Page 222 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

ELECTRONIC CHARGE DENSITY OF QUANTUM SYSTEMS 205

result for expressed in terms of Airy function and its first derivative. The only
applications of the result thus deduced will be restricted to the case of a model of
independent particles moving in a quadratic potential while simultaneously acted by a
static electric field: the predicted electric dipole moment induced by the field in the
system is shown to be the exact value, despite the fact that the electronic charge density
resulting from the approach is only an approximation to the correct one.

2. An approximate approach to the electronic density

The system under study is assumed to consist of electrons, possibly in the presence
of a nuclear framework. An orbital picture of the quantum behaviour of the system is
then introduced on accepting the validity of an independent-particle model where each
electron moves in the field of an effective potential which afterwards is left
substantially unspecified. We emphasize, however, that the choice of is an essential
step of any modeling. Besides semiempirical forms, effective potentials
functionally dependent on the electron numeral density are intuitively bound to play
a prominent role in applications.
The one-electron Hamiltonian operator kinetic energy operator,
generates a complete spectrum of orbitals according to the Schrödinger equation

The ground state of the system corresponds to electrons occupying the lowest-
energy levels, so that the electron numeral density is

To attain an expression of which does not make explicit reference to the occupied
orbitals, we rewrite eq. (2.2) in the form

where we have introduced the Heavisidefunction and the Fermi level energy If
we make use of the following standard representation of

from eq. (2.1) and the completeness of the spectrum of orbitals supported by the
electron density n (2.3), can be expressed as

(atomic units with are used throughout this paper). Eq. (2.5) is a well
known result [12,14-17,20,24,25], that makes evident the key-role played by the diagonal