Abstract
In this article, we present a new decomposition approach for the efficient 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. \(\mathbb{R}^{3(M+N)}\), where M is the number of nuclei and N 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 finite 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, is necessary to approximate the total ground state energy. To determine the required subsystems, we employ molecular graph theory combined with molecular bonding knowledge. In principle, the local electronic subproblems may be approximately evaluated with whatever technique is appropriate, e.g. HF, CC, CI, or DFT. From these local energies, the total energy of the overall system is then approximately put together in a telescoping sum 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. We discuss the details of our new approach and apply it to both, various small test systems and interferon alpha as an example of a large biomolecule.
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Notes
- 1.
Ultimately, the aim would be a decomposition of \(\mathbb{R}^{3(M+N)}\), the space where the full Schrödinger equation lives in.
- 2.
Note however that our approach should work equally well also in the non-charge neutral case.
- 3.
The ratio of the velocity v K of a nucleus to the velocity of an electron v e is in general smaller than 10−2.
- 4.
This excludes in general metallic systems, whose electrons may be delocalized due to a vanishing band gap.
- 5.
Furthermore, the notion of the locality of the wave function is important as it leads to the general chemical understanding of molecules from the general bond structure up to nucleophilic sites.
- 6.
The ANOVA decomposition of a M-dimensional function \(f: [0, 1]^{M} \rightarrow \mathbb{R}\) reads \(f =\sum _{u\subseteq \{1,\ldots,M\}}f_{u}\) with f u depending only on the variables indicated in u. The functions f u satisfy the recurrence relation \(f_{\varnothing } = L_{\{1,\ldots,M\}}(f)\), \(f_{u} = L_{\{1,\ldots,M\}/u}(f) -\sum _{v\subset u}f_{v}\) with \(L_{w}(f) =\int _{[0,1]^{\vert w\vert }}f(x_{1},\ldots,x_{M})\,dx_{w}\). Thus, f is decomposed into a constant, a sum of one-dimensional functions, a sum of two-dimensional functions, and so on. The involved functions are generated by proper partial integration and telescopic corrections according to the recurrence relation.
- 7.
Note that, in practice, the global electronic problem is only solved approximately anyway, by e.g. DFT, CC, CI.
- 8.
As can be seen from Fig. 11.4, the fragmentation times are negligible.
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This research was funded by the Deutsche Forschungsgemeinschaft (DFG) within the framework of the priority program SPP1324.
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Griebel, M., Hamaekers, J., Heber, F. (2014). A Bond Order Dissection ANOVA Approach for Efficient Electronic Structure Calculations. In: Dahlke, S., et al. Extraction of Quantifiable Information from Complex Systems. Lecture Notes in Computational Science and Engineering, vol 102. Springer, Cham. http://doi.org/10.1007/978-3-319-08159-5_11
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