Classically forbidden electronic

The breakdown of the SH scheme in the case of classically forbidden electronic transitions should not come as a surprise, but is a consequence of the rather simplifying assumptions [i.e., Eqs. (37) and (43)] underlying the SH model. On a semiclassical level, classically forbidden transitions may approximately be described within an initial-value representation (see Section VIII) or by introducing complex-valued trajectories [55]. On the quasi-classical... 

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. 

As discussed above, this discrepancy may be caused by classically forbidden electronic transitions, i.e. cases in which a proposed hopping process is rejected due to a lack of nuclear kinetic energy. Figure 4(c) supports this... 

Jasper, A. W., Stechmann, S. N., Truhlar, D. G. (2002). Fewest-switches with time uncertainty A modified trajectory surface-hopping algorithm with better accuracy for classically forbidden electronic transitions. Journal of Chemical Physics, 116(13), 5424-5431. 

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. 

The LSD and PW91-GGA system-averaged holes agree at zero interelec-tronic separation, where both are nearly exact. In Sect. 2.3, we discuss how the near-universality of this on-top hole density provides the missing link between real atoms and molecules and the uniform electron gas. Except very close to the nucleus, the local on-top hole density is also accurately represented by LSD, even in the classically-forbidden tail region of the electron density [18]. 

While the LSD exchange-correlation hole is accurate for small interelec-tronic separations (Sect. 2.3), it is less satisfactory at large separations, as discussed in Sect. 2.5. For example, consider the hole for an electron which has wandered out into the classically-forbidden tail region around an atom (or molecule). The exact hole remains localized around the nucleus, and in Sect. 2.5 we give explicit results for its limiting form as the electron moves far away [19]. The LSD hole, however, becomes more and more diffuse as the density at the electron s position gets smaller, and so is quite incorrect. The weighted density approximation (WDA) and the self-interaction correction (SIC) both yield more accurate (but not exact) descriptions of this phenomenon. 

Fig. 4A The vibrational levels of the H2 ground electronic state. The levels are drawn between the classical limits of vibration, but there is a small probability for vibration to extend into the classically forbidden region.

This equation (3) is to be used provided p— V(r)>0, while p(r) is zero if p— V(r) =S0. This is readily recognized to be a condition stemming from the semi-classical nature of the TF theory. Electrons are not allowed to occupy regions of negative kinetic energy, i.e. there are no electrons in classically forbidden regions. 

The electronic coupling Hah between the reactant with the donor reduced, and the product with the acceptor reduced, depends on the ability of the electron wavefunction to penetrate the classical forbidden insulating barrier between the donor and acceptor. 

Figure 13. The electron density distribution inside SET displays the Friedel oscillations on the Fermi edges and the peak of resonance tunneling in classically forbidden region in the inter-electrode gap.

For the nonuniform electron gas at a metal surface, the Slater potential has an erroneous asymptotic behavior both in the classically forbidden region as well as in the metal bulk. In the vacuum region, the Slater potential has the analytical [10] asymptotic structure [35,51] V r) = — Xs(p)/x, with the coefficient otsiP) defined by Eq. (103). In the metal bulk this potential approaches [35] a value of ( — 1) in units of (3kp/27r) instead of the correct Kohn-Sham value of ( — 2/3). Further, in contrast to finite systems, the Slater potential V (r) and the work W,(r) are not equivalent [31, 35, 51] asymptotically in the classically forbidden region. This is because, for asymptotic positions of the electron in the vacuum, the Fermi hole continues to spread within the crystal and thus remains a dynamic charge distribution [34]. 

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