Many-body analytic potential energy function

Pair-additive interactions continued to be used in most materials-related simulations for over 20 years after Vineyard s work despite well-known deficiencies in their ability to model surface and bulk properties of most materials. Quantitative simulation of materials properties was therefore very limited. A breakthrough in materials-related atomistic simulation occurred in the 1980s, however, with the development of several many-body analytic potential energy functions that allow accurate quantitative predictions of structures and dynamics of materials.These methods demonstrated that even relatively simple analytic interatomic potential functions can capture many of the details of chemical bonding, provided the functional form is carefully derived from sound physical principles. 

In 1988, Tersoff [7] introduced an analytical expression for a many-body potential energy function based on bond order, which was able to accommodate reactive dynamies in a straightforward manner. In this formalism, the interaction energy Eij between a pair of atoms i and j is given by... 

The analytic form of the first two terms in the Kohn-Sham effective potential (Vrff [p](r)) is known. They represent the external potential (vext which is the nuclear attraction potential in most cases) and Coulomb repulsion between electrons. The second term is an explicit functional of electron density. The last term, however, represents the quantum many-body effects and has a traditional name of exchange-correlation potential. vxc is the functional derivative of the component of the total energy functional called conventionally exchange-correlation energy (Exc[p]) ... 

Attempts to represent the three-body interactions for water in terms of an analytic function fitted to ab initio results date back to the work of dementi and Corongiu [191] and Niesar et al. [67]. These authors used about 200 three-body energies computed at the Hartree-Fock level and fitted them to parametrize a simple polarization model in which induced dipoles were generated on each molecule by the electrostatic field of other molecules. Thus, the induction effects were distorted in order to describe the exchange effects. The three-body potentials obtained in this way and their many-body polarization extensions have been used in simulations of liquid water. We know now that the two-body potentials used in that work were insufficiently accurate for a meaningful evaluation of the role of three-body effects. 

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