DOI:10.1002/chem.201705952 Concept

& Physical Chemistry Quantum Crystallography:CurrentDevelopments and Future Perspectives Alessandro Genoni,*[a] Lukas Bucˇinsky´,[b] NicolasClaiser,[c] JuliaContreras-García,[d] Birger Dittrich,[e] PaulinaM.Dominiak,[f] Enrique Espinosa,[c] Carlo Gatti,[g] Paolo Giannozzi,[h] Jean-Michel Gillet,[i] Dylan Jayatilaka,[j] Piero Macchi,[k] AndersØ.Madsen,[l] Lou Massa,[m] ChØrifF.Matta,[n] Kenneth M. Merz,Jr. ,[o] Philip N. H. Nakashima,[p] Holger Ott,[q] Ulf Ryde,[r] Karlheinz Schwarz,[s] Marek Sierka,[t] and SimonGrabowsky*[u]

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Abstract: Crystallographyand quantum mechanics have Nevertheless, many other active andemerging research always been tightly connected because reliable quantum areas involving quantum mechanics andscattering experi- mechanical models are neededtodetermine crystal struc- ments are not covered by the originaldefinition although tures. Due to this natural synergy,nowadays accurate distri- they enable to observeand explain quantum phenomenaas butions of electrons in space can be obtained from diffrac- accurately and successfully as theoriginalstrategies. There- tion and scattering experiments.Inthe original definition of fore, we give an overview over current research that is relat- quantum crystallography (QCr) given by Massa,Karle and ed to abroader notion of QCr,and discuss optionshow QCr Huang, direct extraction of wavefunctions or density matri- can evolve to become acomplete and independent domain ces from measured intensities of reflectionsor, conversely, of natural sciences. The goal of this paper is to initiate dis- ad hoc quantum mechanical calculations to enhancethe ac- cussionsaround QCr,but not to find afinal definition of the curacy of the crystallographic refinement are implicated. field.

[a] A. Genoni [m] L. Massa UniversitØ de Lorraine,CNRS, LaboratoireLPCT HunterCollege&the Ph.D. Programofthe Graduate Center 1Boulevard Arago, F-57078, Metz (France) City University of New York, New York (USA) E-mail:[email protected] [n] C. F. Matta [b] L. Bucˇinsky´ Department of Chemistry and Physics, MountSaint Vincent University Institute of Physical Chemistryand Chemical Physics Halifax, Nova Scotia B3M 2J6 (Canada) Slovak University of Technology, FCHPT SUT and RadlinskØho 9, SK-812 37 Bratislava(Slovakia) Department of Chemistry,Dalhousie University [c] N. Claiser,E.Espinosa Halifax, Nova Scotia B3H 4J3 (Canada) UniversitØ de Lorraine,CNRS, LaboratoireCRM2 and Boulevard des Aiguillettes, BP 70239,F-54506, Vandoeuvre-l›s-Nancy Department of Chemistry,Saint Mary’s University (France) Halifax, Nova Scotia B3H 3C3 (Canada) and [d] J. Contreras-García DØpartement de Chimie, UniversitØ Laval Sorbonne UniversitØs, UPMC UniversitØ Paris 06 QuØbec, QC G1V 0A6 (Canada) CNRS, Laboratoire de ChimieThØorique (LCT) 4Place Jussieu, F-75252 Paris Cedex 05 (France) [o] K. M. Merz,Jr. Department of Chemistry and Department of Biochemistry and Molecular [e] B. Dittrich Biology,Michigan State University Anorganische und StrukturchemieII, Heinrich-Heine-UniversitätDüsseldorf 578 South Shaw Lane, East Lansing, Michigan 48824(USA) Universitätsstraße 1, 40225 Düsseldorf (Germany) and [f] P. M. Dominiak Institute for Cyber Enabled Research, Michigan State University Biological and Chemical Research Centre,Department of Chemistry 567 WilsonRoad, Room 1440, East Lansing, Michigan 48824(USA) University of Warsaw [p] P. N. H. Nakashima ul. Z˙wirki iWigury 101, 02-089, Warszawa (Poland) Department of Materials Science and Engineering, Monash University [g] C. Gatti Victoria 3800 (Australia) CNR-ISTMIstituto di ScienzeeTecnologie Molecolari [q] H. Ott via Golgi 19, Milano, I-20133 (Italy) Bruker AXS GmbH and ÖstlicheRheinbrückenstraße 49,76187 Karlsruhe (Germany) Istituto Lombardo Accademia di ScienzeeLettere via Brera 28, 20121Milano (Italy) [r] U. Ryde Department of Theoretical Chemistry,Lund University [h] P. Giannozzi Chemical Centre, P.O. Box 124, SE-22100Lund (Sweden) Department of Mathematics, ComputerScience and Physics University of Udine [s] K. Schwarz Via delle Scienze 208, I-33100 Udine (Italy) Technische UniversitätWien, Institut fürMaterialwissenschaften Getreidemarkt 9, A-1060 Vienna (Austria) [i] J.-M. Gillet Structure, Properties and Modeling of Solids Laboratory,CentraleSupelec [t] M. Sierka Paris-Saclay University Otto Schott Institute of Materials Research 3rue Joliot-Curie, 91191 Gif-sur-Yvette (France) FriedrichSchiller University Jena Lçbdergraben 32, 07743 Jena (Germany) [j] D. Jayatilaka SchoolofMolecular Sciences, University of Western Australia [u] S. Grabowsky 35 Stirling Highway,Perth,WA6009 (Australia) Fachbereich 2—Biologie/Chemie Institut fürAnorganische Chemieund Kristallographie, UniversitätBremen [k] P. Macchi LeobenerStr.3,28359 Bremen(Germany) Department of Chemistry and Biochemistry,University of Bern E-mail:[email protected] Freiestrasse 3, CH-3012 Bern (Switzerland) The ORCID identification number(s) for the author(s) of this articlecan be [l] A. Ø. Madsen found under:https://doi.org/10.1002/chem.201705952. Department of Pharmacy,University of Copenhagen Universitetsparken2,2100 Copenhagen(Denmark)

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1. Introduction of molecules, which represents akind of magnifying lens of the scatteringofindividual objects. Properties of materials as well as modes of action of drugs are Thus, theoreticalcalculations and scattering experiments are directly related to their electronic structure. Therefore, one of complementary approaches to gain insightinto the electronic the most importantchallenges in modernscience is the accu- structure of compounds. Actually,X-ray diffraction and wave- rate determination of the electronic structure, from which functions have alwaysbeen intimately related, because model- structure–functionrelationships can be derived. One way of ing crystal structures requires atheoretical framework to inter- obtaining information on electronic structure is by calculating pret the measured data, that is, chargedensity is an “observa- wavefunctions of the materials or compounds under investiga- ble” available by means of modeling. The simplest model as- tion using quantum mechanics. Wavefunctions are mathemati- sumesthat diffraction is caused by acombination of non-inter- cal objects that intrinsically contain all the information of acting atoms, each of them represented by aspherically quantum mechanical systems in specific pure states. They can averaged ground-state electron density.[1] This model,univer- be obtained as approximate solutions of the Schrçdinger sally known as Independent AtomModel (IAM),implies calcula- equation(e.g.,through numerical calculations) and allow to tion of atomicwavefunctions, thus relyingonquantum me- determine variousexpectationvalues that can be directly mea- chanics(QM).[2] The vast majority of moderncrystal structure sured through experiments. Another class of experimental “ob- refinementsadopts this model to obtain comparably accurate servables” are availableonly by means of modeling, namely atomic positions and displacement parameters for non-hydro- through optimizations of some parameters that replicate a gen atoms.[3,4] However,already in 1915, Debye pointed out physicalquantity within the assumptionsofagiven theoretical that there is more information in the measuredX-ray diffrac- framework (for example, electron density,magnetization, etc.). tion pattern than the atomic positions. In particular,herecog- At the same time, some modelsbased on electron density nized that “it should be possible to experimentally determine functions may return partial information also on the wavefunc- the special arrangements of the electrons inside an atom”.1[5] tion:for example, an approximate form of the spin part of the More than half acenturylater,this proposal became reality.[6] wavefunctionoranapproximate partofthe atomic or molecu- Today,more accurate crystallographic models are available, lar orbitals. Due to the increasing computational powerand which allow to obtain apicture of the “special arrangements the continuing development of sophisticatedmethods and of the electrons” (namely,the electron density) from X-ray dif- software, wavefunctions can provide profound insights into fractionmeasurements.[7] electronic distributionsand are becoming increasingly impor- In this context, besidesmaximum entropy methods,[8–10] the tant. most populartechnique is the atomic multipolar expansion of Althoughquantum chemistry has reached great maturity the electron density,based on the projection of the electron and abroad base of applications,itisworth bearing in mind density in atomicterms.[11–15] The multipole model is the result that even the most rigorous first-principle calculations for sys- of necessary approximations. In fact, arigorous treatment of tems with more than one electron depend on approximations. two-center scattering would be quite complicated even for Therefore, their predictions must find validations. In quantum simplecompounds,although reported for some diatomic mol- chemistry,this is particularly cogent because the uncertainty ecules.[16] Furthermore, the modeling of atomicdisplacements intrinsically associated with the approximationchosen forthe initially led to additional problems in case of two-center elec- calculation is unknown. Onecan only evaluate the per- tron density functions.[17,18] The one-center expansionled to a formance of agiven theoretical method by using aset of ex- much easier formalism, by whichradial andangular parts de- perimental values as benchmarks. In molecular quantum rived from atomicorbitals are used as atomicdensity func- chemistry,the experimental validations often come from spec- tions, and where pseudo-atoms are naturally defined.[19] The troscopy. However, in materials science and solid-state chemis- radial decay of the pseudo-atomsand the core and valence try,the best “eye” to probe the quantum behavior of matter is scattering factors in multipole modelsare directly calculated the scattering of radiation or particles of sufficient energy,typi- from wavefunctions and hence the analytical shape of the re- cally X-rays, g-rays, electrons and neutrons. The measured dif- fined electron density is significantly influenced by quantum fraction pattern is arepresentation of the charge-density distri- chemistry.Itisimportant to mention that, to agood approxi- bution in the compound under examination. X-rays and g-rays mation, the set of multipolar orbitals may be relatedtoatomic interactwith the thermally smeared electrons in acrystal, so hybridization states[20] and even to some individual orbital oc- that one can model either the dynamic electron density or the cupancies, for example that of d-orbitals in transition metals,[21] positive and negative charge-density distributions independ- and more recently that of f-orbitals in lanthanides.[22] Moreover, ently after deconvolutioninto atomic displacementparameters an extensionofthe traditional multipole model is the spin-po- and static electron density.Neutrons interact with nuclearpar- larized multipole model,adding the spin-density information ticles and, therefore, they mapthe probability distribution of to the chargedensity.[23] atomic nuclei, from which one can easily derivethe positive Cross-fertilized by charge-density research, ways of directly charge density distribution.Electrons interactwith the electric fitting the shapes of orbitals andwavefunctions to the mea- field generated by electrons and nuclei. There is an additional benefitfrom measuring scatteringofordered matter,such as 1 German original:“…muss es dann gelingen, die besondereAnordnung der crystals(or even quasi-crystals), namely the cooperative effect Elektronen im Atom experimentellfestzustellen.”

Chem.Eur.J.2018, 24,10881 –10905 www.chemeurj.org 10883  2018 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Concept sured diffraction pattern were also devised. This is at the heart AtomiqueetMolØculaire), which was organized to discussthe of the originaldefinition of quantum crystallography (QCr) meaningand perspectivesofquantum crystallography in given by Massa,Karle and Huang, which encompasses meth- light of the recentand significant increaseinthe use of this ods where the information resulting from traditional quantum term and relatedtechniques in the scientific literature chemistry calculations is enhanced by externalinformationin- (Figure 1).[55–58] Therefore, in the following section, we will trinsically containedinthe experimental crystallographic show how this increased interestinQCr manifestsitself in data.[24] The first discussion about the perspective of obtaining method developments and applications that are not necessari- wavefunctions from X-ray scattering (here:Compton scatter- ly within the originaldefinition of QCr or in the framework of ing) goes back to 1964 and the first Sagamore conference,[25,26] conventional multipole-based experimental chargedensity re- whereas the first quantum crystallographic method according search as discussed above.Infact, if all fields andapplications to the original definition[24] and based on X-ray diffraction was where quantum chemistry and experimental approachesbased proposed by Clinton and Massa in 1972.[27] Nowadays, Jayatila- on diffraction andscattering mutuallyenrich each other are to ka’s X-ray constrained wavefunction (XCW) fittingap- be accommodated within aunified research area, the original proach[28–32] and its later developments[33–40] are the most popu- definition of QCr is too narrow.Inthis light, in section3 we lar modernversions of theoriginal quantum crystallographic will present anddiscuss the differentpoints of view on QCr as methods based on X-ray diffraction. They practically aim at de- they emerged during the recent CECAM meeting. This will termining wavefunctions that minimize the energy, while re- highlight different ways in whichthe rapid and fruitful scientif- producing, within the limit of experimental errors, X-ray struc- ic evolutions touched upon in section2 could eventuallylead ture factor amplitudes collected experimentally.Asanalterna- to abroadened definition of quantum crystallography and to tive, joint refinement methods for the complete reconstruction the foundation of anew and flourishing researchfield and of N-representable one-electron density matrices exploit both community (see Figure 1). X-ray diffraction and inelastic Compton scattering data.[41–43] More detailed reviewsofmethods can be found in Ref. [44] 2. Current Developments and Ref. [45]. In 1999, Massa, Huang and Karle also pointedout[46] that In this section, the authors of this paper will outline current there is another stream of quantum crystallographic tech- highlightsfrom their own research activities to exemplifythe niques that directly use wavefunctions and orbitals—not multi- wide scope of methods and applicationsthat might be includ- poles—to improve the accuracyand information contentsof ed into abroadened definition of quantum crystallography. crystallographic refinements.Thisisbasically the converse of This section can neither be an exhaustive review nor will it their original definition of QCr.Inthis converse sense, the first cover all possible areas of overlap and interestfor QCr.Itwill developments are associated with Quantitative Convergent- show the diversity of the field, not aunified picture, so that it Beam Electron Diffraction (QCBED),[47–50] for which the knowl- will pave the way fordiscussions about the meaning and use- edge of the wavefunction describing the high-energyelectron fulnessofQCr. passing through the crystal is essential for solving the dynami- In the first three subsections, we will present the two tradi- cal electron scattering equations. The solutionstothese equa- tional ways to conductquantum crystallographic investiga- tions give the scattered intensities in calculated diffraction pat- tions, namely the completely theoretical approach (subsec- terns that are compared to experimentalones in QCBEDrefine- tion 2.1) and the completely experimental approach—the ments.Another area in which the converse definition of QCr latter discussed both in terms of new technical and instrumen- plays an important role is macromolecular crystallography,[51,52] tal developments (subsection 2.2) and in terms of the tradition- where quantum mechanically derived restraints are successful- al experimentalcharge density methods (subsection 2.3). We ly exploited to supplementthe limited resolution and amount will afterwards illustrate techniques in the framework of both of diffractiondatacomparedtothe number of parameters aspects of the originaldefinition of quantum crystallography neededtomodel atomicpositions and displacementsinlarge by introducing density-matrix-based and wavefunction-based systems. Finally,the most recent technique in this framework is refinement strategies (subsections 2.4 and 2.5, respectively), the Hirshfeld Atom Refinement (HAR),[53, 54] which exploits quantum crystallographic techniques to refine crystal struc- ad hoc Hartree–Fock (HF) or DFT computations to derive frag- tures of biological macromolecules (subsection 2.6) as well as ment electron densities for refinements that provide the most the Kernel EnergyMethod (KEM, subsection 2.7). We will con- accurate and precise structuralresults currently attainable from clude the overview of the methods of the original definition of X-ray data. In reference[45] both aspects of the Massa–Huang– QCr by discussing dynamic quantum crystallography (particu- Karle QCr definition werediscussed in detail. From here on, larly the NoMoRe approach, subsection 2.8) and the quantita- these two aspectstogether are termed“original quantum crys- tive convergent-beam electron diffraction (QCBED) technique tallography”. (subsection 2.9). In subsection 2.10, we will present techniques This viewpoint paper has been initiated at the Discussion to derive information on chemical bonding from wavefunc- Meeting “Quantum Crystallography:Current Developments tions and electron densities.Inthe last five subsections, we and Future Perspectives” (Nancy,France, 19–20 June 2017) will showhow quantum crystallographic approaches are al- under the umbrella of the European Centre for Atomic and ready fundamentalfor many interesting applications in differ- Molecular Calculation (CECAM,Centre EuropØen de Calcul ent fields, such as in crystal engineering (subsection 2.11), in

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Figure 1. Most of the participants at the CECAMDiscussion Meeting in Nancy(and co-authorsofthis paper) with aselectionofthe fields that they represent. These fieldsare importantpillars at the outset of adiscussionaboutanew meaning of quantum crystallography.They are not exclusive, but the starting point for abroadening of QCr.

the determination of magnetic properties (subsection 2.12), in ample,configurationinteraction andcoupled cluster methods, the studyofmolecular and extended solids (subsection 2.13), as sketched in Figure 2. However,the additional accuracy in materials science (subsection 2.14) and in crystal structure comes at the price of steeply increasing computational cost. prediction(subsection 2.15). An alternative route is DFT with the much simpler ground- state electron density 1 r as the main variable. The ground ðÞ state energy E of asystem is afunctional of 1 r ,[60] whose 2.1. Theoretical quantum crystallography ðÞ mathematical form is however unknown, thus requiring ap- Quantum mechanical methodsare one of the main ingredients proximations. Kohn andSham[61] (KS) proposed ascheme to of QCr.Webriefly discuss here the most important approxima- make DFT calculations feasible by mapping the interacting tions involved and some of the available computer codes. system of electrons onto anon-interacting one that leads to Any quantum mechanicalmethod starts with an idealization the true density.The searchfor better DFT energy functionals of the atomic structure. Isolated molecules can be accurately is an activefield of research (see, e.g.,Ref. [59, 62]). The main represented by aset of atomiccoordinates. Solids are typically categories of functionals are, in order of increasing complexity represented as perfect, infinite crystals, defined by aunit cell (see Figure 2): the Local Density Approximations (LDA),[61,63] the and alattice,under periodic boundary conditions. However, Generalized Gradient Approximation (GGA)[64,65] and meta- real crystals differ from this ideal situation, due to defects, im- GGAs.[66] These functionals depend upon the local values of 1, purities, surfacerelaxations, non-stoichiometry,and disorder.A 1 , 21 and/or t (the kinetic energy density). It is interesting jr j r strength of theory is that model systems can be simulated irre- to note that these same quantities are also used in the topo- spectiveoftheir existence in nature,allowing the investigation logical analysis of the chemical bond (see section2.10). Hybrid of effects associated with different modificationsonsystem functionals[67] additionally include acertain fractionofHFex- properties. change. Unfortunately,there is not asingle optimal DFTfunc- The direct solutionofthe many-particle Schrçdingerequa- tional that works well for all cases and properties. Hence,com- tion is an intractable task for most systemsofinterest in QCr. promises are necessary.Werefer the readertoarecent Simplifying the electronic wavefunction to asingle Slater de- paper[68] critically analyzing DFT functionals in terms of accura- terminant leads to the Hartree–Fockmethod.[59] The often cy and addressing their differences for molecules and solids. severe deviation of the HF solution from the exact one is col- There are severalcomputer codes for molecular and solid- lectively termed as electron correlation. There are many so- state quantum mechanical simulations, for example, Quantum called post-HF methods,[59] which approximate electron correla- ESPRESSO,[69] Turbomole,[70] and WIEN2k,[62,71] implementing tion and yield accurate many-electron wavefunctions, for ex- three very different approaches (plane waves and pseudopo-

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Figure 2. The most importantquantum mechanical approximations and methods. The many-particle Schrçdinger equation is simplified using the Born–Op- penheimer approximation.From there, one route is DFT and the other is the HF method. The most importantpost-HF methods that approximate electron correlation missinginHFare based on Coupled Cluster(CC) and Configuration Interaction (CI) techniques as well as the Møller–Plesset (MP) perturbation theory.

tentials,Gaussian basis-sets, all-electron augmented plane mentalchargedensity research. Data collection times can waves). The variety of computer codes is very useful, since therebybereduced from weeks to days/hours and weakly dif- each code has adifferent focus, but raises the issue of repro- fractingsamples can now be studied more easily.Modern ducibility of resultsobtained by different codes. Recently,the short-wavelength sources(Ag/In Ka)extend the data resolution calculated values forthe equationofstates were compared limit and minimize X-ray absorption and extinction. In parallel, using 40 different DFTmethodtypes showing that deviations large-scale facilities (synchrotrons)are becoming more widely between the accurate codes are smaller than those of experi- accessible. Properties like the extraordinaryX-ray flux and a ments.[72] Acomparison of charge densitiesobtained with dif- tunable wavelength make them attractive, although “only a ferent methods (DFT or HF-based) and different basis sets minor fraction of the published electron density literature” (plane waves,Gaussians) has been done in Ref. [73].Insumma- data is collected there.[78] This may seem surprising but is ex- ry,approximations make simulations feasible butneed to be plained with the fact that there is no dedicated charge density verifiedand improved if necessary.The combinationofQM beam line satisfying the special requirementstocollect high- and diffraction/scattering experiments within QCr is certainly a quality charge density data.[78] promisingroute for future progress in this field. For example, Alongside source development, major accomplishments in there are numerousexamples that demonstrate that experi- the field of X-ray detection have been achieved.Today’s HPADs ments guided by QM are often the only way to an unambigu- (Hybrid Pixel Array Detector) and CPADs (Charge Integrating ous atomic structure determination of low-dimensional sys- Pixel Array Detector) are detectors based on Complementary tems.[74] As afurtherexample, the usage of modernnon-local Metal-Oxide-Semiconductor (CMOS) technologyand these are functionals coupled with molecular dynamics has allowed to capable of fast shutterless data collection. This removes signifi- interpret experimental resultsinmolecular crystalsatfinite cant sources of errors such as read-out overhead, shutter jitter temperature.[75] and goniometer repositioning inaccuracies. The HPAD technol- ogy is presently the standard technologyatprotein synchro- tron end stations. Such detectors are known for their sensitivi- 2.2. Development of experimental techniques and instru- ty,highspeed, and dynamic range. However,HPADs do have ments limitations for charge density experiments. Stronglow-resolu- Outstanding data quality is required for performing atradition- tion reflections, which contain the bondingelectron density al multipole refinement[15] or an X-ray Wavefunction Refine- information,may suffer from count-rate saturation[79] and ment (XWR).[76] In addition to highest quality crystals, ahigh- charge-sharing effects can lead to information loss.[78,80,81] end experimental setup is the key to extracting ameaningful The appearance of X-ray Free Electron Laser (XFEL) facilities crystallographic outcome. led to the developmentofCPADs to overcome theselimita- Coupled to multilayer X-ray optics, high-brilliance microfocus tions.[82] The high-frequency readoutand mixed mode opera- sealed tubes[77] and rotating anode sources dramatically in- tion, combining the advantages of integration for strong sig- crease the available X-ray flux density in comparison to con- nals with the photon counting approachfor weaksignals, re- ventional sealed tubes, amajor benefitinthe field of experi- cently becameavailable for the home laboratory.Although

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CPADs have excellent count-rate linearity,they can suffer from nal electric field.[96,97] Parameters of the multipole model are pixel saturation, leading to missing reflections. Sophisticated usually refinedbyaleast-squares fitting against experimental data-scaling programsaid to ensure data completeness by observations. Thus, first principles of quantum mechanics are handling different frame exposure times or differentprimary not imposed during this process, as they are in the case of beam intensities(e.g. beam attenuation).[83] quantum mechanicalcomputations. Therefore, the wavefunc- Besides sources and detectors, precise multi-axis goniome- tion is not aresult of amultipole refinementand information ters allow the collection of true data multiplicity and keep the contained in the experimental data alone shouldguarantee crystal scattering volume as constant as possible. Together that the obtained model obeys first principles. Nevertheless, with rigid sample mounts, this is akey to data quality. the multipole modelcan provide achargedensity model as Sophisticatedsoftwareisnecessary to accurately process the good as adouble-zeta basis-set representation, is mature and carefullycollected frames. Imagesmeasured with ascan width its application is fast. Moreover,unlike in X-ray wavefunction significantly smallerthan the crystal mosaics are beneficial for refinement,[76,98] the extracted experimental information can be data quality when using modern detectors.[84] Up-to-date inte- easily identified and the danger to “get what you put in” is gration routines apply an incident-angle correction, which smaller (“in data we trust”).[99] takes the X-ray conversion factors of the individual detector With the ever-rising quality of experimental data, more and into account.[85] Absorption and Lorentz/polarization correc- finer details of the electron density are observed with the help tions as well as scaling, the latter being particularly important of the multipole model:chemical bonds, lone pairs, crystal for Gaussian-shaped beam profiles, guarantee high data quali- field effects on electron density of transition metals,[100] core ty.[86] Some programs even offer adedicated charge density region contraction upon chemical bond formation,[89] intermo- mode,[82] which,among other functionalities, prevents data lecular chargetransfer[101] and electron density polarization bias introduced by standard structure error-model settings.[87] upon interaction with neighboring molecules.[92] In addition, In conclusion, an impressive number of recent technological the model providesthe meanstoobserve ahigh degree of innovations support the scientists to successfully determine transferability of atoms in similar chemical environments and charge-density distributions. The on-going collaborationbe- to build pseudoatom databanks.[102–105] tween researchers, hardwareand software providers will drive Therefore, the multipole model allows to study many quan- future developments from which the QCr community,aswell tum phenomena, and to obtain apredominantly experimental as the entiresingle-crystal X-ray diffraction community,will answer to research questionsthat depend on 1 r .However, ðÞ benefit. as with any other model, care must be taken not to interpret the model beyondits limits[106] and not to ask for information absent in the experimental data.[107] An example of asuitable 2.3. Chargedensity from X-ray diffraction researchquestion is measuring the correct covalentbond dis- The charge density of acrystal is aquantum mechanicalob- tance from X-ray diffraction in bonds involving hydrogen servable in an X-ray diffraction experiment. In principle, Fourier atoms. Here the multipolemodel permitted to get abetter summation of measured structure factorsshould provide answer than the IAM early on,[105,107] in rather close agreement direct access to experimental electron densities. In practice, it to Hirshfeld Atom Refinement.[108,109] More recently,the bond is necessary to use amodel of the charge density to minimize distance of hydrogen in the vicinity of ametal atom was char- consequences of the lack of structure factor phases, experi- acterized.[110] The power of the multipole modelisalso illustrat- mental errors and finite resolution. ed by research that aims at understanding interactions in mac- The most widely used Stewart–Hansen–Coppens multipole romolecular complexes of biological importance.Here the model[15,88] relies on the observation that spherically-averaged model provides fast access to electrostatic potential, electro- free-atom electron densities provide avery good zero-th order static energies and the topology of 1 r ,[111,112] while still main- ðÞ approximation of the electron density of acrystal (and amole- taining the accuracy of quantum mechanical computa- cule). The multipole modelconsists of atomicelectron densi- tions.[113, 114] ties, which are pre-computed from first principles of quantum mechanics at ahigh level of theory.The spherical atomic cores 2.4. Density matrix refinement and data combination are supplemented with additional pre-defined radial functions, which are combined with spherical harmonics to account for To this day,quantum crystallography mostly relies on the inter- the asphericity of the atomic electron density.Only selected play between spin or charge densities derived from first-princi- parameters of the multipole model are allowed to vary.Tradi- ples calculations and X-ray Diffraction (XRD) or high-resolution tionally these are populations and expansions/contractions of Polarized Neutron Diffraction (PND) experimental data. Despite valence electron densities. More recently,[89] the corresponding their high quality and the absence of extinctioneffects, few parameters of core densities were allowed to change as well. studies have made sole use of QCBED-derived structure factors The model, starting from free quantum-physical atoms, to reconstructchargedensities (see Introduction and Subsec- allows to “measure” the response of 1 r to the formation of tion 2.9). Their number is usually small and afull reconstruction ðÞ molecules (chemical bonds),[90] supramolecular assemblies[91] process necessitates complementing them with higher-Q Four- and crystals,[92] andtoother physical stimuli, such as different ier components obtainedby“traditional” X-ray methods.[115,116] temperatures,[93] high pressure,[94] light[95] or possibly an exter- This is aclear example of the need of combining two experi-

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mental data sets to take the best of what is available experi- ip r r0 =h JCS q; u 1 r; r0 1 r; r0 eÀ Áð À Þ d p u q dr dr0dp mentally.Asexplained in Subsection 2.12, the problemis ðÞ¼ "ðÞþ#ðÞ ðÞÁ À Z found to be rather similar in the spin density case, although ÀÁ 5 the limitations are not as stringent as those encountered in ð Þ the QCBED technique.[117] However,when such difficulties are and circumvented, electron density reconstruction is still not a

(spin resolved) wavefunction extraction. ip r r0 =h JMCS q; u 1 r; r0 1 r; r0 eÀ Áð À Þ d p u q dr dr0dp: ðÞ¼ "ðÞÀ #ðÞ ðÞÁ À The work carried out on electron spin density can be some- Z what extended by considering the reduceddensity matrix for- ÀÁ 6 ð Þ malism. Obviously,noN-electron wavefunction has yet been recovered this way,but significant advances can be (and have These equations demonstrate the importance of density matrix been) made, including the possibility of tackling the problem determination of experimental quantities derived from scatter- from aphase-space perspective.Inits simplest expression, the ing techniques, whichare at the very heart of crystallography. one-electron reduced density matrix (1-RDM), which was Resultsprovided by coherent elastic scattering of X-rays, polar- strongly promoted by Weyrich and Massa and co-workers ized neutrons, polarized X-rays and electrons are relatedtothe during the 1980s and1990s,[118–121] can be seen as the most Fouriercoefficients of the electron or spin density in crystals, direct object to connect positionand momentum space prop- hence the diagonal elements of the density matrix [see Eqs. (3) erties[122] and thereby offers an efficient means to combine and (4)].Conversely, spectra from X-ray incoherent inelastic XRD, QCBEDand PNDdata with experimental resultsprovided scattering, as well as spectroscopic methods, such as positron by inelastic scattering techniques, such as polarized or non-po- annihilation or (g,eg), dominantly address the off-diagonal part larized X-ray Compton, (g,eg)scattering or positronannihila- of the same 1-RDM [see Eqs. (5) and (6)].Therefore, the 1-RDM, tion.[123] Moyal or Wigner functions (see, for example, as the simplest and closest transcriptionofawavefunction, Ref. [124]),couldequally be considered to play the same role. but adapted to mixed state configurations, can be considered The pure state 1-RDM 1 r; r0 is associated with the N-electron as acommon denominatorofalarge range of experimental ðÞ wavefunctionfrom: techniques, making it possible to check their mutual coher- ence. While the refinement of 1-RDM modelscan still be consid- 1k r; r0 y* r; r2;:::; rN yk r0; r2;:::; rN dr2:::drN: 1 ðÞ¼ k ðÞðÞ ð Þ ered as avery recent research field, the past decade has wit- Z nessed strongadvances thanks to the fruitful joint efforts of This puts rather strongconstraints on the form taken by ex- theoreticians and experimentalists with respective expertise in perimentally derived 1-RDMs which are required to be N-repre- scattering techniques as complementary as high resolution sentable (see also Subsection 2.7) and, in the pure-state case, XRD, PND, CS andMCS. An example of atentative 1-RDM re- idempotent 1 r; r00 1 r00; r0 dr00 1 r; r0 .Nevertheless, no construction from pseudo-dataisgiven in Figure 3. kðÞkðÞ¼ kðÞ such condition can be imposed on experimentally derived 1- R RDMs, which are expectedtooriginate from amixture of states and expressed by means of each microstate canonical 2.5. Wavefunction-based refinement probability pk, The wavefunction is the fundamental entity that intrinsically containsall the information about asystem.Therefore, there is

1 r; r0 1 r; r0 1 r; r0 pk1k r; r0 , " # 2 ðÞ¼ðÞþðÞ¼k ðÞ ð Þ X in which the spin up and down contributions have been made explicit. For data from XRD,PND, non-polarizedCompton Scat- tering (CS) and Magnetic Compton Scattering (MCS), the spin- resolved 1-RDM, for agiven scatteringvector g,isconnected to structure factors through

i2pg r FXRD g 1 r; r 1 r; r eÀ Á dr 3 ðÞ¼ "ðÞþ#ðÞ ð Þ Z ÀÁ and Figure 3. Refined 1-RDM for the toy model triplet-state urea in aperiodic lat- tice with parallel molecules separated by 3 Š.Left:apath through O, C,N i2pg r and Hinthe molecular plane. Right: asimilar path but 0.5 Š above the FPND g 1 r; r 1 r; r eÀ Á dr 4 ðÞ¼ "ðÞÀ #ðÞ ð Þ plane. The upper half is the refined 1-RDM for asimple single molecule Z ÀÁ model (expressedona3-21G(d) basisset).The lower half represents the ref- erence result, namely aperiodicDFT computation with an extended basis- and, for Compton profiles in ascatteringdirection given by u, set (pob-TZVP), from which 12 MCS profilesand 500 PND structure factors through were computed and used for the refinement.[125]

Chem.Eur.J.2018, 24,10881 –10905 www.chemeurj.org 10888  2018 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Concept adirect relationship between the structure factorsofX-ray or 2.6. Quantum refinement of biologicalmolecules electron diffraction—the Fourier transforms of the electron density—andthe wavefunction. With modern computing Quantum refinement is amethod to supplement standard power andincreasing experimental accuracy, this relationship crystallographic refinement with quantum mechanical calcula- can be exploited more and more efficiently for crystallographic tions.[51, 52,137,138] In the refinement process, the model is opti- refinement. mized to provide an ideal fit to the experimental raw data (the On the one hand, the already mentioned X-ray constrained structure factors).[139] For resolutionsobtained for mostbiologi- wavefunctionfitting approach[28–32] (see introduction) plays a cal macromolecules, 1–3 Š,availabledata are not complete prominentrole. It was initially developed in the framework of enough to determinethe exact positions of all atoms. There- the RestrictedHartree–Fock formalism[28,29, 31] and was after- fore, the experimental data are supplemented by empirical wards extended to otherapproaches (e.g.,DensityFunctional chemicalinformation in the form of aMolecular Mechanics Theory,[32] relativistic Hamiltonians,[33,34] ExtremelyLocalized (MM) force field. Consequently,the refinement takes the form Molecular Orbitals (ELMOs),[35–38] etc.). Severalinvestigations of aminimization of the function:[139] have shown that XCW fitting allows not only to obtain reliable charge density distributions, but also to determine physical Ecryst EX-ray wAEMM 7 ¼ þ ð Þ properties of materials[126–129] (e.g.,non-linear opticalproper- [130] ties) and to consistently capture electroncorrelation, polari- where wA is aweight factor that is required because the crys- zation and crystal fieldeffects. Other ongoing studies also tallographic (EX ray)and MM (EMM)energy functions do not À focus on the capability of the method in capturing relativistic have the same units. effects[34] and in reliably determining experimental spin densi- This works fairly well for proteins and nucleic acids, for ties for interesting open-shell systems, such as the cyclic alkyla- which there are plenty of information about the ideal geome- [131] [139] minocarbene radicalcAAC-SiCl3, for which preliminaryre- try so that aMMdescription works well. However,for other sults are available. parts of the structure, for example, metal sites, substrates, in- From the methodological point of view,future challenges hibitors, cofactors and ligands, such information is normally for the technique are its extensions to periodic systemsand to missing or much less accurate. Moreover,the MM description multi-determinant wavefunction approaches, although the is rather inaccurate,omitting electrostatics, polarization and recent X-ray Constrained ELMO-Valence Bond (XC-ELMO-VB) charge transfer.This can be solved by employing amore accu- method[39,40] can be already consideredanattemptinthe latter rate energy function, provided by QM calculations, which in- direction. volve all energy terms and do not requireany parameteriza- On the other hand, tailor-made wavefunctions have already tion.[51,52, 137,138] In the first implementation of this quantum-re- been successfully used to improveX-ray structure refinements finementapproach,QMwas employed for asmall, but interest- by means of Hirshfeld Atom Refinement (HAR).[53,54] In its cur- ing part of the macromolecule, using the energy function[51, 137] rent implementation, molecular electron densities are theoreti- cally calculated and afterwards partitioned using Hirshfeld’s EQM-refine EX-ray12 wA EQM1 EMM12 EMM1 8 ¼ þ ð þ À Þ ð Þ stockholder partitioning method.[132, 133] This enables a“classi- cal” atom-centered crystallographic description and aleast- where EQM1 is the QM energy and the subscripts indicate squaresrefinement of atomicpositions and thermal parame- whether the methodisused for the whole macromolecule (12) ters using quantum atoms. These contain all the information or only for the QM region (1). from the parent wavefunction(or electron density) from which This approach was implemented in 2002 using DFT calcula- they have been partitioned. The HAR procedure is iteratively tions.[51, 137] It was shown to locally improve the geometry of repeated until convergenceisreached. Recent investigations metal sites in proteins.[140] Moreover,the protonation state of have revealed that HAR currently provides the most accurate metal-bound ligands could be determined by comparing quan- and precise structuralresultsfrom X-ray data, even for the po- tum-refined structures optimizedindifferent protonation sitions of hydrogen atoms.[108, 109,134] Afuture challenge forHAR states using real-space R factorsand comparing geometries is the need of extending its applicability to large molecules and energies with those obtained for the QM system in (e.g.,proteins)and to heavy-metalsystems (e.g.,coordination vacuum.[141] In the same way,the oxidation state of metal sites compounds). Coupling HAR with ab initio linear-scaling strat- could be deduced,although it often changes during data col- egies (e.g.,techniques based on the transferability of lection, owing to photoreduction by electrons released in the ELMOs)[135, 136] would allow considerable progress in this re- crystalsbythe X-rays.[142, 143] spect. In 2004, arelated approach was presented, in which a Finally, an intriguing future perspectiveinthisframework is linear-scaling semiempiricalQMmethod was employedfor the thesystematiccouplingofthe twoapproachesdiscussed above entire protein, obtained by simplyreplacing EMM in Equa- [52,138] (i.e., XCWfitting andHAR)togiverisetowhathas been termed tion (7) by EQM. It was shown that the re-refinement could X-rayWavefunction Refinement.[98] Athoroughvalidationof correct structuralanomalies in the originalrefinement of a1Š- XWRhas recently been conductedand showed that thenew resolution protein structure. This approach has later been ex- approach canindeedbeconsideredasamethod forbothstruc- tended to other software, including also QM/MM and ab initio ture and chargedensity determination from experiment.[76] methods.[52,138,144]

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[149] Figure 4. Electron-density maps of two possible protonation states of the homocitrate ligand in nitrogenase. The 2mFo-DFc maps are contoured at 1.0 s and the mF -DF maps are contouredat+3.0 s (green)and 3.0 s (red). The structure to the left fits the experimental databetter,especiallyaround theO1 o c À and O7 atoms. This is also reflected by the real-space difference density Z-scores,which are 3.0 and 3.2, respectively.Reprinted with permission from Ref. [149](Copyright 2017 American Chemical Society).

Quantumrefinement has been applied to many systemsof biological or chemicalinterest, for example, heme proteins, zinc proteins, superoxide dismutase, hydrogenaseand nitroge- nase.[51,52,137,138,141–143,145–149] Typical applications regard the nature,protonation and oxidation state of the active site, com- paring different structural alternatives, as shown in Figure 4. Applications to drug designand ligand refinement[144,150–155] have advanced the accuracyofthe determination of small-mol- ecule parameters in active site pockets over what is possible with standard models, in which the ligand force-field is gener- Figure 5. Sketch depicting the mapping problem associated with wavefunc- tion N-representability of density matrices.[170] ally less well validated than that for the protein. Naturally,the largesteffects are typically seen for low resolutions and at res- olutions better than 1 Š,effects of systematic errors in the For single determinant N-representability,the density matrix  QM method start to be apparent. The method has been ex- shall be aprojector,that is, P2 P,encompassing that of DFT ¼ tended to neutron,[156] NMR[157–163] and EXAFS (Extended X-ray solutionsofthe Kohn–Sham equations.Walter Kohn was fond Absorption Fine Structure) refinement.[164–168] of sayingthat “…the only purpose of the KS orbitals is to de- liver the exact density”. In this regard, one may take notice that orbitals of the experimental X-ray determined density 2.7. X-ray diffractionand N-representability matrixofform P2 P do just that. ¼ The compliance with the N-representabilityisasine qua non This would be satisfactory and deliver acomplete quantum to obtain quantum mechanically rigorous density matrices mechanics for small organic molecules containing afew tens from X-ray diffraction data. In this context, the conversion of X- of atoms. However,this would not work for biological mole- ray diffraction data into an electrondensity matrix cules containing very large numbersofatoms, Nat.That is be- 1 r; r0 TrP y r y r0 (where P is apopulation matrix and y cause the number of unknowns grows more rapidly with in- ðÞ¼ ðÞ ðÞ is acolumn-vector collection of atomicbasis functions) that re- creasing Nat than does the number of experimental data. To flects the antisymmetry of an N-electron wavefunctionisac- surmountthe difficulty arising from insufficient data one may complished by applying iterative Clinton matrix equations[169] invokethe Born–Oppenheimer approximation, using nuclear of the form: positionsobtained from crystallography,and then, instead of an experimental fit, theoretically calculating the density matri- 2 3 n ðÞ Pn 1 3Pn 2Pn lk Ok ces. Nevertheless, at this point the problem is that the difficul- þ 9 ¼ À þ k ð Þ X ty of quantum chemical calculations rises as ahighpowerof n the number of atoms. where lðÞ is the k-th Lagrangian multiplierpertaining to en- À k The KernelEnergy Method (KEM) is asuccessful way of cal- forcement of an X-ray scattering observable represented by culating the ab initio density matrices for molecules of any the matrix O . k size. KEM was devisedbyMassa, Huang and Karle[24] based Figure 5pictures all antisymmetricwavefunctions andall N- upon previouswork relatedtothe Clinton equations and it representable 1-body density matrices. Given avalid wavefunc- has been thoroughly tested against awide variety of biological tion, by integration over its square, one obtains aquantum molecules and also extendedaromatics with and withoutim- mechanically satisfactory density matrix. In general,anarbitrary posed strongexternalelectric fields. 1-body function f r; r0 will not map back to the set of valid The KEM working equation for the total energy is, ðÞ antisymmetric wavefunctions. However, beginningwith an N- m 1 m m À representable density matrix, this will map backtoawavefunc- E total E m 2 E 10 ¼ ab À ðÞÀ c ð Þ tion, which defines it as N-representable. KEM a 1 b a 1 c 1 ðÞ X¼ X¼ þ X¼

Chem.Eur.J.2018, 24,10881 –10905 www.chemeurj.org 10890  2018 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Concept and the corresponding KEM density matrices are

m 1 m m À 12 12ab m 2 12c, 11 ¼ a 1 b a 1 À ðÞÀ c 1 ð Þ X¼ X¼ þ X¼ m 1 m m À 11 11ab m 2 11c, 12 ¼ a 1 b a 1 À ðÞÀ c 1 ð Þ X¼ X¼ þ X¼

m 1 m m À 1 1ab m 2 1c: 13 ¼ a 1 b a 1 À ðÞÀ c 1 ð Þ X¼ X¼ þ X¼

In the above equations, Eab is the energy of adoublekernel of name ab, Ec is the energy of asingle kernel of name c. 12 , 11, 1 symbolize, respectively,two-body density matrices, one-body density matrices and electrondensities. Notice that the QCr/KEM procedure extracts the complete quantum mechanics based on the X-ray experiment. However, there is no mathematical requirement that the KEM summa- tion of kernels must be N-representable. Again, to resolve this QCr/KEM shortcoming, the Clinton equations will prove to be sufficient (see for example illustrations in Ref. [171]). Summarizing, the importance of the Clinton equations within QCr/KEM means that true quantum mechanics can be extracted based on X-ray data (this in KEM form) and then made to be guaranteed single Slaterdeterminant N-represent- able. This paragraph emphasizes discussion of the complete Figure 6. (a) Heat capacity (Cp) of naphthalene:from thermodynamicmeas- quantum mechanics in single determinant form, for both small urements (blue curve), calculatedfrom frequencies obtained after NoMoRe and large molecules, aclose experimental analogy to the theo- (red curve) and estimated by Aree and Bürgi (green curve), (b) difference be- retical single determinantform of DFT. tweenCpfrom calorimetric measurements and Cp estimated by Aree and Bürgi(green curve)and from NoMoRe (red curve). Reprinted from Ref. [173] underthe general license agreement of IUCr journals. 2.8. Dynamicquantum crystallography Aplethora of important bulk solid-state properties depend on rimetricmeasurements is compared with the heat capacities crystal vibrational properties. The atomicand molecular mo- obtained by the NoMoRe procedure, as well as with the relat- tions determine the vibrational entropy of crystals, and are ed modelsofBürgi and Aree.[174] therefore crucial for understandingstabilities and phase The atomicmean square displacements obtained by fitting changes.Mechanical properties, such as the elastic moduli of the normal modes against the diffraction intensities compare the crystal, are also intrinsically linked to the crystal lattice dy- well with the displacements obtained from standard crystallo- namics. graphic models. Additionally,the hydrogen atom anisotropic Information on the correlation of atomic motion is lost in displacements compare well with independent information the standard elastic scattering experiment. However,the from neutrondiffraction experiments.However,bycombining atomicmotion gives rise to changes in the diffraction intensi- aspherical atom refinement and normal mode refinement,itis ties, which are taken into account in the form of the Debye– evident that there is information in the diffraction experiments Waller factor.The temporaland spatialaverage of the atomic that is not captured by the model;[175] there is plentyofroom fluctuations—mean square displacements—can therefore be for improvements. One obvious next step in quantum crystal- retrieved from adiffraction measurement, and in recent work lographic studies of dynamics is to model Thermal Diffuse this information has been combined with lattice-dynamical Scattering (TDS). models derived from periodic DFT calculations.[172, 173] In this ap- Studies of diffusescattering from crystals are experiencing a proach,the amplitudes of the acoustic and lowest-frequency renaissance in these years. This is due to the advent of very opticalphonons are refinedagainstthe diffraction intensities. sensitivelow-noise detectors and because high-performance In the simplest model,thesephonon modes are approximated computing has made it possible to construct ab initio models by the motion at the Gamma point of the Brillouin zone. of crystals that can explain diffuse patterns. Diffuse scattering Despite the very simple lattice dynamical model,these patterns originate from ordering at length scales larger than Normal Mode Refinements (NoMoRe) capture essentialinfor- the unit cell dimensions. The ordering can be of either static or mation aboutthe crystal dynamics from the experiment. In dynamic character, and in both cases, it reveals important in- Figure 6the heat capacity of naphthalene obtained from calo- formation about the physical properties of the crystal.

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The TDS signal can be diminished by cooling the crystalsto spatialconfinement and selectivity meansthat Convergent- very low temperatures, butitcan never be fully removed. If Beam Electron Diffraction (CBED) patterns can be obtained TDS is not accounted for it will give rise to additional systemat- from regions of perfect crystallinity,avoiding grain boundaries, ic changes in the Bragg intensities,and thus create artifacts in dislocations, stacking faults,voids, surface blemishes and other the crystallographic models, as it was recently demonstrated in imperfections. amodel study on silicon and cubic boronnitride.[176] This im- The technique of Quantitative Convergent-Beam Electron plies that even in quantum crystallographic studies in which Diffraction (QCBED)involves the matching of asimulated CBED the dynamics is of secondary interest, it is important to have pattern to an experimental one whilst varying the structure an accurate model of motion in order to properly take these factors of reflectionsatornear the Bragg condition, Vhkl (i.e. contributions into account. Vg ), and the specimen thickness, H,tooptimise the fit. Over the last three decades, QCBED has developed to where it can routinelymeasurebonding-sensitive low-order structurefac- 2.9. Electron density from quantitative convergent-beam tors with uncertainties of the order of 0.1%.[178–186] Further- electrondiffraction more, it can do so without extinction and scale problems be- The electron density, 1 r ,and the electrostatic potential, Vr, cause the analysisuses afull dynamical treatment of electron ðÞ ðÞ at position r within acrystal can be described by the following scattering, necessitated by the strong interaction of electrons Fouriersums: with matter.Figure 7schematically illustrates QCBED from data collection to the determination of adeformation electron den- 1 2pig r 1 r F eÀ Á sity.From the refined values of the structure factors given in W g 14 ðÞ¼ g ð Þ the caption, the very high level of precision within such are- X finementisevident. and

2pig r 2.10. Chemical bonding analysis V r V eÀ Á ðÞ¼ g 15 g ð Þ X Apillar of the emerging field of quantum crystallography is represented by the group of methods aiming at analyzing the The sums are over reciprocal lattice vectors g, W is the unit chemicalinformation contained in experimental and theoreti- cell volume, and Fg and Vg are the Fourier coefficients (struc- cal static electron density distributions.Quantum Chemical ture factors) of the electron density and crystal potential, re- Topology (QCT) does so by analyzing local functions, f,which spectively.Byapplying Poisson’s equation in relating charge yield achemical pictureofthesystem: distribution to electrostatic potential, the Mottformula[177] de- scribes the relationship between Fg and Vg as follows: f : R3 molecular space R chemical picture ð Þ! ð Þ 2 2 2 16p e Ws 2pig rj Bj s 0 Fg ZjeÀ Á eÀ Vg 16 These methods are epitomized by Bader’s Quantum Theory e ¼ j À ð Þ [193] X jj of Atoms in Molecules (QTAIM) whose departing functionis the electron density, f 1 r .The existence of abond is asso- ¼ ðÞ The sum is over all atoms, j,inthe unit cell with atomic num- ciated with the presence of afirst-order Critical Point (CP) in bers Zj and Debye–Waller factors Bj and positions rj.The free- the electron density (or Bond Critical Point,BCP) and of the space permittivity is e , e is the magnitude of the electronic gradientline joining it to the linked atoms (or Bond Path, 0 jj charge and s sinq =l. BP).[194] Figure 8shows the ability of bond paths to reveal both ¼ ðÞ In X-ray diffractionexperiments, Fg are the observables be- covalent and intermolecular interactions in abenzene crystal. cause X-rays interact with the electron distribution, whilst in QTAIM,through integrationsover atomicbasins, yields atomic electron diffraction, Vg are the observables because electrons, properties such as QTAIM charges(carbonand hydrogen being charged, interact with the crystal potential. From Equa- charges, q,shown in Figure 8), atomicmultipoles and volumes. tions (14)–(16), it is evident that X-ray and electron diffraction The examination of BCPs provides insight into the structure measurequantum mechanical features of crystals, namely the and stability of crystalsrevealed in the bonding patterns that electron distribution and the electrostatic potential. They are, can be obtained from either theoreticalorexperimental elec- therefore, complementary techniques in quantum crystallogra- tron densities.[195] phy. Localization and delocalization (sharing)indices have also Electrons, being charged, can be focusedinto nanometer- been defined within QTAIM by integralsofthe Fermi hole den- sized (or smaller) probes using electromagnetic lenses in stan- sity (or exchange-correlation hole density in correlatedcalcula- dard electron microscopes. Additionally,the position of the tions).Delocalization indices d can yield insight on the driver focal point within the specimen can be controlled (using elec- of the binding between monomers of adimer,whether tromagnetic deflection) with sub-nanometer precision.Due to “through bond” or “through space” (i.e.,non-bonded their charge,electrons interactabout104 times more strongly atoms)[196] (see Figures 8and 9). This information can be con- with matter than X-rays and specimensmusttherefore be very densed in matrixformat knownasthe Localization-Delocaliza- thin ( 100 nm) in ordertoavoid strong absorption. This high tion Matrix (LDM).[197] LDMs are of fundamental interestbut  Chem. Eur.J.2018, 24,10881 –10905 www.chemeurj.org 10892  2018 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Concept

25 3 Figure 7. Asummary of QCBED. The first step is CBED data collection from aregion of perfect crystal ( 10À m in volume).Very high spatialselectivity  makesiteasytoavoid crystal imperfections and surface blemishes. The CBEDpatterns are corrected for instrumentalpoint spreadfunction[187] (PSF) and are differentiatedtoremove the inelastic background.[185, 188] Selected reflections are pattern matchedwith full dynamical electronscattering calculations using [189] [190, 191] either the multislice or Bloch-wave formalisms.The structurefactorsofreflections at or nearthe Bragg condition, Vhkl or Vg ,and the crystal thickness, [177] H,are refined to minimizethe mismatch (following the loop of the grey arrow). The refined Vg canbeconverted to Fg via the Mott formula for the deter- mination of the deformationelectron density, D1 r ,asshown here for the example of aluminium[186] (drawn with VESTA).[192] In this example, the CBED pat- ðÞ tern was collected with 160 keV electrons along [001].The refined structure factors were V =5.29 0.01V(F =33.39 0.03e )and V = 3.219 0.006 V 200 Æ 200 Æ À 220 Æ (F =29.35 0.04 e )atT=0K.The refined specimenthickness at the electron probeposition was H= 1056 1 Š. 220 Æ À Æ

Figure 8. CH p and CH Cinteractions in abenzene crystal:(a) QTAIM, (b) NCI, (c) ELF.(d) LDM for isolatedbenzene (see Figure9for LDM of the benzene À À dimer).

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Figure 9. (Top) Localization-Delocalization Matrix of an isolatedbenzene molecule, and (middle andbottom) the two partial LDMs (sub-matrices) of mono- mers Aand Bofabenzene dimer (withoutthe 2”12”12 interaction elements that appear in the full 24”24 matrix of the dimer). Notethat the sum of column or rows of the isolatedbenzene molecule is 42.001 indicating a+0.001 cumulative integration error (numericalnoise),which is 0.002 %ofthe exact electronpopulationofbenzene (N=42). The full 24”24 LDM of the dimer (not shown) sum to 83.996 (with acumulative integrationof 0.004). The differ- À ence between that total dimer population (that includes the numerical noise) and the sum of the populationsofmonomers A(41.936) and B(41.917)within the dimer is 0.143 electrons. Those 0.143 electrons are delocalized between the two rings and hold them together,taking as the most dominant delocaliza- tion bridges the atoms connected by inter-monomer bond paths, namely, d(C4, H22)=0.02, d(C12, H18)=0.01, but also other atoms mainly d(C4, C10)=0.01, and d(C7, H18) =0.01. also have practical applications in Quantitative Structure–Activ- can be derived from the Laplacian of the electron density,an ity Relationship (QSAR) predictions of molecular proper- experimentally accessible quantity,from the Bader-Gatti Source ties.[197,198] LDMs need the first- and second-order density matri- Function (SF)[199] whichcan also be casted in amatrix form and ces for their construction,which for experimental densities re- used in asimilar fashion as LDMs.[200] quires the techniques introduced in the previous sections. Due to their delocalized nature, special f functions have Thus, generally at the time of writing, when only the electron been designed to visualize non-covalent interactions. As an ex- density is available, then LDMs cannotbecalculated from the ample,the reduced density gradient (aka NCI for non-covalent experimentaldata alone. However,acomplete population interactions)[201] has been designed to detectweak interactions analysisincluding localizationand delocalization information such as halogen bonds from the electron density.Ithelps to

Chem. Eur.J.2018, 24,10881 –10905 www.chemeurj.org 10894  2018 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Concept provide more stable pictures that do not change upon the The topological analysis of any scalar functionprovides quality of X-ray refinement.[202] Weak interactions in abenzene richer information than appears from the direct observation of crystal are shown within this approachinFigure 8. The delocal- the function itself. In the widely used QTAIM theory,[193] this ized nature of CH–p vs. CH–C interactionsisapparent. kind of analysishas been deeplydeveloped for 1 r)tocharac- ð Another important set of f-functions are those for the analy- terize many aspects that this functionexhibits in atoms,mole- sis of electron pairing, such as the electron density Lapla- cules and their interactions. As an alternative, analogous topo- cian,[203] the Electron Localization Function (ELF)[204] and the logical concepts to 21 r and V r can be appliedtoget in- r ðÞ ðÞ Electron Localizability Indicator (ELI).[205] This family of functions sights into recognition, assembling and organization of mole- identifies localized electrons, such as those in covalentbonds cules in space. Hence, the study of the topological critical and lone pairs (Figure 8). By integration over bonds and lone points (CP) of 21 r has indicated that charge concentration r ðÞ pairs, properties (e.g. valence populations as shown in (CC) and charge depletion (CD) sites found in the valence shell Figure 8) can be obtained. This analysis allows for example to of atoms are drivinggeometric preferences of molecules in the rationalize the formation of channels in MOFs,clathrates or solid state.[209–211] This is shown in Figure 10a, in which direc- molecular crystals.[206] It is important to note that the ELF and tional nucleophilic-electrophilic interactions between several ELI require the first-order density matrix, so that quantum crys- CC and CD sites are simultaneously involved in the relative ori- tallography developments also hold great potential for these entation of three molecules in acrystal.Additionally,specific analyses.[57] CPs along bonding directions point out the nature of interac- tions.[212, 213] Thus, in the polyiodide chains of the crystal struc- ture of tyrosinium polyiodide hydrate (Figure 10b), iodines are 2.11. Crystal Engineering distinguished from iodides by the type of CPs found in their in- Molecular recognition,molecular assembling and molecular or- teractions with surrounding atoms. In the case of V r ,its gra- ðÞ ganization in space are three of the most important aspects in dient vector field(i.e. the electric field E r V r )and the ðÞ¼Àr ðÞ the study of intermolecular interactions. They have astrong corresponding zero-flux surfaces originating from its topology impact in large domainsofscience such as crystal growth, permittocharacterize the force lines driving the early interac- crystal engineering, supramolecularchemistry and materials tion between molecules and the extension of the influence science, because the physical–chemical properties of supra- zones that molecular electrophilic and nucleophilic sites exhibit molecular entitiesand crystalline solids do not only depend on in intermolecular regions (Figure 10c).[214] Furthermore, the in- the molecules they are constituted of, but also on the way tersection of the gradient vector fields of 1 r and V r has ðÞ ðÞ molecules interact with each other. This is astraightforward shown to help to understand the counterintuitive assembling consequence of the structure-properties relationship. The case of anion-anion and cation-cation aggregates,[215–218] pointing of polymorphism in crystalline solids can be invoked as an ex- out the regions of space that are involved in local attractive ample of the modification of the solid properties induced by electrostatic interactions in hydrogen[219] and halogen[212] bond- differentorganizationsofthe same molecules in space.The ing. To illustrate these features, Figure 10 dshows the phos- variability that these molecules can display in their assembling phate-phosphate aggregation at equilibrium geometry,where can thusbeunderstood in terms of the severalpossibilities ex- attractive electrostatic forces take place in the hydrogen-bond- isting in the recognition betweendifferentmolecular regions. ing region and keep the complex assembled. Indeed, in spite From the Hellmann–Feynman theorem,[207,208] the electron of the global destabilizing interaction due to Coulombic repul- 1 density distribution 1 r in the ground state of anymolecular sion (interaction energy > + 100 kJmolÀ ), the system is ðÞ 1 system only depends on the nuclear positions. Accordingly, trapped in asharp potential well and ~70 kJmolÀ are needed 1 r is straightforwardly relatedtothe molecular structure, be- to overcomethe energetic barrier to dissociate the aggrega- ðÞ cause the latter is defined by the atomic positions that are in tion. turn identified by those of the nuclei. On the other hand, the Hohenberg and Kohn theorem[60] states that the total energy 2.12. Spin densities of asystem can be writteninterms of its electron density dis- tribution. Hence, bringing together both fundamentaltheo- Modeling magnetic properties of crystalline compounds by ex- rems, the 1 r function emerges as aconceptual bridgebe- periments is achallenging task for many reasons. First, the ðÞ tween the structure and the energetic properties of molecular electrons contributing to the magnetism are few compared to systems. Agood strategy to understand the relationship be- the overall content of the unit cell. Asecond problemcomes tween them is thus the analysis of 1 r ,which codes essential from the necessity of obtaining good quality crystalsofsuita- ðÞ features of the molecular organization in space because it re- ble size for neutron diffraction (mm3)including long irradiation flects the molecular interactions in the system.Byextension, and data collection times. Moreover,the magnetic properties these features are not only implicitlycoded in 1 r ,but also in usually appear at very low temperature (helium cooling)and ðÞ its derived properties, as in the Laplacian 21 r and the elec- experiments requiremagnetic fields to be applied on the sam- r ðÞ trostatic potential V(r). These scalar functions characterize the ples, excluding routine experiments. Finally,the number and regions of the space where 1 r is locally concentrated ( 21 r the experimentalresolution of the collected data are limited, ðÞ r ðÞ <0) or depleted ( 21 r >0) and the molecular electrophilic making it necessary to refine models with only alimited r ðÞ V r >0) and nucleophilic (V r <0) regions. number of parameters. ððÞ ðÞ Chem. Eur.J.2018, 24,10881 –10905 www.chemeurj.org 10895  2018 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Concept

Figure 10. (a) L r 21 r experimentalmap in intermolecular regions of the C O H Se crystal structure:CCsites are represented by (3, 3) CPs of L ðÞ ¼ Àr ðÞ 8 2 4 À (yellow spheres), CD sitesare denoted by (3, + 3) (violet), (3, +1) (pink)and (3, 1) (green) CPs of L. CC···CDdirectionsclosely correspond to internucleardi- À rections(dashed lines), which in turn almost superimpose with bondpath directions(bondcritical points are in blue). (b) CPs of L r 21 r in two re- ðÞ¼Àr ðÞ gionsofapoly-iodide chain (same color-code used as in (a)): covalent bondsare identified by (3, 3) CPs in the middle of the bonds of I1-I2, I4-I5 and I9-I10, À whileintermolecularinteractions are characterized by (3, +1) CPs in the middle of I2···I3···I4and I8···I9 regions. I9 is an iodine atom exhibiting atype-II halo- gen bonding interaction with iodine I8 (I8(CC)···(CD)I9) and acovalentinteraction with I10,whereas I3 is an iodide displaying non-covalent interactions with both I2 and I4 iodine. (c) Experimental gradient vector field of V r for the l-histidiniumcation in the crystal structureofl-histidinium phosphate phosphoric ðÞ acid:the influence zone of its nucleophilic COO-group is limitedbyazero-flux surface of V r ,the distance up to the saddle pointinthe surface ((3, + 1) CP, ðÞ red sphere)being approximately 6 Š.(d) At the equilibrium geometry of the phosphate…phosphate complex, the O-atomic basin (grey region)lays within the influence zones(electrostatic basins) of the H-nuclei.Black E r lines crossing the region between the zero flux surfaces of 1 r)(grey border) and V r (red ðÞ ð ðÞ border)are attractiveforce lines,whereas blue E r dashed-lines crossing the sameregion are repulsive force lines. Figure 10areprintedwith permissionfrom ðÞ Ref. [211],Copyright2013 American Chemical Society; Figure 10breproduced from Ref. [212]with permission from the Royal SocietyofChemistry;Figure 10c reprinted with permission from Ref. [214], Copyright 2007 American Chemical Society;Figure 10disadaptedfrom Figure3(upper left, including atomic labels) in Ref. [216], Copyright 2013, with permission from Elsevier.

Recently,new methodological routes and softwarehave To analyze these modeled (or theoretical) spin densities, been developedcombining data obtained by different tech- new methodsare developed or adapted, thus giving rise to niques such as X-ray and polarized neutron diffraction[23,220] or new tools to explore magnetic pathways in crystals. For exam- polarizedneutron and X-ray magnetic diffraction.[221] Combin- ple, the SourceFunction methodcan produce representations ing different experiments is away to overcome limited resolu- that allow for visualizationofthe magnetic pathways and for tion and leads to more precise modelsofspin or spin-resolved conclusions about the specific role played by each atom along electron densities. From these joint refinements, experimental them.[222] charge density or wavefunction-based models can be ob- Another complementary way is to combine theoreticalcal- tained. These experimental modelsare of utmost importance culations and experimental data sets in constrained wavefunc- to test high-level theoretical calculations, as recently shown for tion refinement approaches.[28–30,32] Thismethod has provento aparamagnetic radical.[117] be efficient for combining X-ray data and calculations, but one

Chem.Eur.J.2018, 24,10881 –10905 www.chemeurj.org 10896  2018 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Concept can imagine to include in this approachexperimental magnetic change of geometry leads to an abrupt change in the crystal structurefactors. graph (whichisthe entirety of the bond paths in the crystal). All these promisingdevelopments should not mask several Description in terms of topology is sharply discontinuousin critical aspects. First, neutron diffraction facilities are undergo- such acase, whileitiscontinuous in terms of electron sharing ing significant changes:most reactorswill close (ILL in Greno- between pairs of atoms, regardless of abond path linking ble, LLB in Saclay), but spallation sources appeared around the them existsordoes not exist.[224] This impasse, partly solved by world (Switzerland, Japan,USA…) and anew one is under con- showingthat BPs play the role of privileged electron exchange struction in Sweden (ESS, Lund) for the European researchers. channels,[226] has called for an ever increasing adoption of con- These new facilitiesshould develop beam-linesable to carry tinuous descriptors,directly/indirectly related to electron pair- out polarized neutron diffraction experiments (like MAGIC at ing and able to characterize multi-centerbondingaswell as ESS), but also ensure our community with asufficient beam- the strengthening or weakening of competing interactions in time access. To support this demand, we have to enlargethe series of chemically related systems.[224] The delocalization indi- number of possible users of our methodsand this represents ces (DI),[196,227,228] the ELF,[229] the ELIs,[230] the Domain Averaged the second important point. For this, we have to consider that, Fermi Hole (DAFH),[231, 232] the SourceFunction[233] andthe de- until now,the studied crystals are mainlyorganic or metal-or- scriptorsinherenttothe InteractingQuantum Atom (IQA)[230] ganic compounds.Toopenour community to solid-state phys- energydecomposition supply anon exhaustive list of them. icists, we have to apply our methods to inorganic compounds Their joint use provides complementary,often compelling in- in order to prove their capabilities in exploring the magnetic sight, but warrants thoughtful analyses.[195, 224] Apart from the properties of these materials.Apossible approachconsists in Source Function, all these descriptors require the pair density including delocalized electrons in our models. Indeed, experi- or at least the 1-RDM for their evaluation, calling for QCr tools mental Compton and magnetic Comptonprofiles can be used to retrieve them directly from ab initio methods or,indirectly, to model delocalized spin and charge densities that are of fun- through the XCW approach. Although akin to an orbital pic- damental importance in inorganic materials (for example, in ture, DAFH analysisisfully derived from aquantum observa- conducting properties studies). Proposing this kindofnew and ble. It so establishes alink with the localised single-particle more complete model is acurrentlyongoingeffort.[223] states approaches (ELMOs,[234] Natural Bond Orbitals (NBOs),[235] extremelylocalized Wannier functions).[236] Besides defining the network of bonds, QCT methods serve 2.13. Molecular and extended solids for characterizing their nature.When heavy atoms are in- Molecular and extended solids (M&ES) include quite diverse volved, the task is not as simple as for bonds within light ele- objects: on the one hand molecular crystalsmade by organic, ments.[195,224] It calls for additional descriptorstobeexam- inorganic or metallorganic molecules and, on the other hand, ined[237] and/orconventional ones to be reinterpreted.[195,224] extendedsystemsformed by n-dimensional (n= 1–3) periodic Many properties of M&ES, like those relatedtoelectronic networks, such as ionic solids (salts),covalent/polar solids (e.g. transport, depend directly on reciprocal space properties (e.g. semiconductors;binary compounds:carbides, silicides, ni- energydispersion of electronic bands in k-space). Yet, not trides) and metallic, intermetallic and low-dimensional organo- much is known on how the characteristic features in the band metalliccompounds. Focus here is only on the non-organic structure are reflected in the electron density.[238] This relevant, M&ES, although some of the considerations apply equallyto although seldomly explored QCr aspect, has been tackledina the organic M&ES in specific cases. study on transition metal carbides[238] andfor optimizingther- Most M&ES are of relevance for their technologicalapplica- moelectric compounds via orbitalengineering.[239] tions and are intensely scrutinized, with QCr playing abasic M&ES provide an endlessfieldofapplication and develop- role in the puzzle that relatestheir geometrical structure, ment for QCr approaches. Figure 11 detailstwo examples. chemicalbondingnature and properties to each other. Acommon trait of many M&ES is that the sole knowledge of 2.14. Quantumcrystallography and materials science their crystal structure is not enough to infer their chemical and electronicstructure.[195,224,225] Chemists like to draw dashes, Althoughquantummechanics may appear as acomplicated arrows and dotted lines connecting atoms or groups of them, theorythat is understandable only to experts in the field, a but there are plenty of M&ES where this operation is ambigu- huge array of recent inventions rely on it (opto-electronic com- ous or simplyimpossible, if based on their structure alone. ponents,devices to store and transform energies, thermoelec- Electron density and wavefunction analyses are often used to tric materials, etc.). Even if many phenomena were knownalso provide insights. Nevertheless,even with these tools, the in the pre-quantum mechanics era, only after its development answer may not be unique, as it may dependonthe kind of one could correctly interpret the mechanismsand therefore descriptor that is used and on the physical space with which designmaterials, beyond modelsbased on classicalmechanics. such descriptor is associated (1-electron R3,2-electrons R6, Nonetheless, the quantum mechanical information extracted ! ! basis-function space,local or non local, etc.,comparesection from experiments remains largely unused, and this is true in 2.10).[224] crystallography as well. However, the recent research in quan- There are M&ES in which two or more alternative pairs of tum crystallography opens new perspectivesand potentialap- atoms compete for abond path and where even anegligible plications,asdescribed in the following.

Chem. Eur.J.2018, 24,10881 –10905 www.chemeurj.org 10897  2018 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Concept

Figure 11. Examplesofapplication of QCr methods to the realm of molecular and extended solids. (Left) Twoseparated valence shell charge concentrations, indicative of two lone pairs, are present in the non-bonding region of the central Si atom. They confirm the interpretation of the compound (cAAC)2Si as asi- 2 5 lylone stabilizedbytwo cyclic alkyl aminocarbenes(experimental electrondensity Laplacian isosurfaces, 1= 2.5 eŠÀ ). Adapted from Scheme1and r À Figure 1with permission from Ref. [240],Copyright 2014 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim. (Right) Acombined experimental and theoretical chargedensity study on the quintuply bonded dichromium complex (1)Cr2(dipp)2 (dipp=(Ar)NC(H)N(Ar) and Ar=2,6-iPr2-C6H3). The electron density at the Cr-Cr BCP is essentiallydetermined by the metalions, as shown by the Source Function percentage (SF%) contributions[contrast SF% values for Cr Cr and À Cr X(X= N,O) interactions].Blue and red balls indicatesources and sinks, respectively.Data for compound 2 with formal bond order 4are reported for the À sake of comparison.Adjustedfrom Figure 8with permission from Ref. [241],Copyright 2011American Chemical Society.

Quantummagnets are appealing materials, consisting of time scale, time-resolved experiments (down to femtosecond spin centers coupledthrough exchange interactions. They may resolution) are essential to understand processes that underlie give rise to extendednetworks of different dimensionality as materialbehaviors.Althoughexperimental requirementsdo well as to zero-dimension molecular magnets. Superconduct- not yet allow avery accurate mapping of the charge density, ing quantum interference or muon spin spectroscopy are one may envisage afruitful synergy between theoretical ap- methods to measuremagnetic susceptibility,spatial ordering proachesand experimental measurements to overcome cur- and spin dynamics, and thereafter derive simplemodels of ex- rent limitations. change. However,the quantum informationislimited to the Quantum crystallographic studies on many other kinds of spin states and their relative energies, whereas scattering ex- materials, such as thermoelectric pulses, metals and alloys,su- periments, in particular elasticorinelastic neutron diffraction, perconductors etc.,[250] have appeared with the goal of finding enable to model the magnetic structure of acrystal and its dy- amore robuststructure-property correlation.The advanced namics. Moreover,flipping ratio measurements[242] of polarized methodologies of quantum crystallography may certainly neutrondiffraction allow the refinementofspin density distri- enable to gain more insights into the nature of properties. butionsinsolids,[243,244] even in combinationwith the charge density,asrecently demonstrated.[23] The charge and spin den- 2.15. Crystal structure prediction sity models inform on the electron correlation,the preferential exchange paths and the corresponding strength of spin cou- Crystal Structure Prediction (CSP) wasanalmostimpossible pling. This is vital to understand how the magnetism works challenge for along time,[251, 252] but recent methodological de- and how to optimize the materialdesign.[245] velopments[253] have eventually made CSP manageable and led Dielectric properties are equallyimportant in materials sci- to many successesand impressive discoveries.[254–257] Given a ence. The industry of semiconductors seeks for high- or low-di- chemicalformula, CSP stands forfinding the corresponding electric constantmaterials (with respect to SiO2), necessary for stable crystal structure at agiven pressure (and temperature). the miniaturization of micro-electronic components, to reduce Asimultaneous prediction of all stable stoichiometries and the size of gate dielectrics or to guarantee better separation structures for aset of composing elements is performed using between transistors.[246] In telecommunication industry,trans- the variable-composition variant of CSP methods.[258] parentmaterials with high refractive index are fundamental to CSP is aglobal optimization problem since the stable struc- improvethe performances of fiber optics.[247] The direct obser- ture is associated with the lowest minimum of the free energy vation of atomic and molecular polarizations, due to the inter- surface.[253] The starting point in metadynamics, simulated an- nal electric fields in crystals, gives aclue of the atomic/molecu- nealing, basin hopping, and minima hopping approachesis lar polarizability.Spackmanand Jayatilaka[126] demonstrated the chosen in agood region of configuration space to avoid sam- possibility of calculating polarizabilities of molecules in crystals pling in poor regions,while in the self-improving methods the using X-ray constrained molecular orbitals. Using the quantum best structures are located step by step, using evolutionaryal- theory of atoms in molecules, one can calculate polarizabilities gorithms. The latter are particularly suited and powerful for of atoms, functional groups or molecules embedded in crystal- CSP,being unbiased, fully ab initio (only the exact or variable line matrices and therefore recognize key factors for the tech- chemicalcomposition needs to be known)and able to pro- nological requests.[248] duce, by their own nature,increasingly good structures in sub- Quantumcrystallographic approaches also enable the char- sequent generations.[258] acterization of non-equilibrium phenomenainmolecular crys- Intimate relationshipslink CSP and QCr.Anenthalpy or free tals, due to photo-excitation or X-ray probe pulses.[249] Because energy value needs to be computed for each sampled struc- these phenomena depend on structural changes on awide ture in CSP approaches. Depending on the method, also a

Chem. Eur.J.2018, 24,10881 –10905 www.chemeurj.org 10898  2018 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Concept local structuralrelaxation is derived using quantum mechanical between alkali and transition elements progressively vanish- (periodic electronic structure)calculations or suitable force es.[262] For astudy of chemical bonding in such unconventional fields, calibrated via QM calculations. DFT approaches are gen- cases, this requires completely unbiased approaches, such as erally adopted,but they have their well-known limits—a ten- the QuantumChemical Topology methods,[264] based on QM dency to prefer electron delocalization (HF methods behave observables, the heart of QCr. oppositely)[259] favoring delocalized metal-like structures over One representative example is the recentdiscovery[256] of covalently bonded insulating structures.Inaddition DFT has Na2He, the first neutralthermodynamically stable compound the difficulty in properly evaluating dispersion energy effects of He (at pressure >113GPa), using avariable-composition with auniform accuracy in the whole range of sampled pres- evolutionary CSP approach, followed by experimental valida- sures.[260,261] Therefore predicted structures and phasediagrams tion and rationalization of its geometrical and electronic struc- need to be validated[254–257] through experiments, also inherent- tures in terms of QCT methods (Figure 12). Na was loaded into ly relatedtoQCr (X-ray diffraction, Raman spectroscopy,etc.). aHemedium in alaser-heated diamond anvil cell and com- CSP often involves exploring unbeaten tracks of potential pressed up to 155 GPa, with synchrotronX-ray diffractionused energy surfaces, leading to thediscovery of new structures to monitor sample evolution. Na2He resembles a3Dchecker- where atotally new chemistry emerges, with exotic, unantici- board with Na8 cubes alternatively filled with He or allocating pated bonding situations.[254–257,262,263] Indeed, even at moder- an interstitially localized electron pair, revealed by anon-nucle- ate pressure (<10 GPa), the compressiveenergy,which adds ar attractor electron density maximum and amassive accumu- up to the system internal energy,isakin to the energy of a lation in the deformationelectron density map. Na2He is an moderate strength hydrogen bond and, at high pressure (< electride, with localized electron pairs forming 8-center-2-elec- [262] 100 GPa), is large enough to break normalchemical bonds. tron bonds within the empty Na8 cubes. Our usual chemical knowledge is based on the periodicity of atomic properties (radii), reflecting the periodicity in the elec- 3. Future Perspectives tron configurations of the outermost shell.When pressure in- creases, the variation in atomicradii across the periodic table In the previous sections, it emerged that significant research becomes much less important and thus, e.g.,the distinction based on quantum mechanics encompasses several areas of

Figure 12. Na2He:(Topleft): predicted convex hulls of the Na-Hesystem, based on the theoretical ground states of Na and He at each pressure. The calculat- ed DH (Na He )=xH(Na)+(1-x)H(He) H(Na He )(H=enthalpy) indicates that Na He has alower enthalpy, hence is thermodynamically morestable, formation x 1-x À x 1 > 2 than the mixture of elementalNaand He, or any other mixture, at pressures above 160 GPa. (Top right): SynchrotronXRD data. Below113 GPa only single crystal reflections of elementalNawere observed, whereas abovethis pressure new single crystal reflections appeared,which becameevenstronger after laser heatingtoatemperature >1500 K. Vertical ticks correspondtoexpected positionsand intensitiesofXRD peaksofNa2He, tI19-Na, Re (gasket) and W

(pressure gauge). (Bottom left): Geometric and electronic structure of Na2He. (Bottom right): Interaction density in the (100) plane of Na2He passing through

He atoms.This quantity was obtainedbymaking separate calculations on the Na and He sublattices (i.e. Na2He without He and without Na, respectively) and subtracting the resulting electron densities from that of Na2He. NNAs indicateinterstitialelectronlocalizations. Insertion of He pushes electrondensity out, causing its interstitiallocalization. Adjusted from Figures 1, 3and 4with permission from Ref. [256]and (bottomright) from Figure 1inRef. [265], with per- missionfrom and credit to Artem Oganov, Skolkovo Institute, Moscow,Russia.

Chem. Eur.J.2018, 24,10881–10905 www.chemeurj.org 10899  2018 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Concept crystallography,both experimental and theoretical, with a First option:Preserving the original meaning strong interplay between them. This implies the existence of a field of quantum crystallography,although aconcise and com- According to this view,quantum crystallography would remain prehensive definitionisdifficult, at least at this stage. Thus, the tightly linked to wavefunction or density matrix modeling with question remains:what exactly is quantum crystallography? constraints to scattering experiments and, conversely,tothe According to Feynman,[266] quantummechanics “is the de- improvement of crystallographic information via ad hoc QM scriptionofthe behavior of matter and light in all itsdetails calculations. In this option, the purposeofQCr is making pre- and, in particular, of the happenings on an atomicscale”. dictions of crystal features and properties more reliable than Amongvariousfields, he mentioned also the study of periodi- from pure first-principle calculations or experimentallyderived cally arrangedquantum systems. Often scientists use “quan- models.Anadvantage of this option is that aclear definition is tum mechanics” and “quantum physics” as synonyms, to em- possible that coincideswith the historic development of the phasize the intrinsic quantum nature of all investigations deal- term, thus avoiding misunderstandings. One relevant drawback ing with atoms and molecules. For this reason,inphysics, of this vision is the exclusion of charge density refinements, quantum mechanics is not apurely theoreticaldiscipline, but it which exploit the more traditional multipole model techniques embraces all aspects of anatural science:observations,ration- or the maximum entropy method. alizations, theorizations and applications.Onthe contrary, quantum chemistry and quantum biology,although lacking a Secondoption:Outcome-based definitions. formal definition, are often perceived only as theoretical disci- plines that provide answers by performing computer calcula- OptionI.clearly limits QCr to the instrumental scope of en- tions andexploiting sophisticated software. Nevertheless, both hancingquantum mechanical modelsorcrystallographic infor- of them have experimental counterparts, mainly spectroscopy mationthrough their suitable combination,when, actually,in- or any form of energy conversion. On the other hand, quan- formation on asystem may also be enhanced the other way tum crystallographyismore naturallybound to scattering ex- around. Adifferent definition could indeed be based on the periments, as clearly reflected in the originaldefinition by purpose of the study,without arigid limitation to experimen- Massa,Karle and Huang,[24, 46] as well as in many previous tally constrained wavefunction strategies or refinementsin- works on refinements of electrondensities, density matrices trinsically linked to updated wavefunction information. This im- and wavefunctions.[6,14,15, 27,28,88] In fact, historically,the interplay plies amore classicalapproach:experiments validate and stim- between quantummechanical modelsand experimental scat- ulate theoretical predictionsand vice versa. In fact, crystallo- tering data has been enormousand vital for crystallography in graphic data contain informationonareal system (which is in its entirety (see Introduction). principle “unknown”, defective, at agiven temperature, etc.) We want to highlight that, in general, the term “quantum” and on aspace/time averaged basis, whereas QM calculations does not only refer to computationalapproaches. In fact, ex- refer to amodel system (with acertain composition, geometry, periments by themselves can reveal the quantum nature of Hamiltonian, etc.) andhave both the advantages and the dis- matter and describe quantum mechanical phenomena(e.g. advantages of treating awell-defined, but approximate object. the Stern–Gerlachexperiment, Bose–Einstein condensation, su- In this view,QCr would be the branch of science studying perconductivity,tunneling effects, or,infact, scattering of X- the quantum mechanical functions (and properties derived rays by electrons and of electrons by the electric potential). In from them) in crystals. This includes the investigation,inposi- turn, crystallography,being anatural science, is not only asso- tion or in momentum space, of chargeand spin density,wave- ciated with observations (and not only with scattering tech- functions, density matrices, based on experiments,ontheoreti- niques). Quantum mechanical modelshave always been neces- cal calculations or on acombination of them. sary to interpret the measurements andovercome the loss of the phase information in the X-ray diffraction data. Further- Third option:Crystalsasquantum objects more, crystallographyalso consists of direct applications of first-principle quantummechanical methods with periodic According to this view,QCr is the study of those properties boundary conditions (nowadays implemented in world-wide and phenomena which occur in crystalline matter and can be recognized software, such as QuantumEspresso,[69] explained only by quantum mechanics. Again, acombination WIEN2k,[62,71] Crystal,[267] VASP,[268–271] Turbomole[70])tosolve of experimentalmethodologies andtheoriesdescribe, explain problemsinsolid-state physics/chemistry. and predict those phenomena. In keeping with Feynman, the Nevertheless,while the originaldefinitionofQCr encom- varioustheories concern especially the interaction between passes only the developments andapplications of strategies matter and radiation (which reveals the atomic and electronic that combine quantum mechanics and experimental scattering structure of acompound) as well as the bonding between in astrictly intertwined protocol, it is evident that other facets atoms or molecules(which dictates the electronic structure of quantum crystallographyfoster abroadening of its defini- and explainsthe behavior of amaterial). The experimental tion. In the following subsections, we present the different per- methods include scattering and spectroscopictechniques, with spectives that have emergedsofar and their possible implica- observations enabling to refine quantum mechanical models tions. that reveal structuralorfunctional features. Specific goals of QCr are the determination of quantum relatedfunctions and

Chem. Eur.J.2018, 24,10881 –10905 www.chemeurj.org 10900  2018 Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim Concept quantities (such as wavefunctions, charge and spin densities, Risks and Opportunities density matrices, electric or magnetic moments, etc.), the eval- uation of the properties of materials and the analysisofthe One of our goals is finding anew definition of QCr.However, bondingfeatures between the atoms and moleculesthat con- this includes the inherent risk of confusingthe new meaning/ stitute acrystal. definition of QCr with the original one that hasjust recently This open definition reflects the broad definitions of quan- been reviewed[44,45] or the risk of excluding new facets that will tum biologyand quantum chemistry,that is, the application of naturallyemerge due to acontinuousprogress of crystallogra- quantum mechanical concepts to chemical compoundsand phy and quantum mechanical methods. Therefore, it is impor- biologically relevant objects.This implies that QCr does not tant to dynamically fill the new definition with applications only include determination of wavefunctions, density matrices and examples in order to be acknowledged as useful and or electron densities, but, more generally,itconcerns the study meaningful. This will allow beneficial interactions with neigh- of electronic structures and their changes, including, for exam- boringresearch areas where important innovations are expect- ple, the determination of magnetic structures, photo-induced ed. This article intends to provide afirst attempttoset the processes, electric and magnetic polarizations and polarizabili- domain,toaddress the connections with other disciplines and ties, etc. In this definition, the simple determination of crystal the possible implicationsofquantum crystallographic studies. structures is not per se aquantum crystallographic study,ifthe Strongsynergiesare expected with materials and life sciences, purpose is simply mechanical but not quantum-mechanical. In including many different fields of crystallography. other words,the determination of astructure to provide a “ball and stick” picture of amolecule or asolid cannot be con- sidered quantum crystallography,although quantum crystallo- graphic methodsmay find application also for more general Acknowledgements purposes (see also next section). On the other hand, determi- nations of structures of photo-excited speciesor(electric or The authors thank Hans-Beat Bürgi, Lukµsˇ Palatinus, Wolfgang magnetic) field-polarized compounds clearly belongtoquan- Scherer,Michael E. Wall, Bartolomeo Civalleri, Roberto Dovesi, tum crystallography.This open definition meansthat QCr is Alessandro Erba and Silvia Casassa for their contributionsto the field that bridges structure and functions through the dis- the discussion meetinginterms of their definitions and opin- tribution and dynamics of electrons in space. ions about quantum crystallography.Financial support for the discussion meeting by the Centre EuropØen de Calcul Atomi- que et MolØculaire, Bruker AXS GmbH,WIEN2k, TURBOMOLE GmbH,French ChemicalSociety (SCF), and MØtropole du Fourth option:Quantum crystallography as amultidiscipli- Grand Nancy is gratefully acknowledged. A.G. acknowledges nary field. the French Research Agency (ANR) for financial support If quantum crystallographic studies tackle all quantum me- through the Grant No. ANR-17-CE29-0005. L.B. acknowledges chanics-based problems in crystallography,then the possible financial support from VEGA (grant nos. 1/0598/16 and 1/0871/ connections with other disciplines are enormous. Molecular 16) and APVV (grant nos. APVV-15-0079 and APVV-15-0053). chemists, biochemistsand solid-state chemists may take ad- E.E. acknowledges the French Research Agency (ANR) for finan- vantage of improved modelsfor equilibrium structures at spe- cial support through the grant ANR-08-BLAN-0091-01, CINES/ cific thermodynamic conditions. Materials scientists may appre- CEA CCRT/IDRIS for allocation of computing time (project ciate an improved knowledge of the dynamics of atomicand x2016087449) andhis main collaborators EmmanuelAubert, electronic structures under perturbation. Theoretical chemists Slimane Dahaoui, Ignasi Mata, Ibon Alkorta, Elies Molins and will find new and more precise ways to test their theoretical Marc FourmiguØ.C.G. gratefully acknowledges funding from models. Danmarks Grundforskningsfond(awardNo. DNRF93). P.G. ac- However, this concept of QCr is not limited to solid-state sci- knowledgessupport from the EU through the MaXCentre of ence, butalso includes surfacescience as well as studies of Excellence (Grant No. 676598). J.-M.G. gratefully acknowledges nanoscale materials,for example, via radiation-damage free the collaborationofSaber Gueddida and Zeyin Yan. P. M. ac- femtosecondX-ray protein nanocrystallography[272–274] or via knowledgesfundingform the Swiss National Science Founda- electron diffraction in thin films and monolayers.[275] Since QCr tion (project 160157).C.F.M. acknowledges the Natural Scien- is inherently associated with scattering, an obvious extension ces andEngineering Research Council of Canada (NSERC), of the domain is the scattering and imaging of isolated Canada Foundation for Innovation (CFI), the Lady Davis Fellow- (macro-)molecules or single objects such as cells,[276] which is ship Trusts &The HebrewUniversity of Jerusalem and Mount nowadays feasible with new X-ray lasing or intense electron Saint Vincent University for funding. P.N.H.N.acknowledges the sources.[277–281] In this respect, we note that the term “quantum Australian Research Council for funding through the Future crystallography” has been also used for these kinds of stud- Fellowships Scheme (grant number FT110100427). U.R. ac- ies.[56] knowledgesfinancial support from the Swedish research coun- cil (project 2014-5540). S.G. thanks DFG within EmmyNoether GR 4451/1-1. The frontispiece is the painting “Mehr Licht” by F. Rasma (1991).

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