By elliptical cone

The numerical results for the eigenenergies and eigenfunctions evaluated in Refs. [5] and [6] for molecules with different asymmetry distributions and states are accurate and consistent. The zeros of the individual Lame functions can be determined with high accuracy, and are illustrated in Figure 1 in Ref. [6]. They allow writing the Lame polynomial in product forms presented in Sections 2.1 and 2.2. They are also the key to implement the boundary condition for the rotations of molecules confined by elliptical cones as discussed in Section 3. 

A GUIDE TO ROTATIONS OF THE HYDROGEN ATOM AND ASYMMETRIC MOLECULES CONFINED BY ELLIPTICAL CONES... 

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. 

Because the Hamiltonian of any central potential quantum system, H p/ commutes with the operators and H, they also have common eigenfunctions, including the situation of confinement by elliptical cones. Although Ref. [8] focused on the hydrogen atom. Ref. [1] included the examples of the free particle confined by elliptical cones with spherical caps, and the harmonic oscillator confined by elliptical cones. They all share the angular momentum eigenfunctions of Eqs. (98-101), which were evaluated in Ref. [8] and could be borrowed immediately. Their radial functions and their... 

Readers are invited to do their own reading of Ref. [1], including the effects of the confinement by elliptical cones on the energy spectra and eigenfunctions of the familiar free particle and harmonic oscillator. 

Another property of interesf in the confined hydrogen afom is the pressure distribution on the walls of the confining surface. Reference [8] adapted the method introduced for the case of confinement by paraboloids [41] using Hirschfelder s work on the mechanical properties of quantum systems [42], for the specific case of confinement by elliptical cones. Table V and Figure 3 in Ref. [8] illustrate the variations of the pressure in different radial positions r and angular positions sn(x2 (r) on the walls of some confining cones sn(xiok). 

The topics of this chapter, listed in the Contents and outlined in the Introduction, discuss the rotations of asymmetric molecules and the hydrogen atom in their natural free configurations as reviewed in Section 2 changes in the properties of the same systems in configurations of confinement by elliptical cones are reported, including new results for asymmetric molecules, in Section 3 and some advances in developing the theory of angular momentum in spheroconal harmonic bases, as well as some possible routes under exploration are presented in Section 4. 

The rotations of asymmetric molecules confined by the two alternative families of spheroconal elliptical cone boundaries xi = Xio and X2 = X20 is a richer problem compared with the previous ones. In fact, the boundary... 

Ghapter 4 presents a review of fhe exacf formulafions and evaluafions for the rotations of free asymmefric molecules and for treafment of fhe sysfems when confined by boundary condifions including elliptical cones. In addition, some tools and advances in development in the theory of angular momentum in bases of spheroconal harmonics are discussed. 

The overview of this section is restricted to the geometries previously investigated by the author and his collaborators. It is also being extended in a natural way to the confinement in the complementary geometries of elliptical cones and dihedral angles, already included in Section 3 and previewed in Section 5. 

The problems on new situations of confinement of atoms and molecules previewed in this section are in different stages in their respective investigations. The hydrogen atom confined by an elliptical cone was formulated between the completion of [37] and the invitation to write this Chapter, as described in 5.1. Preliminary numerical results were presented... 

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. 

We formulate the confinement of the hydrogen atom by an elliptical cone defined by xi = Xio- The boundary condition on the complete wave function of Equation (48), with its factors from Equations (39) and (57), is... 

In Figure 2.20b these transition structures locate the energy ridges that separate the IRD valleys located by Mi and Mg. Thus, although there is no analogue of the transition vector for a conical intersection, the simple case of an elliptic cone shows that the IRDs are still uniquely defined in terms of Ml and Mg. 

The conformational space sampled by the two-domain protein calmodulin has been explored by an approach based on four sets of NMR observables obtained on Tb " and Tm -substituted proteins. The observables are the pseudocontact shifts and residual dipolar couplings of the C-terminal domain when lanthanide substitution is at the N-terminal domain. Each set of observables provides independent information on the conformations experienced by the molecule. It was found that not all sterically allowed conformations are equally populated. For example, taking the N-terminal domain as the reference, the C-terminal domain preferentially resides in a region of space inscribed in a wide elliptical cone. 

Amano T, Tamura K (1984) The study of an elliptical cone spinning by the trial equipment. In Proceedings of the 3rd international conference on rotary metalworking processes (ROMP 3). IPS (Publ) Ltd, Kempston, England, pp 213-224... 

In the linear approximation, since the cone is elliptic (see discussion in the preceding section) two steep sides (see Figure 14b) exist in the immediate vicinity of the apex of the cone. As one moves away from the apex along these steep directions, real reaction valleys (as in Figure 14a rather than approximate ones) develop, leading to final photoproduct minima. Thus in reality the first-order approximation will break down at larger distances, and there will be more complicated cross sections and more than two relaxation channels. Also there are symmetric cases (such as H3) in which the tip of the cone can never possibly be described by Eq. [8] because one has three equivalent relaxation channels from the very beginning of the tip of the cone. 

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