Page 226 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

ELECTRONIC CHARGE DENSITY OF QUANTUM SYSTEMS 209

3. An elementary application of the formalism
As a very simple application of the approach presented in sect. 2, we confine our
attention to a model system consisting of independent charged particles
("electrons"), moving in a one-dimensional harmonic effective potential
while simultaneously acted by a static, homogeneous electric field E. An exact treatment
of this standard problem is sketchily reviewed in Appendix D for reasons of
completeness.
The approximate numeral density n(x;E) is that obtained from eqs. (2.17), (2.18) with

Typical properties of the charge distribution are summarized by its various electric
multipole moments. The electric dipole moment induced in the system by the external
field is obviously

For further progress, it is convenient to change integration variable from x to F, eq. (3.1),
so that

with

After noting that we are simply left with

Use of the normalization condition finally leads to

a result coincident with the exact prediction [see Appendix D, eq. (D.6)].
The static electric dipole polarizability of the model system investigated is therefore

while higher-order polarizabilities (hyperpolarizabilities) vanish rigorously.
In view of the result just found, it is interesting to contrast exact and approximate
behaviour of the density n(x;E) [eqs. (D.5) and (2.17), respectively]. Some insight into
the nature of the approximations contained in our treatment is gained through the
inspection of Table 1, which collects Fermi-level energy values calculated for several
electron occupation numbers and two different electric field amplitudes. The entries have