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.
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