New approach for the electronic energies of the hydrogen molecular ion

Abstract

Herein, we present analytical solutions for the electronic energy eigenvalues of the hydrogen molecular ion H + 2 , namely the one-electron two-fixed-center problem. These are given for the homonuclear case for the countable infinity of discrete states when the magnetic quantum number m is zero i.e. for 2 Σ + states. In this case, these solutions are the roots of a set of two coupled three-term recurrence relations. The eigensolutions are obtained from an application of experimental mathematics using Computer Algebra as its principal tool and are vindicated by numerical and algebraic demonstrations. Finally, the mathematical nature of the eigenenergies is identified.

Theory of Three-Electron Bond in the Four Works with Brief Comments

“…Calculations of a hydrogen molecule will actually come to the solution of the movement of one point in the field of two protons, which is similar to the solution of a task for the hydrogen molecular ion H 2 + [22][23][24][25][26][27][28]. And we can expect that finally the two-electron chemical bond will be calculated "on the tip of the pen".…”

“…Let us note for comparison that if we take Lc-c=1.400 Å for the planar cyclooctatetraen, we will have L(1-5)=3.380 Å, L(1-2)=L(8-1)=1.293 Å which vary just slightly from the above mentioned distances between the central electrons at Lс-с=1.410 Å [20][21][22][23][24][25].…”

Canonical reduction for dilatonic gravity in3+1dimensions

“…For example, dimensional scaling helped establish that the mathematical structure of the energy eigenvalues for the three-dimensional hydrogen molecular ion was a generalized Lambert W function [15] from its simpler one-dimensional counterpart, the double-well Dirac-delta function model [16].…”

N-body gravity and the Schrödinger equation

“…Since dimensional scaling applies to H + 2 [22,23,24], it is conceivable that a scheme of dimensional scaling might apply to the gravity problem if dilatons are involved. Though the generalization of the present work has not yet been carried out for higher dimensions, we can nonetheless anticipate the following features.…”

“…This is an N-body generalization of the problem of the H + 2 molecular ion in one dimension [19,20,21]. We note that there exists a D-dimensional version of H + 2 , which can be obtained via a scheme called 'dimensional scaling' [22,23,24]. The 2-body case of the one-dimensional limit of H + 2 is solvable in terms of a generalized Lambert W function [15].…”