Let’s start with crystal structure prediction. Suppose that you would like a crystal with various properties (it’s purple, porous with channels two nanometers wide, nonconductive, etc.). Traditionally, finding such a crystal would involve synthesizing many novel chemicals and annealing them, and hopefully a sequence of increasingly successful combinations would ultimately lead to success. This was the method that the alchemists employed in their pursuit of the Philosopher’s Stone, and whose modern, automated incarnation is called Combinatorial Chemistry. It may have been good enough for Paracelsus and Robert Boyle, but it is expensive and frustrating.
More to the point, this is not what engineers and architects do. In construction and industry, someone composes a set of blueprints specifying the final product and (hopefully) intermediate steps, and then someone (often someone else) uses those blueprints to construct the desired product – which, no coincidence, satisfies the original specifications. Chemists should be able to do that.
The general project is crystal engineering, which Wikipedia defines as “the design and synthesis of molecular solidstate structures with desired properties, based on an understanding and exploitation of intermolecular interactions.” Wikipedia says that the oldest reference to the term is G. M. J. Schmidt, Photodimerization in the Solid State in Pure & Applied Chemistry 27:4 (1971), pp. 647 – 678, but Web of Science says oldest reference they have is an abstract for a paper presented to the American Physical Society meeting in Mexico City, Mexico, August, 1955 by R. Pepinsky entitled Crystal Engineering – A New Concept in Crystallography, abstract published (but not posted!) in the Physical Review 100:3 (1955), p. 971. There being nothing new under the sun, the notion probably has been floating around since the beginning of the 20th century.
In the last few decades, crystal engineering has become of field of its own: on October 23, Web of Science listed 2,520 hits from all databases, and now the subject has its own textbook: Crystal Engineeering : A Textbook by Gautam Desiraju, Jagadese Vittal, and Arunachalam Ramanan.
The growth of the field is a little more problematic. Here are the 2,520 hits in the Web of Science by date:
Notice that after Pepinsky’s 1955 talk, Web of Science has no hits until 1976 (Schmidt’s 1971 paper did not have “crystal engineering” in any of the fields Web of Science checks – a warning to people who rely on such databases), and things really didn’t get going until 1991. Most of the growth was during the 1990s: a linear regression shows that publication growth during 2001 to 2013 has been nearly linear, an average additional 3.3 papers a year, or less than 3 % growth per year. Growth in citations looks more exponential, which shows that crystal engineering has been gaining a higher profile lately, but the number of citations has been flattening out during the last few years:
Let’s take a closer look.
In 1988, John Maddox
wrote an oped in Nature:
A new calculation of the polymorphs of silica appears to have broken new ground in deriving crystal structure from chemical composition. But Xray crystallographers need not worry — yet.

and grumped that “[o]ne of the continuing scandals in physical sciences is that it remains in general impossible to predict the structure of even the simplest crystallographic solids from knowledge of their chemical composition.” This is an intermediate position: he is complaining that as of then, a chemist couldn’t choose a random small molecule and predict the structure (or structures!) exhibited by its crystal.
But even if Maddox was taking an intermediate position, he had launched the verb, and it was quickly associated not only with predicting what crystal a given molecule would produce, but what crystal structures could exist. During the past few decades, chemists have explored the possibility of taking the desired properties as a list of specifications, designing a crystal at the molecular or atomic level that will satisfy those specifications, generating a synthesis process from the design, and synthesizing the crystal – which will satisfy the specifications. As of 22 July 2014, Google Scholar listed 3,450 hits for “crystal structure prediction”, and on 25 June 2015, Web of Science reported rapid growth in the field since the midnineties:
Notice that unlike crystal engineering, there has been no recent growth slowdown. (Although I have heard chemists and crystallographers use the phrases “crystal enumeration” and “crystal structure enumeration”, and I have seen those phrases in print, Web of Science reports no hits for either one.)
As of 19 October 2014, Web of Science listed 702 hits for either “crystal prediction” or “crystal structure prediction”, and the journals that had published at least twenty articles in the subject were:
Journal 
Hits 
Crystal Growth & Design 
64 
CrystEngComm 
47 
Physical Chemisty / Chemical Physics 
33 
Physical Review B 
30 
Journal of Chemical Physics 
25 
Journal of Physical Chemistry 
25 
And this is just those articles for which one of the two phrases is picked up by the Web of Science topics field.
Crystal structure prediction is new enough so that one can’t say what the fundamental mathematical issue is, but one could start with the work of Alexander Wells, whose book on Three dimensional nets and polyhedra presents the notion of a net, i.e. a finite or infinite graph embedded in three dimensional space. The vertices of this graph are points in space, while the edges are line segments (or curves) whose endpoints are vertices; typically, we all edge intersections should be at vertices.
(“Nets” might also be called “Euclidean graphs” or “geometric graphs” as they are graphs embedded in a Euclidean space.)
A net could represent a material structure by having the vertices represent atoms or molecular building blocks and having the edges represent chemical bonds or linkers. (We sidestep the issue of whether we believe in chemical bonds.) Then a (classical) crystal may be represented by a periodic graph, i.e. a graph with translational symmetries in three axial directions. We have reached the formulation of “crystal nets” as described in Michael O’Keeffe and Bruce Hyde’s Crystal Structures I : Patterns and Symmetry.
In mathematics, once you get your paws on a definition you can do things with it. We have a definition of the word “periodic graph”, which is the central mathematical notion in what Omar Yaghi calls reticular chemistry. A mathematician then mimics Stewart Robertson’s agenda and proposes the following:
 Let P be the set of all periodic graphs. Since P itself has a geometric and topological structure, we might call P the space of all periodic graphs.
 We usually regard two periodic graphs as being the “same” if one is the result rigidly moving the other around, so we can define the equivalence relation ≅, where “A ≅ B” means that the periodic graph A is the result of moving B. Given a periodic graph A, its equivalence class is the set [A]_{≅} = { B : A ≅ B}. It is this equivalence class that fixes the size and shape of the structure, so once could say that the space of these equivalence classes, which we denote P/≅, are the models of (classical) crystal structures.
 This is not quite how periodic graphs are currently specified in, say, the Reticular Chemistry Structure Resource (RCSR). Current practice is to specify periodic graphs by isomorphism type. Call two periodic graphs A and B isomorphic if there is a onetoone correspondence between their vertices such that that there is an edge connecting two vertices of A if and only if there is an edge connecting the corresponding two vertices of B. If two periodic graphs A and B are isomorphic, write A ∼ B, and the isomorphism type of A is [A]_{∼} = {B : A ≅ B}. We could denote the entire space of periodic graph isomorphism types by P/∼, and RCSR has a catalogue of periodic graph isomorphism types.
As Igor Barburin observed (see his comment below, which inspired a revision of this post), the situation is a bit more subtle than this. While two periodic graphs are often considered to be the “same” if they are isomorphic, a database like RCSR may have, for each periodic graph listed, a particular periodic graph in its catalogue, listed as a geometric object with geometric properties. For example, while pcu comes in many shapes and sizes, RCSR lists a particular geometric realization of pcu (of maximal symmetry). The infinitely many other geometric realizations of pcu are themselves classifiable by geometric properties (including symmetries) – and topological issues like chirality.
We will discuss issues in classifying and cataloguing periodic graphs in later posts…beginning with the question of exactly why Dr. Baburin’s comment inspired me to replace the word “classify” with the word “specify”.
Anyway, two of the central problems of crystal structure prediction is the generation and classification of periodic graphs by rigid motion equivalence relations, by isomorphism types, and by other criteria. But even that is not enough. The Atlas of Prospective Zeolites Structures has over five million nets, but few of them have been synthesized. If crystal structure prediction is to be more than a recreational activity, it must include designing the synthesis process. After all, the purpose of a blueprint is to provide a roadmap for building the structure.
But at present, most of the activity seems to be in generating periodic graphs. But before we survey that activity, we should follow Socrates’ advice and get a handle on what it is we are talking about. So what do we know about periodic graphs…?