Supermolecular Density Functional Theory DFT

Dispersion corrections to DFT and HF (including semiempirical methods) have recently been reviewed [59, 60] and hence only a brief description is given. Asymptotically correct approaches are either atom-pair-based (e.g., DFT-D3 [42], XDM [61], or Tkatchenko-Scheffler (TS)-vdW [62]) or compute the dispersion energy from the electron density (called vdW-DF [25, 63]). For recent work on the 

In the general vdW-DF framework, such an NL correlation (dispersion) energy takes the form of a double-space integral 

Higher order dipole-quadrupole, quadrupole-quadrupole,. .. coefficients (i.e., Cg, CjQ,. ..) can also be computed by similar formulas [3]. Odd-order terms vanish for spherical systems (free atoms) but occur for nonspherical atoms as found in molecular environments. It is currently not clear how large the effect of including, for example, Cj really is [67], but so far very good results have been obtained by including even-order terms only. 

The Q coefficients (and derived Cg) in the D3 method have been computed using a modified form of this relation, where the a(ia ) are computed nonempirically by TDDFT and A and B are reference molecules from which atomic values are derived [42]. Because the reference system can also be a molecular cluster modeling a solid environment, special coefficients for atoms in the bulk can be derived [68]. The final form for the DFT-D3 two-body part of the dispersion energy employs the so-called Becke-Johnson (BJ) damping [61, 69] and truncates the expansion at Cg 

This form of the damping function ensures that for small interatomic distances the right constant limit for the energy correction is obtained [71] 

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