BOSSANOVA: A bond order dissection approach for efficient electronic structure calculations

Abstract

In this article, we present a new decomposition approach for the eff cient approximate calculation of the electronic structure problem for molecules. It is based on a dimension-wise decomposition of the space the underlying Schrödinger equation lives in, i.e. &#8477;^{3(<em>M</em>+<em>N</em>)}, where <em>M</em> is the number of nuclei and <em>N</em> is the number of electrons. This decomposition is similar to the ANOVA-approach (analysis of variance) which is well-known in statistics. It represents the energy as a f nite sum of contributions which depend on the positions of single nuclei, of pairs of nuclei, of triples of nuclei, and so on. Under the assumption of locality of electronic wave functions, the higher order terms in this expansion decay rapidly and may therefore be omitted. Furthermore, additional terms are eliminated according to the bonding structure of the molecule. This way, only the calculation of the electronic structure of local parts, i.e. small subsystems of the overall system (plus some additional saturation with hydrogen) is necessary to approximate the total ground state energy. To determine the necessary subsystems, we employ molecular graph theory combined with molecular binding knowledge. In principle, the local electronic subproblems may be approximately evaluated with whatever technique is appropriate, e.g. DFT, CC or CI. From these local energies, the total energy of the overall system is then approximately put together in a telescope-like fashion. Thus, if the size of the local subproblems is independent of the size of the overall molecular system, linear scaling is directly obtained. As the size of each subproblem depends on the bond coordination of the involved atoms, we coined the method BOSSANOVA (Bond Order diSSection ANOVA). We discuss the details of our new approach and apply it – based on state-of-the-art graph algorithms – to various test systems and to C- and BN-nanotube structures.