5.7.2 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. ²³⁰ Contreras-García J. et al.
J. Chem. Theory Comput.
(2011), 7, pp. 625. Link (This is the fundamental idea behind the non-covalent interaction plots described in Section 10.5.6; 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, ²⁸³ Dion M. et al.
  Phys. Rev. Lett.
  (2004), 92, pp. 246401. Link ^{, 284} Dion M. et al.
  Phys. Rev. Lett.
  (2005), 95, pp. 109902. Link implemented as described in Ref.  1241 Vydrov O. A., Wu Q., Van Voorhis T.
  J. Chem. Phys.
  (2008), 129, pp. 014106. Link .

• •

  vdW-DF-10 (also known as vdW-DF2), which is a re-parameterization of vdW-DF-04. ⁶⁹⁰ Lee K. et al.
  Phys. Rev. B
  (2010), 82, pp. 081101(R). Link

• •

  VV09, developed ¹²³⁸ Vydrov O. A., Van Voorhis T.
  Phys. Rev. Lett.
  (2009), 103, pp. 063004. Link and implemented ¹²³⁹ Vydrov O. A., Van Voorhis T.
  J. Chem. Phys.
  (2010), 132, pp. 164113. Link by Vydrov and Van Voorhis.

• •

  VV10 by Vydrov and Van Voorhis. ¹²⁴⁰ Vydrov O. A., Van Voorhis T.
  J. Chem. Phys.
  (2010), 133, pp. 244103. Link

• •

  rVV10 by Sabatini and coworkers. ¹⁰⁴⁶ Sabatini R., Gorni T., de Gironcoli S.
  Phys. Rev. B
  (2013), 87, pp. 041108. Link

Each of these functionals is implemented in a self-consistent manner, and analytic gradients with respect to nuclear displacements are available. ¹²⁴¹ Vydrov O. A., Wu Q., Van Voorhis T.
J. Chem. Phys.
(2008), 129, pp. 014106. Link ^{, 1239} Vydrov O. A., Van Voorhis T.
J. Chem. Phys.
(2010), 132, pp. 164113. Link ^{, 1240} Vydrov O. A., Van Voorhis T.
J. Chem. Phys.
(2010), 133, pp. 244103. Link 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.  1241 Vydrov O. A., Wu Q., Van Voorhis T.
J. Chem. Phys.
(2008), 129, pp. 014106. Link ), 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. ¹²⁴¹ Vydrov O. A., Wu Q., Van Voorhis T.
J. Chem. Phys.
(2008), 129, pp. 014106. Link ^{, 1239} Vydrov O. A., Van Voorhis T.
J. Chem. Phys.
(2010), 132, pp. 164113. Link ^{, 1240} Vydrov O. A., Van Voorhis T.
J. Chem. Phys.
(2010), 133, pp. 244103. Link By default, the SG-1 quadrature (described in Section 5.5.3 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 5.5.3.

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. Now, the VV10 codes have been rewrited and the feature of NMR chemical shielding and analytical second derivative are added. If you want to use the old codes, please set USE_LIBNLQ to false.

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.