Understanding the spectra of a few electrons confined in a quasi-one-dimensional nanostructure

The energy spectra and wavefunctions of three electrons confined by a quasi-one-dimensional Gaussian potential have been calculated and analyzed for three regimes of the strength of confinement omega(z), namely large (omega(z) = 5.0), medium (omega(z) = 1.0) and small (omega(z) = 0.1), by using the full configuration interaction method. For large and medium omega(z) the energy spectrum shows a band structure which is characterized by the polyad quantum number upsilon(p), while for small omega(z) it is characterized by the extended polyad quantum number v(p)*. The wavefunctions of the quartet states have been assigned uniquely by counting the number of nodal planes for the three normal modes, namely, the center-of-mass, permutation and breathing modes. The energy levels for small omega(z) form nearly degenerate triplets, each of which consists of two doublet states and one quartet state. The nodal patterns of their wavefunctions in this small omega(z) regime are almost identical to each other except for their phases. The origin of the tripling of energy levels and the similarity of the wavefunctions for different spin states has been rationalized by using the projection of one- and two-electron potentials onto the internal plane. Effects of anharmonicity in the confining potential on the energy spectra and wavefunctions have also been examined.