Redfield theory

Redfield theory [18-20] is a microscopic semi-classical theory of spin relaxation in which the spin system is treated quantum mechanically whilst the coupling of the spins with the lattice is treated classically. In this classical approximation, the spin states are in equilibrium and a correction factor is needed to ensure the spin ensemble relaxes to the correct limits. This problem can be overcome by treating the lattice quantum mechanically, however, the details and nature of the computational details are beyond the scope of this thesis. In this section a brief introduction to Redfield theory for spin relaxation is provided, a more detailed analysis of the theory can be found elsewhere [21]. 

The time evolution of the density matrix is known to be described by the well known Liouville-von Neumann equation [21] 

H (t) = 0, where the bar is used to signify the ensemble average, one obtains a set of linear differential equations of the form [21] 

The prime in the summation indicates only the terms where and copp/ are retained. Transforming Eq. (3.42) to the laboratory frame gives the solution for the evolution of the density matrix to be 

The first term in the above equation provides the frequency of the transition (a a ), whilst the second term gives its relaxation. 

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We begm tliis section by looking at the Solomon equations, which are the simplest fomuilation of the essential aspects of relaxation as studied by NMR spectroscopy of today. A more general Redfield theory is introduced in the next section, followed by the discussion of the coimections between the relaxation and molecular motions and of physical mechanisms behind the nuclear relaxation. 

Szymanski S, Gryff-Keller A M and Binsch G A 1986 Liouville space formulation of Wangsness-Bloch-Redfield theory of nuclear spin irelaxation suitable for machine computation. I. Fundamental aspects J. Magn. Reson. 68 399-432... 

Jean J M, Fleming G and Friesner R 1992 Application of a multilevel redfield theory to electron transfer in condensed... 

A. Highly symmetric systems and the Redfield theory for electron spin relaxation... 

A more general formulation of relaxation theory, suitable for systems with scalar spin-spin couplings (J-couplings) or for systems with spin quantum numbers higher than 1/2, is known as the Wangsness, Bloch and Redfield (WBR) theory or the Redfield theory 17). In analogy with the Solomon-Bloembergen formulation, the Redfield theory is also based on the second-order perturbation approach, which in certain situations (not uncommon in... 

In an alternative formulation of the Redfield theory, one expresses the density operator by expansion in a suitable operator basis set and formulates the equation of motion directly in terms of the expectation values of the operators (18,20,50). Consider a system of two nuclear spins with the spin quantum number of 1/2,1, and N, interacting with each other through the scalar J-coupling and dipolar interaction. In an isotropic liquid, the former interaction gives rise to J-split doublets, while the dipolar interaction acts as a relaxation mechanism. For the discussion of such a system, the appropriate sixteen-dimensional basis set can for example consist of the unit operator, E, the operators corresponding to the Cartesian components of the two spins, Ix, ly, Iz, Nx, Ny, Nz and the products of the components of I and the components of N (49). These sixteen operators span the Liouville space for our two-spin system. If we concentrate on the longitudinal relaxation (the relaxation connected to the distribution of populations), the Redfield theory predicts the relaxation to follow a set of three coupled differential equations ... 

We now come back to the simplest possible nuclear spin system, containing only one kind of nuclei 7, hyperfine-coupled to electron spin S. In the Solomon-Bloembergen-Morgan theory, both spins constitute the spin system with the unperturbed Hamiltonian containing the two Zeeman interactions. The dipole-dipole interaction and the interactions leading to the electron spin relaxation constitute the perturbation, treated by means of the Redfield theory. In this section, we deal with a situation where the electron spin is allowed to be so strongly coupled to the other degrees of freedom that the Redfield treatment of the combined IS spin system is not possible. In Section V, we will be faced with a situation where the electron spin is in... 

Assuming that the lattice can, on the time scale relevant for the evolution of the nuclear spin density operator, be considered to remain in thermal equilibrium, a = a, and applying the Redfield theory to the nuclear spin sub-system allows us to obtain the following expressions for nuclear spin-lattice and spin spin relaxation rates ... 

A. Highly Symmetric Systems and the Redfield Theory for Electron Spin Relaxation... 

Sharp and Lohr proposed recently a somewhat different point of view on the relation between the electron spin relaxation and the PRE (126). They pointed out that the electron spin relaxation phenomena taking a nonequilibrium ensemble of electron spins (or a perturbed electron spin density operator) back to equilibrium, described in Eqs. (53) and (59) in terms of relaxation superoperators of the Redfield theory, are not really relevant for the PRE. In an NMR experiment, the electron spin density operator remains at, or very close to, thermal equilibrium. The pertinent electron spin relaxation involves instead the thermal decay of time correlation functions such as those given in Eq. (56). The authors show that the decay of the Gr(T) (r denotes the electron spin vector components) is composed of a sum of contributions... 

As in Eq. (64), the electron spin spectral densities could be evaluated by expanding the electron spin tensor operators in a Liouville space basis set of the static Hamiltonian. The outer-sphere electron spin spectral densities are more complicated to evaluate than their inner-sphere counterparts, since they involve integration over the variable u, in analogy with Eqs. (68) and (69). The main simplifying assumption employed for the electron spin system is that the electron spin relaxation processes can be described by the Redfield theory in the same manner as for the inner-sphere counterpart (95). A comparison between the predictions of the analytical approach presented above, and other models of the outer-sphere relaxation, the Hwang and Freed model (HF) (138), its modification including electron spin... 

As in paper [5], we start from a system Hamiltonian consisting of three (one ground g) and two nonadiabatically coupled excited (j) ) and 1 states strongly coupled to a reaction mode, which in turn is weakly coupled to a dissipative environment (see Fig. 1). The bath degrees of freedom are integrated out in the framework of Redfield theory, and the signals are calculated according to the explicit formulas derived in [6,7]. 

A more general approach is required to interpret the current experiments, Jean and co-workers have developed multilevel Redfield theory into a versatile tool for describing ultrafast spectroscopic experiments [22-25], In this approach, terms neglected at the Bloch level play an important role for example, coherence transfer terms that transform a coherence between levels i and j into a coherence between levels j and k ( /t - = 2) or between levels k and l ( f - j - 2, k-j = 2) and couplings between populations and coherences. Coherence transfer processes can often compete effectively with vibrational relaxation and dephasing processes, as shown in Fig. 4 for a single harmonic well, initially prepared in a superposition of levels 6 and 7. The lower panel shows the population of levels 6 and 7 as a function of time, whereas the upper panels display off-diagonal density matrix ele-... 

Figure 6. Population decay of the initial state in a barrierless double-well system calculated using multilevel Redfield theory [25]. The vibrational frequency is 60 cm1. (----------)...

G. R. Fleming An interesting feature of the Redfield theory calculations is that attempts to stop coherence transfer by increasing the dephasing rate also increases the coherence transfer rate. In addition, two-state calculations [M. Jean and G. R. Fleming, J. Chem. Phys. 103, 2092 (1995)] show that the coherence transfer can survive reasonable amounts of anharmonicity. It appears to be quite robust. 

G. R. Fleming Yes, the role of orthogonal coordinates could be significant. I believe that Prof. J. Jean (Washington University, St. Louis) is beginning to address this issue via Redfield theory. 

The time-local approach is based on the Hashitsume-Shibata-Takahashi identity and is also denoted as time-convolutionless formalism [43], partial time ordering prescription (POP) [40-42], or Tokuyama-Mori approach [46]. This can be derived formally from a second-order cumulant expansion of the time-ordered exponential function and yields a resummation of the COP expression [40,42]. Sometimes the approach is also called the time-dependent Redfield theory [47]. As was shown by Gzyl [48] the time-convolutionless formulation of Shibata et al. [10,11] is equivalent to the antecedent version by Fulinski and Kramarczyk [49, 50]. Using the Hashitsume-Shibata-Takahashi identity whose derivation is reviewed in the appendix, one yields in second-order in the system-bath coupling [51]... 

Fig. 3 Absorption spectra of one sample of B850 calculated with different methods for an Ohmic spectral density with / = 3.37 and usc = 0.027 eV. Left Methods with Markov approximation, i.e. Redfield theory with and without secular approximation and TL method. Right TL with and without Markov approximation, TNL and modified Redfield method. (Reproduced from Ref. [37]. Copyright 2006, American Institute of Physics.)...

Redfield limit, and the values for the CH2 protons of his- N,N-diethyldithiocarbamato)iron(iii) iodide, Fe(dtc)2l, a compound for which Te r- When z, rotational reorientation dominates the nuclear relaxation and the Redfield theory can account for the experimental results. When Te Ti values do not increase with Bq as current theory predicts, and non-Redfield relaxation theory (33) has to be employed. By assuming that the spacings of the electron-nuclear spin energy levels are not dominated by Bq but depend on the value of the zero-field splitting parameter, the frequency dependence of the Tj values can be explained. Doddrell et al. (35) have examined the variable temperature and variable field nuclear spin-lattice relaxation times for the protons in Cu(acac)2 and Ru(acac)3. These complexes were chosen since, in the former complex, rotational reorientation appears to be the dominant time-dependent process (36) whereas in the latter complex other time-dependent effects, possibly dynamic Jahn-Teller effects, may be operative. Again current theory will account for the observed Ty values when rotational reorientation dominates the electron and nuclear spin relaxation processes but is inadequate in other situations. More recent studies (37) on the temperature dependence of Ty values of protons of metal acetylacetonate complexes have led to somewhat different conclusions. If rotational reorientation dominates the nuclear and/or electron spin relaxation processes, then a plot of ln( Ty ) against T should be linear with slope Er/R, where r is the activation energy for rotational reorientation. This was found to be the case for Cu, Cr, and Fe complexes with Er 9-2kJ mol" However, for V, Mn, and... 

Agreement between theoretical and experimental TJT2 values is much poorer for X = Br and I and indicates a breakdown of the Redfield theory. (32) Isotropic proton shifts for pyridine-N-oxide and y-picoline-A-oxide protons have been reported in the 5-coordinated adducts of these bases with bis(di-p-tolyl-dithiophosphinato)-Co(ii) and -Ni(ii). (70,71) Dipolar shifts have been evaluated and indicate that the Co(ii)-pyridine-N-oxide adducts have a bent structure in solution with Co-O-N angle of 125°. Results indicate that a rr-spin delocalization mechanism is operating, and INDO calculations suggest that the highest bonding orbital is involved in the spin transfer process. 

This Hamiltonian must be put in the form of equation (Al) (See Appendix A ) for the Redfield theory to be applicable, and depending on the origin, different treatments of the perturbation is necessary. How the direct product is handled is determined by the correlation between the different parts. For simple liquids, the dipole-dipole tensor fluctuates on the picosecond to nanosecond time scale and it is thus not correlated with the nuclear spins. 

Under certain assumptions in equation (A3), the time dependence in the lattice interaction tensors is sufficient to describe the relaxation and derive expressions for the relaxation times. This is the basis of Redfield theory, in which first the Master equation is expanded... 

The expressions in the Redfield theory [82] is derived under the assumptions of weak coupling between the spin states and that the correlation of the perturbation, i.e. of Gg-g(r), has decayed to zero for the times of interest. This is formally expressed as... 

Second, we should keep in mind that between the two extreme limits discussed above there exists a regime of intermediate behavior, where dephasing/decoherence and molecular response occur on comparable timescales. In this case the scattering process may exhibit partial coherence. Detailed description of such situations requires treatment of optical response within a formalism that explicitly includes thermal interactions between the system and its environment. In Section 18.5 we will address these issues using the Bloch-Redfield theory of Section 10.5.2. 

An important consequence of the lineshape theory discussed above concerns the effect of the bath dynamics on the linewidths of spectral lines. We have already seen this in the discussion of Section 7.5.4, where a Gaussian power spectrum has evolved into a Lorentzian when the timescale associated with random frequency modulations became fast. Let us see how this effect appears in the context of our present discussion based on the Bloch-Redfield theory. 

Our discussion of motional narrowing has focused so far on the homogeneous spectrum described by the Bloch-Redfield theory, which is valid only when the bath is fast relative to the system timescale. In this case we could investigate the Xc 0 limit, but the opposite case, Tc —> oo, cannot be taken. In contrast, the classical... 

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