Prolate spheroidal coordinates angle

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 ). 

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... 

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. 

When the Gaussian curvature of the surface is not constant (for example, for a prolate spheroid) the wave s tip experiences varying curvature F as it moves over the surface. This results in the systematic drift of spiral waves on the nonuniformly curved surfaces [51 ]. To check this prediction, the numerical simulation of the spiral wave on the surface of a prolate spheroid has been performed in [28] using full reaction-diffusion equations of the model (60)-(62). Figure 13 shows the computed trajectory of motion of the tip of a spiral wave on the coordinate plane (0, ) where 6 and are the spherical angles. We see that the spiral wave drifts approximately along the equator of the spheroid. 

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