CHAPTER 1
Spin-Spin Interactions in Weakly Interacting Dimers
BY C. P. KEIJZERS
1 Introduction
As such, the subject of "Spin-Spin Interactions" has not beenthe subject of discussion in this series. Under different titles, such as "Transition Metal Ions", "Triplet Biradicals" or "Inorganic and Organometallic Radicals", various theoretical and experimental results have been discussed that are related to this subject (see, for instance, reference 1) but an integrated discussion has not been provided. In the past years, several groups have applied themselves specifically to the study of various aspects of the "exchange" phenomenon in order to obtain a better understanding of the physical interactions that are underlying the various terms in the effective spin-Hamiltonian with which the EPR spectra of systems with spin-spin interactions are described. An understanding of the magnetic exchange interactions propagated by multi-atom bridges could give insight into, for instance, the pathways of electron transfer in biological electron transport chains. It could also be used as a guideline for the preparation of new and interesting polymetallic complexes or one-and two-dimensional magnetic exchange systems with magnetic properties that can be predicted, both in nature and in magnitude.
It is not the intention of this contribution to be an all inclusive review of spin-spin interaction studies, this would be impossible in view of the breadth of the field and the vast literature. Instead, the subject is limited to the spin-Hamiltonian
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
which is applied for the description of magnetic exchange in weakly interacting dimers or other discrete (transition) metal complexes. A review is given of the fundamental theory that is necessary for the interpretation of the Hamiltonian (1) and selected papers from the literature are cited in which these interactions are either calculated or experimentally determined. For the experimental work, the attention is focussed mainly to EPR which means that various other techniques which are relevant to the subject (like for instance susceptibility, NMR, Mossbauer, optical spectroscopy) are not discussed. Also lineshape and linewidth studies in (low dimensional) magnetic systems are not discussed. This extensive, complicated but very interesting subject would warrant a separate contribution in this series. For the time being, we refer to some reviews and especially also to the extensive work of Soos (for instance references 5 and 6).
Many reviews, textbooks and conference proceedings are available that have a bearing on the subject of this contribution. Especially the last one will be referenced often: it contains contributions covering a wide range of experimental and theoretical topics in this field.
2 Spin-Hamiltonian
The spin-Hamiltonian (1) is the usual Hamiltonian for the description of the interaction of two S = 1/2 ions in zero magnetic field. The tensor[??] can be decomposed into its trace, a symmetric tensor and an antisymmetrical tensor:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
(One has to pay attention when "J"-values from different authors are to be compared, because for the isotropic part also the formulations [FORMULA NOT REPRODUCIBLE IN ASCII] and [FORMULA NOT REPRODUCIBLE IN ASCII] are used). The effect of J is a separation of the four spin-functions into a singlet and a triplet. [??]s splits the triplet into a doublet and a singlet (in case of axial symmetry) or into three singlets.[??], finally, mixes the singlet with all three triplet functions. The essential requirement for the antisymmetric term,[??].[??]1x[??]2 is the absence of a centre of symmetry between the magnetic sites containing S1 and S2. If an inversion centre would exist, [??]1 and [??]2 would interchange under the inversion operation and [??]1x[??]2 would change its sign. Since the Hamilton operator must be invariant for any symmetry operation of the system, this means that[??] must change sign as well. Hence,[??] = 0.
In the principal axes system of[??]s and in the basis of the eigenfunctions of [??]s, the Hamiltonian matrix is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
where X, Y, z are the eigenvalues of Ds (X + Y + Z = 0), and the basisfunctions are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
For [??]s and/or [??] equal to zero, the energies and eigenfunctions of (3) are:
[ILLUSTRATION OMITTED]
In this diagram, the functions |ψi >are:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The diagram shows that the effect of [??], the antisymmetric exchange interaction (AEI), is an increase of the apparent isotropic exchange interaction (IEI) and, as far as the triplet manifold is concerned, it is equivalent to an axial zero field splitting (ZFS) tensor [??]s, but only for the energies. The AEI mixes also the triplet functions with the singlet and, therefore, the EPR transition probabilities within the triplet manifold are influenced. It is questionable, however, whether this effect can be distinguished from the symmetrical ZFS tensor in an actual EPR spectrum where also other interactions (like nuclear hyperfine coupling) will influence the line intensities. A second consequence of the singlet-triplet mixing is that new transitions, between |ψ1 > and |ψ2-4>, are allowed but even the observation of these transitions is not a direct proof of the presence of an antisymmetric contribution in the spin-spin interaction. The reason is that d exists only for low symmetry dimers (see paragraph 5). For such systems, the g-tensors of the two spins will be different and also this causes singlet-triplet transitions to be allowed (|S0 > is not an eigenfunction of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The symmetry aspects of the antisymmetric exchange and a general formulation for its matrix elements for any pair of S1, S2 values were discussed by Bencini and Gatteschi. These authors conclude also that" ... at the present stage of knowledge the meaningful quantitative determination of the antisymmetric exchange term seems to be impossible in general cases". Their expectation that the misalignment between the g and D axes can give the key to recognize the presence of this exchange term is not justified because even in high(er) symmetry systems g and D do not necessarily coincide as is found in many dimers like, for instance, Cu-Cu and Ag-Ag dimers in [Zn(et2dtc)2]2.
For the case |J | [much greater than] [??]s, [??] (i.e. S is a good quantum number) Scaringe et al. derived expressions for the g-tensor, the ZFS and the nuclear hyperfine interaction tensors of a dimer in relation to the monomer tensors both, for = S1 = S2 and for S1 ≠ S2. The validity of these relations was shown in many examples as, for instance, Cu-Cu, Cu-Mn and Cu-Zn pairs in [Cu(PyNO)Cl2 (H2O)]2 (PyNO = Pyridine-N-oxide).
An application of the relations between monomer and dimer tensors could be the determination of the monomer tensors from the dimer parameters if the monomers themselves do not show an EPR signal. Gatteschi and Bencini discussed the calculation of the g-tensor of Ni(II) ions from the spectra of Cu-Ni pairs and knowing the tensor of the Cu-ion. However, care should be exercised in applying these relations because sometimes they do not hold and the reason for that is not yet found although magnetostriction and/or the presence of antisymmetric exchange have been mentioned.
3 Isotropic Exchange
3.1 Spin-Hamiltonian. -Although the isotropic exchange interaction between magnetic atoms or molecules is usually represented by the Heisenberg Hamiltonian
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
in principle also higher order terms are possible, like for instance biquadratic exchange [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. However, multiple exchange interactions between the atoms or molecules are negligible, then the Hamiltonian can be truncated after the bilinear term. Anderson derived with a perturbation treatment that (a) these higher order terms only appear for ions with more than one electron and (b) that they are of the order of 1% of the bilinear term. Herring used a modified Heitler-London method to prove the validity of the Heisenberg Hamiltonian assuming only that the real Hamiltonian does not contain spin variables and that the separated monomers have no orbital degeneracy. As Anderson, he concluded that the higher order terms are in principle present but that they are negligibly small. The quantitative estimate of Anderson was experimentally verified for a number of systems: Gudel et al. found (with inelastic neutron scattering) that the Lande-interval rule (E(S) -E(S-1) -2JS), which is the result of the Heisenberg Hamiltonian, is not satisfied for dimers of Mn2+ (S = 5/2) ions in CsMnxMg1-xBr3. The magnitude of the biquadratic exchange which they derived was ~0.003% of the bilinear term. A similar ratio between J and j was found in mono and bis(µ-hydroxo)-bridged Cr3+ (S = 3/2) dimers with luminescence and absorption spectroscopy and in a tris (µ-hydroxo)-bridged Cr3+ dimer from a fit of the magnetic susceptibility Gudel warned that if larger j-values are obtained experimentally, they are likely to be artifacts or the result of other physical effects, as e.g. magneto striction. Also an ab-initio calculation of the exchange interaction between O2 molecules showed that the coupling between the monomer triplets can be very well fitted with the Heisenberg Hamiltonian and that the Lande-interval rule is perfectly obeyed.
Drillon and Georges and Leuenberger and Gudelpointed out that the Heisenberg Hamiltonian is appropriate only for ions in orbital singlet states. For the coupling of ions in orbitally degenerate states, as Ti3+ in a trigonal ligand field, Hamiltonians are to be used which contain orbital parameters that cannot be collected in an overall J: for instance for the description of the exchange interaction between two Ti3+ ions, also the local trigonal distortion, spin-orbit coupling and covalency effects are introduced in the Hamiltonian. A study of Ti2x3-9 (X = C1, Br) units showed that the exchange parameters are in agreement with a simple molecular orbital calculation. The most uncertain parameter in the Hamiltonian is the (trigonal) ligand field.
3.2 Analytical Expressions for J. -In this section, the discussion will be limited to orbitally non-degenerate states of the monomers. For systems with orbital degeneracy, Hamiltonians were derived by Fuchikami in collaboration with Tanabe and with Block.
For non-degenerate systems, various analytical expressions for J were derived by a number of authors. All expressions agree in so far as that they contain a ferromagnetic and an antiferromagnetic contribution and that the ferromagnetic contribution (positive in the formulation (6) of the Heisenberg Hamiltonain) is, in first order, a two-electron exchange integral. However, the expressions for the antiferromagnetic contributions are different for different derivations. The reason for this disagreement is that the description of a dimer (the discussion is restricted to the interaction between two monomers) is not straightforward. It is well known that a molecular orbital (MO) treatment of a dimer as a "super molecule" leads to an incorrect asymptotic behaviour of the wave function for large monomer-monomer distances (It contains covalent and ionic configurations with equal weight). But also a localized, Heitler London (or Valence Bond, VB), approach does not lead fast enough to a correct asymptotic result. For that reason, Herringdeveloped a modified Reitler-London method with functions that resemble, in their localization, the free atom functions but which, at the same time, approximate as closely as possible the exact eigenfunctions at large separations of the monomers. However, this approach found little application, as far as we know.
For MO as well as VB descriptions, improvement is to be expected from configuration interaction (CI), but one is forced to a limited CI if an analytical expression for J is to be derived. (However, also the ab-initio calculation of J, as the difference between the energies of different spin states of a dimer, is impossible without approximations for extended systems, because of the very large number of two-centre integrals and the enormous number of possible configurations).
Usually, the first step in a calculation of J is obtaining a wave function of the magnetic monomers which are surrounded by various diamagnetic groups. In this step, the exchange effects of the other magnetic ion(s) are excluded. That these exchange effects in weakly interacting dimers do not disturb the wave function is shown experimentally by the agreement of hyperfine interactions with ligand nuclei in dilute and in concentrated paramagnetic dopes of the same diamagnetic lattice. The next step is to construct dimer functions from the monomer fragments. The most serious question that arises here is that concerning the orthogonality of the monomer wave functions. Already Andersonrealized the importance of this point and more recently Kahnand others drew the attention to the fact that the construction of orthogonal monomer functions does lead to an easier but certainly not to a better description of the dimer. The orthonormalized functions fail to be eigenfunctions of any physical system. Therefore, they cannot be expected to describe reasonably correct the unperturbed states of the monomers. For that reason, a perturbation expansion may have to be carried through to a higher order in order to obtain acceptable values for the parameters in the spin-Hamiltonian.
3.2.1 Localized Method. -In the VB method, the orbital parts of the ground state singlet and triplet functions are constructed with the functions ψA and ψB of the monomers A and B:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
where "+" holds for the singlet and "-" for the triplet; SAB is the overlap integral < ψA|ψP >. If the Hamiltonian is defined
H = h(l) + h(2) + hint (8)
where h(i) are the one-electron Hamiltonians, including the intramonomer interactions hA(l) and hB(2) (of which ψA and ψB are eigenfunctions) and the interactions with the nuclei of the "other" monomer, and hint is e2/r12, then the energies of ψS,T are:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
and where it is assumed that the monomers A and B are identical. The resulting singlet-triplet splitting is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
This localized method, but with orthogonalized orbitals, was used by Anderson and by Hay, Thibeault and Hoffmann. Essentially they followed the same procedure but Anderson applied it for an infinite lattice whereas Hay et al. calculated J for a dimer. Kahn rephrased the Anderson approach for the interaction between two identical single-ion doublet states. The first step is to calculate the two highest singly occupied (sometimes called magnetic) molecular orbitals ψb (bonding) and ψa (antibonding) of the triplet state. In a weakly interacting dimer these MO's are approximately:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
The next step is to determine the orthogonalized magnetic orbitals:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
ψA and [spi]B have metal and ligand character, but they are essentially centered on the monomers A and B, respectively (In the Anderson treatment these steps were 1. calculation of Bloch functions and 2. construction of Wannier functions). Another way of obtaining orthogonal localized orbitals is Lowdin orthogonalization [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], resulting in:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
