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 up.
- $h_{33}$, 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?
