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# 5.7.3 Empirical Dispersion Corrections: DFT-D

(December 20, 2021)

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). 400 Grimme S.
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2011), 1, pp. 211.
, 395 Grimme S. et al.
Chem. Rev.
(2016), 116, pp. 5105.

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 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 Grimme 399 Grimme S. J. Comput. Chem. (2006), 27, pp. 1787. EMPIRICAL_CHG DFT-CHG dispersion correction from Chai and Head-Gordon 183 Chai J.-D., Head-Gordon M. Phys. Chem. Chem. Phys. (2008), 10, pp. 6615. 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. 392 Grimme S. et al. J. Chem. Phys. (2010), 132, pp. 154104. D3_BJ DFT-D3(BJ) dispersion correction from Grimme et al. 394 Grimme S., Ehrlich S., Goerigk L. J. Comput. Chem. (2011), 32, pp. 1456. D3_CSO DFT-D3(CSO) dispersion correction from Schröder et al. 1006 Schröder H., Creon A., Schwabe T. J. Chem. Theory Comput. (2015), 11, pp. 3163. D3_ZEROM DFT-D3M(0) dispersion correction from Smith et al. 1051 Smith D. G. et al. J. Phys. Chem. Lett. (2016), 7, pp. 2197. D3_BJM DFT-D3M(BJ) dispersion correction from Smith et al. 1051 Smith D. G. et al. J. Phys. Chem. Lett. (2016), 7, pp. 2197. D3_OP DFT-D3(op) dispersion correction from Witte et al. 1226 Witte J. et al. J. Chem. Theory Comput. (2017), 13, pp. 2043. D3 Automatically select the “best” available D3 dispersion correction D4 DFT-D4 dispersion correction from Caldeweyher et al. 146 Caldeweyher E., Bannwarth C., Grimme S. J. Chem. Phys. (2017), 147, pp. 034112. , 147 Caldeweyher E. et al. J. Chem. Phys. (2019), 150, pp. 154122. , 148 Caldeweyher E. et al. Phys. Chem. Chem. Phys. (2020), 22, pp. 8499. RECOMMENDATION: Use D4 if the specified functional is avialable. Currently, only a subset of functionals in DFT-D4 is supported. It includes B3LYP, B97, B1LYP, PBE0, PW6B95, M06L, M06, WB97, WB97X, CAMB3LYP, PBE02, PBE0DH, MPW1K, MPWB1K, B1B95, B1PW91, B2GPPLYP, B2PLYP, B3P86, B3PW91, O3LYP, REVPBE, REVPBE0, REVTPSS, REVTPSSH, SCAN, TPSS0, TPSSH, X3LYP, TPSS, BP86, BLYP, BPBE, MPW1PW91, MPW1LYP, PBE, RPBE, and PW91. The oldest of these approaches is DFT-D2, 399 Grimme S. J. Comput. Chem. (2006), 27, pp. 1787. 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 (5.23) This function is damped at short range, where $R_{AB}^{-6}$ diverges, via  $f^{\text{D2}}_{\text{damp}}(R_{AB})=\left[1+e^{-d(R_{AB}/R_{0,AB}-1)}\right]^{% -1}$ (5.24) 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:  $C_{6,AB}=\bigl{(}C_{6,A}C_{6,B}\bigr{)}^{1/2}$ (5.25) 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 5.7.2 seems inconsistent and is not recommended. The global parameter $s_{6}$ in Eq. (5.23) was optimized by Grimme for four different functionals, 399 Grimme S. J. Comput. Chem. (2006), 27, pp. 1787. 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, 183 Chai J.-D., Head-Gordon M.
Phys. Chem. Chem. Phys.
(2008), 10, pp. 6615.
which uses the same form as Eq. (5.23) but with a slightly different damping function:

 $\displaystyle f^{\text{CHG}}_{\text{damp}}(R_{AB})=\bigl{[}1+a(R_{AB}/R_{0,AB}% )^{-12}\bigr{]}^{-1}$ (5.26)

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

DFT_D_A
Controls the strength of dispersion corrections in the Chai–Head-Gordon DFT-D scheme, Eq. (5.26).
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 DFT-D3 method 392 Grimme S. et al.
J. Chem. Phys.
(2010), 132, pp. 154104.
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 (5.27)

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, 392 Grimme S. et al.
J. Chem. Phys.
(2010), 132, pp. 154104.
which is now known as the “zero-damping” version [DFT-D3(0)], uses damping functions of the form

 $f_{\text{damp},n}^{\text{D3(0)}}(R_{AB})=\left[1+6\left(\frac{R_{AB}}{s_{r,n}R% _{0,AB}}\right)^{-\beta_{n}}\right]^{-1}$ (5.28)

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, 394 Grimme S., Ehrlich S., Goerigk L.
J. Comput. Chem.
(2011), 32, pp. 1456.
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

 $f_{\text{damp},n}^{\text{D3(BJ)}}(R_{AB})=\frac{R_{AB}^{n}}{R_{AB}^{n}+\left(% \alpha_{1}R_{0,AB}+\alpha_{2}\right)^{n}}$ (5.29)

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. 394 Grimme S., Ehrlich S., Goerigk L.
J. Comput. Chem.
(2011), 32, pp. 1456.

The $C_{6}$-only (CSO) approach of Schröder et al. 1006 Schröder H., Creon A., Schwabe T.
J. Chem. Theory Comput.
(2015), 11, pp. 3163.
discards the $C_{8}$ term in Eq. (5.27) and uses a damping function with one parameter, $\alpha_{1}$:

 $f_{\text{damp},6}^{\text{D3(CSO)}}(R_{AB})=\frac{C_{AB}^{6}}{R_{AB}^{6}+(2.5% \mbox{\AA})^{6}}\left(s_{6}+\frac{\alpha_{1}}{1+\exp[R_{AB}-(2.5\mbox{\AA})R_{% 0,AB}]}\right)\;.$ (5.30)

The DFT-D3(BJ) approach was re-parameterized by Smith et al. 1051 Smith D. G. et al.
J. Phys. Chem. Lett.
(2016), 7, pp. 2197.
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):

 $f_{\text{damp},n}^{\text{D3M(0)}}(R_{AB})=\left[1+6\left(\frac{R_{AB}}{s_{r,n}% R_{0,AB}}+\alpha_{1}R_{0,AB}\right)^{-\beta_{n}}\right]^{-1}.$ (5.31)

Finally, optimized power approach of Witte et al. 1226 Witte J. et al.
J. Chem. Theory Comput.
(2017), 13, pp. 2043.
treats the exponent, $\beta_{6}$, as an optimizable parameter, given by

 $f_{\text{damp},n}^{\text{D3(op)}}(R_{AB})=\frac{R_{AB}^{\beta_{n}}}{R_{AB}^{% \beta_{n}}+(\alpha_{1}R_{0,AB}+\alpha_{2})^{\beta_{n}}}\;.$ (5.32)

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=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 DFT_D3_S6 The linear parameter $s_{6}$ in eq. (5.27). Used in all forms of DFT-D3. TYPE: INTEGER DEFAULT: 100000 OPTIONS: $n$ Corresponding to $s_{6}=n/100000$. RECOMMENDATION: NONE DFT_D3_RS6 DFT_D3_RS6 The nonlinear parameter $s_{r,6}$ in Eqs. (5.28) and Eq. (5.31). 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 DFT_D3_S8 The linear parameter $s_{8}$ in Eq. (5.27). 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 DFT_D3_RS8 The nonlinear parameter $s_{r,8}$ in Eqs. (5.28) and Eq. (5.31). 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 DFT_D3_A1 The nonlinear parameter $\alpha_{1}$ in Eqs. (5.29), (5.30), (5.31), and (5.32). 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 DFT_D3_A2 The nonlinear parameter $\alpha_{2}$ in Eqs. (5.29) and (5.32). 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 DFT_D3_POWER The nonlinear parameter $\beta_{6}$ in Eq. (5.32). 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, $E^{(3)}$, 392 Grimme S. et al. J. Chem. Phys. (2010), 132, pp. 154104. must be explicitly turned on, if desired. DFT_D3_3BODY DFT_D3_3BODY Controls whether the three-body interaction in Grimme’s DFT-D3 method should be applied (see Eq. (14) in Ref. 392 Grimme S. et al. J. Chem. Phys. (2010), 132, pp. 154104. ). 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 More recently, Grimme published an extended D3 model, D4. 146 Caldeweyher E., Bannwarth C., Grimme S. J. Chem. Phys. (2017), 147, pp. 034112. , 147 Caldeweyher E. et al. J. Chem. Phys. (2019), 150, pp. 154122. , 148 Caldeweyher E. et al. Phys. Chem. Chem. Phys. (2020), 22, pp. 8499. 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  $E_{\text{D4,2-body}}=-\sum^{\text{atoms}}_{A}\sum^{\text{atoms}}_{B (5.33) 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  $C_{6}^{\text{AB}}=\sum^{\text{N}^{A,\mathrm{ref}}}_{A,\mathrm{ref}=1}\sum^{% \text{N}^{B,\mathrm{ref}}}_{B,\mathrm{ref}=1}\frac{3}{\pi}\int_{0}^{\infty}d% \omega\alpha^{A,\mathrm{ref}}(i\omega,z^{A})\times W_{A}^{A,\mathrm{ref}}% \alpha^{B,\mathrm{ref}}(i\omega,z^{B})W_{B}^{B,\mathrm{ref}}\;,$ (5.34) where  $\alpha^{A,\mathrm{ref}}(i\omega,z^{A})=\frac{1}{m}\left[\alpha^{A_{m}H_{n}}(i% \omega)-\frac{n}{2}\alpha^{H_{2}}(i\omega)\times\zeta(z^{H_{A},\mathrm{ref}},z% ^{H_{2}})\right]\zeta(z^{A},z^{A,\mathrm{ref}})\;$ (5.35) and  $\zeta(z^{A},z^{A,\mathrm{ref}})=ba^{[1.47\text{exp}(z^{A}/z^{A,\mathrm{ref}})% \text{log}_{10}(z^{A,\mathrm{ref}}/z^{A})]}\;.$ (5.36) $\alpha^{A_{m}H_{n}}$ 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 $q^{A}$,  $z^{A}=Z^{A}+q^{A}.$ (5.37) The coefficients $a$ and $b$ are parametrized to match cationic static polarzizabilies and TD-DFT derived molecular dispersion coefficients, respectively. DFT_D4_S6 DFT_D4_S6 The linear parameter $s_{6}$. Used in DFT-D4. TYPE: INTEGER DEFAULT: Optimized number for the specified functional OPTIONS: $n$ Corresponding to $s_{6}=n/100000000$. RECOMMENDATION: NONE DFT_D4_S8 DFT_D4_S8 The linear parameter $s_{8}$. Used in DFT-D4. TYPE: INTEGER DEFAULT: Optimized number for the specified functional OPTIONS: $n$ Corresponding to $s_{8}=n/100000000$. RECOMMENDATION: NONE DFT_D4_S10 DFT_D4_S10 The linear parameter $s_{10}$. Used in DFT-D4. TYPE: INTEGER DEFAULT: Optimized number for the specified functional OPTIONS: $n$ Corresponding to $s_{10}=n/100000000$. RECOMMENDATION: NONE DFT_D4_A1 DFT_D4_A1 The nonlinear parameter $\alpha_{1}$. Used in DFT-D4. TYPE: INTEGER DEFAULT: Optimized number for the specified functional OPTIONS: $n$ Corresponding to $\alpha_{1}=n/100000000$. RECOMMENDATION: NONE DFT_D4_A2 DFT_D4_A2 The nonlinear parameter $\alpha_{2}$. Used in DFT-D4. TYPE: INTEGER DEFAULT: Optimized number for the specified functional OPTIONS: $n$ Corresponding to $\alpha_{2}=n/100000000$. RECOMMENDATION: NONE DFT_D4_S9 DFT_D4_S9 The linear parameter $s_{9}$. Used in DFT-D4. TYPE: INTEGER DEFAULT: Optimized number for the specified functional OPTIONS: $n$ Corresponding to $s_{9}=n/100000000$. RECOMMENDATION: NONE DFT_D4_WF DFT_D4_WF Weighting factor for Gaussian weighting. TYPE: INTEGER DEFAULT: 600000000 OPTIONS: $n$ Corresponding to $wf=n/100000000$. RECOMMENDATION: Use default DFT_D4_GA DFT_D4_GA Charge scaling TYPE: INTEGER DEFAULT: 300000000 OPTIONS: $n$ Corresponding to $ga=n/100000000$. RECOMMENDATION: Use default DFT_D4_GC DFT_D4_GC Charge scaling TYPE: INTEGER DEFAULT: 200000000 OPTIONS: $n$ Corresponding to $gc=n/100000000$. RECOMMENDATION: Use default 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