  We derive expressions for classical isothermal and adiabatic elastic
constants for periodic systems with the boundary contributions included
explicitly. The potential-dependent part of these expressions is written in
terms of potential energies of atomic groups that make up the total potential
energy. It is shown that in the thermodynamic limit, the Born term, which
depends on the second derivatives of potential energy, can be expressed exactly
in terms of equilibrium averages that involve two types of atomic-group
virials. As a result, the new form of the Born term involves only first
derivatives of either atomic-group or total potential energies. The derived
elastic constant expressions involving the two forms of the Born terms are
tested and compared using molecular-dynamics simulations of crystalline argon
and silicon. For both materials, the elastic constants obtained using the two
forms of the Born term are in good agreement. In particular, the new form of
the Born term converges to the same value as the original Born term but at a
slower rate. The results for silicon also agree well with the results from the
previous molecular-dynamics studies.
