Paraboloidal Coordinates

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. 

Asag for the S5mimetric paraboloid is known from Eq. 2. To find Agag over the off-axis aperture a coordinate transformation is performed to shift the center of the coordinate system to the center of the off-axis segment and to scale the off-axis aperture radius to a. Then we have... 

For an A-dimensional paraboloid in the space Q= Q 1 .. . , QN let us order the times to reach the point Q, , = tuj 1 cosh 1(<2/ IQ)), as follows t1>t2> - >tn. Then the tunneling path consists of segments separated by takeoff points. At the A th segment, classical motion in the -dimensional space of slow coordinates occurs, while the other N — k degrees of freedom are frozen. The genuine classical motion takes place only at the last segment, when all the times r, are equalized. 

Here Pa(a = 6, e) is the momentum conjugate to Qa. In the absence of spin-orbit interaction, the e vibration does not mix the orbital components of the 4T2 g and we have vibrational potential energy surfaces consisting of three separate ( disjoint ) paraboloids in the two-dimensional (2D) space of the Qe and Qe coordinates of the e vibration. The Jahn-Teller coupling leads only to a uniform shift (—ZsPJX = — V2/2fia>2 = —Sha>) of all vibronic levels. 

Panchenkov s viscosity formula, 89 paper chromatography, 182 Papin s digester, 276 parabola, 421 paraboloid, 429, 430 parachor, 144, 159 atomic, 144 bibliography on, 148 and bond-type, 147 constitutional effect on, 144 and coordinate link, 144 and dipole moment, 146 and entropy, 147 and latent heat of evaporation, 323 and latent heat of fusion, 146 and magnetic susceptibility, 147 for mixtures, 145 negative, 147 and refractivity, 146-7 in solution, 145 and space-filling fraction, 146 and space-filling, 146 ... 

In Figure 3a, the paraboloids correspond to electronic states In, b, 2a, and 2b. This situation arises when the potential along the rp coordinate (shown in Figure 3b) is slightly asymmetric, with a high barrier so that the lowest two vibrational states are virtually pure (i.e., localized) a and b PT states. Note that the minima for the paraboloids corresponding to the a and b states are shifted relative to each other. A transition from la to 2a corresponds to ET, while a transition from la to 2b corresponds to EPT (where both the electron and the proton are transferred). 

A two-dimensional free energy reaction diagram may be described intuitively for a fundamental reaction (Figure 1). The curve at the saddle point and for most of the reaction coordinate can be modelled by the paraboloid Equation (6) but the equation fails to predict the minima in potential energy at the reactant and product positions. The value of will increase as increases and Equation (6) will have a as negative. The Px or p value for this change is defined as -3pNuy pAa . ... 

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 thermal boundary-layer equation, (9-257), also apphes for axisymmetric bodies. One example that we have already considered is a sphere. However, we can consider the thermal boundary layer on any body of revolution. A number of orthogonal coordinate systems have been developed that have the surface of a body of revolution as a coordinate surface. Among these are prolate spheroidal (for a prolate ellipsoid of revolution), oblate spheroidal (for an oblate ellipsoid of revolution), bipolar, toroidal, paraboloidal, and others.22 These are all characterized by having h2 = h2(qx, q2), and either h2/hx = 1 or h2/hx = 1 + 0(Pe 1/3) (assuming that the surface of the body corresponds to q2 = 1). Hence the thermal boundary-layer equation takes the form... 

The ionization limit of the Schrodinger equation and its eigenfunctions for the free hydrogen atom, at a vanishing energy value, corresponds to Bessel functions in the radial coordinate as known in the literature and illustrated in 2.1. The counterparts for paraboloidal [21], hyperboloidal [9], and polar angle [22] coordinates have also been shown to involve Bessel functions. These limits and their counterparts for the other coordinates are reviewed successively in this section. 

Fixed values of f and r] define confocal orthogonal paraboloids with the focus at the origin and their common axis as the z-axis, opening in the directions of increasing and decreasing values of z, respectively. The radial coordinate and the scale factors can be evaluated to be... 

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. 

In general, in the above considerations the coordinate x is presumed to describe nuclear motion normal to the intersection line L of the diabatic.potential energy surfaces of reactants and products. In particular cases, however, the coordinate x can coincide with a dynamically separable reaction coordinate. Then, the whole manydimensional problem of calculating the transition probability for any energy value is simply reduced to a one-dimensional one. Such is, for instance, the situation in a system of oscillators making harmonic vibrations with the same frequency in both the initial and final state /67/ which we considered in Sec.3.1.1. The diabatic surfaces (50.1) then represent two similar (N+1>dimensional rotational paraboloids which intersect in a N-dimensional plane S, and the intersection... 

We now turn to the inner-sphere redox reactions in polar solvents in which the coupling of the electron with both the inner and outher solvation shells is to be taken into account. For this purpose a two-frequency oscillator model may the simplest to use, provided the frequency shift resulting from the change of the ion charges is neglected. The "adiabatic electronic surfaces of the solvent before and after the electron transfer are then represented by two similar elliptic paraboloids described by equations (199.11), where x and y denote the coordinates of the solvent vibrations in the outer and inner spheres, respectively. The corresponding vibration frequencies and... 

This is equivalent to replacing V by a 3iV-dimensional paraboloid with origin J o- The normal mode analysis finds a rotation of the coordinate system such that the new axes coincide with the principal axes of the paraboloid. 

Almost two decades previous to the Doering papers a reasonable model for substituent rate effects was proposed that was based on a geometric model for the MOE-J energy surface for the 3,3-shift. Thus, a hyperbolic paraboloid surface equation could be differentiated to obtain coordinates and the activation free energy for the saddle point (the transition state) cast in terms of the relative free energies for formation of the diyl and the two allyl radicals, the same independent variables of Eqs. (7.1) and (7.2). Equation 7.3, which relates the independent variables by the harmonic mean is based on the simplest hyperbolic paraboloid surface, that is, one with linear edge potentials. Slightly more realistic models were also explored. 

For a steady state when 7 = 0 and A< = constant, the entropy production would be represented by a section of the paraboloid, which would be a parabola with coordinates of vertex given by... 

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