Calculation of Elastic Constants Using Molecular Dynamics

  • John R. Ray
  • Aneesur Rahman


Ray and Rahman have developed a useful method to determine elastic constants in molecular dynamics computer simulations. The adiabatic elastic constants are contained in a formula involving fluctuations in the microscopic stress tensor. Mij, in the microcanonical or EhN ensemble, whereas the isothermal elastic constants are contained in a fluctuation formula of the same form in the canonical or ThN ensemble. Here, E is the system energy, h is a 3x3 matrix constructed from the three vectors spanning the periodically repeating computational cell: h=(a,b,c), N is the particle number, and T is the system temperature. For a potential U which depends only upon the distances between the particles (and is not necessarily pairwise additive) this formula gives the elastic constants as a sum of three terms: a fluctuation term, a kinetic term and the Born contribution which depends upon the potential U. In the static Born method of calculating elastic constants, we have only the Born term evaluated at the static lattice positions of the atoms. The fluctuation equation furnishes a practical method of calculating elastic constants which introduces temperature contributions to the static Born values, producing a significant difference.

We shall give results of our calculations for a nearest neighbor Lennard-Jones system, for which independent Monte Carlo data is available, and for silicon using the 2- and 3-body Stillinger-Weber potential.


Molecular Dynamic Elastic Constant Born Term Fluctuation Term Finite Elasticity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. C. Andersen, J. Chem Phys. 72:2384 (1980).CrossRefGoogle Scholar
  2. 2.
    M. Parrinello and A. Rahman, Phys. Rev. Lett. 45:1196 (1980).CrossRefGoogle Scholar
  3. 3.
    M. Parrinello and A. Rahman, J. Appl. Phys. 52:7182 (1981).CrossRefGoogle Scholar
  4. 4.
    J. Ray and A. Rahman, J. Chem. Phys. 80:4423 (1984).CrossRefGoogle Scholar
  5. 5.
    R. N. Thurston, in Physical Acoustics, W. P. Mason, ed., Academic, New York (1964), Vol 1, Part A.Google Scholar
  6. 6.
    S. Nosé, Mol. Phys. 52:255 (1984)CrossRefGoogle Scholar
  7. 6.
    S. Nosé, J. Chem. Phys. 81:511 (1984).CrossRefGoogle Scholar
  8. 7.
    J. Ray and A. Rahman, J. Chem. Phys. 82:4243 (1985).CrossRefGoogle Scholar
  9. 8.
    M. Parrinello and A. Rahman, J. Chem. Phys. 76:2662 (1982).CrossRefGoogle Scholar
  10. 9.
    J. R. Ray, J. Appl. Phys. 53:6441 (1982).CrossRefGoogle Scholar
  11. 10.
    J. R. Ray and H. W. Graben, “Fundamental Treatment of the Isoenthalpic-Isobaric Ensemble,” to appear in Phys. Rev. A, and references contained therein.Google Scholar
  12. 11.
    E. R. Cowley, Phys. Rev. B 28:3160 (1983).CrossRefGoogle Scholar
  13. 12.
    M. Parrinello and A. Rahman, unpublished results.Google Scholar
  14. 13.
    M. Sprik, R. W. Impey and M. L. Klein, Phys. Rev. B 29:4368 (1984).CrossRefGoogle Scholar
  15. 14.
    J. R. Ray, M. C. Moody and A. Rahman, Phys. Rev. B 32:733 (1985).CrossRefGoogle Scholar
  16. 15.
    J. R. Ray, M. C. Moody and A. Rahman, Phys. Rev. B 33:895 (1986).CrossRefGoogle Scholar
  17. 16.
    F. H. Stillinger and T. A. Weber, Phys. Rev. B 31:5262 (1985).CrossRefGoogle Scholar
  18. 17.
    M. D. Kluge, J. R. Ray and A. Rahman, “Molecular Dynamic Calculation of Elastic Constants of Silicon,” J. Chem. Phys., to appear.Google Scholar
  19. 18.
    K. H. Hellwege, editor, Landolt-Börnstein: Crystal and Solid State Physics, Vol. 11, Springer-Verlag, Berlin (1979); p. 116.Google Scholar

Copyright information

© Plenum Press, New York 1987

Authors and Affiliations

  • John R. Ray
    • 1
  • Aneesur Rahman
    • 2
  1. 1.Department of Physics and AstronomyClemson UniversityClemsonUSA
  2. 2.Supercomputer Institute and School of Physics and AstronomyUniversity of MinnesotaMinneapolisUSA

Personalised recommendations