The Cluster

Unlike more conventional methods, MOPAC does not normally use a fundamental unit cell. Neither does it sample the Brillouin Zone in order to model the electronic structure. Instead, it uses a large unit cell, called a 'cluster', and applies the Born-von Kármán  [68,69] periodic boundary conditions. In this discussion, the term 'solid' is intended to include polymers, layer systems, and true solids, unless otherwise indicated by the context.

If a unit cell of a solid is large enough, then a single point in k-space, the Γ point, is sufficient to specify the entire Brillouin zone. The secular determinant for this point can be constructed by adding together the Fock matrix for the central unit cell plus those for the adjacent unit cells. The periodic boundary conditions are satisfied, and diagonalization yields the correct density matrix for the Γ point.

At this point in the calculation, conventionally, the density matrix for each unit cell is constructed. Instead, the Γ-point density and one-electron density matrices are combined with "Γ-point-like" Coulomb and exchange integral strings to produce a new Fock matrix. The calculation can be visualized as being done entirely in reciprocal space, at the Γ point.

Most solid-state calculations take a very long time. These calculations, called "Cluster" calculations after the original publication, require between 1.3 and 2 times the equivalent molecular calculation. 

A minor 'fudge' is necessary to make this method work. The contribution to the Fock matrix element arising from the exchange integral between an atomic orbital and all atomic orbitals which are more than half a unit cell away must be ignored.

The unit cell must be large enough that an atomic orbital in the center of the unit cell has an insignificant overlap with the atomic orbitals at the ends of the unit cell. In practice, a translation vector of more that about 7 or 8 Ångstrom is sufficient. For one rare group of compounds a larger translation vector is needed. Solids with delocalized π-systems, and solids with very small band-gaps will require a larger translation vector, in order to accurately sample k-space. For these systems, a translation vector in the order of 15-20 Ångstroms is needed.