DSpace at VNU: The role of ligands in controlling the electronic structure and magnetic properties of Mn-4 single-molecule magnets

Available online at www.sciencedirect.com

Computational Materials Science 44 (2008) 111–116
www.elsevier.com/locate/commatsci

The role of ligands in controlling the electronic structure
and magnetic properties of Mn4 single-molecule magnets
Nguyen Anh Tuan a,b, Shin-ichi Katayama a, Dam Hieu Chi a,b,*
a

School of Materials Science, Japan Advanced Institute of Science and Technology, 1-1, Asahidai, Nomi, Ishikawa 923-1292, Japan
b
Faculty of Physics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam
Available online 18 April 2008

Abstract
single-molecule magnets (SMM), i.e,
We present our studies of electronic structure and magnetic properties of Mn4þ Mn3þ
3
4þ
3þ
½Mn4þ Mn3þ
O
Cl
ðOAcÞ
ðpyÞ
Š
(py
=
pyridine)
and

½Mn
Mn
O
ClðOAcÞ
ðdbmÞ
Š
(dbmH
=
dibenzoyl-methane)
molecules by using
3
4
3
3
3
3
3
3
3
a first-principles all-electron relativistic method within spin-polarized density functional theory. To investigate the possibility of ligands
controlling the electronic structure and magnetic properties, we designed and calculated the geometric and electronic structures of twelve
n+
other Mn4þ Mnnþ
ions, and the
3 (n = 2, 3, 4) molecules with different peripheral-ligand configurations. The electronic structure of Mn
4þ
nþ
interatomic distances, electronic structure and magnetic properties of Mn Mn3 molecules display an interesting variation with n.
Ó 2008 Elsevier B.V. All rights reserved.
PACS: 75.50.Xx; 75.75.+a; 31.15.Ar; 33.15.Àe; 33.15.Dj; 75.30.Wx

Keywords: First-principles calculation; Single-molecule magnets; Mn clusters; Nano-piezomagnets; Molecular design

1. Introduction
Single-molecule magnets (SMM) have recently attracted
much interest since they are collections of identical nanomagnets in which quantum phenomena such as step like
hysteresis curves of magnetization are observed [1,2].
Beyond being the actors of fundamental quantum phenomena, molecular magnets are widely studied because various
present and future specialized applications of magnets
require monodisperse, small magnetic particles.
Thestructure of each molecular magnet consists of the
two components: the core which contains transition metal
atoms, and the outer ligand complex. Since each transition
metal atom carries its own spin moment, the core of the
SMM plays the primary role of determining the magnetic
structure of the SMM, and the substitution of the transi*
Corresponding author. Address: School of Materials Science, Japan
Advanced Institute of Science and Technology, 1-1, Asahidai, Nomi,
Ishikawa 923-1292, Japan. Tel.: +81 76 151 1584; fax: +81 76 151 1535.
E-mail address: (D.H. Chi).

0927-0256/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.commatsci.2008.01.060

tion metal elements becomes an important way of controlling the magnetic character of the molecular magnet. Of
course, the outer ligand configuration around the core is
another factor which controls the charge, i.e., valence of
the metal ion and, thereby, its spin. Indeed, rather different
magnetic characteristics are observed in some SMM systems which have the same core structure [1,3–6]. The only
difference lies in their ligand components of the SMM system. Moreover, the outer ligands govern the mutual spatial
arrangement of the metal-oxide core, and thus play an

important role in determining the intermolecular interaction [7]. For example, Mn4O3Cl4(O2CEt)3(py)3, one of
the tetrahedral Mn4 SMM system, forms a dimer structure
in its crystal structure, and shows interesting magnetic
behavior completely different from that of individual molecules [3,8,9]. In other words, the difference in the spatial
arrangement is the primary factor making various Mn4
molecules so different from each other, thereby contributing to the magnetism of the SMM system.
In this paper, we present our studies of the electronic
structure and magnetic properties of trigonal-pyramid


112

N.A. Tuan et al. / Computational Materials Science 44 (2008) 111–116

Mn4þ Mn3þ
single-molecule magnets (SMM), i.e, ½Mn4þ 3
3þ
Mn3 O3 ClðOAcÞ3 ðdbmÞ3 Š (dbmH = dibenzoyl-methane)
and ½Mn4þ Mn3þ
3 O3 Cl4 ðOAcÞ3 ðpyÞ3 Š (py = pyridine) molecules by using a first-principles all-electron relativistic
method within spin-polarized density functional theory.
To investigate the possibility of controlling the electronic
structure and magnetic properties, we designed and calculated the geometric and electronic structures of the four(n = 2, 3, 4) molecules.
teen trigonal-pyramid Mn4þ Mn3þ
3
Our calculations reveal an important role for the ligand
complex in controlling electronic and magnetic properties
of Mn4 SMM as well as in designing new SMM with new
functions.
2. Methodology

We performed cluster calculations using the program
DMOL3 [10] in Materials Studio package, which is
designed for the realization of large-scale density functional theory (DFT) calculations. All-electron relativistic
calculations were performed with the double numerical
basis sets plus polarization functional (DNP). The
DNP basis sets are of comparable quality to 6-31G**
Gaussian basis sets [11]. Delley et al. showed that the
DNP basis sets are more accurate than Gaussian basis
sets of the same size [10]. The RPBE functional [12] is
so far the best exchange-correlation functional [13],
based on the generalized gradient approximation
(GGA), is employed to take account of the exchange
and correlation effects of electrons. The real-space global
˚ . Spin-unrestricted DFT
cutoff radius was set to be 7.0 A
was used to obtain all results presented in this work. For
better accuracy, the octupole expansion scheme is
adopted for resolving the charge density and Coulombic
potential, and a fine grid is chosen for numerical integration. The charge density is converged to 1 Â 10À6 a.u. in
the self-consistent calculation. In the optimization process, the energy, energy gradient, and atomic displacement are converged to 1 Â 10À5, 1 Â 10À4 and 1 Â 10À3
a.u., respectively. In order to explore the full freedom
in the potential energy surface and avoid possible saddle
points, the geometric optimization is performed without
any symmetry restriction. The atomic charge and magnetic moment are obtained by Mulliken population analysis. A Fermi smearing of 0.005 hartree (Ha)
(1Ha = 27.2114 eV) was used to improve computational
performance.
3. Results and discussion

OAc bridges, but differ in the peripheral-ligand L1 and
L2 groups (Fig. 1). Each of them is distinguished from

the other by its peripheral ligands L1 and L2. L1 and L2
make two coordinations to complete the distorted octahedral geometry at each b-site (as shown in the inset of
Fig. 1a), and thus are crucial factor in controlling the
charge of Mn ions at this site without breaking the distorted cubane geometry of the Mn4O3Cl core. A naı¨ve
expectation of the formal charge state of metal ions in
Mn4O3Cl core can be derived from the nominal charge of
the connected ligands. In the case that both L1 and L2
are neutral ligands, the obtained result is Mn4þ Mn2þ
3 molecules. In the case that L1 and L2 are a neutral ligand and a
molecules are
radical anion, respectively, Mn4þ Mn3þ
3
obtained. In the case that both L1 and L2 are radical
anions, Mn4þ Mn4þ
3 molecules are formed. By this means,
molecules, five Mn4þ Mn3þ
molecules,
four Mn4þ Mn2þ
3
3
4þ
4þ
and five Mn Mn3 molecules have been designed. Some
of Mn4þ Mn3þ
3 molecules have been synthesized [1,4]. The
molecules are labeled from (1) to
fourteen Mn4þ Mnnþ
3
(14), being classified into the three groups by the formal
charge of the manganese ions at the b-site (as shown in

z

a

x

O(OAc)
L

O(core)

Mn
y
L

O(core)

Cl(1)

O(6)
O(5)

O(4)

OAc

O(9)

Mn(1)
O(8)

O(3)

O(2)
O(1)

Mn(4)

O(7)

Mn(3)
Mn(2)

L1
Cl(1)

L2

Mn4+, a-site

b

μ3-O2-

μ3-O2-

μ3-O2-

Mnn+
Mnn+

Mnn+

3.1. Designing trigonal-pyramid Mn4þ Mnnþ
3 molecules
In this study, fourteen trigonal-pyramid Mn4þ Mnnþ
3
(n = 2, 3, 4) molecules have been designed or reconstructed.
They have the general chemical formula Mn4O3Cl(OAc)3L13L23 (L1 and L2 are ligand groups). These
molecules consist of the same Mn4O3Cl core and three

μ 3-Cl-

b-site

Fig. 1. (a) The geometric structure of Mn4O3Cl(OAc)3L13L23, with
hydrogen removed for clarity, (b) The geometric structure of the core
Mn4O3Cl.


N.A. Tuan et al. / Computational Materials Science 44 (2008) 111–116
Table 1
The chemical formula and classification of Mn4 molecules by the formal
charge of Mn ions at b-site
Label
(1)
(2)
(3)
(4)
(5)

(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)

Ref.

[4]

[1]

L1

L2

n

Group

CH3CN
NH3
py
CH2O
py
py

NH3
CH2O
dbm
CH3O
CH3O
Br
Cl
Cl

CH3CN
CH2O
CH2O
CH2O
Cl
Br
Cl
Cl

2

I

3

II

4

III

Br
Cl
Br
Br
Cl

the Table 1). Group I consists of the four Mn4þ Mn2þ
3
molecules labeled from (1) to (4). Group II consists of
molecules labeled from (5) to (9).
the five Mn4þ Mn3þ
3
Group III consists of the five Mn4þ Mn4þ
3 molecules from
(10) to (14).
3.2. Equilibrium geometry, electronic and magnetic
properties of Mn4þ Mnnþ
3 molecules
To determine the ground-state atomic structure of each
Mn4 SMM, we have carried out total-energy calculations
with full geometry optimization allowing the relaxation
of all atoms in the cluster. In addition, to investigate the
magnetic properties of the Mn4 SMMs, we probe several
different spin configurations, which were imposed as an initial condition of the self-consistent calculation procedure.
Four possible spin configurations considered in this work
include (i) AFM-HS, (ii) AFM-LS, (iii) FM-HS, and (iv)
FM-LS, where FM and AFM denote the ferromagnetic
and antiferromagnetic couplings between Mn4+ ion at the
a-site with Mnn+ ions at the b-site, respectively. HS and
LS correspond to the high-spin (electrons are distributed

so that all t2g and eg orbitals are singly occupied before
any pairing occurs) and low-spin (electrons are distributed
in t2g and eg orbitals so that they occupy the lowest possible
energy levels) states of Mnn+ ions at the b-site. We confirmed that the full geometry optimization calculation of
all fourteen Mn4 molecules have a similarity in the arrangement of atoms in the core Mn4O3Cl and three bridging
groups OAc (Fig. 1a). Due to the surrounding oxygens
and other ligand structures, one Mn ion at the a-site and
three Mn ions at the b-site are correspondingly labeled as
Mn(1), Mn(2), Mn(3) and Mn(4) to distinguish them.
3.2.1. Equilibrium geometry and magnetic structure
From four initial spin configurations, we obtained different geometric and magnetic structures of the Mn4 molecules
in each group. In the case of group I, we obtained four equilibrium geometric structures corresponding to four different

113

magnetic structures AFM-IS, AFM-LS, FM-IS and FMLS (IS denotes an intermediate-spin state between HS and
LS of Mn ions at the b-site) of each Mn4 molecule. Our calculations showed that there is no difference in atomic
arrangement among the four geometric structures of each
Mn4 molecule in group I. Moreover, the geometric structures corresponding to the magnetic structures AFM-LS
and FM-LS are nearly the same. The geometric structures
corresponding to the magnetic structures AFM-IS and
FM-IS are also nearly the same. Overall bond distances
of the geometric structure corresponding to the magnetic
structures AFM-IS and FM-IS are longer than those of
the geometric structure corresponding to the magnetic
structures AFM-LS and FM-LS. Therefore, we call the
geometric structure corresponding to the magnetic structures AFM-IS and FM-IS as the ‘‘long-structure”, and
the geometric structure corresponding to the magnetic
structures AFM-LS and FM-LS as the ‘‘short-structure”.
In the case of (1), the most stable state is the short-structure

with the magnetic structure AFM-LS, while the most stable
state of the three other Mn4 molecules (2)–(4) is the longstructure with the magnetic structure AFM-IS.
In the cases of groups II and III, we only obtained two
equilibrium geometric structures of each Mn4 molecule
from four initial spin configurations. The two geometric
structures of each Mn4 molecule in groups II and III are
nearly the same. They are only distinguished by difference
in magnetic structure. Their magnetic structures are
AFM-HS and FM-HS. The most stable state of Mn4 molecules of group II corresponds to the magnetic structure
AFM-HS, while the most stable state of Mn4 molecules
of group III corresponds to the magnetic structure FM-HS.
The geometric structures of the most stable state of (5) and
(9) from our calculations are good in agreement with the
experimental data reported in [1] and [4]. Most differences
of interatomic distances and bond angles are below 5%
between our results and the experimental data. Some selected
interatomic distances of the 14 Mn4 molecules are shown in
Fig. 2. They are quite similar within the same group, but
some of them are considerably different between groups.
Previous experimental studies [1,4] reported that each of
the three Mn3+ ions of (5) and (9) exhibit a Jahn–Teller distortion (elongation) along the Cl(1)–Mn3+–O(OAc) axis.
Our results also show the difference between bond distances from each Mn3+ ion to its six surrounding ligands
for all five molecules of group II. In each molecule of group
II, Mn3+–O(OAc) and Mn3+–Cl(1) bond distances are considerably longer than the others. The difference between
Mn3+–O(OAc) bond distances with the other Mn3+–O
bond distances is over 10%. This is evidence of strongly
elongated Jahn–Teller distortions along the Cl(1)–Mn3+–
O(OAc) axes.
No Jahn–Teller distortion is observed in the five molecules of group III. This result is also consistent with the
HS state of all four Mn4+ ions in these molecules.

There is also no Jahn–Teller distortion observed in the
short-structures of SMMs of group I. But, each of the four


114

N.A. Tuan et al. / Computational Materials Science 44 (2008) 111–116
Table 2
The detailed projections of magnetic moments at Mn sites of some selected
Mn4 molecules

Fig. 2. Some selected interatomic distances of the 14 Mn4 molecules.

long-structures of SMMs of this group displays three
strongly elongated Jahn–Teller distortions along three
Cl(1)–Mn(2)/Mn(3)/Mn(4)–O(OAc) axes.
The difference in interatomic distances between the
short- and long-structures of each SMM of group I is a
consequence of the Jahn–Teller distortions. These elongated Jahn–Teller distortions are good evidence for the
existence of an IS state of Mn2+ ions at the b-site in the
long-structure of each SMM in group I, where four electrons occupy in three t2g states (dxy, dyz and dzx), one occupied in the higher energy state (dz2). We will discuss this in
more detail in the next section.
3.2.2. Electronic and magnetic properties
Previous experimental studies [1,4] reported that (5) and
(9) have the ground state spin ST of 9/2, where Mn(1) is
antiferromagnetically coupled to Mn(2), Mn(3) and
Mn(4), and assigned a formal valence charge +4 with corresponding magnetic moment À3 lB. At the same time,
Mn(2), Mn(3) and Mn(4) are ferromagnetically coupled
to each other and have a formal valence of +3 with its magnetic moment 4 lB. From our calculations, the ground
states of (5) and (9) are determined to have ST of 8.92/2

and 8.89/2, respectively, and the antiferromagnetic configuration, in good agreement with the experimental observation [1,4]. Here, it should be noted that these calculated
values are from a Mulliken analysis, so that the values do
not match exactly with the formal valence and spin but
the relative magnitudes compare well. The detailed projections of the calculated magnetic moments for each individual Mn site of Mn4 molecules, as listed in Table 2, also turn
out to be consistent with the formal charges and magnetic
moments of Mn. In the case of (5), these results are also
compared well with those of Han et al. [14]. In generally,

Molecule

Magnetic structure

mMn(1)

mMn(2)

mMn(3)

mMn(4)

(1)

AFM-IS
FM-IS
AFM-LS
FM-LS

À2.859
2.623
À2.949

2.789

3.098
3.179
1.072
1.135

3.095
3.180
1.064
1.119

3.103
3.184
1.070
1.125

(4)

AFM-IS
FM-IS
AFM-LS
FM-LS

À2.758
2.485
À2.866
2.570

3.071

3.183
1.046
1.146

3.071
3.196
1.030
1.125

3.071
3.169
1.050
1.141

(5)

AFM-HS
FM-HS

À2.708
2.905

3.879
3.897

3.873
3.889

3.872
3.888

(9)

AFM-HS
Han et al.
FM-HS

À2.687
À2.540
2.894

3.862
3.690
3.874

3.853
3.710
3.862

3.863
3.680
3.876

(10)

AFM-HS
FM-HS

À2.857
2.893

2.735
2.733

2.720
2.720

2.728
2.727

(14)

AFM-HS
FM-HS

À2.903
2.921

2.778
2.805

2.765
2.793

2.773
2.802

the magnitude of magnetic moment of an Mn4+ ion at
the a-site has a nearly constant value of 3 lB, while the
magnetic moment of Mnn+ ions at the b-site displays an

interesting variation with n.
In the case of n = 4, the magnetic ground state of
Mn4þ Mn4þ
3 molecules are the FM-HS state with a magnetic
moment nearly 3 lB for all Mn ions. These values of magnetic moment are consistent with the formal charge of Mn
ions.
In the case of n = 3, the magnetic ground state of
Mn4þ Mn3þ
3 molecules are the AFM-HS state with a magnetic moment nearly À3 lB for Mn(1) and 4 lB for
Mn(2), Mn(3) and Mn(4). These values of magnetic
moment are also in good agreement with the formal charge
of Mn as well as the existence of the Jahn–Teller distortions
at Mn(2), Mn(3) and Mn(4) sites.
In the case of n = 2 within the long-structure, the magnitude of the magnetic moment of Mn(1) is nearly equal
to 3 lB, and the magnetic moment of Mn(2), Mn(3) and
Mn(4) is nearly 3 lB. In the case of the short-structure,
the magnetic moment of Mn(1) is also nearly equal to
3 lB, but the magnetic moments of Mn(2), Mn(3) and
Mn(4) are smaller by 2 lB than those in the case of the
long-structure. The more detailed analyses show that the
total number of down-spin electron of 3d states of Mn ions
at the b-site in the long-structure and the short-structure is
about 1 and 2, respectively, as listed in Table 3. These
results show that the spin state of Mn2+ ions of
molecules must be IS and LS corresponding
Mn4þ Mn2þ
3
to the long- and short-structures.
As presented in the previous section, the ground state of
(1) is the short-structure with the magnetic structure AFMLS, and the ground state of (2)-(4) is the long-structure

with the magnetic structure AFM-IS. There is no ground
state with the HS state of Mn2+ ions at the b-site of


N.A. Tuan et al. / Computational Materials Science 44 (2008) 111–116

115

Table 3
The calculated down-spin electron, nd; projected at 3d states of Mn ions at
b-site of Mn4þ Mn2þ
3 molecules
Molecule

(1)

Equilibrium
geometry

Magnetic
structure

nd;
Mn(2)

Mn(3)

Mn(4)

Long-structure

AFM-IS
FM-IS
AFM-LS
FM-LS

1.258
1.213
2.299
2.268

1.257
1.211
2.303
2.375

1.255
1.210
2.299
2.273

AFM-IS
FM-HS
AFM-LS
FM-LS

0.998
0.950
2.046
2.009

1.001
0.954
2.050
2.019

0.996
0.948
2.048
2.014

AFM-IS
FM-IS
AFM-LS
FM-LS

1.019
0.965
2.098
2.055

1.019
0.968
2.105
2.062

1.018
0.950
2.094
2.052

AFM-IS
FM-IS
AFM-LS
FM-LS

1.012
0.950
2.068
2.014

1.011
0.943
2.075
2.025

1.010
0.958
2.065
2.016

Short-structure
(2)

Long-structure
Short-structure

(3)

Long-structure

Short-structure

(4)

Long-structure
Short-structure

Mn3þ Mn2þ
3 molecules, while the magnetic ground state of
4þ
4þ
Mn4þ Mn3þ
3 and Mn Mn3 molecules exhibits the HS state
of Mn ions at the b-site. Moreover, no compressed
Jahn–Teller distortions are observed at the b-site of
4þ
3þ
Mn4þ Mn2þ
3 and Mn Mn3 molecules. These results mean
that the dx2-y2-orbital of Mn ions at the b-site of
4þ
3þ
Mn4þ Mn2þ
3 and Mn Mn3 molecules must be empty. This
can be explained in the term of the ligand field.
3.2.3. Magneto-structural correlation in Mn4 molecules of
group I
In this section, we discuss about the relation between
magnetic and geometric structures of Mn4 molecules. The
geometric structures of isomers of each Mn4 molecules of

groups II and III are nearly the same. Therefore, they are
not mentioned further in this section. In the case of group
I, the considerable difference in some interatomic distances
between the short- and long-structures of each Mn4þ Mn2þ
3
molecules is found. In each geometric structure of
molecules, the total energy corresponding to
Mn4þ Mn2þ
3
the IS and LS states of Mn2+ ions has been calculated.
In the short-structure, the LS state of Mn2+ ions is more
favourable than the IS state, while the IS state of Mn2+
ions is more favourable than the LS state in the longstructure.
To investigate the possibility of transitions between the
IS and LS states of Mn2+ ions, we performed calculations
of the total energy corresponding to the two magnetic
structures AFM-IS and AFM-LS of the four linear transition structures from the long-structure to short-structure of
each Mn4þ Mn2þ
3 molecule. The total energy corresponding
to the IS state of Mn2+ ions increases on going from the
long-structure to the short-structure, while the total energy
corresponding to the LS state of Mn2+ ions is decreasing.
Fig. 3 displays the total energy corresponding to the two

Fig. 3. The total energy corresponding to the two magnetic structures
AFM-IS and AFM-LS of the linear transition structures from the longstructure to short-structure of Mn4þ Mn2þ
3 molecules (1) and (4).

magnetic structures AFM-IS and AFM-LS of the linear
transition structures from the long-structure to short-strucmolecules (1) and (4). These

ture of selected Mn4þ Mn2þ
3
results show the existence of a transition structure in which
the two magnetic structures AFM-IS and AFM-LS of each
Mn4+Mn2+ molecules are equal in the total energy. These
results also show the existence of a low barrier about 0.5
eV between the long- and short-structures of each
Mn4þ Mn2þ
3 molecules, therefore the structure with higher
total energy is considered as the meta-stable state. Therefore, the magnetic transition between IS and LS states of
Mn2+ ions accompanied by the transition between the
long- and short-structures of Mn4þ Mn2þ
3 molecules is more
favourable than keeping their geometric structure. By this
particular behavior, Mn4+Mn2+ molecules can become
potential candidates for nano-piezomagnets.
4. Conclusions
We have performed studies of the structural, electronic
and magnetic properties of fourteen Mn4 molecules using
a first-principles method. We found that the peripheral
ligand groups play an important role in controlling charge
and spin states of Mn ions, as well as type of Jahn–Teller
distortion at the b-site octahedrons. Changing peripheral
ligands becomes an effective way to control the electronic
structure and magnetic properties of Mn4 molecules. The


116

N.A. Tuan et al. / Computational Materials Science 44 (2008) 111–116

geometric structure, electronic structure and magnetic
molecules display an interesting
properties of Mn4þ Mnnþ
3
variation with the charge state of Mnn+ ions at the b-site.
In these Mn4 molecules, the magnetic interaction between
Mn ions is FM between ions in the same valence states,
being AF between ions in difference valance states. The
strong magneto-structure correlation of Mn4þ Mn2þ
3 molecules leads to the possibility of these molecules acting as
a nano-piezomagnet.
Acknowledgments
This work was supported by Special Coordination
Funds for Promoting Science and Technology commissioned by MEXT, JAPAN.
References
[1] S. Wang, H.-L. Tsai, E. Libby, K. Folting, W.E. Streib, D.N.
Hendrickson, G. Christou, Inorg. Chem. 35 (1996) 7578.
[2] B. Hammer et al., Phys. Rev. B 59 (1999) 7413.
[3] W. Wernsdorfer, N. Aliaga-Alcalde, G. Christou, Science 302 (2003)
1015.

[4] D.N. Hendrickson, G. Christou, E.A. Schmitt, E. Libby, J.S.
Bashkin, S. Wang, H.-L. Tsai, J.B. Vincent, P.D.W. Boyd, J.C.
Huffman, K. Folting, Q. Li, W.E. Streib, J. Am. Chem. Soc. 114
(1992) 2455.
[5] S.M.J. Aubin, N.R. Dilley, L. Pardi, J. Krzystek, M.W. Wemple, L.C. Brunel, M.B. Maple, G. Christou, D.N. Hendrickson, J. Am.
Chem. Soc. 120 (1998) 4991.
[6] H. Andres, R. Basler, H.-U. Gu¨del, G. Aromı´, G. Christou, H.
Bu¨ttner, B. Ruffle´, J. Am. Chem. Soc. 122 (2000) 12469.

[7] M.R. Pederson, S.N. Khanna, Phys. Rev. B 59 (1999) R693.
[8] S. Hill, R.S. Edwards, N. Aliaga-Alcalde, G. Christou, Science 302
(2003) 1015.
[9] K. Park, M.R. Pederson, S.L. Richardson, N. Aliaga-Alcalde, G.
Christou, Phys. Rev. B 68 (2003) R020405.
[10] B. Delley, J. Chem. Phys. 92 (1990) 508;
B. Delley, Int. J. Quantum Chem. 69 (1998) 423;
B. Delley, J. Chem. Phys. 113 (2000) 7756;
B. Delley, Phys. Rev. B 65 (2002) 085403;
A. Kessi, B. Delley, Int. J. Quantum Chem. 68 (1998) 135.
[11] W.J. Hehre, L. Radom, P.v.R. Schlyer, J.A. Pople, Ab Initio
Molecular Orbital Theory, Wiley, New York, 1986.
[12] B. Hammer, L.B. Hansen, J.K. Nørskov, Phys. Rev. B 59 (1999)
7413.
[13] A. Matveev, M. Staufer, M. Mayer, N. Ro¨sch, Int. J. Quantum
Chem. 75 (1999) 863.
[14] M.J. Han, T. Ozaki, J. Yu, Phys. Rev. B 70 (2004) 184421.