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Here rq is the electron-nucleus separation and we have used effective atomic units. The symmetry of the confining domain determines the form of rq in the set of coordinates compatible with it. In the following, we construct explicitly the trial wavefunctions for the chosen symmetries.

First, if we assume that the hole is located at the center of a confining sphere of radius r0, the trial wavefunction in spherical coordinates can be written as [66]

The continuity of the logarithmic derivative at r = r0, given in Eq. (285), allows one relate a and fi by

in such a way that only a variational parameter needs to be determined, once that the energy functional is minimized in the standard way.

If we now consider the exciton confined within the prolate spheroid

r q with the nucleus located at a foci, the electron-nucleus separation can then be written as

where 2R is the interfocal distance of the spheroid. Following the solutions of Coulson and Robinson [122] for the free hydrogen atom in this coordinate system, the trial wavefunc-tion is of the form and

Once again, the boundary condition given by Eq. (285) allows one to connect a and ( through

As the reader must be aware, this form for the trial wave-function satisfies automatically the boundary condition on its logarithmic derivative with respect to 7, at g = g0. As in the previous case, only a variational parameter needs to be determined when we use the variational approach.

We have to point out that the auxiliary function of the form (q0 - aq), which was used to construct the trial wave-function for two symmetries, is flexible enough to describe the situation in which V0 ^ <x>. In that case a ^ 1 and correspondingly ( ^ & [see Eqs. (295) and (300)] so that the external wavefunction becomes vanishingly small in this limit, as expected.

In Figure 60a and b, the energy for the ground state of the hydrogenic impurity confined within spherical and prolate spheroidal quantum dots is displayed as a function of the volume, for V0 = 2.0 and 8.0 effective Rydbergs respectively. In both figures we have assumed an R = 1 effective Bohr for the case of the prolate spheroidal dot. The effect of considering a different geometry can be noticed immediately from these figures, which supports that, for a given volume of the dot and barrier height, the ground state energy shows a dependence on the shape of the confining box that is particularly stronger for smaller sizes of the dot. The effect of including the internal interaction potential (Coulomb term) of the components of the system (electron-nucleus) in both regions (inside and outside the quantum dot) is also shown in the figures. When the Coulomb term is considered in the exterior of the quantum dot, a remarkable difference can be noticed for the smaller barrier height potential and smaller sizes of quantum dot. The latter is due to the fact that the confining capability of the box decreases and correspondingly, the probability of finding the electron in the exterior of the quantum dot is increased (see Fig. 60a). In Figure 60b, this effect is smaller. In Figure 60a, we can also observe a stronger dependence, by including the Coulomb term in the exterior of the quantum dot, for the spheroidal symmetry than for spherical one. Another observation is that, for the spheroidal symmetry, we have considered the nucleus of the impurity placed in a focus not in the origin, but in the latter case, keeping in mind the work of Marin et al. [65], the resulting geometrical effect would be stronger.

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