Distance Based Screening Procedures

Weak interactions die off quickly with respect to the separation distance between the interacting units. Perhaps unsurprisingly a number of groups have proposed screening methods which are distance based.

FMO00 Screening

The orginal FMO procedure Nakano et al.⁶⁹ for screening dimers involved distance cutoffs. If the shortest inter-atom distance between two monomers \(I\) and \(J\) is less than a threshold (default is 10 Bohr), then the dimer energy is approximated by:

\[ \begin{align}\begin{aligned}\newcommand{\vr}{\vec{r}} \newcommand{\vrp}{\vec{r}'} \newcommand{\density}[2]{rho_{#1}\left(#2\right)}\\E_{IJ} = E_{I} + E_{J} - \int d\vr\int\vrp \frac{\density{I}{\vr}\density{J}{\vrp}} {\mid \vr -\vrp\mid}\end{aligned}\end{align} \]

FMO04 Screening

In the original formulation of three-body FMO Fedorov and Kitaura¹⁷, the FMO00 screening procedure is extended to three-body terms. Instead of one shortest inter-atomic distance we now have three, one for each pair of monomers. The FMO04 screening method screens out a trimer if the second smallest distance is greater than a threshold (default is 2.0, in units of van der Waal radii).

EC-MFCC Screening

As part of the EC-MFCC method, Li et al.⁵¹ introduced a screening method for pairs which relies on two thresholds. The first threshold, :math`R_1` is used for screening van der Waals interactions, whereas the second threshold, :math`R_2`, is used for screening electrostatics. Exactly how these cut-offs are used is a bit unclear from the manuscript. More specifically, the manuscript mentions that :math`R_2` is only used for charged molecules; however, since :math`R_1` and :math`R_2` are specified per molecule it is not clear why one would specify both :math`R_1` and :math`R_2` for a charged molecule if only :math`R_2` is used. Our guess is that :math`R_1` is used when the fragments in the pair are neutral and :math`R_2` is used when one or both fragmnts are charged.

SMF06 Screening

Collins and Deev⁶ introduced a screening method which relies on a distance cut-off, \(d\), to separate the dimers into two groups. Defining \(f_{ij}\) to be the shortest atom-atom distance between an atom in fragment \(i\) and an atom in fragment \(j\), the first group of dimers are those for which \(f_{ij}\) is less than or equal to \(d\). The dimers in the first group are treated normally. The dimers in the second group are those for which \(f_{ij}\) is greater than \(d\). The interactions of the dimers in the second group are approximated with a multipole expansion. Based on the preliminary test set, Collins and Deev⁶ recommend a value of \(d\) around 3 Angstroms. Conceptually this is just the FMO00 Screening method with a multipole expansion.

The screening method introduced by Li et al.⁵⁹ is a variation on this which truncates the multipole expansion at point charges. Li et al. recommended a value of \(d\) of 10 Angstroms.

EE-MB-CE Screening

Proposed by Dahlke and Truhlar⁸, this screening method is the same as FMO00 screening, but uses the center of mass to determine the distance (not shortest inter-atomic distance). For water clusters the recommended cut-off was 5 or 6 Angstroms.