Time-dependent populations

We saw that the time-dependent populations in each of the two levels is given by and... 

While the time dependent populations Pj(t) may generally show a complicated behaviour, certain simple limiting cases can be distmguished and characterized by appropriate parameters ... 

However, TDSE permits an analysis of the time-dependent population of the molecular orbitals during the tunneling, which is unavailable with the TISE. 

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... 

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.

Experimental details for the cross-section measurements were presented in the literature. Briefly, after the irradiation by electron beam pulse for a few nanoseconds, the time-dependent absorption for the atomic line transition Rg Rg -i-/zv was measured to observe the time-dependent population of the excited rare gas atoms Rg. The population of excited Rg was determined using an absorption law for the atomic lines, where the broadening of the absorption profile due to the thermal Doppler effect and due to the attractive interatomic potentials was reasonably taken into consideration. The time-dependent optical emission from energy transfer products, such as ... 

Fig. 5. Time-dependent populations of OH and CH stretching of (a) ethanol, (b) 1-propanol, (c) 1-butanol and (d) 2-propanol. The dashed vertical lines indicate the peak of the von population and the 95% point of the vaCH3 population. Reproduced from ref. [17].

Fig. 2. Pump and probe scheme within a tiers picture (schematic). The zeroth order bright state which is not Franck-Condon (FC) active in the electronic transition is excited via the near IR- laser pulse. FC-active modes m later tiers having no population at t=0 are probed and their time dependent population is a measure for IVR (Vj being matrix elements connecting zeroth order states) in the molecule giving rise to an enhancement of the electronic absorption.

Very recently the formation and vibrational relaxation of OH(X2n , v = 1-9) by 02 and C02 [73] and NH(X3 -, v = 1-3) by N2, Ar, and H2 [72] has been investigated in electron-irradiated gas mixtures. OH( )-m)-1,d-2) and NH(d-n) —1) IR emission was observed using a quasi-SS time-resolved FTIR spectrometer, averaging about 100 samples to characterize the IR transient for each of 1000 sampling positions. Time-dependent populations are shown for OH(t = 1-6) in Figure 24, taken from... 

Fig.7b,d,f. Time-dependent populations of level 2 in Fig. 5 calculated from Eq. (10) using the same kinetic parameters as in Fig. 6 b, d, f The vertical dashed lines indicate the termination of the square laser pulse. The dashed curves show the evolution of (b) and (d) Nf, normalized for comparison... 

In cases where energy migration is a dominant feature of luminescence, as in molecular crystals, various forms of decay are expected depending upon circumstances, but relying upon solutions, usually complex, to the basic rate equations where E(t) is the time-dependent population of the initially excited (exciton) state, T(t) the population of the trap state, kg the decay rate constant for band... 

Ignoring ESA, and the population of/from higher levels in Fig. 24d, then in the case of continuous excitation, assuming the steady state [382], the time dependent populations of levels/and/ in Fig. 24d are given by... 

Figure 4.15 Time-dependent populations of some vibrational states In, l> with 6,0,0> initially populated (labeled 1). The label 2 refers to jS,2,0> and 7 refers to 10,12,0> (Baggott et al., 1985).

Figure 1 shows the time-dependent populations in a typical system of six bridge sites. When the coupling between the donor and bridge was turned on at t = 0, there was a sudden jump of population into the bridge. 

Figure 9.25. Time-dependent population of the target diabatic (a) or adiabatic (b) state obtained from exact solution of the Zener problem for two values of the Massey parameter = 0.01 [(a) nonadiabatic crossing] and = 2.25 [(fo) adiabatic crossing]. Time is in units of Tna =...

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