8. Band gaps, self interaction, and delocalization
Position in the course: Lesson 8 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
1. Purpose and assumptions
A fundamental gap is an electron-addition/removal energy difference. An optical gap involves a neutral excitation, and can differ because the electron and hole interact. A Kohn–Sham orbital gap belongs to an auxiliary one-particle problem. These quantities are related but not interchangeable. When reporting a gap, define the measured or calculated quantity before comparing numbers.
The electron-count formula follows by subtracting the affinity from the ionization energy. In exact density-functional theory, the fundamental gap includes a derivative discontinuity beyond the orbital gap. Semilocal approximations usually miss important discontinuity physics. Hybrid generalized Kohn–Sham eigenvalues can incorporate part of the gap differently, so the simple local-potential interpretation requires care. A scissors shift is an empirical adjustment, not a proof that all excitation physics is corrected.
For a one-electron system, the Hartree energy contains an interaction of the density with itself. Exact exchange cancels this self interaction, while correlation vanishes for the exact one-electron case. Many approximate functionals do not cancel it fully. This error can favor excessive charge spreading, distort dissociation and charge transfer, and place localized orbitals too high. Self-interaction error and delocalization error are related but are not identical labels for every many-electron failure.
Exact ensemble ground-state energy is piecewise linear between adjacent integer electron counts under the usual conditions. Convex curvature in an approximate energy can favor distributing a fractional electron across fragments. Compare energies of localized and distributed charge in carefully controlled settings. Charged periodic cells require finite-size and electrostatic corrections; differences of raw charged-cell energies can produce an apparent gap that mixes model bias with boundary artifacts. Numerical convergence cannot remove missing discontinuity or static correlation.
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. Keep vertical and relaxed energy differences separate
The displayed electron-number gap is vertical if every charge state uses the same geometry. Optimizing each charge state instead gives adiabatic addition/removal energies and includes structural reorganization. Comparing one with the other can falsely attribute a discrepancy to exchange–correlation alone. The electron affinity also depends on whether the added electron is bound; a finite diffuse basis or charged supercell can artificially confine an unbound state. Define the reference at infinity and inspect orbital extent. For solids, finite-size extrapolation and alignment matter. These boundary and geometry questions must be settled before interpreting a missing discontinuity.
3. Worked example
Let E(N−1)=−8, E(N)=−12, E(N+1)=−13 eV. Then I=4 eV, A=1 eV, and Eg=3 eV. If the auxiliary gap is 1.5 eV, the remaining 1.5 eV would represent a discontinuity contribution in an exact local Kohn–Sham interpretation; these invented energies are an algebra example.
4. Exercises with explained solutions
Exercise. For E(η)=Elinear(η)−aη(1−η), a>0, compare two fragments each with η=1/2 to one with η=1 and one with η=0.
Explained solution. Each half-charge fragment gains −a/4 relative to its linear energy, so the pair gains −a/2. Integer endpoints gain zero. This artificial convex curvature favors fractional distribution.
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
Orbital energy differences alone are insufficient for general optical spectra. Excitonic effects, relaxation, and selection rules require appropriate excited-state methods.
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