vdW

- J. Hermann and A. Tkatchenko, Phys. Rev. Lett. 124, 146401 (2020).
- https://link.aps.org/doi/10.1103/PhysRevLett.124.146401

Van der Waals (vdW) interactions originate from nonlocal __correlations__ in the quantum motion of electrons.

- J. Tao, H. Tang, A. Patra, P. Bhattarai, and J.P. Perdew, Phys. Rev. B 97, 165403 (2018).
- https://link.aps.org/doi/10.1103/PhysRevB.97.165403

Graphene on metals has two types of adsorptions according to the distance `d`

between graphene and metals:
1. chemical adsorption, `d`

is about 2 - 2.5 Å.
2. physical adsorption, `d`

> 3 Å.

An accurate exchange correlation functional must be able to capture both types.
This paper focus on the physical adsorption.
The total energy of the graphene on metals is regarded as the sum of DFT energy at GGA level and a vdW (empirical) correction.
$$
E _ {total} = E _ {DFT} + E _ {vdW}
$$
The design of $E _ {vdW}$ is based on Lifshitz-Zaremba-Kohn second-order perturbation theory,
in which metal surfaces must be clean (low-index surface ?).
The form of $E _ {vdW}$ is
$$
E _ {vdW} = \Big[- \dfrac{C _ 3}{(Z - Z _ 0)^3} - \dfrac{C _ 5}{(Z - Z _ 0)^5} \Big] f _ d
$$
where Z is the distance between the centers of the particles and
the planar surface of the outermost metal slab and $f_d$ is the damping function to avoid double counting due to the long-range part.
$E _ {vdW}$ is designed in such a way to reflect the difference between __ the long-range vdW interaction between particles and surfaces__ and different from __the vdW interaction between particles__.
The damped Zaremba-Kohn model (dZK) combined with PBE is called `PBE-vdW-dZK`

.

- RPA is often used as a reference when benchmarking vdW interactions. However, RPA typically underbinds covalent bonds. Therefore, it is not suitable for benchmarking chemical adsorptions.
- The vdW interaction for physisorption arises from instantaneous charge fluctuations of particles and inducedmultipole moments on the surface of a substrate.

This paper has compared the performance a set of vdW functionals in solid properties (lattice constant, bulkmoduli, ...) and proposed a new functional `optB86b-vdW`

.

- Lattice constant error: PBEsol <
`optB86b-vdW`

< optB88-vdW ~ optPBE-vdW < PBE < vdW-DF ~ vdW-DF2

- J. Klimeš, D.R. Bowler, and A. Michaelides, J. Phys.: Condens. Matter 22, 022201 (2009).
- https://iopscience.iop.org/article/10.1088/0953-8984/22/2/022201

This paper introduces two variants (optB88-vdW and optPBE-vdW) of vdW-DF by tuning the exchange and correlation parts of the functional. Parameters for these two functionals are fitted by S22 (a set CCSD(T) reference data of 22 weakly interacting dimers, mostly of biological interests) benchmark energies (no geometry optimizations). The vdW calculations are performed in a non self-consistent manner in this work.

- optPBE: a mixture of x=94.5268% PBE and (100 - x)% RPBE
- optB88: one parameter change in B88

Small changes in PBE or B88 parameters (when being combined with vdW kernel) lead to significant improvements in describing S22 dataset.

- optPBE-vdW: the second best in reproducing S22, the best in water hexamer
- optB88-vdW: the best in S22

- D.C. Langreth, M. Dion, H. Rydberg, E. Schröder, P. Hyldgaard, and B.I. Lundqvist, Int. J. Quantum Chem. 101, 599 (2005).
- http://onlinelibrary.wiley.com/doi/abs/10.1002/qua.20315

The long-range forces are included in the DFT framework, but not with the exchange-correlation approximation of LDA and GGA.
The vdW-DF functional has the correct asymptotic dependence at large distances and are seamless `(smooth)`

at small distances, meaning that short, medium, long-range interactions are well described with the vdW-DF functional.
The development of vdW-DF functional is divided into two parts: one focus on the exchange energy and another targets at the correlation energy which contains the long-range effects.
To say it more clear, van der Waals binding in reality is a correlation effect unrelated to exchange.

It is important to make sure the functional has good descriptions for both short and long ranges and varies smoothly from short to long ranges when developing a vdW functional.

The exchange energy does not have contributions from long-range interactions which only come from correlation effects.

Why is that ? - This is just the design philosophy of vdW-DF functional which needs an ideally vdW-free exchange.

The vdW-DF functional takes the exchange part from GGA ($E_x^{GGA}$), because the GGA functionals like PW91, PBE, RPBE are good at describing covalent bonds and such feature should be kept in the vdW-DF functional. Regarding the specific choice of GGA functional, the goal is that $E_x^{GGA}$ does not produce a binding in the van der Vaals region (and neither does the exact exchange) so that the exchange part of the vdW-DF functional alone is as accurate as possible.

- What does 'binding' refer to here?
- Judged by interaction energy and equilibrium distance.
- Why is the binding not present when the exchange is treated exactly?
- What should the exact exchange look like?
- The calculation from Hartree-Fock is regarded as the exact exchange.

The exchange part of revPBE is chosen for constructing the vdW-DF functional, because it doesn't show binding for a vdW system (Kr dimmer) which is similar to the behavior of Hartree-Fock, while the exchange part of another GGA functional PW91 exhibits binding due to the negative interaction energy ($E_{interaction} = E_{Kr-dimer} - 2E_{Kr}$). This choice provides a conservative staring point for adding vdW corrections, thus guaranteeing that any binding in Kr dimer will not be coming from exchange.

The pool of GGA functionals is so small that more accurate $E_x^{GGA}$ might be missing. This leaves rooms for further optimization of the vdW-DF family like vdW-optB88, vdW-optPBE

The correlation energy is divided into two parts: local term and non-local term: $E _ c ^{vdW-DF}[n] = E _ c ^0[n] + E _ c ^{nl}[n]$