The analytical solution of the Schr\"{o}dinger equation in Born-Oppenheimer approximation for $H_2^+$ molecular ion

Abstract

An analysis of the analytical solution of the Schr\"{o}dinger equation (which is a second order differential equation) for $H_2^+$ shows that the second linear independent solution of this equation is a square integrable function and therefore the ground state total wave function is a linear combination of two linear independent wave functions of different space symmetry: cylindrical and spherical. The wave function of cylindrical symmetry is well known. It has maxima at the positions of nuclei. The wave function of spherical symmetry and the corresponding spherical electron distribution, which exists at $R\neq0$ and locates at the middle of the bond, represents a quasiatom of electron density of non-nuclear united atom. In the light of the new result the qualitative behavior of the ground state wave function and the electron density of $H_2^+$ has been reinvestigated. It is shown analytically that a transformation of the total molecular wave function with two maxima to that one with one maximum passes through a flat wave function. The presented three-dimension figures of the electron density visualize the spherical component of the total wave function and its transformation with increasing internuclear separation.