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200 300

200 300

r0IBohr

Figure 44. Physical properties of H+ enclosed within penetrable spheroidal boxes as a function of box size The continuous lines represent the variational calculations of this subsection. The circles are the results of [109,110], while the squares are those of [125]. 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 44. Physical properties of H+ enclosed within penetrable spheroidal boxes as a function of box size The continuous lines represent the variational calculations of this subsection. The circles are the results of [109,110], while the squares are those of [125]. 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.

results will yield some confidence on the capability of the procedure proposed in this work to account for the behavior of certain physical properties of the system as a function of box size and barrier height. Accordingly, for this system, in addition to the energy we evaluate the Fermi contact term, polarizability, pressure, and magnetic shielding constant.

For the rest of the systems, the estimated energies as a function of the characteristics of the confining potential and a comparison of their qualitative and quantitative behavior with available calculations for the impenetrable case by Ludena and Gregori [107] are presented.

3.9.1. Model and Theory

For penetrable walls, let the ansatz wavefunction in the interior of the box be given as

where t denotes the wavefunction of the free system and f denotes an auxiliary function that guarantees the adequate matching at the boundary with the exterior wavefunction

such that e keeps the proper asymptotic behavior characteristic of the system under study. Furthermore, the choice of the auxiliary function f must be such that it reduces to the familiar cutoff function proposed previously for an infinitely high confining potential. These requirements, together with the fulfillment of the virial theorem [126], constitute the key assumptions to be employed in order to construct our ansatz wavefunctions.

According to the direct variational method, an upper bound to the energy for a particular state may be found by requiring that

<Xi\Hi IXi >o + <Xo\Ho\Xo >o = minimum where

ft and ft are the interior and exterior regions, respectively, and {q} represents the set of coordinates used. V(q) is the acting potential within the interior region and V0 is the height of the confining potential.

In addition to Eqs. (232) and (233), the normalization condition

must be satisfied together with the requirement for continuity of the logarithmic derivative at the boundary (q = q0):

Xi dq q=qo

0 0

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