10. A reproducible DFT investigation
Position in the course: Lesson 10 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
1. Purpose and assumptions
A defensible DFT result begins with a sharply defined question. Specify the observable, reference state, boundary conditions, and required resolution. An adsorption energy, a cohesive energy, and a reaction barrier use different references and require different structures. State the sign convention before presenting a number. Negative adsorption energy in the displayed convention favors binding relative to the separated slab and molecule, without automatically including thermal or solvent effects.
Separate physical modeling from numerical settings in a calculation record. The physical model includes nuclei, charge, spin, exchange–correlation approximation, dispersion, core treatment, and constraints. Numerical settings include basis, meshes, occupation width, SCF criteria, and optimizer criteria. Keep input files, code version, dataset identifiers, logs, and a table linking each reported result to its calculation. A screenshot of a final energy is not enough to reconstruct the investigation.
Converge the target difference with controlled changes. Differences often cancel large systematic contributions, but cancellation is a hypothesis to verify. Refine cutoff and k mesh independently, then check their interaction. For slabs refine thickness and vacuum; for defects refine cell size and charge corrections. The numerical uncertainty should be smaller than the effect being interpreted. Changes between two functionals are a sensitivity estimate, not a statistical confidence interval.
Use checks with distinct failure modes. Verify stoichiometry and units, visualize geometry and spin density, compare analytic and finite-difference forces when needed, and benchmark a related system with known reference data. A paper-like table should distinguish results actually computed from proposed calculations. The worked example here is deterministic arithmetic, not an executed electronic-structure simulation. For practical execution, connect this curriculum to the ABACUS, VASP, Quantum ESPRESSO, and CP2K tutorial courses and follow their input/output validation practices.
2. Derivation step by step
Read each equality with its assumptions. Atomic units are used for DFT equations unless another unit is stated; TB parameters retain explicit energy and length units. The conjugate transpose is denoted by a dagger, and a prime on a coordinate denotes a separate integration variable.
2.1. A result table should support reconstruction
For every energy difference, record its component energies, units, geometry identifiers, and common reference choices. A difference can be right by cancellation even when its components contain mistakes; component records help expose that. Archive failed or unconverged runs with clear status rather than silently treating them as valid samples. Automated extraction should check termination, electronic residuals, and structural convergence before collecting an energy. Software defaults can change across versions, so explicit settings and dataset hashes are useful. A proposed workflow is not an executed study: keep the distinction visible in teaching material and reports.
3. Worked example
Suppose a binding-energy difference is −0.210, −0.202, and −0.201 eV at three increasing cutoffs. The last change is 1 meV; it supports cutoff stability at that scale, but says nothing yet about mesh, slab thickness, or functional bias. If two candidate structures differ by 0.5 meV, the current evidence cannot resolve their ordering.
4. Exercises with explained solutions
Exercise. A slab+molecule energy is −115 eV, slab −100 eV, molecule −14 eV. Find the adsorption energy and list two missing free-energy contributions.
Explained solution. Eads=−115+100+14=−1 eV. Vibrational zero-point/thermal contributions and entropy are missing; solvent or pressure-dependent chemical potentials may also matter. Negative electronic adsorption energy alone does not establish experimental occupancy.
Further check. State the units and the allowed regime for every parameter in the worked example. Change one assumption and identify which derivation step must be revisited. A correct explanation names the affected constraint, operator, or boundary condition rather than merely saying that the answer changes.
5. Misconceptions and limitations
No universal convergence setting fits every material. A reproducible workflow reports evidence and conditions, rather than claiming that a copied parameter guarantees accuracy.
The illustration is an original teaching schematic. It is not output from a numerical materials simulation.
6. Connections and sources
Related: localized-basis theory · Molecular electronic structure
ABACUS · VASP · Quantum ESPRESSO · CP2K
- Hohenberg–Kohn, ground-state density theorem (1964)
- Kohn–Sham, self-consistent orbital equations (1965)
- PBE, constrained GGA construction (1996)
The explanations, algebra, and invented worked examples are original teaching synthesis. The cited papers establish the underlying theories, not the numerical toy values.
Original analytic teaching diagram under the lesson assumptions; no simulation results.
7. Related theory and practice
Quantum mechanics · Molecular methods · Tight binding · Molecular dynamics · VASP · Quantum-Espresso · CP2K · ABACUS