*ab initio*quantum wavepacket dynamics formalism for electronic structure and dynamics in open systems

^{1}and Srinivasan S. Iyengar

^{1,a)}

### Abstract

We propose a multistage quantum wavepacket dynamical treatment for the study of delocalized electronic systems as well as electron transport through donor-bridge-acceptor systems such as those found in molecular-wire/electrode networks. The *full* donor-bridge-acceptor system is treated through a rigorous partitioning scheme that utilizes judiciously placed offsetting absorbing and emitting boundary conditions. These facilitate a computationally efficient and potentially accurate treatment of the long-range coupling interactions between the bridge and donor/acceptor systems and the associated open system boundary conditions. Time-independent forms of the associated, partitioned equations are also derived. In the time-independent form corresponding to the bridge system, coupling to donor and acceptor, that is long-range interactions, is completely accounted. For the time-dependent study, the quantum dynamics of the electronic flux through the bridge-donor/acceptor interface is constructed using an accurate and efficient representation of the discretized quantum-mechanical free-propagator. A model for an electrode-molecular wire-electrode system is used to test the accuracy of the scheme proposed. Transmission probability is obtained directly from the probability density of the electronic flux in the acceptor region. Conductivity through the molecular wire is computed using a wavepacket flux correlation function.

This research is supported by the National Science Foundation Grant No. NSF CHE-0750326 (SSI) and the Arnold and Mabel Beckman Foundation (SSI).

I. INTRODUCTION

II. A MS-AIWD FORMALISM THROUGH OFFSETTING ABSORBING/EMITTING POTENTIALS FOR STUDIES ON DELOCALIZED ELECTRONIC PROBLEMS

III. PROPAGATION SCHEMES FOR MS-AIWD

A. Treatment of stage I and its associated coupling to stage II

B. Stage II: Propagation of the accumulated density from stage I, through the molecular region

1. A time-independent form for Eq. (4b)

C. Stage III/IV: Propagation into the electrode regions and computing the overall transmission and reflection coefficients

IV. BENCHMARKS ON THE ACCURACY OF MS-AIWD

A. Propagation scheme

B. Description of model system

C. Analysis of multistage scheme

D. Computing correlation functions and observable quantities

V. CONCLUSION

### Key Topics

- Electrodes
- 35.0
- Backscattering
- 10.0
- Boundary value problems
- 10.0
- Wave functions
- 7.0
- Correlation functions
- 6.0

## Figures

(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) 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.

(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

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 (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.

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

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

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 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

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 .

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).

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.

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.

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.

## Tables

List of parameters used for propagation, potential, and wavepacket description used for the benchmark calculations.

List of parameters used for propagation, potential, and wavepacket description used for the benchmark calculations.

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