Elliptic cone, equation

Symmetry 50. Intercepts 50. Asymptotes 50. Equations of Slope 51. Tangents 51. Equations of a Straight Line 52. Equations of a Circle 53. Equations of a Parabola 53. Equations of an Ellipse of Eccentricity e 54. Equations of a Hyperbola 55. Equations of Three-Dimensional Coordinate Systems 56. Equations of a Plane 56. Equations of a Line 57. Equations of Angles 57. Equation of a Sphere 57. Equation of an Ellipsoid 57. Equations of Hyperboloids and Paraboloids 58. Equation of an Elliptic Cone 59. Equation of an Elliptic Cylinder 59. 

Equation (standard form) of an Elliptic Cone (Figure 1-50)... 

Both works [2] and [3] show the separations of the eigenvalue equations for H and H, and H and H, in their respective spheroconal coordinates, into Lame differential equations in the individual elliptical cone angular coordinates. The corresponding solutions are Lam6 spheroconal polynomials included in the classic book of Whittaker and Watson [12]. In practice, the numerical evaluation of such Lame functions was not developed in an efficient manner so that the exact formulation of Ref. [2] did not prosper. Consequently, the analysis of rotations of asymmetric molecules took the route of perturbation theory using the familiar basis of spherical harmonics. 

Notice the common radial coordinate and scale factor for both spherical and spheroconal coordinates, Equations (45) and (46). Fixed values of Xi correspond to elliptical cones with a common vertex at the origin, and an axis along the z-axis for i = 1 and along the x-axis for i = 2. 

In fact, the elliptical-cone coordinate dependent part of the Laplacian is identified as the square of the orbital angular momentum, via a direct comparison of Equations (28) and (47) ... 

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 [36,37] the zeros of these functions were determined for specific values of k, corresponding to the asymmetry of the molecule, and for integer values of the angular momentum quantum number l and of its components n and 2 in Equation (58). In general, the zeros of the function in Equation (126), for an elliptical cone with fixed values of k and xio> become non-integer values represented as v, with the corresponding change in the value of l in Equation (58), just like in [22] and Section 4.2 for the circular cone. 

Equation [8] is the equation of an elliptic double cone (i.e., with different axes) with vertex at the origin (it will be a circular cone only for the case k = /). Thus, such crossing points are called conical intersections. Indeed, if we plot the energies of the two intersecting states against the two internal coordinates xx and x2 [whose values at the origin satisfy the two conditions and H1 j = H22 and H12 (= H21) = 0], we obtain a typical double-cone shape (see Figure 5). 

To complete the proof that (31) and (32) constitute an elliptical double cone, let us show that their intersections with % = constant horizontal planes in Qx, Q2, % space are ellipses. Those equations can be put in the form... 

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