5.7.4 Tkatchenko-Scheffler van der Waals Model (TS-vdW)

Tkatchenko and Scheffler⁹²² have developed a pairwise method for van der Waals (vdW, i.e., dispersion) interactions, based on a scaling approach that yields in situ atomic polarizabilities ($\alpha$), dispersion coefficients ($C_6$), and vdW radii ($R_{\text{vdW}}$) that reflect the local electronic environment. These are based on scaling the free-atom values of these parameters in order to account for how the volume of a given atom is modified by its molecular environment. The size of an atom in a molecule is determined using the Hirshfeld partition of the electron density. (Hirshfeld or “stockholder” partitioning, which also affords one measure of atomic charges in a molecule, is described in Section 11.2.1). In the resulting “TS-vdW” approach, only a single empirical range-separation parameter ($s_R$) is required, which depends upon the underlying exchange-correlation functional.

Note:  The parameter $s_R$ is currently implemented only for the PBE, PBE0, BLYP, B3LYP, revPBE, M06L, and M06 functionals.

The TS-vdW energy expression is based on a pairwise-additive model for the dispersion energy,

$$E_{\text{vdW}}^{\text{TS}}=-\frac{1}{2}\sum _A^{\text{atoms}}\sum _{B\ne A}^{\text{A}}\left(\frac{C_{6,AB}^{\text{eff}}}{R_{AB}^6}\right)f_{\text{damp}}(R_{AB}).$$

(5.37)

As in DFT-D the $R^{-6}$ potentials in Eq. (5.37) must be damped at short range, and the TS-vdW model uses the damping function

$$f_{\text{damp}}(R_{AB})=\frac{1}{1+\exp \left[-d(R_{AB}/s_R{R}_{\mathrm{vdW},AB}^{\text{eff}}-1)\right]}$$

(5.38)

with $d=20$ and an empirical parameter $s_R$ that is optimized in a functional-specific way to reproduce intermolecular interaction energies.⁹²² Optimized values for several different functionals are listed in Table 5.4.

The pairwise coefficients $C_{6,AB}^{\text{eff}}$ in Eq. (5.37) are constructed from the corresponding atomic parameters $C_{6,A}^{\text{eff}}$ via

$$C_{6,AB}^{\text{eff}}=\frac{2C_{6,A}^{\text{eff}}{C}_{6,B}^{\text{eff}}}{\left({\alpha }_B^{\text{0,eff}}/{\alpha }_A^{\text{0,eff}}\right)C_{6,A}^{\text{eff}}+\left({\alpha }_A^{\text{0,eff}}/{\alpha }_B^{\text{0,eff}}\right)C_{6,B}^{\text{eff}}},$$

(5.39)

as opposed to the simple geometric mean that is used for $C_{6,AB}$ parameters in the empirical DFT-D methods [Eq. (5.23)]. These are “effective” $C_6$ coefficients in the sense that they account for the local electronic environment. As indicated above, this is accomplished by scaling the corresponding free-atom values, i.e.,

$$C_{6,A}^{\text{eff}}=C_{6,A}^{\text{free}}{\left(\frac{V_{A,\text{eff}}}{V_{A,\text{free}}}\right)}^2$$

(5.40)

where $V_{A,\text{eff}}$ is the effective volume of atom $A$ in the molecule, as determined using Hirshfeld partitioning. Effective atomic polarizabilities and vdW radii are obtained analogously:

$${\alpha }_A^{\text{0,eff}}={\alpha }_A^{\text{0,free}}\left(\frac{V_{A,\text{eff}}}{V_{A,\text{free}}}\right)$$

(5.41)

$$R_{\text{vdW,}A}^{\text{eff}}=R_{\text{vdW,}A}^{\text{free}}{\left(\frac{V_{A,\text{eff}}}{V_{A,\text{free}}}\right)}^{1/3}.$$

(5.42)

All three of these atom-specific parameters are therefore functionals of the electron density.

As with DFT-D, the cost to evaluate the dispersion correction in Eq. (5.37) is essentially zero in comparison to the cost of a DFT calculation. A recent review³⁷⁶ shows that the performance of the TS-vdW model is on par with that of other pairwise dispersion corrections. For example, for intermolecular interaction energies in the S66 data set,⁷⁸⁵ the TS-vdW correction added to PBE affords a mean absolute error of 0.4 kcal/mol and a maximum error of 1.5 kcal/mol, whereas the corresponding errors for PBE alone are 2.2 kcal/mol (mean) and 7.2 kcal/mol (maximum).

During the implementation of the TS-vdW scheme in Q-Chem, it was noted that evaluation of the free-atom volumes affords substantially different results as compared to the implementations in the FHI-aims and Quantum Espresso codes, e.g., $V_{\text{H,free}}$ = 8.68 a.u. (Q-Chem), 10.32 a.u. (FHI-aims), and 10.39 a.u. (Quantum Espresso) for hydrogen atom using the PBE functional.⁶⁰ These discrepancies were traced to different implementations of Hirshfeld partitioning. In Q-Chem, the free-atom volumes are computed from an unrestricted atomic SCF calculation and then spherically averaged to obtain spherically-symmetric atomic densities. In FHI-aims and Quantum Espresso they are obtained by solving a one-dimensional radial Schrödinger equation, which automatically affords spherically-symmetric atomic densities but must be used with fractional occupation numbers for open-shell atoms. Q-Chem’s value for the free-atom volume of hydrogen atom (7.52 a.u. at the Hartree-Fock/aug-cc-pVQZ level) is very to the analytic result (7.50 a.u.), lending credence to Q-Chem’s implementation of Hirshfeld partitioning and suggesting that it probably makes sense to re-parameterize the damping function in Eq. (5.38) for use with Q-Chem, where the representation of the electronic structure is quite different as compared to that in either FHI-aims or Quantum Espresso.

This has not been done, however, and the parameters were simply taken from a previous implementation.⁹²² It was then noted that for S66 interaction energies⁷⁸⁵ the PBE+TS-vdW results obtained using FHI-aims and Quantum Espresso are slightly closer to the benchmarks as compared to results from Q-Chem’s implementation of the same method, with root-mean-square deviations of 0.55 kcal/mol (Quantum Espresso) versus 0.70 kcal/mol (Q-Chem). Comparing ratios $V_{A,\text{eff}}/V_{A,\text{free}}$ between Q-Chem and FHI-aims, and performing linear regression analysis, affords scaling factors that can be applied to these atomic volume ratios, in order to obtain results from Q-Chem that are consistent with those from the other two codes using the same damping function.⁶⁰ Use of these scaling factors is controlled by the $rem variable HIRSHMOD, as described below.

The TS-vdW dispersion energy is requested by setting TSVDW = TRUE. Energies and analytic gradients are available.

$molecule
   0 1
   O
   H 1 0.95
   H 1 0.95 2 104.5
$end

$rem
   BASIS               6-31G*
   METHOD              PBE
   !vdw settings
   TSVDW               TRUE
   !setting the SCF_CONVERGENCE for single
   !atom calculations to 6
   HIRSHFELD_CONV      6
   !Apply modifiers to the free-atom volumes
   !the elements H, C, N, and O
   HIRSHMOD            4

$end