Mayer bond orders

Use the density and overlap matrices to calculate Mayer bond orders. The values depend on the wavefunction and basis. They are not automatic bond labels or formal oxidation states.

Validation status: Validated for the documented everyday-QC reference scope.

For total AO density P, spin density Q = P_alpha - P_beta, and overlap S, openWFN uses the spin-corrected Mayer definition

B_AB = sum_(mu in A,nu in B) [(PS)_mu,nu (PS)_nu,mu + (QS)_mu,nu (QS)_nu,mu].

For a restricted closed shell, the spin term is zero. Unrestricted calculations use the stored total and spin densities. Keep the convention and basis with the result, because different representations can give different indices.

Requirements and safeguards

Required data are an isolated molecule, a supported Gaussian basis, total AO density, and analytical overlap. A confirmed restricted closed shell can derive Q = 0; an open-shell input without the required spin information does not silently invent it. Nonfinite, asymmetric, or dimensionally inconsistent matrices are rejected.

Post-HF inputs retain the density-source warning when the available matrix is the SCF density rather than a correlated density. Charge and spin conservation diagnostics are part of the result. A conservation failure produces a partial result and warning instead of a clean success.

The full matrix is symmetric, dimensionless, and reported with a zero diagonal. Public atom indices are one-based. The compact pair table includes entries satisfying abs(B_AB) >= threshold; filtering never changes the full matrix or row sums. Negative values are retained. bonded_valence is the unfiltered row sum and should be treated as a diagnostic, not as a formal oxidation state or universally integral bond count.

openwfn molecule.molden bondorder mayer --threshold 0.1
openwfn molecule.molden batch --analyses mayer --output-dir results
openwfn molecule.molden report build report.html --analyses mayer

from openwfn import load
record = load("molecule.molden").mayer(threshold=0.1)
print(record.data["bond_order_matrix"])

Structured output includes bond_order_matrix, pairs, bonded_valence, threshold, charge_conservation_error, density-source fields, reference kind, and spin-conservation diagnostics.

Validation evidence

The implementation is checked against analytic two-center cases, including a two-electron bond order of 1.0 and the corresponding one-alpha-electron case of 0.5. Independent cclib 1.8.1 Mayer contractions verify the restricted and unrestricted algebra. Cross-format water regressions verify the normalized input adapters.

For the stable everyday-QC scope, the committed RHF/UHF molecular corpus is compared against independently constructed same-wavefunction references. validation/everyday-qc/pyscf-report.json records the reference matrices, input hashes, dependency versions, and observed errors. validation/manifest.json is the authoritative source for the current capability status and evidence scope.

These checks validate openWFN's post-processing implementation for the named wavefunctions. They do not establish that Mayer bond order is basis-independent, that it reproduces every other program's population-analysis convention, or that any particular numerical value is an automatic chemical bond classification. Diffuse functions, non-idempotent/correlated densities, and unusual AO representations still require careful interpretation.