Background information

BOSSANOVA or a linear-scaling fragmentation scheme

Here, we give some back ground information on the linear-scaling fragmentation scheme, coined Bond Order diSSection in an ANOVA-like fashion (ANOVA). The idea is based on the many-body expansion of the Born-Oppenheimer surface. In my ​doctoral thesis (german) you find extensive derivations on how and why this works for the Hartree-Fock equations and some notes on Post-Hartree-Fock methods such as Moeller-Plesset second order.

Basic idea

Essentially, the idea is to split up a molecular system into many overlapping so-called fragments. Ground-state energy calculation on each fragment is fast and if their number scales only linearly with the overall system size, we automatically obtain a linear-scaling scheme.

In order to just obtain a linear scaling number (and not the full power set of 2^M possible fragments), we make use of the bond graph by constraining the admissible fragments to those whose associated subgraph in this bond graph is connected, i.e. there is a path from each atom to another one by jumping along the (covalent) bonds.

The essential trick then is how to sum up the fragment energies to obtain a good approximation of the ground-state energy of the full molecular system. Naive summation of all does not do this trick, as the fragments overlap and the energy sum is far too large. Instead, the set of index sets of these fragments has to fulfill a certain property. In the above thesis, it is called hierarchical. It means that the intersection of any two subsets of the set of index sets must also be a set in this set of index sets.

What's the use?

MoleCuilder uses BOSSANOVA for many things:

• linear-scaling ab-initio molecular dynamics (combining ground-state energy calculations with derivatives with respect to nuclear coordinates and time integration in discrete steps)
• fitting empirical potential functions (sort of caching the ground-state energy of specific fragments in cleverly guessed functions to save computation time and to allow neglect of explicit electron computations)
• fitting particle nuclei charges (by making partial charges mimic the long-range potential of the fragments and the molecular system as a whole)

Compare against experimental value

In the experimental section of above thesis, many test calculations have been performed to assess speed and accuracy of the BOSSANOVA scheme. In order to let you compare against the values found therein, you can download an archive below, attached to this TRAC page, with all the molecular configurations employed in this experimental section.

Have a look at the UseCases to see how calculation proceeds and read the experimental section, especially the scaling part, where each of the sequential algorithmic is checked to be linear-scaling.