A systematic route through quantum chemistry and simulation
1. Choose a question, then a method
The seven courses form one connected curriculum. Quantum mechanics establishes states, operators and variational reasoning. Molecular methods use antisymmetric many-electron states to approximate molecular energies. Density functional theory organizes the electronic problem around density. Tight binding reduces the basis and interaction structure to an interpretable model. Molecular dynamics and Monte Carlo explain how microscopic models produce statistical observables. Quantum Monte Carlo combines wavefunctions with sampling and imaginary-time projection.
Each course contains a numbered sequence rather than a collection of disconnected definitions. Read the assumptions before the derivation; reproduce the intermediate algebra; work the example without looking at its result; then attempt the exercise before reading its explanation. The software tutorials are companions for implementation. A successful executable run does not establish a correct physical model, converged calculation or trustworthy error bar.
The arrows indicate useful conceptual preparation, not strict enrollment barriers. Statistical mechanics can be studied alongside quantum mechanics. Molecular dynamics is not required to understand every Monte Carlo algorithm, but the shared ensemble concepts help connect them.
2. Prerequisites and a self-check
Bring derivatives and integrals, Taylor series, complex numbers, matrix multiplication and elementary probability. The quantum course introduces inner products and eigenproblems; the sampling courses introduce equilibrium weights and correlated estimators. If an expression uses a concept you cannot explain in words, pause and repair that prerequisite before proceeding.
- Normalize a Gaussian and compute its second moment.
- Diagonalize a real symmetric two-by-two matrix and check orthogonal eigenvectors.
- Differentiate an energy to obtain a force, retaining its sign and units.
- Normalize three probabilities and compute an expectation and variance.
- Explain why the standard error of a mean differs from the spread of individual observations.
Explained example: for weights \(1,2,1\) on values \(-1,0,1\), normalize to probabilities \(1/4,1/2,1/4\). The expectation is zero and variance is \(1/2\). For \(M\) independent draws, the mean's variance is \(1/(2M)\). Correlated draws require covariance terms; the individual variance stays \(1/2\) while the mean may be much less precise than the independent formula predicts.
3. Beginner route: build an operational foundation
Start with the quantum-mechanics course and its finite-basis variational example. Continue through molecular orbitals, Hartree–Fock and SCF; then study density foundations, Kohn–Sham energies and functional approximations. These ideas let you explain why an orbital energy sum is not a total energy and why convergence of an SCF solver is not convergence of the physical approximation.
For molecular or materials simulation, next study MD potentials, periodic boundaries, Verlet integration, thermal degrees of freedom and ensembles. Read probability, Metropolis–Hastings and correlated uncertainty in the Monte Carlo course. Finish by revisiting one software tutorial with an explicit error budget. A beginner completion task is to design a molecular energy difference or liquid structural average and identify every required reference, unit and convergence control.
4. Advanced routes according to the scientific question
Molecular electronic structure: complete angular momentum and perturbation theory, then the full molecular-methods course. Compare MP2, configuration interaction and coupled cluster in terms of reference quality, extensivity and near-degeneracy. Use the Gaussian course to connect basis and method decisions to complete input examples.
Periodic materials: complete DFT and tight binding. Derive Bloch matrices, distinguish eigenvalue gaps from charged excitation gaps, and audit functional, basis, pseudopotential, reciprocal sampling and finite-cell errors. Connect to VASP, Quantum ESPRESSO and ABACUS.
Thermodynamics and rare events: complete MD and Monte Carlo, including free energies, advanced sampling and autocorrelation. Decide whether the goal is an equilibrium average, a free-energy difference or a real-time transport coefficient. Use CP2K for electronic-force MD and examine its timestep/SCF checks before production.
Stochastic electronic structure: complete quantum mechanics, many-electron methods and Monte Carlo before the QMC course. Distinguish variational error, projection bias, fixed-node error, timestep error, population bias and sampling uncertainty. Read the QMCPACK software reference as an implementation entry point; the theory lessons do not claim that a QMC calculation was executed.
5. Notation and units used throughout
| Symbol | Meaning and convention |
|---|---|
| \(\Psi\); \(\psi_i\); \(\chi_\mu\) | Many-electron wavefunction; one-particle orbital; basis function |
| \(r\); \(R\); \(x=(r,\sigma)\) | Electronic position; nuclear configuration (or explicitly defined QMC configuration); spatial-spin coordinate |
| \(H\); \(S\) | Hamiltonian; basis overlap matrix, not entropy unless explicitly stated |
| \(n(r)\); \(\rho\) | Electronic number density; probability density or bulk particle density, defined locally |
| \(i,j\); \(a,b\); \(\mu,\nu\) | Occupied orbitals; virtual orbitals; basis-function indices |
| \(\beta\) | \(1/(k_BT)\) in statistical mechanics |
| \(h\); \(\hbar\) | MD timestep when explicitly defined; reduced Planck constant |
| \(\tau\); \(M\) | Imaginary time or lag (specified in each lesson); sample count or nuclear mass (specified locally) |
Electronic derivations generally use atomic units when stated: \(\hbar=m_e=e=4\pi\epsilon_0=1\). Energies are then Hartree and distances bohr. Nuclear dynamics still requires the correct nuclear masses and consistent time units. A dimensionless teaching example is not a recommended physical parameter set. Always carry units before comparing a code output with an equation.
6. Four layers of evidence
Separate physical-model error (Hamiltonian, force field, functional, boundary conditions), representation error (basis, grid, finite cell), solver error (SCF residual, integration step, constraint tolerance) and sampling uncertainty (finite correlated observations). Refining one layer does not remove the others. A tiny standard error does not imply an accurate functional; stable NVE energy does not prove ergodicity; a low variational energy does not guarantee every property is improved.
For a final project, give the physical question, observable, model assumptions, derivation, one exact limiting check, numerical convergence plan and uncertainty estimator. State the provenance of figures and results. The lesson examples are original analytic constructions; proposed simulation workflows remain unexecuted unless evidence explicitly states otherwise.
7. Course directory and academic sources
- Quantum mechanics
- Molecular methods
- Density functional theory
- Tight binding
- Molecular dynamics
- Monte Carlo
- Quantum Monte Carlo
The following sources provide academic context; explanations, algebra and examples in these lessons are original teaching synthesis.