Quantum crystallography

Quantum crystallography is a branch of crystallography that investigates crystalline materials within the framework of quantum mechanics, with analysis and representation, in position or in momentum space, of quantities like wave function, electron charge and spin density, density matrices and all properties related to them (like electric potential, electric or magnetic moments, energy densities, electron localization function, one electron potential, etc.). Like the quantum chemistry, Quantum crystallography involves both experimental and computational work. The theoretical part of quantum crystallography is based on quantum mechanical calculations of atomic/molecular/crystal wave functions, density matrices or density models, used to simulate the electronic structure of a crystalline material. While in quantum chemistry, the experimental works mainly rely on spectroscopy, in quantum crystallography the scattering techniques (X-rays, neutrons, γ-Rays, electrons) play the central role, although spectroscopy as well as atomic microscopy are also sources of information.

The connection between crystallography and quantum chemistry has always been very tight,[1] after X-ray diffraction techniques became available in crystallography. In fact, the scattering of radiation enables mapping the one-electron distribution[2][3][4] or the elements of a density matrix.[5] The kind of radiation and scattering determines the quantity which is represented (electron charge or spin) and the space in which it is represented (position or momentum space). Although the wave function is typically assumed not to be directly measurable, recent advances enable also to compute wave functions that are restrained to some experimentally measurable observable (like the scattering of a radiation).[6][7]

The term Quantum Crystallography was first introduced in revisitation articles by L. Huang, L. Massa and Nobel Prize winner Jerome Karle,[8][9] who associated it with two mainstreams: a) crystallographic information that enhances quantum mechanical calculations and b) quantum mechanical approaches to improve crystallography information. This definition mainly refers to studies started in the 1960s and 1970s, when first attempts to obtain wave functions from scattering experiments appeared,[10] together with other methods to constrain a wavefunction to experimental observations like the dipole moment.[11][12] This field has been recently reviewed, within the context of this definition.[13][14][15][16][17]

Parallel to studies on wave function determination, R. F. Stewart[18] and P. Coppens[19][20] investigated the possibilities to compute models for one-electron charge density from X-ray scattering (for example by means of pseudoatoms multipolar expansion), and later of spin density from polarized neutron diffraction,[21] that originated the scientific community of charge, spin and momentum density.[22] In a recent review article, V. Tsirelson[23] gave a more general definition: "Quantum crystallography is a research area exploiting the fact that parameters of quantum-mechanically valid electronic model of a crystal can be derived from the accurately measured set of X-ray coherent diffraction structure factors".

The book Modern Charge Density Analysis offers a survey of the research involving Quantum Crystallography and of the most adopted experimental or theoretical methodologies.[24]

The International Union of Crystallography has recently established a commission on Quantum Crystallography, as extension of the previous commission on Charge, Spin and Momentum density, with the purpose of coordinating research activities in this field.[25]

## External links

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The Erice School of crystallography (52nd course): first course on Quantum crystallography (June 2018)
The XIX Sagamore Conference (July 2018)
The CECAM meeting on Quantum crystallography (June 2017)
The IUCr commission on Quantum crystallography
The International Union of Crystallography

## References

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1. ^ Macchi, Piero (2020). "The connubium between crystallography and quantum mechanics". Crystallography Reviews. 26 (4): 209–268\. Bibcode:2020CryRv..26..209M. doi:10.1080/0889311X.2020.1853712. S2CID 229935993.
2. ^ Coppens, Philip (1997). X-Ray Charge Densities and Chemical Bonding. International Union of Crystallography. ISBN 9780195356946.
3. ^ Macchi, Piero; Gillet, Jean-Michel; Taulelle, Francis; Campo, Javier; Claiser, Nicolas; Lecomte, Claude (1 July 2015). "Modelling the experimental electron density: only the synergy of various approaches can tackle the new challenges". IUCrJ. 2 (4): 441–451\. Bibcode:2015IUCrJ...2..441M. doi:10.1107/S2052252515007538. PMC 4491316. PMID 26175903.
4. ^ Tsirelson, V.G.; Ozerov, R.P. (1996). Electron density and bonding in crystals: Principles, theory and X-ray diffraction experiments in solid state physics and chemistry. CRC Press. ISBN 978-0750302845.
5. ^ Gillet, Jean-Michel (1 May 2007). "Determination of a one-electron reduced density matrix using a coupled pseudo-atom model and a set of complementary scattering data". Acta Crystallographica Section A. 63 (3): 234–238\. Bibcode:2007AcCrA..63..234G. doi:10.1107/S0108767307001663. PMID 17435287.
6. ^ Jayatilaka, Dylan; Grimwood, Daniel J. (1 January 2001). "Wavefunctions derived from experiment. I. Motivation and theory". Acta Crystallographica Section A. 57 (1): 76–86\. doi:10.1107/S0108767300013155. PMID 11124506.
7. ^ Weyrich, Wolf (1996). "One-Electron Density Matrices and Related Observables". Quantum-Mechanical Ab-initio Calculation of the Properties of Crystalline Materials. Lecture Notes in Chemistry. Vol. 67\. Springer Berlin Heidelberg. pp. 245–272\. doi:10.1007/978-3-642-61478-1_14. ISBN 9783540616450.
8. ^ Massa, L.; Huang, L.; Karle, J. (25 February 1995). "Quantum crystallography and the use of kernel projector matrices". International Journal of Quantum Chemistry. 56 (S29): 371–384\. doi:10.1002/qua.560560841.
9. ^ Huang, L.; Massa, L.; Karle, J. (1999). "Quantum crystallography applied to crystalline maleic anhydride". International Journal of Quantum Chemistry. 73 (5): 439–450\. doi:10.1002/(SICI)1097-461X(1999)73:5<439::AID-QUA7>3.0.CO;2-5.
10. ^ Clinton, William L.; Galli, Anthony J.; Massa, Louis J. (5 January 1969). "Direct Determination of Pure-State Density Matrices. II. Construction of Constrained Idempotent One-Body Densities". Physical Review. 177 (1): 7–13\. Bibcode:1969PhRv..177....7C. doi:10.1103/PhysRev.177.7.
11. ^ Mukherji, A.; Karplus, M. (January 1963). "Constrained Molecular Wavefunctions: HF Molecule". The Journal of Chemical Physics. 38 (1): 44–48\. Bibcode:1963JChPh..38...44M. doi:10.1063/1.1733493.
12. ^ Rasiel, Yecheskel; Whitman, Donald R. (15 March 1965). "Constrained-Variation Method in Molecular Quantum Mechanics. Application to Lithium Hydride". The Journal of Chemical Physics. 42 (6): 2124–2131\. Bibcode:1965JChPh..42.2124R. doi:10.1063/1.1696255.
13. ^ Grabowsky, Simon; Genoni, Alessandro; Bürgi, Hans-Beat (2017). "Quantum crystallography". Chemical Science. 8 (6): 4159–4176\. doi:10.1039/C6SC05504D. PMC 5576428. PMID 28878872.
14. ^ Massa, Lou; Matta, Chérif F. (14 November 2017). "Quantum crystallography: A perspective". Journal of Computational Chemistry. 39 (17): 1021–1028\. doi:10.1002/jcc.25102. PMID 29135029.
15. ^ Matta, Chérif F. (2010). Quantum Biochemistry. John Wiley & Sons. ISBN 9783527629220.
16. ^ Matta, Chérif F. (15 May 2017). "Guest Editorial: A path through quantum crystallography: a short tribute to Professor Lou Massa". Structural Chemistry. 28 (5): 1279–1283\. doi:10.1007/s11224-017-0961-8. S2CID 125516603.
17. ^ Massa, Lou (2011). Science and the Written Word: Science, Technology, and Society. Oxford University Press. ISBN 9780199831777.
18. ^ Stewart, Robert F. (15 November 1969). "Generalized X-Ray Scattering Factors". The Journal of Chemical Physics. 51 (10): 4569–4577\. Bibcode:1969JChPh..51.4569S. doi:10.1063/1.1671828.
19. ^ Coppens, Philip; Pautler, D.; Griffin, J. F. (March 1971). "Electron population analysis of accurate diffraction data. II. Application of one-center formalisms to some organic and inorganic molecules". Journal of the American Chemical Society. 93 (5): 1051–1058\. doi:10.1021/ja00734a001.
20. ^ Hansen, N. K.; Coppens, P. (1 November 1978). "Testing aspherical atom refinements on small-molecule data sets". Acta Crystallographica Section A. 34 (6): 909–921\. Bibcode:1978AcCrA..34..909H. doi:10.1107/S0567739478001886.
21. ^ Deutsch, Maxime; Gillon, Béatrice; Claiser, Nicolas; Gillet, Jean-Michel; Lecomte, Claude; Souhassou, Mohamed (14 April 2014). "First spin-resolved electron distributions in crystals from combined polarized neutron and X-ray diffraction experiments". IUCrJ. 1 (3): 194–199\. Bibcode:2014IUCrJ...1..194D. doi:10.1107/S2052252514007283. PMC 4086435. PMID 25075338.
22. ^ Frankenberger, C. (1 October 1990). "Report of the Executive Committee for 1989". Acta Crystallographica Section A. 46 (10): 871–896\. Bibcode:1990AcCrA..46..871.. doi:10.1107/s0108767390006109. ISSN 0108-7673. PMC 234214. PMID 16016227.
23. ^ Tsirelson, Vladimir (9 August 2017). "Early days of quantum crystallography: A personal account". Journal of Computational Chemistry. 39 (17): 1029–1037\. doi:10.1002/jcc.24893. PMID 28791717. S2CID 13675932.
24. ^ Gatti, Carlo; Macchi, Piero (2012). Modern Charge-Density Analysis. Springer Science & Business Media. ISBN 9789048138364.
25. ^ "IUCr". www.iucr.org.