Improved Tutorials

[PaulWAyers]

We should add some improved tutorials for the most common use cases, including plain-vanilla Richardson-Gaudin and the picket-fence Richardson-Gaudin model.

[RichRick1]

We can do that! @PaulWAyers do you have any interesting property of the hamiltonian (like energy limit in case of infinite lattice, etc) we can show in the tutorial?

[PaulWAyers]

There are some phase transitions that one can see as a function of the pair-interaction strength, $g$. The issue with Richardson is that the (easy) way to solve it using a CI code is incredibly inefficient compared to the "right" way to solve it (pseudoanalytically). Perhaps just showing that the wavefunction becomes very far from single-reference for sufficiently large systems, $|\braket{\Psi(0)|\Psi(g)}| \rightarrow 0$ is already interesting.

[PaulWAyers]

I think that @JannisErhard may have some ideas on this based on the frustrations of actually getting integrals in and out of PySCF and/or PyCI.

[typhonshambo]

#153 opened a PR related to this

[ijatinydv]

Hi @PaulWAyers and @RichRick1 , I'd love to tackle this! Based on the comments, here is my plan for a PR to overhaul the RG tutorials:

1. Reframe the current RG.ipynb to explicitly state it is the Picket-Fence model (since eps = np.arange(6)).
2. Add a new notebook for the Plain-Vanilla RG model (degenerate $\varepsilon_i = 0$) to fulfill the issue request.
3. Add a Phase Transition / Wavefunction Overlap demonstration plotting $|\langle\Psi(0)|\Psi(g)\rangle| \to 0$ against $g$ to show the loss of single-reference character.
4. Add a brief Scaling demonstration showing the exponential wall of solving this via FCI vs the pseudo-analytical approach.
5. Document the PySCF integral pain points (explaining the einsum chemist notation transformation and the 1*h1, 2*h2_ch scaling factors).

Does this align with your vision for this issue?