Chapter 4

Inversion of Cohesive Energy Curves

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The physicist cannot ask of the analyst to reveal to him a new truth; the latter could at most only aid him to foresee it.

-- Henri Poincare

Having gained insight into interatomic forces mediated by hybrid covalent bonds in ideal lattices, we now ask what truths may be revealed concerning global trends across more complex structures with different bonding character. Quantum approximations are very useful in suggesting qualitative trends and providing physical understanding, but one wonders whether any quantitative information can be extracted directly from ab initio energy data without resorting to the uncontrolled and inconclusive fitting approach. Inspired by Poincare, we may see if pure mathematics can lead us in a fruitful direction.

So, what is the basic mathematical problem we are interested in solving? The answer is, of course, that we wish to reproduce the many-dimensional Born-Oppenheimer energy surface with a relatively simple functional form. Thus stated, the inverse problem is incredibly overdetermined, and we must settle for an approximate solution obtained by some sort of optimization procedure. However, if the dimensionality of the manifold we wish to fit is sufficiently reduced, then we may be left with a nonsingular and tractable inverse problem. For example, we may hope to uniquely determine a force law containing a single, one-variable, continuous function from a one-parameter energy curve.

In 1980, Carlson, Gelatt and Ehrenreich (CGE) showed that this is indeed possible by proving an inversion formula which gives the pair potential that exactly reproduces a given cohesive energy versus volume curve. In spite of its mathematical elegance, the CGE formula has so far not produced potentials of practical use or been connected with theories of chemical bonding, and hence it has only been employed by a handful of authors. Nevertheless, it is such a radically different and aesthetically appealing approach compared to brute-force fitting, that in this chapter we set out to understand its limitations and extend its applicability to more realistic functional forms for covalent solids. In order to make progress toward these goals, it will be necessary to invent a new way to think about the mathematics of inversion. Following this work, fitting will still play the central role in developing potentials because the important regions of the Born-Oppenheimer surface are too vast to permit an exact solution, but inversion will at least provide sorely needed guidance for the fitting process and build our physical intuition.

1. Pair Potentials
   1. The Carlsson-Gelatt-Ehrenreich Formula
   2. The Chen-M\"obius Theorem
   3. Limitations of the CGE Formula
   4. Recursive Inversion
   5. Ab Initio Cohesive Energy Curves for Silicon Crystals
   6. Physical Validity
   7. An Essential Modification
   8. Numerical Stability
   9. A Study of Pair Bonding in Silicon
   10. Discussion
2. Three-Body Cluster Potentials
   1. A Three-Body Inversion Formula
   2. Existence of the Inverse
   3. Numerical Stability
   4. A Study of Angular Forces in Silicon
3. An Ab Initio Three-Body Potential for Silicon
   1. Derivation
   2. Tests of the Inverted Potential
   3. Discussion
4. Conclusion

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© 1997 Martin Zdenek Bazant.