Pressure

There is a pressure field caused by the embedding environment. We represent this pressure field in a continuum fashion, following the concept of electronic enthalpy introduced by Cococcioni et al [1]. The approach exploits the notion of quantum volume, which can be straightforwardly defined in terms of a continuum interface function as

\[V \equiv V[s] = \int s(\mathbf{r})d^3\mathbf{r}.\]

The contribution to the enthalpy of the system is expressed as

\[G^{\text{PV}}[s] = P^{\text{ext}}V[s],\]

with \(P^{\text{ext}}\) the external pressure of the environment. This approach is based on an electronic interface, but equivalent arguments may be adopted for ionic or mixed interfaces. The key ingredient for the derivation of the enthalpy contribution to the Kohn-Sham potential or the inter-atomic forces is the functional derivative of the additional energetic term with respect to the interface function, which for the equation defining the continuum interface function above, is simply

\[\frac{\delta F^{\text{PV}}[s]}{\delta s}(\mathbf{r}) = 1.\]

[1]	Cococcioni M, Mauri F, Ceder G, Marzari N, Phys. Rev. Lett. 2005 4;94(14):145501