Inferring the one‐electron reduced density matrix of molecular crystals from experimental data sets through semidefinite programming

• Published in: Acta crystallographica. Section A, Foundations and advances Foundations and advances, 2020-01, Vol.76 (1), p.1-6
• Main Authors: De Bruyne, Benjamin, Gillet, Jean-Michel
• Format: Article
• Language: English
• Subjects: Compton diffusion; Condensed Matter; Crystals; Density; Dry ice; Electrons; Materials Science; Mathematical analysis; Mathematical models; Matrix methods; one‐electron reduced density matrix; Physics; Semidefinite programming; X‐ray diffraction
• ISSN: 2053-2733; 0108-7673; 2053-2733
• DOI: 10.1107/S2053273319015870

Constructing a quantum description of crystals from scattering experiments is of great significance to explain their macroscopic properties and to evaluate the pertinence of theoretical ab initio models. While reconstruction methods of the one‐electron reduced density matrix have already been proposed, they are usually tied to strong assumptions that limit and may introduce bias in the model. The goal of this paper is to infer a one‐electron reduced density matrix (1‐RDM) with minimal assumptions. It has been found that the mathematical framework of semidefinite programming can achieve this goal. Additionally, it conveniently addresses the nontrivial constraints on the 1‐RDM which were major hindrances for the existing models. The framework established in this work can be used as a reference to interpret experimental results. This method has been applied to the crystal of dry ice and provides very satisfactory results when compared with periodic ab initio calculations. In the inference process of finding the one‐electron reduced density matrix (1‐RDM) of a molecular crystal from diffraction and Compton diffusion scattering data, semidefinite programming can take into account the N‐representability conditions on the 1‐RDM.