1. Van der Waals dispersion in density functional theory

2. Significant problem with DFT
Description of atomic and molecular scale interactions – requires accurate methods.
Density functional Theory
Exact?
In principle, yes, but approximations required for electron – electron interactions.
XC functionals
Standard XC (LDA, GGA, hybrid)– fail to describe long range interactions!
Significant problem with DFT

3. What is Dispersion?
Attractive interaction originating from the response of electrons in one region to instantaneous charge density fluctuations in another.
Many terms – leading term - dipole induced dipole  -1/r6
r -Interatomic separation

4. Why do standard XCs not describe dispersion?
Do not consider Instantaneous density fluctuations
Use only local properties
Give binding ONLY if there is overlap of electron densities between two atoms..
PBE cannot reproduce -1/r6 decay.

5. Dispersion corrected DFT

6. Ground: Binding with incorrect asymptotics
DO NOT describe long range asymptotics.
Incorrect shapes of BE curves.
Binding of well-separated molecules are underestimated.
LDA – still being used – may provide inconsistent accuracies.
Minnesota functionals – uses reference data
accurate around minima but fails at asymptotes. Accurate for chemistry problems.
Specially constructed pseudopotential projector (dispersion corrected atom- centered potentials)

7. Step one – simple C6 corrections
Try to get -1/r6 asymptotic behaviour.
EDFT – total energy computed with some XC functional
Coefficients depend on A and B.
Dispersion -Pairwise additive

8. Grimme : DFT-D2 Merits: low computational cost. Shortcomings:
Represents only leading term.
Where to get coefficients? (ionization potential, polarizabilities – relies on experimental data).
Coefficients are kept constant: no dependence on environment.
Diverges for small r.
Grimme : DFT-D2

9. Grimme DFT-D2 Most widely used.
C6 from a formula which couples ionization potentials and static polarizabilities of isolated atoms.
Data for all elements up to Xe are available.
BUT for some elements arbitrary choices of dispersion coefficients.
Damped dispersion correction:
Must be adjusted to be compatible with XC.
Gradually activates the dispersion correction over a distance characterized by the sum of the two atomic vdW radii.
Includes C8/r8 and C10/r10 terms.
DFT - CC

10. Step two – environment dependent C6 corrections
Carbon - C6 coefficients differ by about 35% between sp2 and sp3 hybridized states.
Three major methods:
DFT-D3 of Grimme
Tkatchenko and Scheffler method (vdW (TS))
Becke-Johnson model
In all three methods, C6 depends on the effective volume of the atom
If the atoms are squeezed, small volume, less polarizability, small C6

11. Grimme – DFT D3
Environmental dependence: coordination number, reference values for pairs of elements in different .
If the hybridization state changes during a simulation, C6 also changes simultaneously.
Simple, yet accurate.

12. TS Method
Relies on reference atomic polarizabilities and reference C6 coefficients.
Dispersion coefficients and damping function are charge-density dependent.

13. Step three – Long-range density functionals
Do not rely on external input parameters for calculating coefficients, but calculate it from the electron density.
Adds dispersion directly within a DFT functional
vdW-DF scheme –
increases the computational time by 50%

14. Higher Steps..