Efficient computation of the dispersion interaction with density-functional theory

One of the major deficiencies of the standard density functionals is their inability to describe dispersion interactions. Becke and Johnson recently proposed a conceptually simple yet accurate dispersion model called the exchange-dipole moment (XDM) model, which allows the calculation of both intermolecular and intramolecular dispersion interactions with density functional theory (DFT). In this paper, we present an efficient self-consistent-field (SCF) solution of the XDM model. We also give detailed analysis of the post-SCF approach in which the dispersion term is added to the Hamiltonian as a perturbation, and show that it has an error on the order of ${10}^{-5}$ in density matrix due to the addition of dispersion to the Hamiltonian. In addition, for gradient calculations with respect to the atomic movement, we introduce a further approximation in which the electronic part of the XDM gradient formula is omitted, and show that it yields an error smaller than ${10}^{-3}\text{a}\text{.u}\text{.}$ These approximations offer a simple and efficient route with good precision for the implementation of XDM model and other dispersion models into existing DFT codes. The effectiveness of our implementation is demonstrated through several examples. The first example shows that inclusion of the XDM model leads to much more accurate prediction of enthalpies of formation of straight-chain alkanes than does the Becke three-parameter Lee-Yang-Parr (B3LYP) functional. The inclusion of the dispersion is also shown to improve the accuracy in the calculation of isomerization energies and bond-dissociation energies of alkanes. The last example shows that the qualitative difference in optimal geometries of a tyrosine-glycine peptide, calculated using the B3LYP functional and the second-order Møller-Plesset method, essentially disappears when XDM is added to the DFT calculation.

Figure 1

(Color online) Optimized structures of the methane cluster and the water-methane clusters.

Figure 2

(Color online) Plots of deviations from experimental formation enthalpies of straight-chain alkanes obtained at G3, B3LYP, $\text{B}3\text{LYP}+\text{XDM}6$, and $\text{B}3\text{LYP}+\text{XDM}10$ levels of theory.

Figure 3

The structures of the linear and the branched conformers of octane.

Figure 4

(Color online) Structures of the optimal geometries of the tyrosine-glycine peptide calculated using the $6\text{−}31+{\text{G}}^{\ast }$ basis set. The calculations are based on (a) B3LYP, (b) MP2, and (c) $\text{B}3\text{LYP}+\text{XDM}6$. The numbers indicate the distance between oxygen of tyrosine and hydrogen of glycine, measured in angstroms.