Molecular Hamiltonian of Helium hydride ion

Question

Currently, I am going through Quantum Simulation of Helium Hydride Cation in a Solid-State Spin Register. I am not a chemist but rather a computer scientist hence having trouble following the paper.

I am using this easy to access note on Hartree-Fock approximation.

So, I am not sure how the authors created the molecular Hamiltonian in Eq. 2 of the supplementary material.

$$ H = \begin{pmatrix} -2.85404 & 0.130671 & 0.0 \\ 0.130671 & -0.760916 & -0.323568\\ 0.0 & -0.323568 & -1.91238 \end{pmatrix} $$

From the note, I understand that the Born-Oppenheimer approximation (Eq. 3) is the starting point.

$$ H = -\sum^{N}_{i = 1} \frac{1}{2} \nabla^2_i - \sum^{N}_{i = 1} \sum^{M}_{A = 1} \frac{Z_A}{r_i A} + \sum^{N}_{i = 1} \sum^{N}_{j > 1} \frac{1}{r_{ij}} $$

What should be my following steps to reach at the molecular Hamiltonian.

I am extremely sorry if I am asking for a really long answer.

UPDATE 1: The molecular Hamiltonian is:

\begin{align} \hat{H} &= \underbrace{-\frac{\hbar^2}{2 M_{He}} \nabla^2_{He} - \frac{\hbar^2}{2 M_H}}_{\text{Nuclear kinetic energy}} \underbrace{- \frac{\hbar^2}{2 m_e} \nabla_1^2 - \frac{\hbar^2}{2 m_e} \nabla^2_2}_{e^-\text{'s kinetic energy}} \nonumber\\ & \underbrace{-\frac{1}{4 \pi \epsilon_0} \frac{2 e^2}{|\overrightarrow{r_1} - \overrightarrow{R_{He}}|} -\frac{1}{4 \pi \epsilon_0} \frac{2 e^2}{|\overrightarrow{r_1} - \overrightarrow{R_{H}}|}}_{\text{Attractive potential energy between } 1\text{\textsuperscript{st} } e^- \text{ and nuclei}} \nonumber\\ & \underbrace{-\frac{1}{4 \pi \epsilon_0} \frac{2 e^2}{|\overrightarrow{r_2} - \overrightarrow{R_{He}}|} -\frac{1}{4 \pi \epsilon_0} \frac{2 e^2}{|\overrightarrow{r_2} - \overrightarrow{R_{H}}|}}_{\text{Attractive potential energy between } 2\text{\textsuperscript{nd} } e^- \text{ and nuclei}} \nonumber\\ & \underbrace{+\frac{1}{4 \pi \epsilon_0} \frac{2 e^2}{|\overrightarrow{r_1} - \overrightarrow{r_2}|} }_{e^-\text{ - } e^- \text{ repulsion}} \underbrace{+\frac{1}{4 \pi \epsilon_0} \frac{2 e^2}{|\overrightarrow{R_{He}} - \overrightarrow{R_H}|} }_{\text{Nuclear - Nuclear repulsion}} \end{align}

After Born–Oppenheimer approximation:

\begin{align} \hat{H} &= \underbrace{- \frac{\hbar^2}{2 m_e} \nabla_1^2 - \frac{\hbar^2}{2 m_e} \nabla^2_2}_{e^-\text{'s kinetic energy}} \nonumber\\ & \underbrace{-\frac{1}{4 \pi \epsilon_0} \frac{2 e^2}{|\overrightarrow{r_1} - \overrightarrow{R_{He}}|} -\frac{1}{4 \pi \epsilon_0} \frac{2 e^2}{|\overrightarrow{r_1} - \overrightarrow{R_{H}}|}}_{\text{Attractive potential energy between } 1\text{\textsuperscript{st} } e^- \text{ and nuclei}} \nonumber\\ & \underbrace{-\frac{1}{4 \pi \epsilon_0} \frac{2 e^2}{|\overrightarrow{r_2} - \overrightarrow{R_{He}}|} -\frac{1}{4 \pi \epsilon_0} \frac{2 e^2}{|\overrightarrow{r_2} - \overrightarrow{R_{H}}|}}_{\text{Attractive potential energy between } 2\text{\textsuperscript{nd} } e^- \text{ and nuclei}} \nonumber\\ & \underbrace{+\frac{1}{4 \pi \epsilon_0} \frac{2 e^2}{|\overrightarrow{r_1} - \overrightarrow{r_2}|} }_{e^-\text{ - } e^- \text{ repulsion}} \underbrace{+\frac{1}{4 \pi \epsilon_0} \frac{2 e^2}{|\overrightarrow{R_{He}} - \overrightarrow{R_H}|} }_{\text{Now a constant since nuclei positions are fixed}} \end{align}

Answer 1

Your provided note is about the HF Hamiltonian in the one-electron basis set (atomic orbitals). Since a minimal STO-3G basis set is used here, the corresponding matrix representation of this Hamiltonian is a $2\times 2$ matrix.

The $3\times 3$ Hamiltonian from the SI is the FCI Hamiltonian in the many-electron basis set. These basis functions are Configuration State Functions, arising from linear combinations of (excited) Slater determinants (electron configurations), which in turn are built from the HF molecular orbitals.

Configuration Interaction calculations (like FCI) are typically done on top of a HF calculation, to correct for missing electron correlation effects. Your linked note is only about HF. For details about CI, I can recommend the book by Szabo and Ostlund which is also cited in your note.