# Theoretical study of the magnetic interaction for M–O–M type metal oxides. Comparison of broken-symmetry approaches

**ABSTRACT** 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 Jab values 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.

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**ABSTRACT:**The electronic structure and magnetic interaction of the active site of pig purple acid phosphatase (PAP, uteroferrin) were investigated using pure DFT (UBLYP) and hybrid DFT methods (UB3LYP and UB2LYP). Uteroferrin catalyzes the hydrolysis of a phosphate ester under acidic conditions and contains a binuclear iron center. The mammalian PAPs are expected to be targets for drug design of osteoporosis. Their active sites are typical examples of the Fe(II)-Fe(III) mixed-valence system. We studied double exchange interaction of the mixed-valence system, using the potential energy difference between the Fe(II)-Fe(III) and the Fe(III)-Fe(II) states. The pathway of the antiferromagnetic coupling between Fe(III) and Fe(II) were also discussed by using chemical indices, which are evaluated by the occupation numbers of singly occupied natural orbitals. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010International Journal of Quantum Chemistry 01/2010; · 1.17 Impact Factor - Synthetic Metals - SYNTHET METAL. 01/2003; 135:663-664.
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**ABSTRACT:**Our theoretical efforts towards molecule-based magnetic conductors and superconductors on the basis of ab initio Hamiltonians and effective model Hamiltonians are summarized in relation to recently developed DNA-based molecular materials. Guanine and adenine derivatives coupling with organic radicals (R) are investigated as possible π–R components. In order to elucidate electronic and magnetic properties of these species, effective exchange integrals (Jab) for magnetic clusters are calculated by ab initio hybrid density functional methods. Theoretical possibilities of organic magnetic conductors and the organic solenoid are elucidated on the basis of these models in self-assembled DNA wires, sheets and related materials. Implications of the calculated results are finally discussed in order to obtain a unified picture of many p–d, π–d and π–R molecule-based systems with strong electron correlations.Polyhedron 01/2005; 24(16):2758-2766. · 2.05 Impact Factor

Page 1

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

Abstract

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

www.elsevier.nl/locate/poly

1. Introduction

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-

sistance(CMR), Bose–Einstein

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[2], while it is 5.4 K for

one-dimensional (1D) Sr2CuO3[3], 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 [6] because of strong

electron correlation. The most recent density functional

condensationof

(DFT) calculations have therefore tended to include

modifications such as self-interaction-corrected (SIC)

LSDA [7] and LSDA+on site Coulomb repulsion (U)

[8]. 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+) [18]. 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: taku@chem.sci.osaka-u.ac.jp (T. Onishi).

0277-5387/01/$ - see front matter © 2001 Elsevier Science Ltd. All rights reserved.

PII: S0277-5387(01)00591-5

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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

[18]. 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 [18], 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

[18]. 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]:

H(HB)= −?

ab

2JabSa·Sb

(1)

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)

state [18,19].

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).

Jab

(1)=

LSE(X)−HSE(X)

Smax

2

(2a)

Jab

(2)=

LSE(X)−HSE(X)

Smax(Smax+1)

(2b)

Jab

(3)=

LSE(X)−HSE(X)

HS?S2?−LS?S2?

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 [28], and Davidson [29] (GND), while the

second scheme Jab

Bencini [30], Ruiz [31] and others. Jab

the overlap of magnetic orbitals is sufficiently small,

while Jab

large. Jab

procedure[22]is close

HS?S2??Smax(Smax+1) andLS?S2??Smax, where Smax

is the spin size for the HS state. Jab

to Jab

[18].

As shown previously [26], our AP scheme can be

extended to trinuclear, and more larger clusters, though

Jab

in Eq. (2c) is applicable to linear trinuclear complexes

and ring form of tetranuclear complexes [18]. On the

other hand, Jab

both linear and ring forms by:

(2c)

(1)has been derived by Ginzberg [27],

(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)

to

Jab

(1)

by GNDif

(3)becomes equivalent

(2)in the strong overlap region, whereLS?S2??0

(1)and Jab

(2)do not work for them. For example, Jab

(3)

(4)without spin projection is given for

Jab

(4)=

LSE(X)−HSE(X)

4(N−1)SaSb

(2d)

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

[14].

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

matrices as:

Page 3

T. Onishi et al. / Polyhedron 20 (2001) 1177–11841179

?(r, r?)=? ni?i(r)?i(r?)*

(3)

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

+

(4)

?i=cos ??i−sin ??i

+

(5)

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.

??i?id?=Ti=cos2?−sin2?=cos 2?

The occupation numbers of NOs are given by the

orbital overlap Tias:

(6)

ni=1+T, ni*=1−Ti

(7)

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:

EXC=C1EX

HF+C2EX

+C5?EC

Slater+C3EX

Becke88+C4EC

VWN

LYP

(8)

where the third and fourth terms in this equation mean

Becke’s exchange correlation [32] involving the gradient

of the density and the Vosko, Wilk and Nusair (VWN)

[33] correlation functional, respectively, and the last

term is the correlation correction of Lee, Yang and

Parr (LYP) [17] 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 [34]

plus Hay’s diffuse basis sets [35] are used for the

transition metals and 6-31G* basis set is used for

oxygen. All calculations were performed by using

GAUSSIAN-94 program [36] 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

BS method.

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

[1–5].

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

?Jab

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

(1)? value

(2)? value is much

(1)? value, indicating that the mag-

(1)? value is equivalent to the ?Jab

(4)? value for 1a (N=2,

(1)?=?Jab

(4)???Jab

(3)?.

(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

M4O4(model 3).

Page 4

T. Onishi et al. / Polyhedron 20 (2001) 1177–1184 1180

Table 1

Effective exchange integarals (Jab)acalculated for transition metal oxides (model 1–3)

UHFUB2VWN US2VWNSchemeUB2LYPExp.

−219.5

−109.7

−218.1

CuOCu

−1800

−899.8

−1666

Jab

Jab

Jab

(1)

−2041

−1021

−1874

−1576

−787.8

−1476

(2)

(3)

−79.71

−80.00

−907.4

−939.2

−1001

−1042

−785.4

−809.0

−765 to −1049 CuOCuOCu

Jab

Jab

(3)

(4)

−108.2

−145.0

−468.8

−652.8

−721.4

−1014

Cu4O4

−514.9

−717.2

Jab

Jab

(3)

−484 to −625

(4)

NiONi

Jab

Jab

Jab

(1)

−49.38

−32.92

−49.28

−278.4

−185.6

−266.7

−373.1

−248.7

−358.3

−238.7

−159.2

−229.9

(2)

(3)

−43.04

−43.10

NiONiONi

−334.1

−339.1

Jab

Jab

(3)

−383.3

−390.7

−362.8

−368.5

(4)

−36.47

−49.05

−140.6

−190.9

−156.5

−212.5

Ni4O4

−145.1

−196.2

Jab

Jab

(3)

−139.0

(4)

aJabare shown in cm−1.

unprojected ?Jab

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[3] by the magnetic susceptibility measurement

and −765 cm−1for Sr2CuO2by the same method [4],

while −1049 cm−1for Sr2CuO3by the midinfrared

optical absorption method [5]. 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].

The ?Jab

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 [1]. 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

(3)?

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].

The ?Jab

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 [18]. 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 [8].

(4)?

(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

[14,20,21].Actually, we

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.

mustperformsymmetry-

Page 5

T. Onishi et al. / Polyhedron 20 (2001) 1177–11841181

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

?Jab

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

clusters.

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

(3)? 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.

Page 6

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

?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.

Spin-projected ?Jab

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 [14], 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-

tions of spinandcharge

UB2VWN.

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 [14] 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

densityobtainedby

Page 7

T. Onishi et al. / Polyhedron 20 (2001) 1177–11841183

UB2VWN, respectively. The CT interaction predicted

by UB2VWN is stronger than that by UHF, indicating

thatSEinteraction increases

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

follows:

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-

Table 2

Spin (charge)adensities of copper and nickel oxides (1–3) by UHF and UB2VWN methods

Model Site numberb

Methods

M1O2O4M5 O6 M7M8 M3

1a

−0.95

(1.61)

−0.79

(1.49)

0.00 0.96

(1.61)

0.79

(1.49)

UHF

(−1.21)

0.00

(−0.97)

UB2VWN

−0.99

(1.33)

−0.91

(1.22)

0.040.96

(1.52)

0.71

(1.35)

2a

0.96

(1.52)

0.71

(1.35)

UHF 0.04

(−1.19)

0.25

(−0.95)

(−1.19)

0.25

(−0.95)

UB2VWN

UHF0.92

(1.15)

0.67

(0.97)

0.000.92

(1.15)

0.67

(0.97)

0.000.92

(1.15)

0.67

(0.97)

0.000.92

(1.15)

0.67

(0.97)

0.00

3a

(−1.15)

0.00

(−0.97)

(−1.15)

0.00

(−0.97)

(−1.15)

0.00

(−0.97)

(−1.15)

0.00

(−0.97)

UB2VWN

UHF1.91

(1.62)

1.62

(1.45)

1b

0.00

−1.91

(1.62)

−1.62

(1.45)

(−1.24)

0.00

(−0.89)

UB2VWN

2b

UHF 1.86

(1.55)

1.05

(0.93)

0.13

−1.97

(1.31)

−1.89

(1.37)

0.13 1.86

(1.55)

1.05

(0.93)

(−1.20)

0.90

(−0.61)

(−1.20)

0.90

(−0.61)

UB2VWN

UHF 1.82

(1.12)

1.41

(0.94)

0.00

−1.82

(1.12)

−1.41

(0.94)

3b

0.001.82

(1.12)

1.41

(0.94)

0.00

−1.82

(1.12)

−1.41

(0.94)

0.00

(−1.12)

0.00

(−0.94)

(−1.12)

0.00

(−0.94)

(−1.12)

0.00

(−0.94)

(−1.12)

0.00

(−0.94)

UB2VWN

aCharge density populations are given in parentheses.

bThe site numbers are shown in Fig. 1.

Page 8

T. Onishi et al. / Polyhedron 20 (2001) 1177–1184 1184

tronic, magnetic and optical properties of the transition

metal oxides elsewhere.

Acknowledgements

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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