Stochastic Liouville Theory

Most detailed studies of spin effects in the literature are based on the Stochastic Liouville equation (SLE), which treats the system as an ensemble and requires the use of the density matrix Pij(r, t) [7-9], The density matrix (i) contains all the information about the ensemble physical observables of the system (ii) describes the distribution of spin states for an ensemble of particles and (iii) is constructed from the vector representation of the spin function (c ) relative to some predefined basis, such that pij(r, t) = c Cj). 

The SLE for a single radical pair (with a mutual diffusion coefficient D ) can be written as 

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Both vibrational and rotovibrational relaxation can be described analyti-caDy as multiplicative stochastic processes. For these processes, RMT is equivalent to the stochastic Liouville equation of Kubo, with the added feature that RMT takes into account the back-reaction from the molecule imder consideration on the thermal bath. The stochastic Liouville equation has been used successfully to describe decoupling in the transient field-on condition and the effect of preparation on decay. When dealing with liquid-state molecular dynamics, RMT provides a rigorous justification for itinerant oscillator theory, widely applied to experimental data by Evans and coworkers. This implies analytically that decoupling effects should be exhibited in molecular liquids treated with strong fields. In the absence of experimental data, the computer runs described earlier amount to an independent means of verifying Grigolini s predictions. In this context note that the simulation of Oxtoby and coworkers are semistochastic and serve a similar purpose. 

In the papers by Berk et al. (42) the EPR linewidths of triplet excitons in single-crystal pyrene at room temperature have been measured in experiments performed at 24 GHz. The data are fitted to a formula first presented by Reineker (43) in a theory based on the Haken-Strobl-Reineker model of exciton motion (the Haken-Strobl-Reineker model can be applied for triplet excitons because they have an exciton bandwidth small in comparison with the thermal energy ksT for more details see the review paper by Reineker (44)). This formula was rederived in the paper by Berk et al. (42) more directly from Blume s stochastic Liouville formalism (45). The agreement was excellent. This result again implied that the dominant spin-relaxation mechanism in pyrene, as in anthracene and presumably in similar molecular crystals, results from hopping between differently oriented molecules in the unit cell. 

Various methods have been developed that interpolate between the coherent and incoherent regimes (for reviews see, e.g. (3)-(5)). Well-known approaches use the stochastic Liouville equation, of which the Haken-Strobl-Reineker (3) model is an example, and the generalized master equation (4). A powerful technique, which in principle deals with all aspects of the problem, uses the reduced density matrix of the exciton subsystem, which is obtained by projecting out all degrees of freedom (the bath) from the total statistical operator (6). This reduced density operator obeys a closed non-Markovian (integrodifferential) equation with a memory kernel that includes the effects of (multiple) interactions between the excitons and the bath. In practice, one is often forced to truncate this kernel at the level of two interactions. In the Markov approximation, the resulting description is known as Redfield theory (7). 

To describe the relaxation phenomenon theoretically, Kowalewski and his coworkers have developed a formalism based on the Stochastic Liouville Equation (SLE). In recent work, Kowalewski et aP have compared theoretical simulations of Ri field dispersion profiles computed using the SLE formalism with the results of more restricted theory developed in Florence that neglects reorientational motion of the zfs tensor. Kruk and KowalewskP describe a theoretical model that incorporates fast reorientational motion of the zfs tensor. These workers also present theoretical calculations describing the situation where collisional distortions of the zfs tensor are much larger than the time-averaged zfs tensor. 

The Marcus equation for nonadiabatic electron-transfer reactions (Eq. B5.3.4), and the Forster theory that we discussed in Chap. 7 apply only to systems with weak intermolecular interactions, which we now can define more precisely as meaning that H21 lh steady-state approximation to the stochastic Liouville equation for a two-state reaction in this limit From Eqs. (BIO.1.15), (10.29a), (10.29b), and (10.30), we have... 

The three-pulse EOM-PMA can be formulated not only in terms of density matrices and master equations but also in terms of wavefunctions and Schrodinger equations [29]. The EOM-PMA can therefore be straightforwardly incorporated into computer programs which provide the time evolution of the density matrix or the wavefunction of material systems. Besides the multilevel Redlield theory, the EOM-PMA can be combined with the Lindblad master equation [49], the surrogate Hamiltonian approach [49], the stochastic Liouville equation [18], the quantum Fokker-Planck equation [18], and the density matrix [50] or the wavefunction [14] multiconfigurational time-dependent Hartree (MCTDH) methods. When using the... 

Of course, as was shown in Section V-A, this latter expression may also be derived starting from the hydrodynamical equations for the pair distribution and the Poisson equation it is also the final result of the theories developed independently by Falken-hagen and Ebeling,9 and by Friedman 12-13 in these two approaches, the starting point is a Liouville equation for the system of ions with an ad hoc stochastic term describing the interactions with the solvent. 

As explained in the Introduction, one needs to distinguish the following kinds of surface hopping (SH) methods (i) Semiclassical theories based on a connection ansatz of the WKB wave function, " (ii) stochastic implementations of a given deterministic multistate differential equation, e.g. the quantum-classical Liouville equation, and (iii) quasiclassical models such as the well-known SH schemes of Tully and others. " In this chapter, we focus on the latter type of SH method, which has turned out to be the most popular approach to describe nonadiabatic dynamics at conical intersections. 

A. THE STOCHASTIC RELAXATION MODEL. The most general theories of magnetic relaxation in Mossbauer spectroscopy involve stochastic models see, for example. Ref. 283 for a review. A formalism using superoperators (Liouville operators) was introduced by Blume, who presented a general solution for the lineshape of radiation emitted (absorbed) by a system whose Hamiltonian jumps at random as a function of time between a finite number of possible forms that do not necessarily commute with one another. The solution can be written down in a compact form using the superoperator formalism. 

The theoretical approach to the interpretation of ESR spectra is based on the solution of the SLE. This is essentially a semiclassical approach based on the Liouville equation for the magnetic probability density of the molecule augmented by a stochastic operator which describes the relevant relaxation processes that occur in the system and is responsible for the broadening of the spectral lines [2]. The SLE approach can be linked profitably to density functional theory (DFT) evaluation of geometry and magnetic parameters of the radical in its... 

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