DispersionCorrected DFT with XDM
Authors: Kyle Bryenton and Erin Johnson
Date: 20241020
What is the XDM Dispersion Correction?
(Description adapted from the FHIaims manual)
The Exchangehole Dipole Moment (XDM) Model is a postSCF dispersion correction created by Drs. Erin Johnson and Axel Becke to calculate the dispersion energy in molecules and solids. The dispersion energy in XDM is calculated as a damped asymptotic expression, $$ E_{\text{disp}} = \sum_{i>j} \frac{C_{6,ij}f_6(R_{ij})}{R_{ij}^6} + \frac{C_{8,ij}f_8(R_{ij})}{R_{ij}^8} + \frac{C_{10,ij}f_{10}(R_{ij})}{R_{ij}^{10}} \,, $$ where \(i\) and \(j\) run over atoms, and \(R_{ij}\) is the interatomic distance. The \(C_{n}\)'s are the socalled dispersion coefficients. \(C_{6}\) accounts for dipoledipole interactions, \(C_{8}\) accounts for dipolequadrupole interactions, and \(C_{10}\) accounts for quadrupolequadrupole and dipoleoctupole interactions. The XDM contribution is used to correct the energy from the base density functional approximation for missing dispersion, where $$ E_{\text{total}} = E_{\text{base}} + E_{\text{DFT}} \,. $$ The dispersion coefficients are calculated from the selfconsistent density and kinetic energy density. For instance, the leading dispersion coefficients are $$ C_{6,ij} = \frac{\alpha_i\alpha_j\langle M_1^2\rangle_i \langle M_1^2\rangle_j}{\langle M_1^2\rangle_i\alpha_j + \langle M_1^2\rangle_j \alpha_i} $$ where \(\langle M_1^2\rangle_i\) are the multipole moments of the electron plus exchangehole dipole distribution and \(\alpha_i\) are the inmolecule atomic polarizabilities. The BeckeJohnson damping functions, \(f_n\) for \(n = 6,8,10\), contain two adjustable parameters \(a_1\) and \(a_2\) (in Å), which are determined for each basisfunctional combination with which XDM is coupled by fitting to a small set of molecular gasphase dimers, KB49. More details can be found in the original reference^{1}, the periodic XDM implementation^{2}, and recent reviews^{3}.
When Should You Use a Dispersion Correction?
It makes more sense to ask "When should you not use a dispersion correction?" In that case:
 If your base functional already accounts for dispersion
 If your system has less than 2 atoms
 If you know you're already overbinding, and you're hoping to cancel that error by neglecting dispersion rather than getting the physics correct
Dispersion is a real force and not accounting for it means you're missing physics in your model. PostSCF dispersion corrections require very little computational time compared to the cost of the SCF, so they're almost "free accuracy".
Tutorial Contents
This tutorial has five sections:
Theory
A discussion on dispersion, postSCF dispersion corrections, and the underlying theory behind the XDM dispersion model.
Usage
Usage instructions, keywords, aliases, and lists of supported functionals and basis sets.
Tutorial 1: Molecular Dimers
This tutorial walks through calculating the binding energy of a CO\(_2\) dimer. You run the singlepoint calculations twice, once with the XDM dispersion correction, and once without. The difference in binding energy between these two sets of calculations is shown to be significant, and the accuracy vastly improves when XDM is included. Further the additional computational cost of using XDM is shown to be near negligible.
Tutorial 2: Graphite Exfoliation
This tutorial manually scans the potential energy surface (PES) as we separate the layers of graphite from equilibrium, with and without the XDM dispersion correction. It is shown that neglecting dispersion in this case will predict zero binding energy between the layers (and thus your pencil lead would fall apart). Further, choosing an appropriate GGA functional paired with a highlyaccurate dispersion correction like XDM allows us to predict the binding energy and equilibrium separation at levels of accuracy comparable to QMC methods.
Tutorial 3: Fitting XDM for New Functional//Basis Combinations
This tutorial will teach the user how to generate the optimal XDM damping parameters for a functional//basis combination that is not natively supported by XDM. For this tutorial, we will use PBEsol0//lightdenser as our example case. The KB49 benchmark inputs are provided, as are the Octave fit scripts used to generate the damping coefficients.
References

Becke, A. D. & Johnson, E. R. Exchangehole dipole moment and the dispersion interaction revisited. J. Chem. Phys. 127, 154108. (2007) https://doi.org/10.1063/1.2795701 ↩

OterodelaRoza, A. & Johnson, E. R. Van der Waals interactions in solids using the exchangehole dipole moment. J. Chem. Phys. 136, 174109. (2012) https://doi.org/10.1063/1.4705760 ↩

Johnson, E. R. The ExchangeHole Dipole Moment Dispersion Model in Noncovalent Interactions in Quantum Chemistry and Physics: Theory and Applications, Edited by Alberto Otero de la Roza and Gino A DiLabio. Elsevier, 169–194. (2017) https://doi.org/10.1016/C20150063833 ↩