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have been included which have a similar contour to the spheroidal confinement box to allow the total wavefunction to cancel out at the boundary of the box. The kind of wave-functions like those given in Eq. (204) have been used to investigate one- and two-electron atomic systems confined by spherical penetrable boxes [78], the H, H+, and HeH2+ quantum systems within soft spheroidal boxes [111], and the helium atom inside boxes with paraboloidal walls [105]. In all of these confined quantum systems such simple functions can describe the symmetry in a qualitative way. Quantitatively, the results are close to those obtained by other methods [106, 107].

The evaluation of the matrix element for the electron-electron Coulomb repulsion with the trial wavefunction [Eq. (204)] involves integration over the azimuthal angles, which can be done immediately, and only the term with m = 0 of Eq. (200) remains. The latter shows the symmetry of system under rotations around the z-axis. Additionally, the associate Legendre polynomials become ordinary Leg-endre polynomials. A complete evaluation of the leading expression for the matrix element is presented in Section 4.7.3.

The ground state energy of He and Li+ for R = 0.1, 0.5, and 1.0 Bohr as a function of volume of the enclosing box is displayed in Figure 38, where a comparison with results of Ley-Koo and Flores-Flores [105] for a He atom within a paraboloidal box is also made.

3.7.2. Pressure, Polarizability, and Dipole and Quadrupole Moments

The computed variational energy and trial wavefunctions allow calculation of some of the properties of the confined atom. The average pressure exerted by the boundary on the atom is given by

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