Van der Waals dispersion in density functional theory

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

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

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.

Dispersion corrected DFT

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)

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

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

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

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

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.

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

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%

Higher Steps..