Molecular science begins with the fact that electrons are quantum particles and nuclei move on coupled potential energy surfaces. The simplest useful description of a molecule therefore starts from the time-independent electronic Schrödinger equation $\hat{H}_e\Psi = E\Psi$, where the Born-Oppenheimer approximation separates fast electronic motion from slower nuclear motion.

In the Born-Oppenheimer picture, the total wavefunction is approximated as a product of electronic and nuclear factors. This works when the electronic energy gap is large relative to nuclear kinetic energy. It fails near conical intersections, avoided crossings, and for strongly vibronically coupled systems, where nonadiabatic effects dominate.

Electron correlation is the failure of single-determinant mean-field theory to capture instantaneous electron-electron avoidance. Post-Hartree-Fock methods such as MP2, CCSD, and CCSD(T) incorporate correlation with increasing accuracy and cost. Basis sets control the quality of the orbital expansion; diffuse and polarization functions are often essential for anions, excited states, and weak interactions.