SPIN-FRUSTRATED TRINUCLEAR CU (II) CLUSTERS WITH
MIXING OF GROUND (2S=1/2) AND EXCITED (S=3/2) STATES
BY ANTISYMMETRIC EXCHANGE: LARGE DZIALOSHINSKYMORIYA EXCHANGE CONTRIBUTION TO ZERO-FIELD
SPLITTING OF S=3/2 STATE
Moisey I. Belinsky
School of Chemistry, Tel-Aviv University, 69978 Tel-Aviv, Israel,
belinski@post.tau.ac.il
Abstract.
The mixing of the S=1/2 and S=3/2 states by the Dzialoshinsky-Moriya (DM)
antisymmetric exchange coupling is considered for the copper (II) trinuclear clusters
with strong Heisenberg and DM exchange coupling. The S=1/2-S=3/2 mixing by the
components Gx , y of the vector DM exchange parameters lying in the plain of the
antiferromagnetic Cu3 trimer (the DM(x) mixing) results in the large positive DM
contribution 2DDM to the axial zero-field splitting (ZFS) 2D of the S=3/2 state of the
clusters with strong DM exchange coupling in spite of the large interval between the
ground S=1/2 and excited S=3/2 states. The DM(x) contribution 2DDM to the ZFS
depends on the orientation, value and sign of the DM exchange parameter. The DM(x)
mixing slightly increases or decreases also an initial ZFS of the spin-frustrated 2(S=1/2)
state, depending on the sign of the DM parameter Gz . In distorted Cu3 clusters, the
DM(x) exchange mixing results in the magnetic anisotropy, the non-linear magnetic
behavior of the Zeeman levels of the S=3/2 state and the field dependence of the
intensities of the EPR transitions. The DM(x) exchange mixing explains large negative
ZFS ( 2 D  5cm 1 ) observed in ferromagnetic Cu3 cluster. In the {Cu3} cluster with the
small Heisenberg exchange and DM exchange parameters, the DM(x) mixing results in
the ~60% contribution 2DDM to the observed ZFS 2D .
1. Introduction
Polynuclear clusters of metal ions have attracted significant interest as active
centers of biological systems [1], the building blocks of molecular magnets [2] and
models for investigation of magnetism at the mesoscopic scale [3]. A large number of
the copper (II) trimers in coordination compounds and biological systems were
investigated in details [4-21]. The isotropic antiferromagnetic (AF) exchange
interaction H 0   J ij Si S j in trigonal clusters ( J ij  J 0 ) leads to the spin-frustrated
2(S=1/2) ( 2 E ) ground state and excited S=3/2 ( 4 A2 ) states [6-9]. As shown in refs
[6,7], the Dzialoshinsky-Moriya (DM) antisymmetric (AS) exchange [22] coupling
H DM   G ij [Si  S j ] results in the zero-filed splitting (ZFS) 0z  GZ 3 of the ground
spin-frustrated 2(S=1/2) state of the trimeric metal M 3 centers,
95
determines magnetic
anisotropy, anisotropy of the EPR characteristics and hyperfine fields on nuclei,
GZ  (G12z  G23z  G31z ) / 3 , Z is the trigonal axis. The value of the antisymmetric DM
exchange parameter was estimated by Moriya | Gij | g / gJ ij [22b]. In the clusters with
different
isotropic
exchange
parameters
(isosceles
triangle
with J13  J 23  J , J12  J ',   J ' J ), the splitting of the 2(S=1/2) state
0   2  3GZ2 is determined by the parameters of the DM exchange GZ and distortion
 [6, 7]. The AF Cu3 clusters investigated in refs [9, 11c] were characterized by the
large Heisenberg AF exchange parameters ( J 0  300cm 1 [9], J 0  150cm 1 [11c]) and
DM parameters ( GZ  6cm 1 , GZ / J av  0.02 [9], GZ  5.5cm 1 ,
GZ / J av  0.037 [11c]). Recently, the [Cu3 ( II )] clusters with strong DM exchange
coupling were synthesized and investigated [18-21]. Ferrer et al [18] described the
magnetic and EPR characteristics of the two {Cu3 ( 3  O)( N  N )3} clusters by the two
different AF parameters of the distorted Heisenberg model and strong DM coupling:
J  191cm1 , J '  156.4cm1 , GZ  27.8cm 1 , GZ / J av  0.155 {cluster 1 [18]};
relatively small
J  175cm1 , J '  153.2cm1 , GZ  31cm 1 , GZ / J av  0.185 {cluster 2 [18]}. Liu et al
[19] demonstrated that magnetic and EPR-spectroscopic properties of the two AF
[Cu3 ( II )] cluster compounds containing [Cu3 ( 3  Me)]5 core can only be interpreted
by taking into account large DM exchange and  distortions: J  186cm1 ,
J '  210cm1 , J av  194cm 1 ,   24cm1 , GZ  33cm 1 ,  0  62cm 1 , GZ / J av  0.170
{cluster 1 [19]}; J  189cm1 , J '  252cm1 , J av  210cm 1 ,   63cm1 , GZ  47cm 1 ,
 0  103cm 1 , GZ / J av  0.224 {cluster 2 19]). Solomon, Stack and coworkers [20a]
describe the magnetic, EPR and MCD data of a D3 symmetric hydroxo-bridged
[Cu3 ( II )] complex by large isotropic AF exchange coupling constant and DM
exchange parameter: J  210cm1 ,   17.5cm1 , GZ  36cm 1 , GZ / J  0.171.
The ferromagnetic (FM) [Cu3 ( II )] cluster ( 3  O system) with J 0  109cm1
has the ground state 4 A2 and excited spin-frustrated state 2 E [21]. The system is
characterized by the large negative experimentally observed axial ZFS parameter
D  2.5cm 1 for the ground FM 4 A2 level, that can be related to the strong DM
exchange coupling, which splits the excited 2 E term ( | Gz | 42cm1 [21a). The ZFS of
the S=3/2 ground state was explained by the anisotropic exchange [21].
The other example of the system with the relatively large DM exchange is the
cluster compound with the {Cu3} trimer [23], which was characterized by the small
(slightly anisotropic) Heisenberg exchange parameters
av
J13av  J 23
 4.04 K  2.81cm 1
and
DM
exchange
J12av  4.52 K  3.14cm 1 ,
parameters
Gijx  Gijy  Gijz 
0.53K  0.37cm1 , GZ / J av  0.12 [23]. This Cu cluster is characterized by the crossing
of the S=3/2 and S=1/2 levels in high magnetic field H= 4-5 T [23].
96
The zero-field splitting of the S=3/2 state of the Cu3 clusters is described by the
standard ZFS Hamiltonian [24] of the trigonal system
0
H ZFS
 D0 [ S Z2  S ( S  1) / 3] .
(1)
The physical origin of the axial ZFS parameter D0 is the anisotropic exchange ( J x  J z )
[20a, 21a] since the single ion contribution to the cluster ZFS parameter is equal to zero
( si  1/ 2 ). The value 2D0 of the ZFS of the S=3/2 state was used to determine the
anisotropy of the Heisenberg exchange parameters [23, 25a].
Recently, the effect of quantum magnetization owing to the spin-frustrated
2(S=1/2) doublets of the AF triangular cluster with si  1/ 2 was observed in the V3
trimeric clusters [25], V3 clusters in the V15 molecular magnet [26-30] and {Cu3} trimer
[23] with the crossing of the S=3/2 and S=1/2 levels in high magnetic field. A half step
magnetization in the V3 and {Cu3} clusters was described by the S  3/ 2  S  1/ 2
mixing in the levels crossing point due to the Gx , G y components of the vector of the
pair DM exchange lying in the plain of the trimeric cluster [23-30]. The {Cu3} trimer
was characterized by the left chirality of the ground state [23].
In the consideration of the Cu3 clusters with the strong Heisenberg exchange
coupling and large interval 3J/2 between S  3/ 2 and S  1/ 2 levels, the zero-field
splitting of the ground spin-frustrated state 2(S=1/2) is only determined by the GZ DM
parameter [6-11c, 18-21] and the sign and direction of the DM vector doesn't influence
the splitting, magnetic behavior and EPR characteristics of the ground state. At the
same time, in the Cu3 (and V3 ) systems with the S  3/ 2  S  1/ 2 levels mixing, the
existence of the components of the DM vector ( Gx , G y ) in the plane of the trimer play
the principal role in an explanation of the low-temperature magnetic behavior in the
region of the levels crossing [26-30].
The aim of the paper is the consideration of the contribution of the
S  3/ 2  S  1/ 2 AS exchange mixing to the ZFS and magnetic anisotropy of the
S=3/2 state in the Cu3 clusters with strong DM exchange coupling and large 
distortions. As will be shown, in the clusters with strong Heisenberg and DM exchange
coupling, the S  3/ 2  S  1/ 2 DM exchange mixing results in a large contribution of
the AS exchange to the ZFS of the S=3/2 state, which depends on the sign of GZ .
2. The DM exchange in the Cu3 trimer
The isotropic Heisenberg exchange, the Kambe eigenvalues and Zeeman
interaction in distorted Cu3 clusters are described by the standard equations [5, 6]
H 0  J (sˆ1sˆ 2  sˆ 2sˆ 3  sˆ1sˆ 3 )   sˆ1sˆ 2 
  sˆ g H,
i 13
i
i
(2)
E1  S12  S   J / 2  S  S  1  3 4    / 2[ S12  S12  1  1],
(3)
where   J12  J , S12 is the intermediate spin. The eigenvalues E1[(0)1/ 2] E1[(1)1/ 2]
and eigenfunctions S12 ( S , M ) of (2) are characterized by S12  0 and 1
0 
1
2
(123  123 ),
0 (
1
2
(123  123 ),
97
(4)
1 
1
6
(2123  123  123 ),
1 
1
6
(123  123  2123 ) ,
where 0,1 ( M  1/ 2)  0,1 , 0,1 (M  1/ 2)  0,1 . In the trigonal cluster with   0 , the
two S=1/2 levels with S12  0 and S12  1 have the same Heisenberg energy (spinfrustration). The theory of the magnetic properties, EPR spectra, including the hyperfine
structure, of the trimers with the DM exchange was developed in refs [6, 7, 9]. The
antisymmetric DM exchange interaction has the form ( G ij  Gij )
H DM   G ij [Si  S j ] .
(5)
The directions of the pair antisymmetric DM vectors in the trimer are determined by the
symmetry conditions. In the case of the trigonal symmetry ( J ij  J 0 ), the z components
of the pair vectors are equal and directed along the trigonal axis Z, G12z  G23z  G31z  Gz .
Z
In the trigonal cluster, the DM(z) exchange ( H DM
  Gijz [Si  S j ]z ) results in the ZFS of
the ground spin-frustrated 2(S=1/2) state [6, 7, 9]:
E0 (1/ 2)  GZ 3 / 2 .
The wave functions
 0( ) (1/ 2, 1/ 2)  1 2 [0 (1/ 2)  ()i1 (1/ 2)] ,
 0( ) (1/ 2, 1/ 2) 
1
2
(6a)
(6b)
[0 ( 1/ 2)  ()i1 ( 1/ 2)]
diagonalize the DM(z) exchange [6]. The sign of GZ determines the chirality of the
lowest (and the first excited) Zeeman S=1/2 level with M=-1/2:  0 (1/ 2, 1/ 2)
 0 (1/ 2, 1/ 2) for GZ  0 . The correlations between the "in-plane"
components of the DM vectors in the pair coordinate system ( xi , yi ) and in the trigonal
cluster coordinate system (X, Y) have the form
Y
G12X  G12x , G12Y  G12y ; G23X   1 2 G23x  3 2 G23y , G23
 3 2 G23x  1 2 G23y ;
(7)
for GZ  0 ,
G31X   1 2 G31x 
3
2
Y
G31y , G31

3
2
G31x  1 2 G31y ,
k
G12k  G23
 G31k  Gk , k  x, y . The Gx , G y components of the DM vector don't
contribute to the ZFS E0 (1/ 2)  GZ 3 / 2 of the 2(S=1/2) ground state of the trigonal
cluster [6, 7, 9] since the matrix elements of the 0 (1/ 2)  1 ( 1/ 2) DM mixing
 0 (1/ 2) | H DM | 1 ( 1/ 2) 
i
2 3
Y
Y
[(G12X  G23X  G31X )  i(G12Y  G23
 G31
)]
(8)
Y
Y
 G31
are proportional to the G12X  G23X  G31X and G12Y  G23
parameters, which are equal to
zero in the trigonal system (7) [6].
In distorted Cu3 clusters [6], the Zeeman splitting is linear in low magnetic field
H  H Z || Z normal to the molecular xy-plane ( g  H Z  hZ  3J / 2 ):
Ez [()1/ 2]  1 2 [ 0  () hZ ] ,
(9a)
The wave functions, which describe the splitting (9a), have the form [6, 7]
1 (1/ 2)  a0 (1/ 2) ia1 (1/ 2),
(9b)
 2 (1/ 2)  a0 (1/ 2)  ia1 (1/ 2),
a 
98
1
2
(1   / 0 ) .
In low magnetic fields, the angle dependence of the Zeeman splitting (   ZH )
(10)
[02  ( g  H )2  () g  H  2  Gz2 cos 2  ] ,
2
the positions of the EPR lines and the intensities of the X-band ( 9.4GHz  0.313cm1 )
and Q-band ( 33.8GHz  1.13cm 1 ) EPR transitions follows the theory [6-9] as was
confirmed experimentally [9,11c, 18-21, 23]. The microscopic origin of the strong DM
AS exchange coupling in trimeric Cu3 clusters was explained by Solomon [20a, 21a] by
the large overlap of the d-functions of the neighboring Cu(II) ions in the ground and
excited states due to the geometry of the metal centers. The DM exchange parameter for
the Cu3 cluster with the non-direct inter-ion exchange was considered in ref [31].

E1(2)
1
3. The DM exchange mixing of the S=1/2 and S=3/2 states in the trigonal clusters
In the cases of the Cu3 [23] and V3 [25] clusters with week Heisenberg exchange
interaction, high magnetic field results in the crossing of the | S  3 / 2, M  3 / 2  and
|1/ 2, 1/ 2  Zeeman levels under condition hZ  3J / 2 ( H C1  4.4T , H C2  5T [23]).
The mixing of the S=3/2 and S=1/2 states of the si  1/ 2 trimers was considered in the
model with the three non-zero components Gijx , Gijy , Gijz [28-30, 23, 25] of the pair DM
coupling parameters. The analytical solutions of the S=1/2-S=3/2 AS mixing for the
trigonal cluster without an initial ZFS ( D0  0 ) were obtained in refs [29, 30c] for
H  H z and in ref. [30b] for H  C3 . We will consider the system with the orientation
of the Gijx component in the trimer plane perpendicular to the Cui  Cu j line
and Gijy  0 . The Hamiltonian of the system has the form
0
H  H 0  H DM  H ZFS
.
(11)
The 8X8 energy matrix of the Hamiltonian (11) were calculated in the basis set
0 (1/ 2), 1 (1/ 2), 0 (3/ 2, M ) . The DM mixing of the 0 (m), 1 (m) states (4) with
the  (3 / 2, M )  ( M ) states of the isosceles trimer is described by eq. (12)
 0 (1/ 2) | H DM | (3/ 2)  3iGx / 4 2;  1 (1/ 2) || (3/ 2)  3Gx / 4 2;
 0 (1/ 2) || ( 1/ 2)  i 3Gx / 4 2;
 1 (1/ 2) || ( 1/ 2)  3Gx / 4 2;
(12)
 0 (1/ 2) || (3/ 2, 1/ 2)  i(G12z  G23z ) / 6.
In the trigonal cluster ( H  H z ), the DM exchange mixing (12) results in the mixed
|S=1/2-S=3/2 states with the energies:
E1  K a  1 ,
E2  K a  1 , E3  Kb   2 ,
E4  Kb   2 ;
(13)
1  ( K1 )2  9Gx2 /16,
2  ( K2 )2  3Gx2 /16 ;
Ka  (3J 0  GZ 3  2D0  4hz ) / 4, K1  (3J 0  GZ 3  2D0  2hz ) / 4,
Kb  (3J 0  GZ 3  2D0 ) / 4, K2  (3J 0  GZ 3  2D0  2hz ) / 4;
and corresponding wave functions
1  b1  0 (1/ 2)  ib1  (3 / 2, 3 / 2),  2  b2  0 (1/ 2) ib2 (3 / 2,3 / 2),
99
(14)
 3  b3  0 (1/ 2) ib3 (3 / 2, 1/ 2),  4  b4  0 (1/ 2)  ib4 (3 / 2, 1/ 2);
b1 
1
2
(1  K1 / 1 ) , b2 
1
2
(1  K1 / 1 ) ,
b3  1 2 (1  K2 / 2 ) , b4  1 2 (1  K2 / 2 ) .
Eqs. (13), (14) show that in the trigonal cluster the excited (3 / 2, 3 / 2) ,  (3 / 2,3 / 2)
x
states are mixed by the DM(x) exchange ( H DM
  Gijx [Si  S j ]x ) with the ground
trigonal 0 (1/ 2) ,  _0 (1/ 2) states (6) ( E IR state of the double group D3 ),
Z
respectively, which diagonalize the Z component H DM
{ E0  GZ 3 } [6]. The excited
of the AS exchange
(3 / 2, 1/ 2) [ (3 / 2,1/ 2) ] state is mixed by the
DM(x) exchange with the  0 (1/ 2) [ 0 (1/ 2) ] state ( A1 , A2 IR states of the double
Z
group D3 ), which diagonalize { E0  GZ 3 } the DM(z) exchange H DM
(eq. (6)).
The DM(x) exchange coupling results in the repulsion of the E1 and E1 levels in
large positive magnetic fields H Z  0 or causes the avoided level crossing structure,
Fig.1. The E 2 and E 2 levels repulse in large negative fields H Z  0 . Fig. 1 represents
the result of the diagonalization of the energy matrix (8X8) of the Hamiltonian (11) for
the cluster with the small parameters of the Heisenberg exchange J 0  4.2 K  2.92cm 1
and AS exchange Gz  Gx  0.53K  0.37cm 1 [23] (Fig.1a), Gz  Gx  0 . 3 7cm1
(Fig. 1b), D0  0 , H  H z . In the low magnetic field H  H z , the Zeeman splitting does
not depend on the sign of the GZ DM parameter (Fig. 1a, 1b) and on the spin chirality
( E1 ,  0 (1/ 2) for GZ  0 or E3 ,  0 (1/ 2) for GZ  0 ). In the case G  0 , the ground
E1 Zeeman level mainly represents the  0 (1/ 2) state in the low magnetic field and
(3 / 2, 3 / 2) state in high magnetic field (eq. (14). The allowed EPR transitions for the
trigonal cluster in low magnetic field H  H z are shown by the vertical arrows in Fig.
1b. There are only two allowed EPR transitions between the S=1/2 levels E1  E4 and
E3  E2 with the same intensities W14  W23  0.25 [6, 7], Wi j |  j | S x | i |2 . The
positions of the EPR lines ( h1  h  0 , h2  h  0 , h2  h1  20 ) show that the pure
trigonal DM model can not describe the EPR spectra (X-band) of the V15 cluster [26b,
26c, 27c]. The cluster distortions should be taken in consideration.
The importance of the spin chirality of the ground state for the magnetization in the
area of the crossing point was discussed for the (VO )36  [25] and {Cu3} [23] clusters.
The repulsion of the | 3 / 2,| M | 3 / 2  Kramers doublet from the ground E
state
{ 0 (1/ 2) , 0 (1/ 2) }
3
times
differs
from
the
repulsion
of
the
| 3 / 2,| M | 1/ 2  doublet from the A1 , A2 states { 0 (1/ 2) , 0 (1/ 2) }, eq. (13). This
leads to the repulsion of the E1 and E1 levels and the minimal interval 3 | Gx | / 2
100
between these levels in the crossing point field in Fig. 1a (H~4.4T), and Fig. 1b
(H~5.2T), and the minimal interval 3 | Gx | / 2 between the excited E 4 and E 4 levels.
101
In the zero magnetic field ( H  0 ), different repulsions of the | 3 / 2,| M | 3 / 2 
and | 3 / 2,| M | 1/ 2  Kramers doublets from the ground trigonal E and A1 , A2 states
(eq. (13)) results in the following DM(x) exchange contribution 2DDM to the zero-field
splitting 2 Deff  2 D0  2 DDM of the S=3/2 level of the trigonal AF cluster ( Gk  J ):
Gx2 
2GZ 
2 DDM 
(15)
 1 
.
4 J 0  J 0 3 
Since the repulsion of the | 3 / 2,| M | 3 / 2   |1/ 2,1/ 2  doublets is 3 times stronger
than the repulsion of the | 3 / 2,| M | 1/ 2   |1/ 2,1/ 2  doublets, the DM(x) exchange
contribution
to
the
ZFS
of
the
S=3/2
excited
state
is
positive: EDM (3/ 2,| M | 3/ 2)  EDM (| 3/ 2,| M | 1/ 2) , DDM  0 . Eq. (15) describes the
dependence of the ZFS on the sign of Gz . The different ZFS of the S=3/2 state induced
by the DM exchange is shown in Fig. 1a ( 2 DDM  0.010cm 1 ) and Fig. 1b
( 2 DDM  0.013cm 1 ) for Gz  0.37cm 1 and Gz  0.37cm 1 , respectively.
The zero-field splitting of the excited S=3/2 state of the V3 center induced by the
DM(  ) exchange was obtained by Tsukerblat et al [30c] in the form 2 D '  G2 / 4 J 0 .
The zero-field splitting of the S=3/2 state induced by the DM(x) exchange
coupling in the Cu3 cluster with strong Heisenberg exchange and AS exchange coupling
is shown in Fig. 2 ( H  H Z ) on an example of the trigonal cluster with J av  168cm 1 ,
Gz  Gx , | G | 31cm1 , | G / J av | 0.185 and D0  0 . In spite of the large S=3/2-S=1/2
interval ( 3J av / 2  252cm 1 ), the DM(x) exchange results in the small different
admixture (eq. (14)) of the ground trigonal S=1/2 states ( 0 (1/ 2) , eq. (9a)) to the
S=3/2 Kramers sublevels in the zero magnetic field:
(3 / 2, 3 / 2)  1 ( H  0)  0.9966 0 (3 / 2, 3 / 2)  i0.0825 0 (1/ 2, 1/ 2),
(16)
(3 / 2, 1/ 2)   3 ( H  0)  0.9982 0 (3 / 2, 1/ 2)  i0.0593 0 (1/ 2, 1/ 2),
that leads to the zero-field gap 2DDM between these states. Different DM(x) repulsion of
the | 3 / 2,| M | 3 / 2  and | 3 / 2,| M | 1/ 2  Kramers doublets (eqs. (12), (13)) from the
ground 2(S=12/) states results in the large DM contribution 2 DDM  1.73cm 1 to the
axial ZFS parameter
gives 2 DDM  1.735cm 1 ).
2 Deff
in the case Gz  Gx  31cm 1 , Fig. 2 (eq. (15)
The DM(x) contribution to the ZFS of the S=3/2 state
depends on the sign of the Gz parameter: the calculations for Gz  Gx  31cm 1 lead
to 2 D 'DM  1.13cm 1 , eq. (15) results in 2 D 'DM  1.1cm1 . In the trigonal cluster, the
Zeeman splitting is linear (Fig.2), the intensities of the allowed EPR transitions follow
W [3 / 2  1/ 2]  0.75,
the standard relations for the S=3/2 state:
W [1/ 2  1/ 2]  1, W [1/ 2  3 / 2]  0.75.
102
In the case of the taking into account of both Gijx and Gijy components of the
DM exchange, the multiplier G x2 in eq. (15b) should be replaced by (Gx2  Gy2  G2 ) . For
av
 2.81cm 1 [23]), the
the {Cu3} cluster with J av  2.92cm 1 ( J12av  3.14cm 1 , J13av  J 23
authors used the three components of the AS exchange parameters
Gz  Gx  Gx  0.53K  0.37cm 1 [23] for the explanation of the magnetic and EPR
data. In this case, the eq. (15) with the ( Gx2  Gy2 ) multiplier may be used to estimate
the
DM
contribution
2DDM
to
the
cluster
ZFS
parameter 2 Deff :
2 DDM  0.0383K  0.0265cm 1 for G  0.37cm 1 . The experimentally observed zero-
field gap for the S=3/2 state of the {Cu3} cluster is 2 De  0.0662 K  0.0460cm 1 [23].
The DM exchange contribution 2DDM gives 58% of the experimentally observed ZFS
gap 2 De . The ZFS 2 De was used in [23] to estimate the exchange
anisotropy A | J x  J z |~ 0.6 K . However, the large DM(x, y) exchange contribution
2DDM to the ZFS parameter 2 Deff should be taken into account in the conclusion about
the exchange anisotropy extracted from the ZFS 2 De of the S=3/2 state in the
system: 2 DAN  2 D0  2 De  2 DDM .
Due to the repulsion of the S=1/2 and S=3/2 levels at H=0, the DM(x) exchange
results also in the common shifts lDM of the S=3/2 and S=1/2 states, respectively, and
the splitting  DDM of the 2(S=1/2) states additional to the standard Gz 3 / 2 AS
exchange splitting. The zero-field energies of the S=3/2 and S=1/2 states are
103
E1  E2  3J 0 / 2  lDM  ( D0  DDM ), E3  E4  3J 0 / 2  lDM  ( D0  DDM ),
E1  E2  Gz 3 / 2  lDM  DDM ,
E3  E4  Gz 3 / 2  lDM  DDM ,
(17)
where the common shift lDM  (Gx2 / 8J 0 )[2  GZ / J 0 3] depends on Gx and Gz . The
different additional shifts of the S=1/2 levels due to the DM(  ) exchange mixing was
described in ref [30c] in the form G2 / 8 J 0 and 3G2 / 8 J 0 . Fig. 3 shows the zero-filed
splitting of the ground 2(S=1/2) state for the trigonal cluster ( J 0  168cm 1 , Gz  Gx ,
| G | 31cm1 ) with the taking into account the DM(x) mixing of the S=1/2 and S=3/2
states. The DM(x) common shift of the levels is lDM  1.51cm1
lDM  1.51cm
1
for S=3/2 and
for S=1/2 states for Gz  0, lDM  3J 0 / 2. In the case Gz , x  0 , the
ground state with S=1/2 is E1  E2  Gz 3 / 2  lDM  DDM (with the strong DM(x)
exchange repulsion) and the excited state with S=1/2 is E3  E4  Gz 3 / 2 
lDM  DDM (with the weak DM(x) exchange repulsion), eq. (71). In the case Gz , x  0 , the
structure of the levels is opposite; the ground state is E3  E4 and the excited state
is E1  E2 . The resulting splitting of the trigonal 2(S+1/2) ground state is different


( E3,4
 E1,2
) | Gz | 3  2 DDM for Gz , x  0 ,
(18)


 '( E1,2
 E3,4
)  Gz 3  2DDM for Gz , x  0
and depends on chirality of the ground state (sign of Gz , x ). In an according with the
DM(x) exchange contribution to the ground-state ZFS (17), (18), the splitting in Fig. 3 is
  52cm 1 for Gz , x  31cm1 and   54.9cm 1 for Gz , x  31cm1 in comparison
with an initial DM(z) splitting 0z | Gz | 3  53.7cm1. An additional DM splitting of
the 2(S=1/2) states is a result of the different DM(x) exchange admixture of the Kramers
levels of the excited S=3/2 state to the ground trigonal states in zero magnetic field. An
additional DM(x) exchange splitting is small for the ground 2(S=1/2) state (-3.2% for
Gz , x  31cm1 and +2.2% for Gz , x  31cm1 ), however the DM(x) exchange splitting of
the S=3/2 state is large (eq. (16), Fig. 2). The considered ZFS induced by the DM
exchange essentially depends on the orientations of the pair G ij vectors. In the
case GZ  0, Gx  0 , the splitting of the ground trigonal spin-frustrated state 2(S=1/2) has


the DM(x) exchange origin: ( E3,4
 E1,2
)  2DDM .
104
In the trigonal Cu3 cluster, the DM(x) exchange mixing of the S=1/2 and S=3/2
states remains the system trigonal: the ZFS Hamiltonian for the S=3/2 has the trigonal
form (1) with the axial ZFS parameter Deff ( D0  DDM ) , the magnetic field dependence
for H  H Z is linear (Fig. 3) and the intensities of the EPR transitions follow the
relations characteristic for the standard S=3/2 state.
4. The DM exchange S=1/2-S=3/2 mixing in distorted Cu3 clusters
4.1. In distorted Cu3 clusters (   0 ), both Ez [ M ] levels (9a) are mixed with both
| 3 / 2, 3 / 2  and | 3 / 2, 1/ 2  Kramers sublevels. Thus, for example, for the M=-1/2
ground states one obtains
 1(2) (1/ 2) | H DM | (3/ 2, 3/ 2  (3iGx / 4)
 1(2) (1/ 2) | H DM | (3/ 2,1/ 2  (iGx / 4)
3
2
1
2
(1 Gz 3 /  0 ) ,
(19)
(1 Gz 3 /  0 ) .
This admixture results in the repulsion of all lowest levels in the area of the crossing
point for the distorted (isosceles) cluster with J12  4.52 K  3.143cm 1 ,
J13  J 23  4.808K   2.808cm1 , Gz , x  0.53K  0.368cm1 (Fig.4). In Fig. 4, the
minimal intervals between the levels in the area of the crossing in magnetic field are
described by equation
min
12(23)
 (3Gx / 2)
1
2
[1  ()Gz 3 / 0 ] ,
 min
45(56)  (Gx / 2)
3
2
[1  ()Gz 3 / 0 ] . (20)
For the cluster with Gz , x  0.37cm1 , the corresponding splitting and intervals are also
determined by eq. (20). In Fig. 4, the ZFS of the ground 2(S=1/2) state
 DM  0.7284cm1 is slightly larger than the interval 0   2  3Gz2  0.72cm1 (only
105
z
induced by the H DM
coupling) due to the repulsion (16). The zero-field ground state
wave
function
of
the
distorted
cluster
  (1/ 2, 1/ 2) H 0 
0.85240  i0.51991  i0.0549 0 (3 / 2, 3 / 2)  i0.075670 (3/ 2, 1/ 2)
demonstrates
the admixture of both |3/2,-3/2> and |3/2, +1/2> excited states. The coefficients of the
0 (1/ 2) and 1 (1/ 2) functions are very close to the coefficients a ( 0.856) and
a ( 0.517) (eq. (9c) [6, 7]) with the considered Gz and  parameters. In the low field,
there are the allowed EPR transitions I ( 1  4, 2  3 ) and II ( 1  3, 2  4 ) in the
S=1/2 states, the calculated intensities of the EPR transitions W ( I )  0.198 ,
W ( I )  0.052 follow the relations
Wz ( I )  3Gz2 / 4 02 , Wz ( II )   2 / 4 02 ,
(21)
( Wz ( I )  0.197, Wz ( II )  0.053) which were obtained [6, 7] for the system without the
DM(x) mixing of the S=1/2 and S=3/2 states. The ZFS induced by the DM(x) exchange
is 2 DDM  0.01cm 1 (Fig. 4).
The ZFS and Zeeman ( H  H Z ) splitting of the S=3/2 state of the distorted Cu3
cluster with strong AF coupling J12  252cm1 , J13  J 23  189cm 1 ,   63cm 1 and
DM exchange
Gz  Gx  47cm 1 [19] are shown in Fig. 5a. The DM(x) exchange
contribution to the ZFS parameter is 2 DDM  2.89cm 1 for the distorted cluster in Fig. 5.
For the cluster with Gz  Gx  47cm1 , the ZFS parameter of the DM(x) exchange
origin is 2 DDM  1.89cm1 and the Zeeman splitting is the same as in Fig. 5a. In
106
comparison with the trigonal cluster (Fig. 2), the new peculiarity of the AS mixing in the
distorted Cu3 cluster is the repulsion of the Zeeman sublevels 6 and 7 (formally |3/2,
107
min
 0.38cm 1 in
1/2> and |3/2, -3/2>). The minimal interval between these levels is  67
the field H z  D / g  . (In negative magnetic fields H   H z , there is the repulsion of
the sublevels 5 and 8.) This non-linear magnetic behavior in the magnetic field H z is
determined by the different admixture of the S=1/2 levels to the E3 (|3/2, +1/2>) and
E 4 (|3/2,-3/2) levels and resulting mixing of the  0 (3/ 2, 1/ 2) and 0 (3/ 2, 3/ 2)
states ( M  2 ). The non-linear magnetic behavior (Fig. 5a) determines the positions of
the EPR lines in magnetic field and leads to the complicated field dependence
Wij ( H z ) of the intensities of the allowed EPR transitions for the S=3/2 state shown in
Fig.
5b.
Wij ( H z )
differs
essentially
from
the
standard
relation:
W [3 / 2  1/ 2]  0.75, W [1/ 2  1/ 2]  1, W [1/ 2  3 / 2]  0.75 (dotted
straight lines in Fig. 5b) characteristic for the S=3/2 EPR transitions in H  H z , Fig. 2.
The mixing of the S=3/2 states with M  2 is described usually by the rhombic
term E ( S x2  S y2 ) in the standard ZFS Hamiltonian [24]. In the considered
isosceles Cu3 cluster with the DM(x, z) exchange, the distortions  lead to the new
rhombic-type anisotropy in the S=3/2 state, which depends essentially on magnetic
filed H z .
4.2. The other effect of the large distortions in the isosceles Cu3 clusters is
connected with the proportionality of the pair DM exchange parameters to the
Heisenberg exchange parameters: | Gij | (g / g ) J ij [22b]. Thus, for example, for the
Cu3
cluster
with
the
Heisenberg
parameters
J13  J 23  J  189cm1 ,
J12  J '  252cm 1 ,   63cm1 , [19] the relation between the DM parameters is
G12 / G23  J12 / J 23  1.33, G31  G23 , and using GZ  47cm 1 [19] as an average value
( GZ  (G12z  G23z  G31z ) / 3 ), one obtains G12  56.4cm 1 , G23  42.3cm 1 . The difference
of the pair DM exchange parameters results in the mixing of the S=1/2 and S=3/2 states
with the same M (eq. (12)) and modification of this equation. This difference results
also in the DM mixing of the S=1/2 state with different M (eq (8)). The DM(z) mixing
of the 0 ( M  1/ 2) and (3 / 2, M  1/ 2) states with the same M (the
term iGz / 6 , eq. (12)) depends on the Gz parameter. This DM(z) exchange mixing
results in an additional magnetic anisotropy and the repulsion of the Zeeman levels 5
and 6 (shown by the vertical arrow) of the |3/2, -3/2> and |3/2, -1/2> states.
108
5. DM exchange mixing in ferromagnetic [Cu3 ] clusters
In the case of the ferromagnetic [Cu3 ] clusters ( J 0  0 ) with the ground S=3/2
state, the DM(x) exchange mixing leads to the negative contribution to the effective ZFS
parameter 2 Deff since the repulsion of the |M|=3/2 Kramers levels is 3 larger than the
repulsion
of
the
|M|=1/2
doublet: EDM (3/ 2,| M | 3/ 2)  EDM (| 3/ 2,| M | 1/ 2) ,
DDM  0. For the FM cluster, the DM contribution to the ZFS of the 4 A2 ground state
has the form 2D 'DM  (Gx2 / 4 | J 0 |)[1  2GZ / | J 0 | 3] . The ZFS of the S=3/2 ground
state induced by the DM(x) exchange mixing is shown in Fig. 7 for the FM slightly
distorted [Cu3 ] cluster with J 0  109cm1 , | Gx, z | 40cm1 [21a]. The ZFS of the
DM(x) exchange mixing origin depends essentially on the values and sign of the AS
exchange parameters: 2 D 'DM  2.16cm1 for Gx, z  40cm1 and 2 D 'DM  5.14cm1
109
for Gx, z  40cm1 . The dependence on the sign is opposite to the case of the S=3/2 state
of the AF cluster (Fig.2). The contribution of the S=1/2-S=3/2 mixing to the ZFS of the
4
A2 state may be of the order of the experimentally observed ZFS 2 De  5cm 1 [21a].
In the ferromagnetic [Cu3 ] trigonal cluster [32] with J 0  1.52cm 1 , an axial ZFS
parameter D3/ 2  74 104 cm 1 was found experimentally, which differs in sign and
value from the possible spin-spin dipolar contribution ( Ddip,3/ 2  15 104 cm1 ) [32]. In
the case of the DM exchange parameter 12% of J 0 , as in the case [21], the DM(x, y)
contribution to the ZFS parameter is DDM  70 104 cm 1 , which is of the same order
as the observed ZFS parameter.
6. Conclusion
The mixing of the S=1/2 and S=3/2 states by the AS exchange results in the large
DM exchange contribution 2DDM to the zero-field splitting of the S=3/2 excited state of
the Cu3 clusters with the strong DM exchange coupling. This 2DDM contribution to
ZFS depends on the value and sign of the DM exchange parameters. The DM(x)
exchange mixing also results in the small contribution to the splitting of the ground
spin-frustrated 2(S=1/2) state and shift of the S-levels. In the distorted Cu3 clusters, the
DM exchange mixing results in the magnetic anisotropy, the non-linear field behavior of
the Zeeman levels of the S=3/2 state, complicated field dependence of the intensities of
the EPR transitions.
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110
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