Foundations

The proton count of an atomic nucleus identifies its chemical element.

Isotopes share a proton count but have different neutron counts.

Atomic-number order reveals recurring patterns in electron structure and chemical properties.

Bonding describes interactions that bind atoms into molecules or extended structures.

Quantum labels specify orbitals and spin states.

Other electrons modify the attraction experienced by an electron in an atom.

Orbital occupations connect atomic electronic structure with valence chemistry.

Localized orbital overlap and resonance are complementary chemical descriptions.

A molecular orbital description distributes electronic states across a molecule.

Dispersion, dipole interactions and hydrogen bonding influence molecular materials.

For the canonical oscillator, a=√(mω/(2ħ))x+ip/√(2mħω) and its adjoint satisfy [a,a†]=1.

In the long-wavelength electric-dipole approximation, atomic transition strength depends on the dipole matrix element.

For a self-adjoint Hamiltonian bounded below, every normalized trial state with a finite energy expectation satisfies ⟨ψ_trial|H|ψ_trial⟩≥E₀.

A weak coupling from an initial state to a sufficiently dense continuum gives a leading transition rate Γ≈(2π/ħ)|V_fi|²ρ(E_f) at the energy-conserving final energy, for slowly varying matrix elements and density of states.

Near a stationary equilibrium, a conservative system with quadratic kinetic matrix M and potential Hessian K has Mq̈+Kq=0.

A nonrelativistic point-Coulomb two-body bound state separates as ψ_nlm=R_nl(r)Y_lm(θ,φ).

For a normalized trial state of the same Hermitian, lower-bounded Hamiltonian, the Rayleigh energy cannot be below its ground-state energy.

Basis convergence must be tested separately for SCF and correlation contributions.

Small energy and orbital-gradient residuals identify a stationary SCF solution, which may be unstable to allowed orbital rotations.

A larger orbital basis does not remove exchange-correlation functional error.

Counterpoise compares the complex energy with fragment energies evaluated in the complex basis, including partner basis functions without partner nuclei.

Near electronic degeneracy, nuclear motion can couple electronic states through derivative couplings, so a single Born–Oppenheimer potential surface can fail.

A computed vertical electronic excitation is not automatically a measured spectral maximum.

Truncated CI can retain a variational upper bound yet fail size consistency because products of fragment excitations may lie outside its excitation space.

CCSD and perturbative-triples CCSD(T) truncate correlation differently from CI; FNO variants also truncate the virtual space using MP2-derived occupations.

CAS-CI solves the configuration problem within a selected active orbital/electron space; CASSCF also optimizes the orbital partition.

EOM-EE describes target electronic states by excitation operators acting on a correlated reference.

Fewest-switches surface hopping propagates classical nuclear trajectories on individual electronic surfaces with stochastic electronic transitions.

Molecular-dynamics convergence requires checking integration step, constraints and conserved quantities for the chosen ensemble.

QM/MM uses quantum treatment in a selected reactive region and a classical environment.