Population probability

Werner Heisenberg stated that the exact location of an electron could not be determined. All measuring technigues would necessarily remove the electron from its normal environment. This uncertainty principle meant that only a population probability could be determined. Otherwise coincidence was the determining factor. Einstein did not want to accept this consequence ("God does not play dice"). Finally, Erwin Schrodinger formulated the electron wave function to describe this population space or probability density. This equation, particularly through the work of Max Born, led to the so-called "orbitals". These have a completely different appearance to the clear orbits of Bohr. 

The beautiful Bohr atomic model is, unfortunately, too simple. The electrons do not follow predetermined orbits. Only population probabilities can be given, which are categorized as shells and orbitals. The orbitals can only accommodate two electrons. Shells and orbitals can also merge ("hybridization"). In the case of carbon, the 2s orbital and the three 2p orbitals adopt a configuration in the shape of a tetrahedron. Each of these sp3 orbitals is occupied by one electron. This gives rise to the sterically directed four-bonding ability of carbon. 

To permit inferences about the target population, probability sampling methods will be used in designing the survey. Sample collection and interview procedures being considered include face to face interviews using field study staff, random digit dialing telephone interviews or some combination of these procedures. The face to face procedure is the most likely method at the present time and will be described here to illustrate the manner in which the probability sample will be selected. 

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... 

In a mixed quantum-classical simulation such as a mean-field-trajectory or a surface-hopping calculation, the population probability of the diabatic state v[/ t) is given as the quasiclassical average over the squared modulus of the diabatic electronic coefficients dk t) defined in Eq. (27). This yields... 

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

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 S2 state shown in Fig. lb exhibits an initial decay on a timescale of 20 fs, followed by quasi-peiiodic recurrences of the population, which are... 

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 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 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.

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... 

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... 

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

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.

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...

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. 

Let us first consider the population probability of the initially excited adiabatic state of Model 1 depicted in Fig. 17. Within the first 20 fs, the quantum-mechanical result is seen to decay almost completely to zero. The result of the QCL calculation matches the quantum data only for about 10 fs and is then found to oscillate around the quantum result. A closer analysis of the calculation shows that this flaw of the QCL method is mainly caused by large momentum shifts associated with the divergence of the nonadiabatic couplings F = We therefore chose to resort to a simpler approximation... 

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.

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. 

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... 

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 24. Diabatic (left) and adiabatic (right) population probabilities of the C (full line), B (dotted line), and X (dashed line) electronic states as obtained for Model II, which represents a three-state five-mode model of the benzene cation. Shown are (A) exact quantum calculations of Ref. 180 mean-field trajectory results [panels (B),(E)] and quasi-classical mapping results including the full [panels (C),(F)] and 60% [panels (D),(G)] of the electronic zero-point energy, respectively.

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.

Figure 27. Diabatic (a) and adiabatic (b) population probabilities for Model IVb. Shown are exact quantum results (thick full lines), mean-field-trajectory results (upper thin full line), quasi-classical mapping results including the full zero-point energy (i.e. y = 1, lower thin full line), as well as ZPE-corrected mapping results corresponding to y = 0.6 (dashed line) and y = 0.8 (dotted line), respectively.

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... 

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