MSINDO-SECCM electronic stability

The semiempirical cyclic cluster model (SECCM) is experimental. Its post-SCF stability audit tests the electronic state of the specified finite cyclic cluster at fixed geometry and embedding. SCF convergence, a passed local stability test, and a global electronic ground state are different claims. See MSINDO for the method and semiempirical methods for supported routes.

What the audit covers

For a converged closed-shell state with occupied and virtual orbitals, the audit searches for a negative eigenvalue of the real restricted orbital Hessian. It rotates occupied and virtual orbitals together for both spins, preserving the electron count and integer occupations in the orthonormal INDO basis. The geometry, cyclic translations, Hamiltonian and embedding remain fixed. The local stability criterion follows Section 10 of Lehtola, Susi, Blockhuys, Frank, and Van Alsenoy, Christian (2020). *An Overview of Self-Consistent Field Calculations Within Finite Basis Sets.* Molecules, **25**(5), 1218. doi:[10.3390/molecules25051218](https://doi.org/10.3390/molecules25051218).

A negative curvature establishes an electronic saddle in this restricted space. A positive lowest eigenvalue, when established with a converged eigensolver at a stationary state, supports a local minimum only in that space. The audit does not test unrestricted spin breaking, complex orbital rotations, another cluster or periodicity, nuclear displacements or phonons. It does not compare all distinct SCF solutions or prove a global minimum.

Read the whole result

run_seccm exposes the following result attributes. Its run_fields record uses the corresponding output names for output integration.

Result attribute	Output field	Interpretation
converged	General SCF convergence record	Electronic iterations converged; this alone says nothing about stability.
stability_checked	seccm_stability_checked	An orbital stability analysis was attempted.
stability_analysis_converged	seccm_stability_analysis_converged	The stability eigensolver converged. Read this before interpreting its eigenvalue.
stability_eigenvalue	seccm_stability_eigenvalue	Lowest computed full orbital-Hessian curvature in Hartree for the implemented rotation normalization.
internal_instability	seccm_internal_instability	A negative mode passed the finite-descent test described below and remains in the returned state.
n_stability_restarts	seccm_stability_restarts	Number of stability restarts; this is not a count of all possible states.

The implementation solves a quarter-Hessian problem and reports four times that eigenvalue. A quarter-Hessian eigenvalue below -1e-6 Hartree triggers a scan in both signs of its normalized mode, at angles 0.1, 0.25, 0.5, 1.0 and 2.0 radians. A scanned energy reduction of at least 1e-6 Hartree is needed to flag the state or attempt a restart. The negative eigenvalue is retained even if the scan does not meet that energy threshold. An absent flag therefore does not by itself prove a nonnegative Hessian.

At a resolved initial frontier gap, the reference policy can return a converged SCF state even when the stability eigensolver did not converge. Such a result has stability_analysis_converged=False; its eigenvalue placeholder and absent flag must not be interpreted as a passed audit.

State-selection policies

Set CCMOptions.stability explicitly when comparing states or geometries:

• "reference" keeps the Hcore/DIIS trajectory when its initial frontier gap is resolved. A detected saddle can be returned with its instability flag. At an unresolved frontier gap, the solver attempts state selection and stability descent within the iteration budget.

• "lowest" attempts descent from detected saddles, including at resolved frontier gaps. The name does not promise the lowest of all electronic states. Descent can fail or exhaust the budget, and independent calculations at neighboring geometries can select different local solutions.

All restarts share the cumulative max_iter budget. The native route raises on SCF nonconvergence; the Python reference route can return an unconverged result. Preserve refusals and convergence diagnostics when auditing states. Do not convert a failed alternative into a claim that the reference state is stable.

Curves, derivatives and equilibrium claims

Retain each converged alternative separately with its density, occupations, energy, convergence and stability diagnostics, actual controls and runtime identity. Compare charge distributions and occupied subspaces, as well as energies, to detect branch changes. A rotated trial density that lowers the energy is evidence of descent, not a converged alternative SCF solution.

An electronic finite-difference check rotates the retained occupied space at fixed geometry and compares the energy curvature with the orbital Hessian. It neither computes nuclear forces nor establishes a structural minimum. For a nuclear finite-difference check, verify that both displaced calculations remain on the intended electronic branch before comparing their energies with an analytic gradient. Using the same policy string at both displacements is insufficient evidence of branch continuity.

A fitted volume minimum is a property of the sampled electronic branch. Before interpreting it as a stable equilibrium, establish electronic stability and branch continuity around that minimum, together with the structural stability appropriate to the claim. A stability result at one representative volume cannot qualify an entire fitted curve. Keep flagged points and fitting exclusions explicit; do not smooth away state changes or replace one branch’s points with a lower alternative without a separate, documented analysis.