Prolate spheroid, hydrogen confinement

This is the title of Chapter 3 in Ref. [9], Advances in Quantum Chemistry, Vol. 57, dedicated to confined quantum systems. The conoidal boundaries involve spheres, circular cones, dihedral angles, confocal paraboloids, con-focal prolate spheroids, and confocal hyperboloids as natural boundaries of confinement for the hydrogen atom. In fact, such boundaries are associated with the respective coordinates in which the Schrodinger equation is separable and the boundary conditions for confinement are easily implemented. While spheres and spheroids model the confinement in finite volumes, the other surfaces correspond to the confinement in semi-infite spaces. 

The chapter contains a review of the free hydrogen atom eigenfunctions in the spherical, spheroconal, parabolic, and prolate spheroidal coordinates an overview of our own works for confinement by most of the above-mentioned boundaries and a preview of problems on confined atoms and molecules of current and future investigations. 

This is the counterpart of Equation (100), which is recovered in the limit of vanishing / and infinite u, so that fu becomes r, the radial coordinate. Correspondingly, it is the reference for the analysis of the ionization of the hydrogen atom confined by prolate spheroids. 

The confinement of a three-dimensional hydrogen atom by a dihedral angle, defined by its meridian half-planes — 0 = 0 and = o in spherical, parabolic and prolate spheroidal coordinates — is the natural extension of the confinement by an angle of the two-dimensional hydrogen atom... 

Our early work on the hydrogen atom confined inside prolate spheroidal boxes also dealt with the molecular hydrogen H+ and molecular HeH++ ions [18]. The investigation of the hydrogen molecular ion was extended recently for confinement in boxes with the same shape with penetrable walls [40]. On the other hand, more than ten years ago, we investigated the ground state of the helium atom confined in a semi-infinite space [46] and inside boxes [47] with paraboloidal boundaries. 

The first one, on the binding of an electron by a confined polar atom, is a challenging alternative to [4,5]. The second one is an alternative in the choice of boundary to [43] for the molecular hydrogen close to a confining plane. Section 5.4 emphasizes the importance of using the complete harmonic expansions, outside and inside the sources, of the electrostatic potential of atoms and molecules. In particular, the dipole fields in spherical, prolate and oblate spheroidal coordinates define other alternatives to [4,5]. 

As a specific experience, we can point out that our works on the helium atom confined by paraboloids [46,47] were preceded by variational calculations for the hydrogen atom in the respective confinement situations [49] and by the construction of the paraboloidal harmonic expansion of the Coulomb potential [50]. The last reference also includes the corresponding expansions in prolate and oblate spheroidal coordinates. 

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