5.7.3 Empirical Dispersion Corrections: DFT-D

A major development in DFT during the mid-2000s was the recognition that, first of all, semi-local density functionals do not properly capture dispersion (van der Waals) interactions, a problem that has been addressed only much more recently by the non-local correlation functionals discussed in Section 5.7.2; and second, that a cheap and simple solution to this problem is to incorporate empirical potentials of the form $-C_6/R^6$, where the $C_6$ coefficients are pairwise atomic parameters. This approach, which is an alternative to the use of a non-local correlation functional, is known as dispersion-corrected DFT (DFT-D). ⁴²⁵ Grimme S.
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2011), 1, pp. 211. Link ^{, 420} Grimme S. et al.
Chem. Rev.
(2016), 116, pp. 5105. Link

There are currently three unique DFT-D methods in Q-Chem. These are requested via the $rem variable DFT_D and are discussed below.

The oldest of these approaches is DFT-D2, ⁴²⁴ Grimme S.
J. Comput. Chem.
(2006), 27, pp. 1787. Link in which the empirical dispersion potential has the aforementioned form, namely, pairwise atomic $-C/R^6$ terms:

This function is damped at short range, where $R_{AB}^{-6}$ diverges, via

which also helps to avoid double-counting of electron correlation effects, since short- to medium-range correlation is included via the density functional. (The quantity $R_{0,AB}$ is the sum of the van der Waals radii for atoms $A$ and $B$, and $d$ is an additional parameter.) The primary parameters in Eq. (5.23) are atomic coefficients $C_{6,A}$, from which the pairwise parameters in Eq. (5.23) are obtained as geometric means, as is common in classical force fields:

The total energy in DFT-D2 is of course $E_{\text{DFT-D2}}=E_{\text{KS-DFT}}+E_{\mathrm{disp}}^{\text{D2}}$.

DFT-D2 is available in Q-Chem including analytic gradients and frequencies, thanks to the efforts of David Sherrill’s group. The D2 correction can be used with any density functional that is available in Q-Chem, although its use with the non-local correlation functionals discussed in Section 5.7.2 seems inconsistent and is not recommended. The global parameter $s_6$ in Eq. (5.23) was optimized by Grimme for four different functionals, ⁴²⁴ Grimme S.
J. Comput. Chem.
(2006), 27, pp. 1787. Link and Q-Chem uses these as the default values: $s_6=0.75$ for PBE, $s_6=1.2$ for BLYP, $s_6=1.05$ for BP86, and $s_6=1.05$ for B3LYP. For all other functionals, $s_6=1$ by default. The D2 parameters, including the $C_{6,A}$ coefficients and the atomic van der Waals radii, can be modified using a $empirical_dispersion input section. For example:

$empirical_dispersion
   S6 1.1
   D 10.0
   C6 Ar 4.60 Ne 0.60
   VDW_RADII Ar 1.60 Ne 1.20
$end

Values not specified explicitly default to the values optimized by Grimme.

Note:  1. DFT-D2 is only defined for elements up to Xe. 2. B97-D is an exchange-correlation functional that automatically employs the DFT-D2 dispersion correction when used via METHOD = B97-D.

An alternative to Grimme’s DFT-D2 is the empirical dispersion correction of Chai and Head-Gordon, ¹⁹¹ Chai J.-D., Head-Gordon M.
Phys. Chem. Chem. Phys.
(2008), 10, pp. 6615. Link which uses the same form as Eq. (5.23) but with a slightly different damping function:

This version is activated by setting DFT_D = EMPIRICAL_CHG, and the damping parameter $a$ is controlled by the keyword DFT_D_A.

Note:  1. DFT-CHG is only defined for elements up to Xe. 2. The $\omega$B97X-D and $\omega$M05-D functionals automatically employ the DFT-CHG dispersion correction when used via METHOD = wB97X-D or wM05-D.

Grimme’s DFT-D3 method ⁴¹⁷ Grimme S. et al.
J. Chem. Phys.
(2010), 132, pp. 154104. Link constitutes an improvement on his D2 approach, and is also available along with analytic first and second derivatives, for any density functional that is available in Q-Chem. The D3 correction includes a potential akin to that in D2 but including atomic $C_8$ terms as well:

The total D3 dispersion correction consists of this plus a three-body term of the Axilrod-Teller-Muto (ATM) triple-dipole variety, so that the total D3 energy is $E_{\text{DFT-D3}}=E_{\text{KS-DFT}}+E_{\text{D3,2-body}}+E_{\text{ATM,3-body}}$

Several versions of DFT-D3 are available as of Q-Chem 5.0, which differ in the choice of the two damping functions. Grimme’s formulation, ⁴¹⁷ Grimme S. et al.
J. Chem. Phys.
(2010), 132, pp. 154104. Link which is now known as the “zero-damping” version [DFT-D3(0)], uses damping functions of the form

for $n=6$ or 8, ${\beta }_6=14$, and ${\beta }_8=16$. ⁴¹⁷ Grimme S. et al.
J. Chem. Phys.
(2010), 132, pp. 154104. Link ^{, 731} Lin Y.-S. et al.
J. Chem. Theory Comput.
(2013), 9, pp. 263. Link The parameters $R_{0,AB}$ come from atomic van der Waals radii, $s_{r,6}$ is a functional-dependent parameter, and $s_{r,8}=1$. Typically $s_6$ is set to unity and $s_8$ is optimized for the functional in question.

The more recent Becke–Johnson-damping version of DFT-D3, ⁴¹⁹ Grimme S., Ehrlich S., Goerigk L.
J. Comput. Chem.
(2011), 32, pp. 1456. Link DFT-D3(BJ), is designed to be finite (but non-zero) as $R_{AB}\to 0$. The damping functions used in DFT-D3(BJ) are

where ${\alpha }_1$ and ${\alpha }_2$ are adjustable parameters fit for each density functional. As in DFT-D3(0), $s_6$ is generally fixed to unity and $s_8$ is optimized for each functional. DFT-D3(BJ) generally outperforms the original DFT-D3(0) version. ⁴¹⁹ Grimme S., Ehrlich S., Goerigk L.
J. Comput. Chem.
(2011), 32, pp. 1456. Link

The $C_6$-only (CSO) approach of Schröder et al. ¹⁰⁶⁸ Schröder H., Creon A., Schwabe T.
J. Chem. Theory Comput.
(2015), 11, pp. 3163. Link discards the $C_8$ term in Eq. (5.27) and uses a damping function with one parameter, ${\alpha }_1$:

The DFT-D3(BJ) approach was re-parameterized by Smith et al. ¹¹¹⁶ Smith D. G. et al.
J. Phys. Chem. Lett.
(2016), 7, pp. 2197. Link to yield the “modified” DFT-D3(BJ) approach, DFT-D3M(BJ), whose parameterization relied heavily on non-equilibrium geometries. The same authors also introduces a modification DFT-D3M(0) of the original zero-damping correction, which introduces one additional parameter (${\alpha }_1$) as compared to DFT-D3(0):

Finally, optimized power approach of Witte et al. ¹²⁹⁴ Witte J. et al.
J. Chem. Theory Comput.
(2017), 13, pp. 2043. Link treats the exponent, ${\beta }_6$, as an optimizable parameter, given by

Note that ${\beta }_8={\beta }_6+2$.

To summarize this bewildering array of D3 damping functions:

• •

  DFT-D3(0) is requested by setting DFT_D = D3_ZERO. The model depends on four scaling parameters ($s_6$, $s_{r,6}$, $s_8$, and $s_{r,8}$), as defined in Eq. (5.28).

• •

  DFT-D3(BJ) is requested by setting DFT_D = D3_BJ. The model depends on four scaling parameters ($s_6$, $s_8$, ${\alpha }_1$, and ${\alpha }_2$), as defined in Eq. (5.29).

• •

  DFT-D3(CSO) is requested by setting DFT_D = D3_CSO. The model depends on two scaling parameters ($s_6$ and ${\alpha }_1$), as defined in Eq. (5.30).

• •

  DFT-D3M(0) is requested by setting DFT_D = D3_ZEROM. The model depends on five scaling parameters ($s_6$, $s_8$, $s_{r,6}$, $s_{r,8}$, and ${\alpha }_1$), as defined in Eq. (5.31).

• •

  DFT-D3M(BJ) is requested by setting DFT_D = D3_BJM. The model depends on four scaling parameters ($s_6$, $s_8$, ${\alpha }_1$, and ${\alpha }_2$), as defined in Eq. (5.29).

• •

  DFT-D3(op) is requested by setting DFT_D = D3_OP. The model depends on four scaling parameters ($s_6$, $s_8$, ${\alpha }_1$, ${\alpha }_2$, and ${\beta }_6$), as defined in Eq. (5.29).

The scaling parameters in these damping functions can be modified using the $rem variables described below. Alternatively, one may simply set DFT_D = D3, and a D3 dispersion correction will be selected automatically, if one is available for the selected functional.

Note:  1. DFT-D3(0) is defined for elements up to Pu ($Z\mathrm{=}\mathrm{94}$). 2. The B97-D3(0), $\omega$B97X-D3, $\omega$M06-D3 functionals automatically employ the DFT-D3(0) dispersion correction when invoked by setting METHOD equal to B97-D3, wB97X-D3, or wM06-D3. 3. The old way of invoking DFT-D3, namely through the use of EMPIRICAL_GRIMME3, is still supported, though its use is discouraged since D3_ZERO accomplishes the same thing but with additional precision for the relevant parameters. 4. When DFT_D = D3, parameters may not be overwritten, with the exception of DFT_D3_3BODY; this is intended as a user-friendly option. This is also the case when EMPIRICAL_GRIMME3 is employed for a functional parameterized in Q-Chem. When any of D3_ZERO, D3_BJ, etc. are chosen, Q-Chem will automatically populate the parameters with their default values, if they available for the desired functional, but these defaults can still be overwritten by the user.

The three-body interaction term, $E^{(3)}$, ⁴¹⁷ Grimme S. et al.
J. Chem. Phys.
(2010), 132, pp. 154104. Link must be explicitly turned on, if desired.

More recently, Grimme published an extended D3 model, D4. ¹⁵² Caldeweyher E., Bannwarth C., Grimme S.
J. Chem. Phys.
(2017), 147, pp. 034112. Link ^{, 153} Caldeweyher E. et al.
J. Chem. Phys.
(2019), 150, pp. 154122. Link ^{, 154} Caldeweyher E. et al.
Phys. Chem. Chem. Phys.
(2020), 22, pp. 8499. Link The main feature of D4 is that the coefficients are generated through Casimir-Polder integration of the dynamic atomic polarizabilities $\alpha (i\omega )$ where electronic density information is employed via atomic partial charges. Benchmark results show that the proposed D4 model yields significantly lower error bars. The DFT-D4 dispersion energy similar to D3 model is given by

The Becke-Johnson damping is utilized as default. The coordination number dependent $C_6^{\text{AB}}$ coefficients are obtained on-the-fly via Casimir-Polder integration

where

and

${\alpha }^{A_m{H}_n}$ denotes the reference polarizailities which represents the molecular polarizability of symmetric hydride systems $A_m{H}_n$. $W_{A/B}^{A,\mathrm{ref}/B,\mathrm{ref}}$ are weighting factors determining the contributions of all element specific reference systems $N^{A,\mathrm{ref}/B,\mathrm{ref}}$. $z^{H_A}$ describes the effective charge of hydrogen connectd to atom A in the reference system $A_m{H}_n$. The effective charge $z^A$ is computed self-consistently via Mulliken charge $q^A$,

The coefficients $a$ and $b$ are parametrized to match cationic static polarzizabilies and TD-DFT derived molecular dispersion coefficients, respectively.