From the standpoint of the electron density, the vdW interaction is a non-local one: even for two non-overlapping, spherically-symmetric charge densities (two argon atoms, say), the presence of molecule B in the non-covalent A$\mathrm{\cdots}$B complex induces ripples in the tail of A’s charge distribution, which are the hallmarks of non-covalent interactions.^{Contreras-Garcia:2011} (This is the fundamental idea behind the non-covalent interaction plots described in Section 10.5.5; the vdW interaction manifests as large density gradients in regions of space where the density itself is small.) Semi-local GGAs that depend only on the density and its gradient cannot describe this long-range, correlation-induced interaction, and meta-GGAs at best describe it at middle-range via the Laplacian of the density and/or the kinetic energy density. A proper description of long-range electron correlation requires a non-local functional, i.e., an exchange-correlation potential having the form
$${v}_{\mathrm{c}}^{\mathrm{nl}}(\mathbf{r})=\int f(\mathbf{r},{\mathbf{r}}^{\prime})\mathit{d}{\mathbf{r}}^{\prime}.$$ | (5.19) |
In this way, a perturbation at a point ${\mathbf{r}}^{\prime}$ (due to B, say) then induces an exchange-correlation potential at a (possibly far-removed) point $\mathbf{r}$ (on A).
Q-Chem includes four such functionals that can describe dispersion interactions:
vdW-DF-04, developed by Langreth, Lundqvist, and coworkers,^{Dion:2004, Dion:2005} implemented as described in Ref. Vydrov:2008.
vdW-DF-10 (also known as vdW-DF2), which is a re-parameterization of vdW-DF-04.^{Lee:2010}
VV09, developed^{Vydrov:2009} and implemented^{Vydrov:2010a} by Vydrov and Van Voorhis.
VV10 by Vydrov and Van Voorhis.^{Vydrov:2010b}
rVV10 by Sabatini and coworkers.^{Sabatini:2013}
Each of these functionals is implemented in a self-consistent manner, and analytic gradients with respect to nuclear displacements are available.^{Vydrov:2008, Vydrov:2010a, Vydrov:2010b} The non-local correlation is governed by the $rem variable NL_CORRELATION, which can be set to one of the four values: vdW-DF-04, vdW-DF-10, VV09, or VV10. The vdW-DF-04, vdW-DF-10, and VV09 functionals are used in combination with LSDA correlation, which must be specified explicitly. For instance, vdW-DF-10 is invoked by the following keyword combination:
$rem CORRELATION PW92 NL_CORRELATION vdW-DF-10 ... $end
VV10 is used in combination with PBE correlation, which must be added explicitly. In addition, the values of two parameters, $C$ and $b$ (see Ref. Vydrov:2008), must be specified for VV10. These parameters are controlled by the $rem variables NL_VV_C and NL_VV_B, respectively. For instance, to invoke VV10 with $C=0.0093$ and $b=5.9$, the following input is used:
$rem CORRELATION PBE NL_CORRELATION VV10 NL_VV_C 93 NL_VV_B 590 ... $end
The variable NL_VV_C may also be specified for VV09, where it has the same meaning. By default, $C=0.0089$ is used in VV09 (i.e. NL_VV_C is set to 89). However, in VV10 neither $C$ nor $b$ are assigned a default value and must always be provided in the input.
Unlike local (LSDA) and semi-local (GGA and meta-GGA) functionals, for non-local functionals evaluation of the correlation energy requires a double integral over the spatial variables, as compared to the single integral [Eq. (5.8)] required for semi-local functionals:
$${E}_{\mathrm{c}}^{\mathrm{nl}}=\int {v}_{\mathrm{c}}^{\mathrm{nl}}(\mathbf{r})\mathit{d}\mathbf{r}=\int f(\mathbf{r},{\mathbf{r}}^{\prime})\rho (\mathbf{r})\mathit{d}\mathbf{r}\mathit{d}{\mathbf{r}}^{\prime}.$$ | (5.20) |
In practice, this double integration is performed numerically on a quadrature grid.^{Vydrov:2008, Vydrov:2010a, Vydrov:2010b} By default, the SG-1 quadrature (described in Section 5.5.2 below) is used to evaluate ${E}_{\mathrm{c}}^{\mathrm{nl}}$, but a different grid can be requested via the $rem variable NL_GRID. The non-local energy is rather insensitive to the fineness of the grid such that SG-1 or even SG-0 grids can be used in most cases, but a finer grid may be required to integrate other components of the functional. This is controlled by the XC_GRID variable discussed in Section 5.5.2.
The two functionals originally developed by Vydrov and Van Voorhis can be requested by specifying METHOD = VV10 or METHOD LC-VV10. In addition, the combinatorially-optimized functionals of Mardirossian and Head-Gordon ($\omega $B97X-V, B97M-V, and $\omega $B97M-V) make use of non-local correlation and can be invoked by setting METHOD to wB97X-V, B97M-V, or wB97M-V.
$molecule 0 1 C 0.000000 -0.000140 1.859161 H -0.888551 0.513060 1.494685 H 0.888551 0.513060 1.494685 H 0.000000 -1.026339 1.494868 H 0.000000 0.000089 2.948284 C 0.000000 0.000140 -1.859161 H 0.000000 -0.000089 -2.948284 H -0.888551 -0.513060 -1.494685 H 0.888551 -0.513060 -1.494685 H 0.000000 1.026339 -1.494868 $end $rem JOBTYPE opt BASIS aug-cc-pVTZ EXCHANGE rPW86 CORRELATION PBE XC_GRID 2 NL_CORRELATION VV10 NL_GRID 1 NL_VV_C 93 NL_VV_B 590 $end
In the above example, the SG-2 grid is used to evaluate the rPW86 exchange and PBE correlation, but a coarser SG-1 grid is used for the non-local part of VV10. Furthermore, the above example is identical to specifying METHOD = VV10.
NL_CORRELATION
Specifies a non-local correlation functional that includes non-empirical dispersion.
TYPE:
STRING
DEFAULT:
None
No non-local correlation.
OPTIONS:
None
No non-local correlation
vdW-DF-04
the non-local part of vdW-DF-04
vdW-DF-10
the non-local part of vdW-DF-10 (also known as vdW-DF2)
VV09
the non-local part of VV09
VV10
the non-local part of VV10
RECOMMENDATION:
Do not forget to add the LSDA correlation (PW92 is recommended) when
using vdW-DF-04, vdW-DF-10, or VV09. VV10 should be used with PBE
correlation. Choose exchange functionals carefully: HF, rPW86, revPBE, and
some of the LRC exchange functionals are among the recommended choices.
NL_VV_C
Sets the parameter $C$ in VV09 and VV10. This parameter is fitted to asymptotic
van der Waals ${C}_{6}$ coefficients.
TYPE:
INTEGER
DEFAULT:
89
for VV09
No default
for VV10
OPTIONS:
$n$
Corresponding to $C=n/10000$
RECOMMENDATION:
$C=0.0093$ is recommended when a semi-local exchange functional is used.
$C=0.0089$ is recommended when a long-range corrected (LRC) hybrid functional is used.
For further details see Ref. Vydrov:2010b.
NL_VV_B
Sets the parameter $b$ in VV10. This parameter controls the short range behavior
of the non-local correlation energy.
TYPE:
INTEGER
DEFAULT:
No default
OPTIONS:
$n$
Corresponding to $b=n/100$
RECOMMENDATION:
The optimal value depends strongly on the exchange functional used.
$b=5.9$ is recommended for rPW86. For further details see Ref. Vydrov:2010b.
USE_RVV10
Used to turn on the rVV10 NLC functional
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE
Use VV10 NLC (the default for NL_CORRELATION)
TRUE
Use rVV10 NLC
RECOMMENDATION:
Set to TRUE if the rVV10 NLC is desired.