Electronic population probability

To illustrate the quality of the MFT method, let us first consider Model 1 describing the S2 — conical intersection in pyrazine. Figure 1 compares quantum-mechanical and MFT results obtained for the adiabatic and diabatic electronic population probabilities P t) and as well as for the mean... 

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. 

As an example. Fig. 18 shows the diabatic electronic population probability for Model I. The quantum-mechanical results (thick line) are reproduced well by the QCL calculations, which have assumed a localization time of to = 20 fs. The results obtained for the standard QCL (thin full line) and the energy-conserving QCL (dotted line) are of similar quality, thus indicating that the phase-space distribution p]](x, p) at to = 20 fs is similar for the two schemes. Also shown in Fig. 18 are the results obtained for a standard surface-hopping calculation (dashed line), which largely fail to match the beating of the quantum reference. 

Figure 18. Diabatic electronic population probability obtained for Model 1. Compared are quantum results (thick line) and standard (thin full line) and energy-conserving (dotted Une) quantum-classical Liouville results.

In addition to the equations of motion, one needs to specify a procedure to evaluate the observables of interest. Within a quasi-classical trajectory approach, the expectation value of an observable A is given by Eq. (16). For example, the expression for the diabatic electronic population probability, which is defined as the expectation value of the electronic occupation operator, reads... 

Let us investigate to what extent this simple classical approximation is able to describe the nonadiabatic dynamics exhibited by our model. To this end, we consider the diabatic electronic population probability defined in... 

Eq. (21), which directly reflects the non-Born-Oppenheimer dynamics of the system. Assuming that the system is initially prepared at x = 3 in the diabatic state /2), the corresponding initial distribution pg mainly overlaps with orbits A and C, since at x = 3 these orbits do occupy the state /2). Similarly, excitation of /i) mainly overlaps with orbits of type B, which at x = 3 occupy the state /i) (see Fig. 32). In a first approximation, the electronic population probability P t) may therefore be calculated by including these orbits in the... 

Figure 33. Time-dependent electronic population probability of Model IVa. Compared...

Furthermore, all four periods found in the quantum calculation can be readily explained in terms of the single-orbit contributions to the electronic population probability. In accordance with the discussion above, the 36-fs period of a and the 46-fs period of reflect quasi-harmonic motion on the upper and lower adiabatic potential, respectively. The 12-fs period of <5 c can be attributed to Rabi-type oscillations between the two diabatic states. The 8-fs... 

In Eq. (46) the diabatic electronic representation has been employed. Of course, adiabatic electronic populations Pf(t) may be defined in a completely analogous manner. These electronic population probabilities and their behavior for typical conical intersection models are further discussed in Chapter 9. 

Early attempts to treat the dynamics at conical intersections were based on the Landau-Zener-Stiickelberg approach.While the Landau-Zener-Stiickelberg modeD provides a transparent picture for one-dimensional avoided-crossing situations, its generalization to multi-dimensional nuclear motion is nontrivial, and no quantitative results for electronic population probabilities or transition rates have been obtained in these early studies. 

Equivalently, these electronic population probabilities may be expressed as the integral of the nuclear probability density over all nuclear degrees of freedom, i.e. 

It should be stressed that we are discussing here numerically exact results obtained by the solution of the time-dependent Schrodinger equation for an isolated system. No assumptions or approximations leading to decay or dissipation have been introduced. The time evolution of the wave function (t) is thus fully reversible. The obviously irreversible time evolution of the electronic population probabilities in Figs. 2 and 3 arises from the reduction process, that is, the integration over part of the system [in this case, the nuclear degrees of freedom, cf. Eqs. (12) and (13)]. 

As an example. Fig. 7 shows the diabatic electronic population probability for Model I. The quantum-mechanical results (thick line) are reproduced... 

Finally, it is instructive to discuss the relation of the integral pump-probe signals derived above from the electronic polarization to the corresponding expressions obtained from the calculation of electronic population probabilities. To this end, we consider the stimulated-emission polarization (33) and rewrite the the integral signal (8) in the suggestive form... 

Calculations of the ion yield in dependence on the pulse delay time At and on the parameters of the laser pulses have been performed in Refs. 86 and 93 for simple one-dimensional models of excited-state vibrational motion and vibronic coupling. It has been found that for the vibronic-coupling examples considered and for suitably chosen pulse parameters, the ion signal as a function of At maps very well the adiabatic electronic population probability. As an example of a molecular system comprising conical intersections. Sec. 5.1 presents a calculation of the time-resolved photoelectron spectrum of pyrazine. 

It has been shown that in the limit of ultrashort laser pulses the stimulated-emission pump-probe signal is proportional to the population probability of the initially excited diabatic state [Tf)) Eq. (59) and Refs. 7, 99 and 141. As has been emphasized in Chapter 9, the electronic population probability P2 t) represents a key quantity in the discussion of internal-conversion processes, as it directly reflects the non-Born-Oppenheimer dynamics (in the absence of vibronic coupling, P2 t) = const ). It is therefore interesting to investigate to what extent this intramolecular quantity can be measured in a realistic pump-probe experiment with finite laser pulses. It is clear from Eq. (33) that the detection of P2(t) is facilitated if a probe pulse is employed that stimulates a major part of the excited-state vibrational levels into the electronic ground state, that is, the probe laser should be tuned to the maximum of the emission band. Figure 4(a) compares the diabatic population probability P2(t) with a cut of the stimulated-emission spectrum for uj2 3.4 eV, i.e. at the center of the red-shifted emission band. Apart from the first 20 fs, where the probe laser is not resonant with the emission [cf Fig. 2(b)], the pump-probe signal is seen to capture the overall time evolution of electronic population probability. Pump-probe experiments thus have the potential to directly monitor electronic populations and thus non-Born-Oppenheimer dynamics in real time. ... 

Fig. 4. Integral stimulated-emission signals for the three-mode model of pyrazine (full lines), obtained for pulse durations (a) t = T2 = 20fs, (b) t = 0,T2 = 20fs, and (c) Ti = T2 = 40 fs. It is seen that laser pulses of 20 fs duration are sufficiently short to monitor the time evolution of the diabatic electronic population probability (dotted lines).

Fig. 10. Diabatic (P2, dotted line) and adiabatic (P, dashed line) electronic population probabilities of the A state for the X Ai-A B2 conical intersection in N02. Pi (full line) represents the population probability of the initially-prepared Franck-Condon...

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