Diabatic population

Let us next turn to Model II, representing the C —> B —> X internal-conversion process in the benzene cation. Figure 2 demonstrates that this (compared to the electronic two-state model, Model I) more complicated process is difficult to describe with a MFT ansatz. Although the method is seen to catch the initial fast C —> B decay quite accurately and can also qualitatively reproduce the oscillations of the diabatic populations of the C- and B-state, it essentially fails to reproduce the subsequent internal conversion to the electronic X-state. Jn particular, the MFT method predicts a too-slow population transfer from the C- and B-state to the electronic ground state. 

Figure 6 shows the results for the more challenging model. Model IVb, comprising three strongly coupled vibrational modes. Overall, the MFT method is seen to give only a qualitatively correct picture of the electronic dynamics. While the oscillations of the adiabatic population are reproduced quite well for short time, the MFT method predicts an incorrect long-time limit for both electronic populations and fails to reproduce the pronounced recurrence in the diabatic population. In contrast to the results for the electronic dynamics, the MFT is capable of describing the almost undamped coherent vibrational motion of the vibrational modes. 

In direct analogy to the adiabatic case, the classical diabatic population probability is given by... 

Let us turn to Model 11 describing the C —> B —> X internal-conversion of the benzene cation. Figure 2 shows the diabatic population probabilities pertaining... 

Figure 13. Comparison of quantum (thick hues), QCL (thin lines), and SH (dashed lines) results as obtained for the one-mode two-state model IVa [205], Shown are (a) the adiabatic excited-state population P i), (b) the corresponding diabatic population probability and (c) the...

Finally, we consider Model V by describing two examples of outer-sphere electron-transfer in solution. Figures 7 and 8 display results for the diabatic electronic population for Models Va and Vb, respectively. Similar to the mean-field trajectory calculations, for Model Va the SH results are in excellent agreement with the quantum calculations, while for Model Vb the SH method only is able to describe the short-time dynamics. As for the three-mode Model IVb discussed above, the SH calculations in particular predict an incorrect long-time limit for the diabatic population. The origin of this problem will be discussed in more detail in Section VI in the context of the mapping formulation. 

In the case of Model II, neither the state-specihc nor the total quantum-mechanical level densities are available. To determine the optimal value of the ZPE correction, therefore criterion (98) was applied, which yielded y = 0.6. The mapping results thus obtained (panels D and G) are seen to reproduce the quantum result almost quantitatively. It should be noted that this ZPE adjustment ensures that the adiabatic population probabilities remain within [0, 1] and at the same time also yields the best agreement with the quantum diabatic populations. 

Figure 39. Initial decay of the diabatic population of the S2 state for the four-mode pyrazine model. Compared are quantum (full line), semiclassical (dashed line), and normalized semiclassical (dotted line) results.

Figure 42. Diabatic population (a) and modulus of the autocorrelation function (b) of the initially prepared state for Model IVa. The full tine is the quantum result, and the dashed line depicts the semiclassical mapping result. The semiclassical data have been normalized. Panel (c) shows the norm of the semiclassical wave function.

The result for the E state population thus obtained is depicted in Fig. 6. It should be pointed out that diabatic populations are given here for simplicity (for two-state problems, also adiabatic populations have been computed [47]). In agreement with... 

The time-dependence of the electronic (diabatic) populations of the B state in the A-B coaled state dynamics is shown in Fig. 6b. The WP is initially (t = 0) located on the B state and therefore, its population starts from 1.0. Since th e ilibrium minimum of the B state nearly coincides with the minimum ofjhe A-B CIs, the population of this state decays (nonradiatively) rapidly to the A state through the CIs, and reaches to a value of 0.05 at longer times. The initial fast decay of the population relates to a decay rate of 30 fs for the B state. 

Fig. 9 Decay of the electronic (diabatic) populations of the A (panel a) and B (panel b) electronic states in the coupled X—A—B—C states dynamics of PA +...

Fig. 10 The vibronic bands of the Di and D2 states of N + are shown in the left and middle of the figure, respectively. The present theoretical results are compared to the available experimental results of [45]. The decay of the diabatic population of the Di and the D2 electronic states in the Dq — D — D2 coupled states dynamics is shown in the right side of the figure. The Dq, D and D2 adiabatic potential energy surfaces of N + along the vy (symmetric C=C stretching) vibrational mode is also shown at top of the diagram...

Figure 4.14 Population on Sq (solid line), Si (dashed line), and S2 (dotted line) as a function of time after photoexcitation of butadiene. Results are averaged over 10 runs. Immediately upon excitation to the bright excited state, there is nearly diabatic population transfer to Sg. Quenching from S2 to Si begins about 70 fs after photoexcitation and quenching to the ground state begins roughly 30 fs later.

Figure 5 shows some computed adiabatic/diabatic populations for the twomode model system illustrated in Fig. 1. They are intended to illustrate different typologies concerning the ground-state equilibrium position with respect to the crossing surfaces, a point which will be further discussed in the next section. What we do, in practice, is to select the different initial...

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