(a) is general schematic depicting a molecular system coupled to a donor and acceptor that may each contain a reservoir of states. This is seen more clearly in (b) which shows the energy levels of a molecular system coupled to electrodes in the presence of an external bias. The arrows display one of the channels available for electron transfer between electrodes and the wire.
(a) displays a schematic for stages I–III. The light gray vertical lines represent absorbing potentials introduced between the various stages. For clarity, (b) displays stages II–IV. As noted in the discussion, the initial wavepacket emanates from stage I, proceeds into stage II, and then either transmits through (stage III) or gets reflected from (stage IV) the molecule that is present in stage II. The red vertical line represents an emitting potential and is placed at the same position as the absorbing potential between stages I and II in (a). Formally we assume that the absorbing (gray vertical line) and emitting (red) potentials on the left side of (b) are infinitesimally close. See Eqs. (4a) and (4b) and associated discussion on the offsetting absorbing/emitting potentials.
Comparison of the probability density of the multistage wavepacket (labeled stage I–IV in the figure) with a full wavepacket (labeled Full) for a 1 fs dynamics trajectory is provided through this image and attached movie clip. (Enhanced online.). As can be seen, the multistage formalism rigorously accounts for coupling between the various stages and accurately reproduces the dynamics noted in the full wavepacket. The multistage wavepackets are shown in different colors, and each of these calculations requires a smaller system size. For example, the “stage I” wavepacket is confined between the two gray stripes on the left (depicting absorbing potentials) and the calculation for stage I is performed using the associated smaller system size. Similarly for other stages. These aspects are discussed in detail in Sec.III. [URL: http://dx.doi.org/10.1063/1.3463798.3]10.1063/1.3463798.3
Plot of (a) propagation error, Eq. (36), and (b) rms deviation of the energy conservation, Eq. (37), as a function of and for propagation steps, , , and . The propagation error is identical for the same value of resulting in identical dynamics irrespective of mass and time step. The energy conservation for the same value of improves by an order of magnitude as time step increases by an order of magnitude. In Ref. 192, for a proton wavepacket and shows energy conservation in the order of
Plot of the analytical potential (black curve) as a function of distance. The electrostatic potential (red curve) along an axis of 0.5 Å above the internuclear axis of an optimized structure of the molecule is superimposed for comparison. The minima in the potentials correspond to the atomic positions.
Comparison between the real [(a)–(c)] and imaginary [(d)–(f)] parts of the wavepacket with for the full and the multistage dynamics as a function of distance at [(a) and (d)] 0.15 fs, [(b) and (e)] 0.35 fs, and [(c) and (f)] 0.60 fs. The shaded areas represent the absorbing region with the absorbing potential as a function of distance. (a) and (d) are hyperlinked to animations showing the comparison over a 1 fs dynamics. These animation can be accessed for the real part and for the imaginary part online, respectively. (Enhanced online.). [URL: http://dx.doi.org/10.1063/1.3463798.2] [URL: http://dx.doi.org/10.1063/1.3463798.1]10.1063/1.3463798.210.1063/1.3463798.1
Same as Fig. 6 for the wavepacket with and . (a) and (d) are hyperlinked to animations showing the comparison over a 1 fs dynamics. These animation can be accessed for the real part and for the imaginary part online, respectively. (Enhanced online.). [URL: http://dx.doi.org/10.1063/1.3463798.5] [URL: http://dx.doi.org/10.1063/1.3463798.4]10.1063/1.3463798.510.1063/1.3463798.4
The time evolution of the error from Eq. (42) for . The full wavepacket constructed as a sum of the multistage wavepackets for [using Eq. (15)] and [using Eq. (29), including backscattering] are also plotted.
The evolution of the probability density for the full and multistage wavepackets in the real space region including backscattering. The probabilities in the stage II and III regions are plotted on the right axis. In (a) and (b), it appears that stage I wavepacket tunnels into stage II at 0.25 fs and stage II wavepacket tunnels into stage III at 0.5 fs. In (c) and (d), the corresponding times are 0.3 and 0.7 fs, respectively
Comparison of the time average of (a) the expectation values of the wavepacket momentum, , (b) the expectation values the kinetic energy, , and (c) average momentum uncertainty, , as a function of initial conditions [, see Eq. (41)]. The momentum and kinetic energy are in atomic units. In (b) and (c), the left axis describes the narrow wavepacket with initial condition [see Eq. (41) for notation], while the right axis describes the wide wavepacket with initial condition .
Evolution of the transmission probability for the calculations outlined in Fig. 10. While the behavior for all calculations in (a) is similar, (b) shows a complex behavior that is representative of the oscillatory nature of the green circles in Fig. 10(b).
Normalized energy-dependent cross-section, , [Eq. (43)] of waves arriving in stage III for wavepackets with varying widths and initial momentum. (a) ; (b) . The wider wavepacket has a more complex behavior as already illustrated in Figs. 10 and 11.
Flux-flux correlation function of the full electrode-wire-electrode system for the wavepackets with varying widths and initial momentum. All wavepackets with narrow width and a few of the wider wavepackets show a peak at which corresponds to a intraband transitions of metallic Al.
List of parameters used for propagation, potential, and wavepacket description used for the benchmark calculations.
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