Skip to main content
SpringerLink
Log in

Scalars, vectors and tensors evolving from slabs to bulk

  • Regular Article
  • Published:
Theoretical Chemistry Accounts Aims and scope Submit manuscript

Abstract

When a slab includes an increasing number of layers, some of its properties evolve naturally toward the corresponding ones of the bulk. In other cases, however, this correspondence must be established, as is the case, for example, of the elastic constants involving the lattice parameter orthogonal to the basal plane in the bulk, which in the latter is not defined. Other properties as the first and second hyperpolarizabilities, of the piezoelectricity and elastic tensors, in both the electronic and ionic contributions, are also included in the present work. In all cases, the corresponding property in the bulk is identified. The good agreement between the slab and bulk results documents the convergence of the former with the number of layers. It represents also an important check of the numerical accuracy of the CRYSTAL code in computing this large set of properties.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. Actually, the CPHF expression for \(\gamma ^e\) is a product of perturbed density matrices (at the first and second order of perturbation) by first (and second) derivatives of the hamiltonian with respect to the field. However, this latter first derivative of the hamiltonian also depends on the perturbed density matrix when orbital relaxation is taken into account as in CPHF and is not equal to the unique dipole moment matrix independent of the polarization as in SOS or which is rather used in the CPHF \(\alpha ^e\) expression (see Ref. [17]). Then, the number of z indices in the component of \(\gamma ^e\), indicating the power of the derivate of the energy with respect to \(F_z\), gives the power of \(\epsilon _{zz}^e\) to be used in the bulk to slab transition.

  2. http://www.pmmp.jussieu.fr/yves/BNslabbulk.

References

  1. Celebi N, Ángyán JG, Dehez F, Millot C, Chipot C (2000) Distributed polarizabilities derived from induction energies: a finite perturbation approach. J Chem Phys 112:2709

    Article  CAS  Google Scholar 

  2. Bucko T, Hafner J, Lebegue S, Angyan JG (2010) Improved description of the structure of molecular and layered crystals: Ab initio dft calculations with van der Waals corrections. J Phys Chem A 114(43):11814–11824

    Article  CAS  Google Scholar 

  3. Bucko T, Lebègue S, Hafner J, Ángyán JG (2013) Tkatchenko–Scheffler van der Waals correction method with and without self-consistent screening applied to solids. Phys Rev B 87:064110

    Article  Google Scholar 

  4. Becke AD (1993) Density-functional thermochemistry. III. The role of exact exchange. J Chem Phys 98:5648

    Article  CAS  Google Scholar 

  5. Dovesi R, Erba A, Orlando R, Zicovich-Wilson CM, Civalleri B, Maschio L, Ferrabone M, Rérat M, Casassa S, Jacopo B, Simone S, Bernard K (2018) Quantum-mechanical condensed matter simulations with crystal. WIREs Comput Mol Sci 8:e1360. https://doi.org/10.1002/wcms.1360

    Article  CAS  Google Scholar 

  6. Hehre WJ, Ditchfield R, Pople JA (1972) Self-consistent molecular orbital methods. XII. Further extensions of Gaussian-type basis sets for use in molecular orbital studies of organic molecules. J Chem Phys 56(5):2257–2261

    Article  CAS  Google Scholar 

  7. Becke AD (1988) A multicenter numerical integration scheme for polyatomic molecules. J Chem Phys 88(4):2547–2553

    Article  CAS  Google Scholar 

  8. Dovesi R, Saunders VR, Roetti C, Orlando R, Zicovich-Wilson CM, Pascale F, Civalleri B, Doll K, Harrison NM, Bush IJ, D’Arco P, Llunell M, Causà M, Noël Y, Maschio L, Erba A, Rérat M, Casassa S (2017) CRYSTAL17 user’s manual. University of Torino. http://www.crystal.unito.it

  9. Buckingham AD, Long DA (1979) Polarizability and hyperpolarizability [and discussion]. Philos Trans R Soc Lond Ser A 293(1402):239. https://doi.org/10.1098/rsta.1979.0093

    Article  CAS  Google Scholar 

  10. Kittel C (1976) Introduction to solid state physics, 5th edn. Wiley, New York

    Google Scholar 

  11. Blount EI (1962) In: Ehrenreich H, Seitz F, Turnbull D (eds) Solid state physics. Academic, New York

    Google Scholar 

  12. Otto P (1992) Calculation of the polarizability and hyperpolarizabilities of periodic quasi-one-dimensional systems. Phys Rev B 45:10876

    Article  CAS  Google Scholar 

  13. Hurst GJB, Dupuis M, Clementi E (1988) Ab initio analytic polarizability, first and second hyperpolarizabilities of large conjugated organic molecules: applications to polyenes c4h6 to c22h24. J Chem Phys 89:385

    Article  CAS  Google Scholar 

  14. Ferrero M, Rérat M, Orlando R, Dovesi R (2008a) The calculation of static polarizabilities of periodic compounds. The implementation in the CRYSTAL code for 1D, 2D and 3D systems. J Comput Chem 29:1450

    Article  CAS  Google Scholar 

  15. Ferrero M, Rérat M, Orlando R, Dovesi R (2008b) Coupled perturbed Hartree–Fock for periodic systems: the role of symmetry and related computational aspects. J Chem Phys 128:014110

    Article  Google Scholar 

  16. Orlando R, Ferrero M, Rérat M, Kirtman B, Dovesi R (2009) Calculation of the static electronic second hyperpolarizability or \(\chi ^{(3)}\) tensor of three-dimensional periodic compounds with a local basis set. J Chem Phys 131:184105

    Article  Google Scholar 

  17. Ferrero M, Rérat M, Kirtman B, Dovesi R (2008) Calculation of first and second static hyperpolarizabilities of one- to three-dimensional periodic compounds. Implementation in the CRYSTAL code. J Chem Phys 129:244110

    Article  Google Scholar 

  18. Maschio L, Kirtman B, Rérat M, Orlando R, Dovesi R (2013a) Ab initio analytical Raman intensities for periodic systems through a coupled perturbed Hartree–Fock/Kohn–Sham method in an atomic orbital basis. I. Theory. J Chem Phys 139:164101

    Article  Google Scholar 

  19. Maschio L, Kirtman B, Rérat M, Orlando R, Dovesi R (2013) Ab initio analytical Raman intensities for periodic systems through a coupled perturbed Hartree–Fock/Kohn–Sham method in an atomic orbital basis. II. Validation and comparison with experiments. J Chem Phys 139:164102

    Article  Google Scholar 

  20. Pascale F, Zicovich-Wilson CM, Lòpez Gejo F, Civalleri B, Orlando R, Dovesi R (2004) The calculation of the vibrational frequencies of the crystalline compounds and its implementation in the crystal code. J Comput Chem 25:888–897

    Article  CAS  Google Scholar 

  21. Zicovich-Wilson CM, Pascale F, Roetti C, Saunders VR, Orlando R, Dovesi R (2004) The calculation of the vibration frequencies of \(\alpha\)-quartz: the effect of hamiltonian and basis set. J Comput Chem 25:1873–1881

    Article  CAS  Google Scholar 

  22. Erba A, Ferrabone M, Orlando R, Dovesi R (2013) Accurate dynamical structure factors from ab initio lattice dynamics: the case of crystalline silicon. J Comput Chem 34:346

    Article  CAS  Google Scholar 

  23. Carteret C, De La Pierre M, Dossot M, Pascale F, Erba A, Dovesi R (2013) The vibrational spectrum of caco3 aragonite: a combined experimental and quantum-mechanical investigation. J Chem Phys 138:014201

    Article  Google Scholar 

  24. Baima J, Ferrabone M, Orlando R, Erba A, Dovesi R (2016a) Thermodynamics and phonon dispersion of pyrope and grossular silicate garnets from ab initio simulations. Phys Chem Miner 43:137–149

    Article  CAS  Google Scholar 

  25. Barrow GM (1962) Introduction to molecular spectroscopy. McGraw-Hill, New York

    Google Scholar 

  26. Hess BA, Schaad LJ, Carsky P, Zahradnik R (1986) Ab initio calculations of vibrational spectra and their use in the identification of unusual molecules. Chem Rev 86:709–730

    Article  CAS  Google Scholar 

  27. Maschio L, Kirtman B, Orlando R, Rérat M (2012) Ab initio analytical infrared intensities for periodic systems through a coupled perturbed Hartree–Fock/Kohn–Sham method. J Chem Phys 137:204113

    Article  Google Scholar 

  28. Maschio L, Kirtman B, Rérat M, Orlando R, Dovesi R (2013c) Comment on “ab initio analytical infrared intensities for periodic systems through a coupled perturbed Hartree–Fock/Kohn–Sham method” [J Chem Phys 137:204113 (2012)]. J Chem Phys 139:167101

    Article  Google Scholar 

  29. Baima J, Erba A, Maschio L, Zicovich-Wilson CM, Dovesi R, Kirtman B (2016b) Direct piezoelectric tensor of 3D periodic systems through a coupled perturbed Hartree–Fock/Kohn–Sham method. Z Phys Chem 230:719–736

    Article  CAS  Google Scholar 

  30. Ashcroft NW, Mermin N (1976) Solid state physics. Holt-Saunders International Editions, Philadelphia

    Google Scholar 

  31. Li Y, Rao Y, Fai Mak K, You Y, Wang S, Dean CR, Heinz TF (2013) Probing symmetry properties of few-layer MoS2 and h-BN by optical second-harmonic generation. Nano Lett 13:3329–3333

    Article  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IPREM, UMR5254, CNRS / Université de Pau et des Pays de l’Adour, 64000, Pau, France

    Michel Rérat & Philippe Carbonnière

  2. Laboratoire de Physique et Chimie Théoriques, UMR 7019, Université de Lorraine – Nancy, CNRS, 54506, Vandœuvre-lès-Nancy, France

    Fabien Pascale

  3. Institut des Sciences de la Terre de Paris (UMR 7193 UPMC-CNRS), Sorbonne Université, Paris, France

    Yves Noël

  4. Dipartimento di Chimica and NIS (Nanostructured Interfaces and Surfaces) Centre, Università di Torino, Via Giuria 5, 10125, Turin, Italy

    Roberto Dovesi

Authors
  1. Michel Rérat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fabien Pascale
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yves Noël
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Philippe Carbonnière
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Roberto Dovesi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michel Rérat.

Additional information

Published as part of the special collection of articles In Memoriam of János Ángyán.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rérat, M., Pascale, F., Noël, Y. et al. Scalars, vectors and tensors evolving from slabs to bulk. Theor Chem Acc 137, 150 (2018). https://doi.org/10.1007/s00214-018-2360-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00214-018-2360-7

Keywords

Search

Navigation