Solving Shroedinger Equation for the electronic energies of the Molecular Ion Hydrogen H2+ in the Elliptic coordinate system
Electronic Energies of Molecular Ion Hydrogen $H_2^{+}$
$r_1$ is the distance between the proton $1$ and the electron.
$r_2$ is the distance between the proton $2$ and the electron.
$R$ is the distance between the two protons, fixed parameter in Born Oppenheimer approximation.
<PERSON>'s Equation:
$$ \hat{H} \Psi = E_{el} \Psi $$
$$ -\frac{{\hbar}^2}{2 m_e} \Delta \Psi -\frac{e^2}{4 \pi \varepsilon_0} (\frac{1}{r_1}+\frac{1}{r_2})\Psi = E_{el} \Psi $$
Atomic Units:
$$ a_0=\frac{4 \pi \varepsilon_0 \hbar^2}{m_e e^2} = 0.5291 *10^{-10} m $$
$$ E_h=\frac{\hbar^2}{m_e a_0^2}=27.21 eV $$
$$ \Delta \Psi + \frac{2}{a_0}(\frac{1}{r_1}+\frac{1}{r_2})\Psi=-2 \frac{E_{el}}{E_h a_0^2} \Psi $$
We introduce the adimentional parameters $ r= \frac{R}{a_0} $ and $ \varepsilon = \frac{E_{el}}{E_h} $
Elliptic coordinate system:
$ \xi = \frac{r_1 + r_2}{R} $ , $ \eta=\frac{r_1 - r_2}{R} $ and $ \phi $ is the angle between the position vector of the electron and the $xy$ plane. 374