  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.
