Stochastic Liouville equation spectra

Similar to fluorescence depolarization and NMR, two limiting cases exist in which the molecular motion becomes too slow or too fast to further effect the ESR lineshape (Fig. 8) (35). At the fast motion limit, one can observe a narrow triplet centered around the average g value igxx + gyy + giz with a distance between lines of aiso = Axx- -Ayy- -A2,z)l3, where gu and Ajj are principal values of the g-tensor and the hyperflne splitting tensor A, respectively. At the slow motion limit, which is also referred to as the rigid limit, the spectrum (shown in Fig. 8) is a simple superposition of spectra for all possible spatial orientations of the nitroxide with no evidence of any motional effects. Between these limits, the analysis of the ESR lineshape and spectral simulations, which are based on the Stochastic Liouville Equation, provide ample information on lipid/protein dynamics and ordering in the membrane (36). 

In order to describe the time evolution of the density matrix Q(t) during some arbitrary pulse sequence, we divide the sequence into regions, where a pulse is present and regions where there is no pulse. The action of the different non-selective puls (including a single 90° pulse for the FID which after FT yields the CW frequency spectrum) is considered by unitary transformations employing Wigner rotation matrices [10, 49]. After the pulse the density matrix is assumed to obey the stochastic Liouville equation [85, 86]... 

In general, to describe the EPR spectrum in a motional regime, a procedure based on the solution of the stochastic Liouville equation should be used [49-52]. However, in low-viscosity media, a very fast molecular tumbling leads to an averaging of the anisotropic part of the spin Hamiltonian to zero, and the spectrum is composed of narrow Lorentzian lines, and is characterized by... 

Furthermore, a cw-EPR spectrum can be simulated based on quantum mechanics. The most widely used approach in EPR spectral simulation is based on the stochastic Liouville equation (SLE), which treats the electronic and nuclear spins quantum mechanically, while the nitroxide re-orientation motion is treated classically and parameterized in terms of rotational diffusion constants. The SLE approach is extremely efficient and capable of computing a spectrum in a fraction of a second. This enables iterative fitting of experimental spectra, including those that fall within the slow-motion regime. " However, SLE-based spectral simulations depend on the physical model used to describe the nitroxide motion, which usually requires a large number of parameters, and unique determination of nitroxide motion from simulation remains challenging. 

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