Stochastic Liouville equation calculation

The stochastic Liouville equation, in the form relevant for the ESR line shape calculation, can be written in a form reminiscent of the Redfield equation in the superoperator formulation, Eq. (19) (70-73) ... 

It has been shown that for the case in which the pumping and probing lasers do not overlap, one can use the GLRT. In this section, it shall be shown how the GLRT can be applied to calculate the ultrafast time-resolved spectra. For this purpose, start from the stochastic Liouville equation to describe the EOM for the density matrix system embedded in a heat bath... 

It is possible to perform more precise calculations that simultaneously account for the coherent quantum mechanical spin-state mixing and the diffusional motion of the RP. These employ the stochastic Liouville equation. Here, the spin density matrix of the RP is transformed into Liouville space and acted on by a Liouville operator (the commutator of the spin Hamiltonian and density matrix), which is then modified by a stochastic superoperator, to account for the random diffusive motion. Application to a RP and inclusion of terms for chemical reaction, W, and relaxation, R, generates the equation in the form that typically employed... 

Quantitative calculations of CIDNP intensities can be performed with the stochastic Liouville equation (Eq. 9.3), ... 

Since the Bia value are less than 10 mT for most organic radical pairs, the magnetically induced changes due to the HFCM are usually saturated below 0.1 T. It is noteworthy that no analytical prediction of the magnetic field dependence of Yc (B) and Ye (B) is possible in the case of the HFCM. On the other hand, the quantitative Yc (B) and Ye (B) values can only be obtained by numerical calculations with the stochastic Liouville equation [27]. 

Quantitative calculations of the CIDNP effect can be performed as described in Section II.B.5. One has to set up the nuclear spin system of the intermediate radical pair or biradical, choose a diffusional model, compute reaction probabilities for every nuclear spin state by solving the stochastic Liouville equation numerically or approximately, establish a correlation between the nuclear spin states in the paramagnetic intermediates and the nuclear spin states in the products to obtain the populations of the latter, and finally apply Eq. 61 or the formalism of the preceding section to get line intensities. This approach, which for all but the simplest systems is impracticable except on a computer, is often necessary with the usual uncertainty of the parameters entering the calculations of the radical pair mechanism, a reasonable accuracy can be expected. However, qualitative relationships between signal intensities, especially signal phases, and parameters of the reaction mechanism as well as magnetic properties of the intermediates are... 

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. 

Today the spectral profiles can be simulated for any motional regime by a numerical integration of the stochastic Liouville equation, as discussed in Chapter 12 and in the references therein. The noticeable improvement in the techniques of calculation of the magnetic parameters and their dependence on the solvent, and of the minimum energy conformation of the molecules, have opened the possibility of an integrated computational approach. Since it gives calculated spectral profiles completely determined by the molecular and physical properties of the radical and of the solvent at a given temperature, this method is a step forward in the direction of a sound interpretation of complex spectra. 

Since the CW-ESR spectra provides structural information and dynamics at different time scales, proper account of fast and slow motion of the labeled molecules is required for correct reproduction of the spectra. While the fast motion can be derived from a fast-motional perturbative model, in the slow-motion regime the effects on the spin relaxation processes exerted by the molecular motions requires a more sophisticated theoretical approach. The calculation of rotational diffusion in solution can be tackled by solving the stochastic Liouville equation (SLE) or by longtime-scale molecular dynamics simulations [94—96]. 

The approach recentiy proposed by Pohmeno and Barone to simulate CW-ESR spectra [93] is composed of several steps. First, state-of-the-art QM calculations provide the structural and local magnetic properties of the investigated molecular system. Second, dissipative parameters such as rotational diffusion tensors are calculated by using stochastic Liouville equation. Third, in the case of multiple-label systems, electron exchange and dipolar interactions are computed. 

The goal of this chapter is twofold. First we wish to critically compare—from both a conceptional and a practical point of view—various classical and mixed quantum-classical strategies to describe non-Born-Oppenheimer dynamics. To this end. Section II introduces five multidimensional model problems, each representing a specific challenge for a classical description. Allowing for exact quantum-mechanical reference calculations, aU models have been used as benchmark problems to study approximate descriptions. In what follows, Section III describes in some detail the mean-field trajectory method and also discusses its connection to time-dependent self-consistent-field schemes. The surface-hopping method is considered in Section IV, which discusses various motivations of the ansatz as well as several variants of the implementation. Section V gives a brief account on the quantum-classical Liouville description and considers the possibility of an exact stochastic realization of its equation of motion. 

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