ZERO-FIELD SPLITTINGS FORMED BY ANTISYMMETRIC DOUBLE EXCHANGE IN MIXED-VALENCE [Fe(II)Fe(III)] CLUSTER
Moshe Belinsky
School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel,
The model of an antisymmetric double exchange (AS DE) interaction is developed for the mixed valence (MV) [Fe(II)Fe(III)] cluster. The spin-orbit coupling effect is considered for the MV [Fe(II)Fe(III)] dimer, in which strong isotropic Anderson-Hasegawa double exchange interaction forms isotropic ground state with maximal total spin S=9/2. The AS double exchange interaction mixes the Anderson-Hasegawa DE states with the same S and of the different parity. The AS double exchange and Dzialoshinsky-Moriya AS exchange mix the DE states with different S and of the same parity. The AS DE mixing of the Anderson-Hasegawa levels results in the AS DE contributions to the zero-field splittings, which depend on S and parity.
1
1. Introduction
The MV [Fe(II)Fe(III)] clusters are structural elements of many ferredoxins, enzymes and their synthetic bioinorganic model compounds [1]. In the localized [Fe(II)Fe(III)] clusters of the high-spin iron ions (and ), the Heisenberg exchange interaction forms the cluster states with the total spin S=9/2, 7/2, 5/2, 3/1, ½;, . The spin-dependent resonance splittings of the S states due to the hopping of the extra electron between the MV ions is described by the Anderson-Hasegawa (AH) [2] model of the double exchange (DE) coupling:
, (1.1)
. In the MV [Fe(II)Fe(III)] clusters, the AH double exchange (1.1) and Heisenberg exchange interactions forms the isotropic DE states
E±0(S) =±(S+1/2)t0/(2s0+1)-S(S+1) . (1.2)
The DE concept is widely used in the theory of the MV [Fe(II)Fe(III)] clusters in bioinorganic chemistry of iron-sulfur proteins and in magnetism of the MV compounds.
The isotropic AH double exchange and Heisenberg exchange in dimeric MV clusters has been the subject of theoretical and experimental investigations [3-25]. Strong double exchange interaction with the DE parameter =1350 cm-1 ( cm-1) destroys the Heisenberg antiferromagnetic ordering (=70 cm-1, ) and results in the delocalized ground state of the [Fe(II)Fe(III)] cluster in [9-12]. The MV [Fe(II)Fe(III)] dimers with strong double exchange (3000-4715 cm-1, =600-943 cm-1, ) and delocalized ground state were found also in the model clusters [13-16]. The centers of the Clostridium pasterianum mutant 2Fe ferredoxins possess the delocalized ground state [18-20] due to strong DE interaction (|t| ~ 2250 cm-1 [19]). Valence delocalized pairs with strong DE were found in a variety of the trimeric and tetrameric iron-sulfur clusters in ferredoxins, enzymes and synthetic models [1].
Zero-field splittings (ZFS) of the delocalized ground state with =9/2 of the [Fe(II)Fe(III)] dimers were determined from the EPR, Mössbauer and MCD data. The ZFS of the delocalized cluster ground state with was described by the standard effective ZFS Hamiltonian [26-29]:
, (1.3)
where and are the axial and rhombic cluster ZFS parameters, respectively. The delocalized ground state of the model {ferredoxin} [Fe(II)Fe(III)] clusters are characterized by large positive [9, 16, 17] {negative [18-20]} axial ZFS parameter. The zero-field splittings and of the individual and ions were considered the origin of the ZFS of the cluster delocalized state of the MV [Fe(II)Fe(III)] cluster [9] ().
In the pure exchange mononuclear dimers, the anisotropic (pseudodipolar) exchange, single-ion ZFS contributions [29, 30], Dzialoshinsky-Moriya antisymmetric exchange [31-34], dipole-dipole interaction strongly contribute to the cluster ZFS parameters [29].
For the MV clusters, it was shown that the antisymmetric double exchange [35, 36] contribute to ZFS of the high-spin cluster S levels of the MV dimer [35].
The aim of this work is the consideration of the antisymmetric double exchange interaction in the [Fe(II)Fe(III)] cluster with strong Anderson-Hasegawa DE splittings and finding the ZFS contributions connected with the antisymmetric double exchange. The taking into account of the spin-orbit coupling in the theory of the Anderson-Hasegawa double exchange for dimeric [Fe(II)Fe(III)] clusters leads to antisymmetric double exchange interaction. An antisymmetric double exchange mixes the AH double exchange states with the same total spin S of the different parity and also the DE states with different total spin S of the same parity. The AS double exchange contributes to the zero-field splittings of the AH levels E±0(S). The AS DE contributions to ZFS depend on the total spin and parity of the DE levels.
2. Hamiltonian of Antisymmetric Double Exchange in [Fe(II)Fe(III)] MV Cluster
The isotropic Anderson-Hasegawa [2] DE interaction in dimeric MV cluster may be described by the effective Hamiltonian of the double exchange or spin-dependent electron transfer (ET):
, (2.1)
where the DE operator is determined by the equation
,
(2.2)
{} are the ground set spin wave functions in the case of the |a*b> {|ab*>} localization of the extra electron on the center a* {b*} without taking SOC into account , The operator (2.1) describes the Anderson-Hasegawa DE splitting (1.1) connected with the ET between the-ion in the ground state and the -ion in the ground state.
We will consider antisymmetric DE in the delocalized MV cluster [Fe(II)Fe(III)] formed by the high-spin iron ions in the non-degenerate ground states. Antisymmetric double exchange interaction originates from the combined effect of the SOC admixture of the excited states on the centers and isotropic DE interaction (ET) between the excited states with the ground state on the center. For the double exchange in the delocalized MV pair, the two-center second-order perturbation antisymmetric DE terms, which describes this combined effect, have the form
(2.3)
The ket represents the ground S state ( ) in the case of localization of the extra electron,. The ket [] represents the cluster excited states ( ) coupled to the cluster ground state [] by the spin-orbit interaction [] on the -center , [ center , ]. and are the energies of the ground and excited states of the center a*, respectively [27, 28]. The first {second} term in eq. (2.3) includes the SOC ( {}) mixture between the ground () and excited () states of the ion on the center a* {b*} in the |a*b> {ab*>} localization and the transfer of the extra d-electron between the SOC admixed k-excited states of the -ion [ {}] to the ground state of the -ion [{}]. is the operator of the direct (Coulomb) or indirect (throw the ligand bridges) inter-ion interaction.
To illustrate the antisymmetric DE interaction in the MV pair we will consider the double exchange and SOC in the slightly distorted bitetrahedral Fe-Fe cluster. In the |a*b> localization ([Fe(II)Fe(III)]), a distortions which flatten each tetrahedron along the local z-axis result in the non-degenerate ground state for the ion [37] and the ground term for the ion [27]. The spin-orbit coupling admixes to the ground term only the excited states of the distorted Fe(II) ion, [27]. In this case, the eq. (2.3) includes the SOC admixture of the excited states to the ground state and ET from the excited state of the -center to the ground state of the -center. The terms of eq. (2.3) may be represented in the form of the effective Hamiltonian of antisymmetric double exchange:
, (2.4)
where is the isotropic DE operator (eqs. (2.1), (2.1)), which describes the Anderson-Hasegawa double exchange coupling (1.1). The ASDE operator includes the scalar product of the real antisymmetric vector constant () of the AS double exchange and spin operator . represents the spin-transfer interaction induced by SOC. In the case of the ground state of the ion, the microscopic calculations show that and
, (2.5)
where , , is the SOC constant of the ion [27], is the energy interval between the excited and the ground states of the ion [37]. is the ET integral between the and neighboring 3d-functions.
The effective AS DE Hamiltonian (2.4) acts between the cluster ground states and of different localizations, The spin operators and the transfer operator don’t commute. The arrows under the spin operators indicate that the operator {} act on the ground state spin functions {} in the |a*b> {|ab*>} localization.
For the AS DE coupling between the states of different localization with the same total spin , the AS DE operator (2.4) of the MV [Fe(II)Fe(III)] cluster may be represented in the form [35],
. (2.6)
The matrix elements of the AS double exchange effective Hamiltonian (2.6) for
(2.7)
include the Anderson-Hasegawa term (eq. (1.1)) as a multiplier. The correlations
take place for. As a result, in the [Fe(II)Fe(III)] cluster, the AS double exchange mixes the Anderson-Hasegawa states with the DE states with the same total spin S of different parity.
The AS DE interaction (2.4) mixes the states of different localization with different total spin and. The operator (2.4) for the casehas the following form of the effective AS DE Hamiltonian
. (2.8)
The effective DE operator for the coupling (2.8)
(2.9)
represents the difference and does not depend on the total spin S [35].
The matrix elements of the AS DE Hamiltonian (2.8) for have the form:
(2.10)
For the AS DE coupling, the correlation takes place. As a result, in the [Fe(II)Fe(III)] cluster, the AS double exchange coupling mixes the Anderson-Hasegawa DE states with the DE states with different total spin of the same parity.
3. Microscopic calculations of the AS double exchange parameter.
We will consider here the model of the centers of ferredoxins and enzymes [1] in the form of the slightly distorted bitetrahedral cluster with the common edge. Each iron ion possesses distorted tetrahedral coordination of the center. For the ion, a distortion, which flattens the tetrahedron along the local z-axis, results in the non-degenerate ground state [37]. The SOC admixture of the excited states to the ground terms of individual Fe(II) and Fe(III) ions determines the ZFS of individual ions [27]. Without the taking SOC into account, the ground and excited Slater determinant wave functions are the following [27, 28], ():
(3.1)
For the center in the distorted tetrahedral coordination, the ground state wave functions with the SOC admixture of the exited states have the form
(3.2)
where the coefficients describe the SOC admixture of the excited states for the ion. These parameters of the SOC admixture determine the local anisotropy of g-factors () and positive ZFS () of the individual ion in the flattened tetrahedral coordination with the ground state [37, 38].
The calculations show that only the terms proportional to the parameters of the SOC admixture for the ion contribute to the antisymmetric double exchange in the [Fe(II)Fe(III)] MV pair. We will consider here only the ground state ( term) of the high-spin ion (without the SOC admixture of the excited states) with the wave functions [28]:
. (3.3)
In the model of the direct interaction between the MV iron ions, one obtains the standard Anderson-Hasegawa DE splittings (1.2), where the DE parameter in the ground states set of the [Fe(II)Fe(III)] cluster has the form
. (3.4)
The parameter for the Heisenberg exchange splittings in the ground S states set of the [Fe(II)Fe(III)] cluster is the following
, (3.5)
where, and .
In the consideration of the [Fe(II)Fe(III)] DE pair with the taking SOC into account, we will calculate the non-diagonal () matrix elements, proportional to for the states of different localization with and with. These terms are equal to zero in the Anderson-Hasegawa model (1.1). We will use the cluster wave functions formed using the SOC admixed wave functions (3.2) and (3.3). The ET integrals between the ground () and excited cluster states () of different localization, which are formed by the excited states of the ion, appier in the consideration of SOC in the DE model. The calculations show that these DE integrals between the ground and excited cluster states follow to the Anderson-Hasegawa type rules:
, (3.6)
,
and don’t depend on M, for. The one-electron transfer integrals and in eq. (3.6) are the ET integrals between the ground cluster state and excited cluster states = (n=x, y).
For the states with the same S, for example, for S=9/2, M=9/2 and, the DE mixing with the taking SOC into account has a form
(3.7)
These transfer integrals and ( and) between the and 3d-functions of the Fe ions on different centers in their local coordinate axis in eq. (3.13) are considered following the model [32, 34], which was used for antisymmetric Dzialoshinsky-Moriya exchange interaction between monovalent ions. In the slightly distorted dimer of the two tetrahedra with the common edge (X-axis), we consider that the local z-axis of the a (b) tetrahedra is tilted in the plane ZY on the angle relative to the cluster Z-axis (axis ). For this distorted dimeric cluster, the 3d-crystal–field orbitals of the individual a (b) Fe centers have the following form in the cluster coordinate system XYZ:
(3.8)
Using these expressions for the DE (ET) integrals between the ground and excited cluster states of different localization, one obtains
(3.9)
where, {} denote the transfer between the neighboring an { and} orbitals. The comparison of the resulting matrix element (3.16) of the DE mixing in the microscopic calculation