This book is dedicated to Jerome Karle (Nobel Laureate 1985) and Isabella Karle (US National Medal of Science 1995), important supporters of quantum crystallography.

Isabella and Jerome as graduate students, sometime during the World War II years, with their Ph.D. mentor Lawrence Brockway (left) (courtesy of the U.S. Naval Research Laboratory).

Acknowledgements

Chemists should naturally be the first and greatest appreciators. Research is appreciation. Willis R. Whitney (1921)

("The Biggest Things in Chemistry", Journal of Industrial & Engineering Chemistry, 13 (2), 161-166).

L. M. thanks his wife Mary for her constant support. C. F. M. is grateful to Maged Matta, Hebatallah Habib, and Sara and Nadine Matta. L. H. thanks her family.

The authors are indebted to too many people to mention here who have helped them to complete this work. Of those many people, the authors are particularly grateful to the following scholars for directly and personally influencing their thoughts over the years: William L. Clinton, Jerome and Isabella Karle, William N. Lipscomb Jr., Ada Yonath, Richard F. W. Bader, John C. Polanyi, Claude Lecomte, Russell J. Boyd, Philip Coppens, Todd A. Keith, André D. Bandrauk, Sason Shaik, Tina A. Harriott, Daniel Majaess, Paul W. Ayers, Leon Cohen, Viraht Sahni, Rafik A. Matta, Ashraf W. El-Miniawy, Bhakta Rath, Samuel Lambrakos, Nagwa El-Badry, Mahmoud Abdel-Aty, Hugo Bohórquez, Eva Knoll, Ángel Martín-Pendás, Labiba K. El-Khordagui, Abdel-Aziz Saleh, Mohamed El-Raey, Ignacy Cukrowski, Thanh-Tung Nguyen-Dang, Jesús Hernández-Trujillo, Fernando Cortés-Guzmán, Aurora Costales Castro, Olimpia Lombardi, Eric Scerri, and James S. M. Anderson.

The authors' appreciation is also directed to Piero Macchi, Alessandro Genoni, Simon Grabowsky, Dylan Jayatilaka, Carlo Gatti, Naganami Sukumar, and Paulina M. Dominiak, and all the members and consultants, past and present, of the Commission on Quantum Crystallography of the International Union of Crystallography (IUCr) for furthering the subject of quantum crystallography.

Much gratitude is due to the publisher de Gruyter, its editorial office, and handling editors, especially Ria Sengbusch, for supporting this monograph. The authors also thank Laura Gamble for helping with typesetting parts of this book.

Most importantly of all, the authors deeply appreciate the many students who have contributed their work to the study of quantum crystallography over the years and whose work this monograph is largely based upon.

Finally, much of the work in this monograph has been enabled through the support of the U.S. Naval Research Laboratory, Hunter College (and its President Jennifer J. Raab) - The City University of New York (CUNY), the Natural Sciences and Engineering Research Council of Canada (NSERC), Canada Foundation for Innovation (CFI), Compute Canada, and Mount Saint Vincent University, a support much appreciated by the authors.

Preface

[T]he task of physics as of all science is found in the coherent description of experience. Robert Bruce Lindsay & Henry Margenau (1936)

(Foundations of Physics, p. 2, Dover Publications, Inc., New York (1963)).

Readers are presented information about the authors' experience in the developing field of quantum crystallography (QCr). Their inspiration which underlies the subject is the fundamental idea of N-representability and its implementation by derivation of the Clinton equations. The purpose of the discussion includes a reference to the historical context in which QCr was developed.

This monograph introduces the reader to the core ideas that led to the development of quantum crystallography (QCr). This field of science lies at the intersection of crystallography, as commonly known, and quantum mechanics. In this monograph, some aspects of this exciting field are reviewed with special focus on the contributions and interests of the writers. It is made clear from the outset that this is not a textbook on crystallography or quantum mechanics, nor on QCr, rather it is a topical review focused on the authors' interests and contributions. This means that several important topics are omitted or alluded to in passing.

Crystallography is very old. But, modern crystallography begins when X-rays were discovered by Wilhelm C. Röntgen in 1895, and their wave nature was exploited by Max von Laue and the Braggs (William and Laurance) in 1912 and 1913, respectively. Quantum mechanics begins in the latter half of the 1920s with Werner Heisenberg, Erwin Schrödinger, and Paul A. M. Dirac.

Soon after the discovery of quantum mechanics, its application to crystallography came about in increments. It was implemented in the solution of crystal structures by using as a model the sum of the spherical atoms that occupy the crystal unit cell. Of course, the overlap of orbitals associated with chemical bonding is not well represented in that way. This led to a representation of directional bonding and lone pair density achieved by placing multipoles at atomic centers. The extraordinary success of such multipole representation placed ever-increasing emphasis upon electron density as the source of scattering. At the same time, it led to looking away from the density matrix, whose diagonal elements are the electron density. And, the density matrix is required to be N-representable in order that it, and the electron density corresponding to it, can both be quantum mechanically valid. For a density matrix to be N-representable simply means it can be derived from an antisymmetric N-body wave function.

What occurred at about the same time as the ever-increasing prominence of multipole representations was the invention of a fully quantum mechanical X-ray formalism. Its goal was that of extracting from the X-ray scattering experiment of the crystallized molecule a representation of exactly the same formalism as would be obtained by solving the Schrödinger equation within a single determinant approximation. Such single determinants underlie the Hartree-Fock approximation and the Kohn-Sham equations of density functional theory. For the first time the density matrix of theory and experimental crystallography would have exactly the same parameters and physical interpretation. They could therefore be compared one to another in the same way.

It was the N-representable density matrix X-ray formalism that came to be called quantum crystallography. It is perhaps a subtle point that an "accurate" electron density can, at the same time, be not quantum-mechanically valid. As discussed in a recent paper [1], a leader in the field of experimental X-ray density opined that to be accurate is to be quantum mechanical. But that misses the point that accuracy in the density does not mean the same as quantum mechanical validity of the density matrix. Fortunately, something is known about the quantum mechanical validity of the density matrix. In particular, the mathematical requirements for N-representability by a single determinant are exactly known. There is another factor sometimes not sufficiently appreciated regarding the difference between having an accurate electron density and a quantum mechanically valid density matrix. The difference is that all quantum properties can be calculated from N-representable density matrices, but only some properties follow from the electron density, no matter how accurate that density may be.

Having "grown up" in the field of quantum mechanics, as it were, it would be natural to look for N-representability of the density matrix in an attempt to extract its elements from the X-ray scattering data. The emphasized N-representability is after all just an acknowledgment of the experimental indistinguishability of the electrons which scatter X-rays. Thus, it was natural evolution that we would introduce N-representability into the formalism of QCr. On the other hand, a crystallographer searching for a representation of chemical bonding might be drawn to increasingly accurate densities represented by, say, multipoles. As it happens, both "N-representability" and accuracy are important.

C. F. Matta, Halifax, NS

L. Huang, Alexandria, VA

L. Massa, New York City, NY

July 2022

Reference

[1]Massa L. A zigzag path through quantum crystallography. Struct. Chem. 28, 1293-1296 (2017).

Foreword

Quantum crystallography, the field of this book, aims at expanding crystallography by combining scattering experiments with quantum mechanics. The field has been launched more than 50 years ago and is now a fast-developing area of research to which dozens of research groups around the world are contributing. The authors, Chérif F. Matta, Lulu Huang, and Lou Massa, are well known among these scientists. They are known to me through their meaningful contributions to understanding ribosome's...