Polyhedron 20 (2001) 1177–1184
Theoretical study of the magnetic interaction for M–O–M type
metal oxides. Comparison of broken-symmetry approaches
Taku Onishi *, Yu Takano, Yasutaka Kitagawa, Takashi Kawakami,
Yasunori Yoshioka, Kizashi Yamaguchi
Department of Chemistry, Graduate School of Science, Osaka Uni?ersity, Toyonaka, Osaka 560-0043, Japan
Received 17 September 2000; accepted 6 December 2000
The unrestricted Hartree–Fock (UHF) and hybrid-density functional theory (DFT) calculations have been carried out for the
metal oxides such as copper oxides and nickel oxides. In order to elucidate magnetic properties of the species, the effective
exchange integrals (Jab) have been obtained by the total energy difference between the highest and lowest spin states in several
computational schemes with and without spin projection. The mixing ratios of the exchange correlation functionals in the hybrid
DFT method have been reoptimized so as to reproduce the Jabvalues for strongly correlated oxides. The natural orbital analysis
has also been performed for elucidation of symmetry and occupation numbers of the magnetic orbitals. From these calculated
results, we discuss characteristics of the magnetic interactions for metal oxides in the strong correlation regime. © 2001 Elsevier
Science Ltd. All rights reserved.
Keywords: Hybrid-DFT; Superexchange interaction; Metal oxides; Effective exchange integrals
Recently, experimental and theoretical studies on
antiferromagnetic transition-metal oxides [1–5] have
attracted much attention in relation to antiferromag-
netism, high-Tcsuperconductivity, colossal magnetore-
excitons, etc. Various experimental results have demon-
strated important roles of both lattice dimensionality
and electron correlation for these species. For example,
the observed Ne ´el temperature (TN) is 410 K for two-
dimensional (2D) YBa2Cu3O6.15, while it is 5.4 K for
one-dimensional (1D) Sr2CuO3, though the effective
exchange integrals for the Cu–O–Cu unit are not so
different between them. Band calculations of transition
metal oxides based on the local-spin-density approxi-
mation (LSDA) have often predicted that these crystals
are metals instead of the antiferromagnetic insulator,
showing the breakdown of LSDA  because of strong
electron correlation. The most recent density functional
(DFT) calculations have therefore tended to include
modifications such as self-interaction-corrected (SIC)
LSDA  and LSDA+on site Coulomb repulsion (U)
. The periodic MO calculations based on the unre-
stricted Hartree–Fock (UHF) approximation have
been also utilized for theoretical studies of magnetism
of transition metal oxides, although UHF usually over-
estimates the band gaps and spin polarization (SP)
effect of the species [9–14].
Both cluster and band models have been used for
DFT calculations [9–14] of effective exchange integrals
of antiferromagnetic insulator, though the latter is cru-
cial for magnetic metals. The hybrid DFT methods
such as B3LYP [15–17] have been used for binuclear
transition metal complexes in the field of quantum
chemistry to include the exchange-correlation effect.
However, the conventional B3LYP method has overes-
timated the magnitude of effective exchange integrals
(Jab) for the K2MX4-type metal halides (M=Cu2+,
Ni2+, Mn2+) . For example, the Jab value by
UB3LYP was −574 cm−1for the Cu4F4cluster, while
it was −92 cm−1for Ni4F4; note that the experimental
values for the K2CuF4and K2NiF4crystals are −132
* Corresponding author. Fax: +81-6-6850-5550.
E-mail address: firstname.lastname@example.org (T. Onishi).
0277-5387/01/$ - see front matter © 2001 Elsevier Science Ltd. All rights reserved.
T. Onishi et al. / Polyhedron 20 (2001) 1177–1184 1178
and −36 cm−1, respectively. In order to improve the
calculated results, we have reoptimized the mixing co-
efficients of the hybrid DFT approach to magnetic
clusters. It was shown that the half-and-half (HH) type
DFT, UB2VWN, can reproduce the experimental re-
sults; namely the calculated Jabvalues are −199 and
−33 cm−1for Cu4F4and Ni4F4clusters, respectively
. This implies the necessity of modifications of
exchange correlation functions for strongly correlated
systems in the hybrid DFT method.
As a continuation of the previous work , we here
examine perovskite-type transition metal oxides [1–5].
The 1D chain and 2D layer of K2NiF4-type copper
oxides are mainly responsible for the magnetic interac-
tions. Then, we have focused on 1D chain for Sr2CuO3
and SrCuO2, and the 2D layer for YBa2CuO6.15,
La2CuO4and related species. The isostructural nickel
oxide, La2NiO4, is also examined in comparison with
La2CuO4. To this end, we have constructed the simple
linear cluster models (M–O–M, M–O–M–O–M) for
1D chain and ring cluster model (M4O4) for 2D layer
(M=Cu2+, Ni2+). For computational methods, UHF
and three types of half-and-half DFT (UB2VWN,
US2VWN and UB2LYP) have been employed to ob-
tain the effective exchange integrals using several com-
putational schemes with and without spin projection
. In order to clarify the magnetic interactions, we
have also calculated spin densities, charge densities and
natural orbitals. From the viewpoint of these results,
we have examined whether hybrid-DFT methods com-
bined with cluster models are applicable to metal oxides
in the strong correlation regime or not.
2. Theoretical background
2.1. Computation of effecti?e exchange integrals (Jab)
Magnetic properties of transition metal complexes
and/or clusters have been investigated by the Heisen-
berg-type spin coupling Hamiltonian [1–5]:
where Jab is the orbital-averaged effective exchange
integral between the a-th and b-th metal sites with total
spin operators Saand Sb. Recent development of the ab
initio computational techniques has made it possible to
calculate the Jabvalues [18–25] for transitional-metal
complexes. These methods are classified into two types.
One is the spin-symmetry adapted (SA) perturbational
and configuration interaction (CI) approach [14,20,21].
Use of the SA method is desirable for quantitative
computations of Jab, but it is hardly applicable to
transition-metal complexes because of high computa-
tional costs. The other is the broken-symmetry (BS)
approach [22–30], which is often used for such systems
because of lower computational costs, though the spin
contamination problem occurs in the low-spin (LS)
Broken-symmetry (BS) calculations of Jabfor binu-
clear transition metal complexes have been performed
on the basis of different computational schemes such as
Eqs. (2a), (2b) and (2c).
whereYE(X) andY?S2?(X) denote, respectively, total
energy and total spin angular momentum for the spin
state Y by the method X (=UHF, UDFT, etc). The
first scheme Jab
Noodleman , and Davidson  (GND), while the
second scheme Jab
Bencini , Ruiz  and others. Jab
the overlap of magnetic orbitals is sufficiently small,
procedure is close
HS?S2??Smax(Smax+1) andLS?S2??Smax, where Smax
is the spin size for the HS state. Jab
As shown previously , our AP scheme can be
extended to trinuclear, and more larger clusters, though
in Eq. (2c) is applicable to linear trinuclear complexes
and ring form of tetranuclear complexes . On the
other hand, Jab
both linear and ring forms by:
(1)has been derived by Ginzberg ,
(2)has been proposed by GND,
(1)is derived when
(2)is applicable when the overlap is sufficiently
(3)by our approximate spin projection (AP)
(2)in the strong overlap region, whereLS?S2??0
(2)do not work for them. For example, Jab
(4)without spin projection is given for
where N is the total spin site number and Scis the size
of spin Sc(c=a, b). This equation is applicable to
periodic systems involving the same cluster unit cells
2.2. Natural orbital analysis
Molecular orbital (MO) pictures of transition metal
complexes are desirable for deeper understanding of the
magnetic interactions between the transition metal
atoms via oxygen dianion. For MO-theoretical explana-
tion of magnetic interaction, the natural orbitals (NO)
are determined by diagonalizing their first-order density
T. Onishi et al. / Polyhedron 20 (2001) 1177–1184 1179
?(r, r?)=? ni?i(r)?i(r?)*
where niand ?idenote the occupation number and
natural orbital, respectively.
The magnetic orbitals for transition metal oxides are
expressed by the NOs:
?i=cos ??i+sin ??i
?i=cos ??i−sin ??i
where ? is the orbital mixing parameter, and ?iand
?i+denote the i-th bonding and corresponding anti-
bonding NOs, respectively. The NO mixing in turn
provides the BS orbitals (?iand ?i) which are more or
less localized on different magnetic sites, respectively.
The orbital overlap between the magnetic orbitals is
defined to elucidate their localizability.
The occupation numbers of NOs are given by the
orbital overlap Tias:
The occupation numbers of bonding and antibonding
NOs are close to 2.0 and 0.0, respectively, except the
magnetic NOs, for which they are close to 1.0. The
bonding NOs with n?2.0 are regarded as the closed-
shell orbitals. From Eqs. (3)–(6) we can elucidate the
close relationship between the BS and symmetry-
adapted (SA) CI approaches via the occupation num-
bers of NOs as discussed in Refs. 18, 19, 26.
3. Calculation methods and structures
3.1. Computational methods
Computational methods employed here are unre-
stricted Hartree–Fock method (UHF) and hybrid-DFT
[18,19] methods (UB2VWN, US2VWN and UB2LYP).
In the hybrid-DFT calculations, exchange-correlation
potentials are generally defined by:
where the third and fourth terms in this equation mean
Becke’s exchange correlation  involving the gradient
of the density and the Vosko, Wilk and Nusair (VWN)
 correlation functional, respectively, and the last
term is the correlation correction of Lee, Yang and
Parr (LYP)  which includes the gradient of the
density. Ci(i=1–5) are the mixing coefficients.
The parameter sets (C1, C2, C3, C4and C5) are (0.5,
0.5, 0.5, 1.0 and 0.0) for UB2VWN, (0.5, 0.5, 0.0, 1.0
and 0.0) for US2VWN and (0.5, 0.5, 0.5, 1.0 and 1.0)
for UB2LYP [18,19]. Tatewaki–Huzinaga MIDI 
plus Hay’s diffuse basis sets  are used for the
transition metals and 6-31G* basis set is used for
oxygen. All calculations were performed by using
GAUSSIAN-94 program  package.
3.2. Model structures of K2NiF4-type metal oxides
In this study, we focused on copper oxides and nickel
oxides. These metal oxides form the 1D chain and 2D
layered perovskite-type structures. First of all, we ex-
amine the simple dimer model 1 in Fig. 1 in order to
elucidate the interrelationships among the computa-
tional schemes in Eqs. (2a–d). The linear cluster model
2 is investigated as the simplest model for 1D Sr2CuO3
and Sr2CuO2crystals. As it is known that the MO2
sheets are mainly responsible for magnetic properties
for K2NiF4-type transition metal oxides [37,38] such as
La2CuO4and YBa2Cu3O6.15, we constructed the square
planar M4O4(model 3) as the simplest model. The full
geometry optimizations of 1–3 were performed by each
4. Results and discussion
4.1. Effecti?e exchange integrals (Jab)
In order to examine the magnetic interaction, we
have calculated effective exchange integrals (Jab) for
1–3 by UHF and hybrid-DFT calculations. Table 1
summarizes the Jabfor copper and nickel oxides (1–3).
All the Jabvalues are negative, showing the antiferro-
magnetic interaction in accord with the experiments
From Table 1, the ?Jab
each BS method is a little smaller than the ?Jab
because of the non-negligible orbital overlap between
the magnetic orbitals, while the ?Jab
smaller than the ?Jab
netic orbital overlap (Ti) is far smaller than 1.0. The
Sa=Sb=1/2). The spin projection effect is rather small
even for US2VWN; note that ?Jab
The spin-projected ?Jab
CuOCuOCu (2a) by each BS method is close to the
(3)? value for CuOCu (1a) by
(2)? value is much
(1)? value, indicating that the mag-
(1)? value is equivalent to the ?Jab
(4)? value for 1a (N=2,
(3)? value for the linear cluster
Fig. 1. The model clusters; the linear dimer M–O–M (model 1), the
linear trimmer M–O–M–O–M (model 2) and the square-planar
T. Onishi et al. / Polyhedron 20 (2001) 1177–1184 1180
Effective exchange integarals (Jab)acalculated for transition metal oxides (model 1–3)
UHF UB2VWNUS2VWN SchemeUB2LYP Exp.
−765 to −1049 CuOCuOCu
−484 to −625
aJabare shown in cm−1.
lap effect is rather small, even in the case of the
HH-type DFT method. The ?Jab
UHF is about one tenth of HH-type DFT method.
Experimentally, the ?Jab? values are −904 cm−1for
Sr2CuO3 by the magnetic susceptibility measurement
and −765 cm−1for Sr2CuO2by the same method ,
while −1049 cm−1for Sr2CuO3by the midinfrared
optical absorption method . From these data, the Jab
value for the 1D chain is in the range −765 to −1049
cm−1. The present HH-type DFT cluster model calcu-
lations well reproduced the experimental results [3,4].
significantly smaller than the ?Jab
method. This implies that the orbital overlap (Ti) effect
is not negligible because of the electron delocalization
(see also below). The ?Jab
about 1/5–1/7 of the HH-type DFT methods. Experi-
mentally [1–5], the Jabvalues are −484, −488, −536
and −625 cm−1for YBa2Cu3O6.15, Pr2CuO4, La2CuO4
and Nd2CuO4, respectively. Then, we may consider that
the Jabvalue is in the range −484 to −625 cm−1,
though the Cu–O–Cu distances are different among
the four oxides, and the correlation between the ob-
served Jabvalue and Cu–O length is complex . From
Table 1, it may be concluded that the HH-type DFT
calculations for 3a can reproduce the experimental Jab
values for the copper oxides, though computations of
more larger clusters are desirable for the refinements of
the calculated results for 3a.
In order to confirm the calculated results for the
copper oxides, we have performed the same computa-
tions of the nickel oxides clusters 1b–3b. The similar
tendencies are also recognized for the computational
results for NiONi (1b) and NiONiONi (2b). The ?Jab
(4)? value, indicating that the orbital over-
(3)? value for the 2a by
(3)? value for the tetranuclear ring Cu4O4(3a) is
(4)? value by each BS
(3)? value for 3a by UHF is
value for Ni4O4(3b) is largely smaller than the ?Jab
value, showing the nonnegligible orbital overlap (Ti)
effect which entails the necessity of the approximate
spin projection (AP). Indeed, the ?Jab
the HH-type DFT methods with the AP procedure are
close to the experimental value [37,38].
are smaller than those of 2b for the model of the 1D
chain. The same tendency is also recognized for the
copper oxides (3a and 2a). This is consistent with the
experimental observation [1–5]. In conclusion, the HH-
type DFT methods such as UB2LYP and US2VWN
are reliable enough for theoretical investigation of the
effective exchange integrals in transition metal oxides,
as in the case of transition metal halides . This in
turn indicates a crucial role of electron correlation in
both species in accord with the LSDA+U and/or SIC
LSDA calculations of NiO .
(3)? values for 3b by
(3)? values of 3b for the model of the 2D sheet
4.2. Natural orbitals and occupation numbers
The CASSCF and CASPT2 calculations are desirable
for further refinements of the BS computational results
adapted CI computation in order to elucidate the mag-
netic interaction in detail. However, high computer
costs do not allow us to carry out these computations
for such large systems. Therefore, we have investigated
the shapes and occupation numbers of natural orbitals
(NO) obtained by Eqs. (3) and (7) using BS solutions.
We can obtain an MO-theoretical explanation of mag-
netic interaction for the transition metal oxide clusters
by the NO analysis. Here, we only discuss the shapes
and occupation numbers of NOs by UHF and
UB2VWN methods, since other HH-type DFT meth-
ods provide similar results.
T. Onishi et al. / Polyhedron 20 (2001) 1177–1184 1181
Fig. 2 shows the shapes and occupation numbers of
singly occupied NO (SONO) for copper oxides 2a and
3a. In the case of copper oxide 2a, it is found that
SONOs by both methods are delocalized over the clus-
ter, and dz2 orbitals of copper sites interact with pz
orbitals of oxygen sites. These results indicate that the
negative Jabvalues of 2a is responsible for the antiferro-
magnetic ?-type superexchange (SE) interaction be-
tween dz2 orbitals of copper atoms via pzorbitals of
oxygen atoms. While, we can find the delocalized
SONOs for 3a on the cluster and the interactions
between dx2−y2orbitals of copper parts and pxor py
orbitals of oxygen parts, also indicating that the nega-
tive Jabvalues of 3a are derived from the antiferromag-
netic ?-type SE interaction between dx2−y2orbitals of
copper sites via pxor pyorbitals of oxygen sites. The
SONOs obtained for 2a and 3a by UB2VWN are more
delocalized over the clusters than those by UHF. Judg-
ing from this result and obtained Jabvalues, UB2VWN
estimates SE interactions appropriately than UHF.
The occupation numbers of SONO−1(SONO+1)
for 2a are 1.06 (0.94) and 1.29 (0.71) by UHF and
UB2VWN, respectively, namely the orbital overlap (Ti)
between the magnetic HOMO−1s (?HOMO−1 and
?HOMO−1) for the up- and down-spins are 0.06 and
0.29, respectively. Since the orbital overlap (Ti) shows
the strength of the interaction between the magnetic
orbitals, it is found from rather small Tivalues that the
NO mixing in Eqs. (4) and (5) is very large and
therefore ?HOMO+1overlaps with ?HOMO−1weakly for
2a. Therefore, ?Jab
between ?HOMO−1and ?HOMO−1values for 3a are 0.12
and 0.43 by UHF and UB2VWN, respectively. The Ti
values for 3a are larger than those for 2a. Therefore, we
(3)? is close to ?Jab
(4)?. While, Tivalues
can find that the interaction between ?HOMO−1and
?HOMO−1for 3a is stronger than that for 2a, namely
UB2VWN are several times larger than those by UHF.
These results are consistent with the fact that ?Jab
UB2VWN is much larger than that by UHF.
Fig. 3 illustrates the shapes and occupation numbers
of SONOs for nickel oxides 2b and 3b by UHF and
UB2VWN. We can find similar tendency that all
SONOs for 2b and 3b are delocalized on the clusters. In
addition, SONO?1 for 2b and SONO?3 for 3b show
?-type interaction between d orbital of nickel sites and
p orbital of oxygen sites, while SONO?2 for 2b and
SONO?2 for 3b shows ?-type interaction. These re-
sults indicate that the negative Jabvalues for 2b and 3b
are responsible for the antiferromagnatic ?- and ?-type
SE interactions between d orbitals of nickel atoms via
the p orbital of oxygen atoms. The SONOs obtained
for 2b and 3b by UB2VWN are more delocalized on
clusters than those of UHF. Judging from this result
and the calculated Jabvalues, UB2VWN estimates SE
interactions appropriately than UHF for nickel oxides
For nickel oxides 2b, the Tivalues between ?-type
?HOMO−2and ?HOMO−2are 0.090 and 0.345 by UHF
and UB2VWN, respectively, while those between ?-
type ?HOMO−1 and ?HOMO−1 are 0.035 and 0.017,
respectively. Judging from the orbital overlap Ti, it is
noteworthy that ?-type interaction by UB2VWN is
much stronger than by UHF, though ?-type interaction
by UB2VWN is as strong as that by UHF. The Ti
values between ?HOMO−3 and ?HOMO−3 for 3b are
0.134 and 0.411 by UHF and UB2VWN, respectively,
while those between ?HOMO−2and ?HOMO−2are 0.069
(3)? for 3a is smaller than ?Jab
(4)?. Tivalues estimated by
Fig. 2. The shapes and occupation numbers of SONO for the low spin state of copper oxides (models 2a and 3a) by UHF and UB2VWN.
T. Onishi et al. / Polyhedron 20 (2001) 1177–11841182
Fig. 3. The shapes and occupation numbers of SONO for he low spin state of nickel oxides (models 2b and 3b) by UHF and UB2VWN.
and 0.205, respectively. These indicate that ?-type in-
teraction is much stronger than ?-type interaction. The
Tivalues for 3b are larger than those for 2b. Therefore,
it is consistent with the fact that ?Jab
As shown in the above results, we can conclude that
copper and nickel oxides show antiferromagnetic inter-
actions because of the SE interaction between d orbital
of transition metal sites via p orbital of oxygen site.
smaller than unprojected ?Jab
should consider the BS correction when we investigate
the magnetic interaction for the model clusters of 2D
layer type transition metal oxides.
(3)? is smaller than
(4)? because of spin delocalization.
(3)? for the model 3 for the 2D layer is
(4)?, indicating that we
4.3. Spin density and charge density populations
In the BS approach, spin densities appear even for
the low-spin singlet state of small clusters. As shown
previously , the spin densities are not strictly related
to real spin populations as in the case of antiferromag-
netic solids, but they are useful indices to express the
magnitude of electron and spin correlations in strongly
correlated finite systems. On the other hand, popula-
tions of charge densities are useful for qualitative anal-
ysis of the superexchange (SE) interaction via the
charge transfers between transition metal ion and oxide
anion. Table 2 summarizes the spin density populations
in the low spin state of copper and nickel oxides (1–3),
while the charge density populations are also given in
parenthesis. Since different HH-type DFT methods
provided the similar results for spin density and charge
density populations, here, we only show the popula-
It is found from Table 2 that spin densities on metal
sites estimated by UB2VWN are smaller than those by
UHF. It is thought that the exchange correlation terms
by UB2VWN, reduce the magnitude of them. UHF
tends to overestimate the spin densities for organic
radicals  and transition metal complexes via the spin
polarization (SP) mechanism, while UB2VWN im-
proves the magnitude of spin density for transition
metal oxides. The sign of spin densities for the nearest
transition metal ions of 1 and 3 is opposite, showing the
antiferromagnetic spin correlation. In the case of 2, the
oxygen atoms (sites 2 and 4) have the positive spin
densities, although oxygen atoms for 1 and 3 have no
spin densities because of the spatial symmetry. The
overlaps between metal and oxygen are found in
SONO+1 of copper oxides and SONO+3 of nickel
oxides. Since these SONOs are strongly delocalized by
the SE interaction, it is thought that positive spin
densities emerge on oxygen. The positive spin densities
on oxygen sites are characteristic of these odd size
cluster. In periodic systems such as solids, we could not
find such effect.
From Table 2, the charge transfer (CT) from oxygen
to metal sites are 0.39 (0.38) and 0.51 (0.55) for 1a (1b)
and 0.85 (0.88) and 1.03 (1.06) for 3a (3b) by UHF and
T. Onishi et al. / Polyhedron 20 (2001) 1177–1184 1183
UB2VWN, respectively. The CT interaction predicted
by UB2VWN is stronger than that by UHF, indicating
UB2VWN, because of the larger orbital overlaps be-
tween metal and oxygen by UB2VWN. Judging from
Jab values, we can find that the CT interaction is
properly estimated by UB2VWN (Table 2).
in order UHF?
4.4. Concluding remarks
We have investigated the magnetic interactions of
metal oxides on the basis of effective exchange integrals
estimated by three different schemes, populations of
spin and charge density, the shape of natural orbitals
and their occupation numbers. We have concluded as
1. Both UHF and hybrid DFT calculations of the
model clusters show the antiferromagnetic interac-
tion via superexchange (SE) mechanism between
metal ion and oxygen dianion.
2. Hybrid-DFT provide reliable Jabvalues compared
with experimental one, due to the proper inclusion
of antiferromagnetic SE effects.
3. The shape of SONO and their occupation numbers
show ?-type and ?-type SE interaction. Only ?-type
SE interaction occurs in copper oxides, while ?-type
and ?-type SE interactions occur in nickel oxides
4. The ?-type SE interaction is weaker than ?-type
from their occupation numbers in nickel oxides.
From the conclusions of (1–4), it is considered that
hybrid-DFT provide the proper Jabvalues for transition
metal oxides via the SE mechanism. Therefore, it is
clear that present hybrid-DFT methods are applicable
to transition metal oxides with strong electron correla-
tions. It is noteworthy that the Jabvalue for the 2D
CuO2plane of copper oxides is closely related to the
energy gap ? of the high-Tcsuperconductivity [22,37–
39], namely ??0.1?Jab?. This relationship is derived
from the exact diagonalization of the t−J model for
finite clusters [37–39], since the mean-field band calcu-
lations are not so reliable for strongly correlation sys-
tems. The 1D chain of Sr2CuO3has also attracted great
interest in relation to the two-photon absorption and
large hyperpolarizabilities [5,40]. The transition metal
oxides are promising in the intersection area between
magnetism and optics. The implications of the present
calculated results are discussed in relation to the elec-
Spin (charge)adensities of copper and nickel oxides (1–3) by UHF and UB2VWN methods
Model Site numberb
M1 O2O4 M5O6 M7M8M3
aCharge density populations are given in parentheses.
bThe site numbers are shown in Fig. 1.
T. Onishi et al. / Polyhedron 20 (2001) 1177–11841184 Download full-text
tronic, magnetic and optical properties of the transition
metal oxides elsewhere.
This work has been supported by Grants-in-Aid for
Scientific Research on Priority Areas (Nos. 10132241,
10149105 and 1224209) from the Ministry of Educa-
tion, Science, Sports and Culture, Japan. Y.T. is also
supported by Research Fellowships of the Japan Soci-
ety for the Promotion of Science for Young Scientists.
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