Time-dependent population probability

Figure 1. Quantum-mechanical (thick lines) and mean-field-trajectory (thin lines) calculations obtained for Model 1 describing the S2 — Si internal-conversion process in pyrazine. Shown are the time-dependent population probabilities Pf t) and Pf (t) of the initially prepared adiabatic and diabatic electronic state, respectively, as well as the mean momenta pi (t) and P2 t) of the two totally symmetric modes Vi and V( of the model.

Figure 3 shows the quantum results (thick full lines) for time-dependent population probabilities of the initially prepared (a) adiabatic and... 

Figure 7. Comparison of SH (thin solid line), MFT (dashed line), and quantum path-integral (solid line with dots) calculations (Ref. 198) obtained for Model Va describing electron transfer in solution. Shown is the time-dependent population probability Pf t) of the initially prepared diabatic electronic state.

Figure 11. Time-dependent population probability of the upper (a) adiabatic and (b) diabatic electronic state of Model 1. The quantum-mechanical results (thick lines) are compared to SH results obtained directly from the electronic coefficients (dashed lines) and to SH results obtained from binned coefficients (thin solid lines), reflecting the percentage N2(t) of trajectories propagating on the upper adiabatic surface. Panel (c) shows the absolute number of successful (thick hue) and rejected (thin line) surface hops occurring in the SH calculation.

The quantity of primary interest for the description of radiationless electronic transitions is the time-dependent population probability of excited electronic states. The population Pn t) of the nth diabatic electronic state is defined as the expectation value of the projection operator... 

In the case of open-system dynamics, assuming weak coupling of the conical intersection with an environment, the time evolution of the system is determined by the Redfield Eq. (7) for the reduced density matrix. In this case, the time-dependent population probabilities of diabatic and adiabatic states are given by... 

Pig. r. Calculated time-dependent population probability of the diabatic state... 

Fig. 15. (a) Time-dependent population probability of the trans conformer, obtained... 

To describe the electronic relaxation dynamics of a photoexcited molecular system, it is instructive to consider the time-dependent population of an electronic state, which can be defined in a diabatic or the adiabatic representation [163]. The population probability of the diabatic electronic state /jt) is defined as the expectation value of the diabatic projector... 

Let us briefly discuss the characteristics of the nonadiabatic dynamics exhibited by this model. Assuming an initial preparation of the S2 state by an ideally short laser pulse. Fig. 1 displays in thick lines the first 500 fs of the quantum-mechanical time evolution of the system. The population probability of the diabatic state shown in panel (b) exhibits an initial decay on a timescale of w 20 fs, followed by quasi-periodic recurrences of the population, which are damped on a timescale of a few hundred femtoseconds. Beyond 500 fs (not shown) the S2 population probability becomes quasi-stationary, fluctuating statistically around its asymptotic value of 0.3. The time-dependent population of the adiabatic S2 state, displayed in panel (a), is seen to decay even faster than the diabatic population — essentially within a single vibrational period — and to attain an asymptotic value of 0.05. The finite asymptotic value of is a consequence of the restricted phase space of the three-mode model. The population Pf is expected to decay to zero for systems with many degrees of freedom. 

Fig. 8.9 Summary of the coupled electron and nuclear dynamics during the dissociation. The black vertical line indicates the time of recollision, 1.7 fs after ionization at the maximum electric field, (a) Temporal evolution of the electric field, (b) Time-dependent populations of the E n, and iPn states of CO+ after recoUision excitation ( solid CEP = 0, dotted CEP = 7r). (c) Temporal evolution of the probability of measuring a C+ fragment Pc+ for the dissociative ionization of CO+ after recoUision (black CEP = 0, gray CEP = it). Reprinted from [66] with copyright permission of APS...

E. Quantitative Aspects of Tq-S Mixing 1. The spin Hamiltonian and Tq-S mixing A basic problem in quantum mechanics is to relate the probability of an ensemble of particles being in one particular state at a particular time to the probability of their being in another state at some time later. The ensemble in this case is the population distribution of nuclear spin states. The time-dependent Schrodinger equation (14) allows such a calculation to be carried out. In equation (14) i/ (S,i) denotes the total... 

Figure 3. Time-dependent simulations of the nonadiabatic photoisomerization dynamics exhibited by Model III, comparing results of the mean-field-trajectory method (dashed lines), the surface-hopping approach (thin lines), and exact quanmm calculations (full lines). Shown are the population probabilities of the initially prepared (a) adiabatic and (b) diabatic electronic state, respectively, as well as (c) the probability Pdsit) that the sytem remains in the initially prepared cis conformation.

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

The probability of finding a new nonlinear crystal with such specific optical characteristics within a reasonable period of time depends on several factors. The first factor is the rate at which the relevant optical properties can be measured with sufficient accuracy to decide if the crystal is likely to be better than the ones already in hand. Second, there must be a reasonably high probability that the set of materials chosen for characterization contains crystals with the desired characteristics. Finally, assuming that crystals with the desired properties are only sparsely distributed among the set, tiie rate at which the crystals can be synthesized or otherwise obtained in characterizable form must be high, so that a large population of crystals can be scrutinized. The object of this paper is to review the strategy we have developed to optimize these factors in our search for improved ICF frequency convertors. 

For the specific case of differential circularly polarized emission, starting with time-dependent perturbation theory, one obtains the following relationship between the time-dependent experimental observable, AI(X, ), the excited state population, Nn, and the differential transition probability, AW = WLeft - Wpjght, following an excitation pulse of polarization, k, at time t = 0... 

Finally, the time-dependent regimes of the emission85 are consistent with this model. The decay time of the narrow lines reflects the lifetime of the main distribution D t = 0.9ns for the three main lines at 25052, 24700, and 23 692cm-1. The satellite lines at 25052cm-1 have slower decay times, t2 = 2 ns they probably reflect the emission duration. The rise and decay times from the second distribution D2 indicate87 that the relaxation becomes faster as the exciton is created nearer the energy (with a population time of a few picoseconds), in agreement with the increase in the polariton density of states. 

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