3.22 $C_6$/$R^6$ corrections for long-range van der Waals interactions

Note that the vdw part of the forces may be separately converged to (if set) sc_accuracy_forces.

This method is commonly referred to as the Tkatchenko-Scheffler method. The procedure is as follows. First, the normal self-consistency cycle is completed for a semilocal or hybrid density functional, most commonly PBE or PBE0. Second, the resulting self-consistent electron density is used to create interatomic (pairwise) $C_6$ coefficients. A simple pairwise van der Waals term is then added once to the self-consistent total energy from the preceding semilocal or hybrid functional. In other words, the Tkatchenko-Scheffler method is normally employed as a post-processing term in a non-self-consistent way, not during the self-consistency cycle. Since it needs to be combined with a different density functional, you would normally use it like this (example for “PBE+vdW”):

  xc pbe
  vdw_correction_hirshfeld

Three more caveats: (1) Do not use this method together with the local-density approximation (LDA) unless you know exactly what you are doing. The LDA already contains a spurious interaction term that will lead to very strange results if added to a pairwise van der Waals term. (2) Do not simply apply this method to a metallic system unless you know what you are doing. (3) This is also not the (very different) functional commonly known as the Langreth-Lundqvist or vdw-DF functional. [81] FHI-aims contains at least two implementations of vdw-DF for those who are interested, but either implementation is much slower than the Tkatchenko-Scheffler pairwise interatomic sum.

This flag adds the Tkatchenko-Scheffler vdW functional as a part of the given exchange-correlation (XC) functional. In a self-consistent scheme, the contribution of the vdW potential, $v_{\mathrm{vdW}}[n(𝐫)]=\delta E_{\mathrm{vdW}}[n(𝐫)]/\delta n(𝐫)$, is added to the XC potential to form the total effective potential in the Kohn-Sham equations. As a result, the van der Waals interatomic contributions affect the total electron density and are computed at each self-consistent cycle, until convergence is reached. In this way, it is possible to evaluate the effects of vdW interactions on the electron density and electronic properties, going beyond the vdW a posteriori correction of the total DFT energy.

Note: Do not use this self-consistent flag during a relaxation. The self-consistent forces are not implemented (yet) for the Tkatchenko-Scheffler vdW functional and this will lead to inconsistency errors.

The vdW radii for the alkali Cs, Rb, K, Na (and Li) in the original Tkatchenko-Scheffler method, vdw_correction_hirshfeld, are too low and drastically overbind these elements. This has been shown for Cs in Ref. [175] (see Figures S7 and S8 in the supporting information of that reference). The vdw_correction_hirshfeld_alkali parameterization is now the default for alkali species, and the code will stop when the original version is requested for a structure containing alkali atoms. A broader benchmark is provided in Ref. [286]. The original version is still available for comparison when the override_vdw_correction_hirshfeld_alkali keyword is set.

Importantly, vdw_correction_hirshfeld_alkali leaves the parameters for all other elements unchanged compared to the original (2009) Tkatchenko-Scheffler method, i.e., it is fully compatible with any other non-alkali containing calculations based on the Tkatchenko-Scheffler method.

This keyword applies updated vdW radii and interatomic van der Waals distances for all elements to the original Tkatchenko-Scheffler method, vdw_correction_hirshfeld. Unlike the keyword
vdw_correction_hirshfeld_alkali, the vdw_correction_hirshfeld_AP2018 approach will also change total energies and forces for other, non-alkali containing compounds.

The approach is benchmarked and described in full in Ref. [286].

Alternative implementation of the Tkatchenko-Scheffler (TS) dispersion energy term [311] via the Libmbd library [146] that provides some extra features compared to the vdw_correction_hirshfeld and vdw_correction_hirshfeld_sc implementations.

Note that this keyword requires the options listed below on the same line and as key=value pairs, e.g.

• •

  vdw_ts
  (default settings)

• •

  vdw_ts vdw_params_kind="ts"
  (default, but stated explicitly)

• •

  vdw_ts vdw_params_kind="tssurf"
  (uses the parameters associated with the vdW${}^{\text{surf}}$ approach [281])

See below for definitions of all options. Multiple concurrent options should be specified on the same single line.

The main difference to vdw_correction_hirshfeld and vdw_correction_hirshfeld_sc is the use of the Ewald summation to sum the $1/R^6$ sum in the periodic case, which makes the use of the vdw_convergence_threshold keyword obsolete. Furthermore, forces and stress are available also in the self-consistent case. This implementation does not honor the vdw_pair_ignore and hirshfeld_param keywords.

• •

  self_consistent=<logical> [default: .false.] turns on self-consistency as explained under vdw_correction_hirshfeld_sc.

• •

  vdw_params_kind=<string> [default: "ts"] specifies the set of free-atom vdW parameters (${\alpha }_0$, $C_6$, $R_{\text{vdW}}$) used. "ts" uses the original set of parameters, "tssurf" uses values from the so-called vdW${}^{\text{surf}}$ approach [281] for some of the elements. The used parameters are listed here.

The user needs to specify the interaction parameters for all atomic pairs in the system (i.e. for CNOH, there are 10 atomic pairs). This is done by putting “vdw_pairs $N$”, where $N$ is the number of pairs. This should be followed by $N$ lines of “vdw_coeff atom_i atom_j $C_{6ij}$ $R_{ij}^0$ $d$”, where $C_{6ij}$ is the $C_6$ coefficient for the interaction between atom_i and atom_j, $R_{ij}^0$ is the corresponding vdW radius and $d$ is the damping function parameter. A choice $d$=20 is suggested for all atomic pairs. An example for C-C interaction is: “vdw_coeff C C 30.00 5.59 20.0”.