Becke and Johnson have proposed an exchange dipole model (XDM) of dispersion.71, 533 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. 584.
The dipole moment of the exchange hole function is given at point by
where . This depends on a model of the exchange hole, and the implementation in Q-Chem uses the Becke-Roussel (BR) model.73 In most implementations the BR model, 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,584 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 coefficient, and its implementation in Q-Chem is denoted as “XDM6”. In this version the dispersion energy is
where is a universal parameter, and is the sum of the absolute values of the correlation energies of the free atoms and . The dispersion coefficients is computed according to
where is the square of the exchange-hole dipole moment of atom , whose effective polarizability (in the molecule) is .
The XDM6 scheme can be further generalized to include higher-order dispersion coefficients, which leads to the “XDM10” model in Q-Chem:
The higher-order dispersion coefficients are computed using higher-order multipole moments of the exchange hole.534 The quantity is the sum of the effective van der Waals radii of atoms and ,
with a critical distance
XDM10 contains two universal parameters, and , whose default values of 0.83 and 1.35, respectively, were fit to reproduce intermolecular interaction energies.533 Becke later suggested several other XC functional combinations with XDM, which employ different values of and . The user is advised to consult the recent literature for details.72, 548
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.
$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.584 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. 548 and 846 for some recent choices in this regard.