# 5.7.1 Non-Local Correlation (NLC) Functionals

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$\cdots$B complex induces ripples in the tail of A’s charge distribution, which are the hallmarks of non-covalent interactions.186 (This is the fundamental idea behind the non-covalent interaction plots described in Section 11.5.5; 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

 $v_{\rm c}^{\rm nl}(\mathbf{r})=\int f(\mathbf{r},\mathbf{r^{\prime}})\,d% \mathbf{r^{\prime}}\;.$ (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,223, 224 implemented as described in Ref. 953.

• vdW-DF-10 (also known as vdW-DF2), which is a re-parameterization of vdW-DF-04.544

• VV09, developed950 and implemented951 by Vydrov and Van Voorhis.

• VV10 by Vydrov and Van Voorhis.952

• rVV10 by Sabatini and coworkers.799

Each of these functionals is implemented in a self-consistent manner, and analytic gradients with respect to nuclear displacements are available.953, 951, 952 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. 953), 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:

 $E_{\rm c}^{\rm nl}=\int v_{\rm c}^{\rm nl}(\mathbf{r})\;d\mathbf{r}=\int f(% \mathbf{r},\mathbf{r^{\prime}})\,\rho(\mathbf{r})\;d\mathbf{r}\;d\mathbf{r^{% \prime}}.$ (5.20)

In practice, this double integration is performed numerically on a quadrature grid.953, 951, 952 By default, the SG-1 quadrature (described in Section 5.5.2 below) is used to evaluate $E_{\rm c}^{\rm nl}$, 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. Example 5.8 Geometry optimization of the methane dimer using VV10 with rPW86 exchange. $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. 952.

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. 952.

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.