Derivation, properties and application of Coulomb Sturmians defined in spheroidal coordinates

Abstract

Coulomb Sturmian amplitude functions are derived in prolate spheroidal coordinates and are presented in a closed algebraic form. Spheroidal Sturnian functions are revealed to be related to the polynomial solutions of Heun's confluent equation. A reduction of symmetry from spherical to axial leads to the coupling of spherical polar orbitals and the formation of hybrid orbitals. The contribution of each spherical orbital into a hybrid orbital depends strongly on distance R from a nucleus to the dummy centre, and substantially alters when R varies. At two limiting cases R = 0 and R → ∞ spheroidal Sturmians are purely atomic orbitals, whereas at intermediate R they contain many features intrinsic to diatomic molecular orbitals. Applications of spheroidal Sturmian basis are discussed; Coulomb spheroidal Sturmians are asserted to be the most appropriate basis functions for diatomic molecular calculations.

Normed Division Algebras Application to the Monopole Physics

“…[71], the interbasis transformation between the parabolic and spherical bases has been constructed. However, a similar transformation between the prolate spheroidal and spherical bases was not built in an analytical approach since it required a complex integration between confluent Heun [75], associated Laguerre, and generalized Jacobi polynomials [36]. These integrations have not been analytically calculated yet, to the best of our knowledge.…”

Highly Accurate Potential Energy Curves for the Hydrogen Molecular Ion

“…[25] The solutions of the H 2 + molecule ion problem have been also analyzed in terms of hypergeometric functions, [26] the Heun confluent functions, [27,28] and Coulomb Sturmians. [29,30] Series solutions have also been given for the spectrum of H 2 + . [31] Upon thorough inspection of the literature surveyed here, it is found that all the tabulated spectra contain at most 15 significant digits, which is a product of the limitation of floating-point calculations.…”

Highly accurate potential energy curves for the hydrogen molecule ion

“…Both the lowerand higher-lying states of the H + 2 have been recently computed using semiclassical approximations [22][23][24], and variational approaches [25]. The solutions of the H + 2 molecule ion problem have been also analyzed in terms of hypergeometric functions [26], the Heun confluent functions [27,28], and Coulomb Sturmians [29,30]. Series solutions have also been given for the spectrum of H + 2 [31].…”