spin–orbit coupling in the double exchange model 2. comparison of the antisymmetric double...

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Chemical Physics 325 (2006) 326–340

Spin–orbit coupling in the double exchange model2. Comparison of the antisymmetric double exchange with the

Dzialoshinsky–Moriya antisymmetric exchange, spin canting and ZFS

Moisey I. Belinsky

School of Chemistry, Sackler Faculty of Exact Sciences, Tel Aviv University, 69978 Tel Aviv, Israel

Received 2 October 2005; accepted 21 December 2005Available online 15 February 2006

Abstract

The Dzialoshinsky–Moriya (D–M) antisymmetric exchange interaction is considered and compared with antisymmetric (AS) doubleexchange in the mixed-valence [Fe(II)Fe(III)] clusters. It is shown that the antisymmetric double exchange interactionðH 1X ¼ iKX ½ðbSaX � bSbX ÞbTab þ bTabðbSbX � bSaX Þ�Þ is stronger than the D–M exchange interaction ðHX

DM ¼ GX ½~Sa �~Sb�X Þ in the valence-delocalized [Fe2.5+ Fe2.5+] cluster with strong double exchange, KX/GX = �5B/2J, KX� GX. The mixing of the ground valence-delocal-ized state U0

�ðS ¼ 9=2;MÞ with the first excited U0�ð7=2;M 0Þ state (S 0 = S � 1) of the same parity depends on both the AS double

exchange and D–M exchange. In the valence-delocalized cluster, the admixture of the excited double exchange states to the groundvalence-delocalized state U0

�ðS ¼ 9=2Þ by the AS double exchange and D–M exchange results in non-zero expectation values of the localspins hsaYi = �hsbYi in Y-direction, that leads to the canting of the spins in the plane (YZ) perpendicular to the direction of the vectorsKX of the antisymmetric double exchange and GX of the D–M exchange. In the valence-delocalized Sgr = (9/2)deloc state, the deviations ofspins from the parallel alignment, anisotropy of hyperfine (internal) fields on the Fe nuclei, anisotropy of the hyperfine constants dependon the parameters KX, GX, B and J. The spin canting induced by the AS double exchange and D–M exchange in the valence-delocalized[Fe2.5+Fe2.5+] cluster (Sgr = (9/2)deloc) is larger than the spin canting induced by the D–M exchange in the ferromagnetic localized [Fe(II)-Fe(III)]loc cluster (Sgr = (9/2)loc). The mixing of the double exchange levels results in the antisymmetric double exchange and D–Mexchange contributions to the ZFS parameters of the ground valence-delocalized state Sgr = (9/2)deloc.� 2006 Elsevier B.V. All rights reserved.

Keywords: Dzialoshinsky–Moriya antisymmetric exchange; Antisymmetric double exchange; Double exchange; Spin canting; Hyperfine fields on nuclei;Hyperfine constants; Zero-field splittings; Magnetic anisotropy; Valence-delocalized [Fe2.5+–Fe2.5+] cluster; Feromagnetic and antiferromagnetic localized[Fe(II)Fe(III)] clusters

1. Introduction

The antisymmetric double exchange (DE) coupling inthe valence-delocalized [Fe2.5+Fe2.5+] cluster was consid-ered in a previous paper 1 (Ref. [1]). Strong Anderson–Hasegawa [2] isotropic DE coupling in this mixed-valencecluster forms the delocalized ferromagnetic ground statewith maximal total spin Sgr = (9/2)deloc. It was shown inRef. [1] that the operator of the antisymmetric double

0301-0104/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.chemphys.2005.12.020

E-mail address: [email protected].

exchange for the valence-delocalized [Fe2.5+Fe2.5+] dimericcluster has the form

H 1X ¼ iKX ½ðbSaX � bSbX ÞbTab þ bTabðbSbX � bSaX Þ�; ð1Þwhere KX ð¼ ð~KabÞX Þ is the antisymmetric ð~Kab ¼ �~KbaÞ realvector parameter of the antisymmetric DE coupling. Theantisymmetric double exchange in the valence-delocalized[Fe2.5+Fe2.5+] cluster is the result of the account of thespin–orbit coupling (SOC) [1] in the Anderson–Hasegawadouble exchange model.

In the exchange coupled mono-valent [dn � dn] clustersof the transition metal ions, the account of SOC results

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M.I. Belinsky / Chemical Physics 325 (2006) 326–340 327

in the Dzialoshinsky–Moriya (D–M) [3–5] antisymmetricexchange interaction HDM ¼ ~Gab½~Sa �~Sb� and anisotropicexchange H AN ¼ ~Sa

eCab~Sb. The Heisenberg exchange inter-

action H 0 ¼ 2J~Sa~Sb results in parallel or antiparallel align-

ment of ~Sa and ~Sb, depending on the ferro- (J < 0) orantiferromagnetic (J > 0) coupling. The D–M coupling~Gab½~Sa �~Sb� tends to orient the spins perpendicular to eachother and ~Gab [3–6]. The D–M antisymmetric exchangeleads to spin canting in magnetic materials with magneti-cally ordered states of extended lattices [6,7].

In the mono-valent [dn � dn] clusters, the D–M antisym-metric exchange HDM results in the mixing of the S andS ± 1 cluster levels [6]. In the clusters, the magnitude ofthe D–M exchange was determined for the [Cr3+–Cr3+]impurity pairs (Gy = �0.26 cm�1) in LiGa5O5 [8], antiferro-magnetic [Fe3+–Fe3+] dimers of methane monooxygenase(G � 2.2 cm�1) and the synthetic complex (G � 1.5 cm�1)[9]. Trimeric antiferromagnetic [M3] clusters (M = Cu(II),Cr(III), Fe(III)) are the ideal objects for directly findingand investigating of the D–M coupling since the antisym-metric (AS) exchange H0DM ¼

P~Gij½~Si �~Sj� splits thespin-frustrated degenerate ground state 2(S = 1/2) of theseclusters [10]. As was shown first in Ref. [10], the D–Mexchange (parameter GZ ¼ ðGZ

12 þ GZ23 þ GZ

31Þ=3) and sym-metry lowering (d) determine the zero-field splittings,magnetic anisotropy, anisotropy of the g-factors and hyper-fine structure of the EPR lines, the hyperfine fields onnuclei in the spin-frustrated trimers [M3] (Refs. [10–17]).Experimental applications of the D–M term H0DM ¼Pð~GijÞZ ½~Si �~Sj�Z and the theory [10–16] have been

reported for various ½Cu2þ3 � trimers [12,14,15,17–23] as well

as the ½V 4þ3 � clusters [24], ½Cr3þ

3 � [10,11,13–15,25], ½Fe3þ3 �

[10,11,13–15,26–28], ½Cr3þ2 Fe3þ� and ½Fe3þ

2 Cr3þ� carboxi-lates [14–16], and ½Co2þ

3 � complexes [29]. The followingD–M parameters (in cm�1) were found for trimeric metalcomplexes: GZ = 6 (GZ = 5.5) for ½Cu2þ

3 � [12,18], GZ = 5–9for ½V 4þ

3 � [24]; GZ = 0.3 (GZ = 0.62) for ½Cr3þ3 � [14,25],

GZ = 1.4 for ½Fe3þ3 � [24]; GZ = 0.4 for [3Fe–4S]+ cluster

[27]; GZ = 0.8–1.1 for [Cr2Fe] and [Fe2Cr] [14,16];GZ = 2.8 for ½Co2þ

3 � [29]. Large antisymmetric exchangewas recently found for the spin-frustrated ½Cu2þ

3 � clusters:GZ = 36 (d = 17.5) [22], GZ = 27.8, 31 [20a], GZ = 47(d = 63) [21], GZ = 35 [23]. Antisymmetric D–M exchangedetermines the spin canting of the 2-D kagome lattice,formed of the ½Fe3þ

3 � triangles [30,31]. Strong D–M coupling(GZ � 10–25) is important for the anisotropy of the Mn12

cluster (single molecular magnet) [32].As recently shown experimentally, the antisymmetric

Dzialoshinsky–Moriya exchange interaction HDM ¼~Gab½~Sa �~Sb� mixes the ground Sgr = (1/2)loc and firstexcited Sexc = 3/2 levels of the localized antiferromagnetic(AF) mixed-valence [Fe(II)Fe(III)]loc cluster of the [2Fe–2S]1+ center of the Rieske protein from Thermus thermophi-

lius [33]. Strong Dzialoshinsky–Moriya exchange coupling(jGabj � 7.2 cm�1, (jGabj/2J = 0.18, 2J � 40 cm�1,H = 2JSaSb) [33]) results in anisotropy of the g-factors[33] of the AF ground state S = 1/2.

All these data show that the D–M antisymmetricexchange may be important in the metal clusters with Hei-senberg exchange inter-ion coupling (jJj � 100 cm�1). Thefinding of relatively strong D–M antisymmetric exchangein the mono-valent clusters and especially in the localizedAF [Fe(II)Fe(III)]loc cluster shows that taking spin–orbitcoupling into account in the double exchange model canresult in a strong antisymmetric DE effect for thevalence-delocalized [Fe2.5+Fe2.5+] cluster with strong iso-tropic Anderson–Hasegawa DE coupling.

Since the antisymmetric double exchange coupling is theresult of taking SOC into account in the valence-delocal-ized [Fe2.5+Fe2.5+] cluster with the Anderson–Hasegawadouble exchange inter-ion interaction [1] and the D–Mantisymmetric exchange is the result of taking SOC intoaccount in the superexchange (Heisenberg) model, theaim of the paper is the consideration of the D–M exchangein the localized [Fe(II)Fe(III)] cluster and comparison ofthe D–M exchange with the antisymmetric doubleexchange.

The origin of the zero-field splitting (ZFS) and magneticanisotropy, anisotropy of the g-factors, the hyperfine(internal) fields on the Fe nuclei and hyperfine constantsis important for the polynuclear clusters. In the spin cou-pling model, the spin coefficients hsizi/hSZi and intrinsicparameters (ai, gi) determine the hyperfine fields on theFe nuclei, the hyperfine constants and the g-factors. Theorigin of anisotropy in this case is the anisotropy ofthe individual parameters of the iron ions. The role ofthe D–M exchange and local ZFS in an anisotropy of thecluster parameters of the antiferromagnetic [2Fe] centerswas considered in Refs. [9,33,65]. The influence of the anti-symmetric double exchange on the deviations of spins fromthe parallel alignment (spin canting), hyperfine fields andhyperfine constants of the [Fe(II)Fe(III)] clusters was notconsidered.

The ground S = 9/2 levels of the MV [Fe(II)Fe(III)]clusters undergo relatively large zero-field splitting�1.5 < D9/2 (cm�1) < 4 [34–44]. Zero-field splitting of theground state Sgr = 9/2 is described by the standard effectiveZFS Hamiltonian H 0

ZFS¼DS ½S2Z�SðSþ1Þ=3� þ ESðS2

X � S2Y Þ

[45–48]. In the spin-coupling model, the cluster ZFSparameters DS, ES are formed by the ZFS contributionsof individual ions (D1(Fe2+), E1(Fe2+); D2(Fe3+),E2(Fe3+)) [34,36,42,44]. The origin of the cluster ZFSparameters for the exchange [dn � dn] clusters was consid-ered in Refs. [48,14,15,49]. The contributions of the anti-symmetric double exchange and D–M exchange to thecluster ZFS of the [Fe(II)Fe(III)] clusters were not consid-ered for the MV [Fe(II)Fe(III)] cluster.

The D–M exchange admixes to the ground state theexcited S levels of the monovalent [dn � dn] clusters, thatresults in the induced hyperfine fields on nuclei [9] and con-tributes to the spin canting [6]. As shown in Refs. [1,50,51]the antisymmetric double exchange mixes the DE levels ofthe valence-delocalized dimers and contributes to ZFS. Wewill consider in this paper the contributions of the antisym-

328 M.I. Belinsky / Chemical Physics 325 (2006) 326–340

metric double exchange and D–M exchange to the spincanting, anisotropy of the hyperfine fields on nuclei, thehyperfine constants and the cluster ZFS parameters inthe MV [Fe(II)Fe(III)] clusters.

As will be shown in the paper for the valence-localized[Fe2.5+Fe2.5+] cluster, the antisymmetric double exchangeinteraction H1X (1) is stronger than the Dzialoshinsky–Moriya antisymmetric exchange H X

DM ¼ ð~GabÞX ½~Sa �~Sb�X ,jð~KabÞX j � jð~GabÞX j. The correlation between the parame-ters KX and GX has the form KX/GX = �5B/2J. Strongdouble exchange leads to the ferromagnetic delocalizedground state with parallel spin alignment. The antisymmet-ric double exchange interaction H1X and Dzialoshinsky–Moriya exchange H X

DM yield deviation of spins from theparallel alignment (spin canting), anisotropy of the hyper-fine fields on nuclei and hyperfine constants. In thevalence-delocalized ground Sgr = (9/2)deloc state of thedelocalized [Fe2.5+Fe2.5+] cluster, the spin canting andanisotropy of hyperfine fields are determined by theparameters KX, GX, B and J. Both the antisymmetric dou-ble exchange and D–M coupling contribute to the clusterZFS parameters.

2. The Dzialoshinsky–Moriya antisymmetric exchange in the

[Fe(II)Fe(III)] localized cluster

The microscopic explanation of the Dzialoshinsky–Moriya antisymmetric (AS) exchange [3–5] interaction

HDM ¼ ~Gab½~Sa �~Sb� ð2Þwas developed by Moriya [4,5] for the monovalent [dn � dn]pair, the vector D–M parameter ~Gab is antisymmetricð~Gab ¼ �~GbaÞ. (The D–M constant was determined as ~Dij

in Refs. [4,5], we use the notation ~Gab ¼ ~Dab for the D–Mconstant since the individual axial ZFS parameters D1

and D2 and the cluster axial ZFS parameters DS, DS(2S+1C)are used in the consideration.) In the case of the superex-change interaction, the parameter of the D–M exchangehas the form [4,5]

~Gab ¼4itUð~Cab � ~CbaÞ ¼ 2iðJ=tÞ~Cab; ð3Þ

where the vector transfer integrals ~Cab were determined in[4], J = 4t2/U is the constant of the superexchange, U theCoulomb interaction constant and t the ET integral[4,5,52–55].

In the case of the direct exchange interaction in the½dn

a � dnb� cluster ðJ ¼ hu0

au0bku0

bu0aiÞ, the D–M parameter

is determined by the Moriya equation [4,5]

Gexab ¼ 2ikf

Xm

ðLam0=emÞJðam; b0Þ

�X

m0ðLbm00=em0 ÞJða0; bm0Þg; ð4Þ

where J(am, b0) {J(a0, bm0)} are the parameters of the di-

rect Heisenberg exchange between the excited states /ma

of the dna ion and the ground state u0

b of the dnb ion {between

the ground state u0a of the dn

a ion and excited states /m0

b ofthe identical dn

b ion}.Let us consider the mixing of the Sloc states of the local-

ized [Fe(II)Fe(III)] cluster by the D–M antisymmetricexchange (2). Strong Anderson–Hasegawa [2] DE couplingforms the lowest ground Sgr = (9/2)deloc ðE0

�ð9=2ÞÞ and firstexcited Sdeloc = 7/2 ðE0

�ð7=2ÞÞ levels (Fig. 1b, [1]),U0�ðSÞ ¼ ½U0

a�bðSÞ þ U0ab� ðSÞ�=

ffiffiffi2p

. We will consider first theD–M mixing of the localized U0

a�bð9=2Þ (Sloc = 9/2) andU0

a�bð7=2Þ (Sloc = 7/2) states in the ja*bi localization,Fig. 1a, [1]. As it follows from the microscopic calculations(Eq. (22), Section 4), only the GX ¼ ½~Gab�X component ofthe Dzialoshinsky–Moriya antisymmetric exchange is dif-ferent from zero (GY = GZ = 0) for the considered localized[Fe(II)Fe(III)] cluster. The matrix elements of the operatorof the D–M antisymmetric exchange

HXDM ¼ GX ½~Sa �~Sb�X ; ð5Þ

which mixes the localized U0a�bðS;MÞ and U0

a�bðS0 ¼ S � 1;M 0 ¼ M 1Þ states of the ½dnþ1

a� ðsaÞ � dnbðsbÞ�loc cluster in

the ja*bi localization, are described by the equation

hU0a�bðS;MÞjH X

DMjU0a�bðS � 1;M 1Þi

¼ ðiGX=4ÞF ðS;M ; sa; sbÞ; ð6Þ

where the spin multiplier F±(S, M, sa, sb) has the form

F ðS;M ; sa; sbÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðS jM j � 1ÞðS jM jÞð2S þ 1Þð2S � 1Þ

s

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½ðsa þ sb þ 1Þ2 � S2�½S2 � ðsa � sbÞ2�

q.

ð7Þ

The matrix elements of HXDM (6) for (�jMj) and

(M 0 = �(jMj 1)) have opposite sign.For example, the mixing by the D–M exchange H X

DM (5)of some of the localized U0

a�b ðS ¼ 9=2;MÞ andU0

a�b ðS0 ¼ 7=2; M 0 ¼ M � 1Þ states of the [Fea(II)Feb(III)]-

loc cluster is described by Eq. (8):

hU0a�bðS ¼ 9=2;MÞjH X

DMjU0a�bðS ¼ 7=2;ðM � 1ÞÞi

¼ 3iffiffiffi5p

GX=2 for M ¼ 9=2;

iffiffiffiffiffi35p

GX=2 for M ¼ 7=2;

iffiffiffiffiffiffiffiffi105p

GX=4 for M ¼ 5=2. ð8Þ

For comparison, the matrix elements of the antisymmet-ric double exchange H1X (1) between the U0

a�bðS;MÞ andU0

ab� ðS0 ¼ S � 1;M 0 ¼ M 1Þ states of different localiza-tions (Eq. (19) [1]) may be represented in the form

hU0a�bðS; jM jÞjH 1X jU0

ab� ðS � 1; jM j 1Þi¼ �i½KX=2Sð2sa þ 1Þ�F ðS;M ; sa; sbÞ; ð9Þ

where F±(S, M, sa, sb) is determined by Eq. (7).The mixing of the S and S 0 = S � 1 localized Heisenberg

states by the D–M antisymmetric exchange HXDM (5) is the

same in the ja*bi (Fig. 1a [1]) and jab*i (Fig. 1c [1]) local-izations of the extra electron:

Fig. 1. The localized ½Fe2þa Fe3þ

b �loc cluster. The scheme of the ligand fieldsplittings, spin–orbit coupling mixing and exchange interactionsbetween the ground and excited states of the Fe2þ

a and Fe3þb ions, which

determine the Dzialoshinsky–Moriya antisymmetric exchange parameterGX. (a,c) The ligand field splittings and the spin–orbit mixing of the statesof the Fe2þ

a {Fe3þb } ion; (b) exchange interactions.


hU0a�bðS;MÞjHX

DMjU0a�bðS0;M 0Þi

¼ hU0ab� ðS;MÞjH X

DMjU0ab� ðS0;M 0Þi. ð10Þ

3. Mixing of the DE states of the valence-delocalized

[Fe2.5+Fe2.5+] cluster by the antisymmetric double exchange

and Dzialoshinsky–Moriya exchange

Table 1 (in the representation of the states of differentlocalizations U0

a�bðS;MÞ and U0ab� ðS0;M 0Þ) demonstrates

the matrix of the Heisenberg exchange H0 (Eq. (2) [1]),antisymmetric Dzialoshinsky–Moriya exchange HX

DM (5)and initial zero-field splittings H 0

ZFS in the localized statesja*bi and jab*i and also the Anderson–Hasegawa doubleexchange H 0

DE (Eq. (6) [1]) and antisymmetric double

Table 1The matrix of the Hamiltonian of the Anderson–Hasegawa double exchange coH1X (1) (KX terms) in the basis of the states U0

a�bðS;MÞ and U0ab� ðS0;M 0Þ of diffe

9/2, 7/2, 5/2), S = 7/2 ð eM ¼ 7=2; 5=2Þ and S 0 = 9/2 (M 0=9/2, 7/2, 5/2), S 0 = 7

U0a�bðS ¼ 9=2;MÞ . . . j U0

a�bð7=2; j eM jÞM ¼ 9

2 72 5

2 . . . eM ¼ 72

eM ¼ 52

24J + 12D0 3iffiffi5p

GX2

24J + 4D0 iffiffiffiffi35p

GX2

24J-2D0iffiffi5p

GX4

15J + 7D 0

15J + D0

�5B�5B c.c.

�5B

�4B�4B

The diagonal matrix elements in the ja*bi ð½Fe2þa� Fe3þ

b �locÞ and j ab�ið½Fe3þa Fe2þ

b� �splittings (H0

ZFS, Eqs. (12), (13)). The Dzialoshinsky–Moriya exchange interactiolocalizations (GX terms).

exchange H1X (1) coupling between the states of differentlocalizationseH ¼ H 0 þ H 0

ZFS þ HXDM þ H 0

DE þ H 1X ð11Þfor the some of the exchange U0

a�bðS;MÞ and U0ab� ðS0;M 0Þ S

levels (S = 9/2,7/2, see Fig. 1 [1]). The diagonal matrix ele-ments of Table 1 represent the valence-localized Heisenberglevels Sloc ¼ ð9=2Þja�bi; Sloc ¼ ð9=2Þjab�i ðE0 ¼ 24JÞ andSloc ¼ ð7=2Þja�bi; Sloc ¼ ð7=2Þjab�i ðE0 ¼ 15JÞ, which aresplit on the localized Kramers doublets (jMj = 9/2, 7/2,5/2 (S = 9/2); j eM j¼ 7=2; 5=2 (S = 7/2), only part of theKramers doublets are represented in Table 1) in the pres-ence of an initial zero-field splitting H 0

ZFS in different local-ized states ja*bi and jab*i. (The left-upper 5 · 5 block {theright-lower 5 · 5 block} of the matrix in Table 1 representsthe ja*bi{jab*i} localization.) We consider, for simplicity,the cluster with axial anisotropy of the individual ionsand cluster states. The standard axial ZFS splittings [45–48]

H 0ZFS ¼ D0

S ½S2Z � SðS þ 1Þ=3� ð12Þ

(ES = 0) of the considered S = 9/2 and S = 7/2 levels aredescribed by the same axial ZFS parameters D0

S¼9=2 ¼ D0

and D07=2 ¼ D0 in different localizations ja*bi and jab*i.

(The valence-delocalized ground state Sgr = (9/2)deloc withE = 0 {E 5 0} was observed experimentally in Refs.[34,36] {[32,40,41,61]}). In the spin-coupling model, theHamiltonian of the localized exchange pair of the two ionswith an initial zero-field splitting has the form[48,34,42,44]:

Hab ¼ 2Jsasbþ eH 0ZFS;eH 0

ZFS ¼ D1½s2az � saðsaþ 1Þ=3� þD2½s2

bz � sbðsb þ 1Þ=3�. ð13Þ

The ZFS parameter D0S of the localized cluster in the spin-

coupling model is formed by the individual ZFS parame-ters D1 = D1(Fe2+) and D2 = D2(Fe3+) of the operatoreH 0

ZFS [34]:

upling H 0DE (15) (B terms) and the antisymmetric double exchange coupling

rent localizations of the mixed-valence [Fe2.5+Fe2.5+] cluster, S=9/2 (M =/2 ð eM 0 ¼ 7=2; 5=2Þj . . . U0

ab� ðS0 ¼ 9=2;M 0Þ . . . j U0ab� ð7=2; j eM 0 jÞ

M 0 ¼ 92 7

2 52 . . . eM 0 ¼ 7

2eM 0 ¼ 5

2

�5B iKX3 2iKX

3ffiffi5p

iKX3 �5B 4iKX

9 14iKX

9ffiffiffiffi35p

4iKX9 �5B iKX

9ffiffi5p

2iKX

3ffiffi5p iKX

9ffiffi5p �4B 44iKX

45ffiffi7p

14iKX

9ffiffiffiffi35p 44iKX

45ffiffi7p �4B

24J + 12D0 3iffiffi5p

GX2

24J + 4D0 iffiffiffiffi35p

GX2

24J-2D0iffiffi5p

GX4

15J + 7D 0

15J + D0

locÞ localizations show the Heisenberg exchange energy (H0) and zero-fieldn HX

DM (Eq. (5)) mixes the localized S = 9/2 and S = 7/2 levels of the same


D0S¼9=2 ¼ D0 ¼ ½0:17D1ðFe2þÞ þ 0:27D2ðFe3þÞ�;

D07=2 ¼ D0 ¼ ½0:05D1ðFe2þÞ þ 0:22D2ðFe3þÞ�. ð14Þ

The individual axial ZFS axes fix the cluster axial Z axis.The mixing of the localized Kramers doublets jS = 9/2,M = ±7/2i{j9/2, ±5/2i} with the state j7/2, ± 7/2i{j7/2,±5/2i} by the ZFS operator eH 0

ZFS (13) (proportional to[4D1 � 3D2]) is not shown in Table 1. There is no eH 0

ZFS

mixing of the j9/2, ±9/2i Kramers doublet with the otherstates.

The Anderson–Hasegawa double exchange H 0DE cou-

pling (Eq. (6) [1])

H 0DE ¼ bTabt0 ð15Þ

only mixes the Kramers doublets of different localizationsU0

a�bðS;MÞ and U0ab� ðS;MÞ with the same S and M (the

non-diagonal terms �5B and �4B for S = 9/2 and 7/2,respectively, in Table 1). Each Kramers doublet of thelocalized S = 9/2 {S = 7/2} level in Table 1 undergoesthe same DE splitting ±5B {±4B} in the valence-delocal-ized cluster since the double exchange interaction only con-nects the states of different localizations with the same S

and M (Table 1). The DE coupling in the presence of aninitial ZFS (D0 and D 0 terms on the diagonals of Table1) forms the same order of Kramers doublets for theE0þð9=2Þ and E0

�ð9=2Þ {E0þð7=2Þ and E0

�ð7=2Þ} DE levelsas those of the localized states (Table 1). The average axialZFS parameter is the same for the E0

þð9=2Þ and E0�ð9=2Þ

{E0þð7=2Þ and E0

�ð7=2Þ} DE states: Dav9=2 ¼ D0 ¼ 0:44Dav

i ¼0:44DifDav

7=2 ¼ D0 ¼ 0:27Davi ¼ 0:27Dig [34].

The antisymmetric double exchange H1X (1) mixes theKramers doublets of different localizations with the sameS and different M, for example, jS ¼ 9=2; M ¼ 9=2ia�band jS 0 = 9/2, M 0 = ±7/2iab*, Eqs. (13)–(15) [1]. Thej 9=2;7=2ia�b Kramers doublet is mixed by the antisym-metric DE H1X with the Kramers doublets j9/2, ±9/2iab*

and j 9=2;5=2iab� of the opposite localization (Table 1,KX terms). The mixing of the states of different localiza-tions with different total spin S

0= S ± 1, M

0= M ± 1

depends on the sign of M (Eqs. (19) and (20) [1], Table 1,±KX terms). The antisymmetric double exchange H1X (1)mixes the Kramers doublets of different localizations withS0= S � 1, M

0= M � 1, for example, j 9=2;M ¼ 9=2ia�b

fj 9=2, M ¼ 5=2ia�bg and j 7=2;7=2iab�fj7=2;5=2iab�g,etc., Table 1, Eq. (20) [1].

In the localized states, the Dzialoshinsky–Moriyaexchange H X

DM (Eqs. (5)–(7)) mixes the Kramers doubletsof the Heisenberg states jS = 9/2, Mi and jS = 7/2,M 0 = M 1i of the same localization (Table 1, GX terms).The mixing of the S and S 0 = S � 1 localized Heisenbergstates by the D–M antisymmetric exchange H X

DM (5) isthe same in the ja*bi and jab*i localizations of the extraelectron (Eq. (10)) (Table 1).

As a result of correlations (6), and (10) for the D–Mexchange and Eqs. (19)–(21), [1] for the AS doubleexchange in the valence-delocalized MV ½Fe2:5þ

a Fe2:5þb �deloc

cluster, both the D–M exchange HXDM (5) and antisymmet-

ric double exchange H1X (1) contribute to the mixing of thedouble exchange states ðE0

ðSÞ and E0ðS0 ¼ S � 1ÞÞ of the

same parity with different total spin. Thus, using Eqs.(6)–(9) and correlations (14), (19)–(21) [1], one obtainsthe following expressions for the matrix elements of theantisymmetric double exchange and D–M exchangeH 1X þ HX

DM between the U0ðS;MÞ and U0

ðS � 1;M 1ÞDE states ðU0

ðSÞ ¼ ½U0a�bðSÞ U0

ab� ðSÞ�=ffiffiffi2pÞ:

hU0ðS;MÞjH 1X þ H X

DMjU0ðS � 1;M � 1Þi

¼ i½KX=2Sð2sa þ 1Þ GX=4�F þðS;M ; sa; sbÞ. ð16Þ

For example, the matrix elements of the AS doubleexchange H1X (Eqs. (19)–(21) [1]) and D–M exchangeHX

DM (Eqs. (6)–(10)) between the ground valence-delocal-ized states U0

�ðS ¼ 9=2;MÞ and first excites U0�ðS ¼ 7=2;

M � 1Þ DE states (Eqs. (8)–(10) and (20), (21) [1]) havethe form

hU0ðS ¼ 9=2;M ¼ 9=2ÞjH 1X þ HX

DMjU0ð7=2;7=2Þi

¼ 2iðKX � 45GX=4Þ=3ffiffiffi5p

;

hU0�ð9=2;7=2ÞkU0

�ð7=2;5=2Þi¼ 2i

ffiffiffi7pðKX � 45GX=4Þ=9

ffiffiffi5p

. ð17Þ

As a result, the antisymmetric mixing of the U0ðS ¼

9=2;MÞ and U0ðS ¼ 7=2;M � 1Þ DE states of the same

parity with different S (Eq. (16)) are determined by theeffective antisymmetric parameter

PX ð9=2Þ ¼ ½KX 45eGX �; ð18ÞeGX ¼ GX=4; which includes both the contributions of theantisymmetric double exchange and Dzialoshinsky–Mor-iya exchange. The KX {GX} terms in Eqs. (16)–(18) and Ta-ble 1 represent the AS mixing of the states of the different{same} localizations. The DE levels E0

ð9=2Þ and E0ð7=2Þ

are separated by the intervals B 9J. For the antisymmet-ric mixing of the U0

ðS ¼ 7=2;MÞ and U0ðS ¼ 5=2;M � 1Þ

DE states (separated by the intervals B 7J) the effectiveparameter of the antisymmetric mixing has the formPX ð7=2Þ ¼ ½KX 35GX=4� (Eq. (16)).

The matrices in Eqs. (19) and (20) show examples of theantisymmetric double exchange mixing of the Kramersdoublets jMj = 9/2, 7/2 of the lowest DE level E0

�ð9=2Þ inthe resonance (DE) representation U0

�ðS;MÞ andU0þðS0;M 0Þ, which diagonalizes the double exchangeðH 0

DEÞ, Heisenberg exchange (H0) and ZFS ðH 0ZFSÞ

operators

U0�ð9=2;9=2Þ U0

�ð7=2;7=2Þ U0þð9=2;7=2Þ

�5Bþ 24J þ 12D0 2iP�X =3ffiffiffi5p

�iKX=3

�4Bþ 15J þ 7D0 0

c.c. 5Bþ 24J þ 4D0

ð19Þ

U0�ð9=2;7=2Þ U0

�ð7=2;5=2Þ U0þð9=2;9=2Þ U0

þð9=2;5=2Þ�5Bþ 24J þ 4D0 2i

ffiffiffi7p

P�X =9ffiffiffi5p

�iKX=3 �i4KX=9

�4Bþ 15J þ D0 0 0

5Bþ 24J þ 12D0 0

c.c. 5Bþ 24J � 2D0

ð20Þ


D0 ¼ Dav9=2 ¼ 0:44Di, D0 ¼ Dav

7=2 ¼ 0:27Di [34] in Eqs. (19)and (20), P�X ¼ P�X ð9=2Þ ¼ ½KX � 45GX=4�. The non-diago-nal matrix elements in the matrices in Eqs. (19) and (20)represent the antisymmetric mixing of the Anderson–Hase-gawa DE levels. In Eqs. (19) and (20), the mixing of the DEKramers doublets U0

�ð9=2; jM jÞ ðE0�ðS ¼ 9=2; jM jÞÞ and

U0þð9=2;M 0 ¼ jM j � 1ÞðE0

þð9=2;M 0 ¼ jM j � 1ÞÞ (the DEinterval 2t0 = 10B) with the same S = 9/2 only dependson the antisymmetric double exchange KX parameter,Eqs. (13)–(15) of Ref. [1]. The D–M antisymmetric ex-change HX

DM (5) does not mix the DE levels E0�ðS;MÞ and

E0þðS;M 0Þ with the same total spin of different parity since

the D–M exchange is active only between the levels withdifferent S in the localized state, Table 1. In the caseS 0 = S ± 1 (Eqs. (19) and (20)), both the antisymmetricdouble exchange H1X and the Dzialoshinsky–Moriya ex-change HX

DM participate in the antisymmetric mixing ofthe DE levels E0

�ð9=2;MÞ and E0�ð7=2;M 0 ¼ M 1Þ with

different total spin in the form of the effective mixingparameter P�X ð9=2Þ ¼ KX � 45GX=4 (Eqs. (16), (19) and(20)).

The Kramers doublet U0�ð9=2;9=2Þ (E0

�ð9=2; j9=2jÞ ¼�5Bþ 24J þ 12D0, Eq. (19)) of the lowest valence-delocal-ized level E0

�ð9=2Þ is mixed with the Kramers doubletsU0�ð9=2;7=2Þ of the next nearest DE state E0

�ð7=2Þ (sepa-rated by the interval (B � 9J) from E0

�ð9=2Þ) and the high-est DE state E0

þð9=2Þ (separated by the interval 10B), Eq.(19). The Kramers doublet U0

�ð9=2;7=2Þ ðE0�ð9=2;

j7=2jÞ ¼ �5Bþ 24J þ 4D0Þ is mixed by the antisymmetricdouble exchange and D–M exchange with U0

�ð7=2;5=2Þand only by the AS double exchange with U0

þð9=2;9=2Þ,U0þð9=2;5=2Þ, Eq. (20), KX terms.

4. Microscopic parameters of the D–M antisymmetric

exchange

Now let us consider, following the Moriya theory [4,5],the microscopic origin of the parameter of the antisymmet-ric D–M exchange in the localized [Fe(II)Fe(III)] for com-parison with the antisymmetric double exchange in thesame delocalized system.

In the localized ½Fe2þa � Fe3þ

b �loc system with directexchange between the different ions, the antisymmetric Dzi-aloshinsky–Moriya exchange interaction originates fromthe combined effect of: 1) the SOC V a

SO admixture of theexcited states (5Ex,y) to the ground state (5A1) of the Fe2þ

a

center and isotropic Heisenberg exchange between these

excited states 5Ex;y ½Fe2þa � and the ground state (6A1) of the

Fe3þb ion; and 2) the SOC V b

SO admixture of the excitedstates 4Ex,y (4T1x,y) to the ground (6A1) state of the Fe3þ

b

ion and isotropic Heisenberg exchange between the excitedstates (4Ex,y(4T1x,y) [Fe3þ

b ]) and the ground state (5A1) ofthe Fe2þ

a ion. This scheme of the excited-ground statesSOC mixing for the Fe2þ

a and Fe3þb ions for the localized

½Fe2þa � Fe3þ

b �loc cluster is represented in Fig. 1a and c,respectively. The exchange interactions between the excitedstates of the Fe2þ

a ðFe3þb Þ ion and the ground state of the

Fe3þb ðFe2þ

a Þ is represented schematically in Fig. 1b. Thesame scheme takes place for the ½Fe3þ

a Fe2þb �loc cluster.

Strong direct Heisenberg exchange couples the Fe2þa and

Fe3þb ions in the ground states 5A1ðdz2Þ½Fe2þ� and

6A1[Fe3+] (Fig. 1b, bottom). In the localized ½Fe2þa Fe3þ

b �loc

cluster, the Fe2þa fFe3þ

b g ion is characterized by the SOCconstant fFe2þðk1ðFe2þ

a ÞÞffFe3þðk2ðFe3þb ÞÞg. In the case of

the direct exchange between the localized Fe2þa and Fe3þ

b

ions, the expression for the D–M parameter (Eq. (4))should be modified in the form

eGexab ¼ 2i

Xm

½ðk1ðFe2þa Þ=emÞ

(� hw0a� ð5EmÞjLajw0

a� ð5A1ÞiFeðIIÞJðð5EmÞam; ð5A1Þb0Þ��X

m0½ðk2ðFe3þ

b Þ=em0 Þ

� hu00bð4T1m0 ÞjLbju0bð6A1ÞiFeðIIIÞJðð6A1Þa0; ð4T1m0 Þbm0 Þ�

�.

ð21Þ

The coefficients (Gab)n of the antisymmetric D–Mexchange (2) may also be found from the exchange matrixelements hUa�bð9=2;MÞjV abjUa�bð7=2;M � 1Þi, calculatedusing the localized cluster wave functions with the SOCaccount and the tilting of the [FeS4] tetrahedra, which wereused in Ref. [1] for consideration of the AS doubleexchange. These matrix elements in the ja*bi localizationwill describe the D–M mixing of the S and S � 1 states.The comparison, for example, of the results of the micro-scopic calculations with the matrix element of the Dzialo-shinsky–Moriya Hamiltonian (2) hUa�bð9=2; 9=2ÞjHDMjUa�bð7=2; 7=2Þi ¼ 3

ffiffiffi5pðiGX þ GY Þ=2 shows that the compo-

nents of the vector of the Dzialoshinsky–Moriya ASexchange (2) have the form

GX ¼#

2ðcxJ I þ �cxJ IIÞ; ð22Þ


GY = 0, GZ = 0. (The parameter GY is proportional to thedifference(Jn � Jg); the exchange parameters Jn and Jg areequal in the cluster ðJ n ¼ 1

5

Pu¼n;g;f;u;vJ nuÞ, that leads to

GY = 0.) The vector of the D–M antisymmetric exchangeis oriented along the X axis (Eq. (22)) in accordance withthe Moriya theory [4,5]. The coefficients

cx ¼ fFe2þ=4Dx; �cx ¼ fFe3þ=�Dx

in Eq. (22) are the parameters of the SOC admixture [1] forthe Fe2+ and Fe3+ ions, Dx ¼ Ex � E0 and �Dx ¼ E0x � E00 inFig. 1, fFe2þ and fFe3þ are the one-electron SOC parametersof the Fe2+ and Fe3+ ions, respectively [46]. According toEq. (21), the exchange parameters in Eq. (22)

J I ¼ 3ðJ u � J nÞ; J II ¼ ½J I þ ðJ f � J nÞ� ð23Þ

describe the Heisenberg exchange interactions I) betweenthe excited state of the Fe2+ ion and the ground stateof the Fe3+ ion, JI (the first term in Eq. (22) and II) be-tween the excited state of the Fe3+ ion and the ground stateof the Fe2+ ion, JII (the second term in Eq. (22)), Fig. 1.

Since we consider the D–M exchange parameters in themodel of direct exchange (Eqs. (21) and (22)) with theexchange parameters JI and JII, we will briefly considerthe Heisenberg exchange in the excited states in the modelof the direct interaction with the account of the non-zerooverlap of the d-functions huajubi5 0 [56–58].

In the MV [Fe(II)Fe(III)] clusters with short Fe–Fe dis-tance (2.51–2.74 A) [10–19,21] and strong double exchangecoupling, which is the result of the direct through-space r-or p-type overlap between d-orbitals, the non-zero overlapof the d-functions huajubi = du 5 0 [56–58] should beincluded in the model of the direct exchange interaction.The overlap in the ground state hU0

a�bð9=2; 9=2ÞjU0ab�

ð9=2; 9=2Þi ¼ �du is characterized by du = hua jubi. In thiscase, the DE wave functions eU0

ðSÞ of the valence-delocal-ized cluster have the form

eU0ðSÞ ¼ ½U0

a�bðSÞ U0ab� ðSÞ�=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2NðSÞ

p;

NðSÞ ¼ 1 AðSÞdu þ fSðd2uÞ. ð24Þ

The normalization factor N±(S) depends on the overlapintegrals huajubi = Sab(u) = du of the d-functions andA(S) = [(S + 1/2)/(2s0 + 1)] is the Anderson–Hasegawa [2]DE spin factor [58]. Thus, for example, the normalizationfactors for eU0

ð9=2; 9=2Þ and eU0ð7=2; 7=2Þ are the

following

NðS ¼ 9=2Þ ¼ 1 du � d2u �

Xu¼n;g;f;v

d2u;

Nð7=2Þ ¼ 1 4

5du � d2

u �11

20

Xu¼n;g;f;v

d2u. ð25Þ

In the model of the direct exchange accounting for thenon-zero overlap of the d-functions [58], the Heisenbergexchange parameter J ¼ J 0

a�b ¼ J 0ab� (H0 = 2J sasb) for the

ground set of the states of the localized [Fe(II)Fe(III)]dimer has the form

J ¼ JðuÞ ¼ 1

4

Xu¼n;g;f;v

½qud2u � ðJu þ

2

5tuduÞ�. ð26Þ

where Ju ¼ 15

Pw¼n;g;f;u;vJðuwÞ, J(wu) = hwaubjVabjubwai is

the standard inter-center exchange integral, qu ¼ Q0=5 ¼Qu þ

PuQu, Q0 the Coulomb integral for the ground set

(Eq. (24) [1]), Qw ¼ 15

PuQðwuÞ and Q(wu) = hwaub

jVabjwaubi. The 25tudu ¼ 2Budu term in Eq. (26) describes

the DE contribution to the Heisenberg parameter J. Thedz2 orbitals do not contribute to the Heisenberg exchangeparameter J (Eq. (26)) in the ground set of levels. This re-sult for the Heisenberg exchange between the consideredtetrahedrally-coordinated Fe(II) and Fe(III) ions with theorbitally non-degenerate ground states 5A1 and 6A1 coin-cides with the result for J of the [Fe2(OH)3(tmtacn)2]2+ cen-ter with the trigonal coordination of the Fe(II) and Fe(III)ions [35]. The origin of antiferromagnetic exchange andsuperexchange pathways in the MV [Fe(II)Fe(III)] clusterwere considered in Ref. [35]. The calculations of the ex-change and DE parameters of the MV [Fe(II)Fe(III)] clus-ter in the broken symmetry DFT model were published inRefs. [7,18–20,32,35–38] of paper 1 [1]. In the model withthe d-overlap, the Heisenberg exchange parameters, whichdetermine the Heisenberg splittings of the excited localizedstates U0a�bðSÞx, have the form

JðnÞ ¼ J ½5Ex� ¼1

4

Xu¼g;f;u;v

½qnd2u � ðJu þ

2

5tuduÞ�;

JðgÞ ¼ J ½5Ey � ¼1

4

Xu¼n;f;u;v

½qgd2u � ðJu þ

2

5tuduÞ�;

JðfÞ ¼ J ½5B1� ¼1

4

Xu¼n;g;u;v

qfd2u � Ju þ

2

5tudu

� �� ;

JðvÞ ¼ J ½5B2� ¼1

4

Xu¼n;g;f;u

qvd2u � Ju þ

2

5tudu

� �� . ð27Þ

All the Heisenberg parameters (27) include the DE contri-bution 2Budu, which may be of the order of 10 cm�1 fordu = 0.01. In the case du = 0 for the model of the direct ex-change, the Heisenberg exchange parameters (27), whichdetermine the Heisenberg splittings of the ground, first, sec-ond, etc. excited states sets are the following:

J ¼ J ½5A1;6A1� ¼ �1=4ðJ n þ J g þ J f þ J vÞ;

J 01 ¼ J ½5B1;6A1� ¼ �1=4ðJ n þ J g þ J u þ J vÞ;

J 02 ¼ J ½5Ex;6A1� ¼ �1=4ðJ g þ J f þ J u þ J vÞ; ð28Þ

J 03 ¼ J ½5Ey ;6A1� ¼ �1=4ðJ n þ J f þ J u þ J vÞ;

J 04 ¼ J ½5B2;6A1� ¼ �1=4ðJ n þ J g þ J u þ J vÞ.

Using Eq. (28), we estimate that the parameters JI and JII

(Eq. (23)) of the exchange between the ground and excitedstates are proportional to the differences between the ex-change in the ground state and exchange in the excitedstates

J I ¼ 12ðJ � J 02Þ; J II ¼ ½J I þ 4ðJ 01 � J 02Þ�. ð29Þ


In the considered model with antiferromagnetic Heisen-berg exchange, J > 0 for H0 = 2JSaSb, (Fig. 1 [1]), we sup-pose J > J 01; J

02; . . . that leads to JI, JII > 0 and GX > 0 (Eq.

(23)) in the case of the direct exchange.In the case of superexchange interaction in the metal

dimer, the D–M vector coefficient ~Gab ¼ 2iðJ=tÞ~Cab wasdetermined by Moriya in Refs. [4,5] by the vector transferintegrals ~Cab for the ½dn

a � dnb� pair

~Cab ¼ �k=2X

m0ð~L�am00=em0 Þtam0;b0 þ

Xm

ð~Lbm0=emÞta0;bm

" #;

ð30Þ~Lbm0 is the matrix element of the orbital operator ~Lb be-tween the mth excited state and ground (0) state of thebth ion, em the energy of the mth 3d orbital and tam,b0 thetransfer integral between the mth excited state of the athcenter and ground (0) state of the bth center [4,5]. Eq.(30) should be modified for the localized cluster with differ-ent SOC constants k1ðFe2þ

a Þ (fFe2þ) and k2ðFe3þb ÞðfFe3þÞ and

different LF intervals em and e0m0 for the Fe2þa and Fe3þ

b ions,

~Cab ¼ �1=2X

m

ðk1ðFe2þa Þ~L�am0=emÞtam;b0

"

þX

m0ðk2ðFe3þ

b Þ~Lbm00=e0m0 Þta0;bm0

#; ð31Þ

where the first term includes the ET integrals between theexcited states (5Ex,y) of the Fe2þ

a ion and ground state(6A1) of the Fe3þ

b ion (Fig. 1a) and the second term includesthe ET integrals between the excited states (4T1x,y) of theFe3þ

b ion and the ground (5A1) state of the Fe2þa ion

(Fig. 1c). These ET integrals were considered in Part 3 ofRef. [1]: tamðnÞ;b0ðuÞ ¼ hnakubi ¼ #

ffiffiffi3pðtu þ tnÞ; tamðgÞ;b0ðuÞ ¼

< gakub >¼ 0, ta0,bm0 = 0 (eqs. (38) and (28) [1]). FromEq. (31) with these ET integrals one obtains in the caseof the superexchange in the [Fe(II)Fe(III)]loc cluster, thatJ in the Moriya equation ð~GabÞX ¼ 2iðJ=tÞð~CabÞX [4,5] rep-resents the superexchange eJ I between the excited state ofthe Fe2þ

a ion and the ground state of the Fe3þb ion and

t = tu is the virtual transfer in the ground state. The rela-tions ð~GabÞX ¼ 2iðJ=tuÞð~CabÞX [5] and iKX ¼ ðC�abÞX (Eq.(46) in Ref. [1,54]) lead to the correlation

GX ¼ �2ðJ=tuÞKX ¼ �2=5ðJ=BÞKX ; ð32Þ

where KX is determined by Eq. (33), GY = 0, GZ = 0,tu = B/5 for the valence-delocalized [Fe2.5+ Fe2.5+] cluster.The signs of GX is opposite to the sign of KX, Eq. (32).The D–M exchange coefficient GX (3) is proportional toJDg?#, in accordance with the Moriya theory [4,5]. Therelation jKXj/GX � J/tu between the AS double exchangeand D–M exchange was obtained in Ref. [50]. The micro-scopic calculations of the D–M exchange parameter ~Gab

(Eqs. (22) and (32)) for the MV [Fe(II)Fe(III)] cluster con-firm the suggestion [33] that the D–M exchange is charac-terized by the GX-component (GX = 7.2 cm�1 [33]) in the

localized antiferromagnetic [Fe(II)Fe(III)]loc cluster withSgr = (1/2)loc.

Eq. (32) KX = �(5B/2J)GX shows that if the MV clusterpossesses of the conditions of the non-zero D–M exchangeinteraction (Dg?# 5 0,GX 5 0), than the antisymmetricdouble exchange is active (KX 5 0) in the valence-delocal-ized system with B 5 0.

Let us compare the parameter

KX ¼ �10BDg?#ð1þ tn=tuÞ ð33Þ

of the AS double exchange H1X (1) in the valence-delocal-ized [Fe2.5+Fe2.5+]deloc cluster (Eq. (41) [1]) with the coeffi-cient GX (Eqs. (22) and (23)) of the D–M exchangeHX

DM ¼ GX ½~Sa �~Sb�X in the localized [Fe(II)Fe(III)] cluster.KX is proportional to the DE parameter B and dependsonly on the parameter cx ¼ ðfFe2þ=4DÞ. The D–M exchangeparameter GX (22) of the localized [Fe(II)Fe(III)] clusterwith direct exchange depends on both the cx(Fe2+) and�cxðFe3þÞ parameters. GX is proportional to the parametersJI and JII, GX � ðJ Icx þ J II�cxÞ#. Since KX � Bcx# andB� J, JI, JII in the delocalized cluster [Fe2.5+Fe2.5+]deloc,we can conclude that the AS double exchange coupling isstronger than the Dzialoshinsky–Moriya antisymmetricexchange interaction, jKXj � jGXj. The D–M antisymmet-ric exchange operator H X

DM (5) acts between the states ofthe same localization. The operator of the AS doubleexchange H1X (1) acts between the states of differentlocalizations.

In the case of the superexchange, one obtains from Eqs.(32), (33) the following correlation:

GX ¼ 4JDg?#ð1þ tn=tuÞ ð34Þand jKXj � jGXj since KX/GX = �5B/2J ðcx ¼ fFe2þ=4Dx ¼�kðFe2þÞ=Dx ¼ Dg?=6Þ. Comparison of the value of theexperimentally observed D–M exchange parameterjGabj � 7.2 cm�1 [33] (for the localized AF [Fe(II)Fe(III)]cluster) and estimated antisymmetric double exchangeparameter (jKXj � 60–100 cm�1 [1]) for the valence-delo-calized [Fe2.5+Fe2.5+]deloc cluster confirms the conclusionjKXj � jGXj.

For the valence-localized AF [Fe2+Fe3+]loc cluster(Sgr = 1/2) of the [Fe2S2]+ active site of ferredoxins, anexperimental value J AF

eff ¼ 105 cm�1 have been reported[59,60], D3/2�1/2 = 315 cm�1 = 3Jeff, H0 = 2JSaSb, JAF > 0.Using in Eq. (34) this Heisenberg exchange parameterJ AF

eff and Dg? = 0.08, # = 0.1, one obtains the estimate ofthe antisymmetric D–M exchange constant GX(loc) ffi3.4 cm�1 for the localized [Fe2+Fe3+]loc cluster withSgr = (1/2)loc. In the case of the valence-delocalized sys-tems, as was shown in Refs. [61,62], the double exchangeinteraction and vibronic coupling reduce essentially theAF Heisenberg exchange parameter J eff ¼ J AF

0 �B2=ðk2=kÞj [61], where k is the vibronic constant and k

the force constant. This DE reduction of the Heisenbergexchange results in the effective antiferromagneticHeisenberg exchange parameter eJ eff ¼ 70 cm�1 [35] forthe valence-delocalized (Sgr = (9/2)deloc) cluster [Fe2.5+


Fe2.5+]deloc with strong double exchange B = 1350 cm�1

[34–36]. Using the effective parameter eJ eff ¼ 70 cm�1 oneobtains from Eq. (34) the estimation of the D–M exchangeparameter GX in the valence-delocalized [Fe2.5+Fe2.5+]deloc

cluster GX(deloc) ffi 2.3 cm�1. The DE reduction of JAF

[61,62,35] leads to reduction of the constant of the antisym-metric Dzialoshinsky–Moriya exchange GX in the valence-delocalized system in comparison with the localized cluster.

One can note that the estimated value of the D–Mexchange parameter for the localized cluster GX(loc) ffi3.4 cm�1 (34), Jeff = 105 cm�1 is of the same order of thevalue as the experimentally observed D–M parameterjGabj � 7.2 cm�1 for the localized [Fe(II)Fe(III)] clusterwith Sgr = 1/2 (J = 20 cm�1, D3/2�1/2 = 60 cm�1 = 3J [33],H0 = 2JSaSb). The relations G/J are: GX(loc)/Jeff = 0.032(Eq. (34)) and jGabj/J = 0.36 [33]. The standard Moriya[4] estimate G/J = Dg/g gives G/J = 0.04 for Dgx(Fe2+) =0.08 [63].

5. Spin canting in the ground state of the valence-delocalized

cluster

The D–M antisymmetric exchange leads to spin cantingin magnetic materials (Erdos [6]). Let’s consider the spincanting induced by the antisymmetric double exchangeand D–M exchange in the [Fe(II)Fe(III)] clusters. In thespin coupling model of these clusters (Refs. [8,9] in [1]),the expectation values of the local spins hsii determinethe magnetic hyperfine (HF) constants Ai = aihsii/hSZi

Fig. 2. Spin canting for the ground jSgr = 9/2, M = 9/2i state of the[Fe(II)Fe(III)] cluster. (a) The expectation values hsaZi and hsbZi for theferromagnetic localized [Fe(II)Fe(III)]loc cluster with the ground statejSgr = (9/2)loc, M = 9/2i, (b) deviations of spins from the parallel(ferromagnetic) alignment for ferromagnetic localized cluster. The non-zero expectation values hsiYiDM along Y direction (Eq. (35)) are inducedby the Dzialoshinsky–Moriya exchange coupling HX

DM, u1 > u2, (c)deviations of spins from the ferromagnetic alignment for the grounddelocalized jSgr = (9/2)deloc9/2i state of the valence-delocalized[Fe2.5+Fe2.5+] cluster. The non-zero expectation values hs0iY i are inducedby the antisymmetric double exchange and D–M exchange (Eq. (38)),u > u1, u2.

and hyperfine (internal) fields on the 57Fe nuclei HiZ ¼�ðai=gnbnÞhsii ¼ h0

i hsii, where a1 = a0 (Fe2+), a2 = a0

(Fe3+) are intrinsic HF constants of the individual ions,gn and bn are the nuclear g-factor and Bohr magneton.The intrinsic HF constants of the [FeS4] centers of ferrousand ferric rubredoxins correspond to the HF fields16T < �a0ðFe2þÞ=gnbn ¼ h0

1 < 20T and �a0ðFe3þÞ=gnbn ¼h0

2 ¼ 22T [41,44,64,65].1) Lets consider first, for comparison, the localized fer-

romagnetic [Fe(II)Fe(III)]loc cluster [41,42,44,66] with theground state Sgr = (9/2)loc, in this case hsaZi = 2,hsbZi = 5/2 (Fig. 2a) for M = 9/2 (D9/2 < 0); A1 = 4a1/9,A2 = 5a2/9 [41,42]; HaZ ¼ 2h0

1, H bZ ¼ 5=2h0b; g9/2 =

4/9g1 + 5/9g2 [44]. The D–M antisymmetric exchangeHX

DM (5) admixes to the ground j(9/2)loc,9/2i state thefirst excited j7/2,7/2i state separated by the interval9jJj: Ua�bð9=2; 9=2Þ ¼ U0

a�bð9=2; 9=2Þ þ ð3iffiffiffi5p

=2ÞðGX=9jJ jÞU0

a�bð7=2; 7=2Þ (Table 1). This admixture leads to non-zerospin expectation values of the local spins hsiY iDM along Y

direction for the localized ferromagnetic ground j9/2,±9/2iloc state

hsaY iDM ¼ �hsbY iDM ¼ 5GX=9jJ j; ð35ÞhsaXiDM = hsbXiDM = 0, where hsiY iDM ¼ hUa�bðS;MÞj siY j Ua�bðS;MÞi. This result is in accordance with the ac-tion of the D–M operator HX

DM ¼ GX ½~Sa �~Sb�X (5), whichtends to orient the spins perpendicular to each other andð~GabÞX (spin canting) [3–6]. The non-zero hsaYiDM =�hsbYiDM values (35) are shown schematically in Fig. 2b,hSYi = hsaYi + hsbYi = 0. The values hsiY iDM (35) (Fig. 2c)correspond to the Y-components of the HF fields on nuclei

HDMaY ¼ �5h0

1ðGX=9jJ jÞ � �11ðGX=jJ jÞT ;HDM

bY ¼ 5h02ðGX=9jJ jÞ � 12ðGX=jJ jÞT

for the j9/2,9/2iloc state, H DMaY =HaZ ¼ �5GX=18jJ j,

HDMbY =H bZ ¼ 2GX=9jJ j. The HF fields H DM

iY depend on thevalue and sign of GX. The resulting HF fields Hi ¼ ½H 2

iZþðH DM

iY Þ2�1=2 are oriented along the hsai and hsbi directions

in Fig. 2b. The non-zero values hsaYiDM = �hsbYiDM (inter-nal fields H DM

aY and HDMbY ) results in the deviation of the ori-

entations of spins ~sa and ~sb from the ferromagnetic(parallel) alignment (spin canting [6]) Fig. 2b. These devia-tions for the localized ferromagnetic ground statej(9/2)loc,9/2i are determined by the ratios hsiYiDM/hsiZi:

tgðuDM1 Þ ¼ hsaY i=hsaZi ¼ �5=18GX=jJ j;

tgðuDM2 Þ ¼ hsbY i=hsbZi ¼ 2=9GX=jJ j. ð36Þ

The angles uDM1 and uDM

2 of the canting are different forthe Fe(II) and Fe(III) ions in the ferromagnetic localized[Fe(II)Fe(III)]loc cluster (Fig. 2b). For jS = 9/2, M = �9/2i,the orientations of all hsaNi are opposite in Fig. 2b.

Using for these ferromagnetic (Sgr = (9/2)loc) clusters,for estimation, the relation GX/j2Jj = 0.18, which wasobtained experimentally [33] for the AF [Fe(II)Fe(III)]loc

cluster, one obtains hsaYiDM = �hsbYiDM = �0.2 (Eq.(35)). This results in different HF fields H DM

aY ffi �4T ,


H DMbY ffi 4:4T for the jS = 9/2, M = 9/2i state (HaZ ffi 40T,

HbZ ffi 55T). Fig. 2 is not drawn to scale; the canting anglesin Fig. 2b are uDM

1 ¼ �5:8�;uDM2 ¼ 4:6� for the set of

parameters used for estimation. In the case of the estima-tion GX/jJj = 0.1 in Eq. (34) (GX(loc) = 7 cm�1, jJj =70 cm�1), one obtains h~saY iDM ¼ �h~sbY iDM ¼ �0:056,~uDM

1 ¼ �1:6�; ~uDM2 ¼ 1:3�; eH DM

aY ¼ �1:1T , eH DMbY ¼ 1:2T .

For the j9/2, Mi Zeeman levels (hsaZi = 4M/9,hsaZi = 5M/9) of the ground S = 9/2 manifold one obtainshsaYiDM = �hsbYiDM = �10GXM/81jJj, in particular,hsaYiDM = �hsbYiDM = 5GX/81jJj for the lowest jS =9/2, M = ±1/2i Kramers doublet in the case D9/2i0. Theangles uDM

1 , uDM2 are determined by Eq. (36). The deviation

of spins from the parallel orientation in the localized MVcluster is induced by D–M coupling and determined bythe (GX/J) relation [6].

For comparison, in ferromagnetic mono-valent cluster[Fe2(III)] (Sgr = 5, sa = sb = 5/2), the D–M exchangeðHX

DMÞ mixing of the ground Sgr = 5 and first excitedS = 4 states yields to non-zero values hsiYi along Y-direc-tion, hsaYiDM = �hsbYiDM = 5GX/8jJj for M = ±5,(D < 0,hsaZi = hsbZi = 5/2, hsiXi = 0). (Compare withhsiYi = 5GX/9jJj for the MV cluster, Eq. (35).) The anglesof the deviation from the ferromagnetic alignment are thesame (tg�u1 ¼ �tg�u2 ¼ �GX=4jJ j for M = 5). Using forestimation, for example, the relation GX/2jJj = 0.11, whichwas obtained for the AF [Fe2(III)] cluster (GX = 2.2 cm�1,2jJj = 20 cm�1 for H0 = 2JSaSb [9]), one obtains ��u1 ¼�u2 ¼ �u ¼ 3:20, hsaYiDM = �hsbYiDM = -0.1375, H DM

aY ¼�HDM

bY � �3T for M = 5.2) In the valence delocalized [Fe2.5+Fe2.5+] cluster with

Sgr = (9/2)deloc, the HF fields are the same H 0aZ ¼ H 0bZ sincehs0aZi ¼ hs0bZi ¼ 9=4 (Fig. 2c). (The Mossbauer experiments[34,37,41,66] for valence-delocalized MV [Fe2.5+Fe2.5+]clusters indicate that the two iron sites contribute identi-cally to the spectra. The average HF constant for the delo-calized ground Sgr = (9/2)deloc state has the form A(9/2) =[(4/9)a1 + (5/9)a2]/2 [41], hdeloc

i ¼ �hi ¼ ðh01 þ h0

2Þ=2 is theaverage h0

i .) In the valence-delocalized [Fe2.5+Fe2.5+] clus-ter with the ground state U0

�ð9=2Þ, the admixture of theexcited DE states U0

�ð7=2; 7=2Þ and U0þð9=2; 7=2Þ (in the

first order of the perturbation theory) to the ground delo-calized U0

�ð9=2; 9=2Þ DE state (Eq. (19)) has the form

U�ð9=2; 9=2Þ ¼ U0�ð9=2; 9=2Þ � ð2i=3

ffiffiffi5pÞ

� ½ðKX � 45GX=4Þ=ðB� 9JÞ�U0�ð7=2; 7=2Þ

� i=3½KX=10B�U0þð9=2; 7=2Þ; ð37Þ

Using the antisymmetric admixture (37) and relationshsaYi = �hsbYi,

hU0�ð9=2; 9=2ÞjsaY jU0

�ð7=2; 7=2Þi

¼ iffiffiffi5p

=3; hU0�ð9=2; 9=2ÞjsaY jU0

þð9=2; 7=2Þi ¼ i=12

for the individual spin operators, we obtain the non-zeroexpectation values of the local spins hs0iY i along Y directionfor the delocalized ground state U�(9/2,9/2)

hs0aY i ¼ �hs0bY i ¼1

9

4KX � 45GX

B� 9Jþ KX

20B

� �; ð38Þ

hs0aX i ¼ hs0bX i ¼ 0. The sign in Eq. (38) is opposite forU�(9/2, � 9/2). In the localized case (B = 0, KX = 0), Eq.(38) leads to jhsaYij = 5GX/9J,hsbYi = �hsaYi (see Eq.(35)). In the simplified limiting case B� J,KX� GX

(J = 0,GX = 0, pure DE model), one obtains

hs0aY i ¼ �hs0bY i ¼ 9KX=20B. ð39ÞAs follows from Eqs. (38) and (39), in the valence-delocal-ized cluster, the antisymmetric double exchange H1X (1) re-sults in non-zero expectation values of the local spinshj s0aY ji ¼ �hj s0bY ji along the Y-direction. H1X (1) leadsto the deviation of spins from the ferromagnetic alignmentin the plane (YZ) perpendicular to the direction of the vec-tor of the AS double exchange (KX). We see that differentoperators: antisymmetric double exchange H1X (1) andD–M exchange H X

DM (5) [with different spin partsðbS bX � bS aX ÞbT ab (1) and ½~Sa �~Sb�X (5), respectively] lead tospin canting (38) in the plane YZ. Indeed, Eqs. (6), (7),(9) show that the matrix elements of both the operatorsH1X (1) and HX

DM (5) include the same spin factor F±

(Eq.(7)) and, as a result, both the coefficients KX and GX

form the effective AS parameter PX ðSÞ (Eqs. (16) and(18)) for the mixing of the DE states with different S

(Eqs. (19) and (20)).The first and second terms in Eq.(38) describe the contribution to hs0iY i of the: 1) mixing ofthe excited U0

�ð7=2; 7=2Þ and the ground U0�ð9=2; 9=2ÞDE

states by the antisymmetric double exchange and D–Mexchange; and 2) U0

þð9=2; 9=2Þexc � U0�ð9=2; 9=2Þgr mixing

only by the AS double exchange.Since KX and GX have opposite sign (Eqs. (22), (32), and

(33)), both the AS double exchange and D–M exchangeenlarge the value of hs0iY i for the ground valence-delocalizedstate (Eq. (38)). Using in Eq. (38), for example, the corre-lation between KX and GX in the superexchange case (Eq.(32)), one obtains the HF fields ðH 0aY ¼ �H 0bY Þ

H 0aY ffi ð�hiKX=180BÞ½80ð1þ 4:5J=BÞ=ð1� 9J=BÞ þ 1�. ð40Þ

The non-zero expectation values hs0aY i ¼ �hs0bY i (38),which are induced by the AS double exchange and D–Mexchange, determine the spin canting (Fig. 2c) in theground valence-delocalized U0

�ð9=2; 9=2Þ state. The devia-tions of the local spins from the parallel (ferromagnetic)alignment (tgua ¼ �tgub ¼ 4hs0aY i=9, Eq. (38)) for theground valence-delocalized U�(9/2,9/2) state is shown inFig. 2c. In the valence delocalized cluster, the deviationsof spins from the parallel alignment depend on the param-eters of the AS double exchange (KX), D–M exchange (GX),double exchange (B) and Heisenberg exchange (J). Thecoefficient KX is determined by Eq. (33), the coefficientGX is determined by Eq. (22) {(32)} in the case of the directexchange {superexchange}.

Using in Eq. (38) the experimental values (B � 9J) =720 cm�1 [34–36], B = 1350 cm�1, jJj = 70 cm�1 [34–36],estimated value KX = �100 cm�1 and GX/2jJj = 0.18 (for


AF cluster 2jJj = 40 cm�1 [33], H = 2JSaSb).), we can esti-mate hs0aY i ¼ �hs0bY i ¼-0.24 (Eq. (38)) for the ground delo-calized U�(9/2,9/2) state in comparison with hsaYiDM =�hsbYiDM = �0.2 (Eq. (35)) for the localized ferromagneticj(9/2)loc,9/2i state with the same relation GX/2jJj = 0.18. Inthis case, the same angles �ua = ub = u of the spin cantingfor j(9/2)deloc,9/2i of the valence-delocalized cluster are lar-ger than those in the case of the D–M exchange spin devi-ation u ¼ 6:1� > �uDM

1 ¼ 5:8�, uDM2 ¼ 4:6�, Fig. 2c) for

the ferromagnetic localized cluster with j(9/2)loc,9/2i (forthe set of parameters used for estimation). In calculationwith the DE reduced GX(deloc) parameter GX(deloc) =4.8 cm�1 (and the same B, J, KX) for the valence-delocal-ized cluster, one obtains the estimate hs0aY i ¼ �hs0bY i ¼�0:1 ðH 0aY ¼ �H 0bY ¼� �2:1T Þ {u 0 = 2.5�} which are alsolarger than the corresponding values h~saY iDM¼�h~sbY iDM¼�0:056 ð eH DM

aY ¼ �1:1T , eH DMbY ¼ 1:2T Þf~uDM

1 ¼ �1:6�;~uDM

2 ¼ 1:3�}, that are obtained for j(9/2)loc,9/2i of the fer-romagnetic cluster with the D–M exchange parameterGX(loc) = 7 cm�1, jJj = 70 cm�1, GX(loc)/jJj = 0.1.

The non-zero expectation values hsaYi = �hsbYi (38) ofspins along the Y-axis induce the Y-components of theHF on nuclei H 0aY ¼ �H 0bY ¼ �hihsaY i (Eq. (38)) in groundstate U-(9/2,9/2) of the valence-delocalized [Fe2.5+Fe2.5+]cluster, that may be found in the Mossbauer experiment.With the parameters �hi ¼ 21T ðH 0aZ ¼ H 0bZ ¼ 9�hi=4 � 47T Þand hsaYi = �0.24 one can estimate the induced hyperfinefields along Y-direction H 0aY ¼ �H 0bY ¼ �hihsiY i � �4:8T .The resulting HF fields j Ha j¼j Hb j¼ ½H 02aZ þ H 02aY �

1=2 ¼j H 0aZ j ½1þ ð4=9hsaY iÞ2�1=2 are oriented along the hs0ai andhs0bi directions in Fig. 2c (angle �u1 = u2 = u). In the caseof the parallel applied magnetic field HZ, the resulting HFfield has the form H a ¼ H b ¼ ½ðHZ þ H 0aZÞ

2 þ H 02aY �1=2,

tgu ¼ H 0aY =ðH Z þ H 0aZÞ.The canting of spins depends on S, M and parity of the

double exchange states. Thus, for example, for the U+(9/2,9/2) DE state, the expectation values of the local spinsalong Y-axis have the form

hs00aY i ¼ �hs00bY i ¼ �1

9

4KX þ 45GX

Bþ 9Jþ KX

20B

� �; ð41Þ

The AS mixing of the DE levels results in the spin cant-ing (Eq. (38)) and anisotropy of the HF fields on nuclei(HiX 5 HiY) and magnetic HF constants (AiX 5 AiY). Thisanisotropy may be found in the MV [Fe(II)Fe(II)] clusters.

In the cases of the investigated clusters, there is noAX,AY anisotropy (Ax,y = �10.6T,Az = �13.5T) in thevalence delocalized [Fe2.5+Fe2.5+] cluster (Sgr = (9/2)deloc)[34] with strong DE (B = 1350 cm�1). This cluster pos-sesses the high symmetry of the MV pair of the trigonallycoordinated Fe ions (D9/2 = 1.08 cm�1, E9/2 = 0.00(1)cm�1, gx = gy = gz = 2.00 [36]). Two different axial HFparameters [Ax,y = �12.7 MHz, Az = �8.9 MHz andAx,y = �13.3 MHz,Az = �9.3 MHz] were observed for thetwo Fe2.5+ ions of the delocalized [Fe2.5+Fe2.5+] MV pairof the ferredoxin [2Fe � 2S]+ cluster with D9/2 =�1.6 cm�1, E/D = 0.115, gx = gy = gz = 2.03 [41]. The

absence of the anisotropy of the HF constants, which isinduced by the antisymmetric mixing of states, may be con-nected in these symmetrical cases with the fact that boththe parameters of the AS double exchange KX (Eq. (33))and D–M exchange GX (Eqs. (22) and (32)) are propor-tional to the parameter Dg?#. In the symmetrical case(# = 0), the parameters KX and GX are equal to zero. Theisotropy of the g-factors (Dg = 0) of these systems is alsoimportant since KX,GX � Dgx.

Anisotropy of the AX,AY components of the HF con-stants was observed in Ref. [37] for the valence-delocalizedground state Sgr = (9/2)deloc of the MV [Fe2.5+Fe2.5+] clus-ter (A/gnbn = �(11.1,11.6,16.0)T; HiY 5 HiX). The originof this anisotropy is unknown and may be connected withthe intrinsic anisotropy of ai, however the valence-delocal-ized clusters with strong DE (mentioned above [34,41]) arecharacterized by the axial HF constants Ax = Ay. If we willsuppose that the anisotropy DHiY = HiY � HiX = �0.5T isconnected with the antisymmetric anisotropy considered inthe paper, we can use the Eq. (40) to estimate the relationKX/B in this cluster. Since the values of B, J and GX are notknown for this system [37], one can use the limiting caseB� J, J = 0 in Eq. (40) for the rough estimate: the equa-tion H 0aY � 9�hiKX=20B ¼ 9:5ðKX=BÞT ¼ DH iY results inthe relation KX/B � �0.05. This rough estimate is in linewith the theoretical estimation (�KX/B > 0.07) and leadsto large negative values of KX since the DE parameter Bis usually large (B > 1300 cm�1).

Since both the coefficients of the AS double exchange KX

and D–M exchange GX are proportional to Dgx and the tilt-ing angle #, it is important that the D–M exchange wasfound in the dimeric AF [Fe(II)Fe(III)], [Fe(III)Fe(III)],clusters and trimeric metal centers [8–32]. Strong D–Mexchange (28 hjG(cm�1)j 6 47) was found recently in tri-meric clusters [20–23] where this interaction leads to linearsplitting of the spin-frustrated levels [10,14,15]. As shownin Ref. [51a], in the valence-delocalized MV trimers anti-symmetric double exchange results in the linear splittingof the degenerate ground state DE levels. One can supposethat the manifestation of the AS double exchange will beessential in trimeric and tetrameric valence-delocalizedclusters and double exchange magnets.

6. Spin canting in the localized ground state Sgr = 1/2 of

antiferromagnetic [Fe(II)Fe(III)] cluster

Since the D–M exchange was found experimentally in theantiferromagnetic (Sgr = (1/2)loc) localized [Fe(II)Fe(III)]loc

cluster [33], we will consider in this part the spin canting inthe Sgr = (1/2)loc state induced by the D–M exchange. Asshown in Ref. [33], the D–M admixture of the first excitedS = (3/2)loc state to the ground Sgr = (1/2)loc state (Fig. 1a[1]) determines the anisotropy of the gZ-component of theg-factor of the ground state of this cluster.

For the pure AF localized ground state Sgr = (1/2)loc,M = 1/2 the spin-coupling model leads to hsaZi = �2/3,hsbZi = 7/6 (Fig. 3a), A1 = �4/3a1, A2 = 7/3a2 [41];

Fig. 3. Spin canting for the ground jSgr = (1/2)loc, M=1/2i state of theantiferromagnetic [Fe(II)Fe(III)]loc cluster. (a) The standard hsaZi andhsbZi values for the antiferromagnetic cluster, (b) deviations of spins fromthe antiparallel (antiferromagnetic) alignment for the AF cluster. The non-zero expectation values hsiYiDM of the local spins along Y-direction (Eq.(42)) are induced by the Dzialoshinsky–Moriya exchange coupling HX

DM,u1 > u2.


H 0a ¼ �2=3h0

1, H 0b ¼ 7=6h0

2; g1/2 = �4/3g1 + 7/3g2. In theAF [Fea(II)Feb(III)]loc cluster (Fig. 1a [1]), the D–Mexchange H X

DM (5) admixes (Eqs. (6) and (7)) to the groundlocalized j1/2,1/2i state the excited states j3/2,3/2i and j3/2,�1/2i, separated by the interval 3J, J > 0, Ua�bð1=2; 1=2Þ ¼U0

a�bð1=2; 1=2Þ � ðiffiffiffiffiffi14p

=4ÞðGX=3JÞ½ffiffiffi3p

U0a�bð3=2; 3=2Þ � U0

a�b

ð3=2;�1=2Þ�. The D–M exchange results in non-zero expec-tation values of the local spins hsiY iDM along Y direction forthe AF ground state Ua�bð1=2;1=2Þ:hsaY iDM ¼ �hsbY iDM ¼ 7=9ðGX=JÞ; ð42ÞhsaXiDM = �hsbXiDM = 0. The deviations of spins from theantiparallel (antiferromagnetic) alignment (Fig. 3b) in-duced by the D–M exchange have the form (for Ua�bð1=2;1=2Þ):tgðuDM

1 Þ ¼ hsaY i=hsaZi ¼ 7=6ðGX=JÞ;tgðuDM

2 Þ ¼ hsbY i=hsbZi ¼ 2=3ðGX=JÞ. ð43ÞUsing the experimental value GX/2J = 0.18 [33] (2J =40 cm�1, H0 = 2JSaSb), one obtains hsaYiDM =�hsbYiDM =�0.28 (Eq. (42)) and the estimation u1 = 22.8�,u2 = 13.5�(Eq. (43)) for the angles of the D–M canting of the spinssa and sb (Fig. 3b). The Y-components hsaYiDM =�hsbYiDM

correspond to H DMaY ¼ �7=9h0

1ðGX=JÞ � �15:5 ðGX=JÞT ,H DM

bY ¼ 7=9h02ðGX=JÞ � 17ðGX=JÞT ; and ADM

aY ¼ 7=6a1

ðGX=JÞ, ADMbY ¼ 2=3a2 ðGX=JÞ. The calculation predicts

the HF fields H DMaY ffi �5:6T , H DM

bY ffi 6:2T along Y-direc-tion for GX/2J = 0.18, HaZ ffi �13.3T, HbZ ffi 25.7T. Theresulting HF fields H i ¼ ½H 2

iZ þ ðH DMaY Þ

2�1=2 (j Ha j¼2=3h0

1½1þ ð7=6GX= j J j Þ2�1=2 �14.4T, j H b j¼ 7=6h01½1þ

ð2=3GX= j J j Þ2�1=2 � 26.4T) are oriented along the hsaiand hsbi directions in Fig. 3b. The local Z axes are deter-mined by the directions of the Z-components of the localZFS tensors D1 and D2.

For comparison, the mixing of the ground stateU0

a�bð1=2; 1=2Þ and excited U0a�bð3=2; 1=2Þ state by the ZFS

operator eH 0ZFS (Eq. (13)) [60] leads to the shift of the

hsiZiexpectation values (hsaZi � �2/3 + (3D1 + 8D2)/J,hsbZi � 7/6 � (3D1 + 8D2)/J) from the standard hsaZi =�2/3, hsbZi = 7/6 set for the j1/2,1/2i state.

As was shown in Ref. [9] for the monovalent antiferro-magnetic ½Fe3þ

2 � cluster (Sgr = 0, sa = sb = 5/2) in syntheticcomplex and diferric methan monooxidase, the D–Mexchange leads to non-zero expectation values hsiYi (Eq.(8) in [9]):

hsaY i ¼ �hsbY i ¼ �35GX ðg0bH ZÞ

6½J 2 � ðg0bH ZÞ2�and HF fields HiY induced in the Y direction by theparallel applied magnetic field H = HZ and D–M exchange.This effect of the non-zero HF fields H int

aY ¼ �H intbY ¼

�hsaY ia2ðFe3þÞ=gnbn on the Fe nuclei in the ? (Y) directionin the ground singlet state, which were induced by the D–Mexchange (GX = 2.2 cm�1, 2J = 20 cm�1 for H0 = 2JSaSb

[9]) and external magnetic field HZkZ, was observed exper-imentally (Mossbauer spectra, v(T)) by Munck andco-workers in Ref. [9].

We will consider here briefly the effect of the externalparallel field HZ for the ground states of the MV [Fe(II)-Fe(III)] clusters.

1) For the lowest Zeeman level jSgr = 1/2, �1/2i(hsaZi = 2/3, hsbZi = �7/6) of the AF cluster [Fe(II)Fe(III)]ðg0bH Z < 3 j J j þeDÞ, the Hamiltonian matrix of the Hei-senberg exchange, ZFS in the excited state ( eH 0

ZFS, Eqs.(12) and (13)), antisymmetric D–M exchange (H X

DM (5))and Zeeman interaction (Hh = g0b HZSZ) has the form

U0a�bð1=2;�1=2Þ U0

a�bð3=2;�3=2Þ U0a�bð3=2;1=2Þ

�g0bHZ=2 iGX

ffiffiffiffiffi42p

=4 �iGX

ffiffiffiffiffi14p

=4

3J þ eD � 3g0bHZ=2 0

c.c. 3J � eDþ g0bH Z=2

ð44Þ

where eD ¼ D3=2 ¼ �1:00D1ðFe2þÞ � 0:27D2ðFe3þÞ is theaxial ZFS parameter of the S = 3/2 state [34]. The D–M ex-change terms (�GX) in Eq. (44) describe the mixing of theground state j1/2, � 1/2i and first excited j3/2,Mi state. Inthis case, the expectation values of spins hsiYi in Y-directionfor the lowest Zeeman level are described by equation

hsaY ðHZÞiDM ¼ �hsbY ðH ZÞiDM

¼ 7GX ½3J þ 2ðg0bHZ � eDÞ�3½9J 2 � ðg0bH Z � eDÞ2� . ð45Þ

In the limiting case 3J � ðg0bHZ � eDÞ, one obtains

hsaY ðHZÞiDM ¼ �hsbY ðH ZÞiDM

ffi ð7GX=9JÞ½1þ ðg0bH Z � eDÞ=3J �2. ð46Þ

For the individual ZFS parameter D1(Fe2+)i0(5 < D1(cm�1) < 10), the positive �eD > D1ðFe2þÞ term inEqs. (45) and (46) and magnetic field HZ enlarge the


expectation values hsiY(HZ)iDM . The dependence of theperpendicular HF fields HDM

iY ¼ h0i hsiY ðHZÞiDM on the par-

allel applied magnetic field HZ for the level j1/2, � 1/2imay be represented in the form:

HDMiY ðHZÞ ffi H iY þ 2H 0

iY ðg0bH Z=3JÞ; ð47Þwhere the HF fields in the zero applied field (HZ = 0) are

HiY ¼ H 0iY ð1� 2eD=3JÞ; H 0

aY ¼ h01ð7GX=9JÞ;

H 0bY ¼ �h0

2ð7GX=9JÞ. ð48Þ

Eq. (47) shows the linear increase of the perpendicularinternal HF fields j HDM

iY ðH ZÞ j (in the H 0iY units (48)) under

the increase of the applied parallel magnetic field HZ. Eqs.(45) and (48) show the influence of the value and sign of theZFS in the excited S = 3/2 state on the HF fields in theground S = 1/2 state. The ZFS in the excited stateð�eD ¼ 10 cm�1Þ results in essential increase (decrease)of H iY (48) since the exchange parameter may by not large(2jJj = 40 cm�1 [33] yields �2eD=3J ¼ 0:33 in Eq. (48)).The estimates of the D–M hyperfine fields result inj H 0

iY j¼ h0i ð7GX=9JÞ � 6T for GX/2J = 0.18 [33], eD ¼ 0.

The increase of the Y expectation values hsiY(HZ)i (andperpendicular HF fields HDM

aY and HDMbY ) under the increase

of the parallel applied magnetic field HZ (Eq. (47)) in theground j1/2, �1/2i state is explained by the different Zee-man splittings in the ground S = 1/2 and excited S = 3/2states and different D–M exchange admixture of thej3/2,-3/2i and j3/2,1/2i excited states (Eq. (44)).

For the ground j1/2, � 1/2i state of the AF [Fe(II)-Fe(III)]loc cluster, the internal HF fields H iY (48) on nuclei,dependence on the ZFS in the excited state and increase ofthe perpendicular HF fields (47) under the increase of theapplied parallel magnetic field HZ may be found in theMossbauer experiment as it was found for the perpendicu-lar HF fields H DM

aY and H DMbY in the singlet ground state of

the AF [Fe(III)Fe(III)] cluster [9].In the case of the localized mixed-valence [Fe(II)Fe(III)]

cluster with Sgr = 1/2, the non-zero expectation values hsiYi(42) and (45) and corresponding HF fields H DM

iY (Eqs. (47)and (48)) in the ground states of the localized antiferro-magnetic [Sgr = 1/2, Eq. (42)] (and ferromagnetic[Sgr = (9/2)loc, Eq. (35)]) MV clusters [Fe(II)Fe(III)] areproportional to the D–M parameter GX and take place inzero applied magnetic field HZ = 0 (Eqs. (42), and (46)–(48)). In the case of the monovalent AF cluster withSgr = 0, hjsiYji and H iY ðHZÞ are proportional to GX andHZ (Eq. (8) in [9]) and is equal to zero for HZ = 0. This dif-ference in the magnetic behavior is explained by the differ-ent D–M exchange mixing between the ground and firstexcited states in the cases of the: 1) singlet–triplet states(monovalent AF cluster) and 2) doublet–quadruplet states(mixed-valent AF cluster). In the monovalent AF cluster[Fe2(III)] with Sgr = 0, the D–M exchange admixture ofthe j1,�1i and j1,1i excited states is the same (in value) thatleads to the absence of the zero-field term in hsiY(HZ)iDM

[9]. In the mixed-valent AF [Fe(II)Fe(III)] cluster with

Sgr = 1/2, M = �1/2, the D–M exchange admixture ofthe j3/2, �3/2i and j3/2, 1/2i states is different (Eq. (44))that leads to the ‘‘zero applied field’’ term hsiY(HZ = 0)iDM =(7GX/9J) (Eqs. (42) and (46)) and HiY (Eqs. (47) and (48)).

2) For the lowest localized ferromagnetic Zeeman levelj9/2, � 9/2iloc (D0

S¼9=2 < 0; hsaZi ¼ �2; hsbZi ¼ �5=2; J �DS), the spin expectation values hsiY iDM along the Y-axisin the presence of the magnetic field HZ have the formhsaY(HZ) iDM = �hsbY(HZ)iDM = 5GX/[9jJj + g0bHZ]. Theincrease of the applied field HZ leads to the decrease of theHF fields on nuclei HDM

aY and HDMbY due to enlarge of the inter-

val between the Zeeman levels mixed by the D–M exchange.3) For the ferromagnetic cluster with D0

S¼9=2 > 0, thelowest localized Zeeman level is j9/2, � 1/2i forg0bHZ < 5D9/2. In the presence of HZ, the expectation val-ues hsiY(HZ)iDM for the ground state have the formhsaY ðHZÞiDM ¼ �hsbY ðH ZÞiDM

¼ 5GX ½9jJ j þ 8D0 � 4ðg0bHZÞ�9½ð9jJ j þ 8D0 � 4D0Þ2 � ðg0bH Z � D0Þ2�

;

ð49Þwhere D0 ¼ D0

S¼9=2 and D0 ¼ D0S¼7=2 Eq. (14), jD0j � jD 0j

[34]. The applied magnetic filed reduces the hsiY(HZ)iDM

values and the HF fields.4) In the case of the valence-delocalized ground state

U�(9/2, �9/2), the increase of HZ reduces the HF fieldsHDM

iY on nuclei.

7. Contributions of the antisymmetric double exchange to the

zero-field splittings of the [Fe(II)Fe(III)] cluster

As shown in Refs. [34,42,44] in the spin-coupling model,the one-center ZFS of the individual Fe2+ and Fe3+ ionscontribute to the cluster ZFS, which is described by thestandard ZFS Hamiltonian H 0

ZFS (Eqs. (12)–(14)). Let usconsider the contributions of the AS double exchange tothe ZFS parameter D of the ground valence delocalizedE0�ð9=2Þ level. As a result of the antisymmetric admixture

of the excited states to the ground E0�ðS ¼ 9=2Þ DE

level (Eqs. (19), (20), and (37)), the AS double exchangecontributes to the axial ZFS parameter D½E0

�ð9=2Þ� of thevalence-delocalized ground state U0

�ð9=2Þ. The axial ZFSparameter D½E0

�ð9=2Þ� of the ZFS Hamiltonian

eH ZFS ¼ D½E0ðSÞ�½S2

Z � SðS þ 1Þ=3� ð50Þ

for the delocalized ground state U0�ð9=2Þ has the form

D½E0�ð9=2Þ� ¼ D0 þ D�K ð9=2Þ;

D�K ð9=2Þ ¼ 1

405½K2

x=B� ðKx � 45Gx=4Þ2=ðB� 9JÞ�;ð51Þ

where D0 ¼ Dav9=2 ¼ 0:44Di > 0 (Eqs. (12)–(14)) is the aver-

aged axial ZFS parameter for the S = 9/2 level [34] of thestandard ZFS Hamiltonian H 0

ZFS (see Eqs. (19) and (20)).For the E0

þð9=2Þ DE level, one obtains


D½E0þð9=2Þ� ¼ D0 þ DþK ð9=2Þ;

DþK ð9=2Þ ¼ 1

405½ðKx þ 45Gx=4Þ2=ðBþ 9JÞ � K2

x=B� ð52Þ

The antisymmetric double exchange contributionsDK ðSÞ to the ZFS parameter D½E0

ðSÞ� depend on the totalspin S.

The DK ð9=2Þcontributions to the axial ZFS parameter ofthe DE levels E0

ðS ¼ 9=2Þ depend on the parameters B, J,KX and GX.

In the case B� J, Kx� Gx, the D�K ð9=2Þ (52) andDþK ð9=2Þ (53) values are reduced, using Eq. (32), to theone ZFS parameter DK ð9=2Þ ¼ DKð9=2Þ, where

DKð9=2Þ � ðK2x=45BÞð5Gx=2Kx � J=BÞ

¼ �2JðKx=BÞ2=45 ¼ �5JðGx=JÞ2=18. ð53ÞFor comparison, the contribution of the D–M exchange

to the ZFS parameter of the ground localized S = 9/2 stateof ferromagnetic cluster [Fe(II)Fe(III)]loc has the form:

DDMð9=2Þ ¼ �5J F ðGX=jJ F jÞ2=144;

Dð9=2Þ ¼ D0 þ DDMð9=2Þ; ð54Þ

where D0 = D9/2 is determined [34] by Eq. (13). The Eqs.(53) and (54) lead to the relation DK(9/2)/DDM(9/2) =8JAF/JF. Using Eq. (32) and the relation J/B = 70/1350= 0.052 [34,35], we can represent the contribution of theAS double exchange and D–M exchange to the clusterZFS parameter in the form

D½E0�ð9=2Þ� ffi D0 � 0:28ðGX=JÞ2J � ðGX=JÞ2D0; ð55Þ

The third term in Eq. (55) represents the contribution ofthe different ZFS in the S = 9/2 and S = 7/2 states (Eqs.(19) and (20)) to the cluster ZFS. Using for estimationthe experimental values JAF = 70 cm�1 [34,35] andjGXj/2J = 0.18 [33] (strong D–M exchange), one obtainsthe estimation DKð9=2Þ � �2:5 cm�1 and �0.13D0 for thethird term (DKð9=2Þ � �0:2 cm�1 for GX/J = 0.1). Theexperimental values of D (in cm�1) are in the interval�1.5 < D < 4 [34–39,41,43]. Negative contributionsDKð9=2Þ to the cluster ZFS parameter D½E0

�ð9=2Þ� may beof the order of the individual Fe3+ ion contributions tothe cluster ZFS parameter DS (0.27D2, Eq. (14)) and inthe case of the strong D–M coupling should be taken intoaccount. The clusters with D1(Fe2+) < 0 should be consid-ered separately since in this case the ground state of theFe2+ ion is 5B1 term, Ref. [57] in [1].

The contribution of the D–M exchange (54) to the clus-ter ZFS in the localized ferromagnetic cluster is determinedby equation Dð9=2Þ ffi D0 � 0:035ðGX=JÞ2J F � 0:06ðGX=JÞ2D0. The estimate of the D–M exchange contribution (54) toD(9/2) is DDM(9/2) ffi �0.3 cm�1 for jGXj/2J = 0.18 andjJFj = 70 cm�1.

We can conclude that spin–orbit coupling leads to theantisymmetric double exchange in the valence-delocalized[Fe2.5+Fe2.5+] clusters with strong double exchangecoupling.

8. Conclusion

The comparison of the antisymmetric double exchangeinteraction H1X (1) (KX � BDg?#) and the Dzialoshin-sky–Moriya antisymmetric exchange interaction H X

DM (5)(the D–M coefficient GX � JDg?#) shows that the antisym-metric double exchange H1X is stronger than the Dzialo-shinsky–Moriya exchange H X

DM in the valence-delocalized[Fe2.5+Fe2.5+] cluster with large double exchange sinceB� J. The parameters of antisymmetric double exchangeand Dzialoshinsky–Moriya exchange are connected bythe relation KX/GX = �5B/2J. The D–M antisymmetricexchange HX

DM mixes the S and S 0 = S ± 1 states of thesame localization. The antisymmetric double exchangeH1X mixes the S and S 0 states of different localizations,S0= S,S ± 1. The mixing of the double exchange states

U0ðS;MÞ with the excited DE states U0

ðS � 1;M 0Þ with dif-ferent total spin of the same parity depends on both theantisymmetric double exchange and D–M exchange (mix-ing parameter [KX � S(2sa + 1)GX/2], Eqs. (16), (18), (19),and (20)). The mixing of the DE states U0

�ðS;MÞ andU0þðS;M 0 ¼ M 1Þ with the same total spin S of different

parity depends only on the antisymmetric double exchange(KX) (eqs. (19) and (20)). This mixing of the doubleexchange levels results in magnetic anisotropy.

In the ground delocalized Sgr = (9/2)deloc state of thevalence-delocalized cluster [Fe2.5+Fe2.5+], the antisymmet-ric double exchange and D–M exchange result in non-zero expectation values of the local spins hsaYi = �hsbYiin direction Y that leads to the canting of spins in theplain (YZ) perpendicular to the direction of the vectorof the AS double exchange KX and D–M exchange GX

(Eq. (38), Fig. 2c). The deviations of spins from the par-allel alignment in the valence-delocalized ground U0

�ð9=2Þstate depend on the parameters KX, GX, B and J. Theantisymmetric double exchange and D–M exchange influ-ence on the hyperfine field on nuclei and hyperfine con-stants in the ground state of the valence-delocalizedclusters.

The calculations show that the observed strong D–Mexchange coupling in the localized antiferromagneticmixed-valence [Fe(II)Fe(III)]loc cluster (Sgr = 1/2) of the[2Fe � 2S]1+ center of the Rieske protein from Thermus

thermophilius [33] leads to large spin canting and inducedhyperfine fields HDM

aY ¼ �H DMbY on nuclei.

The mixing of the double exchange levels by the anti-symmetric double exchange and D–M exchange results inthe AS double exchange contributions D�K ð9=2Þ to the clus-ter ZFS parameter D½E0

�ð9=2Þ� ¼ D0 þ D�K ð9=2Þ of thevalence-delocalized ground state U0

�ð9=2Þ. The antisym-metric double exchange contributions to ZFS depend onthe total spin of the double exchange levels.

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spin–orbit coupling in the double exchange model 2. comparison of the antisymmetric double...

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