Dihedral angle, hydrogen confined

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. 

In particular, for the hydrogen atom, it was recognized that the confinements by elliptical cones and by dihedral angles were pending. Section 5.1 in the Preview of Ref. [9] formulated the problem of the hydrogen atom confined by a family of elliptical cones identified in its Eqs. (123 and 124), with the boundary condition that the wavefunctions vanish in such cones, Eqs. (125 and 126). The corresponding solution [8] is the subject of Section 3.3. 

The experience of the study of confining the two-dimensional hydrogen atom in an angle can be extended to its three-dimensional counterparts of confinement by dihedral angles. This is taken up in Section 5.2. 

Both sets of families span the shapes from circular cones, for ki = 1 and eccentricity zero, to dihedral angles, for ki = 0 and eccentricity 1, keeping in mind the complementarity between their parameters, Equation (50). While in [37] these parameters are related to the asymmetry of the molecules, here they determine the shape of the confining elliptical cones. The spheroconal harmonics borrowed from [37] in Section 3.2 are some of the solutions for the hydrogen atom confined by an elliptical cone, when the latter coincides with one of the elliptical-cone nodes in Equation (57). Of course, just as already discussed for the other conoidal boundaries in Section 4, here there are other solutions for the same elliptical-cone boundary, and their evaluation is discussed next. 

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... 

The discussion has been focussed on the hydrogen atom, but some of the results are applicable also for the confinement by a dihedral angle of any system with a central potential, or simply with rotational symmetry around the z-axis, including isotropic and anisotropic harmonic oscillators. 

The electronic structure of the neutral hydrogen molecule is well-known in its free form [48], and has been investigated under different situations of confinement [43]. Here the confinement by a nodal plane or a dihedral angle are suggested as relatively simple situations susceptible to analysis... 

The two types of remaining coordinates included in the review of Section 3, representing elliptical cones and dihedral angles, complete the list of natural boundaries for the confinement of the hydrogen atom. Their investigation has been incorporated in the first half of the preview in Section 5. It may be reiterated that these types of confinement are also natural for any central potential, and the dihedral one also for any potential with rotational symmetry around the z-axis. 

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