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and r 1

With the recurrence relation k + 1 k vpk(v) = pk+1(^ + 2k+1 pk—1(y the I2 integral can be written as

(l + 1)(l + 2) l+2 , (i2ccR)l (2l2 + 2l — 1) (2l + 1)(2l + 3)' Jl+2(l2aR) + (2l — 1)(2l + 3)

The integrals of Eq. (212) are evaluated numerically.

3.8. Quantum Systems within Penetrable Spheroidal Boxes

In this section the direct variational method is used to study some physical properties of H, H+, and HeH2+ enclosed within soft spheroidal boxes [111]. The ground state energy, polarizability, and pressure are calculated for these systems as a function of the size and penetrability of the boxes. For these systems the ground state energy as a function of box eccentricity for different values of the barrier height is calculated. Furthermore, in order to demonstrate the applicability of the variational method, other properties of physical interest directly involving the wavefunction, such as the polariz-ability and the pressure, are calculated.

3.8.1. H, H++, and HeH2+ within Penetrable Spheroidal Boxes

The Hamiltonian for a one-electron molecular ion in the Born-Oppenheimer approximation can be written as

2 r1 r2 2R

where the units are chosen to make h = e = m*e = 1. The subscripts 1 and 2 denote the nucleus of charges Z1 and Z2, respectively.

The prolate spheroidal coordinates (g, y, p) are defined as [115]

where 2R is the interfocal distance of the prolate spheroids of revolution = const.; — 1 < ^ < 1; 0 < p < 2w}.

If the nuclei are located in the foci of this coordinate system, the Hamiltonian of Eq. (217) can be now expressed as

0 0

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