Tkatchenko and Scheffler913 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 11.2.1). 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. (5.37) 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.913 Optimized values for several different functionals are listed in Table 5.4.
The pairwise coefficients in Eq. (5.37) 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. (5.23)]. 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. (5.37) is essentially zero in comparison to the cost of a DFT calculation. A recent review373 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,777 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.59 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.913 It was then noted that for S66 interaction energies777 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 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.59 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.