb .
(s10)

The low-spin (LS) BSI MO configuration involves both singlet diradical (DR) and ZW configuration as in
the case of the VB CI, but it also includes the pure triplet DR (TD) component, showing the spin-symmetry

breaking
property. The LS BSII MO solution also involves the singlet DR and ZW configurations, together
with the pure triplet component. Thus the spin symmetry breaking is inevitable for diradical species in the
case of the single-determinant (reference) BS solution; the Hartree-Fock and Kohn-Sham DFT models
belong to this category. Nevertheless, BS methods can be regarded as a convenient and handy procedure to
determine both delocalized and localized MOs for open-shell transition-metal clusters such as
CaMn4O5(H2O)4. However, both orbital and spin symmetries should be conserved in finite systems. Then
the recovery of them is performed as shown below.
SI.2 Recovery of symmetry breaking via quantum resonance and valence-bond
theory
The BSI and BSII solutions are degenerate in energy. Then the resonance of them is feasible as
follows:
RBS(+) 


RBS(–) 

1
BSI  BSII 
2
(s11a)
1
 2 cos2SD  sin2ZWa  ZWb ,
2
(s11b)
1
BSI  BSII 
2
(s12a)
 TD.
(s12b)
Thus the in- and out-of-phase resonating BS (RBS) solutions are nothing but the pure singlet and triplet


states wave functions, respectively. The chemical bonding between a and b sites is expressed with the
mixing of the singlet diradical (SD) and ZW configurations under the LMO approximation.
The
valence-bond (VB) CI type explanation of electronic structures becomes feasible under the LMO CI
approximation. For example, the effective bond order becomes zero for the pure SD state, but it increases
with the increase of mixing with the ZW configuration until the ZW/SD ratio becomes 1.0, namely
closed-shell limit. Thus the BS computational results can be utilized to extract the VB CI pictures based on
the LMO CI expressions.
On the other hand, the molecular orbital concepts are also generalized on the basis of the natural
orbital analysis of broken-symmetry solutions. The Coulson’s (first-order) bond order is well-accepted in
the MO method. In order to obtain the Coulson’s effective bond order for the RBS(+) solution, it is
transformed into the symmetry-adapted NMO expression as
RBS() 
1
2(1 Ti 2 )
(1 cos2 )(
i

 


HOMOi HOMOi )  (1 cos2 i )( LUMOi LUMOi )
(s13)
where the first and second terms denote the ground and doubly excited configurations, respectively. This

MO-CI picture is useful for theoretical understanding of labile chemical bonds with moderate orbital
For example, manganese oxo (Mn=O) and -oxo (Mn-O-Mn) bonds in
overlaps in equation (s3).
high-valent manganese oxides such as CaMn4O5(H2O)4 exhibit such behaviors: this is an origin of
chameleonic (radical/electrophilic) reactivity of these bonds. The effective bond order (B) for RBS( ) is
introduced to express the nature of these labile chemical bonds as
Bi 
nHOMOi RBS()  nLUMOi RBS()
2
1 T   1 T 

21 T 
2
2
i
i
2
i



2Ti
1 Ti2
2bi
 bi .
1 bi2
(s14a)

(s14b)
(s14c)
The effective bond order (Bi) after elimination of triplet contamination part is larger than that (bi) of the BS
solution itself. This is not at all trivial, indicating that BS computational results without symmetry recovery

are often biased to radical picture arising from high-spin component. The diradical character (yi) after spin
contamination is defined by the weight of the doubly excited configuration under the NMO approximation as
1  T 

2
yi  2WD
i
1  Ti
 1  Bi
2
 1
2Ti
1  Ti 2
(s15a)
(s15b)
The diradical character y is directly related to the decrease of the effective bond order B: these indices are
used for diagnosis of radical reactivity of manganese oxide species such as CaMn4O5(H2O)4.
In the multi-nuclear transition-metal complexes such as CaMn4O5(H2O)4, the spin polarized orbital pairs
are over ten (i>10), indicating the necessity of total chemical indices such as polyradical character (Y), total
effective bond orders (b and B).
b   bi , B   Bi , Y   yi
i
i
(s15c)
i
SI.3 Variation of diradical character with environmetal effects
theory
The diradical character of the formal high-valent manganese-oxo (Mn(V)=O) bond is not zero even in the
low spin (LS) if the Mn-O bond length is longer than 1.7 A in gas phase.
However, it is highly sensitive to
environmental effects such as hydrogen bonding in biological systems as illustrated in Fig. S1A. On the
other hand, the diradical character in high-spin state is not lost even in such conditions.
Fig. S1A Variation of the metal diradical character of formal Mn(V)=O species with environmental effects
Fig. S1B Orbital energy splitting of broken-symmetry molecular orbitals.  denotes the gap of the
Coulomb integrals () of fragment orbitals a and b under the Huckel-Hubbrad-Hund (HHH) model:  =
a-b
Fig. S1C
Configuration (and state) correlation diagrams for diradical and zwitterionic states with the
ionicity () (see also Fig. S1B). The dotted line denotes the avoided crossing, where the superposition of two
configurations is remarkable: chameleonic behavior is realized in chemical reaction (see text).
Zwitterionic character via symmetry breaking
The closed-shell configurations constructed of the broken-symmetry (BS) MO in eq. (7) are crucial for
the MO-theoretical illumination of the zwitterionic (ZW) character via symmetry breaking. The ZW MO
configurations can be expanded with using LMOs to the VB CI form as
ZWI   i  i


 cos LMOa + sin LMOb cos LMOa + sin LMOb

(s16a)



1
(1 cos2)ZWa  2 sin 2SD  (1 cos2)ZWb ,
2
(s16b)

where the VB configurations, SD, ZWa , and ZWb are expressed by eq. (s9a) and (s10), respectively.
The ZWMO configuration does not involve the triplet diradical ( T D) component. Similarly, the other other
ZW MO configuration is expressed by
 

ZWII   i  i




(s17a)

1
(1 cos2)ZWb  2 sin 2SD  (1 cos2)ZWa .
2
(s17b)
The ZWII MO configuration also involves three singlet VB configurations.
The
ZWI and ZWII MO configurations are degenerated in energy under the homopolar condition that the
energy levels of fragment orbitals are equivalent. However, the energy splitting occurs in several
broken-symmetry conditions: i) a and b are different fragments (atoms) such as Mn and O, ii) a and b have
quite different ligand fields, and so on (see Scheme 1-3). The broken-symmetry MO levels in eq. (7) are
indeed separated under these hoteropolar condition as illustrated in Fig. S1B. Therefore, the up- and downspins enter into the stabilized MO (  i ) if the energy gap exceeds through a certain limit, affording the ZWI
structure (a-b+). The up-spin MO (  i ) in eq. (7) is approximately given by the fragment orbitals at the ionic
limit as

 i  LMOa (  0)
(s18a)
 i  LMOb (  0)
(s18b)

Therefore the ZWI and ZWII MO configurations are also expressed by them as



ZWI   i  i  LMOa LMOa
(ZWI)
ZWII   i  i  LMObLMOb
(ZWII)
(s19a)
(s19b)
The diradical (DR) configuration is the ground state at the homopolar limit, and ZWI(II) configuration is

the excited state because of a strong one-site repulsion (U). However, the ZWI MO configuration becomes
the true ground state in the broken-symmetry ionic limit as illustrated in Fig. S1C. Thus, the diabatic curve
crossing between the SD and ZWI configurations occurs in an intermediary ionic region as shown in Fig.
S1C. Moreover, the avoided crossing is resulted from the configuration mixing of SD and ZWI, affording the
noncrossing potential curves depicted by the dotted line. The superposed state in the intermediary region is
responsible for chameleonic behavior with both diradical and zwitterionic characters as discussed in the text.
SII Electronic structures of manganese-oxo species and possible mechanisms of water splitting
reactions
SII. Electronic structures of manganese-oxo species
Electronic structures of formal high-valent manganese-oxo species (Mn=O) are variable on the basis of
environmental conditions as illustrated in Figs. S2A and S2B.
Fig. S2A Schematic expressions of the low-, intermediate and high spin states of formal Mn(V)=O species
Fig. S2B
Schematic expressions of the low-, intermediate and high spin states of formal Mn(IV)=O
species
SII.2 Possible mechanisms of water splitting reactions
Possible mechanisms of water splitting reactions are derived on the basis of the guiding principles (A-F)
obtained by accumulated experimental results and broken-symmetry computational results.
They are
summarized as follows.
(A) The local high-spin configurations of manganese ions Mn(X) (X=II–V) in oxide compounds
are conserved in the oxidation-reduction processes; Hund rule for local spins of Mn(X) ions.
(B) The nature of the Mn-O bonds is formally regarded as the ionic structure Mn(X)=O2- ; oxygen site is
double anion.
However, back one-electron transfers are feasible on the basis of several
environmental conditions such as oxidation number of manganese ions (X=V, IV) and push-pull
stabilization effects of ligands, etc.
(C) Metal diradical (•Mn(X-1)-O•1-) generated via back one-electron transfer (or spin polarization)
provides the local singlet (LSD) or local triplet (LTD) diradical configuration in manganese oxides;
LSD pair is usually more stable than LTD because of the partial covalent bonding character.
(D) Tetraradical radical intermediate [•Mn(X-1)-O•1-…•OH Mn(Y-1) •] generated via one-electron transfer
from hydroxide anion affords the LSD or LTD configuration in O and OH sites.
The
antiferromagnetic (AF) and ferromagnetic (F) configurations in [Mn(X)-O2-…-OH Mn(Y)] provide,
respectively, LSD and LTD configurations:
LSD is more stable than LTD.
Furthermore, the
zwitterionic (ZW) configuration [•Mn(X-1)-O+0…-OH Mn(Y-1)•] is also feasible in the case of the
AF configuration.
(E) Atomic oxygen and molecular oxygen are exceptional cases since these species are ground-triplet; the
Hund rule in (A) should be applied for the species.
(F) Possible reaction mechanisms for water splitting reaction are derived to satisfy the above
conditions.
Fig.S3 Orbital and spin correlation diagram for the O-O bond formation between hydroxide anion and OH
site of forma Mn(IV)=O at the low-spin state. Both one-electron transfer (OET) and electron pair transfer
(EPT) mechanisms are feasible for the reaction.
Fig.S4 Orbital and spin correlation diagram for the O-O bond formation between hydroxide anion and OH
site of forma Mn(IV)=O at the high-spin state. Both one-electron transfer (OET) and electron pair transfer
(EPT) mechanisms are feasible for the reaction. Spin inversion (SI) and
processes are also necessary in this case.
spin exchange (SE)
Fig. S5 Possible state correlation diagrams for the O-O bond formation between formal Mn(IV)=OH and
hydroxide anion in Figs. S3 and S4.
Fig. S6 Orbital and spin correlation diagram for water splitting reaction for binuclear manganese clusters
[(H2O)Mn(III)O(H2O)Mn(III)].
The first deprotonation is expected for the inner water under the
appropriate design of ligand fields.
Fig. S7
Orbital and spin correlation diagram for the HO-OH bond formation between formal Mn(IV)=OH
and OH anion in the case of binuclear manganese oxides MnO2Mn(L)n.
Fig. S8A
Orbital and spin correlation diagram for the HO-OH bond formation between formal
Mn(IV)=OH and OH anion ligated to Ca(II) in the case of binuclear manganese oxides.
Fig. S8B
Orbital and spin correlation diagram for the HO-OH bond formation between formal
Mn(IV)=OH and O(57)H anion in the case of binuclear manganese oxides.
Fig. S9 Calculated mononuclear Mn complex on the basis of Shen-Kamiya structure (PDB code: 3arc, see
Fig. 7). The atoms marked with asterisks were kept fix during the geometry optimization.
Fig. S10
Ca depleted cluster structures (IIIb, IVb, Vb and VIb) for water splitting reaction. Further O1
depleted structures are given by IIIc, IVc, Vc and VIc. On the other hand, O2 depleted structures are given
by IIId, IVd, Vd and VId.
Fig. S11 Linear and branched cluster structures (VIII-XVII) proposed as active sites for water splitting
reaction.
Fig. S12 Kok cycle for water splitting reaction.
Fig. S13 Possible reaction mechanisms for water splitting reaction proposed on the basis of the UB3LYP
calculations (ref. 51).
Fig. S14 Orbital and spin correlation diagram for the HO-OH bond formation between formal Mn(IV)=OH
and OH anion in the case of tetranuclear manganese oxides with London structure
Fig. S15 Orbital and spin correlation diagram for the HO-OH bond formation between formal Mn(IV)=OH
and OH anion in the case of tetranuclear manganese oxides with a modified Berlin (Loll) structure with the
O5-site.
Table S1 The orbital overlaps for magnetic orbitals and active orbitals for the reactnat (R) and transition
structure (TS) for the O-O bond formation in the LS (S=1/2) state.
ia
R
TS
0
0.000
0.000
1
0.015
0.001
2
0.035
0.023
3
0.178
0.104
4
0.316
0.120
5
0.652b
0.876
6
0.952
0.960
7
0.972
0.993b
8
0.995
0.996
a
highest occupied natural orbital (HONO)-i
a
The magnetic orbital corresponding to the O-O bond formation.