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The geometry of the confining boundary allows one to choose f in a quite simple way, namely, similar to its contours. To exemplify the latter, if the assumed confining boundary is a circle of radius r0, then f = r0 — r (i.e., this function maps all circles of radius 0 < r < r0, that is, the allowed region for the atomic electron). Similarly, if the confining boundary is an ellipse of "size" £0 (eccentricity = 1/£0), then f = £0 — £ and the function would maps all ellipses of size consistent with 1 < £ < £0, thus generating the allowed space for the atomic electron in the plane. The definition of f can be done for other confining symmetries accordingly.

A description on the flexibility of so-constructed trial wavefunctions, to deal with a variety of confining situations, can be found in [65, 66, 78, 111, 121]. In particular, in [78, 111] other physical properties for one- and two-electron confined atoms are calculated to test the goodness of the wavefunctions. The results show good agreement compared with exact or more elaborate approximated methods.

Hence, once that coordinate system (q1, q2) is chosen, all necessary elements for constructing the trial wavefunction are defined and the procedure outlined in the previous section can immediately be used to calculate and minimize the energy functional. Of course, the transformation equations defining the chosen coordinate system are assumed to be known; indeed, it is usually the case and they can be found elsewhere.

It is worth mentioning that the binding energy, defined as the absolute value of the difference between the energies of the electron with and without the Coulombic interaction, has only meaning in the case of closed curves. In the case of open curves, the electron energy belongs to the continuum spectrum and the binding energy is meaningless. Moreover, in the former situation, the binding energy would behave as the total energy, that is, as the confining region becomes smaller or either grows without limit, due to the closed and impenetrable property of the confining boundary.

Figures 54a, 55a, 56a, 57a, 58a, and 59a display the confining geometries considered in this section, while Figures 54b, 55b, 56b, 57b, 58b, and 59b show the behavior of their corresponding ground state energies as a function of confining parameter, respectively.

The ground state energy of the two-dimensional hydrogen atom confined within an ellipse is displayed in Figure 54b as a function of parameter £0 (which defines the degree of confinement) for a = 0.5, 1.0, and 2.0 Bohrs, respectively. Notice that for a given £0 different than 1 and to, when a decreases, the major and minor semiaxes also decrease (i.e., the region of confinement is smaller and the energy increases). If £0 ^ 1, the two-dimensional hydrogen atom becomes a quasi-zero-dimensional system and the ground state energy goes to to (see Fig. 54a). In the case that £0 ^ to, the atom is allowed to occupy the whole plane for any value of a, thus recovering the case of an unconfined two-dimensional hydrogen atom, as expected.

In Figure 55b, the energy for the ground state of the two-dimensional hydrogen atom confined in a region of the plane limited by a hyperbola (see Fig. 55a) is displayed as a function of (which defines the degree of confinement) for a = 0.5,1.0, and 1.5 Bohrs, respectively. As can be observed, for = 0, the hyperbola becomes degenerate and their two 0 0