MAGNETIC EXCHANGE IN METAL CLUSTERS:
MATHEMATICAL MODELLING AND PHYSICAL RESULTS
J.J.Borras-Almenar1, J.M.Clemente-Juan1, E.Coronado1, A.V.Palii1,2, B.S.Tsukerblat3
1
Instituto de Ciencia Molecular,Universidad de Valencia,
46100 Burjassot (Valencia) Spain;
1,2
Institute of Applied Physics of the Academy of Sciences of Moldova,
2028 Kishinev, Moldova;
3
Chemistry, Ben-Gurion University of the Negev,Beer-Sheva 84105,Israel
In this article we present a new conception of the magnetic exchange and double
exchange in clusters containing orbitally degenerate metal ions. The main theoretical
tools for the study of polynuclear clusters (Heisenberg- Dirac- Van-Vleck model of
exchange and Anderson’s model of the double exchange ) are invalid as applied to
the orbitally degenerate systems. As distinguished from the situation in spin-clusters
where the exchange interaction is predominating isotropic, the orbital degeneracy of
the constituent metals is shown to produce new kinds of the interionic interactions
resulting in an anomalously high magnetic anisotropy in these systems. Using the
developed effective Hamiltonian approach combined with the technique of the
irreducible tensor operators and efficient computer programs we have analyzed
orbitally dependent magnetic exchange in localized systems and the double exchange
in mixed-valence metal clusters. This study leads to a new concept of “anisotropic
double exchange”. The energy pattern as well as the character of the magnetic
anisotropy is closely related to the ground terms of the ions in crystal field, electron
transfer pathways and overall symmetry of the systems being affected also by spin –
orbital interaction and vibronic coupling ( Jahn-Teller and pseudo Jahn-Teller
effects). Special attention is paid on the origin of the magnetic anisotropy in the faceshared binuclear unit [Ti2Cl9]3-. The theory was applied to the study of the
ferromagnetic spin alignment in the heterobinuclear Cr(III)Fe(II) cyano-bridged
dimer.
1. INTRODUCTION
Finite molecular clusters of exchange coupled ions are currently important for
many areas of research such as solid state chemistry, magnetism and biochemistry.
They provide new possibilities in design of nanometer size magnets possessing the
unusual magnetic properties of paramagnetic-like behavior and quantum tunneling of
magnetization (1,2). The interplay between the electron delocalization (double
exchange) and magnetic interactions play a crucial role in the properties of many
mixed valence (MV) compounds of current interest in solid state materials science.
Till now the the existed models of the magnetic interactions took into consideration
only orbitally non-degenerate terms of the interacting ions. Anderson and Hasegawa
(3) proposed the theory of the double exchange and the usually accepted model for the
magnetic exchange in the low-dimensional and extended materials was based on the
Heisenberg-Dirac-Van Vleck (HDVV) model (4). In our papers (5-6) we developed
the theories of the magnetic exchange and the double exchange in the materials
containing orbitally degenerate ions in the high-symmetric crystal fields. Hereafter we
3 - 20
discuss the main approaches and the role of the orbital degeneracy in the problem of
the exchange paying attention on the main physical manifestations.
2. MIXED-VALENCE SYSTEMS
The theory of the double exchange proposed by Anderson and Hasegawa (3) is
essentially based on the assumption that the interacting metal ions (in both oxidation
degrees) are orbitally non-degenerate (spin systems). The main conclusion of the
theory is that the delocalization of the extra electron over two spin cores produces a
linear spin dependence of the double exchange splitting resulting thus in the
ferromagnetic ground state of the dimer. At the same time in these spin systems the
double exchange is magnetically isotropic. However, in a variety of compounds the
metal ions possess orbitally degenerate ground states. In this case the conventional
theory of the double exchange proves to be inapplicable.
The electron transfer (double exchange) Hamiltonian can be presented in a
conventional way as:
V   t 
 
a  b   b  a    V






AB
 VBA ,
(1)

where a   creates an electron on the orbital  of site A with the spin projection
  or  and b  
annihilates an electron on the orbital   of site B;
t   t A  B  t B  A are the intersite one-electron transfer integrals. In eq.(1) all
relevant transfer pathways are included. We will consider only transfer processes
with participation of t 2 orbitals, so  ,     ,  ,    yz,   xz,   xy denote the
cubic t 2 basis related to the C4 axes of sites. The pairs
 
 
 
 
T1 t22  2T2 t21
3
and
T1 t 22  4A2 t 23 are considered in three high-symmetric topologies: edge-shared
3
D 2h , corner-shared D 4h , and face shared D 3h bioctahedra. Figure 1 shows the
effective overlap of the d- orbitals giving rise to the most efficient transfer pathways.
The T-P isomorphism (6) allows us to assign the T1(2) basis to the P-states (L=1).The
calculation of the matrix of the double exchange operator gives the following result:
~ ~
~ ~
S A LA , S B LB , S L M S M L VAB S A LA , S B LB , S  L M S M L
~
~
~
2 1
S S
~~
n 1 S  S  S  L  L
2
 1
 S S  M M  S L T1 1 S L 
(2)
~
2
S S
S S S 
~
L L L
LM

~
k m  m
m
 2 L  1
2k  1L L L   t m m  1 C 1 m 1mCL M L k m m ,
L
 k 1 1  m m
k  0, 1, 2



~ ~
where S A L A a1A1 S A L A is the reduced matrix element of the creation operator that
2
can be considered as an irreducible tensor of rank 1(m,m’=0,+1,-1) in the orbital
subspace and as that of rank ½ in spin space, t mm' are the transfer parameters in the
~ ~
angular momentum representation (5). S L for d n1 , and S L for d n -ions, with




~
~
L  1 L  1 for orbital triplets and L  0 L  0 for orbital singlets. One can see that
3 - 21
yA
(a)
EE
 
t'  t a 
yB
D2h
t  t`
xB
(b)
1.0
xB
0.5
E
t
5
; ; 0
2
5
; ;  1
2
3
; ; 0
2
3
, ,  1
2
1
; ; 0
2
1
; ;  1
2
; S;  1
; S ; 0
; S;  1
1
; ; 0
2
1
; ;  1
2
3
; ; 0
2
3
; ;  1
2
5
; ; 0
2
5
; ;  1
2
D4h
zA
zB
0.0
(c)
(c)
yA
yB
t = t  t
-0.5
D3h
A t
a
-1.0
t
B
Fig.1
The overlap patterns related to the most
efficient transfer pathways: (a) D2h, (b)
D4h, (c) D3h.
D2 h D3 h
D4 h
(a)
(b)
Fig.2
Energy diagram for the 3T1( t 22 )-4A2( t 23 )
MV dimers: (a) D2h, D3h; and (b) D4h. A
short notation
 ; S , M S , L  1, M L 
S; M L is used.
the matrix of the double exchange is diagonal with respect to the spin quantum
numbers S and M S . The matrix element of the double exchange proves to be
proportional to S  12 (this dependence is contained in the 6j-symbol in eq.(2)) as
well as in the case of non-degenerate ions (spin systems). The matrix elements of the
double exchange for orbitally degenerate dimers depend also on the orbital quantum
numbers L , L , M L , M L . This dependence produces a magnetic anisotropy in the
orbital subsystem. The character of this anisotropy is closely related to the set of
transfer integrals in eq.(2), reflecting both the point symmetry of the dimer and the
specific choice of physically significant transfer pathways. Strong magnetic
anisotropy of the degenerate double exchange systems is to be considered as the main
physical consequence of this dependence.
Figure 2 shows the energy splitting for a 3T1 t22 4A2 t23 pair with different
overall symmetries. Providing D2 h , D3h symmetries (Fig.2a) the energy pattern
involves three pairs (+ and -) of levels with S=1/2, 3/2, 5/2; the energies are
 13 t  S  1/ 2 . All these levels correspond to M L  0 . The spectrum contains also
one highly degenerate level at E  0 . This level comprises states with all S values,
 
3 - 22
 
each belonging to M L  1 . In the case of D4 h symmetry (Fig.2b) we meet the
reverse situation. The state with E  0 involves all S-values and corresponds to
M L  0 , while all the states with the energies  13 t S  12  possess M L  1 . One
can see that the D4 h system exhibit strong magnetic anisotropy with the C4 easy axis
of magnetization meanwhile D2 h , D3h systems are also anisotropic but possess only
Van Vleck-type paramagnetism in the ground state. More complicated cases of the
“anisotropic double exchange” are considered in refs (6,7).
S=
5
;L =1
2
U(q)
p;S;ML
t
5
+; ;±1
2
5
±; ;0
2
0
5
- ; ;±1
2
-t
q
S=
Electronic Levels
ANISOTROPIC
5
, L = 1, ML = 0, ±1
2
6
P
( )
Minimum of the adiabatic potentials
ISOTROPIC
Fig.3
Suppression of the magnetic anisotropy by the PKS vibration as
illustrated by the singlet-triplet pair in the D4h system.
The vibronic interaction (pseudo Jahn-Teller effect (8,9)) in MV compounds is
usually strong coupled to the electronic motion. In order to illustrate (at least
qualitatively) the influence of the vibronic interaction we employ the Piepho-KrauzsSchatz (PKS) model (9,10) dealing with the so called out-of-phase mode q ( this mode
is constructed from the breathing modes of the octahedral subunits). The main effect
of the vibronic interaction is illustrated in Fig.3 where the adiabatic potentials of a
singlet-triplet pair 3T1 t 22  4A2 t 23 are depicted. The vibronic interaction is operative
within the sets of states with a given full spin, Fig.3 shows the selected S=5/2 levels.
One can see that the gap t between the levels with S=5/2 and ML =0 and |ML|=1 is
strongly decreased in a deep minima of the lower sheet of the adiabatic potential.
Thus the effect of the localization of the extra electron is accompanied by the
reduction of the anisotropy induced by the double exchange in the orbitally
degenerate system.
In the case of orbital degeneracy of the constituent ions, the isotropic spinHamiltonian of the magnetic exchange HDVV model becomes invalid even as a
zeroth order approximation. In our recent papers (11,12) we proposed a new approach
 
 
3 - 23
to the problem of the kinetic exchange between orbitally degenerate multielectron
transition metal ions. Our consideration takes into account explicitly complex energy
spectrum of charge transfer crystal field states exhibited by the Tanabe-Sugano
diagrams. Taking advantage from the symmetry arguments we have deduced the
effective exchange Hamiltonian in its general form for an arbitrary overall symmetry
of the dimer taking into account all relevant electron transfer pathways. The effective
Hamiltonian was constructed in terms of spin-operators and standard orbital operators
(cubic irreducible tensors). All parameters of the Hamiltonian incorporate physical
characteristics of the magnetic ions in their crystal surroundings. In fact, they are
expressed in terms of the relevant (in a given overall symmetry) transfer integrals and
crystal field and Racah parameters for the constituent ions.
Along with the isotropic spin-spin interactions the effective Hamiltonian in the
case of orbital degeneracy contains terms like OA A OB B (orbital matrices) and mixed
terms like S A S B OA A OB B containing both types of operators. All these operators can
be expressed in terms of the irreducible tensor operators acting in the orbital and spin
subspaces. Then the effective Hamiltonian can be represented as a linear combination
of the irreducible products. The last step of the mathematical procedure involves
decoupling of these products and the calculation of the eigenvectors and energy levels
(11). Symmetry properties of the effective Hamiltonian are studied in ref (12).
The results can be illustrated by the application of the developed approach to the
-3
binuclear unit [Ti2Cl9] in Cs3Ti2Cl9 that represents a face-shared 2T2  2 T2 cluster
with D3h overall symmetry. Figure 1c shows the most important electron transfer
pathway t a . The energy levels are obtained as the functions of the ratio t e t a , where
t e is associated with the e-orbitals in the trigonal symmetry. The model also takes into
account local trigonal crystal field (parameter  ) and also spin-orbital coupling.
Figure 4 shows that the calculated magnetic susceptibility is in a good agreement with
the experimental data (13). In the agreement with the experimental data the system
exhibits the magnetic anisotropy arising from the orbitally dependent exchange
interaction.
3. ORBITALLY DEPENDENT KINETIC EXCHANGE IN THE CYANOBRIDGED Cr(III)-Fe(II) PAIR
Magnetic
properties
of
3D
bimetallic
cyano-bridged
systems
.
{M(II)3[Cr(III)(CN)6] 2} 13H2O (M=Co, Fe,Ni) attract considerable interest as
molecule based magnets with promising application and represent interesting systems
for testing different models of the exchange interactions. The early papers devoted to
these kind of bimetallic systems were based on the theoretical model of the potential
exchange interaction dealing with the orbitally nondegenerate localized orbitals (4).
This model predicts short range ferromagnetic interaction through a strict
orthogonality of the magnetic orbitals in a bimetallic system. Using the strategy based
on this theoretical background M.Verdaguer and coworkers performed a rational
design of high Tc 3D ferromagnetic and ferrimagnetic cyanide- bridged systems (14).
During last decade this successful route allowed to raise Tc from 5.5K in Prussian
blues to 315K (14) creating thus the first room-temperature molecule based magnet.
The further progress in this area led to the revealing of the ferromagnetic
properties of two bimetallic cyanide-bridged systems M 3II Cr III CN6  2  13H 2O
 
3 - 24
 
Fig.4
Magnetic behavior of the [Ti2Cl2] unit: comparison with the theoretical curve (solid
line) calculated at te/ta=-0.154, ta=52028 cm-1, =-320 cm-1, =155 cm-1 and orbital
reduction k=0.71. (Inset) Temperature dependence of the degree of the anisotropy,
compared with the theoretical curve (solid line).
-3
with M  Fe, Co (Tc=16K and 23K). This observation turned out to be in a sharp
contrast to the orthogonality rules in the localized model of potential exchange and
motivated the development of the model of kinetic exchange that takes into account
the effect of electron delocalization. In our paper (15) we have extended the
Anderson’s theory of the double exchange to the case of a heterobimetallic pair. The
delocalization was supposed to occur over two non-equivalent spin cores comprising
nondegenerate orbitals. Second order effect of the double exchange is proved to give
rise to a ferromagnetic kinetic exchange in the system under consideration. Although
the proposed orbital model (10) provided an explanation of the ferromagnetic spin
alignment the problem of the kinetic exchange in the named bimetallic compounds is
far from being completed and requires more general consideration with due account
of the orbital degeneracy of the ground terms of Fe(II) and Co(II) ions.
In the orbital angular momentum representation the kinetic exchange Hamiltonian
consists of two commuting parts arising from the second order processes connecting
the ground state with the high spin and low spin CT states:
1 t2 2
1 t2  1 2 
LZ 9  2 S A S B ,
H LS 
(3)
1  LZ  3  S A S B .
6 LS  2 
15 HS
The full Hamiltonian proves to be isotropic in spin subspace and axially symmetric in
orbital subspace, so that S M S and M L are the good quantum numbers.
H HS  
3 - 25
One can easily find two sets of sublevels corresponding to M L  0 and M L  1 :
E S , M L  0  
21 t 2
t2

S S  1 ,
16 LS 12 LS
(4)
 1
21  2  1
1 
E S, M L  1  t 2 

t

S S  1 .
 20 HS 32 LS 
 24 LS 15 HS 




As one can see from Eqs. (4) the energy pattern of Cr(III)Fe(II) contains two
superimposed groups of the energy levels (with M L  0 and M L  1 ) each obeying


the Lande’s rule with the total spin S of the pair taking the values S  12 , 32 , 52 , 72 .
One can see that the Heisenberg-like energy pattern with M L  0 is
antiferromagnetic due to the mixing with the low spin CT states while the levels with
M L  1 form the ferromagnetic pattern appearing as a result of the competition
between the contributions of high and low spin CT states. These results are in
agreement with the theory of superexchange proposed by Anderson (16) and
Goodenough and Kanamori (see (17) and refs. therein). It is to be underlined that
along with the full spin quantum number S the energy levels are enumerated by the
quantum numbers M L . The dependence on M L reflects the magnetic anisotropy of
the whole system arising from the non-vanishing orbital angular momentum of FeII 
ion in its ground state. It is remarkable that the first order axial orbital magnetic
contribution exists only in one of two subsets of the levels, namely in the
ferromagnetic pattern for which M L  1 . This contribution is axial with the C4 axis
of the bioctahedron being the easy axis of magnetization. On the contrary, the
antiferromagnetic pattern is expected to possess only the second order orbital
magnetism (Van Vleck paramagnetism). One might say that although energy subsets
obey the Lande’s rule they can not be considered as the Heisenberg schemes due to
strong orbital contributions to the magnetic moments.
Regarding the spin alignnement it is to be inquired whether the overall effect of
the kinetic exchange is ferro- or antiferromagnetic. The lowest energy level with
|ML|=1 belongs to the maximum spin S=7/2 and one finds that
E S  72 , M L  1   t 2 /  HS ; the lowest level with ML=0 is that possessing the




minimum spin S 1 / 2 for which E S  1/ 2 , M L  0   5t 2 / 4 LS . Comparing these
energies one finds the inequality that assures the ferromagnetic ground state:
 LS  5  HS / 4.
As distinguished from the finite clusters, the treatment of the extended systems
requires some approximate approaches (for example, mean field approximation)
Nevertheless, our conclusion about the nature of the ferromagnetic ground state of the
bimetallic Cr(III)Fe(II) compounds undoubtedly remains valid for 3D-systems. Some
results essentially based on the point symmetry are applicable to the pair of ions only
(axial magnetic anisotropy). At the same time it is to be noted that the orbitally
dependent (anisotropic) interactions of the effective Hamiltonian are of primary
importance for both dimers and 3D-systems. In the last case they are responsible for
the cooperative phenomena in solids.
3 - 26
ACKNOWLEDGMENTS
Financial support of INTAS is highly appreciated (Grant INTAS 00-0651). Authors
indebted to the Supreme Council on Science and Technological Development of
Moldova for the financial support (grant 111). We thank Oleg Reu for his kind help in
the computer artwork.
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3 - 27