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Becke and Johnson have proposed an *exchange dipole model* (XDM) of
dispersion.^{47, 442} 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. 496.

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

$${d}_{\sigma}(\mathbf{r})=-\mathbf{r}-\int {h}_{\sigma}(\mathbf{r},{\mathbf{r}}^{\prime}){\mathbf{r}}^{\prime}\mathit{d}{\mathbf{r}}^{\prime},$$ | (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.^{49} 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,^{496} 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

$$ | (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

$${C}_{6,ij}=\frac{{\u27e8{d}_{\mathrm{X}}^{2}\u27e9}_{A}{\u27e8{d}_{\mathrm{X}}^{2}\u27e9}_{B}{\alpha}_{A}{\alpha}_{B}}{{\u27e8{d}_{\text{X}}^{2}\u27e9}_{A}{\alpha}_{B}+{\u27e8{d}_{\text{X}}^{2}\u27e9}_{B}{\alpha}_{A}}$$ | (5.33) |

where ${\u27e8{d}_{\mathrm{X}}^{2}\u27e9}_{A}$ 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:

$$ | (5.34) |

The higher-order dispersion coefficients are computed using higher-order
multipole moments of the exchange hole.^{443} The quantity
${R}_{\mathrm{vdW},AB}$ is the sum of the effective van der Waals radii of
atoms $A$ and $B$,

$${R}_{\mathrm{vdW},AB}={a}_{1}{R}_{\text{crit},AB}+{a}_{2}$$ | (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.^{442} 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.^{48, 458}

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–${N}_{\mathrm{atoms}}$

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

$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.^{496} 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. 458 and
716 for some recent choices in this regard.