5.7 DFT Methods for van der Waals Interactions 5.7.2 Empirical Dispersion Corrections: DFT-D 5.7.4 Tkatchenko-Scheffler van der Waals Model (TS-vdW)

5.7.3 Exchange-Dipole Model (XDM)

(September 1, 2024)

5.7.3.1 Theory

Becke and Johnson have proposed an exchange dipole model (XDM) of dispersion. ⁸⁰ Becke A. D., Johnson E. R.
J. Chem. Phys.
(2005), 123, pp. 154101. Link ^, ⁸¹ Becke A. D., Johnson E. R.
J. Chem. Phys.
(2005), 122, pp. 154104. Link ^, ⁸² Becke A. D., Johnson E. R.
J. Chem. Phys.
(2006), 124, pp. 014104. Link ^, ⁸³ Becke A. D., Johnson E. R.
J. Chem. Phys.
(2007), 127, pp. 154108. Link ^, ⁶¹³ Johnson E. R., Becke A. D.
J. Chem. Phys.
(2005), 123, pp. 024101. Link ^, ⁶¹⁴ Johnson E. R., Becke A. D.
J. Chem. Phys.
(2006), 124, pp. 174104. Link The attractive dispersion energy arises in this model via the interaction between the instantaneous dipole moment of the exchange hole in one molecule, and the induced dipole moment in another. This is a conceptually simple yet powerful approach that has been shown to yield very accurate dispersion coefficients without fitting parameters. This allows the calculation of both intermolecular and intramolecular dispersion interactions within a single DFT framework. The implementation and validation of this method in the Q-Chem code is described in Ref. 669 Kong J. et al.
Phys. Rev. A
(2009), 79, pp. 042510. Link , with an updated set of damping parameters ⁹⁵⁸ Otero-de-la-Roza A., Johnson E. R.
J. Chem. Phys.
(2013), 138, pp. 204109. Link added in Q-Chem v. 6.1.1. ⁴⁴⁸ Gray M., Herbert J. M.
Annu. Rep. Comput. Chem.
(2024), 20, pp. 1. Link

The dipole moment of the exchange hole function $h_{\sigma}(\mathbf{r},\mathbf{r}^{\prime})$ is given at point $\mathbf{r}$ by

d_{\sigma}(\mathbf{r})=-\mathbf{r}-\int h_{\sigma}(\mathbf{r},\mathbf{r}^{% \prime})\;\mathbf{r}^{\prime}\;d\mathbf{r}^{\prime}\;,

(5.38)

where $\sigma=\alpha,\beta$ . This depends on a model of the exchange hole, and the implementation in Q-Chem uses the Becke-Roussel (BR) model. ⁸⁴ Becke A. D., Roussel M. R.
Phys. Rev. A
(1989), 39, pp. 3761. Link In most implementations the BR model, $h_{\sigma}$ is not available in analytic form and its value must be numerically at each grid point. Q-Chem developed for the first time an analytical expression for this function, ⁶⁶⁹ Kong J. et al.
Phys. Rev. A
(2009), 79, pp. 042510. Link based on non-linear interpolation and spline techniques, which greatly improves efficiency as well as the numerical stability.

Two different damping functions have been used with XDM. One of them relies only the intermolecular $C_{6}$ coefficient, and its implementation in Q-Chem is denoted as “XDM6”. In this version, the dispersion energy is ⁸⁰ Becke A. D., Johnson E. R.
J. Chem. Phys.
(2005), 123, pp. 154101. Link

E_{\text{vdW}}^{\text{XDM6}}=\sum^{\text{atoms}}_{A}\sum^{\text{atoms}}_{B<A}E% _{\text{vdW},AB}=-\sum^{\text{atoms}}_{A}\sum^{\text{atoms}}_{B<A}\frac{C_{6,% AB}}{R_{AB}^{6}+\kappa\,C_{6,AB}/E^{\text{corr}}_{AB}}\;.

(5.39)

The term $\kappa\,C_{6,AB}/E^{\text{corr}}_{AB}$ in the denominator prevents short-range divergence. The quantity $E_{AB}^{\text{corr}}$ is the sum of the absolute values of the correlation energies of the free atoms $A$ and $B$ , whereas $\kappa$ is a damping parameter that is universal in the sense that it is independent of the choice of functional. ⁸⁰ Becke A. D., Johnson E. R.
J. Chem. Phys.
(2005), 123, pp. 154101. Link The dispersion coefficients $C_{6,AB}$ is computed according to ⁸⁰ Becke A. D., Johnson E. R.
J. Chem. Phys.
(2005), 123, pp. 154101. Link

C_{6,ij}=\frac{\langle d_{\rm X}^{2}\rangle_{\!A}\langle d_{\rm X}^{2}\rangle_% {\!B}\;\alpha_{\!A}\,\alpha_{B}}{\langle d_{\text{X}}^{2}\rangle_{\!A}\,\alpha% _{B}+\langle d_{\text{X}}^{2}\rangle_{\!B}\,\alpha_{A}}

(5.40)

where $\langle d_{\rm X}^{2}\rangle_{\!A}$ is the square of the exchange-hole dipole moment of atom $A$ , whose effective polarizability (in the molecule) is $\alpha_{\!A}$ .

The XDM6 scheme can be further generalized to include higher-order dispersion coefficients, ⁸² Becke A. D., Johnson E. R.
J. Chem. Phys.
(2006), 124, pp. 014104. Link ^, ⁶¹⁴ Johnson E. R., Becke A. D.
J. Chem. Phys.
(2006), 124, pp. 174104. Link which leads to the “XDM10” model in Q-Chem:

E_{\rm vdW}^{\text{XDM10}}=-\sum^{\text{atoms}}_{A}\sum^{\text{atoms}}_{B<A}% \left(\frac{C_{6,AB}}{R_{{\rm vdW},AB}^{6}+R_{AB}^{6}}+\frac{C_{8,AB}}{R_{{\rm vdW% },AB}^{8}+R_{AB}^{8}}+\frac{C_{10,AB}}{R_{{\rm vdW},AB}^{10}+R_{AB}^{10}}% \right)\;.

(5.41)

The higher-order dispersion coefficients are computed using higher-order multipole moments of the exchange hole. ⁶¹⁴ Johnson E. R., Becke A. D.
J. Chem. Phys.
(2006), 124, pp. 174104. Link The quantity $R_{{\rm vdW},AB}$ prevents short-range divergence and is nominally equal to the sum of effective atomic radii for atoms $A$ and $B$ . In practice it is determined from the formula ⁶¹⁴ Johnson E. R., Becke A. D.
J. Chem. Phys.
(2006), 124, pp. 174104. Link

R_{{\rm vdW},AB}=a_{1}R_{{\text{crit}},AB}+a_{2}

(5.42)

with a critical distance

R_{{\text{crit}},AB}=\frac{1}{3}\left[\left(\frac{C_{8,AB}}{C_{6,AB}}\right)^{% 1/2}+\left(\frac{C_{10,AB}}{C_{6,AB}}\right)^{1/4}+\left(\frac{C_{10,AB}}{C_{8% ,AB}}\right)^{1/2}\right]

(5.43)

and two parameters, $a_{1}$ and $a_{2}$ . These were originally fitted using Hartree-Fock exchange ⁶¹⁴ Johnson E. R., Becke A. D.
J. Chem. Phys.
(2006), 124, pp. 174104. Link but were later optimized for several different exchange-correlation functionals: PW86PBE, PBE, BLYP, B97-1, B3LYP, B3P86, B3PW91, PBE0, BHHLYP, LRC- $\omega$ PBE, and CAM-B3LYP. ⁹⁵⁸ Otero-de-la-Roza A., Johnson E. R.
J. Chem. Phys.
(2013), 138, pp. 204109. Link Parameters for this set of functionals are implemented in Q-Chem. ⁴⁴⁸ Gray M., Herbert J. M.
Annu. Rep. Comput. Chem.
(2024), 20, pp. 1. Link

Note: For functionals other than the ones specified above, Hartree-Fock values of the parameters $a_{1}$ and $a_{2}$ are used, although this may not be optimal.

5.7.3.2 Job Control

As in DFT-D, the van der Waals energy is added as a post-SCF correction. Analytic gradients and Hessians are available for both XDM6 and XDM10. The dispersion correction is requested by setting XDM = TRUE in the $rem section. All other job control variables belong in the $xdm input section, as described below. Of these, only NAtoms_Mol1 is required, as this determined which part of the $molecule section corresponds to monomer $A$ , with the rest corresponding to monomer $B$ . It is required that all atoms for $A$ are grouped together in the $molecule section.

XDM

XDM
       Controls whether to add XDM dispersion to an SCF calculation
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 Do not apply the XDM scheme. 1 Add XDM dispersion as a correction to the SCF energy (and gradient, if appropriate). 2 Add dispersion as a DFT functional and do full SCF.
RECOMMENDATION:
       The second (self-consistent) option is only available for XDM6.

NAtomsMol1
       Sets the size of monomer $A$ .
INPUT SECTION: $xdm
TYPE:
       INTEGER
DEFAULT:
       NONE
OPTIONS:
       $N$ Monomer $A$ consists of the first $N$ atoms in the $molecule section.
RECOMMENDATION:
       This is the only required keyword in the $xdm section. The two monomers must be grouped together in the $molecule section; note that Q-Chem’s fragment-based input format does not work with XDM.

Method
       Controls which XDM method to use.
INPUT SECTION: $xdm
TYPE:
       STRING
DEFAULT:
       XDM10
OPTIONS:
       XDM6 Use the $C_{6}$ -only model of Eq. (5.39). XDM10 Use the model that includes $C_{8}$ and $C_{10}$ , Eq. (5.41).
RECOMMENDATION:
       The XDM10 is generally more accurate and is now the preferred approach.

Damp_C6_K
       Set the damping parameter $\kappa$ in Eq. (5.39).
INPUT SECTION: $xdm
TYPE:
       FLOAT
DEFAULT:
       0.8
OPTIONS:
       $x$ Set $\kappa=x$ .
RECOMMENDATION:
       The default value is taken from Ref. Becke:2005a and is intended to be universal.

Damp_A1
       Sets the parameter $a_{1}$ in Eq. (5.42).
INPUT SECTION: $xdm
TYPE:
       FLOAT
DEFAULT:
       Various
OPTIONS:
       $x$ Set $a_{1}=x$ .
RECOMMENDATION:
       Functional-specific defaults are available for HF, PW86PBE, PBE, BLYP, B97-1, B3LYP, B3P86, B3PW91, PBE0, BHHLYP, LRC- $\omega$ PBE, and CAM-B3LYP, taken from Ref. 958 Otero-de-la-Roza A., Johnson E. R.
J. Chem. Phys.
(2013), 138, pp. 204109. Link . For other functionals, the HF values (Ref. ⁶¹⁴ Johnson E. R., Becke A. D.
J. Chem. Phys.
(2006), 124, pp. 174104. Link ) are used by default, although Q-Chem will print a warning that this may not be optimal.

Damp_A2
       Sets the parameter $a_{2}$ in Eq. (5.42).
INPUT SECTION: $xdm
TYPE:
       FLOAT
DEFAULT:
       Various
OPTIONS:
       $x$ Set $a_{2}=x$ .
RECOMMENDATION:
       The same comments apply (regarding functional-specific values) as in the case of Damp_A1.

Print
       Controls the print level for the XDM procedure.
INPUT SECTION: $xdm
TYPE:
       INTEGER
DEFAULT:
       1
OPTIONS:
       0 No printing. 1 Minimal printing. 2 Debug-level printing.
RECOMMENDATION:
       None

Use_Elec_Drv
       Specify whether to add the gradient correction to the XDM energy.
INPUT SECTION: $xdm
TYPE:
       LOGICAL
DEFAULT:
       TRUE
OPTIONS:
       TRUE Use the gradient correction. FALSE Do not use the gradient correction.
RECOMMENDATION:
       This is only valid with Becke’s $C_{6}$ damping function using the interpolated BR89 model.

Example 5.12 Sample input illustrating a frequency calculation of a vdW complex He $\cdots$ N ${}_{2}$ using the XDM dispersion correction.

$molecule
   0 1
   He 0.000000  0.00000   3.800000
   N  0.000000  0.000000  0.546986
   N  0.000000  0.000000 -0.546986
$end

$rem
   JOBTYPE             FREQ
   IDERIV              2
   EXCHANGE            B3LYP
   INCDFT              0
   SCF_CONVERGENCE     8
   BASIS               6-31G*
   XDM                 TRUE
$end

$xdm
  method      xdm6
  natoms_mol1 1
$end

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5.7.3 Exchange-Dipole Model (XDM)

5.7.3.1 Theory

5.7.3.2 Job Control