Free-channel continuum completion

Free-channel completion is an optional diagnostic for the quantum continuum. It restores the free-electron contribution omitted by the accepted angular cutoff. The default calculation leaves this option disabled.

Usage

Enable the same correction for the full and external electronic states:

from otter import PlasmaWorkflowConfig, solve_plasma_workflow

cfg = PlasmaWorkflowConfig(
    formula="Be",
    temperature_ev=10.0,
    rho_g_cc=10.0,
    aa_overrides={"cont_free_completion": True},
)
result = solve_plasma_workflow(cfg)
aa = result["electronic"]["species"][0]["result"]
print(aa["meta"]["cont_free_completion"])

The correction also applies to mixtures through the shared aa_overrides. It requires the quantum electronic model. No additional dependency is needed.

The direct FullExternalConfig interface accepts the keyword-only option cont_free_completion=True, preserving existing positional arguments. Direct scattering integrations accept free_completion=True; adaptive and fixed-grid basis metadata record the resolved value under free_completion. Direct KS results record cont_free_completion.

Spectral density

For positive energy \(E\) in atomic units, let \(k=\sqrt{2E}\). With spin included, the free partial-wave density is

\[n_{E\ell}^{0}(r) =\frac{k}{\pi^2}(2\ell+1)j_\ell(kr)^2.\]

The energy-normalized continuum convention follows [Starrett and Saumon, 2014], Eq. A3. Spherical Bessel completeness gives \(\sum_{\ell=0}^{\infty}(2\ell+1)j_\ell(z)^2=1\) [National Institute of Standards and Technology, n.d.], Eq. 10.60.12. After the original angular selection has accepted a cutoff \(L\), the completed spectral density is

\[n_E^{\mathrm{comp}}(r) =\sum_{\ell=0}^{L}n_{E\ell}^{V}(r) +\frac{k}{\pi^2} \left[1-\sum_{\ell=0}^{L}(2\ell+1)j_\ell(kr)^2\right].\]

The free remainder can be evaluated from two orders. With \(z=kr\), the recurrence in [National Institute of Standards and Technology, n.d.], Eq. 10.51.1, gives

\[1-\sum_{\ell=0}^{L}(2\ell+1)j_\ell(z)^2 =z^2\left[j_L(z)^2+j_{L+1}(z)^2\right] -2(L+1)zj_L(z)j_{L+1}(z).\]

The correction is added once, after any native angular retries. The same Fermi–Dirac occupation multiplies the interacting sum and the free remainder before energy integration. Negative roundoff is not clipped, and no density or charge normalization is applied.

Scope

Only the omitted free part is exact. The omitted interacting contribution \(\sum_{\ell>L}(n_{E\ell}^{V}-n_{E\ell}^{0})\) still needs an angular convergence check. The option preserves the original angular acceptance, phase shifts, propagation domain, matching window, energy limits and B3 policy. It does not reduce the number of propagated channels.

Completion supplies no density from omitted energy intervals and does not remove B3. Agreement with an enlarged angular sum at the same matching boundary does not establish convergence of that boundary. Full, external and pseudoatom electron numbers must be checked independently after SCF.

Cache entries are specific to the completion setting. Use separate energy caches for enabled and disabled calculations; mixing these entries raises an error. Reusing a full state with a different completion setting is also rejected before the external solve.

See Free-channel completion: numerical controls for the numerical controls and their scope.