X-ray electron density investigation of chemical bonding in van der Waals materials


Van der Waals (vdW) solids have attracted great attention ever since the discovery of graphene, with the essential feature being the weak chemical bonding across the vdW gap. The nature of these weak interactions is decisive for many extraordinary properties, but it is a strong challenge for current theory to accurately model long-range electron correlations. Here we use synchrotron X-ray diffraction data to precisely determine the electron density in the archetypal vdW solid, TiS2, and compare the results with density functional theory calculations. Quantitative agreement is observed for the chemical bonding description in the covalent TiS2 slabs, but significant differences are identified for the interactions across the gap, with experiment revealing more electron deformation than theory. The present data provide an experimental benchmark for testing theoretical models of weak chemical bonding.

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Fig. 1: Structure of 1T-TiS2.
Fig. 2: Static deformation density and Laplacian for intralayer Ti−S interaction.
Fig. 3: Static deformation density and Laplacian for interlayer S···S interaction.


  1. 1.

    Whittingham, M. S. Lithium batteries and cathode materials. Chem. Rev. 104, 4271–4301 (2004).

    Article  Google Scholar 

  2. 2.

    Imai, H., Shimakawa, Y. & Kubo, Y. Large thermoelectric power factor in TiS2 crystal with nearly stoichiometric composition. Phys. Rev. B 64, 241104 (2001).

    Article  Google Scholar 

  3. 3.

    Li, Y., Wang, H., Xie, L., Liang, Y., Hong, G. & Dai, H. MoS2 nanoparticles grown on graphene: an advanced catalyst for the hydrogen evolution reaction. J. Am. Chem. Soc. 133, 7296–7299 (2011).

    Article  Google Scholar 

  4. 4.

    Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    Article  Google Scholar 

  5. 5.

    Gamble, F. R., DiSalvo, F. J., Klemm, R. A. & Geballe, T. H. Superconductivity in layered structure organometallic crystals. Science 168, 568–570 (1970).

    Article  Google Scholar 

  6. 6.

    Wilson, J. A., Di Salvo, F. J. & Mahajan, S. Charge-density waves and superlattices in the metallic layered transition metal dichalcogenides. Adv. Phys. 24, 117–201 (1975).

    Article  Google Scholar 

  7. 7.

    Coleman, J. N. et al. Two-dimensional nanosheets produced by liquid exfoliation of layered materials. Science 331, 568–571 (2011).

    Article  Google Scholar 

  8. 8.

    Chhowalla, M. et al. The chemistry of two-dimensional layered transition metal dichalcogenide nanosheets. Nat. Chem. 5, 263–275 (2013).

    Article  Google Scholar 

  9. 9.

    Geim, A. K. & Grigorieva, I. V. Van der Waals heterostructures. Nature 499, 419–425 (2013).

    Article  Google Scholar 

  10. 10.

    Berland, K. et al. Van der Waals forces in density functional theory: a review of the vdW-DF method. Rep. Prog. Phys. 78, 066501 (2015).

    Article  Google Scholar 

  11. 11.

    BjörkmanT.., GulansA.., KrasheninnikovA. V.. & NieminenR. M.. Van der Waals bonding in layered compounds from advanced density-functional first-principles calculations. Phys. Rev. Lett. 108, 235502 (2012).

    Article  Google Scholar 

  12. 12.

    Benedict, L. X. et al. Microscopic determination of the interlayer binding energy in graphite. Chem. Phys. Lett. 286, 490–496 (1998).

    Article  Google Scholar 

  13. 13.

    Filsø, M. Ø., Eikeland, E., Zhang, J., Madsen, S. R. & Iversen, B. B. Atomic and electronic structure transformations in SnS2 at high pressures: a joint single crystal X-ray diffraction and DFT study. Dalton Trans. 45, 3798–3805 (2016).

    Article  Google Scholar 

  14. 14.

    Le, N. B., Huan, T. D. & Woods, L. M. Interlayer interactions in van der Waals heterostructures: electron and phonon properties. ACS Appl. Mater. Interfaces 8, 6286–6292 (2016).

    Article  Google Scholar 

  15. 15.

    Coppens, P. X-ray Charge Densities and Chemical Bonding (Oxford University Press, New York, 1997).

  16. 16.

    Koritsanszky, T. S. & Coppens, P. Chemical applications of X-ray charge-density analysis. Chem. Rev. 101, 1583–1627 (2001).

    Article  Google Scholar 

  17. 17.

    Voufack, A. B. et al. When combined X-ray and polarized neutron diffraction data challenge high-level calculations: spin-resolved electron density of an organic radical. Acta Crystallogr. B 73, 544–549 (2017).

    Article  Google Scholar 

  18. 18.

    Jørgensen, M. R. V. et al. Contemporary X-ray electron-density studies using synchrotron radiation. IUCrJ 1, 267–280 (2014).

    Article  Google Scholar 

  19. 19.

    Wahlberg, N. et al. Synchrotron powder diffraction of silicon: high-quality structure factors and electron density. Acta Crystallogr. A 72, 28–35 (2016).

    Article  Google Scholar 

  20. 20.

    Bader, R. F. W. Atoms in Molecules – A Quantum Theory (Oxford University Press, Oxford, 1990).

  21. 21.

    Trevey, J. E., Stoldt, C. R. & Lee, S.-H. High power nanocomposite TiS2 cathodes for all-solid-state lithium batteries. J. Electrochem. Soc. 158, A1282–A1289 (2011).

    Article  Google Scholar 

  22. 22.

    Hansen, N. K. & Coppens, P. Testing aspherical atom refinements on small-molecule data sets. Acta Crystallogr. A 34, 909–921 (1978).

    Article  Google Scholar 

  23. 23.

    Fischer, A. et al. Experimental and theoretical charge density studies at subatomic resolution. J. Phys. Chem. A 115, 13061–13071 (2011).

    Article  Google Scholar 

  24. 24.

    Gatti, C. Chemical bonding in crystals: new directions. Z. Kristallogr. 220, 399–457 (2005).

    Google Scholar 

  25. 25.

    Johnson, E. R. et al. Revealing noncovalent interactions. J. Am. Chem. Soc. 132, 6498–6506 (2010).

    Article  Google Scholar 

  26. 26.

    Saleh, G., Gatti, C., Lo Presti, L. & Contreras-García, J. Revealing non-covalent interactions in molecular crystals through their experimental electron densities. Chem. Eur. J. 18, 15523–15536 (2012).

    Article  Google Scholar 

  27. 27.

    Pike, N. A. et al. Origin of the counterintuitive dynamic charge in the transition metal dichalcogenides. Phys. Rev. B 95, 201106 (2017).

    Article  Google Scholar 

  28. 28.

    Brandenburg, J. G. et al. Geometrical correction for the inter- and intramolecular basis set superposition error in periodic density functional theory calculations. J. Phys. Chem. A 117, 9282–9292 (2013).

    Article  Google Scholar 

  29. 29.

    Rimmington, H. P. B., Balchin, A. A. & Tanner, B. K. Nearly perfect single crystals of layer compounds grown by iodine vapour-transport techniques. J. Cryst. Growth 15, 51–56 (1972).

    Article  Google Scholar 

  30. 30.

    Sugimoto, K. et al. Extremely high resolution single crystal diffractometory for orbital resolution using high energy synchrotron radiation at SPring-8. AIP Conf. Proc. 1234, 887–890 (2010).

    Article  Google Scholar 

  31. 31.

    Blessing, R. H. An empirical correction for absorption anisotropy. Acta Crystallogr. A 51, 33–38 (1995).

    Article  Google Scholar 

  32. 32.

    Blessing, R. H. Outlier treatment in data merging. J. Appl. Cryst. 30, 421–426 (1997).

    Article  Google Scholar 

  33. 33.

    Sheldrick, G. M. Crystal structure refinement with SHELXL. Acta Crystallogr. C 71, 3–8 (2015).

    Article  Google Scholar 

  34. 34.

    Blaha, P., Schwarz, K., Madsen, G. K. H., Kvasnicka, D. & Luitz, J. WIEN2k, an Augmented Plane Wave plus Local Orbitals Program for Calculating Crystal Properties (Technical University of Wien, 2001).

  35. 35.

    Sun, J., Ruzsinszky, A. & Perdew, J. P. Strongly constrained and appropriately normed semilocal density functional. Phys. Rev. Lett. 115, 036402 (2015).

    Article  Google Scholar 

  36. 36.

    Perdew, J. P. & Zunger, A. Self-interaction correction to density-functional approximations for many-electron systems. Phys. Rev. B 23, 5048–5079 (1981).

    Article  Google Scholar 

  37. 37.

    Perdew, J. P. & Wang, Y. Accurate and simple analytic representation of the electron-gas correlation energy. Phys. Rev. B 45, 13244–13249 (1992).

    Article  Google Scholar 

  38. 38.

    Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).

    Article  Google Scholar 

  39. 39.

    Peng, H., Yang, Z.-H., Perdew, J. P. & Sun, J. Versatile van der Waals density functional based on a meta-generalized gradient approximation. Phys. Rev. X 6, 041005 (2016).

    Google Scholar 

  40. 40.

    Vydrov, O. A. & Van Voorhis, T. Nonlocal van der Waals density functional: the simpler the better. J. Chem. Phys. 133, 244103 (2010).

    Article  Google Scholar 

  41. 41.

    Sabatini, R., Gorni, T. & de Gironcoli, S. Nonlocal van der Waals density functional made simple and efficient. Phys. Rev. B 87, 041108 (2013).

    Article  Google Scholar 

  42. 42.

    Hamada, I. Van der Waals density functional made accurate. Phys. Rev. B 89, 121103 (2014).

    Article  Google Scholar 

  43. 43.

    Lee, K., Murray, É. D., Kong, L., Lundqvist, B. I. & Langreth, D. C. Higher-accuracy van der Waals density functional. Phys. Rev. B 82, 081101 (2010).

    Article  Google Scholar 

  44. 44.

    Klimeš, J., Bowler, D. R. & Michaelides, A. Van der Waals density functionals applied to solids. Phys. Rev. B 83, 195131 (2011).

    Article  Google Scholar 

  45. 45.

    Tran, F., Stelzl, J., Koller, D., Ruh, T. & Blaha, P. Simple way to apply nonlocal van der Waals functionals within all-electron methods. Phys. Rev. B 96, 054103 (2017).

    Article  Google Scholar 

  46. 46.

    Otero-de-la-Roza, A., Johnson, E. R. & Luaña, V. Critic2: a program for real-space analysis of quantum chemical interactions in solids. Comput. Phys. Commun. 185, 1007–1018 (2014).

    Article  Google Scholar 

  47. 47.

    Otero-de-la-Roza, A., Blanco, M. A., Pendás, A. M. & Luaña, V. Critic: a new program for the topological analysis of solid-state electron densities. Comput. Phys. Commun. 180, 157–166 (2009).

    Article  Google Scholar 

  48. 48.

    Otero-de-la-Roza, A. & Luaña, V. A fast and accurate algorithm for QTAIM integration in solids. J. Comput. Chem. 32, 291–305 (2011).

    Article  Google Scholar 

  49. 49.

    Volkov, A. et al. XD2006 – a computer program package for multipole refinement, topological analysis of charge densities and evaluation of intermolecular energies from experimental and theoretical structure factors (2006).

  50. 50.

    Abramov, Y. A. On the possibility of kinetic energy density evaluation from the experimental electron-density distribution. Acta Crystallogr. A 53, 264–272 (1997).

    Article  Google Scholar 

  51. 51.

    Sabino, J. R. & Coppens, P. On the choice of d-orbital coordinate system in charge-density studies of low-symmetry transition-metal complexes. Acta Crystallogr. A 59, 127–131 (2003).

    Article  Google Scholar 

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This work was supported by the Danish National Research Foundation (DNRF93), the Danish Center for Synchrotron and Neutron Science (DanScatt), the JSPS Bilateral Open Partnership Joint Research Projects for 2015−2017 and 2017−2019, the International Education and Research Laboratory Program and the International Tenure Track system of Univ. Tsukuba. The synchrotron experiment was performed at SPring-8 BL02B1 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) as a Partner User (Proposal No: 2015A0078). The theoretical calculations were performed at the Center for Scientific Computing, Aarhus.

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B.B.I designed and coordinated this study. M.Ø.F. synthesized the samples; H.K., K.T., M.S., V.R.H. and K.S. performed the experiments; H.K. and K.T. analyzed the data, with discussions with M.S., V.R.H., J.O., E.N. and B.B.I.; J. Z. and S.C. performed the theoretical calculations. H.K. and B.B.I. wrote the manuscript. All authors discussed the experimental results and contributed to the manuscript.

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Correspondence to Bo B. Iversen.

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Supplementary Tables 1–10, Supplementary Figures 1–12, Supplementary References 1–23

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Kasai, H., Tolborg, K., Sist, M. et al. X-ray electron density investigation of chemical bonding in van der Waals materials. Nature Mater 17, 249–252 (2018). https://doi.org/10.1038/s41563-017-0012-2

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