Effective 1D Time-Dependent Schrödinger Equations for 3D Geometrically Correlated Systems

The so-called Born-Huang ansatz is a fundamental tool in the context of ab-initio molecular dynamics, viz., it allows effectively separating fast and slow degrees of freedom and thus treating electrons and nuclei with different mathematical footings. Here, we consider the use of a Born-Huang-like expansion of the three-dimensional time-dependent Schrödinger equation to separate transport and confinement degrees of freedom in electron transport problems that involve geometrical constrictions. The resulting scheme consists of an eigenstate problem for the confinement degrees of freedom (in the transverse direction) whose solution constitutes the input for the propagation of a set of coupled one-dimensional equations of motion for the transport degree of freedom (in the longitudinal direction). This technique achieves quantitative accuracy using an order less computational resources than the full dimensional simulation for a typical two-dimensional geometrical constriction and upto three orders for three-dimensional constriction.