Page 227 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

210 G. P. ARRIGHINI AND C. GUIDOTTI

been obtained by an easily feasible trial-and-trial procedure until satisfying the
normalization requirement [eq. (2.19)] evaluated by numerical quadrature. The field-free
values are seen to be spaced nearly uniformly by according to the

Table 1. Fermi-level energy predicted for the harmonic-well model
for different electron numbers and different electric
field amplitudes E (a.u.).

well known behaviour of the quantum harmonic oscillator spectrum. The lowest energy
value (corresponding to and consequently all the others, however, are shifted

upward about thus suggesting the picture of a ladder spectrum of discrete energy
values dephased with respect to the exact one. Some progressive deterioration seems to
creep slowly into such harmonic picture as the electron number increases more and more.
For a given number of electrons, moreover, the dependence of the Fermi-level energy
shift on the electric field E is in perfect agreement with the exact
prediction,

Figs. 1-4 allow one to gain some further feeling about the quality of the approximation
upon which our derivation of eq. (2.17) for n(x;E) has been founded. Figs. 1 and 2
represent the electron density as a function of the coordinate x, in the absence of external
electric field, for respectively. Excellent overall agreement between
exact and approximate profiles of is immediately recognized. In particular,
there is a perfect reproduction of the electron distribution in the outer region, while for
the central core the approximation leads to a "simulation" of the exact behaviour, able to
represent only in some average manner the typical spatial oscillations of the quantum
density. Such "simulation" becomes seemingly more adequate as the electron number
increases from the two cases actually display a similar behaviour, the
drawing in Fig. 2 being unable to put in evidence details because of the too coarse-grain
scale employed. Fig. 3 refers to the same situation illustrated in Fig. 2, the only change
corresponding to the presence of an external electric field a.u. It should be manifest