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)

Becke and Johnson have proposed an exchange dipole model (XDM) of dispersion.^{62, 437} 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. 488.

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}\;,

(5.31)

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.⁶⁴ 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,⁴⁸⁸ 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

E_{\text{vdW}}=\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% }+k\,C_{6,AB}/E^{\text{corr}}_{AB}}

(5.32)

where $k$ is a universal parameter, and $E_{AB}^{\text{corr}}$ is the sum of the absolute values of the correlation energies of the free atoms $A$ and $B$ . The dispersion coefficients $C_{6,AB}$ is computed according to

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}}

(5.33)

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, which leads to the “XDM10” model in Q-Chem:

E_{\rm vdW}=-\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)\;.

(5.34)

The higher-order dispersion coefficients are computed using higher-order multipole moments of the exchange hole.⁴³⁸ The quantity $R_{{\rm vdW},AB}$ is the sum of the effective van der Waals radii of atoms $A$ and $B$ ,

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

(5.35)

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]\;.

(5.36)

XDM10 contains two universal parameters, $a_{1}$ and $a_{2}$ , whose default values of 0.83 and 1.35, respectively, were fit to reproduce intermolecular interaction energies.⁴³⁷ Becke later suggested several other XC functional combinations with XDM, which employ different values of $a_{1}$ and $a_{2}$ . The user is advised to consult the recent literature for details.^{63, 453}

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. Additional job control and customization options are listed below.

DFTVDW_JOBNUMBER
       Basic vdW job control
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 Do not apply the XDM scheme. 1 Add vdW as energy/gradient correction to SCF. 2 Add vDW as a DFT functional and do full SCF (this option only works with XDM6).
RECOMMENDATION:
       None

DFTVDW_METHOD
       Choose the damping function used in XDM
TYPE:
       INTEGER
DEFAULT:
       1
OPTIONS:
       1 Use Becke’s damping function including $C_{6}$ term only. 2 Use Becke’s damping function with higher-order ( $C_{8}$ and $C_{10}$ ) terms.
RECOMMENDATION:
       None

DFTVDW_MOL1NATOMS
       The number of atoms in the first monomer in dimer calculation
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0– $N_{\rm atoms}$
RECOMMENDATION:
       None

DFTVDW_KAI
       Damping factor $k$ for $C_{6}$ -only damping function
TYPE:
       INTEGER
DEFAULT:
       800
OPTIONS:
       10–1000
RECOMMENDATION:
       None

DFTVDW_ALPHA1
       Parameter in XDM calculation with higher-order terms
TYPE:
       INTEGER
DEFAULT:
       83
OPTIONS:
       10-1000
RECOMMENDATION:
       None

DFTVDW_ALPHA2
       Parameter in XDM calculation with higher-order terms.
TYPE:
       INTEGER
DEFAULT:
       155
OPTIONS:
       10-1000
RECOMMENDATION:
       None

DFTVDW_USE_ELE_DRV
       Specify whether to add the gradient correction to the XDM energy. only valid with Becke’s $C_{6}$ damping function using the interpolated BR89 model.
TYPE:
       LOGICAL
DEFAULT:
       1
OPTIONS:
       1 Use density correction when applicable. 0 Do not use this correction (for debugging purposes).
RECOMMENDATION:
       None

DFTVDW_PRINT
       Printing control for VDW code
TYPE:
       INTEGER
DEFAULT:
       1
OPTIONS:
       0 No printing. 1 Minimum printing (default) 2 Debug printing
RECOMMENDATION:
       None

Example 5.10 Sample input illustrating a frequency calculation of a vdW complex consisted of He atom and N ${}_{2}$ molecule.

$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*
   !vdw parameters settings
   DFTVDW_JOBNUMBER    1
   DFTVDW_METHOD       1
   DFTVDW_PRINT        0
   DFTVDW_KAI          800
   DFTVDW_USE_ELE_DRV  0
$end

The original XDM implementation by Becke and Johnson used Hartree-Fock exchange but XDM can be used in conjunction with GGA, meta-GGA, or hybrid functionals, or with a specific meta-GGA exchange and correlation (the BR89 exchange and BR94 correlation functionals, for example). Encouraging results have been obtained using XDM with B3LYP.⁴⁸⁸ Becke has found more recently that this model can be efficiently combined with the P86 exchange functional, with the hyper-GGA functional B05. Using XDM together with PBE exchange plus LYP correlation, or PBE exchange plus BR94 correlation, has been also found fruitful. See Refs. 453 and 693 for some recent choices in this regard.

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5.7.3 Exchange-Dipole Model (XDM)