Biquadratic contribution

In fact, on close examination of the susceptibility variation, slight discrepancies appear at low temperatures, so that a model involving a biquadratic contribution was postulated in a second step [54]. In this assumption, the exchange Hamiltonian is written as ... 

In the Heitler-London approximation, with allowance made only for biquadratic anharmonic coupling between collectivized high-frequency and low-frequency modes of a lattice of adsorbed molecules (admolecular lattice), the total Hamiltonian (4.3.1) can be written as a sum of harmonic and anharmonic contributions ... 

Now we can see that for 5 > 2, in addition to the second-order equivalent operators, the fourth-order equivalent operators can contribute. These, in fact, correspond to higher-order spin-spin interaction—a biquadratic spin-spin interaction, like... 

Abstract Taking a binuclear copper complex as model system, the isotropic magnetic coupling is decomposed into different contributions. Perturbative expressions of the main contributions are derived and illustrated with numerical examples. An effective Hamiltonian is constructed that incorporates all important electron correlation effects and establishes a connection between the complex A-electron wave functions and the simpler qualitative methods discussed in the previous chapter. Subsequently an outline is given of the analysis of the coupling with a single determinant approach and the biquadratic and four-center interactions are decomposed. The chapter closes with the recently proposed method to extract DFT estimates for these complex interactions. 

Fig. 47. But if there is a strong quadrupolar order it will certainly contribute to increasing the polar order, once we remove the reflection symmetry. This leads to a coupling term in P and 0 which is biquadratic, P 9. If we now only consider those terms in the free energy which contain P and q, we can write...

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