Ellinger Y., Defranceschi M. (eds.) Strategies and applications in quantum chemistry (Kluwer, 200

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 shorttime 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

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

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

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

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

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.