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$\mathrm{\cdots }$B complex induces ripples in the tail of A’s charge distribution, which are the hallmarks of non-covalent interactions.¹⁸⁷ (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

In this way, a perturbation at a point ${𝐫}^{\prime }$ (due to B, say) then induces an exchange-correlation potential at a (possibly far-removed) point $𝐫$ (on A).

Q-Chem includes four such functionals that can describe dispersion interactions:

• •

  vdW-DF-04, developed by Langreth, Lundqvist, and coworkers,^{224, 225} implemented as described in Ref. 964.

• •

  vdW-DF-10 (also known as vdW-DF2), which is a re-parameterization of vdW-DF-04.⁵⁴⁸

• •

  VV09, developed⁹⁶¹ and implemented⁹⁶² by Vydrov and Van Voorhis.

• •

  VV10 by Vydrov and Van Voorhis.⁹⁶³

• •

  rVV10 by Sabatini and coworkers.⁸⁰⁷

Each of these functionals is implemented in a self-consistent manner, and analytic gradients with respect to nuclear displacements are available.^{964, 962, 963} 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. 964), 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:

In practice, this double integration is performed numerically on a quadrature grid.^{964, 962, 963} 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.

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.