Supporting materials
SI Theoretical backgrounds
S.I.1 Symmetry breaking and its recovery via quantum resonance and valence-bond CI
model via LMO in diradicals
The orbital symmetry breaking in the independent particle models such as the
Hartree–Fock (HF) model arises from the strong electron correlation effect.
Moreover,
the symmetry breaking in the model entails the concept of quantum resonance that
recovers the broken symmetry in finite systems. The resonance concept is familiar in
quantum chemistry in relation to the valence-bond (VB) theory; to this end the
delocalized molecular orbital (MO) picture can be transformed into localized MO
(LMO) picture for the VB explanation of diradical species. The BS MOs in eqs. 2 in
the text are indeed re-expressed with LMO (LNO)s in eq. 5 in the text as follows
 i  cos LMOa + sin  LMOb
(s1a)
  cos LMOb +sin  LMOa ,
(s1b)
–
i
where the mixing parameter  is given by    4 .

configuration can be expanded with using LMOs as

BSI   i i


 cos LMOa +sin  LMOb  cos LMOb +sin  LMOa


1
 2 cos2SD  2TD  sin2ZWa  ZWb ,
2
(s2a)
(s2b)
where the pure singlet diradical (SD) and triplet diradical (TD) covalent terms are given
by




(s3a)


(s3b)
SD 
1
LMOa LMOb  LMOb LMOa
2
TD 
1
 LMOa  LMOb   LMOb  LMOa .
2
On the other hand, zwitterionic (ZW) configurations result from the charge transfer
from LMOa to LMOb (vice versa) as follows:

ZWa  LMOa LMOa , ZWb  LMOb LMOb .



Therefore the BS MO
(s4)
The low-spin (LS) BSI MO configuration involves the pure triplet covalent term,

showing the spin-symmetry breaking property.
Similarly, the LS BSII MO
configuration is expressed by
BSII   i– i
(s5)
1
 2 cos2SD  2TD  sin2ZWa  ZWb .
2
The LS BSII MO solution also involves the pure triplet term. Thus the spin symmetry


breaking is inevitable in the case of the single-determinant (reference) BS solution for
diradical species. However, both orbital and spin symmetries are conserved in finite
quantum systems [24]; for example, the error arising from the triplet term in eq. s2 (s5)
is easily eliminated by the AP procedure that eliminate the pure triplet term in eq.
s2(s5).
As shown in eqs. (s2) and (s5), the BSI and BSII solutions are degenerate in
energy. Then the quantum resonance of them is permitted as follows [24]:
RBS(+) 


RBS(–) 



1
BSI  BSII 
2
(s6a)
1
 2 cos2SD  sin2ZWa  ZWb ,
2
(s6b)
1
BSI  BSII 
2
(s7a)
 TD.
(s8b)
where the normalizing factor is neglected for simplicity. Thus the in(+)- and
out-of-phase(-) resonating BS (RBS) solutions are nothing but the pure singlet and
triplet states wave functions, respectively; the broken-symmetries are recovered via the
quantum resonance. The chemical bonding between a and b sites is expressed with the
mixing of the SD and ZW terms under the LMO approximation.
The VB type
explanation of electronic structures becomes feasible under the LMO approximation
[24]. For example, the effective bond order becomes zero for the pure covalent term,
but it increases with the increase of mixing with the ZW term until the ZW/SD ratio
becomes 1.0, namely closed-shell limit.
Therefore the diradical character defined with
the weight of the covalent term in the LMO description in the VB approach is in turn
quite different from our definition of the diradical character based on the delocalized
NO (DNO) model [24].
The VB-type descriptions and explanations of unstable
molecules are obtained using the LMO in our approach. Here LMO(LNO) is utilized
as reference orbitals for MkMRCC in relation to elimination of the size-consistent error
[104, 105].
The mutual transformation from the localized MO picture to the conventional
delocalized MO picture is easy because of the mathematical relations in eqs. 2, 5 and 6
in the text.
In order to obtain the effective bond order for the pure singlet RBS(+)
solution, it is transformed into the symmetry-adapted NO expression as
1 cos2   1 cos2  ,
2(1 T )
1
RBS(+) 
i
2
i
*
i
*
i
(s9a)
i
where Ti is the orbital overlap between the up- and down-spin orbitals in eq. s1, and the

first and second terms denote the ground and doubly excited configurations,
respectively.
In the text, the notation Si is utilized for Ti to avoid the confusion of
notations. The effective bond order (B) for RBS( ) is given by
B
n i RBS ()  n *i RBS ()
2
1 T   1 T 
21 T 
2




2
i
i
2

i

(s10a)
2Ti
1 Ti2
2bi
 bi .
1 bi2
(s10c)
The B value, namely the elimination of the triplet state, is larger than b.

(s10b)
The diradical
character (y) is defined by twice of the weight of the doubly excited configuration (WD)
under the delocalized NO approximation as
1 T 

2
y  2W D
i
1 Ti
 1 B.


2
 1
2Ti
1 Ti2
(s11)
(s12)
Thus the diradical character y (=Y by the MR approach such as MkCC) is directly
related to the decrease of the effective bond order B. The chemical indices, b, B, and
y(=Y), are mutually related in the present BS MO approach; in fact, these are key
conceptual bridges for MR CI, MkCC, BS UHF(UB) and BS DFT approaches in the
present paper. These indices are also useful for elucidation of scope and applicability
of the computational schemes of the effective exchange integrals (J) in eq. 8 in the text.
SI.2 MRCC methods as direct extension of BS methods for quasi-degenerated
electronic systems
The theoretical description of diradicals is closely related to the strong electron
correlation problem that has been investigated during past decades.
In late 1960s and
early 1970s the extended Hückel MO and conventional restricted HF (RHF) methods
had been applied to the elucidation of concerted reactions predicted on the basis of the
orbital-symmetry conservation rules proposed by Woodward and Hoffmann; the
electronic structures of key transition structures in these reactions had been assumed to
be nonradical in nature, namely in weak correlation regime. On the other hand, in
1970s, we had been concerned with theoretical studies on strongly correlated electron
systems that were interesting targets for many body theories developed in late 1950 and
1960s as shown in many references (s1-s27).
The instability of closed-shell RHF
solutions had been investigated in relation to more stable BS HF solutions that are
chemically related to the instability of chemical bonds （namely broken-chemical
bonds）in diradical species and their clusters.
Particularly, as shown in the text, we
were interested in BS unrestricted HF (UHF) and unrestricted HF-Slater (UHFS)
models by the use of different-orbitals-for-different spins (DODS) and more generalized
HF (GHF) and generalized HFS (GHFS) based on the general spin orbitals (GSO); two
component spinors, that arise from the static electron and spin correlations.
In late
1970s the extension and refinement of these independent models (namely post RHF,
UHF and GHF models) have been our interesting theoretical problem.
To this end, we
have performed the natural orbital (NO) analysis of these BS solutions to obtain the
natural orbitals and their occupation numbers as shown in the supporting Fig. S1. The
NO analysis has indeed elucidated active orbitals that are closely related to
nondynamical correlation corrections; partitioning active orbitals have entailed the
necessity of the genuine MR approach in strongly correlated electron systems.
Therefore we considered that the MRCI and MRCC schemes are the most natural
extensions of the BS methods, namely independent particle model, because the
reference MR space can be constructed so as to describe quasi-degenerated electronic
systems related to the BS solutions.
We presented our MRCC scheme at Sanibel 1980
(ref. 23) on the basis of the MRCC by Offermann in nuclear physics, Mukherjee and
Sinanoglu in quantum chemistry; a lot of reference papers on the CC methods are cited
in the text for understanding of historical developments. Roos also proposed CASSCF
at the same Sanibel conference. For the natural extension of the BS computations, the
active space is limited so as to include nondynamical (static) correlations that are
origins of instabilities in the RHF solution.
Under this approximation, only the
minimum reaction NO (MinRNO) = principal active space (PAS), namely CAS in our
definition in Fig. 1 in the text, was considered instead of the maximum RNO
(MaxRNO) = PAS+SAS (secondary active space).
Therefore the reference function in
MinRNO was taken to be UNO(=UHF-NO) and GNO(=GHF-NO) CASCI in our MR
CI and MR CC schemes.
On the other hand, Max RNO is often necessary for
CASPT2 and related PT theories to include the higher-order excitations.
The CC excitation operator was considered for the reference state to obtain the
UNO(GNO) CASCC as
CAS (MR CC) = exp (T) | YNO CASCI >
(Y=U, G or D).
(s13)
where T =  Ti (i=1-4). Before Jeziorski and Monkhorst [29] proposed their CC
scheme in 1981, the uniform excitation operator formalism had been employed in the
MRCC approach.
If we consider only the one-electron excitation operator responsible
for semi-internal correlation for full NO space, UNO (GNO) CASCCS is refined to
UNO(GNO) CASSCF after the convergence of the CC equation because of the
Thouless theorem (ref. s13).
 (CASSCF) = exp (T1) | YNO CASCI > (Y=U, G or D).
(s14)
However, the inclusion of the double excitation operator (D) in UNO (GNO) CASCCS
is crucial for dynamical correlation correction, namely UNO(GNO) CASCCSD, as
CAS(MR CCSD) = exp (T1 + T2) | YNO CASCI > (Y=U, G or D). (s15)
The UNO(GNO) CASCCSD approach starting from UNO(GNO) CASCI was our
chemical picture at that time. However, in the next year (1981), Jeziorski and
Monkhorst in ref. 29 in the text proposed a more general state universal (SU) MR CC
scheme; excitation operator is used for each configuration this scheme as shown below.
The SU MRCC scheme is furthermore specified into the state specific (SS) MRCC
version: which is now employed by several groups; in fact, it is developed by
Mukherjee, Kalláy, Paldus, Evangelists, developers of PSIMRCC program (Crawford et
al) and many research groups cited in references. However, the extension of the SS
MRCC scheme to quasi-degenerated systems with large CAS space is still difficult.
Therefore, several MRCC schemes have indeed been presented as shown in many
references in refs. s1-s76. For example, more detailed formulation of the CASCC-type
scheme can be obtained by eliminating redundant excitations with the Feynmandiagram techniques developed by Adamowicz at al. in refs. s69 and s73. For the
purpose, Adamowicz et al. have divided the CC excitation operators into the internal
(Tint) and external (Text) types like the Silberstone-Oktuz-Sinanoglu classification in refs.
S21 and s22
CAS(CASCC) = exp (Text) exp (Tint) | 0 >
(s16a)
= exp (Text) ( 1 +  Ci (i=1-n) )| 0 >
where |0> is taken as the most doubly occupied determinant.
(s16b)
If active m-orbitals
n-electrons are used for CAS, CASCCSD is constructed by using both Tint and Text
excitation operators as
CAS(CASCC) = exp (Ti (i=1-(n+2)) ( 1 +  Ci (i=1-n) )| 0 >
(s17)
This CC scheme is therefore employed as a direct extension of MRSDCI. The total
excitation operators are described with the pure external and mixed (Tint x Text)
excitations in their scheme. The derivations of the MRCC equations and calculated
results are given in several papers by Adamowizc (ref. s69).
Judging from the
numerical results for diradical species such as monocentric diradical (I), antiaromatic
molecules (II) and 1,3-diradicals (III), SR CCSDTQ and MRCCSD provide similar
results. This implies that the truncation of the excitation operators is possible at the
SD level if the MR part in eq. s17 has been appropriately selected.
However, SR
CCSDTQ is indeed necessary for complex diradical as shown in the text.
Our
selection of the MR part is one of such reasonable procedures starting from the BS
calculations for quasi-degenerated systems.
As mentioned above, Jeziorski and Monkhorst [s27] have developed more general
state universal (SU) MR CC scheme on the basis of the Bloch wave-operator technique.
Their MR CC scheme is given by a simple formula
(SU MR CC) =  Cim (i=1-n)exp (Tim) im .
(s18)
As is apparent from this equation, the CC excitation operator is applied to each
configuration involved in the MR zero-order function as mentioned above.
In fact, the
CI coefficient Ci and amplitude in the excitation operators Ti are determined in an
iterative manner. This in turn means that the CASSCF part may be skipped if the
reference orbitals are appropriately determined.
For example, UNO CAS CI can be
used for the purpose. Therefore diradical character can be also defined even in this SU
MRCC scheme. However, expansion of CAS (RNO) space for polyradicals (IV) is not
so easy in this scheme because of too many amplitude equations, though UHF-CC is
easily applicable to them.
Past decades several groups cited in references in the text have performed further
derivations of the MRCC schemes as shown in refs. s40-s76. Now, the MRCC
approach is classified into (a) state-universal (SU) or Hilbert space type, (b)
valence-universal (VU) or Fock space type and (c) state-specific (SS) type.
The
delocalized UNO (UHF-NO) (GNO=GHF-NO) can be used for reference orbitals of
these CC schemes. The size-inconsistent errors however are not negligible in the case
of Mk-type UNO-MRCC (SS) approach, leading to use of the localized UNO (ULO)
(see eq. 5 in the text) for elimination of such errors; ULO has been introduced to obtain
the VB CI like pictures of the ground and excited states of diradical species; Yamaguchi
K, Fueno T (1977) Chem Phys 23:375.
In fact, ULO-MRCC provided reasonable
potential curves of F2, CH2 and others as shown in this supporting material.
Fig. S1
Computational schemes proposed in the paper:
K. Yamaguchi, Int. J. Quant. Chem. S14, 269 (1980).
The Brueckner double (BD) method often applied to polyenic diradicals with
moderate spin polarization effects.
The BD results are approximately reproduced with
those of hybrid DFT. The hybrid DFT natural orbitals can be used as an alternative to
the UBD natural orbitals, giving the delocalized DFT-NO (DNO) and localized
DFT-LNO (DLO) MRCC (SS) as shown. The excitation energies of these systems can
be calculated by the linear response (LR), equation of motion (EOM) or time-dependent
energy derivative (TDEG) (which has been called as the quasi energy derivative (QED))
method for MRCC if correlation effects involved in the ground state are not drastically
changed upon electronic excitations. As shown in the text, the MRCC results for
parent systems have been used to confirm AP-UHF-CC, AP-UBD and AP hybrid UDFT
approximations that have been applied to much more larger systems with chemical
interests, for example molecule-based magnetic materials.
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SI.3 . Dynamical correlations via genuine multi-reference coupled-cluster method: the
MkCCSD method
Independent particle models incorporate the internal (static) electron correlation effects
via symmetry breaking [s1-s6]; however, beyond the model is crucial for inclusion of
dynamical correlation effect.
As shown previously [24], active space constructed of
UNO and/or DNO can be used for reference functions for MRCI and MRCC
computations of quasi-degenerated systems [24]; here genuine MRCC is employed as a
successive reliable and refined procedure for the BS computations.
This in turn
provides a theoretical background for approximate spin projection scheme for the BS
UHF-CCSD and UBD solutions.
Therefore the Mukherjee’s type MRCC (MkCC)
theories are briefly described for the present purpose: namely spin correction to the BS
methods.
In the Hilbert space MRCC, a model space
M 0   ;  1, ,d containing d determinants is defined to construct a projection
operator to M0 ,

P     ,
d

(s19)
 1

and to its orthogonal space M 0 ,

Q

q
q .
(s20)
q M 0

While P projects the exact eigenstates

wave
 functions
  in the
P

P  P  ,
   of
the Schrödinger’s equation to the
M0 ,

(s21)
one canin turn define 
the inversion of P in order to obtain the exact solution, as

where

   P ,
(s21)
 is a wave operator. Eq. (s20), and the Schrödinger’s equation

( H   E   ) yields an effective eigen-value problem,

(s22)
H eff P  E P ,

eff
where H ( P H ) is a non-Hermitian effective Hamiltonian and E is the
corresponding  -th eigen value. Now, an intermediate normalization for   ,

(s23)
    


is assumed.
Projecting eq. (s21) to the model space M0 yields a matrix


representation of the effective eigenproblem, as

d
(s24)
 Heff c  E c  ,

 =1

where
eff
H
  H eff 
(s25)
and
d

P   c  .
(s26)
 =1
In Hilbert space coupled-cluster theory, the wave operator  is approximated as
d

   eT   ,
(s27)
 =1

to describe the exact wave functions as
d

   c eT  ,
(s28)
 =1
where T is an excitation operator from a related Fermi vacuum  , which is
similar to that of the single-reference CC method. However, all SU MRCC

approaches require a separate cluster operator to each reference configuration; this


entails a lot of cluster amplitude equations that are origins of heavy calculations when
the CAS space (namely d in eq. s19) is larger than [2,2]. Therefore many approaches
have been proposed for good convergence and reduction of computational cost.
In the MkCC, the above SU Hilbert space is reduced to a state specific (SS)
formalism through an amplitude equation to determine the T ,
q () e
T
He
  q () e
T
T
 c 
eff 
eT  H 
c
,
 
In eq. (s29), a q () is an element in Q .

(s29)
This amplitude equation (s29) holds only
for the  -th state,  is usually written 0. Containing only connected terms in eq.

(s29) and eq. (s26) with a complete model space causes the size-extensiveness of MkCC.

 work, excitation operators
In this
T were limited to singles (S) and doubles (D).

Natural molecular orbitals used in MkCC calculations were obtained by four methods:
(i) CASSCF calculations (CNO), (ii) ROHF calculations (RNO) of high-spin states, (iii)

BS calculations (UNO) of low-spin states and (iv) hybrid UDFT calculations (DNO) of
low-spin states. The corresponding localized NO (LNO) are also examined concerned
with the size-consistency as mentioned above.
The MkCC for diradical systems is based on the 4-reference configuration model
space that corresponds to the 2-electron 2-orbital complete active space [2,2] has been
applied to MkCCSD calculations of diradicals species without particular mentions as
c1 exp(T1) (core) 2 (inactive ) 2 x x  c 2 exp(T2 ) (core) 2 (inactive ) 2 y y
 c 3 exp(T3 ) (core) 2 (inactive ) 2 x y  c 4 exp(T4 ) (core) 2 (inactive ) 2 y x
, (s30)
where c1 ~ c4 and T1 ~ T4 were same as eq. (s30), (core) 2 was frozen-core orbitals,

(inactive )2 was inactive occupied (cores) orbitals, x and y were two active orbitals

that are taken to be BS natural orbitals (CNO, RNO, UNO and BNO) and CASSCF-NO.



Although we do not use symmetry restrictions, eq. s30 reduces to 2-reference

) (core)2 (inactive )2 y y
(s31)
c exp(T ) (core)2 (inactive )2 x x c exp(T
1
1
2
2
for singlet diradical states as in the case of eq. s31. Therefore the diradical character
(y) in eq. 15 can be easily defined using the coefficient c2 even in the MkCC approach;

however,
appropriate orbital rotation to quench singlet excited wavefunction is often
necessary for RHF(ROHF)-NO.
For calculations of triplet states, MkCCSD wave
functions (Sz = 0) are expressed as


1
exp(T3 ) (core) 2 (inactive ) 2 x y  exp(T4 ) (core) 2 (inactive ) 2 y x . (s32)
2
The localization of active orbitals was also performed to correct size-consistent errors,
and the related MkCCSD wave function was

c1 exp(T1) (core) 2 (inactive ) 2 u u  c 2 exp(T2 ) (core) 2 (inactive ) 2 v v
 c 3 exp(T3 ) (core) 2 (inactive ) 2 uv  c 4 exp(T4 ) (core) 2 (inactive ) 2 v u
, (s33)
where u and v were
(s34)
u  x  y / 2 , v  x  y / 2

respectively. Here, eq. 34 is taken to be eq. 5. When UNO is used as shown in


section II, we can substitute total density of UHF for averaged Fock operator (which is

RHF type Fock operator), and diagonalize the block of occupied orbitals except two
active orbitals to extract frozen and inactive core orbitals.
This pre-adjustment of
UNO was similar to construct zero-th order Hamiltonian of CASPT2. This means that
the UNO(DNO) CASCI wave functions can be used to express the trial model space for
the CC procedure ; the full-optimization of CI coefficients with the CASSCF procedure
is often skipped because the coefficients are relaxed under the iterative CC processes.
E
ALS eTBS BS

ALS e

TLS(prj)

BS
eTHS HS


(1)
( 2)
1
C1 eTMR 1 1  C2 eTMR 22
2

S2
LS 

AHS eTBS BS
eTBS BS
BS

Fig. S2 The projected CC approach to BS CC method

S2
HS

As shown in the text, CI matrix and amplitude equations are solved in the case of
the true projected UHF-CCSD scheme. However, instead of these procedures, we here
employ the simple approximated spin projection scheme for energy corrections of the
UHF-CCSD and UBD solutions.
In the present paper, the MkCCSD method has been
used for elucidation of scope and applicability of the spin projection AP scheme as
discussed in the text.
Figure S2 illustrates the spin-projection scheme in the
UHF-CCSD scheme that provides the LS and HS states after spin projection (see also
the text).
SI.4 Local spins and spin Hamiltonian models, and first principle calculation of J
values
Löwdin has emphasized the spin contamination error involved in the BS solutions
from the mathematical view point, though he of course has known that the
symmetry-breaking problem is closely related to strong correlation effects in
many-body physics [s1-s6]. In fact, in 1950s, this problem has emerged from the
deformation of nuclear matter as shown in Bohr-Mottelson-Rainwater.
They have
investigated the newly appeared rotational spectra in these matters. On the other hand,
we have interested in spin multiplet spectra newly appeared in diradicals; more
(diradical) is different from mono (free) radical.
In fact, the spin-polarized BS
solutions for strongly correlated systems provide broken-symmetry molecular orbitals
that are more or less localized on different radical sites, respectively.
This electron
localizations via strong electron correlation entail the concept of local spins that are
grasped with the spin Hamiltonian models: Yamaguchi K (1974) Chem Phys Lett 28:93;
the effective exchange integrals (J) have been introduced as the resonance energy
between the BS solutions.
The J value is introduced to express the singlet-triplet
energy gap in conformity with Löwdin definition: 2J = E(singlet)-E(triplet). The total
energy E is obtained by the resonating broken-symmetry (RBS) CI and UNO-CASCI.
The computational scheme of the J values for more general case was discussed in
our several papers, and eq. 8 has been introduced as an effective computational scheme.
The broken-symmetry (BS) solutions are utilized for first-principle computations of J
values and projected total energies using J values (in eq. 9) in our many papers. Eq. 8a
can be also applicable to any symmetry-adaped computational methods because it is
based on the spin-projected scheme.
In fact, it can be applicable to exchange-coupled
systems; symmetry-adapted MR methods are utilized in this formula. Eq. 8a was first
applied to the exchange-coupled binuclear transition-metal complexes: particularly the
transition-metal oxides (ref. s53 in the text) that are typical strongly correlated electron
systems as shown in the supporting Table S1.
It has been applied for molecules-based
magnetic materials. This paper is therefore a direct extension of these papers that have
been mainly investigated on the BS hybrid DFT methods.
Table S1
The effective exchange integrals for transition metal oxides:
Yamaguchi K et al. Appl. Quant. Chem. p155 (1986).
Nowadays, the newly appeared spin multiplet spectra are analyzed on the basis of
the calculated J values; for example the exact diagonalization of the spin Hamiltonian
model involving ab initio J values is utilized as a multi-scale multi-physics approach to
exchange-coupled systems.
MSMP simulations.
Accurate computations of J values are really necessary for
Fortunately, recent developments of various MRCC methods
have enabled us to perform such reliable computations. The MkCC methods have
indeed provided reference data that are very useful for elucidation of scope and
applicability of several BS computational methods as has demonstrated in Tables 1-7 in
the text.
This is not at all trivial because applications of MkCC to multi-nuclear
transition metal systems are still limited because of several reasons.
Alternately,
various BS methods have been utilized for qualitative purpose, though systematic
comparisons between MRCC and BS results had been lacking until now.
Indeed,
several BS approaches were still useful for large systems as shown in refs. 66-69.
However, one of the serious problems in applications of MkCC to
exchange-coupled systems is the size-consistency problem.
The J-value should be
completely zero at the dissociation limit of a diradical into two fragment free radicals.
However, as has already demonstrated by Evangelista et al. (refs. 33-35), this condition
is not satisfied in the case of MkMRCC by the use of delocalized reference orbitals over
the fragment free radicals.
On the other hand, Evangelista et al. (refs. 33-35), have
shown that localized reference orbitals can be used for the purpose.
Here, our
numerical results partly published in the recent paper in refs. 66, 104, 105, 115. The
spin contamination error even at the UHF-CCSD level is also serious for reliable
computations of the J-values for diradicals as illustrated in the supporting Tables. This
is also demonstrated on the basis of the potential curves for methylene.
II
Supporting Figures and Tables for diradical species in the text
Several BS computations and successive MkCC calculations have been performed for
diradical species: (1) antiaromatic molecules (a-c), (2) cyclobutadiene derivatives with
polar substituents (d-f), and (3) cyclobutadiene derivatives with radical groups (g-h).
The optimized geometrical parameters are given.
Spin densities by the BS SR
solutions for a-h are summarized in supporting Tables S2-S4 and S9-S11, and S13.
The occupation numbers and chemical indices for a-c are given in Tables S5-S8, and
S12.
The optimized Cartesian coordinates for antiaromatic molecules
C3H3– (a)
6
0.000000
0.838570
0.000000
6
0.726223
-0.419285
0.000000
6
-0.726223
-0.419285
0.000000
1
0.000000
1.924663
0.000000
1
1.666807
-0.962332
0.000000
1
-1.666807
-0.962332
0.000000
C4H4 (b)
6
0.000000
1.026841
0.000000
6
1.026841
0.000000
0.000000
6
0.000000
-1.026841
0.000000
6
-1.026841
0.000000
0.000000
1
0.000000
2.118166
0.000000
1
2.118166
0.000000
0.000000
1
0.000000
-2.118166
0.000000
1
-2.118166
0.000000
0.000000
6
0.000000
1.218676
0.000000
6
1.159030
0.376592
0.000000
6
0.716320
-0.985930
0.000000
6
-0.716320
-0.985930
0.000000
6
-1.159030
0.376592
0.000000
1
0.000000
2.311326
0.000000
1
2.198201
0.714239
0.000000
1
1.358563
-1.869902
0.000000
1
-1.358563
-1.869902
0.000000
1
-2.198201
0.714239
0.000000
C5H5+ (c)
C4H3(NH2) (d)
6
-0.038842
0.667358
0.013902
6
-0.069293
-0.744519
1.496778
6
-0.093175
0.704750
1.455243
1
-0.055984
-1.492092
2.283548
1
-0.140810
1.501553
2.196308
7
-0.111291
1.614637
-0.985894
1
0.225925
2.536881
-0.720770
1
0.284590
1.322053
-1.875949
6
-0.019711
-0.775095
0.047746
1
0.010883
-1.555606
-0.711331
6
0.120170
-1.528913
-1.196781
1
0.110078
-2.614798
-0.937472
6
0.034538
-0.617525
-0.067594
6
-0.066917
-0.734343
1.394492
6
-0.084239
0.697111
1.491929
1
-0.110658
-1.559580
2.106439
1
-0.147771
1.400027
2.325959
8
0.198782
-1.148509
-2.359053
6
0.014554
0.843001
0.065222
1
0.057411
1.677460
-0.634427
0.548783
0.140341
C4H3(CHO) (e)
C4H3(NH2)(CHO) (f)
6
0.508276
6
-0.453143
-0.514797
-0.120616
6
-0.659160
-0.684423
1.328768
6
0.311523
0.388020
1.546942
1
-1.267485
-1.323931
1.963483
1
0.727466
6
-0.894389
0.878128
-1.049891
2.430626
-1.387035
1
-1.646940
-1.871474
-1.359181
8
-0.462507
-0.617775
-2.459648
7
1.241858
1.321176
-0.696744
1
1.878883
2.023354
-0.352163
1
1.142331
1.164654
-1.691261
1,2-isomer (g)
6
1.708581
-1.087143
0.000000
1
2.764177
-0.812830
0.000000
1
1.458640
-2.148661
0.000000
6
-1.708581
-1.087143
0.000000
1
-1.458640
-2.148661
0.000000
1
-2.764177
-0.812830
0.000000
6
0.755922
-0.161742
0.000000
6
-0.755922
-0.161742
0.000000
6
-0.677500
1.320688
0.000000
6
0.677500
1.320688
0.000000
1
-1.431821
2.106540
0.000000
1
1.431821
2.106540
0.000000
1,3-isomer (h)
6
-2.397122
0.000000
0.000000
1
-2.957776
-0.934934
0.000000
1
-2.957776
0.934934
0.000000
6
2.397122
0.000000
0.000000
1
2.957776
0.934934
0.000000
1
2.957776
-0.934934
0.000000
6
0.000000
-1.026676
0.000000
6
1.036908
0.000000
0.000000
6
0.000000
1.026676
0.000000
6
-1.036908
0.000000
0.000000
1
0.000000
-2.114833
0.000000
1
0.000000
2.114833
0.000000
Table S2 Spin densities on the carbon atoms in the singlet (LS) and triplet (HS) states of
cyclobutadiene (b) by the BS methods.
Method
UHF
UBLYP
UB3LYP
UCAM-B3LYP
UBHandHLYP
ULC-BLYP
UBD
Spin state
C1
C2
C3
C4
LS
-1.100
1.100
-1.100
1.100
HS
0.544
0.544
0.544
0.544
LS
-0.695
0.695
-0.695
0.695
HS
0.525
0.525
0.525
0.525
LS
-0.751
0.751
-0.751
0.751
HS
0.526
0.526
0.526
0.526
LS
-0.780
0.780
-0.780
0.780
HS
0.525
0.525
0.525
0.525
LS
-0.838
0.838
-0.838
0.838
HS
0.528
0.528
0.528
0.528
LS
-0.815
0.815
-0.815
0.815
HS
0.524
0.524
0.524
0.524
LS
-0.846
0.846
-0.846
0.846
HS
0.532
0.532
0.532
0.532
The up- and down-spin densties on the carbon atoms of cyclobutadiene (b) can be
depicted by using spin vector models as shown in Fig. 3A and B in the text.
Table S3 Spin densities on the carbon atoms in the singlet (LS) and triplet (HS) states of
cyclopentadienyl cation (c) by the BS methods.
Method
UHF
UBLYP
UB3LYP
UCAM-B3LYP
UBHandHLYP
ULC-BLYP
UBD
Spin state
C1
C2
C3
C4
C5
LS
0.000
0.620
-0.944
0.944
-0.620
HS
0.429
0.429
0.429
0.429
0.429
LS
0.000
0.332
-0.564
0.564
-0.332
HS
0.418
0.418
0.418
0.418
0.418
LS
0.000
0.368
-0.618
0.618
-0.368
HS
0.418
0.418
0.418
0.418
0.418
LåS
0.000
0.391
-0.648
0.648
-0.391
HS
0.417
0.417
0.417
0.417
0.417
LS
0.000
0.429
-0.703
0.703
-0.429
HS
0.419
0.419
0.419
0.419
0.419
LS
0.000
0.421
-0.685
0.685
-0.421
HS
0.416
0.416
0.416
0.416
0.416
LS
0.000
0.427
-0.702
0.702
-0.427
HS
0.422
0.422
0.422
0.422
0.422
The up- and down-spin densties on the carbon atoms of cyclopentadienyl cation (c) can
be depicted by using spin vector model as shown in Fig. 3C in the text.
Table S4 Spin densities on the carbon atoms in the singlet (LS) and triplet (HS) states of
cyclopropenyl anion (a) by the broken-symmetry (BS) single-reference (SR) methods.
Method
UHF
UBLYP
UB3LYP
UCAM-B3LYP
UBHandHLYP
ULC-BLYP
UBD
Spin state
C1
C2
C3
LS
0.000
-0.549
0.549
HS
0.552
0.552
0.552
LS
0.000
-0.453
0.453
HS
0.598
0.598
0.598
LS
0.000
-0.452
0.452
HS
0.597
0.597
0.597
LS
0.000
-0.464
0.464
HS
0.608
0.608
0.608
LS
0.000
-0.454
0.454
HS
0.597
0.597
0.597
LS
0.000
-0.442
0.442
HS
0.610
0.610
0.610
LS
0.000
-0.378
0.378
HS
0.570
0.570
0.570
The up- and down-spin densties on the carbon atoms of cyclopropenyl anion (a) can be
depicted by using spin vector model as shown in Fig. 3D in the text.
Table S5 Chemical indices for cyclobutadiene (b) by the BS methods.
Method
HONO
HONO-1
nLU
B
Y
nLU
b
Y
UHF
0.933
0.067
87
0.017
0.983
0
UBLYP
0.989
0.011
98
0.002
0.998
0
UB3LYP
0.990
0.010
98
0.004
0.996
0
UCAM-B3LYP
0.995
0.005
99
0.004
0.996
0
UBHandHLYP
0.994
0.006
99
0.007
0.993
0
ULC-BLYP
0.995
0.005
99
0.005
0.995
0
UBD
0.973
0.027
95
0.005
0.995
0
Table S6 Chemical indices for cyclopentadienyl cation (c) by the BS methods.
Method
HONO
HONO-1
nLU
b
Y
nLU
b
Y
UHF
0.995
0.005
99
0.081
0.919
0
UBLYP
0.997
0.003
99
0.006
0.994
0
UB3LYP
0.998
0.002
100
0.014
0.986
0
UCAM-B3LYP
0.999
0.001
100
0.021
0.979
0
UBHandHLYP
0.999
0.001
100
0.030
0.970
0
ULC-BLYP
1.000
0.000
100
0.033
0.967
0
UBD
0.999
0.001
100
0.022
0.978
0
Table S7 Chemical indices for cyclopropenyl anion (a) by the BS methods.
Method
HONO
HONO-1
nLU
B
Y
nLU
B
Y
UHF
1.000
0.000
100
0.111
0.889
1
UBLYP
1.000
0.000
100
0.013
0.987
0
UB3LYP
1.000
0.000
100
0.024
0.976
0
UCAM-B3LYP
1.000
0.000
100
0.033
0.967
0
UBHandHLYP
1.000
0.000
100
0.047
0.953
0
ULC-BLYP
1.000
0.000
100
0.047
0.953
0
UBD
1.000
0.000
100
0.034
0.966
0
Table S8 The Y values (%)a for a-c by the BSb and MRc methods.
C3H3– (a)
C4H4 (b)
C5H5+ (c)
UHF
86.6
100.0
98.9
UBDd
94.7
100.0
99.8
UBLYP
97.8
100.0
99.4
UB3LYP
98.0
100.0
99.6
UBHandHLYP
98.8
100.0
99.8
ULC-BLYP
99.1
100.0
99.9
UCAM-B3LYP
98.9
100.0
99.8
CASSCF
21.0
100.0
21.2
ROHF-MkCCSD
93.3
100.0
100.0
CASSCF-MkCCSD
77.0
100.0
55.7
UNO-MkCCSD
87.7
100.0
99.9
Method
a
The corresponding effective bond order B (%) is give by 100-Y.
b
Eq. (6b) is used.
c
d
Y = 2c 22 is used.
The Y values are calculated in the Fermi vacuum.
Table S9
Spin densities on the carbon atoms in the singlet (LS) and triplet (HS) states
of amino cyclobutadiene (d) by the BS methods.
Method
UHF
UBLYP
UB3LYP
UCAM-B3LYP
UBHandHLYP
ULC-BLYP
UBD
Spin state
C1
C2
C3
C4
N
LS
-0.943
1.087
-1.126
1.086
-0.035
HS
0.233
0.593
0.708
0.589
0.031
LS
-0.506
0.678
-0.700
0.676
-0.115
HS
0.346
0.543
0.532
0.542
0.107
LS
-0.568
0.735
-0.764
0.734
-0.102
HS
0.347
0.545
0.547
0.544
0.093
LS
-0.603
0.766
-0.795
0.764
-0.097
HS
0.347
0.544
0.556
0.542
0.086
LS
-0.668
0.825
-0.859
0.823
-0.082
HS
0.339
0.551
0.578
0.548
0.071
LS
-0.645
0.802
-0.832
0.801
-0.091
HS
0.341
0.545
0.566
0.543
0.078
LS
-0.668
0.830
-0.869
0.829
-0.080
HS
0.353
0.552
0.572
0.550
0.071
Table S10 Spin densities on the carbon atoms in the LS and HS states of formyl
cyclobutadiene (e) by the BS methods.
Method
UHF
UBLYP
UB3LYP
UCAM-B3LYP
UBHandHLYP
ULC-BLYP
UBD
Spin state
C1
C2
C3
C4
C5
O
LS
-1.080
1.042
-1.083
1.020
-0.534
0.547
HS
0.689
0.695
0.674
-0.040
-0.483
0.552
LS
-0.683
0.572
-0.670
0.552
-0.037
0.237
HS
0.554
0.438
0.541
0.337
-0.029
0.230
LS
-0.736
0.639
-0.727
0.619
-0.092
0.264
HS
0.561
0.466
0.548
0.316
-0.076
0.255
LS
-0.765
0.682
-0.759
0.660
-0.124
0.273
HS
0.562
0.497
0.549
0.300
-0.106
0.265
LS
-0.821
0.741
-0.815
0.724
-0.197
0.328
HS
0.578
0.515
0.564
0.264
-0.170
0.321
LS
-0.801
0.731
-0.796
0.710
-0.147
0.271
HS
0.567
0.530
0.554
0.271
-0.127
0.266
LS
-0.830
0.783
-0.824
0.726
-0.140
0.241
HS
0.570
0.531
0.554
0.322
0.530
0.232
Table S11
Spin densities on the carbon atoms in the LS and HS states of 1-amino,
2-formyl cyclobutadiene (f) by the BS methods.
Method
UHF
UBLYP
UB3LYP
UCAM-B3LYP
UBHandHLYP
ULC-BLYP
UBD
Spin state
C1
C2
C3
C4
N
C5
O
LS
-0.813
0.986
-1.132
0.997
-0.122
-0.505
0.538
HS
0.099
0.598
0.833
0.433
0.063
0.434
0.508
LS
0.048
-0.103
0.046
0.039
0.016
0.009
-0.051
HS
0.256
0.403
0.610
0.424
0.144
-0.007
0.234
LS
0.033
-0.087
0.010
0.077
0.015
0.009
-0.053
HS
0.246
0.427
0.637
0.419
0.131
-0.053
0.258
LS
-0.479
0.620
-0.807
0.635
-0.160
-0.101
0.273
HS
0.235
0.451
0.652
0.416
0.123
-0.081
0.267
LS
-0.537
0.679
-0.866
0.699
-0.152
-0.170
0.323
HS
0.223
0.468
0.685
0.412
0.111
-0.142
0.314
LS
-0.525
0.673
-0.841
0.685
-0.159
-0.128
0.275
HS
0.220
0.478
0.671
0.412
0.114
-0.104
0.268
LS
-0.532
0.720
-0.881
0.697
-0.151
-0.126
0.248
HS
0.244
0.494
0.667
0.423
0.120
-0.109
0.240
Table S12 The Y values (%)a for d-f by the BSb and MRc methods
Method
C4H3(CHO) (d)
C4H3(NH2) (e)
C4H2(NH2)(CH(f)
UHF
86.6
100.0
98.9
UBDd
94.7
100.0
99.8
UBLYP
97.8
100.0
99.4
UB3LYP
98.0
100.0
99.6
UCAM-B3LYP
98.9
100.0
99.8
UBHandHLYP
98.8
100.0
99.8
ULC-BLYP
99.1
100.0
99.9
CASSCF
21.0
100.0
21.2
ROHF-MkCCSD
93.3
100.0
100.0
CASSCF-MkCCSD
77.0
100.0
55.7
UNO-MkCCSD
87.7
100.0
99.9
a
The corresponding effective bond order B (%) is give by 100-Y.
b
Eq. (6b) is used.
c
d
is used.
The Y values are calculated in the Fermi vacuum.
Table S13
Spin densities on the carbon atoms in the singlet (LS) and triplet (HS)
states of 1,2-bis(methylene) cyclobutadiene (g) by the BS methods.
Method
UHF
UBLYP
UB3LYP
UCAM-B3LYP
UBHandHLYP
ULC-BLYP
UBD
Spin state
C1
C2
C3
C4
C5
C6
LS
-0.651
0.651
0.647
-0.647
0.754
-0.754
HS
0.652
0.652
-0.538
-0.538
1.037
1.037
LS
0.000
0.000
0.000
0.000
0.000
0.000
HS
0.340
0.340
-0.006
-0.006
0.725
0.725
LS
0.000
0.000
0.000
0.000
0.000
0.000
HS
0.362
0.362
-0.079
-0.079
0.781
0.781
LS
0.000
0.000
0.000
0.000
0.000
0.000
HS
0.387
0.387
-0.142
-0.142
0.821
0.821
LS
0.000
0.000
0.000
0.000
0.000
0.000
HS
0.414
0.414
-0.197
-0.197
0.860
0.860
LS
0.000
0.000
0.000
0.000
0.000
0.000
HS
0.418
0.418
-0.215
-0.215
0.864
0.864
LS
-0.001
0.001
0.001
-0.001
0.001
-0.001
HS
0.393
0.393
-0.152
-0.152
0.843
0.843
Table S14
Spin densities on the carbon atoms in the singlet (LS) and triplet (HS)
states of 1,3-bis(methylene) cyclobutadiene (h) by the broken-symmetry (BS)
single-reference (SR) methods.
Method
UHF
UBLYP
UB3LYP
UCAM-B3LYP
UBHandHLYP
ULC-BLYP
UBD
Spin state
C1
C2
C3
C4
C5
C6
LS
0.000
0.000
-0.990
0.000
0.990
0.000
HS
0.909
0.909
1.098
-0.827
1.098
-0.827
LS
0.000
0.000
-0.622
0.000
0.622
0.000
HS
0.498
0.498
0.744
-0.172
0.744
-0.172
LS
0.000
0.000
-0.692
0.000
0.692
0.000
HS
0.547
0.547
0.791
-0.261
0.791
-0.261
LS
0.000
0.000
-0.746
0.000
0.746
0.000
HS
0.587
0.587
0.823
-0.330
0.823
-0.330
LS
0.000
0.000
-0.787
0.000
0.787
0.000
HS
0.639
0.639
0.865
-0.411
0.865
-0.411
LS
0.000
0.000
-0.803
0.000
0.803
0.000
HS
0.632
0.632
0.858
-0.409
0.858
-0.409
LS
0.000
0.000
-0.790
0.000
0.790
0.000
HS
0.601
0.601
0.885
-0.383
0.885
-0.383
Table S15 Acronyms given in the text.
Acronym
Method
MkCCSD
Mukherjee’s state-specific coupled cluster singles and doubles
UHF-CCSD(T)
Spin-unrestricted Hartree-Fock based coupled cluster singles and doubles (with
perturbative triples corrections)
UBD(T)
Coupled cluster doubles (with perturbative triples corrections) based on spin-unrestricted
Brueckner determinant
AP
Approximate spin projection
CASSCF
Complete active space self-consistent field
DDCI
Difference dedicated configuration interaction
CASPT2
Second-order perturbation theory based on the Complete active space self-consistent field
wave function
NEVPT2
Second-order n-electron valence state perturbation theory
XLSD
Exchange local spin density
CAS-DFT
Complete active space density functional theory
MCOEP
Multiconfigurational optimized effective potential
XGGA
Exchange generalized gradient approximation
UDFT
Spin-unrestricted density functional theory
NO
Natural orbital
ROHF
Spin-restricted open-shell Hartree-Fock
UNO
UHF natural orbital
DNO
UDFT natural orbital
CMO
Corresponding molecular orbital
BLYP
Becke’s 1988 exchange functional with Lee-Yang-Parr correlation functional
B3LYP
Becke’s three parameter hybrid functional with Lee-Yang-Parr correlation functional
BHandHLYP
Becke’s half-and-half hybrid functional
CAM-B3LYP
Becke’s three parameter hybrid functional with Lee-Yang-Parr correlation functional
using the Coulomb-attenuating method
LC-BLYP
Long-range corrected Becke’s 1988 exchange functional with Lee-Yang-Parr correlation
functional
EOM-CCSD
Equation-of-motion coupled cluster singles and doubles
MR-AQCC
Multireference averaged quadratic coupled cluster