Prolate spheroid, coordinates

Another curvilinear coordinate system of importance in two-centre problems, such as the diatomic molecule, derives from the more general system of confo-cal elliptical coordinates. The general discussion as represented, for instance by Margenau and Murphy [5], will not be repeated here. Of special interest is the case of prolate spheroidal coordinates. In this system each point lies at the intersection of an ellipsoid, a hyperboloid and and a cylinder, such that... 

Let us also consider the application of these transformations to diatomic molecules. We note that the density transformations carry the initial prolate spheroidal coordinates (C, t], (j)) into the transformed coordinates (C, j/p, 4>)- The equations that define these transformations are [119] ... 

Cox (C5) and Tchen (Tl) also obtained expressions for the drag on slender cylinders and ellipsoids which are curved to form rings or half circles. The advantages of prolate spheroidal coordinates in dealing with slender bodies have been demonstrated by Tuck (T2). Batchelor (Bl) has generalized the slender body approach to particles which are not axisymmetric and Clarke (C2) has applied it to twisted particles by considering a surface distribution rather than a line distribution. 

While a proper aiming of the atom-probe can be experimentally determined, information on field lines and on equipotential lines is difficult to derive with an experimental method because of the small size of the tip. Yet this information is needed for interpreting quantitatively many experiments in field emission and in field ion emission. We describe here a highly idealized tip-counter electrode configuration which may be useful for describing field lines at a short distance away from the tip surface but far enough removed from the lattice steps of the surface. The electrode is assumed to consist of a hyperboloidal tip and a planar counter-electrode.30 In the prolate spheroidal coordinates, the boundary surfaces correspond to coordinate surfaces and Laplace s equation is separable, so that the boundary conditions can be easily satisfied. 

Instead of expressing the scattering problem in terms of the one-centre polar coordinates rq and 0, it is more appropriate to use the two-centre prolate spheroidal coordinates defined by... 

Figure 6. Prolate spheroidal coordinates used to describe an elongated cell mass.

The governing equation for the steady state mass transport in the fluid phase within the porous medium can be written in prolate spheroidal coordinates and in dimensionless form as ... 

The chapter contains a review of the free hydrogen atom eigenfunctions in the spherical, spheroconal, parabolic, and prolate spheroidal coordinates an overview of our own works for confinement by most of the above-mentioned boundaries and a preview of problems on confined atoms and molecules of current and future investigations. 

In general the Hartree-Fock equations for any molecular system form a set of 3-dimensional partial differential equations for orbitals, Coulomb and exchange potentials. In the case of diatomic molecules the prolate spheroidal coordinate system can be used to describe the positions of electrons and one of the coordinates (the azimuthal angle) can be treated analytically. As a result one is left with a problem of solving second order partial differential equations in the other two variables, (rj and ). 

This is the exact second-order equation for the large component of the DHF a orbital which together with Eq. (3) could be used in the second-order formulation of the DHF method for atoms. It is believed that the corresponding equations for diatomic molecules in the prolate spheroidal coordinates can be solved by the same technique as the one used in the FD HF method. 

The prolate spheroidal coordinates used in this chapter are the same as those in [18,40] represented with different letters (1 < u < oo, — 1 < v < 1,0 < (p < 2jt). They are defined by their transformation equations to cartesian coordinates ... 

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... 

For the parabolic and prolate spheroidal coordinates, the changes in the eigenfunctions and eigenvalues are reduced to the replacement of m by the value of iJb. The respective energy parameters become... 

Let us consider a many-electron atom of nuclear charge Z confined by a hard prolate spheroidal cavity. In this study the nuclear position will correspond to one of the foci as shown in Figure 4. In terms of prolate spheroidal coordinates, the nuclear position then corresponds to one of the foci for a family of confocal orthogonal prolate spheroids and hyperboloids defined, respectively, by the variables f and rj as [73] ... 

As discussed in the previous sections, the classical electron-electron interaction given by the first term in Equation (44) has to be calculated using the multipolar expansion in prolate spheroidal coordinates [74] ... 

The family of confocal ellipsoids and hyberboloids represented by the prolate spheroidal coordinates allows us now to treat the case of a many-electron atom spatially limited by an open surface in half-space. A special case of the family of hyperboloids corresponds to an infinite plane defined by jj = 0 according to Equations (35) and (36). We now treat the specific case of an atom whose nuclear position is located at the focus a distance D from the plane as shown in Figure 4. 

These points all suggest that suitable modes for the system and states considered here should be the elliptical or prolate spheroidal coordinates, in which the H atom can be located in terms of its distances r, r2 from two points... 

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