Anisotropic double exchange in mixed-valence dimeric clusters of transition metal ions

Chemical Physics 308 (2005) 27–42 www.elsevier.com/locate/chemphys

Anisotropic double exchange in mixed-valence dimeric clusters of transition metal ions Moisey I. Belinsky

*

School of Chemistry, Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel Received 7 April 2004; accepted 30 July 2004 Available online 11 September 2004

Abstract In the mixed-valence (MV) [dn–dn + 1] clusters of non-degenerate transition metal ions with the migration of the extra electron, taking the spin–orbit coupling into account in the double exchange (DE) model results in anisotropic double P P exchange interaction or ^ sab S^ Þþ anisotropic spin-dependent electron transfer which is described by the effective Hamiltonian H AN DE ¼ a n¼x;y;z ½An tl ð S
^ !b
n a n 8 9 Bn tv ð S^ ^sab S^ þ S^ ^sab S^ Þ, sab is the one-electron DE operator. For the MV [d –d ] cluster, the coefficients of the double a
n

!an

bn

!b
n

exchange anisotropy Antl, Bntv linearly depend on the DE parameters tl and tv of the excited and ground cluster DE states. The one-center spin operators S^ ; S^
ð S^ Þ act in the states of different localization. The anisotropic spin-transfer interaction H AN DE a
n

!b n !an

is active between the states of different localization of the extra electron. Anisotropic double exchange coupling results in the zero-field splitting (ZFS) of the high-spin DE levels E0 ðS ¼ 3=2Þ. This splitting is described by the effective ZFS Hamiltonian ^ ab ½S 2  SðS þ 1Þ=3 þ Et T ^ ab ðS 2  S 2 Þ, where T ^ ab is the double exchange operator in the S representation. The ZFS H tZFS ¼ Dt T Z X Y parameters Dt and Et of the anisotropic DE origin are linearly proportional to the double exchange parameters tl. In the MV clusters, the ZFS operator H tZFS acts between the S states of different localization and should be added to the standard ZFS Hamiltonian H 0ZFS ¼ DS ½S 2Z  SðS þ 1Þ=3 þ ES ðS 2X  S 2Y Þ, which is active in the localized states. The anisotropic double exchange contributions to the ZFS have different sign for the E0þ ðSÞ and E0 ðSÞ DE states: D½E0 ðSÞ ¼ DS  Dt ; E½E0 ðSÞ ¼ ES  Et . The anisotropic DE contributions Dt, Et to the cluster ZFS parameters (DS ± Dt, ES ± Et) may be larger than the single-ion (Di,Ei) and anisotropic (pseudodipolar) exchange (Dpd, Epd) contributions.
2004 Elsevier B.V. All rights reserved.

1. Introduction The double exchange interaction was introduced to explain the magnetism of the mixed-valence manganates [1,2]. The resonance splitting of the spin levels of the MV [dn + 1–dn] dimers due to the hopping of the extra electron between the dn + 1 and dn ions was described by the Anderson and Hasegawa [2] double exchange model E ðSÞ ¼ ðS þ 1=2Þt0 =ð2s0 þ 1Þ;

ð1Þ

where t0 is the electron transfer (ET) integral. In the MV dimeric systems, the Anderson and Hasegawa [2] double exchange (1) and Heisenberg exchange interaction H 0 ¼ 2J ~ S1~ S2 ðt–J modelÞ form the resulting exchange-resonance 0 states E ðSÞ ¼ JSðS þ 1Þ  BðS þ 1=2Þ, where B = t0/(2s0 + 1) is considered the effective DE parameter. The double

*

Tel.: +97236407942; fax: +97236409293. E-mail address: [email protected].

0301-0104/$ - see front matter
2004 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2004.07.043

28

M.I. Belinsky / Chemical Physics 308 (2005) 27–42

exchange coupling or spin-dependent ET results in isotropic splittings: the Anderson–Hasegawa levels E0 ðSÞ depend only on the total spin S and are not mixed by the DE coupling. The isotropic double exchange and Heisenberg exchange in dimeric MV clusters has been the subject of theoretical and experimental investigations [3–39]. Strong DE interaction (B = 1350 cm1, t0 = 6750 cm1), which destroys the Heisenberg antiferromagnetic ordering (JAF = 70 cm1, JB) and results in the delocalized ground state Sgr = Smax = 9/2, was found first in the synthetic MV compound [Fe2(OH)3(tmtacn)2]2+ [14,15,20–22]. Strong DE interaction (t0 = 2250–6750 cm1 (B = 450–1350 cm1), J = 5–100 cm1, JB) was found also for the MV [Fe(II)Fe(III)] centers with delocalized Sgr = 9/2 in the synthetic compounds [26–28,30,31,39] and [Fe2S2]+ centers of the Clostridium pasterianum mutant 2Fe ferredoxins [23,24,29]. Strong DE results in the Sgr = 3/2 ground state of the MV [Ni(II)Ni(I)] cluster [7]. Valence delocalized [Fe2S2]+ pairs with strong DE were found in a variety of the trimeric [Fe3S4] and tetrameric [Fe4S4] iron–sulfur clusters in ferredoxins, enzymes and synthetic models [40,41]. The localized ferromagnetic [Fe(II)Fe(III)] dimers with Sgr = 9/2 were investigated in the model synthetic compounds [12,25,28,33] and native [Fe2S2]+ R2 center of Escherichia coli ribonucleotide reductase [32]. Zero-field splittings (ZFS) of the delocalized cluster ground state with Sgr = Smax is described by the standard effective ZFS Hamiltonian [42–49]:  
 ð2Þ H 0ZFS ¼ DS S 2Z  SðS þ 1Þ=3 þ ES S 2X  S 2Y with the axial DS and rhombic ES cluster ZFS parameters. The delocalized Sgr = 9/2 ground state of the MV [Fe(II)Fe(III)] clusters of the model compounds {Clostridium pasterianum 2Fe ferredoxins} are characterized by large positive {negative} axial ZFS parameters: D9/2 = +1.7 to +4 cm1, E9/2 = 0–0.13 [15,31,39] {D9/2 = 1.1; 1.5 cm1, E9/2 = 0.16; 0.11 [23,24,29]}. The ground state Sgr = 3/2 of the MV [Ni(II)Ni(I)] cluster [7] {valence-delocalized [Fe(III)Fe(IV)] dimer of the low-spin ions [50–52]} is characterized by large positive axial ZFS parameter D3/2 P 30 cm1 {D3/2 = +38 cm1}. The ZFS contributions of individual ions (D1, E1; D2, E2) to the cluster ZFS were considered the origin of the ZFS of the Sgr = 9/2 state of the delocalized [15] and localized [32,33] [Fe(II)Fe(III)] MV clusters. The correlations D9/2 = 1/6D1 + 5/18D2 and E9/2 = 1/6E1 + 5/18E2 were used for the ZFS parameters of the Sgr = 9/2 ground state with the average of the local ZFS contributions due to delocalization [15], where D1, E1 (D2, E2) are the ZFS parameters of the ferrous (ferric) ion in the MV dimer. The origin of magnetic anisotropy in the monovalent [dn–dn] clusters, role of the spin–orbit coupling (SOC), anisotropic (pseudodipolar (pd)) exchange inter-ion interaction H pd ¼ ax J x S 1x S 2x þ ay J y S 1y S 2y þ az J z S 1z S 2z

ð3Þ

and anisotropic exchange contributions to ZFS were considered by Kanamori [48] (see also [49,57]). The ZFS of the levels S (S P 1) of the [dn–dn] clusters is described by the ZFS Hamiltonian (2) [49]. The origin of the axial DS and rhombic ES ZFS parameters for the exchange [dn–dn] dimers was considered in [48,49]. The dipole–dipole interaction [49] and single-ion ZFS [49,53,54] contribute to ZFS parameters DS, ES of dimers. The single-ion anisotropy and ZFS were discussed in [42–49,55,56]. The antisymmetric Dzyaloshinsky–Moriya exchange and symmetric anisotropic exchange in the exchange [dn–dn] dimers was introduced in [58,59] and discussed in [60,61]. For the MV systems, the account of the spin–orbit coupling for the transfer of hole between the neighboring sites in doped La2CuO4 [61] results in the spin–orbit hopping term and symmetric anisotropic exchange. Anisotropic double exchange for the MV dimers of orbitally degenerate ions was considered in [38]. For the MV [dn–dn + 1] dimers of orbitally non-degenerate ions, taking SOC into account in the DE model (in the second order perturbation theory) results in an antisymmetric double exchange [62,63] interaction. The antisymmetric double exchange mixes the DE levels of the Anderson–Hasegawa model (1) that leads to the antisymmetric DE contributions to ZFS [62]. The anisotropic DE contribution to the axial ZFS parameters of the levels E±(S) was considered in the simplified model in [62]. The investigation of anisotropic effects and ZFS connected with anisotropic DE are of interest for the molecular magnetism, since the zero-field splittings of the high-spin ground states of the clusters are important for the molecular magnets, polynuclear metal centers in biological systems and their synthetic models. The anisotropic effects and ZFS contributions connected with the strong double exchange coupling were not considered in the double exchange model of the MV clusters. The aim of this paper is to consider new anisotropic double exchange interaction, which is specific for the MV [dn– n+1 d ] clusters of orbitally non-degenerate ions with the double exchange coupling, and also the zero-field splittings of the high-spin DE levels, which are connected with the anisotropic DE. As will be shown on an example of the MV [d8– d9] cluster, the combined effect of spin–orbit interaction VSO on the centers a and b and the isotropic double exchange

M.I. Belinsky / Chemical Physics 308 (2005) 27–42

29

in the excited states results in the inter-ion anisotropic double exchange interaction. The coefficients of the double exchange anisotropy are linearly proportional to the DE parameters tl and tv of the excited and ground states. The anisotropic double exchange results in the zero-field splitting of the Anderson–Hasegawa DE levels E0 ðSÞ. For the DE levels E0 ðS ¼ 3=2Þ of the MV [d8–d9] cluster, the ZFS of the anisotropic DE origin is described by the effective ZFS Ham^ ab ½S 2  SðS þ 1Þ=3 þ Et T ^ ab ðS 2  S 2 Þ. The ZFS parameters Dt and Et depend linearly on the DE iltonian H tZFS ¼ Dt T Z X Y ^ parameters tl, Tab is the DE operator in the representation of the total spin. The anisotropic double exchange ZFS Hamiltonian H tZFS is active between the states of different localization in comparison with the standard ZFS Hamiltonian H 0ZFS (2) which acts in the localized states. The anisotropic DE contributions to the cluster ZFS parameters have different sign for the E0þ ðSÞ and E0 ðSÞ DE states: D½E0 ðSÞ ¼ DS  Dt ; E½E0 ðSÞ ¼ ES  Et , where the DS, ES contributions include the dipole–dipole, single-ion and anisotropic (pseudodipolar) exchange Dpd, Epd contributions. Consideration of the anisotropic (pseudodipolar) exchange in the localized MV [d8–d9] cluster shows that the anisotropic double exchange contributions Dt, Et (tl(Dgn)2) to the cluster ZFS parameters may be stronger than the anisotropic (pseudodipolar) exchange contributions Dpd, Epd (Jl(Dgn)2) since t  J.

2. Hamiltonian of anisotropic double exchange in a mixed-valence pair An anisotropic double exchange interaction originates from the combined effect of the isotropic double exchange in the excited states and spin–orbit interaction V aSO on the centers a and b. Let us consider the anisotropic double exchange in the [dn + 1–dn] ¢ [dn–dn + 1] pair of the MV ions with orbital singlet ground states. For the double exchange in the MV [dn + 1–dn] pair, the two-center third-order perturbation anisotropic DE terms, including the SOC admixture of the excited d-ions states and the double exchange in the excited states, have the form: ! ! X X hw0a
u0b jV aSO jwka
u0b i hu0a wkb
jV bSO ju0a w0b
i k 0 ^ 0 k V1 ¼ hwa
ub jV ab jua wb
i ; ð4Þ Eka
 E0a
Ekb
 E0b
a k

V2 ¼

X X hw0a
u0b jV aSO jwka
u0b i a

V3 ¼

Eka
 E0a


X X hw0a
u0b jV aSO jwka
u0b i a

V4 ¼

k;k 0

k;~k

Eka
 E0a


X X hw0a
u0b jV aSO jwka
u0b i a

k

!

Eka
 E0a


0

1

0 0 ~0 ~ k
jV b j~ 0 h~ u0a w SO ua wb
iA b k 0 ~k @ 0 ^ ; ua wb
i hwa
ub jV ab j~ k0 0

~
E ~
E b b

!

a

~

~ hwka
u0b jV^ ab juka w0b
i

huka w0b
jV~ SO ju0a w0b
i ~

Eka  E0a

ð5Þ

! ;

ð6Þ

!2 hw0a
u0b jV^ ab ju0a w0b
i;

ð7Þ

where a(b) = a(b), b(a). The ket jw0a
u0b ifju0a w0b
ig represents the ground S state U0a
b ðS; MÞ ¼ jw0a
u0b jfU0ab
ðS 0 ; M 0 Þ ¼ ju0a w0b
jg in the case ja*bæ{jab*æ} of localization of the extra electron (hole). The ket ~ jwka
u0b i½ju0a wkb
ifjuka w0b
ig represents the cluster excited states U0a
b ½U0ab
fU00ab
g coupled to the cluster jS,Mæ ground state 0 0 0 0 jwa
ub i½jua wb
ifju0a w0b
ig by the spin–orbit interaction V aSO ½V bSO fV aSO g on the dn + 1-center a* [dn + 1 center b*] {dn cen~ ter a}. E0a
fE0a g and Eka
fEka gare the energies of the ground state and excited states of the dn + 1 center a* {dn center a}, respectively. To illustrate the anisotropic double exchange terms, we will consider the double exchange and SOC for the bioctahedral MV [d8–d9] cluster (the [Cu3+–Cu2+] pair, sa = 1, sb = 1/2). We consider the cluster with the common Z-axis; xa, xb (ya, yb) axes are parallel to the cluster X(Y) axes. The clusterpground U0 ðS ¼ 3=2Þ, U0 ðS ¼ 1=2Þ is formed ffiffiffi 0 state set 0 0 by the ground states of the coupled individual ions, U ðSÞ ¼ 1= 2½Ua
b ðSÞ  Uab
ðSÞ. The ground state basis functions in different localizations have the form U0a
b ðS; MÞ ¼ jw0a
u0b j and U0ab
ðS; MÞ ¼ ju0a w0b
j, where the single-ion ground state wave functions are w0 and u0 for the d8 and d9 ions, respectively. In Fig. 1(a), the localized cluster ground and excited states in the ja*bæ{jab*æ} localization of the extra d-hole are represented in the form ½d8a
ð0Þd9b ð0Þ0 f½d9a ð0Þd8b
ð0Þ0 g and ½d8a
ðeÞd9b ð0Þe f½d9a ð0Þd8b
ðeÞe g, respectively. The ground state of the d8a
-ion is the 3 A2(e2) term, w0a
¼ w0a
ð3 A2 Þ ¼ jua va j where v ¼ dx2 y 2 ; u ¼ d3z2 r2 [45]. The ground state of the d9b -center is the 2 A1g term ðu0b ¼ ub Þ in octahedral coordination with tetragonal distortion. The operator V1 (4) includes the SOC V aSO fV bSO g admixture of the excited k ligand field (LF) state 3T2 [45] to the ground 3A2 state of the d8a
fd8b
g ion on the center a{b} in the ja*bæ{jab*æ} localization and isotropic double exchange

30

M.I. Belinsky / Chemical Physics 308 (2005) 27–42

(a)

k

Excited States I 9

8

[d a* (e)d b (0)]e

E +(3/2)

Excited States I

k

9

E +(1/2)

S=1/2

8

[da (0)d b*(e)]e

S=3/2

2tk

tk

k

E -(1/2)

3J1k

k

E1k

a

V

SO

~ ~

E -(3/2) 0

3J0+tv

E +(3/2) 0

E +(1/2)

8

9

[d a* (0)d b(0)]0

tv

0

E -(1/2) E -(S=3/2)

localization |a*b> 8 9 [d -d ]

2tv

3J0

-tv/2

0

SO

Ground State Levels

+tv/2

S=3/2 S=1/2

b

V

9

8

[d a(0)d b*(0)]0

3J0-tv

Double Exchange Levels

d-hole localization |ab*> 9 8 [d -d ]

(b) Excited States I 8

Excited States II

9

[d a* (e)d b (0)]e

8

9

9

8

9

<[d ed 0]e|HDE|[d ed 0]e>~tv

E1k

a

V

E0k

0

SO

E +(3/2)

~ ~

8

[d a(e)d b*(0)]e

a

V

SO

~ ~

0

E +(1/2) S=3/2

3J0 8

9

S=1/2

[d a*(0)d b(0)]0 localization |a*b> 8 9 [d -d ]

0

E -(1/2)

tv

2tv 9

8

[d a(0)d b*(0)]0

Ground State 0 E -(3/2) Levels localization |ab*> Double Exchange 9 8 Levels [d -d ]

Fig. 1. (a) The isotropic exchange and double exchange splittings of the ground and the lowest excited states of the [d8–d9] cluster. The localized pure exchange levels in the case of the ja*bæ (left) and jab*æ (right) localization of the extra d-hole. The SOC V aSO fV bSO g admixture of the excited LF state 3 T2 to the ground 3A2 state of the d8a
fd8b
g ion on the center a{b} in the ja*bæ {jab*æ} localization is shown schematically. The Anderson–Hasegawa E0 ðSÞ double exchange levels of the ground states set and the excited DE Ek ðSÞðk ¼ fÞ levels are represented in the center of (a), t  J. The DE levels with S = 3/2 (S = 1/2) are designated by the solid (dashed) lines. (b) The SOC V aSO fV aSO g admixture of the excited states to the ground state of the d8a
fd9a g ion on the center a in the ja*bæ{jab*æ} localization. The double exchange between the excited states 0 00 Uak
b ðSÞð½d8a
ðeÞd9b ð0Þe and Uabk
ðSÞð½d9a
ðeÞd8b ð0Þe is shown schematically.

interaction between the excited cluster states U0a
b ðS; MÞ and U0ab
ðS; MÞ formed by the excited state 3T2 of the d8-ion and the ground state of the d9-ion (Fig. 1(a)). We consider here the SOC admixture of the lowest excited 3T2 term for the d8-ion. Using the correlations hw0a
ð3 A2 ÞjLjwka
ð3 T2 Þi [43–45,48,49] for the orbital angular momentum for d8-ions, one can represent the operator V1 (4) in the form: i X h   V1 ¼ pn S^a
n hwla
3 T2l ; M u0b ð2 A1g ; mÞjV^ ab ju0a ð2 A1g ; mÞwlb
ð3 T2l ; MÞiS^b
n ; n¼x;y;z ð8Þ h  i 2 a b pn ¼ 4 k1 = D1l D1l ; where n = x(y)[z] corresponds to l = n(g)[f] in the excited states Ula
b , k1 and Da1l are, respectively, the effective SOC constant and the ligand field intervals ½Da1l ¼ Ela
ð3 T2l Þ  E0a
ð3 A2 Þ for the d8a
ion, n = dyz, g = dxz, f = dxy; D1 = E1 in Fig. 1. The operator V1 (8) has the form of the inter-ion spin–transfer–spin interaction. In the DE matrix elements hwa
ub jV^ 1 jua wb
i, the one-center spin operators S^a
n ðS^b
n Þ act on the spin functions wa*(wb*) of the d8a
ðd8b
Þ ion in the Æa*bj (jab*æ) localization. This action will be designated by the arrows under the operator S^an : S^ ð S^
Þ. The matrix a
n !b n

M.I. Belinsky / Chemical Physics 308 (2005) 27–42

31

elements of the DE operator in Eq. (8) between the excited states of the first type Ula
b ð¼ jwla
ð3 T2l Þu0b ð2 A1g ÞjÞ and Ulab
of different localizations are considered in part 4 (Eqs. (27) and (28)). In the case of the SOC admixture of the lowest excited state 3T2 only for the d8-ion, the operator V2 (5) has the same form as the operator V1 (Eq. (8)). Collecting the DE terms in the excited states for the V1 and V2 operators (Eqs. (27)–(29), part 4), one can represent the combined operator V1 + V2 in the following form of the effective symmetric Hamiltonian of anisotropic double exchange interaction or anisotropic spin-dependent electron transfer     X

^ ^ ^ ^ ^ ^ HA ¼ Ax tn S ^sab S
þ Ay tg S ^sab S
þ Az tf S ^sab S
: ð9Þ !b x

a
x

a

a
y

!b y

a
z

!b z

The coefficients of anisotropy Antl in the anisotropic DE operator HA (9) are proportional to the parameters tl [tn = t(dyz), tg = t(dxz), tf = t(dxy)] of the double exchange in the excited Ula
b states (see part 4, Fig. 1(a)). The DE parameters tl describe the transfer of the extra d-hole from the excited ðwla
Þ state of the d8a
-ion to the ground ðu0b Þ state of the d9b -ion. As will be shown in part 5, the coefficients An have the form: h  i
 
  Ax ¼ k21 = Da1n Db1n ; Ay ¼ k21 = Da1g Db1g ; Az ¼ 4 k21 = Da1f Db1f : ð10Þ In the case of the same LF intervals ðDa1l ¼ Db1l Þ for the ions d8a
and d8b
, these coefficients have the form (k1/D1l)2. The effective anisotropic DE Hamiltonian (9) acts between the cluster ground states U0a
b ðS; MÞ and U0ab
ðS 0 ; M 0 Þ of different localizations, S 0 = S, S ± 2, M 0 = M, M ± 2. The anisotropic double exchange Hamiltonian (9) includes the one-electron DE operator ^sab . The one-electron DE operator ^sab is determined by the relation h/a j^sab j/b i ¼ 1 for the one-electron d-functions /a. The one-center spin operators S^ f S^
g act on the ground state spin functions a
n !b n

w0a
ðS a ; M a Þfw0b
ðS b ; M b Þg of the d8a
fd8b
g ion in the Æa*bj{jab*æ} localization. This action in the matrix elements  

0 0  0 0

0 0 0 0



jw
u j ^ ^ ^ ^ jwa
ub j An tl S ^sab S
jua wb
j ¼ jua wb
j An tl S
^sab S ð11Þ a b !b n

a
n

is represented by an example   w0a
ðS a ; M a Þu0b ðsb ; mb Þj S^ ¼



b n

a
n

^sab S^



~ aÞ w0a
ðM a ÞjS^a
n jw0a
ðM

!b
n



ju0a ðsa ; ma Þw0b
ðS b ; M b Þ

!a
n



~ a Þu0 ðsb ; mb Þj^sab ju0 ðsa ; ma Þw0
ðS b ; M
bÞ w0a
ðS a ; M b a b




b ÞjS^b
n jw0
ðM b Þ : w0b
ðM b

ð12Þ

~ a þ mb ¼ ma þ M
b. The one-electron DE exchange operator ^sab connects the states with the same projection M : M The correlation  

0 0  0 0

0 0 0 0

ju w
j ¼ ju w
j S^ ^sab S^

jw
u j ¼ 0 ð13Þ jwa
ub j S^ ^sab S^ a a b a b b bn

!an

8

an

!bn

9

takes place for the [d –d ] MV cluster since the double exchange is forbidden between the excited states ~ U00a
b ð½d8a
ð0Þd9b ðeÞe Þ and U00ab
ð½d9a ðeÞd8b
ð0Þe Þ formed by the d9-ions in the excited ðuk Þ state and d8-ions in the ground 0 (w ) state (part 4, Eq. (30)). Since anisotropic Hamiltonian HA (9) includes the DE operator ^sab and the DE parameter tl, HA represents an anisotropic inter-ion double exchange interaction or anisotropic spin-dependent electron transfer. HA is the operator of anisotropic spin-transfer coupling since HA includes the ET operator ^sab and spin operators. The Hamiltonian of an anisotropic double exchange (9) may be represented in the form



 X ~A ¼ H DðtÞ S^ ^sab S^
 ^sab S2 =3 þ EðtÞ S^ ^sab S^
 S^ ^sab S^
; !b z !b x !b y a
z a
x a
y ð14Þ a DðtÞ ¼ fAz tf  ½Ax tn þ Ay tg =2g; EðtÞ ¼ ½Ax tn  Ay tg =2; a = a{b} corresponds to b = b{a}. The parameters of the DE anisotropy D(t), E(t) linearly depend on the DE parameters tl in the Antl coefficients. In the V3 term (6), the SOC V aSO admixes the 3T2 excited state to the ground 3A2 state of the d8a
ion (the center a) in a the ja*bæ localization (Fig. 1(b)). In the jab*æ localization, the SOC V~ SO at the center a admixes the excited na and ga 9 2 0 states to the ground A1g ðua ¼ ua Þ state of the da -ion (Fig. 1(b)). The DE interaction hU0a
b jV^ ab jU00ab
i in Eq. (6) takes place between the different excited cluster states: U0a
b ðSÞ (formed by the excited states of the d8a
-ion and ground state of the d9b -ion, ½d8a
ðeÞd9b ð0Þe in Fig. 1(b)) and U00ab
ðSÞ (formed by the excited states of the d9a -ion and ground state of the d8b
-ion, ½d9a ðeÞd8b
ð0Þe in Fig. 1(b)), part 4. Using 1) the L mixing of the excited wla states ðwla ð3 T2l ÞÞ with the ground

32

M.I. Belinsky / Chemical Physics 308 (2005) 27–42 0

w0a ðw0a ð3 A2 ÞÞ state for the d8a -ion in the ja*bæ localization and 2) the L mixing of the excited ula states with the ground state u0a for the d9a -ion in the jab*æ localization and corresponding b terms, one can represent the V3 operator in the form X     0 0 V3 ¼ p0an S^a
n wla
u0b jV^ ab jula w0b
S^an þ p0bn S^bn wla
u0b jV^ ab jula w0b
S^b
n ; n¼x;y;z

p0an

ð15Þ

 pffiffiffi 
¼ 2 3k1 k0 =Da1l Da0l ;

where n = x(y)[z] corresponds to l, l 0 = n (g)[f], respectively. In the p0n coefficients, k1(k0) and Da1l ðDa0l Þ are the SOC constant and LF intervals of the d8(d9)-ion, respectively. D0 = E0, D1 = E1 in Fig. 1(b). p0z ¼ 0 in the case of the ground u-state of the d9-ion. In the DE matrix elements hwa
ub jV^ 3 jua wb
i, the one-center spin operators S^a
n ðS^an Þ act on the spin functions wa*(ua) of the d8a ðd9a Þ ion on the a-center in different localizations Æ a*bj(jab*æ). This action will be designated by the arrows under the one-center spin operators S^an : S^ ð S^ Þ. a
n !an 0 The double exchange interaction hwla
u0b jV^ ab jula w0b
i between different excited states U0a
b ðSÞ and U00ab
ðSÞ in Eq. (15) is determined by the DE parameter tv of the double exchange in the ground S states, Fig. 1(b) (Eq. (31)), part 4). Collecting the corresponding DE terms in the excited cluster states, the operator V3 may be represented in the other form of the effective Hamiltonian of anisotropic double exchange H B ¼ tv

X X  Ban S^ a¼a;b n¼x;y;z

a
n

^sab S^



!an

 þ Bbn S^ ^sab S^ bn

 ð16Þ

:

!b
n

For the [d8–d9] MV cluster with the ground state u for the d9-ion, the calculations (part 4) result in the following Ban coefficients   Bax ¼ 3k1 k0 =Da1n Da0n ; Bay ¼ ð3k1 k0 =Da1g Da0g Þ; Baz ¼ 0 ð17Þ in the Ban tv parameters of the DE anisotropy. The effective anisotropic double exchange operator HB (16) is active between the ground states U0a
b ðS; MÞ and U0ab
ðS 0 ; M 0 Þ of different localizations. In the matrix elements hjw0a
u0b jBan tv ð S^ ^sab S^ Þju0a w0b
ji, the one-center spin operators S^ f S^ g act on the ground state spin functions a
n

!an

a
n !an

w0a
ðS a ; M a Þfu0a ðsa ; ma Þg of the d8a
fd9a g ion on the a-center in the ja*bæ{jab*æ} localization. For example, the matrix ele~ a þ mb ¼ m
a þ M bÞ ments of anisotropic DE operator (16) have the form ðM    w0a
ðS a ; M a Þu0b ðsb ; mb Þj S^ ^sab S^ ju0a ðsa ; ma Þw0b
ðS b ; M b Þ a
n

¼



!an



~ aÞ w0a
ðS a ; M a ÞjS^a
n jw0a
ðS a ; M

~ a Þu0 ðsb ; mb Þj^sab ju0 ðsa ; m
a Þw0b
ðS b ; M b Þ w0a
ðS a ; M b a


a ÞjS^an ju0a ðsa ; ma Þ ;  u0a ðsa ; m      0 0 0 0 0 0 ^ ^ jwa
ub jj S ^sab S jjua wb
j ¼ jua wb
jj S^ ^sab S^ a
n

!an

an

!a
n



 0 0 jjwa
ub j :

The anisotropic DE Hamiltonian HB (16) includes the anisotropy operators ð S^

a
n

^sab S^ Þ connected with the anisotropy !an

of different ions ðd8a
and d9a Þ on the a-center in different localizations: ja*bæ and jab*æ. The Hamiltonian of an anisotropic double exchange (16) may be represented in the form

 

   X ~B ¼ H D0a ðtv Þ S^ ^sab S^  ^sab S2 =3 þ E0a ðtv Þ S^ ^sab S^  S^ ^sab S^ ; a¼a;b

az

!az

ax

n h i o D0a ðtv Þ ¼ ðtv =2Þ Baz  Bax þ Bay =2 ;

!ax

ay

!ay

ð18Þ

h i E0a ðt0 Þ ¼ ðtv =4Þ Bax  Bay :

The parameters of anisotropy D 0 (tv), E 0 (tv) depend linearly on the DE parameter tv. The resulting Hamiltonian of the inter-ion anisotropic double exchange interaction H AN DE ¼ H A þ H B has the form  

    X X An tl S^ ^sab S^
þ Bn tv S^ ^sab S^ H AN þ S^ ^sab S^
; ð19Þ DE ¼ a¼a;b n¼x;y;z

a
n

!b n

a
n

!an

bn

!b n

where n = x(y)[z] corresponds to l ¼ nðgÞ½f; a½b ¼ a½b; b½a; Ban ¼ Bbn ¼ Bn .

M.I. Belinsky / Chemical Physics 308 (2005) 27–42

33

3. Zero-field splitting induced by anisotropic double exchange For the double exchange Anderson–Hasegawa E0 ðSÞ with S > 1/2 of the ground states set, the anisotropic double exchange interaction (Eq. (19)) results in the zero-field splittings. For the E0 ðS ¼ 3=2Þ DE levels of the [d8–d9] MV cluster, this zero-field splitting induced by the anisotropic DE (19) is described by the effective ZFS Hamiltonian ^ ab ½S 2  SðS þ 1Þ=3 þ Et T ^ ab ðS 2  S 2 Þ: H tZFS ¼ Dt T Z X Y

ð20Þ

^ ab acts in the representation of the total spin S and describes the Anderson–Hasegawa DE splitting The DE operator T (1): ^ ab jU0 ðSÞi ¼ hU0
ðSÞjT ^ ab jU0
ðSÞi ¼ ðS þ 1=2Þ=ð2S 0 þ 1Þ: hU0 ðSÞjT  a b ab

ð21Þ

^ ab and one-electron DE operator ^sab The correlation between the DE operator T ^ ab jU0
ðSÞi ¼ hU0
ðSÞj^sab jU0
ðSÞi hU0a
b ðSÞjT ab a b ab takes place for the states with maximal total spin S. The axial Dt {rhombic Et} ZFS parameter (Eq. (20)) includes the contributions DA(t) {EA(t)} and DB(tv) {EB (tv)} of anisotropic DE interactions HA (9) and HB (16): Dt ¼ DA ðtÞ þ DB ðtv Þ; Et ¼ EA ðtÞ þ EB ðtv Þ; DA ðtÞ ¼ 12½Az tf  ðAx tn þ Ay tg Þ=2; EA ðtÞ ¼ 14ðAx tn  Ay tg Þ; o Xn X Baz  12½Bax þ Bay  ; EB ðtv Þ ¼ 121 tv ½Bax  Bay : DB ðtv Þ ¼ 16tv a¼a;b

ð22Þ

a¼a;b

In the case Ban ¼ Bbn ¼ Bn , the Ban tv contributions to the ZFS parameters have the form DB ðtv Þ ¼ 16ðtv ÞðBx þ By Þ;

EB ðtv Þ ¼ 16ðtv ÞðBx  By Þ:

The V4 term (7) represents the single-ion contributions to the anisotropic DE. As it will be shown in part 4, the V4 term contributes to the axial parameter Dt of the ZFS Hamiltonian (20) in the form X DC ðtv Þ ¼ C 1 tv ½2paf  pan  pag ; ð23Þ a¼a;b

pal

2 ðk1 =Da1l Þ ,

C1 is the numerical coefficient. The DC(tv) contribution to the anisotropic double exchange ZFS where ¼ is linearly proportional to the DE parameter t0 = tv. The resulting ZFS parameters Dt, Et of the ZFS Hamiltonian H tZFS (Eq. (20)) of the anisotropic double exchange origin have the form Dt ¼ DA ðtÞ þ DB ðt0 Þ þ DC ðt0 Þ;

Et ¼ EA ðtÞ þ EB ðt0 Þ:

ð24Þ

The new ZFS Hamiltonian (20) acts between the states of different localizations. For example, for the S = 3/2 Anderson–Hasegawa states with S 0 = S, M 0 = M, the axial ZFS term of H tZFS (20) results in the following splitting: hU0 ðS; MÞjH tZFS jU0 ðS 0 ; M 0 Þi ¼ hU0a
b ðS; MÞjH tZFS jU0ab
ðS; MÞi ¼ Dt ½M 2  SðS þ 1Þ=3:

ð25Þ

The anisotropic DE contributions Dt (20) to the cluster ZFS have different sign for the Anderson–Hasegawa DE levels E0þ ðSÞ and E0 ðSÞ of different parity. The standard ZFS Hamiltonian H 0ZFS (2) [49] operates in the localized states and is not active between the 0 Ua
b ðSÞ and U0ab
ðSÞ states of different localization: hU0a
b ðSÞjH 0ZFS jU0ab
ðSÞi ¼ 0; hU0a
b ðSÞjH 0ZFS jU0a
b ðSÞi ¼ hU0ab
ðSÞjH 0ZFS jU0ab
ðSÞi ¼ DS ½M 2  SðS þ 1Þ=3:

4. Double exchange splittings in the excited states Since the anisotropic double exchange effects (part 2) are connected with the double exchange in the excited cluster states, we consider in this part the isotropic double exchange in the ground and excited states for the MV [d8–d9] cluster (the [Cu3+–Cu2+] pair).

34

M.I. Belinsky / Chemical Physics 308 (2005) 27–42

For the ground Anderson–Hasegawa states U0 ðSÞ, the double exchange interaction ^ ab t0 ; H 0DE ¼ T between the U0a
b ðSÞ and U0ab
ðSÞ states of the [d8–d9] MV dimer results in the standard Anderson–Hasegawa splitting (Eq. (1)) E0 ð3=2Þ ¼ tv ;

E0 ð1=2Þ ¼ tv =2;

ð26Þ

tv ¼ hva jV^ ab jvb i, Fig. 1, center. The hjV^ ab ji term describes the isotropic Anderson–Hasegawa double exchange interaction of the direct type (the Coulomb interaction) or through the ligand bridges in the cluster between the same S states in the ja*bæ and jab*æ localizations. The double exchange in the cluster ground states set is the spin-dependent transfer of the extra d-hole (va) from the d8-ion in the ground state (w0a
½3 A2 ðe2 Þ ¼ jua va j, center a*) to the d9-ion (u0b ½2 A1g ðeÞ ¼ ub , center b). We will consider the cluster excited states formed by the levels, which are mixed by the SOC with the Pion excited ground state (part 2). For the d8a
-ion, the SOC V aSO ¼ f00 ~ si (f00 is the Griffith SOC parameter, ke ¼ f00 =2s [43]) li~ mixes the ground 3A2(e2) term with the excited 3T2(t2e) and 1T2(t2e) terms [43–45]. The ligand field interval D2(1T2,3A2) between the ground 3A2- and excited 1T2-term is approximately twice D1(3T2,3A2) for the d8-ion (Cu3+, Ni2+ [64,46,47]) and we shall consider here only the SOC admixture of the 3T2 excited term to the ground 3A2 term (Fig. 1). The l = n,g,f components of the wave functions of the excited 3T2 term of the d8-ion are represented in [45], for example, wfa
ð3 T2 ; f; M ¼ 1Þ ¼ jfa ua j. ~ For the d9b -ion in the localized ½d8a
–d9b  cluster, the orbital pffiffiffimomentum l admixes to the ground u-state only the ex0 0 cited states ukb ¼ nb and ukb ¼ gb : hujlx jni ¼ hujly jgi ¼ i 3 [45], which are separated from the ground state by the LF intervals D0n and D0g, respectively. There are two types of the considered lowest excited states of the localized cluster: 1) the U0a
b ½wla
ð3 T2 ; l; MÞu0b  ex0 cited states ð½d8a
ðeÞd9b ð0Þe Þ, Figs. 1, 2 (see part 2) and 2) the U00a
b ½w0a
ð3 A2 ; MÞukb  excited states ð½d8a
ð0Þd9b ðeÞe Þ, Fig. 1(b), 2. The ground ðU0a
b ðSÞÞ and excited ðU0ab
ðSÞ and U00ab
ðSÞÞ cluster states are not mixed by the double exchange: 0 hUa
b jV^ ab jU0ab
i ¼ hU0a
b jV^ ab jU00ab
i ¼ 0. The double exchange is active between the excited states Ufa
b ðSÞ and Ufab
ðSÞ: hUfa
b ð3=2ÞjV^ ab jUfab
ð3=2Þi ¼ tf ;

hUfa
b ð1=2ÞjV^ ab jUfab
ð1=2Þi ¼ tf =2;

ð27Þ

where tf ¼ hfa jV^ ab jfb i. The DE splittings for the excited states (Fig. 1(a), k = f) follow the Anderson–Hasegawa rule (Eq. (1)) Ef ðSÞ ¼ tf ðS þ 1=2Þ=2 þ J 1f ½SðS þ 1Þ  3=4 with the DE parameter tk = tf. Eq. (27) describes the spin dependent transfer of the extra d-hole (fa) from the excited j3 T2 ; fi state of the d8-ion to the ground state of the d9-ion. 0l For the double exchange between the excited states U0l a
b and Uab
(Fig. 2) with Ma = Mb, mb = ma we obtain (l = n,g): Ef

x

Excited States I

E'' +(3/2)

Excited States I

x

E' +(3/2)

9

8

8

9

[d a*(e)db (0)]e

[da (0)d b*(e)]e S=3/2 S=1/2

3J1x

S=1/2

S=3/2 8

9

[d a*(0)db (e)]e Excited States II

E1k ~ ~

Ground State Levels

8

9

x

E'' -(3/2) 0

S=1/2

E -(1/2)

localization |a*b> 8 9 [d -d ]

E0k

8

[da (e)d b*(0)]e Excited States II

~ ~

0

E +(1/2)

9

x

E' -(3/2)

E +(3/2)

S=3/2

[d a*(0)d b(0)]0

3J0x

S=1/2

S=1/2

3J0

0

0

E -(3/2)

Double Exchange Levels

9

8

[d a(0)d b*(0)]0 localization |ab*> 9 8 [d -d ]

Fig. 2. The Anderson–Hasegawa E0 ðSÞ levels of the ground states set and the excited DE Ex ðSÞ states, x = n. The double exchange connects the 0 0 0 00 00 excited states 1) Uan
b ðSÞð½d8a
ðeÞd9b ð0Þe Þ and Uabn
ðSÞð½d9a ð0Þd8b
ðeÞe Þ and 2) Uan
b ðSÞð½d8a
ðeÞd9b ð0Þe Þ and Uabn
ðSÞð½d9a ðeÞd8b
ð0Þe Þ fU0nab
ðSÞ and Ua
nb ðSÞg.

M.I. Belinsky / Chemical Physics 308 (2005) 27–42

hUla
b ð3=2ÞjV^ ab jUlab
ð3=2Þi ¼ tl =4;

35

hUla
b ð1=2ÞjV^ ab jUlab
ð1=2Þi ¼ tl =8;

ð28Þ

where tn ¼ hna jV^ ab jnb i and tg ¼ hga jV^ ab jgb i are the parameters of the double exchange in the excited cluster states ~ b 6¼ M a ; m0 6¼ mb and Un and Ug . The DE coupling mixes also the excited states Ula
b and Ulab
ðl ¼ n; gÞ with M a ~ b , for example M ¼ M a þ mb ¼ m0a þ M pffiffiffiffi l 3    

w
T2l ; M a ¼ 1 ub ðmb ¼ 1=2ÞjjV^ ab jjua ðm0 ¼ 1=2Þwl
3 T2l ; M ~ b ¼ 0 ¼ tl =4 2: ð29Þ a a b The double exchange between the second type ðU00a
b ð½d8a
ð0Þd9b ðeÞe Þ and U00ab
ð½d9a ðeÞd8b
ð0Þe ; l ¼ n; gÞ is forbidden

excited

states

hU00a
b ½w0a
ð3 A2 Þulb jV^ ab jU00ab
½ula w0b
ð3 A2 Þi ¼ 0:

of

different

localization ð30Þ

The double exchange coupling connects the excited cluster states of the first U0a
b ðSÞ and second U0ab
ðSÞ type of different localization pffiffiffi ð31Þ hjwla
ð3 T2l ; 0Þub kV^ ab kla w0b
ð3 A2 ; 0Þji ¼ tv 3=4; l = n,g. In this case, the double exchange interaction in the excited states is described by the DE parameter tv of the ground states (Eq. (26)). In Fig. 1(b), this DE coupling between the excited states I ð½d8a
ðeÞd9b ð0Þe Þ and II ð½d9a ðeÞd8b
ð0Þe Þ is represented schematically (an analogous ½d8a
ð0Þd9b ðeÞe  ½d9a ð0Þd8b
ðeÞe DE coupling is not shown). 0l 0l 00l The DE coupling between the l (=n,g) excited states 1) U0l a
b and Uab
(Eqs. (28) and (29)) and 2) Ua
b and Uab
(Eq. (31)) results in the eight l DE excited levels, for example: Elfþg ð3=2Þ ¼ Dl þ 3J l  18tl  fþg½d2  14dtl þ 641 ð48t2v þ t2l Þ1=2 ; 1 Elfþg ð1=2Þ ¼ Dl  161 tl  fþg½d2  18dtl þ 256 ð48t2v þ t2l Þ1=2 ;

Dl ¼

1 ðD0l 2

þ D1l Þ;

d¼

d0l

¼

1 ½D 2 1l

 D0l þ J 1l  J 0l ;

ð32Þ

Jl ¼

1 ðJ 2 0l

þ J 1l Þ:

The Ex ðSÞ excited DE states (x = n) are shown in Fig. 2, the g excited levels are not shown. The double exchange splittings of the excited states depends on two DE parameters tv and tl.

5. ZFS of the Anderson–Hasegawa levels E 0% ðS ¼ 3=2Þ The zero-field splittings of the high-spin states S of the pure exchange mononuclear clusters are described by the standard ZFS Hamiltonian (2) [49]. The axial DS and rhombic ES terms in the ZFS Hamiltonian (2) are connected with the anisotropy parameters Dnn by the standard relations [42–47,49]: 2 2 2 ^^ ~^ ~ SD S ¼ Dxx S^x þ Dyy S^y þ Dzz S^z ;

DS ¼ Dzz  ðDxx þ Dyy Þ=2;

ES ¼ ðDxx  Dyy Þ=2:

The parameters of anisotropy DS and ES determined by the anisotropic (pseudodipolar) exchange are proportional to the exchange integrals Jn [48,49]. As it was shown in part 2 for the MV clusters with the migration of the extra electron, an anisotropic double ext change coupling H AN DE (19) and corresponding ZFS Hamiltonian H ZFS (20) include the terms proportional to the DE parameters tl and tv. Let us consider the correlations between the parameters Antl, Bntv of anisotropy of the DE interaction (Eqs. (20)), the axial and rhombic ZFS parameters Dt and Et (Eq. (22)) of the anisotropic DE origin and the SOC mixing coefficients for the [d8–d9] MV cluster and the DE parameters tl. As a first example, we consider the [d8–d9] cluster with the ground state d3z2 r2 ¼ u of the d9-ion. For finding the contributions of the anisotropic DE coupling (V1–V4 terms) to the axial Dt and rhombic Et ZFS parameters (Eq. (22)) of the E±(3/2) DE levels, we shall calculate in this part the zero-field splittings of the DE levels in two steps. First, we shall consider the renormalized wave functions of the MV [d8–d9] cluster, which are formed by the SOC admixture of 1) the excited 3T2 state to the ground 3 A2 state of the d8-ion and 2) the excited n, g states to the ground state u of the d9-ion. Second, the SOC admixed renormalized spin functions of the cluster are used to calculate the DE splittings including the anisotropic terms for the DE matrix elements hUa
b ðS ¼ 3=2; MÞjH DE jUab
ð3=2; M 0 Þi with jMj = jM 0 j = 3/2, 1/2 and M 0 = M  2 , which result in contributions to the axial and rhombic ZFS parameters. The results are compared with the ZFS and cluster axial Dt and rhombic Et ZFS parameters of the effective ZFS Hamiltonian H tZFS of the anisotropic DE origin (Eqs. (20), (22) and (24)). The spin–orbit coupling VSO admixes the excited 3T2(te) state (Eq. (7)) to the ground 3A2(e2) state [43–45]. For example, the renormalized wa (3A2, M = 0) function in the first order perturbation has the form

36

M.I. Belinsky / Chemical Physics 308 (2005) 27–42

wð3 A2 ; M ¼ 0Þ ¼ w0 ð3 A2 ; M ¼ 0Þ  ic1n ½w1 ð3 T2n ; M ¼ 1Þ þ w1 ð3 T2n ; M ¼ 1Þ  c1g ½w1 ð3 T2g ; M ¼ 1Þ  w1 ð3 T2g ; M ¼ 1Þ:

ð33Þ p ffiffi ffi The SOC admixture coefficients for the d8-ion have the form c1f ¼ 2k1 =D1f ; c1l ¼ 2k1 =D1l ; l ¼ n; g. The k1/D1f and k1 /D1l parameters determine the axial anisotropy of g-factors and ZFS of the single d8-ions (for example, k1/ D1f = 0.0099, k1/D1n = 0.0097 for Cu3+ and k1/D1f = 0.0245, k1/D1n = 0.0234 for Ni2+ in Al2O3 [64]). The spin–orbit coupling V~ SO ðV~ SO ¼ f00 lsÞ mixes the ground u and excited n,g states of the d9-ion [45] pffiffiffi pffiffiffi huð1=2ÞjV~ s0 jnð 1=2Þi ¼ i 3f00 =2; huð1=2ÞjV~ s0 jgð 1=2Þi ¼ 3f00 =2: The renormalized SOC-admixed ground state function ub of the d9b -ion of the ½d8a –d9b  cluster has the form ub ð1=2Þ ¼ u0b ð1=2Þ  icb0n nb ð 1=2Þ cb0g gb ð 1=2Þ; pffiffiffi where cb0l ¼ ð 3k0 =2Db0l Þ; l ¼ n; g. The renormalized ground state functions w and u of individual ions were used for construction of the cluster wave functions jwuj. Calculating the double exchange for the S = 3/2 cluster ground states U0 ðS; MÞ (Fig. 1) and perturbations from the excited states of the d8- and d9-ions, we obtain the following anisotropic DE splittings of the Anderson–Hasegawa ground set states E0 ðS ¼ 3=2Þ: 0  U ðS ¼ 3=2; jMjÞjV^ ab jU0 ð3=2; jMjÞ ¼ tv  Da ðtÞ½jMj2  SðS þ 1Þ=3; ð34Þ pffiffiffi 0  U ð3=2; 3=2ÞjH DE jU0 ð3=2; 1=2Þ ¼  3Ea ðtÞ: The zero-field splittings induced by the anisotropic double exchange have the opposite sign for the E+(3/2) and E(3/2) Anderson–Hasegawa DE levels in accordance with Eqs. (20) and (25). The calculated axial Da(t) and rhombic Ea(t) parameters (34) of the anisotropic double exchange ZFS for the E+(3/2) DE level of the [d8–d9] MV cluster have the following form: 
 
 Da ðtÞ ¼ Dzz ðtÞ  Dxx ðtÞ þ Dyy ðtÞ =2 ; Ea ðtÞ ¼ Dxx ðtÞ  Dyy ðtÞ =2: ð35Þ The Dnn(t) components (35) of the DE anisotropy tensor are given by the expressions:
 Dzz ðtÞ ¼ tf K0ab ðfÞ  tv xðc1 Þ ;

 Dxx ðtÞ ¼ 12 tn K0ab ðnÞ þ tv r0 ðnÞ ; Dyy ðtÞ ¼ 12 tg K0ab ðgÞ þ tv r0 ðgÞ ; pffiffiffiffiffiffiffiffi X  a a  1 X a 2 c1l c0l ; xðc1 Þ ¼ ½ðc1f Þ  ðca1n Þ2  ðca1g Þ2 ; K0ab ðlÞ ¼ 12ðca1l cb1l Þ; r0 ðlÞ ¼ 2=3 6 a¼a;b a¼a;b

ð36Þ

where l = n, g, f. The axial anisotropic DE parameter of the ZFS Da(t) was obtained in [62] for the case r0(l) = 0. The comparison of the zero-field splittings (34)–(36) with the results of the effective ZFS Hamiltonian H tZFS (Eq. (20)) of the anisotropic DE origin, results in the correlations Dt ¼ Da ðtÞ;

Et ¼ Ea ðtÞ:

The comparison of the tl K0ab ðlÞ terms in Eq. (34) with the DA(t), EA(t) (22) contributions to the cluster ZFS parameters Dt, Et (20) shows the following correspondence for the DE anisotropy coefficients Antl (Eq. (9)): 2

Ax tn ¼ tn Kab ðnÞ ¼ tn ðc1n Þ =2;

2

Ay tg ¼ tg Kab ðgÞ ¼ tg ðc1g Þ =2;

2

Az tf ¼ 2tf Kab ðfÞ ¼ tf ðc1f Þ :

ð37Þ

Using the coefficients of the SOC admixture c1l for the d8-ion, one obtains the coefficients An (10). The Antl coefficients demonstrate the double exchange anisotropy connected with the SOC admixture for the same d8 ions on different centers ðd8a
and d8b
Þ in different configurations (localizations ja*bæ and jab*æ). The comparison of the tvr0(n) terms in (Eq. (36)) with the DB(t), EB(t) (22) contributions to the cluster ZFS parameters Dt, Et results in the correlation: pffiffiffi pffiffiffi ð38Þ Bax tv ¼ tv ð 6ca1n ca0n Þ; Bay tv ¼ tv ð 6ca1g ca0g Þ; Baz tv ¼ 0: Using c1l and c0l for the d8 and d9 ions, one obtains the coefficients Ban (17). The Ban tv parameters demonstrate the DE anisotropy connected with the different SOC admixture for the different ions: d8a
and d9a on the a-center in localizations ja*bæ and jab*æ. The tvx(c1) term (tv) in the Dzz(t) component (Eq. (36)) represents the single-ion contribution DC(tv) (23) of the V4 operator (Eq. (7)) to the axial ZFS parameter Dt (Eqs. (20) and (24). The comparison of DC(tv) (23) with Eq. (34) results in the coefficient C1 = 1/3.

M.I. Belinsky / Chemical Physics 308 (2005) 27–42

37

Anisotropic double exchange contributions to the ZFS of the DE E0 ðSÞ levels (Eqs. (20), (22) and (24)) are proportional to the DE parameters tl (l = n, g, f, v). Anisotropy originates in the difference among the double exchange (ET) parameters tn, tg, tf and tv. In the case of the low point symmetry of the ions in the cluster, the energy LF intervals D1n, D1g and D1f (D0n and D0g) and corresponding c1u(c0u) coefficients of the SOC admixture are different which gives rise to anisotropic DE parameters Antl, Bntv (Eqs. (9), (10), (16), (17) and (19)). In the case tn 6¼ tg and cin 6¼ cig, one obtains Dxx(t) 6¼ Dyy(t) and the rhombic ZFS parameter Ea(t) = [Dxx(t)  Dyy(t)]/2 (35) formed by the anisotropic double exchange is different from zero. To estimate the value of the anisotropic DE contributions to ZFS, letÕs consider the parameters of anisotropy in 8 0 Eqs. (35)–(37). For the single the form: pffiffiffi d -ion with g-factor g , the SOC admixture coefficients c1l have 0 0 c1f ¼ Dgz =4; c1l ¼ Dgn =4 2 since the SOC determines the three g-shift components [42–47]: Dg0n ¼ 8ðke =D1l Þ, 9 2+ where n = x(y){z} for l = n(g){f}. For p the ffiffiffi d -ion (Cu 0 ) with pffiffiffi the d3z2 r2 ground state and g-factor g0, the coefficients 0 of the SOC admixture are c0n ¼ Dgx =4 3; c0g ¼ Dgy =4 3; c0f ¼ 0, since the three g-shift components are given by: Dg0x ¼ 6ðk0 =D0n Þ; Dg0y ¼ 6ðk0 =D0g Þ; Dg0z ¼ 0 [42–47]. Using these correlations in (35), one obtains the following expressions:  2  2  2 K0ab ðnÞ ¼ Dg0x =64; K0ab ðgÞ ¼ Dg0y =64; K0ab ðfÞ ¼ Dg0z =32; ð39Þ     2 2 2 xðc1 Þ ¼ fðDg0z Þ  ½ðDg0x Þ þ ðDg0y Þ =2g=48; r0 ðnÞ ¼ Dg0x Dg0x =24; r0 ðgÞ ¼ Dg0y Dg0y =24: Using the g-shifts of the localized d8- and d9-ions, the coefficients of the DE anisotropy Antl (Eqs. (9) and (10)) and Bntv (Eqs. (16) and (17)) may be represented in the form: 2

Ax tn ¼ 641 tn ðDg0x Þ ;

2

Ay tg ¼ 641 tg ðDg0y Þ ;

Bx tv ¼ 161 tv ðDg0x Dg0x Þ;

2

Az tf ¼ 161 tf ðDg0z Þ ;

By tv ¼ 161 tv ðDg0y Dg0y Þ;

Bz ¼ 0:

ð40Þ

The axial Da(t) = Dt and rhombic Ea(t) = Et contributions (35) (of the anisotropic DE origin) to the ZFS parameters of the DE term E0þ ð3=2Þ are given by equation: h  i  2  0 2 8 0 2 0 0 0 0 0 1 1 1 8 Da ðtÞ ¼ 32 tf ðDgz Þ  32tv xðc1 Þ  8½tn Dgx þ 3tv Dgx Dgx   8½tg Dgy þ 3tv Dgy Dgy  ;   ð41Þ  2  0 2 8 0 0 0 0 0 1 8 Ea ðtÞ ¼ 256 ½tn Dgx þ 3tv Dgx Dgx   ½tg Dgy þ 3tv Dgy Dgy  : The value and sign of the ZFS parameters of the anisotropic DE coupling depend on the relation between the double exchange parameters tf, tn, tg of the excited states and tv of the ground states and anisotropic coefficients (g-shifts). For the E0 ð3=2Þ lowest DE level of the MV [d8–d9] cluster, the calculations of the anisotropic DE contributions 0 Da ðtÞ and E0a ðtÞ to the ZFS parameters result in opposite signs in comparison with Eq. (41) 
 
 D0a ðtÞ ¼  Dzz ðtÞ  Dxx ðtÞ þ Dyy ðtÞ =2 ; E0a ðtÞ ¼  Dxx ðtÞ  Dyy ðtÞ =2: ð42Þ These different signs of the anisotropic DE zero-field splittings for the E0þ ð3=2Þ and E0 ð3=2Þ Anderson–Hasegawa double exchange states are in accordance with the effective ZFS Hamiltonian H tZFS (20) and the action of the DE operator ^ ab (Eq. (21)). The ZFS of different sign of the anisotropic double exchange origin for the Anderson–Hasegawa states T E0 ð3=2Þ are shown in Fig. 3. In the localized system, the ZFS of an axial MV [d8–d9] dimer (ES = Et = 0) is determined by the axial ZFS parameter DS of the ZFS Hamiltonian H 0ZFS (2) (Fig. 3(a)). In the cluster with strong DE splitting t  J, the anisotropic DE contribution Dt (Eq. (20)) to the cluster ZFS parameter D½E0 ðSÞ can enlarge essentially or reduce (and change the sign of) the cluster ZFS parameter D = DS ± Dt (Fig. 3(b)). DS is the axial ZFS parameter, which does not depend on the DE parameters tl and includes the individual, dipole–dipole and anisotropic (pseudodipolar) exchange contributions to DS in the DE dimer. One can estimate the axial ZFS D0a ðtÞ ¼ D0t parameter for the lowest E0 ð3=2Þ DE level of the [d8–d9] cluster in the case of the axial symmetry (tn = tg = tx, tf = tz, gx = gy = g^, gz = gi) n o D0a ðtÞ ¼ 321 14½tx ðDg0? Þ2 þ 83tv Dg0? Dg0?  þ ½32tv xðc1 Þ  tz ðDg0k Þ2  ; E0a ðtÞ ¼ 0; ð43Þ using the Dg0k ; Dg0? values for the Cu3þ ðDg0k ¼ 0:0788; Dg0? ¼ 0:077Þ and Ni2þ ðDg0k ¼ 0:196; Dg0? ¼ 0:187Þ ions [64]. The DE parameters tv and tl may be of the order of 103–104 cm1 (experimentally observed DE parameters are t0 = 2250–4715 cm1 [24,26,27]). In the case tv,tx  tz, for example, one can estimate for the MV [Ni2+–Ni+] dimer D0a ðtÞ  1 cm1 for t ¼ 103 cm1 and D0a ðtÞ  10 cm1 for t ¼ 104 cm1 . These anisotropic DE contributions D0a ðtÞ

38

M.I. Belinsky / Chemical Physics 308 (2005) 27–42 0 E +(3/2)=3J0+tv |M|=3/2

+(DS+Dt)

|M|=1/2

-(DS+Dt)

3J0+DS

|M|=3/2

S=3/2

~ ~ ~~

2(DS+Dt)

S=3/2

2tv

2DS

|M|=1/2

~~

3J0-DS

~ ~ +(DS-Dt)

|M|=3/2 ~~

8

9

2(DS-Dt) S=3/2

[d -d ] 0

(a)

E -(3/2)=3J0-tv

-(DS-Dt)

|M|=1/2 (b)

Fig. 3. The zero-field splittings of the Anderson–Hasegawa E0 ðS ¼ 3=2Þ levels of the MV [d8–d9] cluster with axial anisotropy. (a) The zero-field 0 splitting 2DS of the localized E0 ð3=2Þ pure exchange state. (b) the ZFS D ZFS ¼ 2ðDS  Dt Þ of the ground E ðS ¼ 3=2Þ double exchange level; the ZFS 0 Dþ ¼ 2ðD þ D Þ of the excited E ðS ¼ 3=2Þ DE level D > 0; D < 0. S t S t ZFS þ

may exceed (or be of the same order as) the axial ZFS parameters of the single d8-ion: Dl (Ni2+) = 1.312 cm1 and Dl (Cu3+) = 0.1884 cm1 [64] and have different sign. In the case that the ground FM state Sgr = 3/2 of the [Ni(II)Ni(I)] cluster [7] is formed by strong DE, the anisotropic DE axial ZFS parameter D0a ðtÞ can essentially contribute to the experimentally observed [7] large positive axial ZFS parameter D3/2 P 30 cm1. As a second example, we shall consider the anisotropic double exchange in the [d8–d9] MV cluster with the dx2 y 2 ð¼ vÞ ground state of the d9-ion. In this case, the Anderson–Hasegawa DE parameter t0 (Eq. (1) for the ground state is t0 = tu: E0 ð3=2Þ ¼ tu ; E0 ð1=2Þ ¼ tu =2. For the d9-ion, the SOC admixes the n, g, f excited states, separated by the LF intervals D00n ; D00g ; D00f , respectively, from the ground v state. The spin-admixed ground state has the form: vð1=2Þ ¼ v0 ð1=2Þ  i~c0n nð 1=2Þ  ~c0g gð 1=2Þ  i~c0u fð1=2Þ; k0 =D00f ;

ð44Þ

k0 =2D00l ; l

~c0l ¼ where ~c0f ¼ ¼ n; g. Considering the DE with the taking into account the perturbations from the 3T2 level for the d8-ion and SOC admixture (44) for the d9-ion, one obtains the anisotropic DE ZFS parameters Dt and Et of the ZFS Hamiltonian H tZFS (20): ~ zz ðtÞ  1½D ~ xx ðtÞ þ D ~ yy ðtÞg; D t ¼ fD 2 ~ xx ðtÞ ¼ 1½3tn Kab ðnÞ þ tu r ~ðnÞ; D 2 ~ zz ðtÞ ¼ tu ½r ~ðfÞ  xðc1 Þ; D

~ xx ðtÞ  D ~ yy ðtÞ; Et ¼ 12½D


 ~ yy ðtÞ ¼ 1 3tg Kab ðgÞ þ tu r ~ ð gÞ ; D 2

~ðlÞ ¼ r

pffiffi 2 3

X

ð45Þ

ca1l~ca0l ;

a¼a:b

~ zz ðtÞ anisotropic comwhere the anisotropic DE parameters Kab(/) and x(c1) are given by Eq. (39). In this case, the D ~c0l (45) ponent is proportional to the DE parameter tu of the DE in the ground state. coefficients  TheSOC0admixture   have a form ~c0f ¼ D~ g0z =8; ~c0n ¼ D~ g0x =4; ~c0g ¼ D~ g0y =4, since D~ g0z ¼ 8 k0 =D00f ; D~ gx ¼ 2 k0 =D00n [49]. 6. Anisotropic (pseudodipolar) exchange in the localized [d8–d9] cluster In the pure exchange [dn–dn] clusters, the axial DS and rhombic ES components (2) of the ZFS tensor are the combination of the single ion contribution D0 and E0 (for si > 1/2), dipolar contributions Dd and Ed and anisotropic (pseudodipolar (pd)) exchange contribution [49] DS ¼ D0 þ Dd þ Dpd ; n

ES ¼ E0 þ Ed þ Epd :

ð46Þ

n

For the monovalent [d –d ] dimers, the contributions of the individual ions ZFS parameters Dl and El H 0 ¼ Dl ½s2iz  si ðsi þ 1Þ=3 þ El ðs2ix  s2iy Þ

ð47Þ

to the cluster ZFS parameters (D0 = aDl, E0 = bEl [49]) were considered in [49,53,54]. D0 = (1/3)Da, E0 = (1/3)Ea for the ½d8a  d9b  cluster with Da, Ea 6¼ 0, Db, Eb = 0. The average of the local ZFS contributions due to the delocalization was considered in [15].

M.I. Belinsky / Chemical Physics 308 (2005) 27–42

39

The dipole–dipole contributions Dd and Ed to DS and ES are given by equations [49] Dd ¼ ðb2 =3r3 Þ½2g2z þ ðg2x þ g2y Þ=2;

Ed ¼ ðb2 =2r3 Þðg2x  g2y Þ:

ð48Þ

The pseudodipolar (anisotropic) exchange interaction originates from the combined effect of the spin–orbit admixture of the excited d-ion states and the isotropic Heisenberg exchange Hex in the excited states ([48,49]). For the monovalent [dn–dn] exchange dimer of the two identical ions with the orbital singlet ground state w0a , the third-order perturbation term in the Hamiltonian of an anisotropic (pseudodipolar) exchange has the form [48,49]: E1 E1 0D 0D



D E wka w0b V aS0 w0a w0b w0a w0b V aS0 wka w0b X @ A wk w0 jH ex jwk w0 @ A: H 0pd ¼ ð49Þ a b a b Eka  E0a Eka  E0a k The operator of anisotropic (pseudodipolar) exchange (49) includes the terms of the SOC admixture of the excited states wka to the ground state w0a on the same center a. The calculations of the pseudodipolar (anisotropic) exchange for the [Cu2+–Cu2+] ([d9–d9]) [49] pair and [Ti3+–Ti3+] ([d1–d1]) pair [65] result in the anisotropic (AN) exchange interaction in the form [49] H 0pd ¼ ax J x S 1x S 2x þ ay J y S 1y S 2y þ az J z S 1z S 2z ;

2

ax ¼ 8c20n ¼ 16ðDg0x Þ ;

2

ay ¼ 8c20g ¼ 16ðDg0y Þ ;

In this case, the axial Dpd and rhombic Epd parameters in the ZFS Hamiltonian (2) are [49]   Dpd ¼ ð2az J z  ax J x  ay J y Þ=4; Epd ¼ ax J x  ay J y =4:

az ¼ 0:

ð50Þ

ð51Þ

The Jn parameters in the AN (pseudodipolar) exchange Hamiltonian (50) for the monovalent ions represent the Heisenberg isotropic exchange J n ¼ hu0a unb jjunb u0a i between one d-electron in the ground state u0a ðcenter aÞ and the other electron in the excited state unb ðcenter bÞ [49]. The pseudodipolar (AN) exchange Hamiltonian (49) for the [Ni2+–Ni2+] pair was considered in [48]. For comparison of the AN (pseudodipolar) exchange and anisotropic double exchange, let us consider the AN (pseudodipolar) exchange in the localized ðt ¼ 0Þ½d9a –d8b  cluster. In this case of the Heisenberg exchange between different ions d9a and d8b with different ground states u0a and w0b , the third-order pseudodipolar terms have the following form:

0 0 12

b

!2 hu0a w0b V~ S0 u0a wkb i X X u0a w0b V aS0 uka w0b

0 k0  0 k 0 k 0 @ A hu0 wk0 H ^ ex u w i: H pd ¼ ua wb jH ex jua wb þ ð52Þ a b a b k 0 k0 0 Ea  Ea Eb  Eb k k0 The notations in Eq. (52) are the same as in Eqs. (4)–(6) (see part 2, Fig. 1). Following [48,49], one obtains the Hamiltonian of anisotropic (pseudodipolar) exchange interaction for the localized [d9–d8] cluster in the form:   ~ pd ¼ ax ðJ x ÞS 1x S 2x þ ay J y S 1y S 2y þ az ðJ z ÞS 1z S 2z ; H  2  2 2 2 ax ðJ x Þ ¼ 6J 1n c21n þ 4J 0n c20n ¼ 3½4J 1n ðke =D1n Þ þ J 0n ðk0 =D0n Þ  ¼ 163 ½J 1n Dg0x þ 49J 0n Dg0x ;  2  2 ð53Þ   2 2 ay J y ¼ 6J 1g c21g þ 4J 0g c20g ¼ 3½4J 1g ðke =D1g Þ þ J 0g ðk0 =D0g Þ  ¼ 163 ½J 1g Dg0y þ 49J 0g Dg0y ;  2 2 az ðJ z Þ ¼ 3J 1f c21f ¼ 12J 1f ðke =D1f Þ ¼ 163 J 1f Dg0z : The anisotropic exchange coefficients an(Jn) (Eq. (53)) may be represented in the form ax ðJ x Þ ¼ 12½a0x J 0n þ a1x J 1n ;

ay ðJ y Þ ¼ 12½a0y J 0g þ a1y J 1g ;

az ðJ z Þ ¼ 12a1z J 1f ;

ð54Þ

where a0n (50) and a1n are the AN exchange coefficients of the [d9–d9] and [d8–d8] pairs, respectively. The coefficients an(Jn) of the exchange anisotropy are proportional to Jn(Dgn)2. The exchange integrals J0l{J1l} describe the Heisenberg exchange coupling between the ground state w0 {u0 = u} of the d8 {d9} ion and excited state ul {wl} of the d9 {d8} ion. The exchange integrals J0l{J1l} (l = n,g) determine the Heisenberg exchange splittings for the localized cluster excited states II {I}, Fig. 2:   J 1f ¼ J ½wfa ð3 T2f Þu0b  ¼ 12ðJ uu þ J uf Þ; J 1l ¼ J ½wla ð3 T2l Þu0b  ¼ 12 J ul þ 14J uu þ 34J vv ; ð55Þ J 0n ¼ J 0x ¼ J ½w0a ð3 A2 Þnb  ¼ 12ðJ un þ J vn Þ; J 0g ¼ J 0y ¼ J ½w0a ð3 A2 Þgb  ¼ 12ðJ ug þ J vg Þ; J 0f ¼ 0: The parameter of the Heisenberg exchange interaction ðH 0 ¼ 2J 0~ S1~ S2 Þ between the d8 and d9 ions in the localized 0 3 0 ~ pd and E ~ pd ground states is J 0 ¼ J ½wa ð A2 Þub  ¼ 1=2ðJ uu þ J uv Þ The pseudodipolar (AN) exchange contributions D to the axial DS and rhombic ES ZFS parameter (2) have the form:

40

M.I. Belinsky / Chemical Physics 308 (2005) 27–42

~ pd ¼ 1½2az ðJ z Þ  ax ðJ x Þ  ay ðJ y Þ; D 6

~ pd ¼ 1½ax ðJ x Þ  ay ðJ y Þ; E 6

~ ZFS ¼ D ~ pd ½S^2  S ðS þ 1Þ=3 þ E ~ pd ðS 2  S 2 Þ: H z x y

ð56Þ

Comparison of the AN (pseudodipolar) exchange Hamiltonian Hpd (53) for the localized [d8–d9] pair with the anisotropic double exchange operator H AN DE (Eq. (19)) for the delocalized cluster demonstrates the following essential differences. The Hamiltonian of the anisotropic DE H AN DE (Eq. (19)) acts between the different localized states and includes the DE operator ^sab . The AN (pseudodipolar) exchange operator Hpd (53) acts in the localized states and is not active between the states of different localization. The coefficients of anisotropy Antl, Bntv in the anisotropic DE Hamiltonian H AN DE (Eq. (19)) are proportional to the double exchange parameters tn, tg, tu and tv. In comparison, the coefficients of anisotropy an(Jn) in the Hamiltonian of the AN (pseudodipolar) exchange interaction Hpd (53) are proportional to the exchange parameters J0l, J1l, J0. 2 The estimate of the anisotropic double exchange is An ðtl Þ  tl ðDgn Þ ; Bn tv  tv ðDgn Dg0n Þ; the estimate of the AN 2 (pseudodipolar) exchange is an(Jn)  Jn(Dgn) . Since usually the DE interaction is stronger than the Heisenberg exchange coupling (t  J), the contributions of the anisotropic DE interaction H AN DE (Eq. (19)) to the resulting anisotropy may be stronger than the exchange (pseudodipolar) anisotropy in the MV dimers with the extra electron migration. ~ ZFS (56) of the AN (pseudodipolar) exchange origin for the pure exchange Let us compare the ZFS Hamiltonian H localized system with the ZFS Hamiltonian H tZFS (20) of the anisotropic double exchange origin for the delocalized MV system: (1) The ZFS operator H tZFS (20) acts between the states of different localization and includes the DE operator ^ ab . The ZFS Hamiltonian H ~ ZFS (56) for the exchange levels S acts in the localized S states and is not active between T the states of different localization. (2) The anisotropic DE contributions to the ZFS parameters have different sign: ±Dt (±Et) for the Anderson–Hasegawa E0þ ðSÞ and E0 ðSÞ DE levels of the ground set (Eqs. (25), (41), (42)). The contribu~ ZFS (56) to the resulting ZFS parameters are the same for the E0 ðSÞ and E0 ðSÞ DE levtion of the ZFS Hamiltonian H þ  els. (3) The anisotropic DE ZFS parameters DA(t) and EA(t); DB(t) and EB(t) (22), (41)–(44) are proportional to the ~ pd and E ~ pd in H ~ ZFS (56) are proportional to the double exchange parameters tl. The AN exchange ZFS parameters D exchange integrals Jn (Eqs. (53) and (56)). The estimates of the coefficients of the exchange anisotropy an(Jn) and the DE anisotropy Antl, Bnt0 are an(Jn)  Jn(Dgn)2, Antn  tn (Dgn)2, Bnt0  tv(Dgn)2. Since t  J, the anisotropic double exchange contributions Dt and Et (Eqs. (22), (41)–(44)) to the resulting ZFS parameters D and E may be stronger than ~ pd ðJ Þ and E ~ pd ðJ Þ (Eqs. (56) and (53)) or individual ZFS the pure anisotropic (pseudodipolar) exchange contributions D contributions D0, E0. In the resulting cluster ZFS parameters of the DE levels E0 ðSÞ , the anisotropic double exchange contributions Dt and Et (Eqs. (22), (41)–(44)) should be taken into account for the MV dimers with strong double exchange inter-ion interaction D½E0 ðSÞ ¼ DS  Dt ;

E½E0 ðSÞ ¼ ES  Et ;

ð57Þ

where DS and ES are the ZFS parameters, which include the individual, dipole–dipole, and anisotropic (pseudodipolar) exchange contributions which do not depend on the DE parameters tl. P~ ~
ij Sj considered Let us compare the anisotropic double exchange H AN Si C DE (19) with the anisotropic exchange term 2
for the MV system in [61]. This AN exchange term is proportional to the exchange parameter J : Cij  J ð#DgÞ [61] and describes the exchange anisotropy. The anisotropic double exchange H AN DE (19) differs from this AN exchange since the coefficients of the DE anisotropy are linearly proportional to the double exchange parameters Antl  tl(Dgn)2. As was shown in [62], the antisymmetric DE   ~ab T ^ ab ~ H ASDE ¼ 2iK Sa Sb  ~ mixes the Anderson–Hasegawa DE states E0þ ðSÞ and E0 ðSÞ with the same S of different parity. The antisymmetric DE mixes also the DE levels E0þ ðSÞ½E0 ðSÞ and E0þ ðS 0 Þ½E0 ðS 0 Þ with different S (S 0 = S + 1) of the same parity. The mixing of the DE levels by the antisymmetric DE results in the contribution to the ZFS splitting 2 DK  K 2l J =tðt  nJ Þ  J n ð#Dgn Þ [62], where Kl  Jc1l(tv + tl), J is the angle of the tilt of the CuL6 octahedra, 2 DK < Dt since Dt  tl ðDgn Þ and tl > J n :

7. Conclusion The anisotropic double exchange coupling and zero-field splittings determined by the anisotropic double exchange were considered on an example of the MV [d8–d9] cluster.

M.I. Belinsky / Chemical Physics 308 (2005) 27–42

41

The third-order perturbation terms (Eqs. (4)–(7)), including the spin–orbit coupling admixture of the excited states and the double exchange in the excited states, results in an anisotropic double exchange interaction or anisotropic spindependent electron transfer H AN DE (Eq. (19)). The coefficients of the DE anisotropy are linearly proportional to the DE 2 parameters tl and tv: An tl  tl ðDg0n Þ ; Bn tv  tv ðDg0n Dg0n Þ (Eq. (40)). The anisotropic double exchange is active between ~ S 0 ¼ S; S  2; the ground states of different localization of the extra electron ðU0a
b ðS; MÞ and U0ab
ðS 0 ; MÞ; AN ~ M ¼ M; M  2Þ. The operator H DE (19) describes the anisotropic spin–transfer interaction induced by the spin–orbit coupling. The anisotropic double exchange H AN DE and antisymmetric double exchange HASDE [62] should be added to the isotropic Anderson–Hasegawa double exchange Hamiltonian H 0DE ~ DE ¼ H 0 þ H ASDE þ H AN ; H DE DE as the terms which describe the spin–orbit coupling in the double exchange model. For the double exchange Anderson–Hasegawa levels E0 ðSÞ, the anisotropic DE interactions H AN DE (19) results in the zero-field splittings. This ZFS of the S > 1/2 DE states induced by the anisotropic DE is described by the effective ZFS ^ ab . The axial Dt and rhombic Et ZFS parameters of the ZFS Hamiltonian H tZFS (20), which includes the DE operator T t Hamiltonian H ZFS (20) are linearly proportional to the DE parameters tl, tv (Eqs. (22), (41)–(43)); Dt  t(Dgn)2, Dt  1 ~ pd ð J n ðDgn Þ2 Þ (56) contribucm1 may exceed the single-ion contribution and anisotropic (pseudodipolar) exchange D tions to the ZFS parameters. The effective ZFS Hamiltonian H tZFS (20) of the anisotropic DE origin operates in the basis of the ground states U0a
b ðS; MÞ and U0ab
ðS; M 0 Þ ðM 0 ¼ M; M  2Þ of different localization (Eq. (25)). The ZFS Hamiltonian H tZFS (20) differs from the ZFS Hamiltonian H 0ZFS (2). The standard ZFS Hamiltonian H 0ZFS (2) is not active between the states of different localization and acts only in the localized states. The anisotropic DE ZFS Hamiltonian H tZFS (20) should be added to the standard ZFS Hamiltonian H 0ZFS (2) in the MV clusters with the double exchange coupling ~ ZFS ¼ H 0 þ H t : H ZFS ZFS The anisotropic double exchange contributions to the cluster ZFS parameters have different sign for the Anderson– Hasegawa DE levels E0þ ðSÞ and E0 ðSÞ: D½E0 ðSÞ ¼ DS  Dt ; E½E0 ðSÞ ¼ ES  Et (Eqs. (25), (41) and (42), Fig. 3). The effect of the anisotropic DE in ZFS may be observed in the MV clusters with the high-spin ground spin, which may be prepared in the localized and delocalized form. The comparison of the ZFS parameters of the localized and delocalized cluster allows determining the delocalized zero-field splitting of the anisotropic double exchange origin. ~ pd (53). The anisotropic (pseudodipolar) exchange in the localized cluster [d8–d9] is described by the Hamiltonian H 2 2 The coefficients of the pseudodipolar exchange anisotropy an(Jn) are proportional to ½J 1n ðDg0n Þ þ cJ 0n ðDg0n Þ . In the MV [dn–dn + 1] dimers with the double exchange inter-ion coupling, the anisotropic inter-ion coupling is determined ~ by the anisotropic double exchange interaction H AN DE (19) and anisotropic (pseudodipolar) exchange coupling H pd ~ AN ¼ H ~ pd þ H AN . Since An tl  tl ðDg0 Þ2 ; Bn tv  tv ðDg0 Dg0 Þ (Eq. (40)) and an ðJ n Þ  ½J 1n ðDg0 Þ2 þ cJ 0n ðDg0 Þ2  (56): H DE n n n n n (Eq. (53)) and t  J, the double exchange anisotropy may be stronger than the pseudodipolar exchange anisotropy.

Acknowledgement I thank reviewer for helpful comments.

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