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Figure 41. Electric dipole moment versus major semiaxis of the spheroidal box. The atomic unit for the dipole moment is equivalent to 2.54 Debye. Reprinted with permission from [205], J. L. Marin et al., in "Handbook of Advanced Electronic and Photonic Materials and Devices" (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

-2.847656 Hartree (corresponding to "free" helium atom) to a large positive value as a consequence of volume reduction of the confinement box. It can be also noted that for large volumes of spheroidal boxes there is a weak confinement of the atom and the eccentricity of spheroid becomes irrelevant, but for small volumes of the spheroidal box there is a strong confinement and the eccentricity plays a very important role in the energy value as can be corroborated in Figure 38. For R = 0.5 Bohr, the energy curve coincides in both boxes. The latter is an important situation because it has been compared to the energy of an atom enclosed within two boxes of different shape, where additionally the nucleus of the atom for a paraboloidal box is located in the

Figure 42. Electric quadrupole moment versus major semiaxis of the spheroidal box. The atomic unit for the quadrupole moment is equivalent to 1.34 x 10-26 esu x cm2. Reprinted with permission from [205], J. L. Marin et al., in "Handbook of Advanced Electronic and Photonic Materials and Devices" (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

Figure 42. Electric quadrupole moment versus major semiaxis of the spheroidal box. The atomic unit for the quadrupole moment is equivalent to 1.34 x 10-26 esu x cm2. Reprinted with permission from [205], J. L. Marin et al., in "Handbook of Advanced Electronic and Photonic Materials and Devices" (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

common focus (origin of coordinates) of paraboloids (which corresponds to a symmetric case) while for the spheroidal box the nucleus is located out of the origin, leading to an asymmetric case. It seems that for R = 0.5 Bohr there is no difference for the atom to be in either of these boxes (i.e., boundary shape changes the electron distribution so that both effects combine to give the same energy for the atom). Regarding the same figure, the ground state energy for Li+ increases from its asymptotic value of -7.222656 Hartree as the volume of the confinement box decreases, showing the same qualitative behavior as in the helium case. For the same confinement volume, the energy of Li+ is smaller than the energy of He due to the Coulomb attraction between the nucleus of Li+ (Z = 3) and the electrons being greater than in helium atoms (Z = 2).

In Figure 39, curves of pressure applied on the atom by the spheroidal box walls are displayed. In this figure we again compare with the results for helium atoms inside impenetrable paraboloidal boxes. As can be noted, the pressure curve for R = 0.5 Bohr is nearest to the corresponding pressure curve for helium enclosed within a paraboloidal box. In same figure, the pressure curves for a lithium ion are also displayed. For the same R and confinement volume, the pressure on Li+ ion is smaller than for He atoms because the Coulomb attraction for the former is greater, which forces the ion electrons to move closer to the nucleus (i.e., they are farther away from the confinement walls than the He electrons).

In Figure 40a-c, the polarizability tensor components axx and azz are shown. When R = 0.1 Bohr, they begin to decrease from their asymptotic common value 0.986540 (corresponding to free helium atom) to zero as the volume of the enclosing box diminishes. It can also be noted that for a nearly spherical box no differences on values of polariz-ability tensor components are found. For R = 0.5 Bohr, the parallel component of polarizability azz (which always takes a value greater than value of transversal component axx) approaches a positive value near zero as the volume of the box diminishes. Moreover, the influence of nonsphericity of the box on values of polarizability tensor components can also be observed. When R = 1.0 Bohr, azz has a minimum at R£0 = 1.52 Bohr, then increases to a maximum located at R£0 = 1.24 Bohr, and finally diminishes approaching a positive value greater than for the other two R values considered before. This effect in polarizability is also present for Li+, where its polarizability decreases from its asymptotic value of 0.153354. The latter is a consequence of the combination of (a) the deformation of the spheroidal box by increasing its eccentricity (given by 1/£0) as the volume diminishes and (b) the proximity of the wall to the nucleus, which makes the electrons align on the z axis; thus the system reaches the maximum of its polarizability. Then, its subsequent decrease (not to zero) is due to the asymmetric reduction of space where the electrons can move.

In Figure 41, the electric dipole moments for R = 0.1, 0.5, and 1.0 Bohr, respectively, are displayed. The feature of this physical quantity is its decreasing behavior from zero for He atoms and from an asymptotic value (different for each value of R) for Li+ ions to limiting negative values as £0 ^ 1. All these asymptotic values can immediately be determined from Eq. (208). The nonzero value of the dipole moment is due to the asymmetric position of the nucleus inside the spheroid, which makes a permanent dipole moment (-ZR) since {z) ^ 0 as the volume approaches zero. This behavior is qualitatively different from [114] where the electric dipole moment of helium atoms in a semi-infinite space limited by a paraboloidal boundary was investigated. In the latter case, the magnitude of the electric dipole moment seems to increase monotonically from zero (free helium atom) to a limiting value as the volume is reduced. These variations show the shifting of the electron distribution away from the boundary as the latter closes in around the nucleus.

Figure 42 displays the electric quadrupole moment of He atoms increasing from its asymptotic value (corresponding to the free atom), to a positive limiting value different for each value of R, as £0 ^ 1. In contrast to the case studied in [105], the positive sign of the quadrupole moment indicates that the electrons preferentially distribute themselves on the major axis of the spheroid. Additionally, for this kind of confinement box, due to the asymmetric nucleus position within the spheroid, there is not a maximum for this property as in the case of paraboloidal walls where the nucleus is located on the symmetry center of the box.

3.7.3. Explicit Evaluation of Coulombic Interaction Term

The evaluation of the expectation value of the electron-electron repulsive potential requires performing integrals of the form

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