Envelope molecular-orbital theory of extended systems I. Electronic states of organic quasi-linear nano-heterostructures

TL;DR

This paper introduces an envelope molecular-orbital theory for extended quasi-linear systems, simplifying complex electronic structure calculations by mapping them onto one-dimensional equations, aiding the design of molecular nanodevices.

Contribution

It presents a novel, computationally efficient envelope MO approach adapted from solid-state physics for quasi-linear molecular heterostructures, bridging atomistic details and mesoscopic modeling.

Findings

Envelope MO energies match atomistic π MO energies.

Envelope MO's accurately describe long-wavelength variations.

Method is applicable to any dimensionality of extended systems.

Abstract

A conceptually appealing and computationally economical course-grained molecular-orbital (MO) theory for extended quasi-linear molecular heterostructures is presented. The formalism, which is based on a straightforward adaptation, by including explicitly the vacuum, of the envelope-function approximation widely employed in solid-state physics, leads to a mapping of the three-dimensional single-particle eigenvalue equations into simple one-dimensional hole and electron Schr\"odinger-like equations with piecewise-constant effective potentials and masses. The eigenfunctions of these equations are envelope MO's in which the short-wavelength oscillations present in the full MO's, associated with the atomistic details of the molecular potential, are smoothed out automatically. The approach is illustrated by calculating the envelope MO's of high-lying occupied and low-lying virtual $\pi$ states in prototypical nanometric heterostructures constituted by oligomers of polyacetylene and polydiacetylene. Comparison with atomistic electronic-structure calculations reveals that the envelope-MO energies agree very well with the energies of the $\pi$ MO's and that the envelope MO's describe precisely the long-wavelength variations of the $\pi$ MO's. This envelope MO theory, which is generalizable to extended systems of any dimensionality, is seen to provide a useful tool for the qualitative interpretation and quantitative prediction of the single-particle quantum states in mesoscopic molecular structures and the design of nanometric molecular devices with tailored energy levels and wavefunctions.