Page 223 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

206 G. P. ARRIGHINI AND C. GUIDOTTI

matrix element of the QMP in determining the electronic
density. Considering that the QMP knowledge allows one, in principle, to solve the
problem of the time-evolution of any arbitrary initial quantum state, the obtainment of
is to be regarded in general as a true piece of skill. It is a fact that we have at
disposal only very few exact QMP expressions in analytical closed form [29], despite
tremendous advances in quantum dynamics, particularly during the past fifteen years
[31]. Much progress in the QMP evaluation has been realized following mainly the idea
that the propagator for an arbitrary time t can be rigorously expressed in terms of short-
time propagators, for which simple approximations are available [31]. The latter
procedure has actually been developed in some of the papers quoted [10-15,25], which
should therefore be regarded as more rigorous contributions to the problem of
representing the electronic density according to eq. (2.5), even though it is right to say
that the implementation of the formalism to explicit calculations has not kept the pace
with theory.
Our more rudimentary approach is basically founded on an ansatz choice for the quantity
of higher quality with respect to the short-time approximation
which neglects all quantum effects arising from the
noncommutativity of the operators and . In order to appreciate the nature of the
approximation, let us consider the case where the energy potential with
constant quantities. Although the QMP for a particle subjected to a constant
force is one of the few cases explicitly known [32], for our convenience we adopt the
following exact alternative representation of the propagator for a particle moving in a
linear potential [see Appendix A, eq. (A.8)]

The ansatz for the diagonal matrix element of the QMP appearing in eq. (2.5)
corresponds to assume the validity of eq. (2.6) also for potentials other than the
linear one. Taking, moreover, into account that a homogeneous and static electric field
is associated with a potential energy the propagator ansatz for the system subjected
simultaneously to the action of an electric field generalizes in a straightforward way from
eq. (2.6) to yield

Eq. (2.7) is the starting point of the procedure we are going to develop. The neglect of the
exponentials involving and leads to the same result obtainable according to the
Trotter formula [30]; as easily verified, such short-time approximation is the basis for
recovering from eq. (2.5) the same result predicted by the Thomas-Fermi theory.
Thanks to eq. (2.7), the electronic density expression given by eq. (2.5) can be cast into
the (obviously approximate) form