Analytical solution of the Schr\"odinger equation for the hydrogen molecular ion $H_2^+$

TL;DR

This paper presents the first full analytical solution to the Schrödinger equation for the hydrogen molecular ion $H_2^+$, revealing its wave function as a two-term linear combination and visualizing electron density variations.

Contribution

It provides the first complete analytical solution for the $H_2^+$ Schrödinger equation, demonstrating the two-term nature of its wave function.

Findings

Wave function is a two-term linear combination.

Electron density varies with internuclear separation.

Visualization confirms the wave function structure.

Abstract

The full analytical solution of the Schr\"{o}dinger equation for the hydrogen molecular ion $H_2^+$ (special case of the quantum tree-body problem with the Coulomb interaction) is obtained first. The solution shows that the total wave function is a two-term function in the sense that it is a linear combination of the two linear independent wave functions. The two-term character of the total wave function was visualized in calculations of the total electron density of $H_2^+$ at different internuclear separations.