Scientific methods

Use this overview to understand the main calculations. Each linked method page explains the required data, equations, units and validation limits.

Geometry

Distances use Euclidean Cartesian separation. Angles use the normalized dot product. Dihedrals use a signed atan2 convention. Coordinates are converted to ångströms at the parser boundary.

Gaussian basis and density

Cartesian contracted Gaussian functions are evaluated with primitive and contraction normalization. The electron density is

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

Alpha and beta matrices are derived from total and spin matrices as \(P^\alpha=(P+P^s)/2\) and \(P^\beta=(P-P^s)/2\).

Population analysis

Mulliken AO populations are the diagonal elements of \(PS\). Symmetric Löwdin populations use the diagonal of \(S^{1/2}PS^{1/2}\). Atomic charge is nuclear charge minus the assigned electron population.

ESP

Nuclear ESP is \(\sum_A Z_A/|\mathbf r-\mathbf R_A|\). Mulliken and Löwdin point-charge potentials use their respective atomic charges. Electronic and total point ESP use Gaussian Coulomb integrals by default, with validation for the documented reference cases. The optional spatial-grid method remains Experimental. See Population and ESP.