Diatomic Molecules According to the Wave Mechanics I: Electronic Levels of the Hydrogen Molecular Ion

The electronic energies ${W}_{\ensuremath{\rho}}({n}_{y}, {n}_{\ensuremath{\varphi}}, {n}_{x})$ of the hydrogen molecular ion are calculated by means of the wave mechanics as functions of the nuclear separation $c=2\ensuremath{\rho}$, for several values of the quantum numbers ${n}_{y}$, ${n}_{\ensuremath{\varphi}}$ and ${n}_{x}$. The wave function is separable in the elliptical coordinates $y=\frac{({r}_{1}+{r}_{2})}{2\ensuremath{\rho}}$, $\ensuremath{\varphi}$ and $x=\frac{({r}_{1}\ensuremath{-}{r}_{2})}{2\ensuremath{\rho}}$. A qualitative idea of the behavior of these energies as $\ensuremath{\rho}$ changes from infinity to zero is gotten by an investigation of the behavior of the nodal surfaces. The number of these surfaces in any coordinate equals the quantum number in that coordinate. When $\ensuremath{\rho}=\ensuremath{\infty}$ the resulting system is that of a hydrogen atom and a separated nucleus, the nodes are paraboloids and planes with quantum numbers ${n}_{\ensuremath{\eta}}$, ${n}_{\ensuremath{\varphi}}$ and ${n}_{\ensuremath{\xi}}$, and the electronic energy is ${W}_{\ensuremath{\infty}}=\frac{R}{{({n}_{\ensuremath{\eta}}+{n}_{\ensuremath{\varphi}}+n\ensuremath{\xi}+1)}^{2}}$ where $R$ is the lowest energy of the hydrogen atom. When $\ensuremath{\rho}=0$ the system is that of a helium ion, the nodes are spherically symmetric with quantum numbers ${n}_{r}$, ${n}_{\ensuremath{\varphi}}$ and ${n}_{\ensuremath{\theta}}$, and the electronic energy is ${W}_{0}=\frac{4R}{{({n}_{r}+{n}_{\ensuremath{\varphi}}+{n}_{\ensuremath{\theta}}+1)}^{2}}$. As $\ensuremath{\rho}$ changes from zero to infinity it is shown that the quantum numbers are related in the manner ${n}_{r}\ensuremath{\rightarrow}{n}_{y}\ensuremath{\rightarrow}{n}_{\ensuremath{\eta}}$; ${n}_{\ensuremath{\varphi}}\ensuremath{\rightarrow}{n}_{\ensuremath{\varphi}}\ensuremath{\rightarrow}{n}_{\ensuremath{\varphi}}$; ${n}_{\ensuremath{\theta}}\ensuremath{\rightarrow}{n}_{x}\ensuremath{\rightarrow}2{n}_{\ensuremath{\xi}}$ or $2{n}_{\ensuremath{\xi}}+1$. Therefore ${W}_{0}=\frac{4R}{{{n}_{\ensuremath{\eta}}+{n}_{\ensuremath{\varphi}}+2{n}_{\ensuremath{\xi}}+1)}^{2}} or =\frac{4R}{{({n}_{\ensuremath{\eta}}+{n}_{\ensuremath{\varphi}}+2{n}_{\ensuremath{\xi}}+2)}^{2}}$. By this rule it is possible to check the following quantitative calculations. The first order perturbations of the various electronic energies of the first three degenerate levels of the helium ion resulting when $\ensuremath{\rho}=0$ were calculated; the perturbation being the slight separation of the nuclei ($\ensuremath{\rho}>0$). The first order perturbations of the various electronic energies of the first two degenerate levels of the hydrogen atom resulting when $\ensuremath{\rho}=\ensuremath{\infty}$ were calculated when the perturbation was the diminution of the separation ($\ensuremath{\rho}<\ensuremath{\infty}$). The first method is not valid for $\ensuremath{\rho}>\frac{a}{2}$, where $a$ is the radius of the first Bohr orbit of the hydrogen atom, and the second is not valid for $\ensuremath{\rho}<\frac{3a}{2}$. The gap between was extrapolated by means of the nodal reasoning above. These electronic energies plus the energy of nuclear repulsion give the molecular potential energies. A calculation of these shows that of the eight curves obtained only three, the $1s\ensuremath{\sigma}$, $3d\ensuremath{\sigma}$ and $4f\ensuremath{\sigma}$ states show minima, and therefore are stable configurations to this order of approximation (the Hund molecular notation is used for the states). The numerical results check with previous calculations and with the data available.