# Reproducing Hydrogen Molecule Hamiltonian in OpenFermion

I am learning quantum chemistry at the moment and I'm trying to understand the Hamiltonian generated by the OpenFermion package. I'm now stuck at understanding how openfermion calculates the coefficients in second quantization Hamiltonians.

Take the for Hydrogen molecule for example. My understanding is that the coefficients

• $$h_{00}$$, represents electron 1 in 1s orbit with spin up.
• $$h_{11}$$, represents electron 1 in 1s orbit with spin down.
• $$h_{22}$$, represents electron 2 in 1s orbit with spin down.
• $$h_{33}$$, represents electron 2 in 1s orbit with spin down.

As the wavefunction of 1s orbits takes the form

$$\psi_{1s}(r) = \frac{1}{\sqrt{\pi}}e^{-r}$$

The $$h_{ii}$$ should be

$$\begin{equation} 2\pi \int_0^{\infty}\int_0^{\pi}\psi_{1s}(r)\left(-\frac{1}{2}\nabla^2-\frac{1}{r}-\frac{1}{\sqrt{r^2+R^2-2r R\cos\theta}}\right)\psi_{1s}(r)r^2 \sin\theta d\theta dr = -1.07123 \end{equation}$$ when $$R = 0.74$$ Bohr radius, which does not match with openfermion's \begin{align} h_{00} &= h_{11} = -1.2524635735648988\\ h_{22} &= h_{33} = -0.47594871522096416\\ \end{align}

Can anyone tell me why I am wrong?

• Would Computational Science be a better home for this question? Nov 17, 2022 at 15:47
• @Qmechanic No, Quantum Computing is the home to questions on OpenFermion (explicit instructions on the github page under the "How to contribute" heading).
– Kyle Kanos
Nov 17, 2022 at 18:29