Convergence

Every Solution carries a ConvergenceReport. The report is honest about what the method can certify.

Saturation

Stochastic methods report criterion = :saturation when the energy change over the last window committed additions is below tol. This means the current sampler and scale stopped finding improvements. It is not a proof of the exact eigenvalue.

That distinction matters. A single-scale H2+ run can plateau above the physical energy because proton-proton and electron coordinates live on very different length scales. Increase basis, change scale, add Refine, or follow with Variational before treating the plateau as physical.

Stationarity

Gradient methods report criterion = :stationarity when the optimizer meets its gradient tolerance. This is a local stationarity statement in the Gaussian-parameter landscape. The variational upper-bound property still holds: computed energies do not go below the exact eigenvalue for the chosen Hamiltonian.

Early stops

criterion = :early_stop means no admissible candidate was found, often because the basis became nearly singular. Try a smaller scale, fewer basis functions, or a different warm start.

Conditioning

ECG bases are non-orthogonal. Large overlap condition numbers are normal. The stochastic solver uses a whitened incremental eigensolver, and the dense power-user solver regularizes ill-conditioned overlap matrices when needed. The report includes cond_S so you can distinguish physical convergence from linear-algebra stress.

Plots

plot(sol) shows the stage energy history. In a pipeline, each stage gets its own curve. plot(sol, reference) adds a horizontal reference line. A flat curve is a useful diagnostic, but always read it together with the method and the report criterion.