3. Exchange–correlation approximations and convergence logic
Position in the course: Lesson 3 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
1. Purpose and assumptions
The exact exchange–correlation functional is unknown for general systems. Approximate forms encode different physical information. LDA uses the uniform electron gas at the local density. GGA additionally uses density gradients; meta-GGA can depend on the positive orbital kinetic-energy density. Hybrids mix a fraction of orbital-dependent exact exchange with other terms. These categories describe ingredients, not a universal accuracy ranking for every observable.
The local exchange expression illustrates a functional derivative without invoking a black box. Apply ordinary differentiation to n to the four-thirds power at each spatial point, giving four-thirds times n to the one-third power. For a gradient-dependent functional, vary both n and its gradient. The second contribution contains a gradient of δn. Integrating it by parts moves the derivative to the coefficient, giving a minus divergence plus a boundary term. Periodic conditions or suitable decay remove that boundary term.
Numerical convergence and model validation answer different questions. Increasing the basis cutoff or refining integration grids tests whether the chosen approximate equations are solved accurately. Changing the functional tests part of the modeling uncertainty. A converged PBE result is still a PBE result. Basis and pseudopotential choices should be kept compatible; convergence comparisons should track the observable of interest, such as a reaction energy or lattice parameter, rather than only a large absolute total energy.
Build a controlled sequence: fix the geometry and functional, refine one numerical setting, record energy differences and forces, then refine another setting. Repeat a coupled check if two settings interact. Metals can require careful k-point and occupation convergence; isolated systems need vacuum and electrostatic checks. A practical tolerance should be tighter than the physical difference being interpreted. Naming a functional without its version, dispersion correction, spin treatment, and pseudopotential specification does not make a result reproducible.
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. Local density is not local physics in every sense
Even an LDA calculation couples distant regions through Hartree electrostatics and orbitals. Local refers to the exchange–correlation energy approximation at a point, not to a collection of independent atoms. GGA introduces gradients but still lacks general long-distance correlation between nonoverlapping fragments. The reduced gradient compares density variation with a local Fermi-wavevector scale, allowing a dimensionless descriptor. A functional can satisfy useful exact constraints yet still fail where the physical density or correlation pattern differs from its intended regime. For validation, choose an observable sensitive to the known failure mode rather than assuming agreement in geometry guarantees correct charge transfer or excitation energies.
3. Worked example
For a uniform density multiplied by eight, the LDA exchange potential magnitude doubles because 8^(1/3)=2; the exchange energy density magnitude grows by sixteen because 8^(4/3)=16. This is a local scaling check, not a change at fixed particle number unless volume is adjusted.
4. Exercises with explained solutions
Exercise. For f=An²+B|∇n|², derive the potential.
Explained solution. The density derivative is 2An. The gradient derivative is 2B∇n, whose divergence is 2B∇²n. Therefore v=2An−2B∇²n, with boundary terms assumed zero. The Laplacian term penalizes rapid density variation when B is positive.
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
A larger ingredient set can improve one property and worsen another. Numerical error bars obtained by convergence tests do not quantify unknown functional bias.
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