The first model for isotopic exchange was developed by Anderson6 who used an unrestricted Hartree–Fock(HF) formalism in his treatment of superexchange in insulators. In his derivation, Anderson considered singlet and triplet states
ẳ 1 ffiffiffi2
p j’Að1ị’Bð2ị ’Að2ị’Bð1ịj ð3ị where þ holds for the singlet and for the triplet, and ’A and ’B are orthogonal magnetic orbitals. ’A and ’B are derived from a linear combination of the two highest singly occupied molecular orbitals (mostlyd-orbital character for transition metal ions). The energy gapJbetween triplet and singlet states was then calculated by a perturbation treatment, using the effective Hamiltonian,
Hẳhð1ị ỵhð2ị ỵ 1
r12 ð4ị
yielding
Jẳ2KAB2b2
U ð5ị
where the ferromagnetic term 2KAB is the potential exchange, representing pairing energy, andb is the transfer integral, representing the energy released when electrons are able to delocalize into molecular orbitals. Uis the difference in energy between states in which two unpaired electrons are on separate metal ions and paired electrons are on one metal ion.
Hay, Thibeault, and Hoffmann7 showed that JAF can be analyzed in terms of pairwise inter- actions of dimeric molecular orbitals with the square of the splitting in energy between the members of a pair being a measure of the stabilization of the low-spin state. Their final result
Jẳ2KAB"2
U ð6ị
where "is the difference in energy between the linear combinations of the two highest singly occupied molecular orbitals, was shown to be equivalent to Anderson’s treatment of exchange coupling. Extended Hu¨ckel theory was used to evaluate the effect of geometry, electronegativity, and substituents on the splitting energy ".
Girerd, Journaux, and Kahn8re-examined exchange coupling and their solution for the energy between singlet and triplet states
Jẳ2KAB4ðbỵ lị2
U ð7ị
is essentially the Anderson model; however, it included the two-electron ionic integral l for the MMCT event (see below).
It is possible to describe the antiferromagnetic term in Equation (5) in the same conceptual basis as that for Marcus electron-transfer theory9–11 and the Hush model for intervalence transi- tions.12–14Bertrand considered the case of electron transfer between biological molecules coupled by an exchange interaction.15This is represented schematically by two metal sites separated by a bridging medium L (Scheme 1).
MA L MB MA L MB
EMMCT
ground state excited state
Scheme 1
590 Metal–Metal Exchange Coupling
In the ground state, the antiparallel alignment of unpaired electrons is the most stable because this arrangement permits a configuration interaction with the metal–metal charge transfer excited state. The potential energy diagram representing this interaction is shown in Figure 1. This is essentially identical to the MMCT case for an asymmetrical mixed-valence complex16except that the energy of the excited MMCT state includes a contribution from pairing energy. The singlet ground state is stabilized by antiferromagnetic exchange,15
Jẳ2KAB 2b2 EMMCT
ð8ị Herebrepresents the resonance exchange integral for electron transfer between the metal centers.
Tuczek and Solomon17 examined both molecular orbital and valence bond configuration interaction (VBCI) models of magnetic exchange and showed that the VBCI model accounts for both the sign and magnitude of charge transfer state splitting observed in Cu azide systems. In this model, ground state antiferromagnetism is derived from configuration interaction with metal- to-metal and double charge transfer states and has the form
JAFẳchML4 E2
1 U þ 2
EDCT
ð9ị wherecis a constant,hML, the transfer integral (metal–ligand coupling element) corresponds to a one-electron charge transfer between the bridging ligand and the metal (i.e., a LMCT transition), Eis the energy difference between the ground state and the LMCT excited state associated with hML, andEDCTis the difference in energy between the ground and double charge transfer states (Scheme 2).
MA L MB MA L MB
EDCT
ground state excited state
Scheme 2
The singlet double charge transfer state corresponds to the simultaneous transition of one bridging ligand electron to each metal ion. It is not accounted for by Anderson theory and is expected to have a significant contribution to ground state antiferromagetism if the bridging ligand is very polarizable.18
Weihe, Gu¨del, and Toftlund,19 have extended the above valence bond treatment to three additional cases where a bridging ligand orbital simultaneously interacts with a half-filled orbital localized on metal A and an empty orbital on metal B, a full orbital localized on metal A and a half-filled orbital on metal B and finally a full orbital localized on A and an empty orbital
Figure1 Potential energy diagram of metal-to-metal charge transfer.
localized on metal B. All of the treatments demonstrated the need to include higher-order effects in order to properly account for magnetic properties.
Configuration interaction treatments must invoke orthogonal magnetic orbitals and, as a consequence, there is no direct antiferromagnetic contribution to the singlet state. Instead, the stabilization of the singlet state arises through an interaction between the ground state config- uration and an excited state configuration. An alternative approach using nonorthogonal magnetic orbitals was suggested by Kahn.
In Kahn’s approach,8,20magnetic exchange was likened to the formation of a weak bond or, in other words, the overlap of localized singly occupied orbitals. Incorporating nonorthogonal magnetic orbitals in the description of triplet and singlet states (Equation (3)) and applying the effective Hamiltonian (Equation (4)) to solve for the energy gap between these two states yielded
Jẳ2ðKABkS2ị ỵ4tS4jt ỵl ðkỵ KABịSj2
k0k ð10ị
where S is the overlap integral of two magnetic orbitals, k, k0, and t are two-center Coulomb repulsion, one-center Coulomb repulsion, and resonance exchange integrals, respectively. It is important to recognize that the integralsKAB,k,t, andldo not have the same values according to whether the orthogonalized or nonorthogonalized magnetic orbitals are used as a basis set.20
If one assumes that configuration interaction is small, simplification of Equation (10)leads to
Jẳ2ðKABkS2ị 2S ð11ị
where is the energy gap between the two molecular orbitals built from the bonding and antibonding combinations of the magnetic orbital for the triplet state.21The term is formally equivalent to " which appears in Equation (6). If asymmetric systems are studied where magnetic orbitals have different energies, singlet–triplet splitting energy has the final form22
Jẳ2ðKABkS2ị 2Sð22ị ð12ị
where is the difference in energy between the magnetic orbitals.
Quantitative calculations of the magnetic exchange interaction require an accurate description of the electronic structure of the system in both ground and excited states. This is achieved by invoking HF molecular orbital theory and by using extensive configuration interaction (CI) to correlate the magnetic electrons. Calculations at theab initiolevel are possible for large molecules, if the pseudopotential approach can properly model the effect of core electrons. De Loth et al.23 were the first to perform such calculations on the singlet–triplet state energy gap of dinuclear copper(II) acetate. Their results showed that direct superexchange mechanism only cancels the direct exchange (the terms in Equation (5)). Double spin polarization, and higher order contributions, involving superexchange and polarization of ligand and 3d-orbitals, had to be invoked to provide enough stabilization of the singlet state to match with experiment. This work has been followed by a number of similar calculations of J for other dinuclear copper(II) complexes24–26 and complete active-space multiconfigurational calculations of oxo-bridged systems.27,28
A list of second-order contributions and corrections in the ab initio calculation of J is avail- able.20 The spin polarization mechanism of exchange is different from the charge transfer mechanisms discussed so far and is perhaps best illustrated by the exchange between metals ions bridged by the azide ligand (Scheme 3).
The highest occupied molecular orbital of the azide ligand is nonbonding and its molecular orbital can schematically represented as shown in Scheme 3 (the center nitrogen has little contribution to the MO).21To minimize repulsion between electrons in this nonbonding orbital, the most probable electron distribution is one in which an electron withspin is localized at one terminal nitrogen while the spin electron is localized on the other terminal nitrogen. For the end-on bonding case, the bonding nitrogen electron density (say with spin) will delocalize towards the d-orbitals, with the extent of delocalization depending on the overlap integral and the relative energy differences between azide and metal orbitals. This gives an instantaneous density in the metald-orbitals, causing each mostly metal unpaired electron to adoptspin. Thus, a triplet ground state is favored by end-on coordination of the azide bridging ligand. Similar arguments21lead to the prediction of a singlet ground state for terminal coordination of the azide.
592 Metal–Metal Exchange Coupling
M M NNN
M M
N N N
terminal end-on
Scheme 3
Noodleman and Norman suggested that the low spin state could be described by a broken spin and space symmetry single-determinate wave function models and would therefore require less computational effort.29,30In this regard, Rappe´ et al. comparedab initiofull CI and HF broken symmetry (HF-BS) calculations and found that HF-BS gave results in close agreement with large-CI calculations.31–33 More recently, Benciniet al. examined broken symmetry (BS/DFT) and single- determinant density (SD/DFT) functional theory of magnetic exchange and found that BS/DFT provided acceptable semi-quantitative results and was therefore a useful tool for the rationalization of magneto-structural correlations.34Further theoretical studies of magnetic exchange interactions in copper(II) azido bridged complexes have shown remarkable agreement with experiment.35