GEBF Fragmentation Methods

GEBF07 Fragmentation Method

With the original GEBF fragmentation method, Li et al.59 presented a fragmentation method. Using our terminology, this method is to:

  1. Define pseudoatoms

    • In the orignal presentation no general criteria was specified; however, it is assumed that for molecular clusters pseudoatoms are individual molecules.

  2. For each pseudoatom, define a fragment which is the pseudoatom and all pseudoatoms close to it.

    • The original method uses two criteria for “close”, one for covalent systems and one for molecular clusters.

    • For covalent systems two pseudoatoms are close if they are bonded either covalentally or with a hydrogen bond

    • For clusters pseudoatoms within some distance \(\zeta\) are considered close (default is 3 Angstroms)

  3. Remove fragments which are subsets of other fragments

  4. Determine intersections

Of note an algorithm for assigning multiplicities to intersections was also presented namely:

  • Assume largest fragment contains \(n\) pseudoatoms.

  • Look for a set of \(n-1\) pseudoatoms which appears in more than one fragment. The multiplicity of this intersection is \(1-k\), where :math`k` is the number of fragments it appears in.

  • Repeat the previous step for sets of :math`n-2, n-3, ldots, 1` pseudoatoms

GEBF08 Fragmentation Method

Hua et al.33 introduced a new fragmentation method for the GEBF method:

  1. Define \(\eta\), the maximum number of pseudoatoms in a fragment.

  2. Form pseudoatoms.

  3. For each pseudoatom combine it with the \(\left(\eta -1\right)\) closest pseudoatoms to form a fragment.

GEBF10 Fragmentation Method

Hua et al.30 introduced a new fragmention method for the GEBF method. The first steps are just the GEBF07 method (minus the distinction between molecular clusters and covalent systems):

  1. Define a distance threshold \(\zeta\)

  2. Determine pseudoatoms, the \(i\)-th pseudoatom is denoted \(P_i\)

  3. The \(i\)-th fragment, \(F_i\), include \(P_i\) and all pseudoatoms within \(\zeta\) of it.

The remainder of the algorithm amounts to “extension rules” which extend the fragments:

  1. For each fragment check for incomplete rings. If the fragment contains an incomplete ring, complete it by adding the pseudoatoms spanning the rest of the ring. If the resulting structure still has incomplete rings repeat this step one more time. Do not repeat the step beyond the second time.

  2. For each fragment consider pseudoatoms with dangling bonds. Say \(P_i\) has dangling bonds, then if \(P_i\) contains three or more heavy atoms, or if \(P_i\) was added by the previous step ignore it. If \(P_i\) was not ignored then add to the fragment each pseudoatom bonded to \(P_i\). If \(P_i\) contains only 1 heavy atom then for each pseudoatom bonded to \(P_i\), \(P_j\), additionaly add the pseudoatoms bonded to \(P_j\).

At this point caps are added (Hua et al.30 use Standard Distance).

GEBF12 Fragmentation Method

Hua et al.31 built on the GEBF10 Fragmentation method to create the GEBF12 fragmentation method.

  1. Create fragments subject to the GEBF10 Fragmentation Method.

  2. Find the pairs of pseudoatoms, \(P_i\) and \(P_j\), separated by a distance less than \(2\zeta\).

  3. For each pair from step 2, determine if a pair from step 1 contains the pair, if not create a new fragment which contains the \(P_i\) and \(P_j\) and all other pseudoatoms which are within \(\zeta\) of the geometric center of \(P_i\) and \(P_j\).

  4. Apply extension rules (see GEBF10 Fragmentation Method) to each fragment formed in step 3.

GEBF14 Fragmentation Method

Wang et al.89 combined the GEBF08 and GEBF12 fragmenation methods to create the GEBF14 method. In their presentation, Wang et al.89 did not discuss the extension rules; however, because the application was water clusters, the extension rules were not relevant. For clarity the GEBF14 procedure is:

  1. Define a distance threshold \(\zeta\)

  2. Define a maximum number of pseudoatoms \(\eta\).

  3. Determine pseudoatoms, the \(i\)-th pseudoatom is denoted \(P_i\)

  4. The \(i\)-th fragment, \(F_i\), include \(P_i\) and all pseudoatoms within \(\zeta\) of it. If more than \(\eta\) pseudoatoms fall within a radius of \(\zeta\) only the \(\eta\) closest to \(P_i\) are included.

  5. Presumably extension rules are applied here.

  6. Find the pairs of pseudoatoms, \(P_i\) and \(P_j\), separated by a distance less than \(2\zeta\).

  7. For each pair from step 2, determine if a pair from step 1 contains the pair, if not create a new fragment which contains the \(P_i\) and \(P_j\) and all other pseudoatoms which are within \(\zeta\) of the geometric center of \(P_i\) and \(P_j\). If more than \(\eta\) pseudoatoms fall within a radius of \(\zeta\) only the \(\eta\) closest to the geometric center of \(P_i\) and \(P_j\) are included.

  8. Apply extension rules (see GEBF10 Fragmentation Method) to each fragment formed in step 3.

GEBF19 Fragmentation Method

Li et al.56 proposed a new fragmentation method designed to produce smaller fragments then GEBF14 when applied to host/guest complexes.

  1. Define a distance threshold \(\zeta\)

  2. Define a mximum number of pseudoatoms \(\eta\).

  3. Determine pseudoatoms, the \(i\)-th pseudoatom is denoted \(P_i\).

  4. For \(P_i\) form a fragment, \(F_i\), which contains \(P_i\) and all pseudoatoms that are within \(\zeta\) of \(P_i\). \(P_i\) is considered colored in \(F_i\).

  5. For each fragment \(F_i\) with more pseudoatoms than \(\eta\), create new fragments for each un-colored pseudoatom \(P_j\) in \(F_i\). The fragment resulting from \(P_j\) is \(F_j\) and includes \(P_j\), the colored pseudoatoms, and the un-colored pseudoatoms in \(F_i\) which are within \(\zeta\) of \(P_j\). \(P_j\) is considered c olored in \(F_j\). Discard \(F_i\).

  6. Repeat the previous step until no fragments contain more pseudoatoms than \(\eta\) or all pseudoatoms have been colored.