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Evolutionary approach for determining first-principles hamiltonians

Nature Materials volume 4, pages 391–394 (2005)Cite this article

Abstract

Modern condensed-matter theory from first principles is highly successful when applied to materials of given structure-type or restricted unit-cell size. But this approach is limited where large cells or searches over millions of structure types become necessary. To treat these with first-principles accuracy, one 'coarse-grains' the many-particle Schrödinger equation into 'model hamiltonians'1,2,3 whose variables are configurational order parameters (atomic positions, spin and so on), connected by a few 'interaction parameters' obtained from a microscopic theory3,4. But to construct a truly quantitative model hamiltonian, one must know just which types of interaction parameters to use, from possibly 106–108 alternative selections. Here we show how genetic algorithms5, mimicking biological evolution ('survival of the fittest'), can be used to distil reliable model hamiltonian parameters from a database of first-principles calculations. We demonstrate this for a classic dilemma6 in solid-state physics7, structural inorganic chemistry8 and metallurgy9: how to predict the stable crystal structure of a compound given only its composition. The selection of leading parameters based on a genetic algorithm is general and easily applied to construct any other type of complex model hamiltonian from direct quantum-mechanical results.

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Figure 1: The first few MBITs for a b.c.c. lattice.
Figure 2: Flowchart of a genetic algorithm for choosing the terms to retain in a model hamiltonian.
Figure 3: Genetic algorithm-based identification of the optimally predictive sets of MBITs.
Figure 4: 'Usual suspect' structures and actual ground state lines for Ta–W and Ti–N.

References

  1. Landau, L. D. & Lifshitz, E. M. Statistical Physics (Pergamon, Oxford, 1980). Transl. from the Russian by J. B. Sykes & M. J. Kearsley.

    Google Scholar 

  2. Rado, G. T. & Suhl, H. (eds) Magnetism Vol. 2B (Academic, New York, 1965).

    Google Scholar 

  3. Zunger, A. In Statics and Dynamics of Alloy Phase Transformations (eds Turchi, P. & Gonis, A.) 361–419 (Plenum, New York, 1994).

    Book  Google Scholar 

  4. de Fontaine, D. Cluster approach to order-disorder transformations in alloys. Solid State Phys. 47, 33–176 (1994).

    Article  Google Scholar 

  5. Michalewicz, Z. & Fogel, D. B. How to Solve it: Modern Heuristics (Springer, Berlin, 2000).

    Book  Google Scholar 

  6. Maddox, J. Crystals from first principles. Nature 335, 201 (1988).

    Article  Google Scholar 

  7. Phillips, J. C. Bonds and Bands in Semiconductors (Academic, New York, 1973).

    Google Scholar 

  8. Pauling, L. The Nature of the Chemical Bond (Cornell Univ. Press, Ithaca, 1960).

    Google Scholar 

  9. Hume-Rothery, W. & Raynor, G. The Structure of Metals and Alloys (Institute of Metals, London, 1954).

    Google Scholar 

  10. Connolly, J. & Williams, A. Density-functional theory applied to phase transformations in transition-metal alloys. Phys. Rev. B 27, 5169–5172 (1983).

    Article  CAS  Google Scholar 

  11. Kikuchi, R. In Noble Metal Alloys (eds Massalski, T. B., Pearson, W. B., Bennet, L. H. & Chang, Y. A.) (The Metallurgical Society, Warrendale, PA, 1986).

    Google Scholar 

  12. van der Walle, A. & Ceder, G. Automating first-principles phase diagram calculations. J. Phase Equil. 23, 348–359 (2002).

    Article  Google Scholar 

  13. Zarkevich, N. A. & Johnson, D. D. Reliable first-principles alloy thermodynamics via truncated cluster expansions. Phys. Rev. Lett. 92, 255702 (2004).

    Article  Google Scholar 

  14. de Gironcoli, S. & Baroni, S. Effects of disorder on the vibrational properties of SiGe alloys: failure of mean-field approximations. Phys. Rev. Lett. 69, 1959–1962 (1992).

    Article  CAS  Google Scholar 

  15. Deaven, D. M. & Ho, K. M. Molecular geometry optimization with a genetic algorithm. Phys. Rev. Lett. 75, 288–291 (1995).

    Article  CAS  Google Scholar 

  16. Ho, K. M. et al. Structures of medium-sized silicon clusters. Nature 392, 582 (1998).

    Article  CAS  Google Scholar 

  17. Stucke, D. P. & Crespi, V. H. Predictions of new crystalline states for assemblies of nanoparticles: perovskite analogues and 3-D arrays of self-assembled nanowires. Nano Lett. 3, 1183–1186 (2003).

    Article  CAS  Google Scholar 

  18. Morris, J. R., Deaven, D. M. & Ho, K. M. Genetic algorithm energy minimization for point charges on a sphere. Phys. Rev. B 53, R1740 (1996).

    Article  CAS  Google Scholar 

  19. Johannesson, G. H. et al. Combined electronic structure and evolutionary search approach to materials design. Phys. Rev. Lett. 88, 255506 (2002).

    Article  CAS  Google Scholar 

  20. Klimeck, G. & Bowen, R. C. Si tight-binding parameters from genetic algorithm fitting. Superlattices Microstruct. 27, 77–88 (2000).

    Article  Google Scholar 

  21. Blum, V. & Zunger, A. Structural complexity in binary bcc ground states: the case of bcc MoTa. Phys. Rev. B 69, 020301 (2004).

    Article  Google Scholar 

  22. Predel, B. (ed.) Phase Equilibria, Crystallographic and Thermodynamic Data of Binary Alloys. Landolt-Börstein, New Series, Group IV Vol. 5H (Springer, Berlin, 1997).

    Google Scholar 

  23. Sigli, C. & Sanchez, J. M. Electronic Structure calculation of ordering and segregation energies of transition metal alloys. Acta Metall. Mater. 36, 367–375 (1988).

    Article  CAS  Google Scholar 

  24. Turchi, P. E. A., Gonis, A., Drchal, V. & Kurdnovsky, J. First-principles study of stability and local order in substitutional Ta–W alloys. Phys. Rev. B 64, 085112 (2001).

    Article  Google Scholar 

  25. Baumann, K. Cross validation as the objective function for variable selection. Trends Anal. Chem. 22, 395–406 (2003).

    Article  CAS  Google Scholar 

  26. Kresse, G. & Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 6, 15–50 (1996).

    Article  CAS  Google Scholar 

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Acknowledgements

We acknowledge financial support from the NSF through DMR-0244183, and from DOE-SC-BES-DMS, as well as the Intramural Grant Program at Northern Arizona University.

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Author notes

  1. Volker Blum

    Present address: Fritz-Haber Institut der Max-Planck-Gesellschaft, Faradayweg 4–6, 14195, Berlin, Germany

Authors and Affiliations

  1. Department of Physics and Astronomy, Northern Arizona University, Flagstaff, 86011-6010, Arizona, USA

    Gus L. W. Hart

  2. National Renewable Energy Laboratory, Golden, 80401, Colorado, USA

    Volker Blum & Alex Zunger

  3. Computer Science Department, Northern Arizona University, Flagstaff, 86011-5600, Arizona, USA

    Michael J. Walorski

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Correspondence to Gus L. W. Hart.

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Hart, G., Blum, V., Walorski, M. et al. Evolutionary approach for determining first-principles hamiltonians. Nature Mater 4, 391–394 (2005). https://doi.org/10.1038/nmat1374

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