ORBITALLY DEPENDENT SUPEREXCHANGE IN MIXED-VALENCE
CYANO-BRIDGED Mn(III)-Mn(II) DIMER. A NEW PERSPECTIVE FOR
SINGLE MOLECULE MAGNETS
A.V.Palii
Institute of Applied Physics,
Academy of Sciences of Moldova, Academy Str. 5, MD-2028 Kishinev, Moldova
e-mail: andrew.palii@uv.es
The model of the orbitally dependent magnetic exchange in the mixed-valence
bioctahedral Mn(III)-CN-Mn(III) dimer is developed. The kinetic exchange
mechanism involves the electron transfer from the single occupied t2 orbitals of the
Mn(II) ion ( 6 A1 t 23 e 2 ground state) to the single occupied t2 orbitals of the Mn(III)
 
3
  ground state) resulting in the charge transfer
ion ( T 1 t
4
2
5

T2 t 22 e 2

Mn(III)
 
 2 T2 t 25
Mn(II)
state of the pair. The deduced effective exchange Hamiltonian leads to an essentially
non-Heisenberg energy pattern. The energy levels are shown to be dependent on both
spin and orbital quantum numbers providing thus the direct information about the
magnetic anisotropy of the system. Along with the magnetic exchange the model
includes the axial component of the crystal field and the spin orbit coupling operating
within the ground 3T1(t24) cubic term of the Mn(III) ion. We have shown that under
some conditions the interplay between these three interactions leads to the appearance
of the barrier for the reversal of magnetization, so the results obtained can be regarded
as a first step in the explanation of the magnetic bistability exhibiting by the recently
synthesized trigonal bipyramidal cyanide cluster {[MnII(tmphen)2]3[MnIII(CN)6]2}
(tmphen = 4,5,7,8-tetramethyl-1,10-phenantroline).
1. INTRODUCTION
Molecules that exhibit magnetic bistability, commonly referred to as SingleMolecule Magnets (SMM), are of high interest due to their unusual physical properties
and potential importance for high-density data storage and quantum computing (1). To
date, almost all the molecules firmly established as displaying SMM behavior
incorporate oxide-based bridging ligands that mediate the magnetic exchange coupling
between metal centers. A remarkable feature of these systems is that in them all orbital
angular momenta are quenched by the local low-symmetry crystal fields, so the oxobridged SMMs can be referred to as pure spin systems. Such molecules possess a large
total spin ground state (S) formed by the isotropic Heisenberg magnetic exchange,
which, when combined with a negative axial zero-field splitting DS S Z2 (DS <0), leads
to the appearance of the energy barrier for spin reversal.
Recently in the interest of producing clusters with larger spin reversal barriers,
the trigonal bipyramidal cyano-bridged cluster [MnIII(CN)6]2[MnII(tmphen)2]3
(tmphen = 4, 5, 7, 8 – tetramethyl–1, 10–phenanthroline) was synthesized and
characterized, ref. (2). The observed ac-susceptibility signal indicates that this cluster
(hereunder abbreviated as Mn5-cyanide cluster) represents a new SMM.
The Mn(III) ions in the Mn5-cyanide cluster occupy almost perfect octahedral
sites. As a result the strong cubic crystal field produced by six carbon ions leads to the
3 - 95
 
orbitally degenerate 3T1 t 24 ground term of the Mn(III) ion. Insofar as this state
possesses the unquenched orbital angular momentum the system under consideration
is drastically different from the classical oxo-bridged SMMs consisting of orbitally
non-degenerate ions. The first difference is that the conventional Heisenberg-DiracVan-Vleck Hamiltonian fails when one deals with the Mn5-cyanide cluster containing
the Mn(III) ions with unquenched orbital angular momenta. In fact, the exchange
Hamiltonian of the Mn5-cyanide cluster should involve not only spin but also orbital
operators, i. e., such magnetic exchange proves to be orbitally dependent. The most
important feature of the orbitally dependent exchange is that it is highly anisotropic,
refs. (3-5). The second difference is a significant (first order) single ion anisotropy
that can be expected in the Mn5-cyanide cluster. Such anisotropy represents a first
order effect with respect to the spin-orbit coupling and axial component of the crystal
field acting on each Mn(III) ion, i. e., this single ion anisotropy can not give rise to a
global anisotropy described by the second order Hamiltonian DS S Z2 . One can expect
that both exchange anisotropy and single ion anisotropy contributes to the global
magnetic anisotropy responsible for the formation of the barrier for the reversal of
magnetization in the Mn5-cyanide cluster.
In this paper we endeavor to develop a model that would be able to qualitatively
explain the existence of the barrier for the reversal of magnetization in the Mn5cyanide cluster. In order to avoid complications implied by the consideration of
polynuclear the cluster entire and to make the results more clear and transparent we
will restrict ourselves by considering the Mn(III)-CN-Mn(II) pair, that seems to retain
the main peculiarities inherent in the entire Mn5-cyanide cluster. In fact, the analysis
of the structure of the Mn5-cyanide cluster shows that only the superexchange
interaction between Mn(II) and Mn(III) ions through the cyanide bridges can be
significant, meanwhile the interactions between two Mn(II) ions and two Mn(III) ions
are negligible due to the large intermetallic distances. The model includes the orbitally
dependent superexchange mediated by cyanide bridge, as well as the spin-orbit
coupling and axial crystal field operating within the ground 3T1 t 24 state of the Mn(III)
ion. The results obtained in the framework of this model can be considered as the first
step in the understanding of the magnetic behavior of the Mn5-cyanide cluster.
 
2. HAMILTONIAN OF THE KINETIC
ORBITALLY DEGENERATE IONS
EXCHANGE
BETWEEN
Let us consider the kinetic exchange between two octahedrally coordinated
transition metal ions A and B assuming that both ions are in the ground states
2 s 1
2 s 1
( A g  A g d n A and B g  B g d nB terms). We focus on the particular case when one
 
 
or both ground terms are orbitally degenerate. Kinetic exchange appears as a second
order contribution with respect to the following intercenter one-electron transfer
operator playing a role of perturbation:
V 
  t 

Aγ A
B γB
γ , B γ B  CB γBσ C Aγ Aσ  h.c. ,
A A

where the operator CB γBσ C Aγ Aσ


(1)
σ

creates (annihilates) electron on the orbital
φ B γB φ Aγ A of the ion B (A) with spin projection σ ,  A ,  B  t 2 or e, γ A , γ B label the
3 - 96
one-electron basis, and t    is the hopping integral. Hereunder we will consider a
corner shared bioctahedral dimer and use the real one-electron cubic basis related to
the cubic local coordinate frames, with zA (zB) axes being directed along C4 axis of
the pair. This means that  A and  B will run over ξ  yz , η  xz, ζ  xy (t2basis) and u  3z 2  r 2 , υ  3 x 2  y 2  (e-basis). The operator, eq. (1), connects the
ground state with the excited charge transfer (CT) states arising from the electronic
configurations d n A 1  d nB 1 in which one electron is transferred from the site A(B) to
the site B(A).
The general approach to the problem of the magnetic exchange between
orbitally degenerate ions is outlined in our recent papers, refs. (3)-(5) and will not be
repeated here. Application of this approach leads to the following general expression
for the kinetic exchange Hamiltonian operating within the ground
2 s A g 1
2 s 1
 A g d nA  B g  B g d nB manifold of the pair:
 
 
H ex  A, B   2 
 t 

γ  A γ A  A γ A
 A B γ A γ B γA γB
γ


 γ
γ , B γ B  t  B γ B , A γ A 
A A
 γ   B γ B  B γ B OAγ OBγ

(2)

 F0   A , B   F1   A , B s A s B .
In eq. (2) γ  A γ A  A γA are the Clebsch-Gordan coefficients for the Oh group, Oi γ
are the cubic irreducible tensor operators acting within the orbital  i g  i g manifold of
the center i (i = A, B ) and s i are the single ion spin operators. The operators Oi γ are
defined in such a way that their reduced matrix elements  ig Oi  ig   ig  2 ,
1
where  i g  is the dimension of  i g , so the matrix elements of these operators
coincide with the Clebsch-Gordan coefficients appearing in the Wigner-Eckart
theorem , ref. (12). Finally, the parameters F0  and F1  are expressed in terms of
the energies of the CT states, the general formulas for these parameters are given in
ref. (4).
3. EXCHANGE MODEL FOR A Mn(III)-CN-Mn(II) PAIR
Hereunder we will apply the general formalism outlined in the previous Section
to the cyano-bridged Mn(III)-CN-Mn(II) pair in which the Mn(III) ion is surrounded
by six carbon ions, and the Mn(II) ion is surrounded by six nitrogen ones. As a first
step we assume that both metal ions are in a perfect octahedral ligand fields. The
ground term of the Mn(III) ion in a strong cubic field produced by the carbon atoms is
expected to be the low-spin orbital triplet 3T1(t24). On the contrary, weak crystal field
induced by the nitrogen atoms gives rise to a ground high-spin orbital singlet
6
A1 t 23 e 2 of the Mn(II) ion, so the ground state of the bioctahedral corner-shared
 

dimer (overall C4v symmetry) will be 6 A1 t 23e 2

Mn  II 
 
3T1 t 24
Mn  III 
.
In order to apply the general formula for the exchange Hamiltonian, eq.(2), to
the Mn(III)Mn(II) pair we will assign the indices A and B to the Mn(II) and Mn(III)
3 - 97
ions,
respectively,
 
so
that
    

S Ag  Ag 6A1 t23 4 A2 e 2 3 A2  6A1 t23e 2

and
SB g B g 3T1 t24 . Insofar as the transfer of the electron from the site B to the site A
leads to the CT states with very high excitation energies we will neglect such
processes and consider only the A  B electron transfer. There are two possibilities
for the A  B electron transfer, namely, the transfer from the single occupied t 2
orbitals of the Mn(II) ion to the single occupied t2 orbitals of the Mn(III) ion through
the bonding  and antibonding  * orbitals of the cyanide ion, and the transfer from
the single occupied e orbitals of Mn(II) to the empty e orbitals of Mn(III) through the
cyanide  -orbitals (the hopping parameters corresponding to the t 2  e transfer are
expected to be negligible due to the orthogonality of t2 and e orbitals). At the same
time recent density functional theory calculations of the exchange parameters in
cyano-bridged species (6) demonstrated that the interaction through the the cyanide
 -orbitals was significantly smaller compared to the interaction through the  and
 * orbitals (see also (7) and refs. therein). That is why only t 2A  t 2B transfer
processes are assumed to be important and will be taken into account in our exchange
model. It is easy to see that the overlap between  -type t2 orbitals of Mn(II) and
Mn(III) through the  and  * orbitals of cyanide bridge is strong, and the same
overlap takes place between  orbitals. So there are two equivalent hopping
parameters t   t  t associated to these overlaps (Fig.1). At the same time the
integral t   can be omitted
because there is no effective overlap between 
orbitals. Note that the t 2A  t 2B transfer can not affect
X
XB
A
ZB
ZA
Mn(I
I)
Mn(II
I)
Fig.1. Scheme of overlap between t2 orbitals of Mn(III) and Mn(II) through
π orbitals of cyanide bridge
 
the e 2 subshell of the ion A ( 3 A2 e 2 -state). At the same time this transfer decreases
the spin of the ion A by 1 2 . The analysis of the Tanabe-Sugano diagrams (8) for d 5
and d 4 ions shows that the only appropriate state for the oxidized t 22 e 2 configuration
of the ion A is the state 5T2 t 22 3T1 e 2 3 A2  5T2 t 22 e 2 . Analogously, the reduced t 25
    


 
configuration of the ion B gives rise to the only state 2T2 t 25 . We thus arrive at the
conclusion that the t 2A  t 2B transfer results in the only CT state 5T2 t 22 e 2 A  2 T2 t 25 B .
It is remarkable that the single-ion states involved in this CT state are the “pure”
3 - 98
states, each resulting from the only electronic configuration. For this reason, no
complications implied by the Coulomb mixing of different electronic configuration
can appear in the kinetic exchange problem under consideration.
The orbital schemes for the [Mn(II)]A[Mn(III)]B pair (ground state) and
[Mn(III)]A[Mn(II)]B pair (CT state), and the electron transfer process connecting these
states are shown in Fig.2. It is
Excited (CT)
to be noted that each orbital
scheme depicts only one Slater
determinant (microstate) of the
B
many-electron open shell waveu
υ
A
e
function. For example, the only
u
υ
e
determinant ξ η ζ ζ involved
t2
ζ
t2
η
ξ
ζ
 
5
T2 t22e2
 
2
in the two-determinant wave3
T1 t 24 , γ , ms  0
function
η
ξ

T2 t25
1
Ground
t2
ζ
6
υ
υ
u
e
B
u
e
η
ξ
 
A1 t25e2
t2
ζ
ξ
 
3
T1 t24
Mn(II) Mn(III)
Fig.2.
Image representation
Gordan
coefficients of the kinetic
exchange
mechanism for the Mn(III)-CN-Mn(II) pair



 ξηζ ζ

of
η
is represented by the only
microstate,
so
the
corresponding orbital scheme in
this case shows the full wavefunction of the high-spin Mn(II)
ion. The jumping electron does
not change its spin projection
and selects the initial and final
microstates as exemplified in
Fig.2.
Now
it
is
a
straightforward work to adapt
the general expression for the
kinetic exchange Hamiltonian,
eq.(2), to the Mn(III)-CNMn(II) pair under consideration.
Substituting the
relevant
values
of
the Clebschinto eq.(2) one finds:


4
H ex  A, B    t 2 FA10A1 t 2 ,t 2  OAB1  1 2 FA10E t 2 ,t 2  OEBu
3
4
 t 2 FA11A1 t 2 ,t 2  OAB1  1 2 FA11E t 2 ,t 2  OEBu s A s B .
3

 ξ η ζ ζ
the low-spin Mn(III) ion is
shown in Fig.2. On the
contrary,
the
state
6
3 2
A2 t 2 e , ms  5 2  ξ η ζ u υ
Mn(III) Mn(II)
A
2


3 - 99
(3)
While deducing eq. (3) it has been taken into account that
A1 A1 A1  1 and
A1 A1  γ  0 for all   A1 , so O γ  O  1 (only spin operators act within the
A
A
A1
orbitally non-degenerate ground state of the Mn(II) ion). The orbital operators OAB1 and
OEAu are represented by the following matrices in the cubic T1 basis α ,β ,γ  :
1 0 0


OAB1   0 1 0  ,
0 0 1


OEBu
  12

 0
 0

0

0 .
1 
0
 12
0
(4)
The parameters Fk in eq. (3) can be found with the aid of the approach developed in
(3-5); the results are the following:
FA10A1 t 2 ,t 2  
F A1 E t 2 ,t 2   
1
0 
where ε A B
FA11A1 t 2 ,t 2   
1
,
2 ε A B
2 2 ε A B
,
F A1 E t 2 ,t 2  
1
,
5 ε A B
1
1
is the energy of the
5 2 ε A B
6

A1 t 23e 2

A
(5)
,
 
3T1 t 24
B
 5T2 t 22 e 2 A  2 T2 t 25 B
excitation.
The T-P-isomorphism makes it possible to consider the ground 3T1 t 24 term of
the Mn(III) ion (ion B) as a state possessing fictitious orbital angular momentum
lB  1 (8). This allows us to express the cubic irreducible tensor OEBu in eq. (3) in
 
terms of the orbital angular momentum operator
l BZ
acting within the
lB  1, m l B  0,  1 basis ( m l B is the projection of the fictitious orbital angular
momentum) as follows:
2
O EBu  1 3 2 l ZB
.
(6)
Substituting eqs. (5) and (6) into eq. (3) we arrive at the following final formula for
the kinetic exchange Hamiltonian of the Mn(III)-CN-Mn(II) pair:


t2
 5  2 s A s B  2  3 l ZB2 .
H ex  A, B  
30 ε A B
(7)
This Hamiltonian is essentially non-Heisenberg and includes both spin and orbital
angular momenta operators (orbitally-dependent exchange).
4. RESULTS AND DISCUSSION
The Hamiltonian, eq. (7), proves to be isotropic in the spin subspace and
axially symmetric in the orbital subspace, so that S , M S (total spin of the pair and its
3 - 100
projection) and ml B are the good quantum numbers. The eigenvalues of H ex  A, B 
are calculated as follows:
E  S , ml B  0  
E  S ,| ml B
t2
 63 4  S S  1 ,
15 ε A B
(8)
t2
 63 4  S S  1 .
| 1  
6 ε A B
The energy pattern (formed by the magnetic exchange) of the Mn(III)-CNMn(II) pair contains two superimposed groups of the energy levels with ml B  0 and
| ml B |  1 (see Fig.3). The total spin S of the pair takes the values S  3 2 , 5 2 , 7 2 ,
and the energy levels within each group obey the Lande’s rule. The exchange splitting
of both ml B  0 and | ml B |  1 multiplet proves to be antiferromagnetic S gr  3 2 .
The conclusion about the antiferromagnetic exchange splitting in each group of the
energy levels is in agreement with the underlying ideas of Anderson, ref. (9) and
Goodenough and Kanamori (see ref. (10) and refs. therein). In fact, these authors
indicated that the electron hopping between the half-occupied orbitals should result in
the antiferromagnetic exchange coupling.
0.5
t
E
2
ε AB

S  7 2 , ml B  0 ,  1
0
S  5 / 2, ml B  0
0.5
S  3 / 2 , ml B  0
1
S  5 / 2, ml B  1
1.5
S  3 / 2, ml B  1
2
0.02
0.04
0.06
0.08
0.1
Fig. 3. Energy pattern formed by the magnetic exchange
It is to be underlined that the Lande’s rule is not valid for the whole energy
pattern, particularly, the non-monotonic alternation of the levels with S  3 2 and
S  5 2 takes place. Another important result is that the energy levels depend not
only on the total spin
quantum number S but also on | ml B | . This leads to the interesting peculiarities in
the magnetic behavior of the system. In the magnetic field applied parallel to the C4
3 - 101
axis of the bioctahedron the orbital contribution to the Zeeman splitting of the ground
level is significant
(first order effect) because the operator  κ β l Z B H Z
possesses the following
nonvanishing matrix elements within the ground level:
S  3 2 , M S , ml B  1  κ β l Z B H Z l Z B S  3 2 , M S , ml B  1   κ β H Z
(9)
In eq. (9) κ is the orbital reduction factor, sign “minus“ appears due to the fact that
the matrix elements of l B within T1 and P bases are of the opposite signs, ref. (8).
On the contrary, the orbital contribution to the Zeeman splitting of the ground level in
a perpendicular field is much smaller because it appears as a second order effect due
to the mixing of the ground level with the second excited level ( S  3 / 2 , ml B  0 )
by the operator  κ β l X B H X  lY B H Y  (Van Vleck paramagnetism). Therefore, as
distinguished from the Heisenberg magnetic exchange, the orbitally dependent
exchange interaction described by the Hamiltonian, eq. (7), produces the strong
magnetic anisotropy of the pair.
Now in order to make our consideration more realistic we will take into
account the fact that the nearest surrounding of the Mn(III) ion is axially distorted. In
this case the operator of the axial crystal axial crystal field H ax  acting on the Mn(III)
ion should be added to the magnetic exchange Hamiltonian. The operator H ax can be
defined as follows:


H ax   l Z2 B  1 ,
(10)
where  is the parameter of the axial field. This interaction splits the ground 3T1 term
of the Mn(III) ion into the orbital doublet 3 E (orbital basis ml B  1 ) and the orbital
singlet 3 B2 ( ml B  0 ) in such a way that the orbital singlet (orbital doublet) becomes
the ground state providing   0   0 . Finally, one should also take into account
the spin orbit (SO) coupling acting within the 3T1 term of the Mn(III) ion, the
corresponding operator is given by
H SO  κ λ sB l B ,
(11)
 
where λ  180 cm 1 is the many-electron SO coupling parameter for the 3T1 t24 term ,
ref. (11).
We will consider the most typical situation when the axial field is strong
significantly exceeding both the magnetic exchange and the SO coupling (
|  | t 2 ε A B , κ | λ | ). Provided that   0 the strong axial crystal field totally
removes the orbital degeneracy
giving rise to the pure spin system with the Heisenberg-type pattern of the low-lying
energy levels. We will not consider this trivial situation and focus on the analysis of
more interesting
3 - 102
(from the point of view of the barrier for the reversal of magnetization) case when
  0 . Strong negative axial field that results in a strong destabilization (by the value
|  | ) of the subset of the energy levels (formed by the magnetic exchange) with
ml B  0 . The SO coupling splits the levels belonging to the low-lying subset with
ml B  1 and mixes the levels with different S values belonging to this subset. In
addition the SO coupling mixes the
| MJ |
↓
9/2
7/2
5/2
E
κ| λ |
3/2
1/2
0
1/2
3/2
5/2
2
7/2
5/2
3/2
1/2
4
1/2
3/2
1/2
1/2
6
3/2
5/2
0.1
0.2
0.3
0.4
0.5
t 2 6 ε A B 
κ λ
Fig.4. The low-lying energy levels of the Mn(III)-CN-Mn(II)
pair in the limit of strong negative axial crystal field
subsets of the levels with ml B  1 and ml B  0 , but in the strong axial crystal field
limit such mixing can be neglected, and the effective SO coupling operator acting
within the low-lying group of levels with ml B  1 becomes axial
( H SO  κ λ s ZB l ZB ). The low-lying energy levels as the functions of the magnetic
exchange are shown in Fig. 4. One can see that providing t 2 ε A B  0 (exchange
interaction is switched off) the energy pattern contains three equidistant levels with the
κ | λ | (eigenvalues of the operator  κ λ sZB lZB ). When
energies  κ | λ | , 0 and
3 - 103
the exchange interaction is switched on ( t 2 ε A B  0 ) the energies of the levels
become dependent on M J , where M J  ms A  ms B  ml B is the projection of the total
angular momentum ( ms A and ms B are the spin projections of the ions A and B ). It
should be emphasized that the energies of three low-lying levels (with M J   5 2 ,
M J   3 2 and M J  1 2 ) monotonically increases with the decrease of M J , so
these three levels form the barrier for the reversal of magnetization. The magnitude of
this barrier monotonically increases with the increase of the exchange interaction.
These results qualitatively explain the formation of the potential barrier in the entire
Mn5-cyanide SMM. The application of the ideas of this paper to a quantitative
description of the SMM properties of the Mn5-cyanide cluster will be considered in the
forthcoming publications.
ACKNOWLEDGMENTS
The author thanks Professor B. S. Tsukerblat for many useful discussions on the
subject of the paper.
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