Page 225 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

208 G. P. ARRIGHINI AND C. GUIDOTTI

and successively

After carrying out the integrations involved in eq. (2.15) [34], we finally obtain the result

which allows one to calculate the numerical electronic density in terms of both the
potential characterizing the one-electron model assumed and the electric field
polarizing the electron distribution itself. It should be evident from the derivation that the
effect of the field has not been taken into account according to a perturbative
treatment; eq. (2.16) is an approximate result for the electron density that includes at
infinite order the polarization distortion caused by the external field.
Eq. (2.16) is not an entirely new result. After this work had been concluded and we were
looking around in search of bibliographical material, we came upon a paper by Englert
and Schwinger [24] dealing with the introduction of quantum corrections to the Thomas-
Fermi statistical atom. These authors attain the same result expressed by eq. (2.16) (for
the case by resorting to a somewhat more general assumption about the adopted
QMP as compared to our choice.
For a quantum system with a single degree of freedom (dimensionality a procedure
parallel to that sketched above leads to the following result

where

The Fermi level energy appearing in eq. (2.16) [or eq. (2.17)] through the argument b
of the Airy function and its derivative is fixed by the normalization requirement

[or the analogous one-dimensional stemming from eq. (2.17)]. Obviously depends on
the external field amplitude.
Unperturbed electron densities descend naturally from the above formalism by letting
vanish.