ZERO-FIELD SPLITTINGS FORMED BY ANTISYMMETRIC
DOUBLE EXCHANGE IN MIXED-VALENCE [Fe(II)Fe(III)]
CLUSTER
Moshe Belinsky
School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel, belinski@post.tau.ac.il
The model of an antisymmetric double exchange (AS DE) interaction is
developed
for
the mixed valence (MV) [Fe(II)Fe(III)] cluster. The spin-orbit
coupling effect is considered for the MV [Fe(II)Fe(III)] dimer, in which strong isotropic
Anderson-Hasegawa double exchange interaction forms isotropic ground state with
maximal total spin S=9/2. The AS double exchange interaction mixes the AndersonHasegawa DE states with the same S E0 ( S , M ) and E 0 ( S , M '  M  1) of the different
parity. The AS double exchange and Dzialoshinsky-Moriya AS exchange mix the DE
states with different S E0 ( S , M ) and E0 ( S '  S  1, M '  M  1) of the same parity. The
AS DE mixing of the Anderson-Hasegawa levels results in the AS DE contributions to
the zero-field splittings, which depend on S and parity.
1. Introduction
The MV [Fe(II)Fe(III)] clusters are structural elements of many ferredoxins,
enzymes and their synthetic bioinorganic model compounds [1]. In the localized
[Fe(II)Fe(III)] clusters of the high-spin iron ions ( Fe2 (3d 6 ) and Fe3 (3d 5 ) ), the
Heisenberg exchange interaction H0  2 J 0 (SaSb ) forms the cluster states with the total
spin S=9/2, 7/2, 5/2, 3/1, ½; sa  2 , sb  5 / 2 . The spin-dependent resonance splittings
of the S states due to the hopping of the extra electron between the MV ions is described
by the Anderson-Hasegawa (AH) [2] model of the double exchange (DE) coupling:
0  S   (S  1/ 2)t0 /  2s0 1 ,
(1.1)
s0  sa  smin . In the MV [Fe(II)Fe(III)] clusters, the AH double exchange (1.1) and
Heisenberg exchange interactions forms the isotropic DE states
E±0(S) =±(S+1/2)t0/(2s0+1)- J 0 S(S+1) .
(1.2)
The DE concept is widely used in the theory of the MV [Fe(II)Fe(III)] clusters in
bioinorganic chemistry of iron-sulfur proteins and in magnetism of the MV compounds.
The isotropic AH double exchange and Heisenberg exchange in dimeric MV
clusters has been the subject of theoretical and experimental investigations [3-25].
Strong double exchange interaction with the DE parameter B0  t0 /(2s0  1) =1350 cm-1
( t0  6750 cm-1) destroys the Heisenberg antiferromagnetic ordering ( J AF =70 cm-1,
J AF << B0 ) and results in the delocalized ground state S gr  Smax  9 / 2 of the
1
[Fe(II)Fe(III)] cluster in [ Fe2 (OH )3 (tmtacn) 2 ]2 [9-12]. The MV [Fe(II)Fe(III)] dimers
with strong double exchange ( t0  3000-4715 cm-1, B0 =600-943 cm-1, J AF << B0 ) and
delocalized S gr  9 / 2 ground state were found also in the model clusters [13-16]. The
[ Fe2 S 2 ] centers of the Clostridium pasterianum mutant 2Fe ferredoxins possess the
delocalized S gr  9 / 2 ground state [18-20] due to strong DE interaction (|t| ~ 2250 cm-1
[19]). Valence delocalized [ Fe2 S 2 ] pairs with strong DE were found in a variety of the
trimeric [ Fe3 S4 ] and tetrameric [ Fe4 S4 ] iron-sulfur clusters in ferredoxins, enzymes and
synthetic models [1].
Zero-field splittings (ZFS) of the delocalized ground state with S gr  S max =9/2 of
the [Fe(II)Fe(III)] dimers were determined from the EPR, Mössbauer and MCD data.
The ZFS of the delocalized cluster ground state with S gr  S max was described by the
standard effective ZFS Hamiltonian [26-29]:
0
H ZFS
 DS [ S Z2  S ( S  1) / 3]  ES ( S X2  SY2 ) ,
(1.3)
where DS and ES are the axial and rhombic cluster ZFS parameters, respectively. The
delocalized S gr  9 / 2 ground state of the model {ferredoxin} [Fe(II)Fe(III)] clusters are
characterized
by large
D9 / 2  1.1  1.5cm
2
1
positive
D9 / 2  1.7  4cm 1
[9, 16, 17]
{negative
[18-20]} axial ZFS parameter. The zero-field splittings
D1 ( Fe ) and D2 ( Fe ) of the individual Fe2 and Fe3 ions were considered the
3
origin of the ZFS of the cluster delocalized S gr  9 / 2 state of the MV [Fe(II)Fe(III)]
i
  i 1,2 {Di [ Siz2 Si ( Si  1) / 3]  Ei (Six2  Siy2 )} ).
cluster [9] ( H ZFS
In the pure exchange mononuclear [d n  d n ] dimers, the anisotropic
(pseudodipolar) exchange, single-ion ZFS contributions [29, 30], Dzialoshinsky-Moriya
antisymmetric exchange [31-34], dipole-dipole interaction strongly contribute to the
cluster ZFS parameters [29].
For the MV clusters, it was shown that the antisymmetric double exchange [35,
36] contribute to ZFS of the high-spin cluster S levels of the MV [d 1  d 2 ] dimer [35].
The aim of this work is the consideration of the antisymmetric double exchange
interaction in the [Fe(II)Fe(III)] cluster with strong Anderson-Hasegawa DE splittings
and finding the ZFS contributions connected with the antisymmetric double exchange.
The taking into account of the spin-orbit coupling in the theory of the AndersonHasegawa double exchange for dimeric [Fe(II)Fe(III)] clusters leads to antisymmetric
double exchange interaction. An antisymmetric double exchange mixes the AH double
exchange states with the same total spin S of the different parity and also the DE states
with different total spin S of the same parity. The AS double exchange contributes to the
zero-field splittings of the AH levels E±0(S). The AS DE contributions to ZFS depend
on the total spin and parity of the DE levels.
2
2. Hamiltonian of Antisymmetric Double Exchange in [Fe(II)Fe(III)] MV Cluster
The isotropic Anderson-Hasegawa [2] DE interaction in dimeric MV cluster may
be described by the effective Hamiltonian of the double exchange or spin-dependent
electron transfer (ET):
0
ˆ t ,
H DE
 T
ab 0
(2.1)
ˆ is determined by the equation
where the DE operator T
ab
ˆ )   S , M   [(S  1/ 2) /(2s  1)] ,
  S , M  (T
ab

0
ˆ )   S , M   [(S  1/ 2) /(2s  1)].
a*b  S , M  (T
ab
ab*
0
(2.2)
 0a*b ( S ) {  0ab* ( S ) } are the ground set spin wave functions in the case of the |a*b>
{|ab*>} localization of the extra electron on the center a* {b*} without taking SOC into
0
account , 0  S   [0a*b  S , M   0ab*  S , M ]/ 2. The operator H DE
(2.1) describes
the Anderson-Hasegawa DE splitting (1.1) connected with the ET between the 3d n 1 -ion
in the ground state and the 3d n -ion in the ground state.
We will consider antisymmetric DE in the delocalized MV cluster [Fe(II)Fe(III)]
formed by the high-spin iron ions in the non-degenerate ground states. Antisymmetric

double exchange interaction originates from the combined effect of the SOC VSO
admixture of the excited states on the centers  and isotropic DE interaction (ET)
between the excited states with the ground state on the center  . For the double
[ Fea3  Feb2 ] pair, the two-center
exchange in the delocalized MV [ Fea2  Feb3 ]
second-order perturbation antisymmetric DE terms, which describes this combined
effect, have the form
W1   [


 0*0 VSO
 k *0
k
  0* 0 Vˆ 0 k *
The ket | 0*0 
0 k * VSO 0 0 *
Ek *  E0*
electron,  *  S , M  
(2.3)
],
{| 0 0* } represents the ground S state ( 0 *  S , M 
0
 
| * 
{
* ( S , M )} ) in the case
0
 k * 0 Vˆ 0 0 *
Ek *  E0*
C
SM
sa ma sb mb
{|  * } of localization of the extra
|  * (sa ma )0 (sb mb ) | . The ket | k*0  [ | 0 k *  ]
0
ma , mb
'
represents the cluster excited states ( ' * (S ', M ') [
coupled to the
* ( S ", M ")] )

cluster | S , M  ground state | 0*0  [ | 0 0*  ] by the spin-orbit interaction VSO
3
[ VSO ] on the d 6 -center  * , Fe2* [ d 6 center  * , Fe2* ].
E0* and Ek * are the
energies of the ground and excited states of the Fe2 (3d 6 ) center *, respectively [27,

28]. The first {second} term in eq. (2.3) includes the SOC ( VSO
{ VSO }) mixture between
the ground ( 0 ) and excited ( k ) states of the Fe2 (3d 6 ) ion on the center a* {b*}
in the |a*b> {ab*>} localization and the transfer of the extra d-electron between the
SOC admixed k-excited states of the d 6 -ion [ | k*0  { | 0 k *  }] to the ground state
of the d 5 -ion [ |  0 0  { | 0  0  }]. Vˆ is the operator of the direct (Coulomb) or

*
* 
ab
indirect (throw the ligand bridges) inter-ion interaction.
To illustrate the antisymmetric DE interaction in the MV
[ Fe  Feb3 ]
[ Fea3  Feb2 ] pair we will consider the double exchange and SOC in
the slightly distorted bitetrahedral Fe-Fe cluster. In the |a*b> localization
([Fe(II)Fe(III)]), a distortions which flatten each tetrahedron FeL4 along the local z-axis
2
a
result in the non-degenerate 5 A1 (d z 2 ) ground state for the Fea2 ion [37] and the 6 A1
ground term for the Feb3 ion [27]. The spin-orbit coupling admixes to the ground 5 A1
term only the 5 E (d yz , d xz ) excited states of the distorted Fe(II) ion,   o | L |  '  i 3
[27]. In this case, the eq. (2.3) includes the SOC admixture of the excited 5 E states to
the ground 5 A1 state and ET from the excited 5 E state of the Fe 2  -center to the
ground state of the Fe3  -center. The terms of eq. (2.3) may be represented in the
form of the effective Hamiltonian of antisymmetric double exchange H ASDE  H1 :
ˆ T
ˆ (Sˆ - Sˆ ) ] ,
H1  i  (K ab )n [(Sˆ a - Sˆ b )n T
ab
ab
b
a n
(2.4)
n x, y
ˆ is the isotropic DE operator (eqs. (2.1), (2.1)), which describes the Andersonwhere T
ab
Hasegawa double exchange coupling (1.1). The ASDE operator H1 includes the scalar
product of the real antisymmetric vector constant K ab ( K ab  K ba ) of the AS double
exchange and spin operator (Sb  Sa ) .
H1 represents the spin-transfer interaction
induced by SOC. In the case of the A1 ground state of the Fe2 ion, the microscopic
calculations show that (K ab ) y  0, K z  0 and
5
K x  2 3 x [ ua || b    a || ub ] ,
(2.5)
where K x  ( K ab ) x ,  x   ( Fe 2 ) /  x ,  ( Fe2 )   is the SOC constant of the Fe2 ion
[27],  x is the energy interval between the excited 5 Ex and the ground 5 A1 states of the
Fe2 ion [37].  ua || b  is the ET integral between the ua (3d z 2 ) and b (3d yz )
neighboring 3d-functions.

0
a*b
The effective AS DE Hamiltonian (2.4) acts between the cluster ground states
( S , M ) and  0ab* ( S ', M ') of different localizations, S '  S , S  1, M '  M  1. The
4
spin operators and the transfer operator don’t commute. The arrows under the spin
operators indicate that the operator ( Sˆa  Sˆb ) x { ( Sˆb  Sˆa ) x } act on the ground state spin
functions  0a*b ( S , M ) {  0ab* ( S ', M ') } in the |a*b> {|ab*>} localization.
For the AS DE coupling between the states of different localization with the
same total spin S '  S , M '  M  1 , the AS DE operator (2.4) of the MV [Fe(II)Fe(III)]
cluster may be represented in the form [35], K x  (K ab ) x
ˆ (Sˆ - Sˆ ) .
H1  2iK x T
ab
b
a x
(2.6)
The matrix elements of the AS double exchange effective Hamiltonian (2.6) for S '  S
 S  1/ 2   sb  sb  1  sa  sa  1 
 0a*b  S , M  H1  0ab*  S , M  1  iK x 


S ( S  1)
 2sa  1 
(2.7)
( S  M )( S  M  1),
include the Anderson-Hasegawa [( S  1/ 2) /(2sa  1)] term (eq. (1.1)) as a multiplier.
The correlations
 0a*b  S , M  H1 0ab*  S , M '    0ab*  S , M  H1 0a*b  S , M ' 
take place for S '  S , M '  M  1 . As a result, in the [Fe(II)Fe(III)] cluster, the AS
double exchange mixes the Anderson-Hasegawa states E0 ( S , M ) with the DE states
with the same total spin S E 0 ( S , M '  M  1) of different parity.
The AS DE interaction (2.4) mixes the states of different localization with
different total spin S , M and S '  S  1, M '  M  1 . The operator (2.4) for the
case S '  S 1 has the following form of the effective AS DE Hamiltonian
ˆ ' (Sˆ - Sˆ ) .
H1  iK x T
ab
b
a x
(2.8)
ˆ ' for the S '  S coupling (2.8)
The effective DE operator T
ab
ˆ ' | 0 (S 1, M )  1/(2s  1)
 0a*b (S , M ) | T
ab
ab*
a
(2.9)
ˆ | 0 ( S )    0 (S 1) | T
ˆ | 0 (S 1) 
represents the difference  0a*b (S ) | T
ab
ab*
a*b
ab
ab*
 1/(2sa  1) and does not depend on the total spin S [35].
The matrix elements of the AS DE Hamiltonian (2.8) for S '  S 1 have the form:
 0a*b  S , M  H1  0ab*  S  1, M 1  i
( S  M  1)  S  M 
Kx

2S (2sa  1)
(2S  1)(2S  1)
[( sa  sb  1)2  S 2 ][ S 2  ( sa  sb ) 2 ].
5
(2.10)
For the S '  S AS DE coupling, the correlation   0a*b  S , M  H1  0ab*  S  1, M ' 
  0ab*  S , M  H1  0a*b  S  1, M '  takes place. As a result, in the [Fe(II)Fe(III)] cluster,
the AS double exchange coupling mixes the Anderson-Hasegawa DE states E0 ( S , M )
with the DE states with different total spin E0 ( S  1, M '  M  1) of the same parity.
3. Microscopic calculations of the AS double exchange parameter K x .
We will consider here the model of the [ Fe2 S2 ] centers of ferredoxins and
enzymes [1] in the form of the slightly distorted bitetrahedral
[ Fea2  Feb3 ]
[ Fea3  Feb2 ] cluster with the common edge. Each iron ion possesses
distorted tetrahedral coordination of the FeL4 center. For the Fe2 ion, a distortion,
which flattens the tetrahedron along the local z-axis, results in the non-degenerate
5
A1 (d z 2 ) ground state [37]. The SOC admixture of the excited states to the ground A1
terms of individual Fe(II) and Fe(III) ions determines the ZFS of individual ions [27].
Without the taking SOC into account, the ground [ 0 ( 5 A1 , M )] and excited
[ ' ( 5 En , M )] Slater determinant wave functions are the following [27, 28],
M  M max  2 (  d yz ,   d xy , v  d x2  y2 ):
 0 ( 5 A1 , m  2) |  uuv |,
 ' ( 5 Ex , 2) |  uv |,
 ' ( 5 E y , 2) |  uv | .
(3.1)
For the Fe2 center in the distorted tetrahedral coordination, the ground state wave
functions   with the SOC admixture of the exited 5 E (d yz , d xz ) states have the form
  ( 5 A1 , m  2)   0 ( 5 A1 , 2)  3[i x ' ( 5 Ex , 1)  y  ' ( 5 E y , 1)],
(3.2)
  ( 5 A1 , m  1)   0 ( 5 A1 , 1)  3 / 2{i x [ ' ( 5 Ex , 2)  3 ' ( 5 Ex , 0)]
 y [ ' ( 5 E y , 1)  3 ' ( 5 E y , 0)]},
  ( 5 A1 , m  0)   0 ( 5 A1 , 0)  3 / 2{i x [ ' ( 5 Ex , 1)
 ' ( 5 Ex , 1)]   y [ ' ( 5 E y , 1)  ' ( 5 E y , 1)]},
where the coefficients  n   ( Fe 2 ) / n describe the SOC admixture of the excited
5
E (d yz , d xz ) states for the Fe2 ion. These parameters of the SOC admixture determine
the local anisotropy of g-factors ( g  2.00, g   2(1  3 x ) ) and positive ZFS
( D1 ( Fe 2 )  3 2 /  n ) of the individual Fe2 ion in the flattened tetrahedral coordination
with the 5 A1 ground state [37, 38].
6
The calculations show that only the terms proportional to the  n   ( Fe 2 ) /  n
parameters of the SOC admixture for the Fe2 ion contribute to the antisymmetric
double exchange in the [Fe(II)Fe(III)] MV pair. We will consider here only the ground
state ( 6 A1 term) of the high-spin Fe3 ion (without the SOC admixture of the excited
states) with the wave functions  0 [28]:
 0 ( 6 A1 , m  5 / 2)   |  uv | .
(3.3)
In the model of the direct interaction between the MV iron ions, one obtains
the standard Anderson-Hasegawa DE splittings (1.2), where the DE parameter t0 in the
ground states set of the [Fe(II)Fe(III)] cluster has the form
ˆ | u  0 (S ) | V
ˆ | 0 ( S )  .
t0    u a | V
ab
b
a*b
max
ab
ab*
max
(3.4)
The parameter J 0 for the Heisenberg exchange splittings in the ground S states set of
the [Fe(II)Fe(III)] cluster is the following
J 0  14 ( J   J  J   J v ) ,
where J  
1
5
 J  ,
(3.5)
J   ab | Vab | b a  and   ξ,η,ς,u,v .
In the consideration of the [Fe(II)Fe(III)] DE pair with the taking SOC into
account,
we
will
calculate
the
non-diagonal
(M ' M )
matrix
ˆ
elements  a*b (S , M ) | Vab | ab* (S ', M ')  , proportional to  for the states of different
localization with S '  S and with S '  S  1 . These terms are equal to zero in the
Anderson-Hasegawa model (1.1). We will use the cluster wave functions formed using
the SOC admixed wave functions   (3.2) and 0 (3.3). The ET integrals between
the ground (  0a*b ( S , M ) ) and excited cluster states (  'ab* ( S , M ) n ) of different
localization, which are formed by the excited states of the Fe2 ion, appier in the
consideration of SOC in the DE model. The calculations show that these DE integrals
between the ground  0a*b ( S , M ) and excited  'ab* ( S , M ) n cluster states follow to the
Anderson-Hasegawa type rules:
  0a*b ( S , M ) | Vab |  'ab* ( S , M ) n  ua || b  [( S  1/ 2) /(2 sa  1)] ,
(3.6)
  'a*b ( S , M ) n ||  0ab* ( S , M )  a || ub  [( S  1/ 2) /(2 sa  1)] ,
and don’t depend on M, n  x( y ) for    ( ) . The one-electron transfer integrals
 ua || b  and  ua || b  in eq. (3.6) are the ET integrals between the ground cluster
state  0a*b ( S , M ) |  a0* ( 5 A1 , 2)b0 ( 6 A1 ,5 / 2) | and excited cluster
| a0 ( 6 A1 ,5 / 2) b' * ( 5 En , 2) | (n=x, y).
7
states  'ab* ( S , M ) n =
For the states with the same S, for example, for S=9/2, M=9/2
and S '  S  9 / 2, M '  7 / 2 , the DE mixing with the taking SOC into account has a
form
  a*b (9 / 2,9 / 2) | Vab |  ab* (9 / 2, 7 / 2) 
2
3
[i( xb  ua || b   xa   a || ub )
(3.7)
( yb  ua || b   xa  a || ub )]
These transfer integrals  ua || b  and   a || ub  (  ua || b  and  a || ub  )
between the u and   (  ) 3d-functions of the Fe ions on different centers in their
local coordinate axis in eq. (3.13) are considered following the model [32, 34], which
was used for antisymmetric Dzialoshinsky-Moriya exchange interaction between
monovalent ions. In the slightly distorted dimer of the two FeL4 tetrahedra with the
common edge (X-axis), we consider that the local z-axis of the a (b) FeL4 tetrahedra is
tilted in the plane ZY on the angle  ( ) relative to the cluster Z-axis (axis ( ab) ). For
this distorted dimeric cluster, the 3d-crystal–field orbitals  of the individual a (b) Fe
centers have the following form in the cluster coordinate system XYZ:
a (b )    () ( 3u  v),
ua (b )  u  () 3 ,
a (b)  ()   ,
 a (b )  ()   ,
v a (b )  v  () .
(3.8)
Using these expressions for the DE (ET) integrals  ua || b  between the ground and
excited cluster states of different localization, one obtains
 ua || b     a || ub  (tuu  t ) 3,
(3.9)
 ua || b    a || ub  0,
where tuu  tu  ua | Vab | ub  , tu  t0 { t  t   a | Vab | b  } denote the transfer
between the neighboring ua an ub { a and b } orbitals. The comparison of the resulting
matrix element (3.16) of the DE mixing in the microscopic calculation
 a*b (9 / 2,9 / 2) | Vab | ab* (9 / 2,7 / 2)  4i x(tu  t )
(3.10)
with the same matrix element  a*b (9 / 2,9 / 2) | H1 | ab* (9 / 2,7 / 2)  (iK x  K y ) / 3 of
the effective Hamiltonian of the AS double exchange H1 (2.4) results in the following
correlations for the Kn (S  9 / 2)  [ K ab (9 / 2)]n AS DE vector coefficients (2.5):
K x  2 3[ xb  ua || b   xa   a || ub ]  12 x (tu  t ),
K y  0, K z  0.
(3.11)
The vector of the AS double exchange coupling is directed along the X-axis for the
[ Fe( II ) Fe( III )] MV pair. We will use these K x  12t0 x (1  t / tu ) coefficients for the
S '  S and S '  S  1 matrix elements of part 2.
8
4. The Dzialoshinsky-Moriya antisymmetric exchange in the [Fe(II)Fe(II)]
localized cluster
In the localized [ Fe( II ) Fe( III )] cluster (|a*b>) with the pure Heisenberg
exchange inter-ion coupling (without DE), the Dzialoshinsky-Moriya (DM)
antisymmetric exchange [31, 32] interaction
H DM  G ab S a  Sb 
(4.1)
results in the mixing of the total spin levels S and S '  S 1 . G ab is an antisymmetric
( Gab  Gba ) DM parameter [32-34]
G ab  2iCab J / t .
(4.2)
where the vector transfer integrals Cab were determined by Moriya [32]:
Cab    [ (L*am0 /  m )tam,b0   (Lbm0 /  m )ta 0,bm ] ,
2
m
(4.3)
m
Lbm0 is the matrix element of Lb between the excited ( m ) state and ground (0) state of
the ion on the center “ b ”,  m is the energy of the excited m -3d orbital, tam,b 0 is the
transfer integral between the excited ( m ) state of the ion on the center “a” and ground
(0) state of the same ion on the center “ b ” [32].
For the considered [Fe(II)Fe(III)] cluster, the microscopic calculations
show that only Gx  (Gab ) x component of the Dzialoshinsky-Moriya AS exchange is
different from zero ( G y  Gz  0 )
Gx  2 (3 x J1   x J 2 ) ,
J1  ( Ju  J ),
(4.4)
J 2  [3J1  ( J v  J )] ,
where  x   ( Fe 2 ) /  x and  x   ( Fe3 ) /  x ,  x, y   z are the energy intervals
between the excited 4 Ex , 4 Ey ; 4 A1 states and ground 6 A1 state of the Fe3 ion,  ( Fe3 )
is the SOC constant of the Fe3 ion. In the localized [Fe(II)Fe(III)] cluster, the
Dzialoshinsky-Moriya AS exchange (4.1), (4.4) mixes the Heisenberg localized states
S, M
and S '  S  1, M '  M  1 with different S. The Dzialoshinksy-Moriya AS
exchange (4.1) is not active between the states of different localiztion. The mixing of
the S and S '  S 1 localized states by the DM AS exchange (4.1) doesn’t depend on
the localization
x
x
  0a*b ( S , M ) | H DM
|  0a*b ( S ', M ')   0ab* ( S , M ) | H DM
|  0ab* ( S ', M ')  .
9
(4.6)
As a result, in the delocalized MV [Fe(II)Fe(III)] cluster, the AS DE (2.8) and
Dzialoshinsky-Moriya AS exchange H DM  Gx [Sa  Sb ]x together contribute to the
mixture of the Anderson-Hasegawa DE states E0 ( S ) and E0 ( S '  S  1) of the same
parity with different total spin. The Dzialoshinsky-Moriya AS exchange doesn’t mix the
Anderson-Hasegawa DE states E0 ( S , M ) and E 0 ( S , M ') with the same total spin of the
different parity.
The comparison of the coefficient K x (3.11) of the AS double exchange
Hamiltonian H1 (2.4) with the coefficient Gx (4.4) of the Dzialoshinsky-Moriya AS
exchange H DM  Gx [Sa  Sb ]x in the [Fe(II)Fe(III)] MV cluster shows that K x is
proportional to the DE (ET) parameter t0 and depends only on the  x ( Fe2 )
coefficient. The DM parameter Gx of the [Fe(II)Fe(III)] cluster depends on the both
 x ( Fe2 ) and  x ( Fe3 ) factors. Gx is proportional to the parameter J1 { J 2 } of the
Heisenberg exchange between the excited state of the Fe2 { Fe3 } ion and the ground
Gx ~ ( J1 x  J 2 x )
state of the Fe3 { Fe2 } ion. Since K x ~ t0 x ,
and t0  J 0 , J1 , J 2 , we can conclude that in the delocalized MV [Fe (II)Fe(III)] cluster
the AS double exchange coupling is stronger than the Dzialoshinsky-Moriya AS
exchange interaction.
5. Contributions of antisymmetric double exchange to zero-field splittings of
the [Fe(II)Fe(III)] cluster
The one-center zero-field splittings of the individual Fe2 and Fe3 ions
contribute to ZFS of the cluster S levels [9], which is described by the standard ZFS
Hamiltonian (1.3). For example, for the localized |a*b> state of the [Fe(II)Fe(III)] cluster
with S=9/2, the contribution of the individual axial ZFS parameters to the cluster axial
ZFS parameter D0 in the localized system has the form [9]
DS 9 / 2  16 D1 ( Fe2 )  185 D2 ( Fe3 ) ,
(5.1)
where D1 ( Fe 2  ) and D2 ( Fe3 ) are the axial ZFS parameters of individual Fe2 and
i
Fe3 ions, respectively, H ZFS
 Di [siz2  si (si  1) / 3]  Ei (six2  siy2 ) . In the delocalized
cluster, the averaged individual ZFS contributions were considered [9] as the origin of
the cluster ZFS. The ZFS of the mononuclear Fe2 and Fe3 centers was analyzed in
details in refs. [37, 40].
Since antisymmetric double exchange mixes the DE states with the same total
spin S and with different spins ( S '  S  1 ) the AS double exchange will contribute to
the cluster ZFS parameters. Let’s consider the ZFS parameters of the AndersonHasegawa E0 ( S  9 / 2) level with the DE energy t0 . The mixture of the double
exchange levels with S=9/2 E0 ( S  9 / 2, M ) and E 0 ( S  9 / 2, M '  M  1) by the AS
10
DE coupling (2.7) is determined only by the

0

 S  9 / 2, M  H1   9 / 2, M  1
0
=
Kx
coefficient, for example,
iK x / 3 for M=9/2,
4iK x / 9 for M=7/2.
These AS DE mixed Anderson-Hasegawa levels of different parity are separated by the
DE interval 2t0 .
The DE levels with different total spin are mixed by the AS double exchange
x
(2.4), (2.11) and Dzialoshinsky-Moriya AS exchange H DM
. The both antisymmetric
double exchange and DM AS exchange form the effective parameter Keff
mixing of the states with different S=9/2 and S=7/2, for example
of the AS
   (9 / 2,9 / 2) | Vab |   (7 / 2, 7 / 2)  2iKeff / 3 5 ,
Keff  K x  454 Gx .
(5.2)
These mixed Anderson-Hasegawa DE levels E0 ( S  9 / 2) and E0 ( S  7 / 2) of the
same parity are separated by the interval ( 15 t0  9 J 0 ) . As a result of the AS DE mixture
of the E0 ( S  7 / 2) and E0 ( S  9 / 2) DE levels with the E0 ( S  9 / 2) level, the AS
double exchange contribute to the axial ZFS parameter D[ E0 (9 / 2)] of the ZFS
Hamiltonian H ZFS  D[ E0 ( S )][ S Z2  S ( S  1) / 3] :
D[ E0 (9 / 2)]  D9 / 2  DK (9 / 2),
(5.3)
DK ( S  9 / 2)   811 [( K x  45Gx ) 2 /(t0  45 J 0 )  K x2 / t0 ] ,
0
where D9 / 2 is the axial ZFS parameter of the standard ZFS Hamiltonian H ZFS
(1.3),
Gx  Gx / 4 . In the part DK (9 / 2) of (5.3), the parameters K x and Gx of the
[Fe(II)Fe(III)] cluster are determined by eqs. (3.11) and (4.4), respectively.
Since K x  12t0 x (1  t / tu ) , Gx   (3 x J1   x J 2 ) / 2 and we suppose that the DE
coupling is stronger than the Heisenberg exchange interaction: t  J 0 , J1 , J 2 the K x
contribution to ZFS is stronger than the DM Gx contribution.
The ZFS parameter D[ E0 (9 / 2)] of the E0 ( S  9 / 2) Anderson-Hasegawa DE
level has the form
D[ E0 (9 / 2)]  D9 / 2  DK (9 / 2),
(5.4)
DK (9 / 2)  811 [( K x  45Gx ) 2 /(t0  45 J 0 )  K x2 (9 / 2) / t0 ] .
The contributions of the AS DE coupling to the axial ZFS of the Anderson-Hasegawa
levels E0 ( S  9 / 2) and E0 ( S  9 / 2) of different parity are different and depend on t0
and J 0 . In the limiting case t0  J 0 , we estimate the multiplier (1  J 0 / t0 ) as 1 and the
11
AS DE contributions to ZFS of the E0 (9 / 2) Anderson-Hasegawa DE levels may be
represented as
DK (9 / 2)
5 2 [16 J 0 x2  ( J1 x2  13 J 2 x x )] .
(5.5)
In this limiting case, the AS contributions to ZFS are the same for the E0 (9 / 2) DE
levels. The first {second}term in DK (9 / 2) (5.5) represents the AS double exchange
{Dzialoshinsky-Moriya AS exchange} contribution to the ZFS parameter, which is
proportional to the Heisenberg exchange parameter ( J 0 ) for the ground states set {( J1 ,
J 2 ) between the ground and excited states). The both contributions depend on the
cluster distortion (parameter  ).
The AS DE contributions DK ( S ) to the cluster ZFS parameter depend on
the total spin S. For example, for the E0 ( S  7 / 2) DE levels the axial ZFS parameter
has the form
D[ E0 (7 / 2)]  D7 / 2  DK (7 / 2),
DK ( S  7 / 2) 
( 79 ) 2 ( K x
1
81
[( K x
45Gx ) 2 /(t0
(5.6)
45 J 0 )
35Gx ) 2 / 5(t0 35 J 0 )  ( 227 ) 2 K x2 / 5t0 ].
6. Conclusion
The taking into account of the spin-orbit coupling in the first order of the
perturbation theory in the double exchange model for the MV [Fe(II)Fe(III)] cluster
results in the effective antisymmetric double exchange Hamiltonian
ˆ T
ˆ (Sˆ - Sˆ ) ] . The AS double exchange Hamiltonian has a form
H1  iK x [(Sˆ a - Sˆ b ) x T
ab
ab
b
a x
of the spin-transfer interaction.
The antisymmetric double exchange H1 mixes the
Anderson-Hasegawa DE states E0 ( S , M ) and E 0 ( S , M '  M  1) of the [Fe(II)Fe(III)]
cluster with the same total spin S of the different parity. The AS double exchange H1
x
and Dzialoshinsky-Moriya AS exchange H DM
 Gx [Sa  Sb ]x mix the AH states
0
0
E ( S , M ) and E ( S '  S  1, M '  M  1) with different total spin S of the same parity.
This mixing of the double exchange levels results in the AS DE contributions DK (9 / 2)
to the zero-field splitting parameters D[ E0 (9 / 2)]  D9 / 2  DK (9 / 2) of the ground state
DE states with S=9/2. The AS DE contributions to ZFS depend on the total spin and
parity of the DE levels: D[ E0 ( S )]  DS  DK ( S ) .
12
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