6. State counting, density of states, and filling
Position in the course: Lesson 6 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
1. Purpose and assumptions
A density of states counts available levels per energy interval. It does not count occupied electrons until multiplied by an occupation. Normalize it explicitly: per unit cell, per volume, or for the entire system; per spin or including spin degeneracy. Different normalizations can make identical bands appear to have different densities of states.
For the orthogonal nearest-neighbor chain, convert the uniform k measure to an energy measure. The delta-function change-of-variable identity divides by the absolute slope at every root. Interior band energies normally have two roots, one on each side of the band extremum. The derivative is −2ta sin ka; replacing sin² by one minus cos² gives the displayed inverse-square-root expression. The endpoints are integrable singularities, not an infinite number of states.
Integrating the DOS across the band gives one state per cell per spin. A spin-degenerate band can hold two electrons per cell. At half filling for the particle-hole-symmetric chain, the chemical potential lies at the onsite energy. A narrow band increases DOS per energy but does not increase the total number of states. Flat bands can produce a delta peak in an ideal model; finite temperature, disorder, and broadening change the observed profile.
Projected DOS distributes spectral weight among selected orbitals. In a nonorthogonal basis it requires an explicit projection/metric convention and need not be a unique atomic observable. Numerical histogram or Gaussian broadening trades visual smoothness against energy resolution. The integrated weight should still satisfy the expected count. A visually smooth curve can hide an insufficient k mesh, and an arbitrarily small broadening can create spikes that are sampling artifacts rather than physical singularities.
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. Broadening should preserve spectral weight
Replace each delta peak by a normalized kernel, such as a Gaussian whose integral is one. The DOS integral then remains the number of states, provided the plotted energy interval includes the kernel tails. Truncating that interval can create an apparent loss of states. Broadening alters peak heights, so compare integrated windows as well as pointwise curves. A van Hove singularity reflects small energy gradients and depends on dimension and dispersion; the one-dimensional square-root form should not be copied into two or three dimensions. A projected DOS sum rule also depends on a complete consistent set of projectors.
3. Worked example
For |t|=1 eV and ε0=0, the band spans −2 to 2 eV and ρ(0)=1/(2π) eV^-1 per cell per spin. Doubling |t| doubles the bandwidth and halves the central DOS, while the total integral remains one.
4. Exercises with explained solutions
Exercise. Verify the total integral by substituting E−ε0=2|t| sin θ.
Explained solution. The square root becomes 2|t|cos θ and dE=2|t|cos θ dθ. The integral is ∫dθ/π from −π/2 to π/2, hence one. Omitting the two k roots would incorrectly give one half.
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
DOS alone cannot specify wavefunction localization or conductivity; models with similar DOS can have very different eigenstates and response.
The illustration is an original teaching schematic. It is not output from a numerical materials simulation.
6. Connections and sources
Related: density-functional foundations · Interacting Monte Carlo methods
- Slater–Koster, original orbital-geometry construction (1954)
- Su–Schrieffer–Heeger, solitons in polyacetylene (1979)
- Marzari et al., maximally localized Wannier functions review (2012)
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 · Density functional theory · ABACUS · DFTB+