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).^{329, 324}
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 Grimme^{328}
EMPIRICAL_CHG
DFT-CHG dispersion correction from Chai and Head-Gordon^{147}
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.^{321}
D3_BJ
DFT-D3(BJ) dispersion correction from Grimme et al.^{323}
D3_CSO
DFT-D3(CSO) dispersion correction from Schröder et al.^{854}
D3_ZEROM
DFT-D3M(0) dispersion correction from Smith et al.^{894}
D3_BJM
DFT-D3M(BJ) dispersion correction from Smith et al.^{894}
D3_OP
DFT-D3(op) dispersion correction from Witte et al.^{1056}
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,^{328} in which the empirical dispersion potential has the aforementioned form, namely, pairwise atomic $-C/{R}^{6}$ terms:
$$ | (5.21) |
This function is damped at short range, where ${R}_{AB}^{-6}$ diverges, via
$${f}_{\text{damp}}^{\text{D2}}({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}={\left({C}_{6,A}{C}_{6,B}\right)}^{1/2}$$ | (5.23) |
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 5.7.1 seems inconsistent and is not recommended. The global parameter ${s}_{6}$ in Eq. (5.21) was optimized by Grimme for four different functionals,^{328} 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,^{147} which uses the same form as Eq. (5.21) but with a slightly different damping function:
${f}_{\text{damp}}^{\text{CHG}}({R}_{AB})={\left[1+a{({R}_{AB}/{R}_{0,AB})}^{-12}\right]}^{-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 method^{321} 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:
$$ | (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,^{321} 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,^{323} 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
$${f}_{\text{damp},n}^{\text{D3(BJ)}}({R}_{AB})=\frac{{R}_{AB}^{n}}{{R}_{AB}^{n}+{\left({\alpha}_{1}{R}_{0,AB}+{\alpha}_{2}\right)}^{n}}$$ | (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.^{323}
The ${C}_{6}$-only (CSO) approach of Schröder et al.^{854} discards the ${C}_{8}$ term in Eq. (5.25) and uses a damping function with one parameter, ${\alpha}_{1}$:
$${f}_{\text{damp},6}^{\text{D3(CSO)}}({R}_{AB})=\frac{{C}_{AB}^{6}}{{R}_{AB}^{6}+{(2.5\text{\xc5})}^{6}}\left({s}_{6}+\frac{{\alpha}_{1}}{1+\mathrm{exp}[{R}_{AB}-(2.5\text{\xc5}){R}_{0,AB}]}\right).$$ | (5.28) |
The DFT-D3(BJ) approach was re-parameterized by Smith et al.^{894} 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.^{1056} 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\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.
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_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_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, ${E}^{(3)}$,^{321} 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. 321).
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
$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 read $end $rem JOBTYPE sp EXCHANGE B3LYP BASIS 6-311++G** DFT_D D3_ZERO DFT_D3_S6 100000 DFT_D3_RS6 126100 DFT_D3_S8 170300 DFT_D3_3BODY FALSE $end