LESSON NOTES · 09

9. Dispersion, hybrids, and DFT+U

Position in the course: Lesson 9 of 10. Complete the preceding derivation and use the explained exercises to check understanding.

1. Purpose and assumptions

Several important DFT failures require different remedies. Long-range dispersion comes from correlated charge fluctuations, even between neutral nonpolar fragments. Semilocal density ingredients cannot generally reproduce the correct asymptotic attraction between separated fragments. Pairwise dispersion corrections add distance-dependent terms with damping near short range; nonlocal correlation functionals use a density-based interaction kernel. The correction method and parameterization must be reported with the base functional.

The R to the minus sixth law is an asymptotic two-fragment result under its assumptions. At short distance the overlap region and base functional already contain part of the interaction, so damping prevents inappropriate extrapolation and reduces double counting. Many-body screening can matter in extended systems. Adding dispersion improves some binding problems but does not cure charge-transfer or strongly correlated electronic structure.

Hybrids introduce orbital exact exchange. In a simple global hybrid, the fraction a weights that term; range-separated hybrids distribute exchange by distance. Exact exchange removes one-electron self interaction in its exchange component but does not make a hybrid exact for all correlation regimes. Cost, periodic sampling, and screening become additional practical considerations. A functional chosen because it improves a molecular benchmark may require separate validation for metallic screening.

DFT+U adds an occupation-dependent correction in a selected localized subspace. For the simplified rotationally invariant expression shown, diagonalizing the occupation matrix yields a sum n(1−n). Integer occupations contribute zero; fractional occupations are penalized for positive Ueff. Differentiation gives opposing shifts of nearly empty and nearly full orbitals. The projectors, U value, and double-counting convention define the model. U is not a universal element-specific constant independent of basis, oxidation state, and screening.

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.

\[ \begin{aligned} E_{\mathrm{disp}}&=-\sum_{A<B}\frac{C_{6,AB}}{R_{AB}^6}f_{\mathrm{damp}}(R_{AB}),\\ E_{xc}^{\mathrm{hyb}}&=aE_x^{\mathrm{HF}}+(1-a)E_x^{\mathrm{approx}}+E_c^{\mathrm{approx}},\\ E_U&=\tfrac{U_{\mathrm{eff}}}{2}\sum_{I\sigma}\operatorname{Tr}[n^{I\sigma}(1-n^{I\sigma})],\\ \frac{dE_U}{dn}&=U_{\mathrm{eff}}(\tfrac12-n)\quad\text{for one occupation eigenvalue}. \end{aligned} \]

2.1. A correction changes the question being solved

Each correction introduces a definition and a domain. A dispersion pair parameter may depend on coordination; an occupation matrix depends on the chosen projector; a screened hybrid depends on its screening length. Comparing calculations while changing these definitions can mix several effects. Make a short model card listing the base functional, added terms, parameters, and reference justification. Then test at least one property that was not used to choose the parameter. For DFT+U, occupation eigenvalues and magnetic order offer checks beyond the gap. For dispersion, intermolecular distances and different binding geometries test transferability beyond one energy minimum.

3. Worked example

For Ueff=4 eV, one occupation n=1/2 contributes 4×(1/2)×(1/2)/2=0.5 eV; n=0 or 1 contributes zero. Its potential shift is zero at half filling and ±2 eV at the endpoints. These are correction terms, not a complete level-splitting prediction.

4. Exercises with explained solutions

Exercise. If an asymptotic pair distance doubles, what happens to its undamped dispersion energy?

Explained solution. R^-6 becomes (2R)^-6=R^-6/64, so the attraction magnitude falls by a factor of 64. This scaling applies only where the asymptotic expression is valid; it should not be extrapolated into overlapping fragments.

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

Stacking corrections can create double counting. Test a physically defined observable against an independent benchmark and disclose every correction and projector choice.

Self-consistent density cycle with residual check

The illustration is an original teaching schematic. It is not output from a numerical materials simulation.

6. Connections and sources

ABACUS · VASP · Quantum ESPRESSO · CP2K

The explanations, algebra, and invented worked examples are original teaching synthesis. The cited papers establish the underlying theories, not the numerical toy values.

Quantum mechanics · Molecular methods · Tight binding · Molecular dynamics · VASP · Quantum-Espresso · CP2K · ABACUS

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