Multi-center decomposition of molecular densities: a mathematical perspective

The aim of this paper is to analyze from a mathematical perspective some existing schemes to partition a molecular density into several atomic contributions, with a specific focus on Iterative Stockholder Atom (ISA) methods. We provide a unified mathematical framework to describe the latter family of methods and propose a new scheme, named L-ISA (for linear approximation of ISA). We prove several important mathematical properties of the ISA and L-ISA minimization problems and show that the so-called ISA algorithms can be viewed as alternating minimization schemes, which in turn enables us to obtain new convergence results for these numerical methods. Specific mathematical properties of the ISA decomposition for diatomic systems are also presented. We also review the basis-space oriented Distributed Multipole Analysis method, the mathematical formulation of which is also clarified. Different schemes are numerically compared on different molecules and we discuss the advantages and drawbacks of each approach.