Coupling cases

As in the case of H electronic states of tetraatomic molecules, because of generally high degeneracy of zeroth-order vibronic leves only several particular (but important) coupling cases can be handled efficiently in the framework of the pertnrbation theory. We consider the following paiticnlai" cases ... 

In this model there is a quantitative difference between RLT and electron transfer stemming from the aforementioned difference in phonon spectra. RLT is the weak-coupling case S < 1, while for electron transfer in polar media the strong-coupling limit is reached, when S > 1. In particular, in the above example of ST conversion in aromatic hydrocarbon molecules S = 0.5-1.0. 

Solving now the Heisenberg equations of motion for the a operators perturbatively in the same way as in the weak-coupling case, one arrives (at = 0) at the celebrated non-interacting blip approximation [Dekker 1987b Aslangul et al. 1985]... 

The symmetric coupling case has been examined by using Sethna s approximations for the kernel by Benderskii et al. [1990, 1991a]. For low-frequency bath oscillators the promoting effect appears in the second order of the expansion of the kernel in coj r, and for a single bath oscillator in the model Hamiltonian (4.40) the instanton action has been found to be... 

Fig. 4.5. The degree of approximation for the increase of current in time for uncoupled and weakly coupled solutions for impact-loaded, x-cut quartz and z-cut lithium niobate is shown by comparison to the numerically predicted, fully coupled case. In the figure, the initial current is set to the value of 1.0 at the measured value (after Davison and Graham [79D01]).

Couplings shall conform to the dimensions and tolerances shown in Tables 4-148 and 4-149. Unless otherwise specified, threaded and coupled casing and tubing shall be furnished with regular couplings. 

Figure 19. The predicted low T heat conductivity. The no coupling case neglects phonon coupling effects on the ripplon spectrum. The (scaled) experimental data are taken from Smith [112] for a-Si02 (AsTj/ScOd 4.4) and from Freeman and Anderson [19] for polybutadiene (ksTg/Hcao — 2.5). The empirical universal lower T ratio l /l 150 [19], used explicitly here to superimpose our results on the experiment, was predicted by the present theory earlier within a factor of order unity, as explained in Section lllB. The effects of nonuniversaUty due to the phonon coupling are explained in Section IVF.

For the strong coupling case J2jSj 1, we can use the short-time approximation... 

In summary, for displaced oscillators, absorption and emission spectra show a mirror image relation and for the strong coupling case, a(oo) will exhibit a Gaussian band shape, absorption maximum independent of temperature, and bandwidth increasing with temperature. It should be noted that the distortion effect and Duschinsky effect have not been considered in this chapter, but these effects can be treated similarly. 

In the strong coupling case, the transfer of excitation energy is faster than the nuclear vibrations and the vibrational relaxation ( 10 12 s). The excitation energy is not localized on one of the molecules but is truly delocalized over the two components (or more in multi-chromophoric systems). The transfer of excitation is a coherent process9 the excitation oscillates back and forth between D and A and is never more than instantaneously localized on either molecule. Such a delocalization is described in the frame of the exciton theory10 . 

The transfer rate is fast compared to vibrational relaxation but slower than nuclear motions, in contrast to the strong coupling case. It can be approximated as... 

The quantity bk has a physical meaning as the spectral density of photons emitted by the dressed excited particle, and its explicit form is rather complicated [11]. However, for weak coupling case it reduces to a simple form ... 

For the weak coupling case with Eq. (32), our master equation reduces to the well-known quantum master equation, obtained through the approximation, widely used in quantum optics. This equation describes, among other things, quantum decoherence due to Brownian motion. Hence, we have derived an exact quantum master equation for the transformed density operator p that describes exact decoherence. Furthermore, our master equation cannot keep the purity of the transformed density matrix. Indeed, one can show that if p(t) is factorized into a product of transformed wave functions at t = 0, it will not be factorized into their product for t > 0. This is consistent the nondistributivity of the nonunitary transformation (18). 

Application of the symmetry correlation scheme to reaction (12) is summarized in Table 4 where N is the Himd s coupling case (b) rotational quantum number for O2 and - 5 is the difference of the Hund s coupling case (a) quantum numbers of total angular momentum and electron spin (5 = 1/2) angular momentum, respectively. To consider the high-symmetry isotopomer system first, the results in Table 4 indicate that only odd / collisions - 5 = odd) with 2 can lead... 

The bilinear coupling case (i.e. position-independent friction) corresponds to G (s) = 1, or equivalently, to Y s=s . The position-dependent friction Eq. (39) can then be rewritten as... 

Figure 1 A matrix representation of two possible coupling schemes in the WiGLEformalism. The rows correspond to n, the index of a particular realization of the ensemble, and the columns correspond to the index of the other realization of the ensemble which may or may not be in the set Sw,n, depending on whether the matrix element is fill or empty, respectively. The matrix on the left (a) corresponds to the banded coupling case, in which a given particle is coupled to the nearest w particles (for a specified ordering) through the friction. The matrix on the right (b) corresponds to the block-diagonal case, in which a given particle is always coupled to a prespecified set of w particles.

The resulting expression is especially simple in the weak coupling case. In this case, the two propagators in Eq. (12) can be approximated by their first order (i.e, single hop) terms. (The zeroth order term makes no contribution of Kjf as long as i 5 f) In this weak coupling limit, the expression for Pif (t) can be expressed as ... 

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