Mode-truncation

Successive orders H(n ) can be shown to correspond to successive orders in a moment (or cumulant) expansion of the propagator, which takes one to increasing times. Truncation of the chain at a given order n (i.e., 3 + 3n modes) leads to an approximate, lower-dimensional representation of the dynamical process, which reproduces the true dynamics up to a certain time. In Ref. [51], we have demonstrated explicitly that the nth-order (3n+3 mode) truncated HEP Hamiltonian exactly reproduces the first (2n + 3)rd order moments (cumulants) of the total Hamiltonian. A related analysis is given in Ref. [73],... 

FIGURE 5.16 The effect of desensitization on stop-time mode measurements. Bottom panels show the time course of response production for a system with no desensitization, and one in which the rate of response production fades with time. The top dose response curves indicate the area under the curve for the responses shown. It can be seen that whereas an accurate reflection of response production is observed when there is no desensitization the system with fading response yields an extremely truncated dose-response curve. 

The key point is that the underdetermined system of linear equations is rendered soluble by an assumption of the prior probabilities of the unknown coefficients. It is important to realize that truncating the number of modes creates... 

The reason is truncation of dynamics in low-resolution modes in a 14-bit-detector each pixel in low-resolution mode can contain 0 to 16382 counts. With the next photon an arithmetic overflow will occur and the pixel is saturated. In high-resolution mode the same area of the detector is represented by 4 pixels, and if the intensity is evenly distributed it takes 4 times longer before the pixels will be saturated. If the high resolution is not required and the cycle time is 30 s or longer, it is good practice to store away the big files on a spacious USB hard-disk and afterwards to bin the data. 

Fig. 4) is physically more reasonable. As a practical matter, as k is increased, the calculation becomes increasingly unreliable because of the severely truncated basis set. The important point is that inclusion of a low frequency effective solvent mode can nicely rationalize the observed characteristics of the C T ion. Similar improvements may be anticipated in analyzing intervalence bands in other systems. 

In Refs. [55, 79], the truncation at the level of Heg has been tested for several molecular systems exhibiting an ultrafast dynamics at Coin s, and it was found that this approximation can give remarkably good results in reproducing the short-time dynamics. This is especially the case if a system-bath perspective is appropriate, and the effective-mode transformation is only applied to a set of weakly coupled bath modes [55,72]. In that case, the system Hamiltonian can take a more complicated form than given by the LVC model. 

Fig. 8 shows time-dependent state populations as obtained from quantum dynamical (MCTDH) calculations. While the full (here, 24 dimensional) model exhibits an ultrafast XT decay, no net decay is observed for the reduced 3-mode model truncated at the lowest level of the effective mode hierarchy. The dynamics is strongly diabatic if confined to the high-frequency subspace (Heff ) and involves repeated coherent crossings [51]. The dynamical interplay between the high-frequency and low-frequency modes is apparently a central feature of the process. To account for these effects, a treatment at the level of is necessary, i.e., a six-mode model including the low-frequency modes. At the level of the dynamics is found to be essentially exact. 

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