The volume exclusion term comes in two versions, an exclusion list version (this version nbn-) and an inclusion list version (see nbn+ above). In the exclusion list version, every pair of atoms is assumed to have a volume-exclusion interaction unless a pair is explicitly excluded. In the inclusion list version, every atom pair that has a volume-exclusion interaction is explicitly listed. The nbn- term is a good choice if the majority of atom pairs have some kind of volume-exclusion interaction; using the nbn+ term would require a very long inclusion list. The nbn+ term is a better choice if there are few volume-exclusion interactions; using the nbn- term would require a very long exclusion list in this case. Only one version or the other should be used in a descriptor; it is unlikely to be useful (but it is not impossible) to use both versions in a single descriptor.

For a system containing N_atom atoms, the nbn- energy is the sum of a maximum of N_atom(N_atom-1)/2 terms where each term has the form:

knb ( dij - do )n, if dij < do

0, everywhere else

---

where n is an integer greater than one, k_nb is a force constant, d_ij is the current distance between atom_i and atom_j and d_o is the exclusion radius for this pair of atoms. The values of k_nb and d_o depend on the atom types of atom_i and atom_j (and therefore a type record must also be provided). When the pair of atoms are more than d_o apart, there is no interaction. The interaction comes into play only when the atoms are closer together than d_o.

Although there are N_atom(N_atom-1)/2 possible atom pairs, not all pair interactions are evaluated. Atom pairs can be explicitly excluded in an exclusion list. Interactions are evaluated from a list containing pairs of atoms that are not in the exclusion list and which are within a distance smaller than r_cutoff d_o where the cutoff ratio r_cutoff is a number greater than one. This list is updated at regular intervals. In between updates, some of the candidate pairs move apart while other pairs that are not in the candidate list move closer together. The list update interval and the value of r_cutoff are user controlled and have to be chosen so that r_cutoff is not too large (which requires more calculations) and the update interval is not too frequent but is frequently enough that the candidate list is always accurate.

The nbn- record in the descriptor file has the form

nbn-	Nexclude	n	Ntype	rcutoff
	k11	k12	...
		k22	k23	...
			...
	d11	d12	...
		d22	d23	...
			...
	atomi	Nij	atomij	...
	...	...	...	...

The record starts with the keyword nbn- followed by the number of exclusion lists N_exclude, the power n, the number of atom types N_type and the cutoff ratio r_cutoff. N_type must be at least as large as the largest atom type number in the type record. Following the header line are N_type(N_type+1)/2 force constants k₁₁, k₁₂, k₂₂, ... for interactions between all possible pairs of different atom types (not atom numbers). Since k₂₁ has the same values as k₁₂, only one of these values need to be specified, and these are listed in the order shown above. The values for the dissimilar atom type pairs must be listed because mimesis does not have any built-in combination rules. This also means the user has the flexibility to use any combination rule.

Following the force constants are the same number of exclusion distances, d₁₁, d₁₂, d₂₂, ...

Finally, N_exclude exclusion lists finishes off the record. Each exclusion list consists of an atom number atom_i, the number of atoms that are to be excluded from interacting with this atom N_ij and then the N_ij atom numbers atom_ij, ... The exclusion lists follow a strict rule. The atom numbers must be strictly increasing. This means that the atom number that begins each exclusion list atom_i must be greater (not equal and not less) than the previous atom_i. Then, atom_ij must be larger than atom_i and within the list atom_ij each atom number must be larger than the previous one.

Example

nbn-   3   2   3   1.5

20000.0   24490.0   22360.0
          30000.0   27390.0
                    25000.0
30.0   20.0   32.5
       10.0   22.5
              35.0
1   3
    2   3   4
2   2
    3   4
3   1
    4

There are three exclusion lists and three atom types. A power of 2 is chosen and the cutoff ratio is 1.5. With three atom types, six force constants and six exclusion distances are required. Following the header line are the force constants; notice that the constants for the dissimilar atom type pairs (the off-diagonal elements) are derived from a geometric average of the constants for the pure atom types (the diagonal elements). On the other hand, the six exclusion distances that follow are derived from an arithmetic averaging combination rule.