Spin-dynamics models

D. Electron spin dynamics in the equilibrium ensemble Spin-dynamics models Outer-sphere relaxation... 

D. P. Landau, M. Krech. Spin dynamics simulations of ferro- and antiferromagnetic model systems comparison with theory and experiment. J Phys Condens Matter 77 R175-R213, 1999. 

Often the electronic spin states are not stationary with respect to the Mossbauer time scale but fluctuate and show transitions due to coupling to the vibrational states of the chemical environment (the lattice vibrations or phonons). The rate l/Tj of this spin-lattice relaxation depends among other variables on temperature and energy splitting (see also Appendix H). Alternatively, spin transitions can be caused by spin-spin interactions with rates 1/T2 that depend on the distance between the paramagnetic centers. In densely packed solids of inorganic compounds or concentrated solutions, the spin-spin relaxation may dominate the total spin relaxation 1/r = l/Ti + 1/+2 [104]. Whenever the relaxation time is comparable to the nuclear Larmor frequency S)A/h) or the rate of the nuclear decay ( 10 s ), the stationary solutions above do not apply and a dynamic model has to be invoked... 

We have shown in this chapter that the major electronic features that determine the spin dynamics of SIMs based on lanthanides can be directly correlated with the local coordination environment around the 4f metal ions. By using an effective point-charge model that accounts for covalent effects, we have shown that the splitting of the ground state,/, of the lanthanide into Mj sublevels, caused by the influence of the CF created by the surrounding ligands, is consistent with... 

Then, there are model Hamiltonians. Effectively a model Hamiltonian includes only some effects, in order to focus on those effects. It is generally simpler than the true full Coulomb Hamiltonian, but is made that way to focus on a particular aspect, be it magnetization, Coulomb interaction, diffusion, phase transitions, etc. A good example is the set of model Hamiltonians used to describe the IETS experiment and (more generally) vibronic and vibrational effects in transport junctions. Special models are also used to deal with chirality in molecular transport junctions [42, 43], as well as optical excitation, Raman excitation [44], spin dynamics, and other aspects that go well beyond the simple transport phenomena associated with these systems. 

We call the second category of theoretical tools for dealing with rapidly rotating systems the spin-dynamics methods . The models within this category will he reviewed in Section VI. 

Fig. 13. Predicted magnetic field dependence of the electron spin lattice relaxation time. Solid line pseudorotation model dashed line spin dynamics calculation. Reproduced with permission from Odelius, M. Ribbing, C. Kowalewski, J. J. Chem. Phys. 1996,104, 3181-3188. Copyright 1996 American Institute of Physics.

The quantum alternative for the description of the vibrational degrees of freedom has been commented by Westlund et al. (85). The comments indicate that, to get a reasonable description of the field-dependent electron spin relaxation caused by the quantum vibrations, one needs to consider the first as well as the second order coupling between the spin and the vibrational modes in the ZFS interaction, and to take into account the lifetime of a vibrational state, Tw, as well as the time constant,T2V, associated with a width of vibrational transitions. A model of nuclear spin relaxation, including the electron spin subsystem coupled to a quantum vibrational bath, has been proposed (7d5). The contributions of the T2V and Tw vibrational relaxation (associated with the linear and the quadratic term in the Taylor expansion of the ZFS tensor, respectively) to the electron spin relaxation was considered. The description of the electron spin dynamics was included in the calculations of the PRE by the SBM approach, as well as in the framework of the general slow-motion theory, with appropriate modifications. The theoretical predictions were compared once again with the experimental PRE values for the Ni(H20)g complex in aqueous solution. This work can be treated as a quantum-mechanical counterpart of the classical approach presented in the paper by Kruk and Kowalewski (161). 

L. F. Cugliandolo and J. Kurchan, Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model. Phys. Rev. Lett. 71, 173-176 (1993). 

In applying the above model to the interpretation of our EPR results we have to keep in mind that the spin dynamics of the coupled Mn-Cu system experiences a strong bottleneck regime [9]. In the bottleneck regime the... 

In contrast to the subsystem representation, the adiabatic basis depends on the environmental coordinates. As such, one obtains a physically intuitive description in terms of classical trajectories along Born-Oppenheimer surfaces. A variety of systems have been studied using QCL dynamics in this basis. These include the reaction rate and the kinetic isotope effect of proton transfer in a polar condensed phase solvent and a cluster [29-33], vibrational energy relaxation of a hydrogen bonded complex in a polar liquid [34], photodissociation of F2 [35], dynamical analysis of vibrational frequency shifts in a Xe fluid [36], and the spin-boson model [37,38], which is of particular importance as exact quantum results are available for comparison. 

The chapter is organized as follows The quantum-classical Liouville dynamics scheme is first outlined and a rigorous surface hopping trajectory algorithm for its implementation is presented. The iterative linearized density matrix propagation approach is then described and an approach for its implementation is presented. In the Model Simulations section the comparable performance of the two methods is documented for the generalized spin-boson model and numerical convergence issues are mentioned. In the Conclusions we review the perspectives of this study. 

As noted earlier, the fundamental equations of the QCL dynamics approach are exact for this model, however, in order to implement these equations in the approach detailed in section 2 the momentum jump approximation of Eq.(14) is made in addition to the Trotter factorization of Eq.(12). Both approximations become more accurate as the size of the time step 5 is reduced. Consequently, the results presented below primarily serve as tests of the validity and utility of the momentum-jump approximation. For a discussion of other simulation schemes for QCL dynamics see Ref. [21] in this volume. The linearized approximate propagator is not exact for the spin-boson model. However when used as a short time approximation for iteration as outlined in section 3 the approach can be made accurate with a sufficient number of iterations [37]. 

The asymmetric spin boson model presents a significantly more challenging non-adiabatic condensed phase test problem due to the asymmetry in forces from the different surfaces. Approximate mean field methods, for example, will fail to reliably capture the effects of these different forces on the dynamics. 

Fig. 2 B(t) = (az)(t) versus time for the asymmetric spin-boson model with (3 = 25, = 0.13 and Q = 0.4, e = 0.4. (Top) Comparison of exact quantum results (filled circles), ILDM simulations (small open circles), and QCL dynamics (filled triangles). Both ILDM and QCL simulations were carried out for an ensemble of 2 X 106 trajectories and no filters are employed. (Bottom) Convergence of TQCL dynamics with ensemble size 2 X 104 (filled squares) and 1 x 106 (filled triangles). Exact quantum results (filled circles). A filter parameter of Z = 500 is used for these calculations. ...

To determine orientation of an adsorbed molecule relative to the mineral surface, one would need to employ dipolar recoupling techniques to extract the distance constraints between the selected spin species of the molecule and of the surface. For polypeptide-HAp systems, 13C 31P REDOR has been carried out for uniformly 13C labeled molecules, where numerical simulations show that the effect of 13C dipole-dipole interaction is relatively minor.125 For a study of bone sample, o-phospho-L-serine was taken as the model compound for the setup of the 13C 31P REDOR experiments, where the data can be well analyzed by a l3C-3lP spin-pair model with the intemuclear distance equal to 2.7 A.126 Concerning the effect of 31P homonuclear dipolar interaction on the spin dynamics, Drobny and co-workers have carried out a detailed REDOR NMR study of polycrystalline diammonium hydrogen phosphate ((NH4)2F1P04).127,128 The results show that the 15N 31P REDOR data can... 

The frequencies assigned to the individual coherences are different for any two conformers present in dynamic spin systems. The product functions, that is, the states the terms were assigned to, are identical. The single spin vector model of the previous section can be extended to weakly coupled systems by changing the precession frequencies of the single quantum coherences during the exchanges of the nuclei instead of... 

The impact of this is tremendous. No long-range order (LRO) can exist at finite temperature in one dimension no crystals, no magnets, no superconductors. Only special transitions are possible in two dimensions. The Ising model (n = 1 component) is an example [7]. The Kosterlitz-Thouless transition [8], without LRO, is another case for d = 2 and n = 2, discussed in Section V.C. The thermal fluctuations are very destructive in lower dimensions. Quantum fluctuations (i.e., those associated with the dynamics of a system) also tend to suppress LRO and can sometimes destroy it even at 0 K when the Mermin-Wagner theorem does not apply. Such is the case of the quantum spin- antiferromagnetic models [9] in one dimension. 

Theoretical expressions for spin-lattice relaxation of 2H nuclei (determined by locally axially symmetric quadrupolar interactions modulated by molecular motions) can be derived for specific dynamic processes, allowing the correct dynamic model to be established by comparison of theoretical and experimental results [34,35]. In addition, T, anisotropy effects, which can be revealed using a modified inversion recovery experiment, can also be informative with regard to establishing the dynamic model [34,35]. 

The 2-site 120° jump motion for the basal molecules switches between these two hydrogen bonding arrangements and clearly requires correlated jumps of the hydroxyl groups of all three basal molecules. On the assumption of Arrhenius behaviour for the temperature dependence of the jump frequencies, the activation energies for the jump motions of the apical and basal deuterons were estimated to be 10 and 21 kj mol-1, respectively. This dynamic model was further supported by analysis of the dependence of the quadrupole echo 2H NMR lineshape on the echo delay and consideration of 2H NMR spin-lattice relaxation time data. 

T), as well as to the influence of an internal heavy atom. These facts strongly support a model in which geminal combinations are controlled by molecular (rather than spin) dynamics. 

Analogies between replication dynamics and spin lattice models were investigated in recent publications by Demetrius [45] and Leuthausser [46,47]. Both approaches are based on a common concept, and we shall discuss them here together. Replication dynamics is considered as a dynamical system in discrete time and modeled by the difference equation... 

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