Molecular orbitals and electron density

Inspect orbital energies and evaluate electron density when your file contains the required basis, coefficients and density records. Grid settings affect density integrals, so check convergence before reporting a value.

Molecular orbitals

An orbital is expanded in atom-centered basis functions:

\[ \psi_p(\mathbf r)=\sum_\mu C_{\mu p}\phi_\mu(\mathbf r). \]

Frontier analysis uses the available orbital occupations, electron-count records and chosen spin channel to identify occupied and unoccupied orbitals. HOMO–LUMO gaps are orbital-energy differences, not direct predictions of optical excitation energies.

Density

For an AO density matrix \(P\),

\[ \rho(\mathbf r)=\sum_{\mu\nu}P_{\mu\nu}\phi_\mu(\mathbf r)\phi_\nu(\mathbf r). \]

For spin-resolved data, alpha and beta matrices can be derived from compatible total and spin matrices. The spin density is the alpha–beta difference.

Numerical grids

openWFN evaluates density on a rectangular grid around the molecule. --spacing sets the distance between points. --padding extends the box beyond the atoms. Both use bohr. Smaller spacing costs more time and memory; larger padding helps include diffuse density tails.

Electron conservation is the principal numerical check:

\[ N_\mathrm{grid}\approx\int\rho(\mathbf r)\,d\mathbf r. \]

Converge both spacing and padding for each chemical system. The active validation set does not establish universal accuracy.

For spin-resolved calculations, check alpha and beta integrals separately as well as their difference. In an exploratory Linux run of the IOData O₂ UHF WFN fixture, a grid with 0.20 bohr spacing and 4.0 bohr padding integrated 9.03171659 alpha and 7.03163594 beta electrons, versus occupation counts of 9 and 7. Their difference was 2.00008065 versus 2 expected: the small spin error partly reflects cancellation of larger channel errors. The input SHA-256 was 5405b851afbe7d55347a5c4ece0bfca2b9bbc1508e9374129693e0ed46aebb96. This is a grid-convergence illustration, not an independent validation benchmark.

Status: frontier analysis is Stable. Total-density integration and cube export are Validated only for the active provenance-backed cases. See the validation matrix.