vdW-DF functional of Langreth and Lundqvist et al.

This method is available since the 5.2.12.26May2011 version of VASP for the calculation of
total energies and forces. The stress calculation for the cell optimisation (`ISIF=3`) is
available since the VASP 5.2.12.11Nov2011 version for spin unpolarised systems and VASP 5.3.1
for spin polarised systems.

**N.B.**: This feature has been implemented by J. Klimeš.
If you make use of the vdW-DF functionals presented in this section, we ask you to cite Ref. [147].
Please also cite the original vdW-DF paper of Dion et al. [143] and the paper of Roman-Perez and Soler [144].
In addtion, cite the paper of Lee et al. [146] if you use the vdW-DF2 functional,
the paper of Klimeš et al. [145] if you use the optB88-vdW or optPBE-vdW functionals,
and any other appropriate references, such as Ref. [148].

**Correlation functionals**

The method is invoked by setting

`LUSE_VDW = .TRUE.`

Moreover, the PBE correlation correction needs to be removed since only LDA correlation is used in the functionals. This is done by setting

`AGGAC = 0.0000`

The two tags above need to be used for all of the following functionals.

**Exchange functionals**

The `GGA` tag is further used to choose the appropriate exchange functional.
The original vdW-DF of Dion et al uses revPBE, therefore the vdW-DF can be chosen
by setting

GGA = RE LUSE_VDW = .TRUE. AGGAC = 0.0000

More accurate exchange functionals for the vdW correlation functional have been proposed in [145] and [147]. To use these functionals set:

GGA = OR LUSE_VDW = .TRUE. AGGAC = 0.0000

for optPBE-vdW,

GGA = BO PARAM1 = 0.1833333333 PARAM2 = 0.2200000000 LUSE_VDW = .TRUE. AGGAC = 0.0000

for the optB88-vdW functional, or

GGA = MK PARAM1 = 0.1234 PARAM2 = 1.0000 LUSE_VDW = .TRUE. AGGAC = 0.0000

for the optB86b-vdW functional.

In the vdW-DF2 functional the rPW86 exchange functional is used

`GGA = ML`

moreover, the vdW functional needs to be changed to the vdW2 correlation which requires only a change of a parameter:

`Zab_vdW = -1.8867`

Therefore to use vdW-DF2, set:

GGA = ML LUSE_VDW = .TRUE. Zab_vdW = -1.8867 AGGAC = 0.0000

An overview of the performance of the different approaches can be found for example in [145,146] for gas phase clusters and in [147] for solids.

**Important remarks**:

- The method needs a precalculated kernel which is distributed via the VASP
download portal (
`VASP -> src -> vdw_kernel.bindat`) and on the ftp server (`vasp5/src/vdw_kernel.bindat`). If VASP does not find this file, the kernel will be calculated. This, however, is rather demanding calculation. The kernel needs to be either copied to the VASP run directory for each calculation or can be stored in a central location and read from there. The location needs to be set in routine PHI_GENERATE. This does not work on some clusters and the kernel needs to be copied into the run directory in such cases. The distributed file uses little endian convention and won't be read on big endian machines. The big endian version of the file is available from the VASP team. - There are no special POTCARs for the vdW-DF functionals and the PBE or LDA POTCARs can be used. Currently the evaluation of the vdW energy term is not done fully within the PAW method but the sum of the pseudo-valence density and partial core density is used. This approximation works rather well, as is discussed in [147], and the accuracy generally increases when the number of valence electrons is increased or when harder PAW datasets are used. For example, for adsorption it is recommended to compare the adsorption energy obtained with standard PAW datasets and more-electron POTCARs for both PBE calculation and vdW-DF calculation to assess the quality of the results.
- The spin polarised calculations are possible, but strictly speaking the non-local vdW correlation is not defined for spin-polarised systems. For spin-polarised calculation the non-local vdW correlation energy is evaluated on the sum of the spin-up and spin-down densities.
- The evaluation of the vdW energy requires some additional time. Most of it is spent on performing FFTs to evaluate the energy and potential. Thus the additional time is determined by the number of FFT grid points in the calculation, basically size of the simulation cell. It is almost independent on the number of the atoms in the cell. Thus the relative cost of the vdW-DF method depends on the ``filling" of the cell and increases with the amount of vacuum in the cell. The relative increase is high for isolated molecules in large cells, but small for solids in smaller cells with many k-points.