Bath fluctuations

To investigate the relationship between the reaction driven v7 mode and the subsequent protein motions along the dissociative pathway, further modulations of the frequency of the v7 mode by the surrounding intramolecular and protein bath fluctuations were found using an instantaneous frequency (IF) analysis. The IF was derived from the data by applying a Gaussian filter around the v7 mode in the Fourier spectrum. An inverse Fourier transform produced the time trace TT(t) given by ... 

This describes the case that the force exerted by the bath fluctuates rapidly compared to the free motion of S. 

The alkali hydroxide solution produced in this electrolyzer had a concentration of 110 to 130 grams of NaOH per litre and the average current efficiency amounted to 88 per cent. The voltage across the bath fluctuated between 3.7 and 4.2 V. 

In order to calculate the response function of the Feynman diagram R3, it is further assumed that the transition frequency co 12 is anharmonically shifted with respect to the ground states transition frequency so that, u>n = >oi -A. Another assumption that can be made (see later for a discussion of these assumptions) is that the fluctuations between both level pairs are strictly correlated <5co12 = <5 j0i. This implies that only the harmonic part of the potential surface is perturbed by the bath fluctuations and the anharmonicity of the vibrator is unaffected. We then obtain for R3 ... 

The exact value of T, of the run is then determined as the arithmetic mean of a series of digital outputs of T, recorded for 2 d prior to the insertion of the sample cell assembly into the aluminium block bath. Fluctuations of the values of T, during this time are within 0.05 K. 

The fast modulation (homogeneous) limit is obtained when the correlation time of the bath fluctuations is very fast compared with their magnitude, that is, fc 1. In this case, the exp(-Ax) on the right-hand side of Eq. (114) vanishes very rapidly and may be ignored. We then get... 

Because the dynamics in this particular bridge is underdamped, the population that enters the bridge oscillates rapidly among all the bridge sites. The oscillations are gradually dampened to a steady level over the first hundred picoseconds, due both to the dephasing of the quantum coherences by bath fluctuations and to destructive interference with additional population amplitude coming from the donor. 

Both in the strong (Fig. 4.6) and the weak (Fig. 4.7) eases, we find Q(co = 0) 0, which is expected in the slow modulation case eonsidered here. Physically, when the laser detuning frequency is exactly in the middle of two frequency shifts, + v, the rate of photon emissions is identical whether the molecule is in the up state ( + v) or in the down state (—v). Therefore, the effect of bath fluctuation on the photon counting statistics is negligible, which leads to Poissonian counting statistics at o)f = 0. 

We further analyze the case that the bath fluctuation is both strong and fast, case 3, r V R. The results shown in Figure 4.10 correspond to this case. We find that Eq. (4.77) is further simplified in two different limits, F T and r. 

The only remaining case is case 6 (T J v), where the bath fluctuation is fast compared with the radiative decay rate but not compared with the fluctuation amplitude. Because T R,v in this case we can approximate the exact results for and Q by their limiting expressions corresponding to r 0, yielding Eqs. (A.8) and (A.49), and an important relation holds in this limit. 

We investigate the overall effect of the bath fluctuation on the photon statistics for the steady-state case as the fluctuation rate R is varied from slow to fast modulation regime. To characterize the overall fluctuation behavior of the photon statistics, we define an order parameter q. 

When the bath fluctuation becomes extremely fast such that R v7r, the splitting behavior of Q is observed, as discussed in Eq. (4.79) for Ey E, and then q cc l/R similar to the slow modulation regime. The approximate value of q based on fast modulation approximation, Eq. (4.77) (dotted line) shows good agreement with the exact calculation found using Eqs. (A.47) and (A.48). Finally, when R CO, <2 = 0. As mentioned before, this is expected since the molecule cannot interact with a very fast bath hence the photon statistics becomes Poissonian. 

The first quantum-mechanical consideration of ET is due to Levich and Dogonadze [7]. According to their theory, the ET system consists of two electronic states, that is, electron donor and acceptor, and the two states are coupled by the electron exchange matrix element, V, determined in the simplest case by the overlap between the electronic wave functions localized on different redox sites. Electron transfer occurs by quantum mechanical tunneling but this tunneling requires suitable bath fluctuations that bring reactant and product energy levels into resonance. In other words, ET has... 

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