Quantum Chemistry: Computational Methods & Molecular Modeling

1. Schrödinger Equation & Approximations

The time-independent Schrödinger equation ĤΨ = EΨ is exactly solvable only for hydrogen-like atoms. Born-Oppenheimer approximation: separates nuclear and electronic motion (nuclei are ~1836× heavier). Electronic Hamiltonian: Ĥ_el = T̂_e + V̂_ee + V̂_eN. Variational principle: E[Φ] ≥ E₀ for any trial wavefunction Φ — basis for all approximate methods. Antisymmetry requirement (Pauli principle): Ψ must be antisymmetric under electron exchange → Slater determinants. LCAO: molecular orbitals as linear combinations of atomic orbitals φ_i = Σc_μiχ_μ.

2. Hartree-Fock Method

Hartree-Fock (HF) approximates the N-electron wavefunction as a single Slater determinant. Fock equation: F̂φ_i = ε_iφ_i where Fock operator includes Coulomb (Ĵ) and exchange (K̂) operators. Self-consistent field (SCF) procedure: guess orbitals → build Fock matrix → diagonalize → new orbitals → repeat until convergence. Roothan-Hall equations: FC = SCε (matrix form). Restricted HF (RHF): paired electrons, Unrestricted HF (UHF): separate α/β orbitals. HF captures ~99% of total electronic energy but misses correlation energy (typically 1% but chemically crucial). Koopmans' theorem: -ε_i ≈ ionization potential. Basis set superposition error (BSSE): counterpoise correction.

3. Density Functional Theory

DFT uses electron density ρ(r) instead of wavefunction. Hohenberg-Kohn theorems: (1) ground-state energy uniquely determined by ρ(r); (2) variational principle for density. Kohn-Sham equations: non-interacting reference system with same density. Exchange-correlation functional E_xc[ρ] contains all quantum many-body effects. Jacob's ladder of functionals: LDA → GGA (PBE, BLYP) → meta-GGA (TPSS, SCAN) → hybrid (B3LYP: 20% HF exchange, most popular in organic chemistry; PBE0: 25% HF exchange) → double hybrid (B2PLYP). Dispersion corrections: D3(BJ) for van der Waals interactions. DFT scales as O(N³) vs. O(N⁴) for HF, making it practical for large systems (100–1000 atoms). Limitations: self-interaction error, band gap underestimation, strongly correlated systems.

4. Post-Hartree-Fock Methods

Correlation energy: E_corr = E_exact - E_HF. Møller-Plesset perturbation theory: MP2 (O(N⁵), doubles), MP3, MP4. Coupled cluster: CCSD (singles+doubles, O(N⁶)), CCSD(T) "gold standard of quantum chemistry" (O(N⁷), perturbative triples). Full CI: exact within basis set, exponential scaling — only feasible for <20 electrons. CASSCF: complete active space for multireference problems (transition metals, bond breaking). MRCI: multireference CI for excited states. Explicitly correlated methods: F12 accelerates basis set convergence. DLPNO-CCSD(T): local correlation approximation, near-linear scaling, enabling CCSD(T) quality for 100+ atoms. Composite methods: G4, W1, CBS extrapolation for thermochemistry (±1 kcal/mol accuracy).

5. Applications & Software

Drug design: molecular docking + DFT optimization of binding poses. pKa prediction: CCSD(T)/CBS accuracy ±0.5 pKa units. Reaction mechanisms: intrinsic reaction coordinate (IRC), transition state theory. Spectroscopy: TD-DFT for UV-Vis, harmonic/anharmonic frequencies for IR. Materials: band structure calculations (VASP, Quantum ESPRESSO), defect formation energies. Catalysis: microkinetic modeling with DFT energetics. Software: Gaussian (commercial, most citations), ORCA (free for academia, DFT/WF), VASP (periodic DFT), Q-Chem, Psi4 (open-source), NWChem. Machine learning potentials: ANI, SchNet, MACE enable molecular dynamics at DFT accuracy with force-field speed.

Try it live

Everything above runs in your browser — open Quantum Chemistry: Molecular Orbitals and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.