Page 228 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

ELECTRONIC CHARGE DENSITY OF QUANTUM SYSTEMS 211

that the effect of switching on the external field is simply to translate uniformly the
whole electron distribution toward more negative x values, the shift amounting to
in accordance with the exact prediction [eq. (D.5)].

The behaviour of the approximate density n(x;E = 0) at both large and small x values
can be understood considering the analytical properties of the function
, eq. (2.17) [24,33]. As , in fact,
and exponentially, since asymptotically as .
On the other hand, from the asymptotic behaviour valid for
, as , according to , so that we deduce
, a divergent result. The density at x=0, however, is finite, its value
from eq. (2.17) being Very rapid small oscillations
which characterize both Ai(z) and Ai'(z) at large negative values of their arguments
become concentrated in the region of small x values around x=0. Such unphysical
oscillations, which arise from the approximate nature of the QMP utilized in our
approach, do not result evident in the figures because of scale reasons, but can be
partially perceived. Exact and approximate physical oscillations exhibited by n(x; E) are
compared on a magnified scale in Fig. 4 for the case and E=1. a.u.