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 , where
the 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).
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2011), 1, pp. 211. , 386 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.
The oldest of these approaches is DFT-D2,
J. Comput. Chem.
(2006), 27, pp. 1787. in which the empirical dispersion potential has the aforementioned form, namely, pairwise atomic terms:
This function is damped at short range, where 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 is the sum of the van der Waals radii for atoms and , and is an additional parameter.) The primary parameters in Eq. (5.23) are atomic coefficients , 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 .
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 in
Eq. (5.23) was optimized by Grimme for four different
J. Comput. Chem.
(2006), 27, pp. 1787. and Q-Chem uses these as the default values: for PBE, for BLYP, for BP86, and for B3LYP. For all other functionals, by default. The D2 parameters, including the 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,
Phys. Chem. Chem. Phys.
(2008), 10, pp. 6615. 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 is controlled by the keyword DFT_D_A.
Note: 1. DFT-CHG is only defined for elements up to Xe. 2. The B97X-D and M05-D functionals automatically employ the DFT-CHG dispersion correction when used via METHOD = wB97X-D or wM05-D.
Grimme’s DFT-D3 method
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 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
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
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
for or 8, , and . The parameters come from atomic van der Waals radii, is a functional-dependent parameter, and . Typically is set to unity and is optimized for the functional in question.
The more recent Becke–Johnson-damping version of DFT-D3,
J. Comput. Chem.
(2011), 32, pp. 1456. DFT-D3(BJ), is designed to be finite (but non-zero) as . The damping functions used in DFT-D3(BJ) are
where and are adjustable parameters fit for each
density functional. As in DFT-D3(0), is generally fixed to unity and
is optimized for each functional. DFT-D3(BJ) generally outperforms the
original DFT-D3(0) version.
J. Comput. Chem.
(2011), 32, pp. 1456.
The DFT-D3(BJ) approach was re-parameterized by Smith 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 () as compared to DFT-D3(0):
Finally, optimized power approach of Witte et al.
J. Chem. Theory Comput.
(2017), 13, pp. 2043. treats the exponent, , as an optimizable parameter, given by
Note that .
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 (, , , and ), as defined in Eq. (5.28).
DFT-D3(BJ) is requested by setting DFT_D = D3_BJ. The model depends on four scaling parameters (, , , and ), as defined in Eq. (5.29).
DFT-D3(CSO) is requested by setting DFT_D = D3_CSO. The model depends on two scaling parameters ( and ), as defined in Eq. (5.30).
DFT-D3M(0) is requested by setting DFT_D = D3_ZEROM. The model depends on five scaling parameters (, , , , and ), as defined in Eq. (5.31).
DFT-D3M(BJ) is requested by setting DFT_D = D3_BJM. The model depends on four scaling parameters (, , , and ), as defined in Eq. (5.29).
DFT-D3(op) is requested by setting DFT_D = D3_OP. The model depends on four scaling parameters (, , , , and ), 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 (). 2. The B97-D3(0), B97X-D3, 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, ,
J. Chem. Phys.
(2010), 132, pp. 154104. must be explicitly turned on, if desired.
More recently, Grimme published an extended D3 model,
J. Chem. Phys.
(2017), 147, pp. 034112. , 145 J. Chem. Phys.
(2019), 150, pp. 154122. , 146 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 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 coefficients are obtained on-the-fly via Casimir-Polder integration
denotes the reference polarizailities which represents the molecular polarizability of symmetric hydride systems . are weighting factors determining the contributions of all element specific reference systems . describes the effective charge of hydrogen connectd to atom A in the reference system . The effective charge is computed self-consistently via Mulliken charge ,
The coefficients and are parametrized to match cationic static polarzizabilies and TD-DFT derived molecular dispersion coefficients, respectively.
$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