### Abstract

The so-called driven Liouville−von Neumann equation is a
dynamical formulation to simulate a voltage bias across a molecular system and to
model a time-dependent current in a grand-canonical framework. This approach
introduces a damping term in the equation of motion that drives the charge to a
reference, out of equilibrium density. Originally proposed by Horsfield and coworkers, further work on this scheme has led to different coexisting versions of
this equation. On the other hand, the multiple-probe scheme devised by Todorov
and collaborators, known as the hairy-probes method, is a formal treatment based
on Green’s functions that allows the electrochemical potentials in two regions of
an open quantum system to be fixed. In this article, the equations of motion of
the hairy-probes formalism are rewritten to show that, under certain conditions,
they can assume the same algebraic structure as the driven Liouville−von
Neumann equation in the form proposed by Morzan et al. (J. Chem. Phys. 2017, 146, 044110). In this way, a new formal ground
is provided for the latter, identifying the origin of every term. The performances of the different methods are explored using
tight-binding time-dependent simulations in three trial structures, designated as ballistic, disordered, and resonant models. In
the context of first-principles Hamiltonians, the driven Liouville−von Neumann approach is of special interest, because it does
not require the calculation of Green’s functions. Hence, the effects of replacing the reference density based on the Green’s
function by one obtained from an applied field are investigated, to gain a deeper understanding of the limitations and the range
of applicability of the driven Liouville−von Neumann equation.

Original language | English |
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Pages (from-to) | 12542-12555 |

Number of pages | 14 |

Journal | The Journal of Physical Chemistry C |

Volume | 123 |

DOIs | |

Publication status | Published - 30 Apr 2019 |

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## Cite this

Ramirez, F., Dundas, D., Sanchez, C. G., Scherlis, D. A., & Todorov, T. N. (2019). Driven Liouville−von Neumann Equation for Quantum Transport and Multiple-Probe Green’s Functions.

*The Journal of Physical Chemistry C*,*123*, 12542-12555. https://doi.org/10.1021/acs.jpcc.8b12319