Adiabatic population

Sensitivity Enhancement Techniques by Adiabatic Population Transfer. 134... 

In complete analogy, the adiabatic population probability is defined as the expectation value of the adiabatic projector quasi-... 

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. 

As a consistency test of the stochastic model, one can check whether the percentage Nk t) of trajectories propagating on the adiabatic PES Wk is equal to the corresponding adiabatic population probability Pf t). In a SH calculation, the latter quantity may be evaluated by an ensemble average over the squared modulus of the adiabatic electronic coefficients [cf. Eq. (22)], that is. 

As a first example, we again consider Model 1 describing a two-state three-mode model of the Si nn ) and 52(7171 ) states of pyrazine. Figure 11a shows the quantum-mechanical (thick line) and the SH (thin lines) results for the adiabatic population probability of the initially prepared electronic state /2). As... 

As discussed above, this discrepancy may be caused by classically forbidden electronic transitions—that is, cases in which a proposed hopping process is rejected due to a lack of nuclear kinetic energy. Figure 11c supports this idea by showing the absolute numbers of successful (thick fine) and rejected (thin line) surface hops. In accordance with the initial decay of the adiabatic population, the number of successful surface hops is largest during the first 20 fs. For larger times, the number of rejected hops exceeds the number of successful surface hops. This behavior clearly coincides with the onset of the deviations between the two classically evaluated curves Nk t) and P t). We therefore conclude that the observed breakdown of the consistency relation (42) is indeed caused by classically forbidden electronic transitions. 

Although the systems discussed so far exhibit fairly complex vibrational and diabatic electronic relaxation dynamics, their adiabatic population dynamics is relatively simple and thus is well-suited for a SH description. To provide a challenge for the SH method, we next consider various spin-boson-type models, which may give rise to a quite comphcated adiabatic population dynamics. 

Let us begin with the one-mode electron-transfer system. Model IVa, which still exhibits relatively simple oscillatory population dynamics [205]. SimUar to what is found in Fig. 5 for the mean-field description, the SH results shown in Fig. 13 are seen to qualitatively reproduce both diabatic and adiabatic populations, at least for short times. A closer inspection shows that the SH results underestimate the back transfer of the adiabatic population at t 50 and 80 fs. This is because the back reaction would require energetically forbidden electronic transitions which are not possible in the SH algorithm. Figure 13 also shows the SH results for the electronic coherence which are found to... 

To summarize, it has been found that the SH method is able to at least qualitatively describe the complex photoinduced electronic and vibrational relaxation dynamics exhibited by the model problems under consideration. The overall quality of SH calculations is typically somewhat better than the quality of the mean-field trajectory results. In particular, this holds in the case of several curve crossings (see Fig. 2) as well as when the dynamics and the observables of interest are essentially of adiabatic nature— for example, for the calculation of the adiabatic population dynamics associated with a conical intersection (see Figs. 3 and 12). Furthermore, we have briefly discussed various consistency problems of a simple quasi-classical SH description. It has been shown that binned electronic population probabilities and no momentum adjustment for classically forbidden transitions help us to improve this matter. There have been numerous suggestions to further improve the hopping algorithm [70-74] however, the performance of all these variants seems to depend largely on the problem under consideration. 

In order to study the origin of the deviations observed, we first consider the statistical convergence of the QCL data. As a representative example. Fig. 14 shows the absolute error of the adiabatic population as a function of the number of iterations N—that is, the number of initially starting random walkers. The data clearly reveal the well-known 1/Vn convergence expected for Monte Carlo sampling. We also note the occurrence of the sign problem mentioned above. It manifests itself in the fact that the number of iterations increases almost exponentially with propagation time While at time t = 10 fs only 200 iterations are sufficient to obtain an accuracy of 2%, one needs N = 10 000 at t = 50 fs. 

Due to the large-level density of the lower-lying adiabatic electronic state, the chances of a back transfer of the adiabatic population are quite small for a multidimensional molecular system. To a good approximation, one may therefore assume that subsequent to an electronic transition a random walker will stay on the lower adiabatic potential-energy surface [175]. This observation suggests a physically appealing computational scheme to calculate the time evolution of the system for longer times. First, the initial decay of the adiabatic population is calculated within the QCL approach up to a time to, when the... 

Figure 17. Initial decay of the adiabatic population probability obtained for Model I. Compared are quantum results (thick line) and standard (thin full line) and energy-conserving (dotted line) quantum-classical Liouville results.

To obtain the optimal quantum correction from the requirement (98), in general several trajectory calculations with varying values of 7 need to be performed. It turns out, however, that often a single simulation is already sufficient. To explain this, we note that the quantum correction 7 affects the adiabatic population probability =j Xl(y,t) +Pl(y,t) —7) directly... 

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 25. Diabatic and adiabatic population probabilities of the C (fuU line), B (dotted hne), and X (dashed line) electronic states as obtained for a five-state 16-mode model of the benzene cation.

X + P )/4, which by construction varies between 0 (system is in /i)) and 1 (system is in /2)). Describing, as usual, the nuclear motion through the position X, the vibronic PO can then be drawn in the (A dia,v) plane. Here, the subscript dia emphasizes that we refer to the population of the diabatic states which are used to define the molecular Hamiltonian H. For inteipretational purposes, on the other hand, it is often advantageous to change to the adiabatic electronic representation. Introducing the adiabatic population A ad. where Nad = 0 corresponds to the lower and A ad = 1 to the upper adiabatic electronic state, the vibronic PO can be viewed in the (Nad,x) plane. Alternatively, one may represent the vibronic PO as a curve N d i + (1 — A ad)IFi between the... 

Apart from POs that virtually evolve on a single adiabatic potential-energy surface, there are numerous orbits that propagate on several or in between adiabatic surfaces. Orbit C (referred to as la in Table VI) is the shortest PO of this type, with a period of 39.2 fs. While the adiabatic population stays around Vad = 0.5, the PO oscillates between the diabatic states with a Rabi-type... 

The idea exploited by Demirplak and Rice is very simple Given a fleld that would generate adiabatic population transfer from a selected initial state to a selected final state if strong enough, but which does not generate complete population transfer because of lack of intensity, find another fleld, called the CDF, that when combined with the first field generates the state that would have been generated by a strictly adiabatic transformation. Demirplak and Rice show that the CDF exists, and they show how it can be calculated. 

For reference purposes, we consider first adiabatic population transfer in a subset of three states decoupled from the full manifold of states. This adiabatic transfer can be driven by STIRAP. The subset of states we consider consists of 1200000), 1300000) and 200020), and the population transfer is from 200000) to 1200020). In the following paragraph, we refer to these states as 11), 5 ), and 6), respectively. We note that state 210011) with energy 5658.1828 cm is nearly degenerate with state 1200020) with energy 5651.5617 cm . We refer to 1210011) as state 9). Since the transition moment coupling states 11) and 6) is one order of... 

Suppose now that either the pulsed field duration or the field strength must be restricted to avoid exciting unwanted process that compete with the desired population transfer, with the consequence that condition (3.71) cannot be met. In an actual SCCI2 molecule //jg is nonzero, although it is one order of magnitude smaller than //j 9 and //5a 9 as mentioned. This situation prompts us to seek a CDF that restores the adiabatic population transfer. The counter-diabatic Hamiltonian for //rwa(0 ill Eq- (3.68) is obtained by using Eqs. (3.7) and (3.70) it is [11]... 

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