The Stochastic Liouville Equation

What becomes of the various elements of the averaged reduced density matrix p as the ensemble approaches equilibrium Consider the diagonal elements, which represent the populations of the basis states of the quantum systems. The equilibrium populations should depend on the Boltzmann factors for these states  

Here k n and k m are the microscopic classical rate constants for conversion of state n to state m and vice versa. The subscript stochastic indicates that we are considering relaxations that depend on random fluctuations of the surroundings, not the oscillatory, quantum-mechanical phenomena described by Eq. (10.23). The ensemble will relax to a Boltzmann distribution of populations if the ratio k Jk m is given by tJ. —EnJk ). According to Eq. (10.27), relaxations of the diagonal elements toward thermal equilibrium do not depend on the off-diagonal elements of p, which is in accord with classical treatments of kinetic processes simply in terms of populations. 

For a two-state system, any changes of the populations of the two states must always be equal and opposite, which means that and — P22 must both 

The relaxation time constant for a two-state system is the reciprocal of the sum of the forward and backward rate constants k 2 and k2i (1/Ti = ki2+ki ). In NMR and EPR spectroscopy, Ti is called the longitudinal relaxation time or the spin-lattice 

Assuming that the basis states used to define p are stationary, the off-diagonal elements of p must go to zero at equilibrium. There are several reasons for this. First, stochastic fluctuations of the diagonal elements will cause an ensemble to lose coherence. This is because stochastic kinetic processes modify the coefficients c/ of the individual systems at unpredictable times, imparting random phase shifts, WeTl show in Sect. 10.5 that a relaxation of and occurring with rate constant l/Ti causes the off-diagonal elements p and to decay to zero with a rate constant of l/(2Ti). 

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 

The stochastic Liouville equation is highly useful when applied at high field, as techniques exist to reduce in size the typically large matrices it produces, and it has thus been used to simulate electron and nuclear spin polarizations in magnetic resonance experiments.A relatively recent book describes the approach in detail. However, for determining field dependences, such reductions are not possible, meaning that the sizes of the matrices are too large for even modern computers, and so this approach is seldom used for the simulation of field effects. 

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The dynamical behaviors of p(At) v and p(At)av av, have to be determined by solving the stochastic Liouville equation for the reduced density matrix the initial conditions are determined by the pumping process. For the purpose of qualitative discussion, we assume that the 80-fs pulse can only pump two vibrational states, say v = 0 and v = 1 states. In this case we obtain... 

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) ... 

Consider the non-adiabatic transition a — b shown schematically in Figure 5.2, where a and b may denote the electronic states of D A and D A, respectively. To describe the dynamic processes of the system, one starts with the stochastic Liouville equation... 

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... 

In order to better model the effects of J and D without the complexity of the stochastic Liouville equation, Monte Carlo approaches have recently been 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]. 

Theoretical Analysis with the Stochastic Liouville Equation... 

The time dependence of p is given by the stochastic Liouville equation (SLE) ... 

Dekker has studied multiplicative stochastic processes. In his work the stochastic Liouville equation was solved explicitly through first order in an expansion in terms of correlation times of the multiplicative Gaussian colored noise for a general multidimensional weakly non-Markovian process. He followed the suggestions of refs. 17 and 18 and applied, Novikov s theorem. In the general multidimensional case, however, he improved the earlier work by San Miguel and Sancho. ... 

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. 

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). 

Photoinduced charge separation processes in the supramolecular triad systems D -A-A, D -A -A and D -A-A have been investigated using three potential energy surfaces and two reaction coordinates by the stochastic Liouville equation to describe their time evolution. A comparison has l n made between the predictions of this model and results involving charge separation obtained experimentally from bacterial photosynthetic reaction centres. Nitrite anion has been photoreduced to ammonia in aqueous media using [Ni(teta)] " and [Ru(bpy)3] adsorbed on a Nafion membrane. 

The model proposed by Stillman and Freed (SF) in their 1980 paper [33] is very versatile. By choosing carefully (i) the coupling forces between molecule variables (x,) and augmented ones (x,), and (ii) the potential function in the final equilibrium distribution, one can easily recover a variety of mathematical forms, reflecting different physical cases. The SF procedure starts from considering a system coupled to a second one in a deterministic way (interaction potential) the latter, in the absence of any coupling is described by a FP operator. The first step to obtain a description of the full system is to write the stochastic Liouville equation (SEE), according to Kubo [44] and Freed [45]... 

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). 

It is at this point that S-To-type CIDNP and S-T i-type CIDNP diverge. Because there are no flip-flop transitions in the first case, a separation of nuclear and electronic subspaces is possible and leads to a drastic simplification of the problem, since only four elements of the density matrix have to be retained. This even affords a closed-form analytical solution if some simplifying assumptions are introduced. In the second case, no such separation is possible, and numerical solutions of the stochastic Liouville equation are the only feasible procedure this subject is obviously beyond the scope of this review. 

Transient EMR has also been reported on the triplet state of retinal dissolved in liquid crystalline phase (Munzenmaier et al., 1992). The simulation of the transients with the stochastic Liouville equation provides the motional and order parameters of the pigment. The anisotropy ofmotional correlation times is high as expected for such an extended linear molecule and the correlation times couldbe followed with temperature over a range of two orders of magnitude in the nematic and smectic phase. 

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... 

In a study of phenacylphenylsulfone photolysis, CIDNP data were taken as evidence that the primary radical pairs cannot recombine to regenerate the starting material because the micelle forces a certain orientation of the radicals [63], From low-field 13C CIDNP and SNP measurements on cleavage of benzylic ketones in sodium dodecyl sulfate micelles, it was inferred [64] that the exchange interaction in these systems is several orders of magnitude smaller ( 10lorads 1 at a reduction distance of 6 A cf. the values in Section IV.B) and the distance dependence is much weaker (a x 0.5 A" cf. the discussion of Eq. 10) than generally assumed for radical pairs. By numerical solutions of the stochastic Liouville equation for a model of the micelle where one of the radicals is kept fixed at the center of the micelle while the other radical is allowed to diffuse, the results of MARY experiments, 13C CIDNP experiments at variable fields, and SNP experiments could be reproduced with the same set of parameters [65],... 

An approximative analytical treatment of S-T+-type CIDNP of radical pairs in micelles has recently been given [66]. Comparison with numerical solutions of the stochastic Liouville equation obtained by a finite difference technique showed the accuracy of the approximate solution to be quite good. 

The subsequent time evolution of the density matrix elements pAt) is governed by the stochastic Liouville equations (Section I.B). In the electric dipole approximation, the intensity of fluorescence subsequently emitted by such a dimer with polarization e, will be... 

The stochastic Liouville equations are readily solved for the time-dependent density matrix elements pu (e.g., through Laplace transforms) the latter may then be used in turn to develop expressions for the polarized fluorescence or absorption difference signals. The initial values of the density matrix elements under 5-function pulsed excitation are given by... 

To treat slow motion spectra in the rotational correlation time regime of nanoseconds, the time dependence of the Hamiltonian has to be taken explicitly into account. This is most conveniently done by the Stochastic Liouville Equation approach, which contains explicit superoperator expressions for the rotational diffusion of the molecule [54]. 

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]... 

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