# 5.7.2 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 5.7.1; 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).Grimme:2011b, Grimme:2016

There are currently three unique DFT-D methods in Q-Chem. These are requested via the $rem variable DFT_D and are discussed below. DFT_D Controls the empirical dispersion correction to be added to a DFT calculation. TYPE: LOGICAL DEFAULT: None OPTIONS: FALSE (or 0) Do not apply the DFT-D2, DFT-CHG, or DFT-D3 scheme EMPIRICAL_GRIMME DFT-D2 dispersion correction from GrimmeGrimme:2006b EMPIRICAL_CHG DFT-CHG dispersion correction from Chai and Head-GordonChai:2008b EMPIRICAL_GRIMME3 DFT-D3(0) dispersion correction from Grimme (deprecated as of Q-Chem 5.0) D3_ZERO DFT-D3(0) dispersion correction from Grimme et al.Grimme:2010 D3_BJ DFT-D3(BJ) dispersion correction from Grimme et al.Grimme:2011a D3_CSO DFT-D3(CSO) dispersion correction from Schröder et al.Schroder:2015 D3_ZEROM DFT-D3M(0) dispersion correction from Smith et al.Smith:2016 D3_BJM DFT-D3M(BJ) dispersion correction from Smith et al.Smith:2016 D3_OP DFT-D3(op) dispersion correction from Witte et al.Witte:2017b D3 Automatically select the "best" available D3 dispersion correction RECOMMENDATION: Use the D3 option, which selects the empirical potential based on the density functional specified by the user. The oldest of these approaches is DFT-D2,Grimme:2006b in which the empirical dispersion potential has the aforementioned form, namely, pairwise atomic $-C/R^{6}$ terms:  $E^{\text{D2}}_{\text{disp}}=-s_{6}\sum^{\text{atoms}}_{A}\sum^{\text{atoms}}_{% B (5.21) This function is damped at short range, where $R_{AB}^{-6}$ diverges, via  $f^{\text{D2}}_{\text{damp}}(R_{AB})=\left[1+e^{-d(R_{AB}/R_{0,AB}-1)}\right]^{% -1}$ (5.22) 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. (5.21) are atomic coefficients $C_{6,A}$, from which the pairwise parameters in Eq. (5.21) are obtained as geometric means, as is common in classical force fields:  $C_{6,AB}=\bigl{(}C_{6,A}C_{6,B}\bigr{)}^{1/2}$ (5.23) The total energy in DFT-D2 is of course $E_{\text{DFT-D2}}=E_{\text{KS-DFT}}+E^{\text{D2}}_{\rm disp}$. 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 5.7.1 seems inconsistent and is not recommended. The global parameter $s_{6}$ in Eq. (5.21) was optimized by Grimme for four different functionals,Grimme:2006b 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,Chai:2008b which uses the same form as Eq. (5.21) but with a slightly different damping function:

 $\displaystyle f^{\text{CHG}}_{\text{damp}}(R_{AB})=\bigl{[}1+a(R_{AB}/R_{0,AB}% )^{-12}\bigr{]}^{-1}$ (5.24)

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

DFT_D_A
Controls the strength of dispersion corrections in the Chai–Head-Gordon DFT-D scheme, Eq. (5.24).
TYPE:
INTEGER
DEFAULT:
600
OPTIONS:
$n$ Corresponding to $a=n/100$.
RECOMMENDATION:
Use the default.

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 more recent DFT-D3 methodGrimme:2010 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:

 $E_{\text{D3,2-body}}=-\sum^{\text{atoms}}_{A}\sum^{\text{atoms}}_{B (5.25)

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,Grimme:2010 which is now known as the “zero-damping” version [DFT-D3(0)], uses damping functions of the form

 $f_{\text{damp},n}^{\text{D3(0)}}(R_{AB})=\left[1+6\left(\frac{R_{AB}}{s_{r,n}R% _{0,AB}}\right)^{-\beta_{n}}\right]^{-1}$ (5.26)

for $n=6$ or 8, $\beta_{6}=12$, and $\beta_{8}=14$. 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,Grimme:2011a DFT-D3(BJ), is designed to be finite (but non-zero) as $R_{AB}\rightarrow 0$. The damping functions used in DFT-D3(BJ) are

 $f_{\text{damp},n}^{\text{D3(BJ)}}(R_{AB})=\frac{R_{AB}^{n}}{R_{AB}^{n}+\left(% \alpha_{1}R_{0,AB}+\alpha_{2}\right)^{n}}$ (5.27)

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.Grimme:2011a

The $C_{6}$-only (CSO) approach of Schröder et al.Schroder:2015 discards the $C_{8}$ term in Eq. (5.25) and uses a damping function with one parameter, $\alpha_{1}$:

 $f_{\text{damp},6}^{\text{D3(CSO)}}(R_{AB})=\frac{C_{AB}^{6}}{R_{AB}^{6}+(2.5% \mbox{\AA})^{6}}\left(s_{6}+\frac{\alpha_{1}}{1+\exp[R_{AB}-(2.5\mbox{\AA})R_{% 0,AB}]}\right)\;.$ (5.28)

The DFT-D3(BJ) approach was re-parameterized by Smith et al.Smith:2016 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):

 $f_{\text{damp},n}^{\text{D3M(0)}}(R_{AB})=\left[1+6\left(\frac{R_{AB}}{s_{r,n}% R_{0,AB}}+\alpha_{1}R_{0,AB}\right)^{-\beta_{n}}\right]^{-1}.$ (5.29)

Finally, optimized power approach of Witte et al.Witte:2017b treats the exponent, $\beta_{6}$, as an optimizable parameter, given by

 $f_{\text{damp},n}^{\text{D3(op)}}(R_{AB})=\frac{R_{AB}^{\beta_{n}}}{R_{AB}^{% \beta_{n}}+(\alpha_{1}R_{0,AB}+\alpha_{2})^{\beta_{n}}}\;.$ (5.30)

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. (5.26).

• 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. (5.27).

• 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. (5.28).

• 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. (5.29).

• 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. (5.27).

• 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. (5.27).

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=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. DFT_D3_S6 The linear parameter $s_{6}$ in eq. (5.25). Used in all forms of DFT-D3. TYPE: INTEGER DEFAULT: 100000 OPTIONS: $n$ Corresponding to $s_{6}=n/100000$. RECOMMENDATION: NONE DFT_D3_RS6 The nonlinear parameter $s_{r,6}$ in Eqs. (5.26) and Eq. (5.29). Used in DFT-D3(0) and DFT-D3M(0). TYPE: INTEGER DEFAULT: 100000 OPTIONS: $n$ Corresponding to $s_{r,6}=n/100000$. RECOMMENDATION: NONE DFT_D3_S8 The linear parameter $s_{8}$ in Eq. (5.25). Used in DFT-D3(0), DFT-D3(BJ), DFT-D3M(0), DFT-D3M(BJ), and DFT-D3(op). TYPE: INTEGER DEFAULT: 100000 OPTIONS: $n$ Corresponding to $s_{8}=n/100000$. RECOMMENDATION: NONE DFT_D3_RS8 The nonlinear parameter $s_{r,8}$ in Eqs. (5.26) and Eq. (5.29). Used in DFT-D3(0) and DFT-D3M(0). TYPE: INTEGER DEFAULT: 100000 OPTIONS: $n$ Corresponding to $s_{r,8}=n/100000$. RECOMMENDATION: NONE DFT_D3_A1 The nonlinear parameter $\alpha_{1}$ in Eqs. (5.27), (5.28), (5.29), and (5.30). Used in DFT-D3(BJ), DFT-D3(CSO), DFT-D3M(0), DFT-D3M(BJ), and DFT-D3(op). TYPE: INTEGER DEFAULT: 100000 OPTIONS: $n$ Corresponding to $\alpha_{1}=n/100000$. RECOMMENDATION: NONE DFT_D3_A2 The nonlinear parameter $\alpha_{2}$ in Eqs. (5.27) and (5.30). Used in DFT-D3(BJ), DFT-D3M(BJ), and DFT-D3(op). TYPE: INTEGER DEFAULT: 100000 OPTIONS: $n$ Corresponding to $\alpha_{2}=n/100000$. RECOMMENDATION: NONE DFT_D3_POWER The nonlinear parameter $\beta_{6}$ in Eq. (5.30). Used in DFT-D3(op). Must be greater than or equal to 6 to avoid divergence. TYPE: INTEGER DEFAULT: 600000 OPTIONS: $n$ Corresponding to $\beta_{6}=n/100000$. RECOMMENDATION: NONE The three-body interaction term, $E^{(3)}$,Grimme:2010 must be explicitly turned on, if desired. DFT_D3_3BODY Controls whether the three-body interaction in Grimme’s DFT-D3 method should be applied (see Eq. (14) in Ref. Grimme:2010). TYPE: LOGICAL DEFAULT: FALSE OPTIONS: FALSE (or 0) Do not apply the three-body interaction term TRUE Apply the three-body interaction term RECOMMENDATION: NONE Example 5.11 Applications of B3LYP-D3(0) with custom parameters to a methane dimer. $comment
Geometry optimization, followed by single-point calculations using a larger
basis set.
$end$molecule
0 1
C       0.000000    -0.000323     1.755803
H      -0.887097     0.510784     1.390695
H       0.887097     0.510784     1.390695
H       0.000000    -1.024959     1.393014
H       0.000000     0.001084     2.842908
C       0.000000     0.000323    -1.755803
H       0.000000    -0.001084    -2.842908
H      -0.887097    -0.510784    -1.390695
H       0.887097    -0.510784    -1.390695
H       0.000000     1.024959    -1.393014
$end$rem
JOBTYPE         opt
EXCHANGE        B3LYP
BASIS           6-31G*
DFT_D           D3_ZERO
DFT_D3_S6       100000
DFT_D3_RS6      126100
DFT_D3_S8       170300
DFT_D3_3BODY    FALSE
$end @@@$molecule
$end$rem