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The Complete Wavefunction Stability Algorithm

Putting every piece together, a Stable=Opt calculation in Gaussian proceeds as:

  1. Converge the reference wavefunction. Run the ordinary SCF/DFT procedure at the requested level of theory until the energy gradient with respect to orbital rotation vanishes (Brillouin's theorem), giving a stationary — but not necessarily minimal — determinant.
  2. Assemble the response quantities. Compute the molecular-orbital two-electron integrals needed to form \(A\) and \(B\) matrix–vector products on demand, without ever building the full matrices.
  3. Choose the instability channel(s). By default, test the real, internal channel — the combined alpha-alpha/beta-beta (<AA,BB:AA,BB>) rotation space, covering RHF→RHF and RHF→UHF together; extend to complex rotations if Stable=Complex is requested.
  4. Generate initial trial vectors spanning a modest subspace of candidate orbital rotations.
  5. Iterate the Davidson solver: form Hessian–vector products, diagonalize the reduced matrix, check residual norms against the convergence threshold, and expand the subspace with new trial vectors from unconverged residuals — repeating until the lowest eigenvalues stop changing.
  6. Classify each converged eigenvector by its spin expectation value \(\langle S^2\rangle\) (singlet vs. triplet) and its spatial symmetry label.
  7. Apply the stability criterion. If every eigenvalue examined is non-negative, report the wavefunction as stable and stop.
  8. Follow an instability, if found. If the lowest eigenvalue is negative, rotate the orbitals along that eigenvector to form an improved guess, restart the SCF from it, and return to step 1 — repeating the entire cycle until a stable solution is reached or an internal iteration limit is hit.

The end result is a wavefunction that is not just self-consistent, but verified to be a genuine local minimum in orbital space — safe to use as the reference for geometry optimizations, frequency calculations, or excited-state methods built on top of it.