Foundations

Electronic-Structure Theory

The common physical and numerical ideas behind Gaussian, PySCF, Psi4, ORCA, and real-space analyses.

1. Wavefunction and electron density

The many-electron wavefunction contains the full quantum state, while the electron density is a three-dimensional observable obtained by integrating over all but one electron coordinate.

The wavefunction depends on all spatial and spin coordinates. The general density definition therefore uses the absolute square of the wavefunction and a sum over the retained electron's spin:

ρ (r) = N \sum_{σ_{1}}^{spin} \int {|Ψ (r, σ_{1}, x_{2}, \dots, x_{N})|}^{2} d x_{2} \dots d x_{N}

Its integral over all space is the number of electrons. Density, spin density, orbital amplitude, electrostatic potential, and reduced density gradient are distinct fields even when all are stored in cube files.

2. Born-Oppenheimer approximation

Electronic motion is solved at fixed nuclear coordinates. The resulting electronic energy defines a potential-energy surface on which geometry optimization and reaction-path calculations are performed.

The approximation separates electronic and nuclear motion, but it does not make nuclei stationary in reality. Breakdown can matter near degeneracies, conical intersections, and strongly nonadiabatic processes.

Every energy is conditional

Geometry, charge, multiplicity, method, basis, solvent, numerical grid, and convergence settings are part of its definition.

3. Hartree-Fock and SCF

SCF methods repeatedly build a Fock or Kohn-Sham matrix from the current density, solve for orbitals, and update the density until energy and density changes satisfy convergence thresholds.

F C = S C ε

Hartree-Fock uses one Slater determinant: exchange is included within that determinant, while dynamical electron correlation is omitted. Kohn-Sham DFT uses orbitals that reproduce the density and places unknown many-body effects in an approximate exchange-correlation functional.

Build the matrix from a trial density.
Solve the generalized eigenvalue problem.
Construct a new density from occupied orbitals.
Mix the update and iterate to convergence.

Check occupations, the intended state, spin contamination for unrestricted references, and wavefunction stability when relevant.

4. Basis sets

Atomic orbitals are expanded in finite basis functions. Basis size, polarization, diffuse functions, effective core potentials, and basis-set superposition error affect both accuracy and cost.

Choice	Purpose	Caution
Polarization	Angular deformation of density	Important for bonding changes and anisotropic interactions
Diffuse functions	Spatially extended density	Important for anions and weak binding; may cause linear dependence
Effective core potentials	Replace core electrons	Pair with the intended valence basis and relativistic treatment

Interaction energies can suffer from basis-set superposition error. Counterpoise correction and basis enlargement address different aspects of basis incompleteness and must be documented.

5. Density-functional theory

Kohn-Sham DFT maps the interacting system onto orbitals that reproduce the density. Functional choice, dispersion treatment, numerical integration grids, and delocalization error must be considered for the target chemistry.

Hybrid, range-separated, meta-GGA, and double-hybrid functionals make different approximations and have different costs. Dispersion corrections, integration grids, delocalization error, and the chemical domain all affect reliability. Benchmark the protocol when small selectivity differences are the target.

6. Potential-energy surfaces

Minima have no imaginary vibrational modes, while a first-order saddle point normally has one mode along the reaction coordinate. Frequencies and intrinsic reaction coordinates verify the character and connectivity of stationary points.

An optimization locates a nearby stationary point rather than a guaranteed global minimum. A minimum normally has no significant imaginary mode; a first-order saddle point normally has one. Visualize that mode and use IRC or another path-following method to verify both endpoints.

Thermal corrections depend on the harmonic approximation, low-frequency treatment, temperature, pressure, standard state, and conformer populations.

7. Weak interactions

Hydrogen bonding, dispersion, electrostatics, polarization, charge transfer, and Pauli repulsion are not interchangeable concepts. NCI and related fields visualize spatial patterns but are not themselves energy decompositions.

Real-space plots are sensitive to density source, grid, isovalue, color range, and fragment definition. A surface may support a mechanistic interpretation, but it does not uniquely label an interaction or provide its energy.

For reaction selectivity, connect rate ratios to activation free-energy differences under a stated sign and temperature convention, include relevant conformers and pathways, and compare against independent energetic or experimental evidence.

8. Reproducibility

Report software version, geometry, charge, multiplicity, method, basis, solvent, grid, convergence settings, thermal conditions, standard state, and conformer treatment. Compare energies only under consistent definitions.

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Confirm geometry, atom order, charge, multiplicity, and intended state.
Confirm SCF and geometry convergence.
Classify stationary points with frequencies.
Verify transition-state connectivity.
Check conformer and pathway coverage.
Preserve inputs, outputs, versions, and post-processing commands.