Tkatchenko and Scheffler^{Tkatchenko:2009} 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}}$) that reflect
the local electronic environment. These are based on scaling the free-atom
values of these parameters in order to account for how the volume of a given
atom is modified by its molecular environment. The size of an atom in a
molecule is determined using the Hirshfeld partition of the electron density.
(Hirshfeld or “stockholder” partitioning, which also affords one measure of
atomic charges in a molecule, is described in Section 10.2.1).
In the resulting “TS-vdW” approach, only a single empirical range-separation
parameter (${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

$${f}_{\text{damp}}({R}_{AB})=\frac{1}{1+\mathrm{exp}\left[-d({R}_{AB}/{s}_{R}{R}_{\mathrm{vdW},AB}^{\text{eff}}-1)\right]}$$ | (5.38) |

with $d=20$ and an empirical parameter ${s}_{R}$ that is optimized in a
functional-specific way to reproduce intermolecular interaction
energies.^{Tkatchenko:2009} Optimized values for several different
functionals are listed in Table 5.4.

PBE | PBE0 | BLYP | B3LYP | revPBE | M06L | M06 | |

${s}_{R}$ | 0.94 | 0.96 | 0.62 | 0.84 | 0.60 | 1.26 | 1.16 |

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

$${C}_{6,AB}^{\text{eff}}=\frac{2{C}_{6,A}^{\text{eff}}{C}_{6,B}^{\text{eff}}}{\left({\alpha}_{B}^{\text{0,eff}}/{\alpha}_{A}^{\text{0,eff}}\right){C}_{6,A}^{\text{eff}}+\left({\alpha}_{A}^{\text{0,eff}}/{\alpha}_{B}^{\text{0,eff}}\right){C}_{6,B}^{\text{eff}}},$$ | (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 review^{Hermann:2017} 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,^{Rezac:2011}
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. 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.^{Tkatchenko:2009} It was then noted that for S66
interaction energies^{Rezac:2011} 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.^{Barton:2018} 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.

TSVDW

Flag to switch on the TS-vdW method

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

0
Do not apply TS-vdW.
1
Apply the TS-vdW method to obtain the TS-vdW energy.
2
Apply the TS-vdW method to obtain the TS-vdW energy and corresponding gradients.

RECOMMENDATION:

Since TS-vdW is itself a form of dispersion correction, it should *not* be used in conjunction with any of the
dispersion corrections described in Section 5.7.2.

HIRSHFELD_CONV

Set different SCF convergence criterion for the calculation of the single-atom
Hirshfeld calculations

TYPE:

INTEGER

DEFAULT:

same as SCF_CONVERGENCE

OPTIONS:

$n$
Corresponding to ${10}^{-n}$

RECOMMENDATION:

5

HIRSHMOD

Apply modifiers to the free-atom volumes used in the calculation of the scaled
TS-vdW parameters

TYPE:

INTEGER

DEFAULT:

4

OPTIONS:

0
Do not apply modifiers to the Hirshfeld volumes.
1
Apply built-in modifier to H.
2
Apply built-in modifier to H and C.
3
Apply built-in modifier to H, C and N.
4
Apply built-in modifier to H, C, N and O

RECOMMENDATION:

Use the default

$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