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

Tkatchenko and Scheffler¹¹²⁴ 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 10.2.2). 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,

As in DFT-D the $R^{-6}$ potentials in Eq. (5.44) must be damped at short range, and the TS-vdW model uses the damping function

with $d=20$ and an empirical parameter $s_R$ that is optimized in a functional-specific way to reproduce intermolecular interaction energies.¹¹²⁴ Optimized values for several different functionals are listed in Table 5.4.

The pairwise coefficients $C_{6,AB}^{\text{eff}}$ in Eq. (5.44) are constructed from the corresponding atomic parameters $C_{6,A}^{\text{eff}}$ via

as opposed to the simple geometric mean that is used for $C_{6,AB}$ parameters in the empirical DFT-D methods [Eq. (5.25)]. 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.,

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:

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. (5.44) is essentially zero in comparison to the cost of a DFT calculation. A recent review⁴⁶³ 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,⁹⁶⁰ 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.^Barton:2018 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. These differences could likely be ameliorated by reparameterizing the damping function in Eq. (5.45) for use with atomic volumes calculated self-consistently using Q-Chem, wherein 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.¹¹²⁴ In order to reproduced TS-vdW dispersion energies obtained with FHI-aims or Quantum Espresso, it is possible to use this code in Q-Chem with scaling factors for the atomic Hirshfeld volumes, recommended values for which are obtained by linear regression, comparing Q-Chem atomic volumes to those obtained in FHI-aims. For full self-consistency, however, these scaling factors should not be used.

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