Three-dimensional hydrogen atom

As is well known, in the case of the hydrogen atom interacting with a monochromatic field, one-dimensional model provides an excellent description of the experimental chaotization thresholds for real three-dimensional hydrogen atom (Jensen, 1984). 

Common generating function for three-dimensional hydrogen atom complete wavefunctions... 

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... 

In [1] we made a detailed study of the Wigner function s dependence on r, k and u for the three-dimensional hydrogen atom. We mentioned, in particular, that for several purposes it is sufficient to know the function obtained from (28) by integrating over u, and this was in fact the only kind of function that we considered for the polyelectron case. Hence, we shall also limit ourselves to this function in the present work. Actually, we shall also integrate over the 2D — 3 angles that define the x, k plane, but this merely introduces a constant factor. In addition, we multiply by so that the result... 

In the continuum limit a K) —> 0) eqn (E.3) is identical to the effective one-dimensional equation for the radial part of the three-dimensional hydrogen atom wavefunction, u r) = r (r), for the case of zero angular momentum, where 4> r) is the radial wavefunction (see Cohen-Tannoudji et al. (1977, p. 792)). This equation was studied in detail by Loudon (1959). It is useful to treat the even and odd parity solutions separately. 

Fig. 15. A view, perpendicular to the aromatic ring, of the three-dimensional hydrogen-bonded network formed by p-(HOMe2Si)2C6H4, with hydrogen atoms omitted for clarity. Drawn using coordinates taken from the Cambridge Crystallographic Database.

The basic crystal structure consists of P04 (or As04 ) tetrahedra alternating with the K" " (or NH4 ) ions along the c-axis. The P04 units are connected by 0-H...0 hydrogen bonds in the ab plane, forming a three-dimensional hydrogen-bonded lattice [2]. In the ferroelectric phases, the H atoms are localized such that the two close protons are both on the top of the oxygen ions of the XO4 units, as depicted in Fig. 4b. In the antiferroelectric... 

Figure 4. (a) ORTEP diagram (at 50% probability) of IV showing the atom-labeling scheme, (b) Polyhedral representation of the twisted chain, (c) Three-dimensional hydrogen-bonded network of IV. 

In water, there are two hydrogen atoms and two lone pairs in each molecule, allowing for a three-dimensional hydrogen bonded structure. 

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. 

The confinement of the two-dimensional hydrogen atom by angles and hyperbolas [3,10] and 2.3, has its counterparts with circular cones [22] and 4.2, and hyperboloids [9] and 4.5 in the three-dimensional case. The "almost-free" hydrogen atom limit happens at the other end of the domain of the... 

An example of a molecule of this type is 2-propanol carbon atoms 1 and 3 bear three identical hydrogen atoms and the central atom bears two identical methyl groups. If we write three-dimensional formulas for 2-propanol, we find (Fig. 5.6) that one structure can be superposed on its mirror image. 

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