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Note: Energy units: Hartrees, distance units: Bohrs. a Results of this subsection.

b Results of Gorecki and Byers Brown [85] obtained by boundary perturbation theory.

c Results of Brownstein [86] obtained by a variational method in which the trial wavefunction does not satisfy the boundary conditions.

Figure 21. Ground state energy for an off-center hydrogen atom enclosed within an impenetrable spherical box as a function of the position of the nucleus (relative to center of the sphere), for various radii of the confining box. Note that when a/r0 ^ 1 and r0 ^ 1, the energy approaches -1/8 Hartree which corresponds to the ground state of the hydrogen atom close to an infinite planar surface. 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 21. Ground state energy for an off-center hydrogen atom enclosed within an impenetrable spherical box as a function of the position of the nucleus (relative to center of the sphere), for various radii of the confining box. Note that when a/r0 ^ 1 and r0 ^ 1, the energy approaches -1/8 Hartree which corresponds to the ground state of the hydrogen atom close to an infinite planar surface. 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.

variational calculations, using this trial wavefunction, are displayed in Table 2. A comparison with the results from [87] is also shown. In Figure 23, we show the ground state energy as a function of (b/p0) for different values of p0.

The trial wavefunction is accomplished by referring the electron-nucleus distance for the free atom to the origin placed at the center of symmetry of the confining surface, without further assumptions.

The results obtained by applying the direct variational method to compute the ground state energy of the hydrogen atom enclosed within spherical or cylindrical surfaces show good agreement with more elaborate calculations as can be seen from Tables 1 and 2. Furthermore, the trial wavefunc-tions are flexible enough to describe the case when the size of the confining surface becomes infinite and the nucleus is close to the boundary. The latter corresponds to the case when the hydrogen atom is close to a plane.

Figure 22. Coordinates for the off-axis hydrogen atom relative to the axis of the confining cylinder.
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