Order-Theoretic Combination Techniques and the Electronic Schrödinger Equation

Abstract

Most standard constructions of the <em>combination technique</em> [M. Griebel et al., <em>Iterative Methods in Linear Algebra</em>, Elsevier, North Holland, p. 263] manipulate families of functions organised by downward-closed subsets of N^d. We introduce instead an alternative formulation, with functions indexed from a more general kind of partially ordered set (poset). The combinatorial and order-theoretic machinery of <em>Möbius inversion</em> helps us to construct combination sums of functions organised by <em>order ideals</em> of a <em>poset grid</em>. An adaptive algorithm is given for the quasi-optimal assembly of such an order ideal. This <em>order-theoretic combination technique</em> (OTCT) formalism is applied in the quantum-chemical setting of the high-dimensional <em>electronic Schrödinger equation</em>. Here, the OTCT allows us to connect, understand, and improve on a number of existing approaches.

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We consider first a selection of existing extrapolative <em>composite methods</em>. Extending on the idea of the CQML approach [P. Zaspel et al., <em>J. Chem. Theory Comput.</em>, 15(3), 2018], an application of basically just the standard version of the combination technique leads to a <em>generalised composite method</em> (GCM). This approach is systematically improvable and appears comparable, if not yet truly competitive with standard composite methods from the perspectives of both accuracy and of cost.

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We turn then to <em>energy-based fragmentation methods</em>, which are often founded upon a truncated <em>many-body expansion</em> (MBE). It is well-known that Möbius inversion can provide non-recursive expressions for the individual MBE terms, and so the OTCT delivers by construction a framework for the adaptive truncation of MBE-like formulae. The same also functions for a class of related graph-based decompositions described in the existing literature. We term these in our context as SUPANOVA (<em>SUbgraph Poset ANOVA</em>) decompositions, and motivate them as extensions to the BOSSANOVA decomposition [M. Griebel et al., <em>Extraction of Quantifiable Information from Complex Systems</em>, Springer, Cham, 2014, p. 211; F. Heber, Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, 2014]. We identify a subtle technical issue that afflicts BOSSANOVA in certain cases, and apply instead an adaptive SUPANOVA decomposition defined for <em>convex subgraphs</em>.

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Finally, we combine the GCM and SUPANOVA ideas to obtain a poset grid that recovers many existing <em>multilevel fragmentation methods</em>. We extend the ML-BOSSANOVA method [S. R. Chinnamsetty et al., <em>Multiscale Model. Simul.</em>, 16(2), 2018], exploring now also a hierarchy of <em>ab initio</em> theories. Although an initial assessment is inconclusive, this ML-SUPANOVA formulation appears well-founded and paves the way to a number of interesting possible applications in the future.