Chemical Science View Article Online MINIREVIEW View Journal | View Issue Quantum crystallography† Simon Grabowsky,*a Alessandro Genoni*bc and Hans-Beat Burgi*de Cite this: Chem. Sci.,2017,8,4159 ¨ Approximate wavefunctions can be improved by constraining them to reproduce observations derived from diffraction and scattering experiments. Conversely, charge density models, incorporating electron-density distributions, atomic positions and atomic motion, can be improved by supplementing diffraction Received 16th December 2016 experiments with quantum chemically calculated, tailor-made electron densities (form factors). In both Accepted 3rd March 2017 cases quantum chemistry and diffraction/scattering experiments are combined into a single, integrated DOI: 10.1039/c6sc05504d tool. The development of quantum crystallographic research is reviewed. Some results obtained by rsc.li/chemical-science quantum crystallography illustrate the potential and limitations of this field. 1. Introduction today's organic, inorganic and physical chemistry. These tools are usually employed separately. Diffraction and scattering Quantum chemistry methods and crystal structure determina- experiments provide structures at the atomic scale, while the Creative Commons Attribution 3.0 Unported Licence. tion are highly developed research tools, indispensable in techniques of quantum chemistry provide wavefunctions and aUniversitat¨ Bremen, Fachbereich 2 – Biologie/Chemie, Institut fur¨ Anorganische eUniversitat¨ Zurich,¨ Institut fur¨ Chemie, Winterthurerstrasse 190, CH-8057 Zurich,¨ Chemie und Kristallographie, Leobener Str. NW2, 28359 Bremen, Germany. Switzerland E-mail: [email protected] † Dedicated to Prof. Louis J. Massa on the occasion of his 75th birthday and to bCNRS, Laboratoire SRSMC, UMR 7565, Vandoeuvre-l`es-Nancy, F-54506, France Prof. Dylan Jayatilaka on the occasion of his 50th birthday in recognition of their cUniversit´e de Lorraine, Laboratoire SRSMC, UMR 7565, Vandoeuvre-l`es-Nancy, F- pioneering contributions to the eld of X-ray quantum crystallography. 54506, France. E-mail: [email protected] This article is licensed under a dUniversitat¨ Bern, Departement fur¨ Chemie und Biochemie, Freiestr. 3, CH-3012 Bern, Switzerland. E-mail: [email protected] Open Access Article. Published on 27 March 2017. Downloaded 9/28/2021 4:47:23 PM. Simon Grabowsky studied Alessandro Genoni has been chemistry at Free University of CNRS Researcher at the Labo- Berlin and received his doctoral ratory “Structure and Reactivity degree from the same institution of Complex Molecular Systems” in 2010 before he went to the (SRSMC) of the University of University of Western Australia Lorraine (France) since 2011. (UWA) in Perth for a post- Aer earning his PhD in Chem- doctoral stay. He became Assis- ical Sciences from the University tant Professor at UWA in early of Milan (Italy) in 2006 (Maur- 2014, but le later in the same izio Sironi's group), he worked year in order to take on an as post-doctoral fellow in the Emmy Noether fellowship of the group of Kenneth M. Merz Jr at German Research Foundation the Quantum Theory Project of (DFG) which allows him to be head of a research group at the the University of Florida (USA) and in the group of Giorgio University of Bremen, Germany. In 2015 he received the title Colombo at the ICRM Institute of the Italian CNR (Milan, Italy). “Professor” from the University of Bremen for the duration of the His current research mainly focuses on the extension of Jayatila- fellowship. His group's major interests lie in the principles of ka's X-ray constrained wavefunction tting approach and in the chemical bonding in inorganic molecular chemistry related to development of novel quantum chemistry-based approaches to method development in physical chemistry. The group applies rene charge densities of macromolecules. spectroscopic, crystallographic and theoretical methods of struc- ture determination and advances them, especially quantum crystallography. This journal is © The Royal Society of Chemistry 2017 Chem. Sci.,2017,8,4159–4176 | 4159 View Article Online Chemical Science Minireview Fig. 1 Quantum crystallography, first aspect: crystallographic data are Fig. 2 Quantum crystallography, second aspect: quantum chemical integrated into quantum chemical calculations to enhance the infor- calculations are integrated into crystal structure determination to mation content of the wavefunction. The resulting, so-called “exper- improve the dynamic charge density, i.e. the thermally smeared imental wavefunction” represents an improved approximation to the electron and nuclear densities. true wavefunction. obtained from diffraction experiments by combining them in properties derived from them. In this contribution we review a self-consistent way with information from quantum chemical different efforts towards combining tools of these two elds into calculations. integrated quantum crystallographic strategies. The term quantum crystallography was rst introduced by Massa, Huang and Karle in 1995 for methods that exploit 2. Quantum crystallography, first “crystallographic information to enhance quantum mechanical definition: enhancing quantum calculations and the information derived from them”.1 The basic idea is to compensate shortcomings and limitations of chemical calculations with quantum mechanical models, e.g. incomplete consideration of Creative Commons Attribution 3.0 Unported Licence. experimental information electron correlation, with experimental data that are not 2.1 Pioneering “experimental” wavefunction techniques suffering from the same limitations (Fig. 1). In 1999 the same authors also suggested the converse possibility: “. quantum All modern quantum crystallography techniques according to the “ ” mechanics . can greatly enhance the information available rst de nition originate from the pioneering experimental from a crystallographic experiment” (Fig. 2).2 wavefunction strategies originally proposed in the 1960s by 4 – Mutually subsidiary combinations of quantum chemistry Mukherji and Karplus. They perturbed unconstrained Hartree and X-ray structure determination suggest themselves.3 Fock molecular orbitals until they produced satisfactory agree- Quantum chemical models are usually based on some approx- ment with experimental dipole moments or electric eld gradi- This article is licensed under a imations of the true wavefunction. X-ray structure determina- ents at a minimal increase in the energy of the system. tion aims at the true charge density, which is related to the Agreement improved not only for the constraining experimental square of the true wavefunction, but is affected to a smaller or values, but also for other properties, such as diamagnetic and paramagnetic susceptibilities. Rasiel and Whitman5 introduced Open Access Article. Published on 27 March 2017. Downloaded 9/28/2021 4:47:23 PM. larger extent by vibrational motion and experimental errors. ‡ Here the topic of quantum crystallography is presented in experimental dipole moment constraints into the variational two parts. The rst part summarizes the development of minimization of the energy with Lagrange multipliers. Byers 6 increasingly sophisticated methods to combine information Brown and Chong proposed to implicitly introduce the experi- ff mental constraints by adding proper quantum mechanical from quantum chemical calculations with di raction and other ^ operators Oi (multiplied by a Lagrange multiplier li)tothe experimental data (see Fig. 3 for a summarizing scheme). The ^ second part describes ways of improving structural models starting and unconstrained Hamiltonian H: X ^0 ^ ^ H ¼ H þ liOi (1) i Hans-Beat Burgi¨ is Professor emeritus at the Department of The modied Schrodinger¨ equation can be solved varia- Chemistry and Biochemistry of tionally as shown by Fraga and Birss.7 the University of Berne, Switzer- In 1969 Clinton et al. published a series of groundbreaking land. He happily enjoys his ideas that may be considered the direct precursors of today's retirement at the Universities of – quantum crystallography methods.8 11 Although X-ray diffrac- Zurich,¨ Western Australia and tion data were not considered explicitly, the proposed theoret- Berkeley where he collaborates ical framework forms the basis for the methods developed in with his colleagues and friends in the 1970s and 1980s that combine quantum mechanical areas of common interest: struc- ture determination of disordered ‡ In all reviewed methods according to the rst denition, the authors refer to the materials from diffuse scattering, use of external parameters as “constraints”. Hence, we will use this term quantum crystallography, throughout Chapter 2, although we believe that, in a crystallographic sense, the dynamics and thermodynamics of crystalline polymorphs. term “restraints” would be more appropriate. 4160 | Chem. Sci.,2017,8,4159–4176 This journal is © The Royal Society of Chemistry 2017 View Article Online Minireview Chemical Science Fig. 3 Scheme summarizing the features (framed in red) of the main methods (framed in light blue) according to the first definition of quantum crystallography. The lower the position of the method is in the scheme, the lower is the quantum chemistry contribution in it. Each family of techniques is associated with the corresponding bibliographical references in the paper. Creative Commons Attribution 3.0 Unported Licence. calculations with crystallographic data. In the rst paper of the idempotency condition (i.e., P2 ¼