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 (), dispersion coefficients (), and vdW radii () 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 () is required, which depends upon the underlying exchange-correlation functional.
Note: The parameter 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 potentials in Eq. (5.44) must be damped at short range, and the TS-vdW model uses the damping function
with and an empirical parameter 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 in Eq. (5.44) are constructed from the corresponding atomic parameters via
as opposed to the simple geometric mean that is used for parameters in the empirical DFT-D methods [Eq. (5.25)]. These are “effective” 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 is the effective volume of atom 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., = 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.
$molecule 0 1 O H 1 0.95 H 1 0.95 2 104.5 $end $rem BASIS 6-31G* METHOD PBE TSVDW TRUE !vdw settings HIRSHFELD_CONV 6 ! sets SCF_CONVERGENCE for single atom calculations HIRSHMOD 4 ! Apply modifiers to the free-atom volumes for H, C, N, and O $end