Generic Pair Potentials

As said earlier, almost the entire business of the present chapter will be to examine the various ways in which the overwhelming complexity presented by the full Hamiltonian of eqn (4.3) has been tamed. Perhaps the simplest example of simplifying the total energy is via an approximation of the form 

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Note that the expressions we have written thus far are entirely generic. All that we have assumed is the existence of a total energy function that is dependent upon the set of ionic positions. Our analysis is thus indifferent to the microscopic energy functional used to compute Etot and hence the results are equally valid for total energy methods ranging from pair potentials to fully quantum mechanical Etot. On... 

Early atomistic simulations employed pair potentials, usually of the Morse or Lennard-jones type (Figure 11.6). Although such potentials have been and still are a useful model for fundamental studies of generic properties of materials, the agreement between simulation results and experiment can only be quantitative at best. While such potentials can be physically justified for inert elements and perhaps some ionic solids, they do not capture the nature of interatomic bonding even in simple metals, not to mention transition metals or covalent solids. 

Using the generic definition of the total energy as defined within the pair functional formalism, derive an expression for the elastic moduli. Once this result is in hand, use the explicit expressions associated with the Johnson potential to obtain the elastic moduli for Cu. 

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