Interatomic Forces in Covalent Solids

by

Martin Zdenek Bazant




Chapter 5

The Environment-Dependent Interatomic Potential

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I have restricted my work to ideal crystals though I am aware that the theory of the defects in real crystals is practically far more important. This I have left to a younger generation.

-- Max Born


It is ironic that in answering Born's challenge to model defect structures we begin with analytic results for ideal crystals (from the preceding two chapters), some of which can be traced back to Born himself. In order to develop a model for the complex bonding in real crystals, however, a modern computer is required to search efficiently through the myriad of possible parameterizations, but not without significant human direction. Although we have seen that reasonable interatomic potentials can be derived analytically from experimental or ab initio data, inversion schemes become most powerful when used as theoretical guidance for fitting, for two basic reasons. The first is that inversion necessarily involves a restricted set of ab initio data. While the input data can be perfectly reproduced (unless it is overdetermined), it is desirable to allow an imperfect description of the inversion data in order to achieve a better overall fit of a wider ab initio database that includes low symmetry defect structures. The second drawback of inversion is that the class of tractable functional forms is rather limited due to issues of invertability, numerical stability, and physical validity. With the fitting approach, although there is less connection with first principles, we can explore the possibility of functional forms of greater complexity and sophistication. On the other hand, complex fitting schemes are difficult to implement; large parameter sets make it hard to judge transferability; and cumbersome functional forms obscure principles of chemical bonding and reduce the ease of force evaluation. A better approach is to incorporate the theoretically derived features of the previous chapters directly into a functional form, and then to fit the potential to a carefully chosen ab initio database with a minimal number of parameters. In this way, a reliable potential for bulk properties can be derived systematically, while keeping the functional form simple enough to allow for efficient computation of forces as well as intuitive interpretation of chemical bonding.

The results of this chapter are the product of almost ten years of hard work by many people, including E. Kaxiras, J. F. Justo, V. V. Bulatov, S. Yip, S. Ismail-Beigi, E. Chung and K. C. Pandey. In this chapter the current state of our empirical model for Si is presented with emphasis on the author's contributions. In Section 5.1, the theoretical results of the previous chapters are incorporated into a general functional form for interatomic forces in bulk covalent solids, called the "Environment-Dependent Interatomic Potential" (EDIP), and in Section 5.2 the fitting and testing of an EDIP for bulk silicon is described.

  1. Functional Form
    1. Scalar Environment Description
    2. Coordination-Dependent Chemical Bonding
    3. Discussion
  2. Fitting and Tests for Bulk Silicon
    1. Fitting to Defect Structures
    2. Tests for Bulk Properties and Defect
    3. Cohesive Energy Curves
    4. Discussion

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