5. Basis sets, pseudopotentials, and PAW
Position in the course: Lesson 5 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
1. Purpose and assumptions
An orbital representation is a numerical choice distinct from the density functional. Plane waves are systematic for periodic smooth functions; atom-centered Gaussian or numerical orbitals can be economical for localized states. Plane waves truncate by kinetic energy. Local bases truncate by orbital content and radial flexibility. Neither a cutoff value nor a named basis alone proves convergence for a selected property.
Core orbitals vary rapidly near nuclei and are costly in a smooth basis. A pseudopotential replaces the core-region potential while matching selected valence properties. Norm-conserving constructions preserve a core-region norm under specified reference conditions; ultrasoft constructions relax that requirement and introduce augmentation and overlap structure. PAW reconstructs all-electron-like states from smooth functions plus local corrections. Its transformation is conceptually different from simply adding an empirical core energy.
The plane-wave cutoff condition follows directly from the kinetic operator acting on an exponential: −∇² exp(iq·r)/2=|q|² exp(iq·r)/2. A larger cutoff retains more reciprocal vectors. Roughly, their count grows as cell volume times cutoff to the three-halves power. Enlarging vacuum in an isolated-molecule supercell therefore increases cost even if no new atoms are added. Density and augmentation grids can require higher cutoffs than the orbital basis.
Pseudopotential transferability means preserving relevant behavior beyond the generating atomic configuration. Test oxidation states, bonding environments, semicore participation, and relevant energy differences. Absolute energies from distinct core treatments are not generally interchangeable. When calculating a reaction energy, use compatible datasets for every occurrence of each element. A basis convergence test cannot repair an inappropriate frozen-core choice. Forces and stresses can converge differently from energies; local bases can also produce basis-set superposition and Pulay effects.
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. Test the property, not only the basis size
A variational plane-wave total energy may decrease as cutoff rises under suitable fixed-model conditions, yet a difference of two such energies need not change monotonically. Forces are derivatives and can expose errors hidden in the energy. Norm conservation is a construction property under reference conditions, not a universal guarantee of agreement with an all-electron calculation. Compare reference logarithmic derivatives or known atomic excitations where available, then test chemically relevant environments. For local bases, enlarging radial range may change interfragment overlap and apparent binding, so compare both basis content and spatial extent. Keep numerical quadrature convergence separate from orbital completeness.
3. Worked example
At fixed cell volume, doubling a plane-wave cutoff increases the approximate basis count by 2^(3/2)≈2.83. This geometrical estimate is not a wall-time prediction because diagonalization, FFTs, parallelism, and storage enter differently.
4. Exercises with explained solutions
Exercise. Why should a reaction-energy convergence test use the same cutoff for all species?
Explained solution. Consistent settings allow partial cancellation of basis errors. Mixing independently loose tolerances can create a spurious reaction energy. Test the complete stoichiometric energy difference as cutoff increases, while also checking individual forces if structures are relaxed.
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
Convergence is conditional on a fixed dataset. Changing valence/core partition, relativistic treatment, or functional compatibility creates a new model requiring its own checks.
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)
The explanations, algebra, and invented worked examples are original teaching synthesis. The cited papers establish the underlying theories, not the numerical toy values.
7. Related theory and practice
Quantum mechanics · Molecular methods · Tight binding · Molecular dynamics · VASP · Quantum-Espresso · CP2K · ABACUS