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

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 exchange interaction or anisotropic spin-dependent electron transfer which is described by the effective Hamiltonian HDEAN=∑α∑n=x,y,z[Antμ(Sˆ←α∗nτˆαβSˆ→β∗n)+Bntv(Sˆ←α∗nτˆαβSˆ→αn+Sˆ←βnτˆαβSˆ→β∗n)], τab is the one-electron DE operator. For the MV [d8-d9] cluster, the coefficients of the double exchange anisotropy Antμ, Bntv linearly depend on the DE parameters tμ and tv of the excited and ground cluster DE states. The one-center spin operators Sˆ←α∗n,Sˆ→β∗n(Sˆ→αn) act in the states of different localization. The anisotropic spin-transfer interaction HDEAN 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 E±0(S=3/2). This splitting is described by the effective ZFS HamiltonianHZFSt=DtTˆab[SZ2-S(S+1)/3]+EtTˆab(SX2-SY2), where Tˆab is the double exchange operator in the S representation. The ZFS parameters Dt and Et of the anisotropic DE origin are linearly proportional to the double exchange parameters tμ. In the MV clusters, the ZFS operator HZFSt acts between the S states of different localization and should be added to the standard ZFS Hamiltonian HZFS0=DS[SZ2-S(S+1)/3]+ES(SX2-SY2), which is active in the localized states. The anisotropic double exchange contributions to the ZFS have different sign for the E+0(S)andE-0(S) DE states: D[E±0(S)]=DS±Dt,E[E±0(S)]=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.